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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 163 records · Page 9

Two-dimensional radiative transfer in cloudy atmospheres - The spherical harmonic spatial grid method

A new two-dimensional monochromatic method that computes the transfer of solar or thermal radiation through atmospheres with arbitrary optical properties is described. The model discretizes the radiative transfer equation by expanding the angular part of the radiance field in a spherical harmonic series and representing the spatial part with a discrete grid. The resulting sparse coupled system of equations is solved iteratively with the conjugate gradient method. A Monte Carlo model is used for extensive verification of outgoing flux and radiance values from both smooth and highly variable (multifractal) media. The spherical harmonic expansion naturally allows for different levels of approximation, but tests show that the 2D equivalent of the two-stream approximation is poor at approximating variations in the outgoing flux. The model developed here is shown to be highly efficient so that media with tens of thousands of grid points can be computed in minutes. The large improvement in efficiency will permit quick, accurate radiative transfer calculations of realistic cloud fields and improve our understanding of the effect of inhomogeneity on radiative transfer in cloudy atmospheres.

Evans, K. F.↗

Meshfree simulation and experimental validation of extreme thermomechanical conditions in friction stir extrusion

Abstract Friction stir extrusion (FSE) is a novel solid-phase processing technique that consolidates and extrudes metal powders, flakes, chips, or billets into high-performance parts by plastic deformation, which has the potential to save substantial processing time and energy. Currently, most studies on FSE are experimental and only a few numerical models have been developed to explain and predict the complex physics of the process. In this work, a meshfree simulation framework based on smoothed particle hydrodynamics (SPH) was developed for FSE. Unlike traditional grid-based methods, SPH is a Lagrangian particle-based method that can handle severe material deformations, capture moving interfaces and surfaces, and monitor the field variable histories explicitly without complicated tracking schemes. These aspects of SPH make it attractive for the FSE process, where in situ evolution of field variables is difficult to observe experimentally. To this end, a 3-D, fully thermomechanically coupled SPH model was developed to simulate the FSE of aluminum wires. The developed model was thoroughly validated by comparing the numerically predicted material flow, strain, temperature history, and extrusion force with experimental results for a certain set of process parameters. The validated SPH model can serve as an effective tool to predict and better understand the extreme thermomechanical conditions during the FSE process.

99 GENERAL AND MISCELLANEOUS↗

Robust verification of stochastic simulation codes

We introduce a robust verification tool for computational codes, which we call Stochastic Robust Extrapolation based Error Quantification (StREEQ). Unlike the prevalent Grid Convergence Index (GCI) [1] method, our approach is suitable for both stochastic and deterministic computational codes and is generalizable to any number of discretization variables. Building on ideas introduced in the Robust Verification [2] approach, we estimate the converged solution and orders of convergence with uncertainty using multiple fits of a discretization error model. In contrast to Robust Verification, we perform these fits to many bootstrap samples yielding a larger set of predictions with smoother statistics. Here, bootstrap resampling is performed on the lack-of-fit errors for deterministic code responses, and directly on the noisy data set for stochastic responses. This approach lends a degree of robustness to the overall results, capable of yielding precise verification results for sufficiently resolved data sets, and appropriately expanding the uncertainty when the data set does not support a precise result. For stochastic responses, a credibility assessment is also performed to give the analyst an indication of the trustworthiness of the results. Furthermore, this approach is suitable for both code and solution verification, and is particularly useful for solution verification of high-consequence simulations..

97 MATHEMATICS AND COMPUTING↗

Matrix-based, finite-difference algorithms for computational acoustics

A compact numerical algorithm is introduced for simulating multidimensional acoustic waves. The algorithm is expressed in terms of a set of matrix coefficients on a three-point spatial grid that approximates the acoustic wave equation with a discretization error of O(h exp 5). The method is based on tracking a local phase variable and its implementation suggests a convenient coordinate splitting along with natural intermediate boundary conditions. Results are presented for oblique plane waves and compared with other procedures. Preliminary computations of acoustic diffraction are also considered.

Davis, Sanford↗

Multigrid calculation of three-dimensional viscous cascade flows

A 3-D code for viscous cascade flow prediction was developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full multigrid method. The Baldwin-Lomax eddy viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Arnone, A.↗

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗

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.↗

Computational Fluid Dynamics Uncertainty Analysis for Payload Fairing Spacecraft Environmental Control Systems

Spacecraft thermal protection systems are at risk of being damaged due to airflow produced from Environmental Control Systems. There are inherent uncertainties and errors associated with using Computational Fluid Dynamics to predict the airflow field around a spacecraft from the Environmental Control System. This proposal describes an approach to validate the uncertainty in using Computational Fluid Dynamics to predict airflow speeds around an encapsulated spacecraft. The research described here is absolutely cutting edge. Quantifying the uncertainty in analytical predictions is imperative to the success of any simulation-based product. The method could provide an alternative to traditional"validation by test only'' mentality. This method could be extended to other disciplines and has potential to provide uncertainty for any numerical simulation, thus lowering the cost of performing these verifications while increasing the confidence in those predictions. Spacecraft requirements can include a maximum airflow speed to protect delicate instruments during ground processing. Computationaf Fluid Dynamics can be used to veritY these requirements; however, the model must be validated by test data. The proposed research project includes the following three objectives and methods. Objective one is develop, model, and perform a Computational Fluid Dynamics analysis of three (3) generic, non-proprietary, environmental control systems and spacecraft configurations. Several commercially available solvers have the capability to model the turbulent, highly three-dimensional, incompressible flow regime. The proposed method uses FLUENT and OPEN FOAM. Objective two is to perform an uncertainty analysis of the Computational Fluid . . . Dynamics model using the methodology found in "Comprehensive Approach to Verification and Validation of Computational Fluid Dynamics Simulations". This method requires three separate grids and solutions, which quantify the error bars around Computational Fluid Dynamics predictions. The method accounts for all uncertainty terms from both numerical and input variables. Objective three is to compile a table of uncertainty parameters that could be used to estimate the error in a Computational Fluid Dynamics model of the Environmental Control System /spacecraft system. Previous studies have looked at the uncertainty in a Computational Fluid Dynamics model for a single output variable at a single point, for example the re-attachment length of a backward facing step. To date, the author is the only person to look at the uncertainty in the entire computational domain. For the flow regime being analyzed (turbulent, threedimensional, incompressible), the error at a single point can propagate into the solution both via flow physics and numerical methods. Calculating the uncertainty in using Computational Fluid Dynamics to accurately predict airflow speeds around encapsulated spacecraft in is imperative to the success of future missions.

Groves, Curtis E.↗

Dissertation Defense Computational Fluid Dynamics Uncertainty Analysis for Payload Fairing Spacecraft Environmental Control Systems

Spacecraft thermal protection systems are at risk of being damaged due to airflow produced from Environmental Control Systems. There are inherent uncertainties and errors associated with using Computational Fluid Dynamics to predict the airflow field around a spacecraft from the Environmental Control System. This paper describes an approach to quantify the uncertainty in using Computational Fluid Dynamics to predict airflow speeds around an encapsulated spacecraft without the use of test data. Quantifying the uncertainty in analytical predictions is imperative to the success of any simulation-based product. The method could provide an alternative to traditional "validation by test only" mentality. This method could be extended to other disciplines and has potential to provide uncertainty for any numerical simulation, thus lowering the cost of performing these verifications while increasing the confidence in those predictions. Spacecraft requirements can include a maximum airflow speed to protect delicate instruments during ground processing. Computational Fluid Dynamics can be used to verify these requirements; however, the model must be validated by test data. This research includes the following three objectives and methods. Objective one is develop, model, and perform a Computational Fluid Dynamics analysis of three (3) generic, non-proprietary, environmental control systems and spacecraft configurations. Several commercially available and open source solvers have the capability to model the turbulent, highly three-dimensional, incompressible flow regime. The proposed method uses FLUENT, STARCCM+, and OPENFOAM. Objective two is to perform an uncertainty analysis of the Computational Fluid Dynamics model using the methodology found in "Comprehensive Approach to Verification and Validation of Computational Fluid Dynamics Simulations". This method requires three separate grids and solutions, which quantify the error bars around Computational Fluid Dynamics predictions. The method accounts for all uncertainty terms from both numerical and input variables. Objective three is to compile a table of uncertainty parameters that could be used to estimate the error in a Computational Fluid Dynamics model of the Environmental Control System /spacecraft system. Previous studies have looked at the uncertainty in a Computational Fluid Dynamics model for a single output variable at a single point, for example the re-attachment length of a backward facing step. For the flow regime being analyzed (turbulent, three-dimensional, incompressible), the error at a single point can propagate into the solution both via flow physics and numerical methods. Calculating the uncertainty in using Computational Fluid Dynamics to accurately predict airflow speeds around encapsulated spacecraft in is imperative to the success of future missions.

CFD↗

Dissertation Defense: Computational Fluid Dynamics Uncertainty Analysis for Payload Fairing Spacecraft Environmental Control Systems

Spacecraft thermal protection systems are at risk of being damaged due to airflow produced from Environmental Control Systems. There are inherent uncertainties and errors associated with using Computational Fluid Dynamics to predict the airflow field around a spacecraft from the Environmental Control System. This paper describes an approach to quantify the uncertainty in using Computational Fluid Dynamics to predict airflow speeds around an encapsulated spacecraft without the use of test data. Quantifying the uncertainty in analytical predictions is imperative to the success of any simulation-based product. The method could provide an alternative to traditional validation by test only mentality. This method could be extended to other disciplines and has potential to provide uncertainty for any numerical simulation, thus lowering the cost of performing these verifications while increasing the confidence in those predictions.Spacecraft requirements can include a maximum airflow speed to protect delicate instruments during ground processing. Computational Fluid Dynamics can be used to verify these requirements; however, the model must be validated by test data. This research includes the following three objectives and methods. Objective one is develop, model, and perform a Computational Fluid Dynamics analysis of three (3) generic, non-proprietary, environmental control systems and spacecraft configurations. Several commercially available and open source solvers have the capability to model the turbulent, highly three-dimensional, incompressible flow regime. The proposed method uses FLUENT, STARCCM+, and OPENFOAM. Objective two is to perform an uncertainty analysis of the Computational Fluid Dynamics model using the methodology found in Comprehensive Approach to Verification and Validation of Computational Fluid Dynamics Simulations. This method requires three separate grids and solutions, which quantify the error bars around Computational Fluid Dynamics predictions. The method accounts for all uncertainty terms from both numerical and input variables. Objective three is to compile a table of uncertainty parameters that could be used to estimate the error in a Computational Fluid Dynamics model of the Environmental Control System spacecraft system.Previous studies have looked at the uncertainty in a Computational Fluid Dynamics model for a single output variable at a single point, for example the re-attachment length of a backward facing step. For the flow regime being analyzed (turbulent, three-dimensional, incompressible), the error at a single point can propagate into the solution both via flow physics and numerical methods. Calculating the uncertainty in using Computational Fluid Dynamics to accurately predict airflow speeds around encapsulated spacecraft in is imperative to the success of future missions.

Fluid Dynamics↗

Numerical Investigation and Optimization of a Flushwall Injector for Scramjet Applications at Hypervelocity Flow Conditions

An investigation utilizing Reynolds-averaged simulations (RAS) was performed in order to demonstrate the use of design and analysis of computer experiments (DACE) methods in Sandia’s DAKOTA software package for surrogate modeling and optimization. These methods were applied to a flow- path fueled with an interdigitated flushwall injector suitable for scramjet applications at hyper- velocity conditions and ascending along a constant dynamic pressure flight trajectory. The flight Mach number, duct height, spanwise width, and injection angle were the design variables selected to maximize two objective functions: the thrust potential and combustion efficiency. Because the RAS of this case are computationally expensive, surrogate models are used for optimization. To build a surrogate model a RAS database is created. The sequence of the design variables comprising the database were generated using a Latin hypercube sampling (LHS) method. A methodology was also developed to automatically build geometries and generate structured grids for each design point. The ensuing RAS analysis generated the simulation database from which the two objective functions were computed using a one-dimensionalization (1D) of the three-dimensional simulation data. The data were fitted using four surrogate models: an artificial neural network (ANN), a cubic polynomial, a quadratic polynomial, and a Kriging model. Variance-based decomposition showed that both objective functions were primarily driven by changes in the duct height. Multiobjective design optimization was performed for all four surrogate models via a genetic algorithm method. Optimal solutions were obtained at the upper and lower bounds of the flight Mach number range. The Kriging model predicted an optimal solution set that exhibited high values for both objective functions. Additionally, three challenge points were selected to assess the designs on the Pareto fronts. Further sampling among the designs of the Pareto fronts may be required to lower the surrogate model errors and perform more accurate surrogate-model-based optimization.

Shenoy, Rajiv R.↗

A General Framework for Error-controlled Unstructured Scientific Data Compression

Data compression plays a key role in reducing storage and I/O costs. Traditional lossy methods primarily target data on rectilinear grids and cannot leverage the spatial coherence in unstructured mesh data, leading to suboptimal compression ratios. We present a multi-component, error-bounded compression framework designed to enhance the compression of floating-point unstructured mesh data, which is common in scientific applications. Our approach involves interpolating mesh data onto a rectilinear grid and then separately compressing the grid interpolation and the interpolation residuals. This method is general, independent of mesh types and typologies, and can be seamlessly integrated with existing lossy compressors for improved performance. We evaluated our framework across twelve variables from two synthetic datasets and two real-world simulation datasets. The results indicate that the multi-component framework consistently outperforms state-of-the-art lossy compressors on unstructured data, achieving, on average, a 2.3 − 3.5× improvement in compression ratios, with error bounds ranging from 1 × 10 the −6 to 1×10−2. We further investigate impact of hyperparameters, such as grid spacing and error allocation, to deliver optimal compression ratios in diverse datasets.

Gong, Qian↗

Using IKONOS Imagery to Estimate Surface Soil Property Variability in Two Alabama Physiographies

Knowledge of surface soil properties is used to assess past erosion and predict erodibility, determine nutrient requirements, and assess surface texture for soil survey applications. This study was designed to evaluate high resolution IKONOS multispectral data as a soil- mapping tool. Imagery was acquired over conventionally tilled fields in the Coastal Plain and Tennessee Valley physiographic regions of Alabama. Acquisitions were designed to assess the impact of surface crusting, roughness and tillage on our ability to depict soil property variability. Soils consisted mostly of fine-loamy, kaolinitic, thermic Plinthic Kandiudults at the Coastal Plain site and fine, kaolinitic, thermic Rhodic Paleudults at the Tennessee Valley site. Soils were sampled in 0.20 ha grids to a depth of 15 cm and analyzed for % sand (0.05 - 2 mm), silt (0.002 -0.05 mm), clay (less than 0.002 mm), citrate dithionite extractable iron (Fe(sub d)) and soil organic carbon (SOC). Four methods of evaluating variability in soil attributes were evaluated: 1) kriging of soil attributes, 2) co-kriging with soil attributes and reflectance data, 3) multivariate regression based on the relationship between reflectance and soil properties, and 4) fuzzy c-means clustering of reflectance data. Results indicate that co-kriging with remotely sensed data improved field scale estimates of surface SOC and clay content compared to kriging and regression methods. Fuzzy c-means worked best using RS data acquired over freshly tilled fields, reducing soil property variability within soil zones compared to field scale soil property variability.

Sullivan, Dana↗

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

Meade, Andrew J., Jr.↗

A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗

Assessing thermal comfort and participation in residential demand flexibility programs

Residential space-conditioning-based demand flexibility (DF) has become an increasingly sought-after method for demand-side load management to enhance grid reliability and facilitate integration of renewable energy generation. However, predicting the effectiveness and flexibility of residential DF resources is challenging due to the variability in household energy use behaviors. Current estimates show that only 50 % of projected savings from DF resources are actualized due to regulatory, technological, and social barriers. From a household perspective, concerns over thermal comfort during space conditioning-based DF events significantly impact participation decisions. Currently, there is a very limited understanding of how thermal comfort during space-conditioning-based DF events in real-world settings impacts household energy use behaviors and, consequently, the success of DF programs in achieving targeted savings. This paper proposes a method to comprehensively assess the thermal comfort implications of DF strategies and presents results of their impacts on DF event participation decisions and demand savings. Here, the proposed method was applied to a heat pump DF field study in Cordova, Alaska. The study’s key findings are: 1) DF event setpoint offsets that maintain indoor operative temperatures between 18 to 22 °C (65 to 71°F) may be preferred in Cordova, Alaska; 2) Household-level thermal comfort is more sensitive to the duration of the DF event than to the degree of temperature offset from baseline conditions; 3) The delayed impact of changes in indoor operative temperature in response to setpoint offsets, both during and after a DF event, influences occupants’ thermal comfort perceptions and willingness to persistently participate in events. The findings from application of the proposed method can help inform future larger-scale occupant-centric DF programs as it can capture information not readily available through utility and device-level energy use data. Thus, it can supplement these sources and help program administrators develop occupant-centric DF strategies, enabling more accurate predictions of participation rates and savings estimates for space-conditioning-based DF programs.

Demand side management↗

The Spatiotemporal Structure of 20th Century Climate Variations in Observations and Reanalyses: Pacific Pan-Decadal Variability - Part 2

The dominant interannual El Nino-Southern Oscillation phenomenon (ENSO) and the short length of climate observation records make it difficult to study long-term climate variations in the spatiotemporal domain. Based on the fact that the ENS0 signal spreads to remote regions and induces delayed climate variation through atmospheric teleconnections, we develop an ENSO-removal method through which the ENS0 signal can be approximately removed at the grid box level from the spatiotemporal field of a climate parameter. After this signal is removed, long-term climate variations, namely, the global warming trend (GW) and the Pacific pan-decadal variability (PDV), are isolated at middle and low latitudes in the climate parameter fields from observed and reanalyses datasets. In this study, we show that one of several PDV interdecadal regime shifts occurred during the 1990s. This significant change in the Pacific basin is comparable but opposite in phase to the 1976 climate regime shift, which results persisting warming in the central-eastern Pacific, and cooling in the North and South Pacific. The 1990s PDV regime shift is consistent with observed changes in ocean biosphere and ocean circulation. A comprehensive picture of PDV as manifested in the troposphere and at the surface is described. In general, the PDV spatial patterns in different parameter fields share some similarities with the patterns associated with ENSO, but important differences exist. First, the PDV atmospheric circulation pattern is shifted westward by about 20deg and its zonal extent is limited to approx.60deg compared to approx.110deg for ENS0 pattern. The westward shift of the PDV wave train produces a different, more west-east oriented, North American teleconnection pattern. The lack of a strong PDV surface temperature (ST) signal in the western equatorial Pacific and the relatively strong ST signal in the subtropical regions are consistent with an atmospheric overturning circulation response that differs from the one associated with ENSO.

Chen, Junye↗

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Scott, R. K.↗