2.1 Coefficients of variation

The coefficient of variation of a flow quantity is (Pham Reference Pham2006, (48.1), p. 906) the r.m.s. of the relative fluctuation, i.e.

(2.1a ) $$\begin{eqnarray}\displaystyle \text{CV}_{(\cdot )^{\prime }}:={\displaystyle \frac{\sqrt{\overline{{(\cdot )^{\prime }}^{2}}}}{\overline{(\cdot )}}}=\left[{\displaystyle \frac{(\cdot )^{\prime }}{\overline{(\cdot )}}}\right]_{rms} & & \displaystyle\end{eqnarray}$$

(2.1b-d ) $$\begin{eqnarray}\displaystyle \text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}:={\displaystyle \frac{\sqrt{\overline{{\unicode[STIX]{x1D70C}^{\prime }}^{2}}}}{\bar{\unicode[STIX]{x1D70C}}}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{rms}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}};\quad \text{CV}_{T^{\prime }}:={\displaystyle \frac{\sqrt{\overline{{T^{\prime }}^{2}}}}{\bar{T}}}={\displaystyle \frac{T_{rms}^{\prime }}{\bar{T}}};\quad \text{CV}_{p^{\prime }}:={\displaystyle \frac{\sqrt{\overline{{p^{\prime }}^{2}}}}{\bar{p}}}={\displaystyle \frac{p_{rms}^{\prime }}{\bar{p}}}.\qquad \quad & & \displaystyle\end{eqnarray}$$

Although definitions (2.1) are not in general use, their introduction greatly simplifies notation in the equations developed in the paper. An important observation from available compressible DNS data (Donzis & Jagannathan Reference Donzis and Jagannathan2013; Gerolymos & Vallet Reference Gerolymos and Vallet2014; Jagannathan & Donzis Reference Jagannathan and Donzis2016) is that, generally,

(2.2) $$\begin{eqnarray}\displaystyle O\Bigg(\underbrace{{\displaystyle \frac{\sqrt{\overline{{T^{\prime }}^{2}}}}{\bar{T}}}}_{\displaystyle \text{CV}_{T^{\prime }}}\Bigg)=O\Bigg(\underbrace{{\displaystyle \frac{\sqrt{\overline{{\unicode[STIX]{x1D70C}^{\prime }}^{2}}}}{\bar{\unicode[STIX]{x1D70C}}}}}_{\displaystyle \text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}\Bigg)=O\Bigg(\underbrace{{\displaystyle \frac{\sqrt{\overline{{p^{\prime }}^{2}}}}{\bar{p}}}}_{\displaystyle \text{CV}_{p^{\prime }}}\Bigg)=O\Bigg(\!\underbrace{{\displaystyle \frac{\sqrt{\overline{{s^{\prime }}^{2}}}}{R_{g}}}}_{\displaystyle R_{g}^{-1}s_{rms}^{\prime }}\!\Bigg). & & \displaystyle\end{eqnarray}$$

The last term in (2.2) represents the non-dimensional level of entropy fluctuations (§ 2.5), which was shown in Gerolymos & Vallet (Reference Gerolymos and Vallet2014, figure 5, p. 720) to be of the same order of magnitude and to follow a similar $\bar{M}_{CL}$ -dependency as $\{\text{CV}_{p^{\prime }},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }}\}$ .

Condition (2.2) is central in the asymptotic expansions worked out in the paper. In weakly compressible turbulence, the defining assumption $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\ll 1$ (Bradshaw Reference Bradshaw1977) is directly extended to the other thermodynamic variables by (2.2).

2.2 Correlation coefficients

The correlation coefficient (CC) between any 2 flow quantities $[\cdot ]$ and $(\cdot )$ is defined by

(2.3a ) $$\begin{eqnarray}\displaystyle c_{(\cdot )^{\prime }[\cdot ]^{\prime }}:={\displaystyle \frac{\overline{(\cdot )^{\prime }[\cdot ]^{\prime }}}{\sqrt{\overline{{(\cdot )^{\prime }}^{2}}}\sqrt{\overline{{[\cdot ]^{\prime }}^{2}}}}}\,\in [-1,1]. & & \displaystyle\end{eqnarray}$$

These correlation coefficients pertain to the 2-momenta between fluctuating quantities. Definition (2.3a ) can be extended to higher-order correlations as

(2.3b ) $$\begin{eqnarray}\displaystyle c_{(\cdot )^{\prime }\cdots [\cdot ]^{\prime }}:={\displaystyle \frac{\overline{(\cdot )^{\prime }\cdots [\cdot ]^{\prime }}}{\sqrt{\overline{{(\cdot )^{\prime }}^{2}}}\cdots \sqrt{\overline{{[\cdot ]^{\prime }}^{2}}}}}. & & \displaystyle\end{eqnarray}$$

However, in the multiple correlation case, the correlation coefficient ( $n$ CC,  $n\geqslant 3$ ) is not limited in a particular interval, unlike the 2-moment case (2.3a ). Notice that by (2.3b ) skewness and flatness are

(2.3c,d ) $$\begin{eqnarray}\displaystyle S_{(\cdot )^{\prime }}:={\displaystyle \frac{\overline{{(.)^{\prime }}^{3}}}{\left[\sqrt{\overline{{(.)^{\prime }}^{2}}}\right]^{3}}}\stackrel{\text{(2.3b)}}{=}c_{(\cdot )^{\prime }(\cdot )^{\prime }(\cdot )^{\prime }};\quad F_{(\cdot )^{\prime }}:={\displaystyle \frac{\overline{{(.)^{\prime }}^{4}}}{\left[\sqrt{\overline{{(.)^{\prime }}^{2}}}\right]^{4}}}\stackrel{\text{(2.3b)}}{=}c_{(\cdot )^{\prime }(\cdot )^{\prime }(\cdot )^{\prime }(\cdot )^{\prime }}.\quad & & \displaystyle\end{eqnarray}$$

2.3 Fluctuating equation of state and correlations

The basic thermodynamic variables ( $p$ , $\unicode[STIX]{x1D70C}$ , $T$ ) are related by the equation of state (1.1), implying

(2.4) $$\begin{eqnarray}\displaystyle \text{(1.1)}\stackrel{\text{(2.3)}}{\;\Longrightarrow \;}\left\{\begin{array}{@{}l@{}}\bar{p}=\bar{\unicode[STIX]{x1D70C}}R_{g}\bar{T}(1+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }})\\[6.0pt] (1+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}){\displaystyle \frac{p^{\prime }}{\bar{p}}}={\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}}+{\displaystyle \frac{T^{\prime }}{\bar{T}}}+{\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}{\bar{\unicode[STIX]{x1D70C}}\bar{T}}}-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Often, approximations for weakly compressible turbulence are constructed directly from (2.4), by dropping all nonlinear terms (i.e.  $\unicode[STIX]{x1D70C}^{\prime }T^{\prime }$ and $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}$ ). It is nonetheless useful to consider a more systematic approach based on exact relations between correlation coefficients. Multiplying (2.4) by $p^{\prime }$ , $\unicode[STIX]{x1D70C}^{\prime }$ or $T^{\prime }$ , we obtain, upon averaging the exact relations,

(2.5a ) $$\begin{eqnarray}\displaystyle (1+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }})\text{CV}_{p^{\prime }}\stackrel{\text{(2.4)},\,\text{(2.3)}}{=}c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+c_{p^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}+c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }} & & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle (1+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }})c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\text{CV}_{p^{\prime }}\stackrel{\text{(2.4)},\,\text{(2.3)}}{=}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}+c_{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }} & & \displaystyle\end{eqnarray}$$

(2.5c ) $$\begin{eqnarray}\displaystyle (1+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }})c_{p^{\prime }T^{\prime }}\text{CV}_{p^{\prime }}\stackrel{\text{(2.4)},\,\text{(2.3)}}{=}c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+\text{CV}_{T^{\prime }}+c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}. & & \displaystyle\end{eqnarray}$$

2.4 Entropy fluctuations

For any bivariate substance, entropy, as a state variable, is defined (Liepmann & Roshko Reference Liepmann and Roshko1957, p. 338) by the differential equation

(2.6) $$\begin{eqnarray}\displaystyle T\text{d}s=\text{d}h-{\displaystyle \frac{\text{d}p}{\unicode[STIX]{x1D70C}}}\stackrel{\text{(1.1)}}{\;\Longrightarrow \;}\left\{\begin{array}{@{}l@{}}{\displaystyle \frac{\text{d}s}{R_{g}}}={\displaystyle \frac{c_{p}(T)}{R_{g}}}{\displaystyle \frac{\text{d}T}{T}}-{\displaystyle \frac{\text{d}p}{p}}\\[12.0pt] ={\displaystyle \frac{c_{v}(T)}{R_{g}}}{\displaystyle \frac{\text{d}T}{T}}-{\displaystyle \frac{\text{d}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}}}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Integrating (2.6) between the reference states $(\,\bar{p},\bar{T})$ or $(\bar{\unicode[STIX]{x1D70C}},\bar{T})$ and the corresponding instantaneous turbulent flow conditions $(p,T)=(\bar{p}+p^{\prime },\bar{T}+T^{\prime })$ or $(\unicode[STIX]{x1D70C},T)=(\bar{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D70C}^{\prime },\bar{T}+T^{\prime })$ , respectively, readily yields

(2.7a ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{s-s_{(\,\bar{p},\bar{T})}}{R_{g}}}\stackrel{\text{(2.6)}\,}{=}{\displaystyle \frac{1}{R_{g}}}\int _{\bar{T}}^{\bar{T}+T^{\prime }}{\displaystyle \frac{c_{p}(T^{\ast })}{T^{\ast }}}\,\text{d}T^{\ast }-\ln \left(1+{\displaystyle \frac{p^{\prime }}{\bar{p}}}\right), & & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{s-s_{(\bar{\unicode[STIX]{x1D70C}},\bar{T})}}{R_{g}}}\stackrel{\text{(2.6)}}{=}{\displaystyle \frac{1}{R_{g}}}\int _{\bar{T}}^{\bar{T}+T^{\prime }}{\displaystyle \frac{c_{v}(T^{\ast })}{T^{\ast }}}\,\text{d}T^{\ast }-\ln \left(1+{\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}}\right), & & \displaystyle\end{eqnarray}$$

$s_{(\,\bar{p},\bar{T})}$

$s_{(\bar{\unicode[STIX]{x1D70C}},\bar{T})}$

$O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$

$\bar{s}$

$s=\bar{s}+s^{\prime }$

(2.8a ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{s^{\prime }}{R_{g}}} & \stackrel{\text{(2.7a)}}{{\sim}} & \displaystyle {\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}-\frac{1}{2}\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{2}\right)-\left({\displaystyle \frac{p^{\prime }}{\bar{p}}}-\frac{1}{2}\left({\displaystyle \frac{p^{\prime }}{\bar{p}}}\right)^{2}\right)+\frac{1}{2}\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{2}\,{\displaystyle \frac{\bar{T}}{R_{g}}}\left.{\displaystyle \frac{\text{d}c_{p}}{\text{d}T}}\right|_{\bar{T}}\nonumber\\ \displaystyle & & \displaystyle -\,{\displaystyle \frac{\bar{s}-s_{(\,\bar{p},\bar{T})}}{R_{g}}}+O\Bigg(\!\left({\displaystyle \frac{p^{\prime }}{\bar{p}}}\right)^{3},\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{3}\Bigg)\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \stackrel{\text{(2.7b)}}{{\sim}} & \displaystyle {\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}-\frac{1}{2}\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{2}\right)-\left({\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}}-\frac{1}{2}\left({\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}}\right)^{2}\right)+\frac{1}{2}\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{2}\,{\displaystyle \frac{\bar{T}}{R_{g}}}\left.{\displaystyle \frac{\text{d}c_{p}}{\text{d}T}}\right|_{\bar{T}}\nonumber\\ \displaystyle & & \displaystyle -\,{\displaystyle \frac{\bar{s}-s_{(\bar{\unicode[STIX]{x1D70C}},\bar{T})}}{R_{g}}}+O\Bigg(\!\left({\displaystyle \frac{\unicode[STIX]{x1D70C}^{\prime }}{\bar{\unicode[STIX]{x1D70C}}}}\right)^{3},\left({\displaystyle \frac{T^{\prime }}{\bar{T}}}\right)^{3}\Bigg),\end{eqnarray}$$

where we defined for brevity

(2.8c ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D6FE}}:=\unicode[STIX]{x1D6FE}(\bar{T}) & & \displaystyle\end{eqnarray}$$

following the convention that $\breve{\cdot }$ is a function of averaged quantities which cannot be identified with a Reynolds or Favre average (Gerolymos & Vallet Reference Gerolymos and Vallet1996, Reference Gerolymos and Vallet2014). Notice the presence of the constants $\bar{s}-s_{(\,\bar{p},\bar{T})}\neq 0\neq \bar{s}-s_{(\bar{\unicode[STIX]{x1D70C}},\bar{T})}$ , which are $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ (Lele Reference Lele1994, p. 224), and are directly related to the nonlinearity of (2.7). The influence of variable $c_{p}(T)=R_{g}-c_{v}(T)$ (1.1) also induces a quadratic $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ term in the expressions for $s^{\prime }$ (2.8). Therefore, leading-order approximations related to $s^{\prime }$ are valid independently of the variability of $c_{p}(T)$ , and depend only on the local value of $\unicode[STIX]{x1D6FE}(\bar{T})$ (1.1). Notice that by (1.1) $d_{T}c_{p}=d_{T}c_{v}$ .

2.5 Entropy variance and correlations

It is straightforward from (2.8) to calculate expansions for the entropy variance $s_{rms}^{\prime }$ and for correlation coefficients containing $s^{\prime }$ . Squaring and averaging (2.8b ) and introducing definitions (2.1), (2.3) yields after simple calculations using (4.4), (A1)

(2.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \left({\displaystyle \frac{s_{rms}^{\prime }}{R_{g}}}\right)^{2}\stackrel{\text{(2.8b)},\,\text{(4.4)},\,\text{(A 1)},\text{(2.2)}}{{\sim}}{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad -\,{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,S_{T^{\prime }}\,\text{CV}_{T^{\prime }}^{3}-{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,S_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3}+{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,S_{p^{\prime }}\,\text{CV}_{p^{\prime }}^{3}\nonumber\\ \displaystyle & & \displaystyle \quad +\,c_{s^{\prime }T^{\prime }T^{\prime }}\,{\displaystyle \frac{s_{rms}^{\prime }}{R_{g}}}\,\text{CV}_{T^{\prime }}^{2}\,{\displaystyle \frac{\bar{T}}{R_{g}}}\left.{\displaystyle \frac{\text{d}c_{p}}{\text{d}T}}\right|_{\bar{T}}+O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{4}).\end{eqnarray}$$

Notice that (2.9) implies that the correct non-dimensional expression for the entropy variance is $R_{g}^{-1}s_{rms}^{\prime }$ and is of the same order of magnitude as the coefficients of variation of the basic thermodynamic quantities. The reason why $R_{g}^{-1}\,s_{rms}^{\prime }$ should be considered in the order-of-magnitude relation (2.2) instead of $\text{CV}_{s^{\prime }}$ is because, by definition (2.6), entropy is defined with respect to an arbitrary reference state, so that the precise value of $\bar{s}$ that appears in the definition of $\text{CV}_{s^{\prime }}$ (2.1) has no physical significance: only entropy differences have physical meaning.

3.1 Sustained compressible HIT

Donzis & Jagannathan (Reference Donzis and Jagannathan2013) and Jagannathan & Donzis (Reference Jagannathan and Donzis2016) have performed DNS of sustained compressible HIT with solenoidal forcing at the large scales (Eswaran & Pope Reference Eswaran and Pope1988). The flow is modelled by the compressible Navier–Stokes equations (Jagannathan & Donzis Reference Jagannathan and Donzis2016, (3.1–3.5), p. 673), without bulk viscosity $\unicode[STIX]{x1D707}_{b}=0$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (2.1e), p. 706), a power law for the dynamic viscosity $\unicode[STIX]{x1D707}(T)\propto \sqrt{T}$ and constant Prandtl number $Pr=0.72$ (Donzis & Jagannathan Reference Donzis and Jagannathan2013, p. 224). The working medium follows the equation of state (1.1) with a constant ratio of specific heats (1.1) $\unicode[STIX]{x1D6FE}=1.4$ (Donzis & Jagannathan Reference Donzis and Jagannathan2013, figure 7, p. 234). The representative parameters in this configuration are the turbulent Mach number $M_{T}$ (Jagannathan & Donzis Reference Jagannathan and Donzis2016, p. 670) and the Taylor-microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ (Jagannathan & Donzis Reference Jagannathan and Donzis2016, p. 671)

(3.1a ) $$\begin{eqnarray}\displaystyle M_{T}:={\displaystyle \frac{\sqrt{\overline{u_{i}^{\prime }u_{i}^{\prime }}}}{\bar{a}}}={\displaystyle \frac{u_{rms}^{\prime }\sqrt{3}}{\bar{a}}}, & & \displaystyle\end{eqnarray}$$

(3.1b,c ) $$\begin{eqnarray}\displaystyle Re_{\unicode[STIX]{x1D706}}:={\displaystyle \frac{\bar{\unicode[STIX]{x1D70C}}\,\sqrt{u_{rms}^{\prime }}\,\unicode[STIX]{x1D706}_{\Vert }}{\bar{\unicode[STIX]{x1D707}}}};\quad \unicode[STIX]{x1D706}_{\Vert }:=\sqrt{{\displaystyle \frac{\overline{{u^{\prime }}^{2}}}{\overline{\left({\displaystyle \frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}x}}\right)^{2}}}}}. & & \displaystyle\end{eqnarray}$$

These data were made available with a precision of 4 significant digits (D. A. Donzis, 2016, Private communication, lower precision was found inadequate for use in the approximate relations developed in the paper), and cover the range $Re_{\unicode[STIX]{x1D706}}\in [35,430]$ and $M_{T}\in [0.1,0.6]$ , with a corresponding range of $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\in [0.004,0.157]$ (Donzis & Jagannathan Reference Donzis and Jagannathan2013, table 1, p. 225). The ratios $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\text{CV}_{p^{\prime }}$ and $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\text{CV}_{T^{\prime }}$ vary slightly with $M_{T}$ and $Re_{\unicode[STIX]{x1D706}}$ (figure 2). This dependency notwithstanding, in this flow, relative variations of temperature are weaker than relative variations of density ( $0.35\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\lessapprox \text{CV}_{T^{\prime }}\lessapprox 0.4\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ ; figure 2) whereas relative variations of pressure are stronger ( $1.3\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\lessapprox \text{CV}_{p^{\prime }}\lessapprox 1.4\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ ; figure 2).

Figure 2.

Sustained homogeneous isotropic turbulence (HIT) DNS data (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016) for the magnitude of the ratios of the coefficients of variation of temperature $\text{CV}_{T^{\prime }}$ and pressure $\text{CV}_{p^{\prime }}$ to the coefficient of variation of density $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ , as a function of the turbulent Mach number $M_{T}\in [0.1,0.6]$ , for different values of $Re_{\unicode[STIX]{x1D706}}\in [35,430]$ .

3.2 Compressible turbulent plane channel flow

The DNS data for compressible turbulent plane channel flow were obtained (Gerolymos & Vallet Reference Gerolymos and Vallet2014) using a high-order solver (Gerolymos, Sénéchal & Vallet Reference Gerolymos, Sénéchal and Vallet2010). The flow is modelled by the compressible Navier–Stokes equations (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (2.1), p. 706), without bulk viscosity $\unicode[STIX]{x1D707}_{b}=0$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (2.1e), p. 706). The working medium is air, following the equation of state (1.1) with a constant ratio of specific heats (1.1) $\unicode[STIX]{x1D6FE}=1.4$ , and Sutherland-like laws for the dynamic viscosity $\unicode[STIX]{x1D707}(T)$ and heat conductivity $\unicode[STIX]{x1D706}(T)$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, p. 706). The data used in the paper include both data from these computations (Gerolymos & Vallet Reference Gerolymos and Vallet2014) and new unpublished data (table 1). The relevant Mach number is the mean centreline Mach number (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 1, p. 713)

(3.2a ) $$\begin{eqnarray}\displaystyle \bar{M}_{CL}:=\overline{\left({\displaystyle \frac{u_{CL}}{a_{CL}}}\right)}, & & \displaystyle\end{eqnarray}$$

$u$

$a$

$(\cdot )_{CL}$

$y=\unicode[STIX]{x1D6FF}$

$\bar{\unicode[STIX]{x1D70F}}_{w}$

$\bar{\unicode[STIX]{x1D70C}}(y)$

$\bar{\unicode[STIX]{x1D707}}(y)$

(3.2b,c ) $$\begin{eqnarray}\displaystyle y^{\star }:={\displaystyle \frac{\bar{\unicode[STIX]{x1D70C}}(y)\sqrt{{\displaystyle \frac{\bar{\unicode[STIX]{x1D70F}}_{w}}{\bar{\unicode[STIX]{x1D70C}}(y)}}}(y-y_{w})}{\bar{\unicode[STIX]{x1D707}}(y)}}={\displaystyle \frac{\sqrt{\bar{\unicode[STIX]{x1D70C}}^{+}(y)}}{\bar{\unicode[STIX]{x1D707}}^{+}(y)}}y^{+};\quad Re_{\unicode[STIX]{x1D70F}^{\star }}:=\unicode[STIX]{x1D6FF}^{\star }={\displaystyle \frac{\sqrt{\bar{\unicode[STIX]{x1D70C}}_{CL}^{+}}}{\bar{\unicode[STIX]{x1D707}}_{CL}^{+}}}\unicode[STIX]{x1D6FF}^{+}, & & \displaystyle\end{eqnarray}$$

where $(\cdot )_{w}$ denotes wall values and

(3.2d-g ) $$\begin{eqnarray}\displaystyle y^{+}:={\displaystyle \frac{\bar{\unicode[STIX]{x1D70C}}_{w}\sqrt{{\displaystyle \frac{\bar{\unicode[STIX]{x1D70F}}_{w}}{\bar{\unicode[STIX]{x1D70C}}_{w}}}}(y-y_{w})}{\bar{\unicode[STIX]{x1D707}}_{w}}};\quad Re_{\unicode[STIX]{x1D70F}_{w}}:=\unicode[STIX]{x1D6FF}^{+};\quad \bar{\unicode[STIX]{x1D70C}}^{+}:={\displaystyle \frac{\bar{\unicode[STIX]{x1D70C}}}{\bar{\unicode[STIX]{x1D70C}}_{w}}};\quad \bar{\unicode[STIX]{x1D707}}^{+}:={\displaystyle \frac{\bar{\unicode[STIX]{x1D707}}}{\bar{\unicode[STIX]{x1D707}}_{w}}} & & \displaystyle\end{eqnarray}$$

are the usual incompressible-flow inner-scaled variables (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (3.2), p. 714).

Table 1.

Parameters of the DNS computations ( $L_{x}$ , $L_{y}$ , $L_{z}$ ( $N_{x}$ , $N_{y}$ , $N_{z}$ ) are the dimensions (number of grid points) of the computational domain ( $x=$ homogeneous streamwise, $y=$ wall-normal, $z=$ homogeneous spanwise direction); $\unicode[STIX]{x1D6FF}$ is the channel half-height; $(\cdot )^{+}$ denotes wall units; $\unicode[STIX]{x0394}x^{+}$ , $\unicode[STIX]{x0394}y_{w}^{+}$ , $\unicode[STIX]{x0394}y_{CL}^{+}$ , $\unicode[STIX]{x0394}z^{+}$ are the mesh sizes; $(\cdot )_{w}$ denotes wall and $(\cdot )_{CL}$ centreline values; $N_{y^{+}\leqslant 10}$ is the number of grid points between the wall and $y^{+}=10$ ; $\bar{M}_{CL}$ is the centreline Mach number (3.2a );  $Re_{\unicode[STIX]{x1D70F}^{\star }}:=\sqrt{\bar{\unicode[STIX]{x1D70C}}_{CL}\,\bar{\unicode[STIX]{x1D70F}}_{w}}\unicode[STIX]{x1D6FF}\bar{\unicode[STIX]{x1D707}}_{CL}^{-1}$ is the friction Reynolds number in HCB-scaling (3.2b,c );  $Re_{\unicode[STIX]{x1D70F}_{w}}:=\sqrt{\bar{\unicode[STIX]{x1D70C}}_{w}\,\bar{\unicode[STIX]{x1D70F}}_{w}}\unicode[STIX]{x1D6FF}\bar{\unicode[STIX]{x1D707}}_{w}^{-1}$ is the friction Reynolds number (3.2d–g ); $\unicode[STIX]{x0394}t^{+}$ is the computational time step; $t_{\text{OBS}}^{+}$ is the observation period over which single-point statistics were computed; $\unicode[STIX]{x0394}t_{s}^{+}$ is the sampling time step for the single-point statistics).

The coefficients of variation $\{\text{CV}_{p^{\prime }},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }}\}$ are of the same order of magnitude everywhere in the channel (figure 3), and for all $(Re_{\unicode[STIX]{x1D70F}^{\star }},\bar{M}_{CL})$ that were investigated, even at the low Mach number limit. Notice that previously published results (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 6, p. 721) indicate that the correlation coefficients between thermodynamic variables plotted against $y^{\star }$ (3.2b,c ) show little dependence on $\bar{M}_{CL}$ , except perhaps very near the wall ( $y^{\star }<10$ ). Nonetheless, in the neighbourhood of the near-wall $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ peak (Gerolymos & Vallet Reference Gerolymos and Vallet2014, $7\lessapprox y^{\ast }\lessapprox 20$ , figure 5, p. 720), $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\approxeq 5\text{CV}_{p^{\prime }}$ (figure 3), so that the assumption $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})=O(\text{CV}_{p^{\prime }})$ (2.2) is stretched to the limit. On the contrary, $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})=O(\text{CV}_{T^{\prime }})$ is satisfied practically everywhere, except very near the wall ( $y^{\star }\lessapprox 1$ ; figure 3) where the strictly isothermal wall boundary condition enforces $[\text{CV}_{T^{\prime }}]_{w}=0$ at the wall ( $y^{\ast }=0$ ). The question is naturally raised, how a less stringent isothermal-in-the-mean wall boundary condition, where $T_{w}^{\prime }\neq 0$ but instead only $\bar{T}_{w}=\text{const.}$ in the mean is enforced, would modify the viscous sublayer thermodynamic turbulence, and this will be the subject of a future study. Nonetheless, DNS data (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 5, p. 720) suggest that the boundary condition effect is confined very near the wall ( $y^{\ast }\lessapprox 1$ ), and this is further confirmed in § 6.

Figure 3.

Compressible fully developed turbulent plane channel (TPC) flow DNS data (Gerolymos & Vallet Reference Gerolymos and Vallet2014) for the magnitude of the ratios of the coefficients of variation of temperature $\text{CV}_{T^{\prime }}$ and pressure $\text{CV}_{p^{\prime }}$ to the coefficient of variation of density $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ , as a function of the HCB-scaled wall distance $y^{\star }$ (3.2b,c ), for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [78,341]$ and $\bar{M}_{CL}\in [0.3,2.5]$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, the shaded region corresponds to values $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\geqslant 2\text{CV}_{p^{\prime }}$ observed in the neighbourhood of the $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ -peak; figure 5, p. 720).

3.3 Shear turbulence is not polytropic

Near the centreline, where the wall influence is weak, channel DNS data (figure 3) indicate that the relative magnitudes of the coefficients of variation $\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$ approach those observed in the HIT DNS data (figure 2), the more so as $Re_{\unicode[STIX]{x1D70F}^{\star }}=\unicode[STIX]{x1D6FF}^{\star }$ (3.2b,c ) increases. On the other hand, close to the wall ( $y^{\star }\lessapprox 100$ ) there are substantial differences. Whereas in HIT (figure 2) and near the centreline in channel flow (figure 3), $\text{CV}_{p^{\prime }}>\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}>\text{CV}_{T^{\prime }}$ , near the solid wall $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\approxeq \text{CV}_{T^{\prime }}>\text{CV}_{p^{\prime }}$ ( $5\lessapprox y^{\star }\lessapprox 50$ ; figure 3). This difference is easily explained by the approximate leading-order relation (Donzis & Jagannathan Reference Donzis and Jagannathan2013, (3.4), p. 226) $\text{CV}_{p^{\prime }}^{2}\approxeq \text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+\text{CV}_{T^{\prime }}^{2}+2\,c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}$ (4.4). In HIT (Donzis & Jagannathan Reference Donzis and Jagannathan2013, table 1, p. 225) and in channel flow near the centreline (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 6, p. 721), the correlation coefficient (2.3a ) $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}>0\;\Longrightarrow \;\text{CV}_{p^{\prime }}^{2}>\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+\text{CV}_{T^{\prime }}^{2}$ , whereas near the solid wall (Gerolymos & Vallet Reference Gerolymos and Vallet2014, $y^{\star }\lessapprox 100$ , figure 6, p. 721) $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}<0\;\Longrightarrow \;\text{CV}_{p^{\prime }}^{2}<\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+\text{CV}_{T^{\prime }}^{2}$ .

Figure 4.

Estimates $n_{P}(\text{CV}_{p^{\prime }},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})\approxeq n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}$ and $n_{P}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }})\approxeq n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}$ by (3.3) of an eventual representative polytropic exponent, from DNS data of sustained homogeneous isotropic turbulence (HIT) simulations (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016) versus the turbulent Mach number $M_{T}\in [0.1,0.6]$ and for compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014) versus the HCB-scaled wall distance $y^{\star }$ (3.2b,c ) (for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [78,341]$ and $\bar{M}_{CL}\in [0.3,2.5]$ ).

The difference in behaviour between HIT and near-wall turbulence is further highlighted by considering polytropic exponents, in line with Donzis & Jagannathan (Reference Donzis and Jagannathan2013, figure 7, p. 234). If turbulence were polytropic ( $p\propto \unicode[STIX]{x1D70C}^{n_{P}}$ ), then, to leading order, specific relations should hold between CVs (Barre & Bonnet Reference Barre and Bonnet2015, p. 331)

(3.3) $$\begin{eqnarray}\displaystyle p\propto \unicode[STIX]{x1D70C}^{n_{P}}\stackrel{\text{(1.1)}}{\;\Longrightarrow \;}n_{P}\approxeq \underbrace{{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}}_{\displaystyle n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}}\approxeq \underbrace{1+{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}}_{\displaystyle n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}>1}. & & \displaystyle\end{eqnarray}$$

Notice that expression (3.3) which is used in compressible turbulence, both for instantaneous fluctuations (Rubesin Reference Rubesin1976) or on the average (Blaisdell et al. Reference Blaisdell, Mansour and Reynolds1993) is a leading-order approximation of a polytropic process. In practice, $n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}$ and $n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}$ are never equal (3.3). Gatski & Bonnet (Reference Gatski and Bonnet2009, p. 74) suggest considering $n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}$ and $n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}$ (3.3) as independent parameters, but still $n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}\neq n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}$ implies that turbulence does not follow a polytropic behaviour.

In HIT both expressions (3.3) are reasonably consistent one with another (figure 4), and although the process is not isentropic the departure from the isentropic value $\unicode[STIX]{x1D6FE}=1.4$ is not very large. On the contrary, wall turbulence is very far from isentropic (figure 4). Furthermore, for $y^{\star }\lessapprox 100$ , the polytropic exponent estimate $n_{P}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }})\approxeq n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}>1$ is substantially higher than the isentropic value (figure 4), whereas the polytropic exponent estimate $n_{P}(\text{CV}_{p^{\prime }},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})\approxeq n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}$ is substantially lower than the isentropic value (figure 4). Therefore, not only is wall turbulence not isentropic, but it cannot be approximated by a polytropic process, which would require the two estimates of $n_{P}$ (3.3) to be approximately the same. It is not always recognized that the equality $n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}\approxeq n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}$ can only be achieved under very specific conditions. To leading order, straightforward calculation of the difference $n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}^{2}-n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}^{2}$ (3.3), readily yields, using (4.4),

(3.4) $$\begin{eqnarray}\displaystyle \text{(3.3)},\text{(4.4)}\;\Longrightarrow \;n_{P_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}}^{2}-n_{P_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}^{2}\approxeq 2(1-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}){\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\stackrel{\text{(2.3b)}}{{>}}0. & & \displaystyle\end{eqnarray}$$

Therefore, turbulence can only be approximated by a polytropic process iff $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\approxeq +1$ , since the alternative condition $\text{CV}_{T^{\prime }}\ll \text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ contradicts conjecture (2.2) and DNS data (figures 2 and 3). Consistent with (3.4), the observed non-polytropic behaviour of thermodynamic fluctuations (figure 4) is explained by the $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ -data. In HIT (Donzis & Jagannathan Reference Donzis and Jagannathan2013, table 1, p. 225) $0.6\lessapprox c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\lessapprox 1$ (approaching 1 with increasing $M_{T}$ ): the 2 polytropic exponent estimates are close one to another, and their difference decreases with increasing $M_{T}$ (figure 4). In channel flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 6, p. 721) $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}<0\;\forall y^{\star }\lessapprox 100$ (it might approach 1 at high $Re_{\unicode[STIX]{x1D70F}^{\star }}$ at the centreline): the 2 polytropic exponent estimates are largely different, except when approaching the centreline for the higher $Re_{\unicode[STIX]{x1D70F}^{\star }}$ (figure 4). This discussion highlights the importance of the correlation coefficient $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ on thermodynamic turbulence structure, and is further investigated in § 6.

4.1 Approximation of correlation coefficients $\{c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\}$

Using again the order-of-magnitude relation (2.2), the system (2.5) is readily expanded as

(4.1a ) $$\begin{eqnarray}\displaystyle \text{CV}_{p^{\prime }}-c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}-c_{p^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}\stackrel{\text{(2.5a)}}{{\sim}}c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}+O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\text{CV}_{T^{\prime }}\text{CV}_{p^{\prime }}), & & \displaystyle\end{eqnarray}$$

(4.1b ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{p^{\prime }}-\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}\stackrel{\text{(2.5b)}}{{\sim}}c_{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}+O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\text{CV}_{T^{\prime }}\text{CV}_{p^{\prime }}), & & \displaystyle\end{eqnarray}$$

(4.1c ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }T^{\prime }}\,\text{CV}_{p^{\prime }}-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}-\text{CV}_{T^{\prime }}\stackrel{\text{(2.5c)}}{{\sim}}c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}+O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\text{CV}_{T^{\prime }}\text{CV}_{p^{\prime }}), & & \displaystyle\end{eqnarray}$$

$O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\text{CV}_{T^{\prime }}\text{CV}_{p^{\prime }})=O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3})$

(4.2a ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\stackrel{\text{(4.1)},\,\text{(A 1a)}}{{\sim}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{p^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{p^{\prime }}}}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }T^{\prime }}{\displaystyle \frac{\text{CV}_{T^{\prime }}^{2}}{\text{CV}_{p^{\prime }}}}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}\right), & & \displaystyle\end{eqnarray}$$

(4.2b ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }T^{\prime }}\stackrel{\text{(4.1)},\,\text{(A 1a)}}{{\sim}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{p^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{T^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{p^{\prime }}}}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{T^{\prime }}}}-c_{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}}{\text{CV}_{p^{\prime }}}}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}\right), & & \displaystyle\end{eqnarray}$$

(4.2c ) $$\begin{eqnarray}\displaystyle c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\stackrel{\text{(4.1)},\,\text{(A 1a)}}{{\sim}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{T^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{T^{\prime }}}}-c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{p^{\prime }}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}\right), & & \displaystyle\end{eqnarray}$$

$O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3})$

$O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}\right)$

Retaining only $O(1)$ terms in (4.2) yields the approximations

(4.3a ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\stackrel{\text{(4.2a)}}{\approxeq }+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{p^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{p^{\prime }}}}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+O\left({\displaystyle \frac{\text{CV}_{T^{\prime }}^{2}}{\text{CV}_{p^{\prime }}}}\right), & & \displaystyle\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }T^{\prime }}\stackrel{\text{(4.2b)}}{\approxeq }+\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{p^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{T^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{p^{\prime }}}}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{T^{\prime }}}}+O\left({\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}}{\text{CV}_{p^{\prime }}}}\right), & & \displaystyle\end{eqnarray}$$

(4.3c ) $$\begin{eqnarray}\displaystyle c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\stackrel{\text{(4.2c)}}{\approxeq }-\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}{\text{CV}_{T^{\prime }}}}-\frac{1}{2}{\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+\frac{1}{2}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{T^{\prime }}}}+O\left(\text{CV}_{p^{\prime }}\right). & & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle \text{CV}_{p^{\prime }}^{2}\approxeq \text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+\text{CV}_{T^{\prime }}^{2}+2\,c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}+O(\text{CV}_{p^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}). & & \displaystyle\end{eqnarray}$$

4.2 Approximation of entropy variance and correlations

Expansion (2.9) can be readily truncated to yield the approximations

(4.5a ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{s_{rms}^{\prime }}{R_{g}}} & \stackrel{\text{(2.9)},\,\text{(2.2)}}{\approxeq } & \displaystyle \sqrt{{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,O\left(\text{CV}_{T^{\prime }}^{2},{\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3},\text{CV}_{T^{\prime }}^{3},\text{CV}_{p^{\prime }}^{3}\right)}{R_{g}^{-1}\,s_{rms}^{\prime }}}\right)\end{eqnarray}$$

(4.5b ) $$\begin{eqnarray}\displaystyle & \stackrel{\text{(4.4)},\,\text{(2.2)}}{\approxeq } & \displaystyle \sqrt{{\displaystyle \frac{1}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{2}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}}\nonumber\\ \displaystyle & & \displaystyle +\,O\left(\text{CV}_{T^{\prime }}^{2},{\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3},\text{CV}_{T^{\prime }}^{3},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}\,\text{CV}_{p^{\prime }}\right)}{R_{g}^{-1}\,s_{rms}^{\prime }}}\right).\end{eqnarray}$$

${\geqslant}0$

$\left|c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\right|\leqslant 1$

$O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3},\text{CV}_{T^{\prime }}^{3},\text{CV}_{p^{\prime }}^{3},\text{CV}_{T^{\prime }}^{2}\,R_{g}^{-1}\,s_{rms}^{\prime })$

$R_{g}^{-2}\,{s^{\prime }}_{rms}^{2}$

$c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$

$s_{rms}^{\prime }$

$\sqrt{\cdot }$

Multiplying (2.8b ) by $\unicode[STIX]{x1D70C}^{\prime }$ or $T^{\prime }$ yields, upon averaging, using definitions (2.1), (2.3) and truncating to leading order

(4.6a ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}{\displaystyle \frac{s_{rms}^{\prime }}{R_{g}}}\stackrel{\text{(2.8b)}}{{\sim}}{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}-\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2},\text{CV}_{T^{\prime }}^{2}\right), & & \displaystyle\end{eqnarray}$$

(4.6b ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }T^{\prime }}{\displaystyle \frac{s_{rms}^{\prime }}{R_{g}}}\stackrel{\text{(2.8b)}}{{\sim}}{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{T^{\prime }}-c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2},\text{CV}_{T^{\prime }}^{2}\right). & & \displaystyle\end{eqnarray}$$

Multiplying (2.4) by $s^{\prime }$ yields, upon averaging, using definitions (2.1), (2.3) and truncating to leading order

(4.6c ) $$\begin{eqnarray}\displaystyle & & \displaystyle c_{s^{\prime }p^{\prime }}\,\text{CV}_{p^{\prime }}\stackrel{\text{(2.4)}}{{\sim}}c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}+c_{s^{\prime }T^{\prime }}\,\text{CV}_{T^{\prime }}+O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}\right)\end{eqnarray}$$

(4.6d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \stackrel{\text{(4.6a)},\,\text{(4.6b)}}{{\sim}}{\displaystyle \frac{1}{R_{g}^{-1}\,s_{rms}^{\prime }}}\left(-\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+{\displaystyle \frac{2-\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}+{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{T^{\prime }}^{2}\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,O\left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }},{\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3},\text{CV}_{T^{\prime }}^{3}\right)}{R_{g}^{-1}\,s_{rms}^{\prime }}}\right).\end{eqnarray}$$

Using the leading-order approximations (4.3c ) for $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ and (4.5b ) in (4.6) yields the leading-order approximations

(4.7a ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }} & \stackrel{\text{(4.6a)},\,\text{(4.5)},\,\text{(4.2c)}}{\approxeq } & \displaystyle \,\frac{1}{2}{\displaystyle \frac{-{\displaystyle \frac{2\breve{\unicode[STIX]{x1D6FE}}-1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\sqrt{{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}}}}\nonumber\\ \displaystyle & & \displaystyle +\,O\left({\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2},\text{CV}_{T^{\prime }}^{2},\text{CV}_{T^{\prime }}\,\text{CV}_{p^{\prime }}\right)}{R_{g}^{-1}\,s_{rms}^{\prime }}}\right),\end{eqnarray}$$

(4.7b ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }T^{\prime }} & \stackrel{\text{(4.6b)},\,\text{(4.5)},\,\text{(4.2c)}}{\approxeq } & \displaystyle \,\frac{1}{2}{\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}+1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{T^{\prime }}^{2}-\text{CV}_{p^{\prime }}^{2}}{\text{CV}_{T^{\prime }}\,\sqrt{{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}}}}\nonumber\\ \displaystyle & & \displaystyle +\,O\left({\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2},\text{CV}_{T^{\prime }}^{2},\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }\,}\text{CV}_{p^{\prime }}\right)}{R_{g}^{-1}\,s_{rms}^{\prime }}}\right),\end{eqnarray}$$

(4.7c ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }p^{\prime }} & \stackrel{\text{(4.6c)},\,\text{(4.7a)},\,\text{(4.7b)}}{\approxeq } & \displaystyle \,\frac{1}{2}{\displaystyle \frac{-{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{2-\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\text{CV}_{p^{\prime }}^{2}}{\text{CV}_{p^{\prime }}\,\sqrt{{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{(\breve{\unicode[STIX]{x1D6FE}}-1)^{2}}}\,\text{CV}_{T^{\prime }}^{2}+{\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2}-{\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}-1}}\,\text{CV}_{p^{\prime }}^{2}}}}\nonumber\\ \displaystyle & & \displaystyle +\,O\left({\displaystyle \frac{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\,\text{CV}_{T^{\prime }}}{\text{CV}_{p^{\prime }}}},{\displaystyle \frac{\max \left(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3},\text{CV}_{T^{\prime }}^{3}\right)}{\text{CV}_{p^{\prime }}\,R_{g}^{-1}\,s_{rms}^{\prime }}}\right)\end{eqnarray}$$

$\{c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{s^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$

$\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$

$\breve{\unicode[STIX]{x1D6FE}}$

5.1 Approximation accuracy in compressible turbulent plane channel flow

The leading-order approximations (4.3) are assessed by comparison with DNS data (Gerolymos & Vallet Reference Gerolymos and Vallet2014) for channel flow (figures 5–7), but also by comparison with the higher-order expansions (4.2), to analyse the eventual importance of the leading error terms and when necessary to identify the origin of the nonlinearities.

Regarding $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ (figure 5), the leading-order approximation (4.3c ) is in excellent agreement with DNS data $\forall \,y^{\star }$ and for the entire available range of $(Re_{\unicode[STIX]{x1D70F}^{\star }},\bar{M}_{CL})$ . The higher $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ expansion of $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ (4.2c ) is practically identical to the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation (4.3c ) and to the DNS data (figure 5), implying that the leading error $-c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\text{CV}_{p^{\prime }}$ is indeed negligible, even for the highest $\bar{M}_{CL}\approxeq 2.5$ .

Regarding the correlation coefficients involving $p^{\prime }$ , $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ (figure 6) and $c_{p^{\prime }T^{\prime }}$ (figure 7), the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ) are excellent for the subsonic $\bar{M}_{CL}\approxeq 0.8$ case, and remain quite satisfactory at $\bar{M}_{CL}\approxeq 1.5$ , the more so with increasing $Re_{\unicode[STIX]{x1D70F}^{\star }}$ (figures 6, 7). For the higher $\bar{M}_{CL}\in \{2,2.5\}$ cases, the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ) are satisfactory near the wall and in the outer part of the flow, but discrepancies with DNS data appear in the region $10\lessapprox y^{\star }\lessapprox 40$ (figures 6, 7). Both the magnitude of the error and the $y^{\star }$ -range where discrepancies are observed increase with increasing $\bar{M}_{CL}\gtrapprox 2$ . These discrepancies are fully accounted for by the leading error of the approximations (4.3a ), (4.3b ), because the corresponding by (2.2) $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ expansions (4.2a ), (4.2b ), which use DNS data for the 3CCs $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }T^{\prime }}$ and $c_{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ , are in excellent agreement with DNS data (figures 6, 7) $\forall \,y^{\star }$ and for the entire $(Re_{\unicode[STIX]{x1D70F}^{\star }},\bar{M}_{CL})$ -range studied, a very slight discrepancy observed at $\bar{M}_{CL}\approxeq 2.5$ notwithstanding. The leading error in the approximations (4.3a ) for $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ and (4.3b ) for $c_{p^{\prime }T^{\prime }}$ is more important than that for the approximation (4.2c ) for $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ because of the presence of the ratios $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{T^{\prime }}$ and $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ which become ${\geqslant}2$ around $5\leqslant y^{\star }\leqslant 25$ (figure 3), whereas the leading error in (4.2c ) for $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ does not contain ratios between relative amplitudes.

Figure 8.

Comparison of the leading $O(\text{CV}_{{\unicode[STIX]{x1D70C}^{\prime }}^{2}})$ approximation (4.5a ) of $s_{rms}^{\prime }(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ with DNS data for compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014), plotted against the HCB-scaled wall distance $y^{\star }$ (3.2b,c ), for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [98,341]$ and $\bar{M}_{CL}\in [0.8,2.5]$ , and with the higher $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{3})$ expansion (2.9) which uses DNS data for the skewnesses $\{S_{p^{\prime }},S_{\unicode[STIX]{x1D70C}^{\prime }},S_{T^{\prime }}\}$ .

Figure 9.

Comparison of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation (4.7a ) of $c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ with DNS data for compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014), plotted against the HCB-scaled wall distance $y^{\star }$ (3.2b,c ), for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [98,341]$ and $\bar{M}_{CL}\in [0.8,2.5]$ .

Figure 10.

Comparison of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation (4.7b ) of $c_{s^{\prime }T^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ with DNS data for compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014), plotted against the HCB-scaled wall distance $y^{\star }$ (3.2b,c ), for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [98,341]$ and $\bar{M}_{CL}\in [0.8,2.5]$ .

Figure 11.

Comparison of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation (4.7c ) of $c_{s^{\prime }p^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ with DNS data for compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014), plotted against the HCB-scaled wall distance $y^{\star }$ (3.2b,c ), for different values of $Re_{\unicode[STIX]{x1D70F}^{\star }}\in [98,341]$ and $\bar{M}_{CL}\in [0.8,2.5]$ .

Figure 12.

Comparison of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation (4.3c ) of $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ with sustained homogeneous isotropic turbulence (HIT) DNS data (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016), plotted against the turbulent Mach number $M_{T}\in [0.1,0.6]$ , for different values of $Re_{\unicode[STIX]{x1D706}}\in [35,430]$ (solid symbols: DNS, open symbols: approximation).

Figure 13.

Leading-order estimates (4.5a ), (4.3), (4.7) for $\{R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime },c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }},c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{s^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$ with input DNS data for $\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$ from sustained compressible HIT computations (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016), for different values of $Re_{\unicode[STIX]{x1D706}}\in [35,430]$ , plotted against the turbulent Mach number $M_{T}\in [0.1,0.6]$ .

The approximations (4.5a ), (4.7) are assessed (figures 8–11) against DNS data for compressible channel flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014). These data were obtained by computing the instantaneous entropy field $s(\boldsymbol{x},t)$ exactly (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (2.5), p. 710) and sampling (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2010, § 4.4, p. 791) the appropriate moments: they involve no approximation or truncation.

The leading-order approximations (4.5a ) of $R_{g}^{-1}\,s_{rms}^{\prime }$ (figure 8), (4.7a ) of $c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ (figure 9), and (4.7b ) of $c_{s^{\prime }T^{\prime }}$ (figure 10) are in excellent agreement with DNS data $\forall \,y^{\star }$ and for every available $(Re_{\unicode[STIX]{x1D70F}^{\star }},\bar{M}_{CL})$ . On the contrary the leading-order approximation (4.7c ) of $c_{s^{\prime }p^{\prime }}$ (figure 11) behaves in a manner similar to the leading-order approximations (4.3a ), (4.3b ) of $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ (figure 6) and of $c_{p^{\prime }T^{\prime }}$ (figure 7). It is excellent for $\bar{M}_{CL}\approxeq 0.8$ and satisfactory for $\bar{M}_{CL}\approxeq 1.5$ , but presents discrepancies for the higher $\bar{M}_{CL}\in \{2,2.5\}$ (figure 11), in the range $10\lessapprox y^{\star }\lessapprox 40$ . Again, the leading error in (4.7c ) involves division by $\text{CV}_{p^{\prime }}$ and suffers from the high values of $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ and $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{T^{\prime }}$ (figure 3).

As a conclusion, nonlinear effects of compressibility seem to influence some features of the thermodynamic correlations containing $p^{\prime }$ when $\bar{M}_{CL}\gtrapprox 2$ (figures 5–7): in this range, the quadratic terms in the exact relations (2.5) cannot be neglected. Nonetheless, the region where these nonlinear compressibility effects are important remains confined in the buffer zone of the wall layer (Smits & Dussauge Reference Smits and Dussauge2006, p. 203).

5.2 Approximation accuracy in sustained compressible HIT

Only $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ was available in the HIT DNS database (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016). The leading-order approximation (4.3c ) compares very well with the DNS data (figure 12), corroborating the conclusion (§ 5.1) that (4.3c ) is a very robust approximation. Notice also that the ratios $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{T^{\prime }}$ and $\text{CV}_{p^{\prime }}^{-1}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}$ which multiply the leading error of the approximations (4.3a ) for $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ and (4.3b ) for $c_{p^{\prime }T^{\prime }}$ are invariably ${<}1$ in HIT (figure 2). Therefore, we expect that the approximations (4.3a ), (4.3b ) of $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ and $c_{p^{\prime }T^{\prime }}$ should be quite accurate for the HIT case. Of course data for these correlations are needed to fully substantiate this conjecture.

The leading-order approximations (4.3), (4.5a ), (4.7) can be used to discuss the behaviour of compressible HIT, in terms of correlations that were not available in the database (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016). At the highest $M_{T}\approxeq 0.6$ sustained compressible HIT data (Donzis & Jagannathan Reference Donzis and Jagannathan2013, table 1, p. 225), $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}\approxeq 0.16$ , $\text{CV}_{p^{\prime }}\approxeq 0.22$ and $\text{CV}_{T^{\prime }}\approxeq 0.06$ . Based on the assessment of the leading-order approximations (4.3), (4.5a ), (4.7) for compressible channel flow (figures 5–11) we expect these approximations to be reasonably accurate, and use them to investigate the behaviour of compressible HIT with increasing $M_{T}$ (figure 13). The non-dimensional ratio of entropy and density fluctuations $R_{g}^{-1}\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}s_{rms}^{\prime }$ decreases with increasing $M_{T}$ and shows significant $Re_{\unicode[STIX]{x1D706}}$ -dependency (figure 13), which is not surprising since $s_{rms}^{\prime }$ is largely the result of dissipative phenomena. The various correlation coefficients depend mainly on $M_{T}$ , with a weaker $Re_{\unicode[STIX]{x1D706}}$ -dependency (figure 13). We observe in particular that $c_{s^{\prime }T^{\prime }}\approxeq 0.2\;\forall \;(M_{T},Re_{\unicode[STIX]{x1D706}})$ remains practically constant with little scatter (figure 13). This observation is further confirmed in § 6 and will be used in § 7.1 to suggest a simple model for the thermodynamic turbulence structure of subsonic sustained compressible HIT.

7.1 Thermodynamic turbulence structure in sustained subsonic HIT

It appears from DNS data (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016) that, as $M_{T}$ increases, compressible subsonic HIT thermodynamic turbulence structure parameters (1.2) tend to a slowly varying state (figures 2, 12, 8, 13) with a $Re_{\unicode[STIX]{x1D706}}$ -dependency, which is clearly visible e.g. in the evolution of $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ versus  $M_{T}$ (figure 12) and even more so for $R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$ versus  $M_{T}$ (figure 13). However, when considering relations between elements of the set TTS (1.2) only, such as mapping on the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$ -plane (figure 14), the DNS data collapse with little scatter on a single curve, following the $[c_{s^{\prime }T^{\prime }}=0.2]$ -locus (6.2), this value corresponding to the available $\unicode[STIX]{x1D6FE}=1.4$ DNS data (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016). Zoom on the HIT data in the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$ -plane (figure 15) confirms that there is little scatter. The DNS data follow the

$[c_{s^{\prime }T^{\prime }}=0.2]$

$[R_{g}^{-1}\,s_{rms}^{\prime }=\text{CV}_{T^{\prime }}]$

$[c_{s^{\prime }T^{\prime }}=0.2]$

$Q_{2}$

$c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}=0$

$[c_{s^{\prime }T^{\prime }}=0.2]$

$[s_{rms}^{\prime }=0]$

$M_{T}$

$R_{g}^{-1}\,s_{rms}^{\prime }$

$c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$

$[c_{s^{\prime }T^{\prime }}=0.2]$

$Q_{2}$

$[c_{s^{\prime }T^{\prime }}=0.2]$

$[c_{s^{\prime }p^{\prime }}=0]$

$Q_{1}$

$M_{T}$

$M_{T}>1$

$M_{T}$

$M_{T}<1$

We defer for the moment the quest of a $(Re_{\unicode[STIX]{x1D706}},M_{T})$ -correlation of the data and concentrate instead on thermodynamic fluctuations. The data on the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$ -plane, along the $[c_{s^{\prime }T^{\prime }}=0.2]$ -line (figures 14, 15) show that $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }}$ varies little with $M_{T}$ , contrary to $R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$ (figure 13). Therefore, $R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$ is an appropriate structure parameter to be used as independent variable in describing thermodynamic turbulence (1.2), along with the constraint that $c_{s^{\prime }T^{\prime }}=\text{const.}$ ( $\approxeq 0.2$ for $\unicode[STIX]{x1D6FE}=1.4$ ), which is tantamount to mapping the elements of the set TTS (1.2) on the $(R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime },c_{s^{\prime }T^{\prime }})$ -plane. Straightforward calculations yield the following model

(7.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \breve{\unicode[STIX]{x1D6FE}}=1.4\;\Longrightarrow \;c_{s^{\prime }T^{\prime }}\approxeq 0.2, & \displaystyle\end{eqnarray}$$

(7.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{CV}_{T^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\stackrel{\text{(4.6b)},\,\text{(4.5b)}}{\approxeq }\left[\!(\breve{\unicode[STIX]{x1D6FE}}\!-\!1)\!\left(\!c_{s^{\prime }T^{\prime }}{\displaystyle \frac{s_{rms}^{\prime }}{R_{g}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}+\sqrt{1-(1\!-\!c_{s^{\prime }T^{\prime }}^{2})\left(\!{\displaystyle \frac{s_{rms}^{\prime }}{R_{g}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\!\right)^{2}}\right)\!\right]_{\text{(7.1a)}}, & \displaystyle\end{eqnarray}$$

(7.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}\stackrel{\text{(4.6b)}}{\approxeq }\left[\!\sqrt{1-(1-c_{s^{\prime }T^{\prime }}^{2})\left({\displaystyle \frac{s_{rms}^{\prime }}{R_{g}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\right)^{2}}\right]_{\text{(7.1a)}}. & \displaystyle\end{eqnarray}$$

$(R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime },c_{s^{\prime }T^{\prime }})$

$(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$

$\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{p^{\prime }}$

$O(\text{C}V_{\unicode[STIX]{x1D70C}^{\prime }})$

(7.2a ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\approxeq \text{(4.4)}\big|_{\text{(7.1)}}, & & \displaystyle\end{eqnarray}$$

(7.2b ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\approxeq \text{(4.3a)}\big|_{\text{(7.1)},\,\text{(7.2a)}}, & & \displaystyle\end{eqnarray}$$

(7.2c ) $$\begin{eqnarray}\displaystyle c_{p^{\prime }T^{\prime }}\approxeq \text{(4.3b)}\big|_{\text{(7.1)},\,\text{(7.2a)}}, & & \displaystyle\end{eqnarray}$$

(7.2d ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\approxeq \text{(4.7a)}\big|_{\text{(7.1)},\,\text{(7.2a)}}, & & \displaystyle\end{eqnarray}$$

(7.2e ) $$\begin{eqnarray}\displaystyle c_{s^{\prime }p^{\prime }}\approxeq \text{(4.7c)}\big|_{\text{(7.1)},\,\text{(7.2a)}}. & & \displaystyle\end{eqnarray}$$

$R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$

$[c_{s^{\prime }T^{\prime }}=0.2]$

$Re_{\unicode[STIX]{x1D706}}$

$M_{T}$

$c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$

$\breve{\unicode[STIX]{x1D6FE}}$

$c_{s^{\prime }T^{\prime }}(\breve{\unicode[STIX]{x1D6FE}})$

$\unicode[STIX]{x1D6FE}$

$c_{s^{\prime }T^{\prime }}(\breve{\unicode[STIX]{x1D6FE}})$

$R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$

$(Re_{\unicode[STIX]{x1D706}},M_{T})$

Figure 16.

Predictions of the $[c_{s^{\prime }T^{\prime }}=0.2]$ -model (7.1), (7.2) representing thermodynamic turbulence structure (1.2) of sustained solenoidally forced subsonic aerodynamic ( $\unicode[STIX]{x1D6FE}=1.4$ ) HIT as a unique function of the structure parameter $R_{g}^{-1}\,\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,s_{rms}^{\prime }$ , compared with leading-order estimates (4.5a ), (4.3), (4.7) with input DNS data for $\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$ (Donzis & Jagannathan Reference Donzis and Jagannathan2013; D. A. Donzis, 2016, Private communication; Jagannathan & Donzis Reference Jagannathan and Donzis2016,  $0.1\lessapprox M_{T}\lessapprox 0.6$ , $35\lessapprox Re_{\unicode[STIX]{x1D706}}\lessapprox 430$ ).

Figure 17.

Comparison of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ) (4.7c ) of $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ , of $c_{p^{\prime }T^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ and of $c_{s^{\prime }p^{\prime }}(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }})$ , with the $[c_{s^{\prime }p^{\prime }}=0]$ -approximations (7.3) and with DNS data for supersonic compressible fully developed turbulent plane channel (TPC) flow (Gerolymos & Vallet Reference Gerolymos and Vallet2014,   $1.50\lessapprox \bar{M}_{CL}\lessapprox 2.48$ , $98\lessapprox Re_{\unicode[STIX]{x1D70F}^{\star }}\lessapprox 341$ ), plotted against the HCB-scaled wall distance $y^{\star }$ (3.2b,c ).

7.2 The $[c_{s^{\prime }p^{\prime }}\approxeq 0]$ -approximation in compressible wall turbulence

Comparison (figures 6, 7, 11) with compressible channel DNS data (Gerolymos & Vallet Reference Gerolymos and Vallet2014) indicates that the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ), (4.7c ) for the correlation coefficients involving $p^{\prime }$ , $\{c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$ , become, progressively with increasing $\bar{M}_{CL}\gtrapprox 2$ , inaccurate in the region $10\lessapprox y^{\star }\lessapprox 40$ . Furthermore, it was shown that higher $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ expansions for the correlation coefficients $c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}$ (4.2a ) and $c_{p^{\prime }T^{\prime }}$ (4.2b ), which require DNS data for the 3CCs $c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }T^{\prime }}$ and $c_{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }T^{\prime }}$ , are in excellent agreement with DNS data (figures 6, 7), identifying the $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{2})$ terms to be responsible for the observed discrepancies. Therefore, if we need to approximate $\{c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$ with inputs only $\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$ , improvement of the $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ), (4.7c ) in the region of discrepancy ( $10\lessapprox y^{\star }\lessapprox 40$ ) and at high $\bar{M}_{CL}$ , is desirable. The mapping on the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$ -plane (figure 14) revealed (§ 6) that precisely in that region DNS data follow the $[c_{s^{\prime }p^{\prime }}=0]$ -line, and this is confirmed by the plots (figures 6, 7, (1.2)) of $c_{s^{\prime }p^{\prime }}$ versus  $y^{\star }$ $\forall \,(Re_{\unicode[STIX]{x1D70F}^{\star }},\bar{M}_{CL})$ . Further away from the wall, the DNS data show that $c_{s^{\prime }p^{\prime }}$ slightly increases and then returns to $0$ near the centreline (figures 6, 7, (1.2), 14). It seemed therefore worthwhile to check whether a $[c_{s^{\prime }p^{\prime }}=0]$ -approximation (with unique inputs $\{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }},\text{CV}_{T^{\prime }},\text{CV}_{p^{\prime }}\}$ ) can reduce the aforementioned discrepancies. Simple leading-order calculations yield

(7.3) $$\begin{eqnarray}\displaystyle c_{s^{\prime }p^{\prime }}=0\stackrel{\text{(A2a)},\,\text{(4.2a)}}{\;\Longrightarrow \;}\left\{\begin{array}{@{}l@{}}c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\approxeq {\displaystyle \frac{1}{\breve{\unicode[STIX]{x1D6FE}}}}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}}}\\[12.0pt] c_{p^{\prime }T^{\prime }}\approxeq {\displaystyle \frac{\breve{\unicode[STIX]{x1D6FE}}-1}{\breve{\unicode[STIX]{x1D6FE}}}}{\displaystyle \frac{\text{CV}_{p^{\prime }}}{\text{CV}_{T^{\prime }}}}\\[6.0pt] c_{s^{\prime }p^{\prime }}\approxeq 0.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Comparison of the $[c_{s^{\prime }p^{\prime }}=0]$ -approximation (7.3) with the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ), (4.7c ) and DNS data for supersonic channel flows (figure 17), show that (7.3) is a good approximation $\forall \,y^{\star }\gtrapprox 7$ , particularly in the region $10\lessapprox y^{\star }\lessapprox 40$ where the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations are not satisfactory when $\bar{M}_{CL}\gtrapprox 2$ . We may therefore construct a composite $C^{0}$ -continuous approximation using for supersonic $\bar{M}_{CL}>1$ , the $[c_{s^{\prime }p^{\prime }}=0]$ -approximation (7.3) in the interval between the two $0$ -crossing points around the global negative minimum of the leading $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximation of $c_{s^{\prime }p^{\prime }}$ (figure 17), and the $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations (4.3a ), (4.3b ), (4.7c ) elsewhere. For subsonic $\bar{M}_{CL}<1$ flows the $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ approximations are excellent and require no modification.