Time integration for complex fluid dynamics - Christopher, Joshua C., author; Gao, Xinfeng, advisor; Guzik, Stephen M., committee member; Marchese, Anthony J., committee member; Bangerth, Wolfgang, committee member (Colorado State University. Libraries, 2021)
Efficient and accurate simulation of turbulent combusting flows in complex geometry remains a challenging and computationally expensive proposition. A significant source of computational expense is in the integration of the temporal domain, where small time steps are required for the accurate resolution of chemical reactions and long solution times are needed for many practical applications. To address the small step sizes, a fourth-order implicit-explicit additive Runge-Kutta (ARK4) method is developed to integrate the stiff chemical reactions implicitly while advancing the convective and diffusive physics explicitly in time. Applications involving complex geometry, stiff reaction mechanisms, and high-order spatial discretizations are challenged by stability issues in the numerical solution of the nonlinear problem that arises from the implicit treatment of the stiff term. Techniques for maintaining a physical thermodynamic state during the numerical solution of the nonlinear problem, such as placing constraints on the nonlinear solver and the use of a nonlinear optimizer to find valid thermodynamic states, are proposed and tested. Verification and validation are performed for the new adaptive ARK4 method using lean premixed flames burning hydrogen, showing preservation of 4th-order error convergence and recovery of literature results. ARK4 is then applied to solve lean, premixed C3H8-air combustion in a bluff-body combustor geometry. In the two-dimensional case, ARK4 provides a 70× speedup over the standard explicit four-stage Runge-Kutta method and, for the three-dimensional case, three-orders-of-magnitude-larger time step sizes are achieved. To further increase the computational scaling of the algorithms, parallel-in-time (PinT) techniques are explored. PinT has the dual benefit of providing parallelization to long temporal domains as well as taking advantage of hardware trends towards more concurrency in modern high-performance computing platforms. Specifically, the multigrid reduction-in-time (MGRIT) method is adapted and enhanced by adding adaptive mesh refinement (AMR) in time. This creates a space-time algorithm with efficient solution-adaptive grids. The new MGRIT+AMR algorithm is first verified and validated using problems dominated by diffusion or characterized by time periodicity, such as Couette flow and Stokes second problem. The adaptive space-time parallel algorithm demonstrates up to a 13.7× speedup over a time-sequential algorithm for the same solution accuracy. However, MGRIT has difficulties when applied to solve practical fluid flows, such as turbulence, governed by strong hyperbolic partial differential equations. To overcome this challenge, the multigrid operations are modified and applied in a novel way by exploiting the space-time localization of fine turbulence scales. With these new operators, the coarse-scale errors are advected out of the temporal domain while the fine-scale dynamics iterate to equilibrium. This leads to rapid convergence of the bulk flow, which is important for computing macroscopic properties useful for engineering purposes. The novel multigrid operations are applied to the compressible inviscid Taylor-Green vortex flow and the convergence of the low-frequency modes is achieved within a few iterations. Future work will be focused on a performance study for practical highly turbulent flows.
Keywords: computational fluid dynamics; multigrid reduction-in-time; turbulent reacting flows; implicit-explicit time integration; additive Runge-Kutta; parallel in time