Wall-modeled large-eddy simulation based on spectral-element discretization

TL;DR

The paper assesses wall-stress WMLES using a spectral-element discretization implemented in Nek5000, combining algebraic wall models with explicit SGS modeling (Vreman and Sigma) and exploring Neumann and viscosity-based boundary conditions. It demonstrates state-of-the-art accuracy for channel flow and flat-plate TBL while highlighting SEM-specific issues such as jumps in derivatives across element boundaries and imperfect global momentum balance on coarse grids. The results show that a Neumann boundary condition with explicit SGS modeling yields reliable wall-stress predictions and outer-layer statistics, whereas the viscosity-based boundary condition can compromise momentum balance on coarse meshes. Overall, SEM-based WMLES emerges as a competitive, massively-parallel framework for high-order WMLES, with open-source implementation and clear avenues for further stabilization and accuracy improvements.

Abstract

This article analyses the simulation methodology for wall-modeled large-eddy simulations using solvers based on the spectral-element method (SEM). To that end, algebraic wall modeling is implemented in the popular SEM solver Nek5000. It is combined with explicit subgrid-scale (SGS) modeling, which is shown to perform better than the high-frequency filtering traditionally used with the SEM. In particular, the Vreman model exhibits a good balance in terms stabilizing the simulations, yet retaining good resolution of the turbulent scales. Some difficulties associated with SEM simulations on relatively coarse grids are also revealed: jumps in derivatives across element boundaries, lack of convergence for weakly formulated boundary conditions, and the necessity for the SGS model as a damper for high-frequency modes. In spite of these, state-of-the-art accuracy is achieved for turbulent channel flow and flat-plate turbulent boundary layer flow cases, proving the SEM to be a an excellent numerical framework for massively-parallel high-order WMLES.

Figures (21)

• Figure 1: Solid lines: Mean velocity profiles obtained in channel flow simulations, using the Neumann boundary condition. Dotted lines: The relative errors in the profiles with respect to the reference DNS, in the outer layer.
• Figure 2: Solid lines: Reynolds stress profiles obtained in channel flow simulations, using the Neumann boundary condition. Dotted lines: The relative errors in the profiles with respect to the reference DNS.
• Figure 3: Left: Inner-scaled mean velocity profiles obtained in channel flow simulations, using the Neumann boundary condition. Right: The profile of the indicator function $\Phi$ from the same simulations.
• Figure 4: Left: The dependency of $\epsilon[\langle |\tau^{apri}_w| \rangle]$ and $\epsilon[|\langle \tau^{apri}_{w} \rangle|]$ on $\Delta t_\mathrm{avrg}$. Right: The temporal power spectral density of the streamwise velocity fluctuations at $y=h$.
• Figure 5: Errors in the mean wall shear stress and streamwise velocity as a function of the time averaging length of the sampled velocity signal.