A Spectral Collocation Method for Confined Unsteady Flows with Oscillating Boundaries

This paper presents a spectral collocation method especially developed for the study of confined unsteady flows with periodically oscillating boundaries. It is based on suitable expansions of the fluid-dynamic parameters, using Chebyshev polynomials and Fourier functions for the spatial expansions and exponential functions of time, and containing a priori unknown coefficients which are determined using a collocation method. As a special feature, complex exponential functions of time and frequency are used in this method for the time discretization. The method is validated by comparison with analytical solutions for several test problems of typical steady and unsteady confined flows with oscillating boundaries. Excellent agreement has been found in all cases. The solution of nonlinear problems using this spectral method is exemplified by the calculation of the unsteady viscous flow in an eccentric annular space, generated by the oscillatory rotation of the inner boundary.