2.1 Jump conditions and shock adiabats

We consider an infinitely long shock tube in which a normal shockwave is propagating at constant velocity $\bar{u}_{s}$ in a gas modelled with an arbitrary equation of state. We assume a pressure and temperature field to be defined from a canonical equation of state of the form $e_{s}(s,\unicode[STIX]{x1D717})$ , where $e_{s}$ is the specific internal energy of the fluid (Menikoff & Plohr Reference Menikoff and Plohr1989). For simplicity, however, we will express the specific internal energy in terms of the pressure and the specific volume throughout the paper, i.e. $e(p,\unicode[STIX]{x1D717})$ . We are interested in the effect of a non-ideal equation of state on the refraction properties of the compression shockwaves emanating from the one-dimensional Euler equations. The integral form of these equations leads to the well-known Rankine–Hugoniot relations, satisfied across every discontinuity,

(2.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u_{1}/\unicode[STIX]{x1D717}_{1}=u_{2}/\unicode[STIX]{x1D717}_{2},\\ p_{1}+{u_{1}}^{2}/\unicode[STIX]{x1D717}_{1}=p_{2}+{u_{2}}^{2}/\unicode[STIX]{x1D717}_{2},\\ e(p_{2},\unicode[STIX]{x1D717}_{2})-e(p_{1},\unicode[STIX]{x1D717}_{1})=(p_{2}+p_{1})(\unicode[STIX]{x1D717}_{1}-\unicode[STIX]{x1D717}_{2})/2,\end{array}\right\}\end{eqnarray}$$

where the subscripts $(\cdot )_{1}$ and $(\cdot )_{2}$ stand for the uniform upstream and downstream states around the shock observed in a reference frame moving with the shock. The last equation has been obtained by combining both the momentum-balance and mass-conservation principles into the energy equation. Let us define a function $f$ such that $f(\unicode[STIX]{x1D717}_{1},p_{1},\unicode[STIX]{x1D717}_{2},p_{2})=e(p_{2},\unicode[STIX]{x1D717}_{2})-e(p_{1},\unicode[STIX]{x1D717}_{1})-(p_{2}+p_{1})(\unicode[STIX]{x1D717}_{1}-\unicode[STIX]{x1D717}_{2})/2$ . From the implicit function theorem, we know that if $\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}p_{2}(\unicode[STIX]{x1D717}_{1},p_{1},\unicode[STIX]{x1D717}_{2},p_{2})\neq 0$ , there exists a regular function $p_{h}$ such that $p_{2}=p_{h}(\unicode[STIX]{x1D717}_{2},\unicode[STIX]{x1D717}_{1},p_{1})$ . The existence condition can be expressed as

(2.2) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}e}{\unicode[STIX]{x2202}p}}(p_{2},\unicode[STIX]{x1D717}_{2})-{\displaystyle \frac{1}{2}}(\unicode[STIX]{x1D717}_{1}-\unicode[STIX]{x1D717}_{2})\neq 0.\end{eqnarray}$$

Introducing the Grüneisen parameter $G\equiv \unicode[STIX]{x1D717}\left(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}e\right)_{\unicode[STIX]{x1D717}}$ , the condition becomes

(2.3) $$\begin{eqnarray}G_{2}\neq -2\unicode[STIX]{x1D717}_{2}/\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7},\end{eqnarray}$$

where $\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7}\equiv \unicode[STIX]{x1D717}_{2}-\unicode[STIX]{x1D717}_{1}$ is the jump of specific volume across the discontinuity and $G_{2}$ is the post-shock Grüneisen parameter. This condition defines the usual vertical asymptote of the Hugoniot curves in a single-phase gas for which the pressure goes to positive infinity in the $(p,\unicode[STIX]{x1D717})$ -diagram (see Cramer Reference Cramer1989). For a perfect gas, this singularity occurs at $\unicode[STIX]{x1D717}_{2}=\unicode[STIX]{x1D717}_{1}(\unicode[STIX]{x1D6FE}-1)/(\unicode[STIX]{x1D6FE}+1)$ . In what follows, we study shocks described by the right-hand-side branch of this vertical asymptote (defined by $G_{2}+2\unicode[STIX]{x1D717}_{2}/\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7}>0$ ). Introducing the mass flow rate $j\equiv u_{1}/\unicode[STIX]{x1D717}_{1}=c_{1}M_{1}/\unicode[STIX]{x1D717}_{1}=c_{2}M_{2}/\unicode[STIX]{x1D717}_{2}$ (where $c$ and $M$ are the local sound speed and Mach number), the solution of (2.1) can then be expressed for a given upstream state as the intersection between the Rayleigh line ( ${\mathcal{R}}$ -line) and the Hugoniot line ( ${\mathcal{H}}$ -line), defined in the $(p,\unicode[STIX]{x1D717})$ -diagram respectively as

(2.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}p_{2}=p_{1}-j^{2}(\unicode[STIX]{x1D717}_{2}-\unicode[STIX]{x1D717}_{1}),\\ p_{2}=p_{h}(\unicode[STIX]{x1D717}_{2},\unicode[STIX]{x1D717}_{1},p_{1}).\\ \end{array}\right\}\end{eqnarray}$$

These two curves have a simple interpretation in the $(p,\unicode[STIX]{x1D717})$ -diagram. For a given upstream state, the Hugoniot curve is the locus of all downstream thermodynamic states reachable via a discontinuity across which the total enthalpy can be conserved. For a given upstream state and Mach number, the ${\mathcal{R}}$ -line is the locus of all downstream thermodynamic states satisfying the mass-conservation and momentum-balance principles. The slope of the Hugoniot curve on its right-hand-side branch is always negative for single-phase gases with $G>0$ (Cramer Reference Cramer1989). The hypothesis that $G>0$ will be made throughout this paper; this is representative of a substance that expands upon isobaric heating and is relevant for most of the single-phase gas behaviour of interest here. An important outcome is that isentropes do not cross each other in the $(p,\unicode[STIX]{x1D717})$ -diagram and entropy increases towards higher pressures (Menikoff & Plohr Reference Menikoff and Plohr1989). For weak shocks, the curvature of the ${\mathcal{H}}$ -line in the $(p,\unicode[STIX]{x1D717})$ -diagram is given by the local sign of $\unicode[STIX]{x1D6E4}$ (Cramer Reference Cramer1989). Graphical solutions of the RH relations, read in the $(p,\unicode[STIX]{x1D717})$ -diagram, can then be obtained (see figure 1). In most cases, there exist multiple weak solutions for a given upstream Mach number, and additional admissibility criteria must be expressed to select the waves that will not degenerate from the initial-value problem (scale-invariant initial data). This scale-invariant property is of fundamental importance in view of the perturbation analysis performed below. In the process of finding such a solution, the curvature of the ${\mathcal{H}}$ -line plays a key role (Cramer & Kluwick Reference Cramer and Kluwick1984; Menikoff & Plohr Reference Menikoff and Plohr1989; Zamfirescu et al. Reference Zamfirescu, Guardone and Colonna2008).

Figure 1. Admissibility problem of classical and non-classical gas dynamics discussed in the $(p,\unicode[STIX]{x1D717})$ -diagram.

2.2 Geometrical formulation of the admissibility problem in dense gases

Zamfirescu et al. (Reference Zamfirescu, Guardone and Colonna2008) have demonstrated the uniqueness of the Riemann problem in the context of dense gases using theoretical results from the general theory of conservation laws (Liu Reference Liu1975). In what follows, we detail the admissibility selection process for compression shockwaves in dense gases. We consider a gas with positive coefficient of thermal expansion, which implies $G>0$ , and arbitrary sign of $\unicode[STIX]{x1D6E4}$ . For classical gas dynamics (positive nonlinearity), the ${\mathcal{H}}$ -lines are convex and the selection of self-similar solutions (only functions of $x/t$ ) between weak solutions of the Euler equations is operated by the use of the Lax (Reference Lax1957) characteristic criterion (equivalent to the entropy-increase criterion), which also implies the Landau stability constraint on the wave orientation (i.e. $M_{2}\leqslant 1\leqslant M_{1}$ ). This criterion is still sufficient in the case of negative nonlinearity (the Hugoniot curves are then concave everywhere). However, when a finite region of mixed nonlinearity exists in the $(p,\unicode[STIX]{x1D717})$ -diagram and is crossed by the Hugoniot locus, there usually are multiple solutions satisfying both the Landau and Lax conditions (Menikoff & Plohr Reference Menikoff and Plohr1989). The formulation of admissibility criteria for weak solutions of systems involving mixed nonlinearities has been studied in the context of general conservation laws; the corresponding wave solutions are then called locally linearly degenerated (Oleinik Reference Oleinik1959; Liu Reference Liu1976). The resulting criterion states that a shock is admissible if and only if its speed in the laboratory reference frame is greater than or at least equal to the speed of the fastest shock located on the Hugoniot branch joining the upstream thermodynamic state and the tested final state on the ${\mathcal{H}}$ -line. The Liu–Oleinik condition is a necessary condition for a dissipative structure to exist (Liu Reference Liu1976). It was used by Menikoff & Plohr (Reference Menikoff and Plohr1989) and Zhao et al. (Reference Zhao, Mentrelli, Ruggeri and Sugiyama2011) in the context of mixed nonlinearities in BZT fluids. Another admissibility criterion was also formulated by Lax (Reference Lax1971) (Lax’s generalised entropy condition) and used by Cramer (Reference Cramer1989) and Cramer & Crickenberger (Reference Cramer and Crickenberger1991) for dense gases. This condition asks for the segment of the ${\mathcal{R}}$ -line connecting the upstream and downstream states of the shock to lie entirely above (compression shocks) or below (expansion shocks) the ${\mathcal{H}}$ -line. It is straightforward to see the equivalence between the two conditions in the context of shocks in BZT fluids.

Remarkably, all admissibility criteria can be conveniently read in the $(p,\unicode[STIX]{x1D717})$ -diagram. A useful relationship between the ratio of the slopes of the Hugoniot and Rayleigh lines at the downstream state and the downstream Mach number provides a way of graphically checking the Landau evolution criterion. This relation is obtained by combining a differentiation along the ${\mathcal{H}}$ -line at constant upstream conditions with the Gibbs thermodynamic identity (Cramer Reference Cramer1989),

(2.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}p_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}_{2}}}-{\displaystyle \frac{\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}}{\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7}}}={\displaystyle \frac{j^{2}}{{M_{2}}^{2}}}{\displaystyle \frac{{M_{2}}^{2}-1}{\left(1+{\displaystyle \frac{\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7}}{2\unicode[STIX]{x1D717}_{2}}}G_{2}\right)}}.\end{eqnarray}$$

Therefore, on the right-hand-side branch of the ${\mathcal{H}}$ -line (where $1+\unicode[STIX]{x27E6}\unicode[STIX]{x1D717}\unicode[STIX]{x27E7}G_{2}/(2\unicode[STIX]{x1D717}_{2})>0$ from earlier assumptions), the post-shock Mach number is subsonic if the slope of the ${\mathcal{H}}$ -line is smaller (steeper down) than the slope of the ${\mathcal{R}}$ -line and supersonic if it is greater. A sonic post-shock state is reached when the two curves are tangent. Figure 1 gives a picture of the admissibility problem for weak solutions of compression shocks (with upstream condition on point $\unicode[STIX]{x2460}$ ) considered on Hugoniot curves representative of classical and non-classical gas dynamic. For the positive nonlinearity with convex ${\mathcal{H}}$ -line (or the negative nonlinearity with concave ${\mathcal{H}}$ -line), the compression shocks starting at $\unicode[STIX]{x2460}$ (or expansion shock starting at $\unicode[STIX]{x2461}$ ) are the only discontinuities satisfying the entropy-increase criteria (and thus Landau’s wave orientation stability constraint); see figure 1(i). When a mixed nonlinearity is involved (see figure 1(ii–iii)), which is typical for dense gases near the TCP (Thompson & Lambrakis Reference Thompson and Lambrakis1973; Cramer & Kluwick Reference Cramer and Kluwick1984; Cramer & Sen Reference Cramer and Sen1986), Lax’s entropy criterion is no longer guaranteed by the convexity property of the ${\mathcal{H}}$ -line. Let us consider the admissibility problem of compression shocks from upstream states with $\unicode[STIX]{x1D6E4}>0$ (point $\unicode[STIX]{x2460}$ in figure 1(iii)), from which the ${\mathcal{H}}$ -lines cross a $\unicode[STIX]{x1D6E4}<0$ region and have their convexity changed. Two different situations may occur. If there is no sonic point on the ${\mathcal{H}}$ -line, then all compression shocks are admissible. If there exists a sonic point on the concave part of the ${\mathcal{H}}$ -line (point a in figure 1(iii) for which $\unicode[STIX]{x1D6E4}<0$ ), there exists a branch (from a to b) that violates Landau’s evolution criterion (the slope of the ${\mathcal{R}}$ -line being steeper down than that of the ${\mathcal{H}}$ -line). This branch corresponds to unstable waves that will not remain scale invariant. At b, the post-shock condition is again sonic. Branch b to c corresponds to compression shocks that satisfy the entropy-increase criterion (classical Lax criterion) and therefore Landau’s stability criterion but are still inadmissible by failure to fulfil the Liu–Oleinik criterion (or equivalently the general Lax criterion, see Lax Reference Lax1971). The solution of the Riemann problem with initial data taken on this branch is found to degenerate into a non-classical composite wave of type sonic shock/compression fan/shock (Cramer Reference Cramer1989) and is therefore not a suitable base flow. Branches $\unicode[STIX]{x2460}$ to a and c to the highest pressure limit are admissible and remain scale invariant. A limiting case is found when a sonic point with $\unicode[STIX]{x1D6E4}=0$ exists on the ${\mathcal{H}}$ -line, in which case all compression shocks are still admissible (see figure 1(ii)).

Admissibility criteria for compression shockwaves are readily checked when plotting the post-shock Mach number against the pre-shock Mach number (see figure 1). Let us assume a positive value of the fundamental derivative of gas dynamics at the pre-shock state. For low enough pre-shock Mach number $M_{1}$ , the post-shock state is still with $\unicode[STIX]{x1D6E4}>0$ and the post-shock Mach number $M_{2}$ is a decreasing function of $M_{1}$ (classical behaviour). If there exists a finite section on the Hugoniot curve with $\unicode[STIX]{x1D6E4}<0$ , it is possible to observe a non-monotonic variation of the post-shock Mach number with increasing shock strength, for post-shock states that lie after the first inflexion point on the ${\mathcal{H}}$ -line, see (2.5) (non-ideal-gas effect). The non-admissible branches a–b and b–c from figure 1(iii) are part of a loop in $M_{2}(M_{1})$ , which is consistent with the expression of the Liu–Oleinik maximum wave speed criterion (for each point on the b–c branch there exists an intermediate point on the ${\mathcal{H}}$ -line with higher wave speed). In fact, it is the consequence of a change of variation of the ${\mathcal{R}}$ -line slope (i.e. the mass flow rate $j$ ) when the post-shock state point is moving from sonic condition on the non-convex part of the ${\mathcal{H}}$ -line in the $(p,\unicode[STIX]{x1D717})$ -diagram (red segment in figure 1(iii)). For the limiting case when a post-shock sonic state is reached for $\unicode[STIX]{x1D6E4}_{2}=0$ , the post-shock Mach number reaches a maximum of one with no inadmissible loop.

These different scenarios are illustrated in figure 2 using a van der Waals model of the PP10 gas for which $c_{v}/R=78.2$ (see appendix A). Pre-shock states at critical temperature but different upstream pressures are investigated. A behaviour in agreement with that of an ideal gas is obtained at low Mach numbers for low pressures (e.g. $p_{1}/p_{c}=0.13$ ). A progressive departure from the ideal-gas-law behaviour is observed when the post-shock state reaches the TCP region. The divergence occurs at lower shock speeds ( $M_{1}$ ) when the pre-shock state is brought closer to the TCP, corresponding to higher $p_{1}/p_{c}$ values on the critical isotherm (e.g. $p_{1}/p_{c}=0.65$ ). A local maximum in $M_{2}(M_{1})$ emerges above a given upstream pressure $p_{1}/p_{c}$ (due to the mixed nonlinearity along the ${\mathcal{H}}$ -line) but does not necessarily imply the existence of a sonic post-shock state (see $p_{1}/p_{c}=0.45$ in figure 2 b). When a non-admissible path is found on the ${\mathcal{H}}$ -line (thin lines in figure 2 b), the post-shock Mach number $M_{2}$ becomes a discontinuous function of $M_{1}$ . Sonic-shock formation in van der Waals gases has been derived analytically in Cramer & Sen (Reference Cramer and Sen1987) and has been shown to be a direct consequence of the occurrence of states with $\unicode[STIX]{x1D6E4}<0$ on the ${\mathcal{H}}$ -line. A line joining all double-sonic rarefaction pre-shock states has been derived for an arbitrary equation of state by Zamfirescu et al. (Reference Zamfirescu, Guardone and Colonna2008). Their discussion is restricted to rarefaction shockwaves which are very different as far as the admissibility problem is concerned. Indeed, double-sonic compression shocks are not admissible in general in single-phase gases with positive Grüneisen index (decreasing ${\mathcal{H}}$ -line in $(p,\unicode[STIX]{x1D717})$ -diagram); they fail to fulfil the Liu–Oleinik admissibility condition (Kluwick Reference Kluwick1993); while double-sonic rarefaction shocks are admissible. Therefore, post-shock sonic states necessarily occur when $M_{1}\neq 1$ . Thus, the picture given by the four selected reduced pressures is representative of the general evolution of the post-shock Mach number for compression shockwaves with pre-shock states approaching the TCP. Thus, for a given upstream thermodynamic condition with ${\mathcal{H}}$ -line featuring a discontinuous admissible path, there exists a mass flow rate for which the post-shock conditions (e.g. in a nozzle) change abruptly for arbitrarily small changes in upstream conditions.

Figure 2. Non-ideal-gas effects on the post-shock Mach number (for which $T_{1}=T_{c}$ ) (a) and detailed view of the Mach number range up to $1.5$ (b). The equivalent ideal-gas configuration is shown by dashed lines. The thin lines indicate the non-admissible paths.

Scale-invariant admissible compression shocks originating from point $\unicode[STIX]{x2460}$ in figure 1 are used in the next section as the base flow in a small-perturbation analysis. As is first inferred here, a discontinuous admissible Hugoniot locus is a typical feature of non-ideal-gas dynamics and will have a significant impact on the refraction properties of shocks in non-ideal gases (e.g. BZT fluids).

2.3 Isothermal-shock limit

This paper makes frequent references to a hypothetical infinitely dense gas (i.e. a fluid with $c_{v}/R\rightarrow +\infty$ ). This section provides a few comments relating to such a limit. For simplicity, fluids with constant isochoric heat capacities (polytropic gas) are considered.

Let us assume that a finite change of internal energy is supplied, for instance through a shockwave (i.e. equal to the work done by the average pressure), to a ‘fluid’ parcel modelled with a theoretical infinite $c_{v}/R$ value (such a ‘fluid’ would be made of molecules with an infinite number of active degrees of freedom). It is clear from the definition $c_{v}\equiv (\unicode[STIX]{x2202}e/\unicode[STIX]{x2202}T)_{\unicode[STIX]{x1D717}}$ that a necessary condition for the internal energy to remain finite is that such change be following an isothermal process. Hence, the shock becomes an isothermal process. It should be noted that we recover the unphysical isothermal shockwave theoretical model of Stokes (Reference Stokes1848) (with the noticeable difference that the speed of sound in arbitrary media may change even at constant temperature). It is also well known that, in this limiting case, isentropes converge towards the isotherms in the $(p,\unicode[STIX]{x1D717})$ -diagram (Thompson & Lambrakis Reference Thompson and Lambrakis1973; Cramer Reference Cramer1989). Thus, such ‘fluids’ have their ${\mathcal{H}}$ -lines coinciding with the isotherms, which are also the isentropes, in the $(p,\unicode[STIX]{x1D717})$ -diagram.

Of course, isentropic or isothermal compression shocks have no physical meaning (Rankine Reference Rankine1870), and the limits should be seen as qualitatively inferring the behaviour of shocks for large and finite values of $c_{v}/R$ for which the entropy increases along the Hugoniot paths. A similar asymptotic argument was used by Thompson & Lambrakis (Reference Thompson and Lambrakis1973) to infer the existence of a negative $\unicode[STIX]{x1D6E4}$ region in the vapour-phase near-critical conditions for fluids with large molecular heat capacities.

3.1 Small perturbation of 1D inviscid flows with arbitrary equations of state

The eigenmode basis of the Euler equations was first derived by Kovásznay (Reference Kovásznay1953) in the context of weak turbulence generated in a supersonic flow modelled with an ideal-gas equation of state. Small perturbations were found to propagate along an entropy mode, two acoustic (irrotational) modes and a divergence-free vortical mode. This idealised picture was used by many authors (Ribner Reference Ribner1954; Lee et al. Reference Lee, Lele and Moin1997; Larsson & Lele Reference Larsson and Lele2009; Sinha Reference Sinha2012) to address the question of compressibility effects on turbulence. A weakly nonlinear formulation of these interactions has also been provided (Chu & Kovásznay Reference Chu and Kovásznay1958).

For simplicity, a uniform one-dimensional flow is assumed here. A consequence is that no vortical mode will be considered and most of our attention focuses on non-ideal-gas effects on the generation of the acoustic field and the amplification of entropy waves at the shock. A small perturbation ( $\unicode[STIX]{x1D700}\ll 1$ ) is added to the piecewise uniform base flow ( $\bar{\unicode[STIX]{x1D741}}$ ),

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D741}(x,t)=\bar{\unicode[STIX]{x1D741}}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D741}^{\prime }(x,t)+o(\unicode[STIX]{x1D700}\bar{\unicode[STIX]{x1D741}}).\end{eqnarray}$$

The flow perturbation satisfies the one-dimensional linearised Euler equations in non-dimensional form. Reference thermodynamic variables are taken at the TCP (see appendix A) and the velocity scale is chosen in accordance ( $u_{ref}^{\ast }\equiv \sqrt{p_{c}^{\ast }/\unicode[STIX]{x1D70C}_{c}^{\ast }}$ ), while a characteristic length scale is provided by the flow perturbation ( $\unicode[STIX]{x1D706}^{\ast }$ ), resulting in

(3.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D741}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{\boldsymbol{A}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D741}^{\prime }}{\unicode[STIX]{x2202}x}=\mathbf{0}+o(\bar{\unicode[STIX]{x1D741}}\unicode[STIX]{x1D700}),\quad \text{where }\unicode[STIX]{x1D741}^{\prime }\equiv \left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D70C}^{\prime }\\ u^{\prime }\\ p^{\prime }\end{array}\right],\bar{\boldsymbol{A}}\equiv \left[\begin{array}{@{}ccc@{}}\bar{u} & \bar{\unicode[STIX]{x1D70C}} & 0\\ 0 & \bar{u} & 1/\bar{\unicode[STIX]{x1D70C}}\\ 0 & \bar{\unicode[STIX]{x1D70C}}\bar{c}^{2} & \bar{u}\end{array}\right].\end{eqnarray}$$

Solutions to (3.2) are sought in the form of plane travelling waves,

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D741}^{\prime }(x,t)=[\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D70C}},\unicode[STIX]{x1D709}_{u},\unicode[STIX]{x1D709}_{p}]^{\text{T}}\exp [\text{i}(kx-\unicode[STIX]{x1D714}t)],\end{eqnarray}$$

where $k=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}$ is the wavenumber and $\unicode[STIX]{x1D714}$ is a prescribed constant angular velocity (see § 3.2). Equation (3.2) reduces to the eigenvalue problem

(3.4) $$\begin{eqnarray}\bar{\boldsymbol{A}}\boldsymbol{\cdot }\unicode[STIX]{x1D741}^{\prime }=\unicode[STIX]{x1D70E}\unicode[STIX]{x1D741}^{\prime },\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D714}/k$ the eigenvalues of $\bar{\boldsymbol{A}}$ . Each eigenvalue will have an associated unit eigenvector, denoted $\boldsymbol{e}$ ,

(3.5a,b ) $$\begin{eqnarray}\left\{\unicode[STIX]{x1D70E}_{s}=\bar{u},\boldsymbol{e}_{s}=\left[\begin{array}{@{}c@{}}1\\ 0\\ 0\end{array}\right]\right\},\quad \left\{\unicode[STIX]{x1D70E}_{a^{\pm }}=\bar{u}\pm \bar{c},\boldsymbol{e}_{a^{\pm }}={\displaystyle \frac{1}{\sqrt{1+\bar{c}^{2}(\bar{c}^{2}+1/\bar{\unicode[STIX]{x1D70C}}^{2})}}}\left[\begin{array}{@{}c@{}}1\\ \pm \bar{c}/\bar{\unicode[STIX]{x1D70C}}\\ \bar{c}^{2}\end{array}\right]\right\}.\qquad\end{eqnarray}$$

Figure 3. Dense-gas degeneracy of the eigenmode basis close to the TCP.

We recognise Kovásznay’s entropy eigenmode denoted by $(\cdot )_{s}$ and the right/left running acoustic eigenmodes $(\cdot )_{a^{\pm }}$ . The general wave solution of (3.2) can be represented as any linear combination of these three modes. The dependence on the equation of state is embedded in the speed of sound. It is of particular interest to notice that for fluids with molecules featuring a high number of degrees of freedom (typical for dense gas), the speed of sound decreases along an isentropic compression in the dense-gas region, down to a very low value near the TCP (Thompson & Lambrakis Reference Thompson and Lambrakis1973). This may come with a degeneracy of the basis, in which case a strong bias towards density fluctuation is expected. This is illustrated in figure 3, which represents the evolution of the unit eigenmode basis $(\boldsymbol{e}_{s},\boldsymbol{e}_{a^{+}},\boldsymbol{e}_{a^{-}})$ expressed in terms of the primitive variables $(\unicode[STIX]{x1D70C},u,p)$ for an isentropic process following the critical isentrope (the quarter unit sphere is represented for convenience). Thompson & Lambrakis (Reference Thompson and Lambrakis1973) have demonstrated the convergence of isentropes towards isotherms for large values of $c_{v}/R$ . In figure 3, both the critical isentrope and isotherm lines are represented for the PP10 gas with $c_{v}/R=78.2$ , using a van der Waals equation of state (see appendix A). For an isentropic process in the dilute-gas limit (i.e. high specific volume) along the critical isentrope, the speed of sound is low (it vanishes for infinite specific volume) and the acoustic eigenvectors tend to be aligned with $[0,\pm u,0]$ . Thus, in the limit of infinite specific volume, the basis degenerates. The pressure and density fluctuations are then expected to be negligible compared with the velocity fluctuations. For an isentropic process in the high-pressure limit along the critical isentrope, the acoustic eigenvectors tend to align towards $[0,0,p]$ and the velocity and density fluctuations are expected to be negligible compared with the pressure fluctuations. The basis also degenerates. These two limits are found in every equation of state; they match their ideal-gas asymptotic limits. For a dense gas close to the TCP where the sound speed is very small, the acoustic eigenvectors tend to align towards $[\unicode[STIX]{x1D70C},0,0]$ and both the pressure and velocity fluctuations are negligible relative to the density fluctuations. Once again, the eigenmode basis degenerates (for $c_{v}/R\rightarrow +\infty$ ) but with a triple-valued eigenvalue which is not possible in ideal gases. The black thick lines in figure 3 (right) illustrate paths of the eigenmode basis along the critical isentrope (from the dilute-gas to the high-pressure limit) for PP10. The linear interaction of these modes with a shockwave is investigated in the next section.

Figure 4. Sketch of the entropy-mode refraction configuration. A one-dimensional shock travels at constant speed $\bar{u}_{s}$ through a fluid at rest ( $u_{R}=0$ ) on which a periodic perturbation (in space, with periodicity $\unicode[STIX]{x1D706}_{s_{1}}$ ) is superimposed. The shock path $\unicode[STIX]{x1D701}(t)$ oscillates around the mean shock position $\bar{u}_{s}t$ . The reference frame attached to the mean shock is denoted $\unicode[STIX]{x1D702}(t)$ , and the actual shock position in that reference frame is denoted $\unicode[STIX]{x1D702}_{s}$ . (a) Spatio-temporal ( $x$ – $t$ ) diagram and (b) density profile in the reference frame moving with the mean shock position. The refracted density profile is the sum of entropic (with periodicity $\unicode[STIX]{x1D706}_{s_{2}}$ ) and downstream-propagating acoustic (with periodicity $\unicode[STIX]{x1D706}_{a_{2}^{+}}$ ) perturbations.

3.2 Linear shock-refraction properties with arbitrary equations of state

Let us consider a base flow consisting of an isolated one-dimensional compression shock whose compressed state is taken on the admissible branch defined in § 2 (upstream state $\bar{\unicode[STIX]{x1D741}}_{1}$ with $M_{1}>1$ and downstream state $\bar{\unicode[STIX]{x1D741}}_{2}$ with $M_{2}\leqslant 1$ ). The shock is moving at a constant speed $\bar{u}_{s}$ in the laboratory reference frame. Its position at time $t$ is set by the unique parameter $x_{s}(t)=\bar{u}_{s}t$ (assuming $x_{s}(0)=0$ ). For simplicity, we introduce a new coordinate axis attached to the mean shock position $\unicode[STIX]{x1D702}=x-x_{s}$ and work in the reference frame attached to the moving shock. The uniform uncompressed side of the shock is perturbed with a prescribed small-amplitude wave, projected on the aforementioned eigenmode basis, which propagates through the shock discontinuity in the chosen reference frame. The compressed side and the shock front are therefore both perturbed. The fluctuating flow field satisfies the linearised Rankine–Hugoniot relations at the perturbed shock front. The linear-wave/shock interaction problem is solved by considering the most widely accepted methodology (D’yakov Reference D’yakov1954; Kontorovich Reference Kontorovich1957; McKenzie & Westphal Reference McKenzie and Westphal1968; Bates Reference Bates2004; Tumin Reference Tumin2008), which consists of taking as independent linear modes the incoming mode (entropy, upstream-propagating acoustic or downstream-propagating or acoustic), the refracted linear modes (one entropy and one downstream-propagating acoustic mode) and the unsteady shock displacement. The classical linearised Rankine–Hugoniot relations are solved at the shock front without additional surface tension terms (originated from inertial effects), as suggested by Lubchich & Pudovkin (Reference Lubchich and Pudovkin2004).

The Rankine–Hugoniot conditions (2.1) are expressed at the shock position in the reference frame moving with the shock. A perturbation of this base flow (denoted with $\bar{.}$ ) is now performed (with characteristic length scale $\unicode[STIX]{x1D706}$ ). The instantaneous flow field is defined by parts as

(3.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D741}_{1}(\unicode[STIX]{x1D702},t)=\bar{\unicode[STIX]{x1D741}}_{1}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D700}\unicode[STIX]{x1D741}_{1}^{\prime }(\unicode[STIX]{x1D702},t)+o(\unicode[STIX]{x1D700}\bar{\unicode[STIX]{x1D741}}_{1}(\unicode[STIX]{x1D702})),\quad \unicode[STIX]{x1D702}\in ]-\infty ,\unicode[STIX]{x1D702}_{s}(t)[,\\ \unicode[STIX]{x1D741}_{2}(\unicode[STIX]{x1D702},t)=\bar{\unicode[STIX]{x1D741}}_{2}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D700}\unicode[STIX]{x1D741}_{2}^{\prime }(\unicode[STIX]{x1D702},t)+o(\unicode[STIX]{x1D700}\bar{\unicode[STIX]{x1D741}}_{2}(\unicode[STIX]{x1D702})),\quad \unicode[STIX]{x1D702}\in ]\unicode[STIX]{x1D702}_{s}(t),+\infty [,\\ \unicode[STIX]{x1D702}_{s}(t)=\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}_{s}^{\prime }(t)+o(\unicode[STIX]{x1D700}\unicode[STIX]{x1D706}),\end{array}\right\}\end{eqnarray}$$

and $\bar{\unicode[STIX]{x1D741}}_{1}$ and $\bar{\unicode[STIX]{x1D741}}_{2}$ are solutions to the algebraic system of (2.1).

In what follows, the shock is perturbed by an entropy mode (density fluctuations without velocity and pressure fluctuations) and a boundary value problem is solved at the shock front. This approach is different from the linear stability analysis of the shock front (D’yakov Reference D’yakov1954; Kontorovich Reference Kontorovich1957; Bates Reference Bates2004) in the literature. Here, the linear refraction properties (wave transmission coefficients) of compression shocks are solved when the shocks are continuously forced with an entropy mode of prescribed angular velocity ( $\unicode[STIX]{x1D714}$ is real and constant). The shock is considered to be a true discontinuity and to be continuously forced, consistent with the choice of a constant $\unicode[STIX]{x1D714}$ value. The general form of the problem can be summarised as follows.

1. (i) On the upstream supersonic side of the shock lies the continuous forcing by a unitary entropy mode,

   (3.7) $$\begin{eqnarray}\unicode[STIX]{x1D741}_{1}^{\prime }=\unicode[STIX]{x1D741}_{s_{1}}^{\prime }=\left[\begin{array}{@{}c@{}}1\\ 0\\ 0\end{array}\right]\text{e}^{\text{i}\left(k_{s_{1}}\unicode[STIX]{x1D702}-\unicode[STIX]{x1D714}t\right)}\quad (\unicode[STIX]{x1D702}>\unicode[STIX]{x1D702}_{s}),\quad \text{with }k_{s_{1}}=\frac{\unicode[STIX]{x1D714}}{\bar{u}_{1}}.\end{eqnarray}$$

2. (ii) On the downstream subsonic side of the shock lie both the post-shock entropy and acoustic modes leaving the shock,

   (3.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D741}_{2}^{\prime }=\unicode[STIX]{x1D741}_{s_{2}}^{\prime }+\unicode[STIX]{x1D741}_{+_{2}}^{\prime }=\unicode[STIX]{x1D712}_{s}\left[\begin{array}{@{}c@{}}1\\ 0\\ 0\end{array}\right]\text{e}^{\text{i}\left(k_{s_{2}}\unicode[STIX]{x1D702}-\unicode[STIX]{x1D714}t\right)}+\unicode[STIX]{x1D712}_{a^{+}}\left[\begin{array}{@{}c@{}}1\\ \bar{c}_{2}/\bar{\unicode[STIX]{x1D70C}}_{2}\\ \bar{c}_{2}^{2}\end{array}\right]\text{e}^{\text{i}\left(k_{+_{2}}\unicode[STIX]{x1D702}-\unicode[STIX]{x1D714}t\right)}\quad (\unicode[STIX]{x1D702}<\unicode[STIX]{x1D702}_{s}),\nonumber\\ \displaystyle & & \displaystyle \quad \text{with }k_{s_{2}}=\frac{\unicode[STIX]{x1D714}}{\bar{u}_{2}},\quad k_{+_{2}}=\frac{\unicode[STIX]{x1D714}}{\bar{u}_{2}+\bar{c}_{2}}.\end{eqnarray}$$

3. (iii) The velocity of the shock oscillations is given by

   (3.9) $$\begin{eqnarray}u_{s}^{\prime }\equiv {\displaystyle \frac{\text{d}\unicode[STIX]{x1D702}_{s}^{\prime }}{\text{d}t}}=\unicode[STIX]{x1D6FF}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}.\end{eqnarray}$$

The boundary conditions at the shock front are used to obtain analytic expressions for both the entropy-mode amplification $\unicode[STIX]{x1D712}_{s}$ and the acoustic-mode generation $\unicode[STIX]{x1D712}_{a^{+}}$ coefficients for a general equation of state (see appendix B).

4.1 Dense-gas shocks with an infinite $c_{v}/R$ value

It can be shown that the value of $\unicode[STIX]{x1D712}_{s}$ for a post-shock state at the critical point (with $c_{v}/R\rightarrow +\infty$ ) goes to positive infinity (see appendix C). It is conjectured that for finite and large values of $c_{v}/R$ , compression shocks with near-TCP post-shock states featuring finite and large values of $\unicode[STIX]{x1D712}_{s}$ can exist. This is illustrated hereafter using a van der Waals model. It is important to stress that this limiting case has no physical meaning. The question of shock admissibility must be addressed before commenting on the post-shock transmission coefficients. For $c_{v}/R\rightarrow +\infty$ , the horizontal tangent point at the TCP on the isothermal ${\mathcal{H}}$ -line cannot be a Landau-admissible compression post-shock state as the post-shock Mach number is then singular (due to the vanishing sound speed as a result of the isentropes coinciding with the isotherms). A subset of (arbitrarily) large but finite values of $\unicode[STIX]{x1D712}_{s}$ is instead expected in the subset of admissible shocks with post-shock states near the critical point when $c_{v}/R$ is made larger and larger.

The sensitivity of $\unicode[STIX]{x1D712}_{s}$ to near-critical post-shock conditions when the limit $c_{v}/R\rightarrow +\infty$ is reached is provided for a van der Waals gas (see figure 5). The diagrams represent contour plots of $\unicode[STIX]{x1D712}_{s}$ obtained for shocks with low (figure 5 a) and higher (figure 5 b) upstream reduced pressures. Hugoniot lines (black lines) are drawn for shocks with a constant upstream reduced pressure, set to $p_{1}/p_{c}=0.13$ (figure 5 a) and $p_{1}/p_{c}=0.55$ (figure 5 b), and for different reduced specific volumes starting from the critical temperature to supercritical temperatures. Isolines of upstream Mach numbers are also shown. They correspond to the required shock speed to reach the corresponding post-shock thermodynamic state on the ${\mathcal{H}}$ -lines. All ${\mathcal{H}}$ -lines (which coincide with the isotherms here) start from the horizontal $M_{1}=1$ line for the two given upstream pressures. For large values of the upstream specific volume, the linear isotherms in the $(\log p,\log \unicode[STIX]{x1D717})$ -diagram for the ideal-gas model are recovered. Along these dilute-gas isotherms, $\unicode[STIX]{x1D712}_{s}$ is a monotonically increasing function of $M_{1}$ (for this range of upstream Mach numbers). A large local maximum of $\unicode[STIX]{x1D712}_{s}$ is then observed along every isotherm when the post-shock states reach near-TCP conditions. The value of the local maximum along each isotherm increases with lower pre-shock pressures and can be made arbitrarily large in the infinitely dense gas limit. The upstream Mach numbers ( $M_{1}$ ) corresponding to these local maxima in $\unicode[STIX]{x1D712}_{s}$ decrease for increasing upstream pressures. For $p_{1}/p_{c}=0.55$ , large entropy-amplification coefficients are found for relatively weak shocks ( $M_{1}\approx 1.3$ ). For both reduced pressures, $\unicode[STIX]{x1D712}_{s}$ is singular at the TCP; the maximum values obtained in the subset of admissible post-shock states (points not enclosed by the green line) are of order $10^{3}$ and $10^{1}$ respectively.

Figure 5. The divergence of the entropy-amplification factor at the TCP; $(p,\unicode[STIX]{x1D717})$ -diagram of $\unicode[STIX]{x1D712}_{s}$ isocontours and ${\mathcal{H}}$ -lines (solid black lines) for $c_{v}/R\rightarrow +\infty$ using the van der Waals model (appendix A). Upstream Mach numbers ( $M_{1}$ ) indicating the shock speed needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines are represented by the black dashed lines with labels. Ideal-gas ${\mathcal{H}}$ -lines (thick grey lines) are represented in the dilute-gas limit. The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively. The near-TCP non-admissible region is enclosed within the region delimited by the green line. Strong non-ideal-gas amplifications of the incoming entropy modes are found for realistic pre-shock Mach numbers ( $M_{1}<3.0$ ).

The use of the van der Waals analytical equation of state (EOS) in the isothermal infinitely dense gas limit is convenient to isolate the TCP compressibility effects from other TCP behaviours (e.g. large change of heat capacities). Indeed, it is clear from figure 5 that the stronger the local deformation of the isotherm in the $(p,\unicode[STIX]{x1D717})$ -diagram is the higher the entropy-amplification factor is. At the TCP, the critical isotherm eventually features a saddle point and the transmission factor diverges. This plateauing of the isotherms close to the TCP, and thus of the isentropes, indicates a decrease of the speed of sound (eventually vanishing at the TCP in this limit). This typical dense-gas compressible property comes with a tendency towards degeneracy of the linearised Kovásznay eigenmode basis (see the general discussion in § 3.1). Figure 6 illustrates the ‘trajectory’ of normalised slow and fast post-shock acoustic eigenvectors in a quarter unit sphere representation for increasing pre-shock Mach numbers along the critical isotherm. The two upstream pressures (blue lines) from figure 5 are considered together with the equivalent ideal-gas ‘trajectories’ (black). The non-admissible post-shock region is indicated in light blue. A larger non-admissible part is expected as the upstream state is approaching the TCP and the negative- $\unicode[STIX]{x1D6E4}$ region. The singularity for which the eigenbasis degenerates towards a density-fluctuation orientation is found at the TCP, as expected. The ideal-gas eigenmode basis only degenerates towards velocity (for low upstream reduced pressures) and pressure (for post-shock pressures going to infinity) fluctuations. Once again, these two singularities are also recovered for a non-ideal gas considered in the same limits. Finally, the eigenmode basis corresponding to a pre-shock Mach number that brings the post-shock state close to a maximum of $\unicode[STIX]{x1D712}_{s}$ on the admissible part of the ${\mathcal{H}}$ -lines is illustrated for the two upstream pressures. These results are investigated for finite values of the parameter $c_{v}/R$ in the following two sections.

Figure 6. Post-shock normalised Kovásznay eigenmode basis evolution along the critical ${\mathcal{H}}$ -line for $c_{v}/R\rightarrow +\infty$ using the van der Waals model. The admissible path is shown in dark blue while the non-admissible path is represented in light blue. The corresponding ideal-gas eigenmode basis evolution is shown in black.

4.2 Dense-gas shocks with finite $c_{v}/R$ values

Results for a van der Waals gas with a large value of $c_{v}/R=78.2$ (corresponding to the PP10 gas) are reported for ${\mathcal{H}}$ -lines featuring at most one or two sonic points.

4.2.1 Hugoniot lines with at most one sonic point: a selectivity effect

A key feature of the result obtained in the infinite dense-gas limit is the plateauing of the ${\mathcal{H}}$ -line in the $(p,\unicode[STIX]{x1D717})$ -diagram when the post-shock state approaches the TCP. In this diagram, the local changes of both the ${\mathcal{H}}$ -line slope and the post-shock Mach number (which appear explicitly in the amplification coefficient, see (4.1)) are indeed directly linked to the sound speed.

For a given ideal gas (e.g. a given $\unicode[STIX]{x1D6FE}$ ), the density, velocity and pressure jumps can be parametrised by the upstream Mach number $M_{1}$ without any other dependence with respect to the upstream conditions. As a result, ${\mathcal{H}}$ -lines drawn in the $(\log p,\log \unicode[STIX]{x1D717})$ -diagram for a given upstream pressure and different upstream specific volumes are deduced one from another simply by a translation along the $\log \unicode[STIX]{x1D717}$ -axis. When a non-ideal equation of state is used instead, the jump properties are not simply functions of $M_{1}$ but also depend on the upstream thermodynamic state, making the ${\mathcal{H}}$ -lines a two degrees of freedom map in this diagram (see figure 7 a). For high upstream specific volumes, when the gas is in its ideal-gas asymptotic limit, the van der Waals ${\mathcal{H}}$ -lines match those obtained for the equivalent ideal gas (i.e. with $\unicode[STIX]{x1D6FE}=1.013$ for PP10). For post-shock states near the TCP region, the ${\mathcal{H}}$ -lines lean towards the critical isotherm (depicted with a thick blue line passing through the TCP) and depart from the ideal-gas ${\mathcal{H}}$ -lines (this is expected for large $c_{v}/R$ values). The main consequence of this deformation of the ${\mathcal{H}}$ -lines is that results obtained in the infinitely dense-gas limit are recovered even at finite $c_{v}/R$ values, i.e. $\unicode[STIX]{x1D712}_{s}$ has a local maximum along ${\mathcal{H}}$ -lines passing near the TCP.

The same trend towards degeneracy as that for infinitely dense gases is seen for the eigenmode basis along the critical ${\mathcal{H}}$ -line (see figure 7 b). For this finite value of $c_{v}/R$ , the local maximum of $\unicode[STIX]{x1D712}_{s}$ on the critical ${\mathcal{H}}$ -line matches the point on the quarter unit sphere for which the basis is the most density-oriented. Interestingly, there exists another local maximum of $\unicode[STIX]{x1D712}_{s}$ in the dilute-gas limit. Its value is reached for a large value of the pre-shock Mach number ( $M_{1}=12$ ) and matches the pre-shock Mach number for which the equivalent ideal-gas eigenmode basis is the most oriented towards density fluctuations.

For dense gases, the local changes of shock adiabat near the TCP are the result of both the high number of degrees of freedom in the gas molecules (which implies high values of $c_{v}/R$ which in turn brings the ${\mathcal{H}}$ -line close to the isotherm) and the increase of inter-molecular interactions between particles near the TCP (the saddle point of the isotherm at the TCP being a consequence of these interactions). However, the present results are not fundamentally restricted to dense-gas dynamics, and any additional source term in the energy equation that may create a similar deformation of the Hugoniot path can potentially reproduce the conditions observed here for dense gases close to the TCP (e.g. ionisation, see Griffiths, Sandeman & Hornung Reference Griffiths, Sandeman and Hornung1976).

Figure 7. Strong near-critical entropy-mode amplification coefficient for $p_{1}/p_{c}=0.13$ . (a) The $\unicode[STIX]{x1D712}_{s}$ isocontours and ${\mathcal{H}}$ -lines (solid black lines) for the PP10 gas using the van der Waals model (appendix A). The $M_{1}$ values needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines are represented by the black dashed lines with labels. Ideal-gas ${\mathcal{H}}$ -lines in the dilute-gas limit are represented by thick grey lines. The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively. (b) Acoustic eigenmode ‘trajectories’ along the ${\mathcal{H}}$ -line with $T_{1}=T_{c}$ . The vectors represent the basis when $M_{1}=3.10$ . The corresponding ideal-gas path is shown in black.

Dense-gas effects on the entropy-mode amplification are made clear from figure 8, which represents the evolution of $\unicode[STIX]{x1D712}_{s}$ along the critical ${\mathcal{H}}$ -line ( $T_{1}=T_{c}$ ) for increasing upstream Mach numbers. The non-ideal-gas value of $\unicode[STIX]{x1D712}_{s}$ is similar to the corresponding ideal-gas value for small values of the upstream Mach number, which is expected at low upstream reduced pressure and critical temperature. It is in agreement with the results from figure 7, which illustrates the weak dependence of the eigenmode basis with respect to the equation of state at low Mach number for those upstream conditions. When the downstream state reaches near-TCP conditions, the non-ideal-gas value of $\unicode[STIX]{x1D712}_{s}$ can be one order of magnitude higher than the ideal-gas value (for this choice of $c_{v}/R$ and upstream pressure and temperature). This large isolated value of $\unicode[STIX]{x1D712}_{s}$ will have an influence on the (linear) interaction of shocks with arbitrary incoming perturbations. Indeed, in the linear framework, small fluctuations are expected to be strongly amplified by dense-gas shocks with unperturbed Mach number close to the local maximum in figure 8 (i.e. around $M_{1}=3.1$ ). However, for a small change in the base flow (e.g. by modification of the nozzle geometry), any small perturbation would be significantly less amplified due to the sharp drop of $\unicode[STIX]{x1D712}_{s}$ around this local maximum. The resulting refracted perturbed flow can then be strongly selected from the incoming perturbed flow based on a small change in the base flow. This one-dimensional dense-gas selectivity effect is likely to have a significant impact even at higher dimensions, as the one-dimensional results set the behaviour of shock-normal perturbations regardless of the dimension considered (in the linear regime). This non-ideal-gas effect is to be added to the degeneracy of the post-shock eigenmode basis, which forces the post-shock energy to be propagating along a direction more aligned with density fluctuations in the $(\unicode[STIX]{x1D70C},u,p)$ space.

It is worth mentioning that large extrema of $\unicode[STIX]{x1D712}_{s}$ come with strong contractions in wavelengths (see figure 8 b). This contraction effect can be much stronger than the usual contraction observed in ideal gases. The higher values of $\unicode[STIX]{x1D712}_{s}$ also suggest that nonlinear interactions may be more important than in ideal gases. This may have significant consequences on the evolution of the turbulence kinetic energy behind the shock near the TCP. With the decrease of the post-shock characteristic length scales, viscous effects play a greater role in the post-shock evolution of the turbulent kinetic energy, indirectly promoting kinetic energy dissipation.

Figure 8. The TCP effects on the refracted entropy-mode properties ( $p_{1}/p_{c}=0.13$ , $T_{1}=T_{c}$ and $c_{v}/R=78.2$ ). Ideal gas is shown by dashed lines. The thick lines indicate the range of Mach numbers for which the DK instability is expected (to be discussed later).

4.2.2 Hugoniot lines with two sonic points: discontinuous discontinuities

In this section, we investigate the mixed-nonlinearity effects on the evolution of $\unicode[STIX]{x1D712}_{s}$ . Near-TCP upstream pressures are thus considered. Figure 9 illustrates how the ${\mathcal{H}}$ -lines starting at $p_{1}/p_{c}=0.55$ and $T_{1}=T_{c}$ depart from their dilute ideal-gas limit towards the dense-gas isothermal limit close to the TCP. Discrepancies with the ideal-gas model are seen for low supersonic Mach numbers $M_{1}\in [1,2]$ , as expected (see § 2.2). The importance of the critical-point vicinity of the upstream pressure in the discussion of non-admissibility can be appreciated by comparing the two isothermal limits for low upstream pressures (see figure 5 a) and near-critical-point upstream pressures (see figure 5 b). A larger non-admissible region is obtained for near-critical upstream pressures for which the ${\mathcal{H}}$ -line is more likely to feature a sonic point (Zamfirescu et al. Reference Zamfirescu, Guardone and Colonna2008), which is fundamental in the discussion of the admissibility problem (see § 2). In particular, the $p_{1}/p_{c}=0.55$ and $T_{1}=T_{c}$ ${\mathcal{H}}$ -line crosses a region of mixed nonlinearity, which produces two sonic points and consequently a non-admissible path (see figure 9 b). Therefore, a jump in refraction properties is expected on the admissible part of this ${\mathcal{H}}$ -line. This is illustrated with the jump on the quarter unit sphere operated by the post-shock Kovásznay eigenmode basis (see figure 9 b).

Figure 9. Discontinuous entropy-mode amplification factor for $p_{1}/p_{c}=0.55$ . (a) The $\unicode[STIX]{x1D712}_{s}$ isocontours and ${\mathcal{H}}$ -lines (solid black lines) for the PP10 gas using the van der Waals model (appendix A). The $M_{1}$ values needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines are represented by the black dashed lines with labels. Ideal-gas ${\mathcal{H}}$ -lines in the dilute-gas limit are represented by thick grey lines. The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively. The near-TCP non-admissible region is delimited with a green line. (b) Eigenmode basis ‘trajectory’ along the critical ${\mathcal{H}}$ -line ( $T_{1}=T_{c}$ ) with vectors for $M_{1}=1.30$ . The non-admissible path is shown in light blue. The corresponding ideal-gas path is shown in black.

The main consequence on the entropy-amplification factor is the corresponding discontinuity of $\unicode[STIX]{x1D712}_{s}$ with respect to $M_{1}$ when the admissible part of the ${\mathcal{H}}$ -line is considered. This can be seen in figure 10(a,b), where a jump in the entropy-amplification factor is seen for different upstream pressures at $M_{1}=1.29$ and $M_{1}=1.16$ respectively. Thus, for pre-shock conditions close enough to the TCP for the ${\mathcal{H}}$ -line to feature two sonic points, there exists a critical mass flow rate for which a negligible change in operating conditions would lead to significant modifications of the refracted-flow properties. As was shown previously (see figure 2), the discontinuity of the refraction coefficients occurs at lower upstream Mach number $M_{1}$ as the reduced pressure $p_{1}/p_{c}$ approaches the critical point, and the corresponding discontinuity can become larger (corresponding to a larger jump in the density coordinate of the acoustic eigenmode basis). A consequence is that for near-critical conditions, a non-classical selective behaviour is expected for weak shocks. The difference between the classical ideal-gas entropy mode and the non-classical non-ideal-gas one is striking for those weak shocks for which the ideal-gas scenario hardly amplifies incoming entropy waves. This result could have potentially great impact on the transmission properties of eddy shocklets commonly found in weakly compressible turbulent flows, considering that near the TCP, entropy wave/shock interactions are more likely to occur as density perturbations dominate over pressure fluctuations.

Figure 10. Discontinuous entropy-mode amplification factor for $T_{1}=T_{c}$ . The equivalent ideal-gas configuration is shown with dashed lines. The non-admissible path is shown in light blue. The thick lines indicate the range of Mach numbers for which the DK instability is expected (to be discussed later).

5.1 Dense-gas shocks with an infinite $c_{v}/R$ value

Following the same reasoning as for the asymptotic limit of $\unicode[STIX]{x1D712}_{s}$ , it can be shown that the value of $\unicode[STIX]{x1D712}_{a^{+}}$ at the critical point goes to negative infinity. The same precaution as the one taken near the singularity of $\unicode[STIX]{x1D712}_{s}$ has to be considered for the admissibility problem of shocks near the TCP. The isothermal limit suggests two main non-ideal-gas effects on the post-shock acoustic emission produced from incoming entropy perturbations. The first one is a similar selective effect to the one observed in the entropy-mode amplification case. This can be seen in figure 11(a), where large variations of $\unicode[STIX]{x1D712}_{a^{+}}$ are expected for post-shock states near the TCP. The second one is the negative value of the limit, which suggests an anti-correlation (phase shift of $\unicode[STIX]{x03C0}$ ) of the post-shock acoustic emitted field and the post-shock entropy mode (density/temperature fluctuations) at the shock front. This effect is present for both low and high upstream pressure (see figure 11 a,b), but affects a wider area in the $(p,\unicode[STIX]{x1D717})$ -diagram near the TCP when the pre-shock state is closer to the TCP.

Figure 11. The $\unicode[STIX]{x1D712}_{a^{+}}$ isocontours and ${\mathcal{H}}$ -lines (solid black lines) for $c_{v}/R\rightarrow +\infty$ using the van der Waals model (appendix A). Upstream Mach numbers ( $M_{1}$ ) indicating the shock speed needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines are represented by the black dashed lines with labels. Ideal-gas ${\mathcal{H}}$ -lines (thick grey lines) are represented in the dilute-gas limit. The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively. The near-TCP non-admissible region is enclosed within the region delimited by the green line.

5.2 Selective acoustic emission

Results for finite values of $c_{v}/R$ are now considered (see figure 12). A good agreement with the theoretical infinitely dense-gas limit can be obtained for the van der Waals equation with $c_{v}/R=78.2$ (PP10 gas). The local maxima of $\unicode[STIX]{x1D712}_{a^{+}}$ along the ${\mathcal{H}}$ -lines starting at $p_{1}/p_{c}=0.13$ are indeed very similar to their ‘isothermal’ limits apart from the loss of the anti-correlation region and the non-admissible region. For the near-TCP upstream state, however (see figure 12 b), there exist both a non-admissible region (due to the presence of two sonic points on the ${\mathcal{H}}$ -lines when approaching the TCP) and an anti-correlation region. As was observed with the evolution of $\unicode[STIX]{x1D712}_{s}$ for different upstream pressures, non-ideal-gas effects are related to weaker shocks if the upstream pressure is increased from $p_{1}/p_{c}=0.13$ to $p_{1}/p_{c}=0.55$ and are more pronounced at critical temperature (i.e. $T_{1}=T_{c}$ ) for single-phase flows.

Figure 12. Non-ideal selective acoustic emission; $\unicode[STIX]{x1D712}_{a^{+}}$ isocontours and ${\mathcal{H}}$ -lines (solid black lines) for $c_{v}/R=78.2$ using the van der Waals model (appendix A). Upstream Mach numbers ( $M_{1}$ ) indicating the shock speed needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines are represented by the black dashed lines with labels. Ideal-gas ${\mathcal{H}}$ -lines (thick grey lines) are represented in the dilute-gas limit. The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively. The near-TCP non-admissible region is enclosed within the region delimited by the green line.

Figure 13(a) illustrates the evolution of $\unicode[STIX]{x1D712}_{a^{+}}$ for increasing Mach number for a pre-shock state at $p_{1}/p_{c}=0.13$ and $T_{1}=T_{c}$ . The value of $\unicode[STIX]{x1D712}_{a^{+}}$ is plotted alongside the value of the generated acoustic wavelength. The local maximum of $\unicode[STIX]{x1D712}_{a^{+}}$ matches the local maximum of the acoustic wavelength compression ratio (see figure 13 b). The generated acoustic wavelengths are then more likely to be sensitive to wave steepening than their ideal-gas counterparts (for the same value of $M_{1}$ ), which could have a great impact on compressibility effects in turbulent flows. However, these short wavelengths are more prone to viscous damping.

Figure 13. Acoustic-generation coefficient and wavelength (obtained from the $k_{+_{2}}/k_{s_{1}}$ ratio) for $p_{1}/p_{c}=0.13$ , $T_{1}=T_{c}$ . The equivalent ideal-gas configuration is shown with dashed lines. The thick lines indicate the range of Mach numbers for which the DK instability is expected (to be discussed later).

As was seen for $\unicode[STIX]{x1D712}_{s}$ at higher upstream pressure, shocklets can exhibit discontinuous refraction properties with respect to $M_{1}$ . This is also the case as far as the acoustic generation is concerned (see figure 14 a). Another important aspect of near-TCP shocklets is the appearance of an anti-correlation region ( $\unicode[STIX]{x1D712}_{a^{+}}<0$ ), and the generated acoustic mode is thus beating out of phase (phase shift of $\unicode[STIX]{x03C0}$ , see (3.8)) with the refracted entropy mode at the front, making the corresponding shocklets very different from their ideal-gas counterparts, for which $\unicode[STIX]{x1D712}_{a^{+}}$ is always positive. Once again, the admissible path along a given Hugoniot is very important to comment on the refraction properties, and can complicate the evolution of the refraction coefficients. An example of this non-intuitive evolution can be seen for the shocks at $p_{1}/p_{c}=0.65$ , for which the resulting $\unicode[STIX]{x1D712}_{a^{+}}$ discontinuity is small (see figure 14 b) despite a significant jump in $\unicode[STIX]{x1D712}_{s}$ (see figure 10 b).

Figure 14. Discontinuous acoustic-generation coefficients ( $T_{1}=T_{c}$ ). The equivalent ideal-gas configuration is shown with dashed lines. The non-admissible path is shown in light blue. The thick lines indicate the range of Mach numbers for which the DK instability is expected (to be discussed later).

5.3 On the possibility of ‘silent’ interactions normal to the shock front

Figure 15. On the possibility of silent shocks; the $\unicode[STIX]{x1D712}_{a^{+}}=0$ line coloured by the magnitude of $\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{a^{+}}/\unicode[STIX]{x2202}M_{1}$ (high value in red, low value in light blue). The line joining the maximum value of $\unicode[STIX]{x1D712}_{s}$ is also shown (white dashed line). Upstream Mach numbers ( $M_{1}$ ) indicating the shock speed needed to reach the corresponding thermodynamic state on the ${\mathcal{H}}$ -lines starting at $p_{1}/p_{c}=0.55$ (a) and $p_{1}/p_{c}=0.65$ (b) are represented by the thin green lines with labels. The near-TCP non-admissible region is enclosed within the region delimited by the thick green line. The contours represent the local speed of sound normalised by its TCP value ( $c_{v}/R=78.2$ ). The $\unicode[STIX]{x1D6E4}=0$ and $T=T_{c}$ isolines are represented with thick orange and thick blue lines respectively.

The sign change of the acoustic-generation coefficient for some of the ${\mathcal{H}}$ -lines considered implies that there exist mass flow rates (or $M_{1}$ values) for which $\unicode[STIX]{x1D712}_{a^{+}}=0$ . Therefore, shocks with no generated acoustic waves following an entropy-mode refraction are possible. This result contrasts with that of ideal gases (for which perturbations normal to the shock front always generate an acoustic field) and could significantly affect shock/turbulence interaction properties. In most cases, for a given upstream pressure, two values of the upstream Mach number can give $\unicode[STIX]{x1D712}_{a^{+}}=0$ (see figure 15). A broader view is provided by figure 15, in which the effect of the thermodynamic upstream state is investigated by increasing the upstream temperature at constant upstream pressure following the same methodology as for the previous $(p,\unicode[STIX]{x1D717})$ -diagram analysis. The subdomain of the $(p,\unicode[STIX]{x1D717})$ -diagram inside which ${\mathcal{H}}$ -lines feature at least one shock with $\unicode[STIX]{x1D712}_{a^{+}}=0$ is considered. Upstream Mach numbers are indicated only for this subdomain. The post-shock values corresponding to vanishing $\unicode[STIX]{x1D712}_{a^{+}}$ are represented on a line coloured with the local value of $\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{a+}/\unicode[STIX]{x2202}M_{1}$ (low values in light blue, high values in red). The post-shock state giving the maximum value of $\unicode[STIX]{x1D712}_{s}$ is also represented with a dashed white line. As expected from § 4, the local maximum of $\unicode[STIX]{x1D712}_{s}$ is closely related to the local minimum of the sound speed, which in turn strongly deforms the Kovásznay eigenmode basis towards density perturbation. Regions of low values of $\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{a^{+}}/\unicode[STIX]{x2202}M_{1}$ on the isoline of vanishing $\unicode[STIX]{x1D712}_{a^{+}}$ can be identified for both upstream pressures (for instance at $M_{1}\approx 1.5$ and $\unicode[STIX]{x1D717}_{1}/\unicode[STIX]{x1D717}_{c}\approx 4.0$ for $p_{1}/p_{c}=0.55$ ). These shockwaves may be of interest for practical applications aiming at reducing the generation of noise induced by the refraction of a density perturbation (e.g. ORC turbine expanders).