INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 34: 627–650 Solution of the Navier – Stokes equations in velocity – vorticity form using a Eulerian – Lagrangian boundary element method D. L. Young* ,1 , S. K. Yang and T. I. Eldho Department of Ciil Engineering and Hydrotech Research Institute, National Taiwan Uniersity, Taipei, Taiwan, Republic of China SUMMARY This paper describes the Eulerian – Lagrangian boundary element model for the solution of incompress- ible viscous flow problems using velocity – vorticity variables. A Eulerian – Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian – Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier – Stokes problems with low – medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd. KEY WORDS: boundary element method; Eulerian – Lagrangian method; Navier – Stokes equations; velocity – vorticity formulation 1. INTRODUCTION The velocity – vorticity form of the Navier – Stokes equations pioneered by Fasel [1] has been established as an effective formulation for the solution of incompressible viscous flow problems. In the recent times, many researchers used the velocity – vorticity formulation for the calculations of two- and three-dimensional steady and unsteady flows using various numerical methods, such as the finite difference method (FDM) [2], the finite element method (FEM) [3] * Correspondence to: Department of Civil Engineering, Hydraulic Research Laboratory, National Taiwan University, Taipei 10617, Taiwan, Republic of China. 1 E-mail: dlyoung@hy.ntu.edu.tw Copyright © 2000 John Wiley & Sons, Ltd. Receied 13 September 1999 Reised 14 February 2000