3.1 Triple point

First, we consider the region near the triple points, given in figure 11. The classic von Neumann three-shock theory is applied individually to each triple point, which is the point of confluence of four discontinuities: incident shock $i$ , reflected shock $r$ , Mach stem $ms$ and slip line $s$ .

The sign conventions for the angles are as shown in figure 11. Since the vertical axis is inverted for the lower domain, the sign conventions are also reversed. For the general case of DiMR given in figure 11(c), the flow deflection angle $\unicode[STIX]{x1D703}_{3l}$ is taken to be positive for any deviation of the streamline towards the $x$ -axis by a left running shock wave, as in the segment of the Mach stem in this domain. The shock angle $\unicode[STIX]{x1D719}_{3l}$ is acute. On the other hand, the upper domain given in figure 11(b) has an InMR. Following the sign conventions, the flow deflection angle $\unicode[STIX]{x1D703}_{3u}$ (depicted as $-\unicode[STIX]{x1D703}_{3u}$ for clarity) is negative. The segment of the Mach stem in this domain is a left running shock wave (instead of a right running wave if it were a DiMR). Hence, in accordance with the mathematics, the shock angle $\unicode[STIX]{x1D719}_{3u}$ is taken as obtuse. (The angle would be acute for DiMR and $90^{\circ }$ for StMR.)

Figure 11.

(a) Schematic of asymmetric MR; insets (b) upper triple point region, and (c) lower triple point region.

The following equations are used to relate the downstream properties ( $j$ ) to the upstream ones ( $i$ ) across an oblique shock. The details of these relations, which are derived from the conservation equations, can be found in Li & Ben-Dor (Reference Li and Ben-Dor1997) and Ben-Dor (Reference Ben-Dor2007). Here the flow properties are – Mach number $M$ , shock angle $\unicode[STIX]{x1D719}$ , pressure $P$ , flow deflection angle $\unicode[STIX]{x1D703}$ , density $\unicode[STIX]{x1D70C}$ and local acoustic speed $a$ .

(3.1) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}\displaystyle M_{j} & = & F(M_{i},\unicode[STIX]{x1D719}_{j})\\ \unicode[STIX]{x1D703}_{j} & = & G(M_{i},\unicode[STIX]{x1D719}_{j})\\ P_{j} & = & P_{i}H(M_{i},\unicode[STIX]{x1D719}_{j})\\ \unicode[STIX]{x1D70C}_{j} & = & \unicode[STIX]{x1D70C}_{i}E(M_{i},\unicode[STIX]{x1D719}_{j})\\ a_{j} & = & a_{i}A(M_{i},\unicode[STIX]{x1D719}_{j})\end{array}\right\}\unicode[STIX]{x1D70E}_{j}=\unicode[STIX]{x1D70E}_{i}\unicode[STIX]{x1D6F1}(M_{i},\unicode[STIX]{x1D719}_{j}),\end{eqnarray}$$

where, the functions are given as,

(3.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle F(M,\unicode[STIX]{x1D719})=\left[\frac{1+(\unicode[STIX]{x1D6FE}-1)M^{2}\sin ^{2}\unicode[STIX]{x1D719}+\left[\displaystyle \frac{(\unicode[STIX]{x1D6FE}+1)^{2}}{4}-\unicode[STIX]{x1D6FE}\sin ^{2}\unicode[STIX]{x1D719}\right]M^{4}\sin ^{2}\unicode[STIX]{x1D719}}{\left[\unicode[STIX]{x1D6FE}M^{2}\sin ^{2}\unicode[STIX]{x1D719}-\displaystyle \frac{\unicode[STIX]{x1D6FE}-1}{2}\right]\left[\displaystyle \frac{\unicode[STIX]{x1D6FE}-1}{2}M^{2}\sin ^{2}\unicode[STIX]{x1D719}+1\right]}\right]^{1/2} & \displaystyle\end{eqnarray}$$

(3.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle G(M,\unicode[STIX]{x1D719})=\arctan \left[2\cot \unicode[STIX]{x1D719}\frac{M^{2}\sin ^{2}\unicode[STIX]{x1D719}-1}{M^{2}(\unicode[STIX]{x1D6FE}+\cos 2\unicode[STIX]{x1D719})+2}\right] & \displaystyle\end{eqnarray}$$

(3.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle H(M,\unicode[STIX]{x1D719})=\frac{2}{\unicode[STIX]{x1D6FE}+1}\left[\unicode[STIX]{x1D6FE}M^{2}\sin ^{2}\unicode[STIX]{x1D719}-\frac{\unicode[STIX]{x1D6FE}-1}{2}\right] & \displaystyle\end{eqnarray}$$

(3.2d ) $$\begin{eqnarray}\displaystyle & \displaystyle E(M,\unicode[STIX]{x1D719})=\frac{(\unicode[STIX]{x1D6FE}+1)M^{2}\sin ^{2}\unicode[STIX]{x1D719}}{(\unicode[STIX]{x1D6FE}-1)M^{2}\sin ^{2}\unicode[STIX]{x1D719}+2} & \displaystyle\end{eqnarray}$$

(3.2e ) $$\begin{eqnarray}\displaystyle & \displaystyle A(M,\unicode[STIX]{x1D719})=\frac{[(\unicode[STIX]{x1D6FE}-1)M^{2}\sin ^{2}\unicode[STIX]{x1D719}+2]^{1/2}[2\unicode[STIX]{x1D6FE}M^{2}\sin ^{2}\unicode[STIX]{x1D719}-(\unicode[STIX]{x1D6FE}-1)]^{1/2}}{(\unicode[STIX]{x1D6FE}+1)M\sin \unicode[STIX]{x1D719}}. & \displaystyle\end{eqnarray}$$

Across the incident shock $i$ , we have:

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{1}=\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D6F1}(M_{0},\unicode[STIX]{x1D719}_{1}).\end{eqnarray}$$

Then, across the reflected shock $r$ , we have:

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{2}=\unicode[STIX]{x1D70E}_{1}\unicode[STIX]{x1D6F1}(M_{1},\unicode[STIX]{x1D719}_{2}).\end{eqnarray}$$

And, across the Mach stem $ms$ , we have:

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{3}=\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D6F1}(M_{0},\unicode[STIX]{x1D719}_{3}).\end{eqnarray}$$

It is to be noted here that weak shock solutions for (3.3) and (3.4), and strong shock solutions for (3.5), are to be considered.

Finally, we have the following boundary conditions:

(3.6a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D703}_{w},\quad P_{3}=P_{2}\quad \text{and}\quad \unicode[STIX]{x1D703}_{3}=\unicode[STIX]{x1D703}_{1}-\unicode[STIX]{x1D703}_{2}. & & \displaystyle\end{eqnarray}$$

The above set of 18 equations are solved for the 18 unknowns: $M_{1}$ , $M_{2}$ , $M_{3}$ , $P_{1}$ , $P_{2}$ , $P_{3}$ , $\unicode[STIX]{x1D719}_{1}$ , $\unicode[STIX]{x1D719}_{2}$ , $\unicode[STIX]{x1D719}_{3}$ , $\unicode[STIX]{x1D703}_{1}$ , $\unicode[STIX]{x1D703}_{2}$ , $\unicode[STIX]{x1D703}_{3}$ , $\unicode[STIX]{x1D70C}_{1}$ , $\unicode[STIX]{x1D70C}_{2}$ , $\unicode[STIX]{x1D70C}_{3}$ , $a_{1}$ , $a_{2}$ and $a_{3}$ .

For the particular configuration of MR shown in figures 9 and 11, the solution would yield a negative value of $\unicode[STIX]{x1D703}_{3u}$ in the upper domain. This negative value is in accordance with the sign convention and is to be retained as such.

While solving the region around the triple points in the upper and lower domains, it is not particularly required to include equations for $\unicode[STIX]{x1D70C}$ and $a$ in the set to be solved simultaneously; the set of equations would otherwise contain 12 equations and 12 unknowns, and could be solved nevertheless. However, these quantities are being solved for, as they are required in the upcoming subsection.

As evident from the formulation above, each region around a triple point can be solved for, independent of other geometric and/or physical constraints. Consequently, the angle of deflection of the slip line from the $x$ -axis (and hence the classification among DiMR, StMR or InMR) at either of the triple points is independent of the other.

3.2 Subsonic pocket downstream of the Mach stem

3.2.1 Average deflection at the Mach stem

As stated earlier, the inclination of the axis $x_{i}$ is based on an average deflection of the flow by the Mach stem, given as:

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{4}=\unicode[STIX]{x1D703}_{avg}=\frac{(-\unicode[STIX]{x1D703}_{3u})+\unicode[STIX]{x1D703}_{3l}}{2}.\end{eqnarray}$$

The opposite signs of $\unicode[STIX]{x1D703}$ for regions $3_{u}$ and $3_{l}$ are evoked so as to comply with the flipped $y$ -axes (and sign conventions) of the upper and lower domains. In this subsection, a positive sign is given to angles measured anticlockwise, as for $\unicode[STIX]{x1D703}_{4}$ and $\unicode[STIX]{x1D703}_{3l}$ .

With the Mach stem being modelled as a monotonic second-order curve with continuous slope and no inflection point (using (2.1)–(2.3)), there exists exactly a single location on the Mach stem at which the incoming streamline is deflected by $\unicode[STIX]{x1D703}_{4}$ . This location is named $G$ , as given in figure 9. The $x$ -axis passes through this point on the Mach stem, while its location on the Mach stem itself (given by $H_{u}$ and $H_{l}$ ) is a variable as of yet.

To keep in line with the sign conventions of each domain, we henceforward redefine $\unicode[STIX]{x1D703}_{4}$ as:

(3.8a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{4^{\prime }u}=-\unicode[STIX]{x1D703}_{4}\quad \text{and}\quad \unicode[STIX]{x1D703}_{4^{\prime }l}=\unicode[STIX]{x1D703}_{4}. & & \displaystyle\end{eqnarray}$$

3.2.2 Average flow properties behind the Mach stem

With $\unicode[STIX]{x1D703}_{4}$ known, the following 5 equation set is solved for the 5 downstream properties, at some intermediate location $G$ on the Mach stem: $\unicode[STIX]{x1D719}_{4}$ , $M_{4}$ , $P_{4}$ , $\unicode[STIX]{x1D70C}_{4}$ and $a_{4}$ . Again, strong shock solution is taken for this case.

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{4}=\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D6F1}(M_{0},\unicode[STIX]{x1D719}_{4}).\end{eqnarray}$$

Following Li & Ben-Dor (Reference Li and Ben-Dor1997), the average Mach number ( $\overline{M}$ ) behind the Mach stem segments for each domain is defined based on the average velocity ( $\overline{u}$ ) and average acoustic speed ( $\overline{a}$ ), and is given as:

(3.10) $$\begin{eqnarray}\overline{M}=\overline{u}/\overline{a},\end{eqnarray}$$

where a first-order approximation is applied for the following averaging:

(3.11) $$\begin{eqnarray}\displaystyle \overline{u}=\frac{1}{H_{m}\unicode[STIX]{x1D70C}}\int _{0}^{H_{m}}\unicode[STIX]{x1D70C}u.e_{x}\,\text{d}y=\frac{1}{2\overline{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x1D70C}_{3}u_{3}\cos \unicode[STIX]{x1D703}_{3}+\unicode[STIX]{x1D70C}_{4}u_{4}\cos \unicode[STIX]{x1D703}_{4}) & & \displaystyle\end{eqnarray}$$

(3.12a,b ) $$\begin{eqnarray}\displaystyle \overline{a}={\textstyle \frac{1}{2}}(a_{3}+a_{4})\quad \text{and}\quad \overline{\unicode[STIX]{x1D70C}}={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70C}_{3}+\unicode[STIX]{x1D70C}_{4}). & & \displaystyle\end{eqnarray}$$

Hence, for the subsonic region behind the Mach stem, we have:

(3.13) $$\begin{eqnarray}\overline{M}=\frac{2(\unicode[STIX]{x1D70C}_{3}u_{3}\cos \unicode[STIX]{x1D703}_{3}+\unicode[STIX]{x1D70C}_{4}u_{4}\cos \unicode[STIX]{x1D703}_{4})}{(\unicode[STIX]{x1D70C}_{3}+\unicode[STIX]{x1D70C}_{4})(a_{3}+a_{4})},\end{eqnarray}$$

where the velocities are given as,

(3.14a,b ) $$\begin{eqnarray}\displaystyle u_{3}=M_{3}a_{3}\quad \text{and}\quad u_{4}=M_{4}a_{4}. & & \displaystyle\end{eqnarray}$$

The cosine function in the above relations lets us ignore the differing signs (positive or negative) in the two domains for $\unicode[STIX]{x1D703}_{3u}$ , $\unicode[STIX]{x1D703}_{3l}$ and $\unicode[STIX]{x1D703}_{4}$ .

3.3 Expansion-fan interaction region

Figure 12.

Schematic of asymmetric MR: inset; expansion fan interacting with reflected shock wave in upper domain.

Figure 13.

Schematic of asymmetric MR: inset; expansion fan interacting with reflected shock wave in lower domain.

The regions of interaction of the expansion fans with the reflected shocks, $r$ , are given in figures 12 and 13. The inclination of the subsonic pocket (along axis $x_{i}$ ) by angle $\unicode[STIX]{x1D703}_{4}$ is included as an angle correction, in the sense that the streamlines are shown to become parallel to $x_{i}$ rather than $x$ , after interacting with a portion of the expansion fan (Li & Ben-Dor Reference Li and Ben-Dor1997; Ben-Dor Reference Ben-Dor2007). The slip line also follows the curvature of the weak tangential discontinuity (originating from point $B$ ) which is one among an infinite number of such entropy layers. The pressure and flow direction remain constant across each layer. Details of the analytical study of expansion-fan interaction with a shock wave can be found in Li & Ben-Dor (Reference Li and Ben-Dor1996). Beyond the characteristic line $RCD$ of the centred expansion fan, where the flow for the most part assumes a common inclination $\unicode[STIX]{x1D703}_{4}$ , further influence of the expansion fan is ignored as the flow downstream of the throat becomes supersonic and properties there do not influence the upstream flow any further.

Consider the point where the characteristic line $RCD$ interacts with the reflected shock $r$ . The point upstream of the shock is labelled as $C$ , and the point downstream as $C^{\prime }$ . Here, $\unicode[STIX]{x1D707}$ is the Mach angle made by the characteristic line at that point. The angle between the flow direction and axis $x_{i}$ at $C$ , is given as $\unicode[STIX]{x1D6FC}$ .

Firstly, the Prandtl–Meyer relation is applied between a location in region 2 and point $D$ . The net change in the flow direction here is $\unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{4^{\prime }}$ , where $\unicode[STIX]{x1D703}_{4^{\prime }}$ assumes the value of $\unicode[STIX]{x1D703}_{4^{\prime }u}$ or $\unicode[STIX]{x1D703}_{4^{\prime }l}$ as given in (3.8). The pressure remains constant across the parallel entropy layers. Hence, we have the following relations for $M_{D}$ , $P_{D}$ and $P_{C^{\prime }}$ .

(3.15a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{4^{\prime }}=\unicode[STIX]{x1D708}(M_{D})-\unicode[STIX]{x1D708}(M_{2}),\quad P_{D}=P_{2}\unicode[STIX]{x1D712}(M_{2},M_{D}),\quad \text{and}\quad P_{C^{\prime }}=P_{D}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ is the Prandtl–Meyer function and $\unicode[STIX]{x1D712}$ is an isentropic function relating the pressures across an expansion fan, given as follows,

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D708}(M)=\left(\frac{\unicode[STIX]{x1D6FE}+1}{\unicode[STIX]{x1D6FE}-1}\right)^{1/2}\arctan \left[\frac{(\unicode[STIX]{x1D6FE}-1)(M^{2}-1)}{\unicode[STIX]{x1D6FE}+1}\right]^{1/2}-\arctan (M^{2}-1)^{1/2} & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}(M_{i},M_{j})=\left[\frac{2+(\unicode[STIX]{x1D6FE}-1)M_{i}^{2}}{2+(\unicode[STIX]{x1D6FE}-1)M_{j}^{2}}\right]^{\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D6FE}-1)}. & \displaystyle\end{eqnarray}$$

Now, consider a location in region 1, and point $C$ . The Prandtl–Meyer function can be applied again for a change in flow direction by angle $\unicode[STIX]{x1D703}_{1}-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D703}_{4^{\prime }})$ . Standard relations can be applied across shock $r$ at points $C$ and $C^{\prime }$ . Hence, the following set of 6 equations are solved for the 6 remaining unknowns: $M_{C}$ , $M_{C^{\prime }}$ , $P_{C}$ , $\unicode[STIX]{x1D719}_{C}$ , $\unicode[STIX]{x1D703}_{C^{\prime }}$ and  $\unicode[STIX]{x1D6FC}$ .

(3.18a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708}(M_{C})-\unicode[STIX]{x1D708}(M_{1})=\unicode[STIX]{x1D703}_{1}-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D703}_{4^{\prime }}),\quad P_{C}=P_{1}\unicode[STIX]{x1D712}(M_{1},M_{C}), & & \displaystyle\end{eqnarray}$$

(3.19a,b ) $$\begin{eqnarray}\displaystyle M_{C^{\prime }}=F(M_{C},\unicode[STIX]{x1D719}_{C}),\quad P_{C^{\prime }}=P_{C}H(M_{C},\unicode[STIX]{x1D719}_{C}), & & \displaystyle\end{eqnarray}$$

(3.20a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{C^{\prime }}=G(M_{C},\unicode[STIX]{x1D719}_{C}),\quad \text{and}\quad \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D703}_{C^{\prime }}. & & \displaystyle\end{eqnarray}$$

The flow parameters calculated herewith, can now be applied to solve for the geometrical relations.

3.4 Flow in the subsonic pocket and geometrical relations

The boundary conditions at the interface apply to both the upper and lower domains, and hence the set of equations in this subsection have to be solved simultaneously.

Consider the subsonic pocket $TFEKG$ in figure 9. Here, the cross-sectional area of the pocket at the Mach stem is approximated as the vertical height of the Mach stem $H_{m}$ . $H_{s}$ is the perpendicular area of the cross-section at the throat i.e. $EK$ . In order to evaluate $H_{s}$ , certain geometric manipulations are employed. The details of the geometric relations and the set of equations employed for this subsection are detailed in appendix A.

It is to be noted here that the leading edges of the two wedges have been considered to have a horizontal gap $L$ and the same has been included while calculating the corresponding abscissae. $L$ is to be set to zero where the leading edges are required to be at the same horizontal location.

The quasi-one-dimensional analysis and the geometrical relations along with the closing equations, yield a total of 26 equations with 26 unknowns, including $H_{mu}$ and $H_{ml}$ , from which we get the total height of the Mach stem.

(3.21) $$\begin{eqnarray}H_{mt}=H_{mu}+H_{ml}.\end{eqnarray}$$

Finally, the coordinates of the points obtained from the solution are used to reconstruct the entire oMR configuration. Further discussion on this is given in § 4.3.

4.1 Symmetry case verification

Due to the nature of the algorithm used herein, the asymmetric MR formulation is expected to predict symmetric MR cases as well. Hence, the data of non-dimensional Mach stem height for symmetric MR are compared with those of the present model in table 1 and figure 14. It follows from symmetry that for equal upper and lower wedge angles, the shock reflection configuration for any mathematical model would be identical in the upper and lower domains. The comparison has hence been made with symmetric half-models by taking half of the height of the Mach stem $H_{mt}$ and that between the wedges $H_{t}$ .

Table 1.

Comparison of the non-dimensional Mach stem height, $H_{mt}/H_{t}$ , with the values from $^{a}$ Li & Ben-Dor (Reference Li and Ben-Dor1997),  $^{b}$ Mouton & Hornung (Reference Mouton and Hornung2008) and  $^{c}$ Gao & Wu (Reference Gao and Wu2010). All angles are in degrees.

Figure 14.

Comparisons of non-dimensional Mach stem height, $H_{mt}/H_{t}$ , from current theory with those of Gao & Wu (Reference Gao and Wu2010), Li & Ben-Dor (Reference Li and Ben-Dor1997) and Mouton & Hornung (Reference Mouton and Hornung2007), and numerical results from Gao & Wu (Reference Gao and Wu2010);  $w/H_{t}=1.1$ .

The data reported in table 1 and figure 14 are taken from Gao & Wu (Reference Gao and Wu2010). From table 1, it can be seen for cases 5 and 6 that the present model gives better, if not the best, agreement with the numerical data. From figure 14, it is evident that the present model (and that of Li & Ben-Dor (Reference Li and Ben-Dor1997)) shows excellent agreement with numerical results towards higher Mach numbers ( $M_{0}>4.5$ ). In selected cases, the present theory performs even better than that of Gao & Wu (Reference Gao and Wu2010). The Mouton (Reference Mouton2007) theory deviates from the numerical data at such high Mach numbers. At lower Mach numbers, there is considerable under-prediction of the Mach stem height by the present model. Since the primary distinction of the Gao & Wu (Reference Gao and Wu2010) model is the consideration of secondary waves in the MR configuration, the above analysis suggests that ignoring the secondary waves leads to large errors at low Mach numbers. However, the extent of their contribution at higher Mach numbers needs to be further considered, since we see very good agreement with the numerical data for a certain range, and some over-prediction beyond that (figure 14 a,b).

It can also be noted from table 1 and figure 14 that there exists some discrepancy between the present data and those reported in Li & Ben-Dor (Reference Li and Ben-Dor1997). A closer scrutiny revealed a possible error in the algorithm reported in the latter (see Li & Ben-Dor Reference Li and Ben-Dor1997, C9, the appendix). The sign of the angle $\unicode[STIX]{x1D703}_{3}$ is incorrect, and the same has been corrected in the present theory in (A 12). It has separately been verified that for the symmetric case, the present asymmetric formulation gives exactly the same results as the corrected Li & Ben-Dor (Reference Li and Ben-Dor1997) model.

4.2 Variation of Mach stem height

Using the present algorithm, the wedge-angle-variation-induced changes in the Mach stem height have been estimated for a series of cases. Also, the wedge angles at which the Mach stem height becomes zero is in direct conformation to the von Neumann criterion. It is to be noted however that each value calculated using the present theory is a steady-state result. There is no dynamic effect (Naidoo & Skews Reference Naidoo and Skews2014) associated with changing wedge angles in this analytical model.

Figure 15.

Symmetric von Neumann criterion for $M_{0}=4.96$ . (a) The dual-solution domain in the ( $\unicode[STIX]{x1D703}_{wu}$ , $\unicode[STIX]{x1D703}_{wl}$ )-plane (Li et al. Reference Li, Chpoun and Ben-Dor1999); (b) plots of normalised Mach stem height versus lower wedge angle; (c) shock polars for symmetric DiMR–DiMR case; (d) shock polars for asymmetric DiMR-DiMR case; (e) shock polars for asymmetric StMR-DiMR; (f) shock polars for asymmetric InMR-DiMR case.

For all calculations in this subsection, the Mach number $M_{0}$ has been taken to be 4.96. The geometric parameters have been taken following Li et al. (Reference Li, Chpoun and Ben-Dor1999). The vertical height between the wedges is 71 mm and the wedges are of equal slant height at 40 mm each. The total pressure is taken to be 8.5 bar and the total temperature is 473 K. (These parameters are required as input to the algorithm. However, except the Mach number and geometrical parameters, other flow properties are dealt with as ratios across the discontinuities, and can be arbitrarily assumed within the perfect and ideal gas assumptions.) The above parameters ensure that oMR is possible for a range of angles in such a flow field, as determined from experimental results. The horizontal gap $L$ between the wedges is zero.

In figure 15, the transitions from $\text{oMR}\rightarrow \text{oRR}$ have been shown for various types of initial oMR configurations. The cases taken here are symmetric DiMR–DiMR with equal wedge angles changing along the path $(c)$ , asymmetric DiMR–DiMR with unequal wedge angles varying along a straight line path $(d)$ , StMR–DiMR with constant upper wedge angle (symmetric $\unicode[STIX]{x1D703}_{w}^{N}$ ) and varying lower wedge angle along the path $(e)$ and, lastly, InMR–DiMR with unequal wedge angles varying along the straight line path $(f)$ . Figures 15(c) through 15(f) represent shock polars for the paths as indicated in figure 15(a), and the normalised Mach stem heights versus the lower wedge angles for these cases are given in figure 15(b). Solution points have been marked in colour in all the corresponding plots for each of the cases – only oMR, dual-solution (ds) domain and von Neumann (vN) criterion. A global sign convention for wedge angles is adopted for the shock polars here, whereby anticlockwise rotation as for the lower wedge is taken to be positive, and clockwise rotation as for the upper wedge is taken to be negative.

All the plots in figure 15 converge at the symmetric von Neumann point, which for $M_{0}$ of 4.96 occurs at $20.88^{\circ }$ . There is excellent agreement of the analytically predicted value as confirmed from theory from the ( $\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl}$ )-plane and the shock polars. It is interesting to note that both the wedge angles would be equal at the symmetric von Neumann point, and this point hence, is a theoretical StMR–StMR configuration. However, the Mach stem height is zero at this point, and this denotes a transition to oRR.

It is interesting to note from figure 15(b) that the highest Mach stem heights are obtained for the symmetric case, and the Mach stem heights for corresponding lower wedge angles become smaller as one moves away from symmetry. It is easy to extrapolate from symmetry that plots on the other side of symmetry line $(c)$ in the $(\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl})$ -plane would bear the same pattern.

It is also plausible to comment from this analysis that any approximation of an asymmetric MR to a symmetric MR (concept of equivalence shock angle, (Mouton Reference Mouton2007, chap. 5, 9)) would result in estimation of Mach stem height higher than the actual value. To elucidate this, table 2 lists the Mach stem heights for the various types of oMRs on paths $d$ , $e$ and $f$ , and those for the corresponding equivalence shock angle on path $c$ , in figure 15(b). It is seen that the difference increases with asymmetry, with the highest for InMR. For near-symmetric wedge angles as in the DiMR case, however, the difference is quite small, corroborating that proposed in Mouton (Reference Mouton2007).

Table 2.

Comparison of the non-dimensional Mach stem height, $H_{mt}/H_{t}$ , from the present theory for various asymmetric cases with corresponding symmetric case, based on the equivalence shock angle concept (Mouton Reference Mouton2007). All angles are in degrees.

Figure 16.

Asymmetric von Neumann criterion for $M_{0}=4.96$ . (a) The dual-solution domain in the ( $\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl}$ )-plane (Li et al. Reference Li, Chpoun and Ben-Dor1999); (b) plot of normalised Mach stem height versus lower wedge angle for InMR-DiMR; (c) shock polars for asymmetric InMR-DiMR case.

The salient feature of this work is the prediction of the InMR configurations. To see how accurate the present model predicts these configurations, cases of InMR–DiMR have been considered here in figure 16, whereby the upper wedge angle is retained at $18^{\circ }$ and the lower wedge angle is reduced until the von Neumann condition at $23.84^{\circ }$ . The path is shown on the ( $\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl}$ )-plane in figure 16(a), the Mach stem height in figure 16(b), and the corresponding shock polars in figure 16(c). It can be seen that the von Neumann condition is accurately predicted at $\unicode[STIX]{x1D703}_{wl}$ of $23.84^{\circ }$ , as obtained from the shock polar theory.

4.3 Asymmetric MR configurations

The coordinates of various points obtained as solutions from the algorithm are used to render the oMR configuration. Here, (2.1) is used to interpolate points for the curved entities – Mach stem, and portions of the slip lines, expansion-fan characteristic lines and reflected shocks.

The configuration obtained from the present model has been validated with numerical and experimental results in this section. As mentioned earlier, cases of highly asymmetric configurations such as those of InMR, are especially considered.

Figure 17.

Comparison of inverse Mach reflection (DiMR–InMR) configuration at $M_{0}=4.96$ for $(\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl})=(28^{\circ },18^{\circ })$ with present model (line plot). (a) Comparison with colour schlieren (Li et al. Reference Li, Chpoun and Ben-Dor1999); (b) zoomed-in view of the Mach stem region; (c) comparison with data from numerical results, marked with ▫ (Ivanov et al. Reference Ivanov, Ben-Dor, Elperin, Kudryavtsev and Khotyanovsky2002); (d) zoomed-in view of the Mach stem region in plot c.

Figure 18.

Comparison of oMR configurations with present model (line plot) with data from numerical results for $M_{0}=4.96$ , marked with ▫ (Ivanov et al. Reference Ivanov, Ben-Dor, Elperin, Kudryavtsev and Khotyanovsky2002).

In figure 17, the plots corresponding to an inverse Mach reflection, in particular oMR(DiMR $+$ InMR), for an upper wedge angle $\unicode[STIX]{x1D703}_{wu}$ of $28^{\circ }$ and lower wedge angle $\unicode[STIX]{x1D703}_{wl}$ of $18^{\circ }$ , are shown. The wedge angles chosen correspond to an InMR–DiMR in the dual-solution domain on the ( $\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl}$ )-plane, as can be seen from figure 16(a). In figure 17(a), the predicted configuration has been overlapped upon a colour schlieren photograph of an oMR for the same wedge angles, given in Li et al. (Reference Li, Chpoun and Ben-Dor1999). It can be seen that the predicted oMR configuration matches very well with the experimental oMR configuration. The rather small size of the Mach stem makes it difficult to visualise the area close to the stem. A zoomed-in view is therefore presented in figure 17(b). The line plots for various features such as shocks, characteristic lines of the expansion fan, slip line and streamline, are differentiated with colour. (The coordinate system shown in figure 17(b) corresponds to the approach primarily adopted in the present algorithm. For the latter plots, the axes have been shifted vertically down by $H_{l}$ .)

It may be noted that the streamline direction downstream of the Mach stem is fictitious, as the flow is no longer uniform. The streamline represents the general flow direction and inclination of the subsonic pocket, or more correctly, axis $x_{i}$ of the present model. The slip lines become parallel to $x_{i}$ at their respective choking sections (part of the boundary conditions in the model) which is reasonable in the context of the experimental observations of such flow fields.

In figure 17(c,d), the predicted results have been compared with numerical data presented in Ivanov et al. (Reference Ivanov, Ben-Dor, Elperin, Kudryavtsev and Khotyanovsky2002). Here, the trailing edges of the two wedges are at the same horizontal location, unlike the leading edges as in the case of Li et al. (Reference Li, Chpoun and Ben-Dor1999). The two data sets have been scaled and mapped by matching the leading and trailing edges of the wedges. The agreement is seen to be quite good, with a clearer picture being provided by the magnified view.

The present model is known to predict a discontinuity between the upper and lower sections at the choking location. This is likely to become predominant at large asymmetric wedge angles owing the deviation from the quasi one-dimensional assumption used in the present formulation. However, it can be seen from the plots that the agreement between the computational and analytical data is quite good despite the fact that the asymmetry in wedge angles is large, indicating the reliability of the present model even for highly distorted oMR configurations.

Further cases of asymmetric MR have been given in figure 18, to compare the profiles of the present configurations with numerical results of Ivanov et al. (Reference Ivanov, Ben-Dor, Elperin, Kudryavtsev and Khotyanovsky2002). Figure 18(a) shows a direct MR (DiMR–DiMR) case with $(\unicode[STIX]{x1D703}_{wu},\unicode[STIX]{x1D703}_{wl})$ as ( $28^{\circ },30^{\circ }$ ). The oMR configuration is more asymmetric in figure 18(b), but the configuration is still a direct MR. Figure 18(c) shows a case of stationary MR (DiMR–StMR), whereby the slip line for the lower domain emerges parallel to the incoming flow direction at the triple point. Lastly, figure 18(d) shows an inverse MR (InMR–DiMR) with a substantial height of the Mach stem. This configuration clearly reveals the profile of the Mach stem. The agreement with the numerical data is excellent for the above cases, though close inspection reveals that the present model under-predicts the Mach stem height by a small margin for cases ( $a$ ), ( $b$ ) and ( $c$ ), while slightly over-predicting for case ( $d$ ). The Mach stem profile is accurately captured, as seen in cases ( $a$ ) and ( $d$ ).