COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 53 (1985) 223-258 NORTH-HOLLAND NUMERICAL RESULTS ON THE HOMOGENIZATION OF STOKES AND NAVIER-STOKES EQUATIONS MODELING A CLASS OF PROBLEMS FROM FLUID MECHANICS Carlos CONCA Laboratoire d’Analyse Numkrique, Universite’ de Paris VI, 4 Place Jussieu, Tour 55-65, 75230 Paris Cidex OS, France; and Departamento de Matema’ticas, Universidad de Chile, Casilla 5272, Santiago 3, Chile Received 2 January 1984 Revised manuscript received 13 August 1984 The aim of this paper is to present a numerical study of the homogenization theory applied to Stokes and Navier-Stokes equations with nonstandard conditions on the boundary of a periodically multiper- forated domain. From a practical point of view our purpose is the numerical solution of the physical problem of the steam-water condensation in a condenser which contains a periodical structure of pipes. For this problem, we first propose a mathematical model based on theoretical results about the homogenization of Navier-Stokes equations, and then we present a numerical study of this model which allows the effective computation of the homogenized solution. 1. Introduction Dealing with the numerical analysis of an elliptic boundary-value problem in a multiper- forated domain (with a very large number of holes) is well known, and it is clear that any numerical attempt to directly solve the problem is too expensive, and that an averaging or homogenizing method must be considered. The original problem is first approximated by another boundary-value problem governing an average behaviour of the system, and the required numerical approximation of the actual solution is obtained by numerically solving this new problem, which is usually called the homogenized problem. Doing that, we say, in the language of homogenization, that the heterogeneous medium has been replaced by an ‘equivalent’ homogeneous one. In the particular case of a periodical distribution of holes (with period E), the averaging method most currently used in practice to overcome the difficulty discussed above is the two-scale method introduced in the book by Bensoussan, Lions and Papanicolaou [l]. Roughly speaking, this method consists of the following steps: (1) The solution of the original problem is asymptotically expanded in increasing powers of E. (2) The different terms of this expansion are computed by means of a formal calculus, which consists in replacing the equations of the boundary value problem by this expansion. The first term in this expansion corresponds to the homogenized solution, and the next terms are called ‘correctors’. The aim of our paper is to present some numerical experiments on the homogenization of Stokes and Navier-Stokes equations in a periodically multiperforated domain, with Fourier or 00457825/85/$3.30 @ 1985, Elsevier Science Publishers B.V. (North-Holland)