Three-shock Theory Research Articles

Viscous effects in Mach reflection of shock waves and passage to the inviscid limit

The influence of viscosity on the Mach reflection of shock waves in a steady flow of a monatomic gas is studied by solving the Navier–Stokes equations numerically. Based on the nested block grid refinement technique, the flow near the shock wave intersection is simulated, and its behaviour with increasing Reynolds number is studied. The computations are performed for the interaction of both strong (free-stream Mach number$M_\infty = 4$) and weak ($M_\infty = 1.7$) shock waves. In the strong reflection of shock waves at all Reynolds numbers in the examined range, it is found that there exists a small-size zone behind the shock wave intersection where the flow parameters differ from those predicted by the Rankine–Hugoniot relations and hence deviate from the predictions of the inviscid three-shock theory. The structure of this zone is self-similar: in coordinates normalised to the mean free path of molecules in the free stream. The structure is identical at all Reynolds numbers considered in the study. As the Reynolds number increases, the size of this zone in physical coordinates decreases, but the maximum difference between the viscous and inviscid solutions in this zone remains constant, reaching approximately$10\,\%$for pressure. In the weak reflection of shock waves, the flow structure behind the shock wave intersection is not self-similar, i.e. the flow fields at different Reynolds numbers do not coincide in the normalised coordinates, but converge, as the Reynolds number increases, to the parameters predicted by the inviscid three-shock theory.

Analytical model for curved-shock Mach reflection

Mach reflection (MR) is an essential component in the development of the shock theory, as the incident shock curvature is found to have a significant effect on the MR patterns. Curved-shock Mach reflection (CMR) is not yet adequately understood due to the rotational complexity behind curved shocks. Here, CMR in steady, planar/axisymmetric flows is analyzed to supplement the well-studied phenomena caused by oblique-shock Mach reflection (OMR). The solution from the von Neumann's three-shock theory does not fully describe the CMR case. A CMR structure is presented and characterized by an incident shock, reflected shock, Mach stem, and expansion/compression waves over the slipline or occasionally an absence of waves due to pressure equilibrium. On the basis of this CMR structure, an analytical model for predicting the Mach stem in the CMR case is established. The model reduces to the OMR case if the shock curvature is not applicable. Predictions of the Mach stem geometry and shock structure based on the model exhibit better agreement with the numerical results than predictions using previous models. It is found that the circumferential shock curvature plays a key role in the axisymmetric doubly curved CMR case, which results in a different outcome from the planar case.

Prediction of triple point trajectory on two-dimensional unsteady shock reflection over single surfaces

The two-dimensional (2-D) unsteady shock reflection over a single wedge is studied theoretically and numerically, and the emphasis is placed on the trajectory of the triple point (TP). Skews’ relation and the three-shock theory are, respectively, used for determining the trajectory angles of the corner-generated disturbance and the TP, and, subsequently, a model capable of predicting the TP trajectory is established for 2-D unsteady shock reflections over a single wedge. Then, a systematically numerical study is carried out on the 2-D unsteady shock reflection over a single wedge, including five types of shock reflection with the wedge angle increased and five types of shock reflection with the wedge angle decreased. It is found that the Mach stem is always slightly concavely (convexly) curved for reflections with the wedge angle increased (decreased), which should be caused by corner-generated compression (rarefaction) waves propagating along the Mach stem. The new model reasonably predicts the TP trajectory in all the 2-D unsteady shock reflections, and its performance is related not only to the variation trend of the wedge angle (increase or decrease), but also to the type of shock and the initial wedge angle. Specifically, for the shock reflection with the wedge angle increased (decreased), the model generally provides a slightly better (worse) prediction if the TP trajectory angle calculated by the three-shock theory is compulsively modified. The shock-shock/expansion polar analysis presented can partially explain the model performance for predicting the TP trajectory.

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.

Length scale effect on Mach reflection of cellular detonations

An experimental study of the Mach reflection of cellular detonations over the wedge is reported in this paper. Throughout the experiments, high-quality smoked foil is obtained to record cell pattern variation. The initial pressures are varied to yield cellular detonations with varied transverse wave spacing (or cell size). This paper focuses on length scales controlling the deviation and recovery of self-similarity in an unsteady Mach reflection process. The results show that the frozen limit and equilibrium limit both exist for all the mixture compositions. The Mach reflection undergoes a frozen condition in the near field, and then goes through a transition process before asymptotically approaching an equilibrium state in the far field. The cell size variation in the Mach stem region also confirms the transient process. Although the final triple-point trajectory angle in the equilibrium state disagrees with classical three-shock theories, it is in agreement with the reactive three-shock theory when the wedge angle is less than 30°. The triple-point trajectory angle asymptotically approaches zero as the wedge angle increases to approximately 50°, indicating the critical wedge angle from Mach to regular reflection. The transition length associated with the equilibrium limit is found to be dependent on the mixture composition and has the same order of a hydrodynamic thickness, which is approximately a few cell lengths. This means that the hydrodynamic thickness is the characteristic length scale that most significantly dominates the Mach reflection process of cellular detonations.

A specially curved wedge for eliminating wedge angle effect in unsteady shock reflection

A curved wedge with a specific shape is designed and manufactured to guarantee the wedge angle unvaried during the cylindrically converging shock moving along the wedge. Thus the variation of the wedge angle caused by the wedge will be eliminated in unsteady shock reflection. Different initial wedge angles are considered to observe regular reflection and Mach reflection. When Mach reflection occurs, it is found that direct Mach reflection is persisted over the wedge without wave pattern transitions, which differs from our previous work with varied wedge angles [Zhang et al. “Reflection of cylindrical converging shock wave over a plane wedge,” Phys. Fluids 28, 086101 (2016)]. Moreover, the Mach stem is nearly straight when the wedge angle is relatively large, and the trajectory of triple point can be well predicted by three-shock theory. It is believed that the straight Mach stem results from the coupling effect of the converging shock and the convexly curved wedge, which exert opposite effects on the Mach stem curvature. As the wedge angle reduces, the three-shock theory prediction deviates from the present results owing to the curved Mach stem. Stronger vortices are produced near the wall, which are caused by the interaction of two shear layers, and whether the stronger vortices will be generated near the wall depends on the reflection number of the shock wave over the tube wall and wedge. The length of disturbed shock front in the Mach reflection is found to increase nonlinearly due to the unsteady feature of the flow. The growth rate of length reduces as the shock converges because of the geometrical contraction effect. Further the lengths of the Mach stem and the disturbed shock front are compared, and the results show that although the difference exists between them, both of them show a similar variation tendency. Compared with our previous work with varied wedge angles, the variation of the wedge angle has great effects on the Mach stem length and wave pattern transitions. It is believed that the differences between the present study and the pseudo-steady shock reflection are mainly induced by the unsteady feature of the flow because the variation of the shock Mach number is limited and the wedge angle is constant.

The use of optical methods for measuring irregular reflection of weak acoustic shock pulses from a solid surface in air

Irregular reflection ofacoustic weak shockwaves (acoustic Mach numbers less than 0.01) is known as the von Neumann paradox and could not be described by the three-shock theory. In this work, nonlinear reflection regimes were studied experimentally using spark-generated spherically divergent N-waves reflecting from a plane rigid surface. Two optical methods were used in the measurements: a Schlieren system to visualize reflection patterns and a Mach-Zehnder interferometer to reconstruct pressure waveforms. The reconstruction was performed by applying the inverse Abel transform to the phase of the signal measured by the interferometer. The Mach stem formation was observed close to the surface as a result of collision of the incident and reflected front shocks of the N-wave and further away from the surface where the reflected front shock interacted with the incident rear shock. It was shown that irregular reflection occurred in a dynamic way and the length of the Mach stem increased while the N-wave propagated along the surface. In addition, reflection patterns were analyzed for several rough plane surfaces with different roughness size. The height of the Mash stem was found shorter for surfaces with larger dimension of the roughness.

On the transition between regular and irregular shock patterns of shock-wave/boundary-layer interactions

The reflection of strong oblique shock waves at turbulent boundary layers is studied numerically and analytically. A particular emphasis is put on the transition between regular shock-wave/boundary-layer interaction (SWBLI) and Mach reflection (irregular SWBLI). The classical two- and three-shock theory and a generalised form of the free interaction theory are used for the analysis of well-resolved large-eddy simulations (LES) and for the derivation of stability criteria. We found that at a critical deflection angle across the incident shock wave, the perturbations related to the turbulent boundary layer cause bi-directional transition processes between regular and irregular shock patterns for a free-stream Mach number of$\mathit{Ma}_{0}=2$. Computational results show that the mean deflection angle across the separation shock is decoupled from the incident shock wave and can be accurately modelled by the generalised free interaction theory. On the basis of these observations, and the von Neumann and detachment criteria for the asymmetric intersection of shock waves, we derive the critical incident shock deflection angles at which the shock pattern may/must become irregular. Numerical data for a free-stream Mach number of$\mathit{Ma}_{0}=3$confirm the existence of the dual-solution domain predicted by theory.

Mach reflection of a ZND detonation wave

The Mach reflection of a ZND detonation wave on a wedge is investigated numerically. A two-step chain-branching reaction model is used giving a thermally neutral induction zone followed by a chemical reaction zone for the detonation wave. The presence of a finite reaction zone thickness renders the Mach reflection process non-self-similar. The variation of the height of the Mach stem with distance of propagation does not correspond to a straight curve from the wedge apex as governed by self-similar three-shock theory. However, the present results indicate that in the near field around the wedge apex, and in the far field where the reaction zone thickness is small compared to the distance of travel of the Mach stem, the behavior appears to be self-similar. This corresponds to the so-called frozen and equilibrium limit pointed out by Hornung and Sanderman for strong discontinuity shock waves and by Shepherd et al. for cellular detonations. The critical wedge angle for the transition from regular to Mach reflection is found to correspond to the value determined by self-similar three-shock theory, but not by reactive three-shock theory for a discontinuous detonation front.

Geometric criterion for RR↔MR transition in hypersonic double-wedge flows

In a previous research article [Z. M. Hu et al., Phys. Fluids 21, 011701 (2009)], an overall Mach reflection (oMR) configuration with double inverse Mach reflection patterns was computationally confirmed when a double-wedge geometry interacts with a hypersonic flow. Extended computations are conducted in this paper and compared to analytical solutions based on the classical two- and three-shock theories. A geometric criterion is proposed for the transition between regular reflection and Mach reflection occurring inside the parameter space where a type V interaction of shock wave presents in hypersonic double-wedge flows. An oMR solution is allowed by the geometric criterion, while it is theoretically inadmissible. In the vicinity of symmetric condition, regular to Mach reflection transition can also be triggered prior to the theoretical criterion by disturbance generated by a slight increase in the second wedge angle.

Downstream flow condition effects on the RR → MR transition of asymmetric shock waves in steady flows

In this paper, the regular reflection (RR) to Mach reflection (MR) transition of asymmetric shock waves is theoretically studied by employing the classical two- and three-shock theories. Computations are conducted to evaluate the effects of expansion fans, which are inherent flow structures in asymmetric reflection of shock waves, on the RR → MR transition. Comparison shows good agreement among the theoretical, numerical and experimental results. Some discrepancies between experiment and theory reported in previous studies are also explained based on the present theoretical analysis. The advanced RR → MR transition triggered by a transverse wave is also discussed for the interaction of a hypersonic flow and a double-wedge-like geometry.

Computational confirmation of an abnormal Mach reflection wave configuration

For the Mach reflection (MR) of symmetric shock waves of opposite families, only the wave configuration of an overall Mach reflection (oMR) consisting of two direct Mach reflections (DiMR+DiMR) is theoretically admissible. For asymmetric shock waves, an oMR composed of a DiMR and an inverse Mach reflection (InMR) is possible if the two slip layers assemble a converging-diverging stream tube, while an oMR including two inverse Mach reflections (InMR+InMR) is absolutely impossible. In this paper, an overall Mach reflection configuration with double inverse MR patterns is computationally confirmed using the computational fluid dynamics technique. The aerodynamic mechanism behind such an abnormal wave pattern is illustrated. Classical two- and three-shock theories are also applied for the theoretical analysis.

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

The reflection of weak shock waves has been reconsidered analytically using shock polars. Based on the boundary condition across the slipstream, the solutions of the three-shock theory (3ST) were classified as “standard-3ST solutions” and “nonstandard-3ST solutions.” It was shown that there are two situations in the nonstandard case: A situation whereby the 3ST provides solutions of which at least one is physical and a situation when the 3ST provides a solution which is not physical, and hence a reflection having a three-shock confluence is not possible. In addition, it is shown that there are initial conditions for which the 3ST does not provide any solution. In these situations, a four-wave theory, which is also presented in this study, replaces the 3ST. It is shown that four different wave configurations can exist in the weak shock wave reflection domain, a Mach reflection, a von Neumann reflection, a ?R (this reflection is not named yet!), and a modified Guderley reflection (GR). Recall that the wave configuration that was hypothesized by Guderley [“Considerations of the structure of mixed subsonic-supersonic flow patterns,” Air Materiel Command Technical Report No. F-TR-2168-ND, ATI No. 22780, GS-AAF-Wright Field No. 39, U.S. Wright–Patterson Air Force Base, Dayton, OH (October 1947); Theorie Schallnaher Strömungen (Springer-Verlag, Berlin, 1957)] and later termed Guderley reflection did not include a slipstream (see Fig. 7). Our numerical study revealed that the wave structure proposed by Guderley must be complemented by a slipstream (see Fig. 4) in order to be relevant for explaining the von Neumann paradox. Hereafter, for simplicity, this modified GR wave configuration will be also termed Guderley reflection. The domains and transition boundaries between these four types of reflection are elucidated.

Reconsideration of the So-Called von Neumann Paradox in the Reflection of a Shock Wave over a Wedge

Numerous experimental investigations on the reflection of plane shock waves over straight wedges indicated that there is a domain, frequently referred to as the weak shock wave domain, inside which the resulted wave configurations resemble the wave configuration of a Mach reflection although the classical three-shock theory does not provide an analytical solution. This paradox is known in the literature as the von Neumann paradox. While numerically investigating this paradox Colella & Henderson [1] suggested that the observed reflections were not Mach reflections but another reflection, in which the reflected wave at the triple point was not a shock wave but a compression wave. They termed them it von Neumann reflection. Consequently, based on their study there was no paradox since the three-shock theory never aimed at predicting this wave configuration. Vasilev & Kraiko [2] who numerically investigated the same phenomenon a decade later concluded that the wave configuration, inside the questionable domain, includes in addition to the three shock waves a very tiny Prandtl-Meyer expansion fan centered at the triple point. This wave configuration, which was first predicted by Guderley [3], was recently observed experimentally by Skews & Ashworth [4] who named it Guderley reflection. The entire phenomenon was re-investigated by us analytically. It has been found that there are in fact three different reflection configurations inside the weak reflection domain: • A von Neumann reflection – vNR, • A yet not named reflection – ?R, • A Guderley reflection – GR. The transition boundaries between MR, vNR, ?R and GR and their domains have been determined analytically. The reported study presents for the first time a full solution of the weak shock wave domain, which has been puzzling the scientific community for a few decades. Although the present study has been conducted in a perfect gas, it is believed that the reported various wave configurations, namely, vNR, ?R and GR, exist also in the reflection of shock waves in condensed matter.

The physical nature of weak shock wave reflection

For weak shock waves and small wedge angles the application of three-shock (von Neumann) theory gives no physically realistic solutions and yet experiments clearly show a pattern of reflection of three shocks meeting at a triple point. This disagreement is referred to as the von Neumann paradox, and the reflection pattern as von Neumann reflection (vNR). Some recent numerical computations have indicated the existence of an expansion wave immediately behind the reflected wave as originally suggested by Guderley over fifty years ago. Furthermore, a recent solution of the inviscid transonic equations has indicated the possible existence of a very small, multi-wave structure immediately behind the three-shock confluence. A special shock tube has been constructed which allows Mach stem lengths to be obtained which are more than an order of magnitude larger than those obtainable in conventional shock tubes. Schlieren photographs do indeed show a structure consisting of an expansion wave followed by a small shock situated behind the confluence point, with some indication of smaller scale structures in some tests. This indicates that some of the earlier models of vNR, in the parameter space tested, are incorrect. The size of the region influenced by this small wave system is about 2% of the Mach stem length and it is therefore not surprising that it has not been detected before in conventional shock tube facilities.

Comparison between Weak Mach Reflection and von Neumann Reflection.

The purpose of the present paper is to compare the characteristics of the weak Mach reflection (MR) and the so-called von Neumann reflection (vNR). A series of experiments was performed to investigate both reflections. Attention is focused on the flow properties around the triple point. By measuring the location of the triple point and the angle between the incident and reflected shocks, we estimated the approximate flow properties around the triple point. The results show that, while the triple point moves along an almost straight trajectory through a wedge tip, the angles of incidence and reflection vary as the incident shock propagates. As a result, the well-known self-similarity law does not hold for both reflections. For the von Neumann reflection, the present experimental data vary almost along the trivial solution curve of the von Neumann's three-shock theory, and an asymptotic state seems to exist on this curve. In contrast, for weak Mach reflections, these data do not lie on the trivial solution curve but reveal the so-called von Neumann paradox.

Analytical Predictions of Strong Pseudo-Steady Mach Reflections for the Polyatomic Gas: SF6

ABSTRACTStrong pseudo-steady Mach reflections in sulfur hexafluoride (SF6) are analyzed using the three-shock and local three-shock theories, where both the vibrationally-frozen (γ = 1.333) and -equilibrated (γ = 1.093) perfect-gas models are used to compare with existing experiments. The ranges of the incident shock Mach number and reflecting wedge angle studied are 1.49 ≤ Ms ≤ 5.95 and 10° ≤ θω ≤ 42°, respectively. It is found that predicted angles between the incident and reflected shocks from the local three-shock theory using the vibrationally-equilibrated fictitious perfect-gas model (i.e., γ = 1.093) agree closely with those, currently available in literature, measured experimentally; while these predicted angles obtained using the vibrationally-frozen perfect-gas model (i.e., γ = 1.333) differ significantly from the existing experiments. Taking the convex Mach stem curvature at the triple point into consideration, it is shown that both the triple point trajectory angle and the angle between the incident and reflected shocks of strong pseudo-steady Mach reflections in SF6 can be more accurately determined for wide ranges of Ms and θω from the three-shock theory using the vibrationally-equilibrated fictitious perfect-gas model than those without considering this effect.

Non-self-similar behavior of the von Neumann reflection

The purpose of the present paper is to consider the von Neumann reflection (vNR) which takes place for a small incident shock Mach number and a small wedge angle. A series of experiments has been performed with ordinary smooth straight wedges and step-like wedges that simulate the former. The reflection configuration over the step-like wedge has suggested unsteady characteristics of the vNR. Contrary to the established notion of shock reflection phenomena over straight wedges, while the triple point trajectory was approximately a straight line through the apex of the wedge, the vNR showed unsteadiness in the relation between angles of incidence and reflection (ωi,ωr), and thus the flow-field proved to be non-self-similar near the triple point. Based on the measured values, the incident angle for the reflected wave and the Mach number of the flow ahead of the reflected wave were estimated. These values show that the reflected wave is not a Mach wave, but it moves on the (ωi,ωr)-plane almost along a trivial solution. In particular, the flow Mach number ahead of the reflected wave approaches unity, which leads to analytical formulas for angles of incidence and reflection as functions of the incident shock Mach number only. The reflected wave degenerates to a normal Mach wave as the incident shock proceeds. In the case of the vNR, the von Neumann paradox takes place only at an early stage of reflection, and the paradox is resolved later, since the flow properties such as angles of incidence and reflection are given by the trivial solution which is a particular solution for the three-shock theory.

Mach Reflection Wave Configuration in Two-Dimensional Supersonic Jets of Overexpanded Nozzles

The Mach reflection wave configurations and the flowfields associated with two-dimensional supersonic free jets of overexpanded nozzles are analysed. Based on the two- and the three-shock theories along with the classical gasdynamics theory, an analytical model for calculating the height of Mach stem and the cell size of the jet are proposed

A parametric study of Mach reflection in steady flows

The flow fields associated with Mach reflection wave configurations in steady flows are analysed, and an analytical model for predicting the wave configurations is proposed. It is found that provided the flow field is free of far-field downstream influences, the Mach stem heights are solely determined by the set-up geometry for given incoming-flow Mach numbers. It is shown that the point at which the Mach stem height equals zero is exactly at the von Neumann transition. For some parameters, the flow becomes choked before the Mach stem height approaches zero. It is suggested that the existence of a Mach reflection not only depends on the strength and the orientation of the incident shock wave, as prevails in von Neumann's three-shock theory, but also on the set-up geometry to which the Mach reflection wave configuration is attached. The parameter domain, beyond which the flow gets choked and hence a Mach reflection cannot be established, is calculated. Predictions based on the present model are found to agree well both with experimental and numerical results.