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NASA NTRS1990

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Chakravarthy, Sukumar R.↗
NASA NTRS1990

Nonaxisymmetric instabilities in a slender torus - Two- and three-dimensional simulations

Nonaxisymmetric instabilities in accretion disks are investigated in terms of the slender, non-self-gravitating, ideal fluid torus model. Nonlinear simulations are carried out in two and tree dimensions, using Cartesian-grid finite differencing. The fastest growing instability is the principal mode. This mode saturates with the formation of ellipsoidal density distributions, designated as 'planets'. The simulations provide evidence for nonlinear mode-mode coupling. When several wavenumbers m are unstable, multiple 'planets' form that subsequently merge. Sources of numerical error are examined, and the effects are contrasted with those of the physical instability. The same qualitative evolution is seen in both the two- and the three-dimensional simulations, even though strict vertical hydrostatic equilibrium is no longer rigorously maintained after mode saturation.

Hawley, John F.↗
NASA NTRS1991

Analysis of lightning field changes produced by Florida thunderstorms

A new method is introduced for inferring the charges deposited in a lightning flash. Lightning-caused field changes (delta E's) are described by a more general volume charge distribution than is defined on a large cartesian grid system centered above the measuring networks. It is shown that a linear system of equations can be used to relate delta E's at the ground to the values of charge on this grid. It is possible to apply more general physical constraints to the charge solutions, and it is possible to access the information content of the delta E data. Computer-simulated delta E inversions show that the location and symmetry of the charge retrievals are usually consistent with the known test sources.

Koshak, William John↗
NASA NTRS1991

Sonic boom predictions for three generic models using a solution-adaptive full-potential code

The TranAir full-potential code, which utilizes a solution-adaptive, unstructured Cartesian grid, was used to compute near-field pressure signatures of three generic supersonic configurations: a cone-cylinder, a low aspect ratio rectangular wing, and a delta-wing/body. TranAir pressure signatures are extrapolated to near- and mid-field distances using a quasi-linear extrapolation technique. Results are compared with existing wind tunnel data and, where possible, with extrapolated experimental data and quasi-linear Whitham-method results. In general, the extrapolated TranAir data is in good agreement with experimental and Whitham-method pressure signatures, but peak pressures are slightly under-predicted. It is thought that computed data obtained too near to the model was the main reason for the discrepancies. Flow-field grid structure was also found to be important in obtaining accurate offbody data.

Madson, Michael D.↗
NASA NTRS1992

Numerical models for the collapse and fragmentation of centrally condensed molecular cloud cores

The gravitational collapse and fragmentation of centrally condensed molecular cloud cores are investigated using a new hydrodynamical code. The numerical scheme is second-order accurate and uses explicit finite difference methods to advance the fluid variables on a 3D Cartesian grid. Two initial power-law density profiles, rho varies as r exp -1 and rho varies as r exp -2, are considered, as well as two initial density perturbations in the azimuthal coordinate theta, rho-i - rho(1 + a cos 2theta) where a = 0.1 and 0.5. Fragmentation is found to be possible in these centrally condensed cores if the initial conditions also include differential rotation. Models which collapse with initial uniform rotation do not produce fragments. If molecular cloud cores are indeed centrally condensed, as suggested by observations of star-forming regions and by studies of ambipolar diffusion, then differential rotation may be a mechanism for producing binary protostars during gravitational collapse.

Myhill, Elizabeth A.↗
NASA NTRS1992

Sonic boom prediction for the Langley Mach 2 low-boom configuration

Sonic boom pressure signatures and aerodynamic force data for the Langley Mach 2 low sonic boom configuration were computed using the TranAir full-potential code. A solution-adaptive Cartesian grid scheme is utilized to compute off-body flow field data. Computations were performed with and without nacelles at several angles of attack. Force and moment data were computed to measure nacelle effects on the aerodynamic characteristics and sonic boom footprints of the model. Pressure signatures were computed both on and off ground-track. Near-field pressure signature computations on ground-track were in good agreement with experimental data. Computed off ground-track signatures showed that maximum pressure peaks were located off ground-track and were significantly higher than the signatures on ground-track. Bow shocks from the nacelle inlets increased lift and drag, and also increased the magnitude of the maximum pressure both on and off ground-track.

Madson, Michael D.↗
NASA NTRS1993

Ionospheric plasma flow about a system of electrically biased flat plates

The steady state interaction of two electrically biased parallel plates immersed in a flowing plasma characteristic of low earth orbit is studied numerically. Fluid equations are developed to describe the motion of the cold positively charged plasma ions, and are solved using finite-differences in two dimensions on a Cartesian grid. The behavior of the plasma electrons is assumed to be described by the Maxwell-Boltzmann distribution. Results are compared to an analytical and a particle simulation technique for a simplified flow geometry consisting of a single semi-infinite negatively biased plate. Comparison of the extent of the electrical disturbance into the flowing plasma and the magnitude of the current collected by the plate is very good. The interaction of two equally biased parallel plates is studied as a function of applied potential. The separation distance at which the current collected by either plate decreases by five and twenty percent is determined as a function of applied potential. The percent decreases were based on a non-interacting case. The decrease in overall current is caused by a decrease in ionic density in the region between the plates. As the separation between the plates decreases, the plates collect the ions at a faster rate than they are supplied to the middle region by the oncoming plasma flow. The docking of spacecraft in orbit is simulated by moving two plates of unequal potential toward one another in a quasi-static manner. One plate is held at a large negative potential while the other floats electrically in the resulting potential field. It is found that the floating plate does not charge continuously negative as it approaches the other more negatively biased plate. Instead, it charges more and then less negative as ionic current decreases and then increases respectively upon approach. When the two plates come into contact, it is expected that the electrically floating plate will charge rapidly negative to a potential near that of the other plate.

Herr, Joel L.↗
NASA NTRS1993

Implementation of ADI: Schemes on MIMD parallel computers

In order to simulate the effects of the impingement of hot exhaust jets of High Performance Aircraft on landing surfaces a multi-disciplinary computation coupling flow dynamics to heat conduction in the runway needs to be carried out. Such simulations, which are essentially unsteady, require very large computational power in order to be completed within a reasonable time frame of the order of an hour. Such power can be furnished by the latest generation of massively parallel computers. These remove the bottleneck of ever more congested data paths to one or a few highly specialized central processing units (CPU's) by having many off-the-shelf CPU's work independently on their own data, and exchange information only when needed. During the past year the first phase of this project was completed, in which the optimal strategy for mapping an ADI-algorithm for the three dimensional unsteady heat equation to a MIMD parallel computer was identified. This was done by implementing and comparing three different domain decomposition techniques that define the tasks for the CPU's in the parallel machine. These implementations were done for a Cartesian grid and Dirichlet boundary conditions. The most promising technique was then used to implement the heat equation solver on a general curvilinear grid with a suite of nontrivial boundary conditions. Finally, this technique was also used to implement the Scalar Penta-diagonal (SP) benchmark, which was taken from the NAS Parallel Benchmarks report. All implementations were done in the programming language C on the Intel iPSC/860 computer.

Vanderwijngaart, Rob F.↗
NASA NTRS1995

The space-time solution element method: A new numerical approach for the Navier-Stokes equations

This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.

Scott, James R.↗
NASA NTRS1994

A linear method for analyzing lightning field changes

A constrained, least-squares method for analyzing multiple-station measurements of lightning field changes (delta Es) is introduced. Previous methods have attempted to fit the spatial pattern of lightning delta Es using nonlinear models, such as a point charge (Q) or a point dipole (P) model. With the linear method, the delta Es are described not by models but by a general volume charge distribution that is deposited on a large (40 x 40 x 20 cu km) Cartesian grid above the measuring network. A linear system of equations is used to relate the measured delta Es to the charges that are deposited at each grid point. With this approach, the information content of the measurements can be quantified by an eigenanalysis of the covariance matrix of the linear system. Constraints can be used to reduce the infinity of possible solutions to the linear system and also to reduce systematic biases that can be introduced by the method of solution. It is shown that a Landweber iterative method, derived from the general method of steepest descent, can be used to solve the linear system and that the resulting volume charge distributions are generally consistent with computer-simulated charge sources, when these sources are over the measuring network. The Landweber iteration has also provided solutions for natural lightning events that are consistent with Q- and P-model results.

Koshak, William J.↗
NASA NTRS1994

Sonic boom predictions using a solution-adaptive full-potential code

The TranAir full-potential code, which utilizes a solution-adaptive, unstructured Cartesian grid, was used to compute near-field pressure signatures of three generic geometries: (1) a cone-cylinder; (2) a low aspect ratio rectangular wing; and (3) a delta-wing/body. Pressure signatures computed by TranAir are extrapolated to near- and midfield distances using a quasilinear extrapolation technique. Results are compared with wind-tunnel data and, where possible, with extrapolated wind-tunnel data and Whitham-method results. In general, extrapolated TranAir data is in good agreement with experimental and Whitham-method pressure signatures, but peak pressures are slightly underpredicted. It is felt that data computed too close to the model caused some of the discrepancies. The flowfield grid structure was also found to be an important issue in obtaining accurate off-body data. Results were encouraging enough to proceed with the application of TranAir to conceptual configurations in support of the NASA high speed research program (HSRP).

Madson, Michael D.↗
NASA NTRS1996

Multigrid with Overlapping Patches

Solving boundary value problems with optimal efficiency requires adaptivity and multilevel techniques. Previously, an implementation of the AFACx algorithm is presented that is based on rectangular Cartesian grids. This implementation does not allow for the over]ap of grids that lie on the same level of refinement. We investigate the case in which these grids overlap. A standard technique for overlapping grids is the Schwarz algorithm. Some ways of using the Schwarz algorithm in a standard multigrid scheme are presented. Also, a problem that arises in some situations with non-aligned, overlapping grids is described. This situation comes up in a natural way when the Schwarz algorithm is used as a relaxation scheme within a multilevel algorithm. We identify the reason for the bad convergence and show that by more sophisticated interpolation the difficulties can be overcome. Then we present a multiplicative Schwarz algorithm for a large number of grids that has a high potential for parallelization. Finally we give some numerical results for the FACx algorithm with overlapping grids on each refinement level. The implementation of the described codes uses C++ and the array class libraries A++ and P++. Using the A++/P++ programming environment, it was possible to move from a serial code to a parallel code within a few days.

Berndt, Markus↗
NASA NTRS1996

Three-Dimensional Multiscale MHD Model of Cometary Plasma Environments

First results of a three-dimensional multiscale MHD model of the interaction of an expanding cometary atmosphere with the magnetized solar wind are presented. The model starts with a supersonic and super-Alfvenic solar wind far upstream of the comet (25 Gm upstream of the nucleus) with arbitrary interplanetary magnetic field orientation. The solar wind is continuously mass loaded with cometary ions originating from a 10-km size nucleus. The effects of photoionization, electron impact ionization, recombination, and ion-neutral frictional drag are taken into account in the model. The governing equations are solved on an adaptively refined unstructured Cartesian grid using our new multiscale upwind scalar conservation laws-type numerical technique (MUSCL). We have named this the multiscale adaptive upwind scheme for MHD (MAUS-MHD). The combination of the adaptive refinement with the MUSCL-scheme allows the entire cometary atmosphere to be modeled, while still resolving both the shock and the diamagnetic cavity of the comet. The main findings are the following: (1) Mass loading decelerates the solar wind flow upstream of the weak cometary shock wave (M approximately equals 2, M(sub A) approximately equals 2), which forms at a subsolar standoff distance of about 0.35 Gm. (2) A cometary plasma cavity is formed at around 3 x 10(exp 3) km from the nucleus. Inside this cavity the plasma expands outward due to the frictional interaction between ions and neutrals. On the nightside this plasma cavity considerably narrows and a relatively fast and dense cometary plasma beam is ejected into the tail. (3) Inside the plasma cavity a teardrop-shaped inner shock is formed, which is terminated by a Mach disk on the nightside. Only the region inside the inner shock is the 'true' diamagnetic cavity. (4) The model predicts four distinct current systems in the inner coma: the density peak current, the cavity boundary current, the inner shock current, and finally the cross-tail current. (5) The calculated plasma parameters (magnetic field, plasma density, speed, and temperature) are in very good agreement with published Giotto observations.

Gombosi, Tamas I.↗
NASA NTRS1998

Bounded Error Schemes for the Wave Equation on Complex Domains

This paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error.

Abarbanel, Saul↗
NASA NTRS1999

A User's Manual for ROTTILT Solver: Tiltrotor Fountain Flow Field Prediction

A CFD solver has been developed to provide the time averaged details of the fountain flow typical for tiltrotor aircraft in hover. This Navier-Stokes solver, designated as ROTTILT, assumes the 3-D fountain flowfield to be steady and incompressible. The theoretical background is described in this manual. In order to enable the rotor trim solution in the presence of tiltrotor aircraft components such as wing, nacelle, and fuselage, the solver is coupled with a set of trim routines which are highly efficient in CPU and suitable for CFD analysis. The Cartesian grid technique utilized provides the user with a unique capability for insertion or elimination of any components of the bodies considered for a given tiltrotor aircraft configuration. The flowfield associated with either a semi or full-span configuration can be computed through user options in the ROTTILT input file. Full details associated with the numerical solution implemented in ROTTILT and assumptions are presented. A description of input surface mesh topology is provided in the appendices along with a listing of all preprocessor programs. Input variable definitions and default values are provided for the V22 aircraft. Limited predicted results using the coupled ROTTILT/WOPWOP program for the V22 in hover are made and compared with measurement. To visualize the V22 aircraft and predictions, a preprocessor graphics program GNU-PLOT3D was used. This program is described and example graphic results presented.

Tadghighi, Hormoz↗
NASA NTRS2000

An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries

Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS2001

A Regional Model Study of Synoptic Features Over West Africa

Synoptic weather features over West Africa were studied in simulations by the regional simulation model (RM) at the NASA/Goddard Institute for Space Studies. These pioneering simulations represent the beginning of an effort to adapt regional models for weather and climate prediction over West Africa. The RM uses a cartesian grid with 50 km horizontal resolution and fifteen vertical levels. An ensemble of four simulations was forced with lateral boundary conditions from ECMWF global analyses for the period 8-22 August 1988. The simulated mid-tropospheric circulation includes the skillful development and movement of several African wave disturbances. Wavelet analysis of mid-tropospheric winds detected a dominant periodicity of about 4 days and a secondary periodicity of 5-8 days. Spatial distributions of RM precipitation and precipitation time series were validated against daily rain gauge measurements and ISCCP satellite infrared cloud imagery. The time-space distribution of simulated precipitation was made more realistic by combining the ECMWR initial conditions with a 24-hr spin-up of the moisture field and also by damping high frequency gravity waves by dynamic initialization. Model precipitation "forecasts" over the Central Sahel were correlated with observations for about three days, but reinitializing with observed data on day 5 resulted in a dramatic improvement in the precipitation validation over the remaining 9 days. Results imply that information via the lateral boundary conditions is not always sufficient to minimize departures between simulated and actual precipitation patterns for more than several days. In addition, there was some evidence that the new initialization may increase the simulations' sensitivity to the quality of lateral boundary conditions.

Druyan, Leonard M.↗
NASA NTRS2001

Development of a Hybrid RANS/LES Method for Compressible Mixing Layer Simulations

A hybrid method has been developed for simulations of compressible turbulent mixing layers. Such mixing layers dominate the flows in exhaust systems of modem day aircraft and also those of hypersonic vehicles currently under development. The hybrid method uses a Reynolds-averaged Navier-Stokes (RANS) procedure to calculate wall bounded regions entering a mixing section, and a Large Eddy Simulation (LES) procedure to calculate the mixing dominated regions. A numerical technique was developed to enable the use of the hybrid RANS/LES method on stretched, non-Cartesian grids. The hybrid RANS/LES method is applied to a benchmark compressible mixing layer experiment. Preliminary two-dimensional calculations are used to investigate the effects of axial grid density and boundary conditions. Actual LES calculations, performed in three spatial directions, indicated an initial vortex shedding followed by rapid transition to turbulence, which is in agreement with experimental observations.

Georgiadis, Nicholas J.↗