Diffuse interface immersed boundary method for low Mach number flows with heat transfer in enclosures

Abstract

A novel diffuse interface immersed boundary (IB) approach in the finite volume framework is developed for non-Boussinesq flows with heat transfer. These flows are characterized by variable density, large temperature differences, nonzero velocity divergence, and low Mach numbers. The present IB methodology assumes that the solid body immersed in the domain is filled with a “virtual” fluid and constructs a unified momentum equation that is solved everywhere in the domain. The unified momentum equation is obtained as a convex combination of the Navier-Stokes equation and the no-slip boundary condition employing the solid volume fraction. The hydrodynamic pressure (p) that drives the flow is obtained by the solution of a variable density Poisson equation that is constructed by assuming that the velocity field inside the solid always remains solenoidal although the velocity divergence is nonzero in the fluid domain. The unified Poisson equation is also solved everywhere in the domain and has source terms that depend on the solid volume fraction, temperature gradients, and the spatially invariant thermodynamic pressure (P) that vanish in the Boussinesq limit. The thermodynamic pressure in closed domains follows from the principle of global mass conservation and is used to determine the density field everywhere in the domain except inside the solid where the density remains constant. Numerical simulations are carried out for natural and mixed convective flows in enclosures with stationary and moving heated bodies encompassing both Boussinesq and strongly non-Boussinesq flow regimes. The results of these investigations show that the local Nusselt number distribution over the body surface is oscillatory particularly when grid lines are not aligned with the surface of the body. However, the proposed approach can reasonably accurately compute the average heat transfer in both Boussinesq and non-Boussinesq flows. Investigations show that the heat transfer is significantly enhanced in the non-Boussinesq regime as compared to the Boussinesq regime. A comparison of results from the present approach with those obtained using a body-fitted finite volume solver for stationary bodies demonstrates that the proposed IB approach can compute the flow dynamics quite accurately even on Cartesian meshes that do not conform to the geometry. The IB approach presented herein is a generic approach for quasi-incompressible flows and may be applied to other low Mach number flows such as mixing and reacting flows.