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NASA NTRS2018

High-Order Stabilized Finite Elements on Dynamic Meshes

The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered.

Anderson, W. Kyle↗

NASA NTRS1987

Conservatism implications of shock test tailoring for multiple design environments

A method for analyzing shock conservation in test specifications that have been tailored to qualify a structure for multiple design environments is discussed. Shock test conservation is qualified for shock response spectra, shock intensity spectra and ranked peak acceleration data in terms of an Index of Conservation (IOC) and an Overtest Factor (OTF). The multi-environment conservation analysis addresses the issue of both absolute and average conservation. The method is demonstrated in a case where four laboratory tests have been specified to qualify a component which must survive seven different field environments. Final judgment of the tailored test specification is shown to require an understanding of the predominant failure modes of the test item.

Baca, Thomas J.↗

NASA NTRS1997

The Implicit and Explicit alpha-mu Schemes

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

Chang, Sin-Chung↗

NASA NTRS2003

Finite Volume Methods: Foundation and Analysis

Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form. This article reviews elements of the foundation and analysis of modern finite volume methods. The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, non-oscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of diffusion terms and the extension to systems of nonlinear conservation laws.

Barth, Timothy↗

NASA NTRS1999

Acoustic Energy Estimates in Inhomogeneous Moving Media

In ducted fan engine noise research, there is a need for defining a simple and easy to use acoustic energy conservation law to help in quantification of noise control techniques. There is a well known conservation law relating acoustic energy and acoustic energy flux in the case of an isentropic irrotational flow. Several different approaches have been taken to generalize this conservation law. For example, Morfey finds an identity by separating out the irrotational part of the perturbed flow. Myers is able to find a series of indentities by observing an algebraic relationship between the basic conservation of energy equation for a background flow and the underlying equations of motion. In an approximate sense, this algebraic relationship is preserved under perturbation. A third approach which seems to have not been pursued in the literature is a result known as Noether's theorem. There is a Lagrangian formulation for the Euler equation of fluid mechanics. Noether's theorem says that any group action that leaves the Lagrangian action invariant leads to a conserved quantity. This presentation will include a survey of current results regarding acoustic energy and preliminary results on the symmetries of the Lagrangian.

Farassat, F.↗

NASA NTRS2015

Use of Generalized Fluid System Simulation Program (GFSSP) for Teaching and Performing Senior Design Projects at the Educational Institutions

This paper describes the experience of the authors in using the Generalized Fluid System Simulation Program (GFSSP) in teaching Design of Thermal Systems class at University of Alabama in Huntsville. GFSSP is a finite volume based thermo-fluid system network analysis code, developed at NASA/Marshall Space Flight Center, and is extensively used in NASA, Department of Defense, and aerospace industries for propulsion system design, analysis, and performance evaluation. The educational version of GFSSP is freely available to all US higher education institutions. The main purpose of the paper is to illustrate the utilization of this user-friendly code for the thermal systems design and fluid engineering courses and to encourage the instructors to utilize the code for the class assignments as well as senior design projects. The need for a generalized computer program for thermofluid analysis in a flow network has been felt for a long time in aerospace industries. Designers of thermofluid systems often need to know pressures, temperatures, flow rates, concentrations, and heat transfer rates at different parts of a flow circuit for steady state or transient conditions. Such applications occur in propulsion systems for tank pressurization, internal flow analysis of rocket engine turbopumps, chilldown of cryogenic tanks and transfer lines, and many other applications of gas-liquid systems involving fluid transients and conjugate heat and mass transfer. Computer resource requirements to perform time-dependent, three-dimensional Navier-Stokes computational fluid dynamic (CFD) analysis of such systems are prohibitive and therefore are not practical. Available commercial codes are generally suitable for steady state, single-phase incompressible flow. Because of the proprietary nature of such codes, it is not possible to extend their capability to satisfy the above-mentioned needs. Therefore, the Generalized Fluid System Simulation Program (GFSSP1) has been developed at NASA Marshall Space Flight Center (MSFC) as a general fluid flow system solver capable of handling phase changes, compressibility, mixture thermodynamics and transient operations. It also includes the capability to model external body forces such as gravity and centrifugal effects in a complex flow network. The objectives of GFSSP development are: a) to develop a robust and efficient numerical algorithm to solve a system of equations describing a flow network containing phase changes, mixing, and rotation; and b) to implement the algorithm in a structured, easy-to-use computer program. The analysis of thermofluid dynamics in a complex network requires resolution of the system into fluid nodes and branches, and solid nodes and conductors as shown in Figure 1. Figure 1 shows a schematic and GFSSP flow circuit of a counter-flow heat exchanger. Hot nitrogen gas is flowing through a pipe, colder nitrogen is flowing counter to the hot stream in the annulus pipe and heat transfer occurs through metal tubes. The problem considered is to calculate flowrates and temperature distributions in both streams. GFSSP has a unique data structure, as shown in Figure 2, that allows constructing all possible arrangements of a flow network with no limit on the number of elements. The elements of a flow network are boundary nodes where pressure and temperature are specified, internal nodes where pressure and temperature are calculated, and branches where flowrates are calculated. For conjugate heat transfer problems, there are three additional elements: solid node, ambient node, and conductor. The solid and fluid nodes are connected with solid-fluid conductors. GFSSP solves the conservation equations of mass and energy, and equation of state in internal nodes to calculate pressure, temperature and resident mass. The momentum conservation equation is solved in branches to calculate flowrate. It also solves for energy conservation equations to calculate temperatures of solid nodes. The equations are coupled and nonlinear; therefore, they are solved by an iterative numerical scheme. GFSSP employs a unique numerical scheme known as simultaneous adjustment with successive substitution (SASS), which is a combination of Newton-Raphson and successive substitution methods. The mass and momentum conservation equations and the equation of state are solved by the Newton-Raphson method while the conservation of energy and species are solved by the successive substitution method. GFSSP is linked with two thermodynamic property programs, GASP2 and WASP3 and GASPAK4, that provide thermodynamic and thermophysical properties of selected fluids. Both programs cover a range of pressure and temperature that allows fluid properties to be evaluated for liquid, liquid-vapor (saturation), and vapor region. GASP and WASP provide properties of 12 fluids. GASPAK includes a library of 36 fluids. GFSSP has three major parts. The first part is the graphical user interface (GUI), visual thermofluid analyzer of systems and components (VTASC). VTASC allows users to create a flow circuit by a 'point and click' paradigm. It creates the GFSSP input file after the completion of the model building process. GFSSP's GUI provides the users a platform to build and run their models. It also allows post-processing of results. The network flow circuit is first built using three basic elements: boundary node, internal node, and branch.

Majumdar, A. K.↗

NASA NTRS1975

ECASTAR methodology

The methodology used for the ECASTAR study was described. A systems approach was utilized, in conjunction with the following tools: input/output economic analysis, net economics, and net energetics. Some of the considerations between the objective of ECASTAR, which is to assess the potential for and impacts of various energy conservation actions, were listed. These include: data base, constraints and criteria, identification and classification of energy conservation actions, relationship between actions of the conservation sector and requirements of other sectors, evaluation of actions with respect to constraints and criteria, evaluation of impacts of energy conservation actions, and presentation of the results. Each one of these topics was discussed.

Source record↗

NASA NTRS1975

ECASTAR summary and recommendations

A methodology was presented for a systems approach to energy conservation, where conservation was depicted as the result of any action that improves the energy situation of the United States in the present and near future. The relevant constraints and criteria and their application were discussed. Among the most important are the present (capitalistic) structure of the American economy, the lead times necessary for implementation of relevant technologies, and the desire of most policymakers to maintain a reasonable standard of living with a reasonable amount of invulnerability to foreign discretion. The objective of the design group was the assessment of the potential and impact of conservation action in the United States. The U.S. economy was divided into four sectors: energy industry, industry, residential/commercial, and transportation. Each sector was analyzed for conservation actions and their impacts. The sector analysis was characterized as the system design or construction phase.

Source record↗

NASA NTRS1979

Recent progress in finite-volume calculations for wing-fuselage combinations

Progress in the application of finite-volume methods to the calculation of transonic potential flows past general wing-body combinations is reviewed. Two different methods of generating boundary-conforming grids are investigated, and the results compared to provide an estimate of solution sensitivity to grid geometry. Both conservative and quasi-conservative difference schemes are used in one of the coordinate systems. Results show that the error introduced by the quasi-conservative formulation seems to be small, although a one-dimensional analysis suggests that schemes of this type do not necessarily produce mass-conserving shocks. Comparison of calculated results with experimental data for realistic fuselage geometries clearly shows the importance of modelling the effect of fuselage geometry upon the wing pressure distribution.

Caughey, D. A.↗

NASA NTRS1984

Kinematics of the CS method for the treatment of molecular collisions

The problem of the quantum treatment of nonreactive collisions of two simple molecules or of an atom and a molecule is considered mathematically. Kinematical structure theorems analogous to the Wigner-Eckart theorem for scalar operators are derived for the transition operator (T-operator) of a two-molecule system, where, in addition to the usual conservation laws, certain additional conservation laws are presumed satisfied in a collision. The additional conservation laws were examined in order to develop a testable approximation scheme (CS method) for the specification of the kinematics of a system of two rigid diatomic molecules. The rand p-helicity basis states for a two-rigid rotor system are defined and unitary transformations established between these and conventional bases. It is found that only questionable criteria exist for testing the on-the-energy-shell matrix of a T(energy) operator for the presence of properties of two of the additional conservation laws.

Hahne, G. E.↗

NASA NTRS1990

Viscous shock profiles and primitive formulations

Weak solutions of hyperbolic systems in primitive (non-conservation) form for which a consistent conservation form exists are considered. It is shown that primitive formulations, shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two. Scheme-dependent high order correction terms are derived that enforce consistent viscous shock profiles. The resulting primitive algorithm is conservative to the order of approximation. One dimensional Euler calculations of flows containing strong shocks clearly show that conservation errors in primitive flow calculations are of comparable quality.

Karni, S.↗

NASA NTRS1991

Grid generation about complex three-dimensional aircraft configurations

The problem of obtaining three dimensional grids with sufficient resolution to resolve all the flow or other physical features of interest is addressed. The generation of a computational grid involves a series of compromises to resolve several conflicting requirements. On one hand, one would like the grid to be fine enough and not too skewed to reduce the numerical errors and to adequately resolve the pertinent physical features of the flow field about the aircraft. On the other hand, the capabilities of present or even future supercomputers are finite and the number of mesh points must be limited to a reasonable number: one which is usually much less than desired for numerical accuracy. One technique to overcome this limitation is the 'zonal' grid approach. In this method, the overall field is subdivided into smaller zones or blocks in each of which an independent grid is generated with enough grid density to resolve the flow features in that zone. The zonal boundaries or interfaces require special boundary conditions such that the conservation properties of the governing equations are observed. Much work was done in 3-D zonal approaches with nonconservative zonal interfaces. A 3-D zonal conservative interfacing method that is efficient and easy to implement was developed during the past year. During the course of the work, it became apparent that it would be much more feasible to do the conservative interfacing with cell-centered finite volume codes instead of the originally planned finite difference codes. Accordingly, the CNS code was converted to finite volume form. This new version of the code is named CNSFV. The original multi-zonal interfacing capability of the CNS code was enhanced by generalizing the procedure to allow for completely arbitrarily shaped zones with no mesh continuity between the zones. While this zoning capability works well for most flow situations, it is, however, still nonconservative. The conservative interface algorithm was also implemented but was not completely validated.

Klopfer, Goetz H.↗

NASA NTRS1992

Polarimetric signatures of a canopy of dielectric cylinders based on first and second order vector radiative transfer theory

Complete polarimetric signatures of a canopy of dielectric cylinders overlying a homogeneous half space are studied with the first and second order solutions of the vector radiative transfer theory. The vector radiative transfer equations contain a general nondiagonal extinction matrix and a phase matrix. The energy conservation issue is addressed by calculating the elements of the extinction matrix and the elements of the phase matrix in a manner that is consistent with energy conservation. Two methods are used. In the first method, the surface fields and the internal fields of the dielectric cylinder are calculated by using the fields of an infinite cylinder. The phase matrix is calculated and the extinction matrix is calculated by summing the absorption and scattering to ensure energy conservation. In the second method, the method of moments is used to calculate the elements of the extinction and phase matrices. The Mueller matrix based on the first order and second order multiple scattering solutions of the vector radiative transfer equation are calculated. Results from the two methods are compared. The vector radiative transfer equations, combined with the solution based on method of moments, obey both energy conservation and reciprocity. The polarimetric signatures, copolarized and depolarized return, degree of polarization, and phase differences are studied as a function of the orientation, sizes, and dielectric properties of the cylinders. It is shown that second order scattering is generally important for vegetation canopy at C band and can be important at L band for some cases.

Tsang, Leung↗

NASA NTRS1993

Large eddy simulation of shock turbulence interaction

A nonconservative formulation of the energy equation (solving for internal energy) was used to perform large eddy simulations of compressible turbulence in Moin et al. due to its simplicity in implementing SGS models compared to the conservative formulation (solving for total energy). In problems with shocks in the domain, however, the total energy formulation is preferred due to its conservative nature. A conservative set of equations for the LES were derived from the nonconservative equations derived by Moin et al. Performance of the conservative formulation was compared with the experiment on decaying grid-generated turbulence as well as with the filtered DNS field. Various shock-capturing schemes were tested, and an ENO shock-capturing scheme of Shu and Osher was chosen for the simulation of shock/turbulence interaction. The scheme was tested and validated against the data base generated by DNS of weak shock waves. The results obtained with the essentially nonoscillatory (ENO) scheme were within 5 percent from the DNS results, and used less than 25 percent of the CPU time used in the DNS.

Lee, Sangsan↗

NASA NTRS1994

A Coulomb collision algorithm for weighted particle simulations

A binary Coulomb collision algorithm is developed for weighted particle simulations employing Monte Carlo techniques. Charged particles within a given spatial grid cell are pair-wise scattered, explicitly conserving momentum and implicitly conserving energy. A similar algorithm developed by Takizuka and Abe (1977) conserves momentum and energy provided the particles are unweighted (each particle representing equal fractions of the total particle density). If applied as is to simulations incorporating weighted particles, the plasma temperatures equilibrate to an incorrect temperature, as compared to theory. Using the appropriate pairing statistics, a Coulomb collision algorithm is developed for weighted particles. The algorithm conserves energy and momentum and produces the appropriate relaxation time scales as compared to theoretical predictions. Such an algorithm is necessary for future work studying self-consistent multi-species kinetic transport.

Miller, Ronald H.↗

NASA NTRS1995

A Comprehensive Approach to Management of Workplace and Environmental Noise at NASA Lewis Research Center

NASA Lewis Research Center is home to more than 100 experimental research testing facilities and laboratories, including large wind tunnels and engine test cells, which in combination create a varied and complex noise environment. Much of the equipment was manufactured prior to the enactment of legislation limiting product noise emissions or occupational noise exposure. Routine facility maintenance and associated construction also contributes to a noise exposure management responsibility which is equal in magnitude and scope to that of several small industrial companies. The Noise Program, centrally managed within the Office of Environmental Programs at LRC, maintains overall responsibility for hearing conservation, community noise control, and acoustical and noise control engineering. Centralized management of the LRC Noise Program facilitates the timely development and implementation of engineered noise control solutions for problems identified via either the Hearing Conservation of Community Noise Program. The key element of the Lewis Research Center Noise Program, Acoustical and Noise Control Engineering Services, is focused on developing solutions that permanently reduce employee and community noise exposure and maximize research productivity by reducing or eliminating administrative and operational controls and by improving the safety and comfort of the work environment. The Hearing Conservation Program provides noise exposure assessment, medical monitoring, and training for civil servant and contractor employees. The Community Noise Program aims to maintain the support of LRC's neighboring communities while enabling necessary research operations to accomplish their programmatic goals. Noise control engineering capability resides within the Noise Program. The noise control engineering, based on specific exposure limits, is a fundamental consideration throughout the design phase of new test facilities, labs, and office buildings. In summary, the Noise Program addresses hearing conservation, community noise control, and acoustical and noise control engineering.

Cooper, Beth A.↗

NASA NTRS1997

A New Approach to Extreme Value Estimation Applicable to a Wide Variety of Random Variables

Designing reliable structures requires an estimate of the maximum and minimum values (i.e., strength and load) that may be encountered in service. Yet designs based on very extreme values (to insure safety) can result in extra material usage and hence, uneconomic systems. In aerospace applications, severe over-design cannot be tolerated making it almost mandatory to design closer to the assumed limits of the design random variables. The issue then is predicting extreme values that are practical, i.e. neither too conservative or non-conservative. Obtaining design values by employing safety factors is well known to often result in overly conservative designs and. Safety factor values have historically been selected rather arbitrarily, often lacking a sound rational basis. To answer the question of how safe a design needs to be has lead design theorists to probabilistic and statistical methods. The so-called three-sigma approach is one such method and has been described as the first step in utilizing information about the data dispersion. However, this method is based on the assumption that the random variable is dispersed symmetrically about the mean and is essentially limited to normally distributed random variables. Use of this method can therefore result in unsafe or overly conservative design allowables if the common assumption of normality is incorrect.

Holland, Frederic A., Jr.↗

NASA NTRS1998

Satellite Estimation of Spectral Surface UV Irradiance: Effect of Horizontally Homogeneous Clouds - 2

The local variability of UV irradiance at the Earth's surface is mostly caused by clouds in addition to the seasonal variability. Parametric representations of radiative transfer RT calculations are presented for the convenient solution of the transmission T of ultraviolet radiation through plane parallel clouds over a surface with reflectivity R(sub s). The calculations are intended for use with the Total Ozone Mapping Spectrometer (TOMS) measured radiances to obtain the calculated Lambert equivalent scene reflectivity R for scenes with and without clouds. The purpose is to extend the theoretical analysis of the estimation of UV irradiance from satellite data for a cloudy atmosphere. Results are presented for a range of cloud optical depths and solar zenith angles for the cases of clouds over a low reflectivity surface R(sub s) less than 0.1, over a snow or ice surface R(sub s) greater than 0.3, and for transmission through a non-conservative scattering cloud with single scattering albedo omega(sub 0) = 0.999. The key finding for conservative scattering is that the cloud-transmission function C(sub T), the ratio of cloudy-to clear-sky transmission, is roughly C(sub T) = 1 - R(sub c) with an error of less than 20% for nearly overhead sun and snow-free surfaces. For TOMS estimates of UV irradiance in the presence of both snow and clouds, independent information about snow albedo is needed for conservative cloud scattering. For non-conservative scattering with R(sub s) greater than 0.5 (snow) the satellite measured scene reflectance cannot be used to estimate surface irradiance. The cloud transmission function has been applied to the calculation of UV irradiance at the Earth's surface and compared with ground-based measurements.

Krothov, N.↗