2 Regular configurations

An oblique shock traversing a supersonic stream, of Mach number $M$ and adiabatic index $\unicode[STIX]{x1D6FE}$ , with an angle $\unicode[STIX]{x1D70E}$ produces a net deflection of the flow of value $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}(M,\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FE})$ , together with a change of its fluid properties such as temperature, density, pressure and Mach number. Furthermore, two coflowing streams under different upstream conditions are unevenly affected by the same traversing shock. The latter is accommodated in the lower stream by a transmitted angle $\unicode[STIX]{x1D70E}_{t}$ such that the downstream deflection of both currents must match and no pressure variations can be held across their contact surface. This far-field accommodation involves further reflected waves to meet mechanical equilibrium except for the degenerate case of neutral transmission.

For a given coflow configuration, made of the same gas with constant adiabatic index $\unicode[STIX]{x1D6FE}$ , the flow in the interaction region and the curvature of the shock discontinuity through the shear layer both depend on the Mach number distribution upstream of the shock, involving the boundary values $M_{1}$ and $M_{2}$ , and the inclination angle of the incident shock $\unicode[STIX]{x1D70E}_{i}$ . To determine the parametric domain $(M_{1},M_{2},\unicode[STIX]{x1D70E}_{i})$ , under which the resulting solutions correspond to those envisioned in figure 1, the outer-flow structure surrounding the interaction region is considered and assumes that the length scales are much larger than the shear-layer thickness $\unicode[STIX]{x1D6FF}_{u}$ . For these large scales, the shear layer appears infinitesimally thin, so that the problem reduces to that of the interaction of an oblique shock with a contact surface separating two supersonic streams. The outer-flow solution is independent of the Mach number profile across the shear layer ahead of the shock. This seemingly simple assumption admits in general multiple solutions depending on the values of the controlling parameters.

As discussed in Landau & Lifshitz (Reference Landau and Lifshitz1987), the simplest configurations are regular refractions involving straight-line waves, as sketched in figure 1. They involve a transmitted shock propagating from the impingement point into region-2, the lower stream, and a reflected wave propagating into region-3, the postshock stream of region-1. The reflected wave can either be a rarefaction or a shock that leaves different fluid conditions behind, both having been observed experimentally (Jahn Reference Jahn1956; Abd-El-Fattah et al. Reference Abd-El-Fattah, Henderson and Lozzi1976; Abd-El-Fattah & Henderson Reference Abd-El-Fattah and Henderson1978a ,Reference Abd-El-Fattah and Henderson b ; Buttsworth, Morgan & Jones Reference Buttsworth, Morgan and Jones1997). Therefore, a total of five regions of uniform flow are defined, where region-4 and region-5 are the downstream flow regions of the upper and lower streams respectively. In turn, these must coexist under the same deflected angle and pressure. These two types of regular refractions can only occur in certain ranges of the controlling parameters, beyond which more complicated irregular refraction patterns may arise. Theoretical considerations indicate that the ‘atlas’ of possible solutions include, in principle, single and double Mach reflections, 4-wave confluences and continuous expansion bands (Henderson Reference Henderson1966; Henderson & Macpherson Reference Henderson and Macpherson1968; Abd-El-Fattah et al. Reference Abd-El-Fattah, Henderson and Lozzi1976; Abd-El-Fattah & Henderson Reference Abd-El-Fattah and Henderson1978a ,Reference Abd-El-Fattah and Henderson b ), although only some of these solutions have been observed experimentally.

The above-mentioned regular refractions are composed by straight waves that delimit uniform regions of the flow. For the sake of simplicity, a single gas will be considered for both streams here, reducing the mixing-layer problem to a pure shear layer. When the shear-layer thickness is sufficiently small in comparison with any length scale representative of the flow-field domain besides the shock thickness, the regular solution is purely local and lacks a length scale. For example, the angle of the transmitted shock solely depends on the incoming flow conditions $M_{1}$ , $M_{2}$ , the adiabatic index $\unicode[STIX]{x1D6FE}$ and the shock incident angle $\unicode[STIX]{x1D70E}_{i}$ , thereby providing a constant value for $\unicode[STIX]{x1D70E}_{t}$ , i.e. the transmitted shock is a straight line. The solution of regular reflections, corresponding to given values of the input parameters $M_{1}$ , $M_{2}$ , $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D70E}_{i}$ , can be directly determined by application of jump conditions across the different waves in a zero-dimensional problem together with the compatibility conditions across the deflected contact surface. The Rankine–Hugoniot expressions for the pressure jump

(2.1) $$\begin{eqnarray}\displaystyle \hat{p}(M_{u},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FE})=\frac{p}{p_{u}}=\frac{2\unicode[STIX]{x1D6FE}M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}+1-\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}, & & \displaystyle\end{eqnarray}$$

flow deflection angle

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(M_{u},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FE})=\tan ^{-1}\left\{\frac{2(M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}-1)\cot \unicode[STIX]{x1D70E}}{2+M_{u}^{2}[\unicode[STIX]{x1D6FE}+\cos (2\unicode[STIX]{x1D70E})]}\right\} & & \displaystyle\end{eqnarray}$$

and postshock Mach number

(2.3) $$\begin{eqnarray}\displaystyle M^{2}(M_{u},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FE})=\frac{(\unicode[STIX]{x1D6FE}+1)^{2}M_{u}^{4}\sin ^{2}\unicode[STIX]{x1D70E}-4(M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}-1)(\unicode[STIX]{x1D6FE}M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}+1)}{(2\unicode[STIX]{x1D6FE}M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}+1-\unicode[STIX]{x1D6FE})[(\unicode[STIX]{x1D6FE}-1)M_{u}^{2}\sin ^{2}\unicode[STIX]{x1D70E}+2]}, & & \displaystyle\end{eqnarray}$$

are to be used for the shock waves involved in the flow, with the subscript $u$ denoting flow properties immediately upstream from the shock. In fact, for given values of $M_{1}$ and $\unicode[STIX]{x1D70E}_{i}$ , direct evaluation of these expressions provides

(2.4a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{i}=\unicode[STIX]{x1D706}(M_{1},\unicode[STIX]{x1D70E}_{i},\unicode[STIX]{x1D6FE}),\quad p_{3}/p_{u}=\hat{p}(M_{1},\unicode[STIX]{x1D70E}_{i},\unicode[STIX]{x1D6FE}),\quad \text{and}\quad M_{3}^{2}=M^{2}(M_{1},\unicode[STIX]{x1D70E}_{i},\unicode[STIX]{x1D6FE})\quad & & \displaystyle\end{eqnarray}$$

across the incident shock, where $p_{u}=p_{1}=p_{2}$ is the pressure of the incoming streams, equal on both sides of the contact surface. On the other hand, across the transmitted shock

(2.5a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{t}=\unicode[STIX]{x1D706}(M_{2},\unicode[STIX]{x1D70E}_{t},\unicode[STIX]{x1D6FE}),\quad p_{5}/p_{u}=\hat{p}(M_{2},\unicode[STIX]{x1D70E}_{t},\unicode[STIX]{x1D6FE})\quad \text{and}\quad M_{5}^{2}=M^{2}(M_{2},\unicode[STIX]{x1D70E}_{t},\unicode[STIX]{x1D6FE})\quad & & \displaystyle\end{eqnarray}$$

yield the downstream flow properties below the contact surface, involving the unknown value of the transmitted-shock angle $\unicode[STIX]{x1D70E}_{t}$ . When the reflected wave is a shock, the expressions for the jump between region-3 and region-4

(2.6a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{r}=\unicode[STIX]{x1D706}(M_{3},\unicode[STIX]{x1D70E}_{r},\unicode[STIX]{x1D6FE}),\quad p_{4}/p_{3}=\hat{p}(M_{3},\unicode[STIX]{x1D70E}_{r},\unicode[STIX]{x1D6FE}),\quad \text{and}\quad M_{4}^{2}=M^{2}(M_{3},\unicode[STIX]{x1D70E}_{r},\unicode[STIX]{x1D6FE})\quad & & \displaystyle\end{eqnarray}$$

apply to the downstream flow properties above the contact surface. Correspondingly, equations (2.4), (2.5) and (2.6) along with the condition that imposes equal deflection angle and pressure along the streamlines, namely

(2.7a,b ) $$\begin{eqnarray}\displaystyle (p_{4}/p_{3})(p_{3}/p_{u})=(p_{5}/p_{u})\quad \text{and}\quad \unicode[STIX]{x1D706}_{t}=\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{r}, & & \displaystyle\end{eqnarray}$$

provide a set of eleven algebraic equations that determine $M_{3}$ , $M_{4}$ , $M_{5}$ , $\unicode[STIX]{x1D70E}_{r}$ , $\unicode[STIX]{x1D70E}_{t}$ , $\unicode[STIX]{x1D706}_{i}$ , $\unicode[STIX]{x1D706}_{r}$ , $\unicode[STIX]{x1D706}_{t}$ , $p_{3}/p_{u}$ , $p_{4}/p_{u}$ and $p_{5}/p_{u}$ as a function of $M_{1}$ , $M_{2}$ and $\unicode[STIX]{x1D70E}_{i}$ for a regular refraction with a reflected shock wave.

However, when the reflected wave is an expansion, the expressions detailed in (2.6) must be replaced, with $\unicode[STIX]{x1D70E}_{r}=\sin ^{-1}(M_{3})$ , by

(2.8a,b ) $$\begin{eqnarray}\displaystyle \left(\frac{p_{4}}{p_{3}}\right)^{(\unicode[STIX]{x1D6FE}-1)/\unicode[STIX]{x1D6FE}}=\frac{2+(\unicode[STIX]{x1D6FE}-1)M_{3}^{2}}{2+(\unicode[STIX]{x1D6FE}-1)M_{4}^{2}}\quad \text{and}\quad \unicode[STIX]{x1D706}_{r}=F_{PM}(M_{4})-F_{PM}(M_{3}), & & \displaystyle\end{eqnarray}$$

where

(2.9) $$\begin{eqnarray}\displaystyle F_{PM}(M)=\left(\frac{\unicode[STIX]{x1D6FE}+1}{\unicode[STIX]{x1D6FE}-1}\right)^{1/2}\tan ^{-1}\left\{\left[\frac{\unicode[STIX]{x1D6FE}-1}{\unicode[STIX]{x1D6FE}+1}(M^{2}-1)\right]^{1/2}\right\}-\tan ^{-1}[(M^{2}-1)^{1/2}] & & \displaystyle\end{eqnarray}$$

is the Prandtl–Meyer (PM) function for a supersonic expansion. In this case, the computation involves (2.4), (2.5), (2.7) and (2.8), with $-\unicode[STIX]{x1D706}_{r}$ replaced by $+\unicode[STIX]{x1D706}_{r}$ in the second equation of (2.7), which corresponds to the clockwise deflection associated with the reflected rarefaction. The values of the Mach number behind the reflected wave $M_{4}$ and behind the transmitted wave $M_{5}$ bound the shear layer from above and from below, respectively.

Figure 2.

Transmitted-shock angle versus Mach number diagram given by (a) the outer problem for a stream $M_{1}=2$ and incident-shock angle $\unicode[STIX]{x1D70E}_{i}=50^{\circ }$ and (b) the comparison between the outer problem and highly resolved inviscid simulations for $M_{1}=5$ and $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ . Dashed curves represent the unstable branch of the multi-valued solutions.

Different sets of possible regular solutions are shown in figure 2 over the domain of transmitted-shock angle versus stream-2 Mach number. First, a trivial uniform flow field is denoted by a solid circle, where the transmitted solution matches the selected incident-shock conditions $M_{1}=M_{2}$ and $\unicode[STIX]{x1D70E}_{i}=\unicode[STIX]{x1D70E}_{t}$ . The possible locations of the upper-stream incident conditions are bounded by the lower solid line that represents the weak-shock limit, $M_{1}\sin \unicode[STIX]{x1D70E}_{i}=1$ , below which no shock would arise. In addition, cases over the $M_{3}=1$ line would produce subsonic postshock conditions behind the incident shock. Moreover, the colour curves define the collection of transmitted-shock angles $\unicode[STIX]{x1D70E}_{t}$ that comply with the problem equations for different values of $M_{2}$ and the upstream parameters ( $M_{1}$ , $\unicode[STIX]{x1D70E}_{i}$ ).

As soon as $M_{2}\neq M_{1}$ , the solution generally produces either a reflected compression or a reflected expansion wave, due to the combined effects of the convective Mach number and the intensity of postshock Mach waves impinging back on the shock. However, given an infinitesimally small convective Mach $\unicode[STIX]{x1D6FF}M=M_{2}-M_{1}$ , the postshock effects can be neglected as discussed in appendix C. In that limit, and given that $\unicode[STIX]{x1D6FF}M>0$ , compressions leading to a reflected shock are found inside the white island and expansions inside the shaded region of figure 2(a). In turn, the inverse behaviour is found if $\unicode[STIX]{x1D6FF}M<0$ , expansions in the white region and compressions in the shaded region. Nevertheless, for non-negligible values of $\unicode[STIX]{x1D6FF}M$ , the effects of Mach waves reaching the front from behind must be considered. Therefore, the previously mentioned regions are only approximately valid as a cumulative effect of differential increments or decrements of $M_{2}$ with respect to $M_{1}$ . Resolution of the set of equations renders multiple solutions associated with the configurations that provide mechanical equilibrium downstream. Consequently, in figure 2(a) transmitted solutions are plotted in cyan for reflected shocks ( $M_{2}>M_{1}$ ) and in red for reflected expansions ( $M_{2}<M_{1}$ ). However, the differential expansion contributions to the $M_{2}<M_{1}$ points balance out with the differential compressions that appear once into the shaded area to yield a neutral transmission point and, further, reflected-shock solutions.

In addition, three different scenarios with $M_{1}=5$ are represented in figure 2(b), namely $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ . A comparison of the transmitted angle $\unicode[STIX]{x1D70E}_{t}$ is displayed in dependence on $M_{2}$ between numerical simulations (symbols), and the theoretical outer problem (solid lines). The transmitted angle provided by simulations fits reasonably well with Landau’s outer solution, a result that is also extended to any other fluid property such as pressure or deflection angle. Red curves in figure 2(b) represent the solution predicted by a reflected-expansion configuration while cyan curves refer to shock-reflected cases.

Note that the set of solutions given by the regular-problem equations is multi-valued in some regimes, involving further branches of secondary solutions for $M_{2}$ such as the ones computed in the dashed upper curves in figure 2(b), which yield larger angles of the transmitted shock. The two-valued solution suggests two candidates for the final solution in a certain region. Multiplicity in the solutions can be solved by stability criteria (Fowles Reference Fowles1981). This may be more easily seen in the hodograph plane as shown in appendix B. There, the so-called pressure-deflection polar plots are computed by solving the inviscid problem, as proposed by Landau. This method, first introduced by von Neumann (Reference von Neumann1943a ,Reference von Neumann b ) and later extended by Henderson (Reference Henderson1966), Henderson & Macpherson (Reference Henderson and Macpherson1968), is particularly useful to determine all possible solutions that ultimately lead to mechanical equilibrium in the downstream shear layer, i.e. when the upper and lower postshock streams propagate with the same pressure and direction. Additionally, highly resolved inviscid simulations have been used in this work to confirm the unstable nature of the upper branch in figure 2(b). Such numerical computations, whose results have not been displayed here for the sake of conciseness, have been initiated with the results predicted by (2.1)–(2.9). When the lower-branch predictions of (2.1)–(2.9) are employed as initial conditions, the numerical simulations rapidly accommodate to provide the stable results shown in figure 2(b) in good agreement. It must be noted that, in the numerical simulations, the adiabatic index depends on temperature (see appendix A) to meet the correction requirements of the irregular reflections studied below. There, large Mach numbers and flows including multiple shocks combined with expansions produce non-negligible changes in the adiabatic index. However, for regular intersections, small deviations are produced by these effects between the regular theoretical model for a calorically perfect gas (see appendix B) and the numerical solutions.

Table 1.

Critical values ( $M_{2c}^{-}/M_{2c}^{+}$ ) for the lower-stream Mach number predicted by Landau’s outer-problem equations when $\unicode[STIX]{x1D6FE}=1.4=\text{const}$ . Starred values refer to shock-reflected configurations, non-starred values refer to expansion-reflected cases and superscript $N$ refers to neutral transmission. The dash symbol means ‘not found’ within the interval $M_{2}=(1,20)$ .

The limits of the regular configurations are readily identified in the turning points of the curves. Due to the maximum deflection that can be produced by an oblique shock over a supersonic stream of a given Mach number plus a reflected wave, the downstream conditions of parallel isobaric flow cannot be met for any desired combination of $M_{1}$ , $M_{2}$ and $\unicode[STIX]{x1D70E}_{i}$ . Therefore, critical values of the lower stream $M_{2c}^{+}$ and $M_{2c}^{-}$ are found in both panels of figure 2, beyond which no regular solutions can be recovered by the system of (2.1)–(2.9). These limits, and those predicted by the intersection criteria of polar plots are, indeed, the same conditions. For the sake of completeness, lower and upper limits in terms of $M_{2}$ are identified as a function of $M_{1}$ and $\unicode[STIX]{x1D70E}_{i}$ , see table 1. Starred values refer to shock-reflected configurations and non-starred values refer to expansion-reflected cases. The superscript $N$ denotes a case of neutral transmission. The dash symbol means not found within the interval $M_{2}\in [1,20]$ , following the fact that $M>20$ is impractical and not normally encountered in applications. For instance, the curve corresponding to $M_{1}=5$ and $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ in figure 2(b) exhibits two turning points at $M_{2c}^{-}=4.61$ and $M_{2c}^{+}=7.3$ . These limits can be easily anticipated by analysing the hodograph plane, as described in appendix B. The cases where $M_{2}<M_{2c}^{-}$ correspond to situations for which the stream-2 polar in figure 18(b) is sufficiently small so that intersections do not occur either with the stream-3 polar or a Prandtl–Meyer expansion. On the other hand, situations where $M_{2}>M_{2c}^{+}$ correspond to a stream-2 polar sufficiently narrow that it does not intersect with the $M_{3}$ -loop.

3 Unsteady and real-gas considerations

We must accept that small perturbations in the convective Mach number beyond critical limits will produce a lack of equilibrium that fails to deflect the flow consistently. In this fashion, new structures make an appearance after impingement on the shear layer involving curved shocks, triple points, shock intersections, subsonic non-uniform regions and growing finite-length shocks that result from the unsteady flow field. Following Henderson (Reference Henderson1966), Henderson & Macpherson (Reference Henderson and Macpherson1968), the term Mach stem, that was first coined in the context of shock reflection in solid walls, will also be employed here to address the finite-length shocks that emerge in irregular interactions.

Under conditions that render subsonic flow behind the incident shock, $M_{3}<1$ , irregular configurations are found to be strongly affected by the upper-stream local boundaries, which are typically responsible for the incident-shock creation. This was inferred from the experiments in Abd-El-Fattah & Henderson (Reference Abd-El-Fattah and Henderson1978a ,Reference Abd-El-Fattah and Henderson b ), namely bound precursor refraction and lambda shock refraction configurations. Nevertheless, when $M_{3}>1$ and the downstream boundary conditions are sufficiently far to affect the dynamics of the structure through any other subsonic region, the local flow field is self-similar as it lacks characteristic length scales. Therefore, the finite length of the Mach stem must scale with time, growing with the whole interacting structure. The associated movement is performed against the flow, so that the effective Mach number that is perceived at the shock surface is increased sufficiently to provide an equilibrium closure in pressure and deflection. In the hodograph plane, the corrected related polars are slightly larger, such that intersecting solutions independent of time are provided anew. This fact forces the contact point between the shear layer and shock structure to retreat at constant speed. A direct consequence being that self-similar constructions are achieved, where angles and downstream conditions are maintained constant.

Previous works making use of the hodograph method did not include the effects of moving structures or temperature-dependent heat capacities, which we present here. Following this approach, an asymptotic analysis is developed for small perturbations on the post-shock adiabatic index $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}$ and inflow Mach number $\unicode[STIX]{x1D6FF}M$ , which produce necessary second-order corrections to the pressure and deflection equilibrium conditions, as sketched in figure 3(a).

In the absence of viscous and conductive effects, the effect of temperature only enters into play via the heat capacity ratio $\unicode[STIX]{x1D6FE}$ , whose temperature dependence has been taken into account in the numerical simulations through the JANAF polynomials (see appendix A). Direct inspection of the Rankine–Hugoniot equations in the strong-shock limit can be used to provide an upper limit for the maximum temperature achieved for non-reactive cases. As $\unicode[STIX]{x1D6FE}$ may vary down to $6\,\%$ within the range of shock-induced temperature change considered, linear perturbations can be used to predict variations by an amount of order $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{u})\sim -\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D6FE}^{2}-1)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}\sim 10\,\%$ for the mass-compression ratio across one single shock. For a shock-train structure, this effect accumulates, with the value of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}$ being properly modelled with the local temperature.

The mass-compression ratio for a gas that is not calorically perfect reads

(3.1) $$\begin{eqnarray}\displaystyle {\mathcal{R}} & = & \displaystyle \frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{u}}=\frac{\unicode[STIX]{x1D6FE}_{s}(\unicode[STIX]{x1D6FE}_{u}-1)(1+\unicode[STIX]{x1D6FE}_{u}M_{u_{n}}^{2})}{[2+(\unicode[STIX]{x1D6FE}_{u}-1)M_{u_{n}}^{2}]}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\sqrt{(\unicode[STIX]{x1D6FE}_{u}-1)[(\unicode[STIX]{x1D6FE}_{u}-1)\unicode[STIX]{x1D6FE}_{u}^{2}M_{u_{n}}^{4}+2\unicode[STIX]{x1D6FE}_{u}(\unicode[STIX]{x1D6FE}_{u}-\unicode[STIX]{x1D6FE}_{s}^{2})M_{u_{n}}^{2}+\unicode[STIX]{x1D6FE}_{s}^{2}(\unicode[STIX]{x1D6FE}_{u}-1)]}}{[2+(\unicode[STIX]{x1D6FE}_{u}-1)M_{u_{n}}^{2}]},\end{eqnarray}$$

where $M_{u_{n}}=M_{u}\sin \unicode[STIX]{x1D70E}$ , is the normal component of the Mach number, and $\unicode[STIX]{x1D6FE}_{u}$ and $\unicode[STIX]{x1D6FE}_{s}$ are, respectively, the upstream and post-shock values of the adiabatic index. Automatically, it follows that

(3.2a,b ) $$\begin{eqnarray}\displaystyle \hat{p}=1+\unicode[STIX]{x1D6FE}_{u}M_{u_{n}}^{2}(1-{\mathcal{R}}^{-1})\quad \text{and}\quad \unicode[STIX]{x1D706}=\unicode[STIX]{x1D70E}-\tan ^{-1}\left(\frac{\tan \unicode[STIX]{x1D70E}}{{\mathcal{R}}}\right). & & \displaystyle\end{eqnarray}$$

Figure 3.

(a) Sketch of the moving contact point for $M_{2}<M_{2c}^{-}<M_{1}$ in a shock-anchored reference frame. The shock angles and the post-shock deflection angle usually reverse for $M_{2}>M_{2c}^{+}>M_{1}$ . (b) Sketch of the deflected streamline corrected by variable $\unicode[STIX]{x1D6FE}(T)$ in the expansion fan.

Regardless the type of irregular interaction, the contact-point properties are always determined by the local equilibrium downstream, $p_{5}=p_{6}$ and $\unicode[STIX]{x1D706}_{5}=\unicode[STIX]{x1D706}_{6}$ , which yields $\unicode[STIX]{x1D70E}_{m}$ and $\unicode[STIX]{x1D70E}_{t}$ as the local values of the oblique Mach stem and the transmitted shock. Therefore, small variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{t}$ are taken into account with respect to the downstream value $\unicode[STIX]{x1D6FE}_{s}=\unicode[STIX]{x1D6FE}_{m}$ and $\unicode[STIX]{x1D6FE}_{s}=\unicode[STIX]{x1D6FE}_{t}$ behind each shock, yielding the following perturbations to the shock angles

(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}_{m}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}=\frac{\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\left(\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{t}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{t}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{m}\right)+\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\left(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{m}-\hat{p}_{\unicode[STIX]{x1D6FE}}|_{t}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{t}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}\right)}{\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}} & & \displaystyle\end{eqnarray}$$

and

(3.4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}_{t}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}=\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\left(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{m}-\hat{p}_{\unicode[STIX]{x1D6FE}}|_{t}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{t}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}\right)+\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\left(\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{t}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{t}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{m}\right)}{\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}}, & & \displaystyle\end{eqnarray}$$

with the factors

(3.5a-d ) $$\begin{eqnarray}\displaystyle \hat{p}_{\unicode[STIX]{x1D6FE}}|_{m,t}=\frac{1}{\hat{p}}\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{m,t}},\quad \hat{p}_{\unicode[STIX]{x1D70E}}|_{m,t}=\frac{1}{\hat{p}}\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{m,t}},\quad \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{m,t}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{m,t}},\quad \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m,t}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{m,t}}\quad & & \displaystyle\end{eqnarray}$$

defining the variations of the pressure and deflection angle with the adiabatic index and shock angle. The pressure and post-shock deflection angle are also affected by the adiabatic index perturbation. Mechanical equilibrium downstream provides the following deviations with respect to the calorically perfect assumption

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{p}\frac{\unicode[STIX]{x1D6FF}p}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}}=-\frac{\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{m})-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}}=-\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{m})-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}(\hat{p}_{\unicode[STIX]{x1D6FE}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FE}}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}. & \displaystyle\end{eqnarray}$$

In an analogous manner, anticipating that the contact-point speed is relatively small, a perturbation analysis can be carried out to study the effect of the moving structure on the final conditions. As we are considering similar thermodynamical properties in the free streams, a single value of the perturbation parameter $\unicode[STIX]{x1D6FF}M$ is proposed, as shown in figure 3. It can be immediately seen that shock angles get affected in the following proportion for any small $\unicode[STIX]{x1D6FF}M$ ,

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}_{m}}{\unicode[STIX]{x1D6FF}M}=-\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}(\hat{p}_{M}|_{m}-\hat{p}_{M}|_{t})-\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}(\unicode[STIX]{x1D706}_{M}|_{m}-\unicode[STIX]{x1D706}_{M}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}_{t}}{\unicode[STIX]{x1D6FF}M}=-\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}(\hat{p}_{M}|_{m}-\hat{p}_{M}|_{t})-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}(\unicode[STIX]{x1D706}_{M}|_{m}-\unicode[STIX]{x1D706}_{M}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}, & \displaystyle\end{eqnarray}$$

with the factors

(3.10a,b ) $$\begin{eqnarray}\displaystyle \hat{p}_{M}=\frac{1}{\hat{p}}\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}M},\quad \text{and}\quad \unicode[STIX]{x1D706}_{M}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}{\unicode[STIX]{x2202}M} & & \displaystyle\end{eqnarray}$$

defining the variations of the pressure and deflection angle with the upstream Mach number. The pressure and deflection perturbations can be analytically determined by the following expressions

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{p}\frac{\unicode[STIX]{x1D6FF}p}{\unicode[STIX]{x1D6FF}M}=-\frac{\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}(\hat{p}_{M}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{M}|_{m})-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}(\hat{p}_{M}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{M}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}, & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D6FF}M}=-\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}(\hat{p}_{M}|_{m}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}\unicode[STIX]{x1D706}_{M}|_{m})-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}(\hat{p}_{M}|_{t}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}-\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}\unicode[STIX]{x1D706}_{M}|_{t})}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{t}\hat{p}_{\unicode[STIX]{x1D70E}}|_{m}-\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70E}}|_{m}\hat{p}_{\unicode[STIX]{x1D70E}}|_{t}}. & \displaystyle\end{eqnarray}$$

Correspondingly, the intersection of the perturbed $M_{1}$ and $M_{2}$ curves in the polar-plot diagram, which defines the conditions of the contact point, is given by

(3.13a,b ) $$\begin{eqnarray}\displaystyle \frac{p}{p_{u}}=\frac{p_{0}}{p_{u}}+\frac{1}{p_{u}}\frac{\unicode[STIX]{x1D6FF}p}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}+\frac{1}{p_{u}}\frac{\unicode[STIX]{x1D6FF}p}{\unicode[STIX]{x1D6FF}M}\quad \text{and}\quad \unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{0}+\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}+\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D6FF}M}\unicode[STIX]{x1D6FF}M. & & \displaystyle\end{eqnarray}$$

The functions (3.6), (3.7), (3.11) and (3.12) are computed in figure 4 as a function of the free-stream Mach numbers $M_{1}$ and $M_{2}$ for negative values of $\unicode[STIX]{x1D706}$ . The vectors indicate the modified intersection point between stream-1 (upper) and stream-2 (lower) polars by accounting for the pressure and deflection perturbation, where the contour lines denote the isovalues of the modulus. The upper half-domain shows the corrections due to a unitary $\unicode[STIX]{x1D6FF}M$ perturbation while the lower half (shadowed) domain involves unitary $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}$ corrections. As expected, perturbations in high Mach number configurations, which make the polars more slender, produce small flow deflections. Note that the intersection that actually happens for $M_{2}>M_{1}$ is the one placed at $\unicode[STIX]{x1D706}>0$ half-plane, so that perturbations in the deflection angle are the opposite.

Figure 4.

Variation of the intersection point as a function of the free-stream Mach numbers $M_{1}$ and $M_{2}$ for negative deflection angles $\unicode[STIX]{x1D706}$ ( $M_{1}>M_{2}$ ). Iso-contours represent the length of the vectors.

The values of $\unicode[STIX]{x1D6FF}M$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}_{m}$ are, in fact, dependent variables to be calculated with the aid of the ultimate downstream conditions. The effect of the local deviation on the polar-plot intersection points (equilibrium) is propagated through the rest of the contact points. Clearly, a rather similar analysis can be performed for the different wave intersections exhibited in the irregular interaction, but the analytical details are omitted here for the sake of simplicity.

While propagating effects are expected to have a significant role when $|M_{2c}^{\pm }-M_{2}|\gg 1$ , real-gas effects are particularly relevant when temperature changes are sufficiently strong, that is, across the incident shock for $M_{1}>M_{2}$ and along the final expansion. For the latter, equations (2.8) and (2.9) should be conveniently adapted to include the effect of temperature on the adiabatic index. The pressure variation along the streamline in the expansion fan can be approximated through

(3.14) $$\begin{eqnarray}\displaystyle \int _{p_{i}}^{p_{f}}\frac{\text{d}p}{p}=\int _{T_{i}}^{T_{f}}\frac{\unicode[STIX]{x1D6FE}(T)}{\unicode[STIX]{x1D6FE}(T)-1}\frac{\text{d}T}{T},\quad \ln \left(\frac{p_{f}}{p_{i}}\right)\sim \ln \left(\frac{T_{f}}{T_{i}}\right)\frac{\unicode[STIX]{x1D6FE}_{i}}{\unicode[STIX]{x1D6FE}_{i}-1}\left[1-\frac{\unicode[STIX]{x1D6FE}_{f}-\unicode[STIX]{x1D6FE}_{i}}{\unicode[STIX]{x1D6FE}_{i}(\unicode[STIX]{x1D6FE}_{i}-1)}\right],\quad & & \displaystyle\end{eqnarray}$$

with

(3.15) $$\begin{eqnarray}\displaystyle \frac{T_{f}}{T_{i}}=\frac{2+(\unicode[STIX]{x1D6FE}_{i}-1)M_{i}^{2}}{2+(\unicode[STIX]{x1D6FE}_{f}-1)M_{f}^{2}} & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6FE}(T)$ being provided by the corresponding constitutive relationship. Likewise, the flow deflection can be computed with (2.9) by just employing the local value of $\unicode[STIX]{x1D6FE}$ at the head and at the tail of the expansion wave, $F_{PM}(\unicode[STIX]{x1D6FE}_{i},M_{i})$ and $F_{PM}(\unicode[STIX]{x1D6FE}_{f},M_{f})$ , respectively, as a first approximation. Both propagating and real-gas effects are incorporated in the polar plots, with the final result being compared with numerical simulations for the identified structures.

Note that the function $\unicode[STIX]{x1D6FE}(T)$ and the above expressions for the shock wave and the PM expansion implicitly stem from the fact that excitation and relaxation times are negligible, so the properties of the gas particles are assumed to be in local thermodynamic equilibrium everywhere. Small but finite relaxation times would involve the presence of a zone, close to the vertex within the expansion fan, where the residence time compares to the characteristic relaxation time, thereby affecting the final structure. Since such effects are found to be very weak (Hayes & Probstein Reference Hayes and Probstein2004), they will not be considered in this work.

4 Self-similar irregular configurations

Under certain convective Mach numbers, for large enough differences between $M_{1}$ and $M_{2}$ , the interaction with the shock discontinuity is driven to unsustainable conditions. There, regular configurations fail to provide equilibrium downstream and irregular configurations arise involving the formation of additional waves that accommodate the flow by moving upstream in a quasi-steady manner. For the sake of providing a general description of the structures, the focus is placed here in conditions that render supersonic flow behind the incident shock $M_{3}>1$ , mostly yielding complex boundary-independent canonical configurations, such as those depicted in figure 5. For this reason, previously identified structures such as the bound precursor refraction (BPR) and lambda shock refraction (LSR), will not be addressed here.

Figure 5.

Irregular configurations arising from the interaction of an oblique shock with a supersonic shear layer. (a) Free precursor refraction, (b) twin von Neumann reflection, (c) fast–slow single Mach reflection and (d) slow–fast single Mach reflection. Incident $s_{i}$ , reflected $s_{r}$ , transmitted $s_{t}$ and Mach stem $s_{m}$ shocks are labelled.

For sufficiently thin vortex sheets, the evolution of the irregular structure can be split into three different regimes. First, the initial instant at which the shock impacts the shear layer and the transmitted and reflected shock waves get connected through local shocks that grow with time. Second, a mid-time regime in which the structure previously formed evolves in a self-similar form until information of the boundary conditions enters into play. And finally, the boundary-dominated stage where the irregular structure may (or may not) get anchored to the domain walls that are likely present in laboratory set-ups. To facilitate the analysis, we shall focus on the scale-free regime that renders a self-similar flow pattern. There, the irregular structure is much larger than the shear-layer thickness, yet much smaller than the flow-field downstream domain, so the acoustic coupling with post-shock boundary conditions can be neglected. The most characteristic features for the understanding of irregular configurations are developed in the foregoing sections through hodograph and highly resolved inviscid simulation analyses.

4.1 Free precursor refraction (FPR)

The first scenario of $M_{1}>M_{2}$ providing a supersonic shear layer is depicted in figure 5(a). An illustrative combination of parameters $M_{1}=5$ and $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ in air with $\unicode[STIX]{x1D6FE}=1.4=\text{const}.$ , provide a critical value $M_{2c}^{-}=2.18$ as presented in table 1 and shown in figure 2(b). Despite the complex structure that this irregular intersection exhibits, a relatively simple analysis can be carried out to compute the values in the distinguished zones. It implies the use of (2.1)–(2.9) along with the polar-plot diagrams. The regular configuration for $M_{2}>M_{2c}^{-}$ is analogous to that obtained in figure 18(b) showing a reflected PM-expansion wave arising from post-incident-shock conditions that intersects stream-2 polar at a regular equilibrium point. This remains until a regular solution such as $M_{2}$ is progressively diminished to $M_{2c}^{-}$ providing a tangential intersection in the polar-plot diagram. However, the qualitative change from regular to irregular configuration is justified in figure 6(a), where it can be noted that for $M_{2}<M_{2c}^{-}$ the pressure obtained under the expansion is higher than the pressure that could be obtained by a stream-2 transmitted shock at any deflection angle (no intersection). The overpressure behind the interaction point results in a twofold outcome: a push to the transmitted shock, and the formation of a reflected-shock wave instead of the regular expansion. Therefore, secondary shocks are required to accommodate the flow, as found by the numerical simulations. The term secondary is employed here to refer shocks that are not present in the far-field domain, defining Mach stem shocks $s_{m}$ whose bounds are well delimited by other waves and triple points. We have included the polars of $s_{r}$ and $s_{m_{2}}$ given the flow conditions behind $s_{i}$ and $s_{m_{1}}$ , respectively.

Since pressure and deflection angle are constant across the contact surfaces, properties are easily identified there with the crossing points in the $p$ – $\unicode[STIX]{x1D706}$ plane; i.e. the intersection between stream-1 and stream-2 polars for negative $\unicode[STIX]{x1D706}$ provides the flow conditions after the contact point joining $s_{t}$ and $s_{m_{1}}$ . Under these post-shock conditions, a new polar representing the possible outcomes of $s_{m_{2}}$ arises and intersects with the $s_{r}$ polar in a point that establishes the properties at the contact line produced after the four-shock confluence and before the expansion, see figure 5(a). This isentropic expansion, represented as a dashed line in figure 6(a), bends down the reflected shock and accelerates the supersonic flow lowering pressures to fulfil equilibrium with the subsonic flow after the moving transmitted shock. This can be seen in figure 7(b) from the numerical simulations. From the polar-plot viewpoint, the intersection of the Prandtl–Meyer curve with the lower-stream polar would determine the final downstream conditions. As previously deduced, such expansion does not intersect the $M_{2}$ polar indicating that far-field equilibrium cannot be achieved in static conditions. The colour legend for $\unicode[STIX]{x1D70E}$ in figure 6, as well as in figures 8, 10 and 12, refers to the shock angle relative to the local incident streamline slope, which is given by the abscissa.

Figure 6.

FPR polar plots associated with an oblique shock, with incident angle $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ , crossing an upper stream moving at $M_{1}=5$ and lower stream $M_{2}=2<M_{2c}^{-}$ . Standard polar plots do not provide a closure to the analytic problem (a), corrected polars given the perturbations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}M$ for real gases and moving structure (b). Black points denote the numerical solutions evaluated at the different regions with their respective error bars.

Figure 7.

FPR simulations showing Mach (a) and pressure field (b) distribution for $M_{1}=5$ , $M_{2}=2$ and $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ after impingement of the oblique shock with the shear layer.

Figure 8.

TNR polar plots for $\unicode[STIX]{x1D70E}_{i}=50^{\circ }$ and $M_{1}=2$ and $M_{2}=1.5<M_{2c}^{-}$  (a), detail of equilibrium solution for real gases and moving structures involving a subsonic region of expansions (shaded area) (b), including the simulation values (black points) with their respective error bars, and analytical solutions (white circles).

The real-gas effects introduced above must be considered to ensure the closure of the problem with the most sensitive process being the expansion wave. In particular, polars widen substantially for small changes in $\unicode[STIX]{x1D6FE}$ and more-intense less-deflecting expansions are found. The perturbations to the adiabatic index are computed in the hodograph plane as shown above, varying across the different regions that range from $\unicode[STIX]{x1D6FE}_{u}=1.4$ to $\unicode[STIX]{x1D6FE}_{B}=1.36$ in the four-shock confluence. Nevertheless, the moving front is also responsible for the final solution. In the moving transmitted-shock reference frame, the incoming flow in the lower stream moves at a higher Mach number $M_{2}^{\prime }=M_{2}+\unicode[STIX]{x1D6FF}M_{2}$ , so that pressure downstream is increased up to the value that the interaction demands. As a result of the receding transmitted shock, $s_{m_{1}}$ arises to match the static incident shock wave. Consequently, this new shock front must also move with a local inflow Mach number $M_{1}^{\prime }=M_{1}+\unicode[STIX]{x1D6FF}M_{1}$ , with $\unicode[STIX]{x1D6FF}M_{1}=(a_{2}/a_{1})\unicode[STIX]{x1D6FF}M_{2}$ . When the upper and lower streams share thermodynamical properties, and the sound speeds are equal, the Mach number variation $\unicode[STIX]{x1D6FF}M$ is the same in both gas streams. Although the flow behind the incident shock is not affected by the moving structure, the reflected wave does move, varying the pre-shock streamwise Mach number $\unicode[STIX]{x1D6FF}M_{3x}=(a_{1}/a_{3})\unicode[STIX]{x1D6FF}M$ and yielding

(4.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}_{3}=-\text{sin}\,\unicode[STIX]{x1D706}_{3}\frac{\unicode[STIX]{x1D6FF}M_{3x}}{M_{3}}=-\text{sin}\,\unicode[STIX]{x1D706}_{3}\frac{a_{1}}{a_{3}M_{3}}\unicode[STIX]{x1D6FF}M\quad \text{and}\quad \unicode[STIX]{x1D6FF}M_{3}=\cos \unicode[STIX]{x1D706}_{3}\frac{a_{1}}{a_{3}}\unicode[STIX]{x1D6FF}M & & \displaystyle\end{eqnarray}$$

for the flow deflection angle and the total Mach number, respectively. The value of $M_{2}^{\prime }$ is then provided by the minimum lower-stream Mach number for which its associated modified polar reaches the final expansion line. This is marked by the tangent solution (C–D) in figure 6(b), where $\unicode[STIX]{x1D6FF}M=0.015$ , according to polar-plot theory.

There are some implicit assumptions in the use of the polar-plot method to solve the complex structures mentioned above. One of them is that it resolves the final expansion as a Prandtl–Meyer expansion. Although it is true that this transformation can be taken isentropic within the interaction domain, the assumption that streamlines bend according to (2.9) is not rigorously right because the expansion does not depart from a single point, but rather from a finite region in the non-uniform subsonic flow that radiates pressure changes along Mach lines. Such interaction is far from being trivial, and theoretical descriptions have only been able to cover weak cases. The secondary shock impingement on the transonic shear layer resembles the flow structure analytically studied in Huete et al. (Reference Huete, Urzay, Sánchez and Williams2016), where it was found that the impinging weak shock reflects as a pressure ridge followed by an expansion. Something similar applies to the flow right behind the secondary shock $s_{m_{1}}$ , which has been assumed completely homogeneous as the influence from the non-uniform subsonic flow has been neglected. The latter assumption is justified by the linear analysis developed in appendix C, which predicts a very small influence of these pressure waves on $s_{m_{1}}$ . As a matter of example, the value of $\unicode[STIX]{x1D6E5}_{I}$ computed in figure 20(b) represents the shock trajectory deviations when it is only affected by acoustic waves departing from non-uniform pressure fields $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D6E5}_{I}\unicode[STIX]{x1D6FF}p/p$ . For the region of interest behind $s_{m_{1}}$ , linear theory predicts $\unicode[STIX]{x1D6E5}_{I}\sim 0.2$ , rendering a very small value of the shock trajectory deviation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\sim 1.45^{\circ }$ , provided that pressure changes are $\unicode[STIX]{x1D6FF}p/p\sim 0.13$ within the influence zone, according to the numerical simulations.

The equilibrium condition predicted by polar plots calls for constant angles of the shock structure, which dictates equal velocities in the backwards-moving intersection points delimiting $s_{m_{1}}$ . The structure is pseudosteady when these values remain constant in the polar diagram. In addition, when all the local lengths in the irregular structure grow proportionally in time with the same power law, the flow field is self-similar, akin to those found in shock–wall reflections (Jones et al. Reference Jones, Martin and Thornhill1951; Sternberg Reference Sternberg1959; Hornung Reference Hornung1986; Tesdall et al. Reference Tesdall, Sanders and Keyfit2008). Note that, for set-ups made of different gases, such as those carried out by Abd-El-Fattah & Henderson (Reference Abd-El-Fattah and Henderson1978a ,Reference Abd-El-Fattah and Henderson b ), a modified version of figure 6(b) would apply, with the adjusted Mach numbers providing different equilibrium angles.

The numerical simulation for $M_{1}=5$ , $M_{2}=2$ and $\unicode[STIX]{x1D70E}_{i}=30^{\circ }$ is shown in figure 7, which displays a self-similar solution that is maintained through simulation time. Mach number variations can be used to identify the aforementioned regions of the flow, where the subsonic zone is confined in the dark-brown domain in figure 7(a). The slip line originating at the four-shock confluence is not easily distinguished as the Mach numbers across the shear layer are very similar, although it can be tracked over the streamlines in the right panel. The contact surface, placed at $y=0$ , encounters first the backwards-moving transmitted shock, which is curved and generates positive (far bottom-wards) and negative (close to the contact surface) deflection angles. The pressure distribution shown in figure 7(b), behind $s_{t}$ and $s_{m_{1}}$ , remains nearly uniform $p/p_{u}\simeq 5$ as denoted by the equilibrium intersections in figure 6(b). The maximum pressure is placed right behind the four-shock confluence. Although not directly seen because of the wide range in the scale, the subsonic zone is not exactly uniform as deduced from the streamlines deflection in figure 7(b). Spatially averaged values retrieved from the simulations are $\langle p_{num}\rangle /p_{u}=5.11$ and $\langle p_{num}\rangle /p_{u}=4.27$ for the pressure behind the transmitted shock close to and far below the contact surface, (points A and D, respectively) and $\langle p_{num}\rangle /p_{u}=22.3$ for the maximum pressure found behind the four-shock confluence (point B), values in great concordance with the results given by the modified hodograph method. As for the receding velocity, polar-plot theory predicts $5.2~\text{m}~\text{s}^{-1}$ , while numerical simulations renders a smaller value, $2.1~\text{m}~\text{s}^{-1}$ . This may be explained by the transmitted-shock trajectory, which employs a very long distance in accommodating the equilibrium condition, thereby generating a non-uniform subsonic flow.

4.2 Twin von Neumann reflection (TNR)

The following configuration includes the previously identified secondary shocks of the FPR, $s_{m_{1}}$ and $s_{m_{2}}$ , plus an extra Mach stem $s_{m_{3}}$ that joins the latter to the incident and reflected shocks, as shown in figure 5(b). This forms a kind of symmetric composition of $s_{i}$ and $s_{r}$ with $s_{m_{1}}$ and $s_{m_{2}}$ , that gives the name twin von Neumann reflection to this case. This kind of structure arises for weak shocks, although an analogous configuration is possible for strong shocks called twin Mach reflection, TMR, which will not be addressed here.

This distinguished case arises when the reflected and secondary shocks, $s_{r}$ and $s_{m}$ , do not suffice to provide equilibrium thereby originating an expansion wave in the upper stream, defined previously by their intersection B in figure 6(b). A linking shock $s_{m_{3}}$ is produced, defined by the segment of loop between the two intersections B and C shown in figure 8(b), where the Mach stems, the transmitted and the reflected-shock polars are modified in consonance with the relative Mach number $\unicode[STIX]{x1D6FF}M=0.02$ and the adiabatic index ranging from $\unicode[STIX]{x1D6FE}_{u}=1.4$ to $\unicode[STIX]{x1D6FE}_{B}\sim 1.38$ .

Figure 9.

TNR simulations showing Mach (a) and pressure field distribution with streamlines (b) for $M_{1}=2$ , $M_{2}=1.5$ , and $\unicode[STIX]{x1D70E}_{i}=50^{\circ }$ after impingement of the oblique shock with the shear layer.

Unlike in the FPR previous case, where the upper stream is supersonic everywhere, the subsonic flow behind the Mach stem impedes solving the local structure by inspection of the polar curves. The subsonic expansion lies in the shaded region of figure 8(b), limited by the known boundaries of the flow, although not explicitly defined by a curve in the hodograph plane, and responsible for the subsonic link of E and D. Note that further increases in $\unicode[STIX]{x1D6FF}_{M}$ modifies the stream-2’ polar in the closure direction (displacing E towards D), but also widens stream-1’ polar finally moving away of D, so that no tangential solution is found. Nonetheless, polar plots can still provide useful information for the distinguished zones in the shock-reflection structure, except for its actual dimensions. Perforce, the relative lengths are retrieved from the numerical simulations shown in figure 9, which show that non-uniform regions extend over lengths of greater order of magnitude than the Mach stem length, involving the coupling of the subsonic flow with downstream boundaries. Spatially averaged values retrieved from the simulations are $\langle p_{num}\rangle /p_{u}=2.3$ and $\langle p_{num}\rangle /p_{u}=4.5$ for points A and D, respectively. Asymptotic far-field pressure values for the end of the expansion and below the transmitted shock are $\langle p_{num}\rangle /p_{u}=2.48$ and $\langle p_{num}\rangle /p_{u}=2.52$ , respectively. Excluding point A, for which the pressure predicted by theoretical polar plots is lower, the rest of the estimations are definitely consistent with the numerical simulations.

Besides rendering more accurate values, numerical simulations help in providing information about the relative sizes of the secondary shocks, as well as the flow-field properties. From figure 9, a supersonic flow layer separating the subsonic regions formed behind the transmitted shock and the Mach stem is identified. In the absence of boundary constraints, a boundless concave Mach steam would dynamically evolve towards a planar shape, as the downstream mass flux induced parallel to the front would accumulate in the local maximum, pushing the front upwards. The dynamical response could exhibit decaying oscillations until the planar shape is ultimately achieved. In this case, as the Mach stem boundaries are determined by the newly formed triple points, the disequilibrium in the concave Mach stem forces it to move upwards as a whole. For this particular case, however, the value of $\unicode[STIX]{x1D6FF}M\sim 0.01$ is found to be quite small.

4.3 Single Mach reflection (SMR)

4.3.1 Fast–slow SMR

We shall now consider the flow structure shown in figure 5(c), which appears, e.g. for $M_{1}=5$ , $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ and $M_{2}=4<M_{2c}^{-}$ . There, a new shear layer is recognized between subsonic and supersonic regions arising from the triple point, namely the confluence of the incident, reflected and Mach stem shocks. Analogously to solutions of FPR type, when $M_{2}$ is reduced below the critical value $M_{2c}^{-}$ , regular configurations fail to provide downstream equilibrium as shown in figure 10(a), with the overpressure triggering the evolution of quasi-steady structures. Akin to the previous case, the Mach stem curvature impedes static equilibrium so it moves upwards, along with the transmitted shock, causing an increase of the relative Mach number.

Figure 10.

Fast–slow SMR regular polar plot for $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ , $M_{1}=5$ and $M_{2}=4$  (a), detail of corrected equilibrium solution involving a subsonic region of expansions (shaded area) (b) with measured values from the simulations (black points), and analytical solutions (white circles).

Figure 11.

Fast–slow SMR simulations showing Mach (a) and pressure field distribution with streamlines (b) for $M_{1}=5$ , $M_{2}=4$ and $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ .

The complete polars are presented in figure 10(b), where the subsonic flow with the appropriate expansion is shown in a shaded area as in the TNR, with $\unicode[STIX]{x1D6FF}M\sim 0.1$ and adiabatic index ranging from $\unicode[STIX]{x1D6FE}_{u}=1.4$ to $\unicode[STIX]{x1D6FE}_{B}=1.33$ , measured at the moving shocks in the numerical solution. The properties of the Mach stem are given by the $M_{1}^{\prime }$ curve and predict a positive angle for $\unicode[STIX]{x1D70E}_{m}$ in contact with $s_{i}$ and negative with $s_{t}$ as recovered from the simulations, see figure 11. Intersection A of stream-1’ and stream-2’ polars yields the conditions at the impingement of the shock with the shear layer, which provides the local angle of the shocks at each side, rendering subsonic and supersonic flow in the lower and upper stream, respectively. The upper conditions form a small supersonic patch that originates a new slip line in contact with the Mach stem, siting very close to the original slip line of the shear layer. This seems a degenerate version of the previous TNR case, where the supersonic intermediate layer becomes a point with no scale linking the regions.

In turn, the span of the secondary wave that generates the supersonic pocket remains negligible to the progressively growing Mach stem, thus sets the slip lines close together enclosing supersonic post-shock conditions at the confluence point in $y=0$ , see A–B. Consideration of viscous and thermal effects would provide a proper development scale to which these small structures could be compared.

The flow behind the reflected shock is supersonic, with higher pressure than the region behind the transmitted shock. Although it is not noticeable from the framework displayed in figure 11, a closer look at the triple-shock point C reveals a Guderley-type locus, with a well-delimited subsonic patch appearing akin to that predicted by Vasilev, Elperin & Ben-Dor (Reference Vasilev, Elperin and Ben-Dor2008). Such non-idealized flow patterns, whose characteristic length is much smaller than large-scale structures developed in the pseudosteady regime, have been experimentally observed in shock–wall reflection phenomena (Skews & Ashworth Reference Skews and Ashworth2005; Skews, Li & Platon Reference Skews, Li and Platon2009) and have been now found in irregular interactions with contact surfaces. Since this work mainly focuses on the large-scale structures, a thorough analysis on this small-scale formation is not provided in this work. Roughly, the fluid particle departs from the triple shock and crosses the small subsonic patch while expanding isentropically, with a flow deflection function that can be approximated with a Prandtl–Meyer expansion, as observed in the numerical simulations.

Finally, note that the transmitted shock displays a varying angle along the $M_{2}^{\prime }$ curve from the interaction with $M_{1}^{\prime }$ to the value given at equilibrium pressure. Far-field numerical solution shows that pressure equilibrium is not achieved between the upper expansion and the flow right behind the transmitted shock far below the contact surface, such as the region in between (left out in figure 11 for clarity purposes), is not uniform. This is mainly due to the strong pressure and deflection-angle gradients found in the intersection point, which induce strong vortex structures convected downwards. Here, the values predicted by the modified polars are $\unicode[STIX]{x1D70E}_{t}(y=0)=105.1^{\circ }$ , referring to the transmitted shock very close to the contact surface, and $\unicode[STIX]{x1D70E}_{r}=157.6^{\circ }$ for the reflected shock departing from the triple point. The corresponding pressure values are $p/p_{u}\simeq 30$ and $p_{5}/p_{u}\simeq 19$ , whose comparison with numerical simulations stands in good agreement for $\langle p_{num}\rangle /p_{u}=29.51$ and $\langle p_{num}\rangle /p_{u}=19.05$ . However, the gap between the deflection angles provided by D and E in the polar plots is connected via the subsonic vortex sheet with varying deflection angle that is formed at the slip lines. As expected, the supersonic upper region that is less affected by the non-uniform subsonic zone, is reproduced by the polar plots with higher accuracy.

4.3.2 Slow–fast SMR

An irregular flow-field configuration of the SMR kind is also observed when the lower edge supersonic Mach number exceeds a critical value, out of the weak-shock limit. This critical value is easily found in figure 2(b) at the curve bending in the right side of the plot. Beyond this value, there is no solution of the regular kind. The case illustrated in figure 12(a) represents the standard polar plots for critical conditions $M_{1}=5.5$ , $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ , $M_{2}>M_{2c}^{+}$ . It is readily seen that for $M_{2}>M_{2c}^{+}$ the polar for $M_{2}$ no longer intersects the polar $M_{3}$ of the reflected shock, arising from the incident-shock conditions. The resulting configuration is computed in figure 12(b) considering the value of $\unicode[STIX]{x1D6FF}M\sim 0.1$ for the upstream moving structure, according to the simulations, and the effects on $\unicode[STIX]{x1D6FE}$ . A clear picture of the intersection is given by the numerical simulations shown in figure 13. The Mach stem connecting $M_{1}^{\prime }$ – $M_{2}^{\prime }$ and $M_{1}^{\prime }$ – $M_{3}^{\prime }$ crossing points is easily identified, thereby generating a new slip line in the downstream flow. The flow in zones A and B departs with different angles, $\unicode[STIX]{x1D706}_{B}>\unicode[STIX]{x1D706}_{A}$ , which forces the acceleration of the subsonic fluid particles along the stream lines.

Figure 12.

Slow–fast SMR polar plots associated with an oblique shock, with incident angle $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ , crossing an upper stream moving at $M_{1}=5.5$ and a lower stream $M_{2}=10$  (a). Corrected polars including numerical solution measurements (black points) (b).

Figure 13.

Slow–fast SMR simulations showing Mach and pressure field distribution for $M_{1}=5.5$ , $M_{2}=10$ and $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ .

This irregular configuration demands high Mach streams that translate into high pressures downstream. Consequently, temperature changes are alike, with the associated real-gas effects taking a dominant role in providing the equilibrium conditions. As a matter of example, the value of $\unicode[STIX]{x1D6FE}$ in points A and B reaches $\unicode[STIX]{x1D6FE}\sim 1.32$ . The corresponding pressure values of the polar plots are $p/p_{u}\simeq 38$ and $p/p_{u}\simeq 37$ for points A and B, respectively, and indicate a good correspondence with the numerical simulations. They are $\langle p_{num}\rangle /p_{u}=38.24$ and $\langle p_{num}\rangle /p_{u}=37.92$ for the pressure at the contact point A and right behind the triple point B, respectively. According to figure 12(a), calorically perfect predictions yield significantly smaller values of pressure. From figure 12 it is also evidenced that critical conditions are affected by the $\unicode[STIX]{x1D6FE}$ value reduction. In particular, as the polar plots widens with temperature, the value of $M_{2c}^{+}$ is found to be greater and the regular reflection domain broadens. For instance, $M_{1}=5$ , $\unicode[STIX]{x1D70E}_{i}=60^{\circ }$ and $\unicode[STIX]{x1D6FE}(T)$ render $M_{2c}^{+}=8.1$ , which is definitely greater than $M_{2c}^{+}=7.29$ computed in table 1 for $\unicode[STIX]{x1D6FE}=1.4=\text{constant}$ .

6 Concluding remarks

The response of shock waves impinging on contact surfaces, mixing or shear layers, has been addressed here to obtain further understanding on compressible-flow irregular configurations in semi-infinite domains. The critical conditions of convective Mach number and incident shock angle under which these structures arise are delineated from a theoretical zero-dimensional analysis of the inviscid regular problem. These limits to regular solutions are further studied by means of the hodograph plane and high-fidelity numerical simulations of inviscid flows.

In this work, classic polar plots have been extended to include real-gas effects and the effective Mach number at the retreating shocks to be applied to the atlas of irregular configurations. It has been found that the moving set of waves is a result of the lack of mechanical balance in regular configurations, which pushes back the set increasing the relative Mach number and favours the achievement of downstream balanced conditions. Therefore, once the stable conditions are reached, the angles that form the asymptotically growing structure remain unaltered and produce self-similar solutions.

The modified hodograph method provides, with an acceptable degree of accuracy, values of the distinguished complex configurations. In particular, it reproduces remarkably well combined details such as triple shocks, local intersections, shock confluences and expansions. By way of contrast, diagrams do not provide any information about the characteristic size of the structure, which remains a function of time. Nevertheless, the scale of the domain must arise at large times, as it is the only typical length involved and it determines the final stages of the structure.

Numerical simulations using the inviscid Euler equations have been used here to characterize the irregular configurations removing the viscous–diffusive length scale of the problem. This removes the Reynolds number dependency or the classical length scale given as the distance of the impingement to the origin of the mixing layer. These simulations enable the visualization of the complex shock structures, and are able to provide the growth rate of the shock lengths. In addition, they have also been instrumental in confirming the spurious character of particular cases, such as the unstable upper-branch solutions of regular configurations, and discarding them.

From the far-field standpoint, when a single reflected wave cannot yield conditions to provide equilibrium downstream, two mayor events can happen. For fast–slow shear layers, as the transmitted shock pressure is not sufficiently high, the generation of a reflected shock is accompanied with a posterior expansion, and the transmitted shock retreats as a consequence of the pressure mismatch. As a result, secondary shocks must come into play to join the moving transmission and the fixed impinging shock. In this regard, the upper stream and the incoming wave govern the final flow configuration. For slow–fast shear layers, the far-field qualitative picture is similar, but with the absence of the expansion flow downstream since the transmitted shock pressure is higher than that demanded by the mechanical equilibrium in static conditions. In both scenarios, the local structure defined by finite-length shocks are formed in the upper-stream layer.