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NASA NTRS1993

The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows

We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems.

Lufkin, Eric A.↗

NASA NTRS1995

Time accurate application of the MacCormack 2-4 scheme on massively parallel computers

Many recent computational efforts in turbulence and acoustics research have used higher order numerical algorithms. One popular method has been the explicit MacCormack 2-4 scheme. The MacCormack 2-4 scheme is second order accurate in time and fourth order accurate in space, and is stable for CFL's below 2/3. Current research has shown that the method can give accurate results but does exhibit significant Gibbs phenomena at sharp discontinuities. The impact of adding Jameson type second, third, and fourth order artificial viscosity was examined here. Category 2 problems, the nonlinear traveling wave and the Riemann problem, were computed using a CFL number of 0.25. This research has found that dispersion errors can be significantly reduced or nearly eliminated by using a combination of second and third order terms in the damping. Use of second and fourth order terms reduced the magnitude of dispersion errors but not as effectively as the second and third order combination. The program was coded using Thinking Machine's CM Fortran, a variant of Fortran 90/High Performance Fortran, and was executed on a 2K CM-200. Simple extrapolation boundary conditions were used for both problems.

Hudson, Dale A.↗

NASA NTRS1993

An adaptive mesh-moving and refinement procedure for one-dimensional conservation laws

We examine the performance of an adaptive mesh-moving and /or local mesh refinement procedure for the finite difference solution of one-dimensional hyperbolic systems of conservation laws. Adaptive motion of a base mesh is designed to isolate spatially distinct phenomena, and recursive local refinement of the time step and cells of the stationary or moving base mesh is performed in regions where a refinement indicator exceeds a prescribed tolerance. These adaptive procedures are incorporated into a computer code that includes a MacCormack finite difference scheme wih Davis' artificial viscosity model and a discretization error estimate based on Richardson's extrapolation. Experiments are conducted on three problems in order to qualify the advantages of adaptive techniques relative to uniform mesh computations and the relative benefits of mesh moving and refinement. Key results indicate that local mesh refinement, with and without mesh moving, can provide reliable solutions at much lower computational cost than possible on uniform meshes; that mesh motion can be used to improve the results of uniform mesh solutions for a modest computational effort; that the cost of managing the tree data structure associated with refinement is small; and that a combination of mesh motion and refinement reliably produces solutions for the least cost per unit accuracy.

Biswas, Rupak↗

NASA NTRS1994

A comparison of cosmological hydrodynamic codes

We present a detailed comparison of the simulation results of various hydrodynamic codes. Starting with identical initial conditions based on the cold dark matter scenario for the growth of structure, with parameters h = 0.5 Omega = Omega(sub b) = 1, and sigma(sub 8) = 1, we integrate from redshift z = 20 to z = O to determine the physical state within a representative volume of size L(exp 3) where L = 64 h(exp -1) Mpc. Five indenpendent codes are compared: three of them Eulerian mesh-based and two variants of the smooth particle hydrodynamics 'SPH' Lagrangian approach. The Eulerian codes were run at N(exp 3) = (32(exp 3), 64(exp 3), 128(exp 3), and 256(exp 3)) cells, the SPH codes at N(exp 3) = 32(exp 3) and 64(exp 3) particles. Results were then rebinned to a 16(exp 3) grid with the exception that the rebinned data should converge, by all techniques, to a common and correct result as N approaches infinity. We find that global averages of various physical quantities do, as expected, tend to converge in the rebinned model, but that uncertainites in even primitive quantities such as (T), (rho(exp 2))(exp 1/2) persists at the 3%-17% level achieve comparable and satisfactory accuracy for comparable computer time in their treatment of the high-density, high-temeprature regions as measured in the rebinned data; the variance among the five codes (at highest resolution) for the mean temperature (as weighted by rho(exp 2) is only 4.5%. Examined at high resolution we suspect that the density resolution is better in the SPH codes and the thermal accuracy in low-density regions better in the Eulerian codes. In the low-density, low-temperature regions the SPH codes have poor accuracy due to statiscal effects, and the Jameson code gives the temperatures which are too high, due to overuse of artificial viscosity in these high Mach number regions. Overall the comparison allows us to better estimate errors; it points to ways of improving this current generation ofhydrodynamic codes and of suiting their use to problems which exploit their best individual features.

Kang, Hyesung↗

NASA NTRS1996

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

Gerritsen, Margot↗

NASA NTRS1997

An NPARC Turbulence Module with Wall Functions

The turbulence module recently developed for the NPARC code has been extended to include wall functions. The Van Driest transformation is used so that the wall functions can be applied to both incompressible and compressible flows. The module is equipped with three two-equation K-epsilon turbulence models: Chien, Shih-Lumley and CMOTR models. Details of the wall functions as well as their numerical implementation are reported. It is shown that the inappropriate artificial viscosity in the near-wall region has a big influence on the solution of the wall function approach. A simple way to eliminate this influence is proposed, which gives satisfactory results during the code validation. The module can be easily linked to the NPARC code for practical applications.

Zhu, J.↗

NASA NTRS1998

Three-Dimensional Hydrodynamic Simulations of the Postimpact Proto-Earth

Astrophysical fluid configurations are susceptible to a variety of nonaxisymmetric instabilities under the combined effects of rotation, self-gravity, and thermal pressure. When strong enough, they can induce rapid transport of mass and angular momentum. Our own previous studies of nonaxisymmetric instabilities in model protostars and protostellar disks show that significant transport can occur on orbital timescales and that material can be ejected to large distances. In this contribution, we present three-dimensional simulations of the circumterrestrial debris belt that may have resulted from a giant impact. Our three-dimensional hydrodynamics code with self-gravity and artificial viscosity is fully second-order in space and time; the equations of hydrodynamics and the Poisson equation are solved on an Eulerian cylindrical grid. In the preliminary calculations presented here, we use a simplified EOS where the central proto-Earth is treated as an n = 1/2 polytropic fluid, surrounded by a more compressible, rapidly rotating, fluid disk that represents silicate vapor. Our initial disk parameters are generated from the endstate data of recent smoothed particle hydrodynamics giant-impact calculations. Ultimately, we wish to detennine under what conditions nonaxisymmetric instabilities grow in the postimpact disk and whether they facilitate the transport of material outside the proto-Earth's Roche Limit, leading to the formation of the Moon. In future work, we hope to include a more realistic EOS and the consequences of heating, cooling, and phase transitions.

Pickett, B. K.↗

NASA NTRS2000

Computational Analysis for Rocket-Based Combined-Cycle Systems During Rocket-Only Operation

A series of Reynolds-averaged Navier-Stokes calculations were employed to study the performance of rocket-based combined-cycle systems operating in an all-rocket mode. This parametric series of calculations were executed within a statistical framework, commonly known as design of experiments. The parametric design space included four geometric and two flowfield variables set at three levels each, for a total of 729 possible combinations. A D-optimal design strategy was selected. It required that only 36 separate computational fluid dynamics (CFD) solutions be performed to develop a full response surface model, which quantified the linear, bilinear, and curvilinear effects of the six experimental variables. The axisymmetric, Reynolds-averaged Navier-Stokes simulations were executed with the NPARC v3.0 code. The response used in the statistical analysis was created from Isp efficiency data integrated from the 36 CFD simulations. The influence of turbulence modeling was analyzed by using both one- and two-equation models. Careful attention was also given to quantify the influence of mesh dependence, iterative convergence, and artificial viscosity upon the resulting statistical model. Thirteen statistically significant effects were observed to have an influence on rocket-based combined-cycle nozzle performance. It was apparent that the free-expansion process, directly downstream of the rocket nozzle, can influence the Isp efficiency. Numerical schlieren images and particle traces have been used to further understand the physical phenomena behind several of the statistically significant results.

Steffen, C. J., Jr.↗

NASA NTRS2003

Finite Element Method for Capturing Ultra-relativistic Shocks

While finite element methods are used extensively by researchers solving computational fluid dynamics in fields other than astrophysics, their use in astrophysical fluid simulations has been predominantly overlooked. Current simulations using other methods such as finite difference and finite volume (based on finite difference) have shown remarkable results, but these methods are limited by their fundamental properties in aspects that are important for simulations with complex geometries and widely varying spatial and temporal scale differences. We have explored the use of finite element methods for astrophysical fluids in order to establish the validity of using such methods in astrophysical environments. We present our numerical technique applied to solving ultra-relativistic (Lorentz Factor Gamma >> 1) shocks which are prevalent in astrophysical studies including relativistic jets and gamma-ray burst studies. We show our finite element formulation applied to simulations where the Lorentz factor ranges up to 2236 and demonstrate its stability in solving ultra-relativistic flows. Our numerical method is based on the Flowfield Dependent Variation (FDV) Method, unique in that numerical diffusion is derived from physical parameters rather than traditional artificial viscosity methods. Numerical instabilities account for most of the difficulties when capturing shocks in this regime. Our method results in stable solutions and accurate results as compared with other methods.

Richardson, G. A.↗

NASA NTRS2003

Comparison of the AUSM(+) and H-CUSP Schemes for Turbomachinery Applications

Many turbomachinery CFD codes use second-order central-difference (C-D) schemes with artificial viscosity to control point decoupling and to capture shocks. While C-D schemes generally give accurate results, they can also exhibit minor numerical problems including overshoots at shocks and at the edges of viscous layers, and smearing of shocks and other flow features. In an effort to improve predictive capability for turbomachinery problems, two C-D codes developed by Chima, RVCQ3D and Swift, were modified by the addition of two upwind schemes: the AUSM+ scheme developed by Liou, et al., and the H-CUSP scheme developed by Tatsumi, et al. Details of the C-D scheme and the two upwind schemes are described, and results of three test cases are shown. Results for a 2-D transonic turbine vane showed that the upwind schemes eliminated viscous layer overshoots. Results for a 3-D turbine vane showed that the upwind schemes gave improved predictions of exit flow angles and losses, although the HCUSP scheme predicted slightly higher losses than the other schemes. Results for a 3-D supersonic compressor (NASA rotor 37) showed that the AUSM+ scheme predicted exit distributions of total pressure and temperature that are not generally captured by C-D codes. All schemes showed similar convergence rates, but the upwind schemes required considerably more CPU time per iteration.

Chima, Rodrick V.↗

NASA NTRS1985

Stability, accuracy, and efficiency of some underintegrated methods in finite element computations

In an attempt to increase computational efficiency in the numerical solution of highly nonlinear problems in solid and fluid mechanics, underintegrated finite element methods have been employed by many analysts. Underintegration refers to the use of a rule of an order lower than that required to integrate polynomial integrands exactly. The main drawback of this technique is related to the production of rank-deficient stiffness matrices, or equivalently an expanded kernel of the governing linear momentum operators. Such a development can introduce numerical instabilities. In order to overcome this difficulty, artificial stiffness or viscosity methods, or other stabilization methods have been proposed. One approach involves the elimination of spurious modes in a postprocessing operation. The present study is concerned with this a posteriori elimination method, taking into account the results which can be expected from it, and some of its possible extensions.

Jacquotte, O.-P.↗

NASA NTRS1984

Secular rotational motions and the mechanical structure of a dynamical viscoelastic earth

A survey is presented of analytical methods for computing the linear responses of the rotational axis of a layered viscoelastic earth to surface loading. Theoretical research in this area is first summarized, and the differences between the mechanical boundary conditions to be applied at the interface separating the upper and lower mantles for an adiabatically and chemically stratified mantle are discussed. Some examples of polar wander and secular variation of the spin rate from glacial excitation are presented for various types of chemical and viscosity stratifications. The effects of an artificial density jump at the base of the lithosphere in models are examined, and certain issues concerning the fluid tidal Love number for different types of density stratification are addressed. The meaning of effective plate thickness over geological time scales for rotational dynamics is discussed.

Yuen, D. A.↗

NASA NTRS1992

Prediction of airfoil stall using Navier-Stokes equations in streamline coordinates

A Navier-Stokes procedure to calculate the flow about an airfoil at incidence was developed. The parabolized equations are solved in the streamline coordinates generated for an arbitrary airfoil shape using conformal mapping. A modified k-epsilon turbulence model is applied in the entire domain, but the eddy viscosity in the laminar region is suppressed artificially to simulate the region correctly. The procedure was applied to airfoils at various angles of attack, and the results are quite satisfactory for both laminar and turbulent flows. It is shown that the present choice of the coordinate system reduces the error due to numerical diffusion, and that the lift is accurately predicted for a wide range of incidence.

Choi, D. H.↗

NASA NTRS1988

The effects of rotational flow, viscosity, thickness, and shape on transonic flutter dip phenomena

The transonic flutter dip phenomena on thin airfoils, which are employed for propfan blades, is investigated using an integrated Euler/Navier-Stokes code and a two degrees of freedom typical section structural model. As a part of the code validation, the flutter characteristics of the NACA 64A010 airfoil are also investigated. In addition, the effects of artificial dissipation models, rotational flow, initial conditions, mean angle of attack, viscosity, airfoil thickness and shape on flutter are investigated. The results obtained with a Euler code for the NACA 64A010 airfoil are in reasonable agreement with published results obtained by using transonic small disturbance and Euler codes. The two artificial dissipation models, one based on the local pressure gradient scaled by a common factor and the other based on the local pressure gradient scaled by a spectral radius, predicted the same flutter speeds except in the recovery region for the case studied. The effects of rotational flow, initial conditions, mean angle of attack, and viscosity for the Reynold's number studied seem to be negligible or small on the minima of the flutter dip.

Reddy, T. S. R.↗

NASA NTRS1988

The effects of rotational flow, viscosity, thickness, and shape on transonic flutter dip phenomena

The transonic flutter dip phenomena on thin airfoils, which are employed for propfan blades, is investigated using an integrated Euler/Navier-Stokes code and a two degrees of freedom typical section structural model. As a part of the code validation, the flutter characteristics of the NACA 64A010 airfoil are also investigated. In addition, the effects of artificial dissipation models, rotational flow, initial conditions, mean angle of attack, viscosity, airfoil thickness, and shape on flutter are investigated. The results obtained with an Euler code for the NACA 64A010 airfoil are in reasonable agreement with published results obtained by using transonic small disturbance and Euler codes. The two artifical dissipation models, one based on the local pressure gradient scaled by a common factor and the other based on the local presure gradient scaled by a spectral radius, predicted the same flutter speeds except in the recovery region for the case studied. The effects of rotation flow, initial conditions, mean angle of attack, and viscosity for the Reynolds number studied seem to be negligible or small on the minima of the flutter dip.

Reddy, T. S. R.↗

NASA NTRS1992

Viscosity determinations of some frictionally generated silicate melts: Implications for slip zone rheology during impact-induced faulting

Analytical scanning electron microscopy, using combined energy dispersive and wavelength dispersive spectrometry, was used to determine the major-element compositions of some natural and artificial glasses and their crystalline equivalents derived by the frictional melting of acid to intermediate protoliths. The major-element compositions are used to calculate the viscosities of their melt precursors using the model of Shaw at temperatures of 800-1400 C, with Fe(2+)/Fe(tot) = 0.5 and for 1-3 wt percent H2O. These results are then modified to account for suspension effects in order to determine viscosities. The results have implications for the generation of pseudotachylitic breccias as seen in the basement lithologies of the Sudbury and Vredefort structures and possibly certain dimict lunar breccias. Many of these breccias show similarities with the more commonly developed pseudotachylite fault and injection veins seen in endogenic fault zones that typically occur in thicknesses of a few centimeters or less. The main difference is one of scale: Impact-induced pseudotachylite breccias can attain several meters in thickness. This would suggest that they were generated under exceptionally high slip rates and hence high strain rates and that the friction melts generated possessed extremely low viscosities.

Spray, John G.↗

NASA NTRS2004

Waves on Seas of Mars and Titan: Wind-Tunnel Experiments on Wind-Wave Generation in Extraterrestrial Atmospheres

The generation of waves by winds across Earth's water oceans is a topic of enduring fascination. However, the physics of the problem are rather forbidding and thus the relationships between real-world windspeed and sea state tend to be empirical. Such empirical relations are of limited utility in environments where the physical parameters are different, such as the surfaces of other planets. These environments have only recently come to oceanographers attention, with the discovery of ancient shorelines and lakes on Mars, and the prospects for and recent evidence of lakes and seas of liquid hydrocarbons on Saturn's moon Titan. We are aware of only one other published experimental wind-water tunnel study where the fluid parameters have been varied. This used artificially-generated mm-scale waves at 3.8-7.6 Hz in water, glycerol solutions (higher viscosity) and surfactant solutions (lower surface tension). Lower viscosity solutions had higher wave growth rates: surprisingly, higher surface tension led to more rapid wave growth. The liquid density was not appreciably varied, and 1 bar air was used throughout.We used the MARSWIT (Mars Wind Tunnel) operated by ASU at NASA Ames. A fiberglass tray (5 cm x 120 cm x 75 cm) was installed in the tunnel, with an approx. 1:5 ramp to prevent strong flow separation. The tray was filled to a depth of about 4 cm. Sensors were clamped to the tray itself or held by a steel and aluminium frame just above the water level. A towel was draped on the water surface at the downwind end of the tray to act as a damper to suppress wave reflection. Position-sensitive infrared (IR) reflection sensors (Sharp GP12D02) and ultrasonic rangers (Devantech DF-04) used in mobile robotics were used as water level sensors. The tray was observed with a video camera, whose output could be viewed on a monitor and recorded on VHS tape.

Lorenz, R. D.↗

NASA NTRS1997

The Implicit and Explicit alpha-mu Schemes

Artificial numerical dissipation is an important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The method of space-time conservation element and solution element, currently under development, is a nontraditional numerical method for solving conservation laws. The method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. Explicit solvers for model and fluid dynamic conservation laws have previously been investigated. In this paper, we introduce a new concept in the design of implicit schemes, and use it to construct two highly accurate solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to purely inviscid (convective) equations. The stability and consistency of the schemes are analyzed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme. We also construct an explicit solver with the treatment of diffusion being based on that in the implicit solvers. The explicit solver has only a CFL stability limitation on the Courant number, yet it retains the second-order spatial accuracy of the implicit schemes.

Chang, Sin-Chung↗