Dochan Kwak

Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme, based on flux-difference splitting, is used to compute the convective terms. The upwind differencing is biased, based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. Both third-order and fifth-order differencing schemes are used on the convective fluxes throughout the grids interior. The equations are solved using an implicit line relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudo-time, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, and good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at —90° angle of attack is computed.

Steady and Unsteady Solutions of the Incompressible Navier-Stokes Equations

An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented. 28 refs.

A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

Abstract A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with a consistent approximation of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

A three-dimensional incompressible Navier-Stokes flow solver using primitive variables

An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system. The pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds Stresses, which are uncoupled from the implicit scheme, are lagged by one time step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computer and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.

An upwind differencing scheme for the incompressible Navier-Stokes equations

Abstract The steady-state incompressible Navier–Stokes equations in two dimensions are solved numerically using the artificial compressibility formulation. The convective terms are upwind differenced using a flux-difference split approach that has uniformly high accuracy throughout the interior grid points. The viscous fluxes are differenced using second-order accurate central differences. The numerical system of equations is solved using an implicit line relaxation scheme. The scheme is applicable to both steady-state and unsteady flow computations. In the current work steady-state applications are emphasized. Characteristic boundary conditions are formulated and used in the solution procedure. The overall scheme is capable of being run at extremely large pseudo-time steps, leading to fasr convergence. Three test cases are presented to demonstrate the accuracy and robustness of the code. These are the flow in a square driven cavity, flow over a backward facing step, and flow around a two-dimensional circular cylinder.

The Overgrid Interface for Computational Simulations on Overset Grids

Computational simulations using overset grids typically involve multiple steps and a variety of software modules. A graphical interface called OVERGRID has been specially designed for such purposes. Data required and created by the di erent steps include geometry, grids, domain connectivity information and ow solver input parameters. The interface provides a uni ed environment for the visualization, processing, generation and diagnosis of such data. General modules are available for the manipulation of structured grids and unstructured surface triangulations. Modules more speci c for the overset approach include surface curve generators, hyperbolic and algebraic surface grid generators, a hyperbolic volume grid generator, Cartesian box grid generators, and domain connectivity pre-processing tools. An interface provides automatic selection and viewing of ow solver boundary conditions, and various other ow solver inputs. For problems involving multiple components in relative motion, a module is available to build the component/grid relationships and to prescribe and animate the dynamics of the di erent components.

Three-dimensional incompressible Navier-Stokes solver using lower-upper symmetric-Gauss-Seidel algorithm

A numerical method based on the pseudocompressibility concept is developed for solving the three-dimensional incompressible Navier-Stokes equations using the lower-upper symmetric-Gauss-Seidel implicit scheme. Very high efficiency is achieved in a new flow solver, INS3D-LU code, by accomplishing the complete vectorizability of the algorithm on obIique planes of sweep in three dimensions.

Numerical solution of the incompressible Navier-Stokes equations for steady-state and time-dependent problems

The current work is initiated in an effort to obtain an efficient, accurate, and robust algorithm for the numerical solution of the incompressible Navier-Stokes equations in two- and three-dimensional generalized curvilinear coordinates for both steady-state and time-dependent flow problems. This is accomplished with the use of the method of artificial compressibility and a high-order flux-difference splitting technique for the differencing of the convective terms. Time accuracy is obtained in the numerical solutions by subiterating the equations in psuedo-time for each physical time step. The system of equations is solved with a line-relaxation scheme which allows the use of very large pseudo-time steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. Numerous laminar test flow problems are computed and presented with a comparison against analytically known solutions or experimental results. These include the flow in a driven cavity, the flow over a backward-facing step, the steady and unsteady flow over a circular cylinder, flow over an oscillating plate, flow through a one-dimensional inviscid channel with oscillating back pressure, the steady-state flow through a square duct with a 90 degree bend, and the flow through an artificial heart configuration with moving boundaries. An adequate comparison with the analytical or experimental results is obtained in all cases. Numerical comparisons of the upwind differencing with central differencing plus artificial dissipation indicates that the upwind differencing provides a much more robust algorithm, which requires significantly less computing time. The time-dependent problems require on the order of 10 to 20 subiterations, indicating that the elliptical nature of the problem does require a substantial amount of computing effort.An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a higher-order flux-difference splitting technique for the convective terms and a second-order central difference for the viscous terms. The steady-state solution of flow through a square duct with a 90 deg bend is computed and the results are compared with experimental data. Good agreement is observed. A comparison with an analytically known exact solution is then performed to verify the time accuracy of the algorithm. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Computational Approach for Probing the Flow Through Artificial Heart Devices

Computational fluid dynamics (CFD) has become an indispensable part of aerospace research and design. The solution procedure for incompressible Navier-Stokes equations can be used for biofluid mechanics research. The computational approach provides detailed knowledge of the flowfield complementary to that obtained by experimental measurements. This paper illustrates the extension of CFD techniques to artificial heart flow simulation. Unsteady incompressible Navier-Stokes equations written in three-dimensional generalized curvilinear coordinates are solved iteratively at each physical time step until the incompressibility condition is satisfied. The solution method is based on the pseudocompressibility approach. It uses an implicit upwind-differencing scheme together with the Gauss-Seidel line-relaxation method. The efficiency and robustness of the time-accurate formulation of the numerical algorithm are tested by computing the flow through model geometries. A channel flow with a moving indentation is computed and validated by experimental measurements and other numerical solutions. In order to handle the geometric complexity and the moving boundary problems, a zonal method and an overlapped grid embedding scheme are employed, respectively. Steady-state solutions for the flow through a tilting-disk heart valve are compared with experimental measurements. Good agreement is obtained. Aided by experimental data, the flow through an entire Penn State artificial heart model is computed.

On numerical errors and turbulence modeling in tip vortex flow prediction

The accuracy of tip vortex flow prediction in the near-field region is investigated numerically by attempting to quantify the shortcomings of the turbulence models and the flow solver. In particular, some turbulence models can produce a ‘numerical diffusion’ that artificially smears the vortex core. Low-order finite differencing techniques of the convective and pressure terms of the Navier–Stokes equations and inadequate grid density and distribution can also produce the same adverse effect. The flow over a wing and the near-wake with the wind tunnel walls included was simulated using 2.5 million grid points. Two subset problems, one using a steady, three-dimensional analytical vortex, and the other, a vortex obtained from experiment and propagated downstream, were also devised in order to make the study of vortex preservation more tractable. The method of artificial compressibility is used to solve the steady, three-dimensional, incompressible Navier–Stokes equations. Two one-equation turbulence models (Baldwin–Barth and Spalart–Allmaras turbulence models), have been used with the production term modified to account for the stabilizing effect of the nearly solid body rotation in the vortex core. Finally, a comparison between the computed results and experiment is presented. Published in 1999 by John Wiley & Sons, Ltd.