Abstract
We describe an approach for efficient solution of large scale convective heat transfer problems, formulated as coupled unsteady heat conduction and incompressible fluid flow equations. The original problem is discretized in time using classical implicit methods, while stabilized finite elements are used for space discretization. The algorithm employed for the discretization of the fluid flow problem uses Picard's iterations to solve the arising nonlinear equations. Both problems, heat transfer and Navier-Stokes quations, give rise to large sparse systems of linear equations. The systems are solved using iterative GMRES solver with suitable preconditioning. For the incompressible flow equations we employ a special preconditioner based on algebraic multigrid (AMG) technique. The paper presents algorithmic and implementation details of the solution procedure, which is suitably tuned, especially for ill conditioned systems arising from discretizations of incompressible Navier-Stokes equations. We describe parallel implementation of the solver using MPI and elements of PETSC library. The scalability of the solver is favourably compared with other methods such as direct solvers and standard GMRES method with ILU preconditioning.
Highlights
We describe an approach for efficient solution of large-scale convective heat transfer problems that are formulated as coupled unsteady heat conduction and incompressible fluid-flow equations
For a selected time step and non-linear iteration, we report the characteristics for the solution of a single linear system for the Navier–Stokes equations, which always takes more than 80% of the execution time
We have presented an efficient solution procedure for simulating large-scale convective heat transfer problems
Summary
An efficient numerical solution for convective heat transfer problems has remained a challenge for many decades [22,29]. Following the approach of decoupling the velocity and pressure parts of the discretized fluid-flow equations [22], a similar decoupling for linear systems (the so-called block decomposition) can be introduced [13] This leads to the application of a preconditioner in several steps where smaller decoupled linear systems are employed. The speed of the convergence of the whole procedure depends on many factors, with the size of the time step (and its related CFL number) as well as the Reynolds number of the considered flow being most important [11] This depends on the details of the discrete problem formulation – especially the method that is used to deal with the numerical instabilities that appear in the standard formulations of the convection dominated equations.
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