\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.