Effective Boundary Conditions: A General Strategy and Application to Compressible Flows over Rough Boundaries (Communications in Computational Physics, 2018)
Determining the drag of a flow over a rough surface is a guiding example forthe need to take geometric micro-scale effects into account when computing a macro-scalequantity. A well-known strategy to avoid a prohibitively expensive numericalresolution of micro-scale structures is to capture the micro-scale effects through someeffective boundary conditions posed for a problem on a (virtually) smooth domain. Thecentral objective of this paper is to develop a numerical scheme for accurately capturingthe micro-scale effects at essentially the cost of twice solving a problem on a(piecewise) smooth domain at affordable resolution. Here and throughout the paper"smooth" means the absence of any micro-scale roughness. Our derivation is basedon a "conceptual recipe" formulated first in a simplified setting of boundary valueproblems under the assumption of sufficient local regularity to permit asymptotic expansionsin terms of the micro-scale parameter.The proposed multiscale model relies then on an upscaling strategy similar inspirit to previous works by Achdou et al. [1], Jäger and Mikelic [29, 31], Friedmannet al. [24, 25], for incompressible fluids. Extensions to compressible fluids, although withseveral noteworthy distinctions regarding e.g. the "micro-scale size" relative to boundarylayer thickness or the systematic treatment of different boundary conditions, arediscussed in Deolmi et al. [16,17]. For proof of concept the general strategy is applied tothe compressible Navier-Stokes equations to investigate steady, laminar, subsonic flowover a flat plate with partially embedded isotropic and anisotropic periodic roughnessimposing adiabatic and isothermal wall conditions, respectively. The results are comparedwith high resolution direct simulations on a fully resolved rough domain.