Communications in Mathematical Sciences

Volume 20 (2022)

Number 8

Heterogeneous multiscale methods for rough-wall laminar viscous flow

Pages: 2069 – 2106



Sean P. Carney (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Björn Engquist (Department of Mathematics and Oden Institute for Computational Engineering and Sciences, University of Texas, Austin, Tx., U.S.A.)


We develop numerical multiscale methods for viscous fluid flow over a rough boundary. The goal is to derive effective boundary conditions, or wall laws, through high resolution simulations localized to the boundary coupled to a coarser simulation in the domain interior following the framework of the heterogeneous multiscale method. Rigorous convergence of the coupled system is shown in a simplified setting. Numerical experiments illustrate the utility of the method for more general roughness patterns and far field flow conditions.


rough boundaries, multiscale methods, fluid dynamics

2010 Mathematics Subject Classification

65N22, 65N30, 76Dxx

Received 12 October 2021

Received revised 1 February 2022

Accepted 16 February 2022

Published 29 November 2022