Implicit heat flux correction-based immersed boundary-finite volume method for thermal flows with Neumann boundary conditions

Abstract

In the frameworks of immersed boundary method (IBM) and finite volume method (FVM), an implicit heat flux correction-based IB-FVM is proposed for thermal flows with Neumann boundary conditions. With the use of a fractional-step technique, the preconditioned Navier–Stokes (N–S) equations are solved by the FVM to obtain the intermediate solution in the prediction step and the heat flux is corrected by enforcing the Neumann condition in the correction step. Different from existing IBMs, the cell face centers are defined as the Eulerian points due to the heat flux computation at each face in the FVM. The Neumann condition is implemented in such a way that the interpolated temperature gradient is equal to the specified boundary value at the same point when the corrected gradient field is interpolated from the face centers to the Lagrangian points. To achieve an implicit algorithm, the temperature derivative corrections at the Lagrangian points are set as unknowns and a system of algebraic equations is established by constructing hybrid thin-plate splines (TPS) interpolation/delta function distribution. In the derivative interpolation process, the much more accurate TPS is introduced because the use of cosine delta function yields a less accurate solution. After the distribution process, the heat flux correction of a fluid cell is evaluated by using the solved temperature derivative corrections at the face centers, but that of a solid cell is calculated by using their additive inverses to supplement the same amount of heat flux into the solid domain as that flowing into the fluid domain across the boundary. Finally, the heat flux of a cell is corrected by adding the correction to the intermediate value and the corrected heat flux is utilized to solve the N–S equations in the prediction step. As compared with the available implicit IBMs for Neumann conditions, the present method avoids the introduction of auxiliary layers of Lagrangian points as well as the approximate conversion from the Neumann to Dirichlet condition and thus is suitable for an arbitrary geometry. The proposed method is verified by simulating several benchmark thermal flows with Neumann conditions, the natural convection in an annulus and the steady or unsteady forced convection over a stationary or oscillating cylinder. All the computed results agree well with the literature data.