A global method of generalized differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.
KEY WORDS Generalized differential quadrature Incompressible flows Navier-Stokes equations
Introduction
Most engineering problems can be currently simulated by finite difference and finite element methods. Usually, these methods require a large number of grid points for accurate results. More recently, spectral and pseudospectral methods have provided attractive techniques for the solution of smooth engineering problems, using only a few grid points. Amongst the family of these methods, the Chebyshev pseudospectral method is commonly used. This method usually requires a transformation between the physical space and the computational space since Chebyshev collocation points lie in the domain [l, -13, leading to some inconveniences in use. In seeking a more efficient numerical method, the present authors have developed a method, based on the work of Bellman et a/.,' of generalized differential quadrature (GDQ), which can be considered as a global method. G D Q has overcome the difficulty of differential quadrature (DQ) in obtaining the weighting coefficients for the first-order derivative discretization with arbitrary distribution of grid points, and is easier to apply than spectral methods. It is shown in Reference 2 that GDQ can be considered as the highest-order finite difference scheme, and both G D Q and Chebyshev pseudospectral method provide exactly the same weighting coefficients of the firstorder derivative when the co-ordinates of grid points are chosen as the roots of a Chebyshev polynominal. This demonstrates that GDQ may have a considerable scope for development since it can be used with arbitrary distributions of grid points. In GDQ, the weighting coefficients of the first-order derivative are given by a simple algebraic formulation, and the weighting coefficients of the second-and higher-order derivative are determined by a recurrence relation. Some basic features of GDQ, such as the error estimations of the derivatives approximation and the influence of distribution of grid points on the stability, have also been analysed in Reference 2. The successful application of GDQ for the solution of a partial differential equation has been shown in