Three-dimensional shock wave reﬂection transition in steady ﬂow
This work details ﬁndings of an analytical, numerical and experimental investigation into the physical nature of three-dimensional shock reﬂection transition. Steady ﬂow shock reﬂections comprise two types: regular reﬂection and Mach reﬂection. Reﬂection studies have previously been conducted using double-wedge symmetrical test piece conﬁgurations. It had been found by previous researchers that the expansion waves resulting from the side edges of the wedges would inﬂuence the reﬂection plane. The three-dimensional nature of real experimental ﬂows gives rise to there generally being a coexistence of regular reﬂection (at the central portions) and Mach reﬂection (towards the outer peripheral portions) in between which transition occurs. It is the object of this work to understand three-dimensional transition for ﬂow ﬁelds in which edge eﬀects do not inﬂuence the reﬂection plane. Specially modiﬁed test piece geometry was developed for this purpose. Experimental tests were required for validation of the numerical models of the ﬂow ﬁeld. This was achieved by obtaining oblique shadowgraphs with optical orientation in both yaw and roll to assist in visualising the three-dimensional features of the ﬂow ﬁeld. These were compared with numerically reconstructed images at the same oblique orientations using a novel reconstruction technique. The main objective of this work was to identify the degree of correspondence of the threedimensional transition conditions to those of two-dimensional ﬂows. This was facilitated by means of reducing the three-dimensional analysis to an eﬀective two-dimensional one. It was found that the three-dimensional transition points occur at a higher eﬀective angle than predicted by twodimensional criteria, and tend towards two-dimensional criteria at reduced free-stream Mach numbers and increased model geometrical spreads. Another important aspect of this work was the nature of the intersection line in the vicinity of the transition point, i.e., the point of impingement of the incident wave and its Mach surface on the horizontal symmetry plane in between the test pieces. Here it was found that a cusp exists in the sweep proﬁle of the intersection line at the transition point. This was proved from a theoretical standpoint based on a model developed for the analysis of the ﬂow in the vicinity of transition. Evidence of this from the numerical and experimental results is given as well. Higher geometrical spreads and lower free-stream Mach numbers were found to create ﬂatter intersection line proﬁles at the horizontal symmetry plane on which the transition points were located further forward towards the apex of this line and which gave rise to greater transverse deﬂections for the streamlines passing through the regular reﬂection portions. Further discussion revolves around the nature of the shear and Mach surfaces. The Mach surface heights (representative of the triple line trajectories) are shown to increase monotonically. The shear layer edge trajectory, which originates at the sweep cusp, was found to show considerable transverse divergence but in keeping with the nature and extent of the transition cusp sweep differential, which in some cases was found to be large enough to cause a strong shock solution for the Mach reﬂection portion. In this case the shear surface edge trajectory diverted from trends seen for other models. The nature of the shear surface as a whole revealed interesting insights into the negative triple conﬁguration of the Mach reﬂection portions which comprised of convex Mach surfaces. This is in contrast to the geometry obtained for the Mach surface for full Mach reﬂection numerically studied with a highly-spread geometry. Here, the ﬂat Mach surface was found to increase monotonically towards the periphery in contrast to what was found for ﬂows with edge inﬂuences on the Mach surface. It is suggested that this is what precludes complex reﬂection (central Mach reﬂection, transitioning to regular reﬂection further out, with a further transition to Mach reﬂection at the periphery) from being obtained in such ﬂow ﬁelds.
A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulﬁlment of the requirements for the degree of Doctor of Philosophy,2018