High order finite difference methods for unsteady incompressible flows in multi-connected domains Jian-Guo Liu a , Cheng Wang b, * a Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA b Department of Mathematics, Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, IN 47405-5701, USA Received 23 July 2002; received in revised form 3 January 2003; accepted 4 February 2003 Abstract Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the ‘‘holes’’. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Incompressible flow; Vorticity–stream function formulation; Multi-connected domain; Finite difference method 1. Introduction The homogeneous, incompressible Navier–Stokes equations (NSE) in velocity–pressure for- mulation with no-slip boundary condition can be written as * Corresponding author. Tel.: +1-812-855-9831. E-mail addresses: jliu@math.umd.edu (J.-G. Liu), cwang@indiana.edu (C. Wang). 0045-7930/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0045-7930(03)00037-9 Computers & Fluids 33 (2004) 223–255 www.elsevier.com/locate/compfluid