Unified wall-resolved and wall-modeled method for large-eddy simulations of compressible wall-bounded flows

We present a general strategy to unify wall-resolved and wall-modeled large-eddy simulation (LES) approaches for turbulent wall-bounded compressible flows. The proposed technique allows one to impose the proper wall stress and heat flux, preserving the no-slip and the isothermal and adiabatic conditions for the velocity and temperature fields, respectively. The approach results in a minimal intrusive algorithm that automatically switches between wall-resolved and wall-modeled LES according to the local near-wall resolution. The methodology is discussed and implemented in a flow solver based on high-order finite difference schemes, the application of which in the context of wall-modeled LES has been less explored in the available literature. Numerical simulations of canonical turbulent channel flow and spatially evolving boundary layer are performed in a wide range of Mach and Reynolds numbers. The results highlight the ability of the present method to accurately reproduce the outer layer turbulent dynamics, with a minimal influence of the near-wall grid resolution. In particular, velocity statistics and two-point spatial correlations are in good agreement with reference direct numerical simulation and wall-resolved LES, confirming the potential of the proposed approach for predictive analysis of wall-bounded flows at high-Reynolds number.

Figure 1

Sketch of representative cases for the location of the wall-model–LES interface: (a) fully wall-resolved case, (b) mixed wall-resolved and wall-modeled case, and (c) fully wall-modeled case.

Figure 2

WRLES of turbulent channel flow at $M_b=0.1$ and ${\text{Re}}_{\tau }=590$. Distribution of (a) mean velocity profiles and (b) Reynolds stresses as a function of $y^{+}$, compared with reference DNS data by Vreman and Kuerten [46] (gray circles).

Figure 3

Mean streamwise velocity profiles $u^{+}=\tilde{u}/u_{\tau }$ for channel flow cases at $M_b=0.1$ and various Reynolds numbers. Present results (symbols) are compared with DNS data from Vreman and Kuerten [46], Bernardini et al. [47], and Lee and Moser [48] (solid lines).

Figure 4

Density scaled Reynolds stress components $\overline{\rho }\widetilde{u_i^{\prime \prime }{u}_j^{\prime \prime }}/{\tau }_w$ as a function of the inner scaled wall distance $y^{+}$ for channel flows at $M_b=0.1$ and different Reynolds numbers. Present results (symbols) are compared with DNS data from Vreman and Kuerten [46], Bernardini et al. [47], and Lee and Moser [48] (solid lines).

Figure 5

Mean velocity profiles (a) and Reynolds stresses (b) from unified WR and WMLES simulations of turbulent channel flow at $M_b=0.1$ and ${\text{Re}}_{\tau }=590$, compared with incompressible DNS data of Vreman and Kuerten [46].

Figure 6

Contours of instantaneous streamwise velocity in wall-parallel planes for unified WR and WMLES simulations of the turbulent channel at ${\text{Re}}_{\tau }=590$ and $M_b=0.1$.

Figure 7

Contours of the two-point spanwise correlation as a function of the spanwise shift and wall-normal location for unified WR and WMLES simulations of the turbulent channel at ${\text{Re}}_{\tau }=590$ and $M_b=0.1$. Solid lines denote WRLES data.

Figure 8

Effect of the wall-parallel resolution in unified WR and WMLES simulations of turbulent channel flow. Figures 8 and 8 show mean velocity profiles as a function of $y^{+}$ with progressively coarser grid in the span- and streamwise direction, respectively. Simulation details are reported in Table 1, cases 1000c–1000e and 1000f–1000h. DNS results refer to Bernardini et al. [47].

Figure 9

Mean streamwise velocity profile (a) and mean temperature (b) as a function of $y^{+}$ for channel flow at $M_b=1.5$ and ${\text{Re}}_{\tau }=500$. Blue circles indicate the solution of the modeled region, red circles indicate the resolved LES region, while black solid lines denote reference DNS data by Modesti and Pirozzoli [44].

Figure 10

WRLES of the supersonic boundary layer at $M_{\infty }=2$ and ${\text{Re}}_{\tau }=250$. Distribution of (a) mean velocity profile and (b) density-scaled Reynolds stresses as a function of $y^{+}$, compared with reference DNS data by Vreman and Kuerten [46] (gray circles).

Figure 11

Mean streamwise velocity profiles (top) and density scaled Reynolds stress components (bottom) from WMLES of the spatially developing turbulent boundary layer at $M_{\infty }=2$ and various friction Reynolds numbers. Present results are compared with DNS data by Pirozzoli and Bernardini [35].

Figure 12

Distribution of the skin friction coefficient as function of the incompressible momentum thickness Reynolds number from WMLES of the supersonic boundary layer at $M_{\infty }=2$, compared with the Kármán-Schoenherr ($\mathrm{K}\text{−}\mathrm{S}$) empirical correlation.

Figure 13

Unified WR and WMLES computations of the supersonic turbulent boundary layer at $M_{\infty }=2$ and ${\text{Re}}_{\tau }=250$. Mean streamwise velocity profile (a) and density scaled Reynolds stress components (b) as a function of $y^{+}$. Simulation details are reported in Table 2, cases 250b–250f. DNS results refer to Pirozzoli and Bernardini [35].