Physics and modeling of trailing-edge stall phenomena for wall-modeled large-eddy simulation

The wall-resolved large-eddy simulation (WRLES) database of the flow around the A-airfoil at the near-stall condition [K. Asada and S. Kawai, Phys. Fluids 30, 085103 (2018)] is analyzed to understand the mechanism of the boundary layer development and to examine the predictability of the trailing-edge stall phenomena using the wall-modeled LES (WMLES). The analysis based on the integral relation for the boundary layer indicates that the skin friction has dominant effects on the boundary layer development in the mild-adverse pressure gradient region ($x/c>0.6$). The effects of the skin friction accumulate along the airfoil upper surface, which determines the growth of the momentum thickness in the downstream and the consequent flow separation near the trailing edge. Therefore, this analysis indicates that the wall modeling in $x/c<0.6$ is important for the prediction of the stall phenomena, while that near and downstream of the separation location little affects the stall phenomena. Also, the budget analysis reveals that the eddy viscosity in the mild-adverse pressure gradient regions increases compared to that of the equilibrium boundary layer, which should be incorporated properly in the wall model. The same flowfield is simulated using the WMLES, and the results show an overall good agreement with the WRLES. The analysis based on the integral relation indicates that the WMLES can predict the stall phenomena with reasonable accuracy because the outer layer turbulence, whose effects are dominant near the separation point, is directly resolved in the WMLES. Despite the overall agreement with the WRLES, the WMLES results also suggest that potential issues remain in the underresolution near the leading edge and the eddy viscosity modeling of the wall model for flows with adverse pressure gradient.

Figure 1

Overview of the WRLES flowfield around A-airfoil at the near-stall condition (isosurfaces of the $Q$ criterion colored by the streamwise velocity, reproduced from Ref. [4]).

Figure 2

Statistical quantities of the boundary layer along the airfoil upper surface obtained by the WRLES (reproduced from Ref. [4]).

Figure 3

Integral relation for the boundary layer. (a) Validity of Eq. (1) ($\square$, left-hand side; $—$, right-hand side); (b) components of the right-hand side of Eq. (1) ($┄$, $I$; $—$, $II$); (c) term-by-term integrals [Eq. (2). $┄$, integral of $I$; $—$, integral of $II$; $\square$, $\theta$].

Figure 4

Budget of the wall-parallel momentum equation ($—$, nonequilibrium terms $C+P$; , viscous stress term $V$; , Reynolds stress term $R$; $┄$, residual). For the integrated budget [(c) and (d)], the total shear stress ${\mathcal{T}}_{\xi \eta }$ ($—$) is also shown.

Figure 5

Comparisons of the RANS eddy viscosity estimations [$—$, Eq. (8); $┄$, Eq. (9); , Eq. (10); $\square$, effective RANS eddy viscosity, Eq. (11), calculated by the WRLES data].

Figure 6

Computational grid for the WMLES. (a) Grid around the airfoil (every 10th grid points are shown). (b) Comparison of the chordwise grid spacing $\mathrm{\Delta }\xi$ along the airfoil surface ($—$, WMLES; $+$, WRLES [4]).

Figure 7

Wall normal grid resolution: (a) $┄$, grid spacing at the wall (using the right vertical axis); $—$, ${\eta }^{+}$ at the matching location (using the left vertical axis). (b) Matching height $h_{\mathrm{wm}}$ relative to the 99% boundary layer thickness ${\delta }_{99}$. The vertical dash-dotted line denotes the laminar-turbulent transition location ($x/c=0.14$).

Figure 8

Overall validity of the WMLES: (a) mean pressure coefficient and (b) skin friction coefficient ($—$, WMLES with the EQBL model; $┄$, WMLES with the NonEQBL model; $+$, WRLES [4]; $\square$, wind-tunnel experiment [45]).

Figure 9

Comparison of the probabilities of the flow reversal and negative skin friction ($—$, probability of flow reversal at the first WMLES grid point from the wall; $┄$, probability of negative skin friction obtained by the EQBL model).

Figure 10

Development of the boundary layer along the airfoil upper surface. Lines and symbols are as in Fig. 8.

Figure 11

Nondimensional velocity profile in the wall unit. The black dash-dotted line shows the log-law $u^{+}=log\left({\eta }^{+}\right)/0.41+5.1$, and the other lines and symbols are as in Fig. 8.

Figure 12

Total shear stress in the wall model. The lines and symbols are as in Fig. 8.

Figure 13

Effective eddy viscosity profile in the wall models. The lines and symbols are as in Fig. 8.

Figure 14

Integral relation using the WMLES results ($—$, $I$; $┄$, $II$. The WRLES result is also plotted with symbols ($▫$, $I$; $▵$, $II$).

Figure 15

Influences of the difference in skin friction near the leading edge on the WMLES. (a) momentum thickness and (b) skin friction coefficient near the trailing edge ($—$, Case A; $┄$, WMLES with EQBL model; $+$, WRLES).

Figure 16

Influences of the outer domain size on the mean pressure coefficient and skin friction coefficient along the airfoil [$┄$, WMLES (EQBL) with $r=160c$; $—$, WMLES (EQBL) with $r=20c$; $+$, WRLES [4] with $r=20c$].