443 | IGPM443.pdf January 2016 |

TITLE | Effective boundary conditions: a general strategy and application to compressible flows over rough boundaries |

AUTHORS | Giulia Deolmi, Wolfgang Dahmen, Siegfried Müller |

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 propose a “conceptual recipe” for the derivation of such effective boundary conditions first in a general setting of boundary value problems under the assumption of sufficient regularity to permit asymptotic expansions in terms of the micro-scale parameter. The proposed multiscale model relies then on an upscaling strategy based on homogenization techniques. It is similar in spirit to previous works by Achdou et al. [1], Jäger and Mikelic [29, 31], Friedmann et al. [24, 25] for incompressible fluids and Deolmi et al. [16, 17] for 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. 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 simulations on a fully resolved rough domain. |

KEYWORDS | homogenization, upscaling strategy, effective boundary conditions, Navier wall law, compressible flow |

DOI | 10.4208/cicp.OA-2016-0015 |

PUBLICATION | Communications in computational physics CiCP 21(2), 358-400 (2017) |

CORRESPONDING AUTHOR | Siegfried Müller |