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NASA NTRS2000

Accuracy Improvement in Magnetic Field Modeling for an Axisymmetric Electromagnet

This paper examines the accuracy and calculation speed for the magnetic field computation in an axisymmetric electromagnet. Different numerical techniques, based on an adaptive nonuniform grid, high order finite difference approximations, and semi-analitical calculation of boundary conditions are considered. These techniques are being applied to the modeling of the Variable Specific Impulse Magnetoplasma Rocket. For high-accuracy calculations, a fourth-order scheme offers dramatic advantages over a second order scheme. For complex physical configurations of interest in plasma propulsion, a second-order scheme with nonuniform mesh gives the best results. Also, the relative advantages of various methods are described when the speed of computation is an important consideration.

Ilin, Andrew V.↗

NASA NTRS2001

B-spline Method in Fluid Dynamics

B-spline functions are bases for piecewise polynomials that possess attractive properties for complex flow simulations : they have compact support, provide a straightforward handling of boundary conditions and grid nonuniformities, and yield numerical schemes with high resolving power, where the order of accuracy is a mere input parameter. This paper reviews the progress made on the development and application of B-spline numerical methods to computational fluid dynamics problems. Basic B-spline approximation properties is investigated, and their relationship with conventional numerical methods is reviewed. Some fundamental developments towards efficient complex geometry spline methods are covered, such as local interpolation methods, fast solution algorithms on cartesian grid, non-conformal block-structured discretization, formulation of spline bases of higher continuity over triangulation, and treatment of pressure oscillations in Navier-Stokes equations. Application of some of these techniques to the computation of viscous incompressible flows is presented.

Botella, Olivier↗

NASA NTRS1971

Errors in the use of nonuniform mesh systems

Two nonlinear coupled second order differential equations finite difference solution, using nonsquare and nonuniform grids for minimum error

Crowder, H. J.↗

NASA NTRS1981

A three-dimensional simulation of transition and early turbulence in a time-developing mixing layer

The physics of the transition and early turbulence regimes in the time developing mixing layer was investigated. The sensitivity of the mixing layer to the disturbance field of the initial condition is considered. The growth of the momentum thickness, the mean velocity profile, the turbulence kinetic energy, the Reynolds stresses, the anisotropy tensor, and particle track pictures of computations are all examined in an effort to better understand the physics of these regimes. The amplitude, spectrum shape, and random phases of the initial disturbance field were varied. A scheme of generating discrete orthogonal function expansions on some nonuniform grids was developed. All cases address the early or near field of the mixing layer. The most significant result shows that the secondary instability of the mixing layer is produced by spanwise variations in the straining field of the primary vortex structures.

Cain, A. B.↗

NASA NTRS1985

Interpolation Methods for GTD Analysis of Shaped Reflectors

The finding of smooth analytic representations for antenna reflector surfaces which are prescribed only by discretized data obtained by various synthesis methods is examined. Frequently the data are distributed in a nonuniform grid and contain noise. The smoothness required is to C sub 1 for physical optics diffraction analysis and to C sub 2 for geometrical theory of diffraction (GTD) analysis. The GTD analysis approach requires a surface description which returns data very rapidly. Two methods of interpolation, the global and the local methods, are discussed. They each have advantages and disadvantages; characteristics are discussed and examples are presented.

Galindo-Israel, V.↗

NASA NTRS1986

Thermodynamic evaluation of transonic compressor rotors using the finite volume approach

The finite volume explicit time marching method was refined and improved. Previously, extension had been made to the finite volume method to improve the accuracy of the calculation of total pressure in inviscid flow, extend the method to allow the calculation of laminar and turbulent boundary layers in internal flows, and improve the shock capturing properties of the method by introducing a Mach number dependent interpolation scheme for the pressure used in the calculating the density. The current work extends these developments by using the new pressure interpolation scheme in two dimensional viscous calculations, including a more complete description of the viscous stresses, introducing a criteria for the transverse upwind differencing which is a function of the ratio of transverse and streamwise mass fluxes, and allowing the calculation of internal flow where boundary layers are present on both walls of the duct. The manner in which the viscous stresses are evaluated in the nonorthogonal, nonuniform grid is detailed. The convergence is investigated and results for calculations of laminar flow in a converging duct are presented. Results for calculations of transonic flow in a converging-diverging nozzle are presented and the results are compared with Sajben's measurements and calculations by others.

Nicholson, S.↗

NASA NTRS1986

Accuracy of schemes for the Euler equations with non-uniform meshers

The effect of nonuniform grids on the solution of the Euler equations is analyzed. A Runge-Kutta type scheme is considered based on a finite volume formuation. It is shown that for arbitrary grids the scheme can be inconsistent even though it is second-order accurate for uniform grids. An improvement is suggested which leads to at least first-order accuracy for general grids. Test cases are pesented in both two- and three-space dimensions. Applications to finite difference and impicit algorithms are also given.

Turkel, E.↗

NASA NTRS1985

MHD modelling of the earth's magnetosphere

Attention is given to the shock capturing technique, nonuniform grid system, and multiple time scale problem aspects of a global MHD model of the earth's magnetosphere that uses numerical methods. Because of the fast Alfven waves near the earth, the model is more difficult to solve than problems of hydrodynamic flow past bodies. The Rusanov (1962) scheme, which is a variant of the Lax (1954) method that maintains viscosity at a minimum value except where a large one is required, is used. The magnetosphere shape obtained by these means is noted.

Wu, C. C.↗

NASA NTRS1986

Efficient solutions of two-dimensional incompressible steady viscous flows

A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established.

Morrison, J. H.↗

NASA NTRS1988

Efficient solutions of two-dimensional incompressible steady viscous flows

Morrison, J. H.↗

NASA NTRS1993

The hydrodynamics of galaxy formation on Kiloparsec scales

Two dimensional numerical simulations of Zeldovich pancake fragmentation in a dark matter dominated universe were carried out to study the hydrodynamical and gravitational effects on the formation of structures such as protogalaxies. Preliminary results were given in Yuan, Centrella and, Norman (1991). Here we report a more exhaustive study to determine the sensitivity of protogalaxies to input parameters. The numerical code we used for the simulations combines the hydrodynamical code ZEUS-2D (Stone and Norman, 1992) which was modified to include the expansion of the universe and radiative cooling of the gas with a particle-mesh code which follows the motion of dark matter particles. The resulting hybrid code is able to handle highly nonuniform grids which we utilized to obtain a high resolution (much greater than 1 kpc) in the dense region of the pancake.

Norman, Michael L.↗

NASA NTRS1993

Higher-order accurate Osher schemes with application to compressible boundary layer stability

Two fourth order accurate Osher schemes are presented which maintain higher order accuracy on nonuniform grids. They use either a conservative finite difference or finite volume discretization. Both methods are successfully used for direct numerical simulations of flat plate boundary layer instability at different Mach numbers. Results of growth rates of Tollmien-Schlichting waves compare well with direct simulations of incompressible flow and for compressible flow with results obtained by solving the parabolic stability equations.

Van Der Vegt, J. J. W.↗

NASA NTRS1990

Shape design sensitivities using fully automatic 3-D mesh generation

Previous work in three dimensional shape optimization involved specifying design variables by associating parameters directly with mesh points. More recent work has shown the use of fully-automatic mesh generation based upon a parameterized geometric representation. Design variables have been associated with a mathematical model of the part rather than the discretized representation. The mesh generation procedure uses a nonuniform grid intersection technique to place nodal points directly on the surface geometry. Although there exists an associativity between the mesh and the geometrical/topological entities, there is no mathematical functional relationship. This poses a problem during certain steps in the optimization process in which geometry modification is required. For the large geometrical changes which occur at the beginning of each optimization step, a completely new mesh is created. However, for gradient calculations many small changes must be made and it would be too costly to regenerate the mesh for each design variable perturbation. For that reason, a local remeshing procedure has been implemented which operates only on the specific edges and faces associated with the design variable being perturbed. Two realistic design problems are presented which show the efficiency of this process and test the accuracy of the gradient computations.

Botkin, M. E.↗

NASA NTRS1993

Higher-order accurate Osher schemes with application to compressible boundary layer stability

Vandervegt, J. J. W.↗

NASA NTRS1994

Nonlinear hydrodynamics of cosmological sheets. 1: Numerical techniques and tests

We present the numerical techniques and tests used to construct and validate a computer code designed to study the multidimensional nonlinear hydrodynamics of large-scale sheet structures in the universe, especially the fragmentation of such structures under various instabilities. This code is composed of two codes, the hydrodynamical code ZEUS-2D and a particle-mesh code. The ZEUS-2D code solves the hydrodynamical equations in two dimensions using explicit Eulerian finite-difference techniques, with modifications made to incorporate the expansion of the universe and the gas cooling due to Compton scattering, bremsstrahlung, and hydrogen and helium cooling. The particle-mesh code solves the equation of motion for the collisionless dark matter. The code uses two-dimensional Cartesian coordinates with a nonuniform grid in one direction to provide high resolution for the sheet structures. A series of one-dimensional and two-dimensional linear perturbation tests are presented which are designed to test the hydro solver and the Poisson solver with and without the expansion of the universe. We also present a radiative shock wave test which is designed to ensure the code's capability to handle radiative cooling properly. And finally a series of one-dimensional Zel'dovich pancake tests used to test the dark matter code and the hydro solver in the nonlinear regime are discussed and compared with the results of Bond et al. (1984) and Shapiro & Struck-Marcell (1985). Overall, the code is shown to produce accurate and stable results, which provide us a powerful tool to further our studies.

Anninos, Wenbo Y.↗

NASA NTRS1996

Further Development of a New, Flux-Conserving Newton Scheme for the Navier-Stokes Equations

This paper is one of a series of papers describing the development of a new numerical approach for solving the steady Navier-Stokes equations. The key features in the current development are (1) the discrete representation of the dependent variables by way of high order polynomial expansions, (2) the retention of all derivatives in the expansions as unknowns to be explicitly solved for, (3) the automatic balancing of fluxes at cell interfaces, and (4) the discrete simulation of both the integral and differential forms of the governing equations. The main purpose of this paper is, first, to provide a systematic and rigorous derivation of the conditions that are used to simulate the differential form of the Navier-Stokes equations, and second, to extend our previously-presented internal flow scheme to external flows and nonuniform grids. Numerical results are presented for high Reynolds number flow (Re = 100,000) around a finite flat plate, and detailed comparisons are made with the Blasius flat plate solution and Goldstein wake solution. It is shown that the error in the streamwise velocity decreases like r(sup alpha)(Delta)y(exp 2), where alpha approx. 0.25 and r = delta(y)/delta(x) is the grid aspect ratio.

Scott, James R.↗

NASA NTRS2000

On the Treatment of Electric and Magnetic Loss in the Linear Bicharacteristic Scheme for Electromagnetics

The upwind leapfrog or Linear Bicharacteristic Scheme (LBS) has previously been extended to treat lossy dielectric and magnetic materials. This paper examines different methodologies for treatment of the electric loss term in the Linear Bicharacteristic Scheme for computational electromagnetics. Several different treatments of the electric loss term using the LBS are explored and compared on one-dimensional model problems involving reflection from lossy dielectric materials on both uniform and nonuniform grids. Results using these LBS implementations are also compared with the FDTD method for convenience.

Beggs, John H.↗

NASA NTRS2002

A Two-Dimensional Linear Bicharacteristic FDTD Method

The linear bicharacteristic scheme (LBS) was originally developed to improve unsteady solutions in computational acoustics and aeroacoustics. The LBS has previously been extended to treat lossy materials for one-dimensional problems. It is a classical leapfrog algorithm, but is combined with upwind bias in the spatial derivatives. This approach preserves the time-reversibility of the leapfrog algorithm, which results in no dissipation, and it permits more flexibility by the ability to adopt a characteristic based method. The use of characteristic variables allows the LBS to include the Perfectly Matched Layer boundary condition with no added storage or complexity. The LBS offers a central storage approach with lower dispersion than the Yee algorithm, plus it generalizes much easier to nonuniform grids. It has previously been applied to two and three-dimensional free-space electromagnetic propagation and scattering problems. This paper extends the LBS to the two-dimensional case. Results are presented for point source radiation problems, and the FDTD algorithm is chosen as a convenient reference for comparison.

Beggs, John H.↗