Commun. Comput. Phys. doi: 10.4208/cicp.OA-2016-0015 Vol. 21, No. 2, pp. 358-400 February 2017 Effective Boundary Conditions: A General Strategy and Application to Compressible Flows Over Rough Boundaries Giulia Deolmi, Wolfgang Dahmen and Siegfried M¨ uller ∗ Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany. Received 25 January 2016; Accepted (in revised version) 26 May 2016 Abstract. Determining the drag of a flow over a rough surface is a guiding example for the need to take geometric micro-scale effects into account when computing a macro- scale quantity. A well-known strategy to avoid a prohibitively expensive numerical resolution of micro-scale structures is to capture the micro-scale effects through some effective boundary conditions posed for a problem on a (virtually) smooth domain. The central objective of this paper is to develop a numerical scheme for accurately cap- turing the 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 based on a “conceptual recipe” formulated first in a simplified setting of boundary value problems under the assumption of sufficient local regularity to permit asymptotic ex- pansions in terms of the micro-scale parameter. The proposed multiscale model relies then on an upscaling strategy similar in spirit to previous works by Achdou et al. [1], J¨ ager and Mikelic [29, 31], Friedmann et al. [24, 25], for incompressible fluids. Extensions to compressible fluids, although with several noteworthy distinctions regarding e.g. the “micro-scale size” relative to bound- ary layer thickness or the systematic treatment of different boundary conditions, are discussed in Deolmi et al. [16,17]. For proof of concept the general strategy is applied to the compressible Navier-Stokes equations to investigate steady, laminar, subsonic flow over a flat plate with partially embedded isotropic and anisotropic periodic roughness imposing adiabatic and isothermal wall conditions, respectively. The results are com- pared with high resolution direct simulations on a fully resolved rough domain. AMS subject classifications: 74Q15, 76G25, 35Q30 Key words: Homogenization, upscaling strategy, effective boundary conditions, Navier wall law, compressible flow. ∗ Corresponding author. Email addresses: deolmi@igpm.rwth-aachen.de (G. Deolmi), dahmen@igpm.rwth-aachen.de (W. Dahmen), mueller@igpm.rwth-aachen.de (S. M ¨ uller) http://www.global-sci.com/ 358 c 2017 Global-Science Press