2.1. Simulation parameters

As for the study in Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021), but in contrast to the adiabatic DNS results presented in Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2019), the temperature at the solid wall is fixed according to a prescribed distribution ${T}_w={T}_w(x)$. Except for this condition the numerical set-up of the non-adiabatic cases is identical to that of the adiabatic ones; see Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2019). The reference thermodynamic flow properties are the inflow far-field temperature $T_{\infty,0}=288.15$ K, inflow far-field density $\rho _{\infty,0}=1.225$ kg m$^{-3}$, Prandtl number $Pr=0.71$, specific gas constant $R=287$ J kg$^{-1}$ K$^{-1}$ and ratio of specific heats $\gamma =1.4$ and are set equal for all cases. Data averaging is performed over both time and the spanwise direction and does not start before the flow has passed the whole domain at least twice. Time averages were computed over a flow-through time corresponding to at least $250$ local boundary-layer thicknesses $\delta _{99}$ ($\Delta t\,u_e/\delta _{99}=250$, where $\Delta t$ is the averaging period), which corresponds to at least $8$ eddy-turnover times $\Delta t\,u_\tau /\delta _{99}$ for the most restricting APG cases. Both the appropriate convergence of the statistics and the initial transients have been assessed as described in Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018) and Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2019).

3.1. The FIK identity

The FIK identity (Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002)) is based on a threefold repeated integration of the Navier–Stokes $x$-momentum equation in the wall-normal direction. This introduces weight functions like $(y_b-y)$ to terms such as the $\bar {\rho }\widetilde {u'' v''}$ stress,

(3.2)\begin{equation} -\intop_0^{y_b}\intop_0^{y}\intop_0^{y}\frac{\partial\bar{\rho}\widetilde{u''v''}}{\partial y} \, {\textrm{d} y}\,{\textrm{d} y}\,{\textrm{d} y}={-}\intop_0^{y_b}\left(y_b-y\right)\left[\bar{\rho}\widetilde{u''v''}\right]\,{\textrm{d} y} \neq 0,\end{equation}

where $y_b$ is the non-dimensionalized upper integration bound. These weights enhance the integrand near the wall, $(y_b-y)|_{y=0}=y_b$, and reduce it in the outer region, $(y_b-y)|_{y=y_b}=0$. According to its original formulation, the resulting terms of the FIK identity can be associated with four different skin-friction generating mechanisms $c_f=c_f^L+c_f^{T}+c_f^I+c_f^t$, a laminar one $c_f^L$, a turbulent one $c_f^{T}$, an inhomogeneous one $c_f^I$ and a transient one $c_f^t$, of which the second is the weighted integral of the Reynolds shear-stress distribution (3.2); see Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002). In recent years, the incompressible FIK identity has been further developed and modified, such as by Mehdi & White (Reference Mehdi and White2011), Mehdi et al. (Reference Mehdi, Johansson, White and Naughton2014), Peet & Sagaut (Reference Peet and Sagaut2009), Bannier, Garnier & Sagaut (Reference Bannier, Garnier and Sagaut2015), Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016) and Modesti et al. (Reference Modesti, Pirozzoli, Orlandi and Grasso2018), and has become a common tool used in turbulent flow analysis and control; see e.g. Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005), Kametani & Fukagata (Reference Kametani and Fukagata2011), Kametani et al. (Reference Kametani, Fukagata, Örlü and Schlatter2015) and Stroh et al. (Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016). In contrast to the incompressible regime, only a few detailed studies are available for compressible flat-plate TBLs. In Gomez, Flutet & Sagaut (Reference Gomez, Flutet and Sagaut2009), a possible compressible form of the FIK identity has been derived and corresponding effects on the mean skin-friction drag generation were studied, but without a full clarification of the compressibility effects on the various contributions as mentioned in Li et al. (Reference Li, Fan, Modesti and Cheng2019).

It is a drawback of the FIK identity, however, that some of the contributing terms can easily be misinterpreted (especially for spatially evolving TBLs); see Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014), Renard & Deck (Reference Renard and Deck2016) and Fan, Cheng & Li (Reference Fan, Cheng and Li2019). One of the key controversies is that there is no simple justification for the threefold repeated integration and thus no physics-based explanation for the linearly weighted Reynolds shear stress (3.2). In the authors’ opinion, the problem in justifying the threefold repeated integration is further evidenced by the fact that the threefold repeated integration is not necessary for deriving a valid integral identity for $c_f$. According to Cauchy's formula for repeated integration, $n$ repeated integrations of a continuous function can be rewritten into a single integral,

(3.3)\begin{align} f^{{-}n}\left(y_b\right)=\intop_{y_u}^{y_b}\intop_{y_u}^{\xi}\cdots\intop_{y_u}^{\xi_{n-1}}f\left(\xi_n\right)\,\textrm{d}\xi_n \,\ldots\, \textrm{d}\xi_2 \,\textrm{d}\xi_1=\frac{1}{\left(n-1\right)!}\intop_{y_u}^{y_b} \left(y_b-y\right)^{n-1}f\left(y\right) \,{\textrm{d} y}. \end{align}

This means that any number of repetitions greater than one would result in a FIK-like identity, since $c_f$ already appears as a variable in the equations after the first integration, as shown in Appendix A; variations in the number of repetitions essentially only affect the exponent of the respective weighting factors, as in (3.3).

In the authors’ opinion, there are major advantages of using only a twofold repeated integration. For a clear argument, the resulting integral identities for $c_f$ and $c_h$ are introduced first, before this choice is justified in § 3.4.

3.2. The compressible integral identities for $c_f$ and $c_h$

In this study, the same mix of Reynolds-averaged and Favre-averaged mean quantities is used as in Huang, Coleman & Bradshaw (Reference Huang, Coleman and Bradshaw1995). The Favre decomposition is applied to the convective terms; for all other terms, the Reynolds-averaged decomposition is applied. To allow for a unique interpretation of the effects of Reynolds number, Mach number and heat transfer, the integral identities are presented in dimensionless form with $f=\hat {f}/f_n$, where $\hat {f}$ is the dimensional quantity. Reference values for non-dimensionalization of $f_n$ are defined as $\rho _n=\rho _e$, $p_n=p_e$, $\mu _n=\bar {\mu }_w$, $u_n=u_e$, $L_n=\delta _e$, $T_n=\bar {T}_{r}-T_w$ and $h_n=h_{0,n}=c_p(\bar {T}_{r}-T_w)$ to ensure the appropriate non-dimensionalization of $c_f \equiv \bar {\tau }_w/(\rho _e u_e^2/2)$ and $c_h\equiv \bar {q}_{y,w}/(\rho _e u_e c_p (T_w-\bar {T}_r))$. With $\tilde {h}_{0}=\tilde {h}+({Ec}/{2})\tilde {u}_{i}\tilde {u}_{i}+({Ec}/{2})\widetilde {u_i^{''} u_i^{''}}$, and with the Reynolds number, Mach number and Eckert number defined as

(3.4a–c)\begin{equation} Re=\frac{\rho_{e}u_{e}\delta_e}{\bar{\mu}_{w}},\quad M_e^{2}=\frac{\rho_{e}u_{e}^{2}}{\gamma p_{e}},\quad Ec=\frac{u_{e}^{2}}{c_p(\bar{T}_{r}-T_w)}=\frac{\left(\gamma -1\right)M_e^2T_e}{\left(\bar{T}_r-T_w\right)}, \end{equation}

the simplified two-dimensional (2-D) boundary-layer equations for the $x$-momentum and total-enthalpy can be written as

(3.5)\begin{align} \bar{\rho}\tilde{u}\frac{\partial \tilde{u}}{\partial x}+\bar{\rho}\tilde{v}\frac{\partial \tilde{u}}{\partial y}&={-}\frac{1}{\gamma M_e^{2}}\frac{\partial\bar{p}}{\partial x}+\frac{1}{Re}\left[\frac{\partial \bar{\tau}_{xx}}{\partial x}+\frac{\partial \bar{\tau}_{xy}}{\partial y}\right] -\frac{\partial \bar{\rho}\widetilde{u''u''}}{\partial x}-\frac{\partial \bar{\rho}\widetilde{u''v''}}{\partial y},\end{align}

(3.6)\begin{align} \bar{\rho}\tilde{u}\frac{\partial \tilde{h}_{0}}{\partial x}+\bar{\rho}\tilde{v}\frac{\partial \tilde{h}_{0}}{\partial y}&= \frac{Ec}{Re}\left[\frac{\partial}{\partial y}\left(\bar{\tau}_{xy}\bar{u}+\bar{\tau}_{yy}\bar{v}\right)\right]+\frac{Ec}{Re}\left[\frac{\partial}{\partial y}\left(\overline{\tau'_{xy}u^{'}}+\overline{\tau'_{yy}v^{'}}+\overline{\tau'_{zy}w^{'}}\right)\right]\nonumber\\ &\quad -\frac{\partial \bar{\rho} \widetilde{v^{''}h^{''}}}{\partial y}-\frac{\partial \bar{\rho} \widetilde{u^{''}h^{''}}}{\partial x}-\frac{Ec}{2}\left[\frac{\partial}{\partial y}\left(\bar{\rho} \widetilde{u^{''^{2}}v^{''}}+\bar{\rho} \widetilde{v^{''3}}+\bar{\rho} \widetilde{w^{''^{2}}v^{''}}\right)\right]\nonumber\\ &\quad -Ec\left[\frac{\partial}{\partial y}\left(\tilde{u}\bar{\rho} \widetilde{u^{''}v^{''}}+\tilde{v}\bar{\rho} \widetilde{v^{''^{2}}}\right)\right]-Ec\frac{\partial \tilde{u}\bar{\rho} \widetilde{u^{''^{2}}}}{\partial x}-\frac{1}{RePr}\frac{\partial\bar{q}_{y}}{\partial y}. \end{align}

3.3. Integral identities

Using the assumptions for quasi-2-D boundary-layer flow and a twofold repeated integration of (3.5), the integral identity for $c_f$ yields

(3.7) \begin{align} c_{f}&=\underbrace{\frac{2}{Re\,y_{b}}\intop_{0}^{y_{b}}\left[\bar{\mu}\frac{\partial\bar{u}}{\partial y}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{f}^{L}\\ \text{boundary-layer term}\end{array}} - \underbrace{\frac{2}{y_{b}}\intop_{0}^{y_{b}}\left(\,y_{b}-y\right) \left[\tilde{v}\bar{\rho} \frac{\partial \tilde{u}}{\partial y }\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{f}^{M}\\ \text{mean-convection term}\end{array}}\nonumber\\ &\quad - \underbrace{\frac{2}{y_{b}}\intop_{0}^{y_{b}}\left[\bar{\rho}\widetilde{u''v''}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{f}^{T}\\ \text{turbulent-convection term} \end{array}}\! + \underbrace{\frac{2}{y_{b}}\intop_{0}^{y_{b}}\left(\,y_{b}-y\right)\left[-\tilde{u}\bar{\rho}\frac{\partial \tilde{u} }{\partial x}-\frac{1}{\gamma M_{e}^{2}}\frac{\partial\bar{p}}{\partial x}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{f}^{D}\\ \text{spatial-development term}\end{array}}. \end{align}

The non-dimensionalized upper integration bound $y_b$ becomes equal to one if the integration is performed up to the boundary-layer edge $\delta _e$; see also § 3.4.

For the integral identity of the Stanton number $c_h$, a twofold repeated integration of the total-enthalpy equation in (3.6) results in

(3.8) \begin{align} c_{h}&= \underbrace{\frac{1}{Re \,y_{b}}\intop_{0}^{y_{b}}\left[\frac{1}{Pr}\bar{\mu}\frac{\partial\bar{T}}{\partial y}+ Ec\,\bar{u}\, \overline{ \tau}_{xy}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{h}^{L}\\ \text{boundary-layer term}\end{array}} -\underbrace{\frac{1}{y_b}\intop_{0}^{y_{b}}\left(y_b-y\right)\left[\bar{\rho}\tilde{v}\frac{\partial\tilde{h}_{0}}{\partial y}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{h}^{M}\\ \text{mean-convection term} \end{array}}\nonumber\\ &\quad -\underbrace{\frac{1}{y_b}\intop_{0}^{y_{b}}\left[\bar{\rho} \widetilde{v^{''}h^{''}}+Ec\left(\tilde{u}\,\bar{\rho} \widetilde{u^{''}v^{''}}+\tilde{v}\,\bar{\rho} \widetilde{v^{''^{2}}}\right)+\frac{Ec}{2}\left(\bar{\rho} \widetilde{u''^{2}v''}+\bar{\rho} \widetilde{v''^{3}}+\bar{\rho} \widetilde{w''^{2}v''}\right)\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{h}^{T}\\ \text{turbulent-convection term} \end{array}}\nonumber\\ &\quad -\underbrace{\frac{1}{y_b}\intop_{0}^{y_{b}}(y_b-y)\left[\bar{\rho}\tilde{u}\frac{\partial\tilde{h}_{0}}{\partial x}+\frac{\partial \bar{\rho} \widetilde{u^{''}h^{''}}}{\partial x}+Ec\frac{\partial \tilde{u}\,\bar{\rho} \widetilde{u^{''^{2}}}}{\partial x}\right]\,{\textrm{d} y}}_{\begin{array}{c} c_{h}^{D} \\ \text{spatial-development term} \end{array}}. \end{align}

A detailed derivation of the $c_f$ and $c_h$ identities for an arbitrary number $n$ of repeated integrations is given in Appendix A. In contrast to the more complete forms of the identities in (A2) and (A4), both the viscous-stress fluctuation terms $c_f^{VF}$ and $c_h^{VF}$ are neglected in (3.7) and (3.8) for simplification, respectively. However, these terms have a noticeable share near the inflow regions of the simulation domain, where the flow is not yet fully turbulent. For the composition of $c_h$, for instance, $c_h^{VF}$ can contribute up to $30\,\%$. For boundary layers rapidly changing in the streamwise direction, these terms therefore should be accounted for. For all figures shown hereafter, the inlet regions are cropped.

3.4. Remarks

The following remarks should be made about the benefits and limitations of the integral identities.

3.4.1. General remarks on the FIK identity

The first remark is related to the threefold repeated integration used in the original derivation of the FIK identity; see (3.2). As mentioned, the three integrations are not a requirement in deriving an integral condition for $c_f$ or $c_h$, since an arbitrary number of repeated integrations $n$ provides a valid decomposition of $c_f$ or $c_h$. Exemplarily evaluated for $n=2$, $3$, $4$ and $10$ for the subsonic $iZPG_{M=0.5}^{+10{\rm K}}$ case, the streamwise evolution of the integral identity (A2) is evaluated in Appendix B.2. Briefly, increasing the number of integrations $n$ increases the weights of terms that are dominant near the wall. Nevertheless, even though $n$ can be freely chosen from a mathematical point of view, not every choice of $n$ allows a physically plausible interpretation. In the original formulation of the FIK identity, the motivation for the threefold repeated integration is based on the viscous mean stress term, which reads in the dimensional formulation as

(3.8)\begin{equation} \intop_0^{y_b} \intop_0^y \intop_0^y \frac{\partial}{\partial y}\left(\bar{\mu} \frac{\partial \bar{u}}{\partial y}\right)\, {\textrm{d} y} \, {\textrm{d}y} \, {\textrm{d} y}.\end{equation}

By an appropriate choice, the viscosity can be normalized to one by introducing the Reynolds number into the dimensionless representation, and the integrand simplifies to $\partial /\partial y(\partial \bar {u}/\partial y)$. According to Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), the first integration then results in a force balance, the second leads to the mean velocity profile and the third is akin to obtaining the flow rate. As discussed in Renard & Deck (Reference Renard and Deck2016), however, this interpretation is only a consequence of the dimensionless formulation of the identity and therefore not intuitive, since the product of a force (first integral) and a length (due to the second integral) has the dimension of energy rather than velocity in dimensional representation.

From the authors’ point of view, there are several reasons why using a twofold repeated integration is more appropriate. (i) From a physical point of view, the first integration gives a force/energy balance between the wall and all locations within the boundary layer. The second integration represents its average in the wall-normal direction. In other words, both integral identities simply quantify the wall-normal average of how much momentum/energy is transported towards the wall (positive) or away from the wall (negative) by each term. (ii) Compared with higher-fold integrations, the result is easier to interpret since all terms in the boundary-layer equations appearing with a derivative in the wall-normal direction are integrated without being weighted by a wall-distance-dependent term as in the classical FIK identity. In the present formulation chosen, this holds both for the $c_f^{T}$ and the $c_f^L$ terms in (3.7). The resulting values for $c_f^L$ and $c_f^{T}$ can be therefore simply associated with the area enclosed by the $\bar {\mu }\partial \bar {u}/\partial y$ and $\bar {\rho }\widetilde {u''v''}$ distributions, respectively. Note that the main results derived below concerning the effects of pressure gradients, heat transfer and compressibility are qualitatively identical for $n=2$ and $n=3$.

4.1. Streamwise evolution of $c_f$

According to (3.7), the $c_f$ distributions depicted in figure 1($a$–$d$) are dominated by $c_f^{T}$, $c_f^M$ and $c_f^{D}$; $c_f^L$ only provides a small portion of the overall $c_f$ (especially for high $Re$; see e.g. Fan et al. Reference Fan2020). For both the sub- and supersonic ZPG cases in figure 1($a$,$c$), respectively, the spatial-development term $c_f^{D}$ constitutes the largest portion. For the sub- and supersonic APG cases in figure 1($b$,$d$), the $c_f^D$ distributions are influenced in a different manner by the pressure gradient in regions of approximated self-similarity, increased in panel ($b$) and strongly decreased in panel ($d$). The mean-convection term $c_f^{M}$ essentially balances the changes in $c_f^D$. A comparison of the turbulence terms $c_f^{T}$ (blue) reveals almost identical behaviour for the sub- and supersonic ZPG cases in panels ($a$,$c$) and the sub- and supersonic APG cases in panels ($b$,$d$), respectively. The percentage contribution of $c_f^{T}$ is therefore expected to be predominantly influenced by the pressure-gradient parameter $\beta _K$, and only marginally by the local Mach number.

4.2. Streamwise evolution of $c_h$

According to (3.8), the results of the $c_h$ distributions are depicted in figure 1($e$–$h$). For two reasons, the results are more complex to describe than for the $c_f$ distributions. First, the total enthalpy describes the overall change in enthalpy, mean kinetic energy and mean turbulent kinetic energy. Hence, more terms have to be taken into account to capture a complete representation of $c_h$. Second, the $c_h$ distributions are strongly influenced by the pressure-gradient parameter $\beta _K$, the local Mach number and heat transfer. For the subsonic $iZPG_{M=0.5}^{+10{\rm K}}$ case in figure 1($e$), comparable trends can be found to those for the $c_f$ contributions in panel ($a$); only the relative proportions of $c_h^{T}$ and $c_h^D$ differ. A comparison between the subsonic $iZPG_{M=0.5}^{+10{\rm K}}$ and $iAPG_{\beta _K=0.6}^{+10{\rm K}}$ cases in panels ($e$,$f$), respectively, shows only a small influence of the streamwise pressure gradient. For the supersonic cases in panels ($g$,$h$), the blue $c_h^{T}$ contributions are noticeably increased in comparison with the subsonic cases. For the supersonic $cZPG_{M=2.0}^{+20{\rm K}}$ case in panel ($g$), $c_h^{T}$ is increased from approximately $60\,\%$ of $c_h$ to approximately $120\,\%$ of $c_h$ at $Re_\tau =450$, and for the supersonic $cAPG_{\beta _K=0.6}^{+20{\rm K}}$ case in ($h$) to approximately $140\,\%$ of $c_h$. The increase in $c_h^{T}$ is mainly balanced by a decrease of the spatial evolution term $c_h^D$ to negative values. Note that, in addition to the overall terms $c_h^L$, $c_h^M$, $c_h^T$ and $c_h^D$ shown here, the portions of the individual terms also show a complex behaviour; see figure 14.

5.1. Evaluation of $c_f^x/c_f$

First the ratio $c_f^x/c_f$ in panel ($a$) is discussed, quantifying the portion of every component $c_f^x$ in the overall $c_f$. A comparison of the various Mach-number cases shows almost identical behaviour for the blue subsonic and the red supersonic cases. Hence, to allow for a simple argument, the influence of the pressure gradient is discussed based on the subsonic cases first, and then the implications of the Mach number/heat transfer are examined.

All in all, the blue subsonic results depicted in panel ($a$) reflect the trends expected from figure 1. The boundary-layer term $c_f^L$ is only slightly influenced by the pressure gradient's strength and contributes only approximately 2 %–3 % to $c_f$. For ZPG cases at $\beta _K=0$, the turbulent-convection term $c_f^{T}$ contributes approximately $44\,\%$ to $c_f$. For increasing APG strength this portion increases steeply, while an extrapolation for favourable-pressure-gradient cases ($\beta _K<0$) suggests a decrease. The sum of $c_f^D$ and $c_f^{M}$ constitutes the remaining part and thus essentially behaves in a contrary manner to $c_f^{T}$ with respect to $\beta _K$. In principle, the explanation of the trends shown is straightforward: the normalization of the individual components $c_f^x$ with $c_f$ essentially corresponds to a normalization with the wall shear stress $\bar {\tau }_w$ (inner scaling). Hence, the share of $c_f^L/c_f$ corresponds to the average value of $(\bar {\mu } \partial \bar {u}/\partial y)/\bar {\tau }_w$ between $0\leq y \leq y_b$, and the share of $c_f^T/c_f$ to the average value of $(\bar {\rho }\widetilde {u''v''})/\bar {\tau }_w$. As known, the turbulence term $(\bar {\rho }\widetilde {u''v''})/\bar {\tau }_w$ increases considerably for APG cases compared with ZPG cases, explaining the increasing share of $c_f^T/c_f$ for increasing $\beta _K$. As shown in Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2019), the distributions of $(\bar {\mu } \partial \bar {u}/\partial y)/\bar {\tau }_w$ are almost unaffected by $\beta _K$.

Next, the effects of Mach number and heat transfer are assessed. Indicated by the clustering of both sub- and supersonic cases, only a small influence of the local Mach number is visible in panel ($a$). According to Morkovin's transformation, the distributions of $\bar {\rho }\widetilde {u''v''}/\bar {\tau }_w$ – and hence also its integral version $c_f^T/c_f$ – are Mach-number invariant; compare the good agreement of the sub- and supersonic cases (coloured blue and red) in panels ($a$) and ($a^*$) for cases with the same $\beta _K$. To assess the influence of heat transfer, a comparison between the supersonic near-adiabatic $cZPG_{M=2.0}^{+20{\rm K}}$ case (in red) and the strongly cooled supersonic $ZPG_{M=2.0}^{-308{\rm K}}$ case (in orange) does not reveal a recognizable influence of the wall temperature. The same holds for the near-adiabatic pressure-gradient cases; however, it cannot be excluded that this changes for strongly heated or cooled conditions. In summary, the influences of Mach number and heat transfer can be seen to act comparably on the individual $c_f^{x}$ terms for the cases considered. This causes their relative contributions to the overall $c_f$ always to remain approximately the same, even if $c_f$ changes significantly; compare the $c_f$ and $c_h$ distributions given in figure 10.

For the remaining part of this study, it is useful to approximate the shown trends as a function of the local pressure gradient $\beta _K$. As it is simpler than a direct dependence on $\beta _K$, an empirical approximation as a function of the Reynolds analogy factor $s=f(\beta _K,\ldots )$ is used:

(5.1a–c)\begin{equation} \frac{c_f^{T}}{c_f}\approx 0.3315s^2,\quad\frac{c_f^L}{c_f}\approx 0.02s,\quad\frac{c_f^D+c_f^{M}}{c_f}\approx 1-0.3315s^2-0.02s; \end{equation}

see the solid black lines in panel ($a$). As introduced in more detail in Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021), the distribution for $s$ is determined from the analytically derived relation by So (Reference So1994), which is based on classical, incompressible self-similarity assumptions; see also panel ($b$):

(5.2)\begin{equation} s=\frac{2c_h}{c_f}=\frac{\kappa^{{-}1} \ln{\left(Re_{\delta_{K,w}^*}\right)}+B+A\left(\beta_K\right)}{\kappa_\theta^{{-}1} \ln{\left(Re_{\delta_{K,w}^*}\right)}+B_\theta+A_\theta\left(\beta_K, Pr_t\right)}, \end{equation}

where $Re_{\delta _{K,w}^*}=16\,000$, $\kappa _\theta =0.41/Pr_t$, $B=4.9$ and $B_\theta =3.8$. Both $A(\beta _K)$ and $A_\theta (\beta _K,Pr_t)$ are obtained from the solutions of differential equations for the wall-normal velocity and temperature profiles; see Mellor & Gibson (Reference Mellor and Gibson1966), So (Reference So1994) and Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021) for details. The depicted approximations further allow a qualitative assessment of the non-self-similar regions of flow represented by the transparent lines in figure 2($a$). Both the $c_f^{T}$ and $(c_f^D+c_f^{M})$ terms deviate notably from the approximation curves for self-similar regions, for both sub- and supersonic cases. In APG regions, the turbulent contribution $c_f^{T}$ has a delayed reaction to the pressure gradient compared with a self-similar case at the same $\beta _K$; the sum of $(c_f^D+c_f^{M})$ reacts prematurely. A closer look into the history effects in non-self-similar incompressible TBLs can be found in Tanarro, Vinuesa & Schlatter (Reference Tanarro, Vinuesa and Schlatter2020).

5.2. Evaluation of $c_f^x/(2c_h)$

Next, the results in figure 2($b$) are discussed. With $c_f^x/(2c_h)=(c_f^x/c_f)/s$, panel ($b$) essentially represents a scaling of the $c_f^x/c_f$ distributions depicted in panel ($a$) with $s$, and hence the approximations (5.1a–c) become

(5.3a–c)\begin{equation} \frac{c_f^{T}}{2c_h}\approx 0.3315s,\quad\frac{c_f^L}{2c_h}\approx 0.02,\quad\frac{c_f^D+c_f^M}{2c_h}\approx \frac{1}{s}-0.3315s-0.02; \end{equation}

compare the black solid lines in panel ($b$). As the sum of all contributions results in the inverse of the local Reynolds analogy factor $1/s=c_f/(2c_h)$ as depicted in Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021), the individual portions $c_f^x/(2c_h)$ quantify their relative contributions to $1/s$.

As shown in panel ($a$), the normalized contributions $c_f^x/c_f$ correspond to the integral version of Morkovin's transformation and therefore are almost Mach-number invariant. Since the Reynolds analogy factor $s$ is almost Mach-number invariant as well, this also holds for the components $c_f^x/(2c_h)=(c_f^x/c_f)/s$. Therefore, in self-similar regions, the deviations of the single $c_f^x$ terms from the black approximations are nearly independent of whether they are scaled with $c_f$ as in panel ($a$) or with $2c_h$ as in panel ($b$).

Concluding from panel ($b$), the strong pressure-gradient dependence of the inverse of the Reynolds analogy factor $1/s$ (or of $s$ as shown in Wenzel et al. Reference Wenzel, Gibis, Kloker and Rist2021) can be attributed to the fact that the $c_f^T/(2c_h)$ component increases more slowly with APGs than the $(c_f^D+c_f^M)/(2c_h)$ component decreases. For incompressible cases, this reasoning is straightforward as $c_f$ is more affected by pressure gradients than $c_h$, making the variations of the $c_f^x$ contributions the driving factor of the variation in $s$. For the compressible cases, in contrast, the influence of $\beta _K$ on $c_h$ also can be stronger than on $c_f$, making this result a priori not obvious; compare Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021).

Finally, if the non-self-similar flow regions are assessed for the subsonic cases, the blue transparent lines for $1/s$ closely follow the black dotted reference curve for self-similar cases. This indicates a synchronous response of both $c_f$ and $c_h$ on the imposed pressure gradient also in regions of non-self-similar flows, where the individual contributions $c_f^x/c_f$ clearly have not yet converged to a self-similar state. For the supersonic cases (red transparent lines), in contrast, the values for $1/s$ are higher than the settled values in regions of non-self-similar flows, implying an asynchronous response of $c_f$ and $c_h$ to the imposed pressure gradient; see also Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021).

6.1. Turbulent contribution $c_h^{T}$

For understanding of the integral composition of $c_h$, see (3.8), the turbulent-convection term $c_h^{T}$ is of central importance and will be considered first:

(6.1) \begin{align} c_{h}^{T}&= \underset{\text{ $c_{h}^{T,1}$}}{\underbrace{-\frac{1}{y_b}\intop_{0}^{y_{b}}\left[\bar{\rho} \widetilde{v^{''}h^{''}}\right]\,{{\rm d} y}}}\,\underset{\text{ $c_{h}^{T,2}$}}{\underbrace{-\frac{1}{y_b}\intop_{0}^{y_{b}}\left[Ec\left(\tilde{u}\,\bar{\rho} \widetilde{u^{''}v^{''}}+\tilde{v}\,\bar{\rho} \widetilde{v^{''^{2}}}\right)\right]\,{{\rm d}y}}} \nonumber\\ &\quad \underset{\text{ $c_{h}^{T,3}$}}{\underbrace{-\frac{1}{y_b}\intop_{0}^{y_{b}}\left[\frac{Ec}{2}\left(\bar{\rho} \widetilde{u''^{2}v''}+\bar{\rho} \widetilde{v''^{3}}+\bar{\rho} \widetilde{w''^{2}v''}\right)\right]\,{{\rm d} y}}}. \end{align}

The enthalpy fluctuations are denoted by $c_h^{T,1}$ and the fluctuations of the mean kinetic and turbulent kinetic energy (triple fluctuations) by $c_h^{T,2}$ and $c_h^{T,3}$, respectively. Following the idea that the SRA essentially describes the analogy between enthalpy transfer and kinetic-energy transfer, both kinetic-energy terms $c_h^{T,2}$ and $c_h^{T,3}$ are mainly considered as only a single term $c_h^{T,2+3}$ in the following.

6.1.1. Implications from the SRA

Since the individual terms of the derived integral identities stay formally close to the terms of the momentum and total-enthalpy equations, implications derived from the SRA (Morkovin relations) can be directly transferred to the turbulence terms of the integral identity of $c_h$. Thus, by comparison with the DNS data, the validity of selected implications of the SRA can be evaluated from an integral perspective, depending on where a selected case is located in the parameter space considered.

As introduced in § 1, the SRA assumes a direct proportionality between the streamwise velocity fluctuations $u''$ and the total-enthalpy fluctuations $h_0''$ according to

(6.2)\begin{equation} Ec\,k_1u''\approx h_0''=h''+Ec\,\tilde{u}_iu_i''+\frac{Ec}{2}\widetilde{u_i''u_i''}, \end{equation}

where $k_1=-1/(Pr\,Ec)(\bar {q}_w/\bar {\tau }_w)$ takes into account wall heat fluxes in the case of non-adiabatic wall conditions. Multiplication by $\bar {\rho }v''$, time averaging and rearrangement yields

(6.3)\begin{equation} \bar{\rho}\widetilde{v''h''}\approx{-}Ec\left(\tilde{u}_i\bar{\rho}\widetilde{u_i''v''}+\frac{1}{2}\bar{\rho}\widetilde{u_i''^2v''}\right) - \frac{1}{Pr}\frac{\bar{q}_w}{\bar{\tau}_w}\bar{\rho}\widetilde{u''v''},\end{equation}

which finally predicts the ratio between enthalpy-associated and kinetic-energy-associated fluctuations to be

(6.4)\begin{equation} \frac{1}{Ec}\frac{\bar{\rho}\widetilde{v''h''}}{\tilde{u}_i\bar{\rho}\widetilde{u_i''v''}+\frac{1}{2}\bar{\rho}\widetilde{u_i''^2v''}}\approx \underset{\text{adiabatic}}{\underbrace{ -1 \vphantom{\frac{\widetilde{u''v''}}{\frac{1}{2}\widetilde{u_i''^2v''}}} }} -\underset{\text{non-adiabatic}}{\underbrace{ \frac{1}{Pr\,Ec}\frac{\bar{q}_w}{\bar{\tau}_w}\frac{\bar{\rho}\widetilde{u''v''}}{\tilde{u}_i\bar{\rho}\widetilde{u_i''v''}+\frac{1}{2}\bar{\rho}\widetilde{u_i''^2v''}} }}.\end{equation}

In terms of the integral identity, (6.3) becomes

(6.5)\begin{equation} \underset{c_{h}^{T,1}}{\underbrace{\frac{1}{y_{b}}\intop_{0}^{y_{b}}\left[\bar{\rho}\widetilde{v''h''}\right]\,{\textrm{d} y}}} \approx{-}\underset{c_{h}^{T,2+3}}{\underbrace{\frac{Ec}{y_{b}}\intop_{0}^{y_{b}}\left[\tilde{u}_i\bar{\rho}\widetilde{u_i''v''}+\frac{1}{2}\bar{\rho}\widetilde{u_i''^2v''}\right]\,{\textrm{d} y}}} - \underset{c_{h}^{SRA}}{\underbrace{\frac{1}{Pr}\frac{\bar{q}_w}{\bar{\tau}_w}\frac{1}{y_{b}}\intop_{0}^{y_{b}}\left[\bar{\rho}\widetilde{u''v''}\right]\,{\textrm{d} y}}} ,\end{equation}

where the term including the wall heat flux $\bar {q}_w$ is denoted by $c_h^{SRA}$. Hence, the turbulent heat-flux term $c_h^{T,1}$ predicted by the SRA can be considered as the sum of a non-adiabatic contribution $c_h^{SRA}$ and the contribution of the kinetic-energy fluctuations $c_h^{T,2+3}$. As the $c_h^{SRA}$ term becomes zero for adiabatic flows, $c_h^{T,2+3}$ is associated with the adiabatic portion of $c_h^{T,1}$. Division of (6.5) by $c_h^{T,2+3}$ finally yields the integral version of (6.4),

(6.6)\begin{equation} \frac{c_h^{T,1}}{c_h^{T,2+3}} \approx \underset{\text{adiabatic}}{\underbrace{ -1 \vphantom {\frac{c_h^{SRA}}{c_h^{T,2+3}}} }} - \underset{\text{non-adiabatic}}{\underbrace{\frac{c_h^{SRA}}{c_h^{T,2+3}}}}.\end{equation}

6.1.2-1. Remarks on the functional forms of $c_h^{T,1}$, $c_h^{T,2+3}$ and $c_h^{SRA}$.

Below, the turbulent terms $c_h^{T,1}$, $c_h^{T,2+3}$ and $c_h^{SRA}$ are evaluated normalized by $c_h$, allowing for a rough estimation about their functional form. As the wall-normal integrals of both $\bar {\rho }\widetilde {u''v''}/\bar {\tau }_w$ and the non-dimensionalized $\tilde {u}=\hat {\tilde {u}}/u_e$ only have a weak $Ec$-number dependence in the case of self-similar flows, the normalized contributions $2c_h^{T,2+3}/c_f$ and hence $c_h^{T,2+3}/c_h$ suggest an almost linear $Ec$-number dependence of the form

(6.7)\begin{equation} \frac{c_h^{T,2+3}}{c_h}\approx C_1(\beta_K)Ec.\end{equation}

Here, $C_1(\beta _K)$ denotes a constant mainly depending on the pressure gradient's strength applied. With the same argument for the integrand of $c_h^{SRA}$ and the additional assumption that the ratio $\bar {q}_w/\bar {\tau }_w$ essentially resembles the $Ec$-number-independent Reynolds analogy factor $s$, the course of $c_h^{SRA}/c_h$ is expected to behave as a constant $C_2(\beta _K)$ only depending on the local pressure-gradient strength,

(6.8)\begin{equation} \frac{c_h^{SRA}}{c_h}\approx{-}C_2(\beta_K). \end{equation}

Using (6.5) and the modelling approaches (6.7) and (6.8), the $c_h^{T,1}/c_h$ term should depend linearly on the $Ec$ number:

(6.9)\begin{equation} \frac{c_h^{T,1}}{c_h}\approx C_2(\beta_K)-C_1(\beta_K)Ec.\end{equation}

Finally, with (6.7) and (6.9), the ratio $c_h^{T,1}/c_h^{T,2+3}$ should behave according to

(6.10)\begin{equation} \frac{c_h^{T,1}}{c_h^{T,2+3}}\approx{-}1 + \frac{1}{Ec}\frac{C_2(\beta_K)}{C_1(\beta_K)}.\end{equation}

Thus, the SRA predicts the ratio between the enthalpy and kinetic-energy fluctuations $c_h^{T,1}/c_h^{T,2+3}$ to behave like a hyperbola: for adiabatic conditions ($Ec\rightarrow \pm \infty$), it predicts a ratio of $c_h^{T,1}/c_h^{T,2+3}\approx -1$; for strongly heated or cooled cases ($Ec\rightarrow \pm 0$), it predicts a ratio of $c_h^{T,1}/c_h^{T,2+3}\rightarrow \pm \infty$.

6.1.2. Evaluation of the $c_h^T$ contribution

All following distributions are plotted over the $Ec$ number. Evaluated are the non-adiabatic contribution $c_h^{SRA}/c_h$ in figure 4($a$), the contributions of $c_h^{T,1}/c_h$ and $c_h^{T,2+3}/c_h$ in panel ($b$) and its ratio $c_h^{T,1}/c_h^{T,2+3}$ in panel ($c$); the contribution of $c_h^{T,3}$ will be discussed afterwards. Contrary to figure 2, ZPG cases are plotted in red, APG cases in blue, and the strongly cooled supersonic ZPG case in orange. The red and orange coloured ZPG cases are discussed first, the blue coloured pressure-gradient cases afterward.

Figure 4. Evaluation of the turbulent contribution $c_h^{T}$ according to (6.1) as a function of the Eckert number $Ec=(\gamma -1)M_e^2T_e/(\bar {T}_r-T_w)$. Depicted are the $c_h^{SRA}/c_h$ term according to (6.5) in panel ($a$), the portions $c_h^{T,1}/c_h$ and $c_h^{T,2+3}/c_h$ in panel ($b$) and their ratio $c_h^{T,1}/c_h^{T,2+3}$ in panel ($c$). Symbols represent results evaluated at every 20th numerical grid point in the streamwise direction in regions of self-similarity. ZPG cases are coloured red, APG cases blue and the strongly cooled supersonic case yellow. Blue and red lines are linear best fits according to (6.12a,b)–(6.17).

6.1.4-1. ZPG cases.

Six ZPG cases are considered at $M_e=0.3$, $0.5$, $0.95$ and $2.0$ with various relative wall temperatures of $(\bar {T}_r-T_w)=+2{\rm K}$, $+10{\rm K}$, $+20{\rm K}$ and $-308{\rm K}$. As previewed in the previous section, the heat-flux modelling term of the SRA $c_h^{SRA}/c_h$ is independent of $Ec$ for all ZPG cases in panel ($a$). According to (6.8), it is approximated as

(6.11)\begin{equation} \left.\frac{c_h^{SRA}}{c_h}\right|_{ZPG}\approx{-}0.44.\end{equation}

Next, the ratios of the enthalpy fluctuations $c_h^{T,1}/c_h$ and the kinetic-energy fluctuations $c_h^{T,2+3}/c_h$ depicted in panel ($b$) are discussed. As predicted in § 6.1.1, these two terms show opposite behaviour and scale in good approximation linearly with the $Ec$ number for all cases. The trends are approximated by the red lines according to (6.7) and (6.9) with

(6.12a,b)\begin{equation} \left.\frac{c_h^{T,1}}{c_h}\right|_{ZPG}\approx{-}0.32696Ec + 0.48\quad \text{and}\quad \left.\frac{c_h^{T,2+3}}{c_h}\right|_{ZPG}\approx 0.29565Ec. \end{equation}

While the slope of $c_h^{T,2+3}/c_h$ is positive, the slope of $c_h^{T,1}/c_h$ is negative and approximately $10\,\%$ steeper. The latter differs from the SRA, which predicts an equal absolute slope of $c_h^{T,1}/c_h$ and $c_h^{T,2+3}/c_h$ (limit of the SRA for $c_h^{T, 1}/c_h^{T,2(+3)}(Ec\rightarrow \pm \infty )=-1$). The good agreement of all data shown with the linear approximations confirms the functional forms introduced, as well as the definition of the $Ec$ number introduced. Of particular importance is the behaviour of (6.12a,b) for small absolute $Ec$ numbers. For $Ec$ tending towards $\pm 0$ ($Ec=0$ is only possible if $M_e = 0$), the turbulent contribution of the kinetic energy $c_h^{T,2+3}$ becomes increasingly dominated by the turbulent heat transfer $c_h^{T,1}$, which would contribute $48\,\%$ to the overall $c_h$ for a hypothetical $Ec$ number of $0$. For the ZPG cases, the non-zero contribution of $c_h^{T,1}$ at $Ec=0$ approximately equalizes the constant $c_h^{SRA}/c_h\approx -0.44$. This suggests that the non-zero enthalpy fluctuations $c_h^{T,1}$ at $Ec\rightarrow \pm 0$ are adequately represented by the non-adiabatic contribution $c_h^{SRA}$ in the SRA; see (6.9).

Next, the $c_h^{T,1}/c_h^{T,2+3}$ ratio depicted in figure 4($c$) is discussed. Using the approximations for $c_h^{T,1}/c_h$ and $c_h^{T,2+3}/c_h$ from (6.12a,b), the approximation for the DNS data results in

(6.13)\begin{equation} \left.\frac{c_h^{T,1}}{c_h^{T,2+3}}\right|_{ZPG} \approx{-}1.1059 + \frac{1.6235}{Ec};\end{equation}

its distribution is depicted as a red solid line in panel ($c$). Using the approximations for $c_h^{SRA}/c_h$ and $c_h^{T,2+3}/c_h$ in (6.11) and (6.12a,b), yielding $C_1(\beta _K)\approx 0.29565$ and $C_2(\beta _K)\approx -0.44$, the $c_h^{T,1}/c_h^{T,2+3}$ ratio predicted by the SRA (6.10) results in

(6.14)\begin{equation} \left.\frac{c_h^{T,1}}{c_h^{T,2+3}}\right|^{SRA}_{ZPG} \approx{-}1 + \frac{1}{Ec}\frac{C_2(\beta_K)}{C_1(\beta_K)} ={-}1 + \frac{1.4882}{Ec};\end{equation}

its distribution is depicted as a black solid line in panel ($c$). While there are some differences in the constants between (6.13) and (6.14), the characteristic behaviour of the resulting hyperbolas is the same: for $Ec \rightarrow \pm \infty$, the ratio tends towards a constant value; with $-1.1059$ for the DNS data, this value is approximately $10\,\%$ lower than predicted by the SRA. For these cases, both the turbulent heat transfer and the turbulent kinetic-energy transfer are of the same order and act in the opposite wall-normal direction. For $Ec\rightarrow \pm 0$, both distributions exhibit a singularity, when the turbulent heat transfer greatly exceeds the turbulent kinetic-energy transfer. The different constants in the numerators of the $Ec$-number-dependent terms ($1.6235$ in (6.13) and $1.4882$ in (6.14)) only have a minor influence on the slope of the respective curves. Particular attention should be paid to the $Ec$-number range $0< Ec \lesssim 1.5$. Due to the offset of $c_h^{T,1}$ at $Ec=0$, both terms $c_h^{T,1}$ and $c_h^{T,2+3}$ are positive in the considered range in panel ($b$), causing also a positive ratio in panel ($c$). Hence, for cases with $0< Ec \lesssim 1.5$, both the turbulent heat transfer and the turbulent kinetic-energy transfer act in the same wall-normal direction; see § 7.1 for further discussion.

6.1.5-1. Pressure-gradient cases.

Next, the blue pressure-gradient cases in figure 4 are considered. Depicted are cases with $\beta _K=0.2$, $0.6$ and $1.0$. For $\beta _K=0.2$ and $1.0$ only a single subsonic case is shown. For $\beta _K=0.6$ four cases are shown, three subsonic ones with $(\bar {T}_r-T_w)=+2{\rm K}$, $+10{\rm K}$ and $-10{\rm K}$, and a supersonic one with $(\bar {T}_r-T_w)=+20{\rm K}$. Implied by their equal $\beta _K$ value, the four $\beta _K=0.6$ cases are expected to exhibit similar behaviour; they are approximated by a single set of linear fits depicted as blue lines in figure 4. In panel ($a$), the $c_h^{SRA}/c_h$ term is approximated as a constant,

(6.15)\begin{equation} \left.\frac{c_h^{SRA}}{c_h}\right|_{\beta_K=0.6}\approx{-}0.72, \end{equation}

which is notably smaller than the value of $-0.44$ found for the ZPG cases in (6.8). Depicted in panel ($b$), the approximations for $c_h^{T,1}/c_h$ and $c_h^{T,2+3}/c_h$ are

(6.16a,b)\begin{align} \left.\frac{c_h^{T,1}}{c_h}\right|_{\beta_K=0.6}\approx{-}1.27\times 0.32696 Ec + 0.48 \quad \text{and} \quad \left.\frac{c_h^{T,2+3}}{c_h}\right|_{\beta_K=0.6}\approx 1.27\times 0.29565Ec, \end{align}

yielding the $c_h^{T,1}/c_h^{T,2+3}$ ratio depicted in panel ($c$),

(6.17)\begin{equation} \left.\frac{c_h^{T,1}}{c_h^{T,2+3}}\right|_{\beta_K=0.6} \approx{-}1.1059 + \frac{1.2784}{Ec}. \end{equation}

With the approximations for $c_h^{SRA}/c_h$ and $c_h^{T,2+3}/c_h$ in (6.15) and (6.16a,b), yielding $C_1(\beta _K)\approx 1.27\times 0.29565$ and $C_2(\beta _K)\approx -0.72$, the $c_h^{T,1}/c_h^{T,2+3}$ ratio predicted by the SRA (6.10) becomes

(6.18)\begin{equation} \left.\frac{c_h^{T,1}}{c_h^{T,2+3}}\right|^{SRA}_{\beta_K=0.6} \approx{-}1 + \frac{1}{Ec}\frac{C_2(\beta_K)}{C_1(\beta_K)} ={-}1 + \frac{1.9174}{Ec};\end{equation}

see the black dashed line in panel ($c$).

As shown in figure 4($b$), the slope of the approximations (6.16a,b) differs distinctly between the pressure-gradient and ZPG cases; for the APG cases with $\beta _K=0. 6$, both slopes are approximately $27\,\%$ steeper than in the ZPG cases; compare (6.12a,b) and (6.16a,b). Considering $Ec=0$, the zero offset of the $c_h^{T,1}/c_h$ contribution (${\approx }0.48$) is approximately the same for $\beta _K=0.6$ as for the ZPG cases. In contrast to the ZPG cases discussed before, this offset is not correctly balanced by the heat-transfer modelling term $c_h^{SRA}$ by the SRA in (6.15) (${\approx}{-}0.72$). Due to the large schematic similarity between the APG and ZPG cases in panel ($b$), however, the ratio $c_h^{T,1}/c_h^{T,2+3}$ is very similar for the APG and ZPG cases in panel ($c$); compare the red and blue lines. For $Ec\rightarrow \pm \infty$, both the APG and ZPG cases tend to a comparable value of approximately $-1.1$. For small $Ec$ numbers with $|Ec|\lesssim 5$, in contrast, the deviations can be large. Note that pressure-gradient influences on $c_h^{T,1}/c_h^{T,2+3}$ behave qualitatively differently between DNS data and the prediction of the SRA. For the DNS data, the influence of APGs moves the vertex of the respective hyperbola closer to the origin compared with the ZPG cases, while the SRA predicts the opposite behaviour; compare the red and blue lines and the black dashed and solid black lines. This behaviour can be partly attributed to the mismatch between the contributions of $c_h^{T,1}$ and $c_h^{SRA}$ at $Ec=0$, implying that pressure-gradient effects are not adequately incorporated in the present formulation of the SRA chosen; see § 7 for a more detailed discussion. Nevertheless, from an integral point of view, the underlying schematic, predicting the trends depicted in panel ($c$), can be assessed to apply to both the ZPG and APG cases considered. It is noteworthy that this holds in good approximation for both regions with self-similar flow (symbols) and regions with non-self-similar flow (thick blue lines).

Briefly summarizing the main result for both the ZPG cases and the APG cases, figure 4 provides a schematic to evaluate how the turbulent transfer processes within the total-enthalpy equation are related to each other for a wide parameter range as a function of the local $Ec$ number. In § 7 some further remarks are made on how far the behaviour shown for the integral progressions of $c_h^{T,1}/c_h^{T,2(+3)}$ depicted can be transferred to the local profiles of its integrands $\bar {\rho }\widetilde {v''h''}$ and $Ec\,\tilde {u}\bar {\rho }\widetilde {u''v''}$.

6.1.6-1. Sum of $c_h^{T}$ and influence of the $c_h^{T,3}$ term.

In the previous section, both the triple-fluctuation terms of the mean kinetic energy $c_h^{T,2}$ and the turbulent kinetic energy $c_h^{T,3}$ are considered as a single term $c_h^{T,2+3}$. This differs from the classical approach where the term of the fluctuating turbulent kinetic energy ($c_h^{T,3}$) is usually neglected, raising the question of how large the influence of $c_h^{T,3}$ is and on which factors its influence depends.

For the $c_h^{T,2+3}$ distribution depicted in figure 4($b$), the contribution of $c_h^{T,3}$ is only minor. If only $c_h^{T,2}$ is evaluated instead of $c_h^{T,2+3}$, the slope of the $c_h^{T,2+3}/c_h$ approximation in (6.12a,b) is reduced by approximately $3\,\%$. For the sum of all turbulent terms $c_h^{T}=c_h^{T,1}+c_h^{T,2}+c_h^{T,3}$, however, which represents the contribution of the total-enthalpy fluctuations in (3.8), $c_h^{T,3}$ can make up a substantial part, since the counteracting contributions of $c_h^{T,1}$ and $c_h^{T,2}$ largely cancel. To illustrate the share of $c_h^{T,3}$ in $c_h^T$, both the sum $c_h^{T}/c_h=(c_h^{T,1}+c_h^{T,2}+c_h^{T,3})/c_h$ (red dotted approximation) and its contribution $c_h^{T,3}/c_h$ (red solid approximation) are depicted as functions of the $Ec$ number in figure 5. The red ZPG trend of $c_h^{T}/c_h$ is computed from the sum of its components (6.12a,b), and the red trend for $c_h^{T,3}/c_h$ is approximated by a linear fit:

(6.19a,b)\begin{equation} \frac{c_h^{T}}{c_h}\approx{-}0.03131Ec + 0.48,\quad \frac{c_h^{T,3}}{c_h}\approx{-}0.01Ec. \end{equation}

As both the $c_h^{T}/c_h$ and the $c_h^{T,3}/c_h$ contributions exhibit an $Ec$-number dependence, the percentage share of $c_h^{T,3}$ in the overall $c_h^{T}$ depends on the local $Ec$ number. At a hypothetical $Ec=0$, $c_h^{T,3}$ makes no contribution to $c_h^T$. At $Ec=-25$, $c_h^{T,3}$ contributes approximately $20\,\%$ to the overall $c_h^{T}$ and thus $25\,\%$ to the overall $c_h$. For the subsonic pressure-gradient cases in figure 5, both $c_h^{T}/c_h$ and $c_h^{T,3}/c_h$ are almost unaffected by the pressure-gradient strength in regions of self-similar flow; all results are in good agreement with the red approximations for the ZPG cases. The supersonic pressure-gradient case at $Ec\approx -17$, in contrast, greatly deviates from the approximation shown. However, it is emphasized that $c_h^T$ is the sum of two large opposite terms, which makes its distribution very sensitive. Although Mach-number influences have been almost completely eliminated in previous representations by scaling with $c_f$ and/or $c_h$, they seem to be of relevance for the quantities considered here, at least for the supersonic pressure-gradient case.

Figure 5. Contribution of $c_h^{T,3}$ to $c_h^T=c_h^{T,1+2+3}$ according to (6.1) as a function of the Eckert number $Ec=(\gamma -1)M_e^2T_e/(\bar {T}_r-T_w)$. Symbols represent results evaluated at every 20th numerical grid point in the streamwise direction in regions of self-similarity. ZPG cases are coloured red, APG cases blue and the strongly cooled supersonic case yellow. The red solid line is the sum of the ZPG approximations according to (6.12a,b), and the dashed red line approximates the portion of the $c_h^{T,3}$ term.

6.2. Evaluation of the overall $c_h$ identity

In the following, the overall integral identity of $c_h$ (3.8) is evaluated, putting the turbulent-convection term $c_h^{T}$ in its overall context. All quantities evaluated are normalized with $c_h$ in figure 6($a$) and with $1/2c_f$ in panel ($b$).

Figure 6. Evaluation of the $c_h$ identity according to (3.8) as function of the Eckert number $Ec=(\gamma -1)M_e^2T_e/(\bar {T}_r-T_w)$. The ratio of $c_h^x/c_h$ is given in panel ($a$), $2c_h^x/c_f$ in panel ($b$). Symbols represent results evaluated at every 20th grid point in the streamwise direction in regions of self-similarity. ZPG cases are coloured red, APG cases blue and the strongly cooled supersonic case yellow. Red and blue lines linearly approximate ZPG and $\beta _K=0.6$ cases.

First, figure 6($a$) is discussed. Using (6.19a,b) for $c_h^{T}/c_h$, the remaining ZPG approximations depicted are fitted by

(6.20a,b)\begin{equation} \frac{c_h^{L}}{c_h}\approx{-}0.003913Ec+0.03,\quad \frac{c_h^{D}+c_h^{M}}{c_h}\approx 0.03522Ec+0.49. \end{equation}

The boundary-layer term $c_h^L$ is negligibly small for small absolute $Ec$ numbers, but increases notably with decreasing $Ec$ number; for example, it contributes with approximately $12\,\%$ at $Ec=-25$, a considerable part of $c_h$. Hence, the dominant behaviour occurring with varying $Ec$ number is the variation of $c_h^{T}/c_h$ and $(c_h^D+c_h^{M})/c_h$. Like $c_h^{T}/c_h$, both $c_h^L/c_h$ and $(c_h^D+c_h^{M})/c_h$ are virtually unaffected by the strength of the pressure gradient for the subsonic cases; compare the blue dots and red lines.

With $2c_h^x/c_f=(c_h^x/c_h)s$, panel ($b$) represents a multiplication of the $c_h^x/c_h$ distributions depicted in panel ($a$) by $s$; the same holds for the approximations (6.19a,b) and (6.20a,b). As the sum of all contributions results in the Reynolds analogy factor $s=2c_h/c_f$, see Wenzel et al. (Reference Wenzel, Gibis, Kloker and Rist2021), the individual portions $2c_h^x/c_f$ quantify their relative contributions to $s$. While in panel ($a$) all distributions add up to 100 per cent and thus allow a simple comparison between the different cases, the pressure-gradient and ZPG cases are fanned out in panel ($b$) due to the strong pressure-gradient dependence of $s$. This results in a separate set of curves for each $\beta _K$, adding up to the respective $s$ value; again, only the sets for $\beta _K=0$ and $\beta _K=0.6$ are approximated by red and blue lines, respectively. Since the Reynolds analogy factor is only weakly influenced by the local $Ec$ number, the sum of the respective curves is also virtually invariant to the $Ec$ number for cases with the same $\beta _K$ in near-adiabatic conditions. To what extent the course of the pressure-gradient cases would be affected under strongly non-adiabatic conditions cannot be evaluated based on the available data, however. At least for the strongly cooled supersonic ZPG case, no influence of the wall temperature can be determined.