Higher-Order Correlations Between Thermodynamic Fluctuations in Compressible Aerodynamic Turbulence

Abstract

This paper studies the exact and approximate relations between fluctuations in thermodynamic variables (pressure, density and temperature) that are imposed by the dilute-gas ($\mathcal{Z}=1$) equation-of-state ($\mathrm{EoS}$), which is a satisfactory approximation of air thermodynamics for a wide range of pressures and temperatures. It focuses on triple- and higher-order correlations, extending previous studies that concentrated on second-order moments, with emphasis on the mathematical relations, which are generally valid independently of the particular flow configuration. Exact equations are developed both involving only single-variable moments and relating the correlations between variables. These contain nonlinear terms generated by the density-temperature fluctuation product in the fluctuating $\mathrm{EoS}$. The importance of the nonlinear terms in the 6 exact equations between the 10 third-order moments is assessed using $\mathrm{DNS}$ (direct numerical simulation) data for compressible turbulent plane channel ($\mathrm{TPC}$) flows and analyzed using general statistical inequalities involving third-order and fourth-order moments. The corresponding linearized system between third-order moments is studied to determine approximate relations and 4-tuples of linearly independent moments. These mathematical tools are then used to analyze $\mathrm{TPC}$ flow $\mathrm{DNS}$ data on the triple correlations between the thermodynamic variables.

1. Introduction

The analysis of turbulence-induced fluctuations in thermodynamic variables is essential for advancing our understanding of the complex interactions occuring in compressible (high-speed) aerodynamic flows. Direct numerical simulation ($\mathrm{DNS}$) can provide detailed information on the variances of and correlations between thermodynamic fluctuations [1,2,3,4]. Such data can be found in several studies [1,2,3,4,5,6,7,8,9,10,11]. Systematic tabulation of the profiles of all correlations up to third-order and of all single-variable central moments up to sixth-order for compressible turbulent plane channel ($\mathrm{TPC}$) flows was recently released in [12]. The availability of detailed $\mathrm{DNS}$ data for the correlations between thermodynamic fluctuations renews the interest in developing the exact theoretical relations needed to analyze these results.

Generally, $\mathrm{DNS}$ calculations [1,2,3,4,5,6,7,8,9,10,11] adopt the dilute-gas equation-of-state ($\mathrm{EoS}$) ([13], pp. 1–38)

$$\begin{array}{c}\hfill \mathcal{Z}(\rho ,T):={\displaystyle \frac{p}{\rho R_{g}T}}=1\end{array}$$

(1)

where p is the pressure, $\rho$ is the density, T is the temperature, $R_g$ is the gas constant and $\mathcal{Z}$ is the compressibility factor. $\mathrm{EoS}$ (1) implies relations between mean values $\{\overline{p},\overline{\rho },\overline{T}\}$ (6a) and instantaneous fluctuations $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ (7), where $(·)=\overline{(·)}+{(·)}^{\prime }=\widetilde{(·)}+{(·)}^{\prime \prime }$ denote Reynolds averages $\overline{(·)}$, Reynolds fluctuations ${(·)}^{\prime }$, Favre averages $\widetilde{(·)}$ and Favre fluctuations ${(·)}^{\prime \prime }$. The present work assumes that the dilute-gas $\mathrm{EoS}$ (1) is a valid approximation for the flows studied. Equally important, this $\mathrm{EoS}$ is compatible with the $\mathrm{DNS}$ calculations providing the data. The analytical relations between the variances of and correlations between $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ that are developed in the paper are not affected by the eventual temperature dependence of the specific heat capacity $c_p$, recalling ([13], pp. 1–38) that under the assumption of local thermodynamic equilibrium, $\mathcal{Z}=1\Rightarrow c_p={\mathsf{c}\mathsf{p}}_{\mathrm{EoS}}\left(T\right)$. On the other hand, analytical relations for entropy fluctuations $s^{\prime }$ can be obtained [14] as infinite power series of $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ and include the temperature derivatives of $c_p\left(T\right)$ at the mean temperature $\overline{T}$. Notice, however, that these derivatives influence only the higher-order nonlinear terms [4,14].

Identities involving the correlations between the thermodynamic variables can be obtained starting directly from the fluctuating $\mathrm{EoS}$ (7). This approach was followed in [4,14] for second-order (2-order) correlations between $\{p^{\prime },{\rho }^{\prime },T^{\prime },s^{\prime }\}$. In compressible aerodynamic flows, because of the relations implied by the fluctuating $\mathrm{EoS}$ (7), the relative fluctuation intensities $\{p_{\mathrm{rms}}^{\prime }/\overline{p},{\rho }_{\mathrm{rms}}^{\prime }/\overline{\rho },T_{\mathrm{rms}}^{\prime }/\overline{T}\}$ are of the same order-of-magnitude [2,4], independently of the Mach number of the flow. The nondimensional level of entropy fluctuations $s_{\mathrm{rms}}^{\prime }/R_g$ is also of the same order-of-magnitude [4,14] as the coefficients of variation $\{p_{\mathrm{rms}}^{\prime }/\overline{p},{\rho }_{\mathrm{rms}}^{\prime }/\overline{\rho },T_{\mathrm{rms}}^{\prime }/\overline{T}\}$ for the basic thermodynamic variables. Exact relations between 2-order correlations of $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ are readily obtained ([4], (2.5), p. 452) from (7), and, as usual in the statistical analysis of turbulence [15,16], also contain higher, third-order (3-order) correlations, which are generated by the product ${\rho }^{\prime }{T}^{\prime }$ appearing in the fluctuating $\mathrm{EoS}$ (7). They are therefore nonlinear in fluctuation amplitudes. The term weakly compressible turbulence will be used to characterize flow conditions for which the linearization of such expressions, by dropping terms of higher-than-leading-order in the relative fluctuation amplitudes $\{p_{\mathrm{rms}}^{\prime }/\overline{p},{\rho }_{\mathrm{rms}}^{\prime }/\overline{\rho },T_{\mathrm{rms}}^{\prime }/\overline{T},s_{\mathrm{rms}}^{\prime }/R_g\}$, provides an accurate approximation of the exact relations. For compressible turbulent plane channel ($\mathrm{TPC}$) flows at centerline streamwise Mach number ${\overline{M}}_{{\mathrm{CL}}_x}\le 1.5$, the weakly compressible turbulence linearized approximation for $\mathrm{2CCs}$ (correlation coefficients involving 2-order moments) is very accurate [4] and remains reasonably accurate up to ${\overline{M}}_{{\mathrm{CL}}_x}\le 2$. At a higher ${\overline{M}}_{{\mathrm{CL}}_x}$, some approximations remain robust (very accurate), whereas others are more fragile in the near-wall region (buffer region $5⪅y^{★}⪅40$ where $y^{★}$ is the $\mathrm{HCB}$-scaled nondimensional distance from the wall [17]) where the peaks of $\{{\rho }_{\mathrm{rms}}^{\prime }/\overline{\rho },T_{\mathrm{rms}}^{\prime }/\overline{T}\}$ are located [2,4,11,14]. It was shown in [4] that using higher-order approximations by including the leading term of the linearization error restores the accuracy up to the highest available in the $\mathrm{DNS}$ database [12] ${\overline{M}}_{{\mathrm{CL}}_x}\approxeq 2.5$. It is not obvious to simply define the limit of validity of the linearized approximation, as this depends on the specific correlation studied and also depends on wall-distance, suggesting that a local criterion is necessary. Regarding entropy, the leading terms of the infinite power series expansions of $s^{\prime }/R_g$ [14] were used in [4] to obtain relations for $\{\overline{s^{\prime 2}},\overline{s^{\prime }{p}^{\prime }},\overline{s^{\prime }{\rho }^{\prime }},\overline{s^{\prime }{T}^{\prime }}\}$, whose accuracy with an increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ follows the same behavior as that of the $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ relations. These analyses led to mapping the thermodynamic turbulence structure onto a plane [4]. Notice that approximations valid in specific flows and flow regions can substantially extend these limits [4,18] but are not generally valid, especially very near the wall.

Although second-order moments quantify the average intensity of the fluctuations and provide an initial understanding of the level of correlation between variables [1,4,18], higher-order moments are necessary to provide information on the skewness and flatness (kurtosis) of the probability density function (pdf) and on rare (extreme) events [19]. The exact relations between second-order correlation coefficients ($\mathrm{2CCs}$) [4] involves third-order correlation coefficients ($\mathrm{3CCs}$) in the nonlinear terms that control the fluctuation amplitude effects on the thermodynamic turbulence structure. Furthermore, the difference between Favre-averaged $\widetilde{{(·)}^{\prime \prime 2}}$ and Reynolds-averaged $\overline{{(·)}^{\prime 2}}$ amplitudes, for any flow quantity $(·)$, is controlled by the 3-order correlation coefficient $c_{{\rho }^{\prime }{(·)}^{\prime }{(·)}^{\prime }}$, and fourth-order statistics provide bounds on $\mathrm{3CCs}$ [20,21,22].

The purpose of the paper is to provide mathematical tools that will help to better understand how the dilute-gas $\mathrm{EoS}$ (1) couples the moments and correlations of thermodynamic fluctuations, both for use in the analysis of experimental and $\mathrm{DNS}$ data and for the development of models of the thermodynamic turbulence structure. Of course, thermodynamic fluctuations are the result of the flow, and their interaction with the velocity field can be described in a statistical sense using specific transport equations [2], but the study of these interactions is outside of the scope of this paper. The focus is on the exact mathematical relations between the moments and correlations of thermodynamic fluctuations that are implied by the dilute-gas $\mathrm{EoS}$ (1) and on their linearized approximations, with emphasis on triple correlations. Analysis of $\mathrm{DNS}$ data for specific flows serves to illustrate these relations and to quantify the increase in fluctuation-amplitude-dependent nonlinear effects with an increasing Mach number.

In Section 2, we explain why Reynolds decomposition is preferred in the study of thermodynamic correlations (Section 2.2) and discuss possible approaches to the study of the constraints implied by the $\mathrm{EoS}$ (1) on the thermodynamic turbulence structure. In Section 3, we briefly provide a summary of $\mathrm{DNS}$ data for compressible $\mathrm{TPC}$, which are used to illustrate the mathematical relations developed in the paper. In Section 4, we develop exact relations between arbitrary n-order correlation coefficients (n$\mathrm{CCs}$; Section 4.1) and $p^{\prime }$-moments (Section 4.2) and study the linearization error (Section 4.4) and the system of linearized equations between $\mathrm{3CCs}$ (Section 4.5). In Section 5, we develop identities and inequalities involving $\mathrm{3CCs}$ and $\mathrm{4CCs}$ between arbitrary random variables which are applied in Section 6, in conjunction with the equations of Section 4, to $(p^{\prime },{\rho }^{\prime },T^{\prime })$-correlations under the $\mathrm{EoS}$ (1). Comprehensive $\mathrm{DNS}$ data for the $\mathrm{3CCs}$ of thermodynamic fluctuations for compressible $\mathrm{TPC}$ flows are presented in Appendix B. A summary of the findings of this paper is provided in Section 7.

2. Fluctuating Equation-of-State

We introduce nondimensional coefficients of variation and correlation coefficients (Section 2.1). Favre fluctuations are not practical for the analysis of higher-order correlations between thermodynamic variables (Section 2.2). Therefore, we use Reynolds fluctuations to express the fluctuating $\mathrm{EoS}$ (Section 2.3). Conceptually, the bivariate joint $(\rho ,T)$-pdf contains, under the $\mathrm{EoS}$ (1), all necessary information regarding thermodynamic fluctuations (Section 2.4).

2.1. Definitions

In line with previous work [4,14], we will describe the thermodynamic turbulence structure using coefficients of variation $\mathrm{CVs}$ ([23], (48.1), p. 906) and correlation coefficients (2-order $\mathrm{2CC}$s or higher n-order n$\mathrm{CCs}$):

$$\begin{array}{cccccccc}\hfill {\mathrm{C}\mathrm{V}}_{{(·)}^{\prime }}& :=\hfill & & {\displaystyle \frac{{(·)}_{\mathrm{rms}}^{\prime }}{\overline{(·)}}}\hfill & \hfill ;& {(·)}_{\mathrm{rms}}^{\prime }\hfill & \hfill :=& \sqrt{\overline{{(·)}^{\prime 2}}}\hfill \end{array}$$

(2a)

$$\begin{array}{cccccccc}\hfill c_{{(·)}^{\prime }{[·]}^{\prime }}& :=\hfill & & {\displaystyle \frac{\overline{{(·)}^{\prime }{[·]}^{\prime }}}{{(·)}_{\mathrm{rms}}^{\prime }{[·]}_{\mathrm{rms}}^{\prime }}}\in [-1,1]\hfill & \hfill ;& c_{{(·)}^{\prime }\cdots {[·]}^{\prime }}\hfill & \hfill :=& {\displaystyle \frac{\overline{{(·)}^{\prime }\cdots {[·]}^{\prime }}}{{(·)}_{\mathrm{rms}}^{\prime }\cdots {[·]}_{\mathrm{rms}}^{\prime }}}\hfill \end{array}$$

(2b)

where $\mathrm{rms}$ denotes the root-mean-square (2b), and $\mathrm{2CCs}$ are bounded (2b) by the covariance (Schwarz) inequality ([24], (4.11), p. 71), $|c_{{(·)}^{\prime }{[·]}^{\prime }}|\le 1$. Using the terminology of Frenkiel and Klebanoff [19], skewness $S_{{(·)}^{\prime }}:={}^3{S}_{{(·)}^{\prime }}$, flatness (kurtosis) $F_{{(·)}^{\prime }}:={}^4{F}_{{(·)}^{\prime }}$, superskewness ${}^5{S}_{{(·)}^{\prime }}$ and superflatness ${}^6{F}_{{(·)}^{\prime }}$ are identified to $\mathrm{CC}$s of the corresponding order:

$$\begin{array}{cccccc}\hfill S_{{(·)}^{\prime }}:={}^3{S}_{{(·)}^{\prime }}& :={\displaystyle \frac{\overline{{(·)}^{\prime 3}}}{{(·)}_{\mathrm{rms}}^{\prime 3}}}:=c_{{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }}\hfill & \hfill ;& & \hfill F_{{(·)}^{\prime }}:={}^4{F}_{{(·)}^{\prime }}& :={\displaystyle \frac{\overline{{(·)}^{\prime 4}}}{{(·)}_{\mathrm{rms}}^{\prime 4}}}:=c_{{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }}\hfill \end{array}$$

(2c)

$$\begin{array}{cccccc}\hfill {}^5{S}_{{(·)}^{\prime }}& :={\displaystyle \frac{\overline{{(·)}^{\prime 5}}}{{(·)}_{\mathrm{rms}}^{\prime 5}}}:=c_{{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }}\hfill & \hfill ;& & \hfill {}^6{F}_{{(·)}^{\prime }}& :={\displaystyle \frac{\overline{{(·)}^{\prime 6}}}{{(·)}_{\mathrm{rms}}^{\prime 6}}}:=c_{{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }{(·)}^{\prime }}\hfill \end{array}$$

(2d)

Frenkiel and Klebanoff [19] also considered hyperskewness ${}^7{S}_{{(·)}^{\prime }}$ and hyperflatness ${}^8{F}_{{(·)}^{\prime }}$, but these are not used in this paper.

The turbulence intensity of the thermodynamic fluctuations is quantified by the coefficients of variation $\{{\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$. Often, equations contain products of powers of the $\mathrm{CVs}$, and we will define as n-order terms ($n\mathrm{oTs}$)

$$\begin{array}{c}\hfill n\mathrm{oTs}\sim O\left({\mathrm{C}\mathrm{V}}_{p^{\prime }}^i{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^j{\mathrm{C}\mathrm{V}}_{T^{\prime }}^k\right);i,j,k\in {\mathbb{Z}}_{\ge 0}:i+j+k=n\ge 1\end{array}$$

(3)

2.2. Favre- and Reynolds-Averaging

Entropy s, enthalpy h or velocity components $u_i$ are Favre-decomposed in transport equations for the mean flow or moments [2]. For any Favre-decomposed flow quantity $(·)$,

$$\begin{array}{cccc}& \widetilde{(·)}\hfill & \hfill :=& {\displaystyle \frac{\overline{\rho (·)}}{\overline{\rho }}}=\overline{(·)}+{\displaystyle \frac{\overline{{\rho }^{\prime }{(·)}^{\prime }}}{\overline{\rho }}}\stackrel{\left(2\right)}{=}\overline{(·)}\left(1+c_{{\rho }^{\prime }{(·)}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{(·)}^{\prime }}\right)\hfill \end{array}$$

(4a)

$$\begin{array}{cccc}& {(·)}^{\prime \prime }\hfill & \hfill :=& (·)-\widetilde{(·)}={(·)}^{\prime }+\left(\overline{(·)}-\widetilde{(·)}\right)={(·)}^{\prime }+\overline{{(·)}^{\prime \prime }}\hfill \end{array}$$

(4b)

$$\begin{array}{cccc}& {\displaystyle \frac{\overline{{(·)}^{\prime \prime }}}{\overline{(·)}}}\hfill & \hfill =& -{\displaystyle \frac{\overline{{\rho }^{\prime }{(·)}^{\prime }}}{\overline{\rho }\overline{(·)}}}=-{\displaystyle \frac{\overline{{\rho }^{\prime }{(·)}^{\prime }}}{\sqrt{\overline{{\rho }^{\prime 2}}}\sqrt{\overline{{(·)}^{\prime 2}}}}}{\displaystyle \frac{\sqrt{\overline{{\rho }^{\prime 2}}}}{\overline{\rho }}}{\displaystyle \frac{\sqrt{\overline{{(·)}^{\prime 2}}}}{\overline{(·)}}}\stackrel{\left(2\right)}{=}-c_{{\rho }^{\prime }{(·)}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{(·)}^{\prime }}\hfill \end{array}$$

(4c)

$$\begin{array}{cccc}& \overline{{(·)}^{\prime }{[·]}^{\prime \prime }}\hfill & \hfill =& \overline{{(·)}^{\prime \prime }{[·]}^{\prime }}=\overline{{(·)}^{\prime }{[·]}^{\prime }}\hfill \end{array}$$

(4d)

including the important identity (4d) for mixed 2-moments [25,26]. Notice that although using Favre decomposition for the temperature $T=\tilde{T}+T^{\prime \prime }$ greatly simplifies the expression of the fluctuating $\mathrm{EoS}$, replacing (7) with (A1b), it does not lead to a simplification of the relations between the moments of thermodynamic fluctuations, as explained using the counterexample in Appendix A. Furthermore, the single-variable statistics and pdfs that correspond to Reynolds fluctuations offer considerable insight into the thermodynamic turbulence structure [1,4,14]. Therefore, Reynolds fluctuations $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ are invariably considered in this paper.

Notice the following useful relation (A4c) obtained during these calculations

$$\begin{array}{c}\hfill \widetilde{{(·)}^{\prime \prime 2}}\stackrel{\left(\mathrm{A}4\mathrm{c}\right)\mathrm{and} \left(2\mathrm{a}\right)}{=}\overline{{(·)}^{\prime 2}}\left(1+c_{{\rho }^{\prime }{(·)}^{\prime }{(·)}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}-c_{{\rho }^{\prime }{(·)}^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2\right)\end{array}$$

(5)

showing that for any flow variable $(·)$, the departure of the ratio of Favre-to-Reynolds fluctuation amplitudes $\widetilde{{(·)}^{\prime \prime 2}}/\overline{{(·)}^{\prime 2}}$ involves the $\mathrm{3CC}$ $c_{{\rho }^{\prime }{(·)}^{\prime }{(·)}^{\prime }}$.

2.3. Dilute-Gas Equation-of-State

Straightforward manipulation of the $\mathrm{EoS}$ (1), using the basic properties of Favre averaging (4), yields

$$\begin{array}{cc}\hfill \mathcal{Z}(\rho ,T)=1\stackrel{\left(1\right)}{\Rightarrow }& \overline{p}\stackrel{\left(4\mathrm{a}\right)}{=}\overline{\rho }{R}_g\tilde{T}\stackrel{\left(4\mathrm{a}\right)}{=}\overline{\rho }{R}_g\overline{T}+R_g\overline{{\rho }^{\prime }{T}^{\prime }}\hfill \end{array}$$

(6a)

Using the general relations

$$\begin{array}{cc}& \tilde{T}\stackrel{\left(4\mathrm{b}\right)}{=}\overline{T}-\overline{T^{\prime \prime }}\stackrel{\left(4\mathrm{c}\right)}{=}\overline{T}\left(1+{\displaystyle \frac{\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\right)\stackrel{\left(2\right)}{=}\overline{T}\left(1+c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\hfill \end{array}$$

(6b)

$$\begin{array}{cc}& \overline{T^{\prime \prime }}\stackrel{\left(4\mathrm{c}\right)}{=}-{\displaystyle \frac{\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }}}\stackrel{\left(2\right)}{=}-c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\overline{T}\hfill \end{array}$$

(6c)

and substracting (6a) from the instantaneous $\mathrm{EoS}$ yields the fluctuating $\mathrm{EoS}$

$$\begin{array}{c}\hfill \left(1\right)\mathrm{and} \left(6\right)\Rightarrow {\displaystyle \frac{\tilde{T}}{\overline{T}}}{\displaystyle \frac{p^{\prime }}{\overline{p}}}={\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}+{\displaystyle \frac{T^{\prime }}{\overline{T}}}+{\displaystyle \frac{{\rho }^{\prime }{T}^{\prime }-\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\end{array}$$

(7)

Because of the nonlinearity contained in the product $\rho T$ in the dilute-gas $\mathrm{EoS}$ (1), the fluctuating $\mathrm{EoS}$ written in terms of the fluctuations divided by the mean values $\{p^{\prime }/\overline{p},{\rho }^{\prime }/\overline{\rho },T^{\prime }/\overline{T}\}$ contains not only the nonlinear term ${\rho }^{\prime }{T}^{\prime }$ but also the ratio $\tilde{T}/\overline{T}$ (6b). Both these nonlinear terms are related to the correlation coefficient $c_{{\rho }^{\prime }{T}^{\prime }}$, which, weighted by the product of the coefficients of variation ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}$, quantifies the relative difference between Favre-averaged $\tilde{T}$ and Reynolds-averaged $\overline{T}$ (6b), induced by the nonlinear (strong) compressible turbulence effects.

Using $\mathrm{CVs}$ (2a) and $\mathrm{CCs}$ (2b), the fluctuating dilute-gas $\mathrm{EoS}$ (7) can be rewritten as

$$\begin{array}{c}\hfill \left(7\right)\stackrel{\left(2\mathrm{a}\right),\left(2\mathrm{b}\right),\left(4\mathrm{a}\right)}{\iff }\left(1+c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right){\displaystyle \frac{p^{\prime }}{\overline{p}}}={\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}+{\displaystyle \frac{T^{\prime }}{\overline{T}}}+{\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}{\displaystyle \frac{T^{\prime }}{\overline{T}}}-c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(8)

Notice that the term $\tilde{T}/\overline{T}\stackrel{\left(4\mathrm{a}\right)}{=}(1+c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }})$ appears on the $\mathrm{LHS}$ of the fluctuating $\mathrm{EoS}$ (7) and (8) and will therefore appear in all exact relations between the moments and correlation coefficients. To simplify the notation, we will define the auxiliary weighted coefficient

$$\begin{array}{c}\hfill \mathrm{C}||{\mathrm{V}}_{p^{\prime }}:=(1+c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}){\mathrm{C}\mathrm{V}}_{p^{\prime }}\stackrel{\left(6\mathrm{b}\right)}{=}{\displaystyle \frac{\tilde{T}}{\overline{T}}}{\mathrm{C}\mathrm{V}}_{p^{\prime }}>0\Rightarrow \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\sim {\mathrm{C}\mathrm{V}}_{p^{\prime }}\left(1+O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\right)\end{array}$$

(9)

which includes the compressible ratio $\tilde{T}/\overline{T}$ (4a) and whose difference from ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$ is of the second order in fluctuation amplitudes (9).

The fluctuating $\mathrm{EoS}$ (7) is the generating equation for the relations developed in this paper. Additionally, we report in Appendix C the identities obtained from an infinite series expansion of the instantaneous $\mathrm{EoS}$ (1), which also illustrate the increasing importance of high-order moments with an increasing fluctuation intensity.

2.4. A Conceptual Joint $(\rho ,T)$-pdf Approach

Although we will not use an approach based on joint probability density functions (pdfs) in this paper, it seems worthwhile to discuss, conceptually, why the knowledge of joint statistics of only two thermodynamic variables, e.g., density and temperature $(\rho ,T)$, suffices to calculate exactly the statistics and correlations of all of the other thermodynamic variables. A bivariate $\mathrm{EoS}$ [27] describes the thermodynamic behavior of (dry) air, for a wide range of temperatures ($60\mathrm{K}\le T\le 2000\mathrm{K}$) and pressures ($p\le 2000\mathrm{MPa}$), quite accurately. Because of the equation-of-state ($\mathrm{EoS}$), there are only two independent thermodynamic variables, e.g., density and temperature $(\rho ,T)$. Therefore, the knowledge, at a point $\overrightarrow{x}$ in the flow, of the joint probability density function (pdf) $f_{\rho T}$ of the instantaneous density $\rho$ and temperature T suffices to determine any correlation coefficient between thermodynamic variables at this point. Regarding, e.g., pressure p, the knowledge of an $\mathrm{EoS}$ relation $T={\mathsf{T}}_{\mathrm{EoS}}(\rho ,p)$ suffices, formally, to calculate ([28], (10), p. 26) the joint pdf $f_{\rho p}(\rho ,p)$ or the single-variable pdf $f_p\left(p\right)$ from the joint pdf $f_{\rho T}(\rho ,T)$. Adopting the usual thermodynamic formalism ([29], pp. 97–102) for the transformation-of-variables Jacobian, the joint $(\rho ,p)$-pdf reads as

$$\begin{array}{cc}\hfill f_{\rho p}(\rho ,p)=& |{\displaystyle \frac{\partial (\rho ,T)}{\partial (\rho ,p)}}|f_{\rho T}\left(\rho ,{\mathsf{T}}_{\mathrm{EoS}}(\rho ,p)\right)=\left|{\left({\displaystyle \frac{\partial T}{\partial p}}\right)}_{\rho }\right|f_{\rho T}\left(\rho ,{\mathsf{T}}_{\mathrm{EoS}}(\rho ,p)\right)\hfill \end{array}$$

(10a)

where the partial derivative is of course calculated at state $(p,\rho ,T)=(p,\rho ,{\mathsf{T}}_{\mathrm{EoS}}(\rho ,p))$, and the single-variable pdf $f_p\left(p\right)$ is readily obtained through integration

$$\begin{array}{cc}\hfill f_p\left(p\right)=& {\int }_0^{+\infty }{f}_{\rho p}(\rho ,p)d\rho \hfill \end{array}$$

(10b)

with the positivity of density $\rho$ setting the lower bound of the integral. For the dilute-gas ($\mathcal{Z}=1$) $\mathrm{EoS}$ (1), the Jacobian in (10a) simplifies to

$$\begin{array}{c}\hfill \left(1\right)\stackrel{\left(10\right)}{\Rightarrow }{f}_{\rho p}(\rho ,p)={\displaystyle \frac{1}{\rho R_g}}{f}_{\rho T}\left(\rho ,{\displaystyle \frac{p}{\rho R_g}}\right)\end{array}$$

(11)

However, relations (11) would require very careful numerical implementation, (a) sampling $f_{\rho T}(\rho ,T)$ on a sufficiently large domain of $({\rho }^{\prime },T^{\prime })$-events ensuring that extreme $p^{\prime }$-events are included and (b) the use of a 2-D reconstruction procedure [30] to transform the bin-sampled pdf into a continuous function (this is necessary in order to create a homogeneous set of p-bins from a homogeneous grid of 2-D $(\rho ,T)$-bins). However, this general relation does not provide insight into the basic $p^{\prime }$-moments ($\overline{p^{\prime 2}}$, $\overline{p^{\prime 3}}$, $\overline{p^{\prime 4}}$, ⋯) or coefficients of correlation with the other thermodynamic variables ($c_{p^{\prime }{\rho }^{\prime }}$, $c_{p^{\prime }{T}^{\prime }}$, $c_{s^{\prime }{p}^{\prime }}$, ⋯). Furthermore, the available joint-pdfs in the compressible turbulent flow database [12] which could be used to illustrate the relations developed in this paper were not sampled on a sufficiently large domain of $({\rho }^{\prime },T^{\prime })$-events for the above procedure (11) to be applicable with sufficient accuracy. A similar analysis applies for alternative choices of the independent variables, e.g., $(p,T)$ or $(p,\rho )$, although the dilute-gas $\mathrm{EoS}$ (1) clearly advocates in favor of $(\rho ,T)$ as independent variables, and probably so does the general bivariate thermodynamic behavior of air near the dew/bubble boundary [27] as well.

Because of these difficulties, we examine correlations between the thermodynamic variables starting directly from the fluctuating $\mathrm{EoS}$ (7).

3. Thermodynamic Turbulence Structure in Compressible Plane Channel Flow

The identities and approximations presented in the paper are assessed or illustrated using $\mathrm{DNS}$ data [12] for a turbulent plane channel flow [10,11]. Knowledge of the general behavior of the profiles of the high-order correlations between thermodynamic variables that are considered in the paper is necessary in order to understand these examples. The data for representative flows (${Re}_{{\tau }^{★}}\in [113,983]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.81,2.49]$) are briefly reviewed (Figure 1 and Figure 2). ${Re}_{{\tau }^{★}}$ is the $\mathrm{HCB}$ friction Reynolds number [17], ${\overline{M}}_{{\mathrm{CL}}_x}$ is the streamwise Mach number at the channel centerline, and $y^{★}$ is the $\mathrm{HCB}$-scaled nondimensional wall-distance [17]. A systematic examination of the data is reported in Appendix B. One should bear in mind that the compressible $\mathrm{TPC}$ flow corresponds to very-cold-wall ($\mathrm{VCW}$) conditions (relative to the centerline) [11] and that the thermodynamic turbulence structure ($\mathrm{TTS}$) as determined from the $\mathrm{2CCs}$ and $\mathrm{CVs}$ [4] changes with the thermal wall conditions.

Figure 1. Profiles of coefficients of variation $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\left(\mathbf{a}\right),{\mathrm{C}\mathrm{V}}_{p^{\prime }}\left(\mathbf{b}\right),{\mathrm{C}\mathrm{V}}_{T^{\prime }}\left(\mathbf{c}\right)\}$ (scaled by ${\overline{M}}_{{\mathrm{CL}}_x}^2$ [2]), of skewness $\{S_{{\rho }^{\prime }}\left(\mathbf{d}\right),S_{p^{\prime }}\left(\mathbf{e}\right),S_{T^{\prime }}\left(\mathbf{f}\right)\}$ and of kurtosis $\{F_{{\rho }^{\prime }}\left(\mathbf{g}\right),F_{p^{\prime }}\left(\mathbf{h}\right),F_{T^{\prime }}\left(\mathbf{i}\right)\}$ for thermodynamic fluctuations, plotted against the $\mathrm{HCB}$-scaled nondimensional wall-distance $y^{★}$ (logscale), using $\mathrm{DNS}$ data [12] for compressible turbulent plane channel flows in the range ${Re}_{{\tau }^{★}}\in [113,985]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.81,2.49]$.

Figure 2. Profiles of $\mathrm{3CCs}$ $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\left(\mathbf{a}\right),c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}\left(\mathbf{b}\right),c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}\left(\mathbf{c}\right),c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\left(\mathbf{d}\right),c_{p^{\prime }{T}^{\prime }{T}^{\prime }}\left(\mathbf{e}\right),c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\left(\mathbf{f}\right),c_{p^{\prime }{p}^{\prime }{T}^{\prime }}\left(\mathbf{g}\right)\}$ between thermodynamic fluctuations, plotted against the $\mathrm{HCB}$-scaled nondimensional wall-distance $y^{★}$ (logscale), using $\mathrm{DNS}$ data [12] for compressible turbulent plane channel flows in the range ${Re}_{{\tau }^{★}}\in [113,985]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.81,2.49]$.

The profiles of $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ (Figure 1a,c) exhibit a peak close to $y^{★}\approxeq 10$, which extends over most of the buffer region ($5⪅y^{★}⪅40$) (Figure 1a,c). These peaks scale approximately with ${\overline{M}}_{{\mathrm{CL}}_x}^2$ [2], although the exponent is not exacly 2 but probably increases slightly with ${\overline{M}}_{{\mathrm{CL}}_x}$ [14] and also has a weak $Re$ dependence. The presence of near-peak values of $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$, that increase approximately with ${\overline{M}}_{{\mathrm{CL}}_x}^2$, in the entire buffer region ($5⪅y^{★}⪅40$) explains the increasing importance of higher-order terms with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ (Appendix C). The level of ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$ in the buffer zone is lower than that of $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ (Figure 1b). It also scales approximately with ${\overline{M}}_{{\mathrm{CL}}_x}^2$ but additionally has marked $Re$ dependence, in line with the detailed analysis in [10]. The decrease in ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$ towards the center of the channel is slower than that for $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ (Figure 1a–c) so that ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$ becomes larger than $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ in the outer part of the flow ([4], Figure 3, p. 458).

The profiles of skewness $\{S_{{\rho }^{\prime }},S_{p^{\prime }},S_{T^{\prime }}\}$ are generally $\in [-\frac{3}{2},+\frac{3}{2}]$ (Figure 1d–f). $\{S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ vary with both $({Re}_{{\tau }^{★}},{\overline{M}}_{{\mathrm{CL}}_x})$ (Figure 1d,f) and exhibit important changes, both in level and sign, with wall-distance. Except near the center of the channel where they are both negative ($<0$), $\{S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ are generally of opposite signs (Figure 1d,f). $S_{p^{\prime }}$ is close to 0 near the wall ($y^{★}⪅10$) and then drops to a negative value and forms a plateau at ($y^{★}\approxeq 30$) extending up to the wake region (Figure 1e). The $({Re}_{{\tau }^{★}},{\overline{M}}_{{\mathrm{CL}}_x})$-effects on $S_{p^{\prime }}$ are less pronounced than those on $\{S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ (Figure 1d–f). Notice that the drop in $S_{p^{\prime }}$ from near-0 to negative values at $y^{★}\approxeq 10$ (Figure 1e) occurs very near the location of the $({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})$-peaks (Figure 1a,c).

The profiles of kurtosis $\{F_{{\rho }^{\prime }},F_{p^{\prime }},F_{T^{\prime }}\}$ vary with both $({Re}_{{\tau }^{★}},{\overline{M}}_{{\mathrm{CL}}_x})$ (Figure 1g–i). $\{F_{{\rho }^{\prime }},F_{T^{\prime }}\}$ exhibit a region of platykurtic distributions (negative excess kurtosis, $F_{{(·)}^{\prime }}<3$) at approximately $5⪅y^{★}⪅15$ (the boundaries of the $(F_{{\rho }^{\prime }},F_{T^{\prime }})$-platykurtic region vary with ${Re}_{{\tau }^{★}}$), contrary to $F_{p^{\prime }}$ which is invariably leptokurtic (Figure 1h). Knowledge of kurtosis is important, as it provides bounds for the $\mathrm{3CCs}$ (Section 5). Determination of the precise variation in $\{F_{{\rho }^{\prime }},F_{p^{\prime }},F_{T^{\prime }}\}$ with wall-distance and with $({Re}_{{\tau }^{★}},{\overline{M}}_{{\mathrm{CL}}_x})$ requires a more systematic examination of the $({Re}_{{\tau }^{★}},{\overline{M}}_{{\mathrm{CL}}_x})$-matrix of datasets (Figure A3 and Figure A4).

Regarding the profiles of $\mathrm{3CCs}$, $\{c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\}$, which do not contain $p^{\prime }$, are $\in [-\frac{3}{2},+\frac{3}{2}]$ (Figure 2b,d). The remaining $\mathrm{3CCs}$, which involve $p^{\prime }$, $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }}\}$, are smaller $\in [-1,+1]$ (Figure 2a,c,e–g), with the trivariate $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ being even smaller $\in [-\frac{1}{2},+\frac{1}{2}]$ (Figure 2a). In the near-wall and buffer regions ($y^{★}⪅40$), all $\mathrm{3CCs}$ containing $p^{\prime }$, $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }}\}$, are close to 0, dropping progressively to negative values towards the centerline (Figure 2a,c,e–g). Notice, however, the specific behavior of the ${\overline{M}}_{{\mathrm{CL}}_x}=2.49$ data in the near-wall region (Figure 2a,c,e–g), consistent with the observed near-wall compressibility effects induced by organized streamwise pressure waves [10] which increase with an increasing Mach number. Finally, the larger $\mathrm{3CCs}$, $\{c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\}$, have generally opposite signs everywhere except very near the centerline (Figure 2b,d).

The above discussion of the data is mostly descriptive. The exact and approximate relations and bounds developed in the following (Section 4 and Section 5) provide the mathematical framework for the analysis of the data that helps explain some of the aforementioned observations.

4. Exact and Approximate Relations Between Moments of the Thermodynamic Fluctuations

The fluctuating $\mathrm{EoS}$ (7) implies a system of exact relations between trivariate n-order moments (Section 4.1) and exact expressions for $\overline{p^{\prime n}}$ in terms of $({\rho }^{\prime },T^{\prime })$-moments (Section 4.2). Under weakly compressible turbulence conditions, $max({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})\ll 1$, these exact relations can be linearized (Section 4.3), leading to approximate structure relations between $\mathrm{3CCs}$ (Section 4.5).

4.1. Correlation Coefficients of Order n (n$\mathit{CCs}$)

Previous work [1,4,14] has concentrated on 2-order moments, which are of major importance for practical turbulent flow analysis. However, the relations between 2-order moments of thermodynamic correlations also include 3-order moments ([4], (2.5), p. 452), which quantify the truncation error of the weakly compressible approximation. Selected 3-order moments relations were also reported in [4] with respect to the study of the approximation error, and these contained 4-order moments. A more systematic approach is followed here.

It is straightforward, starting from the fluctuating $\mathrm{EoS}$ (7), to obtain relations for the n-moments (and n$\mathrm{CCs}$) between the fluctuations in the basic thermodynamic variables $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$. Multiplying (7) by $p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m}$ (with $k+\ell +m=n-1\ge 1;k,\ell ,m\in {\mathbb{Z}}_{\ge 0}$) yields, upon averaging the exact relations,

$$\begin{array}{c}\hfill {\displaystyle \frac{\tilde{T}}{\overline{T}}}{\displaystyle \frac{\overline{p^{\prime k+1}{\rho }^{\prime \ell }{T}^{\prime m}}}{\overline{p}}}\stackrel{\left(7\right)}{=}{\displaystyle \frac{\overline{p^{\prime k}{\rho }^{\prime \ell +1}{T}^{\prime m}}}{\overline{\rho }}}+{\displaystyle \frac{\overline{p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m+1}}}{\overline{T}}}+{\displaystyle \frac{\overline{p^{\prime k}{\rho }^{\prime \ell +1}{T}^{\prime m+1}}-\overline{p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m}}\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\\ \hfill \forall k,\ell ,m\in {\mathbb{Z}}_{\ge 0}:k+\ell +m=n-1\ge 1\end{array}$$

(12)

connecting the n-moments and also including a higher $(n+1)$-moment and a lower $(n-1)$-moment. Introducing $\mathrm{CVs}$ (2a) and (9) and $\mathrm{CCs}$ (2b), we obtain the equivalent relation

$$\begin{array}{c}\hfill \underset{\mathrm{nCC}}{\underbrace{c_{p^{\prime k+1}{\rho }^{\prime \ell }{T}^{\prime m}}\mathrm{C}||{\mathrm{V}}_{p^{\prime }}}}\stackrel{\left(12\right),\left(2\right),\left(9\right)}{=}\underset{\mathrm{nCCs}}{\underbrace{c_{p^{\prime k}{\rho }^{\prime \ell +1}{T}^{\prime m}}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+c_{p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m+1}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}}+\underset{O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\mathrm{HoTs}}{\underbrace{(\underset{(\mathrm{n}+1)\mathrm{CC}}{\underbrace{c_{p^{\prime k}{\rho }^{\prime \ell +1}{T}^{\prime m+1}}}}-\underset{(\mathrm{n}-1)\mathrm{CC}}{\underbrace{c_{p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m}}}}{c}_{{\rho }^{\prime }{T}^{\prime }}){\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}}\\ \hfill \forall k,\ell ,m\in {\mathbb{Z}}_{\ge 0}:k+\ell +m=n-1\ge 1\end{array}$$

(13)

This is the generating equation for the study of the constraints imposed by the dilute-gas $\mathrm{EoS}$ (7) on the correlations between fluctuations in the basic thermodynamic variables $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$. It contains a linear $O(\mathrm{C}||{\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})$ leading-order expression relating n$\mathrm{CCs}$ (n-order moments) and quadratic $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ terms containing an $(n+1)$$\mathrm{CC}$ and an $(n-1)$$\mathrm{CC}$. For the lowest $n=2$ case, we have $k+\ell +m=n-1=1$, so that for $n=2$, the $(n-1)$-moment is $\overline{{(·)}^{\prime }}=0$. For $n>2$, the $(n-1)$-moment in (12) and (13) is generally $\ne 0$.

Of course, since equations between turbulent statistics are an open system, there are $\forall n\ge 2$ more unknown n$\mathrm{CCs}$ than equations (13). For the lowest $n=2$ case, the moments are $\{\overline{p^{\prime 2}},\overline{{\rho }^{\prime 2}},\overline{T^{\prime 2}},\overline{p^{\prime }{\rho }^{\prime }},\overline{p^{\prime }{T}^{\prime }},\overline{{\rho }^{\prime }{T}^{\prime }}\}$ so that the nondimensional unknowns include the $\mathrm{CVs}$, i.e., they are $\{{\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }},c_{p^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }}\}$.

The products of order $n\ge 2$ which appear on the $\mathrm{LHS}$ of (12) are of the form $\overline{p^{\prime k+1}{\rho }^{\prime \ell }{T}^{\prime m}}$ (with $k+\ell +m=n-1\ge 1;k,\ell ,m\in {\mathbb{Z}}_{\ge 0}$). These are combinations with repetition of $n-1$ objects from a list of three, $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$. There are therefore ([31], p. 744), by construction, $\left({\displaystyle \genfrac{}{}{0pt}{}{3+n-2}{n-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{n-1}}\right)=\frac{n(n+1)}{2}$ equations. On the other hand, n-order moments contain products of the form $\overline{p^{\prime k}{\rho }^{\prime \ell }{T}^{\prime m}}$ (with $k+\ell +m=n\ge 2;k,\ell ,m\in {\mathbb{Z}}_{\ge 0}$). These are combinations with repetition of n objects from a list of three. There are therefore ([31], p. 744) $\left({\displaystyle \genfrac{}{}{0pt}{}{3+n-1}{n}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{n}}\right)=\frac{(n+1)(n+2)}{2}$n-order moments (Table 1). Solving the system of Equation (13), we can therefore determine $\frac{n(n+1)}{2}$n-order moments from the remaining $\frac{(n+1)(n+2)}{2}-\frac{n(n+1)}{2}=n+1$ independent n-order moments (Table 1). The n$\mathrm{CC}$ terms in the exact Equation (13) between n$\mathrm{CCs}$ are $O(\mathrm{C}||{\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})$, and each equation between n$\mathrm{CCs}$ also involves an $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ term containing moments of orders $(n+1)$ and $(n-1)$. This $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ term is nonlinear in the sense that it is generated by the nonlinear term ${\left({\rho }^{\prime }{T}^{\prime }\right)}^{\prime }:={\rho }^{\prime }{T}^{\prime }-\overline{{\rho }^{\prime }{T}^{\prime }}$ appearing in the fluctuating $\mathrm{EoS}$ (7).

Table 1. Number of n-order moments between $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$-fluctuations, number of Equations (12) and (13) implied by the fluctuating $\mathrm{EoS}$ (7) between n-order moments and number of independent n-order moments (at the weakly compressible limit).

n	$\frac{(\mathit{n}+1)(\mathit{n}+2)}{2}$ Moments	$\frac{\mathit{n}(\mathit{n}+1)}{2}$ Equation (13)	$(\mathit{n}+1)$ Independent
2	$6$	$3$	3
3	10	$6$	4
4	15	10	5
5	21	15	6
6	28	21	7

At the weakly compressible turbulence limit, $max({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})\ll 1$, the quadratic $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ terms in (13) can be dropped, thus obtaining a linear system between n-moments only. The linear system between 2-moments has been studied in [4]. However, for higher-order $(n>2)$ moments, not every choice of $(n+1)$ among the $\frac{(n+1)(n+2)}{2}$n-moments forms a linearly independent set, as shown for $\mathrm{3CCs}$ in Section 4.5.

The study of (13) for $\mathrm{3CCs}$ is reported in Section 4.4, and that for the corresponding linear system at the asymptotic limit of weakly compressible turbulence is reported in Section 4.5.

4.2. Central Moments of $p^{\prime }$

Expressions of the central moments of $p^{\prime }$ in terms of other n$\mathrm{CCs}$ are readily obtained by setting $\ell =m=0$ in (12). Central moments of $p^{\prime }$, in line with Section 2.4, are also readily expressed in terms of $(\rho ,T)$-statistics simply by raising the fluctuating $\mathrm{EoS}$ (7) to the power n. Straightforward calculations, using the multinomial theorem ([32], p. 17) and standard multiple-summation techniques [33,34], yield

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\tilde{T}}{\overline{T}}}\right)}^n{\displaystyle \frac{\overline{p^{\prime n}}}{{\overline{p}}^n}}\stackrel{\left(7\right)}{=}& \overline{{\left({\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}+{\displaystyle \frac{T^{\prime }}{\overline{T}}}+{\displaystyle \frac{{\rho }^{\prime }{T}^{\prime }-\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\right)}^n}\hfill \\ \hfill =& \sum _{\begin{array}{c}0\le i,j,k,\ell \le n\\ i+j+k+\ell =n\end{array}}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i,j,k,\ell }}\right)\overline{{\left({\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}\right)}^i{\left({\displaystyle \frac{T^{\prime }}{\overline{T}}}\right)}^j{\left({\displaystyle \frac{{\rho }^{\prime }{T}^{\prime }}{\overline{\rho }\overline{T}}}\right)}^k{\left(-{\displaystyle \frac{\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\right)}^{\ell }}\right]\hfill \\ \hfill =& \sum _{i=0}^n\sum _{j=0}^{n-i}\sum _{k=0}^{n-i-j}\left[{\displaystyle \frac{n!}{i!j!k!(n-i-j-k)!}}\overline{{\left({\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}\right)}^i{\left({\displaystyle \frac{T^{\prime }}{\overline{T}}}\right)}^j{\left({\displaystyle \frac{{\rho }^{\prime }{T}^{\prime }}{\overline{\rho }\overline{T}}}\right)}^k}{\left(-{\displaystyle \frac{\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\right)}^{n-i-j-k}\right]\hfill \\ \hfill =& \sum _{i=0}^n\sum _{j=0}^{n-i}\sum _{k=0}^{n-i-j}\left[{\displaystyle \frac{{(-1)}^{n-i-j-k}n!}{i!j!k!(n-i-j-k)!}}\overline{{\left({\displaystyle \frac{{\rho }^{\prime }}{\overline{\rho }}}\right)}^{i+k}{\left({\displaystyle \frac{T^{\prime }}{\overline{T}}}\right)}^{j+k}}{\left({\displaystyle \frac{\overline{{\rho }^{\prime }{T}^{\prime }}}{\overline{\rho }\overline{T}}}\right)}^{n-i-j-k}\right]\hfill \end{array}$$

(14)

which using $\mathrm{CCs}$ and $\mathrm{CVs}$ (2) reads as

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\tilde{T}}{\overline{T}}}\right)}^n{\displaystyle \frac{\overline{p^{\prime n}}}{{\overline{p}}^n}}\stackrel{\left(14\right)\mathrm{and}\left(2\right)}{=}\sum _{i=0}^n\sum _{j=0}^{n-i}\sum _{k=0}^{n-i-j}\left[{\displaystyle \frac{{(-1)}^{n-i-j-k}n!}{i!j!k!(n-i-j-k)!}}{c}_{{{\rho }^{\prime }}^{i+k}{T}^{\prime j+k}}{c}_{{\rho }^{\prime }{T}^{\prime }}^{n-i-j-k}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^{n-j}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^{n-i}\right]\end{array}$$

(15a)

where we extend the general definitions (2), valid for $ij\ge 1$, by re-expressing $c_{{{\rho }^{\prime }}^i{T}^{\prime j}}$ in terms of standardized variables

$$\begin{array}{c}\hfill c_{{{\rho }^{\prime }}^i{T^{\prime }}^j}:=\overline{{\left({\displaystyle \frac{{\rho }^{\prime }}{\sqrt{\overline{{\rho }^{\prime 2}}}}}\right)}^i{\left({\displaystyle \frac{T^{\prime }}{\sqrt{\overline{T^{\prime 2}}}}}\right)}^j}\Rightarrow \left\{\begin{array}{c}{c}_{{{\rho }^{\prime }}^0{T}^{\prime 0}}=1\hfill \\ c_{{\rho }^{\prime }{T}^{\prime 0}}=c_{{{\rho }^{\prime }}^0{T}^{\prime }}=0\hfill \end{array}\right.\end{array}$$

(15b)

and by (9),

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\tilde{T}}{\overline{T}}}\right)}^n{\displaystyle \frac{\overline{p^{\prime n}}}{{\overline{p}}^n}}\stackrel{\left(9\right)\mathrm{and} \left(2\right)}{=}{\displaystyle \frac{\overline{p^{\prime n}}}{p_{\mathrm{rms}}^{\prime n}}}{\mathrm{C}||\mathrm{V}}_{p^{\prime }}^n\end{array}$$

(15c)

where $\overline{p^{\prime n}}/p_{\mathrm{rms}}^{\prime n}$, for $n\ge 3$, are identified with skewnesses (2c) and (2d) for n odd and flatnesses (2c) and (2d) for n even. According to (15), the exact general expression of $\overline{p^{\prime n}}$ in terms of $({\rho }^{\prime },T^{\prime })$-correlations involves up to $2n$$\mathrm{CCs}$ appearing in terms of $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^n{\mathrm{C}\mathrm{V}}_{T^{\prime }}^n\right)$, consistently with the conceptual approach of Section 2.4. This approach requires therefore many higher-than-n-order $\mathrm{CCs}$ to exactly calculate $\overline{p^{\prime n}}$ compared to (12), which only includes an $(n+1)$-order correlation, but where the other n$\mathrm{CCs}$ include $p^{\prime }$. The alternative approach (12) is simpler and was followed in [4] to develop linearized approximations for the $\mathrm{2CCs}$ in terms of the coefficients of variation.

It is straightforward to calculate from (15) the relations for ${\mathrm{C}||\mathrm{V}}_{p^{\prime }}^2$ (9)

$$\begin{array}{cc}\hfill {\mathrm{C}||\mathrm{V}}_{p^{\prime }}^2\stackrel{\left(15\right)\mathrm{and} \left(2\right)}{=}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2+{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+2c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\hfill \\ \hfill +& 2c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2+2c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\hfill \\ \hfill +& (c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-c_{{\rho }^{\prime }{T}^{\prime }}^2){\mathrm{C}\mathrm{V}}_{T^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2\hfill \end{array}$$

(16)

and for the skewness $S_{p^{\prime }}={}^3{S}_{p^{\prime }}$ (2c)

$$\begin{array}{cc}\hfill S_{p^{\prime }}{\mathrm{C}||\mathrm{V}}_{p^{\prime }}^3\stackrel{\left(15\right)\mathrm{and} \left(2\right)}{=}& S_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+S_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+3c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2+3c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\hfill \\ \hfill +& 3(c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-c_{{\rho }^{\prime }{T}^{\prime }}){\mathrm{C}\mathrm{V}}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+6(c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-c_{{\rho }^{\prime }{T}^{\prime }}^2){\mathrm{C}\mathrm{V}}_{T^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2+3(c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-c_{{\rho }^{\prime }{T}^{\prime }}){\mathrm{C}\mathrm{V}}_{T^{\prime }}^3{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\hfill \\ \hfill +& 3(c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-2c_{{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}){\mathrm{C}\mathrm{V}}_{T^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+3(c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-2c_{{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}){\mathrm{C}\mathrm{V}}_{T^{\prime }}^3{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2\hfill \\ \hfill +& (c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-3c_{{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}+2c_{{\rho }^{\prime }{T}^{\prime }}^3){\mathrm{C}\mathrm{V}}_{T^{\prime }}^3{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3\hfill \end{array}$$

(17)

illustrating the rapid increase in the number of higher-order correlations when using (15). Each line in (16) and (17) corresponds to an increasingly smaller order of magnitude, with the first line containing the leading-order expression which is approximately used at the weakly compressible limit. Of course, the asymptotic expressions at the weakly compressible turbulence limit are identical for both approaches (13) and (15).

4.3. Unscrambling the Exact Relations

It is straightforward to expand relations (13). The three relations between the six second-order moments in Table 1 were given in [4], (2.5), p. 452, and are reproduced for completeness:

$$\begin{array}{cccccccccc}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }}& {\mathrm{CV}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{T^{\prime }}+\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}\hfill \end{array}$$

(18a)

$$\begin{array}{cccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & & {\mathrm{CV}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{T^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}\hfill \end{array}$$

(18b)

$$\begin{array}{cccccccccc}\hfill c_{p^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{{\rho }^{\prime }}+\hfill & & {\mathrm{CV}}_{T^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}\hfill \end{array}$$

(18c)

These equations were the basis of the linearized analysis of the thermodynamic turbulence structure in [4], which focused on 2-order moments. The origin of the terms containing the product ${\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}$ on the $\mathrm{RHS}$ of (18) is the nonlinear term ${\left({\rho }^{\prime }{T}^{\prime }\right)}^{\prime }$ in the fluctuating $\mathrm{EoS}$ (7). The linearized approximation drops terms of $O\left({\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}\right)$ or higher in (18). Therefore, the linearization error of the weakly compressible approximation depends on the $\mathrm{3CCs}$ $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\}$ in (18).

The six distinct relations between the 10 third-order moments (Table 1) read as (13)

$$\begin{array}{cccccccccccc}\hfill S_{p^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19a)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19b)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19c)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill S_{{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19d)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill S_{T^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19e)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{T}^{\prime }}^2)& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(19f)

Similarly to (18) for the $\mathrm{2CCs}$, the origin of the terms containing the product ${\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}$ on the $\mathrm{RHS}$ of (19) is the nonlinear term ${\left({\rho }^{\prime }{T}^{\prime }\right)}^{\prime }$ in the fluctuating $\mathrm{EoS}$ (7). The linearized approximation drops terms of $O\left({\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}\right)$ or higher in (19). Therefore, the linearization error of the weakly compressible turbulence approximation for $\mathrm{3CCs}$ depends on the differences between $\mathrm{4CCs}$ and $\mathrm{2CCs}$ in (19). This approximation error determines in particular the limit of the fluctuation intensities $\{{\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ for which the identities (19) can be linearized with acceptable accuracy. Since the magnitudes of $\mathrm{4CCs}$ are generally larger than those of $\mathrm{3CCs}$ (e.g., skewnesses squared are smaller than flatnesses [35]), we would expect the linearized approximation for $\mathrm{3CCs}$ to become inaccurate at lower Mach numbers (lower fluctuation amplitudes) than the linearized approximation for $\mathrm{2CCs}$. Furthermore, as the approximation error depends on $\mathrm{CCs}$, it obviously differs from one flow configuration to another.

Similarly, the 10 distinct relations between the 15 fourth-order moments (Table 1) read as (13)

$$\begin{array}{cccccccccccc}\hfill F_{p^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{p}^{\prime }{p}^{\prime }{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{p}^{\prime }{p}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{p}^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & S_{p^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20a)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{p}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{p}^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20b)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{p}^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20c)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{T}^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20d)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20e)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20f)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill F_{T^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & S_{T^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20g)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20h)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20i)

$$\begin{array}{cccccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& \mathrm{C}||{\mathrm{V}}_{p^{\prime }}\hfill & \hfill \stackrel{\left(13\right)}{=}& & \hfill F_{{\rho }^{\prime }}& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}+\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\mathrm{C}\mathrm{V}}_{T^{\prime }}+\hfill & & (c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & S_{{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(20j)

In this case, the nonlinear terms containing the product ${\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}$ on the $\mathrm{RHS}$ of (20) depend on differences between $\mathrm{5CCs}$ and $\mathrm{3CCs}$.

The leading terms in (18)–(20), and generally for the n$\mathrm{CCs}$ in (13), are $O(\mathrm{C}||{\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})$, with the three $\mathrm{CVs}$ being approximately of the same order-of-magnitude The $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ nonlinear terms on the $\mathrm{RHS}$ of (18)–(20) are obviously $\mathrm{2oTs}$ (3), with the coefficient depending on $(n+1)$$\mathrm{CCs}$. The nonlinearity included in $\mathrm{C}||{\mathrm{V}}_{p^{\prime }}$ is of a higher order (21).

4.4. Linearization of the Equations Between $\mathit{3CCs}$

For sufficiently weak fluctuation amplitudes, $max({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})\ll 1$, the six exact equations (19) for the $\mathrm{3CCs}$ can be linearized by dropping $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ and higher-order terms. The term $\mathrm{C}||{\mathrm{V}}_{p^{\prime }}$ is easily linearized by replacing

$$\begin{array}{c}\hfill \mathrm{C}||{\mathrm{V}}_{p^{\prime }}={\mathrm{C}\mathrm{V}}_{p^{\prime }}+\left({\mathrm{C}||}_{p^{\prime }}-{\mathrm{C}\mathrm{V}}_{p^{\prime }}\right)\stackrel{\left(9\right)}{=}{\mathrm{C}\mathrm{V}}_{p^{\prime }}+c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(21)

on the $\mathrm{LHS}$ of (19) and transferring the $c_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}$ term to the $\mathrm{RHS}$. Upon division by ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$, (19) yields the, formally $O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$-accurate, linearized approximations

$$\begin{array}{ccccccccc}\hfill S_{p^{\prime }}& \stackrel{\left(19\mathrm{a}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22a)

$$\begin{array}{ccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}& \stackrel{\left(19\mathrm{b}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22b)

$$\begin{array}{ccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}& \stackrel{\left(19\mathrm{c}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22c)

$$\begin{array}{ccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}& \stackrel{\left(19\mathrm{d}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill S_{{\rho }^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22d)

$$\begin{array}{ccccccccc}\hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}& \stackrel{\left(19\mathrm{e}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill S_{T^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22e)

$$\begin{array}{ccccccccc}\hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}& \stackrel{\left(19\mathrm{f}\right)\mathrm{and} \left(21\right)}{\approxeq }\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +& & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\hfill & \hfill +O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)\end{array}$$

(22f)

This is a linear system of six equations between the 10 $\mathrm{3CCs}$ (Table 1) whose coefficients are the ratios $\{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}/{\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}/{\mathrm{C}\mathrm{V}}_{p^{\prime }}\}$ of the fluctuation amplitudes. The approximation errors in (22) are determined by (19) and (21) as

$$\begin{array}{ccccccc}& \mathrm{error}\left(S_{p^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }})& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill S_{p^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23a)

$$\begin{array}{ccccccc}& \mathrm{error}\left(c_{p^{\prime }{p}^{\prime }{T}^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & \hfill c_{p^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23b)

$$\begin{array}{ccccccc}& \mathrm{error}\left(c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }})& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23c)

$$\begin{array}{ccccccc}& \mathrm{error}\left(c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{{\rho }^{\prime }{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}-\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }})& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23d)

$$\begin{array}{ccccccc}& \mathrm{error}\left(c_{p^{\prime }{T}^{\prime }{T}^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }})& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23e)

$$\begin{array}{ccccccc}& \mathrm{error}\left(c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\right)\hfill & \hfill \stackrel{\left(22\right),\left(21\right),\left(19\right)}{=}& (c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-\hfill & \hfill c_{{\rho }^{\prime }{T}^{\prime }}^2)& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-\hfill & \hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(23f)

with a leading $O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ term and a higher $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ term whose origin is (21). This higher $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ term is proportional to the $\mathrm{3CC}$ that is approximated (the corresponding approximation error is implicit) and was found to be negligibly small compared to the leading $O\left({\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ terms in (23) for all available $\mathrm{TPC}$ flow data [12]. In compressible $\mathrm{TPC}$ flows, the ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$-peak in the buffer zone increases roughly with ${\overline{M}}_{{\mathrm{CL}}_x}^2$ (Figure 1c). Therefore, the linearized approximations (22) are expected to lose accuracy with an increasing ${\overline{M}}_{{\mathrm{CL}}_x}$.

An indication of the flow’s Mach number beyond which the linearized approximation is no longer accurate is obtained by assessing the linearized expressions (22) for the $\mathrm{3CCs}$ using $\mathrm{DNS}$ data for representative compressible $\mathrm{TPC}$ flows (Figure 3 and Figure 4). This limit ${\overline{M}}_{{\mathrm{CL}}_x}$ depends on the specific correlation that is considered. Approximations (22a)–(22c), expressing the $\mathrm{3CCs}$ $\{S_{p^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\}$, are very robust and remain very accurate up to the highest available ${\overline{M}}_{{\mathrm{CL}}_x}=2.49$ (Figure 3). On the contrary, approximations (22d)–(22f) expressing the $\mathrm{3CCs}$ $\{c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$ are less robust (Figure 4). The excellent accuracy observed at ${\overline{M}}_{{\mathrm{CL}}_x}=0.32$ (Figure 4a,f,k) remains satisfactory at ${\overline{M}}_{{\mathrm{CL}}_x}=0.81$ (Figure 4b,g,l), but noticeable discrepancies appear already at ${\overline{M}}_{{\mathrm{CL}}_x}=1.51$ (Figure 4c,h,m), in the buffer region ($5⪅y^{★}⪅40$). These discrepancies grow with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ and become unacceptable at ${\overline{M}}_{{\mathrm{CL}}_x}=1.96$ (Figure 4d,i,n) and ${\overline{M}}_{{\mathrm{CL}}_x}=2.49$ (Figure 4e,j,o). Hence, for compressible $\mathrm{TPC}$ flows, the system (23) remains an accurate approximation of the system of exact relations (19) only for ${\overline{M}}_{{\mathrm{CL}}_x}⪅1.5$. This ${\overline{M}}_{{\mathrm{CL}}_x}⪅1.5$ limit is more stringent than the corresponding ${\overline{M}}_{{\mathrm{CL}}_x}⪅2.0$ limit for $\mathrm{2CCs}$ [4], for which a similar accuracy is obtained for the linearization of system (18), suggesting that the nonlinear effects associated with the product ${\left({\rho }^{\prime }{T}^{\prime }\right)}^{\prime }$ in the fluctuating $\mathrm{EoS}$ (7) are stronger in relations between $\mathrm{3CCs}$ compared to relations between $\mathrm{2CCs}$.

Figure 3. Assessment of the approximate linearized expressions (22a)–(22c) for the $\mathrm{3CCs}$ $\{S_{p^{\prime }}\left(\mathbf{a}–\mathbf{e}\right),c_{p^{\prime }{p}^{\prime }{T}^{\prime }}\left(\mathbf{f}–\mathbf{j}\right),c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\left(\mathbf{k}–\mathbf{o}\right)\}$, evaluated using $\mathrm{DNS}$ data [12] for compressible turbulent plane channel flows in the range ${Re}_{{\tau }^{★}}\in [113,985]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.32,2.49]$, plotted against the $\mathrm{HCB}$-scaled nondimensional wall-distance $y^{★}$ (logscale).

Figure 4. Assessment of the approximate linearized expressions (22d)–(22f) for the $\mathrm{3CCs}$ $\{c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}\left(\mathbf{a}–\mathbf{e}\right),c_{p^{\prime }{T}^{\prime }{T}^{\prime }}\left(\mathbf{f}–\mathbf{j}\right),c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\left(\mathbf{k}–\mathbf{o}\right)\}$ and also of (26) for $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\left(\mathbf{k}–\mathbf{o}\right)$, evaluated using $\mathrm{DNS}$ data [12] for compressible turbulent plane channel flows in the range ${Re}_{{\tau }^{★}}\in [113,985]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.32,2.49]$, plotted against the $\mathrm{HCB}$-scaled nondimensional wall-distance $y^{★}$ (logscale).

An intuitive explanation of the difference in accuracy with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ between the robust approximations (22a)–(22c) for $\{S_{p^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\}$ and the fragile approximations (22d)–(22f) for $\{c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$ is that the former express $\mathrm{3CCs}$ containing $p^{\prime }$ in terms of $\mathrm{3CCs}$ also containing $p^{\prime }$, whereas the latter express $\mathrm{3CCs}$ containing $p^{\prime }$ in terms of $\left({\rho }^{\prime }{T}^{\prime }\right)$-statistics. Exact expressions for $\overline{p^{\prime n}}$ in terms of $({\rho }^{\prime },T^{\prime })$-statistics (15) involve higher $2n$-order correlations, e.g., (17) for $S_{p^{\prime }}$ involves 6-order moments. However, this explanation, although plausible, remains conjectural.

A quantitative assessment is obtained by considering (Figure 5) the coefficients multiplying ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$ in the leading-order term of the approximation error (23) for each of the linearized relations (22). Invariably, these coefficients contain the ratio ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}/{\mathrm{C}\mathrm{V}}_{p^{\prime }}$ which reaches values $\in [2,4]$ in the buffer region ([4], Figure 3, p. 458), where the peak of ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}$ occurs (Figure 1a), whereas ${\mathrm{C}\mathrm{V}}_{p^{\prime }}$ has a much lower inner peak (Figure 1b). Therefore, differences in the increase in the approximation error of the linearized relations (22) with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ (increasing ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$) are essentially controlled by the $\mathrm{4CCs}$ in the leading-order term of the approximation error (23).

Figure 5. Coefficient multiplying ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$ in the expression for the leading-order term of the linearization error (23) in the approximate expressions (22) for the $\mathrm{3CCs}$, $S_{p^{\prime }}$ (a), $c_{p^{\prime }{p}^{\prime }{T}^{\prime }}$ (b), $c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}$ (c), $c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}$ (d), $c_{p^{\prime }{T}^{\prime }{T}^{\prime }}$ (e) and $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ (f), evaluated using $\mathrm{DNS}$ data [12] for compressible turbulent plane channel flows in the range ${Re}_{{\tau }^{★}}\in [113,985]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.81,2.49]$, plotted against the $\mathrm{HCB}$-scaled nondimensional wall-distance $y^{★}$ (logscale).

The $\mathrm{DNS}$ data for compressible $\mathrm{TPC}$ flows indicate that in the buffer region ($5⪅y^{★}⪅40$), the coefficients of ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$ in the leading-order term of the approximation error (23a)–(23c) for the $\mathrm{3CCs}$ $\{S_{p^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}\}$, for which the linearized approximation is robust (Figure 3), are small (Figure 5a–c), $<1$ in absolute value. On the contrary, for the $\mathrm{3CCs}$ $\{c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$, for which the linearized approximation is fragile with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ (Figure 4), the coefficients of ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$ in the leading-order term of the approximation error (23d)–(23f) are large (Figure 5d–f), $\sim 5$ in absolute value. Furthermore, their absolute value increases with increasing ${\overline{M}}_{{\mathrm{CL}}_x}$ to reach $\sim 10$ for ${\overline{M}}_{{\mathrm{CL}}_x}=2.49$ (Figure 5d–f). Further analysis requires the determination of bounds for the quadruple correlation coefficients ($\mathrm{4CCs}$), which are studied in Section 5.

4.5. Linearized Relations Between $\mathit{3CCs}$

Despite the rather low $M_{{\mathrm{CL}}_x}⪅1.5$ limit of validity of the linearized approximations for $\mathrm{3CCs}$, the study of the linear system (22) provides insight into the interdependencies that the fluctuating $\mathrm{EoS}$ (7) imposes on the $\mathrm{3CCs}$.

There are $\left({\displaystyle \genfrac{}{}{0pt}{}{3+3-1}{3}}\right)=10$ combinations with repetitions ([31], p. 744) of the fluctuations of the three basic thermodynamic variables $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ into distinct triple correlations $\{\overline{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$, $\overline{p^{\prime }{p}^{\prime }{\rho }^{\prime }}$, $\overline{p^{\prime }{p}^{\prime }{T}^{\prime }}$, $\overline{p^{\prime 3}}$, $\overline{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}$, $\overline{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}$, $\overline{T^{\prime 3}}$, $\overline{p^{\prime }{T}^{\prime }{T}^{\prime }}$, $\overline{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}$, $\overline{{\rho }^{\prime 3}}\}$, and only six linear equations (22) are obtained by truncating (19) to $O({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})$. We can therefore approximate 6 of the 10 $\mathrm{3CC}$s in terms of 4 linearly independent ones (it is assumed that the coefficients of variation $\{{\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }}\}$ are known).

The last three relations (22d)–(22f) express $\{c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$ in terms of $({\rho }^{\prime },T^{\prime })$-statistics only. The higher-order $O\left({\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\right)$ terms that were dropped from (19d)–(19f) to obtain (22d)–(22f) are also $({\rho }^{\prime },T^{\prime })$-statistics. The system (22) can be solved for all third-order correlations containing $p^{\prime }$, $\{S_{p^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$, thus replacing (19a)–(19c) with

$$\begin{array}{cccccccccc}\hfill S_{p^{\prime }}& \stackrel{\left(22\right)}{\approxeq }\hfill & \hfill 3c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3}}\hfill & \hfill +& & \hfill 3c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3}}\hfill & \hfill +& S_{{\rho }^{\prime }}{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3}}+S_{T^{\prime }}{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3}}+\mathrm{HoTs}\hfill \end{array}$$

(24a)

$$\begin{array}{cccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}& \stackrel{\left(22\right)}{\approxeq }\hfill & \hfill 2c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}\hfill & \hfill +& & \hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}\hfill & \hfill +& S_{{\rho }^{\prime }}{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}+\mathrm{HoTs}\hfill \end{array}$$

(24b)

$$\begin{array}{cccccccccc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}& \stackrel{\left(22\right)}{\approxeq }\hfill & \hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}\hfill & \hfill +& & \hfill 2c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}& {\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}\hfill & \hfill +& S_{T^{\prime }}{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2}}+\mathrm{HoTs}\hfill \end{array}$$

(24c)

The approximate linearized expression (24a) for the skewness $S_{p^{\prime }}$ corresponds to the leading terms (first line) of the exact expression (17) for $S_{p^{\prime }}$ in terms of $({\rho }^{\prime },T^{\prime })$-statistics which contains up to $\mathrm{6oTs}$ (6-order terms) involving $\mathrm{6CCs}$, highlighting the fact that all of these extra terms in the exact expression represent effects dependent on the fluctuation intensity.

The system consisting of the three relations (24) and of the three relations (22d)–(22f) expresses $\{S_{p^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$ in terms of the four independent variables $\{S_{{\rho }^{\prime }},S_{T^{\prime }},c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\}$, in line with the conceptual approach of expressing all correlations in terms of $({\rho }^{\prime },T^{\prime })$-statistics (Section 2.4). More generally, there are ([31], p. 744) $\left({\displaystyle \genfrac{}{}{0pt}{}{2+n-1}{n}}\right)=n+1$ bivariate $({\rho }^{\prime },T^{\prime })$ n-moments, constituting a linearly independent $(n+1)$-tuple (Table 1).

However, not all 4-tuples of $\mathrm{3CC}$s are linearly independent. In particular, $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ is linearly dependent on several symmetric combinations of the other $\mathrm{3CCs}$. Solving each of the relations (22f), (22b), (22c) for $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ yields

$$\begin{array}{cc}\hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\stackrel{\left(22\mathrm{f}\right)}{\approxeq }& c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{{\rho }^{\prime }}}{{\mathrm{CV}}_{p^{\prime }}}}+c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{T^{\prime }}}{{\mathrm{CV}}_{p^{\prime }}}}+\stackrel{\sim \frac{{\mathrm{CV}}_{{\rho }^{\prime }}}{{\mathrm{CV}}_{p^{\prime }}}O\left({\mathrm{CV}}_{T^{\prime }}\right)}{\overbrace{O\left({\displaystyle \frac{{\mathrm{CV}}_{{\rho }^{\prime }}{\mathrm{CV}}_{T^{\prime }}}{{\mathrm{CV}}_{p^{\prime }}}}\right)}}\hfill \end{array}$$

(25a)

$$\begin{array}{cc}\hfill \stackrel{\left(22\mathrm{b}\right)}{\approxeq }& c_{p^{\prime }{p}^{\prime }{T}^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{p^{\prime }}}{{\mathrm{CV}}_{{\rho }^{\prime }}}}-c_{p^{\prime }{T}^{\prime }{T}^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{T^{\prime }}}{{\mathrm{CV}}_{{\rho }^{\prime }}}}+O\left({\mathrm{CV}}_{T^{\prime }}\right)\hfill \end{array}$$

(25b)

$$\begin{array}{cc}\hfill \stackrel{\left(22\mathrm{c}\right)}{\approxeq }& c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{p^{\prime }}}{{\mathrm{CV}}_{T^{\prime }}}}-c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}{\displaystyle \frac{{\mathrm{CV}}_{{\rho }^{\prime }}}{{\mathrm{CV}}_{T^{\prime }}}}+O\left({\mathrm{CV}}_{{\rho }^{\prime }}\right)\hfill \end{array}$$

(25c)

Combining (24a) and (25a), we have

$$\begin{array}{c}\hfill c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\stackrel{\left(22\right),\left(24\mathrm{a}\right),\left(25\mathrm{a}\right)}{\approxeq }{\displaystyle \frac{\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}}+O\left({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }},{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}},{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}},{\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}\right)\end{array}$$

(26)

where the order of magnitude of the approximation error is obtained by including in the calculations the orders of magnitude of the approximation errors in (22). The accuracy of (26) with an increasing fluctuation amplitude is similar to the accuracy of (22f), which approximates $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ with $({\rho }^{\prime },T^{\prime })$-statistics (Figure 4k–n). According to (26), $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},S_{p^{\prime }},S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ are not linearly independent because at the weakly compressible limit, the individual skewnesses and $\mathrm{CVs}$ determine the triple correlation coefficient $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$. Therefore, to express the $\mathrm{3CCs}$ in terms of the individual skewnesses, we cannot choose $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ as the fourth independent variable. Choosing, e.g., as independent variables $\{c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }},S_{p^{\prime }},S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ and solving the system (22) for $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}\}$ yields (26) and five other equations:

$$\begin{array}{cc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\stackrel{\left(22\right)}{\approxeq }& +c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3+\frac{2}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(27a)

$$\begin{array}{cc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2\stackrel{\left(22\right)}{\approxeq }& +c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}+S_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+\mathrm{HoTs}\iff \left(22\mathrm{d}\right)\hfill \end{array}$$

(27b)

$$\begin{array}{cc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}+\frac{2}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{2}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(27c)

$$\begin{array}{cc}\hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+\frac{2}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(27d)

$$\begin{array}{cc}\hfill c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\stackrel{\left(24\mathrm{a}\right)}{\iff }\left(22\mathrm{f}\right)\hfill \end{array}$$

(27e)

Alternatively, choosing $\{c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }},S_{p^{\prime }},S_{{\rho }^{\prime }},S_{T^{\prime }}\}$ as independent variables and solving the system (22) for $\{c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }},c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }},c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }},c_{p^{\prime }{p}^{\prime }{T}^{\prime }},c_{p^{\prime }{T}^{\prime }{T}^{\prime }},c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}\}$ yields (26) and five other equations:

$$\begin{array}{cc}\hfill c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+\frac{2}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3+\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{2}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(28a)

$$\begin{array}{cc}\hfill c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3+\frac{2}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(28b)

$$\begin{array}{cc}\hfill c_{p^{\prime }{p}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}\stackrel{\left(22\right)}{\approxeq }& +c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3+\frac{2}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\hfill \end{array}$$

(28c)

$$\begin{array}{cc}\hfill c_{p^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2\stackrel{\left(22\right)}{\approxeq }& +c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+S_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\iff \left(22\mathrm{e}\right)\hfill \end{array}$$

(28d)

$$\begin{array}{cc}\hfill c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^2{\mathrm{C}\mathrm{V}}_{T^{\prime }}\stackrel{\left(22\right)}{\approxeq }& -c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^2+\frac{1}{3}{S}_{p^{\prime }}{\mathrm{C}\mathrm{V}}_{p^{\prime }}^3-\frac{1}{3}{S}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}^3-\frac{1}{3}{S}_{T^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}^3+\mathrm{HoTs}\iff \left(27\mathrm{e}\right)\hfill \end{array}$$

(28e)

Notice that there are $\left({\displaystyle \genfrac{}{}{0pt}{}{10}{4}}\right)=210$ possible 4-tuples that can be formed from the 10 $\mathrm{3CCs}$ (combinations without repetitions ([31], p. 744)), but it is outside the scope of the paper to systematically investigate which 4-tuples are linearly independent. The approximate equations derived in this subsection (24)–(28) are useful for analyzing the $\mathrm{DNS}$ results for the triple correlations (Section 6).

5. Statistical Identities and Bounds

Contrary to the relations developed in the previous sections, which are consequences of the fluctuating $\mathrm{EoS}$ (7), we consider here general identities for third-order and fourth-order correlation coefficients between the fluctuations in three random variables $\{{\mathsf{a}}^{\prime },{\mathsf{b}}^{\prime },{\mathsf{c}}^{\prime }\}$. These identities can be readily combined with the Schwarz inequality to determine bounds for the $\mathrm{3CCs}$.

Lumley [20] remarks that the general bounds obtained by the Schwarz inequality are not very restrictive. Progress to determining sharper bounds has been made by several authors [22]. Ogasawara’s lemma ([22], Lemma 1, p. 13) provides such slightly sharper bounds, valid for general pdfs, as a multidimensional extension of Pearson’s inequality [35]. This approach considers Schwarz’s inequality for fluctuations in functions (products) of the fluctuating variables, formalizing and systematically extending earlier approaches ([36], (2), p. 477).

We reinterpret here Ogasawara’s lemma ([22], Lemma 1, p. 13) in terms of the Schwarz inequality for the correlation coefficient

$$\begin{array}{c}\hfill c_{{(·)}^{\prime }{[·]}^{\prime }}:={\displaystyle \frac{\overline{{(·)}^{\prime }{[·]}^{\prime }}}{{(·)}_{\mathrm{rms}}^{\prime }{[·]}_{\mathrm{rms}}^{\prime }}};c_{{(·)}^{\prime }{[·]}^{\prime }}^2\le 1\end{array}$$

(29)

written in the formalism of Reynolds averages and fluctuations $(·)=\overline{(·)}+{(·)}^{\prime }$.

Functions (products) of fluctuations can be separated into an average and a fluctuating part:

$$\begin{array}{cccccccccccccc}\hfill {\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }& :=\hfill & \hfill {\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}& \Rightarrow \hfill & \hfill \overline{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime 2}}& =\hfill & & \overline{{({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }})}^2}\hfill & \hfill =& \overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime 2}}-{\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}}^2\hfill & \stackrel{\left(2\right)}{\Rightarrow }& {\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }& =& {\mathsf{a}}_{\mathrm{rms}}^{\prime }{\mathsf{b}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\end{array}$$

(30a)

$$\begin{array}{cccccccccccccc}\hfill {\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }& :=\hfill & \hfill {\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}}& \Rightarrow \hfill & \hfill \overline{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime 2}}& =\hfill & & \overline{{({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}})}^2}\hfill & \hfill =& \overline{{\mathsf{a}}^{\prime 4}}-{\overline{{\mathsf{a}}^{\prime 2}}}^2\hfill & \stackrel{\left(2\right)}{\Rightarrow }& {\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }& =& {\mathsf{a}}_{\mathrm{rms}}^{\prime 2}\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\end{array}$$

(30b)

Notice that (30a) also implies that for any pair of random variables $({\mathsf{a}}^{\prime },{\mathsf{b}}^{\prime })$,

$$\begin{array}{c}\hfill \overline{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime 2}}\ge 0\stackrel{\left(30\mathrm{a}\right)}{\Rightarrow }{c}_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2\ge 0\end{array}$$

(30c)

Using the results (30) for the variances, the following identities for the correlation coefficients between fluctuations of products-of-fluctuations are readily obtained:

$$\begin{array}{cccccccc}\hfill c_{{\mathsf{a}}^{\prime }{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }}:=& {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}})}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime }{\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 3}}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime 3}\sqrt{F_{{\mathsf{a}}^{\prime }}-1}}}\hfill & \hfill \Rightarrow & S_{{\mathsf{a}}^{\prime }}\hfill & \hfill =& c_{{\mathsf{a}}^{\prime }{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }}\hfill & & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\hfill \end{array}$$

(31a)

$$\begin{array}{cccccccc}\hfill c_{{\mathsf{a}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }})}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime }}}{{\mathsf{a}}^{\prime 2}{\mathsf{b}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}}}\hfill & \hfill \Rightarrow & c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}\hfill & \hfill =& c_{{\mathsf{a}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }}\hfill & & \sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\hfill \end{array}$$

(31b)

$$\begin{array}{cccccccc}\hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\mathsf{b}}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}}){\mathsf{b}}^{\prime }}}{{\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }{\mathsf{b}}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime }}}{{\mathsf{a}}^{\prime 2}{\mathsf{b}}_{\mathrm{rms}}^{\prime }\sqrt{F_{{\mathsf{a}}^{\prime }}-1}}}\hfill & \hfill \Rightarrow & c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}\hfill & \hfill =& c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\mathsf{b}}^{\prime }}\hfill & & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\hfill \end{array}$$

(31c)

$$\begin{array}{cccccccc}\hfill c_{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }{\mathsf{c}}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}){\mathsf{c}}^{\prime }}}{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime }{\mathsf{b}}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}}}\hfill & \hfill \Rightarrow & c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}\hfill & \hfill =& c_{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }{\mathsf{c}}^{\prime }}\hfill & & \sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\hfill \end{array}$$

(31d)

$$\begin{array}{cccccccc}\hfill c_{{\mathsf{a}}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }-\overline{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }})}}{{\mathsf{a}}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime }{\mathsf{b}}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2}}}\hfill & \hfill \Rightarrow & c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}\hfill & \hfill =& c_{{\mathsf{a}}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}\hfill & & \sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(31e)

$$\begin{array}{cccccccc}\hfill c_{{\mathsf{b}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{{\mathsf{b}}^{\prime }({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }})}}{{\mathsf{b}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}{\displaystyle \frac{\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime }{\mathsf{b}}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}}}\hfill & \hfill \Rightarrow & c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}\hfill & \hfill =& c_{{\mathsf{b}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}\hfill & & \sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(31f)

These identities express all $\mathrm{3CCs}$ as products of a correlation coefficient of order 2 multiplied by a square root of the $\mathrm{4CCs}$. Straightforward application of the Schwarz inequality (29) provides upper bounds for the absolute values of the $\mathrm{3CCs}$:

$$\begin{array}{cccc}\hfill \left(31\mathrm{a}\right)\mathrm{and} \left(29\right)\Rightarrow & |S_{{\mathsf{a}}^{\prime }}|\hfill & \hfill \le & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\hfill \end{array}$$

(32a)

$$\begin{array}{cccc}\hfill \left(31\mathrm{b}\right),\left(31\mathrm{c}\right),\left(29\right)\Rightarrow & |c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}|\hfill & \hfill \le & min\left(\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2},\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\right)\hfill \end{array}$$

(32b)

$$\begin{array}{cccc}\hfill \left(32\mathrm{b}\right)\Rightarrow & |c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}|\hfill & \hfill \le & min\left(\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2},\sqrt{F_{{\mathsf{b}}^{\prime }}-1}\right)\hfill \end{array}$$

(32c)

$$\begin{array}{cccc}\hfill \left(31\mathrm{d}\right),\left(31\mathrm{e}\right),\left(31\mathrm{f}\right),\left(29\right)\Rightarrow & |c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}|\hfill & \hfill \le & min\left(\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2},\sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2},\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}\right)\hfill \end{array}$$

(32d)

where (32c) is obtained from (32b) by interchanging the random variables $\mathsf{a}$ and $\mathsf{b}$. Analogous relations were given in [22], where a different notation was used. They were redemonstrated here for completeness and notational consistency. Inequality (31a) is Pearson’s inequality [35]. The inequalities (32d) that were obtained from identities (31d)–(31f) correspond to ([22], (2.8), p. 13), and the inequalities (32b) that were obtained from identities (31b) and (31c) correspond to ([22], (2.10), p. 14).

There are two upper bounds (32b) for $|c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}|$, either from joint $({\mathsf{a}}^{\prime },{\mathsf{b}}^{\prime })$-statistics or from the single-variable kurtosis $F_{{\mathsf{a}}^{\prime }}$, while there are three upper bounds (32d) for $|c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}|$ from the joint statistics of the three couples of variables. Many of these bounds (31) are of the form $c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2\ge 0$ (30c). An upper bound for $c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}$ can be obtained in terms of the individual kurtoses from

$$\begin{array}{cccc}\hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime 2}\right)}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}})({\mathsf{b}}^{\prime 2}-\overline{{\mathsf{b}}^{\prime 2}})}}{{\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }{\left({\mathsf{b}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}\hfill & & {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime 2}}-\overline{{\mathsf{a}}^{\prime 2}}\overline{{\mathsf{b}}^{\prime 2}}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime 2}{\mathsf{b}}_{\mathrm{rms}}^{\prime 2}\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{F_{{\mathsf{b}}^{\prime }}-1}}}\hfill \end{array}$$

(33a)

$$\begin{array}{cccc}\hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}})({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }})}}{{\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}\hfill & & {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 3}{\mathsf{b}}^{\prime }}-\overline{{\mathsf{a}}^{\prime 2}}\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime 3}{\mathsf{b}}_{\mathrm{rms}}^{\prime }\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}}}\hfill \end{array}$$

(33b)

$$\begin{array}{cccc}\hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime 2}-\overline{{\mathsf{a}}^{\prime 2}})({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }-\overline{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }})}}{{\left({\mathsf{a}}^{\prime 2}\right)}_{\mathrm{rms}}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}\hfill & & {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-\overline{{\mathsf{a}}^{\prime 2}}\overline{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime 2}{\mathsf{b}}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2}}}\hfill \end{array}$$

(33c)

$$\begin{array}{cccc}\hfill c_{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}:=& {\displaystyle \frac{\overline{({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }})({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }-\overline{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }})}}{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}_{\mathrm{rms}}^{\prime }}}\stackrel{\left(30\right)}{=}\hfill & & {\displaystyle \frac{\overline{{\mathsf{a}}^{\prime 2}{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-\overline{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}\overline{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}}{{\mathsf{a}}_{\mathrm{rms}}^{\prime 2}{\mathsf{b}}_{\mathrm{rms}}^{\prime }{\mathsf{c}}_{\mathrm{rms}}^{\prime }\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}}}\hfill \end{array}$$

(33d)

leading to the identities

$$\begin{array}{cccccc}\hfill \left(33\mathrm{a}\right)\mathrm{and} \left(2\right)& \Rightarrow \hfill & \hfill c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-1& =\hfill & \hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime 2}\right)}^{\prime }}& \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{F_{{\mathsf{b}}^{\prime }}-1}\hfill \end{array}$$

(34a)

$$\begin{array}{cccccc}\hfill \left(33\mathrm{b}\right)\mathrm{and} \left(2\right)& \Rightarrow \hfill & \hfill c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}& =\hfill & \hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }}& \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\hfill \end{array}$$

(34b)

$$\begin{array}{cccccc}\hfill \left(33\mathrm{c}\right)\mathrm{and} \left(2\right)& \Rightarrow \hfill & \hfill c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}& =\hfill & \hfill c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}& \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(34c)

$$\begin{array}{cccccc}\hfill \left(33\mathrm{d}\right)\mathrm{and} \left(2\right)& \Rightarrow \hfill & \hfill c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}{c}_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}& =\hfill & \hfill c_{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}& \sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(34d)

Identity (34a) can be used in (34b)–(34d) to express the $\mathrm{RHSs}$ in terms of individual kurtoses. The above identities (34) contain the $\mathrm{2CCs}$ $\{c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime 2}\right)}^{\prime }},c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }},c_{{\left({\mathsf{a}}^{\prime 2}\right)}^{\prime }{\left({\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }},c_{{\left({\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }\right)}^{\prime }{\left({\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }\right)}^{\prime }}\}$ which are bounded by the covariance inequality (29), readily providing bounds for the $\mathrm{4CCs}$:

$$\begin{array}{cccccccc}\hfill \left(34\mathrm{a}\right)\mathrm{and} \left(29\right)& \Rightarrow \hfill & \hfill c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2\stackrel{\left(30\mathrm{c}\right)}{\le }& c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}\hfill & \hfill \le & 1+\hfill & & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{F_{{\mathsf{b}}^{\prime }}-1}\hfill \end{array}$$

(35a)

$$\begin{array}{cccccccc}\hfill \left(34\mathrm{b}\right)\mathrm{and} \left(29\right)& \Rightarrow \hfill & & |c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}|\hfill & \hfill \le & & & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\hfill \end{array}$$

(35b)

$$\begin{array}{cccccccc}\hfill \left(34\mathrm{c}\right)\mathrm{and} \left(29\right)& \Rightarrow \hfill & & |c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}|\hfill & \hfill \le & & & \sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{c_{{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(35c)

$$\begin{array}{cccccccc}\hfill \left(34\mathrm{d}\right)\mathrm{and} \left(29\right)& \Rightarrow \hfill & & |c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}{c}_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}|\hfill & \hfill \le & & & \sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2}\sqrt{c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }{\mathsf{c}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{c}}^{\prime }}^2}\hfill \end{array}$$

(35d)

where again, (34a) and (35a) can be used to express these bounds in terms of individual (single-variable) kurtoses, but this would render the bounds less sharp. Inequalities (35) are related to ([22], Theorem 3, p. 14).

Quantities of the form $c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2$ appear very frequently in the above identities and inequalities. Through construction as the average of a product of squares $c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}\ge 0$ is positive (30c). However, a stricter lower bound can be obtained by combining the inequalities (32b)–(32d) with (35a), viz

$$\begin{array}{c}\hfill \left(32\right)\mathrm{and} \left(35\mathrm{a}\right)\Rightarrow 0\le c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2+max\left(c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2,c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}^2,c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2\right)\le c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}\le 1+\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{F_{{\mathsf{b}}^{\prime }}-1}\end{array}$$

(36a)

where the lower bound includes the maximum for any random variable ${\mathsf{c}}^{\prime }$ other than $\{{\mathsf{a}}^{\prime },{\mathsf{b}}^{\prime }\}$. Notice that (36a) also implies

$$\begin{array}{c}\hfill \left(36\mathrm{a}\right)\Rightarrow 0\le max\left(c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2,c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}^2,c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{c}}^{\prime }}^2\right)\le c_{{\mathsf{a}}^{\prime }{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }{\mathsf{b}}^{\prime }}-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2\le 1+\sqrt{F_{{\mathsf{a}}^{\prime }}-1}\sqrt{F_{{\mathsf{b}}^{\prime }}-1}\end{array}$$

(36b)

since invariably, $-c_{{\mathsf{a}}^{\prime }{\mathsf{b}}^{\prime }}^2\le 0$.

These upper bounds, although sharper than more naive approaches, are still not very restrictive, while the sharper bounds obtained by Lumley [20] are only valid for particular cases. Notice that in the univariate case, sharper estimates are obtained when the pdf is unimodal, e.g., Pearson’s inequality [35] for unimodal distributions is sharpened to $S_{{(·)}^{\prime }}^2\le F_{{(·)}^{\prime }}-\frac{189}{125}$ [21]. This suggests that the above results can probably be slightly sharpened in the case of unimodal trivariate distributions [37].

6. Implications for $({\mathit{p}}^{\prime },{\mathit{\rho }}^{\prime },{\mathit{T}}^{\prime })$-Correlations

The general statistical inequalities (Section 5) and the exact and approximate relations derived from the fluctuating $\mathrm{EoS}$ (Section 4) provide explanations for some of the $\mathrm{DNS}$ results for $\mathrm{TPC}$ flows (Figure 1 and Figure 2) and for the approximation errors (Figure 3, Figure 4 and Figure 5) of the linearized relations (22).

6.1. Profiles of $\mathit{3CCs}$

The analysis in this subsection applies to the specific case of canonical compressible $\mathrm{TPC}$ flow [5]. In a large part of this class of flows, except in the wake region, $c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}$ and $c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}$ are of opposite signs (Figure 2b,d). According to (25a), this is consistent with the fact that $|c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}|$ takes small values everywhere except in the wake region (Figure 2a). In the buffer region ($5⪅y^{★}⪅40$) of compressible $\mathrm{TPC}$ flows, where $|c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}|\ll 1$ (Figure 2a) and ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\approxeq {\mathrm{C}\mathrm{V}}_{T^{\prime }}$ (Figure 1a,c), (25a) implies $c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}\approxeq -c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}$, in agreement with $\mathrm{DNS}$ data (Figure 2b,d). Further away from the wall, as long as $|c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}|\ll 1$ (Figure 2a), (25a) implies the more general relation $c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}\approxeq -c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}$ which takes into account the differences in fluctuation intensities ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}$ and ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$. Nearer to the centerline, $|c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}|$ increases (Figure 2a), modifying this relation, with $c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}$ and $c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}$ being of the same sign in the centerline region (Figure 2b,d). Similarly, $|c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}|\ll 1$ in the buffer zone (Figure 2a) justifies according to (25b) that $c_{p^{\prime }{T}^{\prime }{T}^{\prime }}$ and $c_{p^{\prime }{p}^{\prime }{T}^{\prime }}$ are of the same sign (Figure 2e,g) there, and (25c) explains why $c_{p^{\prime }{\rho }^{\prime }{\rho }^{\prime }}$ and $c_{p^{\prime }{p}^{\prime }{\rho }^{\prime }}$ are of the same sign (Figure 2c,f).

6.2. Linearization Error in Relations Between $\mathit{3CCs}$

The breakdown of the linearized approximation (22f) for $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$, for $\mathrm{TPC}$ flows at ${\overline{M}}_{{\mathrm{CL}}_x}⪆2$ (Figure 4n,o), is easily explained from inequality (36), written for $c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}$:

$$\begin{array}{c}\hfill \left(36\right)\Rightarrow 0\le max\left(c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}^2,c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}^2,c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}^2\right)\le c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}-c_{{\rho }^{\prime }{T}^{\prime }}^2\le 1+\sqrt{F_{{\rho }^{\prime }}-1}\sqrt{F_{T^{\prime }}-1}\end{array}$$

(37)

which provides a lower bound for the linearization error (23f) of the approximate relation for $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ (22f)

$$\begin{array}{c}\hfill \mathrm{error}\left(c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\right)\stackrel{\left(23\mathrm{f}\right)\mathrm{and} \left(37\right)}{\ge }max\left(c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}^2,c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}^2,c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}^2\right){\displaystyle \frac{{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}}{{\mathrm{C}\mathrm{V}}_{p^{\prime }}}}{\mathrm{C}\mathrm{V}}_{T^{\prime }}-c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}{c}_{{\rho }^{\prime }{T}^{\prime }}{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}{\mathrm{C}\mathrm{V}}_{T^{\prime }}\end{array}$$

(38)

The error (23f) is maximal (Figure 4n,o) in the buffer region ($5⪅y^{★}⪅40$) where the peak of ${\mathrm{C}\mathrm{V}}_{T^{\prime }}$ occurs (Figure 1c). In that region, $max(c_{{\rho }^{\prime }{\rho }^{\prime }{T}^{\prime }}^2,c_{{\rho }^{\prime }{T}^{\prime }{T}^{\prime }}^2,c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}^2)\in [1,2.25]$ (Figure 2a,b,d), and the ratio ${\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }}/{\mathrm{C}\mathrm{V}}_{p^{\prime }}\in [3.5,4]$ ([4], Figure 3, p. 458). Combining these values with the very approximate relation ${\mathrm{C}\mathrm{V}}_{T^{\prime }}\approxeq 0.02{\overline{M}}_{{\mathrm{CL}}_x}^2$ (Figure 1c) determines through (38) an estimate of the lower bound of the leading term of the error, $\mathrm{error}\left(c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}\right)⪆\alpha {\overline{M}}_{{\mathrm{CL}}_x}^2$, with $\alpha \in [0.07,0.18]$, which is coherent with the observed errors (Figure 4n,o) of the linearized approximation (23f).

However, the general statistical inequalities (Section 5) do not provide such lower bounds for the other 4CC expressions that determine the leading-order term of the approximation error (23) of the linearized relations (22). Further work is needed to determine whether the difference in the linearized approximation accuracy (Figure 3 and Figure 4) between the robust relations (22a)–(22c) and the fragile relations (22d)–(22f) is specific to the turbulence structure of canonical $\mathrm{TPC}$ flows or more general.

7. Conclusions

This paper provides the mathematical framework for the study of the exact equations satisfied by n-order correlations between fluctuations in the basic thermodynamic variables $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ that are imposed by the dilute-gas $\mathrm{EoS}$, which describes with satisfactory accuracy most aerodynamic flows. All correlations between $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ can be expressed in terms of $({\rho }^{\prime },T^{\prime })$-statistics, but with an increasing correlation order, progressively, a larger number of higher-order $({\rho }^{\prime },T^{\prime })$-statistics are involved.

Systems of $\frac{n(n+1)}{2}$ equations between the $\frac{(n+1)(n+2)}{2}$ correlation coefficients of order n (n$\mathrm{CCs}$) are readily derived from the fluctuating $\mathrm{EoS}$ ($\forall n\in {\mathbb{N}}_{\ge 2}$). The nonlinear terms in the equations for n$\mathrm{CCs}$ are $\propto {\mathrm{C}\mathrm{V}}_{T^{\prime }}$ and contain an $(n+1)$$\mathrm{CC}$. At the weakly compressible turbulence limit, $max({\mathrm{C}\mathrm{V}}_{p^{\prime }},{\mathrm{C}\mathrm{V}}_{{\rho }^{\prime }},{\mathrm{C}\mathrm{V}}_{T^{\prime }})\ll 1$, these systems can be linearized and solved in terms of $n+1$ linearly independent n$\mathrm{CCs}$, implying that a Mach-independent thermodynamic turbulence structure is reached, at a sufficiently low Mach number, for a given type of flow. Nonetheless, the study of the system for $\mathrm{3CCs}$ shows that not all 4-tuples of $\mathrm{3CCs}$ are linearly independent, as illustrated by several relations expressing $c_{p^{\prime }{\rho }^{\prime }{T}^{\prime }}$ in terms of various couples of other 3CCs or in terms of the individual skewnesses. Several possible choices of 4-tuples of linearly independent $\mathrm{3CCs}$ were explored.

However, the Mach number limit, below which the weakly compressible turbulence linearized approximation holds, depends not only on the particular flow but also on the order of correlations that is considered (diminishing with an increasing correlation order). For instance, for compressible turbulent plane channel ($\mathrm{TPC}$) flows, the $\mathrm{2CC}$ system can be linearized with acceptable (respectively, very good) accuracy for ${\overline{M}}_{{\mathrm{CL}}_x}\le 2$ (respectively, ${\overline{M}}_{{\mathrm{CL}}_x}\le 1.5$), whereas for the $\mathrm{3CCs}$ system, these limits are ${\overline{M}}_{{\mathrm{CL}}_x}\le 1.5$ (respectively, ${\overline{M}}_{{\mathrm{CL}}_x}\le 1$). This is easily explained by the fact that the linearization error for the system of $\mathrm{2CCs}$ is proportional to $\mathrm{3CCs}$, whereas the linearization error for the system of $\mathrm{3CCs}$ is proportional to $\mathrm{4CCs}$, which are generally larger. This effect generalizes to the linearization error of the system of n$\mathrm{CCs}$, which is proportionnal to $(n+1)$$\mathrm{CCs}$. With an increasing correlation order n, the thermodynamic turbulence structure of n$\mathrm{CCs}$ becomes Mach-dependent at progressively lower Mach numbers.

Recent results on the identities and bounds of multivariate skewnesses and flatnesses were applied to $\mathrm{3CCs}$ and $\mathrm{4CCs}$ of $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ and used in the analysis of the linearization error of the equations between $\mathrm{3CCs}$ of $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$.

$\mathrm{DNS}$ data for $\mathrm{3CCs}$ of $\{p^{\prime },{\rho }^{\prime },T^{\prime }\}$ in $\mathrm{TPC}$ flows, covering the range ${Re}_{{\tau }^{★}}\in [97,983]$ and ${\overline{M}}_{{\mathrm{CL}}_x}\in [0.32,2.49]$, were analyzed to determine the influence of Mach and Reynolds numbers. These data reveal that the pressure skewness $S_{p^{\prime }}$ robustly defines various zones of the flow and that the temperature pdf ($S_{T^{\prime }}$ and $F_{T^{\prime }}$ data) is not very sensitive to ${\overline{M}}_{{\mathrm{CL}}_x}$, except perhaps near the centerline. The data also show that a canonical $\mathrm{TPC}$ flow reaches a Mach-independent thermodynamic turbulence structure when ${\overline{M}}_{{\mathrm{CL}}_x}\le 0.8$. On the other hand, even at ${Re}_{{\tau }^{★}}=1000$, some statistics, in particular the skewnesses of density $S_{{\rho }^{\prime }}$ and of temperature $S_{T^{\prime }}$, have not reached a Reynolds-asymptotic inner/overlap law yet, highlighting the need for computations at higher Reynolds numbers.