Efficient spectral implementation of ODE wall model and the extension of integral wall model to unstructured LES solvers

Abstract

While wall modeling has become an indispensable tool for scale-resolving simulation of high Reynolds numbers wall-bounded turbulent flows, discussion of efficient implementation of wall models in parallel unstructured-grid solvers is lacking. In this paper, we discuss physical and numerical improvements to two zonal wall models based on integral form of the boundary layer differential equations within an unstructured-grid cell-centered finite-volume LES solver. The first model is a novel implementation of the ODE equilibrium wall model, where the velocity profile is expressed in the integral form using the constant shear-stress layer assumption and the integral is evaluated using a spectral quadrature method, resulting in a local and algebraic (grid-free) formulation. The second model is the modified form of the integral wall model of Yang et al. (2015) [14], which is based on the vertically-integrated thin-boundary-layer PDE along with a prescribed composite velocity profile in the wall-modeled region. A modification to the inner layer velocity profile which fixes the coordinate invariance issue of the original model is proposed. Additionally, several numerical challenges unique to the implementation of these integral models in unstructured mesh environments, such as the exchange of wall quantities between wall faces and LES cells, and the computation of surface gradients, are identified and possible remedies are proposed. The computational performance of the wall models is assessed both in a priori and a posteriori settings against the traditional finite-volume based ODE equilibrium wall model, showing a comparable computational cost for the integral wall model, and superior cost scaling for the spectral implementation over the finite-volume based approach. Load imbalance among the processors in parallel simulations seems to severely degrade the parallel efficiency of finite-volume based ODE wall model, whereas the spectral implementation is remarkably agnostic to these effects. The integral wall model, however, is even more prone to parallel efficiency degradation than the finite-volume based ODE wall model due to communication overhead among processors in highly parallelized settings.