Efficient Simulations of Large Scale Convective Heat Transfer Problems

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.

Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018