High order multi-scale wall-laws, Part I: The periodic case

Bresch, Didier; Milisic, Vuk

doi:10.1090/S0033-569X-10-01135-0

Authors: Didier Bresch and Vuk Milisic
Journal: Quart. Appl. Math. 68 (2010), 229-253
MSC (2000): Primary 76D05, 35B27, 76Mxx, 65Mxx
DOI: https://doi.org/10.1090/S0033-569X-10-01135-0
Published electronically: March 10, 2010
MathSciNet review: 2663000
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Abstract: In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a Newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelić, W. Jäger, J. Diff. Eqs, 170, 96–122, (2001)] and [Y. Achdou et al, J. Comput. Phys., 147, 1, 187–218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to $\epsilon$ (the roughness’ thickness). We establish mathematically a poor convergence rate for averaged second order wall-laws as it was illustrated numerically for instance in [Y. Achdou, et al]. In comparison, we establish exponential error estimates in the case of an explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that their rate of convergence is at least of order $\epsilon ^{\frac {3}{2}}$. We provide a numerical assessment of the claims as well as a counterexample that makes evident the impossibility of an averaged second order wall-law. Our paper may be seen as the first one to derive efficient high order wall-laws boundary conditions.

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