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Effective Boundary Conditions: A General Strategy and Application to Compressible Flows Over Rough Boundaries

Part of: Homogenization, determination of effective properties General aerodynamics and subsonic flows Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  07 February 2017

Giulia Deolmi ,
Wolfgang Dahmen  and
Siegfried Müller
Giulia Deolmi*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Wolfgang Dahmen*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Siegfried Müller*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
*
*Corresponding author.Email addresses:deolmi@igpm.rwth-aachen.de (G. Deolmi), dahmen@igpm.rwth-aachen.de (W. Dahmen), mueller@igpm.rwth-aachen.de (S. Müller)
*Corresponding author.Email addresses:deolmi@igpm.rwth-aachen.de (G. Deolmi), dahmen@igpm.rwth-aachen.de (W. Dahmen), mueller@igpm.rwth-aachen.de (S. Müller)
*Corresponding author.Email addresses:deolmi@igpm.rwth-aachen.de (G. Deolmi), dahmen@igpm.rwth-aachen.de (W. Dahmen), mueller@igpm.rwth-aachen.de (S. Müller)
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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.

Keywords

Homogenizationupscaling strategyeffective boundary conditionsNavier wall lawcompressible flow

MSC classification

Secondary: 74Q15: Effective constitutive equations 76G25: General aerodynamics and subsonic flows 35Q30: Navier-Stokes equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 358 - 400
Copyright
Copyright © Global-Science Press 2017 

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