An Unsplit, Cell-Centered Godunov Method for Ideal MHD

We present a second-order Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. vol. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergence-free condition of the magnetic fields, we apply a discrete projection to the intermediate values of the field at cell faces, and apply a filter to the primary dependent variables at the end of each time step. We test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions. The test suite includes rotated planar linear waves, MHD shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For all of these cases, we observe that the algorithm is second-order accurate for smooth solutions, converges to the correct weak solution for problems involving shocks, and exhibits no evidence of instability or loss of accuracy due to the possible presence of non-solenoidal fields.Comment: 37 Pages, 9 Figures, submitted to Journal of Computational Physic

Computational methods for internal flows with emphasis on turbomachinery

Current computational methods for analyzing flows in turbomachinery and other related internal propulsion components are presented. The methods are divided into two classes. The inviscid methods deal specifically with turbomachinery applications. Viscous methods, deal with generalized duct flows as well as flows in turbomachinery passages. Inviscid methods are categorized into the potential, stream function, and Euler aproaches. Viscous methods are treated in terms of parabolic, partially parabolic, and elliptic procedures. Various grids used in association with these procedures are also discussed

Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm

A multiple-grid algorithm for use in efficiently obtaining steady solutions to the Euler and Navier-Stokes equations is presented. The convergence of the explicit MacCormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids. Both the fine and coarse grid schemes are readily vectorizable. The combination of multiple-gridding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems. Results are presented for subsonic, transonic, and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows. Work reduction factors over a scalar, single-grid algorithm range as high as 76.8

Simulation of time-dependent compressible viscous flows using central and upwind-biased finite-difference techniques

Four time-dependent numerical algorithms for the prediction of unsteady, viscous compressible flows are compared. The analyses are based on the time-dependent Navier-Stokes equations expressed in a generalized curvilinear coordinate system. The methods tested include three traditional central-difference algorithms, and a new upwind-biased algorithm utilizing an implicit, time-marching relaxation procedure based on Newton iteration. Aerodynamic predictions are compared for internal duct-type flows and cascaded turbomachinery flows with spatial periodicity. Two-dimensional internal duct-type flow predictions are performed using an H-type grid system. Planar cascade flows are analyzed using a numerically generated, capped, body-centered, O-type grid system. Initial results are presented for critical and supercritical steady inviscid flow about an isolated cylinder. These predictions are verified by comparisons with published computational results from a similar calculation. Results from each method are then further verified by comparison with experimental data for the more demanding case of flow through a two-dimensional turbine cascade. Inviscid predictions are presented for two different transonic turbine cascade flows. All of the codes demonstrate good agreement for steady viscous flow about a high-turning turbine vane with a leading edge separation. The viscous flow results show a marked improvement over the inviscid results in the region near the separation bubble. Viscous flow results are then further verified in finer detail through comparison with the similarity solution for a flat plate boundary-layer flow. The usefulness of the schemes for the prediction of unsteady flows is demonstrated by examining the unsteady viscous flow resulting from a sinusoidally oscillating flat plate in the vicinity of a stagnant fluid. Predicted results are compared with the analytical solution for this flow. Finally, numerical results are compared with flow visualization and experimental data for the unsteady flow resulting from an impulsively started cylinder. Each algorithm demonstrates unique qualities which may be interpreted as either advantageous or disadvantageous, making it difficult to select an optimum scheme. The preferred method is perhaps best chosen based on the experience of the user and the particular application

Summary of research in applied mathematics, numerical analysis, and computer sciences

The major categories of current ICASE research programs addressed include: numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; control and parameter identification problems, with emphasis on effective numerical methods; computational problems in engineering and physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and computer systems and software, especially vector and parallel computers

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure