3.1 Set-up and definitions

Numerical experiments are conducted for the plane channel flow configuration, i.e. the flow between two infinite parallel flat plates. This is reproduced by applying periodic conditions in the homogeneous streamwise ( $x$ ) and spanwise ( $z$ ) directions. Isothermal no-slip wall conditions are applied on the lower and upper walls. In order to counteract viscous friction and maintain a target bulk mass flow, the approach described in Gerolymos, Sénéchal & Vallet (Reference Gerolymos, Sénéchal and Vallet2010) and Gerolymos & Vallet (Reference Gerolymos and Vallet2014) is used to compute the forcing term. Specifically, at the end of each Runge–Kutta stage, the bulk density is explicitly constrained to maintain a fixed target value, whereas a source term $f_{u_{1}}$ is injected in the streamwise momentum equation to enforce a constant mass flow. It should be noted that the density correction is equivalent to including a source term in the continuity equation. Nevertheless, this term is very small and can be neglected (Coleman et al. Reference Coleman, Kim and Moser1995; Huang et al. Reference Huang, Coleman and Bradshaw1995; Lechner et al. Reference Lechner, Sesterhenn and Friedrich2001; Foysi et al. Reference Foysi, Sarkar and Friedrich2004; Morinishi et al. Reference Morinishi, Tamano and Nakabayashi2004; Wei & Pollard Reference Wei and Pollard2011; Gerolymos & Vallet Reference Gerolymos and Vallet2014).

In the following, the subscripts $(\cdot )_{B}$ , $(\cdot )_{w}$ and $(\cdot )_{c}$ denote averaging over the whole computational domain, the walls and centreline, respectively; $\overline{(\cdot )}$ indicates Reynolds averaging and $(\cdot )^{\prime }$ are Reynolds fluctuations; similarly, $\widetilde{(\cdot )}$ and $(\cdot )^{\prime \prime }$ denote Favre averaging and Favre fluctuations. In the following, we also use the centreline Reynolds and Mach numbers, respectively defined as $Re_{c}=\overline{\unicode[STIX]{x1D70C}}_{c}\overline{u}_{c}h/\overline{\unicode[STIX]{x1D707}}_{c}$ and $\overline{M}_{c}=\overline{u_{c}/c_{c}}$ .

The DNS operating point is defined by a bulk Reynolds number $Re_{B}$ and a bulk Mach number $M_{B}$ , defined as:

where $\unicode[STIX]{x1D70C}_{B}$ is the bulk density, $\overline{\unicode[STIX]{x1D707}}_{w}$ is the dynamic viscosity and $\overline{c}_{w}$ the sound speed at the walls. In the definition of the bulk reference numbers, wall values for viscosity and speed of sound are usually considered since this greatly simplifies the imposition of the reference conditions for isothermal walls. Indeed, for a perfect-gas model, a fixed wall temperature implies $\overline{\unicode[STIX]{x1D707}}_{w}=\unicode[STIX]{x1D707}(T_{w})$ and $\overline{c}_{w}=c(T_{w})=\sqrt{\unicode[STIX]{x1D6FE}RT_{w}}$ . However, this is no longer true when using more complex thermodynamic models, since the speed of sound and the transport properties depend on both temperature and density values. In these cases, $\overline{\unicode[STIX]{x1D70C}}_{w}$ is not known a priori and it is not possible to fix directly $Re_{B}$ and $M_{B}$ . To enforce the reference conditions, an iterative procedure has been followed: first, a preliminary calculation is run by choosing a reasonable value for the ratio $\overline{\unicode[STIX]{x1D70C}}_{w}/\unicode[STIX]{x1D70C}_{B}$ . Once the solution is converged, the $\overline{\unicode[STIX]{x1D70C}}_{w}$ obtained from the simulation is used to compute the updated values of $\overline{\unicode[STIX]{x1D707}}_{w}$ and $\overline{c}_{w}$ , and the flow field is interpolated onto a new grid (adapted to the updated value of the Reynolds number). In order to achieve convergence to the desired state, about three to four iterative cycles are needed.

We recall the definition of the standard wall coordinates and friction Reynolds number

where $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\overline{\unicode[STIX]{x1D70C}}_{w}}$ is the friction velocity, $\unicode[STIX]{x1D70F}_{w}=\overline{(\unicode[STIX]{x1D707}(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y))}_{w}$ being the shear stress at the wall. It is widely known that the Reynolds number based on the friction velocity does not correctly represent the effect of rapid wall-normal variations for density and viscosity profiles in presence of high compressibility effects. Huang et al. (Reference Huang, Coleman and Bradshaw1995) proposed an empirical semi-local scaling based on both wall and local mean quantities:

with $u_{\unicode[STIX]{x1D70F}}^{\ast }=\sqrt{\unicode[STIX]{x1D70F}_{w}/\overline{\unicode[STIX]{x1D70C}}(y)}$ the semi-local friction velocity. This mixed scaling has proven to give quite satisfactory results in collapsing first- and second-order moments (Foysi et al. Reference Foysi, Sarkar and Friedrich2004; Morinishi et al. Reference Morinishi, Tamano and Nakabayashi2004; Modesti & Pirozzoli Reference Modesti and Pirozzoli2016) obtained from a wide range of $M_{B}$ and $Re_{B}$ . Recently, Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015) and Trettel & Larsson (Reference Trettel and Larsson2016) have developed a mathematical framework giving a theoretical basis to the semi-local scaling. Even if this scaling was initially introduced in order to take into account the effects of compressibility, Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015) assessed its validity also for variable-property flows, since it inherently considers the evolution of transport properties and their relation with thermodynamic quantities.

3.2 Description of the computed cases

Dense-gas effects are evaluated by means of a parametric study at three bulk Reynolds numbers (namely, $Re_{B}=3000$ , 7000 and 12 000) and three bulk Mach numbers ( $M_{B}=1.5$ , 2.25 and 3.0). For PP11 cases, the bulk density was imposed to be $\unicode[STIX]{x1D70C}_{B}=0.618\unicode[STIX]{x1D70C}_{cr}$ and the wall temperature was fixed to $T_{w}=1.01T_{cr}$ . The density value ensures that the flow evolves in the dense-gas region, whereas the slightly supercritical wall temperature condition avoids the occurrence of a two-phase flow. It should be noted that for these conditions it is not possible to obtain negative values of the derivative of gas dynamics $\unicode[STIX]{x1D6E4}$ , so we should not expect BZT phenomena.

In the following, we refer to each simulation with a unique tag of the form $\text{XM}\unicode[STIX]{x1D6FC}\text{R}\unicode[STIX]{x1D6FD}$ , where the first letter indicates the fluid (A for air and P for PP11), $\unicode[STIX]{x1D6FC}$ refers to $M_{B}$ ( $\unicode[STIX]{x1D6FC}=1,2,3$ for $M_{B}=1.5,2.25,3$ , respectively) and $\unicode[STIX]{x1D6FD}$ to $Re_{B}$ ( $\unicode[STIX]{x1D6FD}=3,7,12$ for $Re_{B}=3000,7000,12\,000$ , respectively). For all the cases, the computational domain has dimensions $L_{x}\times L_{y}\times L_{z}=8\unicode[STIX]{x03C0}h\times 2h\times 2\unicode[STIX]{x03C0}h$ , with $h$ the channel half-height. Note that the dimension in the streamwise direction is greater than the one used in the previous compressible channel DNS (Coleman et al. Reference Coleman, Kim and Moser1995; Lechner et al. Reference Lechner, Sesterhenn and Friedrich2001; Morinishi et al. Reference Morinishi, Tamano and Nakabayashi2004; Wei & Pollard Reference Wei and Pollard2011) in order to ensure uncorrelated inlet and outlet quantities for high- $M_{B}$ cases. The computational grids are chosen in order to provide a good spatial resolution in all directions. Specifically, the chosen spatial resolutions, expressed in semi-local units, are in the ranges $\unicode[STIX]{x0394}x^{\ast }\in [10,16]$ , $\unicode[STIX]{x0394}y_{w}^{\ast }\in [0.5,0.8]$ , $\unicode[STIX]{x0394}z^{\ast }\in [4,6]$ , according to the considered flow conditions. The spatial resolution is also evaluated with respect to the wall-normal distribution of the Kolmogorov length scale $\unicode[STIX]{x1D702}=((\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D70C}})^{3}\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D700}})^{0.25}$ , where $\overline{\unicode[STIX]{x1D700}}$ is the turbulent kinetic energy dissipation. According to Zonta et al. (Reference Zonta, Marchioli and Soldati2012) and Lee et al. (Reference Lee, Jung, Sung and Zaki2013), the resolution requirements are $\unicode[STIX]{x0394}x<12\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x0394}y<2\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x0394}z<6\unicode[STIX]{x1D702}$ .

The resolutions obtained for the present cases are reported in table 3. Since a complex equation of state is used for real gases and higher values of the effective Reynolds number are reached, a grid sensitivity study has been conducted and is reported in appendix D. Simulations with PP11 require higher resolution than air to achieve grid convergence of second-order statistics, and the chosen grids in table 3 have quite satisfactory resolution. In our simulations, the overall number of grid points varies between $3\times 10^{6}$ and $1.2\times 10^{9}$ .

Table 3 summarizes the conditions used for the parametric study. Wall friction Reynolds numbers range from 218.7 to 1017 for air, and from 191.3 to 692.8 for PP11. The semi-local scaling drastically changes the predictions in terms of friction Reynolds number. For air, the friction Reynolds number at the centreline $Re_{\unicode[STIX]{x1D70F},c}^{\ast }=Re_{\unicode[STIX]{x1D70F}}^{\ast }(h)$ goes from 88.7 to 324.7, the drop for each case being roughly proportional to $M_{B}$ . For PP11, an opposite behaviour is observed: $Re_{\unicode[STIX]{x1D70F},c}^{\ast }$ is higher than $Re_{\unicode[STIX]{x1D70F}_{w}}$ in each case, and the deviations increase with $M_{B}$ . This is due to the different behaviour of the transport properties as it will be explained later. Thus, contrarily to air, the resolution requirements for PP11 are more severe in semi-local scaling than in the classical wall scaling. However, as reported in table 3 and appendix D, the chosen grids ensure a good resolution in each case. Once $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ is converged, statistics are collected every ten computational time steps, with $\unicode[STIX]{x0394}t^{+}\approx 0.02{-}0.03$ in wall units for all the simulations and during an observation time $t_{obs}^{+}$ in the range 1000–3000.

Table 2.

Line styles for the employed DNS cases.

4.1 Global flow properties

Table 4 summarizes the main characteristic values of the simulations. Figure 1 provides a first insight into the behaviour of PP11 cases compared to air cases. In this figure, $M_{B}$ and ${\overline{T}_{c}}^{+}=\overline{T}_{c}/T_{w}$ are plotted as functions of the average centreline Mach number $\overline{M}_{c}$ . Values obtained for air cases are in good agreement with reference DNS (Coleman et al. Reference Coleman, Kim and Moser1995; Foysi et al. Reference Foysi, Sarkar and Friedrich2004; Gerolymos & Vallet Reference Gerolymos and Vallet2014). For $M_{B}\geqslant 1.5$ , $\overline{M}_{c}$ is systematically lower than $M_{B}$ , since the walls are cooler than the channel core. Indeed for perfect gases, $\overline{M}_{c}$ increases nonlinearly since $\overline{T}_{c}$ grows rapidly and $\overline{c}\propto \sqrt{\overline{T}}$ . The average centreline temperature in figure 1(b) is approximately 1.4 times higher than the wall temperature for AM1R3 and more than 2.5 times higher for AM3R3 due to the significant increase of the viscous heating $\overline{\unicode[STIX]{x1D70F}_{ij}\unicode[STIX]{x1D61A}_{ij}}$ . The dependence on the Reynolds number is weak on the considered range, with slightly lower values of ${\overline{T}_{c}}^{+}$ for higher $Re_{B}$ , which is more clearly appreciated for the highest Mach number conditions. PP11 flows exhibit a rather different behaviour since the molecular complexity of dense gases implies high values of the specific heat $c_{v}/R$ and, therefore, the coupling between the thermal and dynamic fields becomes smaller and smaller. Inspection of figure 1(b) indeed shows that the centreline temperature is almost equal to the wall value for any choice of the Mach and Reynolds numbers. Nevertheless, the evolution of $\overline{M}_{c}$ changes significantly because in dense gases the speed of sound depends also on density, which decreases moving from the wall to the centreline. In turn, $\overline{c}_{c}$ increases leading to lower values of $\overline{M}_{c}$ .

For practical applications, it is interesting to investigate the modifications of momentum and heat transfer coefficients when working with a dense gas. For that purpose, we consider the evolutions of the skin friction coefficient $C_{f}$ and the bulk Nusselt number $Nu_{B}$ , respectively defined as

which are depicted in figure 2. $D_{h}$ denotes the hydraulic diameter, $D_{h}=4h$ for a flat channel. The values predicted by the correlation of Dean (Reference Dean1978), i.e. $C_{f}^{D}=0.073Re_{m}^{-1/4}$ , are superimposed in figure 2(a). The correlation has been established in the incompressible regime for $Re_{m}>6000$ , where $Re_{m}$ is the Reynolds number based on bulk quantities and channel height, i.e. $Re_{m}=2Re_{Bb}=\unicode[STIX]{x1D70C}_{B}\widetilde{u}_{B}2h/\unicode[STIX]{x1D707}_{B}$ . For the dense-gas cases, the computed skin friction coefficients are slightly lower than the values predicted by Dean’s correlation, but follow closely the trend with respect to the bulk Reynolds number. The deviations are greater for air ideal flows with an underestimation more pronounced for higher Mach numbers. Some authors (Moneghan Reference Moneghan1953; Huang, Bradshaw & Coakley Reference Huang, Bradshaw and Coakley1993) have proposed compressibility corrections, which work well for turbulent boundary layers over adiabatic walls. A similar extension for channel flows is not without ambiguities since the definition of the freestream values and of the dynamic pressure for normalization are more arbitrary. It can however be concluded that in dense-gas flow decoupling of dynamic and thermal effects dominates, so that Mach number effects are less influential on the skin friction coefficient than in air flows, and Dean’s formula yields a reasonable correlation.

Figure 2.

Evolutions of $C_{f}$ (a) and $Nu_{B}$ (b) as a function of $Re_{D_{h}}$ for different cases. Same legend as figure 1.

In a similar manner, the Nusselt number is compared in figure 2(b) with a classical empirical correlation, commonly used to estimate heat transfer in hydraulic pipes (Incropera & DeWitt Reference Incropera and DeWitt2007; Zonta Reference Zonta2013). Specifically, we use the Sieder–Tate correlation, $Nu_{B}^{ST}=0.027Re_{D_{h}}^{4/5}Pr_{B}^{1/3}(\unicode[STIX]{x1D707}_{B}/\unicode[STIX]{x1D707}_{w})^{0.14}$ (Sieder & Tate Reference Sieder and Tate1936). Here, the Reynolds number at bulk conditions is based on the hydraulic diameter $D_{h}$ , i.e. $Re_{D_{h}}=2Re_{m}=4Re_{Bb}$ . This formula is an extension of the Dittus–Boelter/Colburn correlations including the viscosity dependence due to temperature change between the bulk average temperature and wall temperature. Figure 2(b) shows that the power-law exponent of 0.8 captures well the Reynolds number dependence for both fluids. The higher values obtained for dense-gas cases are explained by the Prandtl number dependence, which is constant equal to 0.7 for air and varies between 2.3 and 2.4 for PP11. The enhanced heat capacities of dense gases reduce the heat conduction. A Mach number effect can also be noticed, leading to lower Nusselt numbers when the Mach number increases. This compressibility effect, more visible for air flows, is not taken into account by the empirical correlation. The predicted values of $C_{f}$ and $Nu_{B}$ are reported in table 4. The heat flux towards the walls $B_{q}=\overline{q}_{w}/(\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}\overline{h}_{w})$ , with $\overline{h}_{w}$ the specific enthalpy at the wall, is also given. Its order of magnitude is $O(10^{0}{-}10^{1})$ for air and $O(10^{-4})$ for PP11 flows. Hence, for dense gases, isothermal walls translate also into quasi-adiabatic conditions. Also given in table 4, the wall and centreline mean values of the fundamental derivative $\overline{\unicode[STIX]{x1D6E4}}_{w}$ and $\overline{\unicode[STIX]{x1D6E4}}_{c}$ are globally lower than unity. Their increase with $M_{B}$ is mainly associated with a higher dispersion of the instantaneous thermodynamic states.

4.2 Scalings and first-order statistics

Figure 3.

Scalings for the law of the wall for air (top) and PP11 (bottom). (a,d) Classical incompressible scaling $\overline{u}^{+}$ ; (b,e) Van Driest transformation $\overline{u}_{VD}$ ; (c,f) Trettel–Larsson transformation $\overline{u}_{TL}$ in semi-local coordinates. For the meaning of the line styles in this and subsequent figures, see table 2.

Figure 3 displays the mean streamwise velocity profiles for air (a–c) and PP11 flows (d–f). It is known that the classical incompressible scaling (a,d) fails in collapsing the velocity profiles when compressibility effects are present. In particular, predictions get worse in presence of a substantial heat flux towards the walls. Several attempts have been made in the past in order to derive a transformation able to collapse the compressible velocity profiles into an universal law of the wall for a wide range of Mach numbers. For non-hypersonic flows, Van Driest (Reference Van Driest1951) developed the transformation:

which takes into account mean-density variations and translates the compressible profile into an equivalent ‘constant-density’ profile keeping the same wall-normal coordinate $y^{+}$ . The Van Driest transformation provides a substantial improvement of the scaling (b,e), even if it departs from the log law as $M_{B}$ increases. The size of the buffer layer increases rapidly for air since the beginning of the logarithmic zone moves towards higher $y^{+}$ values (from $y^{+}\approx 20$ for AM1R7 to $y^{+}\approx 70$ for AM3R12). On the other hand, for dense gases, the heat flux is greatly reduced and $\overline{u}_{VD}$ -scaling provides quite satisfactory results. Indeed, this scaling was found to behave well for adiabatic walls (Guarini et al. Reference Guarini, Moser, Shariff and Wray2000; Pirozzoli, Grasso & Gatski Reference Pirozzoli, Grasso and Gatski2004; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011). Modifications for the value of the intercept of the log-law velocity profile, $u^{+}=1/\unicode[STIX]{x1D705}_{VK}\log y^{+}+C$ were proposed by Bradshaw (Reference Bradshaw1977), yielding empirical laws that relate $C$ to the friction Mach number $M_{\unicode[STIX]{x1D70F}}$ and to the heat flux $B_{q}$ . Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016) have recently provided a thorough review of the main scalings available in the literature, and have shown that the transformation proposed by Trettel & Larsson (Reference Trettel and Larsson2016),

accurately reproduces the mean velocity profile even for cases with high wall heat flux. Panels (c,f) confirm its validity even for dense gases. Note that, with this transformation, the wall-normal distance is normalized according to the semi-local scaling $y^{\ast }$ .

Figure 4.

Profiles of density $\overline{\unicode[STIX]{x1D70C}}^{+}$ , temperature $\widetilde{T}^{+}$ and pressure $\overline{p}/\overline{p}_{w}$ for air (top) and PP11 (bottom) cases. Same legend as in table 2.

Figure 4 shows the density, temperature and pressure profiles normalized with wall quantities. The semi-local scaling $y^{\ast }$ is retained from hereafter as the standard scaling for comparing perfect- and dense-gas cases. The rise of wall density values for increasing $M_{B}$ is significant for air and less pronounced for PP11 flows. For instance, $\overline{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x1D70C}_{B}=2.51$ for AM3R12 and 1.25 for PM3R12 (see values in table 4). Mean density $\overline{\unicode[STIX]{x1D70C}}^{+}$ decreases towards the centreline, with minimum values $\min _{y}\{{\overline{\unicode[STIX]{x1D70C}}_{c}}^{+}\}\approx 0.39$ for AM3R3 and ${\approx}0.77$ for PM3R3. An eventual influence of the Reynolds number is hidden by the strong dependence on $\overline{M}_{c}$ , noticeable for both $\overline{\unicode[STIX]{x1D70C}}^{+}$ and $\widetilde{T}^{+}$ . As prefigured by figure 1(b), temperature profiles are quite different, with increases lower than 1 % of the wall temperature for PP11 flows. Pressure is roughly constant across the channel. More precisely, it exhibits a constant value in the viscous sublayer, then it slightly decreases in the buffer layer reaching a minimum in the logarithmic zone, and afterwards increases in the outer region. For air, the relationship $\overline{p}/\overline{p}_{w}=\overline{\unicode[STIX]{x1D70C}}^{+}\widetilde{T}^{+}\lessapprox 1$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014) holds in each case. The minimum values for air flows are up to three times higher than for PP11 flows, with $\max _{y}\{(\overline{p}-\overline{p}_{w})/\overline{p}_{w}\}\approx 1.5\,\%$ for AM3R12 and 0.5 % for PM3R12. Although pressure is nearly constant, its mean value can be directly related to the turbulent fluctuations. In fact, by applying the Favre averaging to the Navier–Stokes equations, one obtains for the mean equation in the $y$ -direction (Huang et al. Reference Huang, Coleman and Bradshaw1995):

For compressible flows, the second term is not equal to zero but remains much lower than the other two. As a consequence, the minimum value of $\overline{p}$ coincides with the maximum of the wall-normal Reynolds stresses, as it will be shown later.

Figure 5.

Mean profiles of the speed of sound $\overline{c}/\overline{c}_{w}$ for air (a) and PP11 (b) flows.

Figure 5 shows the mean profiles of the speed of sound across the channel. Sound speed variations across the channel are related to temperature changes for air ( $\overline{c}/\overline{c}_{w}=\sqrt{\overline{T}^{+}}$ ), whereas they follow density variations for PP11. We notice that relative variations of the speed of sound are much smaller for PP11 (approximately 20 % against 60 % for air), leading to higher values of $\overline{M}$ compared to air flows.

Figure 6.

Mean profiles of the viscosity $\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}$ , the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ and the Reynolds number $\overline{Re}$ for air (a–c) and PP11 (d–f) flows.

The most striking differences between perfect- and dense-gas flow cases are highlighted in figure 6, presenting the profiles of viscosity, semi-local friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ and average Reynolds number $\overline{Re}$ . For air flows, $\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}$ increases towards the centreline (gas-like behaviour), with a maximum deviation in AM3R3 for which $\max _{y}\{\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}\}\approx 2$ . For PP11 flows, the ratio decreases (following approximately the $\overline{\unicode[STIX]{x1D70C}}^{+}$ profile) up to $\min _{y}\{\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}\}\approx 0.75$ for PM3R3, which is a liquid-like behaviour. Viscosity largely influences the profiles of $\overline{Re}$ and in particular the centreline values, that tend to be considerably larger for dense gases. The evolution of $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ follows opposite behaviours for the two fluids, due to the aforementioned evolutions of transport properties. For air flows, it decreases rapidly up to the logarithmic layer and then more slowly in the outer region. For a given $Re_{B}$ , a higher $M_{B}$ leads to higher values of $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ near the wall, since $\overline{\unicode[STIX]{x1D707}}_{w}$ is the same whereas $\overline{\unicode[STIX]{x1D70C}}_{w}$ and $\overline{(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)}_{w}$ are increasing functions of the Mach number. At greater distance from the wall, $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ decreases even more quickly due to the strong temperature gradients in the wall-normal direction. Outside of the viscous sublayer, $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ is considerably lower than the near-wall one for the higher $M_{B}$ cases, due to the combined effect of a rapid decrease of local density and an increase of local viscosity. Specifically, the centreline value may drop below the one obtained for lower Mach number cases, for which the fluid properties vary less. For PP11 flows, $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ exhibits an opposite behaviour, namely, it increases when moving from the wall to the centreline. The overall variations, of the order of 12 % at most for high Mach number cases, are anyway much smaller than in the perfect gas (variations up to 60 %). When increasing the wall distance, $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ slightly increases up to the buffer layer and remains roughly constant afterwards. Due to the increasing reduction of centreline viscosity for higher Mach number cases, the growth of $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ is enhanced when $M_{B}$ becomes higher. The liquid-like viscosity behaviour clearly affects the profile of the local Reynolds number $\overline{Re}$ (panel f) compared to the perfect gas (panel c). For instance, at the centreline, $\overline{Re}_{c}$ is about 2.5 times higher for PM3R12 than for AM3R12 results.

Figure 7.

Mean profiles of the Prandtl number $\overline{Pr}$ (a), specific heat at constant pressure $\overline{c_{p}}/R$ (b) for PP11 and Eckert number $\overline{Ec}$ (c) for air and PP11.

A direct consequence of the constant Prandtl number assumption in the ideal model is that $\overline{\unicode[STIX]{x1D705}}/\overline{\unicode[STIX]{x1D705}}_{w}=\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w}$ for air. In PP11 simulations, variations of the thermal conductivity across the channel are smaller than the ones of viscosity ( $\min _{y}\{\overline{\unicode[STIX]{x1D705}}/\overline{\unicode[STIX]{x1D705}}_{w}\}\approx 0.95$ ), nevertheless the ratio of the two transport properties remains roughly constant and $\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D705}}\approx 1.25$ throughout the channel. Thus, the Prandtl number profile (figure 7 a) is essentially driven by the normalized specific heat at constant pressure $\overline{c_{p}}/R$ (figure 7 b) which, in turn, follows a trend similar to the density profile (i.e. decreases with the wall distance). As pointed out in the above, the latter exhibit a strong dependency on $M_{B}$ in the near-wall region, which is then fed back to $\overline{Pr}$ . Specifically, the wall Prandtl number increases by approximately 50 % when doubling the bulk Mach number and is almost insensitive to the Reynolds number. In the outer region, $\overline{Pr}\approx 2.3-2.4$ for all dense-gas cases. On the other hand, the high values of $\overline{c_{p}}/R$ for dense gases lead to very small values of the average Eckert number, $\overline{Ec}=\overline{u^{2}/c_{p}T}$ , a parameter that is representative of the degree of coupling between thermal and kinematic effects. We recall that for air flows, $\overline{c_{p}}/R=c_{p}/R=3.5$ , which is two orders of magnitude lower than the mean values obtained with PP11. In figure 7(c), $\overline{Ec}$ exhibits a similar trend for both fluids, with two logarithmic zones connected near $y^{\ast }\approx 15$ , but the curves are shifted by approximately two orders of magnitude. In fact, the centreline values are of the order of $O(1)$ for air and $O(10^{-2})$ for PP11 cases.

4.3 Second-order statistics of the thermodynamic and transport properties

Figure 8.

Profiles of the normalized root mean square density $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}=\sqrt{\overline{\unicode[STIX]{x1D70C}^{\prime 2}}}/\overline{\unicode[STIX]{x1D70C}}$ (a,d), temperature $T_{rms}/\overline{T}=\sqrt{\overline{T^{\prime 2}}}/\overline{T}$ (b,e) and pressure $p_{rms}/\overline{p}=\sqrt{\overline{p^{\prime 2}}}/\overline{p}$ (c,f) for air flows (a–c) and PP11 flows (d–f).

Root mean square (r.m.s.) values for the density, temperature and pressure are shown in figure 8. Gerolymos & Vallet (Reference Gerolymos and Vallet2014) have shown that for air flows $O(T_{rms}/\overline{T})=O(\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}})=O(p_{rms}/\overline{p})$ , regardless of the $y^{\ast }$ location, Mach and Reynolds numbers. The same trend has been observed for both forced (Donzis & Jagannathan Reference Donzis and Jagannathan2013) and decaying (Sciacovelli et al. Reference Sciacovelli, Cinnella, Content and Grasso2016a ) highly compressible homogeneous turbulence. As discussed in Gerolymos & Vallet (Reference Gerolymos and Vallet2014), this behaviour is a direct consequence of the perfect-gas equation of state, and does not depend on the specific flow configuration under investigation. Indeed, writing the differential form of the ideal-gas law $(\text{d}p/p)=(\text{d}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C})+(\text{d}T/T)$ and approximating differential terms by r.m.s. fluctuations, we get:

After an initial growth in the linear region, $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ and $T_{rms}/\overline{T}$ reach a peak in the buffer layer at $y^{\ast }\approx 10$ and then decay. The pressure fluctuations are rather nearly constant close to the walls, slightly peak at the beginning of the logarithmic zone ( $y^{\ast }\approx 40$ ) and finally decrease, reaching a minimum in the core region. Since pressure fluctuations are of smaller order whereas $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ and $T_{rms}/\overline{T}$ have very similar distributions and levels, equation (4.5) permits one to identify temperature with density fluctuations, corresponding to the assumption of Laufer (Reference Laufer1969) of essentially isobaric temperature fluctuations. Some authors (Rubesin Reference Rubesin1990; Huang et al. Reference Huang, Coleman and Bradshaw1995) propose rather that the gas behaves generally in a polytropic manner.

For dense gases, the observed trends are very different. While the overall levels of density and pressure fluctuations remain of the same order of magnitude as for air cases, the relative temperature fluctuations are more than one order of magnitude lower. The fluctuating pressure distribution is much like the perfect-gas one, since it depends on the fluctuating velocity through a Poisson equation. Furthermore, the r.m.s. density decreases monotonically with the wall distance, the highest density fluctuations being observed close to the walls. The r.m.s. density levels decay at higher rates in the linear layer and in the outer region, and exhibit only weak variations in the logarithmic region. This behaviour can also be directly related to the equation of state, namely the Martin–Hou equation (2.10). Following the same reasoning as for the ideal gas, the differential form $dp=(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})_{T}d\unicode[STIX]{x1D70C}+(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}T)_{\unicode[STIX]{x1D70C}}\,\text{d}T$ is derived and the differential terms are approximated by r.m.s. fluctuations, yielding:

where the partial derivatives are evaluated from mean quantities:

The validity of the approximated relation (4.6) is checked in figure 9 for case PM3R7 against DNS results for $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ and appears to be reasonably accurate, especially concerning the observed trends. Then, considering that the thermal variations are weak for the dense-gas temperature fluctuations are neglected, i.e. $T_{rms}=0$ in (4.6). The corresponding curve follows fairly the trends of $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ . The isothermal assumption is then a good approximation for the dense gas, meaning (i) that density fluctuations are correlated with pressure fluctuations and (ii) that $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ is well approximated by $(p_{rms}/\overline{\unicode[STIX]{x1D70C}})/(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})_{T}$ .

Finally, using the definition of the sound speed $c^{2}=(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})_{s}$ , and replacing the partial derivative of $p$ with respect to $\unicode[STIX]{x1D70C}$ at constant $T$ by the same partial derivative at constant entropy yields $(p_{rms}/\overline{\unicode[STIX]{x1D70C}})/\overline{c}^{2}$ . The proportionality with the fluid compressibility (inverse of the speed of sound squared) is also tested in figure 9. A good match is obtained above the viscous sublayer, which means that the assumption that the isothermal approximation is also close to an adiabatic approximation for a dense gas is globally true, except very near to the wall. The preceding discussion shows that the rise of density fluctuations near the wall is related to the peculiar behaviour of the thermal equation of state for a nearly isothermal process. Similar results (not shown) were also obtained for a simpler thermal equation of state (Van der Waals) with the same qualitative behaviour, under the polytropic-gas assumption.

Despite the complex variations, fluctuations in thermophysical quantities globally remains moderate for PP11. As a consequence, the assumption of Morkovin (Reference Morkovin and Favre1961), stating that the entropy and vortical modes are weakly coupled, is still valid. Fluctuation levels up to 18 % of the mean value are noticed very near the wall for the density, but this region is dominated by viscous stresses, so that the conclusion is not altered.

Figure 9.

Approximation of normalized density fluctuations for case PM3R7. —— (red) $\unicode[STIX]{x1D70C}_{rms}/\overline{\unicode[STIX]{x1D70C}}$ ; - - - - - (black) $[p_{rms}-(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}T)_{\unicode[STIX]{x1D70C}}T_{rms}]/(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})_{T}/\overline{\unicode[STIX]{x1D70C}}$ ; $\boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot$ (blue) $p_{rms}/(\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})_{T}/\overline{\unicode[STIX]{x1D70C}}$ ; $\boldsymbol{-}\cdot \cdot \boldsymbol{-}\cdot \cdot \boldsymbol{-}\cdot \cdot$ (orange) $p_{rms}/\overline{c}^{2}/\overline{\unicode[STIX]{x1D70C}}$ .

Figure 10.

Profiles of the correlation coefficients $C_{\unicode[STIX]{x1D70C}^{\prime }p^{\prime }}$ (a,e), $C_{\unicode[STIX]{x1D70C}^{\prime }u^{\prime }}$ (b,f), $C_{\unicode[STIX]{x1D70C}^{\prime }v^{\prime }}$ (c,g) and $C_{u^{\prime }v^{\prime }}$ (d,h) for air (a–d) and PP11 MAH (e–h) flows.

To further clarify the strong relations between density and pressure, correlation coefficients are calculated. Given two flow variables $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , the correlation coefficient $C_{\unicode[STIX]{x1D6FC}^{\prime }\unicode[STIX]{x1D6FD}^{\prime }}$ is defined as:

Figure 10 shows the correlations $C_{\unicode[STIX]{x1D70C}^{\prime }p^{\prime }}$ , $C_{\unicode[STIX]{x1D70C}^{\prime }u^{\prime }}$ , $C_{\unicode[STIX]{x1D70C}^{\prime }v^{\prime }}$ and $C_{u^{\prime }v^{\prime }}$ . Due to the isothermal boundary conditions, the correlation coefficient $C_{\unicode[STIX]{x1D70C}^{\prime }p^{\prime }}$ is equal to unity at the walls for both air and PP11. Afterwards, it decreases reaching a minimum at $y^{\ast }\approx 8$ , approximately corresponding to the temperature variance peak, and then it increases again towards the centreline. For PP11 flows $C_{\unicode[STIX]{x1D70C}^{\prime }p^{\prime }}$ approaches unity already at $y^{\ast }\approx 20$ and then remains constant. On the other hand, it raises slowly for air flows, with an asymptotic value close to 0.9. The correlation coefficients $C_{\unicode[STIX]{x1D70C}^{\prime }u^{\prime }}$ and $C_{\unicode[STIX]{x1D70C}^{\prime }v^{\prime }}$ , which represent the transport of density fluctuations by the streamwise and wall-normal velocities, exhibit a totally different evolution depending on the fluid nature. Specifically, for air the streamwise correlation is negative and the spanwise is positive, whereas for PP11 both coefficients tend rapidly towards zero. Furthermore, $C_{\unicode[STIX]{x1D70C}^{\prime }u^{\prime }}$ exhibits both a Mach and Reynolds dependence for air, whereas for PP11 it varies only with the Mach number. Lastly, the distributions $C_{u^{\prime }v^{\prime }}$ are similar even if a perfect collapse can be noticed for air, whereas a slight Mach dependence is observed for PP11.

Figure 11.

Distribution of $p-v$ states of an instantaneous flow field for PP11 in a Clapeyron’s diagram. (a) Influence of Mach number on the distribution of $p-v$ states for a slice at $y^{\ast }\approx 10$ and $Re_{B}=3000$ : (○, red) $M_{B}=1.5$ ; (●) $M_{B}=2.25$ ; (○) $M_{B}=3$ . The background colour map shows isocontours of the speed of sound. (b) Influence of Reynolds number on the distribution of $p-v$ states for a slice at $y^{\ast }\approx 10$ and $M_{B}=3$ : (○, red) $Re_{B}=3000$ ; (●) $Re_{B}=7000$ ; (○) $Re_{B}=12\,000$ . The background colour map shows isocontours of the viscosity. (c) Influence of $y^{\ast }$ location on the distribution of $p-v$ states for $M_{B}=3$ and $Re_{B}=12\,000$ : (○, red) $y^{\ast }=798$ (centreline); (●) $y^{\ast }=50$ ; (○) $y^{\ast }=10$ . The background colour map shows isocontours of the Prandtl number.

Another way to interpret the peculiar behaviour of the thermodynamic quantities is to look at representations in Clapeyron’s diagrams. Figure 11 shows distributions of the $p-v$ states in the computational box at one instant for PP11 flows. The different panels correspond successively to the influence of $M_{B}$ , $Re_{B}$ and the wall-normal position. The different $p-v$ diagrams are coloured successively with values of the speed of sound, viscosity and Prandtl number in order to highlight the differences with respect to air. The $p-v$ states are recorded on slices parallel to the wall. The location is fixed at $y^{\ast }\approx 10$ where temperature fluctuations peak for figure 11(a,b). It is clear from figure 11(a) that the main effect of increasing $M_{B}$ is a broadening of the region spanned by the $p-v$ states. For $M_{B}=3$ cases, the states characterized by stronger compressions exit the dense-gas region and reach a supercritical zone where strong variations of the thermodynamic properties are observed. Higher values of $M_{B}$ imply a greater dispersion in $\overline{\unicode[STIX]{x1D70C}}_{w}$ values and in variances of the thermodynamic variables near the wall. Changing the Reynolds number $Re_{B}$ for a given $M_{B}$ in figure 11(b) reveals that the distributions of $p-v$ states are almost superimposed for the same $y^{\ast }$ , providing an a posteriori check of the validity of the semi-local scaling. Lastly, distributions for different $y^{\ast }$ values are shown in figure 11(c). The thermodynamic regions spanned by the $p-v$ states are reduced moving away from the wall. Even if most of the states are enclosed in the dense-gas region, none of the present simulations exhibit peculiar dense-gas effects, such as expansion shock waves. This can be explained by the fact that the turbulent Mach number, $M_{t}=\sqrt{u_{i}^{\prime 2}}/\overline{c}$ , does not reach sufficiently high values. For instance, figure 12 indicates that $M_{t}$ approaches 0.4 for $M_{B}=3$ , whereas Sciacovelli et al. (Reference Sciacovelli, Cinnella, Content and Grasso2016a ) have shown that significant dense-gas effects can appear for $M_{t}$ above 0.8. The distributions of $M_{t}$ are similar as the simulations with air with slightly higher values for PP11 flows, showing that, notwithstanding liquid-like characteristics of dense gases, important compressibility is also present. The profiles of the fundamental derivative of gas dynamics $\overline{\unicode[STIX]{x1D6E4}}$ , plotted in figure 12(c), confirm that mean values are indeed lower than 1 for PP11 flows, with minimal values in the channel core. The flows thus evolve in the dense-gas region but with positive values of $\overline{\unicode[STIX]{x1D6E4}}$ , so that BZT phenomena are not expected.

Figure 12.

Profiles of the turbulent Mach number $M_{t}$ for air (a) and PP11 (b) and of the fundamental derivative of gas dynamics $\overline{\unicode[STIX]{x1D6E4}}$ for PP11 (c).

5.1 Reynolds stresses and Fukagata–Iwamoto–Kasagi (FIK) identity

Figure 13.

Reynolds stresses in the streamwise (a,e), wall-normal (b,f) and spanwise (c,g) directions and Reynolds shear stresses (d,h) for air (a–d) and PP11 (e–h) flows with semi-local scaling.

Figure 13 shows the Reynolds stresses $[\overline{\unicode[STIX]{x1D70C}u_{i}^{\prime \prime }u_{j}^{\prime \prime }}]^{+}=\overline{\unicode[STIX]{x1D70C}u_{i}^{\prime \prime }u_{j}^{\prime \prime }}/\unicode[STIX]{x1D70F}_{w}$ for the selected simulations. Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015) have shown that for variable-property turbulent channel flows, gas-like transport properties tend to lower the spanwise, wall-normal and shear Reynolds stresses with respect to the corresponding incompressible evolution, whereas the streamwise component increases. The opposite trend is expected when liquid-like properties are considered. In our cases the strong compressibility adds a complexity in the sense that higher $M_{B}$ increase the velocity gradients near the wall (thus increasing the Reynolds stresses) but also dramatically affect viscosity distributions. Figure 13 provides the trends of Reynolds stresses as a function of both Reynolds and Mach number for selected air and PP11 flows. The figure shows that the Reynolds number dependency is the same for the ideal or dense gas, and is similar to that observed for incompressible flows, even if compressibility effects modify the effective friction Reynolds number for a given bulk Reynolds number. For instance, case AM1R7 ( $Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 315$ ) is characterized by a semi-local friction Reynolds number higher than case AM3R7 ( $Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 200$ ). On the contrary, the dependency on the bulk Mach number is different for the two fluids. In air, the temperature increases with $M_{B}$ , and so does the viscosity, leading to lower values of $\overline{\unicode[STIX]{x1D70C}v^{\prime \prime }v^{\prime \prime }}^{+}$ , $\overline{\unicode[STIX]{x1D70C}w^{\prime \prime }w^{\prime \prime }}^{+}$ , $\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}^{+}$ as $M_{B}$ grows. For PP11, the viscosity (following the density) becomes lower and lower, thus enhancing the spanwise, cross-wise and shear Reynolds stresses. For the streamwise stress, instead, the same behaviour is observed between air and PP11 since the effect of viscosity is now overwhelmed by the increase of the mean velocity shear across the channel.

Concerning the location of the peak values, for both air and PP11 flows they are roughly constant for the streamwise and spanwise components, namely around $y^{\ast }\in [10,13]$ for $\overline{\unicode[STIX]{x1D70C}u^{\prime \prime 2}}^{+}$ , and $y^{\ast }\in [40,50]$ for $\overline{\unicode[STIX]{x1D70C}w^{\prime \prime 2}}^{+}$ . On the other hand, a weak dependence on $Re$ is observed for wall-normal and shear stresses. The slight shift towards higher  $y^{\ast }$ is more pronounced for PP11 flows since much higher $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ are achieved. The $Re$ influence is consistent with high Reynolds number incompressible simulations (Lee & Moser Reference Lee and Moser2015). It can be noticed that wall-normal stresses $\overline{\unicode[STIX]{x1D70C}v^{\prime \prime }v^{\prime \prime }}^{+}$ for PP11 flows present different centreline values, notably for cases PM3R7 and PM3R12 at $M_{B}=3$ . The levels of wall-normal fluctuations become then of the same level as those in the streamwise direction.

The contribution of Reynolds stresses to the skin friction coefficient can be analysed thanks to the FIK identity. This decomposition of sources of friction also reveals important differences between ideal and dense gases. Starting from the incompressible expression of Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002), Gomez, Flutet & Sagaut (Reference Gomez, Flutet and Sagaut2009) derived an exact relationship for the analysis of the contribution of Reynolds stresses to the skin friction coefficient in compressible cases. The skin friction coefficient may be split into four contributions, namely:

where $C_{L}$ is the laminar contribution, $C_{T}$ the turbulent contribution, $C_{C}$ the compressible contribution (in which $\overline{\widehat{\unicode[STIX]{x1D707}}}=\overline{\unicode[STIX]{x1D707}}-1$ ) and $C_{CT}$ the compressible–turbulent interaction term. Note that all the quantities in (5.1) are non-dimensionalized using the channel half-width $h$ as length scale, $\widetilde{u}_{B}$ as velocity scale, $\unicode[STIX]{x1D70C}_{B}$ as density scale and $\overline{\unicode[STIX]{x1D707}}_{w}$ as viscosity scale. As observed by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), the turbulent term is proportional to the Favre-averaged Reynolds stresses, and the weighting term decreases linearly with the distance from the wall. This explains why the main contribution to the frictional drag is due to the turbulent structures close to the wall, rather than at the location of the peak of Reynolds stresses. The compressible term $C_{C}$ is related to the mean viscosity variations and to the mean wall-normal velocity gradient. $C_{CT}$ represents the interaction between compressibility and turbulent effects and is usually very small. Table 5 gives the values of $C_{f}^{FIK}$ for different cases, as well as the relative error with respect to the values obtained in § 4.1 and the relative percentage of each contribution. To investigate the effect of compressibility, we consider the cases with $Re_{B}=7000$ and different $M_{B}$ , such that the laminar contribution $C_{L}$ is approximately constant (i.e. $C_{L}\approx 13\,\%$ – $14\,\%$ ). For air flows, the contribution due to the turbulent fluctuations $C_{T}$ decreases as $M_{B}$ increases, coherently with the decreasing of the Reynolds stress levels. The opposite trend is observed for PP11 flows, for which the relative weight of $C_{T}$ increases of approximately 1.5 %. This can be related to the liquid-like behaviour of dense gases, which also causes substantial changes in the compressible contribution $C_{C}$ . In fact this term takes on higher values as $M_{B}$ increases for both fluids, but it is positive for air (since $\overline{\widehat{\unicode[STIX]{x1D707}}}>0$ ) and negative for PP11 ( $\overline{\widehat{\unicode[STIX]{x1D707}}}<0$ ). The compressible term is indeed related to variable-property effects. Lastly, the compressible–turbulent interaction term is always quite small and contributes to the total skin friction coefficient for approximately 0.1 % in air and 0.02 % in PP11 flows, for which the dynamic and thermal effects are even less correlated.

Table 5.

Contributions to the skin friction coefficient according to the compressible extension of the FIK identity of Gomez et al. (Reference Gomez, Flutet and Sagaut2009) for different cases.

5.2 Turbulent kinetic energy budgets

For fully developed, statistically stationary channel flow, the turbulent kinetic energy budget reads $P_{k}+T_{k}+V_{k}+M_{k}+\unicode[STIX]{x1D60B}_{k}+\unicode[STIX]{x1D700}_{k}=0$ , with

$P_{k}$ is the production, $T_{k}$ and $V_{k}$ the turbulent and viscous transport respectively, $M_{k}$ the mass flux variation, $\unicode[STIX]{x1D60B}_{k}$ the velocity–pressure gradient tensor (composed of the sum of pressure strain correlation and pressure transport term) and $\unicode[STIX]{x1D700}_{k}$ the dissipation term. Figure 14 shows the turbulent kinetic energy budget for cases AM3R12 and PM3R12 normalized by $\unicode[STIX]{x1D70F}_{w}^{2}/\overline{\unicode[STIX]{x1D707}}(y)$ (Foysi et al. Reference Foysi, Sarkar and Friedrich2004). In semi-local scaling, the viscous and turbulent transports behave similarly, whereas the mass flux variation (which is zero for incompressible flows) is much smaller for PP11 case. In the core region, all the budget terms collapse.

Figure 14.

Comparison of turbulent kinetic energy budgets for cases AM3R12 (dashed lines) and PM3R12 (solid lines) in semi-local scaling: —— (red) $P_{k}$ , —— (green) $T_{k}$ , —— (blue) $V_{k}$ , —— (black) $M_{k}$ , —— (purple) $\unicode[STIX]{x1D60B}_{k}$ , —— (orange) $\unicode[STIX]{x1D700}_{k}$ normalized by $\unicode[STIX]{x1D70F}_{w}^{2}/\overline{\unicode[STIX]{x1D707}}(y)$ .

Figure 15.

Profiles of the turbulent production $P_{k}$ , the turbulent dissipation $\unicode[STIX]{x1D700}_{k}$ and the ratio of production to dissipation $P_{k}/\unicode[STIX]{x1D700}_{k}-1$ for air (a,b) and PP11 (c,d) flows.

A closer inspection of the turbulent production and dissipation for the selected simulations is provided in figure 15, which also shows their ratio, plotted both in linear and logarithmic semi-local coordinates. For both fluids, the production peak is located at $y^{\ast }\approx 11$ –12 for all the considered bulk conditions. Sarkar (Reference Sarkar1995) and Lechner et al. (Reference Lechner, Sesterhenn and Friedrich2001) showed that the main effect of compressibility is a reduction in the production and dissipation rate of the turbulent kinetic energy. Nevertheless, this effect can be associated with the strong variations of properties at high Mach numbers. The semi-local scaling $\unicode[STIX]{x1D70F}_{w}^{2}/\overline{\unicode[STIX]{x1D707}}(y)$ is able to collapse, for instance, the budgets for cases M1R7, M2R7 and M3R7, for which substantial differences should be observed due to the doubling of the bulk Mach number. Globally, a common trend is observed for air and PP11 flows, where the peaks of production and dissipation terms slightly increase with $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ . The ratio of the production and dissipation rates $P_{k}/\unicode[STIX]{x1D700}_{k}-1$ exhibits a similar evolution for similar values of $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ (cases PM1R3 and AM3R7). The ratio peaks at the same $y^{\ast }$ location as the production term. Furthermore, for cases with $Re_{\unicode[STIX]{x1D70F}}^{\ast }>200$ , a second peak starts growing in the logarithmic region. For PP11 flows, the inner peak seems roughly unaffected, as found for high Reynolds number turbulent incompressible flows (Lee & Moser Reference Lee and Moser2015), whereas in air a slight decrease is observed with increasing $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ , as shown in the inset with logarithmic scale. Since the $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ values are low in air cases, the ratio always remains negative and no effective net turbulent production is observed in the outer layer. For PP11, the case PM3R12 is the only one with a positive region, extending between $y^{\ast }\approx 180$ and $y^{\ast }\approx 400$ .

5.3 Near-wall turbulent structures

Figure 16.

$\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}}$ fluctuations at $y^{\ast }\approx 12$ for AM3R7 (a) and PM1R3 (b).

Figure 17.

$\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}}$ fluctuations at $y^{\ast }\approx 50$ for AM3R7 (a) and PM1R3 (b).

Figure 18.

$\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}}$ fluctuations at $y^{\ast }\approx 200$ for AM3R7 (a) and PM1R3 (b).

The difference in near-wall turbulent structures is now analysed by means of instantaneous flow visualizations for iso- $Re_{\unicode[STIX]{x1D70F}}^{\ast }$ cases, namely, AM3R7 and PM1R3 ( $Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 200$ at the centreline). Figures 16–18 show a slice of the instantaneous flow field at different values of $y^{\ast }$ (namely, $y^{\ast }=12$ , 50 and centreline) coloured with levels of the normalized streamwise velocity fluctuations $\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}}$ for cases AM3R7 and PM1R3 (hence for $Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 200$ ). The local density value $\unicode[STIX]{x1D70C}$ is used for normalization instead of $\overline{\unicode[STIX]{x1D70C}}$ , since it correlates better with the modulation of the streak magnitude (Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015). In all cases, the computational box is scaled with respect to semi-local coordinates ( $x^{\ast }$ , $z^{\ast }$ ) (which change with the wall distance) and the box size considered is $5000\times 1200$ based on non-dimensional coordinates $x^{\ast }\times z^{\ast }$ for each case. For both fluids, the maps of the instantaneous streamwise velocity close to the wall exhibits the well-known organization in streaks (Coleman et al. Reference Coleman, Kim and Moser1995; Chernyshenko & Baig Reference Chernyshenko and Baig2005; Gerolymos & Vallet Reference Gerolymos and Vallet2014). In past years, the role of compressibility on the modification of near-wall structures has been widely discussed. Initially, it was argued that its effect was to increase the streamwise correlation length, enhancing the coherence of near-wall streaks (Coleman et al. Reference Coleman, Kim and Moser1995; Duan et al. Reference Duan, Beekman and Martin2010; Lagha et al. Reference Lagha, Kim, Eldredge and Zhong2011). This statement was not confirmed when the semi-local scaling was taken into account (Morinishi et al. Reference Morinishi, Tamano and Nakabayashi2004; Patel et al. Reference Patel, Peeters, Boersma and Pecnik2015). A rather universal behaviour was then observed. The latter conclusion holds in the present comparison between air and PP11 flows at the three heights shown in the figures. Figure 16 shows that near-wall layers are dominated by streaks with similar streamwise lengths and spanwise separations. At $y^{\ast }\approx 50$ , in figure 17, the structures become less coherent and less energetic. At the centreline of the channel, as illustrated in figure 18, no prominent coherent structures can be found.

Figure 19.

Pressure fluctuations ${p^{\prime }}^{+}$ at $y^{\ast }\approx 12$ for AM3R7 (a) and PM1R3 (b).

Figure 20.

Density fluctuations ${\unicode[STIX]{x1D70C}^{\prime }}^{+}$ at $y^{\ast }\approx 12$ for AM3R7 (a) and PM1R3 (b).

Differently from velocity fields which have a similar structural behaviour if represented with the semi-local scaling, the thermodynamic quantities are much more sensitive. Figures 19 and 20 show the instantaneous pressure ${p^{\prime }}^{+}$ and density ${\unicode[STIX]{x1D70C}^{\prime }}^{+}$ fluctuations at $y^{\ast }\approx 12$ . Pressure fluctuations are characterized in both cases by long-range interactions (Kim Reference Kim1989; Chang III, Piomelli & Blake Reference Chang, Piomelli and Blake1999), even if the mean square fluctuations mainly have local contributions (Kim Reference Kim1989). Higher deviations from the mean value are observed in air. Contrary to pressure, density fluctuations behave differently between air and PP11 flows. For air, very elongated scales in the streamwise direction are preferentially observed and follow approximately velocity fluctuations. In fact, the correlation $C_{\unicode[STIX]{x1D70C}^{\prime },u^{\prime }}$ is close to $-1$ in the buffer layer. Consequently, negative (positive) density fluctuations are associated with positive (negative) velocity fluctuations. Low-speed streaks are thus characterized by higher-density fluid and vice versa. The behaviour for PP11 flows is rather different, as inferred from the correlation coefficients $C_{\unicode[STIX]{x1D70C}^{\prime },u^{\prime }}$ and $C_{\unicode[STIX]{x1D70C}^{\prime },p^{\prime }}$ in figure 10. $C_{\unicode[STIX]{x1D70C}^{\prime },u^{\prime }}$ being much smaller in the buffer layer, the density modulation for low and high-speed streaks is less marked. Instead, the pressure signature is visible on the instantaneous density since $C_{\unicode[STIX]{x1D70C}^{\prime },p^{\prime }}\approx 1$ across the whole channel height.

Figure 21.

Premultiplied energy spectra for cases AM3R7 (a) and PM1R3 (b).

Figure 21 show the premultiplied kinetic energy spectra $k_{x}E_{\unicode[STIX]{x1D70C}uu}/\unicode[STIX]{x1D70F}_{w}$ , $k_{z}E_{\unicode[STIX]{x1D70C}uu}/\unicode[STIX]{x1D70F}_{w}$ and $k_{z}E_{\unicode[STIX]{x1D70C}vv}/\unicode[STIX]{x1D70F}_{w}$ for cases AM3R7 and PM1R3 on the whole channel height as a function of semi-local streamwise $k_{x}^{\ast }$ and spanwise $k_{z}^{\ast }$ wavenumbers. The semi-local scaling provides an approximate collapse over a wide range of wavenumbers, improving substantially the classical inner scaling. The strongest modifications are obtained for air flows, in which the high gradients of variable properties near the wall alter the effective scaled wavenumbers up to the buffer layer. This corresponds to elongated (shortened) structures for gas-like (liquid-like) transport properties with respect to the incompressible regime (Coleman et al. Reference Coleman, Kim and Moser1995; Duan et al. Reference Duan, Beekman and Martin2010). The locations $y^{\ast }$ and the wavelengths $\unicode[STIX]{x1D706}^{\ast }$ corresponding to the peak values are reported in table 6. Similar values with respect to those typically obtained in incompressible turbulence are found for both fluids. For instance, for the streamwise velocity, an inner peak is detected at $y^{+}\approx 15$ , where $\unicode[STIX]{x1D706}_{x}^{+}\approx 1000$ and $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ (see, e.g. spectra databases of Kim et al. (Reference Kim, Moin and Moser1987) and Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999)). One-dimensional spectra for the three velocity components, given in figure 22, show that the spanwise length scale for the wall-normal component is approximately half of the one associated with $u$ at the same distance from the wall. Table 6 reports the global maximum non-dimensional wavenumber which arises at different wall distances ( $y^{\ast }\approx 13$ –50 respectively).

Figure 22.

One-dimensional spectra for cases AM3R7 (a,c) and PM1R3 (b,d), at $y^{\ast }\approx 10$ (a,b) and $y^{\ast }\approx 200$ (c,d). —— (red) $E_{\unicode[STIX]{x1D70C}uu}^{(x)}/\unicode[STIX]{x1D70F}_{w}$ ; - - - - - (red) $E_{\unicode[STIX]{x1D70C}uu}^{(z)}/\unicode[STIX]{x1D70F}_{w}$ ; $\boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot \boldsymbol{-}\cdot$ (orange) $E_{\unicode[STIX]{x1D70C}vv}^{(x)}/\unicode[STIX]{x1D70F}_{w}$ ; $\cdots \cdots \cdot$ (orange) $E_{\unicode[STIX]{x1D70C}vv}^{(z)}/\unicode[STIX]{x1D70F}_{w}$ ; $\boldsymbol{-}\cdot \cdot \boldsymbol{-}\cdot \cdot \boldsymbol{-}\cdot \cdot$ (blue) $E_{\unicode[STIX]{x1D70C}ww}^{(x)}/\unicode[STIX]{x1D70F}_{w}$ ; - - - - - (blue) $E_{\unicode[STIX]{x1D70C}ww}^{(z)}/\unicode[STIX]{x1D70F}_{w}$ .

Table 6.

Values of $y^{\ast }$ and $\unicode[STIX]{x1D706}^{\ast }$ for the peak values of the spectra shown in figure 21.

Near-wall turbulent structures can be also analysed by means of the quadrant analysis (Wallace Reference Wallace2016). This representation allows a detailed study of the physical mechanisms producing the Reynolds shear stresses and, in a related way, underlying the turbulent kinetic energy production. The $(u,v)$ plane is divided into four quadrants: Q1 ( $+u^{\prime \prime },+v^{\prime \prime }$ ), Q2 ( $-u^{\prime \prime },+v^{\prime \prime }$ ), Q3 ( $-u^{\prime \prime },-v^{\prime \prime }$ ), Q4 ( $+u^{\prime \prime },-v^{\prime \prime }$ ). The contribution of the different quadrants can be obtained by plotting the isocontours of the probability-weighted Reynolds shear stresses $\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }/\unicode[STIX]{x1D70F}_{w}P(\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}},\sqrt{\unicode[STIX]{x1D70C}}v^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}})$ , depicted in figure 23. Near the walls probability-weighted Reynolds shear stresses appear stretched in the streamwise direction and flatter in the spanwise direction, with a strong contribution in the Q2 and Q4 quadrants, corresponding respectively to ejections ( $u^{\prime \prime }<0,v^{\prime \prime }>0$ ) and sweeps ( $u^{\prime \prime }>0,v^{\prime \prime }<0$ ). Moving towards the centreline, the contributions become more isotropic. The most important differences between air and PP11 are found in the buffer layer. Namely, the contribution of ejections events to shear stresses appear lower for PP11 flows with respect to air flows. This is consistent with recent findings of Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015). Gas-like (liquid-like) transport properties tend to stabilize (weaken) the low-speed streaks, which lift less (more) intensely away from the wall.

Figure 23.

Probability-weighted Reynolds shear stress $\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }/\unicode[STIX]{x1D70F}_{w}P(\sqrt{\unicode[STIX]{x1D70C}}u^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}},\sqrt{\unicode[STIX]{x1D70C}}v^{\prime \prime }/\sqrt{\unicode[STIX]{x1D70F}_{w}})$ for air (a–c) and PP11 (d–f).