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All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

Numerical Calculation of the Drag Force Acting on a Solid Particle Pushed by a Solid/Liquid Interface

The distribution of insoluble particles in a metal casting depends primarily on the interaction of the particles with the solid/liquid interface (SLI) during the solidification process. The balance of the forces acting on the particle essentially determines whether a particle will be engulfed or pushed by the SLI. An important component of this force balance is the drag force generated by the particle motion in front of the SLI. Previously developed mathematical models for particle/SLI interaction made use of steady-state solutions of this force provided by the lubrication theory. However, our numerical model based on the SLI tracking approach shows that the steady-state approach is inappropriate to model the interaction process and that at steady-state the theoretical solution underestimates the drag force. It was found that regression analysis of steady-state numerical solutions for cylindrical particles moving normal to a flat SLI gives a relationship of the form: Abstract The distribution of insoluble particles in a metal casting depends primarily on the interaction of the particles with the solid/liquid interface (SLI) during the solidification process. The balance of the forces acting on the particle essentially determines whether a particle will be engulfed or pushed by the SLI. An important component of this force balance is the drag force generated by the particle motion in front of the SLI. Previously developed mathematical models for particle/SLI interaction made use of steady-state solutions of this force provided by the lubrication theory. However, our numerical model based on the SLI tracking approach shows that the steady-state approach is inappropriate to model the interaction process and that at steady-state the theoretical solution underestimates the drag force. It was found that regression analysis of steady-state numerical solutions for cylindrical particles moving normal to a flat SLI gives a relationship of the form: F(sub D, sup num) =sqoare root of 3(pi)(eta)V(sub p)(R(sub p)/d)(sup 10(gamma)/3). This is to compared to the solution provided by the lubrication theory: F(sub D), sup theor) = 3 square root of 2(pi)(eta)V(sub p)(R(sub p)/d)(sub 3/2), where F(sub D) is the drag force, eta is the dynamic viscosity of the fluid, V(sub p) is the particle velocity, R(sub p) is the particle radius, gamma is Euler's constant (gamma approx. = 0.577), and d is the width of the gap between the particle and the SLI. The domain on which the equations proposed by the lubrication theory are relevant is clearly identified in the paper. The numerical model was then validated against the classical lubrication theory within the domain of its validity.

Catalina, Adrian V.↗
NASA NTRS2000

Recommendations for Achieving Accurate Numerical Simulation of Tip Clearance Flows in Transonic Compressor Rotors

The tip clearance flows of transonic compressor rotors are important because they have a significant impact on rotor and stage performance. While numerical simulations of these flows are quite sophisticated. they are seldom verified through rigorous comparisons of numerical and measured data because these kinds of measurements are rare in the detail necessary to be useful in high-speed machines. In this paper we compare measured tip clearance flow details (e.g. trajectory and radial extent) with corresponding data obtained from a numerical simulation. Recommendations for achieving accurate numerical simulation of tip clearance flows are presented based on this comparison. Laser Doppler Velocimeter (LDV) measurements acquired in a transonic compressor rotor, NASA Rotor 35, are used. The tip clearance flow field of this transonic rotor was simulated using a Navier-Stokes turbomachinery solver that incorporates an advanced k-epsilon turbulence model derived for flows that are not in local equilibrium. Comparison between measured and simulated results indicates that simulation accuracy is primarily dependent upon the ability of the numerical code to resolve important details of a wall-bounded shear layer formed by the relative motion between the over-tip leakage flow and the shroud wall. A simple method is presented for determining the strength of this shear layer.

VanZante, Dale E.↗
NASA NTRS1999

Entropy Splitting and Numerical Dissipation

A rigorous stability estimate for arbitrary order of accuracy of spatial central difference schemes for initial-boundary value problems of nonlinear symmetrizable systems of hyperbolic conservation laws was established recently by Olsson and Oliger (1994) and Olsson (1995) and was applied to the two-dimensional compressible Euler equations for a perfect gas by Gerritsen and Olsson (1996) and Gerritsen (1996). The basic building block in developing the stability estimate is a generalized energy approach based on a special splitting of the flux derivative via a convex entropy function and certain homogeneous properties. Due to some of the unique properties of the compressible Euler equations for a perfect gas, the splitting resulted in the sum of a conservative portion and a non-conservative portion of the flux derivative. hereafter referred to as the "Entropy Splitting." There are several potential desirable attributes and side benefits of the entropy splitting for the compressible Euler equations that were not fully explored in Gerritsen and Olsson. The paper has several objectives. The first is to investigate the choice of the arbitrary parameter that determines the amount of splitting and its dependence on the type of physics of current interest to computational fluid dynamics. The second is to investigate in what manner the splitting affects the nonlinear stability of the central schemes for long time integrations of unsteady flows such as in nonlinear aeroacoustics and turbulence dynamics. If numerical dissipation indeed is needed to stabilize the central scheme, can the splitting help minimize the numerical dissipation compared to its un-split cousin? Extensive numerical study on the vortex preservation capability of the splitting in conjunction with central schemes for long time integrations will be presented. The third is to study the effect of the non-conservative proportion of splitting in obtaining the correct shock location for high speed complex shock-turbulence interactions. The fourth is to determine if this method can be extended to other physical equations of state and other evolutionary equation sets. If numerical dissipation is needed, the Yee, Sandham, and Djomehri (1999) numerical dissipation is employed. The Yee et al. schemes fit in the Olsson and Oliger framework.

Yee, H. C.↗
NASA NTRS2001

The Numerical Analysis of a Turbulent Compressible Jet

A numerical method to simulate high Reynolds number jet flows was formulated and applied to gain a better understanding of the flow physics. Large-eddy simulation was chosen as the most promising approach to model the turbulent structures due to its compromise between accuracy and computational expense. The filtered Navier-Stokes equations were developed including a total energy form of the energy equation. Subgrid scale models for the momentum and energy equations were adapted from compressible forms of Smagorinsky's original model. The effect of using disparate temporal and spatial accuracy in a numerical scheme was discovered through one-dimensional model problems and a new uniformly fourth-order accurate numerical method was developed. Results from two- and three-dimensional validation exercises show that the code accurately reproduces both viscous and inviscid flows. Numerous axisymmetric jet simulations were performed to investigate the effect of grid resolution, numerical scheme, exit boundary conditions and subgrid scale modeling on the solution and the results were used to guide the three-dimensional calculations. Three-dimensional calculations of a Mach 1.4 jet showed that this LES simulation accurately captures the physics of the turbulent flow. The agreement with experimental data was relatively good and is much better than results in the current literature. Turbulent intensities indicate that the turbulent structures at this level of modeling are not isotropic and this information could lend itself to the development of improved subgrid scale models for LES and turbulence models for RANS simulations. A two point correlation technique was used to quantify the turbulent structures. Two point space correlations were used to obtain a measure of the integral length scale, which proved to be approximately 1/2 D(sub j). Two point space-time correlations were used to obtain the convection velocity for the turbulent structures. This velocity ranged from 0.57 to 0.71 U(sub j).

DeBonis, James R.↗
NASA NTRS1994

Numerical Generation of Asymmetric Flows about Slender Bodies of Revolution at Extreme Incidence

It is well known that slender bodies of revolution will develop an asymmetric, unsteady flow pattern in experimental tests, if the angle of incidence to the oncoming stream is above a critical value. It has been suggested that the origin of these asymmetric flows may stem from geometric imperfections of the model being tested, or from disturbances in the oncoming stream. In numerical simulations, it is possible to generate bodies of revolution which are perfectly symmetric about their longitudinal axis, and to impose uniform flow conditions which are free from disturbances. The current work presents numerical simulations of the flow about an ogive-cylinder configuration at 40 and 60 degree angle of incidence. These simulations. were performed using numerical algorithms which are also symmetric about the lateral plane of the cylinder body. The flowfields at 40 degree angle of attack were seen to remain symmetric to the round-off accuracy of the computer. At 60 degree angle of attack, a lateral force coefficient developed of O(1) which progressed to an alternate vortex shedding in time. The nature of this lateral force generation and vortex shedding was dependent on the choice of numerical algorithm. The origin of the asymmetries observed in the 60 degree angle of attack computations were traced to round-off errors in the implicit block-matrix inverter. A means of inverting the implicit operator matrices, which maintains the symmetry of the overall numerical algorithm was implemented.

Murman, Scott M.↗
NASA NTRS1995

Some Numerical Experiments on Detonation Wave Propagation

In this paper we present the results of a series of numerical experiments done on the propagation and initiation of a detonation wave. The calculations are performed in one-dimension, with considerable grid resolution. Of particular interest are the following questions: (1) the nature of periodic and chaotic instabilities generated by the wave; (2) the influence of the grid resolution on these instabilities; (3) the influence of the 'quality' of the numerical scheme; and (4) the influence of 'noise'. In the calculations, we use a second-order Total Variation Diminishing (TVD) scheme as the basic numerical method, with grid spacings as low as a fraction of a micron. Detonations waves are generated at the closed end of a tube, and allowed to propagate for approximately 20 cm. The required energy for successful initiation of the detonation will be measured for different cases of grid resolution and numerical schemes. A modified version of the TVD scheme has also been devised, which allows for much lower numerical diffusion of the radical species in the exponentially growing region behind the shock. The effect of this modification will be demonstrated. Oscillations in peak pressure and induction length are seen to develop in some cases: the oscillations can go through a sequence of modes, from a regular, high frequency mode to a low frequency mode with period doubling. A chaotic regime can also be obtained. General conclusions on the quality of algorithms will be presented. We will also discuss the performance of a version of the code developed on the IBM SP2 parallel computer.

Cambier, Jean-Luc↗
NASA NTRS2001

Numerical Simulation of One- and Two-Phase Flows in Propulsion Systems

In this report, we present some results of problems investigated during joint research between the Hampton University Fluid Mechanics and Acoustics Laboratory (HU/FM&AL), NASA GRC, and the LaRC Hyper-X Program. This work is supported by joint research between the NASA GRC and the Institute of Mechanics at Moscow State University (IM/MSU) in Russia under a CRDF grant. The main areas of current scientific interest of the HU/FM&AL include an investigation of the proposed and patented advanced methods for aircraft engine thrust and noise benefits. These methods are based on nontraditional 3D corrugated and composite nozzle, inlet, propeller and screw designs such as a Bluebell and Telescope nozzle, Mobius-shaped screw, etc. This is the main subject of our other projects, of which one is presented at the current conference. Here we analyze additional methods for exhaust jet noise reduction without essential thrust loss and even with thrust augmentation. Such additional approaches are: (1) to add some solid, fluid, or gas mass at discrete locations to the main supersonic gas stream to minimize the negative influence of strong shock waves formed in propulsion systems. This mass addition may be accompanied by heat addition to the main stream as a result of the fuel combustion or by cooling of this stream as a result of the liquid mass evaporation and boiling; (2) Use of porous or permeable nozzles and additional shells at the nozzle exit for preliminary cooling of the hot jet exhaust and pressure compensation for off-design conditions (so-called continuous ejector with small mass flow rate); and (3) to propose and analyze new effective methods of fuel injection into the flow stream in air-breathing engines. The research is focused on a wide regime of problems in the propulsion field as well as in experimental testing and theoretical and numerical simulation analyses for advanced aircraft and rocket engines. The FM&AL Team uses analytical methods, numerical simulations, and experimental tests at the Hampton University campus, NASA, and IM/MSU. The main outcomes during this reporting period are: (1) Publications: The AIAA Paper #01-1893 has been accepted for the AIAA/NAL-NASDA-ISAS 10th International Space Planes and Hypersonic Systems and Technologies Conference, 24-27 April 2001, Kyoto, Japan. The AIAA Paper #01-3204 has been accepted for presentation at the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference being held on 08-11 July, in Salt Lake City, UT; (2) Grants and proposals: The HU/FM&AL was awarded the NASA grant NAG3-2495 in October 2000 and the CRDF award was granted to the NASA GRC-HU FM&AL and IM/MSU (Russia) in July 2000. A solicited proposal was submitted for the NASA NRA-01GRC-02 competition and two unsolicited proposals to NASA are in preparation; (3) Theory and numerical simulations: Analytical theory, numerical simulation, comparison of theoretical with experimental results, and modification of theoretical approaches, models, grids, etc. Such investigations have been conducted for three main problems: (a) Combustion efficiency optimization in the half-duct combustor system; (b) Drag reduction effects for blunt bodies with solid needles; and (c) Solid particle injection from the butt-end against a supersonic flow. The NASA CFL3D, HU/FM&AL, and IM/MSU GODUNOV codes were used and modified for solution of these problems. The codes are based on full Euler and Navier-Stokes solvers with and without nonequilibrium oxygen-nitrogen and air-hydrogen chemical reactions in laminar and turbulent gas flow regimes; (4) Experimental tests: In the IM/MSU supersonic wind tunnel, experimental tests were conducted with different number, location, and geometric parameters of solid needles mounted at the front of the butt-end in supersonic flow. Optimal parameters were determined that provide minimal butt-end drag in the stationary flow regime. Experimental and numerical simulation results are in good agreement; (5) Student Research Activity: Involvement of one graduate and two undergraduate students as research assistants in the current project.

Gilinsky, Mikhail↗
NASA NTRS1995

Numerical Simulations of High Enthalpy Pulse Facilities

Axisymmetric flows within shock tubes and expansion tubes are simulated including the effects of finite rate chemistry and both laminar and turbulent boundary layers. The simulations demonstrate the usefulness of computational fluid dynamics for characterizing the flows in high enthalpy pulse facilities. The modeling and numerical requirements necessary to simulate these flows accurately are also discussed. Although there is a large body of analysis which explains and quantifies the boundary layer growth between the shock and the interface in a shock tube, there is a need for more detailed solutions. Phenomena such as thermochemical nonequilibrium. or turbulent transition behind the shock are excluded in the assumptions of Mirels' analysis. Additionally there is inadequate capability to predict the influence of the boundary layer on the expanded gas behind the interface. Quantifying the gas in this region is particularly important in expansion tubes because it is the location of the test gas. Unsteady simulations of the viscous flow in shock tubes are computationally expensive because they must follow features such as a shock wave over the length of the facility and simultaneously resolve the small length scales within the boundary layer. As a result, efficient numerical algorithms are required. The numerical approach of the present work is to solve the axisymmetric gas dynamic equations using an finite-volume formulation where the inviscid fluxes are computed with a upwind TVD scheme. Multiple species equations are included in the formulation so that finite-rate chemistry can be modeled. The simulations cluster grid points at the shock and interface and translate this clustered grid with these features to minimize numerical errors. The solutions are advanced at a CFL number of less than one based on the inviscid gas dynamics. To avoid limitations on the time step due to the viscous terms, these terms are treated implicitly. This requires a block tri-diagonal matrix inversion along each line of cells normal to the wall. The cost of this inversion is more than offset by the larger allowable time step. The source terms representing the finite-rate chemical kinetics are also treated implicitly. An algebraic turbulence model for compressible flow is used. The flow in a low pressure shock tube is computed and the results are compared with Mirels'analysis. The driven gas is nitrogen at 70 Pa, and the incident shock speed is approximately 2.9 km/sec so that there is little dissociation. The simulations include a laminar boundary layer and are run until the limiting flow regime is achieved. At this limit, the shock and interface travel at the same velocity because the amount of driven gas between these two features remains the same: the mass flow across the shock is equal to the mass of gas being entrained at the interface by the boundary layer. Simulations with several grids are presented to establish the grid independence of the solution, Good agreement is achieved between Mirels' correlations and the computations. This is expected since the flow conditions are chosen to be consistent with the assumptions used in Mirels' analysis. This comparison adds credibility to the numerical approach and highlights some of the differences between the theory and the detailed simulations. In addition, simulations of the HYPULSE expansion tube are presented for two operating conditions and the computations are compared to experimental data. The operating gas for both cases is nitrogen. One test condition is at a total enthalpy of 15.2 MJ/Kg and a relatively low pressure of 2 kPa. This case is characterized by a laminar boundary layer and significant chemical nonequilibrium. in the acceleration gas. The second test condition is at a total enthalpy of 10.2 MJ/Kg and a pressure of 38 kPa and is characterized by a turbulent boundary layer. The simulations compare well with experiment and reveal that the nonuniformity in pressure observed during the test time is related to variations in the boundary layer displacement thickness.

Wilson, Gregory J.↗
NASA NTRS2001

Numerical Simulation of Two Phase Flows

Two phase flows can be found in broad situations in nature, biology, and industry devices and can involve diverse and complex mechanisms. While the physical models may be specific for certain situations, the mathematical formulation and numerical treatment for solving the governing equations can be general. Hence, we will require information concerning each individual phase as needed in a single phase. but also the interactions between them. These interaction terms, however, pose additional numerical challenges because they are beyond the basis that we use to construct modern numerical schemes, namely the hyperbolicity of equations. Moreover, due to disparate differences in time scales, fluid compressibility and nonlinearity become acute, further complicating the numerical procedures. In this paper, we will show the ideas and procedure how the AUSM-family schemes are extended for solving two phase flows problems. Specifically, both phases are assumed in thermodynamic equilibrium, namely, the time scales involved in phase interactions are extremely short in comparison with those in fluid speeds and pressure fluctuations. Details of the numerical formulation and issues involved are discussed and the effectiveness of the method are demonstrated for several industrial examples.

Liou, Meng-Sing↗
NASA NTRS1997

Numerical Simulation of Unsteady Aerodynamic Models

This report documents the results of the numerical simulations of unsteady aerodynamic models. The results focus on numerical accuracy and efficiency, and the robustness of the numerical methods. The aerodynamic models includes the classical Wagner and Kussner functions and the Leishman-Beddoes dynamic stall model. The simulations includes the numerical approximations of the Duhamel's integrals using both indicial (step) and impulse responses, the numerical integrations of the state-space models, and the exact solutions. The report also presents the conversion among different model representations.

Nguyen, Khanh Q.↗
NASA NTRS1998

A Polynomial Time, Numerically Stable Integer Relation Algorithm

Let x = (x1, x2...,xn be a vector of real numbers. X is said to possess an integer relation if there exist integers a(sub i) not all zero such that a1x1 + a2x2 + ... a(sub n)Xn = 0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the a(sub i) given x. The most efficient of these existing integer relation algorithms (in terms of run time and the precision required of the input) has the drawback of being very unstable numerically. It often requires a numeric precision level in the thousands of digits to reliably recover relations in modest-sized test problems. We present here a new algorithm for finding integer relations, which we have named the "PSLQ" algorithm. It is proved in this paper that the PSLQ algorithm terminates with a relation in a number of iterations that is bounded by a polynomial in it. Because this algorithm employs a numerically stable matrix reduction procedure, it is free from the numerical difficulties, that plague other integer relation algorithms. Furthermore, its stability admits an efficient implementation with lower run times oil average than other algorithms currently in Use. Finally, this stability can be used to prove that relation bounds obtained from computer runs using this algorithm are numerically accurate.

Ferguson, Helaman R. P.↗
NASA NTRS2002

Numerical Simulations of Vortex Generator Vanes and Jets on a Flat Plate

Numerical simulations of a single low-profile vortex generator vane, which is only a small fraction of the boundary-layer thickness, and a vortex generating jet have been performed for flows over a flat plate. The numerical simulations were computed by solving the steady-state solution to the Reynolds-averaged Navier-Stokes equations. The vortex generating vane results were evaluated by comparing the strength and trajectory of the streamwise vortex to experimental particle image velocimetry measurements. From the numerical simulations of the vane case, it was observed that the Shear-Stress Transport (SST) turbulence model resulted in a better prediction of the streamwise peak vorticity and trajectory when compared to the Spalart-Allmaras (SA) turbulence model. It is shown in this investigation that the estimation of the turbulent eddy viscosity near the vortex core, for both the vane and jet simulations, was higher for the SA model when compared to the SST model. Even though the numerical simulations of the vortex generating vane were able to predict the trajectory of the stream-wise vortex, the initial magnitude and decay of the peak streamwise vorticity were significantly under predicted. A comparison of the positive circulation associated with the streamwise vortex showed that while the numerical simulations produced a more diffused vortex, the vortex strength compared very well to the experimental observations. A grid resolution study for the vortex generating vane was also performed showing that the diffusion of the vortex was not a result of insufficient grid resolution. Comparisons were also made between a fully modeled trapezoidal vane with finite thickness to a simply modeled rectangular thin vane. The comparisons showed that the simply modeled rectangular vane produced a streamwise vortex which had a strength and trajectory very similar to the fully modeled trapezoidal vane.

Allan, Brian G.↗
NASA NTRS2001

Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations

The recently developed essentially fourth-order or higher low dissipative shock-capturing scheme of Yee, Sandham and Djomehri (1999) aimed at minimizing nu- merical dissipations for high speed compressible viscous flows containing shocks, shears and turbulence. To detect non smooth behavior and control the amount of numerical dissipation to be added, Yee et al. employed an artificial compression method (ACM) of Harten (1978) but utilize it in an entirely different context than Harten originally intended. The ACM sensor consists of two tuning parameters and is highly physical problem dependent. To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed. The new sensors are derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch to the extra numerical dissipation outside shock layers. The non-dissipative spatial base scheme of arbitrarily high order of accuracy can be maintained without compromising its stability at all parts of the domain where the solution is smooth. Two types of redundant non-orthogonal wavelet basis functions are considered. One is the B-spline wavelet (Mallat & Zhong 1992) used by Gerritsen and Olsson (1996) in an adaptive mesh refinement method, to determine regions where re nement should be done. The other is the modification of the multiresolution method of Harten (1995) by converting it to a new, redundant, non-orthogonal wavelet. The wavelet sensor is then obtained by computing the estimated Lipschitz exponent of a chosen physical quantity (or vector) to be sensed on a chosen wavelet basis function. Both wavelet sensors can be viewed as dual purpose adaptive methods leading to dynamic numerical dissipation control and improved grid adaptation indicators. Consequently, they are useful not only for shock-turbulence computations but also for computational aeroacoustics and numerical combustion. In addition, these sensors are scheme independent and can be stand alone options for numerical algorithm other than the Yee et al. scheme.

Sjoegreen, B.↗
NASA NTRS2004

Numerical Modeling and Testing of an Inductively-Driven and High-Energy Pulsed Plasma Thrusters

Pulsed Plasma Thrusters (PPTs) are advanced electric space propulsion devices that are characterized by simplicity and robustness. They suffer, however, from low thrust efficiencies. This summer, two approaches to improve the thrust efficiency of PPTs will be investigated through both numerical modeling and experimental testing. The first approach, an inductively-driven PPT, uses a double-ignition circuit to fire two PPTs in succession. This effectively changes the PPTs configuration from an LRC circuit to an LR circuit. The LR circuit is expected to provide better impedance matching and improving the efficiency of the energy transfer to the plasma. An added benefit of the LR circuit is an exponential decay of the current, whereas a traditional PPT s under damped LRC circuit experiences the characteristic "ringing" of its current. The exponential decay may provide improved lifetime and sustained electromagnetic acceleration. The second approach, a high-energy PPT, is a traditional PPT with a variable size capacitor bank. This PPT will be simulated and tested at energy levels between 100 and 450 joules in order to investigate the relationship between efficiency and energy level. Arbitrary Coordinate Hydromagnetic (MACH2) code is used. The MACH2 code, designed by the Center for Plasma Theory and Computation at the Air Force Research Laboratory, has been used to gain insight into a variety of plasma problems, including electric plasma thrusters. The goals for this summer include numerical predictions of performance for both the inductively-driven PPT and high-energy PFT, experimental validation of the numerical models, and numerical optimization of the designs. These goals will be met through numerical and experimental investigation of the PPTs current waveforms, mass loss (or ablation), and impulse bit characteristics.

Parma, Brian↗
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 NTRSDate not supplied

Numerical Analysis of Cavitation-Induced Two-Phase Fluid Transients in Cryogenic Feedlines Using Finite Volume Procedure

This article presents a numerical investigation of the cavitation induced fluid transient in cryogenic feedline due to sudden closure of the valve A one dimensional numerical model based on finite volume procedure consisting of nodes and branches using Generalized Fluid System Simulation Program ( has been employed for the prediction of pressure variations The unsteady friction formulation accurately determines wall shear stress, and a homogeneous mixture model is used for two phase flow modeling Numerical results obtained with the present formulation are compared with experimental results and Method of Characteristic ( based results It is observed that low residual in flow variable calculation improves the solution providing numerically stable results GFSSP is able to reproduce the fluid transient peak due to vapor collapse Subsequently, it can retain the shape of pressure wave for two phase fluid transients, but it is dampening faster compared to experiments Further investigation is required with set of experiments to validate the present numerical approach for different experimental data.

Fluid Transient↗
NASA NTRS2014

Comment on 'Shang S. 2012. Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step. Hydrological Processes 26: 3338-3343. DOI: 10.1002/hyp.8405'

A previous study analyzed errors in the numerical calculation of actual crop evapotranspiration (ET(sub a)) under soil water stress. Assuming no irrigation or precipitation, it constructed equations for ET(sub a) over limited soil-water ranges in a root zone drying out due to evapotranspiration. It then used a single crop-soil composite to provide recommendations about the appropriate usage of numerical methods under different values of the time step and the maximum crop evapotranspiration (ET(sub c)). This comment reformulates those ET(sub a) equations for applicability over the full range of soil water values, revealing a dependence of the relative error in numerical ET(sub a) on the initial soil water that was not seen in the previous study. It is shown that the recommendations based on a single crop-soil composite can be invalid for other crop-soil composites. Finally, a consideration of the numerical error in the time-cumulative value of ET(sub a) is discussed besides the existing consideration of that error over individual time steps as done in the previous study. This cumulative ET(sub a) is more relevant to the final crop yield.

Time step↗
NASA NTRS1973

Numerical integration routines for near-earth operations

Two general purpose numerical integration schemes were built into the NASA-JSC computer system. The state-of-the-art of numerical integration, the particular integrators built into the JSC computer system, and the use of the new integration packages are described. Background information about numerical integration and the variable-order, variable-stepsize Adams numerical integration technique is discussed. Results concerning the PEACE parameter optimization program are given along with recommendations and conclusions.

Powers, W. F.↗