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NASA NTRS1992

Fast neural solution of a nonlinear wave equation

A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.

Toomarian, Nikzad↗

NASA NTRS1993

Numerical simulation of three-dimensional self-gravitating flow

The three-dimensional flow of a self-gravitating fluid is numerically simulated using a Fourier pseudospectral method with a logarithmic variable formulation. Two cases with zero total angular momentum are studied in detail, a 323 simulation (Run B). Other than the grid size, the primary difference between the two cases are that Run A modeled atomic hydrogen and had considerably more compressible motion initially than Run B, which modeled molecular hydrogen. The numerical results indicate that gravitational collapse can proceed in a variety of ways. In the Run A, collapse led to an elongated tube-like structure, while in the Run B, collapse led to a flatter, disklike structure.

Shebalin, John V.↗

NASA NTRS1993

Multi-grid domain decomposition approach for solution of Navier-Stokes equations in primitive variable form

The new multigrid (or adaptive) pseudospectral element method was carried out for the solution of incompressible flow in terms of primitive variable formulation. The desired features of the proposed method include the following: (1) the ability to treat complex geometry; (2) high resolution adapted in the interesting areas; (3) requires minimal working space; and (4) effective in a multiprocessing environment. The approach for flow problems, complex geometry or not, is to first divide the computational domain into a number of fine-grid and coarse-grid subdomains with the inter-overlapping area. Next, it is necessary to implement the Schwarz alternating procedure (SAP) to exchange the data among subdomains, where the coarse-grid correction is used to remove the high frequency error that occurs when the data interpolation from the fine-grid subdomain to the coarse-grid subdomain is conducted. The strategy behind the coarse-grid correction is to adopt the operator of the divergence of the velocity field, which intrinsically links the pressure equation, into this process. The solution of each subdomain can be efficiently solved by the direct (or iterative) eigenfunction expansion technique with the least storage requirement, i.e. O(N(exp 3)) in 3-D and O(N(exp 2)) in 2-D. Numerical results of both driven cavity and jet flow will be presented in the paper to account for the versatility of the proposed method.

Ku, Hwar-Ching↗

NASA NTRS1992

Parallel computational fluid dynamics - Implementations and results

The present volume on parallel CFD discusses implementations on parallel machines, numerical algorithms for parallel CFD, and performance evaluation and computer science issues. Attention is given to a parallel algorithm for compressible flows through rotor-stator combinations, a massively parallel Euler solver for unstructured grids, a fast scheme to analyze 3D disk airflow on a parallel computer, and a block implicit multigrid solution of the Euler equations. Topics addressed include a 3D ADI algorithm on distributed memory multiprocessors, clustered element-by-element computations for fluid flow, hypercube FFT and the Fourier pseudospectral method, and an investigation of parallel iterative algorithms for CFD. Also discussed are fluid dynamics using interface methods on parallel processors, sorting for particle flow simulation on the connection machine, a large grain mapping method, and efforts toward a Teraflops capability for CFD.

Simon, Horst D.↗

NASA NTRS1994

Analysis of subgrid models using direct and large-eddy simulations of isotropic turbulence

Direct and large eddy simulations of forced and decaying isotropic turbulence have been performed using a pseudospectral and a finite-difference code. Subgrid models that include a one-equation subgrid kinetic energy model with and without a stochastic backscatter forcing term and a new scale similarity model have been analyzed in both Fourier space and physical space. The Fourier space analysis showed that the energy transfer across the cutoff wavenumber k(sub c) is dominated by local interaction. The correlation between the exact and the modeled (by a spectral eddy viscosity) nonlinear terms and the subgrid energy transfer in physical space was found to be quite low. In physical space, a similar correlation analysis was carried out using top hat filtering. Results show that the subgrid stress and the energy flux predicted by the subgrid models correlates very well with the exact data. The scale similarity model showed very high correlation for reasonable grid resolution. However, with decrease in grid resolution, the scale similarity model became more uncorrelated, when compared to the kinetic energy subgrid model. The subgrid models were then used for large-eddy simulations for a range of Reynolds number. It was determined that the dissipation was modeled poorly and that the correlation with the exact results was quite low for all the models. In general, for coarse grid resolution, the scale similarity model consistently showed very low correlation while the kinetic energy model showed a relatively higher correlation. These results suggest that to use the scale similarity model relatively fine grid resolution may be required, whereas, the kinetic energy model could be used even in coarse grid.

Menon, S.↗

NASA NTRS1994

Numerical simulation of three-dimensional self-gravitating flow

The three-dimensional flow of a self-gravitating fluid is numerically simulated using a Fourier pseudospectral method with a logarithmic variable formulation. Two cases with zero total angular momentum are studied in detail, a 32(exp 3) simulation (Run A) and a 64(exp 3) simulation (Run B). Other than the grid size, the primary differences between the two cases are that Run A modeled atomic hydrogen and had considerably more compressible motion initially than Run B, which modeled molecular hydrogen. ('Compressible motion' is that part of the velocity which has zero curl, but non-zero divergence). The numerical results indicate that gravitational collapse can proceed in a variety of ways. In Run A, collapse led to an elongated tube-like structure, while in Run B, collapse led to a flatter, disk-like structure.

Shebalin, J. V.↗

NASA NTRS1995

CCD/CID Processors Would Offer Greater Precision

Charge-coupled-device/charge-injection-device (CCD/CID) data processors of proposed type offer advantages of massively parallel computational architecture and high computational speed typical of older CCD/CID data processors, but with increased precision. Useful in performing matrix vector multiplications in variety of applications, including solving partial differential equations, processing signal and image data, control computations, and neural-network simulations. Greater precision of proposed devices help to ensure accuracy in CCD/CID implementations of pseudospectral neural networks - particular class of artificial neural networks especially suited to solving nonlinear differential equations.

Barhen, Jacob↗

NASA NTRS1996

The Multigrid-Mask Numerical Method for Solution of Incompressible Navier-Stokes Equations

A multigrid-mask method for solution of incompressible Navier-Stokes equations in primitive variable form has been developed. The main objective is to apply this method in conjunction with the pseudospectral element method solving flow past multiple objects. There are two key steps involved in calculating flow past multiple objects. The first step utilizes only Cartesian grid points. This homogeneous or mask method step permits flow into the interior rectangular elements contained in objects, but with the restriction that the velocity for those Cartesian elements within and on the surface of an object should be small or zero. This step easily produces an approximate flow field on Cartesian grid points covering the entire flow field. The second or heterogeneous step corrects the approximate flow field to account for the actual shape of the objects by solving the flow field based on the local coordinates surrounding each object and adapted to it. The noise occurring in data communication between the global (low frequency) coordinates and the local (high frequency) coordinates is eliminated by the multigrid method when the Schwarz Alternating Procedure (SAP) is implemented. Two dimensional flow past circular and elliptic cylinders will be presented to demonstrate the versatility of the proposed method. An interesting phenomenon is found that when the second elliptic cylinder is placed in the wake of the first elliptic cylinder a traction force results in a negative drag coefficient.

Ku, Hwar-Ching↗

NASA NTRS1996

The Role of Vortices in the Formation of the Solar System

To show that robust vortices can exist in the solar nebula, a pseudospectral model has been developed to examine the evolution of the vortex in a Keplerian shear. Calculations show that a vortex can exist for 10(exp 4)yr at Jupiter's radius.

Mehta, Anand V.↗

NASA NTRS1996

The Effect of Strain Rate on the Evolution of Plane Wakes Subjected to Irrotational Strains

Direct numerical simulations of time-evolving turbulent plane wakes developing in the presence of irrotational plane strain applied at three different strain rates have been generated. The strain geometry is such that the flow is compressed in the streamwise direction and expanded in the cross-stream direction with the spanwise direction being unstrained. This geometry is the temporally evolving analogue of a spatially evolving wake in an adverse pressure gradient. A pseudospectral numerical method with up to 16 million modes is used to solve the equations in a reference frame moving with the irrotational strain. The initial condition for each simulation is taken from a previous turbulent self-similar plane wake direct numerical simulation at a velocity deficit Reynolds number, Re, of about 2,000. Although the evolutions of many statistics are nearly collapsed when plotted against total strain, there are some differences owing to the different strain rate histories. The impact of strain-rate on the wake spreading rate, the peak velocity deficit, the Reynolds stress profiles, and the flow structure is examined.

Rogers, Michael M.↗

NASA NTRS1995

The Evolution of Plane Wakes Subjected to Irrotational Strains

Three direct numerical simulations of time-evolving turbulent plane wakes developing in the presence of various irrotational plane strains have been generated. A pseudospectral numerical method with up to 26 million modes is used to solve the equations in a reference frame moving with the irrotational strain. The initial condition for each simulation is taken from a previous turbulent self-similar plane wake direct numerical simulation at a velocity deficit Reynolds number, Re, of about 2000. Three different plane strains (of the same magnitude) are imposed. In the first two simulations the strain is in a plane normal to the streamwise wake direction (the two cases having strain of opposite sign); in the third the wake is compressed in the streamwise direction and stretched in the inhomogeneous cross-stream direction. The two flows that are stretched in the cross-stream direction experience an exponential increase of Re; flow visualization indicates many small-scale vortices with little or no organized large-scale structure. In the flow that is compressed in the cross-stream direction Re decays exponentially and the layer appears to be relaminarizing. The evolution of several turbulence statistics in each of these flows is examined.

Rogers, M. R.↗

NASA NTRS1997

Reynolds Stress Balance in Plane Wakes Subjected to Irrotational Strains

Direct numerical simulations of time-evolving turbulent plane wakes developing in the presence of various irrotational plane strains have been generated. A pseudospectral numerical method with up to 25 million modes is used to solve the equations in a reference frame moving with the irrotational strain. The initial condition for each simulation is taken from a previous turbulent self-similar plane wake direct numerical simulation at a velocity deficit Reynolds number, R(sub e), of about 2,000. All the terms in the equations governing the evolution of the Reynolds stresses have been calculated. The relative importance of the various terms is examined for the different strain geometries and the behavior of the individual terms is used to better assess whether the strained wakes are evolving self-similarly.

Rogers, Miichael M.↗

NASA NTRS1998

Scaling Properties of Particle Density Fields Formed in Simulated Turbulent Flows

Direct numerical simulations (DNS) of particle concentrations in fully developed 3D turbulence were carried out in order to study the nonuniform structure of the particle density field. Three steady-state turbulent fluid fields with Taylor microscale Reynolds numbers (Re(sub lambda)) of 40, 80 and 140 were generated by solving the Navier-Stokes equations with pseudospectral methods. Large scale forcing was used to drive the turbulence and maintain temporal stationarity. The response of the particles to the fluid was parameterized by the particle Stokes number St, defined as the ratio of the particle's stopping time to the mean period of eddies on the Kolmogorov scale (eta). In this paper, we consider only passive particles optimally coupled to these eddies (St approx. = 1) because of their tendency to concentrate more than particles with lesser or greater St values. The trajectories of up to 70 million particles were tracked in the equilibrated turbulent flows until the particle concentration field reached a statistically stationary state. The nonuniform structure of the concentration fields was characterized by the multifractal singularity spectrum, f(alpha), derived from measures obtained after binning particles into cells ranging from 2(eta) to 15(eta) in size. We observed strong systematic variations of f(alpha) across this scale range in all three simulations and conclude that the particle concentration field is not statistically self similar across the scale range explored. However, spectra obtained at the 2(eta), 4(eta), and 8(eta) scales of each flow case were found to be qualitatively similar. This result suggests that the local structure of the particle concentration field may be flow-Independent. The singularity spectra found for 2n-sized cells were used to predict concentration distributions in good agreement with those obtained directly from the particle data. This Singularity spectrum has a shape similar to the analogous spectrum derived for the inertial-range energy dissipation fields of experimental turbulent flows at Re(sub lambda) = 110 and 1100. Based on this agreement, and the expectation that both dissipation and particle concentration are controlled by the same cascade process, we hypothesize that singularity spectra similar to the ones found in this work provide a good characterization of the spatially averaged statistical properties of preferentially concentrated particles in higher Re(sub lambda) turbulent flows.

Hogan, Robert C.↗

NASA NTRS2003

Linearly Forced Isotropic Turbulence

Stationary isotropic turbulence is often studied numerically by adding a forcing term to the Navier-Stokes equation. This is usually done for the purpose of achieving higher Reynolds number and longer statistics than is possible for isotropic decaying turbulence. It is generally accepted that forcing the Navier-Stokes equation at low wave number does not influence the small scale statistics of the flow provided that there is wide separation between the largest and smallest scales. It will be shown, however, that the spectral width of the forcing has a noticeable effect on inertial range statistics. A case will be made here for using a broader form of forcing in order to compare computed isotropic stationary turbulence with (decaying) grid turbulence. It is shown that using a forcing function which is directly proportional to the velocity has physical meaning and gives results which are closer to both homogeneous and non-homogeneous turbulence. Section 1 presents a four part series of motivations for linear forcing. Section 2 puts linear forcing to a numerical test with a pseudospectral computation.

Lundgren, T. S.↗

NASA NTRS2004

Optimal Reconfiguration of Tetrahedral Formations

The problem of minimum-fuel formation reconfiguration for the Magnetospheric Multi-Scale (MMS) mission is studied. This reconfiguration trajectory optimization problem can be posed as a nonlinear optimal control problem. In this research, this optimal control problem is solved using a spectral collocation method called the Gauss pseudospectral method. The objective of this research is to provide highly accurate minimum-fuel solutions to the MMS formation reconfiguration problem and to gain insight into the underlying structure of fuel-optimal trajectories.

Huntington, Geoffrey↗

NASA NTRS2007

Zero-Propellant Maneuver[TM] Flight Results for 180 deg ISS Rotation

This paper presents results for the Zero Propellant Maneuver (ZPM) TradeMark attitude control concept flight demonstration. On March 3, 2007, a ZPM was used to reorient the International Space Station 180 degrees without using any propellant. The identical reorientation performed with thrusters would have burned 110lbs of propellant. The ZPM was a pre-planned trajectory used to command the CMG attitude hold controller to perform the maneuver between specified initial and final states while maintaining the CMGs within their operational limits. The trajectory was obtained from a PseudoSpectral solution to a new optimal attitude control problem. The flight test established the breakthrough capability to simultaneously perform a large angle attitude maneuver and momentum desaturation without the need to use thrusters. The flight implementation did not require any modifications to flight software. This approach is applicable to any spacecraft that are controlled by momentum storage devices.

Bedrossian, Nazareth↗

NASA NTRS2009

Tracking Vector Magnetograms with the Magnetic Induction Equation

The differential affine velocity estimator (DAVE) that we developed in 2006 for estimating velocities from line-of-sight magnetograms is modified to directly incorporate horizontal magnetic fields to produce a differential affine velocity estimator for vector magnetograms (DAVE4VM). The DAVE4VM's performance is demonstrated on the synthetic data from the anelastic pseudospectral ANMHD simulations that were used in the recent comparison of velocity inversion techniques by Welsch and coworkers. The DAVE4VM predicts roughly 95% of the helicity rate and 75% of the power transmitted through the simulation slice. Intercomparison between DAVE4VM and DAVE and further analysis of the DAVE method demonstrates that line-of-sight tracking methods capture the shearing motion of magnetic footpoints but are insensitive to flux emergence - the velocities determined from line-of-sight methods are more consistent with horizontal plasma velocities than with flux transport velocities. These results suggest that previous studies that rely on velocities determined from line-of-sight methods such as the DAVE or local correlation tracking may substantially misrepresent the total helicity rates and power through the photosphere.

Schuck, P.↗

NASA NTRS2016

Low-Thrust Transfers from Distant Retrograde Orbits to L2 Halo Orbits in the Earth-Moon System

This paper presents a study of transfers between distant retrograde orbits (DROs) and L2 halo orbits in the Earth-Moon system that could be flown by a spacecraft with solar electric propulsion (SEP). Two collocation-based optimal control methods are used to optimize these highly-nonlinear transfers: Legendre pseudospectral and Hermite-Simpson. Transfers between DROs and halo orbits using low-thrust propulsion have not been studied previously. This paper offers a study of several families of trajectories, parameterized by the number of orbital revolutions in a synodic frame. Even with a poor initial guess, a method is described to reliably generate families of solutions. The circular restricted 3-body problem (CRTBP) is used throughout the paper so that the results are autonomous and simpler to understand.

trajectory design↗