Simulation Of Three-dimensional Incompressible Flows Research Articles

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier---Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier---Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton---Krylov---Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier---Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.

A Schwarz Preconditioner for the Cubed-Sphere

A spectral element formulation of the atmospheric two-dimensional shallow water equations on the cubed-sphere is described. The equations are written in tensor form using the contravariant and covariant velocity components. A semi-implicit time discretization results in a reduced Schur complement system for the pressure. The Laplacian operator is approximated by the L2 pseudo-Laplacian arising in the PN/PN-2 spectral element formulation of the incompressible Stokes problem. The overlapping Schwarz preconditioner of Fischer, Miller, and Tufo [An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl. 120, Springer, New York, 2000, pp. 159--180] based on the fast diagonalization method, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite element tensor-product approximation within each spectral element, extrema of the inverse metric tensor and its determinant are employed. Convergence rates and parallel CPU timings are compared against a block-Jacobi preconditioner.

Simulation of three-dimensional incompressible flows with a vortex-in-cell method

A new method for the numerical simulation of three-dimensional incompressible flows is described. Our vortex-in-cell (VIC) method traces the motion of the vortex filaments in the velocity field which these filaments create. The velocity field is not calculated directly by the Biot-Savart law of interaction but by creating a mesh-record of the vorticity field, then integrating a Poisson's equation via the fast Fourier transform to generate a mesh-record of the velocity field. The computed scales of motion are assumed to be essentially inviscid. Viscous of subgrid-scale effects are incorporated into a filtering procedure in wave vector space. Results of tracing a periodic array of single vortex rings are compared with a Green's function calculation. The agreement is very good.