Dynamic Shock Wave Reflection Phenomena
 Kavendra Naidoo
 A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Jo-
 hannesburg, in the fulfilment of the requirements for the Degree of Doctor of Philosophy.
 Johannesburg, May 2011
Declaration
 I declare that this thesis is my own, unaided work, except where otherwise acknowledged. It is being
 submitted for the degree of Doctor of Philosophy at the University of the Witwatersrand, Johannesburg. It
 has not been submitted before for any degree or examination at any other university.
 Signed this 16th day of May 2011
 Kavendra Naidoo
 i
To my baby girl, Kiara Prashanthi, my wife, Reshenthi, my parents, Loganathan and Pathmavathie and my
 brother, Mahendra
 ii
Acknowledgements
 Sincere thanks go to my supervisor, Professor B.W Skews, for providing guidance on this new topic and
 especially for exercising extreme patience over the years. It is indeed fulfilling and exciting to make a
 contribution to the fascinating field of supersonic gas dynamics.
 Thanks are extended to South Africa?s CSIR for funding most of the work associated with this thesis
 and for having the vision to support the development of research professionals. Thanks are also extended
 to the Department of Science and Technology for their Research Professional Development Programme,
 administered at the CSIR by Shavanee Maduray, Joseph Tshikomba and Dan Pillay.
 There are many people that contributed to the achievement of this work at the Defence, Peace Safety and
 Security Operating Unit of the CSIR , viz. Andre Nepgen, Johan Strydom, Monique Woodborne and Marlene
 Padavattan. There are many that supported this process at the Aeronautic Systems Competency, viz. Dr
 Igle Gledhill, Glen Snedden, Thomas Roos, Peter Lake, the late Brian Cannoo and Major General Desmond
 Barker. Special thanks are expressed to Dr Gledhill for her encouragement and support, especially in the
 early stages of this work. Sincere gratitude is extended to Beeuwen Gerryts for supporting the allocation of
 funds for most of this work. His patience and encouragement are gratefully acknowledged. Special thanks
 are extended to Mauro Morelli for providing the environment and the expertise without which this work
 would not have been completed. Expert technical assistance was provided on various aspects of this project
 by Bhavya Vallabh, Ndumiso Zwane, Kimal Hiralall, Alesha Saligram, Kaveshan Nayager, Martins Selepe,
 Robert Mokwebo, David Reinecke, Piet Ramaloko, Jimmy Hannan, Martin Mwila and Marius Olivier. Their
 assistance was critical to the completion of this work. Expert maintenance services, provided by Eugene and
 Deon Lemmer on the CSIR supersonic facility, are gratefully acknowledged. The expert machine services
 provided by Louis du Plessis were critical in the achievement of good experimental data. Sincere thanks are
 extended to Mike Woodhead for machine services on the earlier versions of the experimental rig.
 Thanks are extended to various members of university staff for their expert assistance, advice and en-
 couragement, viz. Dr Craig Law, Dr Luke Felthun, Dr Nandkishore Menon, Randall Paton, Gavin Li, Anton
 Meiring, David Maclucas, Dimi, Botie and Mr Cooper.
 iii
My wife deserves special praise for her love, support and patience during this work, especially in the early
 years of our marriage. I am grateful for the constant reminder that there is more to life than the laboratory.
 I am extremely grateful to my baby girl, Kiara Prashanthi, for being such a joyful and welcome distraction
 to this work. Sincere gratitude is expressed to my loving parents for their undying, unconditional support,
 without which this undertaking would not have been possible. A final word of gratitude is expressed to
 my dear friend and brother, Mahendra, for encouraging me to preserve my fondest and dearest aspirations,
 especially during very difficult times.
 iv
Table of Contents
 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
 Published Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii
 Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 1.2 Ideal, Steady, Two-Dimensional Shock Wave Reflection Transition . . . . . . . . . . . . . . . 1
 1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 1.5 Chapter Overviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
 Chapter 2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 2.2 Steady, Two-Dimensional, Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 2.3 Ideal, Steady, Two-dimensional RR ? MR Transition Criteria . . . . . . . . . . . . . . . . . 13
 2.3.1 RR to MR Transition in the Strong-Reflection Range . . . . . . . . . . . . . . . . . . 14
 2.3.2 MR to RR Transition in the Strong-Reflection Range . . . . . . . . . . . . . . . . . . 15
 2.3.3 Summary of Transition Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 2.4 The Persistence of Steady RR with a Length Scale Present at the Reflection Point . . . . . . 16
 2.5 MR Configurations in a Steady Supersonic Free Stream . . . . . . . . . . . . . . . . . . . . . 17
 2.6 Early Origins of Rapid Wedge Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 2.7 Computational Simulation of Wedge Vibration and Impulsive Wedge Rotation . . . . . . . . 20
 2.8 Dynamic Mach Stem Development for a Stationery Wedge . . . . . . . . . . . . . . . . . . . . 25
 2.9 Research Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 Chapter 3 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
 3.2.1 Simulation of the Ground Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
 3.2.2 Evaluation of Free Stream Turbulence Levels in the Supersonic Facility . . . . . . . . 29
 3.2.3 Three-Dimensional Wedge Edge Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 29
 3.3 Supersonic Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
 3.3.1 Mach Number Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
 3.3.2 Stagnation Temperature Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 3.4 Flow Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
 3.5 Image Calibration Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
 3.6 High-Speed Image Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
 3.7 Summary of Measurement Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
 v
3.8 Dynamic Shock Wave Interaction Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
 3.8.1 System Requirements Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
 3.8.2 Design Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 3.8.3 Actuator for Steady State Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 44
 3.8.4 Actuator for Dynamic Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
 3.9 Sample Image and Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
 3.10 Rig Development History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
 3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
 Chapter 4 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
 4.2 Code Description : Euler Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
 4.3 Code Description : Fluent V 12.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
 4.4 Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
 4.5 Dynamic Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
 4.6 Grid Sensitivity Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
 4.7 Fluent Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
 4.8 The Incidence-Induced Hysteresis Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
 4.9 Compensation for Boundary Layer Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
 4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
 Chapter 5 Steady State RR ? MR Transition . . . . . . . . . . . . . . . . . . . . . . . . . 74
 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
 5.2 A Brief Summary: The Three-Dimensional Nature of Wave Systems in an Experiment . . . . 74
 5.3 Steady State Experiment in the Weak-Reflection Region . . . . . . . . . . . . . . . . . . . . . 76
 5.3.1 Three-dimensional Wave Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
 5.3.2 Weak Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
 5.3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
 5.4 Steady State Experiment in the Strong-Reflection Region . . . . . . . . . . . . . . . . . . . . 80
 5.4.1 Theoretical Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
 5.4.2 Three-dimensional Wave Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
 5.4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
 Chapter 6 Dynamic Two-Dimensional Regular to Mach Reflection Transition in an Ideal
 Steady Supersonic Free Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
 6.2 Experimental Results for Dynamic RR to MR Transition . . . . . . . . . . . . . . . . . . . . 89
 6.2.1 Weak-Reflection Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
 6.2.2 Strong-Reflection Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
 6.3 Computational Simulation of Impulsive Rotation at M = 2.98 . . . . . . . . . . . . . . . . . . 95
 6.3.1 Steady Pressure-Deflection Shock Polars . . . . . . . . . . . . . . . . . . . . . . . . . . 98
 6.3.2 Dynamic Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
 6.3.3 Transient Pressure Rise through the Reflection/Triple Point . . . . . . . . . . . . . . . 101
 6.4 Transition Criteria and Mechanism for Dynamic RR to MR Transition . . . . . . . . . . . . . 102
 6.5 Parameter investigation for dynamic RR to MR transition . . . . . . . . . . . . . . . . . . . . 107
 6.5.1 M = 1.93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
 6.5.2 M = 2.98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
 6.5.3 Dynamic Mach Stem Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
 6.7 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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Chapter 7 Dynamic Two-Dimensional Mach to Regular Reflection Transition in an Ideal
 Steady Supersonic Free Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
 7.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
 7.2.1 Weak-Reflection Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
 7.2.2 Strong-Reflection Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
 7.3 Parameter Investigation for Dynamic MR to RR Transition . . . . . . . . . . . . . . . . . . . 123
 7.3.1 Impulsive Rotation About the Wedge Leading Edge at M = 1.93 . . . . . . . . . . . . 124
 7.3.2 Impulsive Rotation About the Wedge Trailing Edge at M = 1.93 . . . . . . . . . . . . 125
 7.3.3 Parameter Investigation for Dynamic MR to RR Transition at M = 1.93 . . . . . . . . 128
 7.3.4 Parameter Investigation for Dynamic MR to RR Transition at M = 2.98 . . . . . . . . 132
 7.4 Thoughts on Three-dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
 7.6 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
 Chapter 8 Conclusions and Recommendations for Future Work . . . . . . . . . . . . . . . 153
 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
 8.1.1 Summary of Results for Dynamic RR to MR Transition . . . . . . . . . . . . . . . . . 154
 8.1.2 Summary of Results for Dynamic MR to RR Transition . . . . . . . . . . . . . . . . . 155
 8.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
 8.2.1 Dynamic RR to MR Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
 8.2.2 Dynamic MR to RR Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
 Appendix A Data Acquisition of Freestream Conditions . . . . . . . . . . . . . . . . . . . 158
 A.1 Mach Number Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
 A.1.1 Pressure Transducer Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
 A.1.2 Pressure Transducer Specfication and Calibration Results . . . . . . . . . . . . . . . . 160
 A.1.3 Mach Number Calculation and Uncertainty Analysis . . . . . . . . . . . . . . . . . . . 162
 A.2 Test Section Acoustic Speed Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
 A.2.1 Stagnation Temperature Probe and Transducer Specification . . . . . . . . . . . . . . 168
 A.2.2 Acoustic Speed Calculation and Uncertainty Analysis . . . . . . . . . . . . . . . . . . 169
 A.3 National Instruments Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . 171
 Appendix B Schlieren System, High-Speed Imaging and Optics . . . . . . . . . . . . . . . 173
 B.1 Schlieren System Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
 B.1.1 Optical Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
 B.2 Technical Specifications of High Speed Camera . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 B.3 Inclinometer Specification and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 B.4 Routine for co-ordination calculation in GNU Octave . . . . . . . . . . . . . . . . . . . . . . . 177
 Appendix C Rig Design Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
 C.1 Maximum Rig Cross Sectional Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
 C.2 Motor Sizing for Servo-driven Actuator for Steady State Experiments . . . . . . . . . . . . . 179
 C.3 Component Sizing for Spring-Based Actuator for Dynamic Tests . . . . . . . . . . . . . . . . 184
 C.4 Finite Element Analysis for Latch Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
 C.5 Description of the Rig Operator Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
 C.6 Photographs of Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
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List of Figures
 1.1 Simplified schematic of (a) steady shock wave reflection with the flight vehicle in steady, level
 flight and (b) dynamic shock wave reflection when the vehicle increases its pitch orientation
 rapidly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
 1.2 Simplified schematics of regular and Mach reflection generated by a wedge in a steady super-
 sonic free stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
 1.3 Typical curvature observed in flow computations by Felthun & Skews [12] for rapid increasing
 wedge incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 2.1 Simplified schematic of idealised regular reflection and flow conditions in the vicinity of the
 reflection point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 2.2 Simplified schematic of idealised Mach reflection and flow conditions in the vicinity of the
 triple point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 2.3 Locus of flow conditions that can be achieved through an oblique shock wave in a M = 3.0
 free stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 2.4 Sample pressure-deflection polars for regular and Mach reflection at M = 3.0 . . . . . . . . . 11
 2.5 Pressure-deflection at the detachment and von Neumann conditions at M = 3.0 . . . . . . . . 12
 2.6 Pressure-deflection polars in the dual solution domain and at the sonic condition at M = 3.0 12
 2.7 Pressure-deflection shock polar for a reflection at the detachment condition at M = 1.93 . . . 13
 2.8 Theoretical ?N and ?D between M = 1.6 and 10.0 for air with ? = 1.4 . . . . . . . . . . . . . 16
 2.9 Pressure deflection shock polar for the asymmetric case at M = 4.96, ?1 = 35.0? and ?2 = 14.58? 18
 2.10 Direct and inverse Mach reflections possible in a steady supersonic free stream . . . . . . . . 18
 2.11 A series of schlieren images from the experiment of Mouton & Hornung [36] demonstrating
 hysteresis in the dual solution domain at M = 4.0 . . . . . . . . . . . . . . . . . . . . . . . . 21
 2.12 Measured transition results from the dynamic experiment by Mouton & Hornung [36] . . . . 21
 2.13 Periodic formation of compression and expansion waves generated by wedge oscillation about
 its leading edge with amplitude = 0.5?; wedge rotation speed, ? = 8 ? 103 deg/s published by
 Markelov et al. [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.14 Curvature on the incident wave of a RR due to rapid wedge rotation, M = 5.0, g/w = 0.42
 and rotation speed MT = 0.1 at ?w = 24.0? [28] . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.15 Computed effect of rapid wedge rotation on ?T for RR ? MR transition published by Felthun
 & Skews [12], M = 3.0, h/w = 0.9, ?wi = 20.0? . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 2.16 Predicted and measured dynamic Mach stem development by Mouton & Hornung [36] at
 M = 4.0, g/w = 0.3907, ?w = 23.0?, ? = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 3.1 Two and three-dimensional Mach stem measurements from experiments and Euler predictions
 published by Ivanov et al. [24] for a static wedge at M = 4.0, g/w = 0.56 . . . . . . . . . . . 30
 3.2 Side view of supersonic wind tunnel facility at the CSIR, South Africa . . . . . . . . . . . . . 31
 3.3 Simplified schematic indicating tunnel nozzle shape control . . . . . . . . . . . . . . . . . . . 32
 3.4 Schematic of stagnation temperature probe in settling chamber . . . . . . . . . . . . . . . . . 34
 3.5 (a) Sample total temperature probe measurement and (b) magnified view of select data range 34
 3.6 Schematic of schlieren flow visualisation setup (colour mask and high-speed camera not shown) 36
 3.7 Sample images obtained with various colour masks . . . . . . . . . . . . . . . . . . . . . . . . 38
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3.8 Image of (a) 5 mm ? 5 mm square calibration grid with locating markers and the (b) test
 image captured with the high-speed camera at 512 x 512 pixel resolution used for all dynamic
 experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
 3.9 Distribution of deviation from target angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
 3.10 Photograph of the Photron Ultima APX-RS high speed camera with a UV filter to protect
 the imaging sensor and an aspherical achromatic lens for focussing. The schlieren colour mask
 is positioned ahead of the focussing lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
 3.11 Schematic of a section of the supersonic tunnel with walls removed showing available tunnel
 support systems for the mounting of the rig (image provided courtesy of the CSIR) . . . . . . 42
 3.12 Envelope of operating conditions in CSIR supersonic wind tunnel in terms of total pressure
 (gauge) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
 3.13 Illustrations of rig installed in the CSIR supersonic wind tunnel . . . . . . . . . . . . . . . . . 45
 3.14 Symmetric wedge arrangement and the drive path highlighted in grey . . . . . . . . . . . . . 46
 3.15 Servo motor driven actuator for steady state, baseline experiments . . . . . . . . . . . . . . . 48
 3.16 The spring driven actuator and latch mechanism for the dynamic experiment. The actuator
 is assembled for the dynamic RR ? MR experiment. . . . . . . . . . . . . . . . . . . . . . . . 50
 3.17 Sectioned view illustrating jacking nut/screw and thrust bearing arrangement to arm the
 actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
 3.18 A series of CAD drawings illustrating the operation of the latch release mechanism . . . . . . 52
 3.19 Actuator arrangement for dynamic MR ? RR experiments . . . . . . . . . . . . . . . . . . . 54
 3.20 Sample data acquisition readings acquired during an experiment . . . . . . . . . . . . . . . . 55
 3.21 Sample image captured during a dynamic experiment and prepared for measurements. The
 image was captured with the high-speed digital camera at 10000 frames per second with a
 1/20000 s exposure time. Image resolution : 512 ? 512 pixels. . . . . . . . . . . . . . . . . . . 56
 3.22 Comparison of frontal area profiles in the streamwise direction . . . . . . . . . . . . . . . . . 58
 3.23 First rig design is considerably larger than the final version of the rig. Blockage was sufficiently
 large to prevent tunnel startup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
 3.24 (a) Illustration and (b) photograph of the second rig design with (c) a schlieren image of the
 reflection pattern indicating poor optical magnification . . . . . . . . . . . . . . . . . . . . . . 59
 3.25 A series of high-speed schlieren images showing the early release of the drive train and wedges
 due to failure of the latch mechanism on tunnel startup. The detached bow wave, after the
 flow conditions stabilised, can be seen on the last frame, well after the latch has been released.
 The new optics have the desired magnification. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
 4.1 (a) Conformal mesh topology in the in-house Euler code compared to (b) the non-conformal
 mesh topology in Fluent for mesh refinement in the region of the incident wave at the wedge
 leading edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
 4.2 Sample spurious flow feature in Fluent flow solution due to poor mesh quality in the vicinity
 of the reflection point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
 4.3 Schematic of computational model for simulation of the experiment . . . . . . . . . . . . . . . 66
 4.4 Series of images illustrating the successive adaption of an initial coarse mesh to establish an
 initial, grid independent, steady solution. Corresponding computed density contours appear
 on the right hand side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
 4.5 Results from CFD grid sensitivity assessment for a static and dynamic simulation . . . . . . . 69
 4.6 Computed density contours showing RR ? MR transition close to the detachment condition
 condition. M = 2.98, ME = +0.001, fixed h/w = 0.91. . . . . . . . . . . . . . . . . . . . . . . 71
 4.7 Computed density contours showing MR? RR transition close to the von Neumann condition.
 M = 2.98, ME = ?0.001, fixed h/w = 0.91. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
 5.1 Typical 3D geometry of shock wave reflections at M = 4.0, computed by Ivanov et al. [24] . . 75
 5.2 Computed and measured spanwise Mach stem height variation in a M = 4.0 free stream
 published by Ivanov et al. [24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
 5.3 Identification of 3D reflection structures on schlieren images from the steady state experiment
 at M = 1.93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
 ix
5.4 View of CAD model of wedge illustrating the location of counterbores on the stream facing
 surface of the wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
 5.5 Isometric and top view identifying location of reflection point on symmetry plane with respect
 to sonic cone from wedge face counterbores at the detachment condition, M = 1.93 . . . . . . 79
 5.6 High-speed images from steady state experiment at M = 1.93 . . . . . . . . . . . . . . . . . . 81
 5.7 Magnified view : schlieren image of MR at M = 2.98, indicating the maximum Mach stem
 height in the wedge vertical plane of symmetry and the shear layer in the plane of the minimum
 Mach stem height (indicated on the bottom half of the reflection only) . . . . . . . . . . . . . 82
 5.8 High-speed images from steady state experiment at M = 2.98 . . . . . . . . . . . . . . . . . . 83
 5.9 Measured and computed Mach stem development at M = 1.93, g/w ? 0.6. The solid lines
 are second-order polynomial fits to each data set used to extrapolate ?T at zero m/w. The
 uncertainty in ?? and ?m/w for the experimental data is omitted to prevent cluttering on the
 graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
 5.10 Measured and computed Mach stem development at M = 2.98, g/w ? 0.6. The solid lines
 are second-order polynomial fits to each data set used to extrapolate ?T at zero m/w. The
 uncertainty in ?? and ?m/w for the experimental data is omitted to prevent cluttering on the
 graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
 6.1 High-speed images for dynamic RR ? MR transition at M = 1.93 . . . . . . . . . . . . . . . 90
 6.2 Measurements from the dynamic experiment at M = 1.93 . . . . . . . . . . . . . . . . . . . . 91
 6.3 High-speed images for dynamic RR ? MR transition at M = 2.98 . . . . . . . . . . . . . . . 92
 6.4 Measurements from the dynamic experiment at M = 2.98 . . . . . . . . . . . . . . . . . . . . 93
 6.5 Experimental and CFD results for steady and dynamic RR ? MR transition at M = 1.93.
 Solid lines are second-order polynomial fits through each data set and are used to predict ?T
 at zero m/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
 6.6 Experimental and CFD results for steady and dynamic RR ? MR transition at M = 2.98.
 Solid lines are second-order polynomial fits through each data set and are used to predict ?T
 at zero m/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
 6.7 Critical pressure-deflection shock polars for steady reflection at M = 2.98 . . . . . . . . . . . 98
 6.8 Computed density contours showing the flow field development for ME = +0.1, ?wi = 19.0?
 at M = 2.98, g/w ? 0.6. The Mach stem is indicated ?ms? only where clearly visible. This is
 not to be mistaken to indicate the point of transition. . . . . . . . . . . . . . . . . . . . . . . 99
 6.9 Closeup views of computed density contours showing the first traces of the shear layer from
 the triple point as the reflection transitions to MR . . . . . . . . . . . . . . . . . . . . . . . . 100
 6.10 Mach stem development for impulsive rotation at ME = +0.1 with ?wi = 19.0? compared to
 results from the experiment and 2D CFD results. M = 2.98, g/w ? 0.6. . . . . . . . . . . . . 101
 6.11 Computed pressure traces through the reflection point as the wedge rotates from ?wi = 19.0?
 at ME = +0.1 about the model pivot point at M = 2.98, g/w ? 0.6 . . . . . . . . . . . . . . 102
 6.12 Computed density contours showing the development of the subsonic region downstream of
 reflection point before transition at M = 2.98, ME = +0.1, ?wi = 19.0?. The subsonic region
 downstream of the reflection point is shaded black. . . . . . . . . . . . . . . . . . . . . . . . . 104
 6.13 Computed density contours showing the development of the subsonic region downstream of
 the reflection point between ?S and ?C at M = 1.93, ME = +0.05, ?wi = 8.0?. The subsonic
 region downstream of the reflection point is shaded black. . . . . . . . . . . . . . . . . . . . . 105
 6.14 Estimated location of length scale information on the shortest line between the leading edge
 of the expansion and the subsonic region at ?WT = 17.6?. The early development of the shear
 layer from the triple point is also visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
 6.15 ?WT and ?T vs. ME at M = 1.93, ?wi = 8.0? . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
 6.16 ?WT and ?T vs. ME at M = 2.98, ?wi = 19.0? . . . . . . . . . . . . . . . . . . . . . . . . . . 111
 6.17 Dynamic Mach stem development for impulsive rotation about the wedge trailing edge at
 M = 1.93. Solid lines are second-order polynomial fits through each data set. . . . . . . . . . 112
 6.18 Dynamic Mach stem development for impulsive rotation about the wedge trailing edge at
 M = 2.98. Solid lines are second-order polynomial fits through each data set. . . . . . . . . . 113
 x
7.1 High-speed images showing the initial, steady, disgorged wave system at M = 1.92 being
 swallowed as the wedge incidence decreases rapidly . . . . . . . . . . . . . . . . . . . . . . . . 118
 7.2 High-speed images from dynamic MR ? RR experiment at M = 3.26 . . . . . . . . . . . . . 120
 7.3 Measurements from the dynamic experiment at M = 3.26. The time of MR ? RR transition
 is estimated from the images and is indicated on each graph with a broken line. . . . . . . . . 121
 7.4 Mach stem development from experiment and CFD for dynamic MR ? RR transition at
 M = 3.26, g/w ? 0.6. The dashed and solid lines represent first and second-order fits
 respectively, only for ? ? 38.0?, to their respective data sets and are used to extrapolate ?T
 at zero m/w. The offset from the steady data due to rapid rotation of the wedge is labelled
 ?A?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
 7.5 Computed variation of m/w with ?w for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, h/w = 0.84. The dashed line represents a linear fit of the data for ?w ? 4.5? and
 is used to estimate ?WT at zero m/w for the rapidly rotating wedge. The solid line represents
 a second order fit of the steady data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
 7.6 Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, h/w = 0.84. The solid line represents a second order fit of the steady data.
 The data points from the unsteady simulation are connected with a dashed line to clarify the
 sequence of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
 7.7 Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, g/w = 0.6. The solid lines represent a second order fit of the steady and
 unsteady data. For the unsteady case, only data for ?w ? 5.0? is used to estimate ?WT . . . . 128
 7.8 Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, g/w = 0.6. The solid line represents a second order fit of the steady data.
 The data points from the unsteady simulation are connected with a dashed line to clarify the
 sequence of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
 7.9 Computed variation of m/w with ?w for rapid, impulsive rotation. M = 1.93, ?wi = 13.4?,
 h/w = 0.84 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing
 edge). Dashed lines represent linear fits used to estimate ?WT for ME = ?0.01 and ?0.05.
 Solid lines are second-order polynomial fits used to estimate ?WT for ME = ?0.1 and the
 steady state case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
 7.10 Computed variation of m/w with ? for rapid, impulsive rotation. M = 1.93, ?wi = 13.4?,
 h/w = 0.84 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing
 edge). The solid line is a second-order polynomial fit used to estimate ?WT for the steady
 state case. The dashed line joins the data points in each data set to aid visualisation. ?T was
 measured from the flow solution at ?WT in each case. . . . . . . . . . . . . . . . . . . . . . . . 131
 7.11 Computed pressure contours for impulsive rotation at ME = ?0.1. M = 1.93, ?wi = 13.4?,
 h/w = 0.84. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
 7.12 ?WT and ?T vs. ME at M = 1.93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
 7.13 ?WT and ?T vs. ME at M = 2.98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
 7.14 Computed variation of Mach stem height with ?w for rapid, impulsive rotation. M = 2.98,
 ?wi = 24.5?, h/w = 1.01 (for rotation about the leading edge), g/w = 0.6 (for rotation about
 the trailing edge). Dashed lines represent linear fits used to estimate ?WT for ME = ?0.01,
 ?0.05 and ?0.01. Solid lines are second-order polynomial fits used to compute ?WT for the
 steady state case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
 7.15 Computed variation of Mach stem height with ? for rapid, impulsive rotation. M = 2.98,
 ?wi = 24.5?, h/w = 1.01 (for rotation about the leading edge), g/w = 0.6 (for rotation about
 the trailing edge). The solid line is a second-order polynomial fit to compute ?T for the steady
 state case. ?T is measured at ?WT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
 7.16 Steady, 3D reflection pattern computed with an Euler code by Ivanov et al. [24]. M = 4.0,
 ? = 35.5?, b/w = 3.75, g/w = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
 7.17 Computed pressure contours at (a) ?wi = 13.4? and (b) ?w = 13.0? for impulsive rotation
 about the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84. . . . . . 142
 xi
7.18 Computed pressure contours at (a) ?w = 11.5? and (b) ?w = 11.0? for impulsive rotation
 about the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84. . . . . . 143
 7.19 Computed pressure contours at (a) ?w = 10.0? and (b) ?w = 8.5? for impulsive rotation about
 the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84. . . . . . . . . . 144
 7.20 Computed pressure contours showing decreasing Mach stem height between ?w = 7.0? and
 ?w = 1.0? for impulsive rotation about the wedge leading edge at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, h/w = 0.84. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
 7.21 Development of flow field in the vicinity of the reflection point after transition to RR between
 (a) ?w = 0.5? and (b) ?w = 0.1? for impulsive rotation about the wedge leading edge at
 ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84. . . . . . . . . . . . . . . . . . . . . . . . . 146
 7.22 Computed pressure contours at (a) ?wi = 13.4? and (b) ?w = 13.0? for impulsive rotation
 about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. . . . . . . 147
 7.23 Computed pressure contours at (a) ?w = 12.5? and (b) ?w = 12.0? for impulsive rotation
 about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The
 disturbance propagates down the length of the incident wave and the compression waves from
 the wedge surface continue propagating away from the surface. The solid red line indicates
 the position of the incident wave at the initial condition. . . . . . . . . . . . . . . . . . . . . . 148
 7.24 Computed pressure contours at (a) ?w = 11.6? and (b) ?w = 10.0? for impulsive rotation
 about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The solid
 red line indicates the position of the incident wave at the initial condition. . . . . . . . . . . . 149
 7.25 Computed pressure contours at (a) ?w = 9.5? and (b) ?w = 8.5? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. Between
 ?w = 9.5? and 7.5? the Mach stem height is constant at approximately ?m below the initial
 Mach stem height. The discontinuity on the incident wave continues to move towards the triple
 point. The solid red line indicates the position of the incident wave at the initial condition. . 150
 7.26 Computed pressure contours at (a) ?w = 7.0? and (b) ?w = 5.0? for impulsive rotation
 about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The
 discontinuity on the incident wave has reached the triple point and the incident wave is curved
 along its entire length. The Mach stem height decreases until transition to RR. The solid red
 line indicates the position of the incident wave at the initial condition. . . . . . . . . . . . . . 151
 7.27 Computed pressure contours at (a) ?WT = 2.4?, (b) ?w = 1.5? and (c) ?w = 1.1? for impulsive
 rotation about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. . 152
 8.1 Measured ?T from experiments compared to analytical steady transition criteria . . . . . . . 155
 A.1 Druck Digital Pressure Indicator 605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
 A.2 Calibration and regression for pressure transducers . . . . . . . . . . . . . . . . . . . . . . . . 161
 A.3 Schematic of stagnation temperature probe in settling chamber . . . . . . . . . . . . . . . . . 169
 A.4 Transducer response supplied by WIKA Instruments . . . . . . . . . . . . . . . . . . . . . . . 170
 A.5 Sample total temperature probe measurement (magnified view of select data range on the
 right hand side) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
 A.6 Data Acquisition Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
 B.1 Schlieren system light source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 B.2 Machined collar for laser pointer to replace slit mount on schlieren light source for system
 alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
 B.3 Custom collimator for adjustment of second mirror . . . . . . . . . . . . . . . . . . . . . . . . 175
 B.4 Wyler and Pro3600 Inclinometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
 C.1 Data used to determine the maximum permissable model cross sectional area extracted from
 a US Naval Ordnance Report [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
 C.2 Schematic showing derivation of actuator force . . . . . . . . . . . . . . . . . . . . . . . . . . 180
 xii
C.3 Lead screw and bearing assembly to convert rotational motion of the DC servo motor to
 horizontal motion of the actuator required to pitch the wedges. The image is taken from the
 Rexroth Bosch Group product catalogue on precision ball screw assemblies. . . . . . . . . . . 183
 C.4 Solution of the one dimensional ordinary differential equation for the spring mass system with
 m = 1.0 kg, k = 72? 103 N/m, xo = 13.0 mm and zero initial speed . . . . . . . . . . . . . . 185
 C.5 Contours showing the distribution of computed stress in the latch and release pillar. The
 region of maximum stress at approximately 354 MPa is indicated. The FEM analysis was
 performed by Ryan Raath at the CSIR, Pretoria. . . . . . . . . . . . . . . . . . . . . . . . . . 187
 C.6 Control and electrical connections interface for actuators . . . . . . . . . . . . . . . . . . . . . 189
 C.7 Electrical circuit for operation of servo-motor showing current direction for wedge pitch up
 and pitch down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
 C.8 (a) Front and (b) top views of control interface for spring-driven actuator . . . . . . . . . . . 191
 C.9 Solenoid circuit for the operation of the safety pin in the spring-driven actuator . . . . . . . . 192
 C.10 Latch release circuit diagram for the spring-driven actuator . . . . . . . . . . . . . . . . . . . 193
 C.11 Actuators for (a) steady state, baseline experiments and (b) dynamic experiments . . . . . . 194
 C.12 Rig used for dynamic shock wave reflection experiments. Cover plates are installed and the
 release actuator has been removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
 C.13 Rig with cover plates removed and release actuator installed. The actuator is assembled to
 execute the dynamic RR ? MR transition experiment. . . . . . . . . . . . . . . . . . . . . . . 195
 C.14 Closeup view of the spring-driven actuator with the release actuator installed. The actuator
 is assembled to execute the dynamic RR ? MR transition experiment. . . . . . . . . . . . . . 195
 C.15 Closeup view of wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
 C.16 Stream wise view of the rig installed in tunnel test section . . . . . . . . . . . . . . . . . . . . 196
 xiii
List of Tables
 3.1 Results from a calibration check on a test image for measurements between ?20.0? and 40.0? 37
 3.2 Summary of statistics for angular measurement error . . . . . . . . . . . . . . . . . . . . . . . 37
 3.3 Calibration check for measurement uncertainty on distance . . . . . . . . . . . . . . . . . . . 38
 3.4 Summary of measurement uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
 3.5 A summary of results from a sample dynamic experiment at M = 3.0 . . . . . . . . . . . . . 56
 4.1 Computed Mach stem heights compared to simulation results published by Ivanov et al. [24]
 for a stationary 2D wedge at M = 4.0, g/w = 0.56 . . . . . . . . . . . . . . . . . . . . . . . . 70
 4.2 Computed values for ?T for the incidence-induced hysteresis test at M = 2.98 in comparison
 to steady state, theoretical values for RR ? MR transition . . . . . . . . . . . . . . . . . . . 71
 5.1 Summary of steady state results from experiment and CFD at M = 1.93 and 2.98, g/w ? 0.6 87
 6.1 Experimental test conditions for dynamic RR ? MR experiments, g/w ? 0.6 . . . . . . . . . 90
 6.2 Summary of ?T from steady and dynamic experiments and CFD at M = 1.93, g/w ? 0.6 . . 95
 6.3 Summary of ?T from steady and dynamic experiments and CFD at M = 2.98, g/w ? 0.6 . . 95
 6.4 Summary of results for dynamic simulations at M = 1.93 and M = 2.98 to investigate the
 dynamic RR ? MR transition mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
 6.5 ?WT and ?T at M = 1.93, ?wi = 8.0? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
 6.6 Effect of initial incidence on ?T and ?WT at M = 1.93, ME = +0.1 . . . . . . . . . . . . . . . 109
 6.7 ?WT and ?T at M = 2.98, ?wi = 19.0? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
 6.8 Effect of initial incidence on ?T and ?WT at M = 2.98, ME = +0.1 . . . . . . . . . . . . . . . 109
 7.1 Experiment test conditions for dynamic MR ? RR transition experiments, g/w ? 0.6 . . . . 119
 7.2 Experimental and CFD results for steady and dynamic MR ? RR transition at M = 3.26,
 g/w ? 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
 7.3 Experimental and CFD results for steady and dynamic MR ? RR transition at M = 2.96,
 g/w ? 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
 7.4 Wedge and shock incidence at transition : M 1.93, ?wi = 13.4? . . . . . . . . . . . . . . . . . 134
 7.5 Wedge and shock incidence at transition : M = 2.98, ?wi = 24.5? . . . . . . . . . . . . . . . . 134
 7.6 Sensitivity of ??WT and ??T to pivot point and ?wi for ME = ?0.1 at M = 2.98 . . . . . . . 136
 A.1 Technical specifications of the Druck DPI605 . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
 A.2 Pressure Transducer Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
 A.3 Total pressure transducer calibration: High Range . . . . . . . . . . . . . . . . . . . . . . . . 160
 A.4 Total pressure transducer calibration: Low Range . . . . . . . . . . . . . . . . . . . . . . . . . 160
 A.5 Static pressure transducer calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 A.6 Atmospheric pressure transducer calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 A.7 Summary of pressure transducer regression statistics . . . . . . . . . . . . . . . . . . . . . . . 162
 A.8 CSIR tunnel test section calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
 B.1 Optical parameters of schlieren system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
 B.2 Technical specifications of high speed camera . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
 xiv
B.3 Technical specifications of the Wyler Bubble Inclinometer . . . . . . . . . . . . . . . . . . . . 177
 B.4 Calibration check of Pro3600 digital Protractor . . . . . . . . . . . . . . . . . . . . . . . . . . 177
 C.1 Technical specifications for DC servo motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
 C.2 Technical specifications for the lead screw and bearing arrangement . . . . . . . . . . . . . . . 183
 C.3 Technical specifications for safety pin solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
 C.4 Technical specifications for release actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
 xv
Abstract
 There have been numerous studies on the steady state transition criteria between regular reflection (RR) and
 Mach reflection (MR) of shock waves for a stationary, two-dimensional (2D) wedge in a steady supersonic free
 stream since the original shock wave reflection research by Ernst Mach in 1878. The steady, 2D transition
 criteria between RR and MR are well established. There has been little done to consider the dynamic effect
 of rapid wedge rotation on RR ? MR transition.
 This thesis presents the results of an investigation of the effect of rapid wedge rotation on transition
 between 2D regular and Mach reflection in the weak and strong-reflection ranges, with experiment and com-
 putational fluid dynamics. A novel facility was designed to rotate a pair of large aspect ratio wedges in a 450
 mm ? 450 mm supersonic wind tunnel at wedge rotation speeds up to 11000 deg/s resulting in wedge tip
 speeds approximately 3.3 % of the free stream acoustic speed. Steady state, baseline experiments, in which
 the wedges were rotated very gradually, were also completed. High-speed images and measurements are pre-
 sented for the steady and dynamic experiments. Numerical solution of the inviscid governing flow equations
 was used to model the steady case and to mimic the experimental motion in the dynamic experiments. The
 two-dimensional, Euler CFD code was developed at the University of the Witwatersrand.
 Steady state experiments were completed in the weak and strong-reflection ranges and transition measure-
 ments were compared to 2D steady, theoretical values and Euler computations. There was close agreement
 between theoretical, computational and experimental transition for the steady case, with the following ex-
 ception. Due to the levels of free stream noise in the supersonic wind tunnel, incidence-induced hysteresis
 was not observed in the strong-reflection region and transition occurred at the von Neumann condition for
 increasing and decreasing incidence. In the ideal case, RR ? MR transition occurs at the detachment
 condition and the reverse transition occurs at the von Neumann condition. Therefore, there is discrepancy
 between steady theory/CFD and experiment for RR ? MR transition in the strong-reflection range only,
 which is consistent with observations in other facilities with sufficient levels of free stream noise.
 Dynamic RR ? MR Transition : Rapid wedge rotation did generate a measurable dynamic effect on RR
 ? MR transition. This thesis presents the first experimental evidence of RR ? MR reflection transition
 xvi
beyond the steady detachment condition in the weak and strong-reflection ranges. In all instances, there
 was good agreement between experiment and 2D CFD, including dynamic RR ? MR transition in the
 strong-reflection region. The agreement between the experiment, in which small perturbations are always
 present in the free stream, and the CFD, in which the free stream is without perturbations, implies that
 RR ? MR transition in the strong-reflection region becomes insensitive to free stream noise above a certain
 critical rotation speed. Due to the close agreement between CFD and experiment, the Euler code was also
 applied to scenarios beyond the limits of the current facility to explore the influence of variables in the
 parameter space, viz. rotation speed, initial incidence and rotation centre. CFD was also used to investigate
 the dynamic transition mechanism over a limited number of simulations. For dynamic RR ? MR transition,
 a distinction is drawn between the sonic, length scale and detachment conditions. The point at which the
 flow downstream of the reflection point goes sonic is not necessarily the point at which the wedge length
 scale, from the wedge trailing edge expansion, is communicated to the reflection point. There is evidence to
 support that the RR ? MR transition criteria for the rapidly rotating wedge is neither the sonic or length
 scale conditions, but rather the condition at which the reflected wave can no longer satisfy the boundary
 condition at the reflection point. Dynamic simulations showed that RR could be maintained with a length
 scale present at the reflection point. Other dynamic simulations showed, for the first time, that transition
 to MR was possible without the wedge length scale being communicated to the reflection point.
 Dynamic MR ? RR transition : Rapid wedge rotation generated a measurable effect on MR ? RR
 transition. The first experimental evidence of MR ? RR transition below the steady von Neumann condition
 is presented. Once again, there was good agreement between experiment and 2D CFD. CFD was used to
 investigate the sensitivity of transition to rotation speed, initial incidence and rotation centre in the strong
 and weak-reflection ranges. Due to impulsive wedge start and rapid wedge rotation, there are very marked
 dynamic effects on the variation of Mach stem height with wedge incidence and the deviation from the
 steady transition conditions is significant. The MR ? RR transition was found to be dependent on the
 initial condition and the transient variation of Mach stem height with wedge incidence.
 xvii
Published Works
 Aspects of this work have been accepted for publication in the Journal of Fluid Mechanics:
 1. Naidoo K, Skews BW. Dynamic effects on transition between two-dimensional regular and Mach re-
 flection of shock waves in an ideal steady supersonic free stream. Original submission 22 October 2010;
 revised 8 January 2011; accepted 29 January 2011. DOI:10.1017/jfm.2011.58.
 Various aspects of this work have been published in the following conference articles:
 1. Naidoo K, Skews BW. Dynamic shock reflection phenomena in ideal, two-dimensional flows. In Pro-
 ceedings 25th International Symposium on Shock Waves, (edited by G Jagadeesh, E Arunan, K P J
 Reddy), Bangalore, India, EAN 9788173715716, Paper 1187, 2005.
 2. Naidoo K, Skews BW. Characterisation of unsteady shock wave reflection phenomena. In Proceedings
 of the 5th South African Conference on Computational and Applied Mechanics (edited by T M Harms),
 Cape Town, South Africa, CD-ROM, ISBN 1-919966-01-3, 292-299, 2006.
 3. Naidoo K, Skews BW. Computational and experimental investigation of dynamic shock reflection
 phenomena. In Proceedings 26th International Symposium on Shock Waves (edited by K Hannemann
 and F Seiler), Go?ttingen, Germany, ISBN 978-3-540-85180-6, 1377-1382, 2007.
 4. Naidoo K, Skews BW. High-speed imaging of dynamic shock wave reflection phenomena. In Proceedings
 28th International Congress on High-speed Imaging and Photonics (edited by H Kleine and M P B
 Guillen), Canberra, Australia, ISBN 9780819473608, Paper 71260E, 2008.
 5. Naidoo K, Skews BW. Experimental investigation of dynamic shock wave reflection phenomena. In
 Proceedings 27th International Symposium on Shock Waves, St. Petersburg, Russia, 289-294, 2009.
 6. Naidoo K, Skews BW. High-speed imaging of dynamic shock wave reflection phenomena. In Proceedings
 29th International Congress on High-speed Imaging and Photonics (edited by E Sato, T G Etoh, K
 Nagayama, H Shiraga, T Saito, N Yokoyama, S Suzuki, T Aoki), Morioka, Japan, ISBN 978-4-905149-
 01-9, Paper C03, 2010.
 xviii
Chapter 1
 Introduction
 1.1 Background
 Consider the shock wave system generated by a flight vehicle in steady, level supersonic flight as illustrated in
 figure 1.1. The regular and Mach reflection possible on the ground plane as well as the transition conditions
 between idealised versions of these two configurations (figure 1.2) has been researched since the early work of
 Ernst Mach in 1878. The primary interest in the last 30 years has been the establishment of the steady state
 transition criteria between regular and Mach reflection in the strong-reflection range, i.e. free stream Mach
 number, M ? 2.202, for air with a ratio of specific heats, ? = 1.4. To date there has been little published
 research that is relevant to the dynamic development of the reflection pattern as the flight vehicle increases
 its pitch orientation rapidly.
 1.2 Ideal, Steady, Two-Dimensional Shock Wave Reflection
 Transition
 Typically, steady shock wave reflection research has considered the shock wave pattern in the reference
 frame of a simplified wedge for the purpose of fundamental analysis and experiment. Since the gas medium
 ahead of the reflection pattern has no velocity relative to the ground plane in reality, there is no boundary
 layer ahead of the reflection point. Therefore, in the reference frame of the wedge, the ground plane is
 approximated by an idealised, frictionless surface. For the range of positive incidence of the wedge for which
 the incident shock wave is attached to the wedge leading edge, there are two possible reflection patterns,
 viz. a two-shock system or regular reflection (figure 1.2(a)) and a three-shock system or Mach reflection
 (figure 1.2(b)). In the case of a regular reflection (RR), the incident wave, ?i?, deflects the free stream flow
 towards the reflection plane and the reflected wave, ?r?, returns the flow parallel to the reflection plane. In
 1
(a)
 (b)
 Figure 1.1: Simplified schematic of (a) steady shock wave reflection with the flight vehicle in steady, level
 flight and (b) dynamic shock wave reflection when the vehicle increases its pitch orientation rapidly
 2
.
 w
 replacements
 ?w
 ?
 ?
 M
 i
 r
 e
 (a) Regular reflection
 .
 ?
 M
 h
 m
 g
 w
 i
 r
 e
 t
 s
 ms
 (b) Mach reflection
 Figure 1.2: Simplified schematics of regular and Mach reflection generated by a wedge in a steady supersonic
 free stream
 the case of Mach reflection (MR), the incident wave, reflected wave, Mach stem, ?ms?, and the shear layer,
 ?s? meet away from the reflection plane at the triple point, ?t?. Figure 1.2 also includes definitions for the
 flow deflection angle, ?, wedge incidence, ?w, shock incidence, ?, Mach stem height, m, wedge chord, w, as
 well as the leading and trailing edge separation from the reflection plane, g and h.
 Steady state transition criteria between RR and MR generated by a wedge of infinite span in an ideal,
 steady supersonic flow are derived by taking into account local flow conditions at the reflection/triple point
 only and were published by Ben-Dor [3]. As the wedge incidence is increased gradually from an initial,
 steady RR, such that the reflection pattern approximates steady state at each point in time, there is a
 critical incidence beyond which transition to MR occurs. As the wedge incidence is decreased gradually from
 an initial, steady MR, such that the reflection is approximately steady at each instant, there is a critical
 incidence below which transition to RR occurs.
 For M ? 2.202, steady RR ? MR transition is predicted at the sonic/detachment condition and the
 reverse transition occurs at the von Neumann condition (see Ben-Dor [3]). This incidence-induced hysteresis
 phenomenon was confirmed experimentally by Ivanov et al. [22] and by various researchers with numerical
 simulation (for example Ben-Dor [2]). In the weak-reflection range, below M = 2.202, there is no von Neu-
 mann condition and transition occurs at the sonic/detachment condition in both directions. The difference
 between the sonic and detachment conditions is very small and is usually neglected in practice. The theoret-
 ical, transition criteria were derived specifically for ideal, steady flows and are invalid for the dynamic case
 of interest.
 3
1.3 Problem Statement
 There have been many experimental, numerical and analytical studies on various aspects of steady, two and
 three-dimensional shock wave reflections. But, there has been little published research to consider the effect
 of rapid wedge rotation on transition and the dynamic development of the wave pattern. There are a handful
 of publications that consider dynamic phenomena on shock wave reflections in steady flows. However, most of
 these focus on the prediction and observation of the incidence-induced hysteresis phenomenon in the steady
 case for M ? 2.202.
 The theoretical, transition criteria were derived specifically for steady flows and are invalid for the dynamic
 case of interest. Felthun & Skews [12] simulated a rapidly rotating, two-dimensional wedge in a M = 3.0
 free stream with an Euler CFD code and predicted significant deviation from the steady state, theoretical,
 transition criteria. Curvature was observed on the incident wave as illustrated in figure 1.3.
 The incident shock wave incidence, ?, at the reflection/triple point at transition was compared to the
 steady transition criteria and they predicted RR ? MR transition beyond the steady detachment condition
 and MR ? RR transition below the steady von Neumann condition. To date there has been no published
 experiment that has explored this finding. The only relevant, published dynamic experiment was conducted
 by Mouton & Hornung [36]. They demonstrated with a single, rapidly rotating wedge in a M = 4.0 free
 stream (asymmetric double wedge arrangement) the persistence of RR further into the dual solution domain
 with increasing wedge rotation speed. However, RR was not observed up to or beyond the steady detachment
 condition.
 An experimental and computational investigation of the dynamic flow field generated by a rapidly rotating
 wedge, with particular attention to two-dimensional RR ? MR transition and Mach stem development in
 the strong and weak-reflection ranges is proposed. Definitions for the strong and weak-reflection ranges are
 presented in Chapter 2.
 For the sake of clarity, when the incident shock is curved in the dynamic case, ? will refer to the shock
 incidence angle at the reflection point as indicated in figure 1.3. The wedge and shock incidence at transition
 are labelled ?WT and ?T respectively. A dimensionless parameter, ME , is defined to quantify the wedge
 rotation speed in terms of the free stream acoustic speed, viz. ME = VE/a?, where VE is the wedge edge
 speed of the leading or trailing edge, depending on the rotation centre, and a? is the free stream acoustic
 speed. A simple convention for the sign of VE is used to indicate the direction of wedge incidence change, viz.
 VE > 0 for increasing wedge incidence and VE < 0 for decreasing wedge incidence. This results in ME > 0
 for increasing wedge incidence and ME < 0 for decreasing wedge incidence. With respect to the experiment,
 the reflection pattern in the streamwise vertical plane of symmetry is of primary interest. All measurements
 4
Steady
 Dynamic
 ?
 Figure 1.3: Typical curvature observed in flow computations by Felthun & Skews [12] for rapid increasing
 wedge incidence
 are made on this vertical symmetry plane and all results will refer to the measurements made in this plane
 unless otherwise stated.
 1.4 Objectives
 The broad objectives of this study are as follows:
 ? To develop an experimental facility and appropriate computational models for the investigation of
 two-dimensional, dynamic shock wave reflection generated by a rapidly rotating wedge in the strong
 and weak-reflection ranges
 ? To measure and compute the dynamic effect of rapid wedge rotation on transition between two-
 dimensional regular and Mach reflection of shock waves in an ideal, steady, supersonic free stream.
 ? To explore the dynamic RR ? MR transition mechanism to identify transition criteria for the rapidly
 rotating wedge
 ? To explore the effect of other critical variables in the relevant parameter space with the aid of compu-
 tational modelling
 More detail on these objectives are presented after the theory and literature review in chapter 2.
 5
1.5 Chapter Overviews
 Chapter 2 presents the current theory for ideal, steady, two-dimensional shock wave reflection. Steady RR
 ? MR transition criteria are derived with pressure-deflection shock polars in the weak and strong-reflection
 ranges. Chapter 2 also summarises the few publications relevant to dynamic effects on shock wave reflection
 in steady supersonic flows. The research gaps are identified and more detail is provided on the objectives
 listed above.
 Chapter 3 describes the experimental setup. The method of measurement for the tunnel flow conditions
 is presented. The optical measurement technique and calibration of the schlieren system for accurate shock
 incidence and Mach stem height measurement are discussed. The design, development and operation of the
 experimental test rig are discussed in some detail. Data reduction from sample measurements are included.
 Chapter 4 documents details of the computational model. It also includes a brief description of the CFD
 codes used and results of a grid sensitivity study.
 Chapter 5 presents results from steady state experiments and two-dimensional computations in the strong
 and weak-reflection ranges. The transition point between RR and MR in a wind tunnel, in the strong-
 reflection range, is facility dependent as documented by Ben-Dor [4] and Ivanov et al. [22] and had to
 be determined for the CSIR tunnel. Transition and Mach stem height measurements for the steady state,
 baseline experiments are compared to theory and CFD calculations.
 Chapter 6 documents measurements and simulation results for the investigation of dynamic RR ? MR
 transition in the strong and weak-reflection ranges. Comparisons are made with steady state, baseline
 measurements. This chapter includes results of computations applied to scenarios beyond the capability of
 the existing experimental setup to explore the effect of other dependent variables in the parameter space. It
 also investigates the mechanism for dynamic RR ? MR transition.
 Chapter 7 documents measurements and simulation results for the investigation of dynamic MR ? RR
 transition in the strong-reflection range for the purpose of code validation. The code is applied to scenarios
 beyond the capability of the existing experimental setup to explore the effect of other dependent variables
 in the parameter space. Some ideas on three-dimensional effects are presented.
 Chapter 8 summarises the significant findings and makes recommendations for future work.
 6
Chapter 2
 Literature Survey
 2.1 Introduction
 This chapter reviews the relevant steady state theory and literature relevant to the dynamic shock wave
 reflection phenomena generated by a rapidly rotating wedge. Critical pressure-deflection shock polars are
 considered briefly to establish the steady state transition criteria for the ideal, steady, two-dimensional case.
 The source of the earliest known requirement for the rapid wedge rotation case is traced back to steady
 state experimental work started more than 30 years ago. The literature relevant to the rapidly rotating
 wedge is summarised, though spread over a few publications. Most of the research relevant to the dynamic
 case of interest was directed to the investigation of the well known incidence-induced hysteresis problem in
 the strong-reflection region. The research gaps and opportunities are identified and motivated. Research
 findings for consideration in the design of experiment and the computational method will be presented in
 chapters 3 and 4.
 2.2 Steady, Two-Dimensional, Theory
 Currently there is no unsteady theory for the dynamic case of interest. However, the steady theory may
 be used to assist in the interpretation of the dynamic flow field. For the range of wedge incidence, ?w, for
 which the incident wave is attached to the wedge, there are two possible reflection configurations, viz. the
 RR shown in figure 2.1 and the MR in figure 2.2.
 The two and three-shock theory of von Neumann (see Ben-Dor [3]) may be used to calculate the flow
 states in the vicinity of the reflection point of a RR and in the vicinity of the triple point of a MR. The
 theory is based on the following simplifying assumptions:
 ? steady flow
 7
? the discontinuities at the reflection/triple point are straight
 ? the flow obeys the equation of state
 ? the flow is inviscid
 ? the flow is thermally nonconductive
 ? the contact discontinuity at the triple point is infinitely thin
 At the reflection point, ?R?, of the idealised RR shown in figure 2.1, the free stream flow in region 1 is
 deflected ?2 towards the reflection surface by the incident wave, ?i?, and the reflected wave, ?r?, returns the
 flow parallel to the reflection surface by deflecting the flow downstream of the reflected wave ?3 away from
 the reflection surface. There is a zero net flow deflection at the reflection point, i.e.
 ?2 ? ?3 = 0 (2.1)
 The idealised MR in figure 2.2 consists of an incident wave, ?i?, a reflected wave, ?r?, a Mach stem,
 ?ms?, and a contact surface or slipstream, ?s?. The point of confluence of the three shocks is the triple
 point, ?t?. The contact surface arises where the flow downstream of the reflected wave and the Mach stem
 meet, i.e. where regions 3 and 4 meet. The flow on either side of the shear layer at the triple point has the
 same direction. Therefore, at the triple point of the MR:
 ?4 = ?2 ? ?3 (2.2)
 The trailing edge expansion introduces a curvature on the reflected wave and the contact surface. The
 curvature on the shear layer generates a converging nozzle in which the flow, bounded by the Mach stem and
 shear layer, accelerates from subsonic to sonic at the minimum nozzle area (see Hornung & Robinson [19]).
 As the wedge incidence is increased gradually from an initial, steady RR, such that the reflection pattern
 approximates steady state at each point in time, there is a critical incidence beyond which transition to MR
 occurs. As the wedge incidence is decreased gradually from an initial, steady MR, such that the reflection
 is approximately steady at each instant, there is a critical incidence below which transition to RR occurs.
 Steady state transition criteria between RR and MR for a steady wedge of infinite span in a steady supersonic
 flow are derived by taking into account local flow conditions at the reflection/triple point only. Ben-Dor [3]
 includes a detailed treatment of the analytical, two and three-shock theory presented by von Neumann [44].
 Pressure-deflection shock polars are a convenient, effective, graphical means of representing the analytical
 solution for ideal, steady, two-dimensional RR and MR. They are presented here to illustrate the derivation
 8
M
 i
 r
 R
 ?w
 e
 (a) Schematic of RR
 1 2 3
 i r
 R
 ?
 ?2
 ?3
 (b) Flow in the vicinity of the reflection point
 Figure 2.1: Simplified schematic of idealised regular reflection and flow conditions in the vicinity of the
 reflection point
 M
 i r
 ms
 t
 s
 e
 ?w
 (a) Schematic of MR
 1 2 3
 4
 ?2
 ?3
 ?4
 ?
 (b) Flow in the vicinity of the triple point
 Figure 2.2: Simplified schematic of idealised Mach reflection and flow conditions in the vicinity of the triple
 point
 of the steady RR ? MR transition criteria. Ben-Dor [3] and Chapman [5] may be consulted for more detail
 on pressure-deflection shock polars. Sample pressure-deflection shock polars for a M = 3.0 free stream are
 presented for illustrative purposes. The polar in figure 2.3 represents the static pressure rise, P/P?, that
 can be achieved through an incident oblique shock wave at M = 3.0 for a range of flow deflections, ?. P?
 is the free stream static pressure and P is the static pressure downstream of the incident wave. In the ideal
 case, the incident wave deflects the flow parallel to the wedge surface and ? equals ?w. The flow deflection
 at point ?M? is the maximum deflection possible by an oblique wave at M = 3.0. For larger values of ?,
 the oblique wave is detached. The largest pressure rise possible through an oblique wave is at zero flow
 deflection, corresponding to the normal shock solution, i.e. the intersection of the incident polar with the
 vertical axis at point ?N?. The sonic point on the polar, labelled ?S?, separates the polar into supersonic and
 subsonic segments where the downstream flow is supersonic or subsonic respectively (?O? to ?S? : supersonic
 segment; ?N? to ?S? : subsonic segment). For smaller values of flow deflection, there are two solutions for
 9
each value of ? and the downstream flow can be either subsonic or supersonic. Usually, in practice, the
 downstream flow is supersonic as in the case of interest here and the segment of the polar between ?S? and
 ?O? is applicable.
  0
  2
  4
  6
  8
  10
  12
 -40 -30 -20 -10  0  10  20  30  40
 ? [degrees]
 P
 /P
 ?
 Subsonic Subsonic
 Supersonic Supersonic
 M M
 S S
 O
 N
 Figure 2.3: Locus of flow conditions that can be achieved through an oblique shock wave in a M = 3.0 free
 stream
 The flow conditions downstream of the reflection/triple point are determined by superimposing pressure-
 deflection polars for the incident and reflected waves. Figure 2.4(a) includes the polar for the incident wave,
 ?IP?, and the polar for the reflected wave, ?RP?, corresponding to a RR at ?w = 12.0? at M = 3.0. Only
 half of the incident and reflected polars are shown for the analysis of a single wedge. Since the incident
 wave turns the incident flow parallel to the wedge surface (? = ?w = ?2), the wedge incidence determines the
 location of the origin of the reflected polar with respect to the incident polar.
 State 3 of the RR is given by the intersection of ?RP? with the y-axis, since there is a zero net flow
 deflection in region 3. State 3 is a weak solution for the reflected wave and the downstream flow is supersonic.
 The strong solution for the reflected wave, where the reflected polar intersects the y-axis on the subsonic
 segment of the polar, though theoretically possible, is not stable as was demonstrated by Hornung [17].
 Figure 2.4(b) includes the incident and reflected polars corresponding to a MR at ?w = 23.0? at M = 3.0.
 Flow states 3 and 4 of the MR are given by the intersection of the reflected polar with the incident polar.
 10
 0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3
 ? [degrees]
 P
 /P
 ?
 IP
 RP
 (a) Regular reflection at ? = 12.0?
  0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3,4
 ? [degrees]
 P
 /P
 ?
 (b) Mach reflection at ? = 23.0?
 Figure 2.4: Sample pressure-deflection polars for regular and Mach reflection at M = 3.0
 RR is theoretically possible as long as the reflected polar intersects the y-axis. There is a maximum flow
 deflection, ? = ?D, beyond which RR is not possible, i.e. the detachment condition shown in figure 2.5(a).
 At the detachment condition, the maximum deflection point on the reflected polar is tangent to the y-axis.
 Beyond ? = ?D the reflected wave of a RR can no longer satisfy equation 2.1 and only MR is possible, i.e.
 the reflected wave is not able to return the flow in region 3 parallel to the reflection plane. The smallest
 flow deflection at which MR is theoretically possible is ? = ?N , corresponding to the steady von Neumann
 condition shown in figure 2.5(b). At the von Neumann condition, the reflected polar intersects the y-axis at
 the normal shock solution of the incident polar. MR is not possible for ? < ?N . In air, with gamma = 1.4,
 there is no von Neumann condition below M = 2.202.
 In summary, for M > 2.202, both RR and MR are possible for ?N < ? < ?D, also termed the dual solution
 domain. Figure 2.6(a) includes a sample polar in the dual solution domain. The intersection of the reflected
 polar with the y-axis is the solution for the flow state in region 3 of the RR. The indicated MR solution for
 the flow in regions 3 and 4, downstream of the reflected wave and Mach stem respectively, is given by the
 intersection of the reflected polar with the incident polar (also applicable to figures 2.5(a) and 2.6(b)). The
 difference between the von Neumann and detachment conditions increases with free stream Mach number
 as shown in figure 2.8, plotted in terms of ?.
 The detachment condition also happens to be very close to the sonic condition shown in figure 2.6(b).
 At the sonic condition, corresponding to ?S , the flow downstream of the reflection point of a RR is sonic.
 For ? < ?S the flow downstream of the reflection point of a RR is supersonic.
 11
 0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3
 ? [degrees]
 P
 /P
 ?
 3,4 of MR
 (a) Detachment condition at ? = ?D = 21.5?
  0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3
 ? [degrees]
 P
 /P
 ?
 (b) von Neumann condition at ? = ?N = 19.7?
 Figure 2.5: Pressure-deflection at the detachment and von Neumann conditions at M = 3.0
  0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3
 ? [degrees]
 P
 /P
 ?
 3,4 of MR
 (a) Sample polar in the dual solution domain at ? = 21.0?
  0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -20 -10  0  10  20  30  40
 1
 2
 3
 ? [degrees]
 P
 /P
 ?
 3,4 of MR
 (b) Sonic condition at ? = ?S = 21.3?
 Figure 2.6: Pressure-deflection polars in the dual solution domain and at the sonic condition at M = 3.0
 12
 0
  1
  2
  3
  4
  5
  0  5  10  15  20  25
 A
 ? [degrees]
 P
 /P
 ?
 Figure 2.7: Pressure-deflection shock polar for a reflection at the detachment condition at M = 1.93
 2.3 Ideal, Steady, Two-dimensional RR ? MR Transition
 Criteria
 In air, with ? = 1.4, there is no von Neumann condition below M = 2.202. A sample reflected polar
 corresponding to the detachment condition at M = 1.93 is shown in figure 2.7. Hornung [16] defines ?weak?
 and ?strong? reflections in terms of the location of the maximum deflection point on the reflected polar at
 the detachment condition with respect to the normal shock solution on the incident polar. For example, in
 figure 2.7, the detachment point ?A?, lies below the normal shock solution on the incident polar and the
 reflection is termed a weak reflection. In contrast, at M = 3.0, the detachment point ?3? in figure 2.5(a)
 lies above the normal shock solution of the incident wave and the reflection is termed a strong reflection.
 At M = 2.202 the point of maximum flow deflection on the reflected wave at the detachment condition is
 co-incident with the normal shock solution on the incident polar. Accordingly, M < 2.202 is referred to as
 the weak-reflection range and M > 2.202 is referred to as the strong-reflection range.
 Perhaps, the significance of whether the reflection is regular or Mach in the dual solution domain is best
 considered at this point. Consider the polars in figure 2.5. The difference in pressure across the reflection
 point between the von Neumann condition and the detachment condition is approximately 3.3? P?. This
 13
difference increases to 12.9 ? P? at M = 4.0. The point of transition is important in determining the
 maximum pressure rise through the reflection and may be applied to investigations of aircraft sonic boom,
 supersonic intake design, etc. Azevedo & Liu [1] highlighted the relatively large contribution of the subsonic
 region behind the Mach stem of a MR to acoustic levels in comparison to the otherwise supersonic flow
 as applicable to supersonic engine intakes and supersonic vehicle design. Perhaps, this condition could be
 avoided or at least taken into account with a knowledge of the point of transition.
 In an experiment, there is a boundary layer on the wedge surface and ? at the reflection/triple point is
 no longer equivalent to ?w. Since flow deflection at the reflection/triple point cannot be measured directly,
 it is far more practical to refer to shock incidence at the reflection/triple point. The remaining discussion on
 transition criteria will refer to shock incidence at the reflection/triple point, ?, rather than flow deflection,
 ?. The shock incidence angle at the von Neumann, sonic and detachment conditions are labelled ?N , ?S and
 ?D respectively.
 Ben-Dor [3] may be consulted for a detailed review of the ideal, steady, two-dimensional RR ? MR
 transition criteria. In the weak-reflection range RR ? MR transition occurs at the sonic (or detachment)
 condition. In the strong-reflection range, both RR and MR are possible in the dual solution domain and and
 the shock incidence at transition, ?T , depends on the direction of wedge incidence change.
 2.3.1 RR to MR Transition in the Strong-Reflection Range
 Both RR and MR are possible in the dual solution domain between the von Neumann and detachment
 conditions. The smallest shock incidence at which MR is theoretically possible is ?N . The largest shock
 incidence at which RR is possible is ?D.
 Consider an initial, steady RR at an initial wedge incidence, ?wi, with an initial shock incidence, ?i,
 such that ?i < ?N (for example figure 2.4(a) at M = 3.0). Increasing incidence gradually from this initial
 incidence, the smallest incidence at which MR is theoretically possible is when ? = ?N at the von Neumann
 condition. If transition to MR were to occur at this point, there would be a smooth pressure change across
 the reflection/triple point through the point of transition. For this reason, the von Neumann condition was
 also termed the mechanical equilibrium condition by Henderson [14].
 Since there is a length scale associated with a Mach stem, Hornung et al. [18] proposed that RR ?
 MR transition only occurs when conditions change such that the wedge length scale is communicated to the
 reflection point (through the expansion fan), also referred to as the ?information? condition. In the ideal,
 steady case, the smallest incidence at which a communication path is established between the wedge and the
 reflection point is when the flow immediately downstream of the reflection point first goes sonic, M = 1.0
 14
at ?S , which is beyond the von Neumann condition. Since the flow downstream of the reflection point is
 supersonic for ? < ?S there can be no communication of the wedge length scale to the reflection point below
 ?S . Although MR is theoretically possible for ? ? ?N , the RR ? MR transition criteria for ideal, steady,
 two-dimensional flows in the strong-reflection region is the sonic condition at ?S .
 The sonic condition happens to be very close to the detachment condition, beyond which RR is not
 possible. It so happens in the ideal, steady case that the earliest incidence at which the length scale is visible
 to the reflection point is negligibly close to the incidence at which RR is no longer possible. In this case
 there has been no need to differentiate between the two as they are so close, e.g. at M = 3.0, ?S = 39.3?
 and ?D = 39.5?. For all practical intents and purposes this difference is usually neglected.
 In reality, if a disturbance in the free stream is strong enough to set up a temporary MR anywhere in
 the dual solution domain, the communication path is established and MR would be maintained since it is
 more stable than RR (see Hornung [17] and Hornung & Sudani [20]).
 In the general case, the RR ? MR transition criteria for steady, two-dimensional flows in the strong-
 reflection region is the length scale or information condition. In ideal flows this corresponds to the sonic
 condition. In real flows, it depends on the level of free stream turbulence and RR ? MR transition could
 occur anywhere between ?N and ?D.
 2.3.2 MR to RR Transition in the Strong-Reflection Range
 Consider an initial, steady MR with ?wi such that ?i > ?D (for example figure 2.4(b)). Decreasing incidence
 gradually from this initial incidence, the wedge length scale is communicated to the triple point as long
 as MR is maintained, since the flow downstream of the Mach stem is subsonic. As the wedge incidence is
 reduced, the Mach stem height decreases and MR is maintained until the von Neumann condition at ?N at
 which point the Mach stem height reduces to zero and the wedge length scale information disappears at the
 reflection point.
 2.3.3 Summary of Transition Conditions
 The steady state, theoretical transition conditions are summarised below.
 ? RR ? MR transition in the weak-reflection range occurs at ?S ? ?D
 ? RR ? MR transition in the strong-reflection range occurs at ?S ? ?D in the ideal case and anywhere
 between ?N and ?D in the real case, depending on the levels of free stream turbulence
 ? MR ? RR transition in the strong-reflection range occurs at ?N
 15
 0
  10
  20
  30
  40
  50
  1  2  3  4  5  6  7  8  9  10
 Regular Reflection Only
 Dual Solution Domain
 Mach Reflection Only
 Detachment
 von Neumann
 Freestream Mach Number, M
 In
 ci
 de
 n
 t
 Sh
 o
 ck
 W
 a
 v
 e
 A
 n
 gl
 e,
 ?
 [de
 gr
 ee
 s]
 Figure 2.8: Theoretical ?N and ?D between M = 1.6 and 10.0 for air with ? = 1.4
 The difference in ?T for increasing and decreasing incidence in the strong-reflection region for the ideal
 case, is the well known hysteresis loop that was first suggested by Hornung et al. [18]. This was confirmed
 with numerical solution by various researchers (for example Ben-Dor [2]) and in an experiment in a low-noise
 supersonic tunnel by Ivanov et al. [22] in 2003.
 2.4 The Persistence of Steady RR with a Length Scale Present
 at the Reflection Point
 Li & Ben-Dor [31] proved with analytical means that RR was stable between the von Neumann and sonic
 conditions, but unstable in the small region between the sonic and detachment conditions. However, as
 discussed previously the difference between the sonic/length scale condition and detachment for the ideal,
 steady case is extremely small and it may be difficult to investigate the stability of RR in this region with
 experiment or computation.
 In an experiment the ideal, horizontal reflection plane is set up with a symmetric wedge configuration.
 If the wedges are arranged asymmetrically, the reflection plane is not horizontal and the flow deflection
 at the reflection point is not parallel to the flow direction. Li et al. [32] considered various asymmetric
 16
configurations with pressure deflection shock polars and identified a very narrow region in which it was
 theoretically possible to achieve an overall regular reflection with a weak reflected wave on one of the
 reflections and a strong reflected wave on the other. An example of such a polar for the overall regular
 reflection at M = 4.96 is shown in figure 2.9 in which ?1 = 35.0? and ?2 = 14.58?, where ?1 is the wedge
 incidence of the bottom wedge and ?2 is the wedge incidence for the top wedge. The reflected polar, ?RP1?,
 corresponds to ?1, ?RP2? corresponds to ?2 and the incident polar is annotated ?IP?. The sonic point on
 ?RP2? is labelled ?S2? and the detachment point on ?RP1? is labelled ?D1?. For the setup shown in figure
 2.9, the two possible RR configurations for RP2 are weak in nature as both potential solutions are below
 its sonic condition. However, there is a weak and strong solution for RP1, labelled ?RRWS? and ?RRSS?
 respectively. If ?2 is increased there is a point beyond which there are two strong solutions for RP1 and
 two weak solutions for RP2. The flow does not have a choice except to set up one strong RR and one weak
 RR. The strong reflected wave would result in subsonic flow that would allow length scale information to be
 communicated to the reflection point from the leading edge of the trailing edge expansion. This demonstrates
 that it is theoretically possible to maintain RR in the presence of length scale information.
 Khotyanovsky et al. [27] confirmed this with an Euler simulation at M = 4.96. They calculated the
 range for which one weak RR and one strong RR would be set up in an asymmetric arrangement, i.e.
 15.595? < ?2 < 15.983? for ?1 = 35.0? (a range of approximately 0.4?). Their simulations with ?2 = 15.98?
 verified that it was indeed possible to set up such an arrangement. The important conclusion is that it is
 possible under specific conditions for RR to exist in the presence of length scale information.
 2.5 MR Configurations in a Steady Supersonic Free Stream
 There are texts that can be considered for details on the various Mach reflection configurations possible in
 steady, pseudo steady and unsteady flows (see Courant & Friedrichs [9] and Ben-Dor [3]). The configurations
 relevant to this investigation will be discussed briefly. The MR solution for ? > ?N in which the shear layer
 at the triple point is directed towards the reflection surface is referred to as Direct Mach reflection as shown
 in figure 2.10(a). In this case the intersection of the reflected and incident polars is at a positive incidence, to
 the right hand side of the y-axis. Stationary MR, in which the shear layer is parallel to the reflection plane
 at the triple point, is only possible at the von Neumann condition since there is a zero net flow deflection at
 this condition. An inverse MR shown in figure 2.10(b), has the slipstream directed away from the reflection
 surface. In this instance the intersection of incident and reflected polars is at a negative incidence. Sudani
 & Hornung [41] showed that it is possible to achieve a stable inverse MR with a steady wedge in a steady
 17
 0
  10
  20
  30
  40
  50
  60
  70
 -60 -40 -20  0  20  40  60
 ? [degrees]
 P
 /P
 ?
 IP
 RP2
 RP1
 D1
 S2
 RRSS
 RRWS
 Figure 2.9: Pressure deflection shock polar for the asymmetric case at M = 4.96, ?1 = 35.0? and ?2 = 14.58?
 supersonic flow by maintaining a permanent deflection in the wall downstream of the reflection point. An
 inverse MR was also observed in an unsteady case investigated by Felthun & Skews [12] for dynamic MR ?
 RR transition in a steady M = 3.0 free stream (see section 2.7).
 2.6 Early Origins of Rapid Wedge Rotation
 There are a handful of publications that investigate dynamic effects on shock wave reflections in steady
 flows. Most focus on the prediction and observation of the hysteresis phenomenon for the steady case in the
 (a) Direct MR (b) Inverse MR
 Figure 2.10: Direct and inverse Mach reflections possible in a steady supersonic free stream
 18
strong-reflection range.
 In 1979, Hornung et al. [18] first proposed the length scale transition criterion for RR ? MR transition
 in steady flows, viz. the sonic condition in ideal flows. This hypothesis was supported by results from
 pseudosteady experiments in a shock tube in which RR ? MR transition was observed close to the sonic
 condition. In their shock tube experiments, an incident planar shock travels through a stationary gas at
 constant speed over a plane wedge. The flow is termed pseudosteady because the RR that develops as the
 leading shock passes over the wedge is self-similar in time and the flow in the immediate vicinity of the
 reflection point in the reference frame of the reflection point is steady.
 Henderson & Lozzi [15] also presented data that showed the persistence of RR beyond the von Neumann
 condition with the diffraction of strong shocks over surfaces in an unsteady flow field. In their shock tube
 experiments the wedge is concave and the shock incidence decreases as the shock passes over the wedge. In
 the reference frame of the reflection point the flow is changing with time and is considered unsteady.
 These observations in pseudosteady and unsteady flows and the compelling physical arguments of Hornung
 et al. [18] led them to propose an incidence-induced hysteresis experiment in the dual solution domain for the
 steady case. It was predicted that RR ? MR transition would occur at the sonic or detachment condition
 and the reverse transition would occur at the von Neumann condition.
 In 1982, Hornung & Robinson [19] conducted a set of experiments in which the wedge incidence of a
 symmetric double wedge configuration, was increased and decreased gradually through the dual solution
 domain in steady, free stream conditions at M = 2.84, 3.49, 3.98 and 4.96. No hysteresis was observed and
 RR ? MR transition was observed repeatedly at the von Neumann condition, irrespective of the direction
 of incidence change. It was suggested that disturbances in the flow were sufficient to cause early RR ? MR
 transition at the von Neumann condition.
 Since 1995, there have been a number of publications on computational prediction and the experimental
 observation of hysteresis in the dual solution domain. Ben-Dor [3] includes a detailed review and bibliography
 of milestone publications in this field. Various computations with numerical solution of the Euler and
 Boltzmann equations supported the feasability of hysteresis (Vuillon et al. [45], Chpoun & Ben-Dor [7],
 Ivanov et al. [21],[25] and Ben-Dor [2]), due to the absence of free stream perturbations in the simulations.
 Flow simulations were successful in predicting hysteresis, but there was a failure to observe this phenomenon
 in an experiment.
 Computations were done to determine the effect of free stream density, pressure and velocity perturbations
 by Ivanov et al. [23], Khotyanovsky et al. [28] and Kudryavtsev et al [30]. Results supported the hypothesis
 that free stream perturbations present in the experiment, not in the flow computations, would cause early
 19
transition to MR.
 In 1997, Hornung [17] suggested that it was possible that RR ? MR transition would not be influenced
 by free stream disturbances in the experiment if the wedges were rotated sufficiently rapidly into the dual
 solution domain. He proposed establishing a steady RR before rotating the wedge rapidly into the dual
 solution domain, terminating the wedge motion just below the wedge incidence corresponding to detachment.
 This could prevent information of the wedge length scale from reaching the reflection point until the wedge
 was well into the dual solution domain. This is the earliest known published requirement for a rapid wedge
 rotation experiment, though its roots lie in the analysis of the steady problem.
 The first experiment to consider this idea was published by Mouton & Hornung [36] in 2008. The
 experimental rig consisted of two wedges, one of which was actuated by a motor to demonstrate the effect of
 rapid pitch on transition. Figure 2.11 includes a series of schlieren images capturing the hysteresis phenomena
 at M = 4.0. The total motion was executed in approximately 90 ms. Though the wedge arrangement was
 asymmetric the transition criteria were recalculated as per the method published by Li et al. [32]. The
 results presented by Mouton & Hornung [36] in figure 2.12 support the hypothesis made by Hornung [17].
 Rapid wedge rotation did indeed delay transition to MR and RR persisted further into the dual solution
 domain for increasing wedge rotation speeds, but not up to the detachment condition. They quantified wedge
 rotation speed with t/? . ? = w/u1 is the characteristic flow time for the lower wedge, where w is the wedge
 chord, u1 is the flow speed behind the incident shock of the lower stationery wedge and t is the time taken
 to rotate the upper wedge 10? from an initial wedge incidence, ?wi = 20?.
 2.7 Computational Simulation of Wedge Vibration and
 Impulsive Wedge Rotation
 By 1999, various experiments were being conducted in blow-down supersonic tunnels around the world
 that were attempting to observe the elusive hysteresis phenomena (see Fomin [13]). The failure to observe
 hysteresis was attributed to tunnel freestream noise. In contrast, continuum and kinetic models were quite
 successful in modelling the hysteresis phenomena, due to the absence of free stream noise (see Vuillon et al.
 [45], Chpoun & Ben-Dor [6], Ivanov et al. [21],[25] and Ben-Dor [2]). Experimental confirmation of hysteresis
 in the dual solution domain was not achieved until much later in 2003 when Ivanov et al. [22] conducted
 a series of experiments in a low-noise supersonic wind tunnel facility. Transition was observed repeatedly,
 close to the steady, theoretical conditions for RR ? MR transition.
 Prior to this result, the difference between results from steady simulation and experiment motivated
 20
Figure 2.11: A series of schlieren images from the experiment of Mouton & Hornung [36] demonstrating
 hysteresis in the dual solution domain at M = 4.0
  30
  32
  34
  36
  38
  40
  42
  44
  46
  30  31  32  33  34  35  36  37  38
 ?lower [degrees]
 ? u
 pp
 e
 r
 [de
 gr
 ee
 s]
 t/? = 1356
 t/? = 847
 t/? = 593
 t/? = 339
 Figure 2.12: Measured transition results from the dynamic experiment by Mouton & Hornung [36]
 21
Markelov et al. [34] to consider the effect of wedge vibration on RR ? MR transition with impulsive and
 periodic wedge oscillations. They simulated a steady initial RR at M 4.96 just below the detachment
 condition at ? = 38.0?, where ?D = 39.33?. The wedge was rotated impulsively at ? = 3 ? 105 deg/s for 1?
 about its trailing edge. Dynamic flow features were observed, but none significant enough to cause transition.
 However, impulsive rotation about the wedge leading edge did generate dynamic effects to the extent that it
 triggered transition. The impulsive start and stop resulted in an increased shock incidence at the reflection
 point and a substantial increase in pressure downstream of the reflected wave due to the interaction between
 the disturbing shock and the reflected wave. This substantial pressure rise was believed to be sufficient to
 trigger RR ? MR transition.
 They also simulated the effect of periodic wedge oscillation to determine the minimum wedge rotation
 speed required to trigger transition for various initial angles. A 1? amplitude oscillation about the wedge
 leading edge was simulated and this resulted in the periodic formation of compression and expansion waves
 as shown in figure 2.13. This led to a larger pressure rise downstream of the reflected wave than in the corre-
 sponding impulsive rotation case. Consequently, the minimum rotation speed required to trigger transition
 was lower than in the case of impulsive rotation. For reduced wedge rotation amplitudes, higher rotation
 speeds were required to trigger transition. The wedge rotation speeds used in the simulations were reported
 to be in the typical range of vibration frequencies of the test section of a typical blow-down wind tunnel
 facility during its operation. The results of this study supported the possibility that wedge vibration could
 be one of the reasons for differences between transition in experiment and simulation.
 Concurrently, Khotyanovsky et al. [28] investigated the effect of continuous rapid wedge rotation on the
 point of transition with Euler CFD on moving meshes. In contrast to the work by Markelov et al. [34],
 Khotyanovsky et al. [28] considered larger movements of the wedge. Rather than to investigate the dynamic
 phenomena generated by a rapidly rotating wedge in particular, the objective of the study was to determine
 the maximum permissable pitch rate that could be used in simulation without introducing dynamic effects.
 It was an exercise to support the investigation of the steady case. Wedges were rotated about the trailing
 edge at MT = 0.0002, 0.002, 0.01, 0.1 in a M = 5.0 free stream, where MT = w?/U? (w = wedge chord; ?
 = wedge rotation speed [rad/s]; U?=free stream speed). At MT = 0.1 there was significant curvature on
 the incident wave as illustrated in figure 2.14. Though the shock incidence at the reflection point did not
 correspond with the steady state shock angle for the same wedge incidence, the shock incidence at RR ?
 MR transition was close to the detachment condition. Shock incidence at the triple point for MR ? RR
 transition was not reported.
 While the work of Markelov et al. [34] and Khotyanovsky et al. [28] investigated the effect of rapid wedge
 22
Figure 2.13: Periodic formation of compression and expansion waves generated by wedge oscillation about
 its leading edge with amplitude = 0.5?; wedge rotation speed, ? = 8 ? 103 deg/s published by Markelov et
 al. [34]
 Figure 2.14: Curvature on the incident wave of a RR due to rapid wedge rotation, M = 5.0, g/w = 0.42 and
 rotation speed MT = 0.1 at ?w = 24.0? [28]
 23
 39
  39.5
  40
  40.5
  41
  41.5
  42
  0  0.02  0.04  0.06  0.08  0.1
 Dimensionless Rotation Speed, ME = VE/a?
 Sh
 o
 ck
 In
 ci
 de
 n
 ce
 a
 t
 Tr
 a
 n
 si
 tio
 n
 ,
 ? T
 [de
 gr
 ee
 s]
 ?D
 Figure 2.15: Computed effect of rapid wedge rotation on ?T for RR ? MR transition published by Felthun
 & Skews [12], M = 3.0, h/w = 0.9, ?wi = 20.0?
 rotation with particular focus on the implications for the steady case, Felthun & Skews [12] specifically
 investigated the dynamics of the rapidly rotating wedge case. The wedge was rotated impulsively about
 its leading edge at various rotation speeds for increasing and decreasing incidence at M = 3.0 using Euler
 computations. Resultant trailing edge speeds, VE , were between 1 and 10 % of the free stream acoustic speed.
 For increasing incidence, RR ? MR transition was delayed beyond ?D. Convex curvature was generated on
 the incident wave and transition was delayed further for increasing rotation speeds as shown in figure 2.15.
 Felthun & Skews [12] also investigated the effect of rapid wedge rotation on MR ? RR transition at a
 single test point with Euler CFD. A detailed parametric study was not conducted, only the general behaviour
 of the flow was observed. A steady MR was established at ?wi = 23.0? at M = 3.0. The wedge was started
 impulsively and rotated at a constant rotation speed with ME = ?0.05 until ?w = 16.0?, by which time
 transition had not occurred. The evolution of the reflection pattern was observed until transition to MR
 while the wedge was maintained at ?w = 16.0?. The rapid rotation resulted in concave curvature on the
 incident wave and MR ? RR transition was observed approximately 1.8? below ?N . During the wedge
 motion, the shear layer was directed away from the reflection surface, indicating an inverse Mach reflection
 configuration. Due to the wedge impulsive startup, the expansion wave from the wedge surface had the
 24
 0
  0.01
  0.02
  0.03
  0.04
  0.05
  0.06
  0  0.5  1  1.5  2  2.5  3  3.5  4
 a?t/w
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Experiment
 Analytical model
 Figure 2.16: Predicted and measured dynamic Mach stem development by Mouton & Hornung [36] at
 M = 4.0, g/w = 0.3907, ?w = 23.0?, ? = 1.4
 initial effect of moving the triple point away from the reflection surface. No Mach stem measurements were
 made and no experiments were done to verify the computations.
 2.8 Dynamic Mach Stem Development for a Stationery Wedge
 Though not directly applicable to the continuously rotating wedge, the following work is highlighted as it
 represents one of the few pieces of research in the field of dynamic shock wave reflection. Mouton & Hornung
 [36] conducted another experiment in which an initial RR was setup within the dual solution domain. RR ?
 MR transition was subsequently triggered with the deposition of laser energy on the wedge surface and the
 dynamic Mach stem development was measured and compared to results of a moving triple point analysis
 published earlier by the same authors [35]. This is the first analytical model for the prediction of time
 dependent Mach stem growth for a stationary wedge. There was close agreement between the experiment
 and the analytical solution shown in figure 2.16 at M = 4.0, g/w = 0.3907, ?w = 23.0?, ? = 1.4.
 25
2.9 Research Gap
 The following research gaps have been identified and will be investigated in this work. All aspects will be
 conducted in the strong and weak-reflection ranges unless specified otherwise.
 1. An experimental investigation into the effect of a rapidly rotating, symmetric wedge configuration on
 two-dimensional RR ? MR transition in the strong and weak-reflection ranges will be conducted. The
 effect of rapid rotation on MR ? RR transition will also be considered. This requires the development
 of a novel facility to generate and measure the dynamic phenomena of interest. An Euler CFD code,
 developed by Felthun [11] at the University of the Witwatersrand, will be used to simulate all experi-
 ments. This experiment will potentially address two gaps in the published literature. Both are listed
 subsequently.
 2. The first, is an experimental verification of the dynamic phenomena published by Felthun & Skews
 [12]. They reported computational results of a rapidly rotating wedge in a steady M = 3.0 free stream
 that showed RR ? MR transition beyond the steady detachment condition at ?D and MR ? RR
 transition below the steady von Neumann condition at ?N . To date there has been no experimental
 confirmation of the predicted dynamic effect.
 3. The proposed experiment may also address aspects of the original idea proposed by Hornung [17] in
 relation to the observation of RR (or lack thereof) in the dual solution domain in a facility where small
 perturbations are always present. Mouton & Hornung [36] demonstrated with a single, rapidly rotating
 wedge that it was indeed possible to maintain RR further into the dual solution domain for increasing
 wedge rotation speeds but RR was not maintained until detachment. If the free stream noise levels
 in the facility employed are indeed sufficient to prevent hysteresis in the strong-reflection range, is it
 possible to maintain RR until ?D with a larger wedge rotation speed? How do Euler computations
 and measurements from experiment compare under these circumstances? On the other hand, if the
 free-stream noise levels in the facility employed are significantly small such that hysteresis is observed
 for the steady case, how does rapid wedge rotation effect transition?
 4. The mechanism for ideal, two-dimensional, RR ? MR transition in the dynamic case with rapid
 wedge rotation will be investigated and compared to the steady transition criteria, viz. the steady
 sonic/detachment condition and the mechanical equilibrium/von Neumann condition.
 5. Mach stem measurements will be made for the rapidly rotating wedge and compared with two-
 dimensional computations. These will be considered in view of the difference in trend observed between
 26
the steady, two and three-dimensional result reported by Ivanov et al. [22] (discussed in chapter 4).
 6. Computational simulations will be used to investigate the effect of pivot point, wedge rotation speed
 and initial incidence not considered previously in the strong and weak-reflection regions.
 27
Chapter 3
 Experimental Method
 3.1 Introduction
 The dynamic effect of rapid wedge rotation on the transition between two-dimensional (2D) regular and Mach
 reflection of shock waves in an ideal, steady, supersonic free stream is of primary interest in this investigation.
 This is explored with experimental and numerical methods. The numerical method is discussed in chapter
 4. Background literature relevant to the experiment is reviewed briefly. This chapter presents details of
 the experimental setup which includes a rig to generate the dynamic phenomena of interest in the weak
 and strong-reflection ranges. All experiments were done in the blow-down supersonic wind tunnel at the
 CSIR. The facility and the measurement of tunnel conditions are discussed. A schlieren flow visualisation
 system was developed for these experiments and images of the dynamic flow field were recorded with a
 high-speed digital camera. The optical measurement technique and its calibration are outlined. The rig
 design, development and operation are described in some detail. Sample experimental data is presented
 to demonstrate the data reduction process. Uncertainties for all measured quantities are calculated and
 summarised.
 3.2 Background
 3.2.1 Simulation of the Ground Plane
 In reality, the gas medium ahead of the supersonic flight vehicle is stationary and there is no boundary layer
 on the ground plane. In an experiment, the ideal, frictionless reflection plane is generated with a double
 wedge configuration arranged symmetrically about a horizontal plane. This is a widely adopted approach
 and has been used extensively in the shock wave community (see Hornung & Robinson [19] for example). The
 symmetric arrangement of the wedges about a horizontal image plane sets up a perfectly rigid, frictionless
 28
and adiabatic wall. This ensures that the reflection point is not contaminated by a boundary layer that
 would develop on a surface in the tunnel. This implies that the rig must consist of two wedges arranged and
 actuated symmetrically about a horizontal reflection plane.
 3.2.2 Evaluation of Free Stream Turbulence Levels in the Supersonic Facility
 Many attempts were made to observe the elusive hysteresis phenomena since the early analytical work by
 von Neumann [44] and the experimental work by Henderson & Lozzi [15], Hornung et al. [18] and Hornung
 & Robinson [19]. Since 1995, there have been a number of publications and considerable debate on the
 computational prediction and observation of hysteresis in the strong-reflection range. Ben-Dor [3] includes a
 detailed review and bibliography of milestone publications in this field. Since the gas medium ahead of the
 flight vehicle is stationary, the transition criteria in a free stream without perturbations are correct.
 In general, Euler codes are able to model hysteresis due to the absence of free stream perturbations (see
 Vuillon et al. [45], Chpoun et al. [6], Ivanov et al. [21], [25] and Ben-Dor [2]). Sudani et al. [42] were
 able to observe both RR and MR in the dual solution domain, though not repeatable, and demonstrated
 that hysteresis could not be observed with a perturbed free stream flow by introducing water droplets into
 the free stream. In 2003, Ivanov et al. [22] published the first set of repeatable, experimental data that
 confirmed the hysteresis phenomena originally proposed by Hornung et al. [18] more than 20 years earlier.
 The experiment was conducted at M = 4.0, with wedges of an aspect ratio of 3.37, in a low noise supersonic
 tunnel.
 All experiments in this work were done in the 450 mm ? 450 mm supersonic wind tunnel at the Council
 for Scientific and Industrial Research, South Africa. As steady state RR ? MR transition is dependent on
 the level of free stream turbulence in the strong-reflection range, the steady transition conditions for the
 CSIR facility must be determined. This is an indirect measurement of the level of free stream turbulence in
 the facility. This baseline was used to evaluate the results from the dynamic experiments.
 3.2.3 Three-Dimensional Wedge Edge Effects
 Skews [40], [39] highlighted the issue of three-dimensional (3D) influences, from the wedge corner signal, on
 the reflection pattern in the streamwise vertical plane of symmetry. Care must be taken to ensure that the
 transition angle, ?T , is not influenced by 3D wedge edge effects, by testing with sufficiently large aspect ratio
 wedges.
 Ivanov et al. [24] published 2D and 3D computational and experimental data. The data presented here
 was estimated from the publication and is shown in figure 3.1. Figure 3.1 shows the close agreement between
 29
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  0.45
  34  36  38  40  42  44  46
 Shock incidence angle, ? [degrees]
 N
 o
 n
 -
 di
 m
 en
 si
 o
 n
 a
 lM
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Euler CFD
 3D Euler CFD : b/w = 3.75
 3D Euler CFD : b/w = 2.0
 Experiment : b/w = 3.75
 Figure 3.1: Two and three-dimensional Mach stem measurements from experiments and Euler predictions
 published by Ivanov et al. [24] for a static wedge at M = 4.0, g/w = 0.56
 the 3D Euler CFD predictions of Ivanov et al. [24] and their measurements from experiments with the finite
 aspect ratio wedge. This agreement established confidence in their 2D Mach stem predictions with Euler
 CFD. Results show that there are 3D effects on the Mach stem height even for a wedge with an aspect ratio
 of 3.75. In an experiment, the flow downstream of the Mach stem is subsonic and 3D downstream influences
 are always present. Though the point of transition for large aspect ratio wedges approaches the theoretical,
 2D transition value, the Mach stem height is always influenced by 3D effects and is always smaller than the
 2D result. The difference in Mach stem height between the 2D and 3D result for the same shock incidence
 must be considered when comparing 2D computed results with measurements from experiments. Figure 3.1
 serves as a useful summary of the expected difference between steady 2D CFD and 3D experimental results.
 3.3 Supersonic Wind Tunnel
 All experiments in this work were conducted in the blow-down, supersonic wind tunnel facility at the Council
 for Scientific and Industrial Research in Pretoria, South Africa illustrated in figure 3.2. The wind tunnel has
 a 450 mm ? 450 mm test section and a free stream speed range of 0.6 ? M ? 4.3. Supersonic flow conditions
 30
in the test section are achieved with a semi-flexible nozzle upstream of the test section (see figure 3.3). The
 nozzle consists of two stainless steel plates (top and bottom wall) fastened to a pair of ?nozzle throat blocks?,
 positioned symmetrically about the tunnel horizontal centre plane, at the nozzle minimum cross sectional
 area. The supersonic test section Mach number is achieved by controlling the vertical position of the throat
 blocks (and hence the nozzle throat area) with two large hydraulic jacks. The steel plates are shaped with 7
 additional hydraulic cylinders per plate to achieve the required nozzle shape at each set point. The shape of
 the throat blocks is fixed and they are re-oriented at each set point to ensure a smooth transition between
 the blocks and the steel plates.
 A 500kW compressor pressurises 4 large tanks, with a total volume of approximately 350m3, to 12.0
 bar before each experiment. This provides approximately 20 - 30 seconds of useful test time and depends
 on the experiment test conditions. This is adequate for the purposes of this experiment as the motion of
 interest requires less than 20 msec of steady flow. An air dryer, installed between the compressor and storage
 tanks, ensures that dry air is delivered to the test section. At tunnel startup a shutoff valve discharges the
 compressed air into the tunnel. A hydraulically actuated pressure control valve controls the stagnation
 pressure in the settling chamber and test section. A single honeycomb mesh, positioned in the settling
 chamber, pre-conditions the flow to reduce the turbulence in the test section. The test section turbulence
 levels are unknown at this stage.
 Air Storage
 Settling Chamber Test Section
 15.5 m
 Figure 3.2: Side view of supersonic wind tunnel facility at the CSIR, South Africa
 The tunnel control is automated and consists of a National Instruments based control system. The wind
 tunnel is operated remotely during the experiments from a Labview interface. Tunnel transducer data is
 acquired at 500Hz with a National Instruments data acquisition system, i.e. total pressure, test section
 static pressure, atmospheric pressure and total temperature. A more detailed description of the acquisition
 system is included in Appendix A. Technical specifications of all the pressure transducers and the pressure
 calibration standard may also be found in Appendix A. Pressure transducer calibrations were conducted
 before the experiments and the calculated calibration coefficients and linear regression statistics are included
 31
J1 - J 14 : Hydraulic actuators to shape nozzle
 J1 J2 J3 J4 J5 J6 J7
 J15
 J15 - J16 : Hydraulic actuators to control throat area
 J8 J9 J10 J11 J12 J13 J14
 J16
 Test Section
 Flow Direction
 Throat Block
 Throat Block
 Figure 3.3: Simplified schematic indicating tunnel nozzle shape control
 in Appendix A. All rig and high-speed imaging operations were conducted remotely from the tunnel control
 room, due to Environment Health and Safety regulations at the CSIR.
 3.3.1 Mach Number Measurement
 Mach number control in the test section is achieved with accurate control of the nozzle throat area through
 accurate positioning of the throat blocks. The test section Mach number is governed, to a large extent, by
 the area ratio between the nozzle throat and the tunnel test section. The following ideal, isentropic relation
 provides an estimate of the required area ratio between the nozzle throat and the test section (see Zukrow
 & Hoffman [46]):
 M = A
 ?
 A
 {
 2
 ? + 1
 (
 1 +
 ? ? 1
 2
 M2
 )} (?+1)
 2(??1) (3.1)
 The following relationship between the stagnation pressure, static pressure and test section Mach number
 is applicable to isentropic flows (Zukrow and Hoffman [46]):
 32
M =
 [{(
 PO
 p
 ) ??1
 ?
 ? 1
 }{
 2
 ? ? 1
 }] 12
 (3.2)
 Assuming isentropic conditions between the settling chamber and the test section, the stagnation pressure
 is constant from settling chamber to test section. A Mach number measurement on the test section wall is
 derived from the stagnation pressure measurement in the settling chamber and a static pressure measurement
 on the wall of the test section upstream of the model of interest. The CSIR tunnel calibration requires the
 application of a correction factor to the wall measurement to derive the test section Mach number. All
 applicable correction factors are included in Appendix A. All tunnel transducers were calibrated by the
 author as part of the experimental setup, but the current CSIR test section calibration is assumed. From
 the uncertainty in the most recent pressure transducer calibrations and the CSIR test section calibration, a
 value for Mach number uncertainty is estimated. The uncertainty calculation is documented in Appendix
 A. A Mach number uncertainty of ?M = ?0.03 is assumed across the range of experimental test conditions.
 3.3.2 Stagnation Temperature Measurement
 In the unsteady case, the propagation speed of information in the gas medium with respect to the rotation
 speed is an important parameter. Since the acoustic speed is dependent on the static temperature in the
 test section, a static temperature measurement was required.
 The flow between the settling chamber and test section is assumed isentropic. Given that:
 TO
 T
 = 1 +
 ? ? 1
 2
 M2 (3.3)
 , the test section static temperature was derived from a settling chamber temperature measurement (close
 to total temperature) and the derived test section Mach number.
 Due to the contraction ratio from the settling chamber to the test section (? 6.22), the Mach number
 in the settling chamber is very small, and the difference in total and static air temperature in the settling
 chamber is also very small. The largest issue for the total temperature measurement was the short duration
 test time. The test time was insufficient for the sensor reading to stabilise in some instances (see figure
 A.5). The stagnation temperature probe is shown in figure A.3. It consists of an entrance hole that faces the
 oncoming flow and a bleed hole on the leeward side. The probe houses a PT100 RTD (resistance temperature
 detector) sensor in the gas path between the entrance and bleed holes. A custom PT100 with an aerated tip
 was acquired for these experiments to maximise the heat transfer rate to the sensor. Details of the sensor
 and calibration data are included in Appendix A.
 33
PT 100
 Sensor Flow
 Flow
 Settling Chamber Wall
 Figure 3.4: Schematic of stagnation temperature probe in settling chamber
  10
  15
  20
  25
  30
  35
  40
  0  5  10  15  20  25
 Total Temperature [deg C
 ]
 Time [s]
 (a)
  28
  30
  32
  34
  36
  38
  7  8  9  10  11  12
 Total Temperature [deg C
 ]
 Time [s]
 (b)
 Figure 3.5: (a) Sample total temperature probe measurement and (b) magnified view of select data range
 A sample probe measurement from an experiment is shown in figure 3.5. Before the experiment the
 temperature reading was relatively constant at approximately 296.6 K. At tunnel startup there was a rapid
 rise in temperature as the startup shock passed through the settling chamber. As the tunnel flow stabilised
 the probe temperature measurement dropped and began to stabilise. It is assumed that the transient is only
 due to the short test time, i.e. the variation in actual total temperature in the settling chamber is small
 in comparison to the probe transients. From the data between the highlighted markers, it is possible to
 extrapolate a settling temperature of approximately 29.0?C. The total temperature measurement quoted
 for each experiment was determined in a similar manner. The sharp drop in temperature at the end of the
 experiment was due to rapid heat transfer between the air in the settling chamber and the relatively cold
 air in the test section as the supply air is turned off.
 With TO = 29.0? 0.5?C and M = 3.0? 0.03, this results in a maximum uncertainty of approximately
 ?0.75% on the acoustic speed. For any value of wedge rotation speed this translates to an uncertainty of
 34
approximately ?0.75% on wedge rotation speed and dimensionless rotation speed, ME = VE/a?. Sample
 calculations may be found in Appendix A. This uncertainty would have a negligible effect on the transition
 point for the dynamic case.
 3.4 Flow Visualisation
 A standard, z-type schlieren system (see Settles [37]) was designed for these experiments. The system is
 shown in figure 3.6. The optical design was done in accordance with the guidelines provided by the Eastman
 Kodak company [8]. There was a limited budget on this project and every attempt was made to minimise
 cost. Parabolic mirrors (6 inch in diameter, f/8) were acquired from a telescope retailer . Old mirror mounts
 were salvaged and new stands were designed. Stands were designed to be sufficiently heavy and were mounted
 on rubber dampers to eliminate the possibility of system vibration during testing. The light source from
 the previous system was used. The arrangement of optical components attempted to maximise coverage on
 the camera imaging sensor. Various colour masks were tested at the cutoff plane (see Fig. 3.7). The multi-
 coloured mask provided more qualitative information on the three-dimensional nature of the flow field than
 the standard three-colour mask (discussed in chapter 5). The technical specification of the schlieren system
 is documented in Appendix B. Alignment of the schlieren system was done in accordance with the guidelines
 of Settles [37]. See Appendix B for more detail on the procedure and alignment equipment employed.
 3.5 Image Calibration Technique
 Accurate measurement of ?w, ? and Mach stem height, m, from images was important to this investigation.
 The calibration of the schlieren optics for angular and co-ordinate measurements involved imaging a 5 mm
 ? 5 mm uniform grid on the test section window (see figure 3.8). There is no visible pin cushion or barrel
 distortion. The grid was generated with CAD software and printed on a transparency with a laser printer.
 The transformation from the image co-ordinate system to the object co-ordinate system is known from an
 image of the calibration grid.
 An image of the calibration grid was captured before each experiment. The maximum image resolution
 during the dynamic experiments was 512 ? 512 pixels at the required frame rate (discussed later in this
 chapter). Absolute orientation of the uniform grid was measured with a calibrated digital inclinometer (see
 Appendix B for technical specifications and calibration). Locating markers were fixed on the test section
 window and were imaged for the calibration and during the experiments. The camera was stationary between
 the time the calibration grid was imaged and the experiment was completed. The locating markers were
 35
Light Source
 Mirror 1
 Mirror 2
 Focussing Optics
 Test Section
 Figure 3.6: Schematic of schlieren flow visualisation setup (colour mask and high-speed camera not shown)
 used to determine if there was any camera movement between calibration and testing.
 Co-ordinates of points not co-incident with the grid were calculated with linear interpolation from the
 four closest grid points. A software routine was written to transform pixel co-ordinates to object space
 co-ordinates. The calculation routine is documented in Appendix B. This procedure enabled the accurate
 measurement of ?, ?w and m from images. A critical and limiting factor in the uncertainty estimation
 of spatial measurements is the available resolution and object magnification. The uncertainty of spatial
 measurements using this method increases with a decrease in image resolution and a decrease in imaging
 sensor coverage.
 The uncertainty in the measurement technique was determined from angle and distance measurements
 on the test image shown in figure 3.8(b). The test pattern consists of lines at known angular orientations.
 With the exception of the lines at ?2.5? and ?177.5?, the remaining lines are oriented at ?5? intervals.
 Pixel co-ordinates of 10 points along lines of interest were transformed to object space co-ordinates to
 determine the orientation of each line with a linear regression of the measured points. Results of a calibration
 check are documented in table 3.1. A summary of statistics on the measurements are documented in table
 3.2. From the summary statistics in table 3.2 the uncertainty in angular measurement is estimated at
 approximately ?0.3?, a value slightly larger than 95% of the calculated errors.
 The uncertainty of distance measurement was estimated by measuring the distance between the ends of
 36
Table 3.1: Results from a calibration check on a test image for measurements between ?20.0? and 40.0?
 Data Point Calculated Angle (0.0?) Error Calculated Angle (10.0?) Error
 1 0.104 -0.104 9.930 0.070
 2 0.066 -0.066 9.800 0.200
 3 0.123 -0.123 9.892 0.108
 4 0.051 -0.051 10.001 -0.001
 5 0.041 -0.041 9.921 0.079
 Data Point Calculated Angle (20.0?) Error Calculated Angle (30.0?) Error
 1 20.000 0.000 30.176 -0.176
 2 20.121 -0.121 30.174 -0.174
 3 19.889 0.111 29.943 0.057
 4 19.956 0.044 30.289 -0.289
 5 19.911 0.089 30.248 -0.248
 Data Point Calculated Angle (40.0?) Error Calculated Angle (?5.0?) Error
 1 40.030 -0.030 -4.925 -0.075
 2 39.978 0.022 -4.967 -0.033
 3 40.061 -0.061 -5.150 0.150
 4 40.130 -0.130 -5.275 0.275
 5 40.166 -0.166 -5.156 0.156
 Data Point Calculated Angle (?10.0?) Error Calculated Angle (?20.0?) Error
 1 -10.076 0.076 -20.093 0.093
 2 -10.283 0.283 -20.196 0.196
 3 -10.283 0.283 -20.015 0.015
 4 -10.222 0.222 -20.316 0.316
 5 -10.143 0.143 -20.291 0.291
 Table 3.2: Summary of statistics for angular measurement error
 Mean Error 0.035?
 Standard Deviation 0.155?
 Maximum Error 0.316?
 95% of Calculated Errors < 0.28?
 37
Figure 3.7: Sample images obtained with various colour masks
 Table 3.3: Calibration check for measurement uncertainty on distance
 Data Point Measured Distance [mm] Error [mm]
 1 4.426 0.064
 2 4.442 0.080
 3 4.301 0.061
 4 4.417 0.055
 5 4.297 0.065
 6 4.422 0.060
 7 4.435 0.073
 8 4.305 0.057
 any two lines bounding a five degree arc on the test image and comparing the result with the true value
 (approximately 4.362 mm). Results are presented in table 3.3. The maximum error was approximately
 0.08 mm. This translates to a measurement uncertainty on m/w of approximately ?0.002. This does not
 necessarily account completely for the uncertainty in locating the triple point on an image of a MR. By
 considering images of a MR, the uncertainty in the non-dimensional Mach stem height measurement is
 estimated at approximately ?m/w = ?0.004, 0.4% of the wedge chord.
 3.6 High-Speed Image Acquisition
 High-speed schlieren images of the dynamic flow field were captured with the Photron Ultima APX-RS
 high-speed digital camera. The camera has a 10-bit CMOS sensor with 1024 ? 1024 pixels, with a pixel size
 38
(a) (b)
 Figure 3.8: Image of (a) 5 mm ? 5 mm square calibration grid with locating markers and the (b) test image
 captured with the high-speed camera at 512 x 512 pixel resolution used for all dynamic experiments
  0
  2
  4
  6
  8
  10
  12
 -0.35 -0.25 -0.1 0 0.1 0.25 0.35 0.4
 Measurement error [degrees]
 N
 u
 m
 be
 r
 o
 fS
 a
 m
 pl
 es
 Figure 3.9: Distribution of deviation from target angle
 39
of 17?m. Image focussing onto the camera CCD chip was achieved with a 100mm focal length aspherical
 achromatic lens. The larger radius of curvature on the achromatic lens faces the larger conjugate on the
 optical axis, i.e. the test object. The smaller radius of curvature faces the camera CCD chip. The camera
 opening to the lens was covered with a UV filter to protect the imaging chip as seen in figure 3.10. Spec-
 ifications of the camera and focussing lens are documented in Appendix B. Images were captured at the
 camera?s maximum resolution (1024 x 1024 pixels) at 250 frames per second for the steady state experiments
 and at 10000 frames per second (512 x 512 pixels) for the dynamic experiments. Shutter speed was reduced
 to 1/20000s for the dynamic experiments to reduce motion blur that resulted from the rapid rotation of the
 wedge.
 Figure 3.10: Photograph of the Photron Ultima APX-RS high speed camera with a UV filter to protect
 the imaging sensor and an aspherical achromatic lens for focussing. The schlieren colour mask is positioned
 ahead of the focussing lens.
 3.7 Summary of Measurement Uncertainties
 Values for ?M , ?TO, ?m/w and ?? (or ??w) are summarised in table 3.4.
 40
Table 3.4: Summary of measurement uncertainties
 Quantity Value
 ?M ?0.03
 ?TO ?0.5K
 ??, ??w ?0.3?
 ?m/w ?0.004
 3.8 Dynamic Shock Wave Interaction Rig
 This section presents various aspects of the rig design. Critical aspects of the system requirements specifi-
 cation, design considerations and constraints are documented. Four versions of the rig were tested in the
 facility before a final, satisfactory design was achieved. A brief summary of the development history of the
 rig is also presented. A limited series of drawings are presented in the body of this work for the purpose of
 illustration. Calculations for the sizing of the actuators are included in Appendix C.
 3.8.1 System Requirements Specification
 The following basic requirements were used to develop the design concept. These include requirements/constraints
 gathered from the literature.
 Functional and Performance Requirements
 1. The rig shall rotate the wedges to achieve increasing and decreasing incidence
 2. The rig shall enable wedge rotation between 0? and approximately 40?
 3. The wedge shall rotate about its trailing edge
 4. The rig shall generate steady state data as well as dynamic data (not in the same experiment)
 5. The wedge rotation speed for the steady state experiments shall not result in |ME | > 0.001
 6. The required order of magnitude for the wedge rotation speed in the dynamic experiments shall result
 in ME ? +0.01
 Interface Requirements
 1. The rig shall be designed for installation and operation in the CSIR supersonic blow-down facility
 illustrated in figure 3.2.
 2. The rig shall be mounted on either the existing tunnel pitch sector and/or the tunnel cart shown in
 figure 3.11.
 41
Test Section
 Schlieren
 Window
 Pitch SectorTunnel Cart
 Flow Direction
 To Tunnel Diffuser
 Section
 Tunnel Second
 Throat Station
 Movement of Pitch Sector
 Hydraulically Operated
 Figure 3.11: Schematic of a section of the supersonic tunnel with walls removed showing available tunnel
 support systems for the mounting of the rig (image provided courtesy of the CSIR)
 3. The rig shall be designed such that the wedges and the reflection pattern are visible to the schlieren
 system through the glass windows in the test section.
 4. Rig and camera operations shall be conducted remotely from the tunnel control room during the
 experiment. A rig control interface shall be developed.
 5. The high speed camera shall be triggered manually or automatically.
 6. Rig installation and operation shall be safe.
 Constraints
 1. The rig shall consist of a double, symmetric wedge configuration
 2. Both wedges shall be arranged and actuated symmetrically about a horizontal, symmetry plane, parallel
 to the tunnel free stream
 3. The vertical separation between the wedge and the reflection plane shall ensure that the reflected wave
 does not intersect the wedge surface and the expansion fan from the wedge trailing edge does not
 intersect the reflection/triple point at M = 2.0 and M = 3.0
 4. Each wedge shall have a minimum aspect ratio, b/w = 4.0, where b is the wedge span and w is the
 wedge chord
 42
5. The rig cross sectional area shall not cause blockage to the extent that the tunnel does not start. Guide-
 lines provided in a Naval Ordnance Report [43] may be used to estimate this maximum permissable
 blockage (see Appendix C)
 6. The rig shall be designed such that shock waves from the test article or rig, on reflection from the test
 section wall, must not interfere with the test article or flow phenomena of interest
 7. Each experiment shall not exceed 15 seconds due to the limit on stored air
 Environmental Conditions
 1. The rig shall be designed to operate between M = 2.0 and M = 3.0
 2. The rig shall operate within the envelope in figure 3.12. The lower boundary of the test envelope
 represents the minimum pressure required for tunnel startup. The upper boundary represents the
 maximum design strength of the schlieren windows in the test section. Curves for PO = 600 to 1000
 kPa are not shown as they would not be applicable for the range of free stream conditions required
 3. The rig and test article shall be designed to withstand the aerodynamic load experienced on tunnel
 startup as the startup shock passes downstream
 4. The total temperature ranges from 0.0?C to 35.0?C
 3.8.2 Design Description
 The rig consists of two large aspect ratio, symmetrically opposed wedges, a support structure and an actuator
 as illustrated in figure 3.13. The wedges are mounted on the support structure and are actuated symmetrically
 about a horizontal plane of symmetry. The use of finite aspect ratio wedges, as opposed to wedges spanning
 the entire test section, are necessary to avoid the complex shock-boundary layer interaction on the test
 section window, which will produce confusing features on the schlieren images. The approach adopted in
 this investigation was to actuate both wedges symmetrically rather than a single wedge with corrected,
 theoretical transition criteria to account for the asymmetry. With the adopted design approach the wedges
 may also be mounted asymmetrically. Each wedge has an aspect ratio of b/w = 4.25 with w=40.0 mm and
 this is sufficient to ensure 2D RR ? MR transition for 2.0 < M < 3.0.
 The wedge pivot point was maintained as close to the trailing edge as possible and was selected to
 minimise the vertical movement of the trailing edge. The vertical movement of the trailing edge between
 wedge incidence, ?w = 2.0? and ?w = 25? is approximately 2.1% of the wedge chord. This variation is small,
 43
 0
  50
  100
  150
  200
  250
  0  0.5  1  1.5  2  2.5  3  3.5  4
 A
 B
 C
 D
 E F G H I
 Test Section Mach Number, M
 Te
 st
 Se
 ct
 io
 n
 G
 a
 u
 ge
 St
 a
 tic
 P
 re
 ss
 u
 re
 [kP
 a
 ]
 Patmosphere ? 87kPa
 Glass Breaking Pressure
 Tunnel Startup Pressure
 A : PO = 25kPa
 B : PO = 50kPa
 C : PO = 75kPa
 D : PO = 100kPa
 E : PO = 150kPa
 F : PO = 200kPa
 G : PO = 300kPa
 H : PO = 400kPa
 I : PO = 500kPa
 Figure 3.12: Envelope of operating conditions in CSIR supersonic wind tunnel in terms of total pressure
 (gauge)
 but its effect on transition is addressed with numerical simulation in Chapters 6 and 7. The trailing edge
 separation from the symmetry plane, g, was calculated to prevent the intersection of the expansion fan with
 the reflection point (g/w ? 0.6 for all experiments) and to prevent intersection of the reflected wave with
 the wedge chord.
 The drive path from the actuator to the wedges is highlighted in grey in figure 3.14 and consists of a
 vertical bar that synchronizes the horizontal motion of two drive shafts that result in synchronized rotation
 of the wedges. The wedges were rotated gradually, with a servo motor, between 5.0 and 10 deg/s to generate
 steady state data (see figure 3.15). This rotation speed is sufficiently small such that the reflection pattern is
 approximately steady at each point in time. The wedges were rotated at larger rotation speeds with a spring-
 driven actuator to investigate the dynamic case (see figure 3.16). The spring-driven actuator is assembled
 to achieve either rapid increasing or decreasing incidence. Photographs of the hardware are included in
 Appendix C.
 3.8.3 Actuator for Steady State Experiments
 As illustrated in figure 3.15, the actuator for the steady state experiments consists of a DC servo motor
 and lead screw arrangement that rotates the wedges gradually between 5 and 10 deg/s. This is sufficiently
 44
A
 Detail View A
 Free Stream
 Test Section
 Tunnel Cart
 Support Structure
 & Tunnel Interface
 Actuator
 SectionWedge
 Figure 3.13: Illustrations of rig installed in the CSIR supersonic wind tunnel
 45
A
 A Synchronization
 Bar
 Actuator
 Motion
 Synchronized
 Horizontal Motion
 Synchronized
 PitchFront View Sectioned Side View
 on Plane A-A
 Figure 3.14: Symmetric wedge arrangement and the drive path highlighted in grey
 46
slow not to generate any dynamic effects on the transition point and the Mach stem development. The
 synchronisation bar has an embedded bearing arrangement that houses the lead screw. Rotation of the lead
 screw through the bearing is translated to horizontal movement of the synchronisation bar and drive shafts
 that result in wedge rotation. At the time of design, available commercial off the shelf DC motors were not
 able to withstand the predicted axial loads at tunnel startup. A custom thrust bearing arrangement was
 designed to transfer the axial load on the drive shafts to the support bracket. The arrangement decouples
 the motor shaft from the lead screw in the axial direction, but maintains rotational coupling. Calculations
 for the motor selection are included in Appendix C. Details and drawings of the lead screw and bearing
 arrangement are also included in Appendix C. A control interface was developed for remote control of the
 servo-motor, which includes control to reverse the motor direction. Illustrations of the control interface and
 circuit diagrams are included in Appendix C.
 3.8.4 Actuator for Dynamic Experiments
 Wedge rotation speeds that were used in simulations by Felthun & Skews [12] resulted in wedge trailing
 edge speeds up to 10% of the free stream acoustic speed. The wedge was started impulsively (with an
 initial, established steady RR) and rotated at a constant rotation speed. In a typical supersonic blow
 down wind tunnel at M = 3.0, using air stored at 300.0 K, with w = 40.0 mm, this required starting the
 wedge impulsively and rotating it at a constant rotation speed of approximately 30000 deg/s. Evidently,
 the validation of this type of numerical simulation with an experiment is not possible as it requires infinite
 acceleration at startup. Only finite acceleration could be considered in the experiment and this has been
 mimicked in the computations.
 Various concepts were considered to realise the rotation speeds required, viz. spring-driven, electric,
 dynamic impact, pyrotechnics, pneumatics and hydraulics. As these experiments were the first of their kind
 in the CSIR tunnel, a spring-driven mechanism was considered primarily for the sake of simplicity and cost.
 The actuator was designed to achieve movement of the wedges in one direction only per experiment, i.e. rapid
 increasing or decreasing incidence per experiment, but not both in the same experiment. The actuator must
 be configured appropriately to achieved the required rotation direction. The actuator for rapid increasing
 incidence is shown in figure 3.16. The required arrangement for the reverse motion is shown in figure 3.19.
 All components in the drive path are highlighted in grey and are coupled to the horizontal movement of the
 synchronisation bar and drive shafts. A pair of compression springs were designed to generate an average
 ME = +0.01 over a 25? rotation range. Calculations are presented in Appendix C. Though this wedge
 rotation speed is smaller than the maximum rotation speed investigated with computational simulation by
 47
A
 Detail View A
 Drive Shaft Actuator
 Bracket
 Lead Screw Lead Screw
 Bearing
 Thrust
 Bearing
 DC Servo
 Motor
 Figure 3.15: Servo motor driven actuator for steady state, baseline experiments
 48
Felthun & Skews [12], it was considered sufficient to provide the necessary validation data.
 In the configuration shown in figure 3.16 the springs are energised by moving the synchronisation bar
 upstream until it is secured with the latch mechanism. During the experiment the latch mechanism is
 released remotely and the drive path accelerates downstream resulting in the required rapid wedge rotation.
 With an initial spring load of 1000 N, the drive train is accelerated at approximately 1000m.s?2 and the
 motion is completed in approximately 6.0 ms. Depending on the experimental test conditions a maximum
 instantaneous ME ? +0.033 was achieved, i.e. approximately 11000 deg/s.
 The linear potentiometer in figure 3.16 is coupled to one of the drive shafts and its signal is acquired
 along with tunnel Mach number. The potentiometer signal is used to identify the start of the wedge motion
 on the Mach number trace as illustrated in figure 3.20.
 Arming the Actuator
 The springs are energised with the jacking nut on the screw thread shown in figure 3.17. The jacking nut
 is turned against the cover (jacking surface) of the thrust bearing arrangement on the synchronisation bar.
 The springs are compressed until the latch mechanism secures the synchronisation bar as shown in figure
 3.16. Before releasing the latch mechanism, the jacking nut is turned away from the synchronisation bar to
 ensure sufficient clearance for the actuator stroke during an experiment.
 Securing the Actuation Load
 The latch mechanism shown in figure 3.18 was designed to hold the required actuation load until released
 remotely from the control room. This was a particularly important aspect of the design as experiments with
 an earlier version of the mechanism resulted in early release of the load as the startup shock moved through
 the test section (see figure 3.25). The current design ensures that the actuation load is secure during tunnel
 startup. When the actuator is armed (figure 3.18(a)), the actuation load is held in position with a latch that
 is engaged against the shoulder of a release pillar. The shoulder of the release pillar and the mating surface
 on the latch were designed with the same radius of curvature. The centre of curvature of both surfaces in
 the locked position is the rotation centre of the release pillar. Therefore, the distributed load on the latch at
 the shoulder acts through the centre of the release pillar, cancelling any moment that may tend to rotate the
 release pillar. In this way, the latch is self-locking by design and will not release the load due to any sudden
 load application. The latch and the release pillar were precision wire cut from 8 mm thick 174 Ph Stainless
 Steel (1000 MPa yield strength after heat treatment) to ensure an ideal mating surface at the shoulder. The
 mechanism includes a redundant safety pin, shown in figure 3.16, that prevents movement of the release
 49
A
 Detail View A
 LatchSafety Pin
 Latch ReleaseCompression
 Springs
 Potentiometer
 Solenoid
 Latch Release
 Actuator
 Figure 3.16: The spring driven actuator and latch mechanism for the dynamic experiment. The actuator is
 assembled for the dynamic RR ? MR experiment.
 50
Rear Isometric View
 Bearing Seat
 Bearing
 Screw
 Jacking Nut
 Bearing Cover/
 Jacking Surface
 Bearing Track
 Sectioned Side View
 Figure 3.17: Sectioned view illustrating jacking nut/screw and thrust bearing arrangement to arm the
 actuator
 51
Frelease actuator
 LatchShoulderRelease Pillar
 1
 2
 3
 4
 F 1000 Nspring=
 Lever
 (a) Latch in locked position under actuator load, Fspring
 3
 4
 2
 1
 Frelease actuator
 F 1000 Nspring=
 (b) 1. Movement of latch release actuator ? 2. Rotation of lever ? 3. Lever acting on release pillar
 ? 4. Rotation of release pillar away from latch
 1
 2
 3
 4
 5
 Frelease actuator
 Drive Train
 Acceleration
 (c) The release pillar is moved sufficiently away from the latch to release the actuator load
 Figure 3.18: A series of CAD drawings illustrating the operation of the latch release mechanism
 52
pillar when engaged. During an experiment the safety pin is disengaged only after the free stream flow
 has stabilised, before the latch mechanism releases the wedge actuation load. The safety pin is disengaged
 remotely by energising a solenoid. Technical specifications of the solenoid are included in Appendix C.
 Releasing the Actuation Load
 Figure 3.18 illustrates the sequence of events, labelled 1 - 5, that describe the operation of the latch mechanism
 to release the spring load, Fspring. The latch may be released by rotating the release pillar about its pivot
 point away from the latch. This is achieved with the actuation of a lever that acts against the release pillar.
 When the shoulder of the release pillar is moved sufficiently far away from the latch mating surface the latch
 is free to rotate about its own rotation centre under Fspring. The lever is actuated with the latch release
 actuator installed at the rear of the rig shown in figure 3.16.
 Dynamic Actuator Control Interface
 The actuator is operated remotely from the tunnel control room. A control interface was designed to:
 ? Indicate the status of the latch mechanism to the rig operator, i.e. latched or unlatched
 ? Indicate the status of the safety pin to the rig operator, i.e. engaged or disengaged
 ? Indicate the status of the solenoid, i.e. power on or off
 ? Indicate the status of the release actuator, i.e. power on or off
 ? Switch power to the safety pin solenoid
 ? Switch power to the release actuator
 Limit switches are installed at appropriate locations on the actuator to indicate the status of the latch
 mechanism and the safety pin. Illustrations of the control interface and circuit diagrams are included in
 Appendix C.
 Operational Test Procedure
 Before tunnel startup, the actuator is armed, the actuation load is secured and the safety pin is engaged.
 The rig cover plates are installed and the tunnel is prepared for the experiment. The tunnel operator, rig
 operator and camera operator are positioned in the control room. After tunnel startup, the rig operator
 waits for the tunnel free stream conditions to stabilise before disengaging the safety pin. When the actuator
 53
A
 Detail View A
 Figure 3.19: Actuator arrangement for dynamic MR ? RR experiments
 control interface indicates that the safety pin is disengaged, the operator drives the latch release actuator
 installed at the rear of the rig shown in figure 3.16, the load is released and the drive train is accelerated
 to achieved the required wedge rotation. The actuator control interface indicates the latch release and the
 camera is triggered manually. The camera is setup to capture an equal number of images on either side of
 the triggering event.
 3.9 Sample Image and Data Reduction
 Data acquisition signals from an experiment are presented in figure 3.20 for illustrative purposes. Images
 were prepared for measurements as shown in figure 3.21. Wedge and shock incidence measurements were
 made as described previously. A summary of results is presented in table 3.5. The start of the wedge rotation
 is indicated by the movement of the potentiometer as discussed earlier.
 54
 50
  100
  150
  200
  250
  300
  350
  400
  450
  500
  0  5  10  15  20  25  30  35  40
 Total Pressure Guage [kPa
 ]
 Time [s]
  10
  20
  30
  40
  50
  60
  70
  80
  90
  0  5  10  15  20  25  30  35  40
 Static Pressure Guage [kPa
 ]
 Time [s]
  86
  86.2
  86.4
  86.6
  86.8
  87
  0  5  10  15  20  25  30  35  40
 Atmospheric Pressure [kPa
 ]
 Time [s]
  0
  0.5
  1
  1.5
  2
  2.5
  3
  0  5  10  15  20  25  30  35  40
 Mach Numbe
 r
 Time [s]
  16
  18
  20
  22
  24
  26
  28
  30
  32
  0  5  10  15  20  25  30  35  40
 Total Temperature [deg C
 ]
 Time [s]
  3.6
  3.8
  4
  4.2
  4.4
  4.6
  4.8
  0  5  10  15  20  25  30  35  40
 Potentiometer Reading [V
 ]
 Time [s]
 Figure 3.20: Sample data acquisition readings acquired during an experiment
 55
Figure 3.21: Sample image captured during a dynamic experiment and prepared for measurements. The
 image was captured with the high-speed digital camera at 10000 frames per second with a 1/20000 s exposure
 time. Image resolution : 512 ? 512 pixels.
 Table 3.5: A summary of results from a sample dynamic experiment at M = 3.0
 Quantity Value
 Mach Number, M 2.956? 0.03
 Total Temperature, TO 27.85? 0.5?C = 301.0? 0.5K
 Top Wedge Angle 21.3? 0.30?
 Bottom Wedge Angle 22.0? 0.30?
 Top Incident Shock Angle 41.2? 0.30?
 Bottom Incident Shock Angle 41.6? 0.30?
 Mach stem height, m 5.8? 0.16mm
 m/w 0.145? 0.004
 56
3.10 Rig Development History
 The current rig design evolved over a number of unsuccessful tunnel tests. The various previous designs
 will be presented briefly. Videos of the previous unsuccessful experiments are included on a data disc
 accompanying this thesis.
 1. The first design shown in figure 3.23 was designed to accelerate larger wedges (chord = 50 mm; span
 = 200mm) at larger rotation speeds (maximum instantaneous wedge rotation speed = 30000 deg/s)
 to achieve an average ME = +0.1. This required actuation loads in the order of 10000N and a larger
 dynamic actuator was designed. Each wedge had four drive shafts to ensure sufficient support along
 the span. The resultant blockage was sufficient to prevent tunnel startup. Also, the actuation loads
 raised concerns of operator safety and this rig was abandoned.
 2. The blockage may have been reduced with a smaller, spring-driven actuator that would achieve smaller
 rotation speeds or with a high-density energy actuation concept that required a smaller volume, e.g
 pyrotechnics or hydraulics. The spring-driven actuator concept was maintained for mechanical sim-
 plicity. As shown in figure 3.24, the rig support structure was streamlined to reduce model frontal
 area and a smaller actuator was designed. The wedge had w = 20mm with b = 93mm. The blockage
 problem was resolved, but the size of the flow field of interest in comparison to the total image area
 was too small. At this time, the larger CSIR schlieren system, with 600 mm mirrors, was used for the
 experiments.
 3. Subsequently, a larger wedge design was considered, i.e. w = 40mm and b = 170mm span. This
 required modifying the support system. This is the current version of wedge and support system
 shown in figure 3.14. A smaller schlieren system was also designed and a focussing lens was selected to
 maximise sensor coverage. From the images in figure 3.25 the magnification and image quality of the
 new system was satisfactory. However, the latch mechanism for the dynamic actuator failed on tunnel
 startup. At this time, there was no safety pin and the mating surfaces on the latch and release pillar
 were not manufactured to the correct specification.
 4. A safety pin was added to the latch mechanism and stringent manufacturing tolerances were specified
 to ensure a self-locking mechanism.
 57
(a) First rig design (b) Current rig design
 Figure 3.22: Comparison of frontal area profiles in the streamwise direction
 Figure 3.23: First rig design is considerably larger than the final version of the rig. Blockage was sufficiently
 large to prevent tunnel startup.
 58
(a) (b)
 (c)
 Figure 3.24: (a) Illustration and (b) photograph of the second rig design with (c) a schlieren image of the
 reflection pattern indicating poor optical magnification
 59
(a) (b) (c) (d)
 (e) (f) (g) (h)
 Figure 3.25: A series of high-speed schlieren images showing the early release of the drive train and wedges
 due to failure of the latch mechanism on tunnel startup. The detached bow wave, after the flow conditions
 stabilised, can be seen on the last frame, well after the latch has been released. The new optics have the
 desired magnification.
 60
3.11 Conclusion
 All experiments were conducted in the 450mm? 450mm blow-down supersonic wind tunnel at the Council
 for Scientific and Industrial Research in South Africa at approximately M = 2.0 and 3.0. Free stream
 tunnel conditions were acquired with a National Instruments data acquisition system, viz. test section static
 pressure, total pressure and total temperature. Test section Mach number and static temperature were
 derived. Flow-visualisation was achieved with a standard z-type schlieren system. High-speed imaging was
 done with a Photron-Ultima APX-RS at 10000 fps for the dynamic experiments and at 250 fps for the steady
 state experiments. The optical measurement technique was presented and the uncertainties in angular and
 distance measurements were quantified, i.e. ??w, ?? = ?0.03? and ?m/w = ?0.004. The rig consists of
 two large aspect ratio wedges with b/w = 4.25, arranged and actuated symmetrically about a horizontal
 image plane. The rig includes a servo-driven actuator that rotates the wedges gradually between 5 and 10
 deg/s to generate steady state data and a spring-driven actuator to generate rapid wedge rotation in the
 dynamic experiments. The actuator for the dynamic experiments accelerated the drive train at approximately
 1000m.s?2 at release and the wedges achieved a maximum instantaneous rotation speed of approximately
 11000 deg/s i.e. ME = +0.033.
 61
Chapter 4
 Computational Method
 4.1 Introduction
 The dynamic effect of rapid wedge rotation on the transition between two-dimensional (2D) regular and
 Mach reflection of shock waves in an ideal, steady, supersonic free stream is of primary interest in this
 investigation. This is explored with experimental and numerical methods. The experimental method was
 discussed in chapter 3. This chapter presents details of the numerical method.
 Numerical solution of the 2D Euler equations are used to simulate the dynamic experiments and to extend
 the investigation beyond the capability of the existing experimental facility to investigate the effect of pivot
 point, initial incidence and rotation speed on RR ? MR transition.
 An Euler code was developed at the University of Witwatersrand by Felthun [11] and was used for all
 transient flow simulations in this work. The use of Fluent V 12.0 was explored to model viscous effects in
 the dynamic case, but was eventually only used for steady state, inviscid simulations where required. This
 chapter describes the relevant aspects of both codes briefly. Relevant modelling issues identified in Fluent
 are discussed. Results of grid sensitivity studies are also presented.
 4.2 Code Description : Euler Code
 The Euler code used in this investigation was developed specifically for the solution of moving boundary
 problems in compressible flows. It was previously used to simulate the rapidly rotating wedge (see Felthun
 & Skews [12]). The code is a vertex centred, arbitrary Lagrangian Eulerian finite volume scheme for un-
 structured triangular meshes. The Euler equations are solved with second-order accuracy. AUSM+ as
 formulated by Liou [33] is implemented for the calculation of convective fluxes across cell interfaces. Node
 redistribution during boundary movement is implemented every time step with Laplacian smoothing. A
 62
mesh adaption routine was implemented to avoid excessive element deformation. The mesh adaption routine
 includes point insertion (for mesh refinement), edge collapsing (for mesh coarsening) and edge swapping (to
 optimise element quality). The in-house code was not optimised for solver speed, and Fluent V 12.0 was
 used for all steady state, inviscid flow calculations. Fluent V 12.0 has a compressible, density-based solver
 for the Euler and Navier-Stokes equations with adaptive refinement for the resolution of flow field gradients.
 First-order accuracy on triangular meshes was used. The Euler code has proved practical for the solution of
 2D problems, but has limitations in solving three-dimensional (3D) problems adequately. Three-dimensional
 computations are beyond the scope of this work.
 4.3 Code Description : Fluent V 12.0
 Fluent has a 2D and 3D compressible, density based solver for the Euler and Navier-Stokes equations on
 structured and unstructured meshes. It has a first, second and third-order accurate solver with a custom
 version of AUSM+, suited to shock capture. Mesh adaption is also available.
 The second and third order schemes proved unstable for the simulation of a steady MR. The solver
 instability arises from the shear layer instability downstream of the triple point. Only the first order scheme
 proved stable in this case. Fluent has been developed for parallel computing and has a faster solver than the
 available in-house code.
 The moving mesh capability was explored for the rapidly rotating wedge case. Fluent incorporates
 spring-based mesh smoothing and a remeshing algorithm for transient, moving boundary problems. In the
 spring-based model of Farhat [10] the entire meshed domain is viewed as a structural system with stiffness
 provided by the element edges. Each edge is modelled as a spring with stiffness inversely proportional to the
 element edge length. As the edge shortens, its stiffness increases, reducing further deformation of the edge.
 This method is successful in cases with small boundary movement, but does not avoid edge crossing for larger
 boundary movement. Fluent rebuilds/remeshes areas of the domain where elements violate a user-specified
 edge size range and skewness value to prevent edge crossing. However, the remeshing algorithm itself does
 not have explicit control of the size and skewness of the new elements.
 On their own, mesh smoothing and remeshing in Fluent were able to redistribute nodes and remesh the
 entire domain appropriately for the rapidly rotating wedge. However, adaptive refinement was necessary
 in conjunction with mesh smoothing and remeshing to resolve the shock wave system adequately, while
 maintaining practical solution times. The addition of mesh refinement introduced a modelling issue. Fluent
 performs mesh adaption with a non-conformal mesh topology as opposed to the conformal topology used
 63
in the in-house Euler code as illustrated in figure 4.1. In the former approach, data must be interpolated
 across non-conformal interfaces in the mesh. Spurious flow features were generated in regions of the mesh
 with excessive skewness, especially those regions in proximity of the non-conformal interfaces (see figure
 4.2). Currently, there is insufficient control of element quality to model the rapidly rotating wedge in Fluent.
 The remeshing algorithm only has implicit control of the element quality as mentioned earlier. Due to
 this limitation, Fluent was used for steady state simulations only and the in-house code for all dynamic
 simulations.
 4.4 Computational Model
 The steady state cases were simulated in Fluent and all dynamic cases were simulated with the in-house
 Euler code developed by Felthun [11]. Felthun & Skews [12] previously demonstrated the ability of the
 in-house Euler code to predict the theoretical RR ? MR transition conditions at M = 3.0.
 A fundamental issue is the modelling of the steady RR ? MR transition experiment in the strong-
 reflection region in a facility with sufficient free stream noise to suppress hysteresis. If the free stream noise
 levels are large enough, RR ? MR transition will occur at the von Neumann condition, whereas an Euler or
 Navier-Stokes CFD code will predict transition at the detachment condition. The effect of rapid rotation on
 RR ? MR transition as well as the validity of the Euler equations under these conditions will be assessed
 and discussed in chapter 6.
 For steady MR ? RR transition, Ivanov et al. [24] demonstrated close agreement in ?T between experi-
 mental measurements and Euler simulation results at M = 4.0. Both results recorded ?T ? ?N , indicating
 that the Euler equations are sufficient to predict ?T for MR ? RR transition. The effect of rapid rotation
 on MR ? RR transition and the validity of the Euler equations here will be evaluated in chapter 7.
 Figure 4.3 illustrates the flow domain boundaries of the grid for the experimental model. As the flow
 at the trailing edge is supersonic, the geometry downstream of the wedge was not modelled. As the wedge
 wake flow was not modelled, the exit dimensions change during the simulation. The flow was modelled as
 inviscid and all solid surfaces were modelled as ?slip? walls.
 The model rotation centre in the experiment is indicated in figure 4.3. The vertical movement of the
 trailing edge between ?w = 2.0? and ?w = 25? is approximately 2.1% of the wedge chord. This variation is
 small, but its effect on transition is addressed in Chapter 6 with the aid of numerical simulation.
 64
(a)
 (b)
 Figure 4.1: (a) Conformal mesh topology in the in-house Euler code compared to (b) the non-conformal
 mesh topology in Fluent for mesh refinement in the region of the incident wave at the wedge leading edge
 65
(a) Spurious flow feature arising due to poor mesh quality in the region of the reflection point
 (b) Mesh near reflection point (c) Flow field contours near reflection point
 Figure 4.2: Sample spurious flow feature in Fluent flow solution due to poor mesh quality in the vicinity of
 the reflection point
 Fl
 o
 w
 In
 le
 t
 Model Rotation Centre
 Reflection Plane
 M
 0.2375 w
 0.1125 w g ? 0.6w 0
 .
 85
 w
 Figure 4.3: Schematic of computational model for simulation of the experiment
 66
4.5 Dynamic Solution Procedure
 For each dynamic simulation, a steady, initial, grid independent solution was computed before the wedge
 was moved. This was achieved by computing a solution on a coarse mesh and performing successive mesh
 refinements (and mesh coarsenings) to resolve the flow field adequately. The flow solver is run between
 successive passes of the mesh adaption routine to recalculate the flow field on the adapted mesh. Figure 4.4
 illustrates how each pass of the adaption routine halves the previous minimum element size, d, in the regions
 of high flow gradients. The routine also coarsens regions of the mesh without strong flow gradients. During
 wedge movement, Laplacian smoothing is executed at each time step and the adaption routine is executed
 at user-specified intervals. For the modelling of flows with an initial Mach stem, long computation times
 and fine mesh resolution was necessary to achieve a steady, grid independent result.
 4.6 Grid Sensitivity Studies
 Fluent was used for all steady state simulations and the in-house Euler code was used for all dynamic
 simulations. A grid sensitivity study was performed on a static, 2D wedge with g/w = 0.56 and ? = 40.0? at
 M = 4.0 in Fluent to determine the sensitivity of computed Mach stem height to grid element size. Ivanov
 et al. [24] published the result of a 2D Euler calculation of the Mach stem height for this configuration
 (see figure 3.1 in Chapter 3). Figure 4.5(a) shows the convergence of computed Mach stem height with
 the reduction in minimum element size. There is also close agreement to the predicted Mach stem height
 published by Ivanov et al. [24].
 In addition, simulations of a rapidly rotating wedge, with varying minimum element size were done with
 the in-house Euler code to determine the dependence of the computed dynamic RR ? MR transition point
 on minimum mesh element size. A steady RR was established at an initial wedge incidence, ?wi = 19.0?,
 in a M = 2.98 free stream and the wedge incidence was increased rapidly at ME = +0.1 until transition to
 MR. The rotation point was the same as the model in the experiment shown in figure 4.3. The variation
 of ?T with minimum element size is shown in figure 4.5(b). The difference in ?T with the two finest grid
 resolutions is approximately 0.1?. At zero element size, the extrapolated ?T ? 40.57?, 0.03? less than ?T
 with the finest grid. Uncertainty in shock incidence measurement from flow field contours is estimated at
 approximately ?? = ?0.2?.
 67
(a) Initial coarse mesh and solution without local refinement, d/w = 0.05
 (b) The first pass of the mesh adaption routine refines the background mesh such that d/w = 0.025 in the background
 and elements within the shocks and expansion fan are halved such that d/w = 0.0125
 (c) The second pass of the mesh adaption routine refines elements within the shocks and expansion fan only, d/w =
 0.00625
 (d) The third pass of the mesh adaption routine refines elements within the shocks and expansion fan only, d/w =
 0.003125
 Figure 4.4: Series of images illustrating the successive adaption of an initial coarse mesh to establish an
 initial, grid independent, steady solution. Corresponding computed density contours appear on the right
 hand side.
 68
 0.1
  0.12
  0.14
  0.16
  0.18
  0.2
  0.22
  0.24
  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5
 Dimensionless minimum element size, (d? 103)/w
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w Ivanov et al. (2001)
 3rd Order Polynomial Fit
 (a) Computed Mach stem height variation with minimum element size in Fluent at ? = 40.0?, M = 4.0,
 g/w = 0.56. The solid line is a third-order polynomial fit to the data and is used to extrapolate m/w at zero
 d.
  38
  38.5
  39
  39.5
  40
  40.5
  41
  41.5
  42
  0  1  2  3  4  5  6  7
 Dimensionless minimum element size, (d? 103)/w
 Sh
 o
 ck
 In
 ci
 de
 n
 ce
 a
 t
 Tr
 a
 n
 si
 tio
 n
 ,
 ? T
 [de
 gr
 ee
 s]
 Steady ?D
 2nd Order Polynomial Fit
 (b) Variation of computed ?T with minimum element size with the in-house Euler code for ME = +0.1 at
 M = 2.98. The solid line is a second-order polynomial fit to the data and is used to extrapolate ?T at zero d.
 Figure 4.5: Results from CFD grid sensitivity assessment for a static and dynamic simulation
 69
Table 4.1: Computed Mach stem heights compared to simulation results published by Ivanov et al. [24] for
 a stationary 2D wedge at M = 4.0, g/w = 0.56
 ? Computed m/w Published m/w | ?m | [% of w] CFD Code
 36.0? 0.048825 0.05 0.1 Fluent
 40.0? 0.196 0.195 0.1 Fluent
 44.0? 0.40415 0.4 0.4 Fluent
 44.0? 0.404 0.4 0.4 In-house
 4.7 Fluent Benchmarking
 In addition to the steady configuration modelled in section 4.6, Ivanov et al. [24] also published 2D, steady
 Mach stem height data at ? = 36.0?, within the dual solution domain, and at ? = 44.0?, outside the dual
 solution domain (see figure 3.1 in Chapter 3). There is confidence in their 2D Euler predictions due to the
 close agreement between results of the 3D simulations and experiments with the finite aspect ratio wedge
 on the same graph. Both cases were modelled in Fluent. The in-house Euler code was used to simulate
 the wedge with ? = 44.0? only. Simulation results are summarised in table 4.1. There is good agreement
 between the predictions made here and the published data, with the maximum deviation, ?m approximately
 0.4% of the wedge chord. The favourable comparison provides confidence in the ability of Fluent to model
 the steady state case of interest.
 4.8 The Incidence-Induced Hysteresis Test
 In the incidence-induced hysteresis test, originally proposed by Hornung et al. [18], a steady RR is established
 below ?N and the wedge incidence is increased gradually until transition to MR. The wedge incidence is
 subsequently decreased until transition to RR. Ideally, RR ? MR transition must occur at ?D as there are
 no free stream disturbances in the flow simulation and the reverse transition must occur at ?N . Felthun &
 Skews [12] previously demonstrated the ability of the in-house Euler code to model the hysteresis test at
 M = 3.0. This was repeated here at M = 2.98 to benchmark the CFD model rather than the code. Results
 are summarised in table 4.2. Computed density contours are included in figures 4.6 and 4.7.
 A steady RR is established at ? < ?N in figure 4.6(a) and ?w is increased at ME = +0.001 such that
 there is no observable dynamic effect on the reflection pattern. RR was maintained through the dual solution
 domain. Figure 4.6(d) shows the earliest traces of a shear layer on the reflection plane. Transition is assumed
 0.1? before the first appearance of the shear layer on the reflection plane at ?w = 21.5?. For RR ? MR
 transition, ?T = 39.7?, 0.2? larger than ?D. The Mach stem development beyond transition can be seen in
 figures 4.6(d) to 4.6(f) as ?w increases to 22.0?.
 70
Table 4.2: Computed values for ?T for the incidence-induced hysteresis test at M = 2.98 in comparison to
 steady state, theoretical values for RR ? MR transition
 Computed ?T Theoretical ?T
 RR ? MR transition 39.7? ?D = 39.5?
 MR ? RR transition 37.3? ?N = 37.5?
 (a) ?wi = 19.7? (b) ?w = 21.0? (c) ?w = 21.5?
 (d) ?w = 21.6? (e) ?w = 21.7? (f) ?w = 22.0?
 Figure 4.6: Computed density contours showing RR ? MR transition close to the detachment condition
 condition. M = 2.98, ME = +0.001, fixed h/w = 0.91.
 The wedge incidence is subsequently decreased at ME = ?0.001 (see figures 4.7(a) to 4.7(f)) and transition
 to RR was observed at ?w ? 19.4? with ?T = 37.3?, 0.2? below ?N . The deviation from ?D and ?N is within
 the value of uncertainty for shock incidence measurement of ??? 0.2?.
 4.9 Compensation for Boundary Layer Deflection
 The measured wedge motion from all dynamic experiments were used as inputs to the CFD. However, due
 to the boundary layer on the wedge surface in the experiment, ? is larger in the experiment than in the
 Euler simulation for the same value of ?w. Since the flow conditions at the reflection point are critical to RR
 ? MR transition, ? at the reflection point is critical rather than ?w. If ? and consequently ?T are sensitive
 to the time history of the flow field, it is crucial that ?i and the time history of ? is the same between the
 experiment and the computation, rather than ?wi and the time history of ?w.
 The value of ?wi required in the inviscid simulation to match ?i in the experiment is calculated from the
 following well-known oblique shock relation for isentropic flow (see Anderson [26]) :
 71
(a) ?wi = 22.0? (b) ?w = 21.5? (c) ?w = 21.0?
 (d) ?w = 20.5? (e) ?w = 20.0? (f) ?w = 19.6?
 Figure 4.7: Computed density contours showing MR ? RR transition close to the von Neumann condition.
 M = 2.98, ME = ?0.001, fixed h/w = 0.91.
 tan ?wi =
 (M21 sin2 ?i ? 1) cot?i
 1 + (12 (? + 1)? sin2 ?i)M21
 (4.1)
 The difference between ?wi derived from equation 4.1 and the measurement from experiment is the
 flow deflection caused by the boundary layer at the start of the motion and is labelled ??BL. As a first
 approximation, it is assumed that there is no significant change in ??BL until transition and the measured
 motion profile (measured wedge incidence with time) is offset by ??BL to ensure that ?i is matched between
 simulation and experiment.
 It must be noted that even with this correction there must still be a difference in initial Mach stem height
 between the 3D experiment and the 2D simulation as discussed in chapter 3. It is incorrect to adjust ?wi to
 achieve the measured initial Mach stem height in the experiment.
 4.10 Conclusion
 The in-house Euler code was used for all dynamic simulations and Fluent V 12.0 (inviscid model only) was
 used for all steady state predictions. The Navier-Stokes solver in Fluent was also considered initially to
 model the dynamic case. Numerical issues were identified that disqualified its use for the dynamic case of
 interest. The Euler equations are sufficient to predict steady RR ? MR transition. Their ability to predict
 dynamic RR ? MR transition will be explored in later chapters.
 A steady and dynamic grid sensitivity study was completed with Fluent and the in-house code respectively
 72
to determine the minimum required mesh element size.
 Fluent was benchmarked against the steady, 2D Mach stem height data published by Ivanov et al. [24]
 for three configurations at M = 4.0. There was close agreement with the published data in all three cases
 with the maximum deviation in predicted Mach stem height, approximately 0.4% of wedge chord.
 The in-house code was successfully benchmarked previously by Felthun [11] to model the incidence-
 induced hysteresis test originally proposed by Hornung et al. [18]. This exercise was repeated to test the
 integrity of the CFD model developed for this work. Steady RR ? MR transition was predicted close to the
 detachment condition and the reverse transition was observed close to the von Neumann condition.
 The Euler simulations do not account for the flow deflection caused by the wedge surface boundary layer.
 A simple method was proposed to correct the measured wedge motion profile used in the CFD simulations
 to match the initial shock incidence measured in the experiment.
 73
Chapter 5
 Steady State RR ? MR Transition
 5.1 Introduction
 This chapter presents results from experiments and computations on steady, two-dimensional RR ? MR
 transition in the weak and strong-reflection regions. The primary objective of the steady experiments were
 to determine if hysteresis could be observed in the strong-reflection region in the CSIR tunnel. Since the
 wedge aspect ratio is larger than 4.0, it was expected that ?T would not be influenced by three-dimensional
 (3D) effects. These experiments were done with the servo-driven actuator described in chapter 3. Wedges
 were rotated symmetrically about a horizontal plane between approximately 5.0 and 10.0 deg/sec, sufficiently
 slow to ensure an approximately steady reflection pattern at each instant. Schlieren images were captured
 with the Photron Ultima APX-RS at 250 frames per second at 1024 ? 1024 pixel resolution. Aspects of the
 3D structure of the reflections will be highlighted where necessary. Movies of the experiments may be found
 on the accompanying data disc. Only selected images are presented in this chapter. Steady, two-dimensional
 (2D), Euler computations were done with Fluent and results were compared with the measurements from
 experiments. In this chapter, the streamwise vertical plane of symmetry will simply be referred to as the
 symmetry or central plane and the horizontal plane of symmetry will be referred to as the reflection plane.
 5.2 A Brief Summary: The Three-Dimensional Nature of Wave
 Systems in an Experiment
 The multi-coloured mask shown in figure 3.7(a) was used at the schlieren cut-off and limited 3D information
 of the reflection pattern could be inferred from the schlieren images. Relevant literature on 3D shock wave
 reflection is reviewed briefly.
 Ivanov et al. [24] computed the 3D reflection pattern generated by a large aspect ratio, steady wedge in
 74
Figure 5.1: Typical 3D geometry of shock wave reflections at M = 4.0, computed by Ivanov et al. [24]
  0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  0  0.5  1  1.5  2
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Dimensionless spanwise location, z/w
 Experiment, ? = 34?
 Experiment, ? = 37?
 Computation, ? = 37?
 Figure 5.2: Computed and measured spanwise Mach stem height variation in a M = 4.0 free stream published
 by Ivanov et al. [24]
 75
a M = 4.0 free stream as illustrated in figure 5.1. Spanwise Mach stem height was measured and compared
 to 3D computations in figure 5.2 and there was good agreement between computation and experiment and
 provided confidence in the 2D computations. These simulations were completed at M = 4.0 and effectively
 illustrated the 3D geometry of the reflection pattern in an experiment. In figure 5.1(a), the reflection in the
 plane of symmetry is RR. As one moves towards the periphery, the pattern transitions to MR close to the
 wedge corner. In figures 5.1(b) and 5.2, the maximum Mach stem height is in the plane of symmetry. As
 one moves towards the periphery there is a decrease in the Mach stem height until the minimum Mach stem
 height is achieved and this is followed by an increase in Mach stem height toward the periphery. There are
 no published results for 3D computations or experiments in the weak shock region.
 5.3 Steady State Experiment in the Weak-Reflection Region
 A steady state experiment was conducted in the weak-reflection range at approximately M = 1.93. Figure
 5.6 presents a series of images from the experiment, showing the development of the reflection pattern. An
 initial, steady RR is established after tunnel startup. The servo motor is initially driven to increase ?w until
 transition to MR. Subsequently, ?w is reduced until transition to RR. In the weak-reflection region there is
 no von Neumann condition and there is a single theoretical transition point between RR and MR for the
 steady case, viz. the detachment condition (see figure 2.7 in chapter 2).
 5.3.1 Three-dimensional Wave Structure
 Figures 5.3(a) - 5.3(c) are sample images from the experiment and highlight particular 3D features (not
 shown in order). In figure 5.3(a), there is no shear layer visible, and there is RR in the plane of symmetry as
 well as in the wedge spanwise direction. For a larger wedge incidence in figure 5.3(c) a shear layer is observed
 in proximity of the reflection plane. The shear layer could emanate from the triple point of a MR at any
 spanwise location. The reflection in the symmetry plane (the most upstream wave front on the image) is still
 RR. Therefore, MR must occur elsewhere in the spanwise direction. At an even larger wedge incidence in
 figure 5.3(b) MR is also evident in the symmetry plane. The variation in Mach stem height in the spanwise
 direction is visible from the shear sheet emanating from the locus of triple points in the spanwise direction
 (?triple curve?). At M = 1.93, unlike the case investigated by Ivanov et al .[24] in figure 5.1, the minimum
 Mach stem height is in the symmetry plane. In this case, identifying the 2D transition point from the first
 appearance of the shear layer for increasing incidence is incorrect. The transition point must be extrapolated
 from the variation of Mach stem height with ? or ?w.
 76
(a) Central RR peripheral RR (b) Central MR peripheral MR
 (c) Central RR peripheral MR
 Figure 5.3: Identification of 3D reflection structures on schlieren images from the steady state experiment
 at M = 1.93
 77
Figure 5.4: View of CAD model of wedge illustrating the location of counterbores on the stream facing
 surface of the wedge
 5.3.2 Weak Surface Waves
 The weak waves from the wedge in figures 5.3(a) to 5.3(c) are in fact generated by surface flaws. The wedge
 supports are attached to the wedge from the stream facing surface with countersunk screws (counterbored) as
 illustrated in figure 5.4. The weak waves arise from the imperfect application of wax to fill the counterbores.
 Figure 5.5 illustrates the boundaries of the sonic cone from the surface disturbances at the detachment
 condition at M = 1.93, i.e. ? ? 12.1?. From the 2D schlieren image, the intersection of the sonic cone on
 the reflection plane outside the symmetry plane can be mistaken as an interaction on the symmetry plane.
 In reality the sonic cone intersects the symmetry plane downstream of the reflection point on the symmetry
 plane and does not interact with the flow at the reflection point. At M = 2.98, the separation between the
 reflection point and the sonic cone intersection on the symmetry plane is larger. The counterbores were filled
 with silver-solder and polished for the dynamic experiments presented in chapters 6 and 7.
 5.3.3 Experimental Results
 In figure 5.6(a) a steady RR was established, before the wedge incidence was increased gradually. RR was
 maintained until MR developed on the periphery in frame 185. The peripheral Mach stem grew until the
 central reflection pattern transitioned to MR. By frame 235 the entire reflection is MR. Eventually the
 reflected wave intersected the deflection surface. As the wedge incidence increased further, the wave system
 disgorged within 3 frames, i.e. frames 256 to 258. There is a small wedge incidence range between transition
 and disgorge (approximately 2.0?). Subsequently, the motor direction was reversed to decrease the wedge
 incidence. Between frames 731 and 732 the wave system is swallowed and only RR can be seen in the plane
 of symmetry and MR on the periphery. By frame 780 the entire pattern transitioned to RR. The change
 78
Reflection Point
 on Symmetry Plane
 Sonic Cone
 on Reflection Plane
 Reflected Wave
 on Reflection Plane
 M = 1.93
 Figure 5.5: Isometric and top view identifying location of reflection point on symmetry plane with respect
 to sonic cone from wedge face counterbores at the detachment condition, M = 1.93
 79
in incidence between frames 731 and 732 is in the order of magnitude of ? 0.1? and it is unlikely that a
 steady MR was established in the central plane in the time between those frames. It is evident that there
 is hysteresis in the phenomena of shock disgorge and shock swallow, i.e. the Mach stem height just prior to
 disgorge is not the same as just after the wave system is swallowed. The hysteresis associated with shock
 disgorge and swallow, though interesting, is beyond the scope of this work and will not be investigated
 further.
 The implication of not having observed MR in the plane of symmetry for decreasing incidence, is that it
 will not be possible to setup an initial, steady MR at this free stream condition with a fixed initial incidence
 with this setup. Due to the way in which the tunnel flow starts, the flow would setup an initial disgorged
 wave system or a steady RR in the plane of symmetry for a fixed initial incidence.
 Fluent was used to simulate the steady case. Results are summarised in table 5.1 and figure 5.9.
 5.4 Steady State Experiment in the Strong-Reflection Region
 A steady experiment was conducted in the strong-reflection region at approximately M = 2.98. Figure 5.8
 presents a series of images from the experiment, showing the development of the reflection pattern from an
 initial steady RR as the wedge incidence is increased beyond transition and decreased subsequently.
 5.4.1 Theoretical Transition
 In the strong-reflection region, the dual solution domain is bounded by the von Neumann and detachment
 conditions as illustrated by the pressure-deflection shock polars in figure 2.5. The early steady state experi-
 ments by Hornung & Robinson [19] observed RR ? MR transition at the von Neumann condition irrespective
 of the direction of incidence change, though the theory supported RR ? MR transition at detachment. The
 observance of hysteresis or lack thereof was postulated to be dependent on the level of free stream turbulence.
 This was confirmed with the experiments of Ivanov et al. [22] in which the elusive hysteresis phenomenon
 was observed in a low noise wind tunnel facility. The RR ? MR transition point in the strong-reflection
 region is dependent on the level of tunnel free stream turbulence and can vary between wind tunnels. In
 an experiment, provided the wedge aspect ratio is sufficiently large, transition to MR may occur anywhere
 between the von Neumann and detachment conditions. In a low turbulence facility, with low levels of vi-
 bration, MR can be maintained until the detachment condition. Apart from the single honeycomb mesh in
 the settling chamber of the CSIR tunnel, there are no additional mechanisms for turbulence or vibration
 reduction.
 80
(a) Frame 1, t = 0.0 s (b) Frame 185, t = 0.736 s (c) Frame 211, t = 0.84 s
 (d) Frame 235, t = 0.936 s (e) Frame 256, t = 1.02 s (f) Frame 257, t = 1.024 s
 (g) Frame 258, t = 1.028 s (h) Frame 334, t = 1.332 s (i) Frame 730, t = 2.916 s
 (j) Frame 731, t = 2.92 s (k) Frame 732, t = 2.924 s (l) Frame 780, t = 3.116 s
 Figure 5.6: High-speed images from steady state experiment at M = 1.93
 81
A
 C D
 A : Incident wave of MR in the vertical plane of symmetry
 B : Triple point of MR in the vertical plane of symmetry
 C : Shear layer from triple point B
 D : Shear layer in plane of minimum Mach stem height
 B
 Figure 5.7: Magnified view : schlieren image of MR at M = 2.98, indicating the maximum Mach stem height
 in the wedge vertical plane of symmetry and the shear layer in the plane of the minimum Mach stem height
 (indicated on the bottom half of the reflection only)
 5.4.2 Three-dimensional Wave Structure
 In figure 5.7, one can identify the leading edge of the incident oblique wave in the plane of symmetry. The
 intersection of the incident oblique wave with the leading edge of the Mach stem identifies the triple point
 in this plane. In contrast to the M = 1.93 case, the Mach stem decreases in the spanwise direction and a
 minimum Mach stem height can be seen. The 3D reflection pattern could be similar to that computed by
 Ivanov et al. [24] in figure 5.1. This may be verified with high-resolution 3D CFD simulations, but is not
 necessary for this investigation. As the maximum Mach stem height occurs in the plane of symmetry it is
 likely, though it cannot be confirmed, that the appearance/disappearance of the shear layer downstream of
 the reflection point, may be an accurate estimate of transition for increasing/decreasing incidence. Transition
 was extrapolated from Mach stem data.
 5.4.3 Experimental Results
 A steady RR can be seen in frame 1 of figure 5.8. The wedge incidence was increased gradually beyond
 transition to MR. Mach stem growth was continuous with an increase in wedge incidence (see frames 429 -
 82
(a) Frame 1, t = 0.0 s (b) Frame 380, t = 1.516 s (c) Frame 429, t = 1.712 s
 (d) Frame 480, t = 1.916 s (e) Frame 579, t = 2.312 s (f) Frame 665, t = 2.656 s
 (g) Frame 806, t = 3.22 s (h) Frame 807, t = 3.224 s (i) Frame 808, t = 3.228 s
 (j) Frame 849, t = 3.392 s (k) Frame 900, t = 3.596 s (l) Frame 979, t =3.912 s
 Figure 5.8: High-speed images from steady state experiment at M = 2.98
 83
579), indicating that the tunnel free stream turbulence is sufficient to trigger transition at the von Neumann
 condition. The wedge incidence was increased until the wave system disgorged in frame 665. As ?w was
 decreased the wave system was swallowed and a steady MR was established in the central plane as seen in
 frames 806-808. Further decrease in ?w resulted in transition to RR. There is hysteresis in the phenomena
 of shock disgorge and shock swallow as observed at M = 1.93. Incidence-induced RR ? MR transition
 hysteresis was not observed in the CSIR facility. Fluent was used to simulate the steady case. Results are
 summarised in table 5.1 and figure 5.10.
 5.5 Results
 Transition results for both experiments are summarised against the steady state transition criteria in table
 5.1. The shock incidence at transition was extrapolated from a second-order polynomial fit of the Mach
 stem growth data. At M = 1.93, there is good agreement in ?T between experiment and CFD at M = 1.93.
 Transition was measured in the experiment and computed at approximately ?T = 43.4?, 0.2? beyond ?D,
 which is within the uncertainty value of ?? for measurements from experimental images and computed flow
 field contours. The difference in Mach stem growth with ? between experiment and CFD in figure 5.9
 was expected as demonstrated previously by Ivanov et al. [24] in figure 3.1, i.e. the Mach stem height is
 always smaller in a 3D experiment than the ideal, 2D case for any ? > ?T . The results of the 2D CFD and
 experiments converge at ?T .
 In the ideal case in the strong-reflection range, RR ? MR transition occurs at the detachment condition
 and the reverse transition occurs at the von Neumann condition. For ? > ?D the Mach stem height is
 independent of the direction of incidence change. Figure 5.10 shows the hysteresis loop predicted by CFD
 at M = 2.98. However, hysteresis was not observed in the experiment and transition occurred close to
 the von Neumann condition in both directions, which indicates that there is sufficient noise in the free
 stream to suppress hysteresis. Due to the level of free stream noise, there is disagreement between the
 steady theory/CFD and experiment for RR ? MR transition, but good agreement for the reverse transition.
 The CFD result and the experiment are within 0.1? of the von Neumann condition for steady MR ? RR
 transition. This is consistent with observations in all supersonic wind tunnels without special noise and
 vibration reduction measures. Once again, the difference in Mach stem height between CFD and experiment
 was observed for ? > ?T . Transition results from both experiments indicate that the wedge aspect ratio is
 sufficient to ensure that ?T in the wedge vertical plane of symmetry is close to the 2D result.
 These steady state experiments employed a lower quality imaging lens (bi-convex) and a coarser calibra-
 84
 0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  43  44  45  46
 Shock Incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD
 Steady Experiment
 Detachment
 Figure 5.9: Measured and computed Mach stem development at M = 1.93, g/w ? 0.6. The solid lines are
 second-order polynomial fits to each data set used to extrapolate ?T at zero m/w. The uncertainty in ??
 and ?m/w for the experimental data is omitted to prevent cluttering on the graph.
 tion grid than that reported in sections 3.5 and 3.6. They were considered sufficient for the purpose of these
 baseline experiments. The finer calibration grid and aspherical achromatic lens documented in sections 3.5
 and 3.6 were used for the dynamic experiments presented in chapters 6 and 7.
 5.6 Conclusion
 Steady state, baseline experiments and computations were done to determine the 2D RR ? MR transition
 points in the weak and strong-reflection ranges. RR ? MR was observed close to the detachment condition
 at M = 1.93 and close to the von Neumann condition in both directions at M = 2.98. The free stream noise
 in the CSIR supersonic tunnel is sufficient to suppress incidence-induced hysteresis in the strong-reflection
 region. There was good agreement between theory, computation and experiment for the transition point at
 both free stream conditions with the exception of RR ? MR transition at M = 2.98 due to the level of free
 stream noise. The wedge aspect ratio was sufficient to ensure that ?T in the wedge vertical plane of symmetry
 approximates the 2D result. The expected difference in Mach stem growth with shock incidence between the
 2D CFD result and the 3D experimental measurement was observed and exhibits identical characteristics to
 85
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  36  37  38  39  40  41  42  43  44  45  46
 Shock Incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD : MR ? RR
 2D Steady CFD : RR ? MR
 Steady Experiment
 Detachment
 von Neumann
 Figure 5.10: Measured and computed Mach stem development at M = 2.98, g/w ? 0.6. The solid lines are
 second-order polynomial fits to each data set used to extrapolate ?T at zero m/w. The uncertainty in ??
 and ?m/w for the experimental data is omitted to prevent cluttering on the graph.
 86
Table 5.1: Summary of steady state results from experiment and CFD at M = 1.93 and 2.98, g/w ? 0.6
 M = 1.93 M = 2.98
 Analytical ?T 43.2? ?N = 37.5?, ?D = 39.5?
 2D Euler CFD : ?T 43.4? MR ? RR : 37.5?; RR ? MR : 39.7?
 Experiment : ?T 43.4? 37.4?
 the data published by Ivanov et al. [24]. Some interesting 3D wave features were identified. At M = 1.93,
 the minimum Mach stem height is in the streamwise vertical plane of symmetry, not on the periphery as
 observed at M = 2.98. The optical calibration and imaging optics, though sufficient for these steady state
 experiments, were improved for the dynamic experiments.
 87
Chapter 6
 Dynamic Two-Dimensional Regular to
 Mach Reflection Transition in an
 Ideal Steady Supersonic Free Stream
 6.1 Introduction
 This chapter presents results from experiments and computations to investigate the effect of rapid wedge
 rotation on two-dimensional RR ? MR transition in an ideal, steady, supersonic free stream. The inves-
 tigation was conducted in the weak and strong-reflection regions. Results for the steady state experiments
 and two-dimensional (2D) Euler computations were presented in chapter 5. The spring-driven actuator was
 installed and configured for dynamic RR ? MR transition experiments. The wedges achieved a maximum
 instantaneous rotation speed of approximately 11000 deg/s resulting in ME ? +0.033. Schlieren images were
 captured with the Photron Ultima APX-RS at 10000 frames per second at 512 ? 512 pixel resolution. The
 counterbores on the wedge surface identified in chapter 5 were filled with silver solder to remove the surface
 disturbances observed in the steady state experiments. The evolution of the reflection pattern was acquired
 over several high-speed images. Selected images are presented here. Movies with the complete sequence of
 images are included on an accompanying data disc. The measured wedge motion was mimicked in the Euler
 code developed by Felthun [12]. The code was also applied to scenarios beyond the capability of the current
 facility to investigate the dependence of dynamic RR ? MR transition on other variables in the parameter
 space. These include pivot point, initial incidence, rotation speed and Mach number. The dynamic RR ?
 MR transition mechanism was also investigated. Reference will only be made to the reflection pattern in the
 streamwise vertical plane of symmetry unless otherwise stated.
 88
6.2 Experimental Results for Dynamic RR to MR Transition
 Experiments to observe the dynamic RR ? MR transition of interest were conducted in the weak and strong-
 reflection ranges at M = 1.93 and 2.98 respectively. The spring-driven actuator was installed to increase
 wedge incidence rapidly on latch release. Table 6.1 includes measured test conditions, viz. M , stagnation
 temperature (TO), stagnation pressure (PO) and the initial shock incidence (?i). Selected high-speed images
 from both experiments are presented in figures 6.1 and 6.3. Zero time corresponds to the image frame just
 before any wedge movement is visible on the high-speed images. The measured wedge motion (?w vs. time),
 variation of shock incidence (? vs. time), variation in Mach stem height (m/w vs. time) and the streamwise
 movement of the reflection/triple point (x/w vs. time) is included in figures 6.2 and 6.4. The motion after
 the reflected wave of the MR intersects the wedge surface is not of interest here and was not analysed. Each
 frame before this time includes a value of time, t, as well as ?w and ?. Images from both experiments at
 M = 1.93 and 2.98 contain a similar sequence of events and the following qualitative description is applicable
 to both experiments, unless otherwise specified.
 After tunnel startup a steady, initial RR is established (see figures 7.1(a) and 6.3(a)) after which time
 the latch mechanism restraining the actuation load is released. Initially, as the wedges rotate, there is little
 streamwise movement of the reflection point as the measurements show in figures 6.2(d) and 6.4(d). Both
 graphs show a distinct upstream movement only after approximately 2.5 - 2.6 ms of wedge motion. In this
 time the wedge has changed incidence by approximately 5.5? at M = 1.93 and 7.0? at M = 2.98. As ?w
 increases, transition to MR occurs and the triple point moves upstream. The Mach stem and the shear layer
 of the MR is visible in figures 7.1(c) and 6.3(c). In particular, at M = 2.98, the streamwise speed of the
 triple point after transition is different from the streamwise speed of the reflection point before transition
 and is evident from the change in gradient after transition in figure 6.4(d). As the ?w increases further,
 the reflected wave of the MR intersects the wedge surface (figure 7.1(e) and 6.3(g)) and the wave system
 disgorges. The wedge motion terminates and a steady disgorged wave is established as seen in figures 6.1(i)
 and 6.3(i). The motion of interest is completed in approximately 4.5 ms at M = 1.93 and in approximately
 5.5 ms at M = 2.98.
 The speed of the reflection point at transition is used to calculate the effective local free stream speed
 at the reflection point at transition and the steady, analytical transition criteria are corrected accordingly.
 The shock incidence at the corrected von Neumann and detachment conditions will be referred to as ?NC
 and ?DC respectively. The measured test conditions in table 6.1 and wedge motion were used as inputs to
 the Euler simulations.
 89
Table 6.1: Experimental test conditions for dynamic RR ? MR experiments, g/w ? 0.6
 M PO [Pa] TO [K] ?i [degrees]
 1.93 232.0 302.7 35.5
 2.98 474.0 302.3 23.2
 (a) t = 0.0 ms, ?w = 2.4?, ? = 35.5? (b) t = 3.4 ms, ?w = 12.4?, ? = 42.6? (c) t = 3.8 ms, ?w = 15.0?, ? = 45.8?
 (d) t = 4.0 ms, ?w = 16.5?, ? = 48.0? (e) t = 4.2 ms, ?w = 18.1?, ? = 49.7? (f) t = 4.4 ms
 (g) t = 4.6 ms (h) t = 4.9 ms (i) t = 5.4 ms
 Figure 6.1: High-speed images for dynamic RR ? MR transition at M = 1.93
 90
 0
  5
  10
  15
  20
  25
  0  1  2  3  4  5
 Time [ms]
 ? w
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 Transition
 Second Order Fit
 (a) ?w vs time
  35
  40
  45
  50
  55
  0  1  2  3  4  5
 Time [ms]
 ?
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 Transition
 Second Order Fit
 (b) ? vs time
  0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0  1  2  3  4  5
 Time [ms]
 m
 /w
 Transition
 (c) m/w vs time
  0
  5
  10
  15
  20
  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5
 Small x/w for first
 2.5 ms of motion
 Time [ms]
 x
 /w
 Transition
 (d) x/w vs time
 Figure 6.2: Measurements from the dynamic experiment at M = 1.93
 91
(a) t = 0.0 ms, ?w = 2.0?, ? = 23.1? (b) t = 4.0 ms, ?w = 18.6?, ? = 35.2? (c) t = 4.7 ms, ?w = 24.3?, ? = 41.7?
 (d) t = 4.9 ms, ?w = 25.6?, ? = 44.0? (e) t = 5.1 ms, ?w = 27.8?, ? = 46.5? (f) t = 5.3 ms, ?w = 30.2?, ? = 49.5?
 (g) t = 5.4 ms, ?w = 31.1?, ? = 51.2? (h) t = 5.7 ms (i) t = 6.0 ms
 Figure 6.3: High-speed images for dynamic RR ? MR transition at M = 2.98
 92
 0
  5
  10
  15
  20
  25
  30
  35
  0  1  2  3  4  5  6
 Time [ms]
 ? w
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 Transition
 Second Order Fit
 (a) ?w vs time
  20
  25
  30
  35
  40
  45
  50
  55
  60
  0  1  2  3  4  5  6
 Time [ms]
 ?
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 Transition
 Second Order Fit
 (b) ? vs time
  0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  0  1  2  3  4  5  6
 Time [ms]
 m
 /w
 Transition
 (c) m/w vs time
  0
  5
  10
  15
  20
  25
  30
  35
  40
  0  1  2  3  4  5  6
 Small x/w for first
 2.6 ms of motion
 Time [ms]
 x
 /w
 Transition
 (d) x/w vs time
 Figure 6.4: Measurements from the dynamic experiment at M = 2.98
 93
6.2.1 Weak-Reflection Range
 The measurements from experiments and computational results are presented in table 6.2 and figure 6.5. The
 solid lines in figure 6.5 are second-order polynomial fits through each data set and are used to extrapolate
 ?T at zero m/w. The experimental and computed ?T for the steady case are within 0.2? of ?D. In the
 dynamic experiment, the wedge achieved an instantaneous rotation speed (? 6300.0 deg/s) that resulted in
 ME = +0.02 at the point of transition. The average ME up to the point of transition was approximately +
 0.011. The rapid rotation delayed transition in the experiment and CFD of the experiment beyond ?DC by
 approximately 1.2?? 1.3?. Values of ?T from the dynamic experiment and simulation of the experiment are
 within 0.1? of each other. As expected (and discussed by Ivanov et al. [24]), there is the difference in Mach
 stem growth between the 2D computed result and the three-dimensional (3D) experimental measurement
 for the steady case. This characteristic is also evident in the dynamic case. The close agreement between
 experiment and computation lends confidence to the ability of the computational method to predict dynamic
 transition.
 6.2.2 Strong-Reflection Range
 The measurements from experiments and computational results are presented in table 6.3 and figure 6.6. As
 the point of transition in the strong-reflection range is dependent on the level of tunnel free stream turbulence
 and is tunnel dependent, baseline measurements for the steady experiment were necessary. In figure 6.6 the
 von Neumann and detachment conditions are indicated ?N and ?D respectively. The dual solution domain
 is ?N < ? < ?D. The solid lines in figure 6.6 are second-order polynomial fits through each data set and
 are used to extrapolate ?T at zero m/w. In the steady experiment, RR ? MR takes place close to the von
 Neumann condition in both directions, indicating that the noise in the CSIR supersonic tunnel is sufficient
 to suppress hysteresis. The steady CFD successfully predicted the incidence-induced hysteresis loop. For the
 steady case, there is disagreement in ?T between the steady experiment, in which free stream perturbations
 are always present, and the steady CFD, in which there are no perturbations in the free stream.
 For the dynamic case the wedge achieved an instantaneous rotation speed (? 9000.0 deg/s) that resulted
 in ME = +0.028 at the point of transition. The average ME up to the point of transition was approximately
 + 0.015. The measured ?T for the dynamic experiment is labelled ?X?. Figure 6.6 shows, very clearly, that
 the rapid wedge rotation was sufficient to maintain RR past ?T ? ?N observed in the steady experiment,
 through the dual solution domain (as hypothesised by Hornung [17]) and even beyond ?D. Both the ex-
 periment and CFD show that RR persisted approximately 0.9? ? 1.2? beyond ?DC . Even though there was
 disagreement between the steady experiment and CFD, there is close agreement between the experimental
 94
Table 6.2: Summary of ?T from steady and dynamic experiments and CFD at M = 1.93, g/w ? 0.6
 Analytical steady detachment condition, ?D 43.2?
 Measured relative Mach number of reflection point at transition + 0.017
 Corrected analytical steady detachment condition, ?DC 43.1?
 Experiment : steady state ?T 43.4?
 Experiment : dynamic ?T 44.4?
 2D Euler CFD : steady state ?T 43.4?
 2D Euler CFD : dynamic ?T 44.3?
 Difference between dynamic ?T and ?DC (Experiment and CFD) ? 1.2? ? 1.3?
 Table 6.3: Summary of ?T from steady and dynamic experiments and CFD at M = 2.98, g/w ? 0.6
 Analytical steady von Neumann condition, ?N 37.5?
 Analytical steady detachment condition, ?D 39.5?
 Measured relative Mach number of reflection point at transition + 0.046
 Corrected analytical steady detachment condition, ?DC 39.5?
 Experiment : steady state ?T 37.4?
 Experiment : dynamic ?T 40.75?
 2D Euler CFD : steady state ?T 37.5?
 2D Euler CFD : dynamic ?T 40.45?
 Difference between dynamic ?T and ?DC (CFD and Experiment) ? 0.9? ? 1.2?
 and computed values of ?T for the dynamic case. The agreement between the dynamic experiment, in which
 small perturbations are always present in the free stream, and the dynamic CFD, in which the free stream is
 without perturbations, implies that RR ? MR transition in the strong-reflection region becomes insensitive
 to free stream noise above a certain critical rotation speed. This critical rotation speed may depend on the
 level of free stream noise and may vary between facilities.
 The characteristic difference between the 2D and 3D result is also seen here. This result provides ex-
 perimental evidence to support the dynamic effect originally presented by Felthun & Skews [12]. Rapid
 wedge rotation introduces a dynamic effect that delays RR ? MR transition beyond the steady, theoretical
 transition condition.
 6.3 Computational Simulation of Impulsive Rotation at M = 2.98
 The close agreement between experiment and computation provides confidence in the use of flow simulations
 to investigate the dynamics of the flow field, including the dynamic RR ? MR transition mechanism. This
 section analyses 2D Euler CFD results from the simulation of a rapidly rotating wedge in a M = 2.98
 free stream. The wedge is started impulsively from a steady, initial RR, at an initial wedge incidence of
 ?wi = 19.0? and rotated continuously at ME = +0.1 (rotation speed = 32644 deg/s with TO = 302.3K
 and w= 40.0 mm) until transition to MR. The wedge is rotated about the model rotation centre in the
 95
 0
  0.05
  0.1
  0.15
  0.2
  42  43  44  45  46  47  48  49  50
 Shock Incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD
 Steady Experiment
 2D Unsteady CFD
 Unsteady Experiment
 Steady Detachment
 Figure 6.5: Experimental and CFD results for steady and dynamic RR ? MR transition at M = 1.93. Solid
 lines are second-order polynomial fits through each data set and are used to predict ?T at zero m/w.
 96
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52
 Shock Incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD:MR ? RR
 2D Steady CFD:RR ? MR
 Steady Experiment
 2D Unsteady CFD
 Unsteady Experiment
 Steady Detachment
 Steady von Neumann
 ?N
 ?D
 X
 X : ?T Unsteady Experiment
 Figure 6.6: Experimental and CFD results for steady and dynamic RR ? MR transition at M = 2.98. Solid
 lines are second-order polynomial fits through each data set and are used to predict ?T at zero m/w.
 97
 0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -10  0  10  20  30  40
 P
 /P
 ?
 ? [degrees]
 IP : Incident Polar
 I
 II III
 A
 B
 C
 D
 IV
 E
 S
 M
 I : ? = 19.0?
 II : ?N = 19.6?
 III : ?D = 21.3?
 IV : ? = 24.0?
 Reflected Polars
 Figure 6.7: Critical pressure-deflection shock polars for steady reflection at M = 2.98
 experiment. The large rotation speed is implemented deliberately to highlight the transient effects.
 6.3.1 Steady Pressure-Deflection Shock Polars
 As a prelude to the dynamic analysis, critical pressure-deflection shock polars for the ideal, steady case
 at M = 2.98 are reviewed briefly in figure 6.7. The incident polar is labelled ?IP?. ?M? and ?S? are the
 detachment and sonic points on ?IP?. At the initial wedge incidence, ?wi = 19.0?, only RR is possible and
 the pressure downstream of the reflection point is given by the intersection of the reflected polar, ?I?, with
 the y-axis, i.e. point A. As ?w is increased from ?wi very gradually, to ensure an approximately steady
 flow, the pressure rise through the reflection point increases as the intersection of the reflected polar with
 the y-axis moves towards point C. Points B and C on reflected polars ?II? and ?III? represent the pressure
 rise through the reflection point at the von Neumann and detachment conditions respectively (?N = 19.6?
 and ?D = 21.3?). As ?w is gradually increased beyond the detachment condition, there is a marked drop
 in pressure rise across the reflection point from point C to point D as the reflection transitions to MR. At
 ?w = 24.0?, the pressure rise through the triple point of the MR is given by point ?E? on polar ?IV?.
 98
(a) t=0.0ms,?w = 19.0?,? = 36.8? (b) t=0.3ms,?w = 28.1?,? = 40.5? (c) t=0.32ms,?w = 28.5?,? = 41.5?
 (d) t=0.33ms,?w = 29.0?,? = 42.5?
 ms
 (e) t=0.37ms,?w = 30.0?,? = 43.3?
 ms
 (f) t=0.4ms,?w = 31.0?,? = 44.9?
 Figure 6.8: Computed density contours showing the flow field development for ME = +0.1, ?wi = 19.0?
 at M = 2.98, g/w ? 0.6. The Mach stem is indicated ?ms? only where clearly visible. This is not to be
 mistaken to indicate the point of transition.
 6.3.2 Dynamic Flow Solution
 Selected images from the flow solution (flow field density contours) of the impulsive rotation case are presented
 in figure 6.8. An initial, steady RR is established at ?wi = 19.0?, before the wedge is started impulsively and
 rotated about its leading edge at a constant rotation speed with ME = +0.1. As ?w increases, curvature
 develops on the incident wave as observed previously by Khotyanovsky et al. [28] and Felthun & Skews [12].
 The curvature and pressure gradient along the incident wave arise from the rapid wedge rotation and the
 interaction of the resultant compression and expansion waves with the incident wave. To date, there have
 been no detailed studies to quantify these effects. The visible Mach stems are indicated in figures 6.8(e) and
 6.8(f). However, the point of transition cannot be identified accurately from the views presented in figure
 6.8. Closeup views of the reflection pattern at ?w = 28.0? and ?w = 28.2? in figures 6.9(a) and 6.9(b) show
 the early development of the shear layer from the triple point as transition to MR occurs.
 Transition is assumed when ?w = 27.9?, i.e. 0.1? before the appearance of the shear layer in figure
 6.9(a). The corresponding ?T ? 40.5? and ?DC = 39.4?. Transition is delayed with respect to ?DC by
 approximately 1.1?.
 The Mach stem development for this case is compared to the experimental/CFD results presented in
 section 6.2.2 in figure 6.10. The previous dynamic experiment at M = 2.98 and associated 2D CFD had a
 99
(a) ?w = 28.0?
 (b) ?w = 28.2?
 Figure 6.9: Closeup views of computed density contours showing the first traces of the shear layer from the
 triple point as the reflection transitions to MR
 100
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  36  38  40  42  44  46  48  50  52
 Shock Incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD
 Steady Experiment
 2D Unsteady CFD
 Unsteady Experiment
 Steady Detachment
 Steady von Neumann
 ME = +0.1
 Figure 6.10: Mach stem development for impulsive rotation at ME = +0.1 with ?wi = 19.0? compared to
 results from the experiment and 2D CFD results. M = 2.98, g/w ? 0.6.
 significantly smaller and non-constant rotation rate. Due to the larger rotation rate here, the Mach stem
 height at any ? > ?T , is smaller than previously observed. However, there is little difference in ?T between
 the previous dynamic experiment and the impulsive rotation case presented here. The effect of rotation
 speed will be investigated in greater detail later in this chapter.
 6.3.3 Transient Pressure Rise through the Reflection/Triple Point
 Pressure traces through the reflection/triple point for the impulsive rotation case with ME = +0.1 at
 M = 2.98 are analysed with respect to the steady state pressure-deflection shock polars presented earlier
 in figure 6.7. Consider the selection of pressure traces through the reflection/triple point in figure 6.11.
 The pressure rise through the reflection point at the steady, initial condition at ?wi = 19.0? is close to the
 steady, analytical solution given by point ?A? on reflected polar ?I? in figure 6.7, i.e. P/P? = 9.6. As the
 wedge incidence increases, the pressure rise through the reflection point increases and peaks at ?w = 27.8?
 with P/P? ? 19.0 in comparison to 13.3 in the steady case (point ?C? at the detachment condition on
 reflected polar ?III?). Beyond ?w = 27.8? there is a significant drop in pressure. The wedge incidence at
 which the maximum pressure rise through the reflection point was observed in figure 6.11 is close to the
 101
Dimensionless Streamwise Ordinate, x/w
 P/
 P
 -0.2 -0.1 0 0.1
 2
 4
 6
 8
 10
 12
 14
 16
 18
 8
 ? = 26.4
 ? =19.0
 ? = 26.8
 ? = 28.0
 ? = 28.1
 ? = 28.3
 ? = 27.8
 Upstream Movement of
 Reflection Point
 ? = 28.5
 ?
 w
 ?
 ? = 25.0?
 ?
 ?
 ?
 ??
 ?
 wi
 w
 w
 w
 ww
 w
 w
 Figure 6.11: Computed pressure traces through the reflection point as the wedge rotates from ?wi = 19.0?
 at ME = +0.1 about the model pivot point at M = 2.98, g/w ? 0.6
 wedge incidence at which transition was assumed. It is reasonable to assume that the significant drop in
 pressure is associated with transition to MR, as in the steady case just beyond the detachment condition.
 Due to this similarity in trend between the dynamic and steady case, pressure-deflection shock polars may
 be useful in identifying the critical trend that highlights the point of transition.
 6.4 Transition Criteria and Mechanism for Dynamic RR to MR
 Transition
 RR and MR are possible in the dual solution domain between the von Neumann and detachment conditions.
 The shock incidence at the von Neumann condition, ?N , is the smallest incidence at which MR is theoretically
 possible. The shock incidence at the detachment condition, ?D, is the largest incidence at which RR is
 possible. The length scale criteria or information condition proposed by Hornung et al. [18] states that RR
 ? MR transition occurs at the point when flow conditions change such that there is communication of the
 wedge length scale to the reflection point (through the expansion fan). For the ideal, steady case, this is
 when the flow immediately downstream of the reflection point first goes sonic, i.e. M 1.0 at ?S . Since the
 102
flow downstream of the reflection point is supersonic for ? < ?S there can be no communication of the wedge
 length scale to the reflection point below ?S . This also happens to be very close to ?D, beyond which RR
 is not possible. The difference between ?D and ?S is very small and is usually neglected, e.g. at M = 4.0,
 ?D = 39.2? and ?S = 39.1?. It so happens in the ideal, steady case that the smallest incidence at which the
 length scale is visible to the reflection point is negligibly close to the incidence at which RR is no longer
 possible. In this case there has been no need to differentiate between the two as they are so close.
 Once again, consider the impulsive rotation case at M = 2.98 in section 6.3 to explore the dynamic RR
 ? MR transition mechanism. Transition was identified at ?w = 27.9? in section 6.3.2. From observations of
 the Mach number contours in the CFD solution, the first time the flow downstream of the reflection point
 goes sonic is when ?w ? 26.2? as shown in figure 6.12(a), approximately 1.7? below transition. The subsonic
 region is highlighted in black as indicated. The leading edge of the expansion intersects the reflected wave
 downstream of this subsonic region and the length scale cannot be communicated to the reflection point.
 For convenience this will be referred to as the sonic condition and is labelled ?S . As the wedge incidence
 increases, the leading edge of the expansion moves closer to the reflection point and the subsonic region
 grows until they interact at approximately ?w = 26.4? in figure 6.12(b). This is the smallest incidence at
 which there is an established communication path between the trailing edge expansion and the reflection
 point and shall be referred to as ?C for convenience. Taking into account the local acoustic speed in the
 subsonic zone, the wedge rotation speed and the finite time it takes the length scale information from
 the expansion fan to traverse the subsonic region, a prediction can be made as to when the length scale
 information reaches the reflection point. The wedge rotates to approximately ?w = 26.8? as the length scale
 information reaches the reflection point, approximately 0.6? after the sonic condition. For convenience this
 will be referred to as the length scale condition and shall be identified by ?L. Transition was identified in
 section 6.3.2 at ?WT = 27.9?, approximately 1.1? beyond the identified length scale condition at ?L. Though
 the wedge length scale is visible at the reflection point from ?L = 26.8?, RR persists until transition at
 ?w = ?WT = 27.9? (Khotyanovsky et al. [27] demonstrated, with CFD, the possibility of maintaining a
 steady overall RR in steady flow in the presence of length scale information - see section 2.4). Therefore,
 transition to MR must occur when the reflected wave can no longer turn the flow downstream of the incident
 wave parallel to the reflection plane at the reflection point, i.e. the dynamic equivalent of the detachment
 condition.
 The difference between ?S and ?L was even more pronounced at M = 1.93 in which the wedge was rotated
 at a constant rotation speed about its trailing edge at ME = +0.05 (? 18909 deg/s at TO = 302.7K) with
 ?wi = 8.0?. Figure 6.13 shows the development of the subsonic region. The sonic condition is identified at
 103
Wedge trailing edge Leading edge of
 expansion fan
 i
 e
 Subsonic region
 r
 (a) Sonic condition at ?w = ?S = 26.2?
 Wedge trailing edge Leading edge of
 expansion fan
 i
 e
 Subsonic region
 r
 (b) Point of first contact between the expansion fan and the
 subsonic region at ?w = ?C = 26.4?
 Figure 6.12: Computed density contours showing the development of the subsonic region downstream of
 reflection point before transition at M = 2.98, ME = +0.1, ?wi = 19.0?. The subsonic region downstream
 of the reflection point is shaded black.
 ?S = 16.2?. The subsonic region downstream of the reflection point grows until it meets the trailing edge
 expansion approximately 1.0? later at ?C = 17.2?. At this point information still has to traverse the subsonic
 patch before it reached the reflection point. Transition to MR was identified from the flow field contours at
 ?WT = 17.5?. The flow solution 0.1? later at ? = 17.6? is shown in figure 6.14. The first signs of the shear
 layer development at the triple point are visible. Taking into account the local sound speed, the length scale
 information only traverses approximately 40% of the subsonic region by the time transition has occurred.
 The red dot in figure 6.14 marks the estimated distance traversed by the length scale information on the
 shortest line between the leading edge of the expansion and the reflection point at transition. RR ? MR
 transition has occurred without the presence of a length scale at the reflection point. As in the case at
 M = 2.98, transition must occur when the reflected wave can no longer maintain the boundary condition
 at the wall, i.e. the dynamic equivalent of the detachment condition. This particular result demonstrates
 that the length scale information from the trailing edge expansion to the reflection point was not necessary
 for RR ? MR transition in this particular dynamic case. This is purely a dynamic effect introduced by the
 rapid wedge rotation. The various critical points in the flow field development for both impulsive test cases
 are summarised in table 6.4.
 In the ideal, steady case the difference between the length scale and detachment conditions is negligible.
 However, these results show that the difference is more significant for the dynamic cases presented. Also,
 in the ideal steady case, the sonic condition and the length scale condition are synonymous. There is a
 difference between the two in the dynamic case due to the transient nature of the flow. As illustrated in
 104
(a) ?w = ?S = 16.2?
 (b) ?w = 16.8?
 (c) ?w = ?C = 17.2?
 Figure 6.13: Computed density contours showing the development of the subsonic region downstream of the
 reflection point between ?S and ?C at M = 1.93, ME = +0.05, ?wi = 8.0?. The subsonic region downstream
 of the reflection point is shaded black.
 105
Table 6.4: Summary of results for dynamic simulations at M = 1.93 and M = 2.98 to investigate the
 dynamic RR ? MR transition mechanism
 M = 1.93, ME = +0.05 M = 2.98, ME = +0.1
 ?S 16.2? 26.2?
 ?C 17.2? 26.4?
 ?L - 26.8?
 ?WT 17.6? 28.1?
 Figure 6.14: Estimated location of length scale information on the shortest line between the leading edge of
 the expansion and the subsonic region at ?WT = 17.6?. The early development of the shear layer from the
 triple point is also visible.
 106
figure 6.13, it is possible to increase the time between the sonic condition and the length scale condition.
 Results for the dynamic case at M = 2.98 show that it is possible to maintain RR with a length scale
 visible at the reflection point. In addition, results for the dynamic case at M = 1.93 show that it is possible
 to achieve RR ? MR transition without length scale information at the reflection point (from the wedge
 trailing edge expansion).
 In summary, for the dynamic cases investigated here, the criterion for dynamic RR ? MR transition is
 neither the sonic or length scale condition, but rather the dynamic equivalent of the detachment condition.
 6.5 Parameter investigation for dynamic RR to MR transition
 Euler simulations were used to determine the effect of various parameters, within a limited range, on ?T
 and ?WT at M = 1.93 and 2.98, viz. rotation speed, pivot point and initial incidence. Unless otherwise
 stated, ?wi = 8.0?, g/w = 0.6 for trailing edge pivot and h/w = 0.74 for leading edge pivot at M = 1.93.
 At M = 2.98, ?wi = 19.0?, g/w = 0.6 for trailing edge pivot and h/w = 0.91 for leading edge pivot unless
 otherwise stated. At M = 1.93 and 2.98, the dimensionless leading edge separation, h/w, for rotation about
 the leading edge was selected to match the value of h/w in the experimental setup at ?w = ?wi. The effect
 of moving the rotation point between the trailing edge and the model rotation centre is also investigated
 briefly.
 The wedge and shock incidence at the steady detachment condition are annotated ?D and ?D respectively.
 Both are corrected to account for the increase in local Mach number at the reflection point due to streamwise
 movement of the latter and are annotated ?DC and ?DC respectively. The speed of the reflection point at
 transition is dependent on the pivot point and hence ?DC and ?DC are also dependent on the pivot point.
 The abbreviations TE, LE and EXP will be appended to labels of quantities to indicate the rotation centre,
 viz. wedge trailing edge (TE), wedge leading edge (LE) and the model rotation centre in the experiment
 (EXP). For example the corrected ?D for rotation about the trailing edge is ?DC TE . The deviation from
 the corrected theoretical transition condition (corrected detachment condition in this case) is labelled ??WT
 and ??T . Comments are only applicable for the range of simulations presented here, unless otherwise stated.
 The uncertainty in shock incidence measurement from flow field contours in the CFD solution was reported
 earlier in chapter 4 as ?? = ?0.2?.
 107
6.5.1 M = 1.93
 Results for Euler simulations at M = 1.93 are presented in figure 6.15 and tables 6.5 - 6.6. Figures 6.15(a) and
 6.15(b) include ?DC and ?DC to account for the streamwise movement of the reflection point at transition, e.g.
 at ME = +0.1, ?DC LE ? 41.8? in comparison to the uncorrected ?D = 43.22?, a difference of approximately
 1.4?.
 The results at M = 1.93 are summarised as follows :
 1. For the range of rotation speeds investigated, ??WT increased with ME for rotation about the wedge
 leading and trailing edges.
 2. Across the range of simulated rotation rates, 1.3? ? ??WT TE ? 8.5? and 0.8? ? ??WT LE ? 5.4?.
 3. ??T also increased with ME , for a given rotation centre and initial incidence.
 4. 1.2? < ??T TE < 1.6? and 0.8? < ??T LE < 1.3?, a similar order of magnitude observed in the experi-
 ment. The variation in ??T across the range of rotation speeds, is small at 0.4? for trailing edge pivot
 and 0.5? for leading edge pivot.
 5. The dependency of ??T on rotation centre, for a given rotation speed, is also small, e.g. for ME = +0.1,
 ??T LE = 1.3? and ??T TE = 1.6?, a difference of 0.3?, just outside the uncertainty in shock incidence
 measurement of ?? = ?0.2?.
 6. The difference between ?T TE and ?T EXP is negligible for all practical purposes and within the
 uncertainty value of ?? = ?0.2?
 7. Table 6.6 indicates, that the dependency of ?WT and ?T on initial incidence, for a given rotation centre
 and rotation speed, is small and also within the uncertainty of ?? = ?0.2?.
 6.5.2 M = 2.98
 Results are presented in figure 6.16 and tables 6.7 - 6.8. The corrections made to the steady detachment
 condition were smaller as the gradient of ?D with Mach number is smaller around M = 2.98. For the range
 of simulations conducted at M = 2.98, there were many similarities in the results observed at M = 1.93. The
 only difference observed at M = 2.98 worth particular mention was in terms of ??T , i.e. 0.6? < ??T TE < 1.4?
 and 0.4? < ??T LE < 1.3?. The variation in ??T across the range of rotation speeds, for a given rotation
 centre and initial incidence, is larger than at M = 1.93 , viz. 0.8? for trailing edge pivot and 0.9? for leading
 edge pivot. So, while the sensitivity of ??T to rotation speed, for a given rotation centre and initial incidence
 is small at M = 1.93, this is not generally true.
 108
Table 6.5: ?WT and ?T at M = 1.93, ?wi = 8.0?
 ME +0.01 +0.03 +0.05 +0.075 +0.1
 ?T TE [degrees] 13.6 15.3 17.5 19.8 21.9
 ?T TE [degrees] 44.2 44.4 44.3 43.9 43.9
 ?T LE [degrees] 13.1 15.0 16.4 18.1 19.8
 ?T LE [degrees] 43.8 43.7 43.7 43.5 43.1
 ?T EXP [degrees] - - 17.8 - 22.8
 ?T EXP [degrees] - - 44.3 - 43.7
 Table 6.6: Effect of initial incidence on ?T and ?WT at M = 1.93, ME = +0.1
 ?wi [degrees] Pivot Point ?T [degrees] ?WT [degrees]
 2.0 Leading Edge 43.3 19.7
 8.0 Leading Edge 43.1 19.8
 2.0 Trailing Edge 43.8 22.0
 8.0 Trailing Edge 43.9 21.9
 Table 6.7: ?WT and ?T at M = 2.98, ?wi = 19.0?
 ME +0.01 +0.03 +0.05 +0.075 +0.1
 ?T TE [degrees] 22.3 23.7 24.9 26.5 27.7
 ?T TE [degrees] 40.1 40.4 40.5 40.6 40.8
 ?T LE [degrees] 22.3 23.4 24.7 25.8 27.0
 ?T LE [degrees] 39.9 40.5 40.4 40.6 40.7
 ?T EXP [degrees] - - 25.3 - 28.1
 ?T EXP [degrees] - - 40.6 - 40.7
 Table 6.8: Effect of initial incidence on ?T and ?WT at M = 2.98, ME = +0.1
 ?wi [degrees] Pivot Point ?T [degrees] ?WT [degrees]
 11.0 Leading Edge 40.6 27.0
 19.0 Leading Edge 40.7 27.0
 11.0 Trailing Edge 40.7 27.7
 19.0 Trailing Edge 40.8 27.7
 109
 10
  12
  14
  16
  18
  20
  22
  24
  0  0.02  0.04  0.06  0.08  0.1  0.12
 Dimensionless edge speed, ME = VE/a?
 W
 ed
 ge
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? W
 T
 Leading Edge Pivot
 Trailing Edge Pivot
 Experimental Pivot Point
 Uncorrected ?D
 Corrected ?D: LE Pivot
 Corrected ?D: TE Pivot
 (a)
  42
  43
  44
  45
  46
  0  0.02  0.04  0.06  0.08  0.1  0.12
 Dimensionless edge speed, ME = VE/a?
 Sh
 o
 ck
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? T
 Leading Edge Pivot
 Trailing Edge Pivot
 Experimental Pivot Point
 Uncorrected ?D
 Corrected ?D: LE Pivot
 Corrected ?D: TE Pivot
 (b)
 Figure 6.15: ?WT and ?T vs. ME at M = 1.93, ?wi = 8.0?
 110
 20
  22
  24
  26
  28
  0  0.02  0.04  0.06  0.08  0.1  0.12
 Dimensionless edge speed, ME = VE/a?
 W
 ed
 ge
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? W
 T
 Leading Edge Pivot
 Trailing Edge Pivot
 Experimental Pivot Point
 Steady ?D
 Corrected ?D: LE Pivot
 Corrected ?D: TE Pivot
 (a)
  39
  39.5
  40
  40.5
  41
  41.5
  42
  0  0.02  0.04  0.06  0.08  0.1  0.12
 Dimensionless edge speed, ME = VE/a?
 Sh
 o
 ck
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? T
 Leading Edge Pivot
 Trailing Edge Pivot
 Experimental Pivot Point
 Steady ?D
 Corrected ?D: LE Pivot
 Corrected ?D: TE Pivot
 (b)
 Figure 6.16: ?WT and ?T vs. ME at M = 2.98, ?wi = 19.0?
 111
 0
  0.05
  0.1
  0.15
  0.2
  42  44  46  48  50  52  54
 Shock Incidence at Triple Point, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD
 CFD of Unsteady Experiment
 Uncorrected Detachment
 ME = +0.01
 ME = +0.05
 ME = +0.1
 Figure 6.17: Dynamic Mach stem development for impulsive rotation about the wedge trailing edge at
 M = 1.93. Solid lines are second-order polynomial fits through each data set.
 6.5.3 Dynamic Mach Stem Development
 The dynamic Mach stem development for the various impulsive rotation cases at M = 1.93 and 2.98 are
 presented in figures 6.17 and 6.18 to visualise the effect of rotation speed on Mach stem growth with respect
 to ? at the triple point. The results from the steady CFD and the simulated experiments presented in
 sections 6.2.1 and 6.2.2 are overlayed. Only results for rotation about the wedge trailing are presented to
 indicate the general trend. At both free stream conditions, the second-order curve fit lies closer to the x-axis
 with an increase in rotation speed, i.e. for a given value of ? > ?T , Mach stem growth is delayed further with
 an increase in rotation speed. At M = 1.93 there is little difference in the trend between the impulsively
 started wedge with ME = +0.01 and the 2D CFD of the experiment (with average ME = +0.011 up to
 transition). This is not the case at M = 2.98. The discrepency in trend between results for impulsive
 rotation with ME = +0.01 and 2D CFD of the experiment (with average ME = +0.015 up to transition) at
 M = 2.98 is rather curious. The trend has a negative second gradient in comparison to a positive second
 gradient for all the other cases. Though interesting, this particular feature lies beyond the scope of this work
 and is recommended for consideration in future investigations.
 112
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  36  38  40  42  44  46  48  50  52
 Shock Incidence at Triple Point, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 2D Steady CFD
 CFD of Unsteady Experiment
 Uncorrected Detachment
 Uncorrected von Neumann
 ME = +0.01
 ME = +0.05
 ME = +0.1
 Figure 6.18: Dynamic Mach stem development for impulsive rotation about the wedge trailing edge at
 M = 2.98. Solid lines are second-order polynomial fits through each data set.
 6.6 Conclusion
 A rig was designed to investigate the dynamic effect of rapid wedge rotation on 2D RR to MR transition
 in a steady supersonic free stream. Experiments were completed in the weak and strong-reflection regions
 in the blow-down supersonic wind tunnel at the CSIR, South Africa. Results of steady state experiments,
 presented previously in chapter 5, showed that RR ?MR transition occurs close to the detachment condition
 at M = 1.93 and close to the von Neumann condition at M = 2.98. With rapid wedge rotation at M = 2.98 it
 was possible to maintain RR through the dual solution domain, as originally proposed by Hornung [17], and
 even beyond steady detachment. Even though there was disagreement between the steady experiment and
 CFD in the strong-reflection region, there is close agreement between the experimental and computed values
 of ?T for the dynamic case. This agreement implies that RR ? MR transition in the strong-reflection region
 becomes insensitive to free stream noise above a certain critical rotation speed. In the dynamic experiments
 and computations of experiments at M = 1.93 and 2.98, RR persisted approximately 0.9? ? 1.3? beyond
 ?DC .
 In all dynamic cases, there was good agreement in ?T between the measurements in the experiments
 and predictions made by the Euler code developed by Felthun [11]. The measurements provide experimental
 evidence to support the dynamic effect originally presented by Felthun & Skews[12]. The close agreement
 113
between experiment and computation provided confidence in the application of the Euler code to investigate
 particular aspects of the dynamic flow field.
 As expected in the steady case (and discussed by Ivanov et al. [24]), for ? > ?T , the difference in Mach
 stem height between the 2D computed result and the 3D experimental measurement was observed. This
 characteristic was also observed in the dynamic case at M = 1.93 and 2.98.
 The dynamic RR ? MR transition mechanism was investigated with 2D Euler CFD applied to the
 simulation of an impulsively rotated wedge at M = 1.93 and 2.98. For the dynamic cases investigated here,
 a distinction is drawn between the sonic, length scale and ?detachment? conditions for dynamic RR ? MR
 transition. Results show that the wedge length scale from the trailing edge expansion is not necessarily
 communicated to the reflection point as the flow downstream of the reflection point first goes sonic.
 Computations at M = 2.98 also show that it is possible for RR to persist even though length scale
 information from the wedge trailing edge expansion is available at the reflection point. It is possible that
 RR is maintained beyond the length scale condition as long as the reflected wave is able to maintain the
 boundary condition at the reflection plane until the dynamic equivalent of the detachment condition.
 Simulations at M = 1.93 show that it is also possible for RR ? MR to occur without the presence of a
 length scale, perhaps due to the failure of RR to maintain the boundary condition at the reflection point.
 This is purely a dynamic effect due to the rapid wedge rotation.
 Pressure traces through the reflection point show that rapid rotation increases the maximum achievable
 pressure rise through the reflection point of a RR in comparison to the steady case. As in the steady case,
 transition to MR in the strong-reflection region, is accompanied by a rapid decrease in pressure rise through
 the reflection/triple point. Due to the similarity in trend between the steady and dynamic cases, steady
 state shock polars may be useful in identifying the critical trend that highlights the point of transition in
 the dynamic case.
 The parameter investigation at M = 1.93 showed, for +0.01 < ME < +0.1, that there was a small
 dependence of ??T on rotation centre. Over the range of rotation speeds investigated, 0.8? < ??T < 1.6?,
 for both rotation centres. There was no significant change in ?WT and ??T between the two values of initial
 incidence tested for rotation about the wedge leading and trailing edges at ME = +0.1. For the range of
 simulations completed at M = 2.98 there is no fundamental difference in the results observed at M = 1.93.
 At M = 2.98, 0.4? < ??T < 1.4?, for both rotation centres. Graphs of dynamic Mach stem development at
 M = 1.93 and 2.98 show that an increase in rotation speed delays the development of the Mach stem with
 respect to ? at the triple point for a given value of ? > ?T .
 114
6.7 Recommendations for Future Work
 The following items, though interesting, fall outside the scope of this thesis and are recommended for future
 work.
 1. Determination of the minimum critical wedge rotation speed required such that transition is indepen-
 dent of free stream perturbations in the CSIR facility. This could be followed by a more general, but
 detailed investigation into the relationship between free stream noise and the minimum critical wedge
 rotation speed required to achieve RR ? MR transition that is independent of the level of free stream
 noise and this could be applied to any facility.
 2. An investigation to quantify the effect of rapid wedge rotation on the incident wave curvature and
 pressure gradient along the incident wave.
 3. A detailed investigation into the ?detachment? condition for the dynamic case.
 4. Execution of the dynamic experiments at M = 4.0 in which the dual solution domain is larger. At
 M = 4.0 the difference between ?D and ?N is 5.8?.
 5. Execution of the experiment described by Hornung [17]. Establish an initial, steady, RR below the
 dual solution domain, preferably at M = 4.0. Rotate the wedges rapidly and terminate the motion
 just below the detachment condition. The development of the reflection pattern would be of primary
 interest. The challenge, from an experimental perspective, lies in the termination of the wedge motion
 at the desired condition in a way that does not introduce vibration that influences transition or the
 development of the reflection pattern.
 6. Investigation of the curious negative second gradient for Mach stem development against ? with ME =
 +0.01 at M = 2.98.
 7. The rig developed here may be applied to various interesting experimental studies, e.g. the effect of
 acceleration of finite aspect ratio wings in a steady, supersonic free stream on the unsteady evolution
 of the wing wake structure.
 115
Chapter 7
 Dynamic Two-Dimensional Mach to
 Regular Reflection Transition in an
 Ideal Steady Supersonic Free Stream
 7.1 Introduction
 This chapter presents results from experiments and computations to investigate the effect of rapid wedge
 rotation on two-dimensional (2D) MR ? RR transition in an ideal, steady, supersonic free stream. A series
 of steady, baseline experiments were conducted with the servo-driven actuator, and results of the experiments
 and steady state computations were presented in chapter 5. The spring-driven actuator was installed and
 configured appropriately. Experiments were conducted at M = 2.96 and 3.26. The maximum rotation speed
 achieved at transition was approximately 2500 deg/s resulting in ME = ?0.008. Schlieren images were
 captured with the Photron Ultima APX-RS at 10000 frames per second at 512 ? 512 pixel resolution. The
 measured wedge motion was mimicked in the Euler code developed by Felthun [11]. The dynamic Mach
 stem development as well as the measured and computed ?T are compared to steady state results previously
 presented. The primary purpose of mimicking the experiment with CFD was to evalute the ability of the
 computational method to predict the gross dynamic effects and ?T .
 The 2D Euler code was also applied to scenarios beyond the experiments to investigate the dependence of
 dynamic MR ? RR transition on other variables in the parameter space. These include pivot point, initial
 incidence, rotation speed at two free stream conditions, i.e. in the weak and strong-reflection ranges. The
 evolution of the reflection pattern and the development of the Mach stem due to impulsive rotation about
 the wedge leading and trailing edges is investigated. Reference will only be made to the reflection pattern
 in the streamwise vertical plane of symmetry unless otherwise stated. Thoughts on three-dimensional (3D)
 effects will be presented for consideration in future work.
 116
7.2 Experimental Results
 The wedge and shock incidence at the steady detachment condition are annotated ?D and ?D respectively.
 In a similar manner the wedge and shock incidence at the steady von Neumann condition are annotated ?N
 and ?N respectively.
 7.2.1 Weak-Reflection Range
 Due to the way in which the tunnel flow starts, the flow sets up an initial disgorged wave system or a steady
 RR in the plane of symmetry for a fixed initial wedge incidence at M = 1.93 (see results of the steady
 experiment in chapter 4). It was not possible to set up an initial, steady MR in the weak-reflection range
 and hence the dynamic MR ? RR transition was not investigated with experiment. However, it was possible
 to set up an initial, steady MR in the strong-reflection region and experiments were completed at M = 2.96
 and 3.26. These will be presented in the next section.
 Figure 7.1 contains a selected sequence of images that show the initial, steady disgorged wave system at
 M = 1.93 and the development of the flow field as the spring-driven actuator decreases the wedge incidence
 rapidly. As the wedge incidence decreases the wave system is swallowed. A MR in which the reflected wave
 does not intersect the wedge surface can be seen in figure 7.1(e). Though visible, it cannot be considered
 steady state at this instant.
 For dynamic RR ? MR there was little dependence of ?T on ?i. However, for a steady, initial, disgorged
 wave system or MR, the flow downstream of the Mach stem is subsonic and the point of transition is
 likely to be very sensitive to ?i. Since the flow field development from a steady, initial MR is of interest, this
 experimental data was not analysed further. Data from experiments in the strong-reflection range, presented
 in the next section, are considered sufficient to evaluate the computational method.
 7.2.2 Strong-Reflection Range
 Experiments for dynamic MR ? RR transition were done at M = 3.26 and 2.96. In both experiments, an
 initial steady MR is set up and this is followed by the rapid decrease in wedge incidence until transition to
 RR. Experiment test conditions are documented in table 7.1. In both experiments RR persisted below ?NC .
 There is no fundamental difference between the two cases in terms of flow physics. Hence, only results for
 the M = 3.26 case are analysed in some detail. Select high-speed images for the experiment at M = 3.26
 are presented in figure 7.2. High-speed video from both experiments are included on the accompanying data
 disc. Zero time corresponds to the image frame just before any wedge movement is visible on the high-speed
 117
(a) t = 0.0 ms, ?wi = 13.2? (b) t = 2.6 ms, ?w = 10.2? (c) t = 3.3 ms, ?w = 9.0?
 (d) t = 3.6 ms, ?w = 8.1? (e) t = 3.9 ms, ?w = 8.0? (f) t = 4.0 ms, ?w = 7.8?
 (g) t = 4.1 ms, ?w = 7.1? (h) t = 4.2 ms, ?w = 6.8? (i) t = 5.9 ms, ?w = 1.2?
 Figure 7.1: High-speed images showing the initial, steady, disgorged wave system at M = 1.92 being swal-
 lowed as the wedge incidence decreases rapidly
 118
Table 7.1: Experiment test conditions for dynamic MR ? RR transition experiments, g/w ? 0.6
 M PO [Pa] TO [K] ?i [degrees]
 3.26 616.0 302.0 40.2
 2.96 474.0 301.0 41.2
 images. Measurements from the images are included in figures 7.3(a)- 7.3(d), i.e. time histories of wedge
 incidence, shock incidence, Mach stem height and the streamwise location of the reflection/triple point.
 After tunnel startup, an initial, steady MR is set up just beyond the dual solution domain (figure
 7.2(a)) after which time the wedge incidence was reduced rapidly. As ?w decreases, the Mach stem moves
 downstream and the Mach stem height decreases (figures 7.2(b) - 7.2(f)) until transition to RR. In figure
 7.2(h) the reflection pattern is clearly RR. The wedge rotation continues well after transition, but the flow
 field after transition is not analysed further here. The instantaneous rotation speed at the point of transition
 was approximately 2500 deg/s with ME = ?0.008, approximately 0.8% of the free stream acoustic speed.
 The tunnel conditions, measured wedge motion and initial shock incidence, ?i, were used as inputs to the
 CFD simulation. Since the Euler equations do not model the flow deflection due to the wedge surface
 boundary layer, the initial wedge incidence, ?wi, in the simulation was corrected to achieve the same ?i in
 the experiment as discussed in chapter 4.
 The experimental and computed Mach stem height variation with shock incidence angle is presented
 in figure 7.4. A linear fit to both data sets, only for ? ? 38.0?, is used to extrapolate ?T at zero m/w.
 As indicated on the graph, the expected, initial difference in Mach stem height between the CFD and
 experiment is evident, though small at this initial condition (labelled m/wiCFD and m/wiexp respectively).
 As the wedge rotates there is a deviation from the steady case as indicated by ?A? on figure 7.4. As the
 Mach stem height decreases, the unsteady CFD and experiment converge to ?T ? 35.5?. The analytical
 transition condition was corrected for the speed of the triple point at transition and results are summarised
 in table 7.2. Transition was observed approximately 0.8? below ?NC and there is close agreement on ?T
 between the experiment and computation. Though the rotation speeds achieved here were not as large as in
 the RR ? MR transition experiments, the dynamic effect of rapid wedge rotation on ?T is still evident.
 The Mach stem development from a simulation with the same free stream condition and ?i, but with a
 larger and constant rotation speed about the wedge trailing edge at ME = ?0.05, is also plotted in figure
 7.4. The deviation from the steady case is significant. In this instance transition to RR was predicted at
 ?T ? 32.7?, 3.5? below ?N and 4.5? below ?NC .
 The experiment at M = 2.96 yielded a similar result to the experiment at M = 3.26. Transition results
 119
(a) t =0.0 ms,?w = 22.8?,? = 40.2? (b) t = 0.6 ms,?w = 22.1?,? = 39.6? (c) t=1.1 ms,?w = 21.4?,? = 39.0?
 (d) t = 1.4 ms,?w = 20.9?,? = 38.4? (e) t=1.7 ms,?w = 20.3?,? = 37.8? (f) t =2.4 ms,?w = 18.9?,? = 36.4?
 (g) t =2.7 ms,?w = 18.2?,? = 35.5? (h) t=2.8 ms,?w = 18.2?,? = 35.1? (i) t=5.2 ms,?w = 9.2?,? = 26.2?
 Figure 7.2: High-speed images from dynamic MR ? RR experiment at M = 3.26
 120
 0
  5
  10
  15
  20
  25
  0  1  2  3  4  5  6
 Time [ms]
 ? w
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 2nd Order Fit
 Transition
 (a) ?w vs time
  34
  35
  36
  37
  38
  39
  40
  41
  0  0.5  1  1.5  2  2.5  3
 Time [ms]
 ?
 [de
 gr
 ee
 s]
 Bottom Wedge
 Top Wedge
 Average
 Transition
 (b) ? vs time
  0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  0.16
  0  0.5  1  1.5  2  2.5  3
 Time [ms]
 m
 /w
 Transition
 (c) m/w vs time
  0
  0.1
  0.2
  0.3
  0.4
  0.5
  0.6
  0  1  2  3  4  5  6
 Time [ms]
 x
 /w
 Transition
 (d) x/w vs time
 Figure 7.3: Measurements from the dynamic experiment at M = 3.26. The time of MR ? RR transition is
 estimated from the images and is indicated on each graph with a broken line.
 Table 7.2: Experimental and CFD results for steady and dynamic MR ? RR transition at M = 3.26,
 g/w ? 0.6.
 Analytical steady von Neumann condition, ?N 36.2?
 Measured relative Mach number of reflection point at transition - 0.03
 Corrected analytical steady von Neumann condition, ?NC 36.3?
 Experiment : dynamic ?T 35.5?
 2D Euler CFD : steady state ?T 36.2?
 2D Euler CFD : dynamic ?T 35.5?
 Difference between dynamic ?T and ?NC (CFD and Experiment) ? 0.8?
 121
 0
  0.05
  0.1
  0.15
  0.2
  32  34  36  38  40  42
 Shock incidence, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Unsteady experiment
 2D Unsteady CFD of experiment
 2D Steady CFD
 2D Unsteady CFD : ME = ?0.05
 Steady von Neumann
 2nd Order fit - steady CFD
 Linear fit - unsteady experiment & CFD
 2nd Order fit - 2D CFD : ME = ?0.05
 ?N
 ?T experiment
 & CFD?T CFD
 ME = ?0.05
 m/wiexp
 m/wiCFD
 A
 A : Deviation from steady behaviour
 Figure 7.4: Mach stem development from experiment and CFD for dynamic MR ? RR transition at M =
 3.26, g/w ? 0.6. The dashed and solid lines represent first and second-order fits respectively, only for
 ? ? 38.0?, to their respective data sets and are used to extrapolate ?T at zero m/w. The offset from the
 steady data due to rapid rotation of the wedge is labelled ?A?.
 122
Table 7.3: Experimental and CFD results for steady and dynamic MR ? RR transition at M = 2.96,
 g/w ? 0.6.
 Analytical steady von Neumann condition, ?N 37.6?
 Measured relative Mach number of reflection point at transition - 0.02
 Corrected analytical steady von Neumann condition, ?NC 37.6?
 Experiment : dynamic ?T 37.0?
 2D Euler CFD : dynamic ?T 36.8?
 Difference between dynamic ?T and ?NC (CFD and Experiment) ? 0.6? 0.8?
 are summarised in table 7.3. Transition was observed approximately 0.6? ? 0.8? below ?NC . Once again,
 there is close agreement between the experiment and 2D Euler CFD result.
 7.3 Parameter Investigation for Dynamic MR to RR Transition
 Having established the necessary confidence in the Euler code to model the dynamic flow field of interest
 adequately for the purpose of this investigation, this section explores the sensitivity of ?T as well as the dy-
 namic flow field development to rotation speed and pivot point at M = 1.93 and 2.98 with CFD simulations.
 The sensitivity of ?T to ?wi is also investigated briefly. These free stream conditions are the same as those
 used in the parameter investigation for dynamic RR ? MR transition in chapter 6.
 When ?w is decreased gradually from an initial, steady MR such that the flow field is approximately steady
 state at each point in time, the Mach stem decreases continuously with a decrease in ?w until transition to
 RR. Transition to MR occurs at the detachment condition, with ?T = ?D, in the weak-reflection region and
 at the von Neumann condition, with ?T = ?N , in the strong-reflection region. MR ? RR transition occurs
 when the triple point reaches the reflection plane, i.e. when the Mach stem height reduces to zero. It stands
 to reason that dynamic MR ? RR transition is dependent on the initial Mach stem height and the vertical
 movement of the triple point (or Mach stem development).
 For a given free stream condition, and wedge chord, the initial Mach stem height is dependent on ?i and
 g/w or h/w. The effect of the initial Mach stem height will be investigated briefly, but the primary focus of
 this parameter investigation is to identify the dynamic effect of rapid rotation on the evolution of the Mach
 stem and hence transition to RR. This also includes the sensitivity of the flow field development to pivot
 point. Results for simulations at M = 1.93 and 2.98 are summarised in subsections 7.3.3 and 7.3.4. Due to
 the complex nature of the dynamic flow field under extreme rotation speeds, impulsive rotation about the
 wedge leading and trailing edges at M = 1.93 are analysed in some detail in subsections 7.3.1 and 7.3.2 to
 highlight particular, curious aspects of the flow field.
 The theoretical transition angles are recalculated to take into account the decrease in local Mach number
 123
at the triple point due to its streamwise movement. The corrected wedge and shock incidence at the
 detachment condition are labelled ?DC and ?DC respectively. The corrected wedge and shock incidence at
 the von Neumann condition are labelled ?NC and ?NC respectively. The speed of the reflection point at
 transition is dependent on the pivot point and hence ?DC and ?DC are also dependent on the pivot point.
 The abbreviations TE, LE and EXP will be appended to labels of quantities to indicate the rotation centre,
 viz. wedge trailing edge (TE), wedge leading edge (LE) and the model rotation centre in the experiment
 (EXP). For example the corrected ?D for rotation about the trailing edge is ?DC TE . The deviation from
 the corrected theoretical transition condition (corrected detachment condition in this case) is labelled ??WT
 and ??T .
 7.3.1 Impulsive Rotation About the Wedge Leading Edge at M = 1.93
 An initial, steady MR is set up at ?wi = 13.4? in a M = 1.93 free stream with h/w = 0.84. The wedge
 is started impulsively and rotated about its leading edge with ME = ?0.075 until ?w = 0.0?. Computed
 pressure contours showing the development of the flow field are presented in figures 7.17 to 7.21 at the end of
 this chapter. Animations of the dynamic flow field are included on the accompanying data disc. The Mach
 stem evolution with respect to ?w and ? is shown in figures 7.5 and 7.6. Four phases of the triple point
 movement are identified in figure 7.5. The steady state, 2D data computed with Fluent are superimposed.
 Phase I : As the wedge rotates about its leading edge, away from the reflection plane, expansion waves
 are generated on the surface and propagate towards the triple point. Due to the rapid rotation speed, the
 wedge rotates approximately 1.9? before the surface expansion reaches the triple point. Up to this time the
 triple point is unaware of the wedge movement and the Mach stem height is approximately constant during
 phase I (see figures 7.17(a), 7.17(b) and 7.18(a)).
 Phase II : When the expansion waves reach the triple point at ?w ? 11.5?, the expansion has the effect of
 ?sucking? the triple point away from the reflection plane and increasing the Mach stem height momentarily
 (see figures 7.18(b), 7.19(a) and 7.19(b)). This effect was first observed and discussed briefly by Felthun &
 Skews [12] for rapid, impulsive rotation about the wedge leading edge at M = 3.0. The Mach stem height
 increases until the end of phase II at ? ? 8.5? (figure 7.19(b)).
 Phase III : Between ?w ? 8.5? and 7.0?, there is very little change in Mach stem height.
 Phase IV : The Mach stem height decreases until transition to RR at ?WT = 1.7? (see figure 7.20). The
 wedge incidence at transition is estimated with a linear extrapolation of the data for ?w ? 4.5?. At transition
 to RR, a shock is generated at the triple point, indicating a discontinuity in the flow conditions at the triple
 point as transition to RR occurs. As the shock propagates downstream the reflected wave incidence changes.
 124
 0
  0.05
  0.1
  0.15
  0.2
  0  2  4  6  8  10  12  14  16
 III
 Arrival of Expansion
 Waves at Triple Point
 ?WT
 Maximum m/w
 (?wi,m/wi)
 IIIIV
 Wedge incidence angle, ?w [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Leading Edge Pivot : ME = ?0.075
 2D Steady State CFD
 Linear Fit
 Second Order Polynomial Fit
 Figure 7.5: Computed variation of m/w with ?w for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, h/w = 0.84. The dashed line represents a linear fit of the data for ?w ? 4.5? and is used to
 estimate ?WT at zero m/w for the rapidly rotating wedge. The solid line represents a second order fit of the
 steady data.
 Consider the variation of m/w with ? in figure 7.6. The data points are connected with a dashed line to
 clarify the sequence of events. Results for the steady, 2D computation with Fluent are superimposed. The
 shock incidence and Mach stem height remain unchanged at ? = ?i = 45.1? and m/w = m/wi = 0.11 in
 phase I. As the triple point moves away from the reflection plane, the suction has the effect of increasing the
 shock incidence to a maximum value of ? = 46.3? at the triple point. The point of maximum shock incidence
 does not coincide with the point of maximum Mach stem height. Before the maximum Mach stem height is
 achieved the shock incidence decreases and continues to do so until transition with ?T ? 39.2?.
 7.3.2 Impulsive Rotation About the Wedge Trailing Edge at M = 1.93
 The steady, initial condition is set up at ?wi = 13.4? in a M = 1.93 free stream (see figure 7.22(a)). The
 wedge is started impulsively and rotated at ME = ?0.075 about its trailing edge with g/w = 0.6. Figures
 7.22 to 7.27 presented at the end of this chapter are selected images of pressure contours from the CFD
 solution between ?wi = 13.4? and ?w = 1.1?. Animations of the computed flow field are included on the
 125
 0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  0.16
  0.18
  38  40  42  44  46  48  50
 (?i,m/wi)
 1. Increasing ? as Triple
 Point Moves Away
 From Reflection Plane
 2. Maximum ?
 3. Maximum m/w
 4. Decreasing m/w and ?
 ?T at ?WT
 Shock incidence angle, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Leading Edge Pivot : ME = ?0.075
 2D Steady State CFD
 Second Order Polynomial Fit
 Figure 7.6: Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, h/w = 0.84. The solid line represents a second order fit of the steady data. The data points
 from the unsteady simulation are connected with a dashed line to clarify the sequence of events.
 126
accompanying data disc. The variation of m/w with ?w and ? is presented in figures 7.7 and 7.8. Four
 phases of the triple point movement are identified in figure 7.7. The steady state, 2D data computed with
 Fluent are superimposed. There is a distinct difference in flow field development in comparison to the results
 for impulsive rotation about the wedge leading edge.
 Phase I : The impulsive movement of the wedge tip generates a disturbance on the incident wave that
 propagates along the incident wave at a speed equal to the sum of the local acoustic speed and the local
 velocity component parallel to the incident wave. Figures 7.22(b), 7.23(a), 7.23(b) and 7.24(a) track the
 movement of the disturbance and downstream pressure wave until they reach the triple point. The red line
 corresponds to the planar incident wave at the initial condition and is shown to highlight the propagation
 of the disturbance on the incident wave. There cannot be any movement of the incident wave downstream
 of the disturbance. In phase I, the triple point is ?unaware? of the movement of the wedge and the Mach
 stem height is constant until the disturbance reaches the triple point at ?w ? 11.6?.
 Phase II : The disturbance travels through the triple point and down the Mach stem, towards the reflection
 plane. As the disturbance passes through the triple point, the Mach stem height decreases rapidly, but only
 momentarily until ?w = 9.5?. By this time the disturbance has passed through the triple point. The pressure
 wave reflects from the reflection plane as seen in figure 7.24(b).
 Phase III : Between ?w = 9.5? and 7.5?, the disturbance has already passed through the triple point and
 there is little further change in Mach stem height until the start of phase IV at ?w = 7.5?.
 Phase IV : After the phase III, in which there was little movement of the triple point, the Mach stem
 height decreases until transition at ?WT = 2.4? (see figures 7.26 and 7.27).
 Consider the computed pressure contours in figure 7.24(b). The pressure contour between the incident
 and reflected waves that is closest to the triple point, is highlighted in a green dashed line. The intersection of
 the highlighted isobar with the incident wave indicates a discontinuity in curvature on the incident wave. The
 incident wave upstream of the discontinuity is curved and there is a pressure gradient along this segment of
 the incident wave. There is no pressure gradient along the planar segment of the incident wave, downstream
 of the discontinuity. As ?w decreases the discontinuity moves towards the triple point and is evident from the
 movement of the pressure contours on the incident wave as seen in figures 7.25(a) and 7.25(b). By ?w = 7.5?
 the incident wave is curved along its entire length. It appears that the start of phase IV in which the Mach
 stem height decreases, is co-incident with the arrival of the discontinuity on the incident wave at the triple
 point. A more detailed investigation is required to identify and quantify the mechanism(s) that marks the
 inception of phase IV. This lies beyond the scope of this thesis and is recommended for consideration in
 future work.
 127
 0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  0.16
  0.18
  0  2  4  6  8  10  12  14  16
 (?wi,m/wi)
 IIIIIIIV
 ?WT
 Wedge incidence angle, ?w [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot : ME = ?0.075
 2D Steady State CFD
 Second Order Polynomial Fit
 Figure 7.7: Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, g/w = 0.6. The solid lines represent a second order fit of the steady and unsteady data. For
 the unsteady case, only data for ?w ? 5.0? is used to estimate ?WT .
 The wedge incidence at transition is extrapolated from a second order polynomial of the data for ?w ?
 5.0?. At transition to RR a shock is generated at the triple point. As it propagates downstream the
 reflected wave incidence changes and this indicates a discontinuity in the flow conditions in the vicinity of
 the reflection/triple point as transition to RR occurs.
 Consider the variation of m/w with ? in figure 7.8. The data points are connected with a broken line
 to clarify the sequence of events. Results for the steady, 2D computation with Fluent are superimposed.
 The shock incidence remains constant until the arrival of the disturbance on the incident wave at the end
 of phase I. As the disturbance passes through the triple point there is a sudden decrease and increase of ?
 in phase II. In phase III, ? and m/w are constant. In phase IV, ? and m/w decrease until transition at
 ?T = 37.9?.
 7.3.3 Parameter Investigation for Dynamic MR to RR Transition at M = 1.93
 The variation of m/w with ?w over a range of rotation speeds as well as the resultant ?WT and ?T is
 investigated with Euler CFD. Results are presented in figures 7.9, 7.10, 7.12(a) and 7.12(b) respectively.
 Figure 7.10 only includes results for ME = ?0.01 to avoid unnecessary clutter. Variation of m/w with ?
 and ?w for ME = ?0.075 were already presented in figures 7.5 to 7.8. Table 7.4 summarises values for ?T
 128
 0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  0.14
  0.16
  0.18
  38  40  42  44  46
 Shock incidence angle, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot : ME = ?0.075
 ?w = 11.6? to 10.0?
 2D Steady State CFD
 Second Order Polynomial Fit
 Decreasing
 m/w & ?
 (?i,m/wi)
 ?T at ?WT
 (?w = 11.6?)(11.5?)
 (11.0?)
 (10.5?) (10.0?)
 Figure 7.8: Computed variation of m/w with ? for impulsive rotation at ME = ?0.075. M = 1.93,
 ?wi = 13.4?, g/w = 0.6. The solid line represents a second order fit of the steady data. The data points
 from the unsteady simulation are connected with a dashed line to clarify the sequence of events.
 and ?T .
 Consider the variation of m/w with ?w and ? in figures 7.9 (for ME = ?0.01, ?0.05 and ?0.1) and
 7.10 (for ME = ?0.01 only). The dynamic effect of rapid rotation on the Mach stem development for
 ME = ?0.075 presented in subsections 7.3.1 and 7.3.2 are not visible at ME = ?0.01. It would appear
 that at this smaller rotation speed, the rotation centre makes little difference to the transient Mach stem
 development or to the values of ?WT and ?T (see also figures 7.12(a) and 7.12(b)).
 Also, consider the results for ME = ?0.1 in figure 7.9. While the Mach stem development is consistent
 with the observations made at ME = ?0.075, the impulsive start and rapid rotation result in a curious
 scenario in which MR is maintained even at ?w = 0.0?. Selected pressure contours from the flow solution
 for rapid rotation about the leading edge are presented in figure 7.11. The wedge is started impulsively at
 ?wi = 13.4? and stopped at ?w = 0.0?. At ?w = 0.0?, the wave system detaches from the wedge tip and
 proceeds to wash downstream. The Mach stem on the residual wave reflection is clearly visible in figure
 7.11(c). Figures 7.11(d), 7.11(e) and 7.11(f) are magnified views in the vicinity of the reflection plane showing
 the transition of the residual reflection from MR to RR as the wave system washes downstream.
 For the range of rotation speeds investigated, ??WT and ??T increased with an increase in rotation
 speed. For the range of rotation speeds investigated, 1.2? ? ??T TE ? 7.1? and 1.0? ? ??T LE ? 6.2?. The
 129
 0
  0.05
  0.1
  0.15
  0.2
  0  2  4  6  8  10  12  14
 Wedge Incidence Angle, ?w [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot, ME = ?0.01
 Trailing Edge Pivot, ME = ?0.05
 Trailing Edge Pivot, ME = ?0.1
 Leading Edge Pivot, ME = ?0.01
 Leading Edge Pivot, ME = ?0.05
 Leading Edge Pivot, ME = ?0.1
 2D Steady State CFD
 Linear Fit
 Second Order Polynomial Fit
 Figure 7.9: Computed variation of m/w with ?w for rapid, impulsive rotation. M = 1.93, ?wi = 13.4?,
 h/w = 0.84 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing edge). Dashed
 lines represent linear fits used to estimate ?WT for ME = ?0.01 and ?0.05. Solid lines are second-order
 polynomial fits used to estimate ?WT for ME = ?0.1 and the steady state case.
 130
 0
  0.02
  0.04
  0.06
  0.08
  0.1
  0.12
  41  42  43  44  45  46
 Shock Incidence Angle, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot, ME = ?0.01
 Leading Edge Pivot, ME = ?0.01
 2D Steady State CFD
 Second Order Polynomial Fit
 Figure 7.10: Computed variation of m/w with ? for rapid, impulsive rotation. M = 1.93, ?wi = 13.4?,
 h/w = 0.84 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing edge). The
 solid line is a second-order polynomial fit used to estimate ?WT for the steady state case. The dashed line
 joins the data points in each data set to aid visualisation. ?T was measured from the flow solution at ?WT
 in each case.
 131
(a) t = 0.0s, ?wi = 13.4? (b) t = 10.6?s, ?w = 2.5?
 (c) t = 23.8?s, ?w = 0.0? (d) Magnified view of residual wave reflection in the vicinity
 of the reflection point at t = 50.2?s while ?w = 0.0?.
 (e) Magnified view of residual wave reflection in the vicinity
 of the reflection point at t = 71.4?s while ?w = 0.0?.
 (f) Magnified view of residual wave reflection in the vicinity
 of the reflection point showing transition to RR at t = 76.7?s
 while ?w = 0.0?.
 Figure 7.11: Computed pressure contours for impulsive rotation at ME = ?0.1. M = 1.93, ?wi = 13.4?,
 h/w = 0.84.
 corresponding deviation of ?T from ?DC ranges as follows: 1.5? ? ??T TE ? 7.4? and 1.7? ? ??T LE ? 7.9?.
 In contrast, the maximum deviation observed at ME = +0.1 for RR ? MR transition was in the order of
 ??T = 1.6?, significantly smaller than the maximum values observed here.
 7.3.4 Parameter Investigation for Dynamic MR to RR Transition at M = 2.98
 The variation of m/w with ?w at ME = ?0.01, ?0.05 and ?0.1 as well as the resultant ?WT and ?T was
 also investigated at M = 2.98. Results are presented in figures 7.14, 7.13(a) and 7.13(b) respectively. Table
 7.5 summarises values for ?T and ?T .
 Consider the variation of m/w with ?w in figure 7.14. In general, the trends are similar to the results at
 132
 0
  2
  4
  6
  8
  10
  12
  14
  16
  18
 -0.1 -0.08 -0.06 -0.04 -0.02  0
 Dimensionless edge speed, ME = VE/a?
 W
 ed
 ge
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? W
 T
 Leading Edge Pivot
 Trailing Edge Pivot
 Uncorrected ?D
 Corrected ?D: Trailing Edge Pivot
 Corrected ?D: Leading Edge Pivot
 Linear Fit
 (a) ?WT vs. ME at M = 1.93
  36
  38
  40
  42
  44
  46
  48
  50
  52
 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01  0
 Dimensionless edge speed, ME = VE/a?
 Sh
 o
 ck
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? T
 Leading Edge Pivot
 Trailing Edge Pivot
 Uncorrected ?D
 Corrected ?D: Trailing Edge Pivot
 Corrected ?D: Leading Edge Pivot
 Second Order Polynomial Fit
 (b) ?T vs. ME at M = 1.93
 Figure 7.12: ?WT and ?T vs. ME at M = 1.93
 133
Table 7.4: Wedge and shock incidence at transition : M 1.93, ?wi = 13.4?
 ME ?0.01 ?0.05 ?0.075
 ?WT TE 10.3 5.2 2.4
 ?T TE 42.1 39.5 37.9
 ?WT LE 10.3 4.9 1.7
 ?T LE 42.0 39.9 39.2
 Table 7.5: Wedge and shock incidence at transition : M = 2.98, ?wi = 24.5?
 ME ?0.01 ?0.05 ?0.1
 ?WT TE 18.2 12.7 7.1
 ?T TE 36.3 32.5 29.0
 ?WT LE 18.3 12.6 5.7
 ?T LE 36.3 32.8 30.2
 M = 1.93. However, a small, but perhaps significant difference, is noted at ME = ?0.1 for rotation about
 the wedge trailing edge (open triangular symbols in figure 7.14). After phase II in which the Mach stem
 height decreases, there is a small but observable increase in Mach stem height whereas the Mach stem height
 is approximately constant in phase III in figure 7.7. This is also noted at ME = ?0.05 for rotation about
 the wedge trailing edge (open square symbols in figure 7.14).
 For the range of rotation speeds investigated, ??WT and ??T increased with an increase in rotation speed.
 For the range of rotation speeds investigated, 1.2? ? ??T TE ? 10.4? and 1.1? ? ??T LE ? 10.0?. The corre-
 sponding deviation of ?T from ?DC ranges as follows: 1.5? ? ??T TE ? 10.9? and 1.5? ? ??T LE ? 10.9?.
 The effect of initial incidence and rotation centre were investigated very briefly. Results are summarised
 in table 7.6. Results in rows 2 and 3 are compared to to results in row 1. Consider the result in row 2
 which shows the sensitivity of transition to changing the pivot point from the trailing edge to the model
 pivot point in the experiment. ?? increases by approximately 1.5? and ?? reduces by approximately 1.5?.
 This difference is likely to be smaller at a smaller rotation rate. Consider the result in row 3 which shows
 the sensitivity of transition to reducing the initial incidence. The change in ?? is very small and ?? reduces
 by approximately 1.8?. Due to the complex nature of the dynamic case, it is not possible to generalise the
 result from two numerical experiments. However, they do prove that MR ? RR transition is sensitive to
 rotation centre and initial condition.
 7.4 Thoughts on Three-dimensional Effects
 The work of Ivanov et al. [24] on steady, 3D shock wave reflection was presented earlier in chapter 4. Their
 findings show, for a given geometry and free stream condition, that the 2D Mach stem height is always
 134
 0
  5
  10
  15
  20
  25
  30
 -0.1 -0.08 -0.06 -0.04 -0.02  0
 Dimensionless edge speed, ME = VE/a?
 W
 ed
 ge
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? W
 T
 Leading Edge Pivot
 Trailing Edge Pivot
 Uncorrected ?N
 Corrected ?N : Trailing Edge Pivot
 Corrected ?N : Leading Edge Pivot
 (a) ?WT vs. ME at M = 2.98
  25
  30
  35
  40
  45
 -0.1 -0.08 -0.06 -0.04 -0.02  0
 Dimensionless edge speed, ME = VE/a?
 Sh
 o
 ck
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? T
 Leading Edge Pivot
 Trailing Edge Pivot
 Uncorrected ?N
 Corrected ?N : Trailing Edge Pivot
 Corrected ?N : Leading Edge Pivot
 (b) ?T vs. ME at M = 2.98
 Figure 7.13: ?WT and ?T vs. ME at M = 2.98
 135
Table 7.6: Sensitivity of ??WT and ??T to pivot point and ?wi for ME = ?0.1 at M = 2.98
 Case Description ??WT ??T
 ?wi = 24.5?, Trailing edge pivot 10.4 10.9
 ?wi = 24.5?, Model pivot in experiment 12.0 9.4
 ?wi = 23.5?, Trailing edge pivot 10.2 9.1
  0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  5  10  15  20  25
 Wedge Incidence Angle, ?w [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot, ME = ?0.01
 Trailing Edge Pivot, ME = ?0.05
 Trailing Edge Pivot, ME = ?0.1
 Leading Edge Pivot, ME = ?0.01
 Leading Edge Pivot, ME = ?0.05
 Leading Edge Pivot, ME = ?0.1
 2D Steady State CFD
 Linear Fit
 Second Order Polynomial Fit
 Figure 7.14: Computed variation of Mach stem height with ?w for rapid, impulsive rotation. M = 2.98,
 ?wi = 24.5?, h/w = 1.01 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing
 edge). Dashed lines represent linear fits used to estimate ?WT for ME = ?0.01, ?0.05 and ?0.01. Solid
 lines are second-order polynomial fits used to compute ?WT for the steady state case.
 136
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  28  30  32  34  36  38  40  42  44
 Shock Incidence Angle, ? [degrees]
 D
 im
 en
 si
 o
 n
 le
 ss
 M
 a
 ch
 st
 em
 he
 ig
 ht
 ,
 m
 /w
 Trailing Edge Pivot, ME = ?0.01
 Trailing Edge Pivot, ME = ?0.05
 Trailing Edge Pivot, ME = ?0.1
 Leading Edge Pivot, ME = ?0.01
 Leading Edge Pivot, ME = ?0.05
 Leading Edge Pivot, ME = ?0.1
 2D Steady State CFD
 Second Order Polynomial Fit
 Figure 7.15: Computed variation of Mach stem height with ? for rapid, impulsive rotation. M = 2.98,
 ?wi = 24.5?, h/w = 1.01 (for rotation about the leading edge), g/w = 0.6 (for rotation about the trailing
 edge). The solid line is a second-order polynomial fit to compute ?T for the steady state case. ?T is measured
 at ?WT .
 137
larger than that set up with a finite aspect ratio wedge. For MR, the flow downstream of the Mach stem is
 subsonic and the reflection pattern in the vertical plane of symmetry is always influenced by 3D spanwise
 effects. In the limit, as the wedge aspect ratio approaches infinity, the 3D effects vanish and the reflection in
 the vertical plane of symmetry is 2D. Though there is a difference in Mach stem height between the 2D and
 finite aspect ratio wedges, ?T is the same, provided the wedge aspect ratio is sufficiently large (see Skews
 [39]).
 In the dynamic experiment presented earlier in this chapter, the difference in initial Mach stem height
 between the wind tunnel experiment and the 2D CFD result was evident. As ? decreased with decreasing
 ?w, the experimental and 2D CFD result (see figure 7.4) converged to similar values of ?T as observed in
 the steady case of Ivanov et al. [24].
 In this experiment, transition to RR occurred approximately 2.6 ms after the wedge motion commenced.
 Given the free stream conditions at M = 3.26 and TO = 302.0K, the local acoustic speed behind the Mach
 stem at the initial condition is approximately 341 m/s. Given the wedge semi-span of 85 mm, information
 of any disturbance can traverse the semi-span approximately 10 times in 2.6 ms. The time taken for any
 acoustic signal to traverse the semi-span is significantly smaller than the time taken to complete the wedge
 motion. This leads to the question of the sensitivity of ?T should the time taken for information to traverse
 the semi-span be much larger than the time taken to rotate the wedge to the point of transition. This
 may be achieved by increasing the wedge span. This will have practical implications for experiments and
 computations, but if implemented successfully may delay transition to RR in comparison to the 2D case due
 to the delayed arrival of spanwise information at the triple point. It implies that increasing the wedge aspect
 ratio in the dynamic case may result in a deviation from the 2D result for ?T . In the steady case, there is no
 difference in ?T between the 2D and 3D case beyond some critical wedge aspect ratio. In the dynamic case,
 a difference in ?T between the 2D and 3D case may be observed and may be found to increase for increasing
 wedge aspect ratios beyond a critical wedge aspect ratio. It may also be possible to delay the 3D transition
 with respect to the 2D transition by increasing the wedge rotation speed. While the time to complete the
 wedge motion will decrease, it is possible that, due to the dynamic effects highlighted in this chapter, the
 wedge reaches zero incidence before transition occurs.
 Consider the 3D reflection pattern in figure 7.16 published by Ivanov et al. [24]. The reflection in the
 vertical plane of symmetry is MR. In the spanwise direction the reflection changes from MR to RR and back
 to MR. This highlights the possibility of the intermediate RR acting as a filter that prevents information
 from the wedge corner from reaching the reflection in the vertical plane of symmetry. The range of scenarios
 that may be possible is ideally investigated with computational fluid dynamics. However, this requires a
 138
Figure 7.16: Steady, 3D reflection pattern computed with an Euler code by Ivanov et al. [24]. M = 4.0,
 ? = 35.5?, b/w = 3.75, g/w = 0.3.
 3D moving mesh capability with the necessary grid refinement algorithms for accurate and efficient shock
 capture. This is recommended for future work.
 7.5 Conclusion
 Experiments were completed in the CSIR supersonic wind tunnel to investigate dynamic MR? RR transition
 at M = 3.26 and 2.96. The experiments at M = 3.26 and 2.96 were completed successfully and provided
 sufficient data to validate the CFD code. The measured motion and the initial shock incidence were used to
 mimic the experiment with a 2D Euler CFD code. The expected difference between the initial Mach stem
 heights in the experiment and the CFD solution was observed. However, there was close agreement in ?T
 between experiment and CFD. This provided sufficient confidence in the ability of the CFD code to model
 the dynamic case of interest and to extend its application to other scenarios beyond the current experiments.
 CFD was used to further investigate the sensitivity of transition to rotation speed, initial incidence and
 rotation centre in the strong and weak-reflection ranges. The flow downstream of the Mach stem is subsonic
 and is influenced by any interaction or disturbance that appears in this subsonic region. Due to impulsive
 wedge start and rapid wedge rotation, there are very marked dynamic effects on the variation of Mach stem
 height with wedge incidence and the deviation from the steady transition conditions is significant. MR ?
 RR transition depends on the initial condition and the transient variation of Mach stem height with wedge
 incidence. In chapter 5, on dynamic RR ? MR transition, the maximum computed deviation from ?DC at
 ME = +0.1 was in the order of ??T = 1.6?. For the reverse transition, the maximum computed deviations
 from the steady, theoretical transition values are significantly larger.
 M = 1.93: For the range of rotation speeds investigated, ??WT and ??T increased with an increase in ro-
 tation speed. For the range of rotation speeds investigated, 1.2? ? ??T TE ? 7.1? and 1.0? ? ??T LE ? 6.2?.
 139
The corresponding deviation of ?T from ?DC ranges as follows: 1.5? ? ??T TE ? 7.4? and 1.7? ? ??T LE ? 7.9?.
 M = 2.98: For the range of rotation speeds investigated, ??WT and ??T increased with an increase in rota-
 tion speed. For the range of rotation speeds investigated, 1.2? ? ??T TE ? 10.4? and 1.1? ? ??T LE ? 10.0?.
 The corresponding deviation of ?T from ?DC ranges as follows: 1.5? ? ??T TE ? 10.9? and 1.5? ? ??T LE ? 10.9?.
 The sensitivity of ?T to changing the rotation point from the trailing edge to the model pivot point was
 investigated briefly at M = 2.98 with ME = ?0.1. ?T increased by 1.5? and ?T reduced by 1.5?, a significant
 variation.
 The effect of initial incidence was also investigated briefly at M = 2.98 at ME = ?0.1. By reducing ?wi
 from 24.5? to 23.5? ??T decreases by approximately 1.8?, also a marked sensitivity.
 The flow field development for impulsive rotation about the wedge trailing and leading edges at M = 1.93
 for ME = ?0.075 was analysed in some detail. The flow field development is very sensitive to the rotation
 centre, more especially at large rotation rates. Four phases of the Mach stem development were identified in
 both cases.
 Rotation about the wedge leading edge at M = 1.93 for ME = ?0.075: The Mach stem height remains
 constant until the expansion wave arrives at the triple point. This is followed by an increase in Mach stem
 height. After the maximum Mach stem height is reached there is little change in Mach stem height for a
 small period after which time the Mach stem height decreases until transition to RR.
 Rotation about the wedge trailing edge at M = 1.93 for ME = ?0.075: The impulsive start of the wedge
 generates a disturbance on the incident wave. The disturbance propagates down the incident wave and
 propagates through the triple point and down the Mach stem towards the reflection plane. The Mach stem
 height is constant until the arrival of the disturbance on the incident wave. The disturbance causes a sudden,
 but momentary decrease in Mach stem height. Subsequently, there is little change in the Mach stem height
 for a period of time, before the Mach stem height decreases until transition to RR. In contrast, at M = 2.98,
 the Mach stem height increases slightly in phase III.
 It was demonstrated that MR can be maintained until zero wedge incidence with a sufficiently large
 rotation rate (ME = ?0.1 at M = 1.93). At small rotation speeds (ME = ?0.01), Mach stem development
 and ?T exhibit little sensitivity to rotation centre.
 There was good agreement in ?T between the 3D experiment and the 2D CFD result. It is possible, for
 a given rotation speed, that the results deviate beyond some critical value of wedge aspect ratio due to the
 delay in arrival of spanwise information to the reflection pattern in the wedge vertical plane of symmetry. If
 true, it would mark a fundamental difference between the steady and dynamic cases. In the steady case, the
 3D and 2D values for ?T converge above a critical wedge aspect ratio. In the dynamic case, the 3D and 2D
 140
values for ?T may diverge above a critical wedge aspect ratio. Intermediate RR in the spanwise direction
 may also influence the propagation of spanwise information to the wedge vertical plane of symmetry and
 this may also influence ?T .
 7.6 Recommendations for Future Work
 The following items were raised during the course of this work. Though they lie beyond the scope of this
 work, they are highlighted for consideration in future work.
 1. A detailed investigation to identify and quantify the mechanism(s) in phases III and IV of the Mach
 stem development at M = 1.93 and M = 2.98.
 2. Development of an appropriate 3D flow solver for the investigation of dynamic, three-dimensional
 effects on MR ? RR transition. The issues of mesh movement, solver accuracy, flow field resolution
 and solver speed are pertinent for consideration here.
 3. Investigation of the effect of increasing wedge aspect ratio on ?T with respect to the 2D result.
 4. Investigation of the effect of increasing the rotation speed of a finite aspect ratio wedge on ?T with
 respect to the 2D result.
 5. Investigation of the effect of intermediate spanwise RR patterns to the propagation of spanwise infor-
 mation and its effect on ?T .
 141
(a) The initial, steady MR at ?wi = 13.4?, t = 0.0 s
 Stationary Triple Point
 Surface Expansion Waves
 Moving Towards Triple Point
 (b) The flow field at ?w = 13.0?, t = 14.1?s. The expansion waves from the wedge surface move toward the triple point.
 At this time the flow in the vicinity of the triple point is ?unaware? of the movement of the wedge and the triple point is
 stationary.
 Figure 7.17: Computed pressure contours at (a) ?wi = 13.4? and (b) ?w = 13.0? for impulsive rotation about
 the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84.
 142
Arrival of Surface Expansion
 Waves at Triple Point
 Stationary Triple Point
 (a) The expansion waves from the wedge surface arrive at the triple point at ?w ? 11.5? at t = 67?s. Up to this time there
 has been no movement of the triple point.
 Movement of Triple Point
 Away From Reflection Surface
 (b) ?w = 11.0?, t = 84.6?s. The first movement of the triple point is observed after the surface expansion waves reach the
 triple point. The expansion waves ?suck? the triple point away from the reflection surface. This increase in Mach stem was
 also reported on briefly by Felthun and Skews [12] for rotation about the wedge leading edge in a M = 3.0 free stream.
 Figure 7.18: Computed pressure contours at (a) ?w = 11.5? and (b) ?w = 11.0? for impulsive rotation about
 the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84.
 143
Movement of Triple Point
 Away From Reflection Surface
 (a) The Mach stem height continues to increase at ?w = 10.0?, t = 119.9?s
 Maximum Mach Stem Height
 (b) The Mach stem height reaches a maximum value at ?w = 8.5?, t = 172.8?s.
 Figure 7.19: Computed pressure contours at (a) ?w = 10.0? and (b) ?w = 8.5? for impulsive rotation about
 the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, h/w = 0.84.
 144
(a) ?w = 7.0?, t = 225.6?s (b) ?w = 6.0?, t = 260.9?s.
 (c) ?w = 5.0?, t = 296.2?s. (d) ?w = 4.0?, t = 331.4?s.
 (e) ?w = 1.5?, t = 419.6?s. (f) ?w = 1.0?, t = 437.2?s.
 Figure 7.20: Computed pressure contours showing decreasing Mach stem height between ?w = 7.0? and
 ?w = 1.0? for impulsive rotation about the wedge leading edge at ME = ?0.075. M = 1.93, ?wi = 13.4?,
 h/w = 0.84.
 145
(a) ?w = 0.5?, t = 454.8?s (b) ?w = 0.1?, t = 468.9?s.
 Figure 7.21: Development of flow field in the vicinity of the reflection point after transition to RR between
 (a) ?w = 0.5? and (b) ?w = 0.1? for impulsive rotation about the wedge leading edge at ME = ?0.075.
 M = 1.93, ?wi = 13.4?, h/w = 0.84.
 146
(a) The initial, steady MR at ?wi = 13.4?, t = 0.0 s
 Pressure Wave Due to
 Impulsive Start
 Incident Shock at Initial
 Incidence Drawn in Red
 Disturbance Propagation
 on Incident Wave
 (b) ?w = 13.0?, t = 14.1?s. The impulsive movement of the wedge tip generates a disturbance that
 propagates down the length of the incident wave. The resultant pressure wave between the incident wave
 and the wedge surface is indicated. At the same time compression waves are generated at the wedge
 surface. The solid red line indicates the position of the incident wave at the initial condition.
 Figure 7.22: Computed pressure contours at (a) ?wi = 13.4? and (b) ?w = 13.0? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6.
 147
Pressure Wave Due to
 Impulsive Start
 Disturbance Propagation
 on Incident Wave
 (a) ?w = 12.5?, t = 31.7?s.
 Pressure Wave Due to
 Impulsive Start
 Disturbance Propagation
 on Incident Wave
 (b) ?w = 12.0?, t = 49.4?s
 Figure 7.23: Computed pressure contours at (a) ?w = 12.5? and (b) ?w = 12.0? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The disturbance propagates
 down the length of the incident wave and the compression waves from the wedge surface continue propagating
 away from the surface. The solid red line indicates the position of the incident wave at the initial condition.
 148
Disturbance Propagation
 on Incident Wave
 (a) The disturbance on the incident wave reaches the triple point at ?w = 11.6?, t = 63.5?s.
 Reflected Pressure Wave
 Pressure Gradient on
 Incident Wave
 (b) Computed flow field at ?w = 10.0?, t = 119.9?s. As the disturbance passes through the triple
 point there is a rapid and momentary decrease in Mach stem height. The pressure wave generated
 on the incident wave by the impulsive start reflects from the reflection plane. The intersection of
 the isobar highlighted in green with the incident wave indicates the location of the discontinuity
 in curvature on the incident wave. The segment upstream of the discontinuity is curved and a
 pressure gradient is evident over this segment of the incident wave. The segment downstream of the
 discontinuity is planar and there is no pressure gradient over this segment.
 Figure 7.24: Computed pressure contours at (a) ?w = 11.6? and (b) ?w = 10.0? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The solid red line indicates
 the position of the incident wave at the initial condition.
 149
Pressure Gradient on
 Incident Wave
 ? m
 (a) ?w = 9.5?, t = 137.5?s
 Pressure Gradient on
 Incident Wave
 ? m
 (b) ?w = 8.5?, t = 172.8?s
 Figure 7.25: Computed pressure contours at (a) ?w = 9.5? and (b) ?w = 8.5? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. Between ?w = 9.5? and 7.5?
 the Mach stem height is constant at approximately ?m below the initial Mach stem height. The discontinuity
 on the incident wave continues to move towards the triple point. The solid red line indicates the position of
 the incident wave at the initial condition.
 150
Pressure Gradient on
 Incident Wave
 Decreasing m
 (a) ?w = 7.0?, t = 225.6?s
 (b) ?w = 5.0?, t = 296.2?s
 Figure 7.26: Computed pressure contours at (a) ?w = 7.0? and (b) ?w = 5.0? for impulsive rotation about
 the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6. The discontinuity on the
 incident wave has reached the triple point and the incident wave is curved along its entire length. The Mach
 stem height decreases until transition to RR. The solid red line indicates the position of the incident wave
 at the initial condition.
 151
(a) ?w = 2.4?, t = 387.8?s
 (b) ?w = 1.5?, t = 419.6?s
 (c) ?w = 1.1?, t = 433.7?s
 Figure 7.27: Computed pressure contours at (a) ?WT = 2.4?, (b) ?w = 1.5? and (c) ?w = 1.1? for impulsive
 rotation about the wedge trailing edge at ME = ?0.075. M = 1.93, ?wi = 13.4?, g/w = 0.6.
 152
Chapter 8
 Conclusions and Recommendations
 for Future Work
 8.1 Conclusions
 There have been numerous studies on the steady state transition criteria between regular reflection (RR) and
 Mach reflection (MR) of shock waves for a stationary, two-dimensional (2D) wedge in a steady supersonic
 flow, since the original shock wave reflection research by Ernst Mach in 1878. The steady, 2D transition
 criteria between RR and MR are well established. There has been little done to consider the dynamic effect
 of rapid wedge rotation on RR ? MR transition.
 This thesis presents the results of an investigation of the effect of rapid wedge rotation on transition
 between 2D regular and Mach reflection in the weak and strong-reflection ranges, with experiment and
 computational fluid dynamics. A novel facility was designed to rotate a pair of large aspect ratio wedges in a
 450 mm ? 450 mm supersonic wind tunnel at wedge rotation speeds up to 11000 deg/s resulting in wedge tip
 speeds approximately 3.3 % of the free stream acoustic speed. Steady state, baseline experiments in which
 the wedges were rotated very gradually were also completed. A schlieren system and optical measurement
 system was developed. High-speed images and measurements were presented for the steady and dynamic
 experiments. Numerical solution of the inviscid governing flow equations was used to model the steady case
 and to mimic the experimental motion in the dynamic experiments. Most steady state simulations were
 completed with Fluent V 12.0 and all dynamic simulations were done with an in-house, 2D Euler Code.
 Steady state, baseline experiments were completed in the weak and strong-reflection ranges and transition
 measurements were compared to 2D steady state, theoretical values and Euler computations. There was close
 agreement between theoretical, computational and experimental transition for the steady case, with the
 following exception. Due to the levels of free stream noise in the supersonic wind tunnel, incidence-induced
 153
hysteresis was not observed in the strong-reflection region and transition occurred at the von Neumann
 condition for increasing and decreasing incidence. In the ideal case, RR ? MR transition occurs at the
 detachment condition and the reverse transition occurs at the von Neumann condition. Therefore, there is
 discrepancy between steady theory/CFD and experiment for RR ? MR transition in the strong-reflection
 range only, which is consistent with observations in other facilities with sufficient levels of free stream noise.
 8.1.1 Summary of Results for Dynamic RR to MR Transition
 Rapid wedge rotation generated a measurable dynamic effect on RR ?MR transition. The first experimental
 evidence of 2D RR ? MR reflection transition beyond the steady detachment condition in the weak and
 strong-reflection ranges was presented (see figure 8.1). In the dynamic experiments and computations of the
 experiments at M = 1.93 and 2.98, RR persisted approximately 0.9??1.3? beyond the corrected detachment
 condition. In all instances, there was good agreement between experiment and CFD, including dynamic RR
 ? MR transition in the strong-reflection region. The agreement between the dynamic experiment, in which
 small perturbations are always present in the free stream, and the dynamic CFD, in which the free stream is
 without perturbations, implies that RR ? MR transition in the strong-reflection region becomes insensitive
 to free stream noise above a certain critical rotation speed.
 Due to the close agreement between CFD and experiment, Euler CFD was also applied to scenarios
 beyond the limits of the current facility to explore the influence of variables in the parameter space. Over
 the range of free stream conditions and rotation speeds investigated, the deviation from the corrected, steady,
 theoretical shock incidence at transition, ??T , ranged between 0.4? at ME = +0.01 and 1.6? at ME = +0.1.
 For a given rotation centre and free stream condition, there was little dependence of transition on initial
 incidence, for the values of initial incidence tested.
 CFD was also used to investigate the dynamic transition mechanism over a limited number of simulations.
 For dynamic RR ? MR transition, a distinction is drawn between the sonic, length scale and detachment
 conditions. The point at which the flow downstream of the reflection point goes sonic is not necessarily
 the point at which the wedge length scale, from the wedge trailing edge expansion, is communicated to the
 reflection point. There is evidence to support that the RR ? MR transition criteria for the rapidly rotating
 wedge is neither the sonic or length scale conditions, but rather the condition at which the reflected wave
 can no longer satisfy the boundary condition at the reflection point. Dynamic simulations showed that RR
 could be maintained for some time with a length scale present at the reflection point. It is possible that
 RR is maintained as long as the reflected wave is able to turn the incoming flow parallel to the reflection
 plane. Other simulations showed that transition to MR was possible without the wedge length scale being
 154
 34
  36
  38
  40
  42
  44
  46
  1.8  2  2.2  2.4  2.6  2.8  3  3.2  3.4
 Free stream Mach number, M
 Sh
 o
 ck
 in
 ci
 de
 n
 ce
 a
 t
 tr
 a
 n
 si
 tio
 n
 ,
 ? T
 [de
 gr
 ee
 s] Unsteady Experiment: RR ? MRSteady Experiment
 Unsteady Experiment: MR ? RR
 ?D
 ?N
 Figure 8.1: Measured ?T from experiments compared to analytical steady transition criteria
 communicated to the reflection point, perhaps due to the failure of RR to maintain the boundary condition
 at the reflection point.
 Pressure traces through the reflection point show that rapid rotation increases the maximum achievable
 pressure rise through the reflection point of a RR in comparison to the steady case. As in the steady case,
 transition to MR in the strong-reflection region, is accompanied by a rapid decrease in pressure rise through
 the reflection/triple point. Due to the similarity in trend between the steady and dynamic cases, steady
 state shock polars may be useful in identifying the critical trend that highlights the point of transition in
 the dynamic case.
 8.1.2 Summary of Results for Dynamic MR to RR Transition
 Rapid wedge rotation generated a measurable effect on MR ? RR transition. The first experimental evidence
 of 2D MR ? RR transition below the steady von Neumann condition is presented (see figure 8.1). In
 experiments and computations for dynamic MR ? RR transition at M = 3.26, transition was delayed
 approximately 0.8? below the corrected von Neumann condition. Once again, there was good agreement
 between experiment and CFD of the experiment. CFD was used to further investigate the sensitivity of
 155
transition to rotation speed, initial incidence and rotation centre in the strong and weak-reflection ranges.
 Due to impulsive wedge start and rapid wedge rotation, there are very marked dynamic effects on the
 variation of Mach stem height with wedge incidence and the deviation from the steady transition conditions
 is significant. MR ? RR transition depends on the initial condition and the transient variation of Mach
 stem height with wedge incidence. For the range of rotation speeds investigated at M = 1.93 and 2.98, ??T
 ranged from 1.2? at ME = ?0.01 to 10.9? at ME = ?0.1. It was demonstrated that MR can be maintained
 until zero wedge incidence with a sufficiently large rotation rate (for example ME = ?0.1 at M = 1.93). For
 the range of simulations completed, the dependence of transition to initial incidence and rotation centre is
 marked, especially at larger rotation speeds, e.g. ME = ?0.1.
 Some ideas on three-dimensional (3D) effects were presented. For this particular experimental setup
 there was good agreement on transition angle between 2D CFD and 3D experiment. It is possible, for a
 given rapid rotation speed, that they deviate beyond some critical wedge aspect ratio due to the delay in
 arrival of spanwise information to the reflection pattern in the wedge vertical plane of symmetry. If true,
 it would mark a significant deviation from the steady case, i.e. in the steady case, there is no difference in
 transition between the 2D and 3D case beyond a critical wedge aspect ratio. In the dynamic MR ? RR
 case, a difference in transition may be observed between the 2D and 3D case beyond a critical wedge aspect
 ratio, and this may increase for increasing wedge aspect ratios.
 8.2 Recommendations for Future Work
 The recommendations for future work summarised in chapters 6 and 7 are listed here:
 8.2.1 Dynamic RR to MR Transition
 1. Determination of the minimum critical wedge rotation speed required such that transition is indepen-
 dent of free stream perturbations in the CSIR facility. This could be followed by a more general, but
 detailed investigation into the relationship between free stream noise and the minimum critical wedge
 rotation speed required to achieve RR ? MR transition that is independent of the level of free stream
 noise and this could be applied to any facility.
 2. An investigation to quantify the effect of rapid wedge rotation on the incident wave curvature and
 pressure gradient along the incident wave.
 3. A detailed investigation into the ?detachment? condition for the dynamic case.
 156
4. Execution of the dynamic experiments at M = 4.0 in which the dual solution domain is larger. At
 M = 4.0 the difference between ?D and ?N is 5.8?.
 5. Execution of the experiment described by Hornung [17]. Establish an initial, steady, RR below the
 dual solution domain, preferably at M = 4.0. Rotate the wedges rapidly and terminate the motion
 just below the detachment condition. The development of the reflection pattern would be of primary
 interest. The challenge, from an experimental perspective, lies in the termination of the wedge motion
 at the desired condition in a way that does not introduce vibration that influences transition or the
 development of the reflection pattern.
 6. Investigation of the curious negative second gradient for Mach stem development against ? with ME =
 +0.01 at M = 2.98.
 7. The rig developed here may be applied to various interesting experimental studies, e.g. the effect of
 acceleration of finite aspect ratio wings in a steady, supersonic free stream on the unsteady evolution
 of the wing wake structure.
 8.2.2 Dynamic MR to RR Transition
 1. A detailed investigation to identify and quantify the mechanism(s) in phases III and IV of the Mach
 stem development at M = 1.93 and M = 2.98.
 2. Development of an appropriate 3D flow solver for the investigation of dynamic, three-dimensional
 effects on MR ? RR transition. The issues of mesh movement, solver accuracy, flow field resolution
 and solver speed are pertinent for consideration here.
 3. Investigation of the effect of increasing wedge aspect ratio on ?T with respect to the 2D result.
 4. Investigation of the effect of increasing the rotation speed of a finite aspect ratio wedge on ?T with
 respect to the 2D result.
 5. Investigation of the effect of intermediate spanwise RR patterns to the propagation of spanwise infor-
 mation and its effect on ?T .
 157
Appendix A
 Data Acquisition of Freestream
 Conditions
 A.1 Mach Number Measurement
 The test section Mach number is calculated from:
 M =
 [{(
 PO
 p
 ) ??1
 ?
 ? 1
 }{
 2
 ? ? 1
 }] 12
 (A.1)
 For isentropic flow, the stagnation pressure between the settling chamber and the test section is constant.
 Stagnation pressure is measured with a stagnation pressure probe in the settling chamber and test section
 static pressure is measured from a static pressure port in the test section wall. Stagnation and static pressure
 measurements are relative to atmosphere. A dedicated atmospheric pressure transducer is used to measure
 the atmospheric pressure. The following sections document the calibration of the stagnation, static and
 atmospheric pressure transducers.
 A.1.1 Pressure Transducer Calibration
 A calibrated Druck Digital Pressure Indicator 605 (See Figure A.1) was used to calibrate all pressure trans-
 ducers, viz. stagnation, static and atmospheric pressure transducers. The technical specifications of the
 calibration standard are tabulated in Table A.1. All pressure transducers were calibrated by applying a
 known pressure from the DPI605 and recording the voltage output on the National Instruments data acqui-
 sition system. The transducer output voltage was recorded when a stable pressure reference was established
 on the DPI605.
 158
Figure A.1: Druck Digital Pressure Indicator 605
 Table A.1: Technical specifications of the Druck DPI605
 Item Details
 Instrument Name Druck DPI 605
 Calibration Date 10 September 2009
 Calibration Authority Unique Metrology
 Calibration Certificate Number 0909P2536-1
 Serial Number 1140/93-4
 Measurement Uncertainty (0 - 10 bar) ?0.0022bar
 Measurement Uncertainty (10 - 20 bar) ?0.0032bar
 Maximum Correction (12 bar) -0.00309 bar
 Pressure range 0 - 20 bar
 159
Table A.2: Pressure Transducer Specifications
 Transducer TP-HI TP-LO Static Press. Atmospheric Press.
 Instrument Name Schaevitz Schaevitz Schaevitz Vaisala PTB101B
 Serial Number 124151 41024 131546 X1250006
 Input Range 0-200 PSI Gauge 0-3.5 Bar Abs. 0 - 20 PSI Gauge 100 - 1060 hPa
 Excitation 5.0 V DC 5.0 V DC 5.0 V DC 10 - 30 V DC
 Output Range -0.35 mV to 25 mV 0 to 25 mV -0.2 to 25 mV 0 - 2.5 V
 Calibration Date 22/10/2009 22/10/2009 22/10/2009 22/10/2009
 Table A.3: Total pressure transducer calibration: High Range
 Applied Pressure (kPa) Data Acquisition Reading (mV)
 0 -0.2828
 255.38 4.4075
 541.8 9.6516
 811.14 14.5623
 1102.56 19.8493
 650.6 11.6464
 295.26 5.1474
 103.96 1.6407
 0.002 -0.2821
 A.1.2 Pressure Transducer Specfication and Calibration Results
 Stagnation pressure measurement in the settling chamber is performed with two transducers. One is ded-
 icated to measurement between 0 - 3.5 bar absolute (TP-LO) and the other between 3.5 bar absolute to
 200 PSI guage (TP-HI). Table A.2 includes the specifications of all the pressure transducers. Table A.7
 summarises the regression statistics from the calibration process. A comparison of calibration coefficients
 for all pressure transducers in the CSIR HSWT indicate a negligible variation since March 2008.
 The values of pressure applied with the DPI 605 per transducer is tabulated in Tables A.3 - A.6. Figure
 A.2 includes the results of the calibration and the linear fit of the data.
 Table A.4: Total pressure transducer calibration: Low Range
 Applied Pressure (kPa) Data Acquisition Reading (mV)
 0 6.9255
 119.51 15.3477
 179.42 19.5724
 239.38 23.787
 180.5 19.6694
 111.32 14.7932
 0 6.9308
 160
Table A.5: Static pressure transducer calibration
 Applied Pressure (kPa) Data Acquisition Reading (mV)
 -0.012 -0.1134
 32.11 5.579
 59.95 10.5233
 89.97 15.8801
 129.98 23.0047
 110 19.4671
 50.03 8.8049
 -0.012 -0.0819
 -65.97 -11.741
 Table A.6: Atmospheric pressure transducer calibration
 Applied Pressure (kPa) Data Acquisition Reading (mV)
 87.167 -1.48
 84.03 -1.3079
 86.01 -1.4165
 88.04 -1.528
 90.003 -1.6353
 87.16 -1.4798
 -200
  0
  200
  400
  600
  800
  1000
  1200
 -5  0  5  10  15  20
 Data Acquisition Output [mV]
 TP-HI
 A
 p
 p
 lie
 d
 P
 r
 e
 ss
 u
 r
 e
 [kP
 a
 ]
 -50
  0
  50
  100
  150
  200
  250
  6  8  10  12  14  16  18  20  22  24
 Data Acquisition Output [mV]
 TP-LO
 A
 p
 p
 lie
 d
 P
 r
 e
 ss
 u
 r
 e
 [kP
 a
 ]
 -80
 -60
 -40
 -20
  0
  20
  40
  60
  80
  100
  120
  140
 -15 -10 -5  0  5  10  15  20  25
 Data Acquisition Output [mV]
 Static
 A
 p
 p
 lie
 d
 P
 r
 e
 ss
 u
 r
 e
 [kP
 a
 ]
  84
  85
  86
  87
  88
  89
  90
  91
 -1.65 -1.6 -1.55 -1.5 -1.45 -1.4 -1.35 -1.3
 Data Acquisition Output [mV]
 Atmospheric
 A
 p
 p
 lie
 d
 P
 r
 e
 ss
 u
 r
 e
 [kP
 a
 ]
 Figure A.2: Calibration and regression for pressure transducers
 161
Table A.7: Summary of pressure transducer regression statistics
 Transducer TP-HI TP-LO Static Press. Atmospheric Press.
 R Squared 0.999992757 0.999996514 0.999992675 0.999998269
 Standard Error [Pa] 1106.03 187.28 177.92 2.94
 Slope, m [Pa/mV] 54733.607 14190.514 5635.744 -18239.333
 ?m[Pa/mV] 55.678 11.849 5.765 11.999
 Intercept, c [Pa] 14406.451 -98374.822 454.327 60172.917
 ?c[Pa] 551.686 194.507 74.863 17.734
 A.1.3 Mach Number Calculation and Uncertainty Analysis
 The uncertainty estimation method documented by Kirkup [29] is implemented. The ratio of total and static
 pressure as a function of free stream Mach number for an isentropic flow may be expressed as:
 PO
 p
 =
 (
 1 +
 ? ? 1
 2
 M2
 ) ?
 ??1 (A.2)
 Given the stagnation and static pressures the Mach number is given by:
 M =
 [(
 PO
 p
 ) ??1
 ?
 ? 1
 ] 1
 2
 (A.3)
 In the CSIR supersonic tunnel, the stagnation and static pressure transducers are used as guage pressure
 transducers. The gauge reading is added to an accurate atmospheric measurement to obtain an absolute
 measurement. Therefore:
 PO,absolute = PO,gauge + patmosphere (A.4)
 and
 pabsolute = pgauge + patmosphere (A.5)
 Since the stagnation and static pressure gauges are used as gauge pressure transducers only the gradient,
 m, of the linear fit through the transducer calibration data is necessary. The gauge pressure (stagnation and
 static) may be calculated from the data acquisition readings from:
 PO,gauge = (mV ?mPO ) (A.6)
 and
 162
pgauge = (mV ?mp) (A.7)
 The atmospheric pressure is calculated in a similar manner, with the exception that the y intercept,c, of
 the linear fit through the tranducer calibration data is also used, i.e.:
 patmosphere = (mV ?matmosphere) + catmosphere (A.8)
 Substituting equations A.6 - A.8 into equations A.4 and A.5 produces:
 PO,absolute = [(mV ?mPO)] + [(mV ?matmosphere) + catmosphere] (A.9)
 and
 pabsolute = [(mV ?mp)] + [(mV ?matmosphere) + catmosphere] (A.10)
 Substituting equation A.9 and A.10 into A.3 yields Mach number from the static and stagnation pressure
 measurements:
 M =
 [( [(mV ?mPO)] + [(mV ?matmosphere) + catmosphere]
 [(mV ?mp)] + [(mV ?matmosphere) + catmosphere]
 ) ??1
 ?
 ? 1
 ] 1
 2
 (A.11)
 From the calibration of the various pressure transducers, the uncertainty in the slope and intercept from
 a least squares fit of the calibration data are denoted as ?m and ?c respectively. If
 PO,gauge |max= (mV ? (mPO +?mPO)) (A.12)
 PO,gauge |min= (mV ? (mPO ??mPO )) (A.13)
 pgauge |max= (mV ? (mp +?mp)) (A.14)
 pgauge |min= (mV ? (mp ??mp)) (A.15)
 163
patmosphere |max= ((mV ?matmosphere +?matmosphere)) + (catmosphere +?catmosphere) (A.16)
 patmosphere |min= ((mV ?matmosphere ??matmosphere)) + (catmosphere ??catmosphere) (A.17)
 The uncertainty may then be used to determine the range of absolute stagnation and static pressures
 (taking into account the uncertainty in the slope and intercept of the transducer calibrations).
 PO,absolute |max = PO,gauge |max +patmosphere |max
 = (mV ? (mPO +?mPO )) + ((mV ?matmosphere +?matmosphere))
 + (catmosphere +?catmosphere) (A.18)
 PO,absolute |min = PO,gauge |min +patmosphere |max
 = (mV ? (mPO ??mPO)) + ((mV ?matmosphere ??matmosphere))
 + (catmosphere ??catmosphere) (A.19)
 pabsolute |max = pgauge |max +patmosphere |max
 = (mV ? (mp +?mp)) + ((mV ?matmosphere +?matmosphere))
 = + (catmosphere +?catmosphere) (A.20)
 pabsolute |min = pgauge |min +patmosphere |min
 = (mV ? (mp ??mp)) + ((mV ?matmosphere ??matmosphere))
 = + (catmosphere ??catmosphere) (A.21)
 164
When substituted into equation A.3 the uncertainty in Mach number only arising from the uncertainties
 in the slope and intercepts of the first order curve fit to the transducer calibration data may be calculated.
 Static and stagnation pressure readings are offset such that they give guage pressure readings. The offset
 is calculated from the first 500 data points before the tunnel start for which the guage pressure readings
 for the static and stagnation tranducers must give an average of 0 Pa. Due to inherent noise in the data
 acquisition system these corrections also have a minimum and maximum value which contribute to the overall
 uncertainty of the derived Mach number. These must also be taken into account (?p? and ?PO,?). Sample
 values from actual data acquisitions during tests were used to calculate the uncertainties at 3 different nozzle
 settings. Take into account the transducer linear regression statistics in A.7 and the fact that TP-LO is only
 used from 0 - 3.5 bar absolute pressure, the uncertainties are calculated as follows:
 Uncertainty calculation at a M 2.0 nozzle setting:
 patmosphere,mV = ?1.422V
 patmosphere = (?1.422V ??18239.333Pa/V ) + 60172.917Pa= 86111.820Pa
 patmosphere |max = (?1.422V ? (?18239.333Pa/V + 11.999Pa/V )) + (60172.917Pa+ 17.734Pa)
 = 86146.618Pa
 patmosphere |min = (?1.422V ? (?18239.333Pa/V ? 11.999Pa/V )) + (60172.917Pa? 17.734Pa)
 = 86077.023Pa
 Taking ?p? = ?847.371Pa, ?p? |max= ?941.862Pa, ?p? |min= ?737.110Pa and ?PO,? = 97139.294Pa,
 ?PO,? |max= 97187.384Pa and ?PO,? |min= 97079.110Pa, the following corrected gauge pressure measure-
 ments are:
 165
PO,gauge,mV = 17.149mV
 PO,gauge = (?17.149mV ? 14190.51Pa/V )??PO,? = 146208.081Pa
 PO,gauge |min = (?17.149mV ? 14190.51Pa/V )??PO,? |min= 145956.791Pa
 PO,gauge |max = (?17.149mV ? 14190.51Pa/V )??PO,? |max= 146471.463Pa
 PO,absolute = 146208.081Pa+ 86111.820Pa= 232319.901Pa
 PO,absolute |min = 145956.791Pa+ 86077.023Pa= 232033.814Pa
 PO,absolute |max = 146471.463Pa+ 86146.618Pa= 232618.081Pa
 pgauge,mV = ?9.497mV
 pgauge = (?9.497mV ? 5635.744Pa/V )??PO,? = ?52674.869Pa
 pgauge |min = (?9.497mV ? 5635.744Pa/V )??PO,? |min= ?52839.881Pa
 pgauge |max = (?17.149mV ? 14190.51Pa/V )??PO,? |max= ?52525.628Pa
 pabsolute = ?52674.869Pa+ 86111.820Pa= 33436.951Pa
 pabsolute |min = ?52839.881Pa+ 86077.023Pa= 33237.142Pa
 pabsolute |max = ?52525.628Pa+ 86146.618Pa= 33620.990Pa
 The derived Mach number is calculated as follows:
 M =
 [(
 PO,absolute
 pabsolute
 ) ??1
 ?
 ? 1
 ] 1
 2
 =
 [(
 232319.901Pa
 33436.951Pa
 ) ??1
 ?
 ? 1
 ] 1
 2
 = 1.923
 Substituting the extreme values of stagnation and static pressures the minimum and maximum derived
 166
Table A.8: CSIR tunnel test section calibration
 Nozzle Setting ?M |Trans.Calib. 2? |TestSect.Calib. ?M |Total Correction Factor
 2.0 0.004 0.002 0.006 1.0029
 3.0 0.01 0.008 0.011 1.0198
 3.3 0.013 0.013 0.026 1.000818
 Mach numbers may be calculated. In the above sample they are:
 M |min = 1.919
 M |max = 1.927
 2??M = M |max ?M |min
 = 0.007
 ?M = 0.004
 The Mach number and its uncertainty from the pressure transducer calibration at this particular data
 point is expressed as M1.923? 0.004. This uncertainty is the uncertainty in the Mach number measurement
 due to pressure transducer calibration process. The uncertainty for the remaining nozzle settings are included
 in table A.8. The uncertainty in the Mach number from test section calibration must also be taken into
 account. The test section calibration done by the CSIR in 2006 correlates the Mach number derived from
 the static pressure measurement on the wall with the Mach number in the test section. The uncertainties of
 the test section calibration and the correction factors for each nozzle setting is included in table A.8. A sum
 of the uncertainties from the test section calibration and the pressure transducer calibrations will provide
 an estimate of the total uncertainty in the Mach number in the test section.
 In view of the results in table A.8, a value of ?M |Total= 0.03 is assumed across the range of test conditions.
 Continuing with the example, application of the correction factor and the test section calibration uncertainty
 results in :
 M |corrected = 1.923? 1.0029 = 1.929
 ?M |Total = ?0.03
 M = 1.929? 0.03
 167
A.2 Test Section Acoustic Speed Measurement
 Given that
 TO
 T
 = 1 +
 ? ? 1
 2
 M2 (A.22)
 for isentropic flows and that the total temperature is constant between the settling chamber and the test
 section, the static temperature and acoustic speed in the test section may be calculated from the test section
 Mach number and a total temperature measurement in the settling chamber as follows :
 T = TO
 1 + ??12 M2
 (A.23)
 a =
 ?
 ?RT =
 ?
 ?R TO
 1 + ??12 M2
 (A.24)
 A.2.1 Stagnation Temperature Probe and Transducer Specification
 The total temperature probe discussed in Chapter 3 is repeated at this point in figure A.3. The probe
 consists of a PT100 Resistance Temperature Detector (RTD) sensor housed in a machined stainless steel
 shroud. The PT100 Platinum RTD was customised for these tests to increase exposure of the sensor to the
 settling chamber flow to increase the rate of heat transfer for the short duration test. PT100?s generally
 have a second order response to temperature through its entire range and the response between 0 and
 50?C is approximately linear. The PT100 has a dedicated current source that supplies the transducer
 with the necessary excitation to produce a DC voltage output. When connected to the power supply, the
 PT100 generates a 10V DC signal at 50?C and a 0V DC signal at 0?C. The tranducer response for the
 range ?200 ? 349?C was supplied by WIKA Instruments (figure A.4). A linear fit to the response in the
 expected operational range (0 ? 50?C is also provided. As can be seen from the regression the response is
 approximately linear in this range.
 In the range 0 ? 50?C the maximum deviation of the predicted temperature (from the slope of a linear
 regression) to the calibrated temperature is approximately 0.06?C. This is negligible in comparison to the
 uncertainty from the measurement technique. For example, consider the sample measurement at Mach 3.0
 presented earlier and repeated in figure A.5, the uncertainty in the measurement is estimated at 0.5 K.
 168
PT 100
 Sensor Flow
 Flow
 Settling Chamber Wall
 Figure A.3: Schematic of stagnation temperature probe in settling chamber
 A.2.2 Acoustic Speed Calculation and Uncertainty Analysis
 The uncertainty in Mach number and stagnation temperature may be used to determine the range of acoustic
 speed values bounded by amin and amax.
 a |max=
 ?
 ?R TO |max
 1 + ??12 M2 |min
 (A.25)
 a |min=
 ?
 ?R TO |min
 1 + ??12 M2 |max
 (A.26)
 For the sample measurement in figure A.5, in which M = 3.0? 0.03 and TO = 29.0?C? 0.5:
 a |max =
 ?
 1.4? 287.06? (29.0 + 0.5 + 273.15)
 1 + ??12 (3.0? 0.03)2
 = 209.8m.s?1
 a |min =
 ?
 1.4? 287.06? (29.0? 0.5 + 273.15)
 1 + ??12 (3.0 + 0.03)2
 = 206.7m.s?1
 Given that
 a =
 ?
 1.4? 287.06? (29.0 + 273.15)
 1 + ??12 (3.0)2
 = 207.9m.s?1
 the uncertainty band is approximately a |max ?a |min= 3.0m.s?1, approximately 3.0/207.9 = 1.5% ? ?0.75%
 169
 0
  50
  100
  150
  200
  250
 -200 -100  0  100  200  300  400
 ? = 100 + 2.3 ? 10?8T3 ? 6.8 ? 10?5T2 + 0.391649T
 R2 = 0.99999889572
 R
 es
 is
 ta
 n
 ce
 [oh
 m
 s]
 Temperature [deg C]
  100
  102
  104
  106
  108
  110
  112
  114
  116
  118
  120
  0  10  20  30  40  50
 ? = 100 + 0.38860128T
 R2 = 0.99999175
 R
 es
 is
 ta
 n
 ce
 [oh
 m
 s]
 Temperature [deg C]
 Figure A.4: Transducer response supplied by WIKA Instruments
 170
 10
  15
  20
  25
  30
  35
  40
  0  5  10  15  20  25
 Total Temperature [deg C
 ]
 Time [s]
  28
  30
  32
  34
  36
  38
  7  8  9  10  11  12
 Total Temperature [deg C
 ]
 Time [s]
 Figure A.5: Sample total temperature probe measurement (magnified view of select data range on the right
 hand side)
 of the nominal value of acoustic speed. Since ME = VE/a?, the contribution to the uncertainty in dimen-
 sionless edge speed is also approximately ?0.75%.
 A.3 National Instruments Data Acquisition System
 Figure A.6 illustrates the 24 channel National Instruments data acquisition system architecture for the CSIR
 supersonic wind tunnel facility. Transducer signals may be logged from the pc during the tests. Excitation
 is supplied from the SCXI-1520 cards. The tranducers interface with the signal conditioning unit through
 the National Instruments 1314 custom terminal block. The SCXI-1520 cards, in addition to providing
 transducer excitation, set channel gains and signal noise filtering. The SCXI-1520 interfaces to a PCI based
 6052e Analog to Digital Data acquisition card. The operator interfaces with the data acquisition system
 through a Labview software interface for configuration setup and data logging.
 171
Figure A.6: Data Acquisition Architecture
 172
Appendix B
 Schlieren System, High-Speed
 Imaging and Optics
 B.1 Schlieren System Specification
 A standard z-type schlieren system was designed and developed specifically for the dynamic tests [[?],[37]].
 Optical parameters of the system are documented in Table B.1 below. At the required 10000 fps only the
 central 512 x 512 pixels of the Photron Ultima APX-RS CCD chip are active. The schlieren system was
 optimised to focus the schlieren image onto this central section of the imaging chip.
 B.1.1 Optical Alignment
 Alignment of the schlieren system was done in accordance with the guidelines documented by Settles [37].
 All elements were levelled with the ground with a bubble level. All centre points of the elements of the
 schlieren system were arranged at approximately the same height above the ground to ensure co-incidence
 with the system optical axis. Alignment was performed with the assistance of a precision manufactured laser
 pointer as shown in Figure B.2. The laser pointer was mounted on a machined collar on the front end of
 the light source and provided an excellent visible marker to locate the system optical axis. The first mirror
 directed the laser light from the light source through the test section glass. As each surface of the two glass
 windows are not perfectly parallel to each other, each surface reflects a small amount of light back to the
 source. The four reflected images of the laser point on the windows were kept in sharp focus in the plane
 of the light source origin (perpendicular to the optical axis). This ensured that the light source was located
 at the focal point of the first mirror. The four reflections of the light source were also arranged about the
 optical axis as symmetrical as possible. This ensured that the first mirror directed the light source image
 perpendicular to the test section. A collimator was also designed to assist the alignment of the system and
 173
Slit
 Lens Housing
 Light Source
 Housing
 Slit Mount
 Figure B.1: Schlieren system light source
 was installed at the location of the cut-off and imaging lens (Figure B.3). The collimator has an entry hole
 for the incoming laser light. If the incoming light is co-incident with the optical axis and the collimator is
 arranged perfectly about the axis, the incoming laser light goes through the centre of the collimator entrance
 and can be seen on the reflection in the middle of the body. The second mirror and stand mount for the
 schlieren cut-off and imaging lens were adjusted to achieve a perfectly collimated beam.
 B.2 Technical Specifications of High Speed Camera
 Technical high speed camera specifications are listed in table B.2.
 B.3 Inclinometer Specification and Calibration
 A Wyler bubble inclinometer and Pro3600 digital inclinometer were used for angular measurements. The
 Wyler bubble inclinometer was calibrated by the National Metrology Institute of South Africa (NMISA) and
 used as a secondary standard to check the calibration of the Pro3600 digital inclinometer.
 174
A
 A
 SECTION A-A
 Figure B.2: Machined collar for laser pointer to replace slit mount on schlieren light source for system
 alignment
 Incoming Laser Beam :
 Reference Optical Axis
 Collimate Laser Beam
 Figure B.3: Custom collimator for adjustment of second mirror
 175
Table B.1: Optical parameters of schlieren system
 Item Description
 Image sensor size at 10000 fps 8.7 mm x 8.7 mm
 Parabolic Mirror Diameter 6 inch = 152.4 mm
 Focal length f/8 mirrors ? 1219.2 mm
 Distance from Second Mirror to Test Article 4000.00 mm
 Distance from First Mirror to Test Article 4000.00 mm
 Imaging Lens Focal Length 100.00 mm
 Imaging Lens Specification Edmund Optics Achromatic Lens with MgF2 coating
 Approximate Size of Object Field 100 mm
 Distance from knife edge to lens 130.8 mm
 Distance from lens to image sensor 80.14 mm
 Table B.2: Technical specifications of high speed camera
 Item Description
 Manufacturer Photron
 Model Photron Ultima APX-RS
 Maximum Image resolution up to 3000 fps 1024 x 1024
 Image resolution at 10000 fps 512 x 512
 Memory 8 GB
 Record time at 10000 fps 2.5 seconds
 Maximum Images recorded at 10000 fps 24576
 Sensor 10 bit CMOS
 Pixel size 17?m
 Shutter speed 16.7 ms to 2?s
 Figure B.4: Wyler and Pro3600 Inclinometers
 176
Table B.3: Technical specifications of the Wyler Bubble Inclinometer
 Item Details
 Instrument Name Wyler Bubble Inclinometer
 Calibration Date 4 August 2009
 Calibration Authority NMISA
 Calibration Certificate Number DM \DIM? 3229
 Serial Number 80/150
 Measurement Uncertainty 0.017?
 Table B.4: Calibration check of Pro3600 digital Protractor
 Data Point Number Wyler Inclinometer [?] Digital Protractor [?] Error
 1 -5.03 -5.09 -0.06
 2 0.05 0.04 0.01
 3 2.55 2.49 0.06
 4 5.08 4.99 0.09
 5 10.17 10.10 0.07
 6 15.03 14.90 0.13
 7 20.18 20.10 0.08
 8 25.45 25.40 0.05
 The bubble inclinometer has a resolution of 1 minute and was calibrated in 10 degree intervals from
 0? 180?. The magnitude of the error against the calibration standard was 0 minutes for most of the angles
 tested and approximately 1 minute (0.017?) between 70to180? and ?80to? 180?. The magnitude of the
 error was also 1 minute at ?10and? 20?. The Pro3600 digital protractor was checked against the Wyler
 inclinometer and since the inclinometer(s) were to be used for angular measurements below 25?, the bubble
 inclinometer served as an excellent secondary calibration standard. Most of the angular measurements during
 test setup were measured with the Pro3600 digital inclinometer primarily for its ease of use. For this reason
 a calibration check against a reliable secondary standard was necessary.
 The magnitude of the maximum error in the digital inclinometer reading in the range of interest is
 approximately 0.13?.
 B.4 Routine for co-ordination calculation in GNU Octave
 The following short script was used to transform image pixel co-ordinates to spatial co-ordinates using a
 known calibration grid. The pixel co-ordinates of the imaged calibration grid were used to calculate the
 location of any point of interest.
 xpixel = [Ordered array of pixel location of points on calibration grid in x direction]
 ypixel = [Ordered array of pixel location of points on calibration grid in y direction]
 xmap = [Ordered array of x-ordinates of calibration grid]
 177
ymap = [Ordered array of y-ordinates of calibration grid]
 Pointxpixel = [Ordered array containing pixel location of point of interest (x direction)]
 Pointypixel = [Ordered array containing pixel location of point of interest (y direction)]
 Pointx = griddata(xpixel,ypixel,xmap,Pointxpixel,Pointypixel,?linear?)
 Pointy = griddata(xpixel,ypixel,ymap,Pointxpixel,Pointypixel,?linear?)
 Comment : Pointxpixel and Pointypixel are the pixel co-ordinates of the point of interest. Pointx and
 Pointy are the required spatial co-ordinates of the pixel of interest.
 178
Appendix C
 Rig Design Calculations
 This chapter documents important calculations to size the servo-driven and spring-driven actuators. The rig
 operator interface for both actuators is presented. Select photographs of the rig and its components are also
 included. The rationale for the selection of the maximum model cross sectional area is also discussed briefly.
 C.1 Maximum Rig Cross Sectional Area
 There is an upper limit on the model frontal cross sectional area that will permit the establishment of the
 required tunnel free stream conditions and is a function of free stream Mach number. A Naval Ordnance
 report [43] documents experimental data that compares the maximum model frontal cross sectional area for
 a few simple geometries (a 60? cone, a 30? and a solid circular disk) against the maximum theoretical area
 calculated from the one-dimensional isentropic relations. The data is reproduced here and the frontal cross
 sectional area of the experimental rig described in Chapter 3 is included. The model cross sectional area
 is labelled, Am, and the tunnel cross sectional area is labelled, At. The data published in the report [43]
 shows a significant deviation from the ideal case and must be considered when undertaking model design for
 supersonic testing. The design limit was specified at Am/At ? 0.045 in an attempt to avoid tunnel blockage
 as experienced with the first version of the rig (presented in Chapter 3). This was adequate to establish the
 required free stream conditions.
 C.2 Motor Sizing for Servo-driven Actuator for Steady State
 Experiments
 The force required to actuate the wedges with a servo-motor in the steady state, baseline experiments is
 calculated at the condition that the wave system disgorges and is detached from the wedge surface. Only the
 179
 0
  0.05
  0.1
  0.15
  0.2
  0.25
  0.3
  0.35
  0.4
  1  1.5  2  2.5  3  3.5  4  4.5
 A
 m
 /A
 t
 Free Stream Mach Number, M
 Theory
 Disk
 60? Cone
 30? Cone
 Experimental Rig
 Figure C.1: Data used to determine the maximum permissable model cross sectional area extracted from a
 US Naval Ordnance Report [43].
 Figure C.2: Schematic showing derivation of actuator force
 180
aerodynamic force perpendicular to the wedge surface is included in the estimation. The component parallel
 to the surface is not included in the calculation. As will be demonstrated later in this section, its inclusion
 reduces the force required to actuate the wedges. The approach adopted here is a conservative one. Figure
 C.2 illustrates the geometry and forces on a single wedge. The pressure rise through a normal shock is used
 to estimate the static pressure, P , at the wedge surface. The design free stream condition is at M = 3.5
 with PO= 787 kPa. Equation A.2 from Appendix A yields PO/P? = 76.27, where P? is the free stream
 static pressure. The static pressure rise across a normal shock at M=3.5 is P/P? = 14.125 and
 P = P
 P?
 ?
 P?
 PO
 ? PO
 = 14.125? 176.27 ? 787kPa
 = 145746kPa
 The resultant force, FAERO, on a single wedge with surface area, A, is
 FAERO = P ?A
 = 145746kPa? 40mm? 170mm
 = 991N
 The moment generated by FAERO about the wedge centre of rotation is
 MAERO = FAERO ? d1 (C.1)
 The force required to balance MAERO at the end of the driving linkage is FLINK and the total actuator
 force for both wedges is FACTUATOR. Given the geometry in figure C.2 and that
 MAERO = FLINK ? d2 (C.2)
 181
the actuator force is related to the normal force on the wedge surface as follows:
 FAERO ? d1 =
 0.5? FACTUATOR
 cos ?1
 ? d2
 and
 FACTUATOR = 2FAERO cos ?1
 d1
 d2
 (C.3)
 for both wedges. The wedge incidence angle, ?w, and the ?1 are related as follows:
 d3sin?1 = d2cos?w ? d4
 Therefore
 sin?1 =
 (d2cos?w ? d4)
 d3
 (C.4)
 Given the values for d1 = 16.5mm, d2 = 25mm, d3 = 83mm, d4 = 10.5mm, ?w and FAERO = 991N ,
 the value for ?1 and FACTUATOR can be determined. Consider the maximum value of ?w beyond which
 the incident wave detaches from the leading edge of the wedge. At M = 3.5, ?w = 34.07? and ?1 = 7.1?,
 FACTUATOR ? 1298N . If one considers figure C.2, the aerodynamic force component parallel to the wedge
 surface will produce an anti-clockwise moment about the wedge centre of rotation, hence reducing the
 actuator force required. Shigley [38] documents the mechanics of power screws. The motor torque, T ,
 required to balance FACTUATOR, with a lead screw of diameter, dm, and thread pitch, l, is
 T = Fdm
 2
 (
 l + pi?dm
 pidm ? ?l
 )
 (C.5)
 where ? is the coefficient of friction. Figure C.3 illustrates the lead screw arrangement used in the serov-
 driven actuator. The lead screw turns in a ball bearing arrangement and ? is very small. A value ? = 0.1
 is assumed. With dm = 8mm and l = 2.5mm, the required motor torque is T = 1.05 N.m. A motor and
 182
Figure C.3: Lead screw and bearing assembly to convert rotational motion of the DC servo motor to
 horizontal motion of the actuator required to pitch the wedges. The image is taken from the Rexroth Bosch
 Group product catalogue on precision ball screw assemblies.
 Table C.1: Technical specifications for DC servo motor
 Supplier Faulhaber, Minimotor SA
 Motor Description DC micromotor series 3557
 Motor Model No. 3557K-024CR 4.3G60
 Power Supply 24V DC
 Maximum Torque 50 mN.m
 Rotational Speed 5000 RPM
 Gearbox Description Planetary Gearhead Series 30/1
 Gearbox Model No. 30/1 S 43:1
 Gear Reduction Ratio 43:1
 Maximum Continuous Torque 4.5 N.m with steel gears
 gearbox combination from Faulhaber Minimotor SA (Switzerland) capable of delivering 4.5 N.m of torque
 was used (see table C.1 with a lead screw and bearing assembly from Rexroth of the Bosch Group (see table
 C.2).
 Table C.2: Technical specifications for the lead screw and bearing arrangement
 Supplier Rexroth, Bosch Group
 Lead Screw Specification Precision-rolled screw SN-R, Diameter = 8mm, Pitch = 2.5 mm
 Bearing Description Ball screw with flanged single nut
 Bearing Model No. 8? 2, 5R? 1, 588? 3
 183
C.3 Component Sizing for Spring-Based Actuator for Dynamic
 Tests
 The following conceptual design calculations are applicable for M? = 2.0. Assume TO? = 300.0K and
 PO? = 250.0? 103Pa. The specific heats ratio for air is ? = 1.4. Also assume a wedge chord of c = 40.0mm
 and a universal gas constant, R = 287.0J.kg?1.K?1. Free stream static temperature is calculated by:
 T? = TO?
 (
 1 +
 ? ? 1
 2
 M2?
 )
 = 166.67K (C.6)
 The free stream acoustic velocity is given by:
 a? = (?RT?) = 258.8m.s?1 (C.7)
 Given that ?? is the wedge pitch rate expressed in radians per second (rad.s?1):
 ME =
 VE
 a?
 =
 c??
 a?
 (C.8)
 Then for ME = 0.1
 ?? = 647.02rad.s?1 = 37071.26?.s?1 (C.9)
 and VE = 25.88m.s?1
 For ME = 0.01, ?? = 64.7rad.s?1 = 3707.13?.s?1 and VE = 2.59m.s?1.
 Assuming a total pitch scan of ?? = 25.0?. Total time to pitch ?? is given by:
 ?t = ???? (C.10)
 ?t = 0.67? 10?3sec for ME = 0.1 and ?t = 6.7? 10?3sec for ME = 0.01
 The second order ordinary differential equation governing the response of an undamped spring mass
 system was used to estimate the required spring stiffness. Assume that m is the mass being accelerated and
 k is the spring stiffness, then
 m
 dx2
 dt2 + kx = 0 (C.11)
 The solution of equation C.11 is given by
 184
 0
  2
  4
  6
  8
  10
  12
  14
  0  0.001  0.002  0.003  0.004  0.005  0.006
 Time [s]
 M
 a
 ss
 di
 sp
 la
 ce
 m
 en
 t,
 x
 [m
 m
 ]
 Figure C.4: Solution of the one dimensional ordinary differential equation for the spring mass system with
 m = 1.0 kg, k = 72? 103 N/m, xo = 13.0 mm and zero initial speed
 x = xo cos (?nt) + dxdt
 sin (?nt)
 ?n
 (C.12)
 where ?n is the natural frequency of the system and is given by :
 ?n =
 ?
 k
 m
 (C.13)
 Assume an initial displacement, xo = 13.0 mm, which is the linear travel of the actuator required to
 achieve 25.0? pitch and m = 1.0 kg, which is the approximate mass of the system connected to the actuator.
 Assume zero initial velocity, i.e.
 dx
 dt |t=0= 0 (C.14)
 The solution of x with a spring stiffness of k = 72? 103 N/m is plotted in figure C.4. The solution shows
 that it is feasible to achieve the required motion in the required time.
 The stiffness of a helical spring in compression is approximated by the following equation ([38]):
 185
Table C.3: Technical specifications for safety pin solenoid
 Supplier BLP, Suffolk, England
 Specification Series 124 Tubular Solenoid
 Description Pull type tubular solenoid
 Model No. 124 420 610 620
 Power Supply 12 V DC
 Approximate Stroke 10 mm
 Approximate Pulling Force 50 N
 Table C.4: Technical specifications for release actuator
 Supplier Phoenix Mecano
 Description Electric Cylinder
 Model No. M10/BGR 010
 Power Supply 24 V DC
 Total Stroke 40 mm
 Travel Speed 4 mm/s
 Actuator Force 200 N
 k = Gd
 4
 8nD3 (C.15)
 where G is the shear modulus of elasticity, d is the wire diameter, D is the mean coil diameter and n is
 the number of turns on the spring. Using two springs in series requires each to have a stiffness, k ? 40000
 N/m each. The spring diameter, D (excluding the wire thickness), takes into account the space available
 normal to the streamwise direction. A value of D=16.2mm was selected. With G = 76.9 GPa for steel, d
 = 3.8 mm, D =16.2 mm and k ? 40000 N/m, n = 11.5 turns. Two springs, installed in series, with these
 mechanical properties are sufficient to achieve the required motion. Two springs with an uncompressed
 length, L = 93mm, were manufactured for the dynamic actuator. The free length takes into account the
 space available within the actuator volume.
 The operation of the latch mechanism was discussed in detail in Chapter 3 and may be consulted on the
 description of the design. The actuator has a safety pin that prevents the latch from releasing the spring
 load on tunnel startup. The safety pin is actuated by a solenoid. Solenoid technical specifications of are
 included in table C.3. Once, the solenoid is energised and the safety pin is disengaged, the latch is opened
 to release the spring load. The lever that opens the latch is actuated by a release actuator. The actuator is
 also a commercial off-the-shelf item and is essentially a linear motion electrical cylinder. It has a 200 N load
 capacity which is sufficient to open the latch. Specification are summarised in figure C.4.
 186
Figure C.5: Contours showing the distribution of computed stress in the latch and release pillar. The region
 of maximum stress at approximately 354 MPa is indicated. The FEM analysis was performed by Ryan
 Raath at the CSIR, Pretoria.
 C.4 Finite Element Analysis for Latch Design
 Due to the large loads involved in loading the spring, care was taken to apply the necessary safety factors
 and a finite element analysis of the latch was performed with SolidWorksExpress. The distribution of
 computed stress in the latch and release pillar is shown in figure C.5. The maximum stress with a 2000 N
 spring load is approximately 354 MPa. The latch and release pillar were manufactured with 174-Ph Stainless
 steel, heat treated at 900? C. In the hardened condition it has a maximum yield strength of approximately
 1170 MPa and a maximum ultimate strength of 1300 MPa, which provides an adequate safety factor (3.3 on
 yield strength and 3.6 on ultimate strength).
 187
C.5 Description of the Rig Operator Interface
 A control and electrical connections interface was built for the servo and spring-driven actuators as illustrated
 in figure C.6 and C.8. This enables the remote operation of either actuator from the control room.
 The interface for the servo-driven actuator consists of a switch to power the motor and a double throw
 double pole switch to rotate the motor clockwise or anti-clockwise. The rotation is translated to linear
 motion of the actuator through the bearing assembly discussed earlier in this chapter. The servo circuit
 diagram is presented in figure C.7.
 The interface for the spring-driven actuator controls the operation of the safety pin and the latch release
 actuator. Indicator lights on the interface indicate the status of the safety pin and the release actuator
 during a test. After tunnel startup, the solenoid is energised and the safety pin is disengaged. When the pin
 is disengaged the indicator LED is illuminated and the rig operator may proceed to open the latch with the
 release actuator. Once the latch is released an indicator light on the panel signals the camera operator to
 trigger the image capture. As the camera is running in ?centre? mode it stores images before and after the
 trigger is activated. The circuit diagram for the solenoid and the release actuator are presented in figures
 C.9 and C.10.
 C.6 Photographs of Rig
 A detailed description of the rig design and operation was presented in Chapter 3 with detailed accompanying
 schematics. Photographs of the rig and its components are presented in figures C.11-C.16.
 188
(a) Control and electrical connections interface for servo-driven actuator
 (b) Control and electrical connections interface for spring-driven actuator
 Figure C.6: Control and electrical connections interface for actuators
 189
Figure C.7: Electrical circuit for operation of servo-motor showing current direction for wedge pitch up and
 pitch down
 190
(a) Front view showing operation panel for operation of the safety pin and release actuator
 (b) Top view showing electrical connections panel for dynamic actuator
 Figure C.8: (a) Front and (b) top views of control interface for spring-driven actuator
 191
Figure C.9: Solenoid circuit for the operation of the safety pin in the spring-driven actuator
 192
Figure C.10: Latch release circuit diagram for the spring-driven actuator
 193
(a) Servo-driven actuator used for steady state baseline experiments
 (b) Spring-driven actuator used for dynamic experiments without the latch release actuator installed
 Figure C.11: Actuators for (a) steady state, baseline experiments and (b) dynamic experiments
 Figure C.12: Rig used for dynamic shock wave reflection experiments. Cover plates are installed and the
 release actuator has been removed.
 194
Figure C.13: Rig with cover plates removed and release actuator installed. The actuator is assembled to
 execute the dynamic RR ? MR transition experiment.
 Figure C.14: Closeup view of the spring-driven actuator with the release actuator installed. The actuator is
 assembled to execute the dynamic RR ? MR transition experiment.
 Figure C.15: Closeup view of wedges
 195
Figure C.16: Stream wise view of the rig installed in tunnel test section
 196
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