2 Simulation details

Direct numerical simulations of fully developed turbulent channel flows, driven by a constant streamwise pressure gradient, are performed using the low-Mach-number approximation of the Navier–Stokes equations. In the low-Mach-number limit, the density and transport properties can be evaluated as a function of temperature only, independent of pressure fluctuations (Majda & Sethian Reference Majda and Sethian1985; Nemati et al. Reference Nemati, Patel, Boersma and Pecnik2015). Different constitutive relations for density and viscosity are used. In order to achieve variations in temperature $T$ , and consequently in density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D707}$ , the flow is uniformly heated with a volumetric heat source, while the temperature at both channel walls is kept constant. This allows a wall heat flux and ensures that the flow is in thermal equilibrium. Other transport properties like thermal conductivity $\unicode[STIX]{x1D705}$ and specific heat $c_{p}$ are constant in all simulations. The Prandtl number based on wall quantities is taken to be unity.

The DNS code discretises the spatial derivatives in the wall-normal direction using a sixth-order staggered compact finite difference scheme (Lele Reference Lele1992; Boersma Reference Boersma2011) and the derivatives in the spanwise and streamwise directions are computed using a Fourier expansion with periodic boundary conditions. The time integration is performed with a second-order Adams–Bashforth method, and a pressure correction scheme, based on the projection method (McMurtry et al. Reference Mcmurtry, Jou, Riley and Metcalfe1986), is used to ensure mass conservation. Additional details on the governing equations can be found in Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015).

Table 1.

Simulation parameters for all cases: CP395, constant-property case with $Re_{\unicode[STIX]{x1D70F}}=395$ ; $\text{C}Re_{\unicode[STIX]{x1D70F}}^{\star }$ , variable-property case with constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ ( $=395$ ) across the channel; GL, case with gas-like property variations; $\text{S}Re_{\unicode[STIX]{x1D70F}GL}^{\star }$ , variable-property case with $Re_{\unicode[STIX]{x1D70F}}^{\star }$ similar to case GL; LL, case with liquid-like property variations; $\text{S}Re_{\unicode[STIX]{x1D70F}LL}^{\star }$ , variable-property case with $Re_{\unicode[STIX]{x1D70F}}^{\star }$ similar to case LL; CP150, constant-property case with $Re_{\unicode[STIX]{x1D70F}}=150$ ; CP550, constant-property case with $Re_{\unicode[STIX]{x1D70F}}=550$ .

Eight cases have been simulated, which are summarised in table 1. The simulations consist of three constant-property (CP395, CP150, CP550) and five variable-property cases ( $\text{C}Re_{\unicode[STIX]{x1D70F}}^{\star }$ , $\text{S}Re_{\unicode[STIX]{x1D70F}GL}^{\star }$ , GL, LL, $\text{S}Re_{\unicode[STIX]{x1D70F}LL}^{\star }$ ). The acronym CP refers to a constant-property case; GL and LL refer to gas-like and liquid-like property variations respectively; $\text{C}Re_{\unicode[STIX]{x1D70F}}^{\star }$ refers to a variable-property case with constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ ( $=395$ ) across the whole channel; $\text{S}Re_{\unicode[STIX]{x1D70F}GL}^{\star }$ refers to a variable-property case that has a similar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ distribution to case GL; and $\text{S}Re_{\unicode[STIX]{x1D70F}LL}^{\star }$ refers to a variable-property case with an $Re_{\unicode[STIX]{x1D70F}}^{\star }$ distribution similar to case LL. The second and third columns show the functional relations for $\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{w}$ and $\unicode[STIX]{x1D707}/\unicode[STIX]{x1D707}_{w}$ as a function of $T/T_{w}$ . The next two columns report the wall-based friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ and the semilocal Reynolds number at the channel centre ${Re_{\unicode[STIX]{x1D70F}}^{\star }}_{c}=Re_{\unicode[STIX]{x1D70F}}\sqrt{(\overline{\unicode[STIX]{x1D70C}}_{c}/\overline{\unicode[STIX]{x1D70C}}_{w})}/(\overline{\unicode[STIX]{x1D707}}_{c}/\overline{\unicode[STIX]{x1D707}}_{w})$ (the subscript $c$ denotes the value at the channel centre). It should be noted that at the wall ${Re_{\unicode[STIX]{x1D70F}}^{\star }}_{w}=Re_{\unicode[STIX]{x1D70F}}$ . The last two columns show the number of mesh points $N$ and the length of the domain $L$ along the streamwise $x$ , wall-normal $y$ and spanwise $z$ directions. The velocity components along the $x$ , $y$ and $z$ directions are denoted as $u$ , $v$ and $w$ respectively.

Figure 1.

(a) Density $\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w}$ , (b) viscosity $\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}$ and (c) semilocal Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{\star }$ .

Figure 1 shows the distributions of density, viscosity and $Re_{\unicode[STIX]{x1D70F}}^{\star }$ for all cases. Cases with variable density are shown as lines and cases with constant density are shown as symbols. As can be seen, we use different combinations of $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D707}$ to obtain quasisimilar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles. It should be noted that the strongest gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ occur at the wall. The cases forming the first pair – CP395 and $\text{C}Re_{\unicode[STIX]{x1D70F}}^{\star }$ (black symbols and line respectively, colour online) – have constant $Re_{\unicode[STIX]{x1D70F}}^{\star }$ in the wall-normal direction ( $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y=0$ ), even though for case $\text{C}Re_{\unicode[STIX]{x1D70F}}^{\star }$ the density and viscosity are varying and decrease away from the wall. For the second pair – $\text{S}Re_{\unicode[STIX]{x1D70F}GL}^{\star }$ and GL (blue symbols and line respectively, colour online) – the profile for $Re_{\unicode[STIX]{x1D70F}}^{\star }$ decreases from 395 at the wall to approximately 150 at the channel centre. Later, we will denote these cases as $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ . For the pair $LL$ and $\text{S}Re_{\unicode[STIX]{x1D70F}LL}^{\star }$ (red symbols and line respectively, colour online), $Re_{\unicode[STIX]{x1D70F}}^{\star }$ increases from 150 at the wall to approximately 540 at the channel centre. These cases will be denoted as $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ . The simulations CP150 (brown symbol, colour online) and CP550 (green symbol, colour online) are cases that bound the $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles of the variable-property cases in order to investigate and distinguish any Reynolds-number effects with respect to effects caused by property gradients. Considerable variations in properties have been obtained for all variable-property cases. However, the relative property fluctuations ${\unicode[STIX]{x1D70C}^{\prime }}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ and ${\unicode[STIX]{x1D707}^{\prime }}_{rms}/\overline{\unicode[STIX]{x1D707}}$ (the prime denotes Reynolds average fluctuations, and the subscript rms indicates the root-mean-square value) are still less than 0.15 for all variable-property cases.

Figure 2.

Kolmogorov length scale normalised by and plotted using (a) outer scales, (b) wall-based viscous length scales and (c) semilocal viscous length scales.

The adequacy of the mesh resolution to resolve the smallest scales is evaluated by means of wall-normal distributions of Kolmogorov length scales, $\unicode[STIX]{x1D702}=((\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D70C}})^{3}\overline{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x1D716})^{0.25}$ ( $\unicode[STIX]{x1D716}$ is the turbulent kinetic energy dissipation per unit volume). Figure 2 shows $\unicode[STIX]{x1D702}$ , normalised by and plotted using outer scales (figure 2 a), wall-based viscous length scales (figure 2 b) and semilocal viscous length scales (figure 2 c) for all cases. Cases with similar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles show similar distributions of Kolmogorov scales, indicating that, similarly to turbulence statistics (Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015), the smallest scales are also strongly governed by $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . The Kolmogorov scale normalised using the half-channel height increases for cases with decreasing $Re_{\unicode[STIX]{x1D70F}}^{\star }$ , while it decreases for cases with increasing $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . The cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ , for which $Re_{\unicode[STIX]{x1D70F}}^{\star }$ varies from 395 at the wall to approximately 140 at the channel centre, have a Kolmogorov length scale that transitions from the values of case CP395 at the wall to CP150 at the channel centre. A similar observation can be made for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ , for which $Re_{\unicode[STIX]{x1D70F}}^{\star }$ varies from 150 at the wall to approximately 540 at the centre. The Kolmogorov scale, when normalised by and plotted using classical wall-based viscous units, shows strong deviations for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y\neq 0$ . On the other hand, the use of semilocal length scales provides a good collapse of $\unicode[STIX]{x1D702}$ for all cases in the near-wall region. Hence, the semilocal scaling also provides a good measure to assess the mesh spacing requirements for simulations with strong fluid property variations. Table 2 lists the maximum grid spacing in terms of $\unicode[STIX]{x1D702}$ for all cases, and the values are within the resolution requirements of $\unicode[STIX]{x0394}x<12\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x0394}y<2\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x0394}z<6\unicode[STIX]{x1D702}$ , as also reported in other DNS studies (Zonta et al. Reference Zonta, Marchioli and Soldati2012; Lee et al. Reference Lee, Yoon Jung, Jin Sung and Zaki2013).

Table 2.

Maximum spatial resolution normalised using the Kolmogorov scale $\unicode[STIX]{x1D702}$ .

The adequacy of the box size to accommodate the large-scale structures can be studied using the one-dimensional premultiplied energy spectrum, defined as

In the above equation, ${\mathcal{F}}(\unicode[STIX]{x1D713})$ is the Fourier coefficient of $\unicode[STIX]{x1D713}$ , ${\mathcal{F}}(\unicode[STIX]{x1D713})^{\ast }$ is the complex conjugate, $E_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}={\mathcal{F}}(\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }){\mathcal{F}}(\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime })^{\ast }$ represents the energy spectrum of the density-weighted streamwise velocity fluctuation, the double prime denotes Favre-averaged fluctuations and $k_{\unicode[STIX]{x1D6FC}}$ is the wavenumber in the streamwise $k_{x}$ or spanwise $k_{z}$ direction. The streamwise Reynolds stress is related to $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ and $E_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ as

where the tilde denotes Favre averaging and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FC}}=2\unicode[STIX]{x03C0}/k_{\unicode[STIX]{x1D6FC}}$ is the wavelength. The area under the premultiplied spectrum, when plotted in lin-log scale, represents the Reynolds stress. A plot of $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ , normalised by $\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ and plotted as a function of streamwise and spanwise wavelength at different wall-normal planes, is shown in figure 3. While the spanwise spectrum indicates that the spanwise box size is sufficient, it can be argued that the length of the box in the streamwise direction could be increased to resolve all of the large-scale contributions. However, as noted in previous studies (Abe, Kawamura & Choi Reference Abe, Kawamura and Choi2004; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014), the influence of these unresolved large-scale structures (due to the moderate box size) on turbulence statistics is negligible, and resolving the peak of the streamwise premultiplied spectrum (e.g. DNS data of Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999)) was considered sufficient for the present work. The spanwise premultiplied spectrum $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}(k_{z})$ will be discussed in more detail in § 5 to highlight its scaling characteristics.

Figure 3.

(a) Normalised premultiplied streamwise spectrum $k_{x}E_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}/\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ as a function of streamwise wavelength $\unicode[STIX]{x1D706}_{x}/h$ and (b) normalised premultiplied spanwise spectrum $k_{z}E_{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}/\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }u^{\prime \prime }}$ as a function of spanwise wavelength $\unicode[STIX]{x1D706}_{z}/h$ , at different wall-normal locations.

3 Mean velocity scaling

This section discusses the effect of near-wall gradients in density and viscosity on the scaling of mean velocity profiles. We first highlight shortcomings of the van Driest transformation for cases with strong near-wall gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . We then derive a semilocally scaled stress-balance equation and show that the viscous stresses for the investigated cases perfectly collapse, if they are plotted as a function of the semilocal wall distance $y^{\star }$ . We will utilise this collapse to derive a velocity scaling that will be applied for the turbulent channel flows discussed above, and for adiabatic supersonic turbulent boundary layers.

The van Driest transformed velocity $\overline{u}^{vD}$ and the diagnostic function $y\,\text{d}\overline{u}^{vD}/\text{d}y$ as a function of $y^{+}$ and $y^{\star }$ for all cases are shown in figure 4. As noted in our earlier study, cases with quasisimilar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles (symbols and lines with the same colour online) exhibit similar $\overline{u}^{vD}$ profiles, irrespective of their individual density and viscosity profiles. However, when comparing cases with different $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients, deviations can be observed. For example, in figure 4(a), the slope of the linear viscous sublayer ( $\overline{u}^{vD}=y^{+}$ ) increases for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ (red line and symbols, colour online) and decreases for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ (blue line and symbols, colour online). Moreover, an increase in the log-law additive constant (commonly $B=5.2$ ) can be seen for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ , while the opposite is observed for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ . Additionally, the log-layer shifts outwards for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ and inwards for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ , as can be seen from the profiles of the diagnostic function $y\,\text{d}\overline{u}^{vD}/\text{d}y$ in figure 4(c). Figure 4(b,d) shows $\overline{u}^{vD}$ and the diagnostic function for all cases as a function of $y^{\star }$ . The deviation of $\overline{u}^{vD}$ for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y\neq 0$ is even more prominent if plotted as a function of $y^{\star }$ . On the other hand, the diagnostic function collapses the onset of the log-layer if plotted as a function of $y^{\star }$ . The reason for this collapse is investigated further using the streamwise stress-balance equation.

Figure 4.

(a,b) van Driest velocity $\overline{u}^{vD}$ and (c,d) diagnostic function $y\,\text{d}\overline{u}^{vD}/\text{d}y$ shown as a function of $y^{+}$ (a,c) and $y^{\star }$ (b,d). The solid horizontal line in (c) and (d) has a value of $1/0.41$ .

The stress-balance relation for the turbulent and viscous stresses can be obtained by integrating the mean streamwise momentum equation, which, for a fully developed turbulent channel flow (neglecting viscosity fluctuations), gives

where $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70F}_{w}$ are the total and wall shear stress respectively. Analogously to Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015), equation (3.1) can be written in terms of semilocally scaled velocity fluctuations (using the semilocal friction velocity $u_{\unicode[STIX]{x1D70F}}^{\star }=\sqrt{\unicode[STIX]{x1D70F}_{w}/\overline{\unicode[STIX]{x1D70C}}}$ )

and the van Driest mean velocity increment

to obtain the semilocally scaled stress-balance equation

It should be noted that the viscous stress term is a function of the van Driest mean velocity gradient, scaled by the inverse of the semilocal Reynolds number. The reason for writing (3.1) in terms of $\overline{u}^{vD}$ and $\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }}$ originates from Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015), where the Navier–Stokes equations were rescaled using the semilocal friction velocity and local mean properties in order to obtain governing equations for the semilocally scaled mean, $\overline{u}^{vD}$ , and fluctuating velocities, $\hat{u} _{i}^{\prime }$ , with $Re_{\unicode[STIX]{x1D70F}}^{\star }$ as a strong parameter. In other words, the combined influence of density and viscosity variations on turbulence statistics can be characterised using a single parameter, $Re_{\unicode[STIX]{x1D70F}}^{\star }$ .

Figure 5.

(a,b) Reynolds shear stress $-\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }}$ and (c,d) viscous shear stress $(h/Re_{\unicode[STIX]{x1D70F}}^{\star })\,\text{d}\overline{u}^{vD}/\text{d}y$ shown as a function of $y^{+}$ (a,c) and $y^{\star }$ (b,d).

The Reynolds shear stress $\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }}$ and viscous stress term $(h/Re_{\unicode[STIX]{x1D70F}}^{\star })\,\text{d}\overline{u}^{vD}/\text{d}y$ in (3.4) are plotted as a function of $y^{+}$ and $y^{\star }$ in figure 5. It is apparent that neither stress collapses if shown as a function of $y^{+}$ (figure 5 a,c) for cases with different $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients. Similar observations have been reported for compressible non-adiabatic flows in Coleman et al. (Reference Coleman, Kim and Moser1995), Foysi et al. (Reference Foysi, Sarkar and Friedrich2004) and Morinishi et al. (Reference Morinishi, Tamano and Nakabayashi2004), and in Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015) for low-Mach-number channel flows with gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ . In comparison to a constant-property case with a similar $Re_{\unicode[STIX]{x1D70F}}$ value, the Reynolds shear stress decreases for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y<0$ and increases for cases with $\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y>0$ . On the other hand, if the stresses are plotted as a function of $y^{\star }$ (figure 5 b,d), two observations can be made. First, the Reynolds shear stress profiles for all cases collapse closely in the inner layer, except in the region very close to the wall where small differences can be seen. Second, and more importantly, the viscous stresses for all cases perfectly collapse over the entire $y^{\star }$ range. Given the last observation, it is evident that the viscous stress term, expressed as $(h/Re_{\unicode[STIX]{x1D70F}}^{\star })\,\text{d}\overline{u}^{vD}/\,\text{d}y$ , forms the basis to develop a proper scaling law for the mean velocity in flows with large density and viscosity gradients. We thus propose that the transformation must be based on the viscous stress

where $\unicode[STIX]{x1D6F7}$ is an unknown function of $y^{\star }$ . Using the chain rule $\text{d}/\text{d}y=(\text{d}y^{\star }/\text{d}y)\,\text{d}/\text{d}y^{\star }$ , we can write (3.5) as

This allows us to replace $\text{d}y^{\star }/\text{d}y$ in (3.6) by taking the derivative of $y^{\star }=yRe_{\unicode[STIX]{x1D70F}}^{\star }/h$ with respect to $y$ , to obtain

Equation (3.7) thus provides the scaling law for the mean streamwise velocity, which we will denote as $\overline{u}^{\star }$ . Hence, $\text{d}\overline{u}^{\star }$ and $\text{d}\overline{u}^{vD}$ are related through

It should be noted that the transformation proposed by Trettel & Larsson (Reference Trettel and Larsson2016) is equivalent when substituting the definitions of $Re_{\unicode[STIX]{x1D70F}}^{\star }$ and $d\overline{u}^{vD}$ into (3.8). Their transformation, which is expressed in terms of density and viscosity gradients, was obtained by equating the transformed log-law velocity gradient with the velocity gradient obtained from the stress-balance equation, assuming that the Reynolds shear stress is similar for compressible and constant-property cases. The present derivation follows an alternative route. It is based on rescaling the Navier–Stokes equations using the local mean properties and semilocal friction velocity, which naturally suggests that the viscous terms are scaled by the semilocal Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{\star }$ to account for changes in viscous scales due to property variations. The transformation expressed in terms of (3.8) thus emphasises the fact that, similarly to turbulence statistics, the transformation is also governed by the $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles, rather than individual density and viscosity profiles.

The derived velocity transformation $\overline{u}^{\star }=\int _{0}^{\overline{u}^{vD}}(1+(y/Re_{\unicode[STIX]{x1D70F}}^{\star })\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y)\,\text{d}\overline{u}^{vD}$ , is shown in figure 6(a) as a function of $y^{\star }$ . It can be seen that $\overline{u}^{\star }$ is perfectly able to collapse the velocity profiles for all cases over the entire $y^{\star }$ range. In our previous work (Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015), we showed that similar turbulence statistics and van Driest mean velocity profiles $\overline{u}^{vD}$ are obtained for cases with similar $Re_{\unicode[STIX]{x1D70F}}^{\star }$ distributions, even if their individual mean density and viscosity profiles substantially differ. Here, we derive a transformation that accounts for gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ in order to extend the van Driest transformation.

Using the derived velocity transformation it is also possible to explain the collapse of the diagnostic function as a function of $y^{\star }$ , as mentioned earlier. The definitions in (3.5) and (3.8) can be expressed as

By substituting $h/Re_{\unicode[STIX]{x1D70F}}^{\star }=y/y^{\star }$ in (3.9) we can obtain the correlation between the diagnostic function and the newly derived velocity scale $u^{\star }$ as

which explains the collapse of the diagnostic function in figure 4(d).

Figure 6.

(a) Transformed velocity $\overline{u}^{\star }$ and (b) mixing length $l_{m}Re_{\unicode[STIX]{x1D70F}}^{\star }/h$ as a function of $y^{\star }$ .

The invariance of $\unicode[STIX]{x1D6F7}$ as a function of $y^{\star }$ can be further quantified by expressing it in terms of the turbulence mixing length. Following Huang & Coleman (Reference Huang and Coleman1994), the Reynolds shear stress can be written in terms of the velocity gradient using the mixing-length theory as

Substitution of the Reynolds shear stress in (3.4) using (3.11) results in a quadratic equation for $\text{d}\overline{u}^{vD}/\text{d}y$ , which, when solved and simplified, gives

The above expression naturally suggests that $l_{m}$ scales with the semilocal length scale $h/Re_{\unicode[STIX]{x1D70F}}^{\star }$ . Figure 6(b) shows $l_{m}Re_{\unicode[STIX]{x1D70F}}^{\star }/h$ as a function of $y^{\star }$ . A satisfactory collapse is obtained for almost the entire inner layer, except very close to wall where small deviations occur. These deviations stem from turbulence modulation caused by strong $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients, which we will discuss in detail in §§ 4 and 5. However, close to the wall the viscous stress dominates and small values of the mixing length do not alter the velocity scaling.

Figure 7.

(a) The $Re_{\unicode[STIX]{x1D70F}}^{\star }$ distribution shown as a function of $y^{\star }$ , (b) the van Driest velocity $\overline{u}^{vD}$ shown as a function of $y^{+}$ and (c) the transformed velocity $\overline{u}^{\star }$ shown as a function of $y^{\star }$ for adiabatic supersonic boundary layers obtained from Bernardini & Pirozzoli (Reference Bernardini and Pirozzoli2011) and Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011), compared with reference boundary layer data from Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010).

Figure 8.

(a,b) Reynolds shear stress $-\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }}$ and (c,d) viscous shear stress $(h/Re_{\unicode[STIX]{x1D70F}}^{\star })\,\text{d}\overline{u}^{vD}/\text{d}y$ shown as a function of $y^{+}$ (a,c) and $y^{\star }$ (b,d) for adiabatic supersonic boundary layers obtained from Bernardini & Pirozzoli (Reference Bernardini and Pirozzoli2011) and Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011), compared with boundary layer data from Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010).

In the following, we will investigate whether the scaling can also be successfully applied for supersonic adiabatic flows, for which the van Driest scaling already shows a satisfying collapse. For this we will use a DNS database for adiabatic supersonic boundary layers from Bernardini & Pirozzoli (Reference Bernardini and Pirozzoli2011) and Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011), and data for an incompressible boundary layer from Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010). The investigated compressible cases have Mach numbers of $Ma=2$ , 3 and 4 and corresponding Reynolds numbers of $Re_{\unicode[STIX]{x1D70F}}=450$ , 400 and 400 respectively. The incompressible boundary layer is at $Re_{\unicode[STIX]{x1D70F}}=450$ . Figure 7(a) shows the $Re_{\unicode[STIX]{x1D70F}}^{\star }$ distribution for these cases as a function of $y^{\star }$ . It should be noted that, in contrast to the heated and cooled channel flows, the gradient of $Re_{\unicode[STIX]{x1D70F}}^{\star }$ at the wall is negligible. The largest $Re_{\unicode[STIX]{x1D70F}}^{\star }$ variation is obtained for the $Ma=4$  case, where $Re_{\unicode[STIX]{x1D70F}}^{\star }$ at the edge of the boundary layer is more than four times larger than at the wall. The van Driest mean velocity $\overline{u}^{vD}$ is plotted as a function of $y^{+}$ in figure 7(b). A satisfying collapse for the compressible and incompressible boundary layers is obtained, except in the wake region, which has been studied in more detail by Zhang et al. (Reference Zhang, Bi, Hussain, Li and She2012). The transformed velocity $\overline{u}^{\star }$ as a function of $y^{\star }$ is shown in figure 7(c). Here also, a reasonable collapse for the $\overline{u}^{\star }$ velocity profiles is obtained. However, small deviations with respect to the incompressible boundary layer occur in the buffer and logarithmic regions. Interestingly, Trettel & Larsson (Reference Trettel and Larsson2016) have also found a similar disagreement for cooled turbulent boundary layers.

The implication of the above mean velocity scaling characteristics on the stress-balance relation for these turbulent boundary layers is discussed next. The Reynolds shear and viscous stresses are shown in figure 8 as a function of $y^{+}$ and $y^{\star }$ . Both stresses show a superior collapse if they are plotted as a function of $y^{\star }$ . Analogously to the plot of $\overline{u}^{\star }$ versus $y^{\star }$ in figure 7(c), the viscous stresses of the compressible cases also deviate slightly from the incompressible case, as can be seen in figure 8(d). On the other hand, if the stresses are plotted as a function of $y^{+}$ , it can be seen that the turbulent shear stress profiles increase and the viscous stress profiles decrease with Mach number; see figure 8(a,c). This Mach number dependence of the stresses as a function of $y^{+}$ can be shown mathematically by writing the stress-balance equation (3.4) for the incompressible and compressible boundary layers as

In the constant-stress layer, $(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{w})_{comp}=(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{w})_{incomp}=1$ , and due to the collapse of $\overline{u}^{vD}$ as a function of $y^{+}$ , equations (3.13) and (3.14) can be equated as

Therefore, if $Re_{\unicode[STIX]{x1D70F}}^{\star }>Re_{\unicode[STIX]{x1D70F}}$ , it follows that $(-\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }})|_{comp}>(-\widetilde{\hat{u} ^{\prime }\hat{v}^{\prime }})|_{incomp}$ , which explains the Mach number dependence of the stresses. The increase in shear stress for the supersonic cases also explains the corresponding increase in turbulence intensities when compared with incompressible cases (Pirozzoli, Grasso & Gatski Reference Pirozzoli, Grasso and Gatski2004; Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011), due to the increase in the turbulent kinetic energy production. The partial success of both $y^{+}$ and $y^{\star }$ in adiabatic supersonic boundary layers thus warrants future studies. The semilocal scaling utilises the invariance of $(h/Re_{\unicode[STIX]{x1D70F}}^{\star })\text{d}\overline{u}^{vD}/\text{d}y$ as a function of $y^{\star }$ to obtain $\overline{u}^{\star }$ , while adiabatic boundary layers exhibit invariance of a more fundamental variable, namely the wall-normalised van Driest transformed mean spanwise vorticity $\overline{\unicode[STIX]{x1D714}}_{z}^{vD}/(\overline{\unicode[STIX]{x1D714}}_{z}^{vD})_{w}=\text{d}\overline{u}^{vD}/\text{d}y^{+}=(h/Re_{\unicode[STIX]{x1D70F}})\text{d}\overline{u}^{vD}/\text{d}y$ , as a function of $y^{+}$ . More discussion on the significance of mean vorticity is provided in the next section.

4 Influence of $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients on near-wall turbulence statistics

Turbulence statistics are strongly influenced by near-wall modifications of turbulence. Near-wall turbulence modifications for variable-property flows can be classified into two main mechanisms: (i) changes in viscous scales and (ii) structural changes of turbulence. The semilocal scaling is successful in collapsing turbulence statistics, such as mean velocity (see figure 6 a), Reynolds and viscous shear stress (see figure 5 b,d), because it accommodates the change in viscous scales using local quantities, and because these quantities are not significantly affected by structural changes and non-local effects very close to the wall. On the other hand, the semilocal scaling fails for turbulence statistics that are sensitive to strong non-local interactions of the buffer layer vortical structures with the sublayer region, e.g. statistics of vorticity fluctuations. Furthermore, the semilocal scaling also fails for turbulence statistics that contain direct information on turbulence structure, e.g. turbulence anisotropy (Foysi et al. Reference Foysi, Sarkar and Friedrich2004) and low-speed streaks (Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015).

We first study the gradient of the van Driest transformed mean velocity $\overline{u}^{vD}$ , which we will refer to as the van Driest transformed mean spanwise vorticity $\overline{\unicode[STIX]{x1D714}}_{z}^{vD}$ . The van Driest transformed mean velocity $\overline{u}^{vD}$ is the analogue to the mean velocity in a variable-density flow, and therefore a study of its gradients allows us to draw conclusions on the stability of the flow. We then discuss the turbulent vorticity fluctuations and the turbulence anisotropy to emphasise the failure of the semilocal scaling due to non-locality and changes in turbulence structure respectively.

Figure 9.

Wall-normalised van Driest transformed mean spanwise vorticity $\overline{\unicode[STIX]{x1D714}}_{z}^{vD}/(\overline{\unicode[STIX]{x1D714}}_{z}^{vD})_{w}$ shown as a function of (a) $y^{+}$ and (b) $y^{\star }$ .

5 Turbulent structures

Here, we investigate the influence of near-wall $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients on the characteristics of near-wall streaks and quasistreamwise vortices, which are both known to be the dominant structures in near-wall turbulence, in order to provide a mechanistic description for the modulated statistics.