3.2.1 Modification of the standard 2ST for convex surfaces

In order to model correctly the reflections over curved surfaces, the curvature of the surface and the rotational velocity, $\dot{\unicode[STIX]{x1D6FC}}$ , must be attended. For the sake of the arguments that follow, the rotational velocity is evaluated as the time derivative of the angle $\unicode[STIX]{x1D6FC}$ (i.e. slopes presented in figure 14). The rate at which the reflected shock wave changes its orientation depends on the radii dimensions. When radii are small, the reflected shock wave is forced to adjust itself faster to the changing conditions and therefore its rotation increases. Considering the above discussion, the fact that the reflected wave continually changes its orientation implies that additional physics, which is influenced by this non-inertial behaviour, must be considered.

Figure 14.

Inclination angle results for the cylindrical (a), (b), (d) models shown in figure 4 and elliptical model (a) shown in figure 5. The initial angle was not varied since it does not affect the transition (Kleine et al. Reference Kleine, Timofeev, Hakkaki-Fard and Skews2014 and § 3.2.2). Inclination angle $\unicode[STIX]{x1D6FC}$ is defined in figure 10.

The standard 2ST, which is suitable for pseudo-steady reflections, is solved when the reference frame is attached to the reflection point. Thus, the shock waves are considered stationary and the flow passes through the waves (figure 1 b). Figure 15 illustrates the difference in the velocities entering the reflected shock waves when pseudo-steady and non-stationary RR are considered. In figure 15(a) the pseudo-steady RR configuration is presented. The flow enters the incident shock wave at $M_{0}$ and then it is deflected and is reduced to $M_{1}$ . So the flow that enters the reflected shock wave is $M_{1}$ (figure 15 b). When considering the RR over convex surface (figure 5 c), in addition to $M_{1}$ , the reflected shock wave advances towards the incident shock wave. As a result of this relative motion, an increase in the effective velocity that enters the reflected shock wave is obtained, $M_{1}^{\prime }$ (figure 15 d). Each point along the reflected shock wave has a different velocity entering the wave at a different angle.

Figure 15.

Schematic description of the velocity field in domains 0 and 1 when the waves are considered stationary: (a) pseudo-steady RR and (b) velocity entering the reflected shock wave in pseudo-steady reflection. (c) Non-stationary RR and (d) velocity entering the reflected shock wave in non-stationary reflection. $a_{1}$ is the speed of sound at domain 1 and $r$ is the distance from the reflection point. $\dot{\unicode[STIX]{x1D6FC}}$ is the rotational velocity obtained in figure 14.

Another difference results from the curvature of the surface. Unlike the pseudo-steady case where the surface angle is constant (figure 16 a). In convex surfaces, the surface angle increases upstream from the reflection point. Since the incoming flow must be parallel to any local surface angle (‘no-penetration’ condition of the flow), the required deflection angle of the reflected shock wave is decreased along the wave (figure 16 b). Hence, the modified 2ST solution requires the modification of the velocity entering the reflected shock wave together with the modification of the required deflection angle. Since both depend of the distance from the reflection point, the reflected shock wave is divided into increments and the modification is calculated along each point.

Figure 16.

Schematic description of the flow field behind the reflected shock wave (a) pseudo-steady reflection and (b) non-stationary reflections.

Modified 2ST. As regards the calculation of the flow conditions across the incident shock wave, the modified 2ST that will be discussed here does not require any different analysis from that used in the standard 2ST. The conditions behind the incident shock are obtained by solving the standard oblique shock-wave relations (setting $j=1$ and $i=0$ ):

where $\unicode[STIX]{x1D719}_{1}=90^{\circ }-\unicode[STIX]{x1D703}_{w}$ , $M_{0}=M_{s}/\sin \unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D6FE}$ is the heat capacities ratio. Figure 17 shows the velocity behind the shock, $M_{1}$ , and its orientation, $\unicode[STIX]{x1D719}_{1}-\unicode[STIX]{x1D703}_{1}$ . As for the reflected shock wave, consider an arbitrary point A along the reflected shock wave (figure 17). The effective velocity that enters the reflected shock wave is increased because of the relative motion of the waves. Consequently, the Mach number of the flow entering the reflected shock wave at an arbitrary point A is then calculated as follows:

where $a_{1}$ is the sound velocity behind the incident shock wave, and the fluid velocity $V_{1}=M_{1}/a_{1}$ is calculated in the reference frame attached to the reflection point. The orientation of $r_{A}$ with respect to the incident shock-wave motion is initially estimated by a random value of  $\unicode[STIX]{x1D6FC}$ . This value is then updated through an iterative procedure until a convergence of the energy equations was obtained. The required deflection angle through the reflected shock wave decreases and is modified as follows (figure 17):

Note that the dependency of the reflecting surface on the curvature is embodied both in $\dot{\unicode[STIX]{x1D6FC}}$ and in the surface angle at point A. Previous analytical approaches, which were presented in the introduction, failed to integrate this dependence. Substituting the modifications of the flow Mach number and the required deflection angle, equation (3.1) with $i=1$ and $j=2$ results in

A better estimation for $\unicode[STIX]{x1D6FC}$ can be found as $\unicode[STIX]{x1D6FC}=90^{\circ }-(\unicode[STIX]{x1D719}_{1}-\unicode[STIX]{x1D703}_{1})-\unicode[STIX]{x1D719}_{2}$ . The values of $M_{1A}^{\prime }$ and $\unicode[STIX]{x1D703}_{2A}^{\prime }$ are updated with the improved estimation of $\unicode[STIX]{x1D6FC}$ . Equation (3.5) can be solved again. One should note that if there is no change in the angle $\unicode[STIX]{x1D6FC}$ , the solution converged and the conservation equations are satisfied. Following the modification, all the flow properties can now be evaluated with respect to point A. The entire flow field properties can be computed across the shock if the above-mentioned procedure is repeated over the entire reflected shock wave. For the reader’s convenience, a numerical example is presented in § A.1.

Figure 17.

Deflection mechanism and flow domains over a convex surface: velocities, wave angles and turn angles related to the oblique shock-wave relations.

Results of modification and discussion. Figure 18 depicts the analytical results (white lines) over the grey-scale results obtained by the numerical simulations. Figure 18(a–c) shows the full frame results for the cylindrical models. While figure 18(d–f) presents the magnification of the corresponding red squares in figure 18(a–c), respectively. Adjacent to the reflection point, a good agreement was obtained between the simulations and the analytical formulation. The reader should bear in mind that the rotational velocity used for the modification was estimated near the reflection point, where the reflected shock was reasonably straight. Maintaining this notion when the radii are small is more difficult because the wave rapidly grows and has a more pronounced curvature. However, the rotational velocity estimation seems to hold for a larger distance along the reflected shock wave as the radii of the model increases.

Figure 18.

Modified 2ST calculated with $\dot{\unicode[STIX]{x1D6FC}}$ obtained in figure 14 (marked by a white line). (a–c) Show the original images from the simulations. (d–f) Denote the magnifications: (a,d) $R=40~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , (b,e) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ ; (c,f) $R=100~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ .

Figure 19 shows a solution at which an optimal rotational velocity was found to yield the best correlation to the numerical results along a 2 mm distance. The rotational velocity that was found was lower than the previously computed $\dot{\unicode[STIX]{x1D6FC}}$ (figure 14). Figure 20 shows the optimal rotational velocities correlating to the numerical results at different distances along the reflected shock wave. The rotational velocity decreases when larger distances from the reflection point are considered. The same trend was obtained when the modified 2ST was applied for the elliptical model. The previously computed rotational velocity (figure 14) yielded a good correlation with the numerical results close to the reflection point (figure 21 a,d). One should use a smaller rotational velocity for the modified solution to be applicable further away. The reflected shock waves presented in figure 21(b,c,e,f) were calculated with optimal velocities. These rotational velocities were plotted against the distance from the reflection point (figure 22). Similar to the cylindrical models, the rotational velocity of the reflected shock decreased with the increased distance from the reflection point. Therefore, applying the rotational velocity that was found based on a small straight part close to the reflection point was not suitable (figure 14). From a physical perspective, the modification corresponding to the rotational speed is necessary. Yet, the mathematical question on the proper evaluation of the rotational speed along the reflected shock remains open. The rotational velocity evaluation is usually performed using continuum mechanics methods. However, applying it here is impossible because it requires the association of points on the reflected shock at different times. The authors believe that the ‘no-penetration’ condition of the flow, which is presented in the following section of this study, can serve as an additional equation to determine the rotational velocity at each point along the reflected shock wave. As long as the rotational velocity is known, the modified 2ST can be applied to any point along the reflected shock wave. Since the modification is based on the fundamental condition of ‘no penetration’ of the flow, it is not restricted to the area near the reflection point. Yet the crucial area that relates to the transition is the region adjacent to the reflection point.

Figure 19.

Results of the modified 2ST (marked with white lines) calculated with the optimal rotational velocity, $\dot{\unicode[STIX]{x1D6FC}}^{opt}$ . (a–c) Show the original images from the simulations. (d–f) Denote the magnifications. (a,d) $R=40~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ ; (b,e) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ ; and (c,f) $R=100~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ .

Figure 20.

Optimum rotational velocity as a function of the distance from the reflection point, $d$ , over cylindrical models.

Figure 21.

Results of the modified 2ST (marked by white lines) applied to the reflection over the ellipse. (a–c) Denote the original images from the simulations. (d–f) Present the magnifications. (a,d) Modified 2ST with computed in figure 14; (b,e) modified 2ST with the optimal rotational velocity for 2 mm length; and (c,f) modified 2ST with the optimal rotational velocity for the 4 mm length.

Figure 22.

Optimum rotational velocity as a function of the distance from the reflection point, $d$ , over the ellipse.

Figure 23(a) presents a comparison between the standard 2ST and the modified 2ST over a 70 mm cylinder. The radial velocity and the deflection angle corrections depend on the distance from the reflection point. Therefore, a very small correction is required. This deviation increases from the reflection point. Figure 23(b) graphically demonstrates the deviation by the shock polars. The $r$ -polar of the modified 2ST starts at $\unicode[STIX]{x1D703}_{2}^{\prime }$ , which is smaller than $\unicode[STIX]{x1D703}_{2}$ . In addition, the flow Mach number of the modified polar, $M_{1}^{\prime }$ , is larger than that of the flow Mach number computed by the standard 2ST, $M_{1}$ . The two-shock solution is obtained at the intersection of the $r$ -polar with the vertical axis. One can see that the $p_{2}^{\prime }/p_{1}<p_{2}/p_{1}$ .

Figure 23.

The difference between the standard and modified 2ST. (a) Resulting reflected shock wave obtained by the standard 2ST and by the modified 2ST over a 70 mm model. (b) Polar representation of the solutions.

These modified conditions make it easier for the flow to be deflected. Consequently, the RR can be retained for smaller surface angles in comparison with pseudo-steady flows. This explanation sheds new light on the transition criterion disagreements between the pseudo-steady and unsteady reflections. As the effective flow entering the reflected shock wave is increased and the required deflection angle is decreased, the reflected shock-wave angle and the induced flow behind the reflected shock are decreased (for further explanation see § A.2). The rotation of the reflected shock wave alters the effective flow Mach number entering the reflected wave. This velocity is determined so the ‘no-penetration’ condition is maintained (as will be shown in the next section). For smaller radii, the rapid change in the surface angles affects both $\dot{\unicode[STIX]{x1D6FC}}$ and the required deflection angle, $\unicode[STIX]{x1D703}_{2}^{\prime }$ . Hence, a significant modification is needed and the delay in the transition will be more noticeable. This finding is consistent with the observation that the radius of curvature is the only available length scale over curved surfaces (Ram et al. Reference Ram, Geva and Sadot2015) and strengthens the claim that the transition angles depend of the radius of curvature (Takayama & Sasaki Reference Takayama and Sasaki1983; Ben-Dor Reference Ben-Dor2007). No difference between the pseudo-steady and unsteady reflections was observed near the reflection point. Nevertheless, these data are insufficient in determining whether the transition criteria are similar (Hakkaki-Fard & Timofeev Reference Hakkaki-Fard and Timofeev2012; Kleine et al. Reference Kleine, Timofeev, Hakkaki-Fard and Skews2014). Basing the transition criterion on the conditions very close to the reflection point is insufficient because the zone behind the reflection point that leads to the transition is unknown. Moreover, their observation that coarser grid numerical results depart from the pseudo-steady results can also be explained now. The shock waves in coarser grids tend to be thicker. Therefore, they force a large artificial modification in the required deflection angles, $\unicode[STIX]{x1D703}_{2}^{\prime }$ . The MR is eventually formed when the RR can no longer sustain the required deflection of the flow. Consequently, in unsteady reflections, the increase in the flow Mach number along with the decrease in the required deflection angle will delay the transition.

3.2.2 The physical mechanism that leads to transition at $90^{\circ }$ inter-shock-waves angle

The RR configuration is formed so that a supersonic flow can be negotiated about an obstacle. Consider the pseudo steady flow that is shown in figure 24. The induced flow behind the reflected shock wave, $u_{2}$ , is parallel to the surface and the velocity component, which is perpendicular to the surface, is therefore zero (‘no-penetration’ condition).

Figure 24.

Schematic description of the flow regions in pseudo-steady RR.

The flow velocity along the surface behind the reflected shock wave in region 2 is given by:

where $\unicode[STIX]{x0394}u_{i}$ and $\unicode[STIX]{x0394}u_{r}$ are the induced flow velocities from the incident and the reflected shock waves, respectively. Due to the ‘no-penetration’ condition, the normal components of the induced flow velocities of the incident and reflected shock waves counter each other:

Hryniewicki, Gottlieb & Groth (Reference Hryniewicki, Gottlieb and Groth2016) used (3.7) and constructed an error function

When this equals zero a solution for a regular reflection occurs. In other words, the reflected shock-wave angle is chosen in order to maintain the geometrical ‘no-penetration’ requirement.

Figure 25.

Unsteady RR configuration in which the reflected and incident shock waves are perpendicular.

The elementary requirement that the flow will not penetrate the surface exists in both pseudo-steady and unsteady flows. The particular moment at which the reflected shock becomes perpendicular to the incident shock wave was found to be the moment at which the RR $\rightarrow$ MR transition occurs. An explanation to this finding is that at this perpendicular angle equation (3.8) becomes identically zero. (This moment is illustrated in figure 25.):

Before the transition, in order for the ‘no-penetration’ condition to be held, a rotational velocity of the reflected shock wave must be introduced. This velocity causes an increase in the effective flow Mach number entering the reflected shock wave. As a result, the reflected shock-wave angle and the induced flow behind the reflected wave are reduced so (3.8) is maintained. The analytical justification is consistent with the findings presented in figure 11. As the radius decreases, higher rotational velocities are involved enabling the configuration to adjust quickly to the changing surface. This in turn is possible only if the angle between the reflected and incident shock waves increases.

The particular $90^{\circ }$ inter angle between the incident and reflected shock waves enables the existence of an RR without the need of an additional rotational velocity. Indeed, it was demonstrated in figure 13 that when the reflected shock wave is perpendicular to the incident shock wave, the rotation of the reflected shock wave reaches zero. Soon after, the reflected shock wave is no longer perpendicular to the incident shock wave. The only way to enable an RR, so that (3.8) is held, is by introducing an additional rotational velocity of the reflected shock wave. Due to the inertial movement of the flow around the shock wave, the buildup of the rotational velocity cannot transpire immediately. The delay in the rotational velocity buildup is demonstrated in the inflections of the rotational velocity shown in figure 13. Since an RR cannot deflect the flow so that ‘no-penetration’ condition is held, MR is then formed. The geometrical fundamental obligation of ‘no penetration’ is the criterion for the RR $\rightarrow$ MR transition in unsteady reflection.

The use of a highly refined computational mesh and high-resolution experiments enabled us to observe that this intriguing behaviour happened near the RR $\rightarrow$ MR transition. It was evident from both methods that an MR could not be formed if the angle between the incident and reflected shock waves was larger than $90^{\circ }$ . It was very difficult to decide if this $90^{\circ }$ angle was the precise angle at transition considering the above listed methods to determined transition. However, based on the high-resolution findings, it was reasonable to assume that the transition occurred very close to this point at moderate incident shock-wave Mach numbers.

Figure 26 depicts the perpendicular behaviour of the reflected and incident shock waves for reflections over models (a–d) near the transition point. Corresponding to the inflections in figure 13, the perpendicularity was observed to sustain over a certain period. The ranges of the corresponding surface angles at which this behaviour was evident are given in table 1. Optical limitations and changing the simulation time increments may slightly vary these ranges. Geometrical dependence of the perpendicularity is obtained in term of a slight increase in the transition angle when the radius of the model increases. For cylindrical model (d), which has a 100 mm radius, the transition was found to be similar to the pseudo-steady criterion. The effect of the initial angle on the transition angle was also examined by using two models having 70 mm radii and initial angles of $60$ and $90^{\circ }$ . Similar to the findings in Skews & Kleine (Reference Skews and Kleine2010), it was found that the initial angle did not influence the transition. Figures 26(f) and 26(h) show the magnification of the red regions shown in figures 26(b) and 26(d), respectively. The horizontal behaviour of the reflected shock waves adjacent to the reflection is apparent at the same surface angle (table 1). This also reinforces the notion that the perpendicularity of the waves and the corresponding transition depends on the geometric scaling.

Figure 26.

Simulation results for reflections over models (a–d) near the transition angles (table 1). The original images from the computations are shown in (a–d). The magnification of the areas marked by red squares of the results and the reflected shock-wave orientation as extracted by the program are shown in (e–h). (a,e) $R=40~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=45.4^{\circ }$ ; (b,f) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=45.8^{\circ }$ ; (c,g) $R=100~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=46.3^{\circ }$ ; (d,h) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=90^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=45.8^{\circ }$ .

Table 1.

Surface angles corresponding to the horizontal behaviour of the reflected shock wave over cylindrical surfaces.

a The upper boundary of the theoretical error is limited by the detachment criterion.

For the case of reflections over the elliptical models, a geometric scaling was also observed. In these cases, due to the varying curvature, a varying rate of change is imposed on the transition process. Table 2 presents the surface angles of the ellipses corresponding to the moment in which the perpendicularity behaviour of the shock waves was observed. Figure 27 demonstrates the perpendicularity of the reflected and incident shock waves over the elliptical models. For the ellipse with a decreasing radius of curvature (model b), the transition angle is closer to the detachment criterion and is similar to the transition angle found for reflections over the 100 mm cylindrical model. For the elliptical model with an increasing radius of curvature (model a), the transition angle is similar to the transition angle obtained for reflections over the 70 mm cylindrical model. This deviation implies that geometrical scaling may be a solution to eliminate this discrepancy. Using Pratt (Reference Pratt1987) method, a circle was fitted to the area after the transition to describe the MR growth (figure 28), and the scale that was found appropriate for the transition was the best-fit circle prior to the transition ( $46^{\circ }\leqslant \unicode[STIX]{x1D703}_{w}\leqslant 60^{\circ }$ ). The radii of the resulting equivalent circles are 68 and 93 mm for models (a) and (b), respectively (figure 28). In fact, as long as a circle can be fitted, the surface curvature does not play a role in the determination of the transition.

Figure 27.

Simulation results for reflections over elliptical models near the transition (table 2). (a,b) The original images obtained by the computations. (c,d) The magnification of the areas marked by red squares and the reflected shock-wave orientation extracted by the program. (a,c) $a=150~\text{mm}$ , $b=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=45.8^{\circ }$ ; (b,d) $a=70~\text{mm}$ , $b=150~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=90^{\circ }$ , $\unicode[STIX]{x1D703}_{w}=46.2^{\circ }$ .

Figure 28.

Best-fit circle to the ellipse in the area prior to the transition at surfaces angles $46^{\circ }\leqslant \unicode[STIX]{x1D703}_{w}\leqslant 60^{\circ }$ : (a) a circle with a 68 mm radius is fitted to an ellipse with an increasing radius of curvature, and (b) a circle with a 93 mm radius is fitted to an ellipse with a decreasing radius of curvature.

Table 2.

Surface angles corresponding to the horizontal behaviour of the reflected shock wave over elliptical surfaces.

Figure 29 demonstrates the perpendicularity of the reflected shock wave near the transition points that was obtained experimentally. The Mach stems were measured in the following frames, and the angles between the incident and reflected shock waves were found to be smaller than $90^{\circ }$ . Figure 29(g) demonstrates the inflection process as was obtained experimentally. The surface angles at which the reflected shock wave was horizontal, i.e. perpendicular to the incident shock wave, were found to be consistent with the transition angles obtained by extrapolation, with the former method having the advantage of producing smaller uncertainties.

Figure 29.

Experimental results for reflections over cylindrical models at angles at which the reflected shock is horizontal. (a–c) The original images from the experiments. (d–f) The magnification of the areas marked by red squares of the results and the reflected shock-wave orientation. (a,d) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}\approx 45^{\circ }$ ; (b,e) $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=90^{\circ }$ , $\unicode[STIX]{x1D703}_{w}\approx 45^{\circ }$ ; (c,f) $R=100~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{w}\approx 46^{\circ }$ ; (g) variation of the reflected shock-wave angle along the surface for $R=70~\text{mm}$ , $\unicode[STIX]{x1D703}_{i}=60^{\circ }$ .

The perpendicularity that was found to exist in both experiments and simulations could be regarded as a geometrical criterion for the RR $\rightarrow$ MR transition. Incidentally, this $90^{\circ }$ condition is similar to the condition suggested by Shirouzu & Glass (Reference Shirouzu and Glass1986) for the transition between a single Mach reflection and transitional Mach reflection in pseudo-steady reflections. Until now, the rotation of the reflected shock wave was not considered significant and its existence was neglected. However, it was demonstrated that the rotation of the reflected shock wave affects the RR evolution and its termination. The RR evolution and transition to MR waves are perpendicular, stem from the ‘no penetration’ of the flow condition. Which is a fundamental condition that must be held for any flow that passes over any solid body. While performing experiments, it is considerably easier to visualise the perpendicularity of the reflected and incident shock waves. Rather than the incorrect determination based on the appearance of a Mach stem or to carry out the more complicated and uncertain task of measuring the Mach stem length and extrapolating it to zero (Kleine et al. Reference Kleine, Timofeev, Hakkaki-Fard and Skews2014; Ram et al. Reference Ram, Geva and Sadot2015).