Introduction

\label{sec:intro}

Computing wall-bounded turbulent flow at high Reynolds number using direct numerical simulation (DNS) and large eddy simulation (LES) is prohibitively expensive due to their severe grid-resolution requirements near the wall.\cite{Chap79,Choi12} Thus, various techniques of wall-modeled LES (WMLES) with coarse-grid resolution near the wall have been suggested so far.\cite{Pio02,Pio08} WMLES methodologies may be classified into three categories: non-zonal and zonal hybrid RANS/LES methods, and LES with wall shear stress model, where RANS denotes the Reynolds-averaged Navier-Stokes equation. A similar categorization was made by Piomelli.\cite{Pio08}

An example of the non-zonal hybrid RANS/LES approach is the detached eddy simulation (DES).\cite{Spalart97,Spalart09} A single turbulence model is adopted in DES and it acts as both RANS and LES closure models for the near-wall and detached regions, respectively, by adjusting the wall distance function. The use of single grid and single turbulence model is a valuable feature for its easy implementation to turbulent flow over various complex geometries.\cite{Forsythe04,Guilmineau11} In DES, the no-slip boundary condition is used at the wall and first off-wall grids should be located at \(y^+ = O(1)\), where \(y^+ = y u_{\tau}/\nu\), \(y\) is the wall-normal direction from the wall, \(u_{\tau}\) is the wall shear velocity, and \(\nu\) is the kinematic viscosity.

The second type of WMLES is the zonal hybrid RANS/LES method. In this approach, an LES with wall shear stress boundary condition at the wall is conducted with very coarse grid resolution near the wall, and the information of this wall shear stress is obtained by solving additional RANS only near the wall with no-slip boundary condition at the wall. The upper velocity boundary condition for RANS is in turn obtained from the instantaneous solution of LES. Balaras et al.\cite{Bala96} obtained good predictions of turbulent channel flow by solving the turbulent boundary layer equation in RANS region. Cabot and Moin\cite{Cabot99} further improved the prediction capability by introducing a dynamic procedure of determining a model coefficient \(\kappa\) (von Kármán constant) in RANS, and Wang and Moin\cite{Wang02} successfully applied it to a trailing edge flow. Recently, Kawai and Larsson\cite{Kawai12} analyzed numerical and subgrid-scale (SGS) errors at the near-wall grid points from LES,\cite{Cabot99,Nic01} and showed that locating the upper boundary for RANS farther away from the first off-wall grid of LES bypasses this error. Kawai and Larsson\cite{Kawai13} developed a new dynamic procedure for the determination of \(\kappa\) in RANS equation, and showed that it accurately predicts a turbulent boundary layer flow having shock/boundary-layer interaction.

The last type of WMLES is the LES with wall shear stress model.\cite{Sch75} In this approach, instantaneous wall shear stress (or slip velocity), instead of no slip, is used as the boundary condition at the wall\cite{Sch75} (or at slightly off-wall location,\cite{Chung09} respectively), and the grids used in this approach resolve only the outer layer (e.g. the wall-normal distance of the first off-wall grid is \(\Delta y^+ \gg O(10)\)). The key issue in this approach is how to supply the wall shear stress using outer layer information and thus various methods have been suggested so far. Schumann\cite{Sch75} obtained an instantaneous wall shear stress in the form of a normalized velocity fluctuation at the first off-wall grid point multiplied by the mean wall shear stress (the mean wall shear stress is directly computed from the prescribed mean pressure gradient for turbulent channel flow). Grötzbach\cite{Gro87} proposed a procedure to determine this mean shear stress from the log-law. By considering the inclined coherent structures in the near-wall region, Piomelli et al.\cite{Pio89} obtained the instantaneous wall shear stress from the downstream velocity fluctuations. Balaras et al.\cite{Bala95} showed that dynamic Smagorinsky model\cite{Germano91,Lilly92} results in better predictions than the Smagorinsky model with a constant model coefficient.\cite{Smago63} Later, a suboptimal control theory is applied to determine the optimal instantaneous wall shear stress which minimizes the error from a target velocity profile (log-law).\cite{Nic01,Temp06,Temp08} These approaches showed fairly good predictions of the logarithmic velocity profile in turbulent channel flow at relatively high Reynolds numbers, although the grids were not dense enough to resolve the near-wall region. On the other hand, Chung and Pullin\cite{Chung09} provided a slip velocity at a ‘virtual’ wall, which locates slightly off the wall (\(y/\delta = O(10^{-3})\), where \(\delta\) is the channel half height), by solving an auxiliary ordinary differential equation in conjunction with a log-like relation for the mean streamwise velocity, and successfully predicted turbulent channel flow. Inoue and Pullin\cite{Inoue11} applied the same approach to turbulent boundary layer flow up to extremely high Reynolds number, \(Re_{\theta} = O(10^{12})\). They successfully predicted various statistics including mean velocity, turbulence intensity, skin friction coefficient, shape factor, and their variations with the Reynolds number. Recently, Marusic et al.\cite{Marusic10} developed a wall model by which an instantaneous streamwise velocity near the wall is predicted using the velocity in the outer region obtained from experiments\cite{Marusic10,Mathis11} or a WMLES.\cite{Inoue12b} This wall model accurately predicted near-wall statistics in turbulent boundary layer flow.\cite{Marusic10,Mathis11,Inoue12b} LES based on the instantaneous wall shear stress boundary condition has been used in geophysical and environmental flows.\cite{Moeng84,Port00,Rad08} In these flows, the Reynolds number is very high and thus the use of wall shear stress boundary condition is highly demanding for faster computations. For LES of these flows, modified log-laws have been adopted to consider the effects of wall roughness, wall heat flux, etc.

As discussed above, LES based on the instantaneous wall shear stress boundary condition has predicted turbulent channel and boundary layer flows quite well. However, in this approach, the first off-wall grids locate far away from the wall (\(y^+ \gg O(10)\)) at very high Reynolds number, and thus the fluctuating component of wall shear stress should not be correlated well with the fluctuating streamwise velocity there.\cite{Bent12} Moreover, Piomelli and Balaras\cite{Pio02} noted that the near-wall grid size in WMLES is so large that the near-wall region can be treated in an averaged sense. Therefore, providing an accurate mean wall shear stress alone may be sufficient to deliver momentum to the velocity at the first off-wall grids. In the present study, we propose mean wall shear stress boundary condition at the wall for LES at high Reynolds number, and examine its performance in predicting turbulent channel and boundary layer flows without resolving near-wall region. The present numerical problems are very similar to what were investigated in Chung and Pullin\cite{Chung09} and Inoue and Pullin.\cite{Inoue11} However, the main difference is the use of mean wall shear stress as the wall boundary condition in our study, as opposed to instantaneous one. Therefore, another objective of the present study is to examine whether or not the fluctuating component of the wall shear stress is necessary for accurate predictions of wall-bounded flows using WMLES.

Mean wall shear stress boundary condition

\label{sec2}

For incompressible flow, the governing equations for LES are the filtered continuity and Navier-Stokes equations:

\[\label{cont} {\frac{{\partial}\bar{u}_{i}}{{\partial}x_{i}}=0},\]

\[\label{ns} {\frac{{\partial}\bar{u}_{i}}{{\partial}t} +\frac{{\partial}\bar{u}_{i}\bar{u}_{j}}{{\partial}x_{j}} = -\frac{{\partial}\bar{p}}{{\partial}x_{i}} +\frac{1}{Re}\frac{{\partial}^{2}\bar{u}_{i}}{{\partial}x_{j}{\partial}x_{j}} -\frac{{\partial}\tau_{ij}}{{\partial}x_{j}}},\]

where \(t\) is the non-dimensional time, \((x_{1},x_{2},x_{3})=(x,y,z)\) are the non-dimensional streamwise, wall-normal, and spanwise directions, respectively, \((u_1, u_2, u_3)=(u, v, w)\) are the corresponding non-dimensional velocity components, \(p\) is the non-dimensional pressure, \(Re = UL/\nu\), \(\nu\) is the kinematic viscosity, and \(\tau_{ij}\equiv\overline{u_{i}u_{j}}-\bar{u}_{i}\bar{u}_{j}\) is the SGS stress tensor. Here, \(\overline{(\cdot)}\) denotes the filtering operation for LES, and \(U\) and \(L\) are the characteristic velocity and length. The governing equations (\ref{cont})-(\ref{ns}) are solved using a semi-implicit fractional step method and finite volume method based on a staggered grid system: the Crank-Nicolson method and a third-order Runge-Kutta method are applied to the diffusion and convection terms, respectively.\cite{Spalart91} The second-order central difference scheme is used for the spatial discretization. The SGS stress tensor is determined using a dynamic global eddy viscosity model.\cite{Lee10,Park06} In particular, the eddy viscosity at the first grid cell above the wall is obtained using the extension method.\cite{Cabot99}

The instantaneous wall shear stress boundary condition for LES, originally proposed by Schumann,\cite{Sch75} is the following (for turbulent channel flow): \[\label{Sch1} \bar{\tau}_{xy,w}(x,z,t) = \frac{\bar{u}(x,y_1,z,t)}{\langle \bar{u} \rangle (y_1)}\langle \bar{\tau}_{xy,w} \rangle,\] \[\label{Sch2} \bar{v} = 0,\] \[\label{Sch3} \bar{\tau}_{zy,w}(x,z,t) = \frac{\bar{w}(x,y_1,z,t)}{\langle \bar{u} \rangle (y_1)}\langle \bar{\tau}_{xy,w} \rangle,\] where \(\bar{\tau}_{xy,w}\) and \(\bar{\tau}_{zy,w}\) are the filtered streamwise and spanwise components of wall shear stress, respectively, \(y_1\) is the wall-normal location of the first off-wall grid point, and \(\langle \cdot \rangle\) represents the mean value. Schumann\cite{Sch75} obtained mean wall shear stress from the momentum balance, i.e., \(\langle \bar{\tau}_{xy,w} \rangle = - h \langle \partial \bar{p}/ \partial x \rangle\), where \(h\) was the channel half height and \(\langle \partial \bar{p}/ \partial x \rangle\) was a priori given. Later, Grötzbach\cite{Gro87} calculated the wall shear stress from the log-law: \[\label{logvel_Gro} \frac{\langle \bar{u} \rangle(y_1)}{u_{\tau}} = \frac{1}{\kappa} \ln (\frac{y_1 u_{\tau} }{\nu}) + B,\] where \(u_{\tau} = \sqrt{\langle \bar{\tau}_{xy,w} \rangle / \rho}\), \(\kappa\) is the von Kármán constant, \(B\) is a constant, and \(\rho\) is the density. As mentioned in Sec. \ref{sec:intro}, we provide the mean (rather than instantaneous) wall shear stress as a boundary condition at the wall: \[\label{mws1} \bar{\tau}_{xy,w}(x,z) = \langle \bar{\tau}_{xy,w} \rangle,\] \[\bar{v}_w = \bar{w}_w = 0.\] The primary motivation for this is that, with very coarse-grid resolution near the wall, providing accurate mean wall shear stress at the wall is the most important for momentum transport to off-wall grid locations. Another motivation is that initial non-physical fluctuating components of wall shear stress may produce numerical instability during computation. Furthermore, the correlation between the wall shear stress and streamwise velocity fluctuations becomes very small for \(y^+ > 100\),\cite{Bent12} implying that fluctuating components of the wall shear stress do not make an important role in momentum transport to off-wall grid locations.

The mean wall shear stress \(\langle \bar{\tau}_{xy,w} \rangle\) in Eq. (\ref{mws1}) is iteratively determined (given \(y_j\), \(\langle \bar{u} \rangle(y_j)\), \(\kappa\), and \(B\)) at every time step based on a logarithmic velocity profile following Grötzbach:\cite{Gro87} \[\label{logvel} \frac{\langle \bar{u} \rangle(y_j)}{u_{\tau}} = \frac{1}{\kappa} \ln (\frac{y_j u_{\tau} }{\nu}) + B,\] where \(y_j\) is the wall-normal location of the \(j\)th off-wall grid point, and \(u_{\tau}=\sqrt{\langle\bar{\tau}_{xy,w} \rangle/\rho}\). Note that \(\langle \bar{\tau}_{xy,w} \rangle\) obtained is directly imposed in a streamwise momentum equation discretized for the first off-wall grid cell in a staggered grid system. Since we consider turbulent channel and boundary layer flows, \(\langle \cdot \rangle\) denotes the averaging over the streamwise and spanwise directions for channel flow and the spanwise direction for boundary layer flow, respectively. To prevent undesirable numerical and SGS modeling errors near the wall,\cite{Cabot99,Nic01,Kawai12} we adopt the third off-wall grid point as the wall-normal location of \(y_j\) in Eq. (\ref{logvel}) for all computations.

Results

\label{sec3}

Turbulent channel flow

In this section, we conduct LES of turbulent channel flow at high Reynolds numbers using the mean wall shear stress boundary condition. The Reynolds numbers considered are \(Re_{\tau} = u_{\tau} h / \nu = 2\times10^3 \sim 2\times10^8\), where \(h\) is the channel half height. Periodic boundary conditions are imposed in the streamwise and spanwise directions. The computational domain size is \(2\pi h (x) \times 2 h (y) \times 2 \pi h / 3(z)\). The number of grid points used is \(64 (x) \times 33 (y) \times 32 (z)\) and uniform grids are used for all directions. The present grid points per channel half height are \(n_x = h / \Delta x = 10.2\), \(n_y = h / \Delta y = 16\), and \(n_z = h / \Delta z = 15\), which are comparable to those of Chapman’s suggestion (\(n_x = 10\), \(n_y = 25\), and \(n_z = 10\))\cite{Chap79} and those used by other studies (\(n_x = 5 \sim 24\), \(n_y = 6 \sim 91\), and \(n_z = 10 \sim 48\)).\cite{Nic01,Chung09,Inoue11,Kawai12} The present grid spacings in wall units are much larger than those used in other LES with no-slip boundary condition: e.g., \(\Delta x^+ = 19635\), \(\Delta y^+ = 12590\), and \(\Delta z^+ = 6545\) for \(Re_{\tau} = 2\times10^5\). According to Nagib and Chauhan,\cite{Nagib08} we use \(\kappa = 0.37\) and \(B = 3.7\) in Eq. (\ref{logvel}) for determining \(u_{\tau}\) during computation. We also conducted LES with \(\kappa = 0.41\) and \(B = 5.2\),\cite{Cabot99,Nic01} but the results showed negligible differences from those of LES with \(\kappa = 0.37\) and \(B = 3.7\) (not shown in this paper).

\label{fig1}

Figure \ref{fig1} shows the mean streamwise velocity profile at \(Re_{\tau} = 2\times10^3\) when no-slip boundary condition is used with coarse-grid resolution. As shown, the velocity is highly overpredicted compared to the logarithmic profile and the result from DNS,\cite{Hoyas06} indicating that LES with no-slip boundary condition results in serious errors when near-wall region is not adequately resolved.

\label{fig2}

Figure \ref{fig2} shows the mean streamwise velocity profiles at \(Re_{\tau} = 2\times10^3\) and \(2\times10^4\) from the present LES with the mean wall shear stress boundary condition, together with those from coarse-grid DNS with the mean wall shear stress boundary condition, the logarithmic profile, and the result from DNS\cite{Hoyas06} at \(Re_{\tau} = 2\times10^3\). The results from present LES show very good agreements with DNS data\cite{Hoyas06} and the logarithmic profile. On the other hand, the coarse-grid DNS (i.e. without SGS model) fails to predict the logarithmic profile at \(Re_{\tau} = 2\times10^4\). This indicates that the SGS model makes an important role in the performance of present LES. It should be noted that the streamwise velocities predicted at the first and second off-wall (cell-center) grid points (\(y^+ = 62.5\) and 187.5 for \(Re_{\tau} = 2\times10^3\), and \(y^+ = 625\) and 1875 for \(Re_{\tau} = 2\times10^4\)) show deviations from the logarithmic profile. These behaviors are attributed to the fact that numerical and SGS modeling errors are prevalent at the first few off-wall grid points of LES.\cite{Cabot99,Nic01,Kawai12}

\label{fig3}

Kawai and Larsson\cite{Kawai12} showed that, for a zonal hybrid RANS/LES approach, one can bypass the numerical and SGS modeling errors at near-wall grid points of LES\cite{Cabot99,Nic01} by locating the upper boundary of RANS farther away from the wall. A similar technique should be applicable to the present LES approach. That is, the choice of \(y_j\) (Eq. (\ref{logvel})) with \(j > 1\) should improve the prediction capability. Figure \ref{fig3} shows the mean velocity profiles for various \(y_j\)’s. As shown, the results with \(y_j = y_1\) and \(y_2\) show some deviations from DNS data,\cite{Hoyas06} but those with \(y_j = y_3\) and \(y_4\) are nearly identical and agree well with DNS data.\cite{Hoyas06} These behaviors are consistent with the observations made by Kawai and Larsson\cite{Kawai12} in their zonal hybrid RANS/LES approach.

\label{fig4}

In Fig. \ref{fig4}, the root-mean-square (rms) velocity fluctuations and Reynolds shear stress at \(Re_{\tau} = 2\times10^3\) are shown, together with those from DNS.\cite{Hoyas06} The peak value of \(u^+_{rms}\) located at \(y^+ \simeq 15\) from DNS,\cite{Hoyas06} is not captured by the present LES because the present grids do not resolve near-wall region. Nevertheless, the turbulence statistics away from the wall are in good agreements with DNS data.\cite{Hoyas06}

\label{fig5}

We perform LES using the mean wall shear stress boundary condition up to \(Re_{\tau} = 2\times10^8\), and the corresponding mean velocity profiles are shown in Fig. \ref{fig5}. In this figure, the velocity at the first off-wall grid point is overpredicted compared to the logarithmic profile due to the numerical and SGS modeling errors there. Except the first grid point, the predicted mean velocity follows the log-law very well. This indicates that the logarithmic profile of the mean velocity at very high Reynolds numbers can be successfully predicted even with very coarse grid resolution near the wall, when the mean wall shear stress boundary condition is used at the wall. There is no experimental data available at these Reynolds numbers to compare with our simulation result, but as is shown in next section we compare our high Reynolds number simulation data with existing experimental data for turbulent boundary layer flow.

To examine if the overprediction of mean streamwise velocity at the first off-wall grid point comes from the second-order accurate discretization method used, we performed another LES using a fourth-order central difference scheme for the convection term. The result showed that the overprediction at the first off-wall grid point still exists, indicating that the use of a higher-order discretization scheme would not completely bypass this error.

Turbulent boundary layer flow

\label{fig6}

\label{fig7}

In this section, we perform LES of turbulent boundary layer flow with the mean wall shear stress boundary condition. The Reynolds numbers considered are \(Re_{\theta_{0}} = U_{\infty} {\theta_{0}} / \nu = 1.5\times10^4 \sim 10^7\), where \(U_{\infty}\) is the free-stream velocity and \(\theta_{0}\) is the momentum thickness at the computational inlet. The computational domain size is \(600\theta_{0} (x) \times 40\theta_{0} (y) \times 50\theta_{0} (z)\). The number of grid points used is \(769 (x) \times 65 (y) \times 64 (z)\) and uniform grids are used in all directions. Like the case of turbulent channel flow, the grid points per inlet boundary layer thickness \(\delta_0\) are \(n_x = \delta_0 / \Delta x \simeq 13\), \(n_y = \delta_0 / \Delta y \simeq 17\), and \(n_z = \delta_0 / \Delta z \simeq 13\), which are comparable to those of Chapman’s suggestion\cite{Chap79} and those used by other studies.\cite{Nic01,Chung09,Inoue11,Kawai12} Also, the grid resolution is very coarse: e.g., \(\Delta x^+ = 2974\), \(\Delta y^+ = 2379\), and \(\Delta z^+ = 2974\) for \(Re_{\theta_{0}} = 1.3\times10^5\). The rescaling method proposed by Lund et al.\cite{Lund98} is used to provide the turbulence inflow data. The recycling location is placed \(480\theta_0\) downstream of the inlet. The boundary conditions at the free-stream are given by \(u = U_{\infty}\) and \({\partial}{v}/{\partial}{y} = {\partial}{w}/{\partial}{y} = 0\), and the periodic boundary condition is imposed in the spanwise direction. A convective outflow boundary condition is used at the exit, \(\partial u_i / \partial t + c\partial u_i / \partial x = 0\), where \(c\) is the plane-averaged velocity at the exit. We use \(\kappa = 0.384\) and \(B =4.17\)\cite{Nagib08} in Eq. (\ref{logvel}) for determining \(u_{\tau}\) during computation.

Figure \ref{fig6} shows the mean streamwise velocity profile at \(Re_{\theta} = 2.026\times10^4\) obtained from present LES, together with the log-law profile\cite{Nagib08} and previous experimental data.\cite{Osterlund99} We also performed LES with no-slip boundary condition, which led to numerical instability and eventually diverged. This is because an inaccurate skin friction resulted from under-resolved near-wall region and no-slip boundary condition was recursively used in the rescaling procedure for obtaining inflow turbulence data. On the other hand, the result from present LES is in excellent agreement with the logarithmic profile and experimental data.\cite{Osterlund99} We perform LES at higher Reynolds numbers up to \(Re_{\theta_{0}} = 10^7\) and the results are shown in Fig. \ref{fig7}, together with those from previous experiments.\cite{Osterlund99,Oweis10} The results from present LES show excellent agreements with previous experimental data.\cite{Osterlund99,Oweis10} In Figs. \ref{fig6} and \ref{fig7}, as observed in LES of channel flow, the velocity at the first off-wall grid point deviates from the logarithmic profile, which is due to the numerical and SGS modeling errors near the wall.\cite{Cabot99,Nic01,Kawai12}

\label{fig8}

\label{fig9}

The rms velocity fluctuations and Reynolds shear stress at \(Re_{\theta} = 1.560\times10^5\) are shown in Fig. \ref{fig8}. As shown, the results from present LES agree very well with previous experimental data.\cite{Winkel12} Figure \ref{fig9} shows the rms streamwise velocity fluctuations at \(Re_{\theta} = O(10^4) \sim O(10^7)\), together with previous experimental data.\cite{Mathis11,Winkel12} The rms streamwise velocity fluctuations from present LES are in good agreement with those of previous experimental data.\cite{Mathis11,Winkel12} Note that, unlike present LES, rms streamwise velocity fluctuations from the experiment by Winkel et al.\cite{Winkel12} are not zero outside the boundary layer, which may be attributed to non-negligible free-stream turbulence intensity in their experiment. It should be metioned that \(u^+_{rms}\) becomes larger at higher Reynolds number, which was also shown at a lower Reynolds number range (\(Re_{\theta} = 1430 \sim 31000\)) by De Graaff and Eaton.\cite{Graaff00} This may indicate that the magnitude of rms streamwise velocity fluctuations depends on the Reynolds number even at a very high Reynolds number range.

\label{fig10}

\label{fig11}

Figure \ref{fig10} shows the variation of \(U_{\infty}/u_{\tau}\) with \(Re_{\theta}\) from present LES, together with previous experimental data\cite{Osterlund99,Oweis10} and the ‘Coles-Fernholz 2’ relation, (\(U_{\infty} / u_{\tau} = (1 / \kappa) \ln(Re_{\theta})+C\); \(\kappa=0.384\) and \(C=4.127\)).\cite{Nagib07} As shown in this figure, the wall shear velocity \(u_{\tau}\) is well captured by present LES and agrees well with the previous experimental data and the Coles-Fernholz 2 relation.\cite{Nagib07} Nagib et al.\cite{Nagib07} also suggested the Reynolds number dependence of the shape factor, \(H=\delta^{*}/\theta\), as follows: \[\label{shapefactor} H=\frac{1}{1-(C^{'}/U_{\infty}^{+})},\] where \(C^{'}=7.135+O(1/Re_{\theta})\) and \(\delta^{*}\) is the displacement thickness. The shape factor from present LES is shown in Fig. \ref{fig11}, together with previous experimental data\cite{Osterlund99,Oweis10} and Eq. (\ref{shapefactor}). Nagib et al.\cite{Nagib07} mentioned that the shape factor decreases with increasing \(Re_{\theta}\) and does not converge to the traditional value of 1.3, which is also observed from the present results.

Conclusions

\label{sec4}

In the present study, we proposed the mean (rather than instantaneous) wall shear stress as the wall boundary condition for LES without dense grid resolution near the wall, and applied it to turbulent channel and boundary layer flows up to \(Re_{\tau} = O(10^8)\) and \(Re_{\theta} = O(10^7)\), respectively. The mean wall shear stress was determined at every time step from the log-law. It was shown that the present LES with very coarse grid resolution accurately predicted the logarithmic velocity profile and low-order turbulence statistics at high Reynolds numbers. This indicated that the mean component of the wall shear stress alone (as the wall boundary condition) is good enough to predict the flow in terms of low-order turbulence statistics even if the first off-wall grid locates far away from the wall. In other words, the fluctuating component of the wall shear stress is not an essential ingredient for the success of WMLES of wall-bounded flow. This certainly goes against the previous and current efforts to provide physically accurate (or mathematically devised) fluctuating components of the wall shear stress\cite{Sch75,Pio89,Nic01} and near-wall streamwise velocity\cite{Chung09,Marusic10,Inoue11} for WMLES. On the other hand, we also note that the present approach based on the mean wall shear stress supports the hypothesis by Piomelli and Balaras\cite{Pio02} that near-wall region in WMLES may be treated in an averaged sense.

Providing the mean wall shear stress boundary condition is a simple and numerically stable way, especially for LES of complex turbulent flow at high Reynolds number. However, in complex flow, the log-law used in the present study may not be valid. At present, we consider flow over a backward-facing step where flow separation, reattachment, and redeveloping flow exist, and apply the present approach (but keeping the log-law) to determine local mean wall shear stress. Our preliminary study showed that the present approach accurately predicts the flow statistics such as the mean velocity profile and the skin friction distribution behind the step.\cite{Cho12} Another aspect to consider is that for three-dimensionally inhomogeneous flow, spatial averaging for obtaining local mean wall shear stress is not allowed. In this case, a Lagrangian approach (temporal averaging) may have to be used to get local mean wall shear stress. This line of research to determine local mean wall shear stress for three-dimensionally complex turbulent flow is currently under way.

We acknowledge the supports by NRF, MEST, Korea [No. 20120008740, No. R312012000100830, and No. 2012055647].

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