Conforming Chebyshev spectral collocation methods for the solution of the incompressible Navier-Stokes equations in complex geometries

Abstract

This paper examines the numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods. The flow domain is divided into a number of conforming rectangular subregions. Within each of the subregions the solution is approximated by a truncated expansion of Chebyshev polynomials. With a judicious choice of collocation strategy we show that the resulting approximations are pointwise C⁰ and C¹ continuous across the interfaces [1].

The governing equations are the incompressible Navier-Stokes equations, which in the stream function formulation reduces to:

$$\nabla ^4 \psi - Re\left[ {\frac{{\partial \psi }}{{\partial y}}\frac{\partial }{{\partial x}}(\nabla ^2 \psi ) - \frac{{\partial \psi }}{{\partial x}}\frac{\partial }{{\partial y}}(\nabla ^2 \psi )} \right] = 0$$

((A))

where Re is the Reynolds number. The introduction of the stream function ensures that mass is conserved identically. Equation (A) is nonlinear in the stream function and is linearized by means of a Newton-type technique [2,3].

At each Newton step the collocation of the linearized equation in conjunction with the interface patching conditions and the imposition of the boundary conditions (in a collocation sense) produces a system of linear equations for the expansion coefficients. The linear systems resulting from spectral discretizations suffer from not being sparse thus requiring large amounts of storage and being costly to invert. To some extent this can be overcome if the block structure of the spectral domain decomposition matrix is exploited. We explore two techniques: a capacitance matrix technique [4,5] and an algorithm for almost block diagonal systems [6,7].

Numerical results are presented demonstrating the convergence of the numerical solution for different numbers of degrees of freedom and for different values of the Reynolds number. For values of Re ≤ 100 the Newton process converges from a zero solution after six steps. For higher values of Re continuation in Re in increments of 50 is used. The use of conforming subregions enables solutions to be obtained for much larger values of Re than is possible for nonconforming subregions [2]. Quantitative agreement is reached with previous work [8] on the salient corner vortex. Further, our spectral discretizations are able to resolve the flow sufficiently enough to detect a vortex downstream of the contraction which appears at a value of Re around 200 and then continues to grow as the Reynolds number is increased further. A full discussion of these results is given. Contours of the stream function are presented in Figs 1 and 2 for Re = 100 and 500, respectively.