Total and static temperature statistics in compressible turbulent plane channel flow

Abstract

The paper studies the statistics of total and static temperature (total $h_t$ and static $h$ enthalpy) in compressible turbulent plane channel flow using direct numerical simulations (DNS) data covering the range of centreline Mach numbers $0.3\lessapprox \bar {M}_{{{CL}}_x} \lessapprox 2.5$ and Huang–Coleman–Bradshaw friction Reynolds numbers $100\lessapprox Re_{\tau ^\star }\lessapprox 1000$. For this class of very-cold-wall flows, the DNS data for correlation coefficients and joint probability density functions (p.d.f.s) show that $h_t'$ is invariably very strongly correlated with the streamwise velocity fluctuation $u'$, in contrast to static temperature (static enthalpy $h'$) whose correlation with $u'$ weakens rapidly with increasing wall distance. We study various correlations and joint p.d.f.s of $h_t'$ and $h'$ in relation to the fluctuating velocity field, including the turbulent Prandtl number $Pr_{{T}}$, and discuss the predictions of Reynolds analogy. The scaling of the mean enthalpy and the fluctuating enthalpy variance and fluxes with respect to inner and outer velocity scales is investigated. The complex behaviour and scaling of different terms in the transport equations for the enthalpy variance and fluxes are discussed.

1. Introduction

Numerous direct numerical simulations (DNS) studies of compressible wall turbulence, in both channels (Coleman, Kim & Moser 1995; Foysi, Sarkar & Friedrich 2004; Modesti & Pirozzoli 2016; Yao & Hussain 2020) and boundary layers (Duan, Beekman & Martín 2011; Pirozzoli & Bernardini 2013; Zhang, Duan & Choudhari 2018) have provided data for turbulent correlations that are difficult to measure with accuracy near the wall (Smits & Dussauge 2006), and in the case of correlations between higher-order derivatives (Gerolymos & Vallet 2014; Suman & Girimaji 2013) are impossible to obtain experimentally. It is somehow surprising that relatively little has been reported on total temperature $T_t'$ (total enthalpy $h_t'$) correlations with the fluctuating velocity field $u_i'$. The aim of this paper is the systematic study of $h_t'$ statistics in compressible turbulent plane channel (TPC) flow using DNS data.

Independently of the particular bivariate thermodynamics model used for the equation of state, total and static enthalpy are related by

(1.1a)\begin{equation} h_t:=h + \tfrac{1}{2}u_iu_i\implies\left\{\begin{array}{@{}l} \bar{h}_t=\bar{h} + \tfrac{1}{2}\bar{u}_i\bar{u}_i +\tfrac{1}{2}\overline{u_i'u_i'}, \\ h_t' =h'+\bar{u}_i u_i' + \tfrac{1}{2}\left(u_i'u_i'-\overline{u_i'u_i'}\right), \end{array}\right. \end{equation}

readily implying relations for variances and fluxes

(1.1b)\begin{align} \overline{h_t'u_i'}&=\overline{h'u_i'}+\bar{u}_j \overline{u_j'u_i'}+\overbrace{\tfrac{1}{2} \overline{u_j'u_j'u_i'}}^{{3oMs}}, \end{align}

(1.1c)\begin{align} \overline{h'^2}&=\overline{h_t'^2}+\bar{u}_i \bar{u}_j\overline{u_i'u_j'}-2\overline{h_t'u_i'} \bar{u}_i-\underbrace{\left(\overline{h_t'u_i'u_i'}-\bar{u}_i\overline{u_i'u_j'u_j'}\right)}_{{3oMs}} +\underbrace{\tfrac{1}{4}\left(\overline{u_i'u_i'u_j'u_j'}-\overline{u_i'u_i'}\ \overline{u_j'u_j'}\right)}_{{4oMs}},\end{align}

(1.1d)\begin{align} &=\overline{h_t'^2}-\bar{u}_i \bar{u}_j\overline{u_i'u_j'}-2\overline{h'u_i'} \bar{u}_i- \underbrace{\left(\overline{h'u_i'u_i'}+\bar{u}_i\overline{u_i'u_j'u_j'}\right)}_{{3oMs}} -\underbrace{\tfrac{1}{4}\left(\overline{u_i'u_i'u_j'u_j'}-\overline{u_i'u_i'}\ \overline{u_j'u_j'}\right)}_{{4oMs}}, \end{align}

where $h$ is the static enthalpy, and $u_i\in \{u,v,w\}$ are the velocity components in a Cartesian reference frame $x_i\in \{x,y,z\}$. The usual notations $({\cdot} )=\overline {({\cdot} )}+({\cdot} )'=\widetilde {({\cdot} )}+({\cdot} )''$ are adopted for Reynolds or Favre averages and fluctuations, and Einstein's summation convention of repeated indices is applied. The presence of the fluctuation of turbulence kinetic energy $\tfrac {1}{2}(u_i'u_i')':=\tfrac {1}{2}u_i'u_i'-\tfrac {1}{2}\overline {u_i'u_i'}$ in the expression for $h_t'$ in (1.1a) introduces higher-order moments (HoMs): 3-order moments (3oMs) for the fluxes (1.1b) and also 4-order moments (4oMs) for the variances (1.1c). The implication of (1.1) is that total $h'_t$ statistics cannot be reconstructed with DNS accuracy from static $h'$ statistics unless all HoMs are also acquired. It is more practical to sample $h'_t$ statistics directly.

In the classic paper on the strong Reynolds analogy (SRA), Morkovin (1962) identified the assumption of negligible root mean square (r.m.s.) $h'_{t_{rms}}$ as the major structural default of the approximation (Gaviglio 1987), in disagreement with measurements (Kistler 1959). Available data (Guarini et al. 2000; Shahab et al. 2011) show that for an adiabatic wall ($\bar {T}_w\approxeq T_r$, where $T_r$ is the adiabatic wall recovery temperature), $h'_{t_{rms}}/h'_{rms}=O(1)$ in a large part of the layer. As the wall temperature cools down from the adiabatic wall temperature $T_r$ (it is more precise and Mach-independent to say as the non-dimensional wall flux gradient $[d_{\bar {u}}\bar {h}]_w/\bar {u}_\delta$, where $\bar {u}_\delta$ is the mean velocity at the wall layer edge, increases) the ratio $h'_{t_{rms}}/h'_{rms}$ increases relative to the adiabatic level (Debiève et al. 1997, figure 11, p. 56). Turbulent boundary layer (TBL) DNS data (Shahab et al. 2011, figure 15, p. 387) for adiabatic ($\bar {T}_w=T_r$) and isothermal ($\bar {T}_w=\bar {T}_\delta =0.67T_r$) walls shows that there is a very important decrease of $h'_{rms}$ with wall cooling, rather than a strong increase of $h'_{t_{rms}}$. The wall is even cooler for compressible TPC flow, the ratio $T_r/\bar {T}_w$ increasing very strongly with centreline Mach number $\bar {M}_{{{CL}}_x}$ (Gerolymos & Vallet 2023, figure 6, p. A19-15), inducing a slight increase of the ratio $h'_{t_{rms}}/h'_{rms}$ (Huang, Coleman & Bradshaw 1995).

In contrast to this relatively small volume of data for $h_t'$ statistics, there is a very large volume of data for the turbulent Prandtl number $Pr_{{T}}$, both experimental (Meier & Rotta 1971; Bagheri, Strataridakis & White 1992; Wardana, Ueda & Mizomoto 1995) and DNS (Coleman et al. 1995; Guarini et al. 2000; Bernardini & Pirozzoli 2011; Duan et al. 2011; Shahab et al. 2011; Zhang, Duan & Choudhari 2017; Huang, Duan & Choudhari 2022). These data, covering a wide range of Mach numbers and wall temperature conditions (from very cold to very hot walls) in TBLs and TPCs, indicate that $Pr_{{T}}$ diminishes from a value ${\sim }1$ in the buffer layer to a value ${\sim }0.7$ in the wake region, showing little sensitivity to the flow conditions. They also show that the wall-normal flux ($\overline {h'v'}$ or $\overline {\rho h''v''}$) is systematically opposed to the temperature gradient $d_y \bar {T}$, whereas the streamwise flux ($\overline {h'u'}$ or $\overline {\rho h''u''}$) is systematically of the same sign as the temperature gradient $d_y \bar {T}$ (Shahab et al. 2011, figure 17, p. 388). The data also agree that the absolute value of the streamwise flux correlation coefficient $\left \lvert c_{h'u'} \right \rvert$ drops from values near ${\sim }0.9$ close to the wall to lower values ${\lessapprox }0.5$ at $\delta$ (Guarini et al. 2000; Bernardini & Pirozzoli 2011; Huang et al. 2022).

These data contradict the SRA, which postulates (Morkovin 1962; Gaviglio 1987; Huang et al. 1995) an instantaneous relation $h'\propto u'\Rightarrow \left \lvert c_{h'u'} \right \rvert =1$, independently of the particular proportionality function. Regarding this proportionality function, the most successful choice is due to Huang et al. (1995), with the $\mathrm {sgn}(d_{\bar {u}}\bar {h})$ fix of Guarini et al. (2000) allowing the accommodation of both cold and hot walls. Gaviglio (1987, (27), p. 915) remarks that truncating to 2-order moments (2oMs), the relations (1.1) imply in parallel flow that $\overline {h_t'^2}\approxeq \overline {h'^2}+\bar {u}^2\overline {u'^2}+2 \bar {u}\,u'_{rms}\,c_{h'u'}$, and show how the value of the correlation coefficient $c_{h'u'}$ (which, as discussed above, admits large variations) impacts the ratio $h'_{t_{rms}}/h'_{rms}$ (Gaviglio 1987, figure 3, p. 916). This relation also implies that when $\left \lvert c_{h'u'} \right \rvert \approxeq 1$, whether $h'_{t_{rms}}/h'_{rms}\lesseqgtr 1$ is decided by $\mathrm {sgn}(c_{h'u'})$. However, the departure of $\left \lvert c_{h'u'} \right \rvert$ from unity observed in the data can be incorporated only in a correlation-based Reynolds analogy that does not postulate an instantaneous relation between $h'$ and $u'$, but uses relations and approximations for correlations and correlation coefficients instead. Decomposing the instantaneous $h'$ into a correlated (with $u'$) and an uncorrelated part (Zhang et al. 2014) does not solve the problem as it requires further assumptions on the uncorrelated part.

To obtain less cold wall conditions while maintaining the computationally advantageous streamwise invariance of the flow, some authors include an artificial heat-sink source term in the energy equation (Coleman et al. 1995; Yu, Xu & Pirozzoli 2020; Modesti et al. 2022) or apply different upper/lower wall temperatures (Morinishi, Tamano & Nakabayashi 2004; Lusher & Coleman 2022). In the present work, we concentrate on canonical TPC flows (Coleman et al. 1995) between symmetric isothermal walls (Song et al. 2022).

The DNS database (Gerolymos & Vallet 2023) used to acquire $h'$ statistics and $h_t'$ statistics in compressible TPC flow is described in § 2. In § 3, we discuss the effect of frictional heating on the mean flow and compare with other DNS data available in the literature. In § 4, we study the $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})$ dependence of the profiles of variances (§ 4.3), correlation coefficients (§ 4.4) and joint probability density functions (p.d.f.s) with the streamwise velocity (§ 4.5). The data can be analysed (§ 4.3) in terms of the correlation coefficients for $h_t'$ and $h'$ transport (§ 4.1) and of the turbulent Prandtl number (§ 4.2). In § 5.1, we investigate the accuracy of the SRA for the class of canonical TPC flows studied, and discuss its inherent limitations (§ 5.2). For the canonical TPC flows studied, the very strong $(h_t',u')$ correlation revealed by the DNS data (§ 4.4) can be exploited to close the truncated to 2oMs relations, and develop correlations specific to this class of very-cold-wall flows (§ 5.2). In § 6, we study briefly the transport equations for the variance and fluxes of $h'$, and discuss their scaling with $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})$. Finally, in § 7, we summarise the conclusions of the present work and discuss possible perspectives for the extension of a correlation-based Reynolds analogy approach to general parallel flows.

2. DNS data

The DNS code (Gerolymos, Sénéchal & Vallet 2009, 2010) used to acquire the data solves the compressible Navier–Stokes equations with perfect gas constant $c_p$ thermodynamics, for which the fluctuations of enthalpy $h'$ are proportional to the fluctuations of temperature $T'$:

(2.1)\begin{equation} \left.\begin{array}{c@{}} p=\rho R_g T,\\[3pt] c_p=\dfrac{\gamma}{\gamma-1}\,R_g \end{array}\right\}\implies\left\{\begin{array}{@{}c} h'=c_p\,T',\\[3pt] a=\sqrt{\gamma R_g T}, \end{array}\right. \end{equation}

where $R_g=287.04\ \mathrm {m}^{2}\ \mathrm {s}^{-2}\ \mathrm {K}^{-1}$ is the gas constant for air, $\gamma =1.4$ is the isentropic exponent for air, $c_p$ is the heat capacity at constant pressure, $p$ is the pressure, $\rho$ is the density, $T$ is the temperature, and $a$ is the sound velocity. Linear molecular constitutive relations are used for the viscous stresses and heat fluxes, with $\mu _{b}=0$ bulk viscosity, and Sutherland laws for the dynamic viscosity $\mu (T)$ and the heat conductivity $\lambda (T)$ (Gerolymos 1990).

Following Coleman et al. (1995), the TPC flow computations apply constant wall temperature $T_w={\rm const.}$ for all $t$, and bulk density $\rho _{B}:=(\int\kern-3.7pt\int\kern-3.7pt\int _{\mathfrak {V}}\rho \,{\rm d}\kern0.06em x\,{\rm d}y\,{\rm d}z)/\left \lvert \mathfrak{V} \right \rvert ={\rm const.}$ for all $t$ (Gerolymos et al. 2010, (47), p. 791), and adjust the body acceleration $f_{{V}_x}$ at every subiteration to obtain at statistical convergence of the computations a target bulk mass flow (Gerolymos et al. 2010, (48), p. 791), which as $\rho _{B}$ is constant is tantamount to applying a target-mass-averaged bulk velocity $\overline {u_{{B}}}=\overline{({\rho u})_B}/\rho _B$ (table 1).

Table 1. Parameters of the DNS computations, where: $L_x,L_y,L_z$ are the dimensions of the computational domain (directions $x=\text {homogeneous streamwise}$, $y=\text {wall-normal}$, $z=\text {homogeneous spanwise}$); $u,v, w$ are are the velocity components along $x,y,z$; $\delta$ is the channel half-height; $({\cdot} )_w$ denotes wall values and $({\cdot} )_{{CL}}$ centreline values; $({\cdot} )_{{B}}$ denotes bulk (volume) averages; $\rho _{{B}}:=(\int _{0}^\delta \bar {\rho }\,{{\rm d}y})/\delta = {\rm const.}$; $\overline {u_{{B}}}:=\overline {(\rho u)_{{B}}}/\rho _{{B}}=(\int _{0}^\delta \bar {\rho }\tilde {u}\,{{\rm d}y})/(\delta \rho _{{B}})$; $\overline {T_{{B}}}:=\overline {(\rho T)_{{B}}}/\rho _{{B}}= (\int _{0}^\delta \bar {\rho }\tilde {T}\,{{\rm d}y})/(\delta \rho _{{B}})$; $Re_{\tau ^\star }:=\surd (\bar {\rho }_{{CL}}\bar {\tau }_w)\delta /\bar {\mu }_{{CL}}$ is the friction Reynolds number in Huang–Coleman–Bradshaw scaling; $\bar {M}_{{{CL}}_x}:= \overline {u_{{CL}}/a_{{CL}}}$ is the centreline Mach number; $Re_{\tau _w}:=\bar {\rho }_wu_{\tau }\delta /\bar {\mu }_w$ is the friction Reynolds number; $u_\tau :=\surd (\bar {\tau }_w/\bar {\rho }_w)$ is the friction velocity; $Re_{\theta _{{CL}}}:=\bar \rho _{{{CL}}}\tilde {u}_{{CL}}\theta /\bar {\mu }_{{CL}}$ is the momentum-thickness Reynolds number at centreline conditions; $\theta :=\int _0^\delta (1-\tilde {u}/\tilde {u}_{{CL}})\overline {\rho u}/\overline {\rho u}_{{CL}}\,{{\rm d}y}$ is the momentum thickness; $Re_{{{B}}_w}:=\rho _{{B}}\overline {u_{{B}}}\delta /\bar {\mu }_w$ is the bulk Reynolds number; $M_{{{B}}_w}:=\overline {u_{{B}}}/\bar a_w$ is the bulk Mach number at wall sound speed; $T_r$ is the theoretical adiabatic wall temperature ($h_r:=\bar {h}_{{CL}}+\tfrac {1}{2}r_f\bar {u}_{{CL}}^2$, $r_f=0.89$); $B_{q_w}:=\bar {q}_w/(\bar {\rho }_w u_\tau \bar {c}_{p_w}\bar {T}_w)$ is the non-dimensional wall heat flux (Coleman et al. 1995); and $(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$ and $(d_{\bar {u}}\bar {h})|_w/\bar {u}_{{CL}}$ are the non-dimensional enthalpy rise and the wall heat flux parameter.

Although the above conditions fix for each computation the target bulk Mach number $M_{{{B}}_w}$ and bulk Reynolds number $Re_{{{B}}_w}$ (table 1), these parameters were chosen carefully during the construction of the database (Gerolymos & Vallet 2023) to obtain systematic variation of the Huang–Coleman–Bradshaw (HCB) friction Reynolds number $Re_{\tau ^\star }$ (Huang et al. 1995) and the streamwise centreline Mach number $\bar {M}_{{{CL}}_x}$:

(2.2)\begin{equation} \left.\begin{array}{ll@{}} \bar{M}_{{{CL}}_x}:=\overline{\left(\dfrac{u_{{CL}}}{a_{{CL}}}\right)}, & 0.3 \lessapprox\bar{M}_{{{CL}}_x}\lessapprox 2.49,\\[12pt] Re_{\tau^\star}:=\dfrac{\bar{\rho}_{{CL}}}{\bar{\mu}_{{CL}}} \sqrt{\dfrac{\bar{\tau}_w}{\bar{\rho}_{{CL}}}}\delta, & 100\lessapprox Re_{\tau^\star} \lessapprox 1000, \end{array}\right\} \end{equation}

instead, as these parameters (2.2) are more generally relevant, especially with regard to TBLs. The computational box size was $L_x\times L_y\times L_z=8{\rm \pi} \delta \times 2\delta \times 4{\rm \pi} \delta$, where $\{x,y,z\}$ are the {streamwise, wall-normal, spanwise} Cartesian coordinates (the lower wall is located at $y=0$) with corresponding velocity components $\{u,v,w\}$. Details on computational resolution and statistics sampling frequency and interval were reported in Gerolymos & Vallet (2023, table 1, p. A19-8). Results with the DNS solver used (statistics for moments of order 2 and 3, budgets for Reynolds stresses and their dissipation tensor, spectra) have been validated systematically by comparison with other available DNS data (Gerolymos et al. 2010; Gerolymos, Sénéchal & Vallet 2013; Gerolymos & Vallet 2016). The present data on mean and fluctuating temperature are in very good agreement with available data in the databases of Modesti & Pirozzoli (2016) and Yao & Hussain (2020). The consistency of the statistical convergence of the computations is examined and compared with other DNS data in § 3.3. Relevant $h_t'$ statistics and p.d.f.s were acquired directly, with sampling at every computational time step. All moments present in the exact relations (1.1) were also acquired, and the budgets of (1.1) were checked to verify convergence of the statistics.

Following (Huang et al. 1995), inner scaling was based on the system of units defined by

(2.3)\begin{equation} (\cdot)^\star -\text{units}: \{\bar{\tau}_w,\bar{\rho}(y),\bar{\mu}(y)\}, \end{equation}

following accepted practice in compressible wall turbulence (Trettel & Larsson 2016), and were plotted against

(2.4)\begin{equation} y^\star:=\dfrac{\bar{\rho}(y)\sqrt{\dfrac{\bar{\tau}_w}{\bar{\rho}(y)}}\,y}{\bar{\mu}(y)} . \end{equation}

It should be stressed here that when presenting terms including derivatives in $({\cdot} )^\star$ units, there is no transformation of the variables implied or used, but simply the term is made non-dimensional in $({\cdot} )^\star$ units.

3. Frictional heating and mean enthalpy

Before analysing the $h_t'$ and $h'$ correlations, it is useful to identify the parameters characterising the thermal turbulence structure (§ 3.1) and to verify the consistency and accuracy of the computations (§ 3.3).

3.1. Frictional heating

The mean flow equations in the case of steady ($\partial _t\overline {({\cdot} )}=0$) streamwise-invariant ($\partial _x\overline {({\cdot} )}=0$) compressible TPC ($\partial _z\overline {({\cdot} )}=0$) flow between two isothermal walls and equation of state (2.1) read (Huang et al. 1995; Gerolymos & Vallet 2014)

(3.1a)$$\begin{gather} \overline{\rho v}=\bar{\rho}\tilde{v}=0, \end{gather}$$

(3.1b)$$\begin{gather}\dfrac{{\rm d}}{{\rm d}y}\left(-\overline{\rho u'' v''}+\bar{\tau}_{xy}\right)+ \overline{\rho f_{{{V}}_x}}=0\implies\rho_{{B}}\bar{f}_{{{V}}_x}= \dfrac{\bar{\tau}_w}{\delta}, \end{gather}$$

(3.1c)$$\begin{gather}\dfrac{{\rm d}}{{\rm d}y}\left(-\overline{\rho v'' h_t''}+\overline{u_i\tau_{iy}}-\bar{q}_y\right)+ \overline{\rho u f_{{{V}}_x}}=0\implies\rho_{{B}}\overline{u_{{B}}}\,\bar{f}_{{{V}}_x}=- \dfrac{\bar{q}_w}{\delta}, \end{gather}$$

(3.1d)$$\begin{gather}\dfrac{{\rm d}}{{\rm d}y}\left(-\overline{\rho v'' h''}-\bar{q}_y\right)+ \overline{\dfrac{{\rm D}p}{{\rm D}t}}+\overline{\tau_{ij}S_{ij}}=0 \stackrel{(3.2)}{\implies} \bar{q}_w=-\int_0^\delta\left(\overline{\tau_{ij}S_{ij}}-\overline{p\varTheta}\right){{\rm d}y}, \end{gather}$$

for the continuity (3.1a), streamwise ($x$ direction) momentum (3.1b), total energy (3.1c), and static enthalpy (3.1d), respectively. The relations implied in (3.1b) and (3.1c) are obtained readily by integration across the channel (Huang et al. 1995), using the fact that the body acceleration $f_{{{V}}_x}$ does not vary in space but only in time, and for (3.1c) that $\rho _{{B}}={\rm const.}$ for all $t$. They are satisfied upon statistical convergence of the averages. To obtain the integral relation implied in (3.1d), we also used the flow symmetries to simplify the substantial derivative ${\rm D}_tp$:

(3.2)\begin{equation} \overline{\dfrac{{\rm D}p}{{\rm D}t}}\,\stackrel{[\partial_t\overline{({\cdot})}=0]}=\,\overline{u_j\, \dfrac{\partial p}{\partial x_j}}=\dfrac{\partial}{\partial x_j}\left(\,\overline{pu_j}\right)-\overline{p\varTheta} \,\stackrel{[\partial_x\overline{({\cdot})}=\partial_z\overline{({\cdot})}=0]}=\, \dfrac{{\rm d}}{{\rm d}y}\left(\,\overline{pv}\right)-\overline{p\varTheta}, \end{equation}

where $\varTheta :=\partial _{x_j}u_j$ is the dilatation, $S_{ij}$ is the rate of deformation, and $\tau _{ij}$ is the viscous stress. This identifies $\overline {\tau _{ij}S_{ij}}\ggg \left \lvert \overline {p\varTheta } \right \rvert$ as the heat production mechanism, as is also seen in the budgets (figure 1a) of the static enthalpy equation (3.1d). Near the wall ($\,y^\star \lessapprox 5$), $\overline {\tau _{ij}S_{ij}}=\bar {\tau }_{ij}\bar {S}_{ij}+\overline {\tau '_{ij}S'_{ij}}$ is balanced principally by molecular heat conduction $-d_y\bar {q}_y$. Notice that at constant $Re_{\tau ^\star }$, near the wall ($\,y^\star \lessapprox 10$), the heat production term follows $({\cdot} )^\star$ scaling with varying Mach number (figure 1b), i.e. $\overline {\tau _{ij}S_{ij}}^\star :=\overline {\tau _{ij}S_{ij}}/(\bar {\tau }_w^2/\bar {\mu }(y))$ depends essentially on $Re_{\tau ^\star }$ and much less on $\bar {M}_{{{CL}}_x}$. Above the buffer layer ($\,y^\star \gtrapprox 60$), and up to the centreline, there is essentially a balance between heat production by the fluctuating field $\overline {\tau '_{ij}S'_{ij}}$ and turbulent mixing $-d_y(\overline {\rho v'' h''})$ (figure 1a), whereas the pressure term $\overline {{\rm D}_tp}$ is negligible everywhere. Turbulent mixing $-d_y(\overline {\rho v'' h''})$ changes sign at $y^\star \approxeq 25$, opposing molecular heat conduction and enhancing heating in the near-wall region (figure 1a).

Figure 1. (a) Budgets of the static enthalpy (temperature) (3.1d), in $({\cdot} )^\star$ units, plotted against $y^\star$ (log scale), with an outer region zoom plotted against $y/\delta$ (linear), at $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})=(965,1.51)$. (b) Frictional heat generation term $\overline {\tau _{ij}S_{ij}}^{\,\star }$ for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{110,340,1000\}$, from the present DNS database (table 1).

3.2. The $\bar {h}(\bar {u})$ relation

Usually, mean thermal field relations, for attached wall turbulent flows (Zhang et al. 2014, 2017, 2018; Song et al. 2022), TPCs or TBLs, are presented in terms of temperature ratio $\bar {T}(y)/\bar {T}_\delta$, which of course depends strongly on the Mach number $\bar {M}_\delta$. This representation misses the point that working instead in terms of non-dimensional enthalpy differences (figure 2), in the TPC case $(\bar {h}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$, absorbs the largest part of the Mach dependence, results at nearly constant $Re_{\tau ^\star }$ (figures 2b,d,f,h) showing quite weak $\bar {M}_{{{CL}}_x}$ dependence, while results at nearly constant $\bar {M}_{{{CL}}_x}$ (figures 2a,c,e,g) show noticeable $Re_{\tau ^\star }$ dependence, which is discussed further in § 3.3.

Figure 2. Non-dimensional mean enthalpy rise $(\bar {h}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$ plotted against the non-dimensional velocity $\bar {u}/\bar {u}_{{CL}}$, for (a,c,e,g) varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at nearly constant centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and (b,d,f,h) varying Mach numbers $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

It is generally verified that a quadratic polynomial of streamwise velocity $\bar {u}$ that satisfies the wall heat flux $(d_{\bar {u}}\bar {T})|_w$ and the wall and centreline values of temperature fits reasonably well the DNS data for attached wall turbulent flows (Zhang et al. 2017, 2018; Song et al. 2022). Using enthalpies instead, it is straightforward to show that the unique one-sided Hermitian quadratic polynomial (Stoer & Bulirsch 1993, p. 53) that satisfies wall and centreline values and wall flux reads

(3.3)\begin{align} \dfrac{\bar{h}-\bar{h}_w}{\bar{u}_{{CL}}^2}\approxeq\dfrac{1}{\bar{u}_{{CL}}}\left. \dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\right|_w \dfrac{\bar{u}}{\bar{u}_{{CL}}}+ \left(\dfrac{\bar{h}_{{CL}}-\bar{h}_w}{\bar{u}_{{CL}}^2}- \dfrac{1}{\bar{u}_{{CL}}}\left.\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\right|_w\right) \left(\dfrac{\bar{u}}{\bar{u}_{{CL}}}\right)^2+\text{approximation error}, \end{align}

and this is precisely the polynomial for the non-dimensional temperature obtained using physical Reynolds analogy arguments by Zhang et al. (2014). Therefore, to the order of the approximation error, the mean enthalpy field depends on two parameters, namely the non-dimensional wall heat flux $(d_{\bar {u}}\bar {h})|_w/\bar {u}_{{CL}}$ and the wall-to-centreline enthalpy rise $-r_h:=(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$, which in adiabatic wall flows should correspond to the adiabatic recovery factor $r_f$. The DNS data (table 1) show clearly that both these parameters depend principally on $Re_{\tau ^\star }$ and only weakly on $\bar {M}_{{{CL}}_x}$. It will be seen that all of the flows in the database have very similar turbulence structure (§ 4), therefore they should be characterised not by temperature ratios like $\bar {T}_{{CL}}/\bar {T}_w$, which depend very strongly on $\bar {M}_{{{CL}}_x}$, but by $-r_h$ instead.

3.3. Consistency check and comparison with available data

Notice that combining the integral relations for $x$ momentum (3.1b) and total energy (3.1c) readily relates the wall heat flux $\bar {q}_w$ and the wall shear stress $\bar {\tau }_w$ by the exact relation (Huang et al. 1995; Song et al. 2022)

(3.4a)\begin{equation} {} (3.1b),~(3.1c) \implies\dfrac{\bar{q}_w}{\bar{\tau}_w}=-\overline{u_{{B}}} \stackrel{(3.4b)}{\implies} \dfrac{1}{\bar{u}_{{CL}}}\left.\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\right|_w \approxeq Pr_w\,\dfrac{\overline{u_{{B}}}}{\bar{u}_{{CL}}}, \end{equation}

where we used

(3.4b)\begin{equation} -\dfrac{\bar{q}_w}{\bar{\tau}_w}\approxeq\dfrac{\bar{\lambda}_w \left.\dfrac{{\rm d}\bar{T}}{{\rm d}y} \right|_w}{\bar{\mu}_w \left.\dfrac{{\rm d}\bar{u}}{{\rm d}y}\right|_w} \approxeq\dfrac{\bar{\lambda}_w \left.\dfrac{{\rm d}\bar{h}}{{\rm d}y}\right|_w} {\bar{\mu}_w\bar{c}_{p_w}\left.\dfrac{{\rm d}\bar{u}}{{\rm d}y}\right|_w} \approxeq\dfrac{1}{\overline{Pr_w}}\left.\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\right|_w. \end{equation}

In the case of strictly isothermal walls ($T_w={\rm const.}$) used in the present computations, there are no fluctuations of the molecular transport coefficients or heat capacity at the wall, so (3.4b) is exact, and so is the final relation in (3.4a).

Checking whether the DNS results satisfy the exact relation (3.4a) is a good test in assessing proper convergence of the computations (and statistics) regarding the thermal field. The integral relation (3.1d) for the wall heat flux shows that $\bar {q}_w$ is affected by the integration of $\overline {\tau _{ij}S_{ij}}$ across the entire channel, and explains the observed slow convergence of centreline temperature $\bar {T}_{{CL}}$, especially with increasing Reynolds number. The present DNS data satisfy the exact relation (3.4a) with accuracy better than $0.25\,\%\equiv 2.5\,\unicode{x2030}$ (figure 3).

Figure 3. Consistency diagnostic of the present DNS computations (table 1) by verification of the exact (at statistical convergence) relation $\bar {q}_w=-\overline {u_{{B}}}\bar {\tau }_w$ (see (3.4a)) obtained from the integration of the mean momentum and energy equations across the channel (Huang et al. 1995; Song et al. 2022), which also provides the heat flux parameter in the quadratic approximation of the $\bar {h}(\bar {u})$ relation (3.3).

Regarding the prediction of the centreline-to-wall temperature ratio $\bar {T}_{{CL}}/\bar {T}_w$ (figure 4a), the present data (table 1) are in very good agreement with the DNS data of Modesti & Pirozzoli (2016) and Yao & Hussain (2020) ($Pr=0.72$), and of Trettel & Larsson (2016), indicating that it increases approximately as

(3.5)\begin{equation} \dfrac{\bar{T}}{\bar{T}_w}\approxeq1+(0.17\pm0.005)M_{{{B}}_w}^2 . \end{equation}

The small variation of the coefficient in the correlation (3.5) determines the limits of the envelope of the data. These variations in the $M_{{{B}}_w}$ dependence around the average curve actually reveal the $Re$ influence on $\bar {T}_{{CL}}/\bar {T}_w$.

Figure 4. (a) Centreline-to-wall temperature ratio versus $M_{{{B}}_w}$ and (b) non-dimensional enthalpy difference $(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\,Pr_w\,\bar {u}_{{CL}}^2)$ versus $\overline {u_{{B}}}/\bar {u}_{{CL}}$, for the present DNS data (table 1) and other available DNS data of Modesti & Pirozzoli (2016), Trettel & Larsson (2016) and Yao & Hussain (2020) (respectively denoted MP (2016), TL (2016), YH (2020)), covering the ranges $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ and $97\leq Re_{\tau ^\star }\leq 1482$, and comparison with the correlation envelope (3.5) in (a) and the correlation (3.6) of Song et al. (2022) in (b).

Song et al. (2022, (14), p. 6) developed the alternative correlation

(3.6)\begin{equation} \dfrac{\bar{T}}{\bar{T}_w}\approxeq1+1.034\,Pr_w\,\dfrac{\bar{u}_{{CL}}}{\overline{u_{{B}}}}\, M_{{{B}}_w}^2\iff\dfrac{\bar{h}_{{CL}}-\bar{h}_w}{\tfrac{1}{2}\bar{u}_{{CL}}^2} \approxeq1.034\,Pr_w\,\dfrac{\overline{u_{{B}}}}{\bar{u}_{{CL}}} \end{equation}

that was rewritten here as (3.6) in terms of the enthalpy rise parameter $-r_h:=(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$, showing clearly that $r_h$ in the correlation of Song et al. (2022) depends not on $M_{{B}_w}$, but on $\overline {u_{{B}}}/\bar {u}_{{CL}}$ instead, which is tantamount to including a $Re$ effect because of the strong sensitivity of the ratio $\overline {u_{{B}}}/\bar {u}_{{CL}}$ on $Re_{\tau ^\star }$, especially in this relatively low $Re_{\tau ^\star }\lessapprox 1482$ range (table 1). Closer examination of the data (figure 4b) reveals that there still exists a slight $\bar {M}_{{{CL}}_x}$ sensitivity. The value of the ratio $\overline {u_{{B}}}/\bar {u}_{{CL}}$ used in (3.6) was calculated carefully by integrating (table 1) the density and mass flux profiles in the databases of Modesti & Pirozzoli (2016) and Yao & Hussain (2020), and is slightly different to the values tabulated in Song et al. (2022).

The empirical correlation coefficient $1.034$ in (3.6) is probably a little too high at the lower $Re$ (lower $\overline {u_{{B}}}/\bar {u}_{{CL}}$) range (figure 4b). The data of Yao & Hussain (2020) at the higher $Re_{\tau ^\star }\gtrapprox 1000$ ($\overline {u_{{B}}}/\bar {u}_{{CL}}\gtrapprox 0.88$) indicate (figure 4b) not only a steepening of the slope but also a crossing between the $\bar {M}_{{{CL}}_x}\approxeq 1.46$ and $\bar {M}_{{{CL}}_x}\approxeq 0.87$ curves, which was not observed in the present database. This highlights the need for higher $Re$ data to fill the gap with incompressible DNS data, which are currently at $Re_{\tau _w}\approxeq 10\,000$ (Hoyas et al. 2022), tenfold higher than the compressible databases (Modesti & Pirozzoli 2016; Trettel & Larsson 2016; Yao & Hussain 2020; Gerolymos & Vallet 2014, 2023).

4. The $h'_t$ and $h'$ statistics

DNS data (Huang et al. 1995) show that TPC flow without any artificial source term in the energy equation (Yu et al. 2020) is a very-cold-wall (VCW) flow, in the sense that

(4.1)\begin{equation} \text{VCW}: \quad \dfrac{{\rm d}\bar{T}}{{\rm d}y}>0\stackrel{(2.1)}{\implies} \dfrac{{\rm d}\bar{h}}{{\rm d}y}>0\implies\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}>0 \implies\left\{\begin{array}{@{}l} \overline{h'u'} >0,\\ \overline{h'v'} <0,\\ \overline{h_t'u'}>0,\\ \overline{h_t'v'}<0, \end{array}\right.\quad \forall\ y\in\,]0,\delta[, \end{equation}

where the usual attached flow change of variables $y=y(\bar {u})$, based on the monotonicity of $\bar {u}(y)$, was made. This relation (4.1) will be used to define VCW conditions, characteristic of the flows investigated in the paper.

For this class of flows, $h'_{t_{rms}}>h'_{rms}$ (§ 4.3), in contrast to adiabatic or isothermal wall conditions (Shahab et al. 2011). The study of correlation coefficients (§ 4.4) and joint p.d.f.s (§ 4.5) reveals a very strong positive correlation between $h_t'$ and $u'$ for all $y$, also verified for higher-order statistics.

We note

(4.2a)$$\begin{gather} (\cdot)'_{rms}:=\sqrt{\overline{({\cdot})'^2}}, \end{gather}$$

(4.2b)$$\begin{gather}c_{({\cdot})'[{\cdot}]'}:=\dfrac{\overline{({\cdot})'[{\cdot}]'}} {\sqrt{\overline{({\cdot})'^2}}\sqrt{\overline{[{\cdot}]'^2}}}\equiv\dfrac{\overline{({\cdot})'[{\cdot}]'}} {(\cdot)'_{rms}\,[{\cdot}]'_{rms}}\,\in[-1,1], \end{gather}$$

i.e. the r.m.s. of $({\cdot} )'$ and the correlation coefficient (CC) between the fluctuations of two flow quantities $[\cdot ]'$ and $({\cdot} )'$.

4.1. Exact and truncated relations in nearly parallel flow

The general expressions (1.1), under the simplification of parallel ($\bar {v}=0$) or nearly parallel ($\left \lvert \bar {v} \right \rvert \lll \bar {u}$) two-dimensional ($\bar {w}=0$) flow lead to the usual shear flow Reynolds analogies (Huang et al. 1995), i.e. the evaluation of thermal ($h'$) correlations from velocity correlations and the mean velocity and enthalpy profiles.

The TPC flow is almost exactly parallel in the mean ($\tilde {v}=0\Rightarrow \bar {v}= \overline {v''}\lll \bar {u}\,\forall y>0$), so the exact relations for $h_t'$ and its correlations simplify to the parallel flow relations

(4.3a)$$\begin{gather} \bar{h}_t =\bar{h} + \tfrac{1}{2}\bar{u}^2 +\tfrac{1}{2}\overline{u_j'u_j'}, \end{gather}$$

(4.3b)$$\begin{gather}h_t' = h'+\bar{u}\, u' + \tfrac{1}{2}\left(u_j'u_j'-\overline{u_j'u_j'}\right) , \end{gather}$$

(4.3c)$$\begin{gather}\overline{h_t'u'}= \overline{h'u'}+\bar{u}\,\overline{u'^2}+\tfrac{1}{2}\overline{u_j'u_j'u'} , \end{gather}$$

(4.3d)$$\begin{gather}\overline{h_t'v'}= \overline{h'v'}+\bar{u}\,\overline{u'v'}+\tfrac{1}{2}\overline{u_j'u_j'v'} , \end{gather}$$

(4.3e)$$\begin{gather}\overline{h'^2}=\overline{h_t'^2}+\overline{u'^2}\,\bar{u}^2-2\overline{h_t'u'} \bar{u}-\left(\overline{h_t'u_j'u_j'}-\bar{u}\overline{u_j'u_j'u'}\right) +\tfrac{1}{4}\left(\overline{u_i'u_i'u_j'u_j'}-\overline{u_i'u_i'}\ \overline{u_j'u_j'}\right) , \end{gather}$$

(4.3f)$$\begin{gather}\quad =\overline{h_t'^2}-\overline{u'^2}\,\bar{u}^2-2\overline{h'u'} \bar{u}-\left(\overline{h'u_j'u_j'}+\bar{u}\overline{u_j'u_j'u'}\right) -\tfrac{1}{4}\left(\overline{u_i'u_i'u_j'u_j'}-\overline{u_i'u_i'}\ \overline{u_j'u_j'}\right) . \end{gather}$$

As usual, these relations can be simplified further by omitting HoMs, to write approximately

(4.4)\begin{equation} \bar{h}_t \approxeq \bar{h} + \tfrac{1}{2}\bar{u}^2 +\text{{HoMs}} \end{equation}

for the mean flow, and

(4.5a)$$\begin{gather} \overline{h_t'u'}\approxeq \overline{h'u'}+\bar{u}\,\overline{u'^2}+\text{{HoMs}}, \end{gather}$$

(4.5b)$$\begin{gather}\overline{h_t'v'}\approxeq \overline{h'v'}+\bar{u}\,\overline{u'v'}+\text{{HoMs}}, \end{gather}$$

(4.5c)$$\begin{gather}\overline{h'^2}\approxeq\overline{h_t'^2}+\overline{u'^2}\,\bar{u}^2-2\overline{h_t'u'} \bar{u}+\text{{HoMs}}, \end{gather}$$

(4.5d)$$\begin{gather}\overline{h_t'^2}\approxeq\overline{h'^2}+\overline{u'^2}\,\bar{u}^2+2\overline{h'u'} \bar{u}+\text{{HoMs}}, \end{gather}$$

for 2oMs. In the following, we will refer to relations (4.5) and to relations that are derived directly from these as truncated-to-2oMs. The HoMs are not always negligibly small, especially near the peaks of the correlation profiles, and also near the centreline (wake region), but (4.4) does remain a satisfactory working approximation to analyse relations between fluctuation amplitudes and CCs, and is the starting point for the development of Reynolds analogies. Introducing CCs

(4.6a)$$\begin{gather} c_{h'u'} :=\dfrac{\overline{h'u'}} { h_{rms}' \,u'_{rms}},\quad c_{h_t'u'}:= \dfrac{\overline{h_t'u'}}{h_{t_{rms}}'\,u'_{rms}}, \end{gather}$$

(4.6b)$$\begin{gather}c_{h'v'} :=\dfrac{\overline{h'v'}} { h_{rms}' \,v'_{rms}},\quad c_{h_t'v'}:= \dfrac{\overline{h_t'v'}}{h_{t_{rms}}'\,v'_{rms}},\quad c_{ u'v'}:=\dfrac{\overline{ u'v'}}{ u'_{rms} \,v'_{rms}} \end{gather}$$

to replace the corresponding correlations, we may rewrite the working relations (4.5) as

(4.7a)$$\begin{gather} {(4.5a)}\iff c_{h_t'u'}\,h_{t_{rms}}'\approxeq c_{h'u'}\,h_{rms}'+\bar{u}\,u'_{rms}, \end{gather}$$

(4.7b)$$\begin{gather}{(4.5b)}\iff c_{h_t'v'}\,h_{t_{rms}}'\approxeq c_{h'v'}\,h_{rms}'+c_{u'v'}\,\bar{u}\,u'_{rms}, \end{gather}$$

(4.7c)$$\begin{gather}{(4.5c)}\iff \overline{h'^2}\approxeq\overline{h_t'^2}+\overline{u'^2}\, \bar{u}^2-2c_{h_t'u'}\,h_{t_{rms}}'\,\bar{u}\,u'_{rms}, \end{gather}$$

(4.7d)$$\begin{gather}{(4.5d)}\iff \overline{h_t'^2}\approxeq\overline{h'^2}+\overline{u'^2} \,\bar{u}^2+2c_{h'u'}\,h_{rms}'\,\bar{u}\,u'_{rms}. \end{gather}$$

These relations readily provide a weak Reynolds analogy, based on CCs (Gaviglio 1987). Notice that (4.7c) was reported in Gaviglio (1987, (27), p. 915). The truncated-to-2oMs relations (4.7c) and (4.7d) are not strictly equivalent, because the HoMs neglected in each of them are different.

4.2. Wall-normal transport and turbulent Prandtl numbers

In nearly parallel flow, the ratio $\overline {h'v'}/\overline {u'v'}$, between wall-normal transport of temperature and streamwise velocity, is related directly to the turbulent Prandtl number

(4.8)\begin{equation} Pr_{{T}}:=\dfrac{\overline{\rho u'' v''}}{\overline{\rho h'' v''}}\,\dfrac{\dfrac{{\rm d}\tilde{h}}{{\rm d}y}}{\dfrac{{\rm d}\tilde{u}}{{\rm d}y}}= \dfrac{\widetilde{u'' v''} }{\widetilde{h'' v''} }\,\dfrac{\dfrac{{\rm d}\tilde{h}}{{\rm d}y}}{\dfrac{{\rm d}\tilde{u}}{{\rm d}y}} . \end{equation}

Except in the near-wall region ($\,y^\star \lessapprox 10$), there is little influence of $\bar {M}_{{{CL}}_x}$ on $Pr_{{T}}$ (figure 5), data for varying $\bar {M}_{{{CL}}_x}$ at nearly constant $Re_{\tau ^\star }$ practically collapsing on a single curve, against both inner-scaled $y^\star$ (figures 5c,g,k,o) and outer-scaled $y/\delta$ wall distance (figures 5d,h,l,p). In contrast, $Re_{\tau ^\star }$ has a strong influence on $Pr_{{T}}$. At $Re_{\tau ^\star }\approxeq 1000$ ($\bar {M}_{{{CL}}_x}\in \{0.81,1.51\}$), as the log region emerges in the mean velocity profile (Krogstad & Torbergsen 2000; Lee & Moser 2015; Hoyas et al. 2022), the $Pr_{{T}}(y/\delta )$ profile changes substantially in the range $0.2\delta \lessapprox y\lessapprox 0.5\delta$, compared to lower-$Re$ data (figures 5f,j). For $Re_{\tau ^\star }\approxeq 250$, the outer part of the flow seems to follow a consistent outer law

(4.9)\begin{equation} Re_{\tau^\star}\gtrapprox 250\implies\left\{\begin{array}{@{}ll} Pr_{{T}}\approxeq0.9-0.32\left(\dfrac{y}{\delta}\right)^2, & 0.5\delta\lessapprox y\lessapprox0.8\delta,\\[10pt] Pr_{{T}}\approxeq0.7, & 0.8\delta\lessapprox y, \end{array}\right. \end{equation}

similar to the relation proposed (from limited data and based on indirect assessment) by Rotta (1964); the present DNS data (figures 5b,f,j,n) suggest a different constant, $0.32$, in lieu of $0.4$ in Rotta (1964). For the higher $Re_{\tau ^\star }\approxeq 1000$, as the buffer region shrinks closer to the wall in the outer-scaled $y/\delta$ profiles (figures 5f,j), the data nearer to the wall ($0.2\delta \lessapprox y\lessapprox 0.5\delta$) are slightly lower than (4.9). Additional DNS data at higher $Re_{\tau ^\star }$ are needed to determine the asymptotic high-$Re$ form of the outer law, eventually modifying both constants in (4.9) to fit a larger $y/\delta$ range. The buffer region seems to follow an inner-scaled profile (figures 5a,e,i,m), but higher $Re_{\tau ^\star }$ data are required to ascertain the high-$Re$ asymptotics of the peak observed at $40\lessapprox y^\star \lessapprox 45$ and of the valley observed at $17\lessapprox y^\star \lessapprox 23$ (figures 5a,e,i,m). Finally in the near-wall region ($\,y^\star \lessapprox 10$), $Pr_{{T}}$ shows a complex dependence on both $Re_{\tau ^\star }$ and $\bar {M}_{{{CL}}_x}$ (figure 5).

Figure 5. Turbulent Prandtl number $Pr_{{T}}$ (4.8), plotted against inner-scaled ($\,y^\star$, log scale) and outer-scaled ($\,y/\delta$, linear) wall distance, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at nearly constant centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

In lieu of the above usual definition based on Favre averages (4.8), it is the ratio of Reynolds-averaged transport that appears in (4.7b), i.e.

(4.10)\begin{equation} Pr_{h'}:=\dfrac{\overline{u'v'}}{\overline{h'v'}}\,\dfrac{\dfrac{{\rm d}\bar{h}}{{\rm d}y}}{ \dfrac{{\rm d}\bar{u}}{{\rm d}y}}\equiv \dfrac{\overline{u'v'}}{\overline{h'v'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}} , \end{equation}

where the attached flow transformation $y=y(\bar {u})$ was used. Generally, $Pr_{h'}$ differs very little from $Pr_{{T}}$ (figure 6), even for the highest available $\bar {M}_{{{CL}}_x}=2.49$ (figure 6f), and this is true for the entire database. Very small differences are observed only around the buffer-region valley (figure 6) which occurs close to the peak of the streamwise Reynolds stress $\overline {\rho u''u''}$.

Figure 6. Comparison of turbulent Prandtl numbers $Pr_{{T}}$ (4.8) using Favre averages, and $Pr_{h'}$ (4.10) using Reynolds averages, plotted against inner-scaled wall distance $y^\star$ (log scale), for selected flows in the database (table 1), covering the ranges $113\leq Re_{\tau ^\star }\leq 985$ and $0.35\leq \bar {M}_{{{CL}}_x}\leq 2.49$.

Introducing CCs (4.6b) in the definition of $Pr_{h'}$ (4.10) yields readily an exact expression for $h'_{rms}$,

(4.11a)\begin{equation} h_{rms}'\stackrel{(4.6b),~(4.10)} = \left\lvert \dfrac{c_{u'v'}} {c_{h'v'}}\,\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}} \right\rvert\,u'_{rms},\end{equation}

which, combined with the truncated-to-2oMs wall-normal transport relation (4.7b), yields an approximate expression for $h_{t_{rms}}'$,

(4.11b)\begin{equation} h_{t_{rms}}'\stackrel{(4.5b),~(4.10)}{\approxeq} \left\lvert \dfrac{c_{ u'v'}} {c_{h_t'v'}}\left(\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}+\bar{u}\right) \right\rvert u'_{rms}+\text{{HoMs}}, \end{equation}

which forms a weak Reynolds analogy, based on the knowledge of the ratios of the wall-normal transport CCs ($c_{ u'v'}/c_{h'v'}$, $c_{u'v'}/c_{h_t'v'}$) and of the Prandtl number $Pr_{h'}$.

4.3. Fluctuation intensities

Outer-scaling $h'_{t_{rms}}$ or $h'_{rms}$ by $\bar {u}_{{CL}}^2$ is very successful in accounting for $\bar {M}_{{{CL}}_x}$ effects (figures 7d,h,l,p), collapsing data for varying $\bar {M}_{{{CL}}_x}$ at nearly constant $Re_{\tau ^\star }$, except for the near-wall ($\,y^\star \lessapprox 20$) region (figures 7c,g,k,o).

Figure 7. Root mean square fluctuation intensities of total $h'_{t_{rms}}$ and static $h'_{rms}$ enthalpy, scaled by $\bar {u}^2_{{CL}}$, plotted against inner-scaled ($\,y^\star$, log scale) and outer-scaled ($\,y/\delta$, linear) wall distance, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at nearly constant centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

On the other hand, there are important differences in $Re_{\tau ^\star }$ scaling between $h'_{t_{rms}}$ and $h'_{rms}$. The peak of $h'_{rms}/\bar {u}_{{CL}}^2$ reaches an asymptotic level, confirmed by the highest $Re_{\tau ^\star }\approxeq 1000$ ($\bar {M}_{{{CL}}_x}\in \{0.81,1.51\}$) data (figures 7e,i). In contrast, the peak of $h'_{t_{rms}}/\bar {u}_{{CL}}^2$ decreases with increasing $Re_{\tau ^\star }$, suggesting a different inner scaling for $h'_{t_{rms}}$ (figures 7e,i). The most important difference is observed for the higher available $Re_{\tau ^\star }\approxeq 1000$ data, where $h'_{t_{rms}}$ starts forming a plateau in the range $100\lessapprox y^\star \lessapprox 400$ (figures 7e,i), suggesting an asymptotic high-$Re$ form. The centreline intensity of $h'_{t_{rms}}/\bar {u}_{{CL}}^2$ also seems to reach an asymptotic level ${\sim }0.04$. In contrast, $h'_{rms}$ does not follow this trend, but instead decays continuously from peak to centreline (figures 7b,f,j,n), to much lower levels than $h'_{t_{rms}}$. As a consequence, the ratio $h'_{t_{rms}}/h'_{rms}=T'_{t_{rms}}/T'_{rms}$ increases from 1 at the wall (where $u_i=0$ by the no-slip condition) to quite high values at the centreline (figure 8). The $h'_{t_{rms}}/h'_{rms}$ data at nearly constant $Re_{\tau ^\star }$ (figures 8d,h,l,p) show very weak $\bar {M}_{{{CL}}_x}$ influence, in contrast with data at nearly constant $\bar {M}_{{{CL}}_x}$ (figures 8b,f,j,n), which show, in the outer part of the flow ($\,y\gtrapprox \tfrac {1}{2}\delta$), noticeable increase of the ratio $h'_{t_{rms}}/h'_{rms}$ with increasing $Re_{\tau ^\star }$.

Figure 8. Ratio of total-to-static enthalpy (temperature) fluctuation intensities $h'_{t_{rms}}/h'_{rms}=T'_{t_{rms}}/T'_{rms}$, plotted against inner-scaled ($\,y^\star$, log scale) and outer-scaled ($\,y/\delta$, linear) wall distance, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at nearly constant centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

The ratio $h'_{t_{rms}}/h'_{rms}$ is a structure parameter of the turbulent flow that can be expressed in terms of correlation coefficients by (4.7). Squaring (4.7a) relating the streamwise fluxes (4.5a) yields

(4.12)\begin{equation} (4.7a) \implies 2c_{h'u'}\,h_{rms}'\,\bar{u}\,u'_{rms}+ \overline{u'^2}\,\bar{u}^2\approxeq c^2_{h_t'u'}\,\overline{h_t'^2} - c^2_{h'u'}\,\overline{h'^2}, \end{equation}

and replacing (4.7d) in (4.12) yields

(4.13)\begin{align}(4.7d),~(4.7a),~(4.12) \implies (1-c^2_{h'u'})\overline{h'^2} \approxeq (1-c^2_{h_t'u'})\,\overline{h_t'^2}\implies \dfrac{h'_{t_{rms}}}{h'_{rms}}\approxeq\sqrt{\dfrac{1-c^2_{h'u'}}{1-c^2_{h_t'u'}}} . \end{align}

The only approximation in (4.13) comes from neglecting HoMs in (4.3) to obtain (4.7). By (4.13), for this particular class of VCW flows (4.1), the observed large values of the $h'_{t_{rms}}/h'_{rms}$ ratio (figure 8) imply that $c^2_{h_t'u'}$ is much closer to $1$ than $c^2_{h'u'}$, i.e. that in VCW turbulence, $h_{t'}$ is expected to be correlated very strongly with $u'$. This is in contrast with adiabatic wall flows (Zhang et al. 2014), where $h'$ is correlated very strongly with $u'$, and the ratio $h'_{t_{rms}}/h'_{rms}$ is small. This analysis suggests examining the CCs of static and total enthalpy–transport in order to gain further insight into the turbulence structure of the flow (§ 4.4).

4.4. Correlation coefficients

Certainly the most important result observed in the DNS data is that $h_t'$ is very strongly correlated with $u'$ (figure 9), for all $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})$ in the database (table 1) everywhere in the channel (for all $y$). This result is specific to the VCW condition of the TPC flows studied in this work, and does not apply to adiabatic wall turbulence. In the wake region (figures 9a,e,i,m), the CC $c_{h_t'u'}$ (4.6) is very close to $1$ ($c_{h_t'u'}\approxeq 0.997$), with slightly lower, $Re_{\tau ^\star }$-dependent, but still extremely close to $1$, values at the wall ($c_{h_t'u'}\gtrapprox 0.98$ for all $Re_{\tau ^\star }$). The very strong $(h_t',u')$ correlation is further confirmed by the close similarity of the profiles of the CCs (4.6) $c_{h_t'v'}$ and $c_{u'v'}$ (figures 9b,f,j,n), whose ratio is invariably very close to $1$ ($c_{u'v'}/c_{h_t'v'}\sim 1\pm 4\,\%$ for all $y^\star \gtrapprox 5$), except closer to the wall ($\kern0.7pt y^\star \lessapprox 5$) where slightly higher values are observed (not plotted). Similarly to the outer-scaled fluctuation levels $h'_{t_{rms}}/\bar {u}_{{CL}}^2$ and $h'_{rms}/\bar {u}_{{CL}}^2$ (figures 7c,g,k,o), $c_{h_t'u'}$ (figures 9c,g,k,o) and the ratio $c_{u'v'}/c_{h_t'v'}$ (figures 9d,h,l,p) show little sensitivity to $\bar {M}_{{{CL}}_x}$, at constant $Re_{\tau ^\star }$, except in the near-wall region ($\kern0.7pt y^\star \lessapprox 10$) for $\bar {M}_{{{CL}}_x}\gtrapprox 2$.

Figure 9. The CC $c_{h_t'u'}$ of streamwise $h_t'$ transport and ratio of CCs $c_{u'v'}/c_{h_t'v'}$ of wall-normal transport of momentum and total enthalpy, plotted against inner-scaled ($\kern0.7pt y^\star$, log scale) wall distance, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at nearly constant centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

The fact that the deviation of the ratio $c_{u'v'}/c_{h_t'v'}$ (figures 9b,f,j,n) from unity is larger than that of the CC $c_{h_t'u'}$ (figures 9a,e,i,m) highlights the existence of an uncorrelated (with $u'$) part of $h_t'$, which correlates differently with $v'$. Experience with the analysis of thermodynamic fluctuations in compressible turbulence (Gerolymos & Vallet 2018) shows that assuming CCs strictly $=\pm 1$ always leads to singular behaviour. The small deviations of $c_{h_t'u'}$ and $c_{u'v'}/c_{h_t'v'}$ from unity are essential in (4.13) and other truncated-to-2oMs relations (4.7), (4.11b), as these would be reduced to SRA relations by setting these coefficients equal to exactly 1 (§ 5.2).

In contrast to $h_t'$ (figure 9), $h'$ is not correlated in any particular way to $u'$ (figure 10), $c_{h'u'}$ decreasing from ${\sim }1$ at the wall to small values (${\sim }0.2$) at the centreline, while the ratio $c_{u'v'}/c_{h'v'}=c_{u'v'}/c_{T'v'}$ increases significantly from ${\sim }1$ near the wall to ${\sim }1.45$ at the centreline. Both $c_{h'u'}$ and the ratio $c_{u'v'}/c_{h'v'}$ vary with $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})$. Therefore, weak Reynolds analogies specific to the class of flows studied in the paper (see (4.1)) could be constructed based on $h_t'$ correlations, rather than on direct instantaneous $h'$ analogies as in the SRA (Huang et al. 1995).

Figure 10. The CC $c_{h'u'}$ of streamwise $h'$ transport and ratio of CCs $c_{u'v'}/c_{h'v'}$ of wall-normal transport of momentum and static enthalpy, plotted against outer-scaled ($\kern0.7pt y/\delta$, linear) wall distance, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

4.5. Joint p.d.f.s

Further insight into the very strong correlation between $h_t'$ and $u'$ (figure 9) is gained by examining the joint p.d.f. $f_{h_t'u'}$ and comparing it with the joint p.d.f. $f_{h'u'}$ between $h'$ and $u'$ (figure 11), whose correlation diminishes significantly moving away from the wall (figure 10). Those p.d.f. bins for which no hits were recorded were left blank, highlighting the extreme events boundaries.

Figure 11. Joint p.d.f.s ($\log _{10}$) of streamwise velocity $u'$ and enthalpy (total $h_t'$ and static $h'$) fluctuations and integrands for the calculation of the CCs $c_{h_t'u'}$ (4.14a) and $c_{h'u'}$ (4.14b), plotted against the standardised variables ($u'/u'_{rms}$, $h'_t/h'_{t_{rms}}$, $h'/h'_{rms}$), at different inner-scaled wall distances $y^\star \in \{1,15,30,100,\delta ^\star \}$, at $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})=(965,1.51)$, from the present DNS database (table 1).

The CCs can be calculated by integration of these joint p.d.f.s:

(4.14a)$$\begin{gather} c_{h_t'u'}\stackrel{(4.2a)}= \int_{h'_{t_{min}}}^{h'_{t_{max}}} \int_{u'_{min}}^{u'_{max}} h_t'u'\,f_{h_t'u'}(h_t',u')\,{\rm d}\left(\dfrac{h_t'}{h'_{t_{rms}}}\right) {\rm d}\left(\dfrac{u'}{u'_{rms}}\right), \end{gather}$$

(4.14b)$$\begin{gather}c_{h'u'}\stackrel{(4.2a)}=\int_{h'_{min}}^{h'_{max}} \int_{u'_{min}}^{u'_{max}}h'u'\,f_{h'u'}(h',u')\,{\rm d}\left(\dfrac{h'}{h'_{rms}}\right) {\rm d}\left(\dfrac{u'}{u'_{rms}}\right). \end{gather}$$

The joint p.d.f. $f_{h_t'u'}$ (figures 11a,e,i,m,q) shows that all $(u',h_t')$ events are very tightly clustered along the diagonal of the positive ($u'h_t'>0$) quadrants, a level of very small probability $f_{h_t'u'}=10^{-4}$ being reached at distances $(\left \lvert u' \right \rvert <\tfrac {1}{2}u'_{rms},\left \lvert h_t' \right \rvert <\tfrac {1}{2}h'_{t_{rms}})$ from the diagonal (figures 11a,e,i,m,q). Furthermore, except very near the wall ($\kern0.7pt y^\star <15$, figure 11a), no events are observed at distances greater than $(u'_{rms},h'_{t_{rms}})$ from the diagonal, these distances shrinking when moving away from the wall (figures 11e,i,m,q). Since events in the negative ($u'h_t'<0$) quadrants occur only very close to the centre $(u',h_t')=(0,0)$, the integrand $h_t'u'\,f_{h_t'u'}(h_t',u')$ for the evaluation of $c_{h_t'u'}$ in (4.14a) shows two positive peaks elongated along the diagonal, one in each of the positive ($u'h_t'>0$) quadrants (figures 11c,g,k,o,s). Notice that very near the wall ($\kern0.7pt y^\star \approxeq 1$, figure 11c), rare events appear much further above the diagonal, albeit with very low probability of occurrence (figure 11a). In the near-wall and buffer regions (figures 11a,e), off-diagonal $(u',h_t')$ events have slightly higher probability $f_{h_t'u'}$ than further away from the wall (figures 11m,q), leading to the observed (figure 9) slightly higher deviation of $c_{h_t'u'}$ from unity.

Very near the wall ($\kern0.7pt y^\star \approxeq 1$, figure 11), where $(u^2)'\ll h'\Rightarrow h_t'\approxeq h'$ by (4.3e), the joint p.d.f.s $f_{h_t'u'}$ and $f_{h'u'}$ are very similar (figures 11a,b). However, moving away from the wall, $f_{h'u'}$ spreads away from the diagonal, the more so with increasing distance from the wall (figures 11f,j,n,r). As a result, events in the negative ($u'h'<0$) quadrants occur with increasing probability (figures 11d,h,l,p,t), contributing negatively to the integrand $h'u'\,f_{h'u'}(h',u')$ for the evaluation of $c_{h'u'}$ in (4.14b), and causing the progressive decorrelation between $h'$ and $u'$ (figure 10) with increasing distance from the wall.

It should be noted that this very strong $(u',h_t')$ correlation is specific to the class of VCW flows studied. Scatter plots in TBL studies with adiabatic and progressively colder walls (Hadjadj et al. 2015, figure 21, p. 435) show clearly that as the wall gets hotter towards adiabatic, the $(u',h_t')$ events scatter progressively further and further away from the diagonal, reducing the CC.

5. Successes and limitations of the SRA

Weak Reynolds analogy forms use CCs to obtain quite accurate relations between turbulent moments (4.7), (4.13). Ultimately, both $h'_{t_{rms}}$ and $h'_{rms}$ are proportional to $u'_{rms}$ (4.11), but with proportionality coefficients that are functions of CCs of velocity and enthalpy transport. The SRAs postulate instead some direct instantaneous proportionality between $h'$ and $u'$ (§ 5.1). Although the SRA is generally successful in predicting $h'_{rms}$, it may be less successful in predicting $h'_{t_{rms}}$ for less-cold-wall flows, or the ratio $h'_{t_{rms}}/h'_{rms}$, especially in the outer part of the flow. This drawback can be traced to the inherent error in the prediction of CCs by the SRA that can be corrected only by using weak Reynolds analogy approaches (§ 5.2).

5.1. HCB-SRA

The SRA proposed by Huang et al. (1995, (4.8)–(4.10), p. 208) reads

(5.1)\begin{equation} \left.\dfrac{T'\,\bar{u}}{\bar{T}\,(\gamma-1)\,\bar{M}^2\,u'}\right|_{{SRA}} \approxeq\dfrac{1}{Pr_{h'}}\,\dfrac{\dfrac{{\rm d}\bar{T}}{{{\rm d}y} }} {\dfrac{{\rm d}\bar{T}_t}{{{\rm d}y}}-\dfrac{{\rm d}\bar{T}}{{{\rm d}y}}} \stackrel{(2.1),~(4.4)}{\implies} h'|_{{SRA}}\approxeq\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\,u', \end{equation}

which is essentially the relation given in Huang et al. (1995, (4.9), p. 208). It is straightforward, from this basic instantaneous HCB-SRA (5.1), to calculate variances and correlations

(5.2a)\begin{gather} \text{{DNS} data:}\quad\{\bar{u},u'_{rms},\overline{u'v'},\bar{h}(\bar{u}),Pr_{h'}\}, \end{gather}

(5.2b)\begin{gather} h'|_{{SRA}}=\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\,u'\!\implies \!\!\left\{\begin{array}{@{}ll@{}} h'_{rms}|_{{SRA}} =\left\lvert \dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}} \right\rvert \,u'_{rms}, & \\[12pt] h'_{t_{rms}}|_{{SRA}}=\left\lvert \left(\dfrac{1}{Pr_{h'}}\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}+ \bar{u}\right) \right\rvert u'_{rms}, & \\[12pt] \overline{h'u'}|_{{SRA}} =h'_{rms}|_{{SRA}}\, u'_{rms} & \implies \left\lvert c_{h'u'} \right\rvert _{{SRA}}=1, \\ \overline{h_t'u'}|_{{SRA}} =h'_{t_{rms}}|_{{SRA}}\, u'_{rms} & \implies \left\lvert c_{h_t'u'} \right\rvert_{{SRA}}=1, \\ \overline{h'v'}|_{{SRA}} =\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}} \, \overline{u'v'} & \implies \left\lvert c_{h'v'} \right\rvert_{{SRA}}=\left\lvert c_{u'v'} \right\rvert\!,\\[12pt] \overline{h_t'v'}|_{{SRA}} =\left(\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}+\bar{u}\right) \overline{u'v'} & \implies \left\lvert c_{h_t'v'} \right\rvert_{{SRA}}=\left\lvert c_{u'v'} \right\rvert\!, \end{array}\right.\end{gather}

where the truncated expression $\text {\eqref {eqn23}}\Rightarrow h_t'\approxeq h'+\bar {u}\,u'$ was used.

This HCB-SRA (Huang et al. 1995) was developed precisely using data for the class of VCW flows studied in the paper, and relations (5.2b) perform reasonably well, for each individual moment (figure 12). Probably the most inaccurate prediction is that of the streamwise flux $\overline {h'u'}$. At low $Re_{\tau ^\star }\lessapprox 180$ (figures 12c,d,o,p), the SRA prediction of $\overline {h'u'}$ is reasonably accurate, both at nearly incompressible $\bar {M}_{{{CL}}_x}=0.35$ (figures 12c,d) and at high supersonic $\bar {M}_{{{CL}}_x}=2.49$ (figures 12o,p). However, at higher $Re_{\tau ^\star }=342$ (figures 12k,l), the agreement of SRA with DNS deteriorates, regarding both the near-wall peak (figure 12k) and the outer part of the flow (figure 12l), and is even worse at higher $Re_{\tau ^\star }=965$ (figures 12g,h). Recalling that by construction, SRA has the structural deficiency $\left \lvert c_{h'u'} \right \rvert _{{SRA}}=1$, these discrepancies can be explained by the actual behaviour of $c_{h'u'}$ in this VCW flow, which is practically $\bar {M}_{{{CL}}_x}$-independent at constant $Re_{\tau ^\star }$ (figures 10c,g,k,o). With increasing $Re_{\tau ^\star }$ (figures 10a,e,i,m), $c_{h'u'}$ takes progressively lower values in the outer part of the flow, and although the centreline value seems to stabilise around $0.2$ (figures 10e,i), the outer region penetrates closer to the wall with increasing $Re_{\tau ^\star }$, thus spreading the discrepancy in the SRA prediction to a larger part of the flow (figures 12g,h,k,l).

Figure 12. Comparison of DNS data with the predictions HCB–SRA (5.2), for $h'_{t_{rms}},h'_{rms},\overline {h'u'}$, plotted against inner-scaled ($\kern0.7pt y^\star$, log scale) and outer-scaled ($\kern0.7pt y/\delta$, linear) wall distance, for selected flows in the database (table 1), covering the ranges $113\leq Re_{\tau ^\star }\leq 965$ and $0.35\leq \bar {M}_{{{CL}}_x}\leq 2.49$.

Regarding fluctuation intensities $h'_{t_{rms}}$ and $h'_{rms}$, the accuracy of the SRA does not depend in any particular way on either $Re_{\tau ^\star }$ or $\bar {M}_{{{CL}}_x}$, both near the wall (figures 12a,e,i,m) and in the outer part of the flow (figures 12b,f,j,n). Generally, for $y\gtrapprox 0.1\delta$, $h'_{t_{rms}}$ is slightly overpredicted and $h'_{rms}$ is slightly underpredicted by the SRA (figures 12b,f,j,n).

5.2. Limitations of the SRA

Although the discrepancies in the individual predictions of $h'_{t_{rms}}$ and $h'_{rms}$ seem small (figure 12), their cumulative effect induces substantial error in the prediction of the ratio $h'_{t_{rms}}/h'_{rms}$, which is evaluated readily from (5.2b) as

(5.3)\begin{equation} \left.\dfrac{h'_{t_{rms}}} {h'_{rms} }\right|_{{SRA}}=\dfrac{\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}+\bar{u}} {\dfrac{1}{Pr_{h'}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}}= 1+\left[\dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}}{{\rm d}\bar{u}}\right]^{-1}, \end{equation}

i.e. increases linearly with the inverse of the non-dimensional quantity (related to the $\bar {h}(\bar {u})$ relation)

(5.4)\begin{equation} \dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}=\dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{\dfrac{{\rm d}\bar{h}}{{\rm d}y}} {\dfrac{{\rm d}\bar{u}}{{\rm d}y}} \stackrel{(2.1)}= \dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{\bar{c}_p\,\dfrac{{\rm d}\bar{T}}{{\rm d}y}} {\dfrac{{\rm d}\bar{u}}{{\rm d}y}} =\dfrac{1}{Pr_{h'}}\,\dfrac{\bar{\mu}\bar{c}_p} {\bar{\lambda}}\,\dfrac{\bar{\lambda}\,\dfrac{{\rm d}\bar{T}}{{\rm d}y}} {\bar{u}\bar{\mu}\,\dfrac{{\rm d}\bar{u}}{{\rm d}y}}\approxeq\dfrac{\overline{Pr}} { Pr_{h'}}\,\dfrac{-\bar{q}_y}{\bar{u}\bar{\tau}_{xy}}, \end{equation}

where $Pr:=\mu c_p/\lambda$ is the molecular Prandtl number, $q_y$ is the wall-normal component of the molecular heat flux, and $\tau _{xy}$ is the molecular shear stress. The approximation in (5.4) comes from neglecting the fluctuation of the molecular transport coefficients in $\bar {q}_y= -\overline {\lambda d_y T}=-\bar {\lambda } d_y\bar {T}-\overline {\lambda ' d_y T'}$, $\bar {\tau }_{xy}=\overline {\mu d_y u}=\bar {\mu }d_y\bar {u}+ \overline {\mu ' d_y u'}$ and $\overline {Pr}\approxeq \bar {\mu }\bar {c}_p/\bar {\lambda }$.

Notice that for the flows studied (table 1), $1/[d_{\bar {u}}\bar {h}/(\bar {u}/Pr_{h'})]$ is $0$ at the wall, reaching high $Re$-dependent values at the centreline region (figure 13a). The HCB-SRA predicts (see (5.3)) that $h'_{t_{rms}}/h'_{rms}$ increases linearly with the non-dimensional parameter $1/[d_{\bar {u}}\bar {h}/(\bar {u}/Pr_{h'})]$ in (5.4), in contrast with DNS data, which show a bounded increase (figure 13b). This is a general structural deficiency of SRA approaches. Notice first that (4.13) provides no information because SRA assumptions lead to $|{c_{h_t'u'}}|=|{c_{h'u'}}|=1$ in (5.2b). The origin of the problem can be traced by using the truncated-to-2oMs relations (4.7) to express the ratio $h'_{t_{rms}}/h'_{rms}$ as a function of the non-dimensional parameter $[d_{\bar {u}}\bar {h}/(\bar {u}/Pr_{h'})]$ and appropriate CCs. As for the class of the VCW flows considered, $c_{h_t'u'}$ and $c_{u'v'}/c_{h_t'v'}$ show little variation (figure 9), except that in the immediate vicinity of the wall, it is practical to use relations involving these CCs. Combining (4.7c) and (4.11) leads, after simple calculations, to the generally valid truncated-to-2oMs relation

(5.5)\begin{equation} \dfrac{h'_{t_{rms}}} {h'_{rms} }\stackrel{(4.7c), (4.11)} {\approxeq}\dfrac{1} {\sqrt{1+\dfrac{1} {\left[\dfrac{c_{u'v'}} {c_{h_t'v'}}\left(\dfrac{1}{Pr_{h'}}\, \dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}+1\right)\right]^2} - \dfrac{2c_{h_t'u'}} { \left\lvert \dfrac{c_{u'v'}} {c_{h_t'v'}}\left(\dfrac{1}{Pr_{h'}}\, \dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}+1\right) \right\rvert}}}, \end{equation}

showing that the dependence of the ratio $h'_{t_{rms}}/h'_{rms}$ on the non-dimensional parameter $[d_{\bar {u}}\bar {h}/(\bar {u}/Pr_{h'})]$ is much more complex than the SRA predicts, and depends on the CCs $\{c_{h_t'u'},c_{u'v'}/c_{h_t'v'}\}$. The fundamental structural default of the SRA is that it invariably predicts $|{c_{h'u'}}|_{{SRA}}=|{c_{h_t'u'}}|_{{SRA}}= |{c_{u'v'}/c_{h'v'}}|_{{SRA}}=1$ (see (5.2b)) because of the postulated instantaneous strict proportionality $h'\propto u$ for all $t$. Notice that using these SRA values in (5.5) reverts readily to (5.3).

Figure 13. Comparison (b) of DNS data with the predictions of the ratio $h'_{t_{rms}}/h'_{rms}$ by HCB-SRA (5.3) and by the VCW correlation (5.6) as a function of the non-dimensional parameter $1/[d_{\bar {u}}\bar {h}/(\bar {u}/Pr_{h'})]$ (5.4), and DNS profiles (a) of this parameter against $\bar {u}/\bar {u}_{{CL}}$; all available DNS data (table 1) in the ranges $97\leq Re_{\tau ^\star }\leq 985$ and $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ are plotted.

For the particular VCW class of flows studied in the paper, DNS data suggest average values $(c_{h_t'u'})_{{VCW}}\approxeq 0.997$ (figures 9a,e,i,m) and $(c_{u'v'}/c_{h_t'v'})_{{VCW}}\approxeq 0.97$ (figures 9b,f,j,n), leading to the specific correlation for these VCW flows

(5.6)\begin{equation} \left.\dfrac{h'_{t_{rms}}} {h'_{rms}}\right|_{{VCW}}\approxeq \dfrac{1} {\sqrt{1+\dfrac{1} {\left[0.97\left(\dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}+1\right)\right]^2} -\dfrac{2\times0.997} { \left\lvert 0.97\left(\dfrac{1}{Pr_{h'}}\,\dfrac{1}{\bar{u}}\,\dfrac{{\rm d}\bar{h}} {{\rm d}\bar{u}}+1\right) \right\rvert}}}, \end{equation}

which corrects quite satisfactorily the SRA prediction, giving close agreement with DNS data (figure 13b). Of course, correlation (5.6) is limited to the present class of flows (table 1) and is not applicable to flows with different enthalpy rise (equivalently wall heat flux) conditions, for which the general relation (5.5) applies. Finally, it is significant to notice how strong is the effect of the very small departures of $(c_{h_t'u'})_{{VCW}},(c_{u'v'}/c_{h_t'v'})_{{VCW}}$ from unity in (5.6).

6. Transport equations of fluctuating enthalpy variance and fluxes

In addition to the structural difficulties of the SRA (§ 5.2), the DNS results revealed a complex dependence of $Pr_{{T}}$ (4.8), in the near-wall $y^\star \lessapprox 10$ region, with respect to both $Re_{\tau ^\star }$ (figures 5a,e,i,m) and $\bar {M}_{{{CL}}_x}$ (figures 5c,g,k,o). We examine the transport equations for $\overline {\rho h''^2}$ and $\overline {\rho h''u_i''}$ to gain insight into the mechanisms behind this near-wall behaviour.

Under equation of state (2.1), the equations for static enthalpy or temperature are the same to a factor $c_p$ (Gerolymos & Vallet 2014, (2.4e), (2.4f), p. 709). The transport equation for the fluctuating enthalpy variance reads (Gerolymos & Vallet 2014)

(6.1a)\begin{align} &\underbrace{\dfrac{\partial\overline{\rho h''^2} }{\partial t} +\dfrac{\partial\overline{\rho h''^2}\,\tilde{u}_\ell} {\partial x_\ell}}_{ C_{(\rho h''^2)}\stackrel{({TPC})}=0}\nonumber\\ &\quad= \underbrace{-\dfrac{\partial}{\partial x_\ell}\left (\overline{\rho h''^2u_\ell''}+2\, \overline{h'q_\ell'}\right)}_{ d_{(\rho h''^2)}} \underbrace{-2\,\overline{\rho u_\ell''h''}\,\dfrac{\partial\tilde{h}} {\partial x_\ell} +2\left (\overline{h'\tau_{m\ell}'}\,\bar{S}_{m\ell}+ \overline{h'S_{m\ell}'}\,\bar{\tau}_{m\ell}\right)}_{ P_{(\rho h''^2)}}\nonumber\\ &\qquad -\underbrace{\left(-2\,\overline{q_\ell'\,\dfrac{\partial h'} {\partial x_\ell}}\right)}_{ \bar{\rho}\varepsilon_{(\rho h''^2)}}+\underbrace{2\,\overline{h''}\left( \overline{\dfrac{{\rm D}p}{{\rm D}t}} +\overline{\tau_{m\ell} S_{m\ell}} - \dfrac{\partial\bar{q}_\ell} {\partial x_\ell}\right)}_{ K_{(\rho h''^2)}} +\underbrace{2\,\overline{h'\tau_{m\ell}'S_{m\ell}'}}_{ \varXi_{(\rho h''^2)}} +\underbrace{2\,\overline{h'\left (\dfrac{{\rm D}p}{{\rm D}t}\right )'}}_{ \varUpsilon_{(\rho h''^2)}}, \end{align}

where $C_{(\rho h''^2)}$ represents convection by the mean flow ($=0$ in TPC flow), $d_{(\rho h''^2)}$ represents turbulent and molecular diffusion, $P_{(\rho h''^2)}$ represents production by mean flow gradients, $\varepsilon _{(\rho h''^2)}>0$ represents destruction by molecular heat conductivity, $K_{(\rho h''^2)}$ are direct compressibility (density fluctuation) effects, $\varXi _{(\rho h''^2)}$ are triple correlations which together with the corresponding terms in $P_{(\rho h''^2)}$ sum up to $2\,\overline {h'\tau _{m\ell }S_{m\ell }}$, and $\varUpsilon _{(\rho h''^2)}$ represents coupling with $({\rm D}_tp)'$ which was identified by Le Ribault & Friedrich (1997) as a compressibility effect in line with (3.2). The influence of $K_{(\rho h''^2)}$ or $\varUpsilon _{(\rho h''^2)}$ in the budgets of (6.1a) is negligibly small (figure 14a). Very close to the wall ($\kern0.7pt y^\star \lessapprox 1$), the budgets of (6.1a) are a balance between diffusion $d_{(\rho h''^2)}$ (essentially molecular $-2d_y\,\overline {h'q_y'}$) and destruction $-\rho \varepsilon _{(\rho h''^2)}$ (figure 14a). In the outer part of the flow ($\kern0.7pt y\gtrapprox 0.7\delta$), $P_{(\rho h''^2)}+\varXi _{(\rho h''^2)}$ cancel out so that the remaining budgets are a balance between diffusion $d_{(\rho h''^2)}$ (essentially turbulent $-d_y\,\overline {\rho h''^2v''}$) and destruction $-\rho \varepsilon _{(\rho h''^2)}$ (figure 14a). In the near-wall region ($4\lessapprox y^\star \lessapprox 100$), production $P_{(\rho h''^2)}+\varXi _{(\rho h''^2)}$ is important and balances diffusion and destruction $d_{(\rho h''^2)}-\rho \varepsilon _{(\rho h''^2)}$ (figure 14a).

Figure 14. Budgets, at $(Re_{\tau ^\star },\bar {M}_{{{CL}}_x})=(965,1.51)$, of the transport equations (6.1a) and (6.1b), for (a) $\overline {\rho h''^2}$, (b) $\overline {\rho h''u''}$ and (c) $\overline {\rho h''v''}$, in $({\cdot} )^\star$ units, plotted against $y^\star$ (log scale), with wall-zoom (against $y^\star$, linear) and centreline-zoom (against $y/\delta$, linear), from the present DNS database (table 1).

The transport equations for the fluxes $\overline {\rho h''u_i''}$,

(6.1b) \begin{align} &\underbrace{\dfrac{\partial\overline{\rho h''u_i''} } {\partial t} +\dfrac{\partial\overline{\rho h''u_i''}\,\tilde{u}_\ell} {\partial x_\ell}}_{ C_{(\rho h''u_i'')}}\nonumber\\ &\quad= \underbrace{-\dfrac{\partial\overline{\rho h''u_i''u_\ell''}} {\partial x_\ell} +\dfrac{\partial\overline{h'\tau_{i\ell}'}} {\partial x_\ell} -\dfrac{\partial } {\partial x_\ell}\left(\overline{u_i'q_\ell'}\right)}_{ d_{(\rho h''u_i'')}} \nonumber\\ &\qquad \underbrace{-\overline{\rho u_\ell''h''}\,\dfrac{\partial\tilde{u}_i} {\partial x_\ell} -\overline{\rho u_i''u_\ell''}\,\dfrac{\partial\tilde{h}} {\partial x_\ell} +\left(\overline{u_i'\tau_{m\ell}'}\,\bar{S}_{m\ell} +\overline{u_i' S_{m\ell}'}\,\bar{\tau}_{m\ell}\right)}_{ P_{(\rho h''u_i'')}}\nonumber\\ &\qquad -\underbrace{\left(-\overline{q_\ell'\,\dfrac{\partial u_i' }{\partial x_\ell}} +\overline{\tau_{i\ell}'\,\dfrac{\partial h' }{\partial x_\ell}}\right)}_{ \bar{\rho}\varepsilon_{(\rho h''u_i'')}}+ \underbrace{\overline{u_i''}\left(\overline{\dfrac{{\rm D}p}{{\rm D}t}} +\overline{\tau_{m\ell} S_{m\ell}} -\dfrac{\partial\bar{q}_\ell}{\partial x_\ell}\right) +\overline{h''}\left(-\dfrac{\partial\bar{p}}{\partial x_i} +\dfrac{\partial\bar{\tau}_{i\ell}} {\partial x_\ell}\right)}_{ K_{(\rho h''u_i'')}}\nonumber\\ &\qquad +\underbrace{\overline{u_i'\tau_{m\ell}'S_{m\ell}'}}_{ \varXi_{(\rho h''u_i'')}} \underbrace{{}-\overline{ h'\,\dfrac{\partial p' } {\partial x_i }}}_{ \varPi_{(\rho h''u_i'')}}\ +\overline{\rho h'' f''_{{{V}}_i}} +\underbrace{\overline{u_i'\left(\dfrac{{\rm D} p}{{\rm D} t}\right )'}}_{ \varUpsilon_{(\rho h''u_i'')}}, \end{align}

in addition to the mechanisms identified in the transport of $\overline {\rho h''^2}$ (6.1a), contain the pressure-scrambling term $\varPi _{(\rho h''u_i'')}$ (Le Ribault & Friedrich 1997), and in the case of the streamwise flux, the term $\overline {\rho h'' f''_{{{V}}_x}}$, which is negligibly small everywhere in the channel (figure 14b). Notice that for TPC flow,

(6.1c)\begin{equation} P_{(\rho h''u'')}+\varXi_{(\rho h''u'')}\stackrel{({TPC})}=-\overline{\rho h''v''}\,\dfrac{{\rm d}\tilde{u}} {{{\rm d}y} } -\overline{\rho u''v''}\,\dfrac{{\rm d}\tilde{h}} {{\rm d}y} + \overline{u'\tau_{m\ell}S_{m\ell}},\end{equation}

so $Pr_{{T}}$ in (4.8) is precisely the ratio of the production of the streamwise flux $\overline {\rho h''u''}$ from the enthalpy gradient on its production from shear, offering an alternative interpretation of $Pr_{{T}}$, and highlighting the connection between the streamwise $\overline {\rho h''u''}$ and wall-normal $\overline {\rho h''v''}$ fluxes.

In the near-wall region ($\kern0.7pt y^\star \lessapprox 100$), the same remarks apply to the budgets of the streamwise flux $\overline {\rho h''u''}$ (figure 14b) as for those of the variance $\overline {\rho h''^2}$ (figure 14a), with the significant difference that the pressure-scrambling term $\varPi _{(\rho h''u'')}$ is comparable to the sum of the others, $d_{(\rho h''u'')}+P_{(\rho h''u'')}+\varXi _{(\rho h''u'')}-\bar {\rho }\varepsilon _{(\rho h''u'')}$, and therefore has a significant influence on the budgets (figure 14b). In the outer part of the flow ($\kern0.7pt y\gtrapprox 0.7\delta$), destruction of the streamwise flux $-\bar {\rho }\varepsilon _{(\rho h''u'')}$ is negligibly small, and, as production $P_{(\rho h''u'')}$ diminishes to $0$ approaching the centreline, the budgets are essentially a balance of the positive contribution of turbulent diffusion ($-d_y\,\overline {\rho h''u''v''}$) and remaining production $d_{(\rho h''u'')}+P_{(\rho h''u'')}$, with the negative contributions of the pressure terms and the triple correlations, $\varXi _{(\rho h''u'')}+\varPi _{(\rho h''u'')}+\varUpsilon _{(\rho h''u'')}$ (figure 14b), highlighting the importance of the pressure terms almost everywhere ($\kern0.7pt y^\star \gtrapprox 10$).

The budgets of (6.1b) for the wall-normal flux (${VCW}\Rightarrow \overline {\rho h''v''}< 0$ so that negative terms in the budgets count as gain) are completely different, dominated by the positive (loss) pressure-scrambling term $\varPi _{(\rho h''v'')}$ that opposes the negative (gain) production $P_{(\rho h''v'')}$, and is the largest term in the outer part of the flow (figure 14c). Notice that $\varPi _{(\rho h''v'')}$ is of the same order of magnitude as its streamwise counterpart $\varPi _{(\rho h''u'')}$ everywhere, and the orders of magnitude of different mechanisms in the centreline region ($\kern0.7pt y\gtrapprox 0.7\delta$) are comparable for both components of $\overline {\rho h''u_i''}$ (figures 14b,c).

Examination of various terms in the budgets of $\overline {\bar {\rho }h''^2}$ (6.1a) at nearly constant $Re_{\tau ^\star }\approxeq 341\pm 1$ shows that only turbulent diffusion $-d_y(\overline {\rho h''^2v''})$ follows $({\cdot} )^\star$ scaling with varying $\bar M_{{{CL}}_x}\in \{0.80,1.51,1.98\}$ (figure 15c). Both molecular diffusion $-2[d_y(\overline {h'q'})]^\star$ and destruction $-[\bar {\rho }\varepsilon _{(\rho h''^2)}]^\star$, in $({\cdot} )^\star$ units, show substantial near-wall ($\kern0.7pt y^\star \lessapprox 10$) variation with $\bar {M}_{{{CL}}_x}$ (figure 15b), and the same applies to the two different mechanisms of the sum

(6.1d)\begin{equation} P_{(\rho h''^2)}+\varXi_{(\rho h''^2)}\stackrel{({TPC})}=-2\,\overline{\rho h''v''}\,\dfrac{{\rm d}\tilde{h}} {{{\rm d}y} } +2\,\overline{h'\tau_{m\ell}S_{m\ell}}, \end{equation}

with distinct variation of the peak of $-2[d_y(\overline {\rho h'' v''}d_y\tilde h)]^\star$ occurring at $y^\star \approxeq 10$ (figure 15a) and that of $2[\overline {h'\tau _{m\ell }S_{m\ell }}]^\star$ occurring much closer to the wall, at $y^\star \approxeq 4$ (figure 15a). This behaviour contrasts with that of $[\overline {\tau _{m\ell }S_{m\ell }}]^\star$, which does follow $({\cdot} )^\star$ scaling near the wall, at constant $Re_{\tau ^\star }$ (figure 1b).

Figure 15. Influence of Mach number $\bar {M}_{{{CL}}_x}\in \{0.80,1.51, 1.98\}$, at nearly constant $Re_{\tau ^\star }\approxeq 341\pm 1$, on various terms in the budgets of the transport equation for the enthalpy variance $\overline {\rho h''^2}$ (6.1a), from the present DNS database (table 1), in $({\cdot} )^\star$ units, plotted against $y^\star$ (log scale), illustrating both (a,b) complex non-$({\cdot} )^\star$ scaling for some terms, and (c) $({\cdot} )^\star$ scaling for others.

The strong near-wall $\bar {M}_{{{CL}}_x}$ effect on several (figures 15a,b) but not all (figure 15c) of the terms in (6.1a) is related to the complex scaling behaviour of the fluxes (figure 16). Recall first that $h'_{rms}$ scales quite well with $\bar {u}_{{CL}}^2$, both near the wall against $y^\star$ (figures 7c,g,k,o), and in the entire channel against $y/\delta$ (figures 7d,h,l,p). The fluxes $\overline {h'u_i'}$ can scale with the inner $u_{\tau ^\star }:=\sqrt{(\bar {\tau }_w/\bar {\rho }(y))}$, outer $\bar {u}_{{CL}}$, or some mixed inner/outer velocity scale. The inner-scaled wall-normal flux $\overline {h'v'}^{\,\star }:=\overline {h'v'}/u_{\tau ^\star }^3$, at constant $Re_{\tau ^\star }$, practically collapses on a single curve with varying $\bar {M}_{{{CL}}_x}$ (figures 16d,h,l,p), showing a general validity of $({\cdot} )^\star$ scaling for wall-normal transport, as it obviously also applies to $\overline {\rho h''^2v''}$ (figure 15c), and it is well known that it applies to $\overline {\rho u''v''}^{\,\star }\equiv \overline {\rho u''v''}^{\,+}$ (van Driest 1951; Trettel & Larsson 2016). However, the value of the $\overline {h'v'}^{\,\star }$ peak varies very strongly with $Re_{\tau ^\star }$ (figures 16d,h,l,p), and, although not plotted, continues to increase (in absolute value) for $Re_{\tau ^\star }\approxeq 1000$. It appears that outer-scaling the wall-normal flux $\overline {h'v'}/\bar {u}_{{CL}}^3$ (figures 16b,f,j,n) behaves much better with respect to an asymptotic high-$Re$ trend in the near-wall region ($\kern0.7pt y^\star \lessapprox y^\star _{{{PEAK}}_{(h'v')}}$). Notice that, similar to $\overline {\rho u''v''}^{\,\star }\equiv \overline {\rho u''v''}^{\,+}$, $y^\star _{{{PEAK}}_{(h'v')}}$ varies strongly with $Re_{\tau ^\star }$. On the other hand, the streamwise flux seems to follow mixed scaling $\overline {h'u'}/(\bar {u}_{{CL}}^2\,u_{\tau ^\star })$, regarding both the $\bar {M}_{{{CL}}_x}$ effect (figures 16c,g,k,o) and the $Re_{\tau ^\star }$ effect (figures 16a,e,i,m). Considering Favre-averaged fluxes $\overline {\rho h''u_i''}$ (not shown) yields exactly the same conclusions as those obtained for the Reynolds-averaged fluxes $\overline {h'u_i'}$ (figure 16), in line with the close agreement between $Pr_{{T}}$ (4.8) using Favre averages and $Pr_{h'}$ (4.10) using Reynolds averages (figure 6).

Figure 16. Fluxes $\overline {h'u'}$ (mixed scaled by $\bar {u}_{{CL}}^2\,u_{\tau ^\star }$) and $\overline {h'v'}$ (both inner-scaled by $u_{\tau ^\star }^3$ and outer-scaled by $\bar {u}_{{CL}}^3$) for turbulent $h'$-transport, for varying HCB Reynolds numbers $97\leq Re_{\tau ^\star }\leq 985$ at centreline Mach numbers $\bar {M}_{{{CL}}_x}\in \{0.33,0.80,1.50, 2.00\}$, and for varying $0.32\leq \bar {M}_{{{CL}}_x}\leq 2.49$ at nearly constant $Re_{\tau ^\star }\in \{100,110,250,340\}$, from the present DNS database (table 1).

7. Conclusions

Thermal turbulence structure of TPC flow depends very little on $\bar {M}_{{{CL}}_x}$. Therefore, the centreline-to-wall temperature ratio $\bar {T}_{{CL}}/\bar {T}_w$, which depends very strongly on $\bar {M}_{{{CL}}_x}$, is not appropriate to parametrise the thermal turbulence structure. In contrast the non-dimensional enthalpy rise $(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$ is very weakly dependent on $\bar {M}_{{{CL}}_x}$ and is probably a much better choice, in line with the relative $\bar {M}_{{{CL}}_x}$-independence of the non-dimensional $(\bar {h}(\bar {u})-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$ relation.

The DNS data for the total and static enthalpy variances and correlations, for turbulent plane channel flow in the ranges $100\lessapprox Re_{\tau ^\star }\lessapprox 1000$ and $0.3\leq \bar {M}_{{{CL}}_x}\leq 2.5$, indicate the following.

1. (i) Joint p.d.f.s of $(u',h_t')$ show a very strong correlation, with events tightly clustered along the diagonal of the positive quadrants, everywhere in the channel, verified by the correlation coefficient (CC) $c_{h_t'u'}$ and by the ratio of wall-normal transport CCs $c_{u'v'}/c_{h_t'v'}$, both of which are very close to 1.

2. (ii) In contrast, for the class of very-cold-wall (VCW) flows studied, static enthalpy fluctuations $h'$ are not well correlated with $u'$, the CC $c_{h'u'}$ rapidly decreasing with increasing wall distance, as more and more events in the joint p.d.f.s of $(u',h')$ occur away from the diagonal of the positive quadrants.

3. (iii) Outside the very-near-wall region ($\kern0.7pt y^\star \gtrapprox 20$), at nearly constant $Re_{\tau ^\star }$, the quantities $h'_{t_{rms}}/\bar {u}_{{CL}}^2,h'_{rms}/\bar {u}_{{CL}}^2, c_{h_t'u'},c_{u'v'}/c_{h_t'v'},Pr_{{T}}$ show little sensitivity to $\bar {M}_{{{CL}}_x}$, also implying that $\bar {u}_{{CL}}^2$ is the correct outer scaling for both $h'_{t_{rms}}$ and $h'_{rms}$.

4. (iv) At constant $Re_{\tau ^\star }$, the inner-scaled wall-normal flux $\overline {h'v'}^{\,\star }=\overline {h'v'}/u_{\tau ^\star }^3$ is practically independent of $\bar {M}_{{{CL}}_x}$ everywhere in the channel, whereas the streamwise flux follows mixed scaling $\overline {h'u'}/(\bar {u}_{{CL}}^2\,u_{\tau ^\star })$.

5. (v) The higher $Re_{\tau ^\star }\approxeq 1000$ data suggest an outer law $(y/\delta )^2$ decay for $Pr_{{T}}$ (which also fits lower $Re_{\tau ^\star }\gtrapprox 250$ data), highlighting at the same time the need for higher-$Re_{\tau ^\star }$ data to elucidate the high-$Re$ asymptotics.

The HCB–SRA predicts reasonably well $h_{t_{rms}}',h_{rms}',\overline {h'u'}$ for the class of flows studied, but predicts incorrectly unbounded increase of the ratio $h_{t_{rms}}'/h_{rms}'$ in the wake region, with increasing $Re_{\tau ^\star }$. This deficiency can be corrected only using a weak Reynolds analogy, i.e. the exact truncated-to-2oMs relation. For all of the flows considered in the present database, appropriate constant values of $c_{h_t'u'},c_{u'v'}/c_{h_t'v'}$ provide a simple correlation that fits well the DNS data, but remains specific to the class of VCW flows.

The study of transport equations for the enthalpy variance $\overline {\rho h''^2}$ and fluxes $\overline {\rho h''u_i''}$ is probably the only way to analyse the observed strong and complex dependence of $Pr_{{T}}$ on both $Re_{\tau ^\star }$ and $\bar {M}_{{{CL}}_x}$ in the near-wall region ($\kern0.7pt y^\star \lessapprox 15$). With the exception of turbulent diffusion which, in $({\cdot} )^\star$ units, is practically $\bar {M}_{{{CL}}_x}$-independent, all of the other terms in the budgets of the transport equation for $\overline {\rho h''^2}$ show strong $\bar {M}_{{{CL}}_x}$-dependence in the near-wall region. It is believed that further study of the budgets of these transport equations will provide insight into the complexity of the very-near-wall flow.

Detailed data of the various moments, correlations and p.d.f.s studied in the paper, for flows with different non-dimensional enthalpy rise $(\bar {h}_{{CL}}-\bar {h}_w)/(\tfrac {1}{2}\bar {u}_{{CL}}^2)$, are necessary to parametrise the findings of this work for general compressible wall turbulence. Another important perspective is obtaining DNS data of compressible wall turbulence at higher $Re$, this being a global need, as currently available Reynolds numbers in the literature lag by almost one decade incompressible flow DNS.