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NASA NTRS1990

Error analysis for reducing noisy wide-gap concentric cylinder rheometric data for nonlinear fluids - Theory and applications

This work discusses the propagation of errors for the recovery of the shear rate from wide-gap concentric cylinder viscometric measurements of non-Newtonian fluids. A least-square regression of stress on angular velocity data to a system of arbitrary functions is used to propagate the errors for the series solution to the viscometric flow developed by Krieger and Elrod (1953) and Pawlowski (1953) ('power-law' approximation) and for the first term of the series developed by Krieger (1968). A numerical experiment shows that, for measurements affected by significant errors, the first term of the Krieger-Elrod-Pawlowski series ('infinite radius' approximation) and the power-law approximation may recover the shear rate with equal accuracy as the full Krieger-Elrod-Pawlowski solution. An experiment on a clay slurry indicates that the clay has a larger yield stress at rest than during shearing, and that, for the range of shear rates investigated, a four-parameter constitutive equation approximates reasonably well its rheology. The error analysis presented is useful for studying the rheology of fluids such as particle suspensions, slurries, foams, and magma.

Borgia, Andrea↗

NASA NTRS2014

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

acceleration of particles↗

NASA NTRS2003

Magnetohydrodynamic Jump Conditions for Oblique Relativistic Shocks with Gyrotropic Pressure

Shock jump conditions, i.e., the specification of the downstream parameters of the gas in terms of the upstream parameters, are obtained for steady-state, plane shocks with oblique magnetic fields and arbitrary flow speeds. This is done by combining the continuity of particle number flux and the electromagnetic boundary conditions at the shock with the magnetohydrodynamic conservation laws derived from the stress-energy tensor. For ultrarelativistic and nonrelativistic shocks, the jump conditions may be solved analytically. For mildly relativistic shocks, analytic solutions are obtained for isotropic pressure using an approximation for the adiabatic index that is valid in high sonic Mach number cases. Examples assuming isotropic pressure illustrate how the shock compression ratio depends on the shock speed and obliquity. In the more general case of gyrotropic pressure, the jump conditions cannot be solved analytically with- out additional assumptions, and the effects of gyrotropic pressure are investigated by parameterizing the distribution of pressure parallel and perpendicular to the magnetic field. Our numerical solutions reveal that relatively small departures from isotropy (e.g., approximately 20%) produce significant changes in the shock compression ratio, r , at all shock Lorentz factors, including ultrarelativistic ones, where an analytic solution with gyrotropic pressure is obtained. In particular, either dynamically important fields or significant pressure anisotropies can incur marked departures from the canonical gas dynamic value of r = 3 for a shocked ultrarelativistic flow and this may impact models of particle acceleration in gamma-ray bursts and other environments where relativistic shocks are inferred. The jump conditions presented apply directly to test-particle acceleration, and will facilitate future self-consistent numerical modeling of particle acceleration at oblique, relativistic shocks; such models include the modification of the fluid velocity profile due to the contribution of energetic particles to the momentum and energy fluxes.

Double, Glen P.↗

NASA NTRS2014

Mission Design and Analysis for Suborbital Intercept and Fragmentation of an Asteroid with Very Short Warning Time

Small near-Earth objects (NEOs) approximately 50-150 m in size are far more numerous (hundreds of thousands to millions yet to be discovered) than larger NEOs. Small NEOs, which are mostly asteroids rather than comets, are very faint in the night sky due to their small sizes, and are, therefore, difficult to discover far in advance of Earth impact. Furthermore, even small NEOs are capable of creating explosions with energies on the order of tens or hundreds of megatons (Mt). We are, therefore, motivated to prepare to respond effectively to short warning time, small NEO impact scenarios. In this paper we explore the lower bound on actionable warning time by investigating the performance of notional upgraded Intercontinental Ballistic Missiles (ICBMs) to carry Nuclear Explosive Device (NED) payloads to intercept and disrupt a hypothetical incoming NEO at high altitudes (generally at least 2500 km above Earth). We conduct this investigation by developing optimal NEO intercept trajectories for a range of cases and comparing their performances. Our results show that suborbital NEO intercepts using Minuteman III or SM-3 IIA launch vehicles could achieve NEO intercept a few minutes prior to when the NEO would strike Earth. We also find that more powerful versions of the launch vehicles (e.g., total delta V of approximately 9.5-11 km/s) could intercept incoming NEOs several hours prior to when the NEO would strike Earth, if launched at least several days prior to the time of intercept. Finally, we discuss a number of limiting factors and practicalities that affect whether the notional systems we describe could become feasible.

near- Earth object↗

NASA NTRS1992

An explicit canopy BRDF model and inversion

Based on a rigorous canopy radiative transfer equation, the multiple scattering radiance is approximated by the asymptotic theory, and the single scattering radiance calculation, which requires an numerical intergration due to considering the hotspot effect, is simplified. A new formulation is presented to obtain more exact angular dependence of the sky radiance distribution. The unscattered solar radiance and single scattering radiance are calculated exactly, and the multiple scattering is approximated by the delta two-stream atmospheric radiative transfer model. The numerical algorithms prove that the parametric canopy model is very accurate, especially when the viewing angles are smaller than 55 deg. The Powell algorithm is used to retrieve biospheric parameters from the ground measured multiangle observations.

Liang, Shunlin↗

NASA NTRS2009

Advanced Computational and Experimental Techniques for Nacelle Liner Performance Evaluation

The Curved Duct Test Rig (CDTR) has been developed to investigate sound propagation through a duct of size comparable to the aft bypass duct of typical aircraft engines. The axial dimension of the bypass duct is often curved and this geometric characteristic is captured in the CDTR. The semiannular bypass duct is simulated by a rectangular test section in which the height corresponds to the circumferential dimension and the width corresponds to the radial dimension. The liner samples are perforate over honeycomb core and are installed on the side walls of the test section. The top and bottom surfaces of the test section are acoustically rigid to simulate a hard wall bifurcation or pylon. A unique feature of the CDTR is the control system that generates sound incident on the liner test section in specific modes. Uniform air flow, at ambient temperature and flow speed Mach 0.275, is introduced through the duct. Experiments to investigate configuration effects such as curvature along the flow path on the acoustic performance of a sample liner are performed in the CDTR and reported in this paper. Combinations of treated and acoustically rigid side walls are investigated. The scattering of modes of the incident wave, both by the curvature and by the asymmetry of wall treatment, is demonstrated in the experimental results. The effect that mode scattering has on total acoustic effectiveness of the liner treatment is also shown. Comparisons of measured liner attenuation with numerical results predicted by an analytic model based on the parabolic approximation to the convected Helmholtz equation are reported. The spectra of attenuation produced by the analytic model are similar to experimental results for both walls treated, straight and curved flow path, with plane wave and higher order modes incident. The numerical model is used to define the optimized resistance and reactance of a liner that significantly improves liner attenuation in the frequency range 1900-2400 Hz. A liner impedance descriptor is used to determine the liner parameters that achieve the optimum impedance.

Gerhold, Carl H.↗

NASA NTRS1977

An approximate method for design and analysis of an ALOHA system

An approximate method for the design and performance prediction of a multiaccess communication system which employs the ALOHA packet-switching technique is developed, based on the use of a diffusion process approximation of an ALOHA-like system (with or without time-slotting). A simple closed-form solution for the variable Q(t), a variant of the number of backlog messages at time t, is given in terms of a few system and user parameters. Final results are expressed in terms of ordinary performance measures such as throughput and average delay. Several numerical examples are given to demonstrate the usefulness of the approximation technique developed.

Kobayashi, H.↗

NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗

NASA NTRS1984

A numerical study of the 2- and 3-dimensional unsteady Navier-Stokes equations in velocity-vorticity variables using compact difference schemes

A compact finite-difference approximation to the unsteady Navier-Stokes equations in velocity-vorticity variables is used to numerically simulate a number of flows. These include two-dimensional laminar flow of a vortex evolving over a flat plate with an embedded cavity, the unsteady flow over an elliptic cylinder, and aspects of the transient dynamics of the flow over a rearward facing step. The methodology required to extend the two-dimensional formulation to three-dimensions is presented.

Gatski, T. B.↗

NASA NTRS1989

On some numerical schemes for transonic flow problems

New second-order-accurate finite-difference approximations for a class of nonlinear PDEs of mixed type, which includes the two-dimensional low-frequency transonic small-disturbance equation (TSD) and full-potential equation (FP), are presented. For the TSD equation, the scheme is implemented via a time-splitting algorithm; the inclusion of flux limiters keeps the total variation nonincreasing and eliminates spurious oscillations near shocks. Global-linear-stability, TVD, and entropy-stability results are proved. Numerical results for the flow over a thin airfoil are presented. Current techniques used to solve the TSD equation may easily be extended to second-order accuracy by this method. For the FP equation, the new scheme requires no subsonic/supersonic switching and no numerical flux biasing. Global linear stability for all values of the Mach number is proved.

Mostrel, Marco Mosche↗

NASA NTRS1994

Evidence for eddy formation in the eastern Arabian Sea during the northeast monsoon

The seasonal formation of a large (500-800 km diameter) anticyclonic eddy in the upper 300-400 m of the eastern Arabian Sea during the northest monsoon period (December-April) is indicated fom hydrographic and satellite altimetry sea level observations, as well as from numerical model experiments. The center of the eddy circulation is approximately 10 deg N, 70 deg E, just to the west of the north-south Laccadive Island chain. In this paper the eddy is called the Laccadive High (LH). In some ways it is like a mirrorlike counterpart to the Great Whirl that develops during the southwest monsoon of the Somali coast (western Arabian Sea). The LH occurs at the same latitude but on the opposite side of the basin during the reversed monsoon. It is different from the Great Whirl, however, in its formation process, its intensity, and its decay. The hydrographic data obtained from surveys all during a single season give sufficiently close station spacing to allow reasonable contouring of the geopotential surfaces and of the properties within and around the LH region with minimum time aliasing. The Geostat altimeter record extends over 4 years, during which the seasonal variability of the LH indicates a dynamic relief of approximately 15-20 cm, which is in good agreement with the hydrographics observations. The altimetry time series also suggests a westward translation of the LH by January with a subsequent dissipation in midbasin. The model used is a wind-forced three-layer primitive equation model which depicts a LH agreement with the timing, position, and amplitude of both the hydrographic and altimetric measurements. The numerical simulation includes a passive tracer located in the Western Bay of Bengal; the western advection of the tracer around the south coasts of Sri Lanka and India in December and January is consistent with the appearance of low-salinity water observed to extend into the Arabian Sea during this period. The modeling studies suggest that both local and remote forcing are important in formation of the LH.

Bruce, John G.↗

NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗

NASA NTRS1984

Numerical solution of Space Shuttle Orbiter flow field including real gas effects

The hypersonic, laminar flow around the Space Shuttle Orbiter has been computed for both an ideal gas (gamma = 1.2) and equilibrium air using a real-gas, parabolized Navier-Stokes code. This code employs a generalized coordinate transformation; hence, it places no restrictions on the orientation of the solution surfaces. The initial solution in the nose region was computed using a 3-D, real-gas, time-dependent Navier-Stokes code. The thermodynamic and transport properties of equilibrium air were obtained from either approximate curve fits or a table look-up procedure. Numerical results are presented for flight conditions corresponding to the STS-3 trajectory. The computed surface pressures and convective heating rates are compared with data from the STS-3 flight.

Prabhu, D. K.↗

NASA NTRS1987

Acoustic gravity waves: A computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.

Hariharan, S. I.↗

NASA NTRS1988

Acoustic gravity waves - A computational approach

Hariharan, S. I.↗

DOE OSTI2020

Application of a Null-Space Monte Carlo Flow Model Set to the Composite Analysis Base Case Fate and Transport Modeling

The Plateau to River Groundwater Model (P2R Model) is a groundwater flow and contaminant fate and transport (F&T) simulation model used to support remedial activities conducted by CH2M HILL Plateau Remediation Company at the Hanford Site in Washington State. Figure 1-1 illustrates the P2R Model extents, discretization, and boundary conditions. The P2R Model is utilized in the Composite Analysis (CA) for the Hanford Site as the computational engine for computing F&T predictions as described in CP-60406, Hanford Site Composite Analysis Technical Approach Description: Groundwater. The model simulates contaminants of concern within the saturated zone of the uppermost aquifer beneath the Central Plateau and downgradient to the Columbia River. CP-57037, Model Package Report for the Plateau to River Model Version 8.3, documents the current version of the P2R Model including a description of the conceptual site model, model development and calibration, and limitations to the model application. The overall objective of the saturated zone modeling effort is to provide a basis for making informed remedial action decisions based on descriptions of current and expected future contaminant concentrations in groundwater at decision points within and downgradient of the Central Plateau of the Hanford Site. Specifically, the purpose of this environmental calculation is to describe the application of the hydraulic property fields and recharge parameters documented in ECF-HANFORD-20-0027, Null Space Monte Carlo Evaluation of the Plateau to River Model, to the CA flow and fate and transport simulation results to quantify the uncertainty in the simulated results due to input parameter selection. Use of numerical groundwater models is always accompanied with uncertainty in the results produced by a model because models are approximations of reality. Thus, by definition, models lack the detail to fully represent observed behavior. Use of numerical techniques, such as a NSMC analysis, can help in identifying and quantifying the potential uncertainties associated with a numerical model such as the P2R Model. The result of NSCM analysis is a set of F&T simulations that provide an estimate of the range of possible outcomes that are used to quantify the uncertainty in simulated concentrations produced using the base case simulations. The simulated concentrations from all simulations will support calculation of the uncertainty of the total dose calculated in a separate calculation.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

NASA NTRS1990

Axions and SN 1987A: Axion trapping

If an axion of mass between about 10(exp -3) eV and 1 eV exists, axion emission would have significantly affected the cooling of the nascent neutron star associated with SN 1987A. For an axion of mass less than about 10(exp -2) eV, axions produced deep inside the neutron star simply stream out; in a previous paper this case has been addressed. Remarkably, for an axion of mass greater than about 10(exp -2) eV axions would, like neutrinos, have a mean-free path that is smaller than the size of a neutron star, and thus would become 'trapped' and radiated from an axion sphere. In this paper the trapping regime is treated by using numerical models of the initial cooling of a hot neutron star that incorporate a leakage approximation scheme for axion-energy transport. The axion opacity is computed due to inverse nucleon-nucleon, axion bremsstrahlung, and numerical models are used to calculate the integrated axion luminosity, the temperature of the axion sphere, and the effect of axion emission on the neutrino bursts detected by the Kamiokande 2 (K2) and Irvine-Michigan-Brookhaven (IMB) water-Cherenkov detectors. The larger the axion mass, the stronger the trapping and the smaller the axion luminosity. The earlier estimate is confirmed and refined of the axion mass above which trapping is so strong that axion emission does not significantly affect the neutrino burst. Based upon the neutrino-burst duration--the most sensitive barometer of axion cooling--it is concluded that for an axion mass of greater than about 0.3 eV, axion emission would not have had a significant effect on the neutrino bursts detected by K2 and IMB. The present work, together with the previous work, strongly suggests that an axion with mass in the interval 10(exp -3) eV to 0.3 eV is excluded by SN 1987A.

Burrows, Adam↗

NASA NTRS1987

Response of joint dominated space structures

An approximate method is developed for estimating the transient response of nonlinear systems in terms of linearized modes of response. Its advantages are that it is computationally more efficient than the time integration method and that it is possible to view the design problem in the more traditional physical terms of modal response. The major drawback of the approximate method is loss of accuracy. It seems that both approximate methods and time integration have their roles in design. Approximate methods provide efficient tools for performing parametric studies and they supply physical insights into how to optimize system performance that are not easily inferred from strictly numerical methods. Time integration provides a method for assessing the accuracy of the approximate solution for key simulations and for fine tuning the final design. In the procedure presented the nonlinear system is approximated by an equivalent linear system in which the system parameters are constant over the range of transient response.

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