Chebyshev spectral collocation methods for laminar flow through a channel contraction

Abstract

A Chebyshev spectral element method is described for solving the Navier-Stokes equations in a channel contraction. The flow region is divided into two semi-infinite elements. The governing equation for the stream function is solved using a Newton linearization. The semi-infinite elements are treated by means of algebraic-type mappings to transform them onto finite domains. The stream function is represented by a double Chebyshev expansion in each element. The coefficients are determined by collocating the linearized equation at each Newton step and imposing C3 Continuity conditions across the element interface, in a collocation sense. Efficient direct methods based on capacitance matrix ideas are described which take advantage of the structure of the spectral element matrix. Results are presented for Reynolds numbers in the range [0, 200] which are in good agreement with previously published work but requiring fewer degrees of freedom. Only several steps of the Newton process are required to achieve a converged solution. For Re {precedes above single-line equals sign} 120 the method converges from a zero initial approximation and thereafter continuation in the Reynolds number is used with increments of 10. © 1989.