Wall modeling for large-eddy simulation on non-body-conforming Cartesian grids

An approach of wall-modeled large-eddy simulation (WMLES) on non-body-conforming Cartesian grids is proposed. The proposed WMLES employs a partial-slip wall-boundary condition at the wall to reduce the numerical errors due to the insufficient grid resolution in the near-wall regions. Also, since the partial-slip wall-boundary condition reduces the velocity gradient in the near-wall regions, a modeled turbulent shear stress is introduced to maintain the balance of the total shear stress in the near-wall region. The proposed modeled stress augments only the shear-stress components and thus minimizes the adverse effects on the resolved turbulence. The proposed approach is crucial for obtaining the logarithmic law of the wall for the mean streamwise velocity correctly on non-body-conforming grids where the conservation laws are not strictly satisfied. The proposed WMLES is validated through the simulation of fully developed flat-plate turbulent boundary layers on non-body-conforming Cartesian grids. The mean streamwise velocity and Reynolds shear-stress profiles obtained by the proposed WMLES show good agreement with experimental data, while those obtained by the conventional nonslip wall-boundary conditions deviate far from the experimental data. The quantification analysis of the conservation errors indicates that the conservation errors are significantly reduced by introducing the partial-slip boundary condition and modeled turbulent shear stress. The proposed WMLES also shows robustness for various conditions, such as the cases with different plate angles relative to the grid lines and those with various Reynolds numbers.

Figure 1

Control volume and balance of wall-parallel momentum fluxes on (a) a body-conforming grid and (b) a non-body-conforming grid.

Figure 2

Concepts of wall modeling for non-body-conforming Cartesian grids: (a) partial-slip wall-boundary condition and (b) balance of total Cartesian grids.

Figure 3

Schematics of wall-boundary conditions for non-body-conforming Cartesian grids: (a) near-wall stencil settings and (b) image point location.

Figure 4

Definition of fluid volumes for conservation-error evaluations corresponding to (a) Eq. (12) and (b) Eq. (24): $\times$, flux definition points (regular points); $\times$, flux definition points (boundary points) $\text{\_}\text{\_}\text{\_}$, finite-volume interfaces; and $\text{\_}\text{.}\text{\_}\text{.}$, near-wall fluid-volume boundary ${\mathrm{\Gamma }}_G$ to quantify the conservation errors.

Figure 5

Computational grid (for $\alpha ={15}^{\circ }$, every fifth grid point is shown).

Figure 6

Effects of the proposed partial-slip wall-boundary condition and modeled shear stress (a) mean streamwise velocity in wall-viscous units and (b) Reynolds shear stress.: $\text{\_\_\_}$, case 15A; $\text{\_}\text{\_}\text{\_}\text{\_}$, case 15S; $\text{̲}\text{.}\text{̲}$, case 15N; and $\text{▫}$, wind-tunnel experiments [42]. The wind-tunnel experiment data shown in (a) and (b) are at ${\mathrm{Re}}_{\delta }=1.2\times {10}^5$ and ${\mathrm{Re}}_{\delta }=8.7\times {10}^5$, respectively. The vertical black dashed lines at $Y^{+}\approx 250$ and $Y/{\delta }_{99}\approx 0.07$ denote the matching heights of case 15A.

Figure 7

Shear-stress balance (5) in (a) case 15A and (b) case 15S. $\text{\_\_\_}$, resolved Reynolds shear stress $-\overline{\rho }\overline{U^{\prime }{V}^{\prime }}$; $\text{\_}\text{\_}\text{\_}\text{\_}$, modeled shear stress $\overline{{\tau }_{XY}^{\mathrm{model}}}$; $\text{̲}\text{.}\text{̲}$, viscous and SGS stress $(\mu +\overline{{\mu }_{t,\mathrm{SGS}}})(d\overline{U}/dY)$; and $\text{......}$, ideal profile of the modeled shear stress ${\overline{{\tau }_{XY}^{\mathrm{model}}}}^{\mathrm{ideal}}\equiv {\overline{\tau }}_w{f}_s$. The matching height $Y_{\mathrm{match}}/{\delta }_{99}$ is approximately 0.07 for both cases.

Figure 8

Comparisons between different plate angle cases for (a) mean streamwise velocity in wall-viscous units and (b) Reynolds shear stress: $\text{......}$, case 0A; $\text{\_\_\_}$, case 15A; $\text{\_}\text{\_}\text{\_}\text{\_}$, case 30A; $\text{̲}\text{.}\text{̲}$, case 40A; $\text{\_}\text{\_}$, case 45A; and $\text{▫}$, wind-tunnel experiments [42] as in Fig. 6. The vertical black dashed lines denote the matching heights of case 15A.

Figure 9

Comparisons between different Reynolds number cases for (a) mean streamwise velocity in wall-viscous units and (b) Reynolds shear stress: $\text{\_\_\_}$, case 15A (${\text{Re}}_{{\delta }_0}=7.5\times {10}^4$); $\text{̲}\text{.}\text{̲}$, case 15Aa (${\text{Re}}_{{\delta }_0}=1.5\times {10}^5$); $\text{\_}\text{\_}\text{\_}\text{\_}$, case 15Ab (${\text{Re}}_{{\delta }_0}=3.8\times {10}^5$); and $\text{▫}$, wind-tunnel experiments [42] as in Fig. 6. The vertical black dashed lines denote the matching heights of case 15A.

Figure 10

Skin friction as a function of Reynolds number based on momentum thickness. Results are shown for: $\text{▫}$, proposed WMLES (cases 0A, 15A, 30A, 40A, 45A, 15Aa, and 15Ab); $\text{▵}$, case 15N; $\text{▿}$, case 15S; $\_\_\_$, the KS empirical relation [47] $C_f={[17.08{log}_{10}{\left({\text{Re}}_{\theta }\right)}^2+25.11{log}_{10}\left({\text{Re}}_{\theta }\right)+6.012]}^{-1}$; $\text{\_}\text{\_}\text{\_}$, power law [48] $C_f=0.0135{\text{Re}}_{\theta }^{-1/6}$; and $\text{̲}\text{.}\text{̲}$, another power law [48] $C_f=0.0253{\text{Re}}_{\theta }^{-1/4}$. The gray shaded region represents the $\pm 15$% range from the KS relation.

Figure 11

Comparisons between different formulations of the modeled turbulent stress. (a) mean streamwise velocity in wall-viscous units and (b) wall-normal velocity fluctuations. $\_\_\_$, case 0A and $\text{\_}\text{\_}\text{\_}\text{\_}$, case 0B. The vertical black dashed lines denote the matching heights of case 0A.

Figure 12

Effects of the grid resolution on (a) mean streamwise velocity in wall-viscous units and (b) Reynolds shear stress: $\_\_\_$, $\mathrm{\Delta }x/{\delta }_0=1/40$ (case 15A); $\text{\_}\text{\_}\text{\_}\text{\_}$, $\mathrm{\Delta }x/{\delta }_0=1/20$; $\text{̲}\text{.}\text{̲}$, $\mathrm{\Delta }x/{\delta }_0=1/10$; and, $\text{▫}$, wind-tunnel experiments as in Fig. 6. The vertical black dashed lines denote the matching heights for the three cases.