Homogenization of a Large Eddy Simulation Model for Turbulent Fluid Motion Near a Rough Wall

Abstract

In this paper, we are interested in the numerical simulation of a turbulent viscous flow in the vicinity of a periodic rough wall. To this aim, we consider a large eddy simulation (LES) model, and perform its asymptotic analysis, letting the period $\varepsilon$ of the roughness tend to zero. Relying on an homogenization method closely related to the two-scale convergence method (Allaire in SIAM J Math Anal 23(6):1482–1518, 1992; Nguetseng in SIAM J Math Anal 20(3):608–623, 1989), we exhibit a critical scaling associated with parameter $\lambda :=\lim _{\varepsilon \rightarrow 0}{a_\varepsilon }/{\varepsilon ^{5/3}}$, where $a_\varepsilon$ stands for the amplitude of the roughness. In the critical regime $0<\lambda <\infty$, the roughness effect is described by an additional, nonlinear friction term, that depends locally on the solution to a nonlinear auxiliary problem of boundary layer type. In the particular case of riblets, which is of major practical interest, we show that the directional invariance of the geometry results in a simplified expression of this extra term, and we propose a numerical method to simulate the homogenized LES model, based on that observation. Finally, we study the influence of parameter $\lambda$ on the flow approximated by the homogenized model, using different commonly used riblet shapes. Our simulations demonstrate that the extra friction term can be handled numerically, and may have an important influence on the results of LES in presence of roughness.