Three-Dimensional Unsteady Viscous Flows Between Oscillating Eccentric Cylinders by an Enhanced Hybrid Spectral Method

Abstract

An enhanced hybrid spectral method is developed for the analysis of three-dimensional (3-D) unsteady viscous flows in an annular channel between two eccentric cylinders, when one of them executes forced axially variable transverse oscillations of specified frequency. A partial spectral formulation is used for the primitive flow variables, based on Chebyshev polynomials and Fourier expansions of the transverse coordinates and on complex exponential functions of the oscillation frequency and time. The Navier-Stokes equations are discretized in this hybrid method by using a collocation approach in a quasi-radial direction, obtained by a coordinate transformation, in conjunction with an efficient Fourier identification procedure in the azimuthal direction and a mixed-type central-upwind finite-difference scheme in the axial direction, which accommodates both finer and coarser grids. This method, developed for small amplitude oscillations, is first validated for several limiting cases, for which the solutions were found in good agreement with previous results. The method is then used to solve the 3-D unsteady viscous flows between eccentric cylinders executing axially variable transverse oscillations, for which no previous solutions are known. The new numerical solutions, obtained for transverse oscillations following the first flexural beam mode, are illustrated by the axial and azimuthal variations of the real and imaginary components of the reduced unsteady pressure, circumferential velocity component and unsteady fluid-dynamic forces.