Direct Numerical Simulation of Rotating Cavity Flows Using a Spectral Element-Fourier Method

A high-order numerical method is employed to investigate flow in a rotor/stator cavity without heat transfer and buoyant flow in a rotor/rotor cavity. The numerical tool used employs a spectral element discretisation in two dimensions and a Fourier expansion in the remaining direction, which is periodic and corresponds to the azimuthal coordinate in cylindrical coordinates. The spectral element approximation uses a Galerkin method to discretise the governing equations, similarly to a finite element method, but employs high-order polynomials within each element to obtain spectral accuracy. A second-order, semi-implicit, stiffly stable algorithm is used for the time discretisation, and no subgrid modelling is included in the governing equations. Numerical results obtained for the rotor/stator cavity compare favourably with experimental results for Reynolds numbers up to Re₁ = 10⁶ in terms of velocities and Reynolds stresses. For the buoyancy-driven flow, the energy equation is coupled to the momentum equations via the Boussinesq approximation, which has been implemented in the code considering two different formulations. Numerical predictions of the Nusselt number obtained using the traditional Boussinesq approximation are considerably higher than available experimental data. Much better agreement is obtained when the extended Boussinesq approximation is employed. It is concluded that the numerical method employed has considerable potential for further investigations of rotating cavity flows.