INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2010; 62:683–708 Published online 23 March 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2039 A transformation-free HOC scheme for incompressible viscous flows on nonuniform polar grids Rajendra K. Ray ∗, † and Jiten C. Kalita ‡ Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India SUMMARY We recently proposed a transformation-free higher-order compact (HOC) scheme for two-dimensional (2-D) steady convection–diffusion equations on nonuniform Cartesian grids (Int. J. Numer. Meth. Fluids 2004; 44:33–53). As the scheme was equipped to handle only constant coefficients for the second-order derivatives, it could not be extended directly to curvilinear coordinates, where they invariably occur as variables. In this paper, we extend the scheme to cylindrical polar coordinates for the 2-D convection– diffusion equations and more specifically to the 2-D incompressible viscous flows governed by the Navier–Stokes (N–S) equations. We first apply the formulation to a problem having analytical solution and demonstrate its fourth-order spatial accuracy. We then apply it to the flow past an impulsively started circular cylinder problem and finally to the driven polar cavity problem. We present our numerical results and compare them with established numerical and analytical and experimental results whenever available. This new approach is seen to produce excellent comparison in all the cases. Copyright 2009 John Wiley & Sons, Ltd. Received 26 May 2008; Revised 7 February 2009; Accepted 9 February 2009 KEY WORDS: HOC; convection–diffusion; nonuniform; N–S equations; circular cylinder; polar cavity 1. INTRODUCTION The steady two-dimensional (2-D) convection–diffusion equation in Cartesian coordinate system (x , y ) for a transport variable  in some continuous domain with suitable boundary conditions can be written as -∇ 2  + c 1 (x , y )  x + c 2 (x , y )   y = f (x , y ) (1) ∗ Correspondence to: Rajendra K. Ray, Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India. † E-mail: rkr@iitg.ernet.in ‡ Associate Professor. Copyright 2009 John Wiley & Sons, Ltd.