Computers&FluidsVol. 13, No. l, pp. 99-113, 1985 0045-7930/85 $3.00+ ,00 Printed in the U.S.A. © 1985 Pergamon Press Ltd. SOLUTIONS OF THE TWO-DIMENSIONAL NAVIER- STOKES EQUATIONS BY CHEBYSHEV EXPANSION METHODS HWAR-CHING KU and DIMITR1 HATZIAVRAMIDIS~ Department of Chemical Engineering, Illinois Institute of Technology, Chicago, 1L 60616, U.S.A. (Received 28 March 1984; in revised form 8 June 1984) Abstract~The steady two-dimensional Navier-Stokes equations in both the vorticity-stream function and the vorticity-velocity formulation are solved by Chebyshev expansion methods. Numerical experiments for the driven flow in a rectangular cavity and the developing flow in a circular tube at low Reynolds numbers are described. 1. INTRODUCTION Two-dimensional laminar flows of incompressible Newtonian fluids have been extensively studied by different numerical schemes applied to the following formulations of the Navier-Stokes equations, in order of increasing popularity: (i) Vorticity-stream function (ii) Primitive variables (iii) Vorticity-velocity [ 1 ]. While all three formulations are prone to inaccuracies arising from the treatment of the nonlinear convective terms, the vorticity-stream function and vorticity-velocity formula- tions share the difficulties of appropriate implementation of boundary conditions for the vorticity. The primitive variables formulation, on the other hand, is associated with problems of determining the pressure either from a Poisson-type equation with Neumann boundary conditions or from an "artificial" equation of state which constitutes the incompressibility constraint [2]. Most of the schemes employed in the past for the solution of flow problems are based on finite difference and finite element approximations. Less experience exists with the solution of flow problems with pseudospectral methods, in spite of their many advantages: (i) Accurate approximation of spatial derivatives. (ii) No need to introduce fictitious points outside the physical domain for the implementation of other than Dirichlet boundary conditions. (iii) No need for the extensive quadratures of the finite element method. (iv) Ability to resolve accurately steep gradients and discontinuities. Recent calculations by Orszag [3] and Morchoisne [4] have sparked more interest in the application of pseudospectral methods to flow problems. This study deals with the application of pseudospectral methods employing Chebyshev expansions [5] to the two-dimensional steady Navier-Stokes equations cast in either vorticity-stream function or vorticity-velocity variables. In addition to their accuracy and computational economy features, the Chebyshev expansion methods are remarkably effective in resolving thin layers where steep changes in the field variables occur and in implementing complex boundary conditions which, together with the equations, are treated on equal footing. The numerical experiments to be presented involve two extensively studied problems: the driven cavity flow [6, 7] and the entrance region flow in a pipe [8, 9], which provide standard tests for numerical schemes. A Chebyshev expansion method which has been described elsewhere [ 11] is used for the solution of the elliptic-type governing equations. This method is of general applicability t Author to whom correspondence should be addressed. 99