Velocity–vorticity formulation for 3D natural convection in an inclined cavity by DQ method

The present work proposes a novel numerical solution algorithm based on a differential quadrature (DQ) method to simulate natural convection in an inclined cubic cavity using velocity–vorticity form of the Navier–Stokes equations. Since the DQ method employs a higher-order polynomial to approximate any given differential operator, the vorticity values at the boundaries can be computed more accurately than the conventionally followed second-order accurate Taylor’s series expansion scheme. The numerical capability of the present algorithm is demonstrated by the application to natural convection in an inclined cubic cavity. The velocity Poisson equations, the continuity equation, the vorticity transport equations and the energy equation are all solved as a coupled system of equations for the seven field variables consisting of three velocities, three vorticities and temperature. Thus coupling the velocity and the vorticity transport equations allows the determination of the vorticity boundary values implicitly without requiring the explicit specification of the vorticity boundary conditions. The present algorithm is proved to be an efficient method to resolve the non-linearity involved with the vorticity transport equations and the energy equation. Test results obtained for an inclined cubic cavity with different angle of inclinations for Rayleigh number equal to 10³, 10⁴, 10⁵ and 10⁶ indicate that the present coupled solution algorithm could predict the benchmark results for temperature and flow fields using a much coarse computational grid compared to other numerical schemes.