IOP PUBLISHING FLUID DYNAMICS RESEARCH Fluid Dyn. Res. 44 (2012) 031414 (10pp) doi:10.1088/0169-5983/44/3/031414 A mixed Fourier–Galerkin–finite-volume method to solve the fluid dynamics equations in cylindrical geometries J´ ose N ´ u˜ nez 1 , Eduardo Ramos 1 and Juan M Lopez 2 1 Centro de Investigaci´ on en Energ´ ıa, Universidad Nacional Aut´ onoma de M´ exico, Ap.P. 34, 62580 Temixco Morelos, Mexico 2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA E-mail: jnegl@cie.unam.mx Received 21 October 2011, in final form 9 February 2012 Published 23 May 2012 Online at stacks.iop.org/FDR/44/031414 Communicated by E Knobloch Abstract We describe a hybrid method based on the combined use of the Fourier Galerkin and finite-volume techniques to solve the fluid dynamics equations in cylindrical geometries. A Fourier expansion is used in the angular direction, partially translating the problem to the Fourier space and then solving the resulting equations using a finite-volume technique. We also describe an algorithm required to solve the coupled mass and momentum conservation equations similar to a pressure-correction SIMPLE method that is adapted for the present formulation. Using the Fourier–Galerkin method for the azimuthal direction has two advantages. Firstly, it has a high-order approximation of the partial derivatives in the angular direction, and secondly, it naturally satisfies the azimuthal periodic boundary conditions. Also, using the finite-volume method in the r and z directions allows one to handle boundary conditions with discontinuities in those directions. It is important to remark that with this method, the resulting linear system of equations are band-diagonal, leading to fast and efficient solvers. The benefits of the mixed method are illustrated with example problems. (Some figures may appear in colour only in the online journal) 1. Introduction Several numerical algorithms for solving the fluid dynamics equations in cylindrical coordinates are currently available in the literature. The well-known discretization methods, finite difference and finite volume, have been used and explained in several reports (Eggels © 2012 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK 0169-5983/12/031414+10$33.00 1