Wavelet-based adaptive wall-modeled large eddy simulation method for compressible turbulent flows

A wavelet-based adaptive wall-modeled large eddy simulation (WA-WMLES) method is proposed for simulations of wall-bounded compressible turbulent flows. The approach utilizes the wavelet-based adaptive large eddy simulation (WA-LES), incorporated into the anisotropic-adaptive wavelet collocation method, to resolve the outer region of turbulent boundary layer, while the inner part is approximated by the equilibrium wall-shear-stress model. Such an approach for modeling the inner layer is crucial for wavelet-based adaptive turbulent flow simulations because the mesh resolution requirement for WA-LES to resolve inner viscous sublayer becomes computationally prohibitively expensive as the Reynolds number increases. In the outer layer region WA-LES computations take advantage of the wavelet-based local mesh refinement, which not only efficiently captures the physical characteristics of flows on a nearly optimal adaptive computational mesh, e.g., massive boundary layer separation, but also actively controls the error of the solution using a priori defined wavelet filtering threshold. A flat plate turbulent boundary layer flow and a separated flow over NASA's wall-mounted hump are tested to verify and validate the WA-WMLES approach. Good agreement of the results predicted by the WA-WMLES method is achieved compared to the reference data from experiments and simulations. The finest effective mesh resolution of the WA-WMLES is consistently higher than the one used in the wall-modeled LES (WMLES) found in literature, but comparable to the wall-resolved LES, while the similar accuracy is achieved with considerably fewer degrees of freedom than in nonadaptive WMLES. These observations demonstrate both accuracy and efficiency of the WA-WMLES method.

Figure 1

An illustration depicting the nonorthogonal LES mesh, the closest LES mesh points (blue) to the exchange locations, and the boundary points for the RANS ODEs on the wall (black) and the exchange layer (red).

Figure 2

Wall-normal profiles at a streamwise location $x/\delta =24$ downstream of the inlet of the flat boundary layer case with uniform and zonal variable $\epsilon$ for the WA-WMLES method. (a) Wavelet threshold $\epsilon$ for momentum and total energy. (b) Time and spanwise averaged $u_{\mathrm{rms}}^{\prime \prime }$. Comparison is made with the data reported by Ref. [40].

Figure 3

Result comparison of the flat plate turbulent boundary layer case with uniform and different zonal variable $\epsilon$ for the WA-WMLES method. (a) Time and spanwise averaged skin friction distributions. (b) Time and spanwise averaged streamwise velocity profiles in the inner-layer scale.

Figure 4

The adaptive mesh colored with levels of resolution for the subsonic flat plate turbulent boundary layer flow. Axis labels are in the unit of the boundary layer thickness at the inflow.

Figure 5

The instantaneous $Q$-criterion isosurface colored by instantaneous spanwise momentum for the subsonic flat plate turbulent boundary layer flow. Axis labels are in the unit of the boundary layer thickness at the inflow.

Figure 6

Zoomed-in view of a slice in the streamwise and wall-normal plane of the adaptive mesh with the streamwise momentum contours for the subsonic flat plate turbulent boundary layer flow in the background. Axis labels are in the unit of the boundary layer thickness at the inflow.

Figure 7

The mean velocity profile and the turbulence statistics at a streamwise location $x/\delta =24$ downstream of the inlet. Comparisons are made with the results of the inflow flat plate turbulent boundary layer in Ref. [40]. (a) Time and spanwise averaged streamwise velocity. (b) Time and spanwise averaged streamwise velocity in wall units. (c) Time and spanwise averaged turbulence statistics. Symbols denote the data reported in Ref. [40], while lines denote the results of the current WA-WMLES method.

Figure 8

Time and spanwise averaged streamwise velocity profile at the inflow plane $x/c=-2.14$ for the hump flow compared with the RANS data from the $k\text{−}\omega$ SST model-based RANS computation using CFL3D.

Figure 9

The adaptive mesh of the WA-WMLES computation colored with levels of resolution for the hump flow.

Figure 10

The instantaneous $Q$-criterion isosurface colored by instantaneous streamwise momentum for the hump flow.

Figure 11

The instantaneous skin friction coefficient on the streamwise and spanwise plane for the hump flow.

Figure 12

The instantaneous streamwise momentum on the streamwise and wall normal plane for the hump flow.

Figure 13

The time and spanwise averaged pressure on the streamwise and wall normal plane for the hump flow.

Figure 14

The time and spanwise averaged streamwise momentum on the streamwise and wall normal plane for the hump flow.

Figure 15

Time and spanwise averaged skin friction and pressure coefficients over the wall for the hump flow. Comparisons are made with the wall-resolved LES data [48], nonadaptive WMLES results [40, 50], the RANS data using the $k\text{−}\omega$ SST model by CFL3D, i.e., the initial condition of WA-WMLES, and the experimental data [43]. (a) Skin friction coefficient $C_f$. (b) Pressure coefficient $C_p$.

Figure 16

Time and spanwise averaged streamwise velocity profile at $x/c=-0.81$ for the hump flow compared with the nonadaptive WMLES case [40] (a) and the velocity profile, scaled by wall units, compared with the RANS data using the $k\text{−}\omega$ SST model by CFL3D(b). (a) Velocity profile scaled by the free-stream velocity. (b) Velocity profile scaled by the wall units.

Figure 17

Time and spanwise averaged profiles (lines) of turbulence fluctuation statistics at $x/c=-0.81$ for the hump flow compared with the nonadaptive WMLES case [40] (symbols).

Figure 18

Time and spanwise averaged velocity profiles at different streamwise locations. The experimental data [43] and those of the nonadaptive WMLES [40] are shown for comparison. (a) Streamwise velocity. (b) Vertical velocity.

Figure 19

Time and spanwise averaged turbulent stress profiles at different streamwise locations. The experimental data [43] and those of the nonadaptive WMLES [40] are shown for comparison. (a) $\left\{u^{\prime \prime }{u}^{\prime \prime }\right\}$, (b) $\left\{v^{\prime \prime }{v}^{\prime \prime }\right\}$, and (c) $\left\{u^{\prime \prime }{v}^{\prime \prime }\right\}$.