Analysis of the equilibrium wall model for high-speed turbulent flows

We perform a priori and a posteriori analyses of the equilibrium wall model for high-speed wall-bounded turbulent flows. The time-averaged flow from various direct numerical simulation (DNS) databases is used as input to the wall model and the accuracy of the predictions in terms of wall shear stress and wall temperature (or heat flux for isothermal wall boundary conditions) are assessed. Two different mixing-length-based eddy-viscosity models and various damping functions are tested in this study. Both mixing-length models involve two adjustable parameters: (i) the von Kármán constant $\kappa$, which varies in the literature from 0.37 to 0.44, and (ii) a viscous damping constant $A^{+}$ (for the van Driest damping function). Also, for compressible flows, multiple scalings can be used for the viscous wall-normal spacing in the damping function. The sensitivity of the results to these model parameters is reported and it is found that the predictions of skin friction and wall temperature (or heat flux) are sensitive to the constants used, damping function scaling, and the wall-model exchange location. Wall-modeled large-eddy simulation (WMLES) is performed for (i) supersonic channel flow with a cold wall and (ii) an axisymmetric supersonic boundary layer for the adiabatic wall condition, to verify the a priori trends observed with respect to the damping functions. A posteriori WMLES results are consistent with the trends observed in the a priori analysis of DNS data, thus indicating the usefulness of the a priori analysis. We introduce a damping function scaling, which works well over a range of Mach numbers and thermal wall conditions.

Figure 1

Effect of the empirical constant $A^{+}$ on the wall-model predictions for the ${\text{Re}}_{\tau }\approx 2000$ channel: (a) Johnson-King-type mixing-length model and (b) Prandtl mixing-length model. In this and the following plots, the abscissa is the exchange location at which data were input to the wall model.

Figure 2

Effect of (a) the von Kármán constant $\kappa$ using the Johnson-King-type mixing-length model and (b) including the pressure-gradient term on the wall-model predictions with $\kappa =0.41$ for both the Johnson-King ($A^{+}=17$) and Prandtl ($A^{+}=26$) models, for the ${\text{Re}}_{\tau }\approx 2000$ channel.

Figure 3

Effect of different mixing-length models and damping functions on the wall-model predictions for the ${\text{Re}}_{\tau }\approx 2000$ channel with (a) a linear scale and (b) a semilogarithmically scaled abscissa. The numbers in the legend indicate the value of $A^{+}$ used in the damping function.

Figure 4

Effect of different mixing-length models and damping functions on the wall-model predictions for (a) the ${\text{Re}}_{\tau }=590$ channel and (b) the ${\text{Re}}_{\tau }=2118$ turbulent boundary layer.

Figure 5

Effect of damping function scalings for the wall model using the Johnson-King-based mixing-length model for (a) and (b) Mach 1.5 and (c) and (d) Mach 3 cold-wall channel flows corresponding to the DNS of Coleman et al. [26]. The legend is as follows:, wall;  , mixed; , semilocal; , mixed2; , local; , SA; , mixedmin; and $\square$, mixedmin2.

Figure 6

Effect of damping function scalings for the wall model using the Johnson-King-based mixing-length model for Mach 1.5 cold-wall channel flow for (a) skin friction and (b) wall heat flux, corresponding to the DNS of Modesti and Pirozzoli [29]. The legend is as follows: , wall; , mixed; , semilocal , mixed2; , local; , SA; , mixedmin; and , mixedmin2.

Figure 7

Effect of damping function scalings for the wall model using the Johnson-King-based mixing-length model for (a) and (b) the Mach 2 and (c) and (d) the Mach 4 boundary layer at adiabatic conditions. The legend is as follows: , wall; , mixed; , semilocal; , mixed2; , local; , SA; , mixedmin; and , mixedmin2.

Figure 8

Effect of damping function scalings for the wall model using the Johnson-King-based mixing-length model for (a) and (b) an adiabatic Mach 2.5 and (c) and (d) a Mach 5.86, $T_w/T_{\text{ad}}=0.76$ boundary layer. The legend is as follows: , wall; , mixed; , semilocal; , mixed2; , local; , SA; , mixedmin; and , mixedmin2.

Figure 9

Effect of damping function scalings for the wall model using the Johnson-King-based mixing-length model for (a) and (b) a Mach 7.86, $T_w/T_{\text{ad}}=0.48$ and (c) and (d) a Mach 13.68, $T_w/T_{\text{ad}}=0.18$ boundary layer. The legend is as follows: , wall; , mixed; , semilocal, , mixed2; , local; , SA; , mixedmin; and , mixedmin2.

Figure 10

Effect of the mixing-length model for (a) and (b) the cold-wall Mach 3 channel flow and (c) and (d) the adiabatic Mach 4 boundary layer. The legend is as follows: , JK wall; , Prandtl wall; , JK semilocal; and , Prandtl semilocal.

Figure 11

Effect of the turbulent Prandtl number for the wall model using the Johnson-King-based mixing-length model and mixedmin damping function scaling for (a) and (b) the cold-wall Mach 3 channel flow and (c) and (d) the adiabatic Mach 4 boundary layer.

Figure 12

Variation of (a) time-averaged velocity ($u/u_b$), (b) temperature ($T/T_b$), (c) turbulent shear stress ($\overline{u^{\prime }{v}^{\prime }}/u_b^2$), and (d) velocity in inner units ($u^{+}$) for different damping function scalings for the Mach 3 channel flow. The legend is as follows: , wall; , mixed; , semilocal and mixedmin; , mixed2 and mixedmin2; and $\diamond$, DNS of Coleman et al. [26].

Figure 13

Variation of $C_f$ in the zero-pressure-gradient region upstream of the compression corner for the different damping function scalings. The legend is as follows: , wall; , mixed, mixedmin, and mixedmin2; , semilocal; and $⊳$, SA RANS.

Figure 14

Variation of time-averaged (a) velocity ($u/u_{\infty }$), (b) density ($\rho /{\rho }_{\infty }$), (c) turbulent shear stress ($\overline{u^{\prime }{v}^{\prime }}/u_{\infty }^2$), and (d) velocity in inner units ($u^{+}$) for different damping function scalings for the Mach 2.85 boundary-layer flow at $x/{\delta }_{\text{in}}\approx 31$. The legend is as follows: , wall; , mixed, mixedmin, and mixedmin2; , semilocal; $⊳$, SA RANS; and $\diamond$, experiments of Dunagan et al. [54].