Mach Stem Height Research Articles

Shock reflection with incident shock–wedge trailing-edge expansion fan interaction

Steady shock reflection where the incident shock is free of interaction with other waves has been well studied. In this paper, we consider the less studied shock reflection problem where the incident shock interacts with the wedge trailing-edge expansion fan, which occurs when the wedge trailing-edge height surpasses a threshold. The influence of this interaction on the advance of transition from Mach reflection to regular reflection is quantified in terms of the wedge trailing-edge height ratio. The wave pattern, including primary and reflected Mach waves, for Mach reflection with interaction is clarified using computational fluid dynamics (CFD) and the method of characteristics. Those reflected Mach waves having an important effect on Mach reflection are identified. A simplified Mach stem model that accounts for the direct role of the interaction on the incident shock and its indirect role on the reflected shock and slipline is built up on a past model without interaction. Both theory and CFD show that the Mach stem height decreases nonlinearly with increasing trailing-edge height.

Mach reflection of detonation waves over a porous concave wedge

An experimental study of the differences in the Mach reflection process over smooth and porous concave wedges was systematically performed based on soot track measurement. Meanwhile, the interaction mechanism among the incident detonation, the Mach stem, and the porous wall was analyzed in detail. The results show that introducing multiple pores on the concave surface induces an additional attenuation region. In contrast, the triple-point over porous wedges starts traveling later and ends up colliding with the concave surface earlier. Moreover, the Mach stem height is shorter at the same wall position. For the transition angle of Mach-to-regular reflection over porous wedges, it is found to be negatively correlated with the length-scale ratio R/λ of the radius of curvature (R) to the cell size (λ), gas instability, and porosity. Although the experimental transition angle disagrees with the Chester–Chisnell–Whitham (CCW) and the reactive CCW theories, it is in agreement with the trend over smooth wedges that the experimental transition angle approaches to reactive CCW theory as R/λ increases. For the triple-point trajectory, it is almost a straight line when wall angle θ≤30° at R = 522 mm. It is also found that the triple-point trajectory is similar at the beginning for different mixture compositions at an equivalent initial pressure. The height of the Mach stem is shorter over a porous wedge with higher porosity, but it reaches a maximum value at a wall angle of 30° for wedges. Moreover, the Mach stem height decreases with increasing initial pressure or gas instability.

Strong shock solutions in symmetric wedge flows: Unphysical or unstable?

This paper investigates the strong shock solutions in a supersonic wedge experimentally, analytically, and numerically. Experiments and computations are conducted on scaled-down models for the two types of shock reflection to be possible. The time-resolved schlieren observation of the flow evolution revealed that the shock formation is a highly dynamic transition of the starting shock from a strong Mach reflection (MR) to a weak regular reflection (RR) via a strong RR reflection over a constant shock wave angle for a wedge angle less than the detachment criterion angle for the shock transition. However, when the wedge angle is greater than the detachment criterion angle, the shock moves over the wedge with the MR structures of diminishing Mach stem height at a constant incident shock wave angle. These intermediate shock reflections are found to be unstable and oscillate at high amplitude and low frequencies to upstream pressure fluctuations. The nature of the intermediate shock reflection during the shock transition over the wedge has also been studied using an unsteady second-order two-dimensional compressible Navier–Stokes solver code with shear stress transport k-ω turbulence modeling. The computed flow parameters around the intermediate shock reflections confirmed that these are indeed strong shock reflections believed to be unphysical in steady wedge flows.

Reflection and transition of planar curved shock waves

In this paper, the reflection of curved shock waves over a symmetry plane in planar supersonic flow is studied. This includes stable Mach reflection (MR) and the regular reflection (RR) to MR transition process. Curved shock theory (CST) is applied to derive the high-order parameters in front of and behind the shock wave. The method of curved shock characteristics is used to establish an analytical model to predict the wave configurations. The shock structures provided by the proposed model agree well with the numerical results. Flow structures, such as the height of the Mach stem and the shape of the shock wave and slip line, are studied by applying the analytical model. Isentropic waves generated from a curved wall are found to significantly influence the flow patterns. It appears that the compression waves obstruct the formation of the sonic throat and increase the Mach-stem height. The expansion waves have the opposite effect. The evolution mechanism of the Mach stem is found in conjunction with the RR-to-MR transition process. The CST is extended to a moving frame and used to model the transition. The time history of the moving triple point illustrates the effects of the incident shock angle and isentropic waves on the transition process.

Mach reflection of detonation wave on porous wall

This study reports the Mach reflection of gaseous detonation on porous wedges experimentally, in which the porous wall is consisted of equidistant inline square columns. The smoked foil technique was utilized to monitor the evolution of the triple-point trajectory and detonation cells in the Mach stem region. In addition to the wedge angle and initial pressure of gaseous mixture, this paper also focuses on the effect of porosity and pore size on the Mach reflection and its mechanism. The results show that the strength of the Mach stem is significantly weakened by the porous media compared with that on the smooth wedge, which is ascribed to the diffraction and reflection waves generated by the interaction of the Mach stem front with the pore. Furthermore, the onset of the triple-point trajectory is delayed, the angle of which is decreased. The porosity and pore size present distinct influence on the Mach stem height. With the increase in the porosity and the decrease in the pore size, the Mach stem region is attenuated more dramatically and the Mach stem is harder to be formed or even could not be observed. Furthermore, the triple-point trajectory on the porous wall exhibits local self-similarity and satisfies the frozen limit in the near field and the equilibrium limit in the far field. However, the lengths of the existence of the frozen limit and the transition to the equilibrium limit on the porous wall are found to be much shorter than the hydrodynamic thickness, and the recovery of the self-similarity depends largely on the porosity and pore size.

Simulation of high-Mach-number inviscid flows using a third-order Runge-Kutta and fifth-order WENO-based finite-difference lattice Boltzmann method

A discrete-velocity Boltzmann equation(DVBE) with Bhatnagar-Gross-Krook (BGK) approximation is discretized in time and space using a third-order Runge-Kutta (RK3) and fifth-order weighted essentially nonoscillatory (WENO) finite-difference scheme to simulate benchmark inviscid compressible flows. The implementation of the WENO ensures that solutions behave with minimal or no oscillations, narrowing the gap between the exact and the numerical results. Discrete-velocity sets given by Kataoka and Tsutahara [Phys. Rev. E 69, 056702 (2004)10.1103/PhysRevE.69.056702] are used. The additional dissipation terms as well as artificial viscosity are incorporated in the formulation to solve the compressible flows at high Mach number. Further, the flows which are subjected initially to a high density ratio are effectively simulated. In this article, one-dimensional benchmarks are simulated at initial Mach number up to 30 and density ratio up to 1000, whereas, the benchmarks in two dimensions are simulated with a Mach number up to 10. The algorithm is assessed by simulating numerous benchmarks, namely, (i) one-dimensional Riemann problem for various shock waves combinations [namely (a) shock-shock waves in the case of different Mach numbers, (b) rarefaction-shock waves for various density ratios, (c) sudden contact shock discontinuity, and (d) shock-rarefaction waves for density ratio 5], (ii) isentropic vortex convection test, (iii) regular shock reflection (RR) for Mach numbers 2.9 and 10, (iv) double Mach reflection (DMR) for inflow Mach numbers as 6 and 10, and (v) forward-facing step for inflow Mach numbers 2 to 5. Further, the effect of change in Mach numbers and wedge angles on the flow structures in the case of DMR are detailed. In the case of a forward-facing step, the variations of flow structure (e.g., the Mach stem height, triple points location, and shock standoff distance) are detailed with respect to Mach number, step height, and specific-heat ratios. Finally, the numerical stability of the proposed formulation is carried out to assess the behavior of the free parameters.

Analytical Model of Strong Mach Reflection

A Mach reflection structure in steady supersonic flow is studied in which a transonic region exists behind the reflected shock wave. Here, the measure of a strong Mach reflection is the total pressure loss of the shock reflection system. A corresponding classification chart provides a method to directly characterize the regions of the strong Mach reflection. An analytical model is proposed for predicting the Mach stem height in an explicit function. The theoretical results are in good agreement with the numerical simulations. Assumptions supporting the analytical model are thus validated. A linear relationship is found between the dimensionless Mach stem height and the dimensionless wedge length, which is validated with both analytical and numerical results. Taking advantage of this feature, a generalized method to determine the domain regarding the dimensionless wedge length in the strong Mach reflection is created. The analytical method provides a quick and precise prediction of boundaries for more detailed simulations and experiments.

Prediction of Mach stem height in compressible open jets. Part 1. Overexpanded jets

The flow from a supersonic nozzle when operated at off-design conditions exhibits a wide variety of complex flow structures. The prominent ones are the shock reflections inside the jet. The two main types of shock interactions are the regular reflection (RR) and Mach reflection (MR). The prominent characteristic change in the shock pattern when an RR transforms into an MR is the appearance of the Mach stem with a subsonic region downstream of it. The estimation of the Mach stem height gives the size of the subsonic domain and is a direct method to predict the transition from RR to MR or vice versa. The present study is carried out to estimate the size of the Mach stem in an inviscid jet in the overexpanded regime. The analytical methods employed here are the extension of the techniques developed to estimate the Mach stem size in the wedge flows by Li & Ben-Dor (J. Fluid Mech., vol. 341, 1997a, pp. 101–125), Mouton & Hornung (AIAA J., vol. 45, issue 8, 2007, pp. 1977–1987) and Bai & Wu (J. Fluid Mech., vol. 818, 2017, pp. 116–140). The results from the analytical formulation have been compared with the high resolution computational and experimental results. The analytical method reveals that the open jet has a unique, stable MR configuration and forms an upper limit for achievable Mach stem height for the wedge flows. Apart from the estimation of Mach stem height, the growth rates of Mach stem where the RR–MR transition takes place are also calculated and compared with the ones corresponding to wedge flows.

Dynamic effects in transition from regular to Mach reflection in steady supersonic flows.

The effect of rapid wedge rotation on the transition from regular (RR) to Mach reflection (MR) is investigated. This unsteady shock reflection transition is compared with the steady-state transition. The dependence of various flow features such as the unsteady Mach stem height, position of the reflection point, and shock angle at the reflection or triple point on the wedge angle for a fixed Mach number is compared at various rotation rates. The study is further extended to compare the dynamic effects for various Mach numbers in the strong shock reflection domain at higher wedge speeds. Transition lines corresponding to different rotation speeds are obtained similar to the detachment transition line in steady cases. It is found that the pivot point has only marginal effect on the transition point, but it substantially affects the Mach stem growth and the movement of the reflection point, specifically at higher Mach numbers. The location of the transition from the inlet also depends on the pivot point and the rate of rotation.

A Study of the Dependence of the Mach Stem Height on the Trailing Edge Height

The Mach stem height is an important parameter in the Mach reflection of steady supersonic flow. Various experimental, numerical, and theoretical works have been conducted to study this parameter in the past. However, much of the established work focuses around a single set of trailing edge heights. Here, we perform a study to show the dependence of Mach stem height on the trailing edge height for a wider range of geometry. Through numerical simulation for a set of trailing edge heights, we found that the normalized Mach stem height is almost linear with respect to the normalized wedge trailing edge height. The parameter used for normalization can be either the inlet height or the length of the lower wedge surface. The observation of this linear trend is justified through a simplified analysis, which leads to an expression of the Mach stem height that linearly depends on the trailing edge height. The present study extends our knowledge about how the geometry affects the Mach stem height, and provides a basis for future work to elaborate analytical models for Mach stem height.

A method for predicting Mach stem height in steady flows

The prediction of Mach stem height can be important in the design of supersonic intake in supersonic and hypersonic flows. It is also important because of the progress in aircraft and rocket engines. An analytical method of predicting the Mach stem height is necessary in theoretical field of shock reflection and is the basis of the comparable computational fluid dynamics (CFD) method. A method for predicting the Mach stem height in steady flows is performed based on the earlier models. In this article, an analytical model for predicting the Mach stem height is improved based on two main assumptions: one is the calculation of the triple point deflection angle when the Mach stem is an oblique shock and the other is about the shape of the free part of the slip line. Under these assumptions, the relations predicting of Mach stem height in two-dimensional steady flow are derived based on the advanced averaging method of the subsonic flow region. The Mach stem heights are decided solely for the incoming flow Mach numbers and the wedge angles by the improved analytical model. As a result, the Mach stem heights by the model of this article are found to agree well with experimental results at lower Mach numbers, but there are relative errors at higher Mach numbers. The convexity of the slip line is also considered.

Mach reflection of a H2-O2-Ar detonation wave on the rough wedge based on soot track measurements

Gaseous detonation wave reflection on the rough wedge was experimentally investigated by soot track measurements. Various wedge angles, step numbers and initial pressures were taken into account. The results show that the Mach reflection can form on the rough surface due to the three-dimensional structure of the cellular detonation wave or the pseudo triple-point configuration. The wedge angle is still a significant parameter to affect the Mach stem height and the triple-point trajectory. The Mach stem on the rough surface is weaker and the triple-point is easily deflected due to the triple-point disappearance from the Mach stem. The wedge with smaller step number leads to the triple-point trajectory starting in the downstream region, instead of its starting from the wedge apex. As the step number increases, the Mach stem height on the rough wedge gradually approaches the one on the smooth wedge. The initial pressure has a more significant effect on the Mach reflection on the rough surface compared to that on the smooth surface. Moreover, the Mach reflection on the rough wedge keeps localized self-similarity, and satisfies the frozen limit in the near field and the equilibrium limit in the far field. The critical wedge angle of the transition from regular reflection to Mach reflection is dependent on the step number.

Effect of surface roughness on nonlinear reflection of weak shock waves.

The authors have recently shown that irregular reflections of spark-generated pressure weak shocks from a smooth rigid surface can be studied using an optical interferometer [Karzova, Lechat, Ollivier, Dragna, Yuldashev, Khokhlova, and Blanc-Benon, J. Acoust. Soc. Am. 145(1), 26-35 (2019)]. The current study extends these results to the reflection from rough surfaces. A Mach-Zehnder interferometer is used to measure pressure waveforms. Simulations are based on the solution of axisymmetric Euler equations. It is shown that roughness causes a decrease of the Mach stem height and the appearance of oscillations in the pressure waveforms. Close to rough surfaces, the pressure was higher compared to the smooth surface.

Mach reflection in steady supersonic flow considering wedge boundary-layer correction

Mach reflection in steady supersonic flow is an important phenomenon having received extensive studies, among which simplified theoretical models to predict the size of Mach stem and other flow structure are of particular interest. Past efforts for such models were based on inviscid assumption while in real cases the flow is viscous. Here in this paper we consider the influence of wedge boundary layer on the Mach stem height. This is done by including a simplified boundary layer model into a recently published inviscid model. In this viscous model, the wedge angle and the trailing edge height, which control the Mach stem height, are replaced by their equivalent ones accounting for the displacement effect of the wedge boundary layer, with the boundary layer assumed to be laminar or fully turbulent. This viscous model is shown to compare well with numerical results by computational fluid dynamics and gives a Mach stem height as function of the Reynolds number and Mach number. It is shown that due to the viscous effect, the Mach stem height is increased, through increasing the effective wedge angle.

Study of asymmetrical shock wave reflection in steady supersonic flow

The asymmetrical Mach reflection configuration is studied analytically in this paper, using an asymmetrical model extended from a recent symmetrical model and accounting for the new features related to asymmetry of the two wedges. It is found that the two sliplines do not turn parallel to the incoming flow at the same horizontal location and the sonic throat locates at the position where the difference of slopes of the two sliplines vanishes. This allows us to define a new sonic throat compatibility condition essential to determine the size of the Mach stem. The present model gives the height of the Mach stem, declined angle of the Mach stem from vertical axis, sonic throat location and shape of all shock waves and sliplines. The accuracy of the model is checked by computational fluid dynamics (CFD) simulation. It is found that the Mach stem height is strongly dependent on asymmetry of the wedge angles and almost linearly dependent on the asymmetry of the wedge lower surface lengths. The Mach stem height is shown to be insensitive to the asymmetry of the horizontal positions of the two wedges. The mechanisms for these observations are explained. For instance, it is demonstrated that the Mach reflection configuration remains closely similar when there is horizontal shift of either wedge.

Numerical study on the jet formation of simple-geometry heavy gas inhomogeneities

The jet formation of gas inhomogeneities under the impact of a planar shock wave is an interesting phenomenon that is closely related to shock convergence. In this study, a series of heavy gas inhomogeneities with very simple geometries (square, rectangle, circle, and triangle) are numerically reproduced to trace the source of the jet formation. Special attention is given to the wave patterns that lead to the formation of specific jet forms. The shock-accelerated multicomponent flow is simulated by solving inviscid compressible Euler equations. An up-wind characteristic space-time conservation element and solution element scheme is adopted, and a five-equation model is used to treat the gas interface. The jet types that emerge in the experimental images are explained based on the numerical results, and a typical shock pattern that ensures the jet formation is uncovered. It is found that, physically, the jet is initiated by the impact of the internal Mach stem, and the jet growth is nourished by the high speed gas flow induced by this Mach stem. The width of the jet is determined by the height of the internal Mach stem. Geometrically, a focal wedge enveloped by slip lines emerges in the gas inhomogeneity, in which the gas is accelerated. It is found that the existence of the focal wedge and the coordinates of the wedge tip can be used as qualitative criteria to illustrate the mechanism of the jet formation. These criteria provide a more intuitive basis for understanding the direction, scale, and process of jet formation.

Numerical study of cellular detonation wave reflection over a cylindrical concave wedge

Numerical simulations were performed to study reflection of a stable detonation wave with regular cellular patterns over a cylindrical concave wedge. The dynamics of this reflection phenomenon was described by the two-dimensional reactive Euler equations with a two-step induction-reaction kinetic model and solved numerically using the adaptive mesh refinement code AMROC. The effects of various parameters on the reflection evolution were analyzed in detail. The results indicate that the reflection-type transition of a stable cellular detonation is similar to that of a planar shock wave over a concave wedge. The triple-point trajectory resulted from the Mach reflection when the cellular detonation first encounters the concave wedge coincides with that of the planar shock propagating for the case with the same incident Mach number. As the effective wedge angle continuously increases, the Mach reflection of cellular detonation deviates from that of a planar shock with a reduced Mach stem height, and the transition from Mach to regular reflection occurs at a smaller angle. This observation is further explored by adopting the length-scale (or “corner-signal”) concept, examining the velocity variation of corner signals generated by fluid particles around the wedge tip. The reflection dynamics is described qualitatively by the ratio of two length scales characterizing the detonation structure, namely, the induction-zone and reaction-zone lengths. The increase of these length scales raises the Mach stem height and transition angle. Apart from the detonation length scales, the wedge curvature radius is found to have an opposite effect since the increase of radius expands the region where the corner signals are generated by the particles behind the induction zone, and makes the corner signals persist in a state with attenuating velocity.

An analytical model for asymmetric Mach reflection configuration in steady flows

An analytical model is presented for the configuration of Mach reflection (MR) due to the interaction of two-dimensional steady supersonic flow over asymmetric wedges. The present asymmetric MR model is an extension of an earlier model for the symmetric MR configuration. The overall Mach reflection (oMR) in the asymmetric wedge configuration is analysed as a combination of upper and lower half-domains of symmetric reflection configurations. Suitable assumptions are made to close the combined set of equations. The subsonic pocket downstream of the Mach stem is taken to be oriented along an average inclination, based on the streamline deflections by the Mach stem at the triple points. This assumption is found to give satisfactory results for all types of oMR configurations. The oMR configuration is predicted for all types of reflections such as direct Mach reflection (DiMR), stationary Mach reflection (StMR) and inverse Mach reflection (InMR). The reflection configuration and Mach stem shape given by the model for various sets of wedge angles, especially those giving rise to InMR, have been predicted and validated with the available numerical and experimental data. The von Neumann criterion for oMR is accurately predicted by this model. The asymmetric Mach stem profile is well captured. The variation of Mach stem height with wedge angle is also analysed and it is found that simplification of an asymmetric MR to a symmetric MR leads to over-prediction of the Mach stem height and hence the extent of the subsonic region.

Steady Mach reflection with two incident shock waves

Mach reflection in steady supersonic flow with two incident shock waves is studied. The second incident shock wave is produced by an additional deflection of the wedge lower surface, at some point ensuring that the two incident shock waves would intersect at the reflecting surface in case of normal reflection. Both theory and computational fluid dynamics (CFD) are used to study the flow structure and the influence of the second incident shock wave. The overall flow configuration, in case of Mach reflection, is shown to be composed of a triple shock structure, a shock/shock interaction structure and a shock/slipline reflection structure. Similar phenomenon, triggered by a high downstream pressure, has been observed before numerically, but not studied theoretically. The second incident shock wave reflects over the slipline to deflect the slipline more towards the reflecting surface, increasing thus the Mach stem height, advancing the transition of regular reflection to Mach reflection of the first incident shock wave, and causing an inverted Mach reflection below the usual von Neumann condition. A Mach stem height model built for a weak second incident shock wave is used to study the influence of the second incident shock wave on the Mach stem height. Both theory and CFD predict a maximum of the Mach stem height at some additional wedge deflection angle.

Control of a Mach reflection-induced interaction using an array of vane-type vortex generators

An experimental investigation was conducted to control a Mach reflection (MR)-induced flow separation in a Mach 2.05 flow using a 18 $$^{\circ }$$ shock generator (SG). The study was extended to four SG exit heights (g / w) of 0.87, 0.81, 0.725, and 0.66 primarily to study its effect on the extent of flow separation as well as on Mach stem height, with and without control. Two vane-type vortex generator configurations, namely the ramp vane (RV) with device heights $$h/\delta = 0.3, 0.5, 0.8$$ , and 1.0 and the rectangular vane (RRV) with $$h/\delta = 0.3$$ and 0.5, were studied for control. Each control device array was implemented 10 $$\delta $$ upstream of the separation location for no control. For stable MR interactions (i.e., $$g/w = 0.87, 0.81$$ ), the extent of separation and the reattachment shock strength are seen to decrease with increase in RV height (with $$h/\delta =1.0$$ device showing 17% reduction). However, for unstable MR condition (i.e., $$g/w = 0.725$$ ), RV devices of $$h/\delta = 0.8$$ and 1.0 become ineffective. The RRV2 device ( $$h/\delta =0.5$$ ), on the other hand, was found to be more effective in reducing the extent of separation in both the stable (31%) and unstable (24%) MR conditions. The effectiveness of each control device is also accompanied with an increase in height of the Mach stem. This is, however, not seen as a serious limitation since in such strong interactions it is more important to prevent or avert an intake unstart condition. The separation shock unsteadiness or the $$\sigma _{\mathrm{max}}/P_{\mathrm{w}}$$ value, on the other hand, is seen to increase considerably with controls and seems to be almost independent for $$h/\delta \ge 0.5$$ .