### Browsing by Subject "Navier-Stokes equations"

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Item Compact Direct Flux Reconstruction for the Navier-Stokes Equations on Dynamic Meshes(AIAA Aviation Forum, 2017) Wang, Lia; Yu, MeilinIn this study, the high-order discontinuous compact direct ﬂux reconstruction (CDFR) method is used to solve the two-dimensional (2D) Navier-Stokes equations on quadrilateral unstructured dynamic meshes. Within a standard element, the CDFR method employs com-pact ﬁnite diﬀerence (FD) techniques to directly construct the nodal spatial derivatives on Gauss-Legendre solution points. In the procedure of constructing an arbitrary CDFR method, the spatial derivatives are approximated with local ﬂuxes on solution points and common ﬂuxes on element interfaces (ﬂux points) in FD forms. No polynomial reconstruction needs to be employed explicitly. It is observed that the CDFR method is identical with the direct ﬂux re-construction (DFR) method and the nodal ﬂux reconstruction-discontinuous Galerkin (FR-DG) method if Gauss-Legendre points are selected as solution points. For simulations with dynamic meshes, the geometric conservation law (GCL) has been incorporated into the Navier-Stokes equations. The performance of CDFR methods has been veriﬁed with various test cases, in-cluding the Euler vortex propagation on deformable meshes, and the Couette ﬂow. Laminar ﬂows of Ma = 0.2 over a static circular cylinder (Re = 100, 185) and an oscillating circular cylinder (Re = 185) have been studied to demonstrate the capability of the solver developed in this study.Item MULTIGRID PRECONDITIONERS FOR THE NEWTON-KRYLOV METHOD IN THE OPTIMAL CONTROL OF THE STATIONARY NAVIER-STOKES EQUATIONS(2018) Soane, Ana Marie; Draganescu, AndreiIn this work we construct multigrid preconditioners to be used in the Newton-Krylov method for a distributed optimal control problem constrained by the stationary Navier-Stokes equations. These preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods use to solve the Navier-Stokes equations.Item On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations(AIAA SciTech Forum, 2018) Wang, Lai; Yu, MeilinThe Rosenbrock-type implicit Runge-Kutta (ROIRK) methods only require one Jaco-bian matrix evaluation per time step rather than per stage as other types of implicit Runge-Kutta (IRK) methods need. This feature makes ROIRK attractive for numerical simulations using implicit methods. We present the parallel implementation of several matrix-based ROIRK methods with ﬂux reconstruction/correction procedure re-construction (FR/CPR) formulations for solving the 3D Navier-Stokes equations. In this study, METIS has been utilized to partition the mesh in the preprocessing. The complex-step derivative approximation is employed to evaluate the Jacobi matrix, ac-curate to machine zero. The GMRES solver in the PETSc library is used to iteratively solve the linear system. The ROIRK methods have demonstrated high order of ac-curacy in numerical simulations. The scalability study reveals that the matrix-based ROIRK methods have good parallel eﬃciency. With the block Jacobi preconditioner, it is observed that the linear systems resulting from ROIRK3-3 are stiﬀer than those from ROIRK2-2 and ROIRK4-6. This makes the scalability of ROIRK3-3 worse than ROIRK2-2 and ROIRK4-6 taking the number of stages into account.Item Variational Problems in Weighted Sobolev Spaces with Applications to Computational Fluid Dynamics(2008-08-19) Soane, Ana Maria; Rostamian, Rouben; Mathematics and Statistics; Mathematics, AppliedWe study variational problems in weighted Sobolev spaces on bounded domains with angular points. The specific forms of these variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations. Specifically, we introduce new variational formulations for the Poisson and Helmholtz problems in what would be a weighted counterpart of H^{2}