2 The SLS TKE equation

As in Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015), we apply a semi-local scaling transformation to the Navier–Stokes equations for density $\unicode[STIX]{x1D70C}$ , dynamic viscosity $\unicode[STIX]{x1D707}$ , velocity $u_{i}$ and pressure $p$ , defined as

(2.1a-d ) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}/\langle \unicode[STIX]{x1D70C}\rangle ,\quad \hat{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D707}/\langle \unicode[STIX]{x1D707}\rangle ,\quad \hat{u} _{i}=u_{i}/u_{\unicode[STIX]{x1D70F}}^{\star }\quad \text{and}\quad \hat{p}=p/(\langle \unicode[STIX]{x1D70C}\rangle {u_{\unicode[STIX]{x1D70F}}^{\star }}^{2}),\end{eqnarray}$$

where $\langle \unicode[STIX]{x1D70C}\rangle$ , $\langle \unicode[STIX]{x1D707}\rangle$ and $u_{\unicode[STIX]{x1D70F}}^{\star }$ are the Reynolds-averaged values of local density, local viscosity and semi-local friction velocity $u_{\unicode[STIX]{x1D70F}}^{\star }=\sqrt{\unicode[STIX]{x1D70F}_{w}/\langle \unicode[STIX]{x1D70C}\rangle }$ , with $\unicode[STIX]{x1D70F}_{w}$ the averaged wall shear stress. The spatial coordinates are normalized as $\hat{x}=x/h$ . Applying the scaling transformation to the continuity equation and assuming that the averaged wall shear stress $\unicode[STIX]{x1D70F}_{w}$ is constant or changes slowly in the streamwise direction, we obtain (cf. appendix A)

(2.2) $$\begin{eqnarray}t_{\unicode[STIX]{x1D70F}}^{\star }\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}}{\unicode[STIX]{x2202}\hat{x}_{i}}+\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}\underbrace{\frac{1}{2\langle \unicode[STIX]{x1D70C}\rangle }\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x2202}\hat{x}_{i}}}_{d_{i}}=0,\end{eqnarray}$$

with $t_{\unicode[STIX]{x1D70F}}^{\star }=h/u_{\unicode[STIX]{x1D70F}}^{\star }$ . The additional term, $d_{i}$ , is the result of the semi-local scaling transformation, which contains the gradient of the Reynolds-averaged density. Accordingly, the SLS momentum equations in non-conservative form are given as (cf. appendix B)

(2.3) $$\begin{eqnarray}t_{\unicode[STIX]{x1D70F}}^{\star }\hat{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\hat{u} _{i}}{\unicode[STIX]{x2202}t}+\hat{\unicode[STIX]{x1D70C}}\hat{u} _{j}\frac{\unicode[STIX]{x2202}\hat{u} _{i}}{\unicode[STIX]{x2202}\hat{x}_{j}}-\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}\hat{u} _{j}d_{j}=-\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}\hat{x}_{i}}+\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}_{ij}}{\unicode[STIX]{x2202}\hat{x}_{j}}-\frac{\unicode[STIX]{x2202}\hat{D}_{ij}}{\unicode[STIX]{x2202}\hat{x}_{j}}+\hat{\unicode[STIX]{x1D70C}}\hat{f}_{i},\end{eqnarray}$$

with the stress tensor $\hat{\unicode[STIX]{x1D70F}}_{ij}=\hat{\unicode[STIX]{x1D707}}/Re_{\unicode[STIX]{x1D70F}}^{\star }[(\unicode[STIX]{x2202}\hat{u} _{i}/\unicode[STIX]{x2202}\hat{x}_{j}+\unicode[STIX]{x2202}\hat{u} _{j}/\unicode[STIX]{x2202}\hat{x}_{i})-2/3(\unicode[STIX]{x2202}\hat{u} _{k}/\unicode[STIX]{x2202}\hat{x}_{k})\unicode[STIX]{x1D6FF}_{ij}]$ . If compared with the conventional form, two additional terms appear, namely $\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}\hat{u} _{j}d_{j}$ and $\unicode[STIX]{x2202}_{\hat{x}_{j}}\hat{D}_{ij}$ , where $\hat{D}_{ij}=\hat{\unicode[STIX]{x1D707}}/Re_{\unicode[STIX]{x1D70F}}^{\star }[(\hat{u} _{i}d_{j}+\hat{u} _{j}d_{i})-2/3(\hat{u} _{k}d_{k})\unicode[STIX]{x1D6FF}_{ij}]$ . It should be noted that the effective viscosity in $\hat{\unicode[STIX]{x1D70F}}_{ij}$ is proportional to $1/Re_{\unicode[STIX]{x1D70F}}^{\star }$ ; $\hat{f}_{i}$ is a normalized arbitrary body force.

Given (2.2) and (2.3), we can now derive the SLS TKE equation using a standard procedure by first multiplying the momentum equation (2.3) with the Favre fluctuating velocity $\hat{u} _{i}^{\prime \prime }$ and then Reynolds averaging the product. To highlight distinct differences in the derivation when using the SLS Navier–Stokes equations, this procedure is outlined for the terms on the left-hand side of (2.3), while the derivation of the other terms closely follows the standard procedure and is thus not shown. The Favre decomposition is used for the velocity, while the Reynolds decomposition is used for all other quantities, which for an arbitrary quantity $\unicode[STIX]{x1D719}$ are given as $\unicode[STIX]{x1D719}=\{\unicode[STIX]{x1D719}\}+\unicode[STIX]{x1D719}^{\prime \prime }$ and $\unicode[STIX]{x1D719}=\langle \unicode[STIX]{x1D719}\rangle +\unicode[STIX]{x1D719}^{\prime }$ respectively. It is important to note that the Favre mean is $\{\unicode[STIX]{x1D719}\}=\langle \unicode[STIX]{x1D70C}\unicode[STIX]{x1D719}\rangle /\langle \unicode[STIX]{x1D70C}\rangle$ , which, with the locally scaled density, can also be expressed as $\{\unicode[STIX]{x1D719}\}=\langle \hat{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D719}\rangle$ ; an identity we will use throughout the derivation of the SLS TKE equation. In addition, $\langle \hat{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D719}^{\prime \prime }\rangle =0$ .

Multiplying the first term in (2.3) by $\hat{u} _{i}^{\prime \prime }$ and Reynolds averaging the product gives

(2.4) $$\begin{eqnarray}t_{\unicode[STIX]{x1D70F}}^{\star }\left\langle \hat{u} _{i}^{\prime \prime }\hat{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\hat{u} _{i}}{\unicode[STIX]{x2202}t}\right\rangle =t_{\unicode[STIX]{x1D70F}}^{\star }\frac{\unicode[STIX]{x2202}\overbrace{\left\langle {\textstyle \frac{1}{2}}\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}^{\prime \prime }\hat{u} _{i}^{\prime \prime }\right\rangle }^{\langle \hat{\unicode[STIX]{x1D70C}}\hat{k}\rangle =\{\hat{k}\}}}{\unicode[STIX]{x2202}t}-\overbrace{\left\langle \hat{k}t_{\unicode[STIX]{x1D70F}}^{\star }\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}t}\right\rangle }^{(\text{I})},\end{eqnarray}$$

with the definition of the TKE $\hat{k}=\hat{u} _{i}^{\prime \prime }\hat{u} _{i}^{\prime \prime }/2$ . For the convection term, we obtain

(2.5) $$\begin{eqnarray}\left\langle \hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}^{\prime \prime }\hat{u} _{j}\frac{\unicode[STIX]{x2202}\hat{u} _{i}}{\unicode[STIX]{x2202}\hat{x}_{j}}\right\rangle =\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{\hat{u} _{i}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\hat{x}_{j}}(\{\hat{u} _{j}\}\{\hat{k}\}+\{\hat{u} _{j}^{\prime \prime }\hat{k}\})-\underbrace{\left\langle \hat{k}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}\hat{u} _{j}}{\unicode[STIX]{x2202}\hat{x}_{j}}\right\rangle }_{(\text{II})}.\end{eqnarray}$$

The first term on the right-hand side of (2.5) represents turbulence production as a function of SLS quantities. As we will see later, it is crucial to express the partial derivative of the SLS mean velocity in terms of the density-weighted partial derivative of velocity using the van Driest transformation given by (1.2). With the additional relation $\sqrt{\unicode[STIX]{x1D70C}_{w}}u_{\unicode[STIX]{x1D70F}_{w}}=\sqrt{\langle \unicode[STIX]{x1D70C}\rangle }u_{\unicode[STIX]{x1D70F}}^{\star }$ , this leads to

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\{\hat{u} _{i}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\frac{\unicode[STIX]{x2202}\displaystyle \frac{\{u_{i}\}}{u_{\unicode[STIX]{x1D70F}}^{\star }}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\frac{\unicode[STIX]{x2202}\sqrt{\displaystyle \frac{\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x1D70C}_{w}}}\displaystyle \frac{\{u_{i}\}}{u_{\unicode[STIX]{x1D70F}_{w}}}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\frac{\sqrt{\displaystyle \frac{\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x1D70C}_{w}}}\unicode[STIX]{x2202}\displaystyle \frac{\{u_{i}\}}{u_{\unicode[STIX]{x1D70F}_{w}}}}{\unicode[STIX]{x2202}\hat{x}_{j}}+\frac{\displaystyle \frac{\{u_{i}\}}{u_{\unicode[STIX]{x1D70F}_{w}}}\unicode[STIX]{x2202}\sqrt{\displaystyle \frac{\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x1D70C}_{w}}}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\frac{\unicode[STIX]{x2202}\{u_{i}^{vD}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}+\{\hat{u} _{i}\}d_{j}.\end{eqnarray}$$

The turbulence production in (2.5) can then be written as the sum of two terms, namely

(2.7) $$\begin{eqnarray}\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{\hat{u} _{i}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\underbrace{\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{u_{i}^{vD}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}}_{-\hat{P}_{k}}+\underbrace{\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\{\hat{u} _{i}\}d_{j}}_{(\text{III})},\end{eqnarray}$$

with $\hat{P}_{k}$ as the product of Reynolds stress and van Driest velocity gradient, and an additional term ( $\text{III}$ ) that can be large in magnitude, as it is the product of Reynolds stress, Favre-averaged velocity and density gradient. However, as we will see later, this term will cancel. The third term in the momentum equation, multiplied by $\hat{u} _{i}^{\prime \prime }$ , gives

(2.8) $$\begin{eqnarray}-\!\langle \hat{u} _{i}^{\prime \prime }\hat{\unicode[STIX]{x1D70C}}\hat{u} _{i}\hat{u} _{j}\rangle d_{j}=-\!\underbrace{\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\{\hat{u} _{i}\}d_{j}}_{(\text{IV})}-2\{\hat{u} _{j}\}\{\hat{k}\}d_{j}-2\{\hat{u} _{j}^{\prime \prime }\hat{k}\}d_{j}.\end{eqnarray}$$

We can now proceed and sum the individual terms. For example, the addition of $(\text{I})+(\text{II})$ allows us to substitute the continuity equation (2.2) and we obtain

(2.9) $$\begin{eqnarray}-\!\left\langle \hat{k}\left(t_{\unicode[STIX]{x1D70F}}^{\star }\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70C}}\hat{u} _{j}}{\unicode[STIX]{x2202}\hat{x}_{j}}\right)\right\rangle =\{\hat{u} _{j}\}\{\hat{k}\}d_{j}+\{\hat{u} _{j}^{\prime \prime }\hat{k}\}d_{j}.\end{eqnarray}$$

As mentioned earlier, term $(\text{III})$ cancels with $(\text{IV})$ since we expressed the velocity gradient in the turbulence production as a function of the van Driest velocity. Summing up all remaining terms, including the conventional decomposition for the pressure and the viscous terms we omitted earlier, results in the SLS TKE equation, given as

(2.10) $$\begin{eqnarray}\displaystyle t_{\unicode[STIX]{x1D70F}}^{\star }\frac{\unicode[STIX]{x2202}\{\hat{k}\}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\{\hat{k}\}\{\hat{u} _{j}\}}{\unicode[STIX]{x2202}\hat{x}_{j}}=\hat{P}_{k}-\hat{\unicode[STIX]{x1D700}}_{k}+\hat{T}_{k}+{\hat{C}}_{k}+\hat{{\mathcal{D}}}_{k}, & & \displaystyle\end{eqnarray}$$

with production $\hat{P}_{k}=-\{\hat{u} _{i}^{\prime \prime }\hat{u} _{j}^{\prime \prime }\}\unicode[STIX]{x2202}\{u_{i}^{vD}\}/\unicode[STIX]{x2202}\hat{x}_{j}$ , dissipation per unit volume $\hat{\unicode[STIX]{x1D700}}_{k}=\langle \hat{\unicode[STIX]{x1D70F}}_{ij}^{\prime }\unicode[STIX]{x2202}\hat{u} _{i}^{\prime }/\unicode[STIX]{x2202}\hat{x}_{j}\rangle$ , diffusion (decomposed into viscous diffusion, turbulent transport and pressure diffusion) $\hat{T}_{k}=\unicode[STIX]{x2202}(\langle \hat{u} _{i}^{\prime }\hat{\unicode[STIX]{x1D70F}}_{ij}^{\prime }\rangle -\{\hat{u} _{j}^{\prime \prime }\hat{k}\}-\langle \hat{p}^{\prime }\hat{u} _{j}^{\prime }\rangle )/\unicode[STIX]{x2202}\hat{x}_{j}$ , compressibility ${\hat{C}}_{k}=\langle \hat{p}^{\prime }\unicode[STIX]{x2202}\hat{u} _{j}^{\prime }/\unicode[STIX]{x2202}\hat{x}_{j}\rangle -\langle \hat{u} _{j}^{\prime \prime }\rangle \unicode[STIX]{x2202}\langle \hat{p}\rangle /\unicode[STIX]{x2202}\hat{x}_{j}+\langle \hat{u} _{i}^{\prime \prime }\rangle \unicode[STIX]{x2202}\langle \hat{\unicode[STIX]{x1D70F}}_{ij}\rangle /\unicode[STIX]{x2202}\hat{x}_{j}$ , and terms related to the mean density gradient $\hat{{\mathcal{D}}}_{k}=(\{\hat{u} _{j}\}\{\hat{k}\}+\{\hat{u} _{j}^{\prime \prime }\hat{k}\})d_{j}-\langle \hat{u} _{i}^{\prime \prime }\unicode[STIX]{x2202}\hat{D}_{ij}/\unicode[STIX]{x2202}\hat{x}_{j}\rangle$ . The result is an evolution equation in which the varying density has been absorbed into the van Driest velocity for the production $\hat{P}_{k}$ , and the semi-local Reynolds number into the dissipation $\hat{\unicode[STIX]{x1D700}}_{k}$ and viscous diffusion. The TKE equation is thus essentially equivalent to its incompressible form, except for the additional terms $\hat{{\mathcal{D}}}_{k}$ and $\hat{{\mathcal{C}}}_{k}$ , which both can be considered to be small, as we will see later. Since the van Driest velocity is not an independent variable (Patel et al. Reference Patel, Boersma and Pecnik2016), this derivation suggests that the ‘leading-order effect’ on turbulence in variable property flows can be characterized by $Re_{\unicode[STIX]{x1D70F}}^{\star }$ .

Another intriguing observation is that the TKE equation can be used in its ‘incompressible’ form to model variable property turbulent channel flows. To do so, the velocity in the TKE production term and the viscosity in the viscous terms have to be replaced by the van Driest transformed velocity and the semi-local Reynolds number respectively. Both hypotheses will be tested on flow cases that will be introduced next.

3 Turbulent channel flows with variable properties

Figure 1. Contour plots of instantaneous density $\unicode[STIX]{x1D70C}$ (top) and dynamic viscosity $\unicode[STIX]{x1D707}$ (bottom) for cases $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ (a), GL (b) and LL (c).

Table 1. The investigated cases: CP, constant property case with $Re_{\unicode[STIX]{x1D70F}}=395$ ; $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ , variable property case with constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ ( $=395$ ) across the channel; GL, case with gas-like property variations; LL, case with liquid-like property variations; T&L, fully compressible turbulent channel flow with a bulk Mach number of 4 from Trettel & Larsson (Reference Trettel and Larsson2016). The columns report the constitutive relations for density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D707}$ as a function of temperature $T$ . The semi-local friction Reynolds numbers at the wall and channel centre are given by ${Re_{\unicode[STIX]{x1D70F}}^{\star }}_{w}$ and ${Re_{\unicode[STIX]{x1D70F}}^{\star }}_{c}$ respectively.

Table 1 summarizes five turbulent channel flows. The first case, CP, corresponds to a reference flow with constant properties at $Re_{\unicode[STIX]{x1D70F}}=395$ . The next three cases have been obtained by solving the low-Mach-number approximation of the Navier–Stokes equations, whereby the flows have been volumetrically heated (constant volumetric heat source in the energy equation) and both walls are kept at a constant temperature. Different constitutive relations for density, $\unicode[STIX]{x1D70C}$ , and viscosity, $\unicode[STIX]{x1D707}$ , as a function of temperature, $T$ , were used. The case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ corresponds to a flow for which density and viscosity are decreasing away from the wall (figure 1 a), such that the semi-local Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{\star }$ is constant across the whole channel height, meaning that $\sqrt{\langle \unicode[STIX]{x1D70C}\rangle /\unicode[STIX]{x1D70C}_{w}}=\langle \unicode[STIX]{x1D707}\rangle /\unicode[STIX]{x1D707}_{w}$ . Although this case has arbitrary thermophysical properties, it is worthwhile to mention that it bears similarities to supercritical fluids, for which both density and viscosity decrease when heated across the pseudo-critical temperature (Nemati et al. Reference Nemati, Patel, Boersma and Pecnik2016; Peeters et al. Reference Peeters, Pecnik, Rohde, van der Hagen and Boersma2016). Cases GL and LL (figure 1 b,c) are flows with gas-like and liquid-like property variations that both have large gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . More details on the governing equations and the numerical scheme can be found in Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015, Reference Patel, Boersma and Pecnik2016). The last case in table 1 (case T&L) is a fully compressible turbulent channel flow with isothermal walls, a bulk Mach number of 4 and a wall-based friction Reynolds number of 1017 (Trettel & Larsson Reference Trettel and Larsson2016).

Figure 2. Averaged profiles for density (a), viscosity (b), semi-local Reynolds number (c), velocity (d), van Driest transformed velocity (e) and universal velocity scaling (f) for the DNS cases presented in table 1.

The largest decrease of density ( $\unicode[STIX]{x1D70C}_{w}/\langle \unicode[STIX]{x1D70C}_{c}\rangle \approx 8.5$ ) is obtained for case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ , while for cases GL and T&L the density decreases approximately by factors of 5 and 3.6 respectively (figure 2 a). The profiles for viscosity are shown for the sake of completeness in figure 2(b). However, the most important parameter for the characterization of variable property flows is the semi-local Reynolds number shown in figure 2(c). It can be seen that the cases GL and T&L show similar decreasing $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles, while $Re_{\unicode[STIX]{x1D70F}}^{\star }$ for case LL increases. The case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ has a constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profile by construction and collapses with the constant property case CP. The streamwise velocity profiles are shown in figure 2(d–f). It should be noted that, even if the velocity $\langle u\rangle$ for case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ is considerably higher than for case CP, the van Driest velocity transformation is capable of providing a collapse with the constant property universal velocity profile. This is not the case for flows that have gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ (GL, LL and T&L) since the viscous scales for these cases are changing. On the other hand, the universal velocity scaling proposed by Trettel & Larsson (Reference Trettel and Larsson2016), and later independently derived by Patel et al. (Reference Patel, Boersma and Pecnik2016), provides a good collapse for all cases (figure 2 f). It should be noted that the normalized wall-normal coordinates are $y^{+}=Re_{\unicode[STIX]{x1D70F}}y/h$ , and $y^{\star }=Re_{\unicode[STIX]{x1D70F}}^{\star }y/h$ . Since in Patel et al. (Reference Patel, Boersma and Pecnik2016) the universal transformation has been derived by rescaling the Navier–Stokes equations using local mean properties (similarly to the SLS TKE equation), the universal velocity transformation can also be expressed in terms of the van Driest velocity and the semi-local Reynolds number, as $\langle u^{\star }\rangle =\int _{0}^{\overline{u}^{vD}}(1+(y/Re_{\unicode[STIX]{x1D70F}}^{\star })\,\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y)\,\text{d}\langle u^{vD}\rangle$ .

Figure 3. The SLS TKE budgets, equation (4.1). (a) Cases CP (symbols) and $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ (lines), (b) case GL and (c) case LL.

4 The SLS TKE budgets

The budget equation for the SLS TKE for fully developed turbulent channel flows can be written as

(4.1) $$\begin{eqnarray}\hat{P}_{k}-\hat{\unicode[STIX]{x1D700}}_{k}+\hat{T}_{k}+{\hat{C}}_{k}+\hat{{\mathcal{D}}}_{k}=0.\end{eqnarray}$$

The budgets for the cases CP, $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ , GL and LL are shown in figure 3, where they have been scaled by $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . Despite the large variations in density and viscosity for case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ , $\hat{P}_{k}$ and $\hat{\unicode[STIX]{x1D700}}_{k}$ are overlapping with case CP (symbols in figure 3 a), since for both cases the $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles are constant and equal. This confirms that also turbulence production and dissipation are similar for cases with similar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles (Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015). However, the diffusion is slightly affected by strong property gradients at the location of the production peak at $y^{\star }\approx 12$ . In general, however, ${\hat{C}}_{k}$ and $\hat{{\mathcal{D}}}_{k}$ are small for cases $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ and GL, and for cases CP and LL they are zero, since the density is constant. Based on this observation, we can assume that the additional terms, $\hat{{\mathcal{D}}}_{k}$ and $\hat{{\mathcal{C}}}_{k}$ , have a minor effect on the evolution of the SLS TKE for the cases presented herein. In general, it is accepted that for wall bounded flows, the compressibility term ${\mathcal{C}}_{k}$ in the traditional budget equation is negligible if compared with the other terms (Morinishi, Tamano & Nakabayashi Reference Morinishi, Tamano and Nakabayashi2004; Duan et al. Reference Duan, Beekman and Martin2010).

It should be noted that the key difference, if compared with the conventional semi-local scaling of the budget terms as given in Foysi, Sarkar & Friedrich (Reference Foysi, Sarkar and Friedrich2004), Morinishi et al. (Reference Morinishi, Tamano and Nakabayashi2004) and Duan et al. (Reference Duan, Beekman and Martin2010), is that here we do not scale the individual terms, but we evaluate the budget terms in the TKE equation using SLS variables (e.g. $\hat{k}$ , $\hat{\unicode[STIX]{x1D70C}}$ , $\hat{\unicode[STIX]{x1D707}}$ , etc.). It is possible to show that for the production, both approaches are equivalent, since $P_{k}/(\langle \unicode[STIX]{x1D70C}\rangle {u_{\unicode[STIX]{x1D70F}}^{\star }}^{3}/\unicode[STIX]{x1D6FF}_{v}^{\star })=\hat{P}_{k}/Re_{\unicode[STIX]{x1D70F}}^{\star }$ , with $\unicode[STIX]{x1D6FF}_{v}^{\star }=h/Re_{\unicode[STIX]{x1D70F}}^{\star }$ and $P_{k}$ the production term in the traditional TKE equation. However, for the other terms, the conventional semi-local scaling approach and the one presented herein are not equivalent. The SLS TKE equation additionally allows us to clearly distinguish effects related to different distributions of $Re_{\unicode[STIX]{x1D70F}}^{\star }$ from effects that are reflected in the terms $\hat{{\mathcal{C}}}_{k}$ and $\hat{{\mathcal{D}}}_{k}$ (in situations where these terms are larger).

5 Turbulence modelling

Most turbulence models are based on the $k$ – $\unicode[STIX]{x1D700}$ model. However, the standard $k$ – $\unicode[STIX]{x1D700}$ model gives unacceptable results for the turbulent shear stress in the near-wall region. Numerous remedies (damping functions, etc.) have been proposed, but these corrections usually negatively affect the accuracy of the modelled TKE. A model that preserves the accuracy of the TKE and also provides accurate results for the turbulent shear stress is the model proposed by Durbin (Reference Durbin1995), Lien & Kalitzin (Reference Lien and Kalitzin2001). Besides the TKE $k$ and the dissipation $\unicode[STIX]{x1D700}$ , this model solves two additional equations, namely a transport equation for the wall-normal velocity fluctuation, ${v^{\prime }}^{2}$ , which is an appropriate velocity scale for turbulent transport towards the wall, and an elliptic relaxation equation that essentially models the pressure strain correlation that appears in the evolution equation of ${v^{\prime }}^{2}$ . For a fully developed turbulent flow in a channel, the equations for $k$ , $\unicode[STIX]{x1D700}$ , ${v^{\prime }}^{2}$ and $f$ read (the notation of the averaging operators is omitted for brevity)

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x2202}_{y}\left[\left(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}_{t}/\unicode[STIX]{x1D70E}_{k}\right)\unicode[STIX]{x2202}_{y}k\right]=P_{k}-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}, & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x2202}_{y}\left[\left(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}_{t}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D700}}\right)\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D700}\right]=\frac{1}{T}\left(C_{\unicode[STIX]{x1D700}1}P_{k}-C_{\unicode[STIX]{x1D700}2}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}\right), & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle L^{2}\unicode[STIX]{x2202}_{y^{2}}^{2}f-f=\frac{1}{T}\left[\left(C_{1}-6\right)\frac{{v^{\prime }}^{2}}{k}-\frac{2}{3}\left(C_{1}-1\right)\right]-C_{2}\frac{P_{k}}{\unicode[STIX]{x1D70C}k}, & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x2202}_{y}\left[\left(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}_{t}/\unicode[STIX]{x1D70E}_{k}\right)\unicode[STIX]{x2202}_{y}{v^{\prime }}^{2}\right]=\unicode[STIX]{x1D70C}kf-6\unicode[STIX]{x1D70C}{v^{\prime }}^{2}\frac{\unicode[STIX]{x1D700}}{k}, & \displaystyle\end{eqnarray}$$

with $T=\max (k/\unicode[STIX]{x1D700},6\sqrt{\unicode[STIX]{x1D707}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700})})$ , $L=0.23\max (k^{3/2}/\unicode[STIX]{x1D700},70\sqrt[4]{(\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C})^{3}/\unicode[STIX]{x1D700}})$ and the eddy viscosity $\unicode[STIX]{x1D707}_{t}=C_{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D70C}{v^{\prime }}^{2}T$ . Using the Boussinesq approximation, the turbulent shear stress is approximated by $\langle \unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }\rangle =-\unicode[STIX]{x1D707}_{t}\unicode[STIX]{x2202}_{y}\langle u\rangle$ and the production can be expressed as $P_{k}=\unicode[STIX]{x1D707}_{t}(\unicode[STIX]{x2202}_{y}u)^{2}$ . The wall boundary condition for the dissipation is $\unicode[STIX]{x1D700}_{w}=(\unicode[STIX]{x1D707}_{w}/\unicode[STIX]{x1D70C}_{w})\unicode[STIX]{x2202}_{y^{2}}^{2}k$ , while all other quantities are set to zero. The model coefficients are $C_{\unicode[STIX]{x1D707}}=0.22$ , $\unicode[STIX]{x1D70E}_{k}=1.0$ , $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D700}}=1.3$ , $C_{1}=1.4$ , $C_{2}=0.3$ , $C_{\unicode[STIX]{x1D700}1}=1.4(1+0.045\sqrt{k/{v^{\prime }}^{2}})$ and $C_{\unicode[STIX]{x1D700}2}=1.92$ .

The corresponding Reynolds/Favre-averaged streamwise momentum equation, using the Boussinesq assumption to approximate the turbulent shear stress, reads

(5.5) $$\begin{eqnarray}\unicode[STIX]{x2202}_{y}[(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}_{t})\unicode[STIX]{x2202}_{y}u]=-\unicode[STIX]{x1D70C}f_{x}.\end{eqnarray}$$

Since the aim of this study is to investigate the effect of variable properties on turbulent velocity scales, we do not consider the energy equation. Instead, we directly prescribe the averaged density and viscosity profiles from DNS. This allows us to study how variable properties affect turbulence, without including compounding errors that originate from modelling the wall-normal turbulent heat flux in the energy equation, commonly approximated by the ratio of the eddy viscosity and the turbulent Prandtl number. It should be noted that if the eddy viscosity is accurately modelled (as we will show below) and the turbulent Prandtl number is constant and known, the energy equation will provide accurate profiles for density and viscosity. Equations (5.1)–(5.5) can be solved to provide approximate solutions of turbulent statistics in variable property channel flows – if the density and viscosity profiles are provided as an input from the DNS.

On the other hand, instead of using the conventional compressible formulation of the turbulence model (5.1)–(5.4), we can solve it in its SLS form. For the channel cases investigated here, we can assume that $\hat{{\mathcal{D}}}_{k}$ and $\hat{{\mathcal{C}}}_{k}$ can be neglected (see § 4). Moreover, following a pragmatic approach, we assume that, analogously to the TKE equation (2.10), the supporting model equations for $\unicode[STIX]{x1D700}$ , ${v^{\prime }}^{2}$ and $f$ can be expressed in their semi-local formulation as well. We additionally make use of common modelling assumptions, e.g. $\unicode[STIX]{x1D707}^{\prime }\ll \langle \unicode[STIX]{x1D707}\rangle$ , and that the molecular and turbulent diffusion can be approximated by the gradient diffusion hypothesis. Then, the only changes that need to be made to solve a turbulence model in its SLS form for fully developed turbulent channel flows are to

1. (i) set $\unicode[STIX]{x1D70C}=1$ ,

2. (ii) replace $\unicode[STIX]{x1D707}$ by $1/Re_{\unicode[STIX]{x1D70F}}^{\star }$ (assuming that $\unicode[STIX]{x1D707}^{\prime }\ll \langle \unicode[STIX]{x1D707}\rangle$ , such that $\hat{\unicode[STIX]{x1D707}}=1+\unicode[STIX]{x1D707}^{\prime }/\langle \unicode[STIX]{x1D707}\rangle \approx 1$ ),

3. (iii) replace $\unicode[STIX]{x2202}u$ in $P_{k}$ by $\unicode[STIX]{x2202}u^{vD}$

4. (iv) and, if a model makes use of $y^{+}$ , replace it by $y^{\star }$ .

The corresponding momentum equation can be solved in either its conventional (5.5) or its SLS form, i.e. $\unicode[STIX]{x2202}_{{\hat{y}}}[(1/Re_{\unicode[STIX]{x1D70F}}^{\star }+\hat{\unicode[STIX]{x1D707}}_{t})\unicode[STIX]{x2202}_{{\hat{y}}}u^{vD}]=-\unicode[STIX]{x1D70C}f_{x}=-1$ . In the latter, it can be seen that, indeed, the only parameter that governs the turbulence model and the momentum equation is $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . If the momentum equation is solved in its conventional form, the SLS eddy viscosity $\hat{\unicode[STIX]{x1D707}}_{t}$ , which is provided by the turbulence model, has to be transformed to the conventionally scaled form by $\unicode[STIX]{x1D707}_{t}=\langle \unicode[STIX]{x1D70C}\rangle hu_{\unicode[STIX]{x1D70F}}^{\star }\hat{\unicode[STIX]{x1D707}}_{t}$ . This relation can be obtained using the same normalization as introduced in (2.1). Nevertheless, it can be shown that both formulations of the momentum equation lead to equivalent results.

Figure 4. Direct numerical simulation results (symbols) compared with conventional (red dash-dotted line) and SLS turbulence model (blue solid line) results for the cases (columns) introduced in table 1. The rows correspond to the universal velocity transformation $\langle u^{\star }\rangle$ , SLS profiles for turbulent shear stress $\{\hat{u} ^{\prime \prime }\hat{v}^{\prime \prime }\}$ , TKE $\{\hat{k}\}$ and dissipation $\hat{\unicode[STIX]{x1D700}}$ . The dissipation $\hat{\unicode[STIX]{x1D700}}$ (SLS) from DNS for case T&L was not available to the authors.

The results of the conventional compressible form and the SLS form are presented in figure 4 and compared with results from the DNS. Evidently, in contrast to the conventional formulation of the turbulence model, the SLS formulation significantly improves the results. For example, the conventional model fails to provide reasonable results, even for a case with constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ (case $\text{CRe}_{\unicode[STIX]{x1D70F}}^{\star }$ ), which, compared with case CP, has quasi-similar profiles of the viscous scales (see Patel et al. Reference Patel, Boersma and Pecnik2016) and the SLS budgets (figure 3). Moreover, for case GL and the supersonic turbulent channel flow case T&L, the results with the SLS formulation improve considerably. In particular, the velocity $\langle u^{\star }\rangle$ and $\{\hat{k}\}$ close to the wall (rows 1 and 3) show a very good agreement with DNS. Since the density is constant for case LL, both approaches give equivalent results and agree well with DNS.