Abstract
In this paper, a gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme is developed for simulating natural convection in a rotating annulus, which arises in many scientific and engineering fields. Different from most existing methods for the solution of the incompressible Navier–Stokes (N–S) equations with the Boussinesq approximation, compressible full N–S equations with allowable density variation are concerned. An appropriate BGK model is constructed for the macroscopic equations defined in a rotating frame of reference. In particular, in order to account for the source (non-inertial) effects in the BGK model, a microscopic source term is introduced into the modified Boltzmann equation. By using the finite volume method and assuming the flow is smooth, the time-dependent distribution function is simply obtained, from which the fluxes at the cell interface can be evaluated. For validation, a closed rotating annulus with differentially heated cylindrical walls is studied. A conventional N–S solver with the preconditioner is used for comparison. The numerical results show that the present method can accurately predict the variation of the Nusselt number with the Rayleigh number, but no preconditioning is required due to its delicate dissipation property. The computed instantaneous streamlines and temperature contours are also investigated, and it is verified that the Rayleigh–Bénard convection in a rotating annulus is very similar to that in a classical stationary horizontal enclosure.
Highlights
Natural convection in rotating annuli has attracted a great deal of attention due to its widespread presence in both scientific and industrial applications [1]
No preconditioning is required in the BGK scheme, while without it the original N–S solver cannot give reasonable results or even produces floating errors
One is that the preconditioning significantly increases the computational cost for unsteady flows due to the necessity to employ a dual-time stepping approach
Summary
Natural convection in rotating annuli has attracted a great deal of attention due to its widespread presence in both scientific and industrial applications [1]. Depending on the direction of the heat flux inside the cavity, there are two major situations [2,3]. One is axial heat flux configuration, which refers to a cavity with insulated cylindrical walls and differentially heated disks, and the other is radial heat flux configuration, which contains insulated disks and differentially heated cylindrical walls. In both situations, the convective flows are induced by the centrifugal buoyancy force and the Coriolis force, leading to diverse flow structures and heat transfer properties. This work mainly focuses on the radial heat flow, and investigations can be made in an annulus provided that the flow is invariant in the axial direction ( referred to as the Taylor–Proudman theorem)
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