A time viscosity-splitting method for incompressible flows with temperature-dependent viscosity and thermal conductivity

Abstract

A fractional-step method is proposed and analyzed for solving the incompressible thermal Navier–Stokes equations coupled to the convection–conduction equation for heat transfer with a generalized source term for which the viscosity and thermal conductivity are temperature-dependent under the Boussinesq assumption. The proposed method consists of four steps all based on a viscosity-splitting algorithm where the convection and diffusion terms of both velocity and temperature solutions are separated while a viscosity term is kept in the correction step at all times. This procedure preserves the original boundary conditions on the corrected velocity and it removes any pressure inconsistencies. As a main feature, our method allows the temperature to be transported by a non-divergence-free velocity, in which case we show how to handle the subtle temperature convection term in the error analysis and establish full first-order error estimates for the velocity and the temperature solutions and 1/2-order estimates for the pressure solution in their appropriate norms. The theoretical results are examined by an accuracy test example with known analytical solution and using a benchmark problem of Rayleigh–Bénard convection with temperature-dependent viscosity and thermal conductivity. We also apply the method for solving a problem of unsteady flow over a heated airfoil. The obtained results demonstrate the convergence, accuracy and applicability of the proposed time viscosity-splitting method.