A high-order and efficient numerical algorithm for conjugate heat transfer problems

Abstract

To improve numerical schemes associated with CHT problem, we first restrict our consideration to CHT problems involving only heat transfer, as for example in solids. Then we describe a new partitioned approach for solving CHT problems where the governing temperature equations in different material domains are time-stepped in an implicit manner, but where the interface coupling is explicit. The new approach, called the CHAMP scheme (Conjugate Heat transfer Advanced Multi-domain Partitioned), is based on a discretization of the interface coupling conditions using a generalized Robin (mixed) condition. The weights in the Robin condition are determined from the optimization of a condition derived from a local stability analysis of the coupling scheme. The interface treatment combines ideas from optimized-Schwarz methods for domain-decomposition problems together with the interface jump conditions and additional compatibility jump conditions derived from the governing equations. For many problems (i.e. for a wide range of material properties, grid-spacings and time-steps) the CHAMP algorithm is stable and second-order accurate using no sub-time-step iterations (i.e. a single implicit solve of the temperature equation in each domain). In extreme cases (e.g. very fine grids with very large time-steps) it may be necessary to perform one or more sub-iterations. Each sub-iteration generally increases the range of stability substantially and thus one sub-iteration is likely sufficient for the vast majority of practical problems.

To alleviate the viscous time-step restriction, the semi-implicit method updates the viscous terms implicitly, while the convection and pressure terms remain explicit. Both the explicit and the IMEX method are updated in a predictor-corrector manner to improve the stability property. The explicit method is based on the Adams' method, and the semi-implicit method is based on the backward difference formula (BDF) method. In order to ensure robustness for large Reynolds number flows, the convective derivatives are discretized using a new class of finite-difference methods, called the BWENO scheme (Banks' Weighted Essentially Non-Oscillatory) that has certain similarities to the more standard WENO scheme. This scheme is fourth-order accurate in well-resolved regions, and smoothly introduce artificial dissipation to solutions in under-resolved regions. In extreme cases, the BWENO scheme reduces to a third-order upwind scheme for poorly resolved solutions. For stagnation points in an under-resolved solution where extra dissipation is needed, an ad-hoc type dissipation is incorporated to improve the stability property. The other spatial derivative terms for the velocity and the pressure are discretized using fourth-order centered finite-difference methods. The complete solution strategy is developed for general curvilinear grids in two and three dimension space using composite overset grids. Numerical convergence studies confirm that the new INS method is fourth-order accurate for velocity, pressure and divergence of velocity in both time and space. For problems with an under-resolved solution, the new discretization scheme for convection terms is found to keep the solution stable. For problems with a well-resolved solution, the new discretization scheme for convection terms is found to behave the same as the centered difference scheme.

Next, we wish to improve the accuracy of the numerical schemes for solving CHT problems. Extending a CHT solver to higher-order accuracy has two aspects, the first is the development of a higher-order coupling scheme, and the second is the development of the underlying solvers for different fluid and/or solid domains. We restrict our consideration to the development of a fourth-order accurate split-step method for fluid problems. This method solves the incompressible Navier-Stokes (INS) equations based on the velocity-pressure formulation where the momentum and pressure are updated separately at each time-step. The momentum equations are advanced in time using a fourth-order accurate explicit time-stepping method for convection dominant problems, or a fourth-order accurate semi-implicit method, also known as IMEX method, for viscous dominant problems.

The CHAMP algorithm is developed first for a model problem and analyzed using normal-mode theory. The theory provides a mechanism for choosing optimal parameters in the mixed interface condition. A comparison is made to the classical Dirichlet-Neumann (DN) method and, where applicable, to the optimized-Schwarz (OS) domain-decomposition method. For problems with different thermal conductivities and diffusivities, the CHAMP algorithm outperforms the DN scheme. For domain-decomposition problems with uniform conductivities and diffusivities, the CHAMP algorithm performs better than the typical OS scheme with one grid-cell overlap. The CHAMP scheme is also developed for general curvilinear grids and CHT examples are presented using composite overset grids that confirm the theory and demonstrate the effectiveness of the approach.

In this study, we describe advanced numerical approaches for solving conjugate heat transfer (CHT) problems where the temperature and heat flux of the involved fluid and/or solid domains are coupled at their interfaces.