Single Mach Reflection Research Articles

Magnetohydrodynamic shock refraction at an inclined density interface

Shock wave refraction at a sharp density interface is a classical problem in hydrodynamics. Presently, we investigate the strongly planar refraction of a magnetohydrodynamic (MHD) shock wave at an inclined density interface. A magnetic field is applied that is initially oriented either perpendicular or parallel to the motion of incident shock. We explore flow structure by varying the magnitude of the magnetic field governed by the non-dimensional parameter β∈(0.5,106) and the inclination angle of density interface α∈(0.30,1.52). The regular MHD shock refraction process results in a pair of outer fast shocks (reflected and transmitted) and a set of inner nonlinear magneto-sonic waves. By varying magnetic field (strength and direction) and inclination interface angle, the latter waves can be slow shocks, slow expansion fans, intermediate shocks, or slow-mode compound waves. For a chosen incident shock strength and density ratio, the MHD shock refraction transitions from regular (all nonlinear waves meeting at a single point) into irregular when the inclined density interface angle is less than a critical value. Irregular refraction patterns are not amenable to an analytical solution, and hence, we have obtained irregular refraction solutions by numerical simulations. Since the MHD shock refraction is self-similar, we further explore by converting the initial value problem into a boundary value problem (BVP) by a self-similar coordinate transformation. The self-similar solution to the BVP is numerically solved using an iterative method and implemented using the p4est adaptive mesh framework. The simulation shows that a Mach stem occurs in an irregular MHD shock refraction, and the flow structure can be an MHD equivalent to a single Mach reflection irregular refraction and convex-forwards irregular refraction that occur in hydrodynamic case. For Mach number M = 2, both analytical and numerical results show that perpendicular magnetics fields suppress the regular to irregular transition compared to the corresponding hydrodynamic case. As Mach number decreased, it is possible that strong perpendicular magnetics promote the regular to irregular transition, while moderate perpendicular magnetics suppress this transition compared to the corresponding hydrodynamic case.

Investigation of shock wave interactions involving stationary and moving wedges

The present study investigates the shock wave interactions involving stationary and moving wedges using a sharp interface immersed boundary method combined with a fifth-order weighted essentially non-oscillatory scheme. Inspired by Schardin’s problem, which involves moving shock interaction with a finite triangular wedge, we study the influence of incident shock Mach numbers and corner angles on the resulting flow physics under both stationary and moving conditions. The present study involves three incident shock Mach numbers (1.3, 1.9, and 2.5) and three corner angles (60°, 90°, and 120°), while its impact on the vorticity production is investigated using the vorticity transport equation, circulation, and rate of circulation production. Furthermore, the results yield that the generation of the vorticity due to the viscous effects is quite dominant compared to the baroclinic or compressibility effects. The moving cases presented involve shock driven wedge problems. The fluid and wedge structure dynamics are coupled using the Newtonian equation. These shock driven wedge cases show that wedge acceleration due to the shock results in a change in reflected wave configuration from single Mach reflection to double Mach reflection. The intermediary state between them, the transition Mach reflection, is also observed in the process. The effect of shock Mach numbers and corner angles on the triple point trajectory, as well as on the drag coefficient, is analyzed in this study.

Effect of the wall temperature on Mach stem transformation in pseudo-steady shock wave reflections

Pseudo-steady shock wave reflections on a compression ramp are recognized as a significant phenomenon in the shock dynamics field. Several investigations have been performed on this phenomenon by shock tube experiments, combined with numerical and theoretical analyses. In addition to the availability of numerous reliable and accurate data on shock reflection patterns in the form of regular reflection, double-Mach reflection, transitional-Mach reflection, and single-Mach reflection, several transitional criteria such as detachment criterion and mechanical-equilibrium criterion are also established. However, none of the research focuses on the influence of the wall temperature on Mach stem transformation in pseudo-steady shock wave reflections. In order to articulate the role played by the wall temperature, a two-dimensional numerical simulation is performed by resolving the unsteady Reynolds-averaged Navier-Stokes equations to investigate the transformation of the Mach stem. Shear-stress transport k-ω turbulence model and third-order Monotonic Upwind Scheme for Conservation Laws scheme are explored to accurately capture various shock structures. Numerical results are initially validated with experimental data in the open literature and it is evident that there is an excellent match between the present simulation and the actual structure. Subsequently, the grid resolution and time step independence analyses are considered to better resolve the shock reflection configuration. Large-scale negative and positive heat fluxes are imposed on the ramp wall, which can simulate the cooling and heating ramps respectively. A new Mach stem transformation that illustrates significant differences in physical structure evolutions caused by the variation of the wall temperature is numerically captured for the first time. The mechanism of wall temperature effects on the transformation of the Mach stem is clearly expounded with the purpose of profound understanding for pseudo-steady shock wave reflections.

Study on the Mach and regular reflections of shock wave

While a moving incident shock wave moves through a sharp compression ramp with the fixed angle, θr, the incident shock wave is reflected by the ramp surface and the induced pseudo-steady flow behind it is deflected by the ramp corner. It is known that four basic shock reflection patterns can be obtained, including regular reflection, single-Mach reflection, transitional-Mach reflection, and double-Mach reflection. Previously, the shock reflections were mainly studied by various experimental methods with different test gases in the shock tubes, which include air, nitrogen, helium, and argon. In this paper, in order to clearly illustrate the transitional properties between Mach and regular reflections, theoretical and numerical analyses were carried out. Detachment and mechanical equilibrium criteria were established by the theoretical analysis based on the two- and three-shock theories. Further, which criterion is more suitable and accurate to explain the transitional processes between the Mach and regular reflections was illustrated. The generation and variation of the reflected shock waves were clearly captured based on the total variation diminishing scheme within Fluidyn software. A series of 2D (two-dimensional) models were simulated at different operating conditions that the incident Mach numbers (1.1, 1.5, 2.03, 3, 4 and 5) and compression ramp angles (27°, 50°, and 60°) were changed, respectively. The detailed properties on Mach and regular reflections were illustrated in the present studies. In addition, some new and constructive conclusions for transitional processes between the Mach and regular reflections of shock wave were obtained.

Investigation of shock focusing in a cavity with incident shock diffracted by an obstacle

Experiments and numerical simulations were carried out in order to investigate the focusing of a shock wave in a test section after the incident shock has been diffracted by an obstacle. A conventional shock tube was used to generate the planar shock. Incident shock Mach numbers of 1.4 and 2.1 were tested. A high-speed camera was employed to obtain schlieren photos of the flow field in the experiments. In the numerical simulations, a weighted essentially non-oscillatory (WENO) scheme of third-order accuracy supplemented with structured dynamic mesh adaptation was adopted to simulate the shock wave interaction. Good agreement between experiments and numerical results is observed. The configurations exhibit shock reflection phenomena, shock–vortex interaction and—in particular—shock focusing. The pressure history in the cavity apex was recorded and compared with the numerical results. A quantitative analysis of the numerically observed shock reflection configurations is also performed by employing a pseudo-steady shock transition boundary calculation technique. Regular reflection, single Mach reflection and transitional Mach reflection phenomena are observed and are found to correlate well with analytic predictions from shock reflection theory.

Planar laser-induced fluorescence imaging in shock tube flows

The planar laser-induced fluorescence (PLIF) imaging method was used to perform flow visualization and quantitative planar thermometry in shock tube flow fields using toluene as a fluorescence tracer in nitrogen. Fluorescence quantum yield values needed to quantify PLIF images were measured in a static cell at low pressures (<1 bar) for various toluene partial pressures in nitrogen bath gas. Images behind incident and reflected shocks were taken in the core flow away from regions affected by boundary layers. Temperature measurements from these images were successfully compared with predicted values using ideal shock equations. Measured temperatures ranged between 296 and 800 K and pressures between 0.15 and 1.5 atm. The average temperature discrepancies between measurements and the predicted values behind the incident and reflected shocks were 1.6 and 3.6%, respectively. Statistical analyses were also conducted to calculate the temperature measurement uncertainty as a function of image resolution. The technique was also applied to the study of more complex supersonic flows, specifically the interaction of a moving shock with a wedge. Measured temperatures agreed well with the results of numerical simulations in all inviscid regions, and all pertinent features of the single Mach reflection were resolved.

Three-dimensional numerical simulation of detonations in coaxial tubes

Three-dimensional numerical simulation of detonations in both a circular tube and a coaxial tube are simulated to reveal characteristics of single spinning and two-headed detonations. The numerical results show a feature of a single spinning detonation which was discovered in 1926. Transverse detonations are observed in both tubes, however, the single spinning mode maintains the complex Mach reflection whereas the two-headed mode develops periodically from the single Mach reflection to the complex one. The calculated cell aspect ratio for the two-headed mode changes from 1.09 to 1.34 as the radius of axial insert increases from r1/R = 0.1 to 0.9. The calculated cell aspect ratio for r1/R = 0.1 is close to the experimental results without an axial insert. The formation of an unreacted gas pocket behind the detonation front was not observed in the single spinning mode; however, the two-headed mode has unreacted gas pocket behind the front near the axial insert.

Blast-Wave Dynamics in a Circular Duct with a Sudden Expansion

In this study, the problem of a planar blast wave in a duct with a sudden expansion is considered. An incident blast wave that contains two shock waves is arranged so that one pair of vortex rings induced by the first shock wave interacts with the second shock wave of the blast wave and with the reflected shock associated with the first shock wave. A high-resolution Euler/Navier-Stokes solver using a fifth-order weighted essentially nonoscillation scheme is employed to investigate the problem. After the Euler/Navier-Stokes solver is validated to be reasonably accurate, the flow fields induced by the shock/vortex interactions are investigated by computational shadowgraph and computational schlieren techniques. It is found that the trajectories formed by major ring-vortex centers may be different case by case, depending on the arrival time sequence of the second shock wave and the reflected shock wave. In particular, a third vortex ring may be developed, causing the trajectory's turn at some critical time in viscous-flow models. Because of viscous dissipation, consequently stronger vortex-ring strength predicted by the inviscid-flow model than that for the viscous-flow model can result in 14-32% lower pressures at the major ring-vortex centers in the inviscid flow compared with the turbulent flow during the time interval investigated for case A, and 20-32% for case B, and 21-38% for case C. Because of the boundary layer development along the duct wall after the sudden expansion, the reflection type of the first shock wave is changed from a regular reflection in the inviscid-flow case to a single Mach reflection that directly affects the arrival time of the reflected shock wave at the sharp corner. Moreover, the mechanism of the vortex ring formation based on the inviscid-flow model is analyzed. It is found that the term due to the dilation effect also plays a more important role in vorticity generation than the baroclinic term.

A State-of-the-Knowledge Review on Pseudo-Steady Shock-Wave Reflections and their Transition Criteria

The distinguished philosopher Ernst Mach published the first known paper on the phenomenon of planar shock-wave reflections over straight wedges over 125 years ago in 1878. In his publication he presented two wave configurations that could result from this reflection process, a regular reflection (RR) and a configuration that was later named after him and called Mach reflection (MR) in the early 1940s. In 1945, Smith reported on an additional wave configuration, which had a reflected shock wave that was slightly different from that of the just-mentioned Mach reflection. Smith (OSRD Rep. 6271, Off. Sci. Res. Dev., 1945) did not ascribe any special importance to the wave configuration that he observed. The wave configuration that was observed and reported by Smith (OSRD Rep. 6271, Off. Sci. Res. Dev., 1945) was recognized as an independent one only about 5 years later when White (Tech. Rep. II-10, Princeton Univ. Dept. Phys., 1951) reported on the discovery of a new wave configuration that was named double-Mach reflections (DMR) because it had similar features to that of the Mach reflection wave configuration but all the features were doubled. For this reason the Mach reflection wave configuration has been re-named single-Mach reflection (SMR). (Until the late 1970s it was called simple-Mach reflection although nothing is simple about it.). The discovery of the double-Mach reflection revealed that the wave configuration that was first observed by Smith was an intermediate wave configuration between the SMR and the DMR wave configurations. For this reason it was named transitional-Mach reflection (TMR) (Until the early 1980s it was called complex-Mach reflection although it is not the most complex one.). Since the discovery of the DMR many investigations were aimed at elucidating the exact transition criteria between the above-mentioned four different wave configurations as well as some additional configurations and sub-configurations that were discovered later. In 1991 Ben-Dor published a monograph, entitled “Shock Wave Reflection Phenomena”, that was, in fact, a state-of-the-knowledge review of the phenomena. This state-of-the-knowledge will be referred to in the followings as the “old”-state-of-the-knowledge (This state-of-the-knowledge existed until the mid 1990s. A few years later Li and Ben-Dor (Shock Wave 5(1/2), 59–73, 1995) modified the analytical approach for evaluating the transition criteria from the single-Mach to the transitional- Mach reflection (SMR, \(\rightleftarrows\),TMR) and from the transitional-Mach to the double-Mach reflection (TMR, \(\rightleftarrows\) ,DMR) and presented some modified and new criteria for the formation and termination of both the TMR and DMR wave configurations. Experimental results from various sources revealed that the transition boundaries between the SMR, TMR and DMR wave configurations that were based on the modified analytical approach were better than those of the “old” state-of-the-knowledge that as mentioned earlier was summarized in Ben-Dor’s (Shock Wave Reflection Phenomena, Springer, 1991) monograph. Unfortunately, however, the results of Li and Ben-Dor’s (Shock Wave 5(1/2), 59–73, 1995) modified analytical approach have not been internalized, and publications by various scientists in the past decade neglected the revised and better transition criteria and kept on referring to the old and wrong criteria that appeared in Ben-Dor’s (Shock Wave Reflection Phenomena, Springer, 1991) monograph. For this reason, a state-of-the-knowledge review that is based on the above-mentioned 10-year-old findings of Li and Ben-Dor (Shock Wave 5(1/2), 59–73, 1995) is presented herein. At the first step, the “old” state-of-the-knowledge is presented.

Unsteady shock generated vorticity

Abstract. A two-dimensional theory is developed for the vorticity just downstream of a curved, unsteady shock wave. By utilizing Crocco's equation, an explicit formula is obtained for the vorticity that does not require a perfect gas and that holds for arbitrary conditions upstream of the shock wave. The analysis is applied to the flow just downstream of the reflected shock that occurs in a single-Mach reflection pattern. Flow conditions are based on an interferometric photograph of Ben-Dor and Glass (1978). In this case, the reflected shock is weak everywhere from its upstream intersection with the wall to the triple point. The vorticity has a singularity and a change of sign near the triple point that indicates the presence of a weak shear layer downstream of this location.

Numerical simulation for nonstationary Mach reflection of a shock wave: A kinetic-model approach

A numerical simulation was performed for the process of formation of single Mach reflection on a wedge by solving a BGK type kinetic equation for the reduced distribution function with a finite difference scheme. The calculations were carried out for a shock Mach number 2.75 and wedge angle 25° in a monatomic gas, which corresponds to the conditions of single Mach reflection in the classical von Neumann theory. The calculations were performed for both diffuse and specular reflection of molecules at the wall surface. It is concluded that the diffuse reflection of molecules at the wall surface or the existence of the viscous or thermal layer is an essential factor for a nonstationary process at the initial stage of Mach reflection. Furthermore, the numerical results for diffuse reflection are found to simulate the experimental results very well, such as a transient process from regular reflection to Mach reflection along with shock propagation.

An update on non-stationary oblique shock-wave reflections: actual isopycnics and numerical experiments

Nonstationary oblique shock-wave reflections over compressive wedges in air and argon were investigated using infinite-fringe interferometric techniques. These allowed direct, continuous and accurate observations of the isopycnics (lines of constant density) of the flow field. The initial pressures for these experiments were made as high as possible (15 to 250 torr) in order to increase the number of isopycnics and to enhance their details and distribution along the wedge surface over a shock-Mach-number range 2.0 &lt; Ms [les ] 8.7. Included in the study were two cases of regular reflection (RR) and one of each single Mach reflection (SMR), complex Mach reflection (CMR) and double-Mach reflection (DMR) for air, and one RR, SMR, CMR and DMR for argon. These particular cases, which we investigated previously in N2 and Ar using a finite-fringe technique, have been used by computational fluid dynamicists to check their finite-difference results against our experimental data. It will be shown that the isopycnic structure previously reported by us differs in detail, in most cases, from that of the present study. The major difference arises from the fact that it was only possible previously to obtain discrete points on isopycnics and along the wedge surface. Consequently, the results obtained before were not as accurate. Comparisons were made of actual wall-density distributions with numerical simulations of the density contours of the various flows obtained by a number of authors. Each numerical method displays its advantages and disadvantages in describing the details of the flow fields. The present experimental results for air are new. They are of great interest from a practical viewpoint. The experiments in argon were redone to provide better data for a gas free from real-gas effects in the range of initial conditions considered, in order to simplify the computations in the numerical simulations. Although the recent numerical simulations are better than those reported previously, additional efforts are required to improve the predictions of the shape, location and values of the isopycnics and other flow isolines in the various regions and along the wall, and to render the predictions free of computer ‘noise’. It is worth noting that real-gas effects did not play any significant role in determining the various wave systems in RR, SMR, CMR and DMR; a different claim was made in our previous work. Relaxation of nitrogen in air can be seen however, at the highest shock Mach numbers (Ms = 7.19 and 8.70), with relaxation lengths in good agreement with accepted predictions.

Diffraction of a Shock Wave by a Compression Corner: Part II - Single Mach Reflection

The two-dimensional time-dependent Euler equations which govern the flow field resulting from the interaction of a planar shock with a compression corner are solved for initial conditions which result in single Mach reflection of the incident planar shock. The Euler equations are first transformed to include the self-similarity of the flow field. A second transformation is employed to normalize the distances between the ramp and the reflected shock and between the wall and the Mach stem. The resulting equations in strong conservation-law form are solved using a second-order discontinuity-fitting finite-difference approach. The results are compared with experimental interferograms and existing first-order shock-capturing numerical solutions.