2 00 7 High Order Multi - Scale Wall - Laws , Part I : the Periodic Case

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., we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to ǫ (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. In comparison, we establish exponential error estimates in the case of explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that its rate of convergence is at least of order ǫ 3 2. We provide a numerical assessment of the claims as well as a counterexample that evidences the impossibility of an averaged second order wall-law. Our paper may be seen as the first stone to derive efficient high order wall-laws boundary conditions. 1. Introduction. The main goal of wall-laws is to remove the stiff part from boundary layers, replacing the classical no-slip boundary condition by a more sophisticated relation between the variables and their derivatives. They are extensively used in numerical simulations to eliminate regions of strong gradients or regions of complex geometry (rough boundaries) from the domain of computation. Depending on the field of applications, (porous media, fluid mechanics, heat transfer, electro-magnetism), wall-laws may be called Beavers-Joseph, Saffman-Joseph, Navier, Fourier, Leontovitch type laws. High order effective macroscopic boundary conditions may also be proposed if we choose a higher degree ansatz, see [7] for applications in microfluidic. In a similar perspective but in the context of fluid mechanics, numerical simulations have shown that second order macroscopic wall-laws provide the same order of approximation as the first order approximation. Recently a generalized wall-law formulation has been obtained for curved rough boundaries [17, 19] and for random roughness [4]. Note that such generalizations are important from a practical point of view when dealing with e.g. coastal effects in geophysical flows. From a mathematical point of view, wall-laws are also interesting. In the proof of convergence to the Euler equations, the 2D Navier-Stokes system is complemented with wall-laws of the Navier type [6]. Recently several papers analyze in various settings the properties of such boundary conditions, see [12], [16], [11], [5], [13]. In this paper, we focus on fluid flows. Starting from the Stokes system, we …