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. Reference Huang, Coleman and Bradshaw1995; Gerolymos & Vallet Reference Gerolymos and Vallet2014)

(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. Reference Huang, Coleman and Bradshaw1995), 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. Reference Zhang, Bi, Hussain and She2014, Reference Zhang, Duan and Choudhari2017, Reference Zhang, Duan and Choudhari2018; Song et al. Reference Song, Zhang, Liu and Xia2022), 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. Reference Zhang, Duan and Choudhari2017, Reference Zhang, Duan and Choudhari2018; Song et al. Reference Song, Zhang, Liu and Xia2022). Using enthalpies instead, it is straightforward to show that the unique one-sided Hermitian quadratic polynomial (Stoer & Bulirsch Reference Stoer and Bulirsch1993, 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. (Reference Zhang, Bi, Hussain and She2014). 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. Reference Huang, Coleman and Bradshaw1995; Song et al. Reference Song, Zhang, Liu and Xia2022)

(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. Reference Huang, Coleman and Bradshaw1995; Song et al. Reference Song, Zhang, Liu and Xia2022), 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 (Reference Modesti and Pirozzoli2016) and Yao & Hussain (Reference Yao and Hussain2020) ($Pr=0.72$), and of Trettel & Larsson (Reference Trettel and Larsson2016), 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 (Reference Modesti and Pirozzoli2016), Trettel & Larsson (Reference Trettel and Larsson2016) and Yao & Hussain (Reference Yao and Hussain2020) (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. (Reference Song, Zhang, Liu and Xia2022) in (b).

Song et al. (Reference Song, Zhang, Liu and Xia2022, (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. (Reference Song, Zhang, Liu and Xia2022) 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 (Reference Modesti and Pirozzoli2016) and Yao & Hussain (Reference Yao and Hussain2020), and is slightly different to the values tabulated in Song et al. (Reference Song, Zhang, Liu and Xia2022).

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 (Reference Yao and Hussain2020) 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. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022), tenfold higher than the compressible databases (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016; Trettel & Larsson Reference Trettel and Larsson2016; Yao & Hussain Reference Yao and Hussain2020; Gerolymos & Vallet Reference Gerolymos and Vallet2014, Reference Gerolymos and Vallet2023).

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. Reference Huang, Coleman and Bradshaw1995), 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 Reference Gaviglio1987). Notice that (4.7c) was reported in Gaviglio (Reference Gaviglio1987, (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 Reference Krogstad and Torbergsen2000; Lee & Moser Reference Lee and Moser2015; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022), 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 (Reference Rotta1964); the present DNS data (figures 5b,f,j,n) suggest a different constant, $0.32$, in lieu of $0.4$ in Rotta (Reference Rotta1964). 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. Reference Zhang, Bi, Hussain and She2014), 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 Reference Gerolymos and Vallet2018) 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. Reference Huang, Coleman and Bradshaw1995).

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. Reference Hadjadj, Ben Nasr, Shadloo and Chaudhuri2015, 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.

5.1. HCB-SRA

The SRA proposed by Huang et al. (Reference Huang, Coleman and Bradshaw1995, (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. (Reference Huang, Coleman and Bradshaw1995, (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. Reference Huang, Coleman and Bradshaw1995) 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).