JOURNAL OF COMPUTATIONAL PHYSICS 136, 546–558 (1997) ARTICLE NO. CP975780 A Spectral Multidomain Method for the Numerical Simulation of Turbulent Flows A. Pinelli,* 1 A. Vacca,² and A. Quarteroni‡ *School of Aeronautics, Polytechnic University of Madrid, Madrid, Spain; ²Department of Civil Engineering, Second University of Naples, Naples, Italy; ‡Department of Mathematics, Polytechnic of Milan, Milan, Italy and CRS4, Cagliari, Italy Received May 1, 1996; revised April 29, 1997 come several intrinsic limitations of spectral methods, allowing for the use of the latter in a wider context [5–8]. The primitive variable formulation of the unsteady incompressible Navier–Stokes equations in three space dimensions is discretized The most obvious application of spectral multidomain with a combined Fourier–Legendre spectral method. A semi-implicit methods is related to the solution of partial differential pressure correction scheme is applied to decouple the velocity from equations over complex geometries (i.e., geometries which the pressure. The arising elliptic scalar problems are first diagonal- cannot be trivially mapped in the standard [21, 1] square). ized in the periodic Fourier direction and then solved by a multido- Another important feature of this class of algorithms is main Legendre collocation method in the two remaining space coor- dinates. In particular, both an iterative and a direct version of the the natural way in which they exploit the architectures of so-called projection decomposition method (PDM) are introduced modern MIMD computers (including clusters of work- to separate the equations for the internal nodes from the ones stations) paving the way for large scale simulations other- governing the interface unknowns. The PDM method, first intro- wise accessible only to supercomputer users. In fact, the duced by V. Agoshkov and E. Ovtchinnikov and later applied to spectral methods by P. Gervasio, E. Ovtchinnikov, and A. Quarteroni parallelization efficiency is extremely favourable to spec- is a domain decomposition technique for elliptic boundary value tral multidomain methods: the ratio of computation time problems, which is based on a Galerkin approximation of the to communication time is larger for this family of methods Steklov–Poincare´ equation for the unknown variables associated to than for others. This is mainly due to the high order accu- the grid points lying on the interface between subdomains. After racy provided by the spectral method combined with the having shown the exponential convergence of the proposed discret- ization technique, some issues on the efficient implementation of fact that the discrete operators involve matrices not as the method are given. Finally, as an illustration of the potentialities sparse as other ‘‘local’’ methods (i.e., finite differences, of the algorithm for the numerical simulation of turbulent flows, finite element or finite volumes) [9]. the results of a direct numerical simulation (DNS) of a fully turbulent The present Navier–Stokes solution algorithm differs plane channel flow are presented. Q 1997 Academic Press from already well established spectral domain decomposi- tion methods (i.e., the spectral element method). The main 1. INTRODUCTION difference consists in the treatment of the elliptic kernels arising after the application of a continuous semi-implicit Spectral methods have been and still remain the method pressure correction scheme. Each scalar elliptic boundary of choice for numerical simulations of fluid phenomena value problem is transformed in a set of analogous prob- where accuracy plays a fundamental role. The projection lems over subdomains whose boundary values are provided decomposition method, first introduced by [1], was later by an abstract equation on the interface between subdo- applied to spectral methods [2]. mains. This procedure allows us to split the equations over Their inherent high-order accuracy and low phase error two sets of geometrical elements: the subdomains them- motivated the development and implementation of several selves, where the solution is approximated by a local Leg- spectral-based direct numerical simulation (DNS) and endre polynomial basis in the framework of a collocation large eddy simulation (LES) codes for the solution of tur- method; and the subdomain interfaces, where the trace of bulent flows [3, 4]. Nevertheless, almost all these methods the solution is approximated through a Galerkin method have been designed to be extremely efficient for very sim- using a set of special basis functions. Such a sharp subdivi- ple geometries on supercomputers (highly vectorized sion introduces a very flexible tool that is able to deal shared memory machines). On the other hand, domain with more general discretizations such as a nonconforming decomposition methods are becoming viable tools to over- multidomain partition of the original computational do- main. Work is in progress in this direction and it will be the subject of a future publication. 1 E-mail: pinelli@torroja.dmt.upm.es. 546 0021-9991/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.