Effective Boundary Conditions: A General Strategy and Application to Compressible Flows Over Rough Boundaries. (7th February 2017)

• Record Type: Journal Article
• Authors: Deolmi, Giulia; Dahmen, Wolfgang; Müller, Siegfried
• Is Part Of: Communications in computational physics. Volume 21:Number 2(2017:Feb.)
• Page Start: 358
• Page End: 400
• Publication Date: 2017-02-07
• Subjects: 74Q15, -- 76G25, -- 35Q30; Homogenization, -- upscaling strategy, -- effective boundary conditions, -- Navier wall law, -- compressible flow; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.OA-2016-0015
• Languages: English
• ISSNs: 1815-2406

Abstract

Abstract: Determining the drag of a flowover a rough surface is a guiding example for the need to take geometric micro-scale effects into account when computing a macroscale 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 capturing 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 expansions 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], Jäger 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 boundary 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 compared with high resolution direct simulations on a fully resolved rough domain.