2.1 New features of oMR

A schematic illustration of an asymmetric oMR wave configuration is displayed in figure 3(a). In the figure, the oMR configuration consists of two incident shocks ( $IS$ and $IS^{\prime }$ ), two reflected shocks ( $RS$ and $RS^{\prime }$ ), two slip streams ( $SL$ and $SL^{\prime }$ ) and an asymmetric Mach stem (MS). $H_{w}$ denotes the vertical distance between point $A$ and point $C$ . Also, $H$ and $H_{g}$ are the Mach stem height and the vertical distance between point $B$ and point $D$ , respectively. In addition, $w_{1}$ ( $w_{2}$ ) is the wedge length and $L$ denotes the relative position between two wedges.

Figure 3(b) shows an analytical Mach reflection configuration by Gao & Wu (Reference Gao and Wu2010). Comparatively speaking, there exist some new characteristics of the oMR configuration. First, the Mach stem MS may be asymmetric, which is induced by the difference in the wedge angles. Second, the flow direction in the stream tube behind the Mach stem is unknown. Third, it is quite difficult to simulate the interactions between the slip streams and several expansion waves, because they are asymmetric.

Hornung & Mouton (Reference Hornung and Mouton2008), Mouton (Reference Mouton2008), Mouton & Hornung (Reference Mouton and Hornung2008) developed a theory to predict the asymmetric Mach stem height, shown in figure 3(c). In the figure, dashed lines denote equivalence curves, i.e. curves which are equidistant from the von Neumann condition (solid line). Then, the equivalence angle (the ○) can be determined, with a given combination of $\unicode[STIX]{x1D703}_{w1}$ and $\unicode[STIX]{x1D703}_{w2}$ . Although they defined an equivalence angle to estimate the Mach stem height in an overall Mach reflection, it should be noted that the effect of the relative position of the wedges on the Mach stem height is still ignored in this theory.

Considering the new features, an improved analytical model is proposed below.

Figure 3.

An overall Mach reflection (a) and Gao & Wu’s symmetric Mach reflection configuration (b) and equivalence curves, with the ○ representing the symmetric cases (c).

2.2 Wave interactions in oMR

Gao & Wu (Reference Gao and Wu2010) pointed out that the flow region between the reflected shock and the slip stream is one filled with wave interactions. Similarly to the symmetric case, the wave interactions in this region can also be divided into two types, shown in figure 4. Both of these can be solved by using the previous theories. With Gao & Wu’s (Reference Gao and Wu2010) analytical approach, we can hypothesize that the flow parameters ahead of the waves are known, i.e. data in region 1 and region 2. Then, the flow parameters behind the waves (data in regions 3, 4 and 5) can be obtained by solving a series of equations.

Figure 4.

Schematic illustration of the wave interactions. (a) Type (a), a reflected shock $r$ interacts with an expansion wave $E$ , (b) type (b), an expansion wave $e$ (compression wave c) interacts with an expansion wave $E$ .

Figure 4(a) presents the interaction between the reflected shock wave $r$ ( $RS/RS^{\prime }$ ) and the expansion wave $E$ that emanates from the trailing edge of the reflecting wedge. Specifically, the reflected shock $r$ interacts with an expansion wave $E$ at point $R$ , resulting in a refracted shock $r^{\prime }$ , a refracted expansion wave $E^{\prime }$ and a slip stream (denoted by dashed line).

By using the Prandtl–Meyer relations, Mach number $M$ , pressure $P$ and flow direction $\unicode[STIX]{x1D703}$ in region 3 can be achieved. Note that subscripts 1, 2, 3, 4 and 5 refer to the flow states in the corresponding regions.

where

Here, $\unicode[STIX]{x1D6FE}$ is the ratio of specific heats.

Similarly, we also can obtain the flow parameters in region 5,

Considering that

the Mach number $M_{4}$ and the shock angle of $r^{\prime }$ $\unicode[STIX]{x1D6FD}_{r^{\prime }}$ can be obtained with the oblique shock relations. These equations are

where

Thus, type (a) of figure 4(a) can be obtained.

Figure 4(b) displays another type of wave interaction. Specifically, the expansion wave $e$ (compression wave $c$ ) induced by the evolution of the slip stream $SL$ ( $SL^{\prime }$ ) interacts with the expansion wave $E$ emanating from the trailing edge of the wedge. Based on the above conclusions, the flow parameters in region 3 satisfy

In the same way, by using the Prandtl–Meyer function we get:

Considering the features of slip stream, an additional relations are supplemented,

By solving the equations (2.10)–(2.12), $M$ , $P$ and $\unicode[STIX]{x1D703}$ in region 3–5 can be obtained. Consequently, similar to the symmetric case, the flow parameters in the region between $RS$ and $SL$ ( $RS^{\prime }$ and $SL^{\prime }$ ) can be solved and obtained.

2.3 Mach stem shape

For MR, Li & Ben-Dor (Reference Li and Ben-Dor1997) pointed out that the Mach stem shape is slightly curved. Tan, Ren & Wu (Reference Tan, Ren and Wu2006) analytically proved, using the small disturbance theory, that the Mach stem is a circular arc. Then, the fine structure of Mach stem was well observed experimentally by Tao et al. (Reference Tao, Fan and Zhao2015), using the nano-tracer planar laser scattering technique, shown in figure 5. In their experiments, the inflow Mach number $M_{\infty }$ is 4, the inflow total pressure $P_{\infty }$ 101 kPa and the wedge angles $25.7^{\circ }$ (upper) and $20.9^{\circ }$ (lower), respectively.

Figure 5.

The fine structure of the oMR configuration (see Tao et al. Reference Tao, Fan and Zhao2015).

As measured from the photograph, the angle of $SL$   $\unicode[STIX]{x1D703}_{up}$ is $-8.4^{\circ }$ and the angle of $SL^{\prime }$   $\unicode[STIX]{x1D703}_{down}$ is $-1.0^{\circ }$ . Further, we assume that the averaged flow deflection angle while fluid passing across the Mach stem, MS, is equal to the flow deflection angle formed across a strong shock (straight line $TT^{\prime }$ ). In the figure, the angle of the straight line connected to two triple points ( $T$ , $T^{\prime }$ ) is $-88.6^{\circ }$ . By using the oblique shock wave relations, the calculated flow deflection angle $\unicode[STIX]{x1D703}_{m}$ across the MS could be obtained and the value is $-4.9^{\circ }$ . It is easy to notice that the calculated flow deflection angle is approximately equal to the averaged flow deflection angle, i.e. $\unicode[STIX]{x1D703}_{m}$   ${\approx}$ $(\unicode[STIX]{x1D703}_{up}+\unicode[STIX]{x1D703}_{down})/2$ . Based on this, a simplified schematic of Mach stem configuration is abstracted and plotted, see figure 6.

Figure 6.

Schematic illustration of the Mach stem configuration.

As shown in figure 6, the interaction between two incident shocks $IS$ , $IS^{\prime }$ would result in two reflected shocks $RS$ , $RS^{\prime }$ , two slip streams $SL$ , $SL^{\prime }$ and a Mach stem, MS. Using the classical three-shock theory, it is easy to describe the flow fields near both of the triple points $T$ , $T^{\prime }$ . Hence, the flow parameters behind the shock waves could be expressed by the following equations:

Here, $\unicode[STIX]{x1D6FD}_{i}$ , $\unicode[STIX]{x1D6FD}_{r}$ and $\unicode[STIX]{x1D6FD}_{m}$ denote the incident shock angle, the reflected shock angle and the angle of Mach stem, respectively. Subscripts 0, 1, 2 and 3 still refer to the flow states in the corresponding region separately. Then, the angles of the slip streams $\unicode[STIX]{x1D703}_{up}$ , $\unicode[STIX]{x1D703}_{down}$ could be obtained. Based on the foregoing discussion, the initial flow deflection angle $\unicode[STIX]{x1D703}_{m}$ across MS can be expressed as:

Consequently, the angle $\unicode[STIX]{x1D6FD}_{m^{\prime }}$ of line $TT^{\prime }$ can be obtained using the oblique shock relation,

In addition, point $A$ and point $C$ are the leading edge points of the reflecting wedges and their coordinates in the ( $x,y$ )-coordinates system are $(x_{A},y_{A})$ and $(x_{C},y_{C})$ , respectively. For a given vertical distance $H$ between two triple points, the coordinates of $T$ and $T^{\prime }$ can be expressed as:

2.4 The flow in the subsonic pocket

For the oMR configurations, it is difficult to describe the flow in the subsonic pocket analytically because of the uncertainty of the flow direction. In order to describe the flow state behind the Mach stem MS, several assumptions are made in our model.

In many instances, it is a valid approximation to neglect viscous effects along the slip stream completely. Therefore, we assume that the flow in the subsonic pocket is isentropic and quasi-one-dimensional, ignoring the flow deflection. It should be noticed that the actual flow parameters at any cross-section along the flow direction are not the same. Considering that the zone of dependence at any position in subsonic flows is the whole subsonic region, we assume that the averaged flow direction at any cross-section is merely related to the flow directions on the two sides of the slip stream.

The subsonic region before the leading characteristic line interacts with the slip streams is sketched, see figure 7. In the figure, symbols $T$ , $T^{\prime }$ denote two triple points and lines $TU_{n}$ , $T^{\prime }V_{n}$ denote slip streams. Based on (2.14), the cross-sections which are perpendicular to the flow direction ( $TT_{1}$ and $T^{\prime }T_{1}^{\prime }$ ) can be obtained. And, $U_{1}$ and $V_{1}$ are midpoints of $TT_{1}^{\prime }$ and $T_{1}T^{\prime }$ , respectively. Therefore, $U_{1}V_{1}$ is defined as the entrance cross-section, and its sectional height is $H_{1}$ ,

Here, $H_{m1}$ and $H_{m2}$ are the length of $TT_{1}$ and $T^{\prime }T_{1}^{\prime }$ , respectively.

Figure 7.

Schematic illustration of the subsonic region that is not affected by expansion fan.

The height of the throat $H_{s}$ can be expressed as

Here, $\overline{M}$ is the averaged Mach number behind MS, i.e.

Besides, the averaged pressure $\overline{P}$ along the $U_{1}V_{1}$ line is

Gao & Wu (Reference Gao and Wu2010) indicated that the pressure decreases along the stream tube, resulting in expansion waves. Subscripts 1–6 still refer to the flow states in corresponding regions. The flow parameters behind $e_{n}$ and $e_{n}^{\prime }$ can be calculated using the following equations,

where

Here, the averaged Mach number $\overline{M_{i}}$ at $U_{i}V_{i}$ can be obtained using the quasi-one-dimensional steady relation,

where $H_{i}$ is the length of $U_{i}V_{i}$ ( $i=1,2,3\ldots$ ) and $P_{H_{i}}$ denotes the pressure at the cross-section $U_{i}V_{i}$ .

Based on the above, the averaged flow directions in region 3 ( $U_{n-1}V_{n-1}V_{n}U_{n}$ ) and region 4 ( $U_{n}V_{n}V_{n+1}U_{n+1}$ ) are expressed as follows,

In the ( $x,y$ )-coordinate system, the coordinates of $U_{i}$ , $V_{i}$ are $(x_{u_{i}},y_{u_{i}})$ , $(x_{v_{i}},y_{v_{i}})$ , $i=1,2,\ldots \,$ . Then, several geometric relations are supplemented as follow,

Here, $\unicode[STIX]{x0394}x$ is an arbitrary finite value.

Combining (2.17) with (2.28), the flow parameters of the region before the leading characteristic line of the expansion fan can be solved.

Figure 8 shows the wave configuration of the subsonic region and the evolution of one slip stream is affected by the expansion fan. On the one side of the stream tube, expansion waves interact with a slip stream, resulting in compression waves $c$ . On the other side, several expansion waves, which are induced by the evolution of the other slip stream, occur. In the same way, the flow states of regions $(3,5,7)$ and the positions of point $U_{n+1}$ and point $V_{n+1}$ can be obtained.

Figure 8.

Schematic illustration of the subsonic region where one side is affected by the expansion fan.

Specifically, the flow parameters in regions 3 and 7 are expressed, using the Prandtl–Meyer relations, as

Based on the features of slip streams, we can obtain,

In the same way, the coordinate of $G(n,2)$ is ( $x_{g_{(n,2)}}$ , $y_{g_{(n,2)}}$ ). When the interaction between the expansion fan and the slip stream occurs in the upper side, the angle of $G_{(n,2)}U_{n+1}$ ( $\unicode[STIX]{x1D6FD}_{n1}$ ) is ( $\unicode[STIX]{x1D703}_{3}-\arcsin 1/M_{3}$ ). And when it occurs in the lower side, the value is ( $\unicode[STIX]{x1D703}_{3}+\arcsin 1/M_{3}$ ). Therefore, the relations between the point $U_{n+1}$ and the points $U_{n}$ , $G_{(n,2)}$ can be established as,

Similarly, the following geometric relations are added,

By solving (2.29)–(2.34), we can perfectly describe the interaction process displayed in figure 8.

In addition, the expansion fans emanating from the trailing edges may have a direct physical impact on the slip streams on both sides of the subsonic pocket, and this interaction is sketched in figure 9. The flow parameters in region 3 and region 6 can still be expressed as follows,

Note that there is a short distance between the cross-sections $V_{j}V_{j}^{\prime }$ and $U_{i}U_{i}^{\prime }$ , the complexity of this problem for solving the downstream flow states is increased greatly. In order to simplify the analytical model, we assume that points $V_{j}^{\prime }$ and $U_{i}^{\prime }$ are located on the extended lines of $U_{i-2}U_{i-1}$ and $V_{j-1}V_{j}$ , respectively.

Figure 9.

Schematic illustration of the subsonic region where both sides are affected by the expansion fans.

When $V_{j}V_{j}^{\prime }$ is the upstream cross-section relative to $U_{i}U_{i}^{\prime }$ , it is necessary to calculate the flow states in region $V_{j}V_{j+1}F_{(j,2)}$ first.

Then, the pressures on both sides of the slip stream $V_{j}V_{j+1}$ satisfy the following condition,

The averaged flow deflection angle $\unicode[STIX]{x1D703}_{v_{j+1}}$ at the cross-section $V_{j+1}V_{j+1}^{\prime }$ is

The angle of line $F_{(j,2)}V_{j+1}$ ( $\unicode[STIX]{x1D6FD}_{j1}$ ) is

Considering that the coordinate of $F(j,2)$ is ( $x_{f_{(j,2)}}$ , $y_{f_{(j,2)}}$ ), several geometric relations are added as follows,

Using (2.36)–(2.41), the position of point $V_{j+1}$ can be obtained. Then, the location of the new cross-section $V_{j+1}V_{j+1}^{\prime }$ is compared with that of $U_{i}U_{i}^{\prime }$ . If $V_{j+1}V_{j+1}^{\prime }$ is still the upstream cross-section relative to $U_{i}U_{i}^{\prime }$ , the forgoing procedure (2.30)–(2.41) is repeated for several cycles until $VV^{\prime }$ becomes the downstream cross-section relative to $U_{i}U_{i}^{\prime }$ . In addition, we have to obtain the flow parameters in region $U_{i}U_{i+1}G_{(i,2)}$ . In the same way, a series of equations can be listed below,

Here, $\unicode[STIX]{x1D6FD}_{i1}$ is the angle of line $G_{(i,2)}U_{i+1}$ and $\unicode[STIX]{x1D703}_{u_{i+1}}$ is the angle of the averaged flow direction at $U_{i+1}U_{i+1}^{\prime }$ .

Finally, the schematic illustration of an oMR can be well presented and is shown in figure 10. In addition, the relation regarding the sonic throat is performed, i.e. the flow directions of slip streams on the two sides of the tube at the sonic throat are the same. Therefore, the minimum height of the cross-section along the flow direction $H_{s}^{\ast }$ can be achieved. If $H_{s}^{\ast }\neq H_{s}$ , we need to update the initial guess value of $H$ .

Figure 10.

Schematic illustration of an overall Mach reflection.