Solving Navier–Stokes equation for flow past cylinders using single-block structured and overset grids Tapan K. Sengupta * , V.K. Suman, Neelu Singh Department of Aerospace Engineering, IIT Kanpur, UP 208 016, India article info Article history: Received 2 March 2009 Received in revised form 22 July 2009 Accepted 23 September 2009 Available online 29 September 2009 Keywords: Circular cylinder Navier–Stokes equation Overset grid method Chimera grid method Interpolation technique Proper orthogonal decomposition abstract Many complex fluid motions are driven by physical processes of instability, transition and turbulence dependent upon nonlinear mechanisms. Here, we solve the flow past cylin- der(s) using single-block structured and overset grids by computing Navier–Stokes equa- tion in two-dimensions. The suitability of a compact scheme in discretizing convection and diffusion terms are investigated first by looking at relevant numerical properties. Also, for the overset grid method, one of the methods is identified that shows the best results in minimizing interpolation error at sub-domain boundaries for an analytical test function. We provide extensive comparisons with experimental and other computational results for flow past a single cylinder, utilizing both single-block structured and Chimera or over- set grids. Apart from showing instability of this flow calculated by these methods, we also compare the computed vorticity and velocity data using these two grids by employing the proper orthogonal decomposition (POD). We have analyzed and developed an overset grid method with compact scheme that does not need any filtering to control error. This has been ascertained by performing POD analysis. To show that the developed method is capa- ble of handling complex geometries, we have computed flow past two cylinders in side-by- side arrangement. Results obtained capture the known flow characteristics for this arrangement well using relatively fewer number of grid points. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Numerical instability for the solution of partial differential equations has many common features with nonlinear insta- bility of hydrodynamics problems [1]. Thus, it is desirable to characterize numerical methods based on their ability to com- pute proto-typical nonlinear flow instability problems. In this context, vortex shedding behind a circular cylinder constitutes an ideal example, which is known for its initial linear instability followed by super-critical nonlinear saturation. In flow past a cylinder started impulsively, one notes the formation of symmetric recirculation regions at the base of the geometry. These wake bubbles grow in width and length with time, retaining the top-down symmetry. In the early stages of flow evolution, increase in time is equivalent to increase in Reynolds number and this is seen in all flows past a cylinder fol- lowing the impulsive start at all Reynolds numbers. For flows above a critical Reynolds number (Re cr ), an asymmetry in the wake bubble develops leading to alternate growth of the bubbles and eventual shedding, forming the Bènard–Kàrmàn street. Experimentally, it is noted that the time at which this asymmetry occurs is a function of Reynolds number. In addition to alternate vortices in the Bènard-Kàrmàn street, researchers in [2–6] have reported experiments in detection of 2D waves in the separated shear layer in the Reynolds number range 1000–50,000 as quoted in [7]. According to Braza et al. [7], these 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.09.026 * Corresponding author. E-mail address: tksen@iitk.ac.in (T.K. Sengupta). Journal of Computational Physics 229 (2010) 178–199 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp