A Fourier-Chebyshev pseudospectral method is used for the numerical simulation of incompressible flows in a three-dimensional channel of square cross-section with rotation. Realistic, non-periodic boundary conditions that impose no-slip conditions in two directions (spanwise and vertical directions) are used. The Navier-Stokes equations are integrated in time using a fractional step method. The Poisson equations for pressure and the Helmholtz equation for velocity are solved using a matrix diagonalization (eigenfunction decomposition) method, through which we are able to reduce a three-dimensional matrix problem to a simple algebraic vector equation. This results in signficant savings in computer storage requirement, particularly for large-scale computations. Verification of the numerical algorithm and code is carried out by comparing with a limiting case of an exact steady state solution for a one-dimensional channel flow and also with a two-dimensional rotating channel case. Two-cell and four-cell two-dimensional flow patterns are observed in the numerical experiment. It is found that the four-cell flow pattern is stable to symmetrical disturbances but unstable to asymmetrical disturbances.
Kt.Y WORDS
two-and four-cell flow paaem rotating flow; three-dimensional rectangular channel; pseudospechal matrix method; eigenvalue decomposition;
I . INTRODUCTION
The study of flow through a rotating rectangular channel is of theoretical and practical importance. Practically, rotating channel flow is common in rotating machines, such as coolant flow within turbine blades, pulp flow in paper refiners, flow in centrifugal pumps and in instruments that measure mass flow based on the Coriolis effect. On the other hand, the theoretical study of rotating flow can lead to a better understanding of secondary flow motion, bifurcations, secondary stability and the transition to turbulence. As the flow undergoes rather complex structural changes, it provides a useful framework for testing computational algorithms to see whether they can capture all the nuances exhibited by the flow. This requires verification with either experiments or cross validation using different computational discretization schemes.