Mach Reflection in Steady Flow. I. Mikhail Ivanov's Contributions, II. Caltech Stability Experiments

To honor the memory of our friend and colleague Mikhail Ivanov a review of his great contributions to the understanding of the various phenomena associated with steady-flow shock wave reflection is presented. Of course, he has contributed much more widely than that, but I will restrict myself to this part of his work, because it is what I understand best. In particular, his computational and experimental demonstration of hysteresis in the transition between regular and Mach reflection, and his resolution of the difficulties associated with the triple point in weak Mach reflection in terms of the effects of viscosity and heat conduction are reviewed. Finally, some experimental results are presented that demonstrate that, in the dual-solution domain, Mach reflection is more stable than regular reflection

Weak shock wave reflections due to transverse waves in a conventional shock tube

Previous experimental work, utilising a unique large scale shock tube, showed that the four-wave shock reflection pattern, known as the Guderley reflection existed for Mach numbers below 1.10 on wedge angles of 10° and 15°. The current study proves for the first time that these rare reflections can be produced in a conventional shock tube for Mach numbers ranging from 1.10 to 1.40 and for various disturbances in the flow. Two shock tube configurations were tested, the first consisted of a perturbation source on the floor of the tube, and the second utilised a variable diverging section (10°, 15°, and 20°). A new principle was applied where the developed Mach reflection undergoes successive reflections off the upper and lower walls of a tube to produce the desired reflection. The high resolution images captured using a sensitive schlieren system showed evidence of the fourth wave, namely the expansion fan, for the majority of the results for both shock tube configurations. A shocklet terminating the supersonic patch behind the reflected wave was interestingly only observed for Mach numbers of approximately 1.20. The wave structures were similar to those observed in previous experimental work, except no evidence of the second shocklet nor the multi-patch geometry was found. Multi-exposure images of the propagating shock superimposed on a single image frame, analysed with oblique shock equations, estimated the velocities near the triple point. It was shown that the reflected wave is very weak, and that the flow behind the Mach stem is supersonic confirming the shock reflections to be indeed Guderley reflections

The physical nature of weak shock reflection

Student Number : 9900131F - MSc (Eng) dissertation - School of Mechanical Engineering - Faculty of EngineeringRecent high-resolution numerical studies of weak shock reflections have shown that a complex flow structure exists behind the triple point which consists of multiple shocks, expansion fans and triple points. This region had not been detected earlier in experimental observations or numerical studies of weak shock reflections due to the small size of this region. New components were designed and built to modify an existing large-scale shock tube in order to obtain experimental observations to validate the numerical results. The shock tube produced a large, expanding cylindrical incident wave which was reflected off a 15° corner on the roof of the section to produce a weak shock Mach reflection with a large Mach stem in the test section. The shock tube was equipped with PCB high-speed pressure transducers and digital scope for data acquisition, and a schlieren optical system to visualise the region behind the triple point. The tests were conducted over a range of incident wave Mach numbers (M12 = 1.060-1.094) and produced Mach stems of between 694 mm and 850 mm in length. The schlieren photographs clearly show an expansion fan centered on the triple point in all the successful tests conducted. In some of the more resolved images, a shocklet can be seen terminating the expansion fan, and in others a second expansion fan and/or shocklet can be seen. A ‘von Neumann reflection’ was not visualised experimentally, and hence it has been proposed that the four-wave reflection found in these tests be named a ‘Guderley reflection’. The experimental validation of Hunter & Tesdall’s (2002) work resolves the ‘von Neumann Paradox’

Irregular self-similar configurations of shock-wave impingement on shear layers

An oblique shock impinging on a shear layer that separates two uniform supersonic streams, of Mach numbers M1 and M2, at an incident angle σi can produce regular and irregular interactions with the interface. The region of existence of regular shock refractions with stable flow structures is delineated in the parametric space (M1,M2,σi) considering oblique-shock impingement on a supersonic vortex sheet of infinitesimal thickness. It is found that under supercritical conditions, the oblique shock fails to deflect both streams consistently and to provide balanced flow properties downstream. In this circumstance, the flow renders irregular configurations which, in the absence of characteristic length scales, exhibit self-similar pseudosteady behaviours. These cases involve shocks moving upstream at constant speed and increasing their intensity to comply with equilibrium requirements. Differences in the variation of propagation speed among the flows yield pseudosteady configurations that grow linearly with time. Supercritical conditions are described theoretically and reproduced numerically using highly resolved inviscid simulation

A parallel, adaptive discontinuous Galerkin method for hyperbolic problems on unstructured meshes

This thesis is concerned with the parallel, adaptive solution of hyperbolic conservation laws on unstructured meshes. First, we present novel algorithms for cell-based adaptive mesh refinement (AMR) on unstructured meshes of triangles on graphics processing units (GPUs). Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. The algorithm is entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high-order solver and that the proportion of total run time spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmarks as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox. We also study the problem of shock disappearance and self-similar diffraction of weak shocks around thin films. Next, we analyze the stability and accuracy of second-order limiters for the discontinuous Galerkin method on unstructured triangular grids. We derive conditions for a limiter such that the numerical solution preserves second order accuracy and satisfies the local maximum principle. This leads to a new measure of cell size that is approximately twice as large as the radius of the inscribed circle. It is shown with numerical experiments that the resulting bound on the time step is tight. We also consider various combinations of limiting points and limiting neighborhoods and present numerical experiments comparing the accuracy, stability, and efficiency of the corresponding limiters. We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge-Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples show that this result extends to two-dimensional problems on triangular meshes. Finally, we propose a moment limiter for the discontinuous Galerkin method applied to hyperbolic conservation laws in two and three dimensions. The limiter works by finding directions in which the solution coefficients can be separated and limits them independently of one another by comparing to forward and backward reconstructed differences. The limiter has a precomputed stencil of constant size, which provides computational advantages in terms of implementation and runtime. We provide examples that demonstrate stability and second order accuracy of solutions

Three-dimensional shock wave configurations induced by two asymmetrical intersecting wedges in supersonic flow

This study explores the three-dimensional (3D) wave configurations induced by 3D asymmetrical intersecting compression wedges in supersonic and hypersonic inviscid flows. By using the &quot;spatial dimension reduction&quot; approach, the problem of 3D steady shock/shock interaction is converted to that of the interaction of two moving shock waves in the characteristic two-dimensional (2D) plane. Shock polar theory is used to analyze the shock configurations in asymmetrical situations. The results show that various shock configurations exist in 3D asymmetrical shock wave interactions, including regular interaction, transitioned regular interaction, single Mach interaction, inverse single Mach interaction, transitional double Mach interaction, weak shock interaction, and weak single Mach interaction. All of the above 3D steady shock/shock interactions have their corresponding 2D moving shock/shock interaction configurations. Numerical simulations are performed by solving the 3D inviscid Euler equations with the non-oscillatory, non-free parameters, dissipative (NND) numerical scheme, and good agreement with the theoretical analysis is obtained. Furthermore, the comparison of results show that the concept of the &quot;virtual wall&quot; in shock dynamics theory is helpful for understanding the mechanism of two-dimensional shock/shock interactions.</p

Shock focusing in a planar convergent geometry: experiment and simulation

The behaviour of an initially planar shock wave propagating into a linearly convergent wedge is investigated experimentally and numerically. In the experiment, a 25° internal wedge is mounted asymmetrically in a pressure-driven shock tube. Shock waves with incident Mach numbers in the ranges of 1.4–1.6 and 2.4–2.6 are generated in nitrogen and carbon dioxide. During each run, the full pressure history is recorded at fourteen locations along the wedge faces and schlieren images are produced. Numerical simulations performed based on the compressible Euler equations are validated against the experiment. The simulations are then used as an additional tool in the investigation. The linearly convergent geometry strengthens the incoming shock repeatedly, as waves reflected from the wedge faces cross the interior of the wedge. This investigation shows that aspects of this structure persist through multiple reflections and influence the nature of the shock-wave focusing. The shock focusing resulting from the distributed reflected waves of the Mach 1.5 case is distinctly different from the stepwise focusing at the higher incoming shock Mach number. Further experiments using CO_2 instead of N_2 elucidate some relevant real-gas effects and suggest that the presence or absence of a weak leading shock on the distributed reflections is not a controlling factor for focusing