Numerical Investigation of Strakes and Strakelets on a Missile at High Angles of Attack Prevani Kistan A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfllment of the requirements for the degree of Master of Science in Engineering. Johannesburg, August 2006 Declaration I declare that this dissertation is my own, unaided work, except where otherwise ac- knowledged. It is being submitted for the degree of Master of Science in Engineering in the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other university. Signed this day of 20 Prevani Kistan. i Acknowledgements Firstly, I would like to thank the Almighty for giving me the strength to get this far. To Dr Craig Law for his advice and for taking over the role of supervisor from Prof L. Ravaglia, who retired. To Dr C.P. Crosby and Mr S.G. Gobey for their technical advice, patience and support throughout my research. To my parents, my siblings and Shahen Naidoo for their patience and unwavering moral support throughout my research, especially during some really trying times in my work. To the School of Mechanical, Industrial and Aeronautical Engineering and Denel Aerospace Sys- tems for allowing me to work towards my Masters in Science in Engineering degree. To Prof D. Degani (Isreal Technicon) and Mr P. Champigny (ONERA Cha^tillon) for providing research papers which are unavailable and for their technical advice. Finally to Mr Rory Yorke for his constant assistance. ii Abstract A computational uid dynamics (CFD) study was carried out to improve the aero- dynamic performance of an agile high angle of attack missile. The normal force generated by the missile strakes had to be increased at the low angles of attack and the large side forces, experienced at high angles of attack due to the formation of steady asymmetric vortices had to be eliminated using strakelets on the missile nose. The flrst objective was achieved by increasing the missile strake span from 0:06D to 0:13D. The larger strake span increased the efiective diameter of the missile body and prevented ow reattachment to the body, a problem that was experienced when the strake span was 0:06D. Due to ow separating further away from the body, strong vortices formed on the missile strakes, resulting in an increase in the normal force generated by the missile strakes at low angles of attack. The second objective was two-fold. Prior to analysing the efiect of the strakelets on a steady asymmetric owfleld, the steady asymmetric owfleld had to flrst be created. This was achieved by placing a permanent, geometric perturbation on the missile nose. The size of the perturbation used in the study, which was determined by an iterative process, did not force ow separation at low angles of attack and resulted in a steady asym- metric owfleld that was representative of that on a blunt-ogive body. The efiect of changing the span of the strakelets and the axial position of the strakelets were then investigated. It was found that the strakelets with a span of 0:09D, placed 1D from the nose tip eliminated the side forces by forcing vortex symmetry. Increasing or decreasing the span of the strakelet, positioned 1D from the nose tip or placing the strakelets with a span of 0:09D closer or further away from the nose tip did not eliminate the steady vortex asymmetry. iii Contents Declaration i Acknowledgements ii Abstract iii Contents iv List of Figures viii List of Tables xiv List of Symbols xv 1 Introduction 1 1.1 Flowfleld Around a Missile Body at Angles of Attack . . . . . . . . . 1 1.1.1 Low angles of attack (fi ? fiSV ) . . . . . . . . . . . . . . . . . 2 1.1.2 Moderate angles of attack (fiSV ? fi ? fiAV ) . . . . . . . . . 2 1.1.3 High angles of attack (fiAV ? fi ? fiUV ) . . . . . . . . . . . . 3 1.1.4 Very high angles of attack (fi ? fiUV ) . . . . . . . . . . . . . 6 1.2 Reduction of Asymmetric Vortex Efiects . . . . . . . . . . . . . . . . 6 1.2.1 Side force control . . . . . . . . . . . . . . . . . . . . . . . . . 7 iv 1.2.2 Side force alleviation . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Document Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Validation Studies 14 2.1 Mesh Size Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Outlet Boundary Position . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Comparison Between Full Model and Half Model Simulations . . . . 28 2.3.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Efiect of Changing the Span of Missile Strakes 45 v 3.1 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Strake Height In uence on Coe?cient Behaviour . . . . . . . . . . . 49 3.3 Strake Height Efiects on the Flowfleld . . . . . . . . . . . . . . . . . 52 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Creation of Steady Asymmetric Vortices in CFD 55 4.1 Size of Geometric Perturbation Required to Simulate Asymmetric Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.2 Results for 5? angle of attack . . . . . . . . . . . . . . . . . . 61 4.1.3 Results for 20? angle of attack . . . . . . . . . . . . . . . . . 63 4.1.4 Results for 40? angle of attack . . . . . . . . . . . . . . . . . 67 4.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.6 Conflrmation of type of instability . . . . . . . . . . . . . . . 74 4.1.7 Full length missile results . . . . . . . . . . . . . . . . . . . . 75 4.2 Efiect of Axial and Circumferential Position of A Perturbation on the Flowfleld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5 Efiect of Strakelets on Steady Asymmetric Vortices 86 5.1 Efiect of Changing the Span of the Strakelets . . . . . . . . . . . . . 89 5.1.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 89 vi 5.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Efiect of Changing the Axial Position of the Strakelets . . . . . . . . 105 5.2.1 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Efiect of Strakelets on a Steady Asymmetric Flowfleld . . . . . . . . 122 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6 Conclusions and Recommendations 126 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 REFERENCES 131 vii List of Figures 1.1 Four angle of attack regions(Ericsson and Reding, 1991) . . . . . . . 2 1.2 Vortex ows on a body of revolution at high angles of attack (ESDU, 1989) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Typical axial distribution of local side force coe?cient due to asym- metric vortex ow (ESDU, 1989) . . . . . . . . . . . . . . . . . . . . 5 1.4 Suggested cross ow patterns generating opposite side forces for `s = 60? and `s = 90? (Rao et al., 1987) . . . . . . . . . . . . . . . . . . . 8 1.5 Efiect of helical and straight body trips on side force of ogive-cylinder body (Ericsson and Reding, 1991) . . . . . . . . . . . . . . . . . . . 10 2.1 A portion of the missile body geometry . . . . . . . . . . . . . . . . 15 2.2 Structured grid used in this study . . . . . . . . . . . . . . . . . . . 16 2.3 A closer view of the structured grid around the body surface shown in Figure 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Change in normal force per iteration . . . . . . . . . . . . . . . . . . 19 2.5 Surface pressure distribution on the three models with difierent grids at fi = 40? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Comparison of surface pressure distribution along length of body (left of the body centre-line) . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Comparison of surface pressure distribution along length of body (right of the body centre-line) . . . . . . . . . . . . . . . . . . . . . . 21 viii 2.8 Helicity density contours at x = 2:5D at fi = 40? . . . . . . . . . . . 23 2.9 Helical density contours at x = 3:5D at fi = 40? . . . . . . . . . . . . 24 2.10 Streamlines ofi the surface of the missile at fi = 40? . . . . . . . . . . 25 2.11 Comparison of the pressure distribution for the two outlet positions 27 2.12 Closer view of the surface pressure distribution near the missile nose for the two outlet positions. . . . . . . . . . . . . . . . . . . . . . . . 27 2.13 Helical density contours at x = 3:5D at fi = 40? . . . . . . . . . . . . 28 2.14 Full geometry of missile body . . . . . . . . . . . . . . . . . . . . . . 29 2.15 A closer view of the grid constructed on the half model . . . . . . . 29 2.16 Comparison of values of of the normal force coe?cients for the half and full models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.17 Comparison of values of the pitching moment coe?cients for the half and full models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.18 Density contours at x = 0:31D at an angle of attack of 40? . . . . . 32 2.19 Density contours at x = 6:3D at an angle of attack of 40? . . . . . . 33 2.20 Comparison of values of the normal force coe?cients for the two tur- bulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.21 Comparison of values of the pitching moment coe?cients for the two turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.22 Side view of the surface pressure distributions for the difierent turbu- lence models at fi = 40? . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.23 Density contour plot at x = 3:1D at fi = 40? . . . . . . . . . . . . . 40 2.24 Density contour plot at x = 5D at fi = 40? . . . . . . . . . . . . . . 41 2.25 Density contour plot at x = 6:3D at fi = 40? . . . . . . . . . . . . . 41 2.26 Comparison of normal force coe?cients . . . . . . . . . . . . . . . . 42 ix 2.27 Ribbon traces ofi the surface of the missile body at fi = 40? . . . . . 43 2.28 Flow development on the missile body in the high-speed wind tunnel (CSIR-Defencetek, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1 Missile body-strake conflguration . . . . . . . . . . . . . . . . . . . . 45 3.2 Normal force coe?cient of strakes in the presence of the body (Mach 0.8) (DAS, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Pitching moment coe?cient of strakes in the presence of the body (Mach 0.8) (DAS, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Comparison of normal force coe?cient between the difierent span conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Comparison of pitching moment coe?cient between the difierent span conflgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 A side view of the cross ow velocity at x = 4:7D at 20? . . . . . . . 52 3.7 A side view of the cross velocity at x = 4:7D at 20? for the strakes orientated in the ?+? roll orientation . . . . . . . . . . . . . . . . . . 53 4.1 Experimental side force coe?cient on a body of revolution at Mach 0.8 (DAS (2004)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Formation of steady asymmetric vortices in the high-speed wind tun- nel at Mach 0.8 at an angle of attack of 30? ((CSIR-Defencetek, 2004)) 58 4.3 Body and geometric perturbation geometry . . . . . . . . . . . . . . 60 4.4 Surface pressure distribution on missile body at fi = 5? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Helicity density contours at x = 6D at fi = 5? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.6 Surface pressure distribution on missile body at fi = 20? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 63 x 4.7 Surface pressure distribution along the length of the missile body at 20? angle of attack for various geometric perturbations . . . . . . . . 64 4.8 Helicity density contours at x = 4D at fi = 20? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.9 Helicity density contours at x = 6D at fi = 20? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.10 Surface pressure distribution on missile body at fi = 40? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.11 Surface pressure distribution along the length of the missile body at 40? angle of attack for various geometric perturbations . . . . . . . . 68 4.12 Helicity density contours at x = 4D at fi = 40? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.13 Helicity density contours at x = 6D at fi = 40? for various geometric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.14 Comparison of side force coe?cient values for the difierent perturbations 72 4.15 Side force variation per iteration . . . . . . . . . . . . . . . . . . . . 74 4.16 Surface pressure distribution on the full length missile body . . . . . 75 4.17 Front view showing the two difierent circumferential positions of the perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.18 Surface pressure distributions on missile bodies with perturbations at difierent axial locations (fi = 40?) . . . . . . . . . . . . . . . . . . . . 78 4.19 Helical density contours at x = 4D at fi = 40? for the perturbations at difierent axial locations . . . . . . . . . . . . . . . . . . . . . . . . 79 4.20 Helical density contours at x = 6D at fi = 40? for the perturbations at difierent axial locations . . . . . . . . . . . . . . . . . . . . . . . . 80 4.21 Streamlines on body at fi = 40? for the perturbations at difierent axial locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 xi 4.22 Surface pressure distribution on missile bodies with geometric pertur- bations at difierent circumferential locations (fi = 40?) . . . . . . . . 82 4.23 Helical density contours at x = 4D at fi = 40? for the geometric perturbations at difierent circumferential positions . . . . . . . . . . 82 5.1 Comparison of experimental side force data for a body-tail conflgu- ration with and without strakelets . . . . . . . . . . . . . . . . . . . 87 5.2 Geometry of missile with nose strakelets . . . . . . . . . . . . . . . . 89 5.3 A close view of the structured grid constructed on the missile-strakelet geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Surface pressure distributions on the strakelet models with difierent spans at 40? angle of attack . . . . . . . . . . . . . . . . . . . . . . . 91 5.5 The surface pressure distribution along the length of the body for the strakelet models with difierent spans . . . . . . . . . . . . . . . . . . 92 5.6 Flowfleld on the missile bodies with strakelets with difierent spans at x = 1:4D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . 94 5.7 Flowfleld on the missile bodies with strakelets with difierent spans at x = 2:5D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . 96 5.8 Flowfleld on the missile bodies with strakelets with difierent spans at x = 2:8D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . 98 5.9 Flowfleld on the missile bodies with strakelets with difierent spans at x = 6D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . 100 5.10 The efiect of the strakelets on the normal force coe?cient . . . . . . 105 5.11 Surface pressure distributions on the difierent strakelet models at dif- ferent axial locations at 40? angle of attack . . . . . . . . . . . . . . 106 5.12 The surface pressure distribution along the length of the missile body with the difierent strakelet models at difierent axial locations . . . . 107 5.13 Flowfleld on the missile bodies with difierent strakelet models at x = 0:55D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . 109 xii 5.14 Flowfleld on the missile bodies with difierent strakelet models at x = 1:2D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . . 111 5.15 Flowfleld on the missile bodies with difierent strakelet models at x = 1:5D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . . 113 5.16 Flowfleld on the missile bodies with difierent strakelet models at x = 2:3D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . . 115 5.17 Flowfleld on the missile bodies with difierent strakelet models at x = 3:4D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . . 117 5.18 Flowfleld on the missile bodies with difierent strakelet models at x = 6:9D at an angle of attack of 40? . . . . . . . . . . . . . . . . . . . . 119 xiii List of Tables 2.1 Percentage difierence in the normal force and pitching moment coef- flcients between the full and half symmetry models . . . . . . . . . . 31 2.2 Percentage difierence in normal force and pitching moment coe?- cients between the two turbulence models . . . . . . . . . . . . . . . 37 3.1 Percentage increase in normal force coe?cient and pitching moment coe?cient for the difierent missile strake spans . . . . . . . . . . . . 50 4.1 Dimensions of geometric perturbations . . . . . . . . . . . . . . . . . 59 5.1 Coe?cients for Efiects in a 23 Experiment . . . . . . . . . . . . . . . 88 5.2 Comparison of normal force coe?cients for the strakelet models with difierent spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 xiv List of Symbols fi Angle of attack ? Dissipation rate of kinetic energy %? Percentage change ? K arm an?s constant ! Speciflc dissipation rate of kinetic energy ` Circumferential location ? Density Cm Pitching moment coe?cient CN Normal force coe?cient Cy Side force coe?cient C? Closure constant D Diameter Fy Side force Hd Helicity density k Turbulent kinetic energy lref Reference length N Normal force Mz Pitching moment xv M Mach number M1 Freestream Mach number MN Cross ow Mach number P static pressure Sref Reference area T Static temperature u? Friction velocity V Freestream velocity y Distance from the surface y+ Dimensionless, sublayer-scaled, distance Subscripts SV Symmetric vortices AV Asymmetric vortices UV Unsteady asymmetric vortices S Circumferential location of strake S1 Circumferential location of primary separation line S2 Circumferential location of secondary separation line A1 Circumferential location of primary attachment line A2 Circumferential location of secondary attachment line Full Full model Half Half model SST Menter-SST k ? ! turbulence model xvi k ? ? k ? ? turbulence model High Strake span of 0:13D Short Strake span of 0:06D xvii 1 Introduction Missiles, unlike aircraft, have cruciform wing (and control) surfaces, which are sub- jected to signiflcant three-dimensional ow. The wing and control surfaces are mounted on a relatively large diameter body, resulting in considerable mutual aero- dynamic interference between these surfaces and the body. In most cases, the wing and control surfaces have similar dimensions and are often placed in fairly close prox- imity to one another. This gives rise to further aerodynamic interference (Dexter, 1993). In order to achieve increasing performance demands modern missiles need to manoeuver and operate at higher angles of attack than previously. 1.1 Flowfleld Around a Missile Body at Angles of At- tack In the high angle of attack ight domain, the ow around a missile body is very complex. It is characterised by the presence of large separated regions that result in the development of nonlinear normal force and pitching moment characteristics (Dexter, 1993). Ericsson and Reding (1991) identifled four ow regimes in terms of angle of attack. These are illustrated in Figure 1.1. The deflnitions of the symbols used in Figure 1.1 are: ? fi - angle of attack, ? fiSV - angle of attack at which symmetric vortices form, ? fiAV - angle of attack at which steady asymmetric vortices form, and ? fiUV - angle of attack at which unsteady asymmetric vortices form. 1 Figure 1.1: Four angle of attack regions(Ericsson and Reding, 1991) 1.1.1 Low angles of attack (fi ? fiSV ) The ow on a body of revolution pitched up slowly from zero angle of attack, re- mains attached to the body. The axial component of the ow dominates, although the transverse component of the ow is already responsible for the boundary layer thickening on the leeward side. Potential ow theory generally accounts for this flrst ow state. The forces that develop on the body, for example normal force, vary linearly with angle of attack (Champigny, 1994). The side force acting on the body is zero. 1.1.2 Moderate angles of attack (fiSV ? fi ? fiAV ) The transverse component of the ow becomes increasingly important. Under the in uence of adverse pressure gradients and the increasing transverse ow component, ow separates on the leeward side and the resulting shear layers roll up into two well-deflned vortices, which are symmetric. The separated ow is well ordered, but does introduce complexity into the aerodynamic model (Dexter, 1993). Lift increases non-linearly with the angle of attack, due to the vortex lift (Champigny, 1994). The 2 side force acting on the body is still zero since the formed vortices are symmetric. 1.1.3 High angles of attack (fiAV ? fi ? fiUV ) The initial steady symmetric owfleld in Section 1.1.2 around a body of revolution becomes steady asymmetric when the angle of attack of the body is further increased, even though the body and the freestream ow are symmetric. Initially several dif- ferent mechanisms were suggested as causes for the formation of steady asymmetric vortices on bodies of revolution at high angles of attack, however it is now generally accepted that microscopic, time-invariant, geometric disturbances on the nose of bodies of revolution are responsible for the formation of steady asymmetric vortices. This occurs because the ow around a body of revolution at high angles of attack is unstable (Champigny, 1994). Above a certain angle of attack it is not possible for two strong, counter-rotating vortices to co-exist symmetrically. A very small per- turbation, caused by geometric imperfections on the nose of the body of revolution, is su?cient to cause the vortex system to go from a metastable symmetric state to a stable, steady asymmetric state (Champigny, 1994). Champigny (1994) highlighted two irregularities of ow, namely a slight body sideslip and turbulence in the ow as also being responsible for the formation of asymmetric vortices. Experiments highlighted by Champigny (1994) show that while both irreg- ularities did result in the formation of asymmetric vortices, the stable and steady nature of asymmetric vortices were not maintained, as both irregularities are tem- poral. In order to maintain the stable and steady nature of asymmetric vortices the per- turbation needs to be a time-invariant one, such as the geometric imperfections on the nose of a body of revolution. These permanent geometric imperfections are unavoidable as they are often due to imperfect manufacturing. The steady asymmetric owfleld is shown in Figure 1.2. The body vortices grow with difiering strengths along the length of the body, due to ow interaction with the microscopic geometric imperfections on the missile nose. The weaker vortex is forced to separate flrst by the the stronger vortex. The stronger vortex separates at a difierent axial location on the body. The weak vortex becomes the outer primary vortex. The stronger vortex that separated second, remains tucked in between the outer primary vortex and the body. The outer primary vortex moves down the length of the body and away from it, until the shear layer feeding the outer vortex 3 is cut. The vortex is shed, trailing ofi and curving into the freestream direction, downstream of the body. A new primary vortex forms inboard on the side from which the shed vortex originated, as shown in Figure 1.2. The formerly inner vortex now becomes the outer vortex and the separation process continues along the length of the body until this vortex is shed and so on (Dexter, 1993). Very long bodies can develop several pairs of asymmetric vortex pairs. Figure 1.2: Vortex ows on a body of revolution at high angles of attack (ESDU, 1989) Many researchers have related the unsteady two-dimensional von K arm an vortex street to the steady three-dimensional vortex array, by using the principle of space- time equivalence (Ericsson and Reding, 1991). By this principle, ow development is related to time, measured either from the beginning of an impulsive two-dimensional motion or from the instant a uid particle makes contact with a three-dimensional body. In the latter case, time is deflned by a distance travelled along the body and the axial component of freestream velocity (Fidler and Bateman, 1975), as shown in section B-B in Figure 1.2. The steady asymmetric ow on the body results in the generation of steady side forces and yawing moments on the body. Since the asymmetric ow is steady, side force distribution along the length of the body is sinusoidal and each maximum cor- responds to the detachment of a vortex sheet from the body, as shown in Figure 1.3. 4 Figure 1.3: Typical axial distribution of local side force coe?cient due to asymmetric vortex ow (ESDU, 1989) Two difierent types of vortex asymmetry can occur: ? On a pointed, slender body steady vortex asymmetry usually begins at the nose and the frequency at which the vortices are shed increases with angle of attack (Champigny, 1994). ? On slightly blunted bodies steady vortex asymmetry usually begins at the aft end of the body and with a further increase in angle of attack, the asymmetry becomes stronger and moves forward until it reaches the nose tip of the body at high angles of attack (Dexter, 1993). Alternate vortex shedding does not occur as readily, and thus side force cells are much larger and can cover the entire cylindrical body (Champigny, 1994). For pointed, slender bodies the angle of attack at which steady asymmetric vor- tices develop is dependent on the cone-half angle ( c) (Ericsson and Reding, 1991). Asymmetric vortex development occurs when the angle of attack is approximately double the total included angle at the apex (fiAV ? 2 A). For slightly blunted bodies the onset angle of attack for steady asymmetric vortices is determined by the overall body flneness ratio. Ericsson and Reding (1991) provided evidence that for blunt ogive bodies, vortex asymmetry began a distance away from the nose when fiAV ? 4:2dl where d is the diameter and l is the overall length of the body. The existence of steady asymmetric vortices at high angles of attack has been a topic of interest since the early 1950s. Flow-visualisation pictures of experimental ows have shown the existence of steady asymmetric vortices (Levy et al., 1996). In the 1970s schilieren pictures conflrmed that vortices curve away from the body on alternate sides, and move downstream at a small angle to the freestream direction (Levy et al., 1996). 5 1.1.4 Very high angles of attack (fi ? fiUV ) Finally, at very high angles of attack, when the body is almost perpendicular to the ow, the previously described stable and steady asymmetric ow, now changes into a time-dependent ow, becoming a two-dimensional ow around the cylinder at 90? angle of attack. This develops gradually from the aft end of the body, manifesting itself as a periodic uctuation superimposed on an asymmetric pressure distribution around the body and moves forward with increasing angle of attack. On a long body, the aft end may be subjected to signiflcant periodic uctuations while a steady ow is present on the front of the body. Increasing the angle of attack causes the uctuating region to spread further upstream, resulting in reduced, mean local side forces (Champigny, 1994). 1.2 Reduction of Asymmetric Vortex Efiects Due to unavoidable microscopic manufacturing imperfections on the nose cones of missiles, steady asymmetric vortices will always occur at high angles of attack. Therefore some form of control or alleviation is required. In the absence of control, a missile maneuvering at high angles of attack, will experience large continuous side forces and moment changes that vary erratically with angle of attack (Ericsson and Reding, 1991). The side on which the side forces will develop cannot be predicted as manufacturing imperfections change from model to model. Advanced aircraft have the same problem and many of the available solutions have been obtained with aircraft applications in mind. Proposed solutions fall into one of the two following groups (Champigny, 1994): To use the asymmetric owfleld for ight control: This requires sophisticated data systems and data processing which are di?cult to implement. Such sys- tems do not afiect the aerodynamics of the missile as there are no external appendages on the body of revolution. To reduce or eliminate asymmetries in the owfleld, by forcing ow symmetry: Examples of such are forebody strakes. They are easier to implement on to a body of revolution but since they are external appendages the aerodynamics of the body of revolution will change. 6 1.2.1 Side force control Much efiort has been directed toward the development of ways by which the forebody ow separation could be controlled. The motivation of this has been to enhance the agility of advanced aircraft and missiles. Rao et al. (1987) used two deployable strakes on either side of a body of revolution to obtain a more gradual control. The strakes were pivoted along their length and deployed at command from their conformal stored position on the forebody. Two difierent strake deployments were studied by Rao et al. (1987): Asymmetric deployment: Only one strake is deployed at a time, forcing a strong asymmetric vortex. This results in a side force, which is controllable by strake de ection. Simultaneous deployment: Both strakes are deployed. A symmetrical pair of augmented vortices is established, from which a controlled sided force is gen- erated by means of asymmetric strake de ection. However, Rao et al. (1987) did not succeed in eliminating the nonlinear variation of side force with angle of attack. The side force control variation with body roll and sideslip is illustrated in Figure 1.4. When the strake is at a circumferential position (`s) of 90?, the strake forces ow separation which results in the indicated positive side force generation of ow model B. However at `s = 60?, the pre-separation efiect allows for reattachment of the boundary layer to withstand flnal separation well past ` = 90?, resulting in a negative side force for ow model A. For dual strake deployment, at `s = 90?=270?, the left vortex is lifted up while the right vortex is drawn closer to the forebody. The asymmetric pressure distributions were very similar to that produced by the single strake at `s = 90? (Rao et al., 1987). While a single strake was efiective in producing large yawing moments, the control was nonlinear and the strake was inefiective in eliminating the naturally occurring asymmetry and associated yawing moments at zero sideslip. Thus a pair of difierentially de ectable strakes was more appropriate since it was able to address the above issues, but vortex asymmetry was not totally eliminated. Ng and Malcolm (1992) showed that rotatable nose-tip strakes were able to produce controlled yawing moments, even at relatively moderate angles of attack. Efiective control of vortex asymmetry could be obtained by controlling ow separation near the tip region. In this way the ow pattern is modifled and the efiective geometry of the tip is changed. The purpose of the rotatable nose-tip strakes is to in uence 7 Figure 1.4: Suggested cross ow patterns generating opposite side forces for `s = 60? and `s = 90? (Rao et al., 1987) only a small region near the tip of the forebody, whereas large strakes afiect a much larger region aft of the tip. The rotatable nose-tip strakes function by creating, in efiect, an asymmetric forebody apex. Bernhardt and Williams (1998) introduced the idea of using unsteady bleed tech- niques for experiments to control the development of steady asymmetric vortices. The basic premise of the proportional control with unsteady bleed technique is that by controlling the initial ow disturbances at the tip, the conflguration of the fore- body vortices can be modifled. The interaction of the unsteady bleed with the external ow produces a highly localised variable amplitude low pressure distur- bance in the mean ow. This dominates the built-in geometric asymmetry of the cone and controls the formation of tip vortices. In addition to the role as actuators, the unsteady bleed technique enables the investigation of the important ow physics by studying the response of the ow to a controlled input. Bernhardt and Williams (1998) found that the most efiective control of vortex conflguration and side force is achieved by placing ow actuators near the tip of the forebody model. Fidler (1981) rotated the nose-tip, nose and a portion of the body surface aft of the nose, to achieve cyclic variation of side force and yawing moment. These portions were rotated, flrst as smooth surfaces and then with artiflcial disturbances flxed to them. This allowed for the examination of small disturbances under controlled conditions. Increasing the spin rate of the nose and the nose tip resulted in a decrease in the maximum values of the sinusoidal side force variation. This was due to the vortices being unable to establish their owflelds quickly enough to produce the full efiect on the body. Varying the number of artiflcial disturbances flxed to the nose and the nose-tip resulted in a change in the magnitude and sign of the side force. Fidler (1981) concluded that the spinning device concept would be successful 8 irrespective of the direction in which the body was pitched or yawed. 1.2.2 Side force alleviation Early efiorts to alleviate side force problems included various forms of geometric changes such as nose bluntness, strakes and boundary layer trips. Many investiga- tions have focussed on jet blowing to obtain the same efiect (Champigny, 1994). Nose bluntness, which is an example of a passive ow control technique, can be an efiective means of reducing the side force. However, it is possible that nose-induced asymmetry on a pointed nose body can be traded for an aft body asymmetry as surface geometric imperfections will still exist on the nose of the body of revolution. A small degree of nose bluntness delays the formation of steady asymmetric vortices on the nose, thus decreasing the induced side force. Eventually, as the nose bluntness is increased, vortex asymmetry begins on the cylindrical part of the body and that results in the formation of side force cells on the aft body (Ericsson and Reding, 1991). A nose boom, added to the forebody, decreases the separation induced side force. The boom?s multi-vortex wake efiectively nullifles the asymmetry generating poten- tial of a slender pointed nose (Ericsson and Reding, 1991). Ng (1992) found that the ow ofi a nose boom was very similar to that ofi a cylindrical, slender body. Initially, at moderate angles of attack, the ow separation is symmetric, but as the angle of attack increases, the separation becomes asymmetric. The asymmetric pattern, which is related to the number of separated asymmetric vortices and the strength of the asymmetric vortices, is not speciflc. The efiect of the nose boom on the forebody is dependent on how the boom is fltted. Ng (1992) also found that at moderate-to-high angles of attack, where the vortex pattern over the nose boom becomes asymmetric, the wake over the nose boom goes over small, naturally present perturbations on the body. This leads to an increase or decrease in forebody vortex asymmetry when the body is at zero-sideslip. At very high angles of attack, unsteady vortex shedding occurs over the nose and the nose boom wake ow is symmetric on a time-average basis. Nose booms are mainly used on the forebodies of flghter aircraft, operating at high angles of attack but not on missiles as it obstructs the missile seeker heads. Ericsson and Reding (1991) provided evidence that helical body trips were more efiective in reducing the side force, than straight body trips. These can be seen in 9 Figure 1.5. Unlike forebody strakes, body trips do not generate their own vortex. The body trip acts on the boundary layer over the forebody, thus its location relative to the natural separation line is crucial. It was shown that the straight boundary trips were e?cient at supercritical ow conditions but at critical ow conditions, it generated close to maximum side force. The helical trips however, were able to alleviate the separation induced side force. Figure 1.5: Efiect of helical and straight body trips on side force of ogive-cylinder body (Ericsson and Reding, 1991) Both Champigny (1994) and Ericsson and Reding (1991) have demonstrated that boundary layer blowing is efiective at controlling forebody ow asymmetry. If blow- ing is tangential to the wall and upstream of the boundary layer separation, it re-energises the boundary layer, delaying separation. If it is performed under a vortex sheet it will modify the vortex position by entrainment. Champigny (1994) noted that keeping the ow symmetric at high angles of attack by means of blowing is di?cult and it may be necessary to add forebody strakes. The most efiective method of controlling forebody vortex ow is by using forebody strakes (Ng and Malcolm, 1992). The use of a flxed pair of forebody strakes, attached symmetrically to the forebody has been efiective in forcing steady asymmetric vortices at high angles of attack to become symmetric. In this way the large side forces and yawing moments are 10 signiflcantly reduced (Ng and Malcolm, 1992). Actively deployed forebody strakes are also a means of enhancing high angle of attack controllability (Ng and Malcolm, 1992). These strakes extended along the length of the forebody and were de ected at difierent angles of attack about a hinge line flxed along a meridian line. While a single strake was efiective in producing large yawing moments, the control was nonlinear and the strake was inefiective in eliminating the steady asymmetry and associated yawing moments at zero sideslip. Ericsson and Reding (1991) also noted that a single strake, a splitter-plate-fln, could be used to force vortex symmetry. From studies conducted by Ng (1990), it was found that the use of a single strake on the forebody drastically reduced the vortex asymmetry. The position of the strake was very critical. Ng (1990) showed that the ow near the apex was highly three-dimensional and had a strong in uence on the axial ow. By reducing the interaction and entrainment between ows on the two sides, the ow asymmetry is not amplifled downstream. Yuan and Howard (1991) studied the efiects of placing four strakes in the cruciform orientation and the efiect of placing eight strakes symmetrically very close to the tip of the forebody of a missile. It was found that the four-strake conflguration was more efiective than the eight-strake conflguration in reducing the yawing moments experienced by the body. However, the four-strake conflguration was not able to reduce the yawing moments at the low angles of attack, since asymmetries formed on the aft body. At high angles of attack though, the induced side forces and yawing moments were virtually eliminated. The four strakes were orientated such that they were aligned with the horizontal and vertical planes of the body, the ?+? conflguration, and were able to force the forebody-generated vortices to become symmetrical. 1.3 Document Layout The work is structured and presented as follows: This chapter (Chapter one) provides a brief background into high angle of attack missile aerodynamics, a motivation as to why this research was carried out and the objectives of the study. Chapter two validates the mesh, the position of the outlet boundary and the turbu- lence model used in this work. In Chapter two the similarity between results for a 11 full and half model are shown, which is used in Chapter three. In Chapter three the efiect of changing the missile strake span on the values of longitudinal aerodynamic coe?cients is investigated. Chapter four studies the creation of steady asymmetric vortices in CFD by placing a geometric perturbation on the missile nose. The iterative process followed in determining the efiect of the size of the perturbation and the efiect of the axial and circumferential positions of the geometric perturbation on the nose of the missile body are shown. Chapter flve investigates the efiect of changing the span and axial positions of the strakelets on the steady asymmetric owfleld, created in Chapter four. Chapter Six summarises the conclusions drawn from this work and makes some suggestions for future work. 1.4 Motivation In 1993, aerodynamic work towards designing an agile, high angle of attack missile with very low aspect ratio wings and control tail flns, started at Denel. Four low aspect ratio wings, referred to as strakes, were orientated such that they were at 45? to the horizontal and vertical planes of the body (???) and the four tail flns were orientated such that they were aligned with the horizontal and vertical planes (?+?). Initial wind tunnel tests were carried out in a high speed wind tunnel in 1995 and was followed by a medium speed wind tunnel test in 1996 (Gobey, 2004). Data obtained from both wind tunnel tests indicated that severe body and strake vortex interaction with the tail flns, in particular the large lateral disturbances or side forces, were as a result of steady asymmetric body vortices. The irregularity of the disturbances made the design of the missile ight control system very di?cult. The large magnitude of the lateral disturbances encountered at various roll orien- tations, even at moderate angles of attack were of particular interest. The strakes were re-orientated such that they were in-line with the tail flns. This alteration reduced the irregularity of the disturbances, but the magnitude of the side forces was still large and were subsequently attributed to geometric imperfections on the missile nose (Gobey, 2004). 12 The following problematic areas needed to be addressed so that the missile?s aero- dynamic performance could be improved: ? Improve the aerodynamics of the missile strakes when in the ??? orientation ? Reduce lateral disturbances by employing four miniature strakes on the fore- body of the missile. The miniature strakes are referred to as strakelets. 1.5 Objectives The objectives of the study are: ? To investigate the efiect of increasing the strake span on normal force when the strakes are orientated in the ??? conflguration. ? To model a steady asymmetric owfleld on a missile body at high angles of attack in CFD. ? To investigate the efiect of changing the span and axial position of strakelets to alleviate the steady vortex asymmetry around a generic missile body. 13 2 Validation Studies Validation studies were carried out in the following areas of importance: ? Mesh size sensitivity ? Outlet boundary position ? Half symmetry ? Turbulence models 2.1 Mesh Size Sensitivity All computational uid dynamics (CFD) models require appropriate grids, with suf- flcient grid density in regions of high ow gradients. The problem lies in determining where these critical regions exist. In a vortical owfleld, high ow gradient regions exist in the boundary layer, regions of shear layer separation and the primary and secondary vortices. A grid resolution study was performed to minimise the error induced by the spatial resolution of the mesh. In order to validate the chosen mesh size, the number of grid points in the circumferential, radial and axial directions were flrst halved and then doubled. The results obtained from these two simulations were then compared to that of the original mesh. 14 The three models that were used in the mesh sensitivity study are identifled as follows: ? Model A : Model with the number of grid points halved in each direction. ? Model B : Model with the number of grid points doubled in each direction. ? Model C : The grid that has been used for the bulk of this study 2.1.1 Grid generation A portion of the missile body was modelled for the mesh sensitivity study. The chosen geometry, shown in Figure 2.1, had a length of 3.8D, where D is the body diameter. Figure 2.1: A portion of the missile body geometry The geometric perturbation on the nose of the missile body acts as trigger for the formation of steady asymmetric vortices. It?s purpose is further discussed in Chapter 4. All meshes in this study were constructed in CFD-GEOM, which is the grid generator for CFD-FASTRAN. Structured grids were created on the geometries. Thomas and Hartwich (1991) found that the structured grid approach leads to the most e?cient algorithms for treating viscous ows, because the grid cells can be highly stretched in the direction normal to the developing shear layers. 15 The height of the flrst cell perpendicular to the body surface is dependent on the the turbulence model used. The Menter-Shear Stress Transport (SST) k?! turbulence model that has been used in this study, is implemented in CFD-FASTRAN without the use of wall functions1. Therefore the height of the flrst cell perpendicular to the body had to be 10 ?m away from the body so that y+ values of 1 were obtained on the body surface. The height of the flrst cell was calculated from Equation 2.1 (CFDRC, 2003). y+ = u?y? (2.1) where: ? u? is the friction velocity (m=s), ? y is the distance from the surface in the boundary layer (m), ? ? is the kinematic molecular viscosity = ?? (m2=s), ? ? is the molecular viscosity (kg=ms), and ? ? is the density (kg=m3) The mesh domain used in this study is shown in Figures 2.2 and 2.3. Figure 2.2: Structured grid used in this study 1Wall functions bridge the extremely thin viscous layer near the surface. In order to adequately resolve the turbulent portion of the boundary layer, at least 8-10 points are required in the turbulent regime (ERCOFTAC, 2000) 16 Figure 2.3: A closer view of the structured grid around the body surface shown in Figure 2.2 The grid shown in Figure 2.2 consisted of 272 equispaced circumferential points extending completely around the body, which was maintained radially out to the farfleld, boundary 52 radial points between the body surface and the computational outer boundary and 90 axial points between the nose tip and the end of the body. This grid was very similar to that created by Degani (1992) and it was found to capture important ow characteristics (Degani, 1992). The flrst cell of the grids for model A and B?s grids were also 10 ?m high, ensuring that a y+ value of 1 was obtained on the body surface. The grid used for model A comprised of 136 circumferential points, 26 radial points and 45 axial points. The grid constructed on Model B consisted of 542 circumfer- ential points, 104 radial points and 180 axial points. Figure 2.2 shows grid cells concentrated close to the body. Since viscous ows over slender body conflgurations are dominated by separated vortical ows, placing a high density of grid cells close to the body surface allows these ow characteristics to be captured. The ow at the farfleld boundaries, located 15 body lengths away from the body surface, is not of interest and thus larger aspect ratio cells are present in that region. 17 Simulations were carried out at the following ight conditions: ? Freestream Mach number (M1) = 0.8 ? Static Pressure (P) = 101.325 kPa ? Static Temperature (T) = 288 K ? Reynolds number = 3? 106 The values of the turbulent kinetic energy (k) and the speciflc dissipation rate of kinetic energy (!) are determined from Equations 2.2 and 2.4 (CFDRC, 2003). k = u? 2 pC? (2.2) ? = C 3 4? k 32 ?y (2.3) ! = k? (2.4) where ? C? is the closure constant, ? ? is the von K arm an?s constant, and ? ? is the dissipation rate of kinetic energy (J=kgs). The values of k and ! used were 3:703 m2=s2 and 5674:2 s?1 respectively. The mesh sensitivity study was only carried out at an angle of attack of 40?, since many of the simulations in this study were carried out at this angle of attack. The boundaries situated in the farfleld were specifled as In ow-Out ow boundaries and were set at freestream conditions since they were located su?ciently far from the body (Thomas and Hartwich, 1991). The farfleld outlet boundary was placed approximately 5 body lengths away from the base of the missile. In Section 1.2, the efiect of the outlet boundary position 18 is discussed and it is shown that placing the outlet boundary 5 body lengths away from the body base is su?cient since the ow on the nose of the missile body is of interest and the objective of this study is to qualitatively investigate the efiects of the steady asymmetric owfleld observed experimentally. The outlet boundary was not placed at the missile base, as simulations were carried out at subsonic Mach numbers and placing the outlet at the base would force a constant pressure solution across the wake which is not physically possible. The Roe?s Flux Difierencing Splitting (FDS) scheme was used in conjunction with the Osher-Chakravarthy (Osher-C) ux limiter to solve the Reynolds Averaged Navier-Stokes (RANS) equations (CFDRC, 2003). The Roe?s FDS scheme is more accurate for separated ows than the Van Leer Flux Vector Splitting (FVS), which is the other spatial difierence scheme implemented in CFD-FASTRAN (Thomas and Hartwich, 1991). High angle of attack ow, at subsonic and transonic Mach num- bers, is characterised by large regions of separation and no shocks are present, thus allowing for the use of Roe?s FDS with the Osher-C ux limiter. The initial conditions were set to the previously deflned freestream conditions and are applicable to all simulations in this study. Since the solution algorithms, for both unsteady and steady ows, require inputs for boundary and initial conditions, they are considered to be time-marching algorithms. The initial conditions, when set to freestream conditions, correspond to an impulsive start (Thomas and Hartwich, 1991). All simulations were run on a Boxx-Dual-Opteron. Each processor had 2Gb of RAM. The simulations were deemed converged once the forces acting on the modelled geometry reached a steady state value. The convergence of the normal force is shown in Figure 2.4. 0 500 1000 1500 2000 2500 3000 0 2000 4000 6000 8000 10000 12000 14000 Number of iterations Variation of Normal Forc e Figure 2.4: Change in normal force per iteration 19 2.1.2 Results Figure 2.5 shows the surface pressure distribution on the geometries for the difierent grids. (a) Model A (b) Model B (c) Model C Figure 2.5: Surface pressure distribution on the three models with difierent grids at fi = 40? From Figure 2.5, it can be seen that the surface pressure distributions for models B and C are very similar, while model A fails to capture certain critical characteristics that are captured by models B and C. Model A has not captured the high pressure region at the centre of the missile body, which is seen in Figures 2.5b and 2.5c. The high pressure region on model B in Figure 2.5b is larger than that on model C in Figure 2.5c since model B had more grid points than model C. This high pressure region is due to the counter-rotating separated vortices. Models B and C show that the low pressure regions on the nose of the missile, indicated by the blue region, are asymmetric. The asymmetry is due to the formed vortices separating at difierent axial positions on the body as a result of the geometric perturbation on the nose triggering vortices of difierent strengths. Due to the coarse grid on model A, this asymmetry is not captured, as shown in Figure 2.5a. The symmetric, low pressure regions on model A indicate that the vortices separated at the same axial position on the body, that is the efiect of the geometric perturbation 20 is not captured. At the rear of the geometry, the large high pressure regions shown in Figures 2.5b and 2.5c for models B and C are not visible in Figure 2.5a. The high pressure region at the rear of model B is larger than that on model C. The surface pressure distributions, 0:06D on either side of the body centre-line, are shown in Figures 2.6 and 2.7. 0 0.5 1 1.5 2 2.5 3 3.5 4 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Model A Model B Model C Figure 2.6: Comparison of surface pressure distribution along length of body (left of the body centre-line) 0 0.5 1 1.5 2 2.5 3 3.5 4 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Model A Model B Model C Figure 2.7: Comparison of surface pressure distribution along length of body (right of the body centre-line) 21 Figures 2.6 and 2.7 are numerical representations of Figure 2.5. The pressure dis- tribution trend, displayed by all three models is very similar. However, the peak surface pressure on model A is signiflcantly lower than the peak surface pressures on models B and C. The difierence in the surface pressure distributions on model B and C is very small, even though model B has eight times the overall number of grid cells in the three directions. Helicity density plots Levy et al. (1990) found that scalar quantities such as pressure and density were insu?cient in describing vortex formation and development. While these quantities could identify the cores of concentrated primary vortices in high-speed ows, they could not identify low-speed phenomena such as secondary vortices or difiused vor- tices. Another major shortfall was that these quantities could not identify the sense of swirl of the vortices, thus the difierence between primary and secondary vortices could not be distinguished. Levy et al. (1990) found that helicity density was able to: ? identify vortices, ? distinguish between primary and secondary vortices, and ? indicate the direction of the swirling motion Helicity density is deflned as the dot product between the velocity vector and the vorticity vector (Levy et al., 1990): Hd = V ?! (2.5) Even though helicity density is a scalar, both its sign and magnitude are meaningful. High magnitudes of helicity density re ect high values of speed and vorticity when the relative angle between the two vector strengths is small. The sign of helicity density, which is determined by the cosine of the angle between the velocity and vorticity vectors, indicates the direction of the swirl of the vortex relative to the streamwise velocity component. Helical density may be graphically displayed, using colour graduation for magnitude and difierent colours for difierent signs in a two colour ood plot. This allows primary and secondary vortices to be clearly distinguished (Levy et al., 1990). 22 The use of helicity density to indicate vortices in low subsonic ows and at low angles of attack is needed as neither density nor pressure mapping are su?ciently sensitive to the changes in density or pressure variations that are a small percentage of full scale in such owflelds. The sensitivity of helicity density always remains high in the vortex region because in this region both velocity and vorticity are high and the angle between the two vectors are small. Therefore helicity density is a better representation of vortices than density mapping, giving a clear representation of both strength and direction. Helicity density contour plots, showing the separation of vortices at two locations along the length of the body, are illustrated in Figures 2.8 and 2.9. (a) Model A (b) Model B (c) Model C Figure 2.8: Helicity density contours at x = 2:5D at fi = 40? Figure 2.8 shows the formation of primary vortices on the missile body. The shades of blue indicate counter-clockwise rotation, with the darkest shade of blue indicating maximum vortex strength and lightest shade of blue indicating the minimum vortex strength. The shades of red indicate clockwise vorticity. The darkest shade of red indicates maximum vortex strength and the lightest shade of red indicates the minimum. 23 The shear layer on the sides of the missile body, that feeds into the formed vortices for models B and C, shown in Figures 2.8b and 2.8c, are smeared in 2.8a, due to the coarse grid on model A. Two well deflned primary vortices, are formed for models B and C. The vortices formed for model A, show that a set of primary vortices have formed but they are not well deflned. The vortex cores in model A are not captured as well as they are captured in models B and C. (a) Model A (b) Model B (c) Model C Figure 2.9: Helical density contours at x = 3:5D at fi = 40? The formation of secondary vortices, rotating in the opposite direction to their asso- ciated primary vortices can be clearly seen in Models B and C, while that displayed for model A are not well deflned. Model B and C show the strength of the secondary vortices. Ribbon traces were plotted on the body so as to show the vortex trajectories ofi the surface of the body. This is shown in Figure 2.10 24 (a) Model A (b) Model B (c) Model C Figure 2.10: Streamlines ofi the surface of the missile at fi = 40? Model A, due to the fewer grid cells is unable to show the spiraling of the ribbon traces, that models B and C are able to predict. Models B and C also show that the vortices have separated from the body. 2.1.3 Discussion Figures 2.5 to 2.7 show that the mesh of model A, predicts signiflcantly difierent surface pressure distributions to that of models B and C due to the coarser mesh. The high surface pressure distribution, which is captured by models B and C, shown in Figures 2.6 and 2.7, is not captured by model A. Model A predicts a surface pressure distribution that is approximately 37% less than that predicted by models B and C. At axial locations of x = 2:5D and x = 3:5D, model A is unable to adequately capture the formation of the primary and secondary vortices. The ribbon plots in Figure 2.10 show that model A is unable to predict ow swirl. The failure of model A to capture regions of high pressure distribution and the strengths of the primary and secondary vortices shows that by using a grid with too few grid cells, important ow features are not captured. Figures 2.6 and 2.7 show similar surface pressures on models B and C. By increasing 25 the number of grid cells, the increase in surface pressures is approximately 2% at x = 2:5D and approximately 3% at x = 2:8D. Thus by using a grid with almost half the density of cells, an answer of su?cient accuracy can be obtained. The improvement in result quality due to an increase in mesh size for model B does not warrant the computational expense of running at the higher resolution. 2.2 Outlet Boundary Position In order to set the outlet boundary at atmospheric pressure, the outlet boundary has to be placed su?ciently far from the base of the body, so that it has no in uence or a very weak in uence on the upstream ow (ERCOFTAC, 2000). 2.2.1 Grid generation The geometry of Figure 2.2 was modifled and the outlet boundary was placed ap- proximately 5 body lengths away from the base of missile body. This decision was based on the fact that the region of interest was the ow on the nose of the missile. In order to check if this position was suitable, the outlet boundary was moved such that it was 20 body lengths away from the base of the missile. This distance was suggested by CFDRC (2003). The structured grid used for the second model was that of Figure 2.2. However, the number of grid points in the outlet region was increased to 40. This study was carried out at an angle of attack of 40? only. 2.2.2 Results Figure 2.11 shows the surface pressure distribution along the top surface of the missile body and the distance from the base of the missile to the outlet boundary. The pressure trace was placed on the body centre-line. 26 0 10 20 30 40 50 60 70 80 3 4 5 6 7 8 9 10 11 x 104 x/D Pressure (Pa ) Outlet = 20 body lengths Outlet = 5 body lengths Figure 2.11: Comparison of the pressure distribution for the two outlet positions Figure 2.11 shows no difierence in the pressure distributions along the length of the body and both models predict the same value for pressure distribution at the outlet boundary. A closer view of the surface pressure distribution near the missile nose in Figure 2.12 shows difierent peak values for the two difierent models. However, this difierence is small enough to be considered negligible. 2 4 6 8 10 12 14 16 18 20 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 x 104 x/D Pressure (Pa ) Outlet = 20 body lengths Outlet = 5 body lengths Figure 2.12: Closer view of the surface pressure distribution near the missile nose for the two outlet positions. 27 Figure 2.13b shows the formation of primary and secondary vortices on the missile body at x = 3:5D. The vortex formation for both outlet boundary positions look identical. (a) Model A (b) Model B Figure 2.13: Helical density contours at x = 3:5D at fi = 40? Figures 2.11 and 2.12 show that the position of the outlet boundary has had a negligible efiect on the surface pressure distribution on the upper surface of the missile body, while Figure 2.13 shows no change in vortex formation. The study is primarily concerned with ow on the nose of the missile and its in- teraction with the strakelets as well as qualitatively reproducing steady asymmetric vortices. Thus placing the outlet boundary 5 body lengths from the base of the missile should not have a signiflcant in uence on the ow at the nose. This is seen in Figure 2.13, where the outlet boundary position has had no visible efiect on the formation of primary and secondary vortices. The outlet boundary can thus be left at 5 body lengths from the rear of the missile. 2.3 Comparison Between Full Model and Half Model Simulations For symmetric owflelds it is possible to model only a portion of the geometry to obtain a solution which is representative of the whole geometry. The advantage of modelling a portion of the geometry is that the number of grid points and thus the size of the overall mesh is reduced. In this way the computational time required to obtain a reasonably accurate solution is also reduced. An investigation was under- taken to determine if the normal force and pitching moment coe?cients obtained from a full model simulation and a half model simulation would be similar. This would justify the use of a half model for further symmetric owfleld simulations. 28 2.3.1 Grid generation The geometry shown in Figure 2.14 was analysed. The length of the missile body was 18D and the aspect ratio2 of the strakes on the body was 3:7? 10?3. Figure 2.14: Full geometry of missile body Since the owfleld is symmetric about the vertical plane, only one half of the model was created in CFD-GEOM. A portion of the structured grid, created on the half model, is shown in Figure 2.15. Figure 2.15: A closer view of the grid constructed on the half model For the half model a symmetry plane was set. The half model grid consisted of approximately 1.2 million grid cells. For the full model the half model was mirrored 2ratio of strake span to strake chord 29 about its symmetry axis and the symmetry boundary was removed. Simulations were carried out at angles of attack from 0? to 50? in increments of 10? at the ight conditions specifled in Section 2.1. The Menter-SST k ? ! turbulence model was used. The values used for k and ! were 3:703 m2=s2 and 5674:2 s?1 respectively. The spatial numerical method used was Roe?s FDS, together with the Osher-C ux limiter to solve the Reynolds averaged form of the Navier-Stokes equations. 2.3.2 Results The normal force coe?cient (CN ) and pitching moment coe?cient (Cm) for the full model were calculated from Equations 2.6 and 2.7: CN = N0:5?V 2Sref (2.6) Cm = Mz0:5?V 2Sref lref (2.7) where: ? N is the normal force (N), ? Mz is the pitching moment about the z-axis (N.m), ? ? is the density = 1:225 kg=m3, ? V = 272:14 m=s at Mach = 0.8, ? Sref is the cross-sectional area = ?D24 (m2), and ? lref is the reference length = D (m) The percentage difierence in the values obtained for the normal force and pitching moment coe?cients, calculated from Equations 2.8 and 2.9, are shown in Table 2.1. The actual values obtained for the two models are shown in Figures 2.16 and 2.17. %?CN = CNFull ? CNHalfCNFull ? 100 (2.8) %?CN = CmFull ? CmHalfCmFull ? 100 (2.9) 30 where: ? CNFull is the normal force coe?cient for the full model ? CNHalf is the normal force coe?cient for the half model ? CmFull is the pitching moment coe?cient for the full model ? CmHalf is the pitching moment coe?cient for the half model Table 2.1: Percentage difierence in the normal force and pitching moment coe?cients between the full and half symmetry models fi (deg) %?CN %?Cm 0 0.000 0.000 10 4.957 1.346 20 4.494 3.420 30 2.379 0.8610 40 1.361 0.0310 50 0.1340 1.343 Figures 2.16 and 2.17 are comparisons of the normal force coe?cient and pitching moment coe?cients respectively. 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 Angle of Incidence (deg) Normal force coefficien t Half model Full model Figure 2.16: Comparison of values of of the normal force coe?cients for the half and full models 31 0 5 10 15 20 25 30 35 40 45 50 ?5 0 5 10 15 20 25 30 35 Angle of Incidence (deg) Pitching moment coefficien t Half model Full model Figure 2.17: Comparison of values of the pitching moment coe?cients for the half and full models Figures 2.16 and 2.17 show that there is very little difierence in the values of the normal force and pitching moment coe?cients, obtained from the full and half sym- metry model simulations. Figures 2.18 and 2.19 show the ow development on the two models at axial locations of 0:31D and 6:3D. (a) Half model (b) Full model Figure 2.18: Density contours at x = 0:31D at an angle of attack of 40? The vortex formed in Figure 2.18a is very similar to the vortices in Figure 2.18b. The vortices shown in Figure 2.18b are the same height above the body. This vortex height is captured by the half model in Figure 2.18a as well. 32 (a) Half model (b) Full model Figure 2.19: Density contours at x = 6:3D at an angle of attack of 40? In Figure 2.19 ow separation occurs on both sides of the body, at the top strakes. This ow separation is also captured by the half model in Figure 2.19a. The owflelds on both halves of the body in Figure 2.19b are identical, indicating that the owfleld is symmetrical. 2.3.3 Discussion Table 2.1 and Figures 2.16 and 2.17 show that the values for the normal force and pitching moment coe?cients from a half model simulation correlate very well with the values obtained from a full model simulation. The maximum percentage difierence obtained is approximately 5% at an angle of attack of 10? for the normal force coe?cient and 3:4% for the pitching moment coe?cient at 20? angle of attack. These difierences are acceptable when weighed against the computational time saved. Figures 2.18 and 2.19 show that important characteristics such as vortex formation and ow separation is captured even if a half model simulation is carried out. The owflelds in Figures 2.18b and 2.19b show that the owflelds on both halves of the body are identical, indicating that no asymmetries exist in the owfleld. This shows that for a perfectly symmetrical body it is possible to model only half of the body and important ow characteristics will be captured. However, if a time-accurate simulation or asymmetrical owfleld is to be modelled, a full model of the geometry must be used. It is possible that for a time-accurate simulation, the ow might not develop symmetrically (if perturbed) and for an asymmetrical model, the ow on either sides of the model is not identical. 33 2.4 Turbulence models Turbulence is one of the key phenomena in uid dynamics. There are various tech- niques for the numerical prediction of turbulent ows ranging from the Reynolds Averaged Navier-Stokes (RANS), large eddy simulation (LES) and direct numerical simulation (DNS) (Cummings et al., 2003). DNS attempts to resolve all scales of turbulence from the largest to the smallest by solving the Navier-Stokes equations directly. LES attempts to model the smaller, more homogenous scales, while resolv- ing the larger energy containing scales, thus making grid reflnement for LES less than that for DNS. For the RANS approach , the equations have been averaged over a time-scale, which is small in relation to the aerodynamic time-scale but large in comparison to the time-scale of the turbulent eddies (Thomas and Hartwich, 1991). The RANS approach attempts to solve the time-averaged ow, which means that all scales of turbulence must be modelled. Turbulence models are semi-empirical formulations that are used to close the RANS equations by approximating the Reynolds stress terms (Cummings et al., 2003). They are generally calibrated on building block ows such as boundary layers, shear layers and wakes. Reynolds stresses are modelled in two ways, namely eddy viscosity models and shear stress transport models (Cummings et al., 2003). Shear stress transport models make no general assumptions about the form of the six components of the Reynolds stress model and unless assumptions have been made, the turbulence model solves for all six unknowns. The more common eddy viscosity models are based on Boussineq?s approximation that the Reynolds stresses are directly proportional to the local strain rate of ow. This assumption reduces the number of unknowns from six to a single unknown which is the turbulent eddy viscosity term (Wilcox, 2000). Most common are zero-, one- and two-equation turbulence models. The zero- equation turbulence models use algebraic relations rather than partial difierential equations. The zero-equation turbulence models avoid the necessity of flnding the edge of the boundary layer and employ a purely algebraic modelling of the eddy viscosity. The simplicity of the zero-equation turbulence model allows for good computational e?ciency but their applicability is limited. The more complex one- and two-equation models are aimed at more closely mimicking the physics of turbu- lent ows. These models assume that the eddy viscosity is a function of a turbulence length and velocity scale. The two-equation models use two partial difierential equa- tions to compute the velocity and length scales. The one-equation models compute 34 the velocity scales via a partial difierential equation but algebraically evaluate the turbulence length scale (Wilcox, 2000). A major challenge in aerodynamic design is the accuracy of turbulence models for simulations of complex turbulent ows for example high angle of attack ows (Bar- dina et al., 1997). Development of improved turbulence models has increased in the last decade due to the technological requirements of present aerodynamic systems, aided by advances in computers and numerical simulation capabilities. A variety of researchers have proposed methods for adapting algebraic turbulence models for high angles of attack. Degani and Schifi (1991) proposed a modiflcation to the Baldwin- Lomax model, an eddy viscosity turbulence model, that predicted ow reasonably accurately in the separated ow region. In this study, while the ow separation and the formation of steady asymmetric vortices are important, it is the interaction of the steady asymmetric vortices with surfaces on the slender body that is of primary importance. Five difierent turbulence models are available for use in CFD-FASTRAN, namely the Baldwin-Lomax model, the standard k? ? (Launder-Spalding) model, the k?! (Wilcox) model, the Spallart-Allmaras model and the Menter-SST k ? ! model. The standard k ? ? turbulence model is a two-equation eddy viscosity model for incompressible and compressible turbulent ows. It is a high Reynolds number model and is not meant to be used in the near wall regions were viscous efiects are greater that the efiects of turbulence (Bardina et al., 1997). The standard k- ? turbulence model has been implemented in CFD-FASTRAN by means of wall functions.(CFDRC, 2003) The Menter-SST k?! turbulence model is also a two-equation eddy viscosity model. The Menter-SST turbulence model is a combination of the standard k?? turbulence model and Wilcox?s k?! turbulence model. It uses the k?! model near solid walls and the standard k ? ? turbulence model near the boundary layer edges. The Spalart-Allmaras turbulence model is a one-equation model, based on the trans- port eddy viscosity and was designed for aerospace applications. It predicts ow separation very well, (Bardina et al., 1997), but was not used in this study, because it has not been properly implemented in CFD-FASTRAN. The LES turbulence model was formulated for solving unsteady cyclic and vortical ows and should be chosen for modelling steady asymmetric vortex ow. The LES turbulence model was not implemented in CFD-FASTRAN at the time the study 35 was conducted. The Menter-SST k ? ! turbulence model and the standard k ? ? turbulence model were used in this investigation to determine which turbulence model would provide acceptable results for complex high angle of attack ows. Although Bardina et al. (1997) and Menter (2003) have shown that the standard k?? turbulence model does not fair well in separated ows, the shortfall of the standard k? ? turbulence model would be investigated alongside the Menter-SST k ? ! turbulence model. 2.4.1 Grid generation In Section 2.3, it was seen that it was su?cient to model only half of a symmetrical model as all important ow fleld characteristics were captured with the half model simulation. Therefore only half the missile body shown in Figure 2.14 was modelled in CFD-GEOM. Two difierent grids had to be constructed since y+ values of between 30 to 100 were required for the standard k ? ? turbulence model and y+ values equal to 1 were required by the Menter-SST k ? ! turbulence model (CFDRC, 2003). To ensure that these values of y+ were obtained, the height of the flrst cell perpendicular to the body surface had to be approximately 0:5 mm and 10 ?m for the standard k? ? turbulence model and Menter-SST turbulence models respectively. A structured grid, such as that shown in Figure 2.15 was constructed on the body surface. The grid for the standard k ? ? turbulence model consisted of 146 circum- ferential grid points, 55 radial grid points and 150 axial grid points. The number of radial grid points had to be increased for the Menter-SST k ? ! turbulence grid as more points were required closer to the body surface. As with the half symmetry study, the investigation was carried out at the ight conditions specifled in Section 2.1 at angles of attack from 0? to 50? in increments of 10?. Roe?s FDS was the spatial numerical method used, together with Osher-C ux limiter, to solve the RANS equations (CFDRC, 2003). 2.4.2 Results The values for the normal force and pitching moment coe?cients were determined from Equations 2.6 and 2.7. The percentage difierence in normal force and pitching 36 moment coe?cients were calculated from Equations 2.10 and 2.11 and are shown in Table 2.2. %?CN = CNSST ? CNk??CNSST ? 100 (2.10) %?CN = CmSST ? Cmk??CmSST ? 100 (2.11) where: ? CNSST is the normal force coe?cient for the Menter-SST k ? ! turbulence model ? CNk?? is the normal force coe?cient for the standard k ? ? turbulence model ? CmSST is the pitching moment coe?cient for the Menter-SST k?! turbulence model ? Cmk?? is the pitching moment coe?cient for the standard k ? ? turbulence model Table 2.2: Percentage difierence in normal force and pitching moment coe?cients between the two turbulence models fi (deg) %?CN %?Cm 0 0.000 0.000 10 1.120 0.4800 20 18.13 10.56 30 21.32 6.680 40 15.84 10.73 50 12.61 10.02 Table 2.2 indicates that the percentage difierence in the values of obtained for the normal force and pitching moment coe?cients for the two turbulence models at 10? angle of attack are low. However, at the higher angles of attack, a signiflcant difierence exists between the coe?cients obtained from the two turbulence models. 37 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 18 Angle of Incidence (deg) Normal force coefficien t k?? Menter?SST Figure 2.20: Comparison of values of the normal force coe?cients for the two tur- bulence models 0 5 10 15 20 25 30 35 40 45 50 ?5 0 5 10 15 20 25 30 35 Angle of Incidence (deg) Pitching moment coefficien t k?? Menter?SST Figure 2.21: Comparison of values of the pitching moment coe?cients for the two turbulence models 38 Figures 2.20 and 2.21 illustrate the difierences in the normal force and pitching moment coe?cients respectively obtained for the two difierent turbulence models. At the lower angles of attack, up to and including 10?, the standard k?? turbulence model and the Menter-SST turbulence model predict very similar values of normal force and pitching moment coe?cients. However, at higher angles of attack, the difierences in values of the normal force and pitching moment coe?cients between the two turbulence models increases. The Menter-SST turbulence model predicts higher values of normal force and pitching moment coe?cients than the standard k ? ? turbulence model. The difierence in surface pressure distributions, predicted by the two turbulence models is shown in Figure 2.22 for an angle of attack of 40?. (a) Standard k ? ? (b) Menter-SST Figure 2.22: Side view of the surface pressure distributions for the difierent turbu- lence models at fi = 40? 39 The low surface pressure distribution, indicated by the blue region, on the side of missile body in Figure 2.22b indicates that the ow moves to the leeward side of the body before it separates. In Figure 2.22a, the low pressure regions indicate that ow remains attached on the body for a longer period of time and does not move towards the leeward side of the body. Sectional cuts, at difierent axial locations are shown in Figures 2.23 to 2.25. These illustrate the difierence in the ow formation along the length of the missile body. (a) Standard k ? ? (b) Menter-SST Figure 2.23: Density contour plot at x = 3:1D at fi = 40? At an axial location of x = 3:1D (Figure 2.23a) the standard k? ? turbulence model predicts that the ow separation vortex that formed as a result of ow interaction with the top strake has separated from the body. The standard k ? ? turbulence model predicts that the ow between the two strakes remains attached to the body. From Figure 2.23b it can be seen that the Menter-SST turbulence model predicts that ow between the two strakes is about to separate. The vortex, which formed due to ow interaction with the top strake is still close to the body. 40 (a) Standard k ? ? (b) Menter-SST Figure 2.24: Density contour plot at x = 5D at fi = 40? At an axial location of x = 5D, the Menter-SST turbulence model predicts that the ow from the lower body strake separates from the body while the standard k ? ? turbulence model does not show this separation (Figure 2.24). A region of recirculating air at the top strake is captured by the Menter-SST turbulence model (Figure 2.24b). The standard k?? turbulence model fails to predict the recirculating air under the top strake (Figure 2.24a). (a) Standard k ? ? (b) Menter-SST Figure 2.25: Density contour plot at x = 6:3D at fi = 40? At x = 6:3D (Figure 2.25b) the Menter-SST shows that a second vortex separa- tion occurs at the lower strake and that the vortex moves towards the top strake. This second separation is not captured by the standard k ? ? turbulence model of Figure 2.25a. 41 2.4.3 Discussion In Figures 2.20 and 2.21, the values of the normal force and pitching moment co- e?cients calculated from the two turbulence models, at the low angles of attack, (between 0? and 10?) are very similar since in this angle of attack range the ow is still attached to the body. However, at the higher angles of attack, the ow begins to separate from the body and areas of recirculating air develop as in Figure 2.24. Therefore at the larger angles of attack, the values calculated for the normal force and pitching moment coe?cients are signiflcantly difierent, with the standard k-? model predicting lower values for the normal force and pitching moment coe?cients. Simulations were run at 40? angle of attack on the geometry at experimental con- ditions. The normal force coe?cients obtained from the CFD simulations for the standard k ? ? and Menter-SST k ? ! turbulence models are compared to the ex- perimental normal force coe?cient in Figure 2.26. The results obtained from the Spalart-Allmaras turbulence model is also shown in Figure 2.26. ?10 0 10 20 30 40 50 60 ?5 0 5 10 15 20 25 Angle of Incidence (deg) Normal force coefficien t k?? Menter?SST Spalart?Allmaras DAS Figure 2.26: Comparison of normal force coe?cients From Figure 2.26 it can be seen that the Menter-SST turbulence model predicts the closest value for the normal force coe?cient to the experiment. The standard k?? turbulence model under-predicts the value for the normal force coe?cient. The Spalart-Allmaras turbulence model predicts a normal force coe?cient much lower than the experimental normal force coe?cient, indicating that the Spalart-Allmaras turbulence model would not be suitable for further use in this study. 42 Bardina et al. (1997) found that the standard k ? ? model under-predicted ow separation. This is indicated by the lower values for CN . Figure 2.22 shows that the Menter-SST turbulence model predicts separation earlier than the standard k-? turbulence model. This is in keeping with the flndings of Bardina et al. (1997) that the standard k ? ? turbulence model delays ow separation. In Figure 2.27 ribbon traces were plotted on the missile body. This flgure shows that the Menter-SST turbulence model predicts that the vortex core moves away from the body while the standard k? ? turbulence model shows that the vortex core remains relatively close to the body. When Figure 2.27 is compared to Figure 2.28, which shows ow development on the body of the missile in the wind tunnel, the Menter-SST turbulence model shows a better prediction of the ow around the missile body. (a) Standard k ? ? (b) Menter-SST Figure 2.27: Ribbon traces ofi the surface of the missile body at fi = 40? Figure 2.28: Flow development on the missile body in the high-speed wind tunnel (CSIR-Defencetek, 2004) 43 2.5 Conclusions From the validation study a number of decisions were made regarding the simulation methods applied in the balance of this study. These are summarised as: ? If a grid with fewer cells than the one used in this study is used, important ow characteristics are not captured properly. There is also a signiflcant difierence in surface pressure distributions. Using a grid with more cells has resulted in a very small increase in accuracy. The grid used for model C is adequate as the flner grid of model B does not justify the added computational expense. ? Placing the outlet boundary a distance 5 body lengths from the base of the missile, results in a similar owfleld as when the outlet boundary is placed 20 body lengths away from the base of the body. Thus placing the out ow boundary 5 body lengths from the base of the missile and at atmospheric conditions is acceptable since the ow does not have a signiflcant efiect the upstream boundaries. ? As long as the geometry of the missile body and the owfleld are symmetrical and a steady solution is sought, it is su?cient to model half of the geometry in order to obtain values for the normal force and pitching moment coe?cients and to gain an understanding of the owfleld. ? Since the Menter-SST k ? ! turbulence model predicted ow separation at high angles of attack better than the standard k ? ? turbulence model, the Menter-SST k ? ! turbulence model will be used in further investigations. 44 3 Efiect of Changing the Span of Missile Strakes A body and strake conflguration like that shown in Figure 3.1 was tested in past wind tunnel test series (DAS, 2004). The length of the missile body was 18D and the aspect ratio of the strakes was 3:7? 10?3. Figure 3.1: Missile body-strake conflguration 45 The values for normal force coe?cient and pitching moment coe?cient, shown in Figures 3.2 and 3.3, were determined from Equations 3.1 and 3.2 (Gobey, 2004). CNStrake = CNBody?Strake ? CNBody (3.1) CmStrake = CmBody?Strake ? CmBody (3.2) where: ? CNStrake is the increment in normal force coe?cient due to the strakes on a body-strake conflguration. It includes the interference factors of the body on the strakes and the strakes on the body. ? CNBody?Strake is the experimental normal force coe?cient acting on the body with the strakes. ? CNBody is the experimental normal force coe?cient acting on the body of revolution. ? CmStrake is the increment in pitching moment coe?cient due to the strakes on a body-strake conflguration. It includes the interference factors of the body on the strakes and the strakes on the body. ? CmBody?Strake is the experimental pitching moment coe?cient acting on the body with the strakes. ? CmBody is the experimental pitching moment coe?cient acting on the body of revolution. The increment due to the the strakes on a body-strake conflguration is often referred to as the efiect of the strakes in the presence of the body. The normal force coe?cient and pitching moment coe?cient for the strakes in the presence of the body are shown in Figures 3.2 and 3.3. 46 ?5 0 5 10 15 20 25 30 35 40 45 ?1 0 1 2 3 4 5 6 7 Angle of Incidence (deg) Normal force coefficien t Experimental ? x Experimental ? + Figure 3.2: Normal force coe?cient of strakes in the presence of the body (Mach 0.8) (DAS, 2004) ?5 0 5 10 15 20 25 30 35 40 45 ?5 0 5 10 15 20 25 Pitching moment coefficien t Incidence angle (deg) Experimental ? x Experimental ? + Figure 3.3: Pitching moment coe?cient of strakes in the presence of the body (Mach 0.8) (DAS, 2004) 47 The experimental data in Figures 3.2 and 3.3 were analysed by Gobey (2004). When the strakes are in the ?+? roll orientation the strakes have an in uence on the nor- mal force coe?cient from an angle of attack of 3? upwards. The pitching moment coe?cient for the strakes in the ?+? roll orientation in Figure 3.3, shows an initial slope and then a sudden increase in slope at a large angle of attack that corresponds to the increase in the normal force coe?cient slope. This indicates that the centre of pressure on the strakes shifts forward between angles of attack of 5? and 10? and then remains constant at the large angles of attack (Gobey, 2004). When the strakes are in the ??? roll orientation the normal force coe?cient remains at almost zero magnitude up to 25? angle of attack. This indicates that the strakes are not forcing the ow to separate from the body when they are in the ??? orientation (Gobey, 2004). The pitching moment coe?cient of the strake in the presence of the body has a very low slope over the entire angle of attack range in Figure 3.3. There is a dramatic difierence in the aerodynamic behaviour of the strakes in the two roll orientations. Gobey (2004) stated that this difierence must be eliminated, if possible, to make the aerodynamics of the conflguration more consistent with roll angle. Since the existing conflguration could not be drastically changed due to design spec- iflcations, it was proposed that the current strake span of 0:06D be increased to 0:13D. The new value for the strake span was based on the fact that by increasing the span of the strakes greater ow separation is forced around the body, increas- ing the normal force produced by the strakes. However due to design constraints, the strake span could not be increased by a large amount. The value chosen for the new span was not the optimal solution but it was a workable one. A CFD study was carried out to determine if increasing the strake span would result in a su?cient increase in normal force coe?cient generated by the strakes when in the ??? orientation, thus allowing for more consistent aerodynamic behaviour with roll angle. 3.1 Grid Generation Since symmetric owflelds were being investigated, only one half of the geometry, such as that shown in Figure 3.1, was modelled. 48 A structured grid, such as that shown in Figure 2.15, was used. The grid for the short strake model comprised of 146 circumferential grid points, 55 radial grid points and 150 axial grid points. For the long strake model the number of radial grid points was increased to 65. The study was carried out at the ight conditions specifled in Section 1.1, at angles of attack from 0? to 50? in increments of 10?. The RANS equations were used to solve the steady ow simulation. The Menter- SST turbulence model was implemented. Roe?s FDS was used together with the Osher-C ux limiter to solve the RANS equations. 3.2 Strake Height In uence on Coe?cient Behaviour The normal force and pitching moment coe?cients for the two difierent strake span conflgurations were calculated by equations 2.6 and 2.7. The percentage increase on the normal force and pitching moment coe?cients carried by the strakes due to the increased strake span was calculated from Equations 3.3 and 3.4. %?CN = CNHigh ? CNShortCNHigh ? 100 (3.3) %?CN = CmHigh ? CmShortCmHigh ? 100 (3.4) where: ? CNHigh is the normal force coe?cient for the strake with span of 0:13D, ? CNShort is the normal force coe?cient for the strake with span of 0:06D, ? CmHigh is the pitching moment coe?cient for the strake with span of 0:13D, and ? CmShort is the pitching moment coe?cient for the strake with span of 0:06D. These values are shown in Table 3.1. 49 Table 3.1: Percentage increase in normal force coe?cient and pitching moment coe?cient for the difierent missile strake spans fi (?) %?CN %?Cm 0 0.000 0.000 10 41.66 46.19 20 16.08 52.59 30 11.28 43.34 40 16.18 35.44 50 19.16 37.45 Table 3.1 shows that by increasing the strake span, there is a signiflcant increase in the normal force and pitching moment coe?cients, in particular at angles of attack between 10? and 30?. This implies that with the larger span, the aerodynamic loads carried by the strakes have increased. The normal force and pitching moment coe?cients of the strakes in the presence of the body are shown in Figures 3.4 and 3.5 along with the experimental data. The experimental data is present to conflrm the trend of the CFD results and not to serve as an exact match. ?10 0 10 20 30 40 50 ?1 0 1 2 3 4 5 6 7 8 9 Angle of Incidence (deg) Normal force coefficien t Short strakes ? b = 0.01m (CFD) Long strakes ? b = 0.02m (CFD) Short strakes ? b = 0.01m (DAS) Figure 3.4: Comparison of normal force coe?cient between the difierent span con- flgurations In Figure 3.4 the trend of the CFD data for the short strakes resembles that of the experimental data for the short strakes. Between angles of attack of 20? and 30? the 50 slope of the experimental data changes. The normal force coe?cient on the large strakes has increased signiflcantly at the lower angles of attack, up to 25?. For the large strakes there is an increase in the slope of normal force coe?cient at low angles of attack compared to the short strakes where the slope of normal force coe?cient was almost zero and the normal force coe?cient remained at almost zero magnitude up to 20?. The large strakes resulted in an increase in normal force coe?cient at the low angles of attack, ranging from 11:28% at 30? angle of attack to 41:66% at 10? angle of attack. ?10 0 10 20 30 40 50 ?1 0 1 2 3 4 5 6 Angle of Incidence (deg) Pitching moment coefficien t Short strakes ? b = 0.01m (CFD) Long strakes ? b = 0.02m (CFD) Short strakes ? b = 0.01m (DAS) Figure 3.5: Comparison of pitching moment coe?cient between the difierent span conflgurations The pitching moment coe?cient has also increased signiflcantly at the lower angles of attack, due to the increase in the normal force coe?cient. The large strakes have resulted in a percentage increase ranging from 43:34% at 30? angle of attack to 52:59% at 20? angle of attack. For the short strakes pitching moment coe?cient varied from 0 to approximately 3. However, in the same angle of attack range, the large strakes? pitching moment coe?cient varies from 0 to 6. This indicates that the larger strakes have improved the aerodynamics of the missile. The slope of the pitching moment of the strake has increased signiflcantly with the increase in the strake span. At angles of attack between 20? and 30? the CFD results do not display the same trend as the experimental data. This is due to the formation of steady asymmetric vortices on the experimental model. The results obtained from the CFD simulations for normal force coe?cient compare 51 well with the experimental data. The trend for both sets of results are very similar. This is an indication that the CFD results obtained for the increment in normal force coe?cient due to the strakes with a span of 0:13D would be reasonably accurate. 3.3 Strake Height Efiects on the Flowfleld Figures 3.4 and 3.5, show that there is a large increase in the normal force coe?cient and pitching moment coe?cient on the larger strakes. This is attributed to the larger strakes causing the ow around the body to separate. This is conflrmed by Figure 3.6. (a) Short strakes (b) Long strakes Figure 3.6: A side view of the cross ow velocity at x = 4:7D at 20? 52 Figure 3.6 show that the strakes with 0:06D span are not large enough to cause the ow around the body to separate. The ow separates on contact with the lower strake but immediately reattaches itself to the body. When the ow does separate, it does not make contact with the top strake. By increasing the strake span to 0:13D, reattachment of ow to the body is prevented and stronger vortices are produced. Figure 3.7: A side view of the cross velocity at x = 4:7D at 20? for the strakes orientated in the ?+? roll orientation Figure 3.2 shows that when the short strakes are orientated in the ?+?, the normal force generated by the strakes is almost twice the normal force generated by the short strakes in the ??? roll orientation at 20? angle of attack. Figure 3.7 shows that the horizontal strakes force ow separation around the body and the separated ow does not reattach onto the body as seen in Figure 3.6a. Since the short strakes, in the ?+? roll orientation prevent ow reattachment, the normal force on the strakes in the presence of the body is much larger. The larger strakes increase the efiective diameter of the body, thus causing ow to separate from the body and to remain separated. This resulted in an increase in normal force coe?cient at the lower angles of attack, ranging from 11:28% at 30? angle of attack to 41:66% at 10? angle of attack. There was a corresponding increase in the pitching moment coe?cient, ranging from 43:34% at 30? angle of attack to 52:59% at 20? angle of attack. This indicates that the overall aerodynamics of the body with the strakes, orientated in ???, has improved. Figure 3.4 shows that between angles of attack of 20? and 30? the normal force coe?cient slope changes. This is due to ow not reattaching to the body, thus increasing the normal force coe?cient at angles of attack greater than 30?. 53 Increasing the span of the strakes even further could increase the pitch-up per- formance of the missile but the size of the strakes are constrained by the aircraft carriage size and the required overall missile mass. The difierence in the CFD data and experimental data for pitching moment coef- flcient is due to the formation of steady asymmetric vortices on the experimental model. This was expected as CFD does not readily predict the formation of steady asymmetric vortices and considerable efiort is required to simulate a steady asym- metric owfleld. 3.4 Conclusion By increasing the strake span from 0:06D to 0:13D, there was an overall increase in the normal force coe?cient ranging from 11:28% at 30? angle of attack to 41:66% at 10? angle of attack. The increase in the normal force coe?cient resulted in a corresponding increase in the pitching moment coe?cient, ranging from 43:34% at 30? angle of attack to 52:59% at 20? angle of attack. The increase in normal force coe?cients, and thus pitching moment coe?cients, is due to the large span strakes forcing the ow to separate su?ciently far from the body, thus preventing reattachment of the separated ow to the body. From this it can be concluded that an increase in the strake span results in greater ow separation around the body. Increasing the strake span to 0:13D has prevented the ow around the body from reattaching to the body, thus improving the aerodynamics of the body-strake con- flguration, for the strakes in the ??? orientation. The aerodynamics of the strakes in the ??? orientation has now been made more consistent with roll orientation since ow reattachment, observed previously only when the strakes were orientated in the ?+? orientation, is now observed when the strakes are in ??? orientation as well. 54 4 Creation of Steady Asymmetric Vortices in CFD As discussed in Section 1.1 and shown in Figure 1.2 steady vortex asymmetry occurs on a missile body at high angles of attack (fiAV ? fi ? fiUV ), due to geometric im- perfections on the missile nose. It is due to this steady asymmetric vortex formation that large side forces develop on the missile body, even at zero sideslip. This results in an uncontrollable missile (Section 1.3). The geometric imperfections provide the initial disturbance which is amplifled along the missile body by a spatial instability (Bernhardt and Williams, 1998). Flow asymmetry is amplifled along the length of the body by one of the following two types of instabilities: (Cummings et al., 2003) Absolute Hydrodynamic Instability The small ow perturbation yields a bi- furcated asymmetry, even after the perturbation has been removed, for exam- ple unsteady von K?arm?an vortices. This is a temporal instability and is not considered in this study. The absolute hydrodynamic instability hypothesis states that as the angle of attack is increased, a bifurcation state occurs at a critical angle of attack that produces one of two ?mirror images?. At any angle of attack greater than the critical angle of attack, only two values of side force exist. Convective Instability Small geometric perturbations are required for steady vor- tex asymmetry to exist and the owfleld is not limited to two bifurcated states. If the geometric perturbations are removed the ow returns to its steady sym- metric state. The convective instability hypothesis states that any level of asymmetry is possible at high angles of attack. The asymmetry is not conflned to only two levels. As the angle of attack is increased, an unstable state is reached where an inflnite number of paths are possible, until the very high angle of attack 55 regime is reached (Section 1.1.4). At this point, the side force has reached a fully bifurcated state but intermediate regions do exist. It is evident from the two difierent instabilities described above that irrespective of the type of instability, a perturbation is required for the formation of asymmetric vortices. In experiments, these perturbations are very common but in numerical calculations these perturbations need to be introduced into the calculation. In nu- merical calculations the owfleld is an ideal one and the body surfaces are perfectly smooth. Therefore it is not possible to obtain an asymmetric owfleld in numerical calculations without an external perturbation. Cummings et al. (2003) provide evidence that certain numerical algorithms that break symmetry preservation, do exist, thus causing the owfleld to become asym- metric. One such algorithm is the diagonalised algorithm which was developed to speed up the vector- ux splitting algorithm. This algorithm was not used as the level of disturbance introduced by the algorithm is uncontrolled. Degani and Schifi (1991) showed numerically, that when an asymmetric perturba- tion, flxed in time and space, was introduced near the apex of an ogive cylinder, the steady owfleld became asymmetric. By explicitly adding a geometric perturbation on the missile body the level of disturbance introduced into the simulation is known (Cummings et al., 2003). Degani and Schifi (1991) suggested the use of a geomet- rical bump on the nose of the body of revolution or a small jet owing normal to the body of revolution. Levy et al. (1996) have shown that, qualitatively, the es- sential steady asymmetric, multi-vortex structure can be captured by the use of a simple, simulated disturbance. The multi-vortex structure of Figure 1.2 is formed by the breakaway of the higher positioned vortices and the generation of new vortices, developing alternately on either side of the body axis (Xueying et al., 1991). The alternating vortex formation is time-independent but changes along the length of the body (ESDU, 1989). When the perturbation is removed the ow returns to it?s symmetric state, demonstrating that the asymmetry was amplifled by a convective instability (Degani, 1992). 56 In order to investigate the efiect of the strakelets on steady asymmetric vortices in CFD, a steady asymmetric owfleld had to be simulated flrst. Once the steady asymmetric owfleld had been simulated on a missile body of revolution (referred to as the missile body), then the strakelets were added to the missile body to investigate the efiects of the strakelets on the steady asymmetric vortices. The size of the geometric perturbation required to trigger the formation of steady asymmetric vortices on the missile body was unknown. Thus an iterative process was carried out to determine: ? the size of geometric perturbation required to trigger the formation of steady asymmetric vortices, ? the efiect of the axial location of the geometric perturbation on the formation of steady asymmetric vortices, and ? the efiect of the circumferential location of the geometric perturbation on the formation of steady asymmetric vortices. A way of conflrming that an appropriately sized geometric perturbation was chosen, was to to simulate the owfleld around the missile body with the geometric per- turbation and at a very low angle of attack. As mentioned in Section 1.1, at low angles of attack the ow on the body remains attached. Therefore the geometric perturbation must not force the attached owfleld to separate from the missile body as this would not be representative of the naturally occurring owfleld on a body of revolution at low angles of attack. From experimental data shown in Figure 4.1 it was observed that the missile body experienced non-zero side forces at moderate angles of attack due to the formation of steady asymmetric vortices. This is in keeping with the evidence provided by Ericsson and Reding (1991) that for blunt ogive noses, the onset angle of attack for steady asymmetric vortices is lower than that for pointed ogive noses. The onset angle of attack is approximately determined by 4:2dl (Ericsson and Reding, 1991). By that criterion, the onset of steady asymmetric vorticity should occur between 10? and 12? angle of attack. Figure 4.1 shows that steady asymmetric vortices begin at approximately 11? angle of attack. The geometric perturbation must thus be able to simulate steady asymmetric vortices at these low angles of attack as well. 57 ?10 0 10 20 30 40 50 60 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 Angle of Incidence (deg) Side force coefficien t Figure 4.1: Experimental side force coe?cient on a body of revolution at Mach 0.8 (DAS (2004)) Figure 4.2 is a schilieren image which shows ow development on the body of revo- lution in the wind tunnel. Figure 4.2: Formation of steady asymmetric vortices in the high-speed wind tunnel at Mach 0.8 at an angle of attack of 30? ((CSIR-Defencetek, 2004)) 58 From the schilieren image in Figure 4.2 it can be seen that the ow on the nose of the missile body is attached. Flow separation occurs on the aft cylindrical part of the missile body. The vortices have separated at difierent axial positions on the missile body. The difierence in height between the steady asymmetric vortices is small. Therefore the difierence in height between the CFD modelled steady asymmetric vortices must also be small. 4.1 Size of Geometric Perturbation Required to Simu- late Asymmetric Vortices Degani and Schifi (1991) suggested the use of a geometric bump to simulate steady asymmetric vortices in CFD. However, Xuei et al. (2000) showed experimentally that irrespective of the shape of the geometric perturbation, steady asymmetric vortices developed. The bistable state of the steady asymmetric vortices was unafiected and the owfleld structure remained regular even if difierent shaped perturbations were used (Xuei et al., 2000). Thus instead of using a geometric bump, a hexahedral block was placed 0:25D away from the nose tip for this study. Degani (1992) used a geometric perturbation with a height of 0.01D and a length of 0.05D to create a steady asymmetric owfleld on a pointed slender body. However the body used in this research is a blunt ogive body and thus the exact dimensions used by Degani (1992) could not be used. These dimensions were used as guidelines in choosing the sizes of the geometric perturbations that were to be investigated. Three difierent sizes of geometric perturbations were studied. The geometric per- turbations had the following dimensions: Table 4.1: Dimensions of geometric perturbations Model Height Length Width Perturbation G 0:06D 0:13D 0:02D Perturbation H 0:03D 0:13D 0:02D Perturbation I 0:03D 0:06D 0:02D The width of the geometric perturbations were chosen to be very small and was not varied in this investigation as Degani (1992) indicated that the height and length of the geometric perturbation were the critical factors in modelling steady asymmetric vortices on a body of revolution. 59 4.1.1 Grid generation Since the area of interest was the missile forebody and the fact that steady asym- metric vortices alternate along the length of the body, only 8:1D of the missile body of revolution was modelled. In this way the number of grid cells required to create a mesh was reduced, thus reducing the number of iterations required for a solution to be obtained. The geometry modelled is shown in Figure 4.3. This flgure shows the size of the geometric perturbation relative to the missile body. Figure 4.3: Body and geometric perturbation geometry . Each structured grid consisted of 272 equispaced circumferential grid points extend- ing completely around the body. In each circumferential plane, the grid consisted of 45 radial points between the body surface and the computational outer boundary and 95 axial points between the nose tip and the base of the body. This grid was very similar to the one constructed by Degani (1992). The simulations were carried out at the conditions specifled in Section 2.1 Simula- tions were only run at three angles of attack, namely, 5?, 20? and 40? at Mach 0.8 and a Reynold?s number of 3 ? 106. The results from the 5? angle of attack simu- lation would indicate whether the geometric perturbation has changed the owfleld on the missile body at low angles of attack. The geometric perturbation also had to introduce a small degree of asymmetry into the owfleld at angles of attack of 20? and 40? as steady vortex asymmetry is low on blunt ogive bodies. A low degree of asymmetry is characterised by a small difierence in surface pressures across the body?s centre-line and a small difierence in height between the two steady asymmet- ric vortices (Degani and Levy, 1992). 60 The Menter-SST k-! turbulence model was used. The ux-splitting algorithm used was Roe?s FDS algorithm with the Osher-C ux limiter. Simulations were run at steady conditions, independent of time-step size since asymmetric vortex formation is time-independent. Hartwich et al. (1990) showed that asymmetric vortical owflelds are steady-state solutions to the Navier-Stokes equations. Simulations were also run on a body of revolution at the three difierent angles of attack. This was to serve as a comparison between the owflelds created with and without the geometric perturbation. The full length missile body was also modelled, to see if the multi-vortex structure of Figure 1.2 could be captured by the use of a simulated disturbance, as indicated by Levy et al. (1996). 4.1.2 Results for 5? angle of attack The surface pressure distributions, shown in Figure 4.4 for the four models, are identical. It is very di?cult to determine whether the geometric perturbation had an efiect on the owfleld as pressure variations at low angles of attack are very small (Champigny, 1986). (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.4: Surface pressure distribution on missile body at fi = 5? for various geometric perturbations 61 Contours of helicity density (discussed in Section 2.1) were plotted on cross-sectional cuts, so as to determine if the difierent geometric perturbations forced vortex sepa- ration. Figure 4.5 shows the ow development at an axial distance of 6D from the nose tip. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.5: Helicity density contours at x = 6D at fi = 5? for various geometric perturbations Figure 4.5a shows the ow formation on the missile body of revolution without the geometric perturbation on the missile nose. The ow is still attached to the body which is characteristic of ow on bodies of revolution at low angles of attack (Ericsson and Reding, 1991). Figure 4.5b shows that the owfleld due to ow interaction with perturbation G has changed. The right vortex, represented by the blue region, appears to be separating from the body. The two vortices have difiering strengths as indicated by the difierent colour intensities. This indicates that perturbation G is forcing asymmetric vortex separation. The ow formation illustrated in Figures 4.5c and 4.5d is very similar to the owfleld on the body of revolution in Figure 4.5a, indicating that perturbations H and I did not have any signiflcant efiect on the attached ow. 62 4.1.3 Results for 20? angle of attack The symmetrical surface pressure distribution in Figure 4.6a shows that a pair of symmetrical vortices have formed on the missile body. However, the experimental data in Figure 4.1 show that a non-zero side force exists at 20? angle of attack, indicating that the owfleld should be asymmetric. Figures 4.6b, 4.6c and 4.6d show that by adding a geometric perturbation on the nose of the missile body, the steady symmetric owfleld is forced to a steady asymmetric state. In Figures 4.6b, 4.6c and 4.6d the asymmetric pressure is prominent at the rear of the missile body. The high surface pressure at the rear of the bodies indicates that side forces developed on one side of the body and that the side forces do not oscillate on the body. The non-oscillating side force distribution at 20? is very similar to that found by Champigny (1994). (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.6: Surface pressure distribution on missile body at fi = 20? for various geometric perturbations 63 Perturbation G, in Figure 4.6b has resulted in a large difierence in surface pressure distribution between the two halves of the missile body, thus the asymmetry present in the owfleld of Figure 4.6b is high. The difierences in the surface pressure dis- tributions between the two halves of the body for perturbations H and I, in Figures 4.6c and 4.6d, are not as large as that which exists for perturbation G, indicating that the asymmetry present in the owfleld, as a result of perturbations H and I, is lower. This is clearly shown in Figure 4.7, which shows the difierence in surface pressures across the two halves of the body. The surface pressures a distance of 0:06D on either side of body centre-line were plotted. 0 1 2 3 4 5 6 7 8 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (a) Body of revolution 0 1 2 3 4 5 6 7 8 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (b) Body with Perturbation G 0 1 2 3 4 5 6 7 8 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (c) Body with Perturbation H 0 1 2 3 4 5 6 7 8 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (d) Body with Perturbation I Figure 4.7: Surface pressure distribution along the length of the missile body at 20? angle of attack for various geometric perturbations As is to be expected the surface pressure distribution across the body of revolution?s centre-line in Figure 4.7a are identical. For perturbation G, in Figure 4.7b, there is a large difierence in surface pressures across both halves of the body, while the difierences in surface pressure for perturbations H and I, in Figures 4.7c and 4.7d are smaller. The asymmetry due to perturbation G occurs closer to the nose of the missile body than the asymmetry forced by perturbations H and I. Perturbation I 64 shows the smallest difierence in surface pressures, indicating that it introduced the lowest degree of asymmetry into the owfleld. The local minimum in Figure 4.7, located approximately 0:2D from the nose tip, is due to ow interaction with the perturbation. The local minima for the two curves in Figures 4.7b, 4.7c and 4.7d are unequal, since the geometric perturbation is located on one side of the missile body. Figures 4.8 and 4.9 show vortex formation at axial locations of 4D and 6D respec- tively by means of helicity density contours. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.8: Helicity density contours at x = 4D at fi = 20? for various geometric perturbations Figure 4.8a shows the development of a symmetrical pair of vortices on the body of revolution, as previously indicated. This prediction by the CFD simulations is not a realistic representation of the owfleld on a body at high angles of attack. The bodies with the perturbations have forced the vortices to separate asymmetrically. In Figures 4.8b, 4.8c and 4.8d, the right vortices, indicated by the blue region, have separated flrst, since the right vortices are larger than the left vortices, indicating that the right vortices are weaker than the left vortices. This is due to vortex dif- fusion along the length of the missile body. This was expected since the geometric 65 perturbations are placed on the right of the body centre-line. This is further con- flrmed by the higher surface pressure on the right of the body centre-line at the corresponding axial location in Figures 4.7b, 4.7c and 4.7d. The higher surface pres- sure on the right indicates that the right vortex is further away from the body than the left vortex. The two formed vortices are the primary vortices. The right vortex in Figure 4.8b is further away from the body than the vortices on right vortices in Figures 4.8c and 4.8d, resulting in a higher surface pressure on the body with perturbation G at an axial location of 4D. This can be clearly seen in Figure 4.7b where the surface pressure on the right of the body centre-line is higher than that in Figures 4.7c and 4.7d at the corresponding axial location on the missile body. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.9: Helicity density contours at x = 6D at fi = 20? for various geometric perturbations Figure 4.9 shows the development of secondary vortices. The vortex formation in Figures 4.9b, 4.9c and 4.9d corresponds to the ow structure identifled by Degani and Levy (1992). The secondary vortices form due to the adverse circumferential pressure gradient encountered by the boundary layer behind the low pressure region created by the primary vortices (Degani and Levy, 1992). The secondary vortices rotate in the opposite direction to their associated primary vortices, as noted by Degani and Levy (1992). This conflrms that the grid used and the CFD simulations 66 are adequately capturing the important features in the steady asymmetric owfleld. In Figure 4.9 the left primary vortex, represented by the the red region, has sep- arated from the body. The asymmetric primary vortices in Figures 4.9b, 4.9c and 4.9d resemble the asymmetric vortices shown in section A-A of Figure 1.2. Figure 1.2, however, does not show the formation of secondary vortices. The secondary vor- tices form once the primary vortices have separated from the body. Therefore the secondary vortices also develop asymmetrically along the length of the missile body. This is clearly shown in Figures 4.9b, 4.9c and 4.9d, where the left secondary vortices are smaller than the right secondary vortices due to the asymmetric separation of the primary vortices. 4.1.4 Results for 40? angle of attack Figure 4.10 shows the surface pressure distribution on the missile bodies with the difierent geometric perturbations. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.10: Surface pressure distribution on missile body at fi = 40? for various geometric perturbations 67 Figure 4.10 indicates that the degree of asymmetry present in the owfleld at 40? angle of attack is greater than the degree of asymmetry present in the owfleld at 20? angle of attack (see Figure 4.6). This is expected given the ow geometry discussed in Section 1.1.3. Figures 4.6 and 4.7 also show that it is only at the rear that there is a prominent difierence in surface pressure distributions. Figures 4.10 and 4.11 show that the difierence in surface pressure distributions alternate on the body, indicating that vortex asymmetry has started earlier on the body at 40? than at 20? angle of attack. Ericsson and Reding (1991) provided evidence that as the angle of attack of the slender body is increased, vortex asymmetry moves closer to the nose tip. The alternating high surface pressures on the body in Figure 4.10 shows the alternate steady asymmetric vortex formation. The alternating high surface pressures at 40? also indicates that the side force distribution alternates along the length of the body as predicted in Figure 1.3. The surface pressures at a distance of 0:06D from the body centre-line are shown in Figure 4.11. 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (a) Body of revolution 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (b) Body with Perturbation G 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (c) Body with Perturbation H 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (d) Body with Perturbation I Figure 4.11: Surface pressure distribution along the length of the missile body at 40? angle of attack for various geometric perturbations 68 There is a large difierence in surface pressure distributions between both halves of the body for perturbation G in Figure 4.11b. This once again illustrates that pertur- bation G has forced the vortices to separate from the body with larger asymmetry than perturbations H and I. Figure 4.11d shows that perturbation I has introduced the smallest degree of asymmetry into the owfleld indicated by the small difierence in surface pressures across the two halves of the body. This indicates that perturba- tion I has forced an asymmetric owfleld which would be more representative of the actual owfleld on a real blunted ogive body of revolution at high angles of attack. In Figures 4.11b, 4.11c and 4.11d the surface pressures on the left side of the missile body becomes higher than that on the right side of the body at approximately x = 5:2D. These alternating surface pressures across the body centre-line are due to the alternating detachment of vortices from the body (Section 1.1.4). Figures 4.12 and 4.13 are helicity density colourmaps, at difierent axial locations along the body of revolution, showing vortex formation for the various geometric perturbations. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.12: Helicity density contours at x = 4D at fi = 40? for various geometric perturbations 69 Figure 4.12a shows the formation of a pair of symmetric primary and secondary vortices that would result in no side force. However, the experimental results in Figure 4.1 shows that at 40? angle of attack a side force does exist on the missile body, indicating the presence of steady asymmetric vortices. Figures 4.12b, 4.12c and 4.12d show the formation of primary and secondary vortices. At the corresponding axial location on the missile body at 20? angle of attack only the right primary vortex had separated from the missile body. This illustrates that at higher angles of attack vortex separation occurs closer to the missile nose. Perturbation G in Figure 4.12b has resulted in a owfleld with a higher degree of asymmetry than the owflelds on the bodies with perturbations H and I due to the large difierence in size between the right and left vortices. The difierence in the vortex heights away from the body surface is indicated in the large difierence in surface pressures at x = 4D in Figure 4.11b. The small difierence in surface pressures at x = 4D in Figures 4.11c and 4.11d shows that that the right and left vortices in Figures 4.12c and 4.12d are not as far from the body as the vortices in Figure 4.12b. (a) Body of revolution (b) Body with Perturbation G (c) Body with Perturbation H (d) Body with Perturbation I Figure 4.13: Helicity density contours at x = 6D at fi = 40? for various geometric perturbations 70 Perturbation G in Figure 4.13b introduces the alternating behaviour of steady asym- metric vortices discussed in Section 1.1.3 and shown in Figure 1.2, along the length of the missile body. The missile body with perturbation G shows that a second pri- mary vortex is about to form on the right, below the red vortex. Since perturbations H and I introduced a low degree of asymmetry into the owfleld, this alternating behaviour of steady asymmetric vortices along the body length cannot be observed in Figures 4.13c and 4.13d. However Figures 4.11c and 4.11d shows that at 6D away from the missile nose tip, the surface pressure on the left side is larger than that on the right side as opposed to the surface pressures at 4D where the surface pressure on the right side is higher than the left side. 4.1.5 Discussion In Figure 4.5 for the missile body at 5? angle of attack, the right vortex is larger than the left vortex indicating that perturbation G is forcing vortex separation at low angles of attack. A possible reason for ow separation only occurring on the body when perturbation G was used, is that perturbation G has a greater height than perturbation H and has a greater height and length than perturbation I. Even though the separation is small, a side force will develop. Thus perturbation G cannot be used due to its larger height since it would not be representative of the experimental results. The owfleld on the body due to perturbations H and I, in Figures 4.5c and 4.5d resembled the attached owfleld of the missile body of revolution in Figure 4.5a. Since the owfleld on the body with perturbations H and I were very similar to the owfleld in Figure 4.5a, the side forces on the bodies with perturbations H and I would be very close to zero. The side force coe?cients for the bodies with the three perturbations were calculated from Equation 4.1. CY = FY1 2?V 2Sref (4.1) The values for ?, V and Sref are the same as those used in Section 1.3. FY , the side force in Newtons, was obtained from the CFD simulations. The side force coe?cients were plotted with the experimental data and with the CFD data for the body of revolution in Figure 4.14. 71 ?10 0 10 20 30 40 50 60 ?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1 Angle of Incidence (deg) Side force coefficien t Body of revolution (CFD) Perturbation G Perturbation H Perturbation I Experimental Figure 4.14: Comparison of side force coe?cient values for the difierent perturba- tions Quantitatively the CFD data and the experimental data do not match since the experimental conditions and the conditions at which the CFD simulations were carried out were difierent. The experimental side force coe?cients are for the full length missile body while the CFD results are only for a portion of the missile body. The qualitative behaviour of the CFD data, for all three geometric perturbations is very similar to the experimental data, though it does not predict the sign reversal at 40? angle of attack. This can be attributed to the fact that only a portion of the missile body was modelled. The three perturbations produced almost zero side force coe?cients at 5? angle of at- tack. Perturbation G resulted in the largest side force coe?cients at angles of attack of 20? and 40?, which are signiflcantly larger than the experimental data. Perturba- tion I resulted in side force coe?cients which were the closest to the experimental data at 20? and 40?, indicating that it resulted in a steady asymmetric owfleld that is most representative of the experimental owfleld. The side force coe?cients due to perturbation H were in between the side force coe?cients for perturbations G and I. Perturbation H had a smaller height than perturbation G but a larger length than perturbation I and thus the side force values obtained from the simulations with perturbation H would lie between the side force values for perturbations G and I. The experimental data shows that at angles of attack of 20? and 40?, non-zero side force coe?cients exist and at low angles of attack, side force coe?cients are zero. The CFD simulations were not carried out to quantitatively match the experimental data 72 but to conflrm that at 20? and 40? steady side forces due to geometric perturbations can be modelled on a blunt-ogive body of revolution. Figure 4.6 and 4.7 show that the asymmetric surface pressure distribution begins at the rear cylindrical part of the missile and not at the nose. This is characteristic of blunt ogive bodies, where vortex shedding begins on the aft cylindrical part of the body (Ericsson and Reding, 1991). This is an indication that geometric perturba- tions on the missile nose simulated a realistic characteristic of steady asymmetric vortex formation on the missile body. When the angle of attack was increased from 20? to 40?, the asymmetry moved forward as shown in Figure 4.7 and 4.11 and described in Section 1.1.3. This owfleld characteristic was observed by Luo et al. (1998), that as the angle of attack increased, the locations of vortex asymmetry and vortex separation propagated upstream toward the nose tip. This characteristic can also be observed in Figures 4.9 and 4.12. At 20? angle of attack, asymmetric primary and secondary vortex formation was identifled at an axial location of 6D on the body, while at 40? angle of attack, asymmetric primary and secondary vortex formation was identifled at an axial location of 4D. Also evident from Figures 4.9 and 4.12 is that the secondary vortices form asymmet- rically. It is this asymmetric development of primary and secondary vortices that results in the asymmetric surface pressure distributions in Figures 4.6 and 4.10 for the missile bodies with perturbations. The larger height of perturbation G increased the efiective local diameter of the body and thus resulted in a greater degree of ow separation. Degani and Levy (1992) found that for small perturbations, the degree of asymmetry present in the owfleld was small but increasing the size of the perturbation increased the asymmetry in the owfleld. A low degree of asymmetry is characterised by the one vortex being slightly higher than the other and consequently the difierence between the surface pressure distribution on opposite sides of the body is small (Degani and Levy, 1992). At 20? and 40? angles of attack, the right primary vortex separates flrst from the body for perturbations H and I. The difierence in height between the left primary vortex, after it has separated and the right primary vortex is very small. This can be clearly seen in Figures 4.9c and 4.13c and Figures 4.9d and 4.13d, which show the owflelds for perturbation H and I respectively. Since perturbation H and I have small heights, ow separation is small. The surface pressures along the length of the body, shown in Figures 4.6c, 4.7c, 4.10c and 4.11c, show a correspondingly small difierence between the opposite sides of the body centre-line. The steady asymmetric owfleld generated by perturbation G is characteristic of a 73 owfleld with a high degree of asymmetry (Degani and Levy, 1992). Flow separation occurs earlier on the body due to the perturbation?s larger height and length. A large difierence in the surface pressures exists across the body centre-line (Figures 4.10b and 4.11b). The three perturbations used in this study were able to force the formation of asym- metric vortices. However, perturbation G forced a high degree of asymmetry into the owfleld due to it?s larger height and length, as indicated by the large difier- ences in surface pressures across both halves of the body centre-line in Figure 4.7b and flgure 4.11b. Perturbation H introduced a low degree of asymmetry into the owfleld but the difierence in surface pressures across the body centre-line is larger in Figures 4.7c and 4.11c due to perturbation H being longer than perturbation I. The side force coe?cients obtained from the simulations with perturbation I were very close to that obtained experimentally (Figure 4.14). Therefore perturbation I will be used in the remainder of the study to determine the efiect of strakelets on asymmetric ow. 4.1.6 Conflrmation of type of instability In order to conflrm the origin and type of the asymmetric ow, perturbation I was removed from the missile body after simulations had converged to a steady asymmetric solution. The simulation was then restarted from this point. This exercise was performed for an angle of attack of 40? only. The change in side force per iteration is shown in Figure 4.15 0 1000 2000 3000 4000 5000 6000 7000 8000 ?400 ?350 ?300 ?250 ?200 ?150 ?100 ?50 0 50 Number of iterations Variation of Side Forc e Perturbation removed Perturbation present Figure 4.15: Side force variation per iteration 74 Figure 4.15 shows that once the geometric perturbation had been removed, the owfleld returned to it?s ideal symmetric state. Thus the steady asymmetric ow had its origin in a convective instability of the original ow that was induced by a geometric perturbation. A large perturbation resulted in a high degree of ow asymmetry and a small perturbation resulted in a low degree of asymmetry. This is consistent with the idea of convective instability due to geometric perturbations where an increase in the perturbation size results in an increase in the degree of asymmetry (Levy et al., 1996). 4.1.7 Full length missile results The full length of the missile body, with perturbation I, was also simulated at 40? angle of attack. The asymmetric surface pressure distribution for the full length missile is shown in Figure 4.16. Figure 4.16: Surface pressure distribution on the full length missile body The asymmetric surface pressure distribution along the full length of the missile body indicates that a simple simulated disturbance can qualitatively capture the asymmetric multi-vortex structure, shown in Figure 1.2. The alternating nature of the asymmetric pressure distribution along the length of the missile body shows that ow characteristics obtained from modelling only a portion of the missile is representative of the owfleld along the full length of the missile body, since the same asymmetric owfleld is repeated along the length of the body (Xueying et al., 1991). 75 4.2 Efiect of Axial and Circumferential Position of A Perturbation on the Flowfleld G.Zilliac et al. (1991) found that at high angles of attack, the owfleld was very sensitive to the axial and circumferential location of surface imperfections. Since asymmetric vortices exhibit a regular bistable state behaviour, the axial and cir- cumferential location of the surface imperfection could trigger either one of the bistable states. Xuei et al. (2000) deflnes the two regular states of asymmetric owflelds as Left Vortex Pattern and Right Vortex Pattern. According to Xuei et al. (2000) the left vortex pattern is deflned by the right vortex separating flrst from the body and the left vortex separating at a difierent axial location on the body. The left vortex remains closer to the body while the right vortex is further away from the body until it detaches. For the right vortex pattern, the opposite occurs (Xuei et al., 2000). 4.2.1 Grid generation Efiect of the axial location of the perturbation Perturbation I, from Section 4.1, was placed at the following axial locations on a 8:1D missile body, such as that shown in Figure 4.3: ? the nose tip ? x = 0:06D ? x = 0:25D Efiect of the circumferential location of the perturbation Perturbation I, flxed at an axial location of x = 0:25D was placed at 2 difierent circumferential locations: ? 120?, that is 30? clockwise from the lateral meridian, (Figure 4.17a). ? 150?, that is 60? clockwise from the lateral meridian, (Figure 4.17b). 76 The two circumferential positions were chosen based on Degani and Schifi (1991) flndings that the most sensitive circumferential locations to place a disturbance on a pointed slender body are between 90?, which is the lateral meridian and 140?, which is 50? to the right of the lateral meridian. The choice of the second circum- ferential location was to determine if blunt ogive bodies were less sensitive to the circumferential position of the geometric perturbation. (a) Geometric perturbation at ` = 120? (b) Geometric perturbation at ` = 150? Figure 4.17: Front view showing the two difierent circumferential positions of the perturbations The grid described in Section 4.1, was used to determine the efiect of the axial and the circumferential location of the geometric perturbation. The investigation was only carried out at an angle of attack of 40? at Mach 0.8 and the initial conditions specifled in Section 2.1. The Menter-SST turbulence model was used along with Roe?s FDS algorithm with the Osher-C ux limiter. 4.2.2 Results Efiect of the axial location of the perturbation The surface pressure distributions for various pertrubation axial locations are shown in Figure 4.18. 77 (a) Geometric perturbation at tip of nose (b) Geometric perturbation at x = 0:06D (c) Geometric perturbation at x = 0:25D Figure 4.18: Surface pressure distributions on missile bodies with perturbations at difierent axial locations (fi = 40?) Figure 4.18a shows that the surface pressure distribution along the length of the body is symmetrical, which is very difierent from the results obtained by Degani (1992) on a pointed slender body. The grid used by Degani (1992) was similar but the nose geometries difiered, which resulted in the geometric perturbation being at an angle on the body, thus forcing a steady asymmetric owfleld. The perturba- tions located at axial locations of x = 0:06D and x = 0:25D, in Figures 4.18b and 4.18c, produce asymmetric surface pressure distributions. Even thought the geo- metric perturbations were placed on the right of the body centre-line, the geometric perturbation, 0:06D from the nose tip, forced the left vortex to separate before the right vortex. This is indicated by the shorter low pressure region on the left of the missile body surface in flgure 4.18b. The perturbation that was placed 0:25D from the nose tip, forced the right vortex to separate flrst, as indicated by the shorter low pressure region on the right of the missile body centre-line in Figure 4.18c. This demonstrates that the perturbations at difierent axial locations were able to trig- ger two difierent stable states of asymmetric vortices. Figure 4.18b shows the high pressure region at the rear of the missile body is not as large as the high pressure region in Figure 4.18c. This indicates that the perturbation at 0:06D produced an asymmetric owfleld which was not as strong as that produced by the perturbation at 0:25D. 78 Figures 4.19 and 4.20 show owfleld development at axial locations of 4D and 6D by means of helicity density contours. (a) Perturbation at tip of nose (b) Perturbation at x = 0:06D (c) Perturbation at x = 0:25D Figure 4.19: Helical density contours at x = 4D at fi = 40? for the perturbations at difierent axial locations Figure 4.19 shows the formation of primary and secondary vortices on the missile body. Figure 4.19a shows that the symmetric primary vortices formed, even though a perturbation was present at the tip of the nose. This indicates that the perturbation located at the nose was inefiective in creating a steady asymmetric vortex system. Figure 4.19b illustrates that the left primary vortex, which is represented by the red region, is larger than the right primary vortex which is an indication that the left vortex separated from the body flrst. This is further conflrmed by the shorter low pressure region on the left of the missile body surface in Figure 4.18b. The left secondary vortex is larger than the right secondary vortex. The asymmetric owfleld in Figure 4.19c is difierent to that shown in Figure 4.19b, even though both perturbations were placed at the same declination. In Figure 4.19c, the right primary and secondary vortices are larger than the left primary and secondary vortices, indicating that the separation occurred on the right side flrst. 79 (a) Geometric perturbation at tip of nose (b) Geometric perturbation at x = 0:06D (c) Geometric perturbation at x = 0:25D Figure 4.20: Helical density contours at x = 6D at fi = 40? for the perturbations at difierent axial locations Figure 4.20 shows vortex formation at an axial location of 6D due to the difierent positioned geometric perturbations. In Figure 4.19 at an axial location of 4D from the missile nose the two difierent steady asymmetric states are not clearly visible. Further downstream on the body, at an axial location of 6D the difierent asymmetric states can be clearly seen. The left vortex in Figure 4.20b is larger than the right vortex and the right vortex in Figure 4.20c is larger than the left vortex. 80 Ribbon traces, in Figure 4.21, show the difierence in the resulting vorticity patterns for the two geometric perturbations. (a) Geometric perturbation at x = 0:06D (b) Geometric perturbation at x = 0:25D Figure 4.21: Streamlines on body at fi = 40? for the perturbations at difierent axial locations In Figure 4.21a, the left vortex separates and is initially higher than the right vortex. Near the base, the right vortex rises above the left. In Figure 4.21b, the right vortex is initially higher than the left vortex while the left vortex is higher near the base. This further demonstrates the alternating nature of the steady asymmetric ow identifled in Figure 4.16. Efiect of the circumferential location of the perturbation The surface pressure distributions for the geometric perturbations at circumferential locations of 120? and 150? are shown in Figure 4.22 81 (a) Geometric perturbation at ` = 120? (b) Geometric perturbation at ` = 150? Figure 4.22: Surface pressure distribution on missile bodies with geometric pertur- bations at difierent circumferential locations (fi = 40?) The surface pressure distributions on both missile bodies are very similar. The asymmetric pressure distribution indicates that the owfleld for both will be asym- metric and since the high surface pressure region is on the same half of the body, both geometric perturbations triggered ofi the same regular state. Helicity density contours were plotted at an axial location of x = 4D from the nose to show the steady asymmetric vortex formation. (a) Geometric perturbation at ` = 120? (b) Geometric perturbation at ` = 150? Figure 4.23: Helical density contours at x = 4D at fi = 40? for the geometric perturbations at difierent circumferential positions Figure 4.23 shows the primary and secondary vortex formation for the two geometric perturbations. The resultant owfleld for both look very similar. The right vortex is larger than the left vortex, indicating that it separated flrst from the body. This is conflrmed by the shorter low pressure region on the right of the missile body centre-line in Figures 4.22a and 4.22b. 82 4.2.3 Discussion Efiect of the axial location of the perturbation Figures 4.19 and 4.20 illustrate that the ow at high angles of attack is sensitive to the axial location of the perturbation. The perturbations, situated at two difierent axial locations, forced two difierent asymmetric states. It was thought that since the perturbation was located on the right side of the body, the right vortex would separate flrst for both geometric perturbations. Figures 4.19b and 4.21a show that the geometric perturbation at 0:06D triggered the right vortex pattern since the left vortex is further away from the body. Figures 4.19c and 4.21b show that the geometric perturbation at 0:25D triggered the left vortex pattern since the right vortex separated from the body before the left vortex as seen in Figure 4.18c. This is illustrative of the evidence provided by Champigny (1994) that two strong, counter-rotating vortices cannot co-exist symmetrically at high angles of attack and that a perturbation is required to force the unsteady, symmetric vortices into a stable, asymmetric state. However, the bistable state that is triggered by the geometric perturbation cannot be predetermined, even if the perturbations are located at the same circumferential location. It was found that distance was an important factor in the degree of asymmetry obtained on a blunt ogive body. When the geometric perturbation was placed at the tip of the nose, symmetric vortices were generated since the perturbation was symmetric. The geometric perturbation at 0:25D resulted in a stronger asymmetric owfleld than the perturbation at 0:06D. This gives the indication that to simulate asymmetric vortices on a blunt ogive body, the geometric perturbation should be placed away from the nose tip. However, the geometric perturbation cannot be placed to far down the length of the missile body as the steady asymmetric vortex formation would no longer be observed on the body but rather further downstream from the body. Since the geometric perturbation located 0:25D away from the nose of the missile resulted in a higher degree of ow asymmetry, this geometric perturbation was chosen for further investigations. The higher degree of asymmetry would help in gaining an understanding of the efiect the strakelets would have on the asymmetric owfleld (mentioned in Section 1.3). 83 Efiect of the circumferential location of the perturbation The surface pressure distributions in Figure 4.22 and the helicity density contours in Figure 4.23 show that the owflelds for the two perturbations, located at difierent circumferential locations are very similar. Both perturbations triggered the same bistable state. Also evident is that the perturbation located at ` = 150? falls outside the limitations specifled by Degani and Schifi (1991). This difierence is probably due to Degani and Schifi (1991) researching asymmetric ow on a pointed, slender body as opposed to the blunt ogive used in this study. The exact methods used by Degani and Schifi (1991) serve as guidelines and cannot be directly used. Since the same asymmetric vortex state was triggered by perturbations located at ` = 120? and ` = 150?, the perturbation located at 120? was chosen for the remain- der of the study, due to it?s position falling within the limit specifled by Degani and Schifi (1991). 4.3 Conclusion Perturbation I had the least efiect on the owfleld and resulted in a negligible value of side force at 5? angle of attack. The presence of perturbation I on the missile forebody also resulted in the formation of steady asymmetric vortices at angles of attack of 20? and 40?. The resultant steady asymmetric owfleld on the body, due to perturbation I, was characteristic of blunt ogive bodies since the degree of asymmetry introduced by the geometric perturbation was low. When the length of the missile body was increased, perturbation I was able to simulate the alternating behaviour of steady asymmetric vortices. It was also shown that the steady ow asymmetry originated from a convective instability in the owfleld as flrst identifled by Degani and Schifi (1991). Therefore perturbation I will be used in Chapter 5 to create a steady asymmetric owfleld so as to determine the efiect of the strakelets on steady asymmetric ow. A geometric perturbation located at the tip of the nose had no efiect on the owfleld and the ow remained symmetric even at 40? angle of attack. It was found that it was possible to excite difierent stable states of steady asymmetric vortices by placing the geometric perturbations at difierent axial locations on the body. It was 84 also found that as the geometric perturbation distance from the nose increased, the degree of asymmetry increased. The axial location of perturbation I was set at 0:25D, as the steady ow asymmetry was the most pronounced of the two geometric perturbation locations tested and the efiect of the strakelets on the asymmetric vortices could be easily observed. The circumferential location of the geometric perturbation was also an important factor in creating a steady asymmetric owfleld. However it was found that for a blunt nose body, the geometric perturbation could be at an angle of 120? or 150? and a steady asymmetric owfleld would still be created. Both yielded the same steady asymmetric vortex regular state as well. The circumferential position of perturbation I was set at 120? as this fell within the circumferential limits set by Degani and Schifi (1991), even though Degani and Schifi (1991) used a pointed slender body. 85 5 Efiect of Strakelets on Steady Asymmetric Vortices As mentioned in Section 1.2, the large side forces on a missile at high angles of attack, due to the formation of steady asymmetric vortices, need to be controlled or alleviated so that the required missile performance can be obtained (Champigny, 1994). Based on the research by Yuan and Howard (1991) that forebody miniature strakes (strakelets) reduced side forces at high angles of attack, four strakelets were added in the ??? conflguration to the nose of a missile body of revolution (referred to as missile body) in an attempt to eliminate the large side forces, by forcing steady vortex symmetry. The strakelets had to be large enough to force vortex symmetry so that symmetric vortices could interact with the control surfaces, placed at the rear of the full length missile body. The strakelets could not, however, contribute to the overall normal force of the missile as this would force the missile nose to pitch-up resulting in an unstable missile. Experimental data in Figure 5.1 shows that the magnitude of the side force coe?cient was reduced with the addition of the strakelets on a full length missile body with tail flns at the rear. 86 Figure 5.1 illustrates the side force coe?cient on a body with 4 control surfaces (body-tail conflguration) with and without strakelets. ?10 0 10 20 30 40 50 60 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 Side force coefficien t Angle of Incidence (deg) Body and Tail Body, Tail and Strakes Figure 5.1: Comparison of experimental side force data for a body-tail conflguration with and without strakelets When the strakelets were not placed on the body-tail conflguration, side forces developed on the missile body at angles of attack less than 10?. The side force coe?cient uctuated non-linearly from approximately 0.2 at 20? angle of attack to approximately -0.4 at 25? to 0.6 at 38? angle of attack. When the strakelets were added to the body-tail conflguration, side forces developed on the body at approximately 30? angle of attack. The highest value for the side force coe?cient was approximately -0.42 at approximately 48?, which is lower than that obtained when the strakelets were absent. As identifled in Section 1.1.3 the onset of steady vortex asymmetry is dependent on the model used as steady vortex asymmetry is as a result of surface imperfections on the model. These imperfections difier from model to model (Champigny, 1994). This study was conducted in order to gain an understanding of the efiect the strakelets had on the steady asymmetric owfleld and is thus of a purely qualitative nature. The strakelets? height, the strakelets? leading edge position and the strakelets? chord length were identifled as the critical factors that would afiect the asymmetric ow- fleld on the body and thus the side force acting on the body (DAS, 2004). Using the 87 full factorial Design of Experiments (DOE) methodology, as described by Barrentine (1999), Table 5.1 was set up. Table 5.1: Coe?cients for Efiects in a 23 Experiment Runs X Y Z 1 - - - 2 + - - 3 - + - 4 + + - 5 - - + 6 + - + 7 - + + 8 + + + In Table 5.1X represents the strakelet height, Y represents the strakelet leading edge position and Z represents the strakelet chord length position. The base 2 represents the two levels that the variable can have, that is the variable can have a high value (?+?) or the variable can have a low value (???) and the exponent 3 represents the number of variables present in the experiment (Barrentine, 1999). The ?+? signs in Table 5.1 indicates a changing variable while the ??? indicates that the variable is held constant. From Table 5.1 it was decided that the following efiects would be investigated: ? The efiect of changing the height of the strakelets on steady asymmetric vor- tices. (Run 2 from Table 5.1) ? The efiect of changing the axial position of the nose strakelets on steady asym- metric vortices. (Run 3 from Table 5.1) The efiect of the strakelet chord length was not considered and thus runs 5 ? 8 are not applicable. Run 1 represents the baseline run with all variables constant. This simulation validates the simulation methodology from Chapter 4. Runs 2 and 3 are discussed Sections 5.1 and 5.2 respectively. Run 4 was not considered as it entails investigating the efiect of varying both the strakelet height and the strakelet leading edge position in a single run. In this way the mutual interaction between the two parameters would be investigated, however in order to perform run 4, runs 2 and 3 are required flrst so that the in uence of the individual parameters can be determined before investigating their mutual in uence. 88 5.1 Efiect of Changing the Span of the Strakelets 5.1.1 Grid generation Figure 5.2 shows the geometry modelled for the CFD simulations. The geometry of the strakelets were similar to that used in the wind tunnel test. Figure 5.2: Geometry of missile with nose strakelets The strakelets used in the experiments had a height of 0:06D, a chord of 0:8D and a width of 0:06D. The leading edge of the strakelets was 1D from the nose tip. Due to the alternating behaviour of steady asymmetric vortices along the length of the missile body, only 8:1D of the missile body was once again modelled. Perturbation I, from Section 4.1 was placed on the nose, a distance of 0:25D away from the nose tip and at a circumferential location of 120?, 30? clockwise from the lateral meridian, to create a steady asymmetric owfleld. 89 Three difierent strakelet heights, denoted by b, were investigated: ? Model A strakelets: b = 0:06D ? Model B strakelets: b = 0:09D ? Model C strakelets: b = 0:13D The leading edge for the three difierent models was flxed at 1D from the nose tip. The structured grid used in this study is shown in Figure 5.3. Figure 5.3: A close view of the structured grid constructed on the missile-strakelet geometry The grid consisted of approximately 280 circumferential grid points extending around the body, 60 radial grid points extending from the body surface to the farfleld bound- ary and 140 axial grid points extending from the tip of the nose to the outlet bound- ary. Grid points in the axial and radial locations near the strakelets were increased so as to capture the owfleld on the strakelets. Simulations were only carried out at an angle of attack of 40? since at this angle of attack steady vortex asymmetries were most pronounced (as shown in Figure 5.1) and thus the efiect of the strakelets on the steady asymmetric owfleld could be easily observed. The Menter-SST turbulence model was used. Roe?s FDS with the Osher-C limiter was used to solve the RANS equations. 90 5.1.2 Results Figure 5.4 shows the pressure distribution on the top surface (leeward side) of the missile bodies. The clean missile body with the geometric perturbation, from Section 4.1, is also included for comparison between the difierent results. (a) Clean missile body with a geometric perturba- tion only (b) Missile body with the model A strakelets (c) Missile body with the model B strakelets (d) Missile body with the model C strakelets Figure 5.4: Surface pressure distributions on the strakelet models with difierent spans at 40? angle of attack The asymmetric surface pressure distributions on the missile bodies with the models A and C strakelets respectively, in Figures 5.4b and 5.4d indicates that the resultant owflelds are asymmetric. The model A strakelets were used in the experimental tests and it resulted in an asymmetric owfleld at 40? angle of attack, as shown in Figure 5.1. This indicates that the CFD simulations do qualitatively match the experimental owflelds. The symmetric surface pressure distribution at the rear of the missile body in Figure 5.4c indicates that the model B strakelets forced the formation of steady symmetric vortices. Figures 5.4c and 5.4d have two sets of low pressure regions on the leeward side of 91 the body. The flrst set of low pressure regions end almost at the trailing edge of the strakelets. This is due to vortex separation ofi the strakelets. The second set of low pressure regions is on the cylindrical part of the missile body. This is where the ow from the bottom strakelets separate on the leeward side of the missile body and feed into the separated primary vortices. In Figure 5.4b the two sets of low pressure regions on the body are joined. Indicating that the two vortices, formed due to ow interaction with the top strakelets, are still attached to the body surface when ow from the bottom strakelets separate on the leeward side and feed into the vortices. Figure 5.5 shows the surface pressure distribution on the leeward side of the missile body, 0:06D on either side of the body centre-line, for the difierent models. This is a numerical representation of the surface pressures, allowing for a clearer identiflcation of the ow asymmetries. 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (a) Clean missile body with a geometric perturba- tion only 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (b) Missile body with the model A strakelets 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (c) Missile body with the model B strakelets 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (d) Missile body with the model C strakelets Figure 5.5: The surface pressure distribution along the length of the body for the strakelet models with difierent spans 92 Figure 5.5b shows that the difierence in surface pressure distribution at the rear of the missile body, between x = 6D and x = 8D, is very similar to the surface pressure difierence at the corresponding axial locations in Figure 5.5a. Flow interacts with the model A strakelets at approximately 1D from the nose tip. The surface pressures on either side of the missile body centre-line are unequal, indicating that the ow is asymmetric when it interacts with the strakelets. The trailing edge of the strakelets are at 1:9D from the nose tip. At this axial location the surface pressures are difierent, indicating that the formed vortices are asymmetric. The difierent surface pressures along the length of the body shows that the model A strakelets were not able to eliminate the asymmetry introduced by the geometric perturbation on the missile nose. In Figure 5.5c it is evident that from an axial location of approximately 3D from the nose tip, the surface pressures on either side of the body centre-line are equal, indi- cating that the resultant owfleld has symmetric vortices. The ow is asymmetric when it interacts with the leading edge of the model B strakelets at 1D, as can be seen by the difierence in surface pressures across the missile body centre-line. At the trailing edge of the strakelets, surface pressures across both halves of the body are difierent but this difierence is smaller and opposite in sign than that in Figure 5.5b at the corresponding axial location. The difierence in surface pressures at the rear of the missile body, between x = 6D and x = 8D in Figure 5.5d is smaller than the difierence at the corresponding axial locations in Figures 5.5a and 5.5b. Even though the ow is asymmetric when it interacts with the leading edge of the model C strakelets, the surface pressures along the length of the body surface from x = 1D to x = 1:9D are almost equal. Thus the ow became symmetric when it interacted with the strakelets. However, as the ow developed along the remainder of the body length, the asymmetry introduced by the geometric perturbation developed, resulting in asymmetric vortices at the rear of the body. Cross-sections of the ow at difierent axial locations on the body are shown in Figures 5.6 to 5.9 using density colour ood maps. These show the ow development along the length of the missile body. Since it is the interaction of the owfleld with the strakelets that is of importance, density colour maps are su?cient. 93 (a )Clea n missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lA stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lC stra kelet s Figur e5.6 :Fl owflel d on th emissil eb odie swit h stra kelet swit h difiere nt span sa tx = 1:4 D at an angl eo fatta ck of 40 ? 94 Figure 5.6 shows ow interaction with the strakelets, 1:4D from the nose tip. The right vortex in Figure 5.6b is larger than the left vortex. Both vortices are still attached to the body. Figure 5.5b shows that at the corresponding axial location, the surface pressure on the left of the body is slightly higher than that on the right, indicating that the left vortex has less contact with the body than the right vortex. Flow is attached to the bottom strakelets. The right vortex on the missile body with the model B strakelets in Figure 5.6c is slightly larger than the left vortex. The higher surface pressure on the left side of the body at x = 1:4D in Figure 5.5c indicates that due to the small size of the left vortex it has less contact with the body than the right vortex, even though both vortices are still attached to the body surface. In Figure 5.6d two vortices of similar size form at the top strakelets. The almost equal surface pressures at x = 1:4D in Figure 5.5d indicates that both vortices are of similar size. The trailing edge of the three difierent strakelets is located at 1:9D. Figures 5.7 to 5.9 show ow development on the missile body behind the strakelets. 95 (a )Clea n missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lA stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lC stra kelet s Figur e5.7 :Fl owflel d on th emissil eb odie swit h stra kelet swit h difiere nt span sa tx = 2:5 D at an angl eo fatta ck of 40 ? 96 In Figure 5.7 the owfleld 0:4D behind the strakelets is shown. In Figure 5.7b the right vortex is smaller than the left vortex and it is further away from the body surface than the left vortex. This results in a higher surface pressure on the right of the missile body centre-line as shown in Figure 5.5b at 2:5D. The ow from the bottom strakelets has moved around the body surface towards the leeward side. The ow on the right is still attached to the side of the body, while the ow on the left has separated and feeds into the left vortex, strengthening it. In Figure 5.7c ow from the bottom strakelets moves around the missile body surface, towards the leeward side. Both vortices that formed at the top strakelets have separated, with the right vortex being larger than the left vortex. The left vortex is slightly further away from the body than the right vortex, resulting in a higher surface pressure on the left side of the body surface, as seen in Figure 5.5c at 2:5D . Flow from the bottom strakelets in Figure 5.7d moves towards the leeward side of the body surface. The right vortex is slightly smaller than the left vortex but both vortices are almost at the same height above the missile body surface, resulting in the almost equal surface pressures on the both halves of the missile body in Figure 5.5d at the corresponding axial location. 97 (a )Clea n missil eb od y wit h a perturbatio n onl y (b )Missil eb od y wit h th em ode lA stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lC stra kelet s Figur e5.8 :Fl owflel d on th emissil eb odie swit h stra kelet swit h difiere nt span sa tx = 2:8 D at an angl eo fatta ck of 40 ? 98 In Figure 5.8b ow separates on the right side of the missile body and feeds into the small, right vortex. The left vortex is larger than the right vortex due to ow separation occurring on the left at x = 2:5D (Figure 5.7b). The owfleld in Fig- ure 5.8b is asymmetric as indicated by the difierent heights of the vortices from the body surface, which results in unequal surface pressures in Figure 5.5b at the corresponding axial location. In Figure 5.8c the right vortex is larger than the left vortex. Both vortices are almost at the same height above the body surface, resulting in the almost equal surface pressures, seen in Figure 5.5c at the corresponding axial location. The ow on both sides of the body separates at the same axial location on the body surface. Therefore both vortices, that formed at the top strakelets will be strengthened at the same axial location on the body surface. Flow around the missile body separates at the same axial position in Figure 5.8d as well. The higher surface pressure on the right of the missile body centre-line in Figure 5.5d, at x = 2:8D indicates that the right vortex is slightly higher than the left vortex, thus the left vortex would be strengthened by the separated ow before the right vortex, resulting in the two vortices growing with difierent strengths along the length of the body. Figure 5.9 shows the ow at the rear of the body, approximately 6D from the nose tip. 99 (a )Clea n missil eb od y wit h a perturbatio n onl y (b )Missil eb od y wit h th em ode lA stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lC stra kelet s Figur e5.9 :Fl owflel d on th emissil eb odie swit h stra kelet swit h difiere nt span sa tx = 6D at an angl eo fatta ck of 40 ? 100 The resultant asymmetric owfleld in Figure 5.9b is very similar to that of the body and geometric perturbation in Figure 5.9a. The left vortex is further away from the missile body than the right vortex, resulting in the unequal surface pressures at x = 6D in Figure 5.5b. In Figure 5.9c the resultant vortices are of similar strengths and are approximately the same distance above the body surface. The surface pressures in Figure 5.5c, at the corresponding axial location, are equal. Thus the model B strakelets were able to eliminate the asymmetry introduced by the geometric perturbation on the nose. The vortex asymmetry in Figure 5.9d is less than the vortex asymmetry in Fig- ure 5.9a. The difierence in height between the two vortices is small, as indicated by the small difierence in surface pressures in Figure 5.5d at x = 6D. 5.1.3 Discussion The ow over the missile nose is forced to a steady asymmetric state by the geometric perturbation, placed 0:25D from the missile nose tip. The leading edge of the three strakelet models are located approximately 0:75D behind the geometric perturbation at x = 1D. Figures 5.5b, 5.5c and 5.5d show that the surface pressures across both halves of the missile body are unequal at x = 1D, thus the ow is asymmetric when it interacts with the leading edges of the three strakelet models. Flowfleld on the missile body with the model A strakelets From the surface pressure distributions on the missile body with the model A strakelets in Figures 5.4b and 5.5b it can be seen that the resultant owfleld at the rear of the missile body, between axial locations of 6D and 8D, is asymmetric. In Figure 5.6b two vortices formed at the top strakelets, with the right vortex larger than the left one. The surface pressures at the corresponding axial location of 1:4D showed that the surface pressure on the left of the missile body centre-line was a little higher than the surface pressure on the right. The lower surface pressure on the right indicates that due to the right vortex being larger than the left vortex, it has more contact with the body surface than the smaller left vortex. The right vortex separates from the body surface flrst, as indicated by the short low pressure line on the right of the body centre-line in Figure 5.4b. In Section 4.2 it was seen that the geometric perturbation on the missile nose at an axial location of 0:25D forced the right vortex to separate flrst from the body surface (Figure 4.20). Thus 101 the model A strakelets have retained the right vortex pattern that was triggered by the geometric perturbation. At x = 1:9D, the trailing edge of the model A strakelets the vortices are asymmetric, with the the right vortex further away from the missile body than the left vortex, as indicated by the higher surface pressure on the right of the missile body centre-line. Behind the trailing edge of the model A strakelets, ow from the bottom strakelets, remains attached to the body surface and moves towards the leeward side of the body surface. In Figure 5.7b, ow separates on the left and feeds into the left separated vortex strengthening it. The ow on the right separates at a difierent axial position on the missile body and feeds into the right separated vortex (Figure 5.8b). The left vortex is strengthened before the right vortex resulting in the asymmetric owfleld in Figure 5.9b. The ow separated on the leeward side, at difierent axial locations because the short model A strakelets were not able to force the ow to separate su?ciently far from the missile body. The owfleld at x = 6D in Figure 5.9b resembles that in Figure 5.9a of the body with the geometric perturbation only. The asymmetric vortices in Figure 5.9b are stronger than the asymmetric vortices in flgure 5.9a, indicated by the difierent colour intensities. Flowfleld on the missile body with the model B strakelets The resultant owfleld on the missile body with the model B strakelets is symmet- ric as indicated by the symmetric surface pressure distributions at the rear of the body in Figures 5.4c and 5.5c. At the trailing edge of the strakelets, x = 1:9D, the difierence in surface pressures across both halves of the body is smaller than the surface pressure difierence at the trailing edge of the the model A strakelets, indi- cating that the resultant owfleld at the trailing edge of the model B strakelets is less asymmetric. The lower surface pressure on the right side of the body centre-line indicates that the right vortex is closer to the body than the left vortex. The short length of the low pressure region on the left of the body centre-line in Figure 5.4c, indicates that the left vortex separated before the right vortex. In Section 4.2, in Figure 4.20 it was seen that the right vortex separated before the left vortex. Thus the model B strakelets were able to produce strong vortices so that the left vortex separated before the right vortex, changing the vortex pattern triggered by the ge- ometric perturbation. For the missile body with the geometric perturbation only, the right vortex separated before the left vortex and with the addition of the model B strakelets, the vortex pattern changed. Flow from the bottom strakelets moves along the body surface, towards the top 102 surface of the body (Figure 5.7c). Both vortices have separated, with the left vortex higher than the right vortex, as indicated by the higher surface pressure on the left in Figure 5.5c at 2:5D. At x = 2:8D the ow on either side of the body centre-line separates and feeds into the separated vortices at the same axial position on the missile body. This is unlike the ow separation for the model A strakelets where the ow on the left separated at x = 2:5D and the ow on the left separated at x = 2:8D. Thus the larger height of the model B strakelets allowed for the bottom strakelet to force the ow to separate su?ciently far from the body, so that ow separation on the leeward side of the missile body would not be in uenced by the asymmetry introduced by the geometric perturbation on the missile nose. The resultant owfleld, 6D away from the nose tip in Figure 5.9c shows that the vortices at the rear of the missile body are symmetric. Flowfleld on the missile body with the model C strakelets It was thought that since the model B with a height of 0:09D was able to force the asymmetrically triggered ow to become symmetric, the model C strakelets, with a height of 0:13D would also force vortex symmetry. The asymmetric surface pressure distribution in Figures 5.4d and 5.5d indicates that this was not the case. However, the difierence in surface pressures across both halves of the body between 6D and 8D is smaller than the difierence in surface pressures on the body and geometric perturbation at the corresponding axial locations in Figure 5.5a. At x = 1:9D, the trailing edge of the model C strakelets, the owfleld is almost symmetric, as indicated by the almost equal surface pressures on the left and right side of the body centre-line at the corresponding axial location in Figure 5.5d. Even though the owfleld was asymmetric when it interacted with the strakelets at ap- proximately 1:3D from the nose tip, the owfleld was made symmetric, as indicated by the equal surface pressures at the corresponding axial location in Figure 5.5d. The ow remained symmetric for the entire length of the model C strakelets up to it?s trailing edge at x = 1:9D. This is unlike the ow on the model A strakelets where surface pressures on the body alternate for the length of the strakelets and the surface pressures on the body with the model B strakelets have an almost constant difierence for the length of the strakelets. The large height of the model C strakelets forced the formation of strong symmetrical vortices only on the strakelets so that the ow on the strakelets remains symmetric. However, at approximately x = 2D the asymmetry, which was not visible on the strakelets, begins to develop. In Figure 5.8d, the ow from the bottom strakelets separates at the same axial 103 position on the body and feeds into the two separated vortices. Due to the larger height of the model C strakelets, the ow from the bottom strakelets were not in uenced by the asymmetry present in the owfleld and thus separated at the same axial position on the body. Since the left vortex is larger and closer to the body than the right vortex, the separated ow on the left will feed into the left vortex before the separated ow on the right feeds into the right vortex. The larger left vortex will be strengthened flrst. Thus the already asymmetric vortices will develop asymmetrically along the remainder of the missile body length, resulting in the asymmetric owfleld seen in Figure 5.9d. Efiect of strakelets on normal force coe?cient An interesting observation was made when the normal force coe?cient, at 40? angle of attack, for the missile bodies with the three strakelet models were compared to that of the missile body with the geometric perturbation only. It was found that the normal force coe?cient for the models with strakelets was less than that of the missile body with the geometric perturbation only. (Table 5.2). Table 5.2: Comparison of normal force coe?cients for the strakelet models with difierent spans The model CN Body and perturbation 4.303 The model A 4.150 The model B 3.879 The model C 4.0550 This phenomenon was also observed in the data obtained experimentally. The result shown below is for a body, strakelets and tail conflguration (Figure 5.10). 104 ?10 0 10 20 30 40 50 60 ?5 0 5 10 15 20 Angle of Incidence (deg) Normal Force Coefficien t Experimental results Body and Tail Body, Tail and Nose Strakelets Figure 5.10: The efiect of the strakelets on the normal force coe?cient This efiect was also noted by Ericsson and Reding (1991). The asymmetric vortex pattern contributed to the normal force on the nose and did not result in it being reduced, as was expected. The vortex induced normal force on the body is due to the movement of steady asymmetric vortices. For steady asymmetric ow one vortex moves further away from the body while the other remains tucked in between the body and the outer vortex, as discussed in Section 1.1.3. This phenomenon results in an increase in normal force and an increase in the interference efiects of the forebody vortices on the aft body tail surfaces. The strakelets force vortex symmetry by lifting the lower asymmetric vortex up and outboard, resulting in a loss of lift (Ericsson and Reding, 1991). The normal force on the missile body with the model B strakelets is lower than that on the missile bodies with the model A or C strakelets, since it was able to lift the inner vortex su?ciently to force vortex symmetry more than the model A and C strakelets. 5.2 Efiect of Changing the Axial Position of the Strakelets 5.2.1 Grid generation The strakelets span was flxed at 0:09D. The leading edge of the strakelets were positioned at three difierent axial (x) locations from the nose. 105 ? The model D: x = 0:5D ? The model B: x = 1D ? The model F: x = 1:3D The model B strakelets from Section 5.1 are used in this study. The structured grid shown in Figure 5.3 was used. As before simulations were only carried out an angle of attack of 40?. 5.2.2 Results Figure 5.11 shows the surface pressure distributions on the missile bodies with the difierent strakelet models. The missile body with the geometric perturbation, from Section 4.1, is also included for comparison between the difierent set of results. (a) Body of revolution with a geometric per- turbation only (b) Missile body with the model D strakelets (c) Missile body with the model B strakelets (d) Missile body with the model E strakelets Figure 5.11: Surface pressure distributions on the difierent strakelet models at dif- ferent axial locations at 40? angle of attack Figures 5.11b and 5.11d show that moving the axial location of the strakelets does not eliminate the asymmetry in the owfleld. In Figures 5.11c and 5.11d small 106 regions of high surface pressure are visible at the trailing edge of the the model D strakelets. These high surface pressure regions are not visible in Figure 5.11b. This indicates that the vortices, that formed as a result of ow interaction with the model D strakelets are still attached to the missile body at the strakelets? trailing edge. The high surface pressure regions in Figures 5.11c and 5.11d indicate that the vortices separate shortly after the trailing edges of the model B and E strakelets. Due to the difierent axial locations of the strakelets? leading edges, the axial location at which the the vortices separate from the missile body difier. Figure 5.12 shows the surface pressure distribution a distance of 0:06D on either side of the body center line for the difierent models. 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 Pressure (Pa ) x/D Left side of centre?line Right side of centre?line (a) [Missile body with a geometric perturbation only 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (b) Missile body with the model D strakelets 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (c) Missile body with the model B strakelets 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 9 10 x 104 x/D Pressure (Pa ) Left side of centre?line Right side of centre?line (d) Missile body with the model E strakelets Figure 5.12: The surface pressure distribution along the length of the missile body with the difierent strakelet models at difierent axial locations Figure 5.12b shows the surface pressure distribution on the body with the model 107 D strakelets. The leading edge of the model D strakelets is located 0:5D from the nose tip. There is a small difierence in surface pressures across both halves of the body, indicating that the ow is asymmetric when it interacts with the model D strakelets. The trailing edge of the the model D strakelets is located 1:3D from the nose tip. At this axial location there is still a difierence in surface pressures across both halves of the body. The surface pressure difierences at the rear of the body, between 6D and 8D on the missile body is smaller than the difierences in surface pressures in Figure 5.12a at the corresponding axial location, indicating that the model D strakelets were able to reduce vortex asymmetry. The difierence in surface pressures at the leading edge of the the model B strakelets is larger than that at the leading edge of the the model D strakelets, indicating that the ow had a higher degree of asymmetry when it interacted with the model B strakelets due to the leading edge of the model B strakelets being 0:75D behind the geometric perturbation. Figure 5.12c is the same as Figure 5:5c in Section 5.1. In Figure 5.12d the difierence in surface pressures across both halves of the body surface at 1:3D is larger than the surface pressure difierences on the bodies with the model D and B strakelets in Figures 5.12b and Figure 5.12c respectively. This indi- cates that the ow had a greater degree of asymmetry before it interacted with the model E strakelets. The ow remains asymmetric along the length of the strakelets and along the length of the body indicating that the model E strakelets were not able to force the owfleld to become symmetric. Density contour plots at difierent axial locations along the body are shown in Figures 5.14 to 5.18 to show ow development at difierent cross-sections along the body length. 108 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.13 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 0:5 5D at an angl eo fatta ck of 40 ? 109 The owflelds on the missile bodies with models B and E in Figures 5.13c and 5.13d respectively, are the same as that in Figure 5.13a. The leading edge of the model B strakelets is located at 1D away from the nose tip and that of the model E strakelets is located 1:3D away from the nose tip. Figure 5.13b shows ow interaction with the the model D strakelets. The ow is asymmetric when it interacts with the model D strakelets as indicated by the difierence in surface pressures in Figure 5.12d at 0:55D. 110 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.14 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 1:2 D at an angl eo fatta ck of 40 ? 111 In Figure 5.14b the right and left vortices are of similar sizes and are still attached to the missile body. The surface pressure on the right of the missile body centre- line in Figure 5.12b, at 1:2D is slightly higher than that on the left of the missile body centre-line. This indicates that the right vortex is in slightly less contact with the missile body surface than the left vortex and will thus separate before the left vortex. Since the geometric perturbation forced the right vortex to separate flrst, as shown in Section 4.2, the model D strakelets have not changed the vortex pattern in the owfleld. Flow from the bottom strakelets move towards the leeward side of the missile body. In Figure 5.14c the owfleld 0:1D after the leading edge is shown. The left vortex is slightly smaller than the right vortex. Thus the left vortex has less contact with the missile body surface, resulting in the higher surface pressure on the left of the missile body centre-line, as shown in Figure 5.12c at x = 1:2D. The geometric perturbation introduces an asymmetry on the right of the body centre-line, as shown in Section 4.2 in Figure 4.20. Since the left vortex has less contact with the body, it will separate before the right vortex. Thus the model B strakelets changed the vortex pattern introduced by the geometric perturbation. The ow in Figure 5.14d resembles that in Figure 5.14a since the leading edges of the model E strakelets are located 1:3D from the nose tip. Both body vortices are still attached to the body. The surface pressure difierences in Figure 5.12d shows that the ow is asymmetric before it interacts with the leading edge of the the model E strakelets. 112 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.15 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 1:5 D at an angl eo fatta ck of 40 ? 113 Figure 5.15b shows the owfleld behind the the model D strakelets. Flow from the bottom strakelets moves around the missile body surface and separates on the lee- ward side of the body. The separated ow feeds into the two attached vortices. Flow separation has occurred on both sides of the missile body at the same axial location, indicating the two attached vortices will be strengthened at the same axial position on the missile body. The asymmetry, introduced by the geometric perturbation, is not evident in Figure 5.15b, however in Figure 5.12b, at x = 1:5D the surface pressure on the right of the body centre-line is higher than that on the left. This asymmetry will develop along the length of the body resulting in the formation of asymmetric vortices. Thus the vortices, produced by the model D strakelets were not able to produce strong vortices to absorb the asymmetry introduced by the geometric perturbation on the missile nose. In Figure 5.15c the left vortex has separated from the missile body before the right vortex, resulting in a higher surface pressure on the left of the body centre-line in Figure 5.12c at the corresponding axial location. In Figure 5.15d two well-deflned body vortices interact with the model E strakelets. Since the vortices are well formed upon interaction with the strakelets, the vortices are forced to separate. The vortex on the right is further away from the body surface than the left vortex in Figure 5.15d. This is further conflrmed by the higher surface pressure on the right of the body centre-line in Figure 5.12d at x = 1:5D which indicates that the right vortex is slightly higher than the left vortex from the missile body surface. The left vortex pattern, introduced by the geometric perturbation, has not changed, since the right vortex will separate flrst from the body surface. 114 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.16 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 2:3 D at an angl eo fatta ck of 40 ? 115 Figure 5.16b shows the formation of two almost symmetric vortices. Figure 5.12b shows that the surface pressure on the right is slightly higher than that on the left at 2:3D from the nose tip, indicating that the right vortex has less contact with the missile body surface than the right vortex. Since the difierence in surface pressures on both halves of the body, in Figure 5.12b, is small it is di?cult to see the asymmetry in Figure 5.16b. Figure 5.16c shows the ow behind the model B strakelets. The left vortex is further away from the missile body surface than the right vortex. This results in the higher surface pressure on the left in Figure 5.12c at 2:3D. Flow from the bottom strakelets move towards the leeward side of the body. The right vortex in Figure 5.16d has less contact with the missile body than the left vortex. Thus the surface pressure on the right of the body centre-line is higher than that on the left in Figure 5.12d at the corresponding axial location. Flow is still attached to the bottom strakelets. 116 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.17 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 3:4 D at an angl eo fatta ck of 40 ? 117 In Figure 5.17b the right vortex is slightly smaller than the left vortex, indicating that it is further away from the body surface than the left vortex. This is conflrmed by the higher surface pressure on the right of the missile body centre-line in flg- ure 5.12b at 5:5D. The two vortices will continue to develop asymmetrically along the length of the missile body resulting in an asymmetric owfleld, as indicated by the surface pressure distribution in Figure 5.11b and 5.12b. Figure 5.8c in Section 5.1 shows that ow from around the body separates at the same axial position of 2:8D on the missile body. At this axial position both vortices are at the same height above the body surface, as indicated by the equal surface pressure on the missile body at 2:8D in Figure 5.12c. Thus the separated vortices are strengthened at the same axial position on the body resulting in the formation of two symmetric vortices, shown in Figure 5.17c. The surface pressures on either side of the missile body centre-line are equal at this axial location in Figure 5.12c. Figure 5.17d shows that ow from around the body separates at the same axial position along the missile body length. However, the asymmetry introduced by the geometric perturbation has in uenced ow separation since the separating ow on the left is stronger than that on the right. Thus the height of the model E strakelets was not large enough to strengthen the ow such that the separating ow would not be in uenced by the asymmetry present in the owfleld. Since the left vortex is closer to the body surface, as indicated by the lower surface pressure on the left in Figure 5.12d, the left vortex will be strengthened before the right vortex. This results in the vortices developing asymmetrically along the remainder of the missile body length. Figure 5.18 shows the owfleld at the rear of the missile body. 118 (a )Missil eb od y wit h a geometri cp erturbatio n onl y (b )Missil eb od y wit h th em ode lD stra kelet s (c )Missil eb od y wit h th em ode lB stra kelet s (d )Missil eb od y wit h th em ode lE stra kelet s Figur e5.18 :Fl owflel d on th emissil eb odie swit h difiere nt stra kele tm odel sa tx = 6:9 D at an angl eo fatta ck of 40 ? 119 In Figure 5.18b the difierence in height between the two vortices is small when compared to the difierence in height between the two vortices in Figure 5.18a. This indicates that the degree of asymmetry present in the owfleld in Figure 5.18b is less than that in Figure 5.18a. Thus moving the strakelets closer to the nose allows for a reduction in vortex asymmetry. The resultant owfleld in Figure 5.18c is symmetric. Both vortices are similar in size and are the same height above the body surface. The asymmetric owfleld in Figure 5.18d is very similar to that in Figure 5.18a. The surface pressure difierence across both halves of the body in Figure 5.12d at 6:9D is very similar to the surface pressure difierence in Figure 5.18d at the corre- sponding axial location. Thus moving the strakelets further back does not result in a symmetric owfleld. 5.2.3 Discussion Since the leading edge of the three strakelet models are located at difierent axial locations on the missile body surface, the ow asymmetry on the body is at difierent stages in its development along the length of the body, when it interacts with the each of the three models. Missile body with the model D strakelets A distance 0:15D after the ow has been triggered to become asymmetric by the geometric perturbation that was placed 0:25D from the nose tip, the ow interacts with the leading edge of the model D strakelets. The surface pressure on the right side of the body centre-line is higher than the surface pressure on the right in Fig- ure 5.12d at 0:5D, indicating that the ow was asymmetric when it interacted with the strakelets. Figure 5.12b shows that the difierence in surface pressure across both halves of the missile body, at the leading edge of the model E strakelets, was less than the surface pressure difierence at the leading edge of the model B strakelets, at x = 1D in Figure 5.12c. The ow remains asymmetric along the length of the strakelets, as indicated by the surface pressure difierences in Figure 5.12b from x = 0:5D to x = 1:3D. At the trailing edge of the strakelets the vortices are still attached to the body, as shown by the low pressure regions in Figure 5.11b. Figure 5.12b shows that the surface pressure on the right is higher than that on the left, thus the right vortex is in less contact with the missile body than the left 120 vortex. Behind the trailing edge of the strakelets, ow from the bottom strakelets moves, along the missile body surface and separates at the same axial position on the leeward side of the body at x = 1:5D, as shown in Figure 5.15d. The sepa- rated ow feeds the two attached vortices on the leeward side of the body. The ow on both sides of the body separated at the same axial position indicating that the height of the model D strakelets was su?cient to strengthen the ow such that it was unafiected by the asymmetry present in the owfleld. At the corresponding axial location in Figure 5.12b, the higher surface pressure on the right of the body centre-line implies that the right vortex is in less contact with the body. The low surface pressure regions on the missile body surface in Figure 5.11b indicates that the right vortex separates shortly before the left vortex. Therefore the asymmetry, introduced by the geometric perturbation on the nose is still present as shown in Figure 5.18b. The vortices produced by the top strakelets of the model D were not able to eliminate the asymmetry present in the owfleld. Missile body with the model E strakelets The leading edge of the model E strakelets was positioned 1:3D from the nose tip. At this axial location the surface pressure on the right of the missile body centre- line is less than the surface pressure on the left of the body centre-line, as seen in Figure 5.12d. Thus the ow is asymmetric when it interacts with the strakelets. For the length of the strakelets, from x = 1:3D to x = 2:1D, the surface pressure on the right of the body centre-line remains higher than the surface pressure on the left, indicating that at the trailing edge of the model E strakelets the right vortex was slightly higher than the left vortex from the missile body. In Figure 5.15d the high density region under the vortex illustrates that the right vortex is forced further away from the missile body. In Figure 5.16d two well deflned vortices form. The right vortex is slightly larger than the left vortex and is slightly higher than the left vortex. In Figure 5.17d the ow from the bottom strakelets separates from both sides of the missile body at 3:4D from the nose tip, illustrating that the model E strakelets were able to strengthen the ow so that the asymmetry, triggered by the geometric perturbation, did not in uence the ow as it did for the model A strakelets, where ow separation occurred at two difierent axial locations (Section 5.1). At 3:4D, in Figure 5.12d the surface pressure on the right is higher than on the left, thus the right vortex is further away from the missile body than the left vortex. The left vortex would be strengthened before the right vortex and the two vortices would develop asymmetrically along the length of the body, resulting in the owfleld in Figure 5.18d. 121 5.3 Efiect of Strakelets on a Steady Asymmetric Flow- fleld The model A strakelets had a height of 0:06D. The model A strakelets on the lee- ward side of the body were not able to produce strong enough vortices to eliminate the asymmetry in the owfleld. The asymmetry, introduced by the geometric per- turbation on the nose in uenced the ow on the missile body surface by forcing the right vortex to separate before the left vortex. At the trailing edge of the model A strakelets the right vortex is higher than the left vortex, and since the geometric perturbation introduced an asymmetry on the right of the body centre-line (Section 4.2), the right vortex is pushed further away from the missile body. The ow from the bottom strakelets was also not strong enough to absorb the efiect of the asym- metry in the owfleld, thus forcing ow separation at two difierent axial locations on the missile body. Since the ow on the left separated before the ow on the right, the left vortex was strengthened before the right vortex. This resulted in the two asymmetric vortices developing asymmetrically along the rest of missile body. The model C strakelets were large enough to produce strong symmetric vortices which were not in uenced by the asymmetry introduced by the geometric pertur- bation on the nose. At the trailing edge of the model C strakelets two symmetric vortices had formed, however, the asymmetry introduced by the perturbation was still present in the owfleld. This resulted in the right vortex being pushed further away from the missile body. The ow from the bottom strakelets separated at the same axial location on the body. However, since the left vortex was closer to the missile body than the right vortex it was strengthened by the left separated ow before the right vortex was strengthened by the ow separating on the right side of the missile body. Since the vortices were strengthened asymmetrically, the vortices developed asymmetrically along the rest of the body length resulting in an asym- metric owfleld. Even though the ow separated from around the body at the same axial position since the vortices were at difierent heights above the missile body surface one vortex was strengthened before the other. Thus the resultant owfleld was made less asymmetric. The model D strakelets were place 0:25D behind the geometric perturbation, thus the asymmetry in the owfleld had not been allowed to develop for a distance on the missile body before it interacted with the leading edge of the model D strakelets. The ow remained attached to the missile body for the length of the the model D strakelets. The asymmetry introduced by the geometric perturbation was still present in the ow at the trailing edge of the model D strakelets even though the 122 ow was still attached to the body. Flow from the bottom strakelets separated at the same axial location on the body and strengthened the two attached leeward vortices symmetrically. However, since the asymmetry introduced by the geometric perturbation was still present the vortices developed asymmetrically along the mis- sile body. The presence of the model strakelets on the missile body did reduce the vortex asymmetry, since the ow interacted with the strakelets shortly after being forced into a steady asymmetric state. The height of the model D strakelets was not su?cient to produce strong vortices to eliminate the asymmetry. The model E strakelets were placed 0:15D behind the perturbation. Thus when the ow interacted with the model E strakelets two well deflned body vortices had already formed, forcing vortex separation close to the leading edge of the strakelets. The asymmetry, introduced by the geometric perturbation was allowed to develop for a distance of 1:3D along the body before it interacted with the model E strakelets. At the trailing edge of the model E strakelets the right vortex was further away from the body than the left vortex. The ow from the bottom strakelets separated at the same axial location on the missile body with difiering strengths. Since the asymmetry was allowed to develop for a considerable distance along the length of the missile body, it in uenced the strength of the ow separating around the missile body. Since the left vortex was closer to the body than the right vortex and the separated ow on the left was stronger than that on the right, the left vortex was strengthened before the right vortex. This resulted in the vortices developing asymmetrically along the length of the body. At the trailing edge of the model B strakelets the left vortex was further away from the missile body than the right vortex. Thus the model B strakelets changed the asymmetry from the right of the missile body to the left, forcing the left vortex to separate before the right one. Since the geometric perturbation introduced the asymmetry on the right, as the two vortices developed along the length of the body, the right vortex that was closer to the body was pushed further away from the body, such that the two vortices were approximately the same distance from the missile body when ow separated from the sides of the missile body. The bottom strakelets had strengthened the ow such that the ow was not in uenced by the asymmetry in the owfleld and thus separated at the same axial location on the body surface. Since the two separated vortices were at the same height above the body when the ow separation took place, both vortices were strengthened at the same axial position, resulting in the formation of two symmetric vortices. The two vortices developed symmetrically along the rest of the body length and at the base of the missile body, both vortices were at the same height above the missile body. 123 5.4 Conclusion The strakelets with a span of 0:06D, at an axial location of 1:1D did not produce strong enough vortices to eliminate the owfleld asymmetry introduced by the geo- metric perturbation. The strakelets with a span of 0:13D, at an axial location of 1:1D did result in a reduction in ow asymmetry, but were unable to eliminate the asymmetry. The strakelets at an axial location of 0:5D, with a span of 0:09D were not large enough to force the formation of strong vortices to eliminate ow asymmetry but the strakelets were able to reduce ow asymmetry. The strakelets at an axial location of 1:8D, with a span of 0:09D did not reduce or eliminate ow asymmetry. Placing the strakelets too far back allows ow asymmetry to develop, thus making it di?cult to eliminate the asymmetry. The strakelets with a span of 0:09D at an axial location of 1:1D was able to eliminate the asymmetry present in the owfleld. The strakelets resulted in the formation of vortices which were strong enough to force symmetry. Placing the leading edge of the strakelets where the pressure difierence is the greatest, yielded the best results. If the leading edge of the strakelets are placed close to the axial position at which the surface pressure distribution is the greatest, the asymmetry in the owfleld is not allowed to develop along the length of the body. The strakelets would produce vortices that would possibly be stronger than the asymmetry introduced by the geometric perturbation on the nose. However, the span of the strakelets also plays a role in determining if the resultant owfleld is symmetric. Increasing the span of the strakelets does result in the strakelets producing suf- flciently strong vortices to absorb the efiect of the asymmetry. However, if the strakelet span is increased by too much, and if the length of the strakelets is too small the asymmetry could develop along the length of the body behind the strakelets trailing edge. Changing the height and the leading edge of the strakelets, and thus indirectly changing the strakelet?s trailing edge position, has shown that the steady asymmetric owfleld at high angles of attack is very sensitive to the three parameters. When the strakelet height was increased to 0:09D, while the leading edge position was 124 flxed at 1:1D and the chord length was flxed at 0:7D the steady asymmetry in the owfleld was eliminated, showing that the three parameters were in the right combination. However, once the strakelet height was increased further to 0:13D the owfleld remained asymmetric, but the steady asymmetry had been reduced. The increase in strakelet height resulted in a change in combination of the three variables and thus the combined efiects of the variables were not able to eliminate the owfleld asymmetry. This shows that if the strakelet height changes the other two parameters must change as well to obtain a symmetric owfleld. The sensitivity of changing the strakelet parameters was once again illustrated when the leading edge position of the strakelets were changed while keeping the strakelet height and strakelet chord length constant. Once the combinations of the three parameters were changed the owfleld became asymmetric. The steady asymmetry of the owfleld was increased when the strakelets were moved further away from the geometric perturbation and reduced when the strakelets were moved closer to the geometric perturbation but was not eliminated since the other two parameters were not in the right proportion. The efiect of changing the strakelet chord length was not investigated in this study but it is recommended that its efiect must be investigated. However, by the flndings in this study that the three parameters need to be in the right proportion to obtain a symmetric owfleld, it is suggested that when investigating the efiect of the strakelet chord length, the strakelet height and the strakelet chord length must be varied, as per run 8 in Table 5.1. Each of the three strakelet geometry parameters contributes in forcing symmetry on a steady asymmetric owfleld. 125 6 Conclusions and Recommendations 6.1 Conclusions Experimental tests conducted by Gobey (2004) on a body-strake conflguration (a body of revolution with a very low aspect ratio wings), revealed that there was a dramatic difierence in normal force and pitching moment coe?cients generated by the missile strakes at difierent roll orientations. When the missile strakes were orientated in the ??? conflguration the normal force coe?cients generated by the missile strakes were very small when compared to that generated by the strakes in the ?+? orientation. The low normal force coe?cients indicated that the missile strakes were not forcing the ow to separate from the body when they were in the ??? orientation. Thus in order to force the ow to separate su?ciently far from the body, and thus improving the aerodynamics of the body-strake conflguration when the missile strakes are in ??? orientation, the missile strake span was increased from 0:06D to 0:13D. When the shorter strakes were orientated in the ??? ow around the body reattached itself to the body once it had interacted with the bottom strakes. This resulted in very low values of normal force coe?cients, and thus pitching moment coe?cients at low angles of attack. The higher normal force coe?cients generated by the strakes when they were in the ?+? orientation was due to the ow not reattaching itself to the body. The higher strakes (strake span = 0:13D) increased the efiective diameter of the missile body and thus ow around the body did not reattach to the body. This resulted in the formation of stronger vortices and thus an increase in the normal force generated by the strakes at the low angles of attack. From the work of Chapter 2 it can be concluded that an increase in the strake span results in greater ow separation around the body. The increase in strake span resulted in an increase in normal force coe?cient ranging from 11:28% at 30? to 42% at 10?. The increase in normal force 126 resulted in a corresponding increase in pitching moment coe?cient ranging from 43% at 30? to 52% at 20? angle of attack. Thus increasing the strake span resulted in an increase in the normal force coe?cient generated by the strakes due to the prevention of ow reattachment. Degani and Schifi (1991) found that to create a steady asymmetric owfleld in CFD it was necessary to introduce a time-invariant, space-flxed perturbation near the apex of an ogive cylinder. Degani and Schifi (1991) suggested the use of a geometric bump on the nose of the body of revolution or a small jet owing normal to the body of revolution. Levy et al. (1990) found that the essential steady asymmetric multi-vortex structure could be qualitatively captured by the use of a simple simu- lated disturbance. The methods suggested by Degani and Schifi (1991) were used as guidelines since they studied the steady asymmetric owfleld on a pointed slender body and a blunt ogive body of revolution was used in this study. A geometric perturbation, as suggested by Degani and Schifi (1991), was placed on the missile nose to create steady asymmetric vortices on the missile body. An iterative process using three difierent sized perturbations was carried out to determine the size of the perturbation required to create an asymmetric owfleld that would be repre- sentative of that on a blunt ogive body. The efiect of the axial and circumferential position of the chosen geometric perturbation on steady asymmetric vortices was also investigated. All three perturbations forced a steady asymmetric owfleld on the missile body. The primary and secondary vortices were adequately captured. It was found that the degree of asymmetry introduced into the owfleld was dependent on the size of the perturbation used, with the largest perturbation resulting in the highest degree of ow asymmetry, as deflned by Levy et al. (1990). The smallest perturbation, perturbation I, with a height of 0:03D, a length of 0:06D and a width of 0:02D resulted in an asymmetric owfleld most representative of that on a blunt ogive body. This perturbation was used in further studies. It was also found that when the geometric perturbation was removed from the missile body once a steady asymmetric owfleld had been obtained, the owfleld returned to its steady symmetric state. This was similar to the flndings of Degani and Schifi (1991) and conflrmed that the asymmetry introduced by the geometric perturbation on the missile nose was amplifled along the length of the body by a convective instability. When the geometric perturbation was placed on a full length missile body, the multi-vortex structure of steady asymmetric vortices was also captured. Placing the geometric perturbation at the tip of the missile body, as suggested by 127 Degani (1992) did not result in an asymmetric owfleld. It was found that by placing the geometric perturbation further away from the missile nose tip, steady asymmetric vortices formed, with the degree of asymmetry increasing with an increased distance from the missile nose tip. However, placing the geometric perturbation too far down the length of the missile body could result in the formation of asymmetric vortices away from the body, which is not realistic. By placing the geometric perturbations at two difierent axial positions at the same circumferential position resulted in the two difierent states of steady asymmetric vortices. The perturbation used in this study introduced the right vortex pattern, as deflned by Xuei et al. (2000). Changing the circumferential position of the perturbation did not have any visible efiect on the state and strength of the formed asymmetric owfleld. In order to eliminate the side forces that develop on a body of revolution due to the formation of steady asymmetric vortices, four forebody strakelets were placed, in the ??? orientation, on the missile nose, behind the geometric perturbation. The objective of the strakelets was to force the formation of symmetric vortices thus eliminating side force development on the missile body. Strakelets with a span of 0:09D, placed 1D from the missile nose tip, forced the formation of symmetric vortices. This is due to the leading edge of the strakelets being placed close to the point at which the largest difierence in surface pressures existed. The strakelets were also large enough to produce strong vortices to eliminate the asymmetry introduced by the geometric perturbation by changing the vortex pattern introduced by the strakelets. Decreasing the span of the strakelets to 0:06D did not change the vortex pattern introduced by the geometric perturbation as the vortices produced by the shorter strakelets were not strong enough to eliminate the asymmetry in the owfleld. In- creasing the span of the strakelets to 0:13D did not have the desired efiect of elimi- nating the asymmetry introduced by the geometric perturbation. For as long as the ow interacted with the strakelets the formed vortices were symmetric. However, the ow behind the strakelets trailing edge developed asymmetrically resulting in the formation of asymmetric vortices. Placing the strakelets with a span of 0:09 closer to the nose tip reduced the asymme- try introduced by the geometric perturbation. This is due to the vortices, produced by the strakelets not being strong enough to absorb the efiect of the geometric per- turbation. Placing the strakelets leading edge further away from the nose tip did not 128 eliminate the asymmetry introduced by the geometric perturbation, as the asym- metry was allowed to develop in the owfleld. Placing the strakelets close to the point at which the difierence in surface pressures is the largest has the best efiect of reducing the asymmetry in the owfleld. It was found that by placing the leading edge of the strakelets close to the axial position at which the largest surface pressure difierence was experienced, resulted in the formation of symmetric vortices. The asymmetry, introduced by the geometric perturbation, was not allowed to develop along the length of the body since the vortices produced by the strakelets absorbed the efiect of the asymmetry. The span of the strakelets also played an important role in determining if the re- sultant owfleld was symmetric. Increasing the strakelet span does result in the strakelets forcing ow to separate su?ciently far from the body, thus resulting in the formation of strong vortices that absorb the efiect of the asymmetry introduced by the geometric perturbation on the missile nose. However, if the strakelet?s span is increased by too much and the length of the strakelets is too small, the asymmetry could develop in the owfleld behind the trailing edge of the strakelets. This study showed that the steady asymmetric owfleld on a body at high angles of attack is very sensitive to the combination of both geometric strakelet param- eters that were tested. An increase in the strakelet height from 0:06D to 0:09D, while keeping the chord length constant at 0:8D and the leading edge position of the strakelet flxed at 1D, resulted in a steady symmetric owfleld. By increasing the strakelet height to 0:09D the strakelet chord length, strakelet height and strakelet leading edge position were in the right proportion to one another and thus were efiective in forcing a symmetric owfleld. However, once the strakelet height was increased further, or the leading edge position of the strakelet was moved, the three parameters were not in the right proportion to each other and the resultant ow- fleld remained asymmetric. This showed that small changes to strakelet parameters resulted in a steady asymmetric owfleld of a higher or lesser degree than when no strakelets were present on the body. Thus each of the three strakelet geometry parameters contributes in forcing symmetry on a steady asymmetric owfleld. 129 6.2 Recommendations In creating the asymmetric owfleld in CFD, a hexahedral block was used. Studies should also be conducted to determine the efiect of using a difierent shaped geometric perturbation. The study conducted in this research has only focused on the efiect of changing the span and axial location of the strakelets, when orientated in the ???. Studies should also focus on changing the chord length and the width of the strakelets. A similar study, with respect to changing the span, the axial position, the chord length and width of the strakelets should also be carried when the strakelets are in the ?+? orientation. A further study should also be carried to determine the efiect of changing the strakelets geometry, possibly investigating the efiect of a delta-shaped strakelet. The Menter-SST k ? ! turbulence model was used in this study. The e?ciency of using the large eddy simulation (LES) turbulence model or the detached eddy simulation (DES) turbulence model should also be investigated. 130 REFERENCES J.E. Bardina, P.G. Huang, and T.J. Coakley. Turbulence Modelling Validation, Testing and Development. NASA Technical Memorandum, 1997. L.B. Barrentine. An Introduction to Design of Experiments: A simplifled ap- proach. ASQ Quality Press, Wisconsin, 1999. J.E. Bernhardt and D.R. Williams. Proportional Control of Aysmmetric Fore- body Vortices. AIAA Journal, 36(11):2087{2093, November 1998. CFDRC. CFD-FASTRAN Theory Manual version 2003. CFD Research Cor- poration, 2003. P.J. Champigny. Stability of Side Forces on Bodies at High Angles of Attack. AIAA Paper 86-2176, 1986. P.J. Champigny. Side Forces at High Angles of Attack. Why, When, How? La Recherche A?erospatiale, 4:269{282, 1994. CSIR-Defencetek. Confldential. Technical report, CSIR, 2004. R.M. Cummings, J.R. Forsthye, S.A. Morton, and K.D. Squires. Computational Challenges in High Angle of Attack Flow Prediction. Progress in Aerospace Sciences, 39:369{384, 2003. DAS. Confldential. Technical report, Denel Aerospace Systems, 2004. D. Degani. Instabilites of Flows over Bodies at Large Incidence. AIAA Journal, 30(1):94{100, January 1992. D. Degani and Y. Levy. Asymmetrical Turbulent Vortical Flows over Slender Bodies. AIAA Journal, 30(9):2267{2273, September 1992. D. Degani and L.B. Schifi. Numerical Simulation of the Efiect of Spatial Dis- turbances on Vortex Asymmetry. AIAA Journal, 29(2):344{352, February 1991. P.C. Dexter. High Angle of Attack Missile Aerodynamics. Journal of Aerospace Engineering, 207:15{19, 1993. 131 ERCOFTAC. Best Practice Guidelines, Version 1.0. Technical report, Eu- ropean Research Community On Flow, Turbulence and Combustion, January 2000. L.E. Ericsson and J.P. Reding. Tactical Missile Aerodynamics, volume 141, chapter Chapter 10: Asymmetric Vortex Shedding on a Slender Body of Revolu- tion, pages 391{452. Progress in Astronautics and Aeronautics, second edition, 1991. ESDU. Normal force, pitching moment and side force of forebody cylinder com- binations for angles of attack up to 90 degrees and Mach numbers up to 5-esdu 89014. Technical report, Engineering Science Data Unit, 1989. J.E. Fidler. Active Control of Asymmetric Vortex Efiects. Journal of Aircraft, 18(4):267{272, April 1981. J.E. Fidler and M.C. Bateman. Asymmetric Vortex Efiects on Missile Conflg- urations. Journal of Spacecraft and Rockets, 12(11):674{681, November 1975. S.G. Gobey. Examination of Historical Wind Tunnel Testing Data. Internal document, Denel Aerospace Systems, 2004. G.Zilliac, D. Degani, and M. Tobak. Asymmetric Vortices on a Slender Body of Revolution. AIAA Journal, 29(5):667|675, May 1991. P.M. Hartwich, R.M. Hall, and M.J. Hemsch. Navier-Stokes Computations of Vortex Asymmetries Controlled by Small Surface Imperfections. Journal of Spacecraft and Rockets, 28(2):258|264, February 1990. Y. Levy, D. Degani, and A. Seginer. Graphical Visualisation of Vortical Flows by Means of Helicity. AIAA Journal, 28(8):1347{1352, August 1990. Y. Levy, L. Hesselink, and D. Degani. Systematic Study of Correlation between the Geometrical Disturbances and Flow Asymmetrices. AIAA Journal, 34(4): 772{777, April 1996. P. Luo, W. Yankai, and L. Degani. Flowfleld around Ogive/Elliptic Tip Cylinder at High Angles of Attack. AIAA Journal, 36(10):560{566, October 1998. F.R. Menter. Ten Years of Experience with the SST Turbulence Model. Turbu- lence, Heat and Mass Transfer, 4, 2003. T.T. Ng. Efiect of a Single Strake on the Forebody Vortex Asymmetry. Journal of Aircraft, 27(9):844{846, September 1990. 132 T.T. Ng. Efiect of a Nose-Boom on Forebody Vortex Control. Journal of Air- craft, 29(4):738{741, July-August 1992. T.T. Ng and G.N. Malcolm. Forebody Vortex Control using Small Rotatable Strakes. Journal of Aircraft, 29(4):671{678, July-August 1992. D.M. Rao, C. Moskovitz, and D.G. Murri. Forebody Vortex Management for Yaw Control at High Angles of Attack. Journal of Aircraft, 24(4):248{254, April 1987. J.L. Thomas and P.M. Hartwich. Tactical Missile Aerodynamics: Prediction Methodology, volume 142, chapter Navier-Stokes Analyses of Flows over Slender Airframes, pages 561{648. Progress in Astronautics and Aeronautics, second edition, 1991. D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, second edition, 2000. C. Xuei, D. Xueying, W. Yankui, L. Peiqing, and G. Zhifu. In uence of Nose Perturbations on Behaviours of Asymmetric Vortices over Slender Body. AIAA Journal, 38(4):358{388, April 2000. D. Xueying, W. Gang, C. Xuerui, W. Yankui, L. Peiqing, and X. Zhongxiang. A Physical Model of Asymmetric Vortices Flow Structure in Regular State over Slender Body at High Angle of Attack. Science in China, 1991. C.C. Yuan and R.M. Howard. The Efiects of Forebody Strakes on a Missile. Journal of Spacecraft and Rockets, 28(4):411{417, July-August 1991. 133