2 Formulation

We consider the problem of steady shock wave reflection in a channel as sketched in figure 1. Uniform supersonic flow is incident on a thin wedge located in a channel wall and the shock wave generated is reflected off the opposite facing wall. The channel is of width $H$ , the wedge has an inclination angle $\unicode[STIX]{x1D703}\approx a\unicode[STIX]{x1D6FF}\ll 1$ and the upstream supersonic uniform flow $\boldsymbol{u}=(u_{\infty },0)$ is aligned with the $x$ axis and the channel axis. The wedge geometry introduces a small asymptotic parameter $\unicode[STIX]{x1D6FF}>0$ for the problem with the order-one parameter $a$ providing, in the asymptotic model, a measure of the inclination of the wedge. The flow generated by the wedge is described in terms of the velocity $\boldsymbol{u}=(u,v)$ non-dimensionalised with the free-stream velocity and involves a perturbed velocity with components ( $\tilde{u} ,\tilde{v}$ ) given by

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u}{u_{\infty }}=1+\frac{\unicode[STIX]{x1D6FF}^{2/3}}{\unicode[STIX]{x1D6E4}_{\infty }^{1/3}}\tilde{u} +o(\unicode[STIX]{x1D6FF}^{1/2}), & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{v}{u_{\infty }}=\unicode[STIX]{x1D6FF}~\text{sgn}(\unicode[STIX]{x1D6E4}_{\infty })\tilde{v}+o(\unicode[STIX]{x1D6FF}), & \displaystyle\end{eqnarray}$$

where we have assumed that $\unicode[STIX]{x1D6E4}$ given by (1.1) evaluated in the free-stream reference state and denoted by $\unicode[STIX]{x1D6E4}_{\infty }$ is $O(\unicode[STIX]{x1D6FF}^{1/2})$ . This perturbed flow $(\tilde{u} ,\tilde{v})$ can be shown to satisfy the transonic small disturbance equations formulated here for BZT fluids as

(2.3a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}j}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{\unicode[STIX]{x2202}\tilde{v}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=0,\quad j=-K_{1}\tilde{u} -\tilde{u} ^{2}+\frac{K_{2}}{3}\tilde{u} ^{3},\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\tilde{u} }{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{\unicode[STIX]{x2202}\tilde{v}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=0,\end{eqnarray}$$

with the scaled non-dimensional channel coordinates

(2.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}^{-1/3}|\unicode[STIX]{x1D6E4}_{\infty }|^{-1/3}\frac{x}{H},\quad \unicode[STIX]{x1D702}=\frac{y}{H},\quad \unicode[STIX]{x1D6FF}\ll 1,\end{eqnarray}$$

which account for rapid variations in the $x$ coordinate compared to $y$ , consistent with the near normal orientation of incident and reflected shocks.

Figure 1.

Flow geometry for reflection of a plane shock generated by a wedge located at $x=0,y=0$ with inclination $a\unicode[STIX]{x1D6FF}>0$ in a channel of width $H$ .

The order-one similarity parameters $K_{1}$ and $K_{2}$ are given by

(2.6a,b ) $$\begin{eqnarray}K_{1}=\frac{M_{\infty }^{2}-1}{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{\infty })^{2/3}},\quad K_{2}=\frac{\unicode[STIX]{x1D6FF}^{2/3}\unicode[STIX]{x1D6EC}_{\infty }}{\unicode[STIX]{x1D6E4}_{\infty }^{4/3}},\end{eqnarray}$$

where $K_{1}$ is a rescaled transonic similarity parameter with $M_{\infty }=u_{\infty }/c_{\infty }$ the Mach number of the unperturbed upstream flow defined using $c_{\infty }$ the upstream unperturbed sound speed. The parameter $K_{2}$ with $\unicode[STIX]{x1D6EC}_{\infty }=\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70C}_{\infty },s_{\infty })=O(1)$ measures the relevant importance of cubic and quadratic nonlinearity, see Cramer & Tarkenton (Reference Cramer and Tarkenton1992) for details. The choice of scalings leading to (2.3)–(2.6) ensure that the formulation reduces to the classical case when $\unicode[STIX]{x1D6E4}_{\infty }=O(1)$ see Cole & Cook (Reference Cole and Cook1986). Equations (2.3)–(2.6) were first derived in Cramer & Tarkenton (Reference Cramer and Tarkenton1992) in the context of steady transonic flow over thin airfoils and more recently in Kluwick & Cox (Reference Kluwick and Cox2018) for flow over a wedge.

To the authors’ knowledge evolution equations exhibiting both quadratic and cubic nonlinearity were first studied for the simpler case of unsteady planar waves, e.g. Cramer & Kluwick (Reference Cramer and Kluwick1984). Extensions to more complex geometries are discussed in Kluwick (Reference Kluwick1991) which also includes an example of purely cubic nonlinearity, see also Lee-Bapty & Crighton (Reference Lee-Bapty and Crighton1987).

Just as in the classical case the modified transonic small disturbance equations (2.3), (2.4) arises as a distinguished asymptotic limit for the Euler equations leading to the estimates

(2.7a-c ) $$\begin{eqnarray}\frac{u-u_{\infty }}{u_{\infty }}=O(\unicode[STIX]{x1D6FF}^{1/2}),\quad \frac{v}{u_{\infty }}=O(\unicode[STIX]{x1D6FF}),\quad M_{\infty }^{2}-1=O(\unicode[STIX]{x1D6FF}),\end{eqnarray}$$

and differing from the classical results due to the assumption that $\unicode[STIX]{x1D6E4}_{\infty }=O(\unicode[STIX]{x1D6FF}^{1/2})$ and $\unicode[STIX]{x1D6EC}_{\infty }=O(1)$ which ensures that both $K_{1}$ and $K_{2}$ are $O(1)$ . Consistent with the asymptotic scalings in (2.7) we construct solutions of (2.3), (2.4) satisfying the asymptotic boundary conditions

(2.8a ) $$\begin{eqnarray}\displaystyle \text{on }\unicode[STIX]{x1D702}=0: & & \displaystyle \displaystyle \quad \tilde{v}=0,\unicode[STIX]{x1D709}<0,\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \quad \tilde{v}=a\text{ sgn}(\unicode[STIX]{x1D6E4}_{\infty }),\unicode[STIX]{x1D709}>0,\end{eqnarray}$$

(2.8c ) $$\begin{eqnarray}\displaystyle \text{on }\unicode[STIX]{x1D702}=1: & & \displaystyle \displaystyle \quad \tilde{v}=0,-\infty <\unicode[STIX]{x1D709}<\infty .\end{eqnarray}$$

$\tilde{v}=a\text{ sgn}(\unicode[STIX]{x1D6E4}_{\infty })$

$\unicode[STIX]{x1D702}=1$

$\tilde{v}=0$

The basic building blocks used to construct the flows generated include shock waves and centred simple waves. If we first consider shock waves: denoting the jump in a flow property $\unicode[STIX]{x1D6FD}$ across a shock as $[\unicode[STIX]{x1D6FD}]=\unicode[STIX]{x1D6FD}_{2}-\unicode[STIX]{x1D6FD}_{1}$ , where subscripts $1$ and $2$ reference upstream and downstream states, then it is readily established that across a shock we have

(2.9a,b ) $$\begin{eqnarray}\frac{[\tilde{v}]}{[\tilde{u} ]}=\pm \sqrt{-\frac{[j]}{[\tilde{u} ]}},\quad j=-K_{1}\tilde{u} -\tilde{u} ^{2}+\frac{K_{2}}{3}\tilde{u} ^{3}.\end{eqnarray}$$

Details can be found in Kluwick & Cox (Reference Kluwick and Cox2018). It is useful to introduce a scaled Mach number for the flows associated with a shock jump. As shown in Kluwick & Cox (Reference Kluwick and Cox2018) the local Mach number $M$ satisfies

(2.10) $$\begin{eqnarray}2\frac{(M-1)}{(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{\infty })^{2/3}}=K_{1}+2\tilde{u} -K_{2}\tilde{u} ^{2}=-\frac{\text{d}j}{\text{d}\tilde{u} }.\end{eqnarray}$$

With supersonic flow upstream, shock jumps described by (2.9) may involve supersonic–supersonic or supersonic–subsonic jump transitions as calculated from (2.10).

The so-called shock polar curve is a plot derived from (2.9) of all possible downstream states ( $\tilde{u} _{2},\tilde{v}_{2}$ ) that can be connected to a given fixed upstream state ( $\tilde{u} _{1},\tilde{v}_{1}$ ) by a shock jump. This is illustrated in figure 2(a) for the case $K_{1}=1,K_{2}=-0.8$ and upstream state $\tilde{u} _{1}=0,\tilde{v}_{1}=0$ i.e. we plot

(2.11) $$\begin{eqnarray}\tilde{v_{2}}(\tilde{u} _{2})=-\tilde{u} _{2}\sqrt{K_{1}+\tilde{u} _{2}-\frac{K_{2}}{3}\tilde{u} _{2}^{2}}.\end{eqnarray}$$

With decreasing values of the downstream velocity $\tilde{u} _{2}$ , i.e. increasing shock strength, we see a shock transition $S_{1}$ from a supersonic–supersonic shock to a supersonic–subsonic shock $S_{2}$ .

If the shock surface is located at $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{s}(\unicode[STIX]{x1D709})$ then integrating (2.3), (2.4) across the shock discontinuity leads to a shock slope given by

(2.12) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D702}_{s}}{\text{d}\unicode[STIX]{x1D709}}=\mp \frac{1}{\displaystyle \sqrt{-\frac{[j]}{[\tilde{u} ]}}},\end{eqnarray}$$

where we observe the useful geometric construction that the slope of the chord in the $(\tilde{u} ,\tilde{v})$ hodograph plane, connecting upstream states ( $\tilde{u} _{1},\tilde{v}_{1}$ ) to downstream states ( $\tilde{u} _{2},\tilde{v}_{2}$ ), given by $[\tilde{v}]/[\tilde{u} ]$ in (2.9) is orthogonal to the shock slope $\text{d}\unicode[STIX]{x1D702}_{s}/\text{d}\unicode[STIX]{x1D709}$ in the $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D709})$ plane with appropriately chosen signs. This construction is illustrated in figure 2(a) and the orientation of the shocks $S_{1},S_{2}$ in the physical $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane is shown in figures 2(b) and 2(c).

Figure 2.

(a) Shock polar for $K_{1}=1,K_{2}=-0.8$ with the construction of shocks $S_{1},S_{2},S_{so},S_{3}$ of increasing strength. The shock $S_{so}$ corresponds to downstream sonic conditions $\tilde{u} _{2}=\tilde{u} _{so}$ and the possible shock $S_{3}$ violates the speed ordering condition. In (b–d) the corresponding shocks in the $\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}$ plane are plotted with the shocks shown in (a) to be orthogonal to the shock chord (Rayleigh line) in the polar plot.

Not all jump discontinuities described by (2.9) lead to physically acceptable shock solutions. We require in addition the imposing of criteria to eliminate non-physical solutions. A fundamental requirement is that that entropy does not decrease across a shock i.e. $[s]\geqslant 0$ which leads to the constraint $\tilde{u} _{2}\leqslant \tilde{u} _{1}$ , see Kluwick & Cox (Reference Kluwick and Cox2018). Only portions of the solution curve of (2.9) satisfying the entropy jump condition are plotted in figure 2(a). It was shown by Cramer & Kluwick (Reference Cramer and Kluwick1984) in the context of BZT fluids with $|\unicode[STIX]{x1D6E4}|\ll 1$ and $\unicode[STIX]{x1D6EC}=O(1)$ that the entropy condition is too weak to rule out certain inadmissible supersonic–supersonic shock discontinuities, where inadmissibility is understood in terms of shocks for which no thermoviscous profile can be constructed. It was shown that in addition a wave speed ordering condition must be imposed.

We develop this condition, applicable to supersonic–supersonic shock jumps, by first noting that characteristics of (2.3), (2.4) have slopes in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane given by

(2.13) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D702}}{\text{d}\unicode[STIX]{x1D709}}=\mp \frac{1}{\displaystyle \sqrt{-\frac{\text{d}j}{\text{d}\tilde{u} }}},\end{eqnarray}$$

and we then require that the shock slope lies between the characteristic slopes of flow states generating the shock i.e.

(2.14) $$\begin{eqnarray}\sqrt{\left.-\frac{\text{d}j}{\text{d}\tilde{u} }\right|_{1}}\geqslant \sqrt{-\frac{[j]}{[\tilde{u} ]}}\geqslant \sqrt{\left.-\frac{\text{d}j}{\text{d}\tilde{u} }\right|_{2}}.\end{eqnarray}$$

Results are stated here for forward orientated waves (i.e. $\text{d}\unicode[STIX]{x1D702}/\text{d}\unicode[STIX]{x1D709}>0$ ) and holding for $-\text{d}j/\text{d}\tilde{u} |_{1,2}>0$ which ensures that the characteristics are real and the flow is supersonic ahead and behind the shock. Shocks where one of the equalities in (2.14) holds are termed ‘sonic shocks’, see Kluwick (Reference Kluwick, Ben-Dor, Igra, Elperin and Lifshitz2001). Equation (2.14) is recognised as the Lax entropy condition and when one of the equalities is satisfied the shocks are also referred to as a left or right contact shocks, see Dafermos (Reference Dafermos2005). The possibility of ‘sonic shocks’ (contact shocks) occurring has no classical gas dynamics counterpart – the shock condition given in (2.14) reduces on setting $K_{2}=0$ to

(2.15) $$\begin{eqnarray}\sqrt{K_{1}+2\tilde{u} _{1}}>\sqrt{K_{1}+\tilde{u} _{1}+\tilde{u} _{2}}>\sqrt{K_{1}+2\tilde{u} _{2}},\end{eqnarray}$$

which is the well-known classical ideal gas case, with strict inequalities applying, see Cole & Cook (Reference Cole and Cook1986). Also and in contrast to the non-classical gas case this wave speed ordering condition when applied is simply equivalent to the selection rule $[s]\geqslant 0$ rather than a further constraint.

In figure 2(a) the shock labelled $S_{3}$ represents jumps to downstream states $\tilde{u} _{2},\tilde{v}_{2}$ (plotted as a dashed curve) constructed from (2.11) which satisfies $[s]>0$ or equivalently $M_{2}<M_{1}$ but violates the speed ordering condition (2.14). There is a limiting shock strength $\tilde{u} _{so}$ where ‘sonic’ conditions hold downstream and represented by shock $S_{so}$ in figures 2(a) and 2(d). The supersonic state $\tilde{u} _{so}$ is determined from the downstream equality in (2.14) i.e. setting

(2.16) $$\begin{eqnarray}\frac{[j]}{[\tilde{u} ]}=\left.\frac{\text{d}j}{\text{d}\tilde{u} }\right|_{2},\end{eqnarray}$$

which leads to $\tilde{u} _{2}=\tilde{u} _{so}=3/2k_{2}$ . The construction in figure 2 also indicates that when sonic conditions hold the shock chord is tangential to the shock polar curve – a straightforward result to show. Inadmissible shocks then are associated with shock chords that cut the shock polar as seen for $S_{3}$ .

The second important class of elementary solutions, aside from shocks, are simple waves – identified as flows associated with only one family of characteristics. Looking for solutions depending on only a single variable $r(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ i.e. $\tilde{u} =\tilde{u} (r),\tilde{v}=\tilde{v}(r)$ we rewrite (2.3), (2.4) as

(2.17a,b ) $$\begin{eqnarray}\frac{\text{d}j}{\text{d}r}\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{\text{d}\tilde{v}}{\text{d}r}\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=0,\quad j=-K_{1}\tilde{u} -\tilde{u} ^{2}+\frac{K_{2}}{3}\tilde{u} ^{3},\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\frac{\text{d}\tilde{u} }{\text{d}r}\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}-\frac{\text{d}\tilde{v}}{\text{d}r}\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=0.\end{eqnarray}$$

Solutions for $(r_{\unicode[STIX]{x1D709}},r_{\unicode[STIX]{x1D702}})$ exist then only if

(2.19) $$\begin{eqnarray}\frac{\text{d}\tilde{v}}{\text{d}r}=\pm \sqrt{-\frac{\text{d}j}{\text{d}\tilde{u} }}\frac{\text{d}\tilde{u} }{\text{d}r}.\end{eqnarray}$$

The curve $\tilde{u} (r),\tilde{v}(r)$ parameterised by the ‘phase’ variable $r$ has slope

(2.20) $$\begin{eqnarray}\left.\frac{\text{d}\tilde{v}}{\text{d}\tilde{u} }\right|_{r}=\left.\pm \sqrt{-\frac{\text{d}j}{\text{d}\tilde{u} }}\right|_{r}\end{eqnarray}$$

which is orthogonal to the straight line of constant phase given by

(2.21) $$\begin{eqnarray}\left.\frac{\text{d}\unicode[STIX]{x1D702}}{\text{d}\unicode[STIX]{x1D709}}\right|_{r}=\mp \frac{1}{\displaystyle \sqrt{-\frac{\text{d}j}{\text{d}\tilde{u} }(r)}},\end{eqnarray}$$

which is the characteristic slope (compare with (2.13)). The similarity with (2.9) and (2.12) is clear. A relevant example of a simple wave is a centred wave fan with rays

(2.22) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D709}}=\frac{1}{\displaystyle \sqrt{-\frac{\text{d}j}{\text{d}\tilde{u} }(r)}},\quad r_{a}\leqslant r\leqslant r_{b},\end{eqnarray}$$

which provides the continuous transition from one uniform flow $(\tilde{u} (r_{a}),\tilde{v}(r_{a}))$ to another uniform flow $(\tilde{u} (r_{b}),\tilde{v}(r_{b}))$ past sharp corners. Details of shock and wave fan constructions can be found in Kluwick & Cox (Reference Kluwick and Cox2018).

With these basic building blocks we can consider the problem of shock reflection. This problem has been the subject of extensive research in ideal gas dynamics with theoretical results going back to Mach (Reference Mach1878), Neumann (Reference von Neumann and Taub1963a ), Neumann (Reference von Neumann and Taub1963b ) and early experiments in Bleakney & Taub (Reference Bleakney and Taub1949). With the wedge angle measured by the parameter $a$ we have for small angles the regular reflection pattern of figure 1. As is well known in the classical case $K_{2}=0$ with increasing angle $a$ the shock reflection undergoes so-called irregular reflection off the channel wall. This involves the detachment of both the incident and reflected shock from the surface and the generation of a third and stronger shock which remains attached to the surface. The point at which the three shocks meet is termed the triple point $(T)$ . As indicated in figure 3 the flow also includes an expansion fan and a slip line (not drawn as not realisable in the irrotational flow model considered here). The inclusion of the wave fan is significant and helps resolve the so-called von Neumann paradox that three intersecting shocks are seen in experiments but unsupported by theoretical analysis. The inclusion of the wave fan is due to Guderley (Reference Guderley1947), Guderley (Reference Guderley1957) a result that was overlooked for many decades until recent work by Hunter & Brio (Reference Hunter and Brio1997), Hunter & Brio (Reference Hunter and Brio2000), Hunter & Tesdall (Reference Hunter and Tesdall2004) on the unsteady transonic flow equations (the time-dependent form of equations (2.3) and (2.4) with $K_{2}=0$ ) and by Vasil’ev (Reference Vasil’ev, Papailiou, Tsahalis, Periaux, Hirsch and Pandolfi1998) and Vasil’ev & Krajko (Reference Vasil’ev and Krajko1999) for the compressible Euler equations. The structure in the neighbourhood of the triple point is complex and revealed in the numerical results of Tesdall & Hunter (Reference Tesdall and Hunter2002) and Tesdall, Sanders & Popivanov (Reference Tesdall, Sanders and Popivanov2015) and the experimental results of Cachucho & Skews (Reference Cachucho and Skews2012). These results are summarised in the sketch of figure 4. Behind the triple point $T=T_{1}$ of figure 3 there is a sequence of supersonic patches each patch terminated by a shock leading to multiple triple points $T_{2},T_{3}\ldots \,$ .

Figure 3.

Flow geometry for irregular reflection of a plane shock generated by a wedge located at $x=0,y=0$ with inclination $a\unicode[STIX]{x1D6FF}>0$ in a channel of width $H$ .

Figure 4.

Sketch of the complex reflection structure due to Tesdall & Hunter (Reference Tesdall and Hunter2002). Near the leading triple point $T_{1}$ there is a cascade of supersonic patches each patch terminated by a shock (in red) leading to multiple triple points $T_{2},T_{3}\ldots \,$ . Blue lines denote centred wave fans generated at each triple point and dashed curves denote sonic curves bounding the supersonic patches.

In this paper we consider shock reflection for non-ideal gases using the modified small disturbance equations (2.3) and (2.4). In a previous paper (Kluwick & Cox Reference Kluwick and Cox2018) the authors consider the flow over a ramp and in this follow-on paper we consider a shock generated by such a flow impinging now on a reflecting surface as encountered in a channel with a wedge.

3 Regular reflection

In Kluwick & Cox (Reference Kluwick and Cox2018) we identified distinctly different flows in the parameter space $(K_{1}>0,K_{2})$ for the parameter ranges $K_{2}<-1/K_{1}$ , $-1/K_{1}<K_{2}<-3/4K_{1}$ , $-3/4K_{1}<K_{2}\leqslant 0$ and $K_{2}>0$ . For each of these intervals we now consider an incident shock reflection problem with the emphasis on regular reflection. We then show in § 4 that when regular reflection breaks down with increased ramp angle (or equivalently incident shock strength) the pure triple shock structure does not exist. This result extends a well-known result for ideal gases.

We note that descriptions of flows in this paper will assume that $\unicode[STIX]{x1D6E4}_{\infty }$ is positive. This is to simplify the interpretation of the results obtained but is not a requirement for the analysis. As a consequence shocks are compressive and wave fans are expansive. The results are easily reinterpreted for an expanding channel with expansion shocks and compression wave fans associated with $\unicode[STIX]{x1D6E4}_{\infty }<0$ .

3.1 Case: $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$

This case is qualitatively similar to the classical case which is included as a special case on setting $K_{2}=0$ . The wave configuration for regular shock reflection is shown in figure 1 where a steady incident shock wave generated by a wedge reflects from a plane boundary. A reflected shock wave, formed at the point of intersection with the incident shock wave and the channel wall, then returns the flow which is deflected towards the wall by the incident shock to a flow parallel again to the wall.

Figure 5.

(a) Shock polars (incident polar $I$ , reflected polar $R$ ) for regular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$ . Results are for $K_{1}=1,K_{2}=-0.7$ . The incident shock $i$ deflects the flow to $\tilde{u} _{1},\tilde{v}_{1}$ and a weak reflected shock $r$ generates supersonic flow $\tilde{u} _{2_{w}},\tilde{v}_{2}=0$ downstream of the shock – flow which is parallel to the channel wall. The incident and reflected shocks are depicted in (b) in the $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ plane using the construction indicated in (a). The shock reflection point is labelled as $(\unicode[STIX]{x1D709}_{s},\unicode[STIX]{x1D702}_{s})$ .

The shock polars describing regular reflection are shown in figure 5(a) and represent the repeated application of (2.9) to the incident and reflected shocks. Without loss of generality we can assume that the upstream perturbed flow for the reflection problem is $\tilde{u} =0,\tilde{v}=0$ . So the incident shock is described by possible downstream states ( $\tilde{u} ,\tilde{v}$ ) satisfying

(3.1) $$\begin{eqnarray}\tilde{v}(\tilde{u} )=-\tilde{u} \sqrt{K_{1}+\tilde{u} -\frac{K_{2}}{3}\tilde{u} ^{2}},\end{eqnarray}$$

and plotted as curve $I$ in figure 5(a). A wedge inclination angle of magnitude $a$ generates, as shown, two possible downstream flows satisfying the wedge boundary condition $\tilde{v}=a\operatorname{sign}\unicode[STIX]{x1D6E4}_{\infty }$ , see (2.8b ). We will consider a weak incident shock jump to a supersonic state $\tilde{u} _{1},\tilde{v}_{1}=a\operatorname{sign}\unicode[STIX]{x1D6E4}_{\infty }$ located on the incident shock polar $I$ in figure 5(a).

The reflected shock then is described by downstream states $(\tilde{u} _{2},\tilde{v}_{2})$ with states $(\tilde{u} _{1},\tilde{v}_{1})$ upstream. This reflected shock polar derived from (2.9) is given by

(3.2) $$\begin{eqnarray}\tilde{v}_{2}(\tilde{u} _{2})=\tilde{v}_{1}+(\tilde{u} _{2}-\tilde{u} _{1})\sqrt{K_{1}+\tilde{u_{1}}+\tilde{u} _{2}-\frac{K_{2}}{3}(\tilde{u} _{1}^{2}+\tilde{u} _{2}^{2}+\tilde{u} _{1}\tilde{u} _{2})}.\end{eqnarray}$$

The reflected shock polar $R$ intersects the $\tilde{v}=0$ axis at two values of $\tilde{u} _{2}$ both corresponding to possible downstream flow states. We distinguish these two downstream states as in one case generating a weak shock connecting typically to supersonic flow downstream $(\tilde{u} _{2_{w}},\tilde{v}_{2}=0)$ and in the other case generating a strong shock connecting always to a subsonic flow downstream $(\tilde{u} _{2_{s}},\tilde{v}_{2}=0)$ . Which state is realised in practice depends critically on the boundary conditions downstream and the reader is encouraged to see the discussion in Kluwick & Cox (Reference Kluwick and Cox2018) for more details. It is the weak shock certainly that is commonly observed in experiments and depicted in figure 5. The incident and reflected shocks have slopes given by (2.12) and are plotted in figure 5(b) in the asymptotic coordinate system $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ . We have indicated in figure 5(a) the construction leading to the determination of the shock slopes in figure 5(b) which implements (2.12).

Figure 6.

(a) Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a_{s}$ for $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$ . Results are for $K_{1}=1,K_{2}=-0.7$ . A weak reflected shock generates sonic flow $\tilde{u} _{2_{w}},\tilde{v}_{2}=0$ which is parallel to the channel wall. The incident ( $i$ ) and reflected shock ( $r$ ) are depicted in (b). The shock reflection point is labelled as $(\unicode[STIX]{x1D709}_{s},\unicode[STIX]{x1D702}_{s})$ .

Figure 7.

(a) Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a>a_{s}$ for $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$ . Results are for $K_{1}=1,K_{2}=-0.7$ . The incident shock $I$ deflects the flow to $\tilde{u} _{1},\tilde{v}_{1}$ and a weak reflected shock generates subsonic flow $\tilde{u} _{2_{w}},\tilde{v}_{2}=0$ which is parallel to the channel wall. The incident and reflected shocks are depicted in (b). The shock reflection point is labelled as $(\unicode[STIX]{x1D709}_{s},\unicode[STIX]{x1D702}_{s})$ .

The regular reflection pattern shown in figure 5 holds for small wedge inclination angles. As the wedge angle is increased the weak downstream shock state associated with the reflected shock eventually becomes sonic with $M=1$ . The value of $a$ at which this occurs is denoted by $a_{s}$ . The shock polars for this case are shown in figure 6(a) and the incident and reflected shocks in the ( $\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}$ ) plane are shown in figure 6(b). With increasing $a>a_{s}$ the weak shock now describes a jump to subsonic downstream states as shown in figure 7 and with further increases in wedge angle $a$ eventually the weak and strong downstream states coalesce. This gives a detachment wedge angle $a=a_{d}$ such that for values of $a>a_{d}$ regular reflection becomes impossible. For $a>a_{d}$ the flow after any reflected shock is still positive towards the wall and the downstream $\tilde{v}=0$ boundary condition cannot be satisfied by any point on the reflected shock polar, see figure 8.

Figure 8.

Shock polar for reflection of a plane shock generated by a wedge with inclination $a>a_{d}$ for $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$ . Results are for $K_{1}=1,K_{2}=-0.7$ . No reflected shock can return the fluid flow to downstream parallel channel flow.

It is possible in the ideal gas case ( $K_{2}=0$ ) to evaluate $a_{s}$ and $a_{d}$ explicitly and a straightforward calculation gives

(3.3a,b ) $$\begin{eqnarray}a_{s}=\frac{(3-\sqrt{5})\sqrt{1+\sqrt{5}}}{8}K_{1}^{3/2}\approx 0.1718K_{1}^{3/2},\quad a_{d}=\frac{2}{5^{3/2}}K_{1}^{3/2}\approx 0.1789K_{1}^{3/2}.\end{eqnarray}$$

3.2 Case: $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$

In the previous case the shock reflection problem led to flows similar to the classical case with $\unicode[STIX]{x1D6EC}_{\infty }=0$ ( $K_{2}=0$ ). With decreasing values of $K_{2}$ and so larger negative values of $\unicode[STIX]{x1D6EC}_{\infty }$ the shock reflection problem begins to show features that have no counterpart in ideal gas dynamics.

For small wedge angles the incident and reflected shock polar plots are given in figure 9(a) for $\tilde{u} _{1}=-0.05$ , $K_{1}=1$ and $K_{2}=-0.8$ . The construction is as already described in the previous case in § 3.1. The polar plots terminate now not with normal orientated shocks but with sonic conditions, i.e. shocks of increased strength would result in the shock chord cutting the shock polar – these are excluded. It is clear that the reflected shock polar intersects the $\tilde{v}=0$ axes now only once to give a unique reflection shock with a supersonic downstream state returning the deflected flow to parallel channel flow. With increasing wedge angle we have a three shock solution shown in figure 10 involving now an additional subsonic downstream flow labelled $\tilde{u} _{3}$ as well as the two-shock solution similar to the classical ideal gas reflections.

Figure 9.

(a) Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1,K_{2}=-0.8$ . The incident shock generated on the incident shock polar $I$ deflects the flow to $\tilde{u} _{1},\tilde{v}_{1}$ with $\tilde{u} _{1}=-0.05$ and a reflected supersonic shock returns the flow parallel to the channel wall. The incident shock $(i)$ and reflected sonic shock $(r)$ are depicted in (b). The shock reflection point is at $(\unicode[STIX]{x1D709}_{s},\unicode[STIX]{x1D702}_{s})$ .

Figure 10.

Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1,K_{2}=-0.8$ . The incident shock generated on the incident shock polar $I$ deflects the flow to $\tilde{u} _{1},\tilde{v}_{1}$ with $\tilde{u} _{1}=-0.12175$ . Reflection shocks are possible to three downstream states a weak shock $\tilde{u} _{2_{w}}$ and as included in the inset two strong subsonic shock states $\tilde{u} _{2_{s}}$ and $\tilde{u} _{3}$ .

As the wedge angle continues to increase the flow scenarios associated with the reflected shock jumps to $\tilde{u} _{3}$ become more complicated as shown in the reflected shock polar plots of figure 11 which should be compared to the inset in figure 10. The downstream state $\tilde{u} _{3}$ with increasing angle (i.e. decreasing $\tilde{u} _{1}$ ) eventually becomes supersonic and then with further increases the reflected shock is a downstream sonic shock and shock state $\tilde{u} _{3}$ is replaced by a sonic shock to $\tilde{u} _{so}$ and a centred fan structure returning the flow to parallel channel flow. This latter scenario is shown in figure 12(a) for $\tilde{u} _{1}=-0.13185$ with the three possible reflection flows one of which is the compound shock/fan structure. The centred wave fan is constructed from (2.20) (with the appropriate sign) which we integrate from $(\tilde{u} _{so},\tilde{v}_{so})$ to $(\tilde{u} _{3},0)$ . In figure 12(b) the reflected sonic shock $r$ and the attached wave fan rays terminating at $\tilde{u} _{3}$ are plotted for this latter case where we have used the shock slope condition (2.12) and the characteristic ray equation (2.22).

Figure 11.

Section of reflection polar $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1,K_{2}=-0.8$ . The transition is shown to a compound shock/fan structure as the wedge angle increases (or $\tilde{u} _{1}$ decreases) leading to the downstream state $\tilde{u} _{3}$ .

Figure 12.

(a) Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1,K_{2}=-0.8$ . The incident shock generated on the incident shock polar $I$ deflects the flow to $\tilde{u} _{1},\tilde{v}_{1}$ with $\tilde{u} _{1}=-0.13185$ . Reflection shocks are possible to three downstream states a weak shock $\tilde{u} _{2_{w}}$ a strong subsonic shock $\tilde{u} _{2_{s}}$ and a compound sonic shock $\tilde{u} _{so}$ and wave fan to $\tilde{u} _{3}$ . The sonic shock ( $r$ ) and rays for the attached fan are shown in (b).

Shock behaviour described by figure 12 does not persist as the wedge angle continues to increase. The compound downstream shock/wave fan structure requires that the sonic shock involves a jump transition to supersonic flow downstream. A shock bifurcation occurs when the wedge angle increases so that the incident shock involves a jump to

(3.4) $$\begin{eqnarray}\tilde{u} _{1}=\frac{1-2\sqrt{1+K_{1}K_{2}}}{K_{2}}.\end{eqnarray}$$

At this point the reflected sonic shock is normal to the channel wall and describes a jump to sonic ( $M=1$ ) downstream flow. The shock polar plot for this bifurcating flow is shown in figure 13. For increasing wedge angles so that $\tilde{u} _{1}<(1-2\sqrt{1+K_{1}K_{2}})/K_{2}$ the compound shock/fan flow is now no longer possible and the shock polars describe a classical reflection pattern involving the transition, with increasing wedge angle, from a strong subsonic and weak supersonic reflected shock to two subsonic reflected shocks as shown by the shock polar curves in figure 14 for $\tilde{u} _{1}=-0.257$ .

Non-classical behaviour involving a compound sonic shock/wave fan occurs over a narrow range of incident shock strengths (or wedge angles). For the particular flow described with $K_{1}=1$ and $K_{2}=-0.8$ the existence of a third solution branch occurs in the interval

(3.5) $$\begin{eqnarray}-0.131966\lesssim \tilde{u} _{1}\lesssim -0.1217425.\end{eqnarray}$$

Figure 13.

Shock polar for strong reflected shocks for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1,K_{2}=-0.8$ . A non-classical reflection shock/fan solution is obtained for $\tilde{u} _{1}=(1-2\sqrt{1+K_{1}K_{2}})/K_{2}$ involving a normal orientated sonic reflection shock and an attached fan with a transition to $\tilde{u} _{3}$ downstream.

Figure 14.

Shock polar for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1$ , $K_{2}=-0.8$ . Two classical subsonic solutions $\tilde{u} _{2_{w}}$ and $\tilde{u} _{2_{s}}$ are obtained for $\tilde{u} _{1}=-0.257$ .

Figure 15.

Shock polar for regular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,K_{2}<-1/K_{1}$ . Results are for $K_{1}=1,K_{2}=-1.2$ . Small wedge angles lead to a classical regular reflection shock. Larger wedge angles lead to a sonic shock/wave fan configuration. Wedge angles with $a$ greater than $a\operatorname{sign}\unicode[STIX]{x1D6E4}_{\infty }=-(1/K_{2})\sqrt{K_{1}+2/3K_{2}}$ result in a reflected wave fan only.

3.3 Case: $K_{1}>0,K_{2}<-1/K_{1}$

In the previous cases the shock reflection problem has multiple solutions. In contrast the shock configuration is unique for $K_{2}<-1/K_{1}$ . We will consider values of $a$ less than

(3.6) $$\begin{eqnarray}a\operatorname{sign}\unicode[STIX]{x1D6E4}_{\infty }=-\frac{3}{2K_{2}}\sqrt{K_{1}+\frac{3}{4K_{2}}},\end{eqnarray}$$

which is the wedge angle, as shown in figure 15, at which an incident shock becomes sonic and beyond which the incident flow consists of a compound sonic shock/wave fan structure. We have for small wedge angles a classical regular reflection shock configuration as seen in figure 15. With increasing wedge angle the reflected shock consists of a sonic shock/wave fan configuration and as the wedge angle increases further the reflected shock strength decreases until with $a$ greater than

(3.7) $$\begin{eqnarray}a\operatorname{sign}\unicode[STIX]{x1D6E4}_{\infty }=-\frac{1}{K_{2}}\sqrt{K_{1}+\frac{2}{3K_{2}}},\end{eqnarray}$$

the incident shock generates a reflected wave fan only. The analysis used to plot figure 15 is that already described in §§ 3.1 and 3.2.

3.4 Case: $K_{1}>0,K_{2}>0$

From a careful reading of Kluwick & Cox (Reference Kluwick and Cox2018) the only relevant flow is associated with negative downstream $\tilde{u}$ values. The shock polars are qualitatively similar to the classical case and the reflection problem can be analysed in a similar manner. We do not provide details of this case.

Figure 16.

(a) Sketch of the shock geometry in the $\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}$ plane associated with a pure triple point construction. The incident shock $(i)$ , reflected shock $(r)$ and Mach stem $(m)$ intersect at the triple point ( $T$ ). (b) Shock polars for irregular reflection of a plane shock generated by a wedge with inclination $a>a_{d}$ for $K_{1}>0,3/4K_{1}<K_{2}<0$ . The incident shock $i$ deflects the flow to $\tilde{u} _{2},\tilde{v}_{2}$ and a weak reflected shock ( $r$ ) generates supersonic flow $\tilde{u} _{3},\tilde{v}_{3}$ . The pure triple point construction in (a) would require that $\tilde{u} _{3},\tilde{v}_{3}$ coincides with the point $D$ on the incident shock polar.

4 Irregular reflection

It is a well-known result that in planar steady flows a pure triple shock structure does not exist i.e. you cannot have three shocks intersecting with constants states between them. This result is discussed, for example, in Courant & Friedrichs (Reference Courant and Friedrichs1948), Neumann (Reference von Neumann and Taub1963a ), Neumann (Reference von Neumann and Taub1963b ) and Henderson & Menikoff (Reference Henderson and Menikoff1998). More recently in Serre (Reference Serre, Friedlander and Serre2007) a proof is presented that is independent of any assumption on the equation of state of the gas. A corresponding result for the unsteady transonic small disturbance equation can be found in Tabak & Rosales (Reference Tabak and Rosales1994). The result presented here is for steady flow of the modified transonic small disturbance equation and is based on the geometry of the shock polar curves. In figure 16(a) we show the shock geometry associated with a pure triple shock with the reflected shock ( $r$ ) and incident shock ( $i$ ) both forward facing. Both shocks intersect a third shock – the so-called Mach stem $(m)$ at the triple point $(T)$ . In figure 16(b) we show the plot of the reflected polar when $a>a_{d}$ as illustrated for $K_{2}$ in the parameter range $-3/4K_{1}<K_{2}<0$ . A shock from state $(\tilde{u} _{1}=0,\tilde{v}_{1}=0)$ to state $(\tilde{u} _{2},\tilde{v}_{2})$ on the incident shock polar is followed by a second shock from $(\tilde{u} _{2},\tilde{v}_{2})$ to $(\tilde{u} _{3},\tilde{v}_{3})$ where $(\tilde{u} _{3},\tilde{v}_{3})$ lies on the reflected shock polar. The requirement that a triple shock arrangement is possible would require that the reflected shock polar intersects the incident shock polar at $(\tilde{u} _{3},\tilde{v}_{3})$ generating a third jump from $(0,0)$ to $(\tilde{u} _{3},\tilde{v}_{3})$ . This does not occur as seen in figure 16(b) – a result that we can prove holds more generally.

The incident shock polar for $(\tilde{u_{1}}=0,\tilde{v_{1}}=0)$ is given by the curve

(4.1) $$\begin{eqnarray}\tilde{v}(\tilde{u} )=-\tilde{u} \sqrt{K_{1}+\tilde{u} -\frac{K_{2}}{3}\tilde{u} ^{2}},\quad \tilde{u} _{n}\leqslant \tilde{u} <0,\end{eqnarray}$$

which gives all flow states ( $\tilde{u} ,\tilde{v}(\tilde{u} )$ ) that can be connected to $(0,0)$ by an incident shock. For a given state $\tilde{v}_{2}=\tilde{v}(\tilde{u} _{2})$ we construct then the reflected shock polar which describes all flows which can be connected to $(\tilde{u_{2}},\tilde{v_{2}})$ by a reflected shock. These are the flow states

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{v}(\tilde{u} )=-\tilde{u_{2}}\sqrt{K_{1}+\tilde{u_{2}}-\frac{K_{2}}{3}\tilde{u_{2}}^{2}}-(\tilde{u} -\tilde{u_{2}})\sqrt{K_{1}+\tilde{u} +\tilde{u_{2}}-\frac{K_{2}}{3}(\tilde{u} ^{2}+\tilde{u} \tilde{u} _{2}+\tilde{u} _{2}^{2})}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \tilde{u} _{e}\leqslant \tilde{u} <\tilde{u} _{2},\hphantom{\tilde{u_{2}}K_{1}+\tilde{u_{2}}-\frac{K_{2}}{3}\tilde{u_{2}}^{2}-(\tilde{u} -\tilde{u_{2}})\frac{K_{2}}{3}(\tilde{u} ^{2}+\tilde{u} \tilde{u} _{2}+\tilde{u} _{2}^{2})} & \displaystyle\end{eqnarray}$$

and depicted as the labelled curve $C_{1}$ in figure 16(b). We have assumed here that the reflected shock is a forward pointing shock. Now we require that the incident shock curve given by (4.1) lies above the reflected shock curve (4.2). It is readily shown that this reduces to the inequality

(4.3) $$\begin{eqnarray}\sqrt{K_{1}+\tilde{u} -\frac{K_{2}}{3}\tilde{u} ^{2}}<\sqrt{K_{1}+\tilde{u} _{2}-\frac{K_{2}}{3}\tilde{u} _{2}^{2}},\quad \tilde{u} _{e}\leqslant \tilde{u} <\tilde{u} _{2},\end{eqnarray}$$

which is true providing

(4.4) $$\begin{eqnarray}1-\frac{K_{2}}{3}(\tilde{u} +\tilde{u} _{2})>0.\end{eqnarray}$$

This inequality which is trivially true for $K_{2}=0$ holds in the two cases $-1/K_{1}<K_{2}<-3/4K_{1}$ and $-3/4K_{1}<K_{2}\leqslant 0$ where we can show that

(4.5) $$\begin{eqnarray}1-\frac{K_{2}}{3}(\tilde{u} +\tilde{u} _{2})>\frac{1}{3}.\end{eqnarray}$$

For the remaining $K_{2}>0$ case as we require that that the flow downstream of the incident shock is supersonic this excludes sonic incident shocks. The remaining flows have $\tilde{u} +\tilde{u} _{2}<0$ and so inequality (4.4) is trivially satisfied.

An important consequence of this result is the von Neumann paradox. For ideal gases numerical results and experiments show that for certain parameters the fluid flow appears to be described by three shocks separated by constant states which contradicts the theoretical results. A resolution of this paradox is the scenario given by Tesdall & Hunter (Reference Tesdall and Hunter2002), Tesdall et al. (Reference Tesdall, Sanders and Popivanov2015) where numerical experiments on a fine grid show that there is a supersonic region behind the triple point, which consists of a sequence of supersonic patches, shocks, wave fans and multiple triple points. The numerical results do not indicate whether there are finitely many such triple points or not.

We can provide analysis that suggests a similar construction holds in the dense gas case. Hunter & Brio (Reference Hunter and Brio2000) proposed the shock configuration shown in figure 3 where as well as an incident shock, reflected shock and Mach stem there is a centred simple wave behind the reflected shock. The incident, reflected and Mach shocks are all forward facing and the wave fan is backward facing. Given the upstream state $(\tilde{u} =0,\tilde{v}=0)$ , the wedge angle $a$ and the flow behind the reflected shock $\tilde{u} _{2}$ this configuration can be uniquely constructed. An argument presented in Guderley (Reference Guderley1957) suggests that the flow behind the reflected shock is sonic. This leads to the shock polar/characteristic construction shown in figure 17 for the case $K_{2}>0$ and $-3/4K_{1}<K_{2}<0$ . The expansion fan is generated by the characteristic curve obtained by integrating

(4.6) $$\begin{eqnarray}\frac{\text{d}\tilde{v}}{\text{d}\tilde{u} }=\sqrt{K_{1}+2\tilde{u} -K_{2}\tilde{u} ^{2}},\end{eqnarray}$$

from the sonic state $\tilde{u} =\tilde{u} _{3}=(1/K_{2})(1-\sqrt{1+K_{1}K_{2}})$ to $\tilde{u} =\tilde{u} _{4}$ where the characteristic curve (4.6) intersects the incident shock polar. While (4.6) can be integrated analytically the point of intersection with the incident shock has to be determined numerically. Figure 17(b) shows the shock and wave fan orientations and the flow deflected through the Mach stem ( $m$ ). The flow downstream of the Mach stem is still not aligned with the channel wall and so cannot describe the full reflection problem which must certainly involve a curved Mach stem eventually orientated to generate a normal shock at the channel wall. The likely solution is that proposed by Tesdall & Hunter (Reference Tesdall and Hunter2002) where the expansion fan reflects off a sonic curve to form a compression wave which in turn generates a shock that incident on the Mach stem generates a new expansion fan. This process can be repeated with a shrinking series of supersonic pockets generated as sketched in figure 4. The recent high resolution results of Tesdall et al. (Reference Tesdall, Sanders and Popivanov2015) confirms this picture for ideal gases.

Figure 17.

(a) Shock polar and characteristic plot for irregular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-3/4K_{1}<K_{2}\leqslant 0$ . Results are for $K_{1}=1,K_{2}=-0.73$ . The incident shock $i$ deflects the flow to $\tilde{u} _{2},\tilde{v}_{2}$ and a weak reflected shock ( $r$ ) generates the sonic flow $\tilde{u} _{3},\tilde{v}_{3}$ downstream. A centred wave fan solution describes a continuous expansion from $\tilde{u} _{3},\tilde{v}_{3}$ to $\tilde{u} _{4},\tilde{v}_{4}$ which intersects the incident shock polar and generates the shock jump across the Mach stem ( $m$ ). This so-called Guderley reflection is plotted in (b) with a sketch appended (dotted line) of the curved Mach stem that then returns the flow to parallel downstream channel flow.

The scenario of a curved Mach stem orientating the flow to normal channel flow at the boundary has to be modified for the case $-1/K_{1}<K_{2}<-3/4K_{1}$ . Observations are based on the shock polar diagram of figure 18. As the Mach stem shock originating at $\tilde{u} _{4},\tilde{v}_{4}$ increases in strength the flow downstream becomes supersonic and eventually the Mach stem shock turns sonic with downstream flow $\tilde{u} _{so}$ . The Mach stem shock remains sonic (and so of fixed orientation) with then decreasing downstream $\tilde{v}$ values now associated with a wave fan along the characteristic curve (dash-dot curve) connecting $\tilde{u} _{so}$ to $\tilde{u} _{e}$ . This wave fan propagates into a subsonic downstream region and will itself form a supersonic pocket by reflection from a sonic boundary. A possible flow pattern is sketched in figure 19 where we note that as for the triple point the wave fan characteristics reflecting off the sonic curve are expected to intersect constructing a new downstream shock. Current research is underway to confirm this proposed scenario.

Figure 18.

Shock polar plot for irregular reflection of a plane shock generated by a wedge with inclination $a$ for $K_{1}>0,-1/K_{1}<K_{2}<-3/4K_{1}$ . Results are for $K_{1}=1$ , $K_{2}=-0.8$ . A proposed curved Mach stem is generated from the shock polar connecting the downstream state ( $\tilde{u} _{4},\tilde{v}_{4}$ ) to $\tilde{u} _{so}$ where the Mach stem shock becomes sonic. Flow described by a characteristic (dash-dot) curve returns the flow through a wave fan to normal orientated channel flow on the boundary with $\tilde{u} =\tilde{u} _{e}$ .

Figure 19.

Sketch of flow scenario associated with a sonic Mach stem generating a supersonic pocket attached to the reflecting channel wall and terminating in an upstream shock.