Fractional step artificial compressibility schemes for the unsteady incompressible Navier–Stokes equations H.S. Tang a , Fotis Sotiropoulos b, * a Naval Research Laboratory, Oceanography Division, Code 7320, Stennis Space Center, MS 39529, United States b St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55414, United States Received 27 September 2004; received in revised form 24 June 2005; accepted 16 January 2006 Available online 24 January 2007 Abstract We propose a novel, time-accurate approach for solving the unsteady, three-dimensional, incompressible Navier–Stokes equations on non-staggered grids. The approach modifies the standard, dual-time stepping artificial-compressibility (AC) iteration scheme by incor- porating ideas from pressure-based, fractional-step (FS) formulations. The resulting hybrid fractional-step/artificial-compressibility (FSAC) method is second-order accurate and advances the Navier–Stokes equations in time via a two-step procedure. In the first step, which is identical to the convection–diffusion step in pressure-based FS methods, a preliminary velocity field is calculated, which is not divergence-free. In the second step, however, instead of deriving a pressure-Poisson equation as in FS methods, the projection of the velocity field into the solenoidal vector space is implemented using a dual-time stepping AC formulation. Unlike the standard dual-time stepping AC formulations, where the dual-time iterations are carried out with the entire non-linear system, in the FSAC scheme the con- vective and viscous terms are computed only once or twice per physical time step. Numerical experiments show that the proposed method provides second-order accurate solutions and requires considerably less CPU time than the widely used standard AC formulation. To demonstrate its ability to compute complicate problems, the method is also applied to a flow past cylinder with endplates. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction A major difficulty in numerical solutions of the unsteady, incompressible, Navier–Stokes (NS) equations arises from the form of the continuity equation, which is a divergence-free constraint on the velocity field that pro- hibits the straightforward integration of the governing equations in time. A commonly used numerical method for solving the unsteady NS equations in primitive vari- ables is the artificial-compressible (AC) method, which modifies the continuity equation by introducing a time derivative for the pressure. The method was first intro- duced by Chorin [1] and Temam [2] for obtaining steady- state solutions and later extended to unsteady flows by Peyret [3] and Merkle and Athavale [4]. To obtain time- accurate solutions, dual- or pseudo-time derivatives for the pressure and velocity fields are introduced into the con- tinuity and momentum equations, respectively, and the resulting modified system is iterated in dual time until con- vergence is reached at every physical time step—i.e. until changes of all variables in dual time become smaller than a prescribed tolerance. Dual-time stepping AC methods have been employed by many authors for studying unsteady flows. Rogers and Kwak [5] proposed a dual-time AC schemes, which uses second-order backward differenc- ing for time derivative along with Euler-implicit temporal discretization of the spatial derivative terms. Mark [6] showed that this type of method is strongly stable and dis- sipative. Sotiropoulos and Ventikos [7] presented a dual- time AC scheme on non-staggered grids for carrying out direct numerical simulations of three-dimensional, swirling flow in a closed cylinder with a rotating lid. Their method employs second-order backward differencing for the physical time derivatives along with a point-wise implicit, 0045-7930/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2006.01.019 * Corresponding author. E-mail addresses: tang@nrlssc.navy.mil (H.S. Tang), fotis@umn.edu (F. Sotiropoulos). www.elsevier.com/locate/compfluid Computers & Fluids 36 (2007) 974–986