1 Introduction
It is generally known that shock reflection phenomena appear in many practical engineering applications, ranging from supersonic aircraft wings to hypersonic vehicles and engines. In steady flows, there exist two classical types of shock reflection configuration. One is called regular reflection (RR) and the other irregular reflection (IR). In general, IR refers commonly to a three-wave shock wave configuration (Mach reflection, MR).
Since Mach firstly observed two different shock reflection configurations in 1878, abundant research in this field has been carried out. Later, in the 1940s, the detachment criterion and the von Neumann criterion were proposed to describe the transition between MR and RR by von Neumann (Reference von Neumann1943, Reference von Neumann1945), respectively. Based on the two transition criteria, the
$(M_{0},\unicode[STIX]{x1D703}_{w})$
-plane (free-stream Mach number
$M_{0}$
for the
$x$
axis and wedge angle
$\unicode[STIX]{x1D703}_{w}$
for the
$y$
axis) could be divided into three domains: the regular reflection domain, Mach reflection domain and dual solution domain, shown in figure 1 (see Hornung, Oertel & Sandeman Reference Hornung, Oertel and Sandeman1979; Chpoun et al.
Reference Chpoun, Passerel, Li and Ben-Dor1995; Ben-Dor Reference Ben-Dor1999, Reference Ben-Dor2007). Here,
$\unicode[STIX]{x1D703}_{w}^{D}(M)$
is the detachment criterion and
$\unicode[STIX]{x1D703}_{w}^{N}(M)$
the von Neumann criterion. Therefore, for a given combination of
$M_{0}$
and
$\unicode[STIX]{x1D703}_{w}$
, the possible types of shock reflection configuration could be predicted theoretically.
Figure 1. Domains of possible shock reflection configurations (data for the curves taken from Ben-Dor Reference Ben-Dor2007, p. 77).
Further, regular reflection configurations could be well described with the inviscid two-shock theory and Prandtl–Meyer expansion theory. However, it is difficult to predict the Mach reflection configuration with these two theories due to the uncertainty in the Mach stem height. Azevedo & Liu (Reference Azevedo and Liu1993) assumed that the sonic throat formed by the converging stream tube behind the Mach stem, the slip stream and the leading expansion wave, which emanates from the trailing edge, meet at one point. Subsequently, Li & Ben-Dor (Reference Li and Ben-Dor1997) developed a modified model, supposing that the flow direction of the slip stream at the sonic throat is parallel to the symmetric line. In this model, the reflected shock and the slip stream are slightly curved, due to the effect of expansion waves. Based on geometric considerations of the flow, Hornung & Mouton (Reference Hornung and Mouton2008), Mouton (Reference Mouton2008), Mouton & Hornung (Reference Mouton and Hornung2008) presented a new approach for estimating the steady-state Mach stem height. Subsequently, taking the expansion waves and compression waves which exist over the slip stream into consideration, Gao & Wu (Reference Gao and Wu2010) proposed a new model which describes the Mach reflection configuration analytically and agrees very well with the computational fluid dynamics (CFD) numerical results. Also, Bai & Wu (Reference Bai and Wu2017) introduced the effect of secondary expansion waves and compression waves over the slip stream and presented the analytical expressions for the slip stream.
However, the above analytical models focus on the interactions of symmetric, rather than asymmetric shock waves that are more likely to occur in reality. Similar to the interaction of symmetric shock waves in steady flows, two types of shock reflection configuration may be formed when the interaction of asymmetric shock waves occurs. One is called overall regular reflection (oRR) and the other overall Mach reflection (oMR). Figure 2 clearly indicates that the oRR wave configuration consists of two RRs and the oMR wave configuration consists of two MRs. Chpoun & Lengrand (Reference Chpoun and Lengrand1997) conducted an initial experimental study on the reflection of asymmetric shock waves and thereby verified the existence of the asymmetric shock reflection hysteresis in the oRR
$\leftrightarrow$
oMR transition. Li, Chpoun & Ben-Dor (Reference Li, Chpoun and Ben-Dor1999) presented the theoretically possible wave configurations with the aid of shock polar combinations, and explained the physical nature of the asymmetric shock reflection hysteresis phenomenon. Then, a series of experiments were carried out and verified their partial analytical results. Also, Ivanov et al. (Reference Ivanov, Ben-Dor, Elperin, Kudryavtsev and Khotyanovsky2002) conducted a detailed numerical study, further verifying Li et al.’s theoretical findings. In addition, Hornung & Mouton (Reference Hornung and Mouton2008), Mouton (Reference Mouton2008), Mouton & Hornung (Reference Mouton and Hornung2008) defined an equivalence shock angle to estimate the Mach stem height of the asymmetric shock reflection. In short, the above-mentioned theories made it possible to predict the type of overall wave configuration. Similarly to the regular reflections in steady flows, the entire flow field of overall regular reflections could be obtained by solving a series of governing equations. Note that owing to the flow complexity of the converging stream tube formed by two slip streams it is extremely difficult to describe the asymmetric overall Mach reflection configuration exactly.
Figure 2. A schematic illustration of the possible overall wave configurations: (a) an overall Mach reflection (oMR), (b) an overall regular reflection (oRR).
Aiming to solve these problems, an analytical model is established, based on the previous contributions of many researchers. In § 2, we form several acceptable hypotheses on the shape of the asymmetric Mach stem according to Tao, Fan & Zhao’s (Reference Tao, Fan and Zhao2015) experimental results. Taking the features of the subsonic flow behind the Mach stem into consideration, we describe the asymmetric overall wave configurations analytically. Then, an analysis of the predicted results and previous results is described.
2 Physical model
The reflection of asymmetric shock waves in steady flows may be divided into two overall wave configurations: oRR and oMR. With a given combination of oncoming flow Mach number
$M_{0}$
and two reflecting wedge angles
$\unicode[STIX]{x1D703}_{w1}$
,
$\unicode[STIX]{x1D703}_{w2}$
, the possible theoretical overall wave configurations could be obtained by using the shock polars. Since on both sides of the slip stream, the flows are parallel and the pressures are the same, the oRR wave configuration could be well established. Because of the existence of a subsonic region behind the asymmetric Mach stem, the difficulty of reappearance of the oMR wave configuration is increased.
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.
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D708}(M_{3})-\unicode[STIX]{x1D708}(M_{1}),\quad P_{3}/P_{1}=\unicode[STIX]{x1D712}(M_{1},M_{3}), & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D708}(M)=\sqrt{\frac{\unicode[STIX]{x1D6FE}+1}{\unicode[STIX]{x1D6FE}-1}}\arctan \left(\sqrt{\frac{(\unicode[STIX]{x1D6FE}-1)(M^{2}-1)}{\unicode[STIX]{x1D6FE}+1}}\right)-\arctan (\sqrt{M^{2}-1}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}(M_{0},M)=\left[\frac{2+(\unicode[STIX]{x1D6FE}-1)M_{0}^{2}}{2+(\unicode[STIX]{x1D6FE}-1)M^{2}}\right]^{\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D6FE}-1)}. & \displaystyle\end{eqnarray}$$
Here,
$\unicode[STIX]{x1D6FE}$
is the ratio of specific heats.
Similarly, we also can obtain the flow parameters in region 5,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{5}-\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D708}(M_{5})-\unicode[STIX]{x1D708}(M_{2}),\quad P_{5}/P_{2}=\unicode[STIX]{x1D712}(M_{2},M_{5}). & & \displaystyle\end{eqnarray}$$
Considering that
$$\begin{eqnarray}\displaystyle P_{4}=P_{5},\quad \unicode[STIX]{x1D703}_{4}=\unicode[STIX]{x1D703}_{5}, & & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle M_{4}^{2}=F(M_{3},\unicode[STIX]{x1D6FD}_{r^{\prime }}),\quad \unicode[STIX]{x1D703}_{4}-\unicode[STIX]{x1D703}_{3}=G(M_{3},\unicode[STIX]{x1D6FD}_{r^{\prime }}),\quad P_{4}/P_{3}=J(M_{3},\unicode[STIX]{x1D6FD}_{r^{\prime }}), & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \displaystyle F(M,\unicode[STIX]{x1D6FD})=\frac{M^{2}+\displaystyle \frac{2}{\unicode[STIX]{x1D6FE}-1}}{\displaystyle \frac{2\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}M^{2}\sin ^{2}\unicode[STIX]{x1D6FD}-1}+\frac{M^{2}\cos ^{2}\unicode[STIX]{x1D6FD}}{\displaystyle \frac{\unicode[STIX]{x1D6FE}-1}{2}M^{2}\sin ^{2}\unicode[STIX]{x1D6FD}+1}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle G(M,\unicode[STIX]{x1D6FD})=\arctan \left(2\cot \unicode[STIX]{x1D6FD}\frac{M^{2}\sin ^{2}\unicode[STIX]{x1D6FD}-1}{M^{2}(\unicode[STIX]{x1D6FE}+\cos 2\unicode[STIX]{x1D6FD})+2}\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle J(M,\unicode[STIX]{x1D6FD})=1+\frac{2\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}(M^{2}\sin ^{2}\unicode[STIX]{x1D6FD}-1). & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D708}(M_{3})-\unicode[STIX]{x1D708}(M_{1}),\quad P_{3}/P_{1}=\unicode[STIX]{x1D712}(M_{1},M_{3}). & & \displaystyle\end{eqnarray}$$
In the same way, by using the Prandtl–Meyer function we get:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D708}(M_{2})-\unicode[STIX]{x1D708}(M_{1}),\quad P_{2}/P_{1}=\unicode[STIX]{x1D712}(M_{1},M_{2}),\\ \displaystyle \unicode[STIX]{x1D703}_{4}-\unicode[STIX]{x1D703}_{3}=\unicode[STIX]{x1D708}(M_{4})-\unicode[STIX]{x1D708}(M_{3}),\quad P_{4}/P_{3}=\unicode[STIX]{x1D712}(M_{3},M_{4}),\\ \displaystyle \unicode[STIX]{x1D703}_{5}-\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D708}(M_{5})-\unicode[STIX]{x1D708}(M_{2}),\quad P_{5}/P_{2}=\unicode[STIX]{x1D712}(M_{2},M_{5}).\end{array}\right\} & & \displaystyle\end{eqnarray}$$
Considering the features of slip stream, an additional relations are supplemented,
$$\begin{eqnarray}\displaystyle P_{4}=P_{5},\quad \unicode[STIX]{x1D703}_{4}=\unicode[STIX]{x1D703}_{5}. & & \displaystyle\end{eqnarray}$$
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:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle M_{1}^{2}=F(M_{0},\unicode[STIX]{x1D6FD}_{i}),\quad \unicode[STIX]{x1D703}_{1}=G(M_{0},\unicode[STIX]{x1D6FD}_{i}),\quad P_{1}/P_{0}=J(M_{0},\unicode[STIX]{x1D6FD}_{i}),\\ \displaystyle M_{2}^{2}=F(M_{1},\unicode[STIX]{x1D6FD}_{r}),\quad \unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D703}_{1}+G(M_{1},\unicode[STIX]{x1D6FD}_{r}),\quad P_{2}/P_{1}=J(M_{1},\unicode[STIX]{x1D6FD}_{r}),\\ \displaystyle M_{3}^{2}=F(M_{0},\unicode[STIX]{x1D6FD}_{m}),\quad \unicode[STIX]{x1D703}_{3}=G(M_{0},\unicode[STIX]{x1D6FD}_{m}),\quad P_{3}/P_{0}=J(M_{0},\unicode[STIX]{x1D6FD}_{m}),\\ \displaystyle P_{2}=P_{3},\quad \unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D703}_{3}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
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:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{m}=(\unicode[STIX]{x1D703}_{up}+\unicode[STIX]{x1D703}_{down})/2. & & \displaystyle\end{eqnarray}$$
Consequently, the angle
$\unicode[STIX]{x1D6FD}_{m^{\prime }}$
of line
$TT^{\prime }$
can be obtained using the oblique shock relation,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{m}=G(M_{0},\unicode[STIX]{x1D6FD}_{m^{\prime }}). & & \displaystyle\end{eqnarray}$$
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:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle x_{T}=\frac{\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}H-\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}y_{A}-\tan \unicode[STIX]{x1D6FD}_{2}H+\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}\tan \unicode[STIX]{x1D6FD}_{1}x_{A}+\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}y_{c}-\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}\tan \unicode[STIX]{x1D6FD}_{2}x_{c}}{\tan \unicode[STIX]{x1D6FD}_{m^{\prime }}(\tan \unicode[STIX]{x1D6FD}_{1}-\tan \unicode[STIX]{x1D6FD}_{2})},\\ \displaystyle y_{T}=\tan \unicode[STIX]{x1D6FD}_{1}(x_{T}-x_{A})+y_{A},\\ \displaystyle x_{T^{\prime }}=x_{T}-H/\tan \,\unicode[STIX]{x1D6FD}_{m^{\prime }},\\ \displaystyle y_{T^{\prime }}=y_{T}-H.\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
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
$H_{1}$
,
$$\begin{eqnarray}\displaystyle H_{1}=\frac{H_{m1}+H_{m2}}{2}. & & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle \frac{H_{1}}{H_{s}}=\unicode[STIX]{x1D70E}(\overline{M}),\quad \unicode[STIX]{x1D70E}(M)=\frac{1}{M}\left(1+\frac{\unicode[STIX]{x1D6FE}-1}{2}M^{2}\right)^{(\unicode[STIX]{x1D6FE}+1)/(2\unicode[STIX]{x1D6FE}-2)}. & & \displaystyle\end{eqnarray}$$
Here,
$\overline{M}$
is the averaged Mach number behind MS, i.e.
$$\begin{eqnarray}\displaystyle \overline{M}^{2}=F(M_{\infty },\unicode[STIX]{x1D6FD}_{m}^{\prime }). & & \displaystyle\end{eqnarray}$$
Besides, the averaged pressure
$\overline{P}$
along the
$U_{1}V_{1}$
line is
$$\begin{eqnarray}\displaystyle \overline{P}=J(M_{\infty },\unicode[STIX]{x1D6FD}_{m}^{\prime })\cdot P_{\infty }. & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{un}=\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D708}(M_{2})-\unicode[STIX]{x1D708}(M_{1}),\quad P_{2}/P_{1}=\unicode[STIX]{x1D712}(M_{1},M_{2}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{dn}=\unicode[STIX]{x1D703}_{6}-\unicode[STIX]{x1D703}_{5}=\unicode[STIX]{x1D708}(M_{6})-\unicode[STIX]{x1D708}(M_{5}),\quad P_{6}/P_{5}=\unicode[STIX]{x1D712}(M_{5},M_{6}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle P_{3}=P_{1}=P_{5}=(P_{H_{n-1}}+P_{H_{n}})/2,\quad P_{4}=P_{2}=P_{6}=(P_{H_{n}}+P_{H_{n+1}})/2, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle P_{H_{i}}=\unicode[STIX]{x1D712}(\overline{M},\overline{M_{i}})\cdot \overline{P}(i=1,2,\ldots n,\ldots ). & & \displaystyle\end{eqnarray}$$
Here, the averaged Mach number
$\overline{M_{i}}$
at
$U_{i}V_{i}$
can be obtained using the quasi-one-dimensional steady relation,
$$\begin{eqnarray}\displaystyle \frac{H_{i}}{H_{s}}=\unicode[STIX]{x1D70E}(\overline{M_{i}})(i=1,2,\ldots n,\ldots ), & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}=(\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{5})/2,\quad \unicode[STIX]{x1D703}_{4}=(\unicode[STIX]{x1D703}_{2}+\unicode[STIX]{x1D703}_{6})/2. & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}\displaystyle x_{u_{n}}=x_{u_{n-1}}+\unicode[STIX]{x0394}x,\quad y_{u_{n}}=y_{u_{n-1}}+\unicode[STIX]{x0394}x\cdot \tan \unicode[STIX]{x1D703}_{1},\\ \displaystyle x_{u_{n+1}}=x_{u_{n}}+\unicode[STIX]{x0394}x,\quad y_{u_{n+1}}=y_{u_{n}}+\unicode[STIX]{x0394}x\cdot \tan \unicode[STIX]{x1D703}_{2},\end{array}\right\} & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{y_{v_{n}}-y_{v_{n-1}}}{x_{v_{n}}-x_{v_{n-1}}}=\tan \unicode[STIX]{x1D703}_{5},\quad \frac{y_{v_{n}}-y_{v_{n+1}}}{x_{v_{n}}-x_{v_{n+1}}}=\tan \unicode[STIX]{x1D703}_{6},\\ \displaystyle \frac{y_{v_{n}}-y_{u_{n}}}{x_{v_{n}}-x_{u_{n}}}=\tan (\unicode[STIX]{x03C0}/2+\unicode[STIX]{x1D703}_{3}),\quad \frac{y_{v_{n+1}-y_{u_{n+1}}}}{x_{v_{n+1}}-x_{u_{n+1}}}=\tan (\unicode[STIX]{x03C0}/2+\unicode[STIX]{x1D703}_{4}).\end{array}\right\} & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D708}(M_{3})-\unicode[STIX]{x1D708}(M_{2}),\quad P_{3}/P_{2}=\unicode[STIX]{x1D712}(M_{2},M_{3}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{7}-\unicode[STIX]{x1D703}_{6}=\unicode[STIX]{x1D708}(M_{7})-\unicode[STIX]{x1D708}(M_{6}),\quad P_{7}/P_{6}=\unicode[STIX]{x1D712}(M_{6},M_{7}). & & \displaystyle\end{eqnarray}$$
Based on the features of slip streams, we can obtain,
$$\begin{eqnarray}\displaystyle P_{1}=P_{3}=P_{6}=(P_{H_{n-1}}+P_{H_{n}})/2,\quad P_{3}=P_{5}=P_{7}=(P_{H_{n}}+P_{H_{n+1}})/2, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{4}=(\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{6})/2,\quad \unicode[STIX]{x1D703}_{5}=(\unicode[STIX]{x1D703}_{3}+\unicode[STIX]{x1D703}_{7})/2. & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle \frac{y_{u_{n+1}}-y_{g_{(n,2)}}}{x_{u_{n+1}}-x_{g_{(n,2)}}}=\tan \unicode[STIX]{x1D6FD}_{n1},\quad \frac{y_{u_{n+1}}-y_{u_{n}}}{x_{u_{n+1}}-x_{u_{n}}}=\tan \unicode[STIX]{x1D703}_{3}. & & \displaystyle\end{eqnarray}$$
Similarly, the following geometric relations are added,
$$\begin{eqnarray}\displaystyle \frac{y_{u_{n+1}}-y_{v_{n+1}}}{x_{u_{n+1}}-x_{v_{n+1}}}=\tan (\unicode[STIX]{x1D703}_{5}+\unicode[STIX]{x03C0}/2),\quad \frac{y_{v_{n+1}}-y_{v_{n}}}{x_{v_{n+1}}-x_{v_{n}}}=\tan \unicode[STIX]{x1D703}_{7}. & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{3}-\unicode[STIX]{x1D703}_{2}=\unicode[STIX]{x1D708}(M_{3})-\unicode[STIX]{x1D708}(M_{2}),\quad P_{3}/P_{2}=\unicode[STIX]{x1D712}(M_{2},M_{3}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{6}-\unicode[STIX]{x1D703}_{5}=\unicode[STIX]{x1D708}(M_{6})-\unicode[STIX]{x1D708}(M_{5}),\quad P_{6}/P_{5}=\unicode[STIX]{x1D712}(M_{5},M_{6}). & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle P_{6}=(P_{H_{n}}+P_{H_{n+1}})/2. & & \displaystyle\end{eqnarray}$$
The averaged flow deflection angle
$\unicode[STIX]{x1D703}_{v_{j+1}}$
at the cross-section
$V_{j+1}V_{j+1}^{\prime }$
is
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{v_{j+1}}=(\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{6})/2. & & \displaystyle\end{eqnarray}$$
The angle of line
$F_{(j,2)}V_{j+1}$
(
$\unicode[STIX]{x1D6FD}_{j1}$
) is
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{j1}=\unicode[STIX]{x1D703}_{6}+\arcsin \frac{1}{M_{6}}. & & \displaystyle\end{eqnarray}$$
Considering that the coordinate of
$F(j,2)$
is (
$x_{f_{(j,2)}}$
,
$y_{f_{(j,2)}}$
), several geometric relations are added as follows,
$$\begin{eqnarray}\displaystyle \frac{y_{v_{j+1}}-y_{f_{(j,2)}}}{x_{v_{j+1}}-x_{f_{(j,2)}}}=\tan \unicode[STIX]{x1D6FD}_{j1},\quad \frac{y_{v_{j+1}}-y_{v_{j}}}{x_{v_{j+1}}-x_{v_{j}}}=\tan \unicode[STIX]{x1D703}_{6}, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{y_{v_{j+1}^{\prime }}-y_{v_{j+1}}}{x_{v_{j+1}^{\prime }}-x_{v_{j+1}}}=\tan (\unicode[STIX]{x1D703}_{v_{j+1}}+\unicode[STIX]{x03C0}/2). & & \displaystyle\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle & \displaystyle P_{3}=(P_{H_{m}}+P_{H_{m+1}})/2, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{u_{i+1}}=(\unicode[STIX]{x1D703}_{3}+\unicode[STIX]{x1D703}_{6})/2, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}_{i1}=\unicode[STIX]{x1D703}_{3}-\arcsin \frac{1}{M_{3}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{y_{u_{j+1}}-y_{g_{(i,2)}}}{x_{u_{i+1}}-x_{g_{(i,2)}}}=\tan \unicode[STIX]{x1D6FD}_{i1},\quad \frac{y_{u_{i+1}}-y_{u_{i}}}{x_{u_{i+1}}-x_{u_{i}}}=\tan \unicode[STIX]{x1D703}_{3}, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{y_{u_{i+1}^{\prime }}-y_{u_{i+1}}}{x_{u_{i+1}^{\prime }}-x_{u_{i+1}}}=\tan (\unicode[STIX]{x1D703}_{u_{i+1}}+\unicode[STIX]{x03C0}/2). & & \displaystyle\end{eqnarray}$$
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.
2.5 Discussion
Table 1 gives the comparison of the Mach stem configuration between the present analytical results and the previous CFD results. For symmetric conditions (row a and row b), the analytical Mach stem height
$H_{the}/H_{w}$
agrees very well with Gao & Wu’s (Reference Gao and Wu2010) CFD result
$H_{num}/H_{w}$
. Note that both of
$H_{the}$
and
$H_{num}$
are the vertical distances between two triple points and
$\unicode[STIX]{x1D6FD}_{the}$
and
$\unicode[STIX]{x1D6FD}_{num}$
indicate the angle of the straight line connected by two triple points. Therefore, it is easy to understand that
$\unicode[STIX]{x1D6FD}_{the}$
and
$\unicode[STIX]{x1D6FD}_{num}$
are both
$90^{\circ }$
. Furthermore, we can expect that the Mach stem height increases when
$w_{1}/H_{g}(w_{2}/H_{g})$
is increased. So
$H_{the}/H_{w}$
in row b is larger than that in row a.
Table 1. Mach stem height for various conditions at
$M_{0}=4.96$
,
$L/H_{w}=0$
.
For the asymmetric conditions, Kudryavstsev, Khotyanosky & Ivanov (Reference Kudryavstsev, Khotyanosky and Ivanov2000) presented the hysteresis phenomenon that exists in the transition between oRR and oMR. During the hysteresis process, a series of asymmetrical shock reflection configurations are displayed numerically. From rows c, d and e in table 1, it is observed that there is a relatively good agreement between the numerical and the theoretical results of the overall Mach reflection configuration. Specifically, it is shown that excellent agreement between
$\unicode[STIX]{x1D6FD}_{the}$
and
$\unicode[STIX]{x1D6FD}_{num}$
can be observed. Furthermore, the non-dimensionalized Mach stem height
$H_{the}/H_{w}$
predicted by this model agrees with results of Kudryavstsev et al. (Reference Kudryavstsev, Khotyanosky and Ivanov2000), though the theoretical value is slightly larger than the numerical value.
Table 2. Mach stem height for various conditions at
$M_{0}=4$
.
The Mach stem height results for various wedges are presented in table 2. From the table, current theoretical results
$H_{the}$
agree with the previous experimental results
$H_{exp}$
of Hornung & Mouton (Reference Hornung and Mouton2008), Mouton (Reference Mouton2008), Mouton & Hornung (Reference Mouton and Hornung2008). The experimental work of Mouton & Hornung shows slightly higher Mach stem heights than the corresponding ones in the current theoretical work.
3 Conclusions
In conclusion, we have studied the asymmetric shock reflection configurations in steady supersonic flows. On one hand, the overall regular reflection wave configurations can be well established by the current model, because the flow directions are consistent and the pressures are the same on both sides of the slip stream.
On the other hand, it is somewhat powerless to predict the overall Mach reflection wave configurations using the previous analytical models for Mach reflection. Accordingly, the symmetrical reflection model has been extended to an asymmetrical one. Specifically, we abstracted the shape of the Mach stem wave MS from Tao et al.’s (Reference Tao, Fan and Zhao2015) experimental results and assumed an average slip angle to fix the direction of the quasi-one-dimensional duct. This model is capable of predicting the Mach stem height with good comparison to the previous numerical and experimental data.
Acknowledgements
This work was supported by National Natural Science Foundation of China NSFC grant 11372347, 11572347, 51406233, which was gratefully acknowledged.
