Shock-tube experiments on the stability of regular reflection in the dual-solution domain

Regular reflection (RR) and Mach reflection (MR) are theoretically both possible in the dual-solution domain of oblique shock reflection. The physical difference between the two types of reflection is the pressure behind the reflected shock wave: that of MR is lower than that of RR for strong shock reflection. The magnitude relation of these pressures is inverted for weak shock reflection. In the present paper, we performed two kinds of experiment, depending on whether the oblique shock reflection is weak or strong. For strong shock reflection, we decreased the pressure behind the reflected wave of RR using a convex double-wedge. For weak shock reflection, we increased this pressure using a concave double-wedge. Thus, we investigated the stability of RR against pressure disturbances. The results indicate that RR in a shock tube is stable, in the dual-solution domain, for both weak and strong incident shocks.     

Existence criteria for reflection configurations in shock–vortex interactions

The length scale criteria is widely accepted as an explanation for transition and hence existence of different shock wave reflection configurations in pseudo-steady flows. However, there has not been any attempt to validate this criteria using information obtained from a time-dependent numerical simulation. A high resolution time-dependent numerical simulation in pseudo-steady flow is carried out in the present work. Time-dependent numerical data is used to calculate flow features in a laboratory frame of reference to verify validity of the length scale criteria for existence of different shock wave reflection configurations in pseudo-steady flow. This analysis is then extended to the study of unsteady shock wave reflection configurations in shock–vortex interactions. It is shown that the existence of regular reflection (RR) and Mach reflection (MR) configurations in an unsteady flowfield resulting from shock–vortex interactions can also be explained locally based on limiting conditions similar to that prescribed by the length scale criteria for pseudo-steady flow.

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An experimental investigation of the sonic criterion for transition from regular to Mach reflection of weak shock waves

The signal speed, namely the local sound speed plus the flow velocity, behind the reflected shocks produced by the interaction of weak shock waves (M _i < 1.4) with rigid inclined surfaces has been measured for several shock strengths close to the point of transition from regular to Mach reflection. The signal speed was measured using piezo-electric transducers, and with a multiple schlieren system to photograph acoustic signals created by a spark discharge behind a small aperture in the reflecting surfaces. Both methods yielded results with equal values within experimental error. The theoretical signal speeds behind regularly reflected shocks were calculated using a non-stationary model, and these agreed with the measured results at large angles of incidence. As the angle of incidence was reduced, for the same incident shock Mach number, so as to approach the point of transition from regular to Mach reflection, the measured values of the signal speed deviated significantly from the theoretical predictions. It was found, within experimental uncertainty, that transition from regular to Mach reflection occurred at the experimentally observed sonic point, namely, when the signal speed was equal to the speed of the reflection point along the reflecting surface. This sonic condition did not coincide with the theoretical value.     

Numerical simulation of the head-on reflection of a regular reflection

The head-on collision and subsequent reflection of a Regular Reflection (RR) from the end-wall of a shock tube has been investigated both experimentally and numerically for two different incident shock wave Mach numbers and two different reflecting wedge angles. The agreement between the double-exposure holographic interferograms and the numerical simulations which were obtained using a GRP based numerical code, was found to be excellent in the RR region and very good behind the head-on reflected RR. The overall good agreement between the computed and experimental constant-density contours (isopycnics) constitutes a validation of the computational method, including the oblique-wall boundary condition.     

Effect of energy addition on MR $\rightarrow$ RR transition

A combined computational and experimental study was performed to investigate the effect of a single laser energy pulse on the transition from a Mach Reflection (MR) to a Regular Reflection (RR) in the Dual Solution Domain (DSD). The freestream Mach number is 3.45 and two oblique shock waves are formed by two symmetric wedges. These conditions correspond to a point midway within the DSD wherein either an MR or an RR is possible. A steady MR was first obtained experimentally and numerically, then a single laser pulse was deposited above the horizontal center plane. In the experiment, the laser beam was focused resulting in a deposition volume of approximately 3 mm³, while in the simulation, the laser pulse was modeled as an initial variation of the temperature and pressure using Gaussian profile. A grid refinement study was conducted to assess the accuracy of the numerical simulations. For the steady MR, the simulation showed the variation of Mach stem height along the span due to side effects. The predicted spanwise averaged Mach stem height was 1.96 mm within 2% of the experimental value of 2 mm. The experiment showed that the Mach stem height decreased to 30% of its original height due to the interaction with the thermal spot generated by the laser pulse and then returned to its original height by s. That the Mach stem returned to its original height was most likely due to freestream turbulence in the wind tunnel. The numerical simulation successfully predicted the reverse transition from a stable MR to a stable RR and the stable RR persisted across the span. This study showed the capability of a laser energy pulse to control the reverse transition of MR RR within the Dual Solution Domain.     

Supersonic Inviscid Corner Flows with Regular and Irregular Shock Interaction

The results of a numerical and analytical investigation of steady-state supersonic inviscid flows in corners formed by intersecting compression wedges are presented. The flows considered are symmetric about the corner bisector. The distinctive features of flow pattern formation related with the reflection of wedge-generated shocks from the bisector plane are studied. The wedge angles at which transition from regular to irregular shock reflection occurs are determined both numerically and analytically using the criteria available for plane flows; the data thus obtained are found to be in agreement. Flow patterns with irregular shock reflection, namely, single, transitional, and double Mach, as well as von Neumann reflection, are identified; they are similar to the known types of reflection for plane quasi-steady-state flows. Varieties of these types not observed in the plane flows are found to exist. The effects of the angle of inclination of the plane surfaces of the corner to the freestream direction, the sweep angle of the leading edges, and the dihedral angle are investigated. Some previously unknown parameters of corner configurations for which transition may occur in accordance with the von Neumann criterion are determined.     

An experimental and theoretical study of converging polygonal shock waves

An experimental investigation was carried out to explore the possibility of producing converging polygonal shocks in an essentially two-dimensional cavity. Previous calculations by Apazidis and Lesser (1996) suggested that such configurations could be produced by reflecting a cylindrical outgoing shock from a smoothly altered circular boundary, the alteration having n-gonal symmetry. In the experiments the outgoing shock was produced by a spark discharge which yielded shocks in the Mach number range from 1.1 to 1.7 at a radius just prior to the reflection. Polygonal shocks were observed as predicted by using a modified form of geometrical shock dynamics, derived in the above paper. In addition, the modified theory was used to calculate the results of an experiment carried out by Sturtevant and Kulkarny (1976). The results of the numerical calculations were found to be in substantial agreement with both experiments, suggesting that the modifications in geometrical shock dynamics for non-uniform flow ahead of an advancing shock are useful in the case of shock focusing. The experiment also showed that the polygonal shapes were stable in the examined range of shock Mach numbers, a result that may be of importance for a number of practical situations in which shock focusing is present. Received 9 October 2001 / Accepted 7 January 2002 – Published online 11 June 2002     

A study of shock wave interaction with a rotating cylinder

Shock wave reflection over a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from the regular reflection to the Mach reflection is promoted on the cylinder surface which rotates in the same direction of the incident shock motion, whereas it is retarded on the surface that rotates to the reverse direction. Numerical calculations solving the Navier-Stokes equations using extremely fine grids also reveal that the reflected shock transition from RR MR is either advanced or retarded depending on whether or not the surface motion favors the incident shock wave. The interpretation of viscous effects on the reflected shock transition is given by the dimensional analysis and from the viewpoint of signal propagation.Received: 24 April 2002, Accepted: 16 August 2002, Published online: 25 March 2003     

Transition between regular and Mach reflection of shock waves: new numerical and experimental results

New numerical and experimental results on the transition between regular and Mach reflections of steady shock waves are presented. The influence of flow three-dimensionality on transition between steady regular and Mach reflection has been studied in detail both numerically and experimentally. Characteristic features of 3D shock wave configuration, such as peripheral Mach reflection, non-monotonous Mach stem variation in transverse direction, the existence of combined Mach-regular-peripheral Mach shock wave configuration, have been found in the numerical simulations. The application of laser sheet imaging technique in streamwise direction allowed us to confirm all the details of shock wave configuration in the experiments. Close agreement of the numerical and experimental data on Mach stem heights is shown. Received 23 November 2000 / Accepted 25 April 2001     

Controlling the form of strong converging shocks by means of disturbances

The influence of artificial disturbances on the behavior of strong converging cylindrical shocks is investigated experimentally and numerically. Ring-shaped shocks, generated in an annular cross sectional shock tube are transformed to converging cylindrical shocks in a thin cylindrical test section, mounted at the rear end of the shock tube. The converging cylindrical shocks are perturbed by small cylinders placed at different locations and in various patterns in the test section. Their influence on the shock convergence and reflection process is investigated. It is found that disturbances arranged in a symmetrical pattern will produce a symmetrical deformation of the converging shockfront. For example, a square formation produces a square-like shock and an octagon formation a shock with an octagonal front. This introduces an alternative way of tailoring the form of a converging shock, instead of using a specific form of a reflector boundary. The influence of disturbances arranged in non-symmetric patterns on the shape of the shockfront is also investigated.      

Merging of two independent diffracting shock waves

A preliminary experimental and numerical investigation into the interaction between two independent shock waves emerging perpendicular to each other into a common space is presented. It is arranged that two shock tubes have a common diffracting edge, so that the two waves arrive at the edge simultaneously. The shock Mach number was 1.31. The merging three-dimensional diffracting shocks reflect regularly off each other, but as they become more curved due to diffraction the angle between them changes and Mach reflection develops. L-shaped vortices are shed at the two free edges of each tube exit. As they meet, they merge and interact in a complex manner with each other.     

Analytical and numerical investigations of the reflection of asymmetric nonstationary shock waves

The reflection of asymmetric nonstationary shock waves is analytically and numerically studied in this paper. An analytical approach, which is a combination of the shock dynamic and shock polar methods, is advanced to predict the reflection wave configurations. The numerical simulations are performed by the finite volume method based on the second-order MUSCL-Hancock scheme and the HLLC approximate Riemann solver, with the self-adaptive unstructured mesh. It is found that the transition between the overall regular reflection and overall Mach reflection in the asymmetric nonstationary reflection agrees with the detachment criterion, which is analogous to the reflection in pseudo-steady flows (i.e. shock reflection over a wedge). Some special reflection wave configurations, which have never been observed in steady or nonstationary shock reflections so far, are found to exist in this asymmetric reflection. Furthermore, the domains and boundaries of various overall reflection wave configurations are analytically predicted, and the effect of mis-synchronization is also discussed.     

Numerical investigations of transition between regular and Mach reflections caused by free-stream disturbances

Numerical simulations have been performed to study the influence of the free-stream disturbances on the alternation of the steady shock wave reflection configurations in the dual solution domain. Different types of disturbances have been considered. The analysis of interaction between disturbances and the incident shock wave can be substantially simplified for the localized density disturbances. It is shown that such disturbances can indeed cause the transition from regular reflection to Mach reflection and back, so that within a certain range of angles of incidence the shock wave reflection configuration can be considered as a bi-stable system. The threshold amplitude of the localized density disturbance, able to induce the transition, has been estimated theoretically. The results of numerical computations convince of higher stability of the Mach reflection in the dual solution domain compared to the regular reflection, which is in accordance with available experimental data. Received 10 May 2001 / Accepted 15 November 2001 Published online 8 July 2002     

Numerical simulation of Mach reflection in steady flows

The structure obtained when two shocks intersect is known to be highly sensitive to various parameters. In the so-called dual solution domain, both regular and Mach reflection patterns are possible, resulting in hysteresis. The phenomenon is important in inlets because of the substantial difference in entropy rise associated with the two manifestations, and the possibility of unstart with Mach reflection. The effect of various numerical and physical parameters on hysteresis are investigated with two-dimensional simulations. The effect of spanwise relief on a three-dimensional situation is also elucidated. It is confirmed that Mach-stem heights determined from inviscid computations are captured relatively accurately by comparison with experimental data and earlier Euler solutions reported in the literature. Near bifurcation points, however, the solution is highly sensitive to the scheme, and the van Leer and Roe schemes can yield converged solutions with different reflection configurations. Viscous terms and downstream conditions are observed to have relatively minor impact on the solution. The three-dimensional simulations reveal that beyond the spanwise limit of the compression surface, the overall shock-structure remains similar in form but the strengths of various shocks are rapidly muted by the expansion from the side surface. Additionally, the flow downstream of the shock that once formed the Mach reflection rapidly becomes supersonic. The Mach-stem height on the symmetry plane and its variation with spanwise position shows reasonable agreement with the experimental data of other investigators.     

On stability of strong and weak reflected shocks

A time-realistic adaptive unstructured Euler code is used to demonstrate the numerical existence and investigate the stability of both weak and strong reflected shocks in regular reflection. For supersonic parallel flow in a channel, impinging on two symmetrical opposing wedges, the weak reflected configuration is insensitive to downstream pressure disturbances and therefore stable. The strong reflected shock configuration is unstable to positive perturbations in back-pressure and neutrally stable to negative perturbations. A unique -shock structure provides the transition mechanism between weak and strong reflected shock configurations. Received 6 September 1999 / Accepted 10 August 2000     

Reconsideration of inviscid shock interactions and transition phenomena on double-wedge geometries in a M ∞ = 9 hypersonic flow

Shock polar analysis as well as 2-D numerical computation technique are used to illustrate a diverse flow topology induced by shock/shock interaction in a M _∞ = 9 hypersonic flow. New flow features associated with inviscid shock wave interaction on double-wedge-like geometries are reported in this study. Transition of shock interaction, unsteady oscillation, and hysteresis phenomena in the RR ↔ MR transition, and the physical mechanisms behind these phenomena are numerically studied and analyzed.     

Multidimensional Shock Interaction for a Chaplygin Gas

A Chaplygin gas is an inviscid, compressible fluid in which the acoustic fields are linearly degenerate. We analyze the multidimensional shocks in such a fluid, which turn out to be sonic. Two shocks in general position interact rather simply. We investigate several two-dimensional Riemann problems and prove the existence of a unique solution. Among them is the supersonic reflection of a planar shock against a wedge; we remark that the solution cannot be a Mach reflection, contrary to what happens for other gases, and that there always exists a solution in the form of a regular reflection.     

Interferometric measurement of density in nonstationary shock wave reflection flow and comparison with CFD

We present density measurements from the application of interferometry and Fourier transform fringe analysis to the problem of nonstationary shock wave reflection over a semicircular cylinder and compare our experimental measurements to theoretical results from a CFD simulation of the same problem. The experimental results demonstrate our ability to resolve detailed structure in this complex shock wave reflection problem, allowing visualization of multiple shocks in the vicinity of the triple point, plus visualization of the shear layer and an associated vortical structure. Comparison between CFD and experiment show significant discrepancies with experiment producing a double Mach Reflection when CFD predicts a transitional Mach reflection.Received: 12 November 2003, Accepted: 21 October 2004, Published online: 31 March 2005[/PUBLISHED]PACS: 47.40.-x, 42.40.Kw     

Effects of a Single-pulse Energy Deposition on Steady Shock Wave Reflection

The effects of energy deposition in the free stream on steady regular and Mach shock wave reflections are studied numerically. A short-duration laser pulse is focused upstream of the incident shock waves. It causes formation of the expanding blast wave and the residual hot-spot interacting in a complex way with the steady shock wave reflection. It was found that the laser energy addition in the free stream may force the transition from regular to Mach reflection in the dual solution domain. In contrast to previously reported numerical results, the transition from Mach to regular reflection has not been reproduced in our refined computations since the Mach reflection is restored after the flow perturbation.