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

Abstract

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 (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 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.

Relevance of the Slip Condition for Fluid Flows Near an Irregular Boundary

“…As discussed in [8], the Navier wall law is the best homogenized boundary condition: the boundary layer oscillations are O(ε 3/2 ) and thus prevent any improvement at Σ.…”

On Generalized Ventcel's Type Boundary Conditions for Laplace Operator in a Bounded Domain

“…We precisely address this question in the present work. To the best of our knowledge, this condition β > 0 appears for the first time in a recent work of D. Bresch and V. Milisic [6].…”

Rigorous Derivation of the Thin Film Approximation with Roughness-Induced Correctors

“…In 1827, Navier [21] was one of the first scientists to note that the roughness could drag a fluid. Since then, numerous studies attempted to prove mathematical results in this direction, see for instance the works of W. Jäger and A. Mikelic [18], Y. Amirat and co-authors [1,2] and more recently the works of D. Bresch and V. Milisic [10]. Note that all these works formulate the roughness using a periodic function (whose amplitude and period are supposed to be small).…”