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At least 811 records · Page 45

A constitutive relation for the viscous flow of an oriented fiber assembly

A constitutive relation for an equivalent, homogeneous fluid is developed for the anisotropic viscous flow of an oriented assembly of discontinuous fibers suspended in a viscous fluid. The anisotropic viscous compliance matrix can be expressed in terms of three constants by assuming the equivalent fluid to be incompressible and the microstructure to have axial symmetry (transversely isotropic). By means of a micromechanics analysis, the three terms of the constitutive relation are expressed in terms of the viscosity of the matrix fluid, the fiber aspect ratio, and the fiber volume fraction. A comparison of the viscosity terms reveals that the elongational viscosity in the fiber direction varies as the square of the fiber aspect ratio and a complex function of the fiber volume fraction. Furthermore, the ratio of the axial elongational viscosity to the transverse elongational viscosity and both axial and transverse shear viscosities was shown to be 10 exp 4 - 10 exp 6 for fiber aspect ratio of 100-1000, except at extreme values of the fiber volume fraction.

Pipes, R. B.↗

Delay of Turbulent Boundary Layer Detachment by Mechanical Excitation: Application to Rearward-facing Ramp

A vane oscillating about a fixed point at the inlet to a two-dimensional 20 deg rearward-facing ramp proved effective in delaying the detachment of a turbulent boundary layer. Flow-field, surface static pressure, and smoke-wire flow visualization measurements were made. Surface pressure coefficient distributions revealed that two different effects occurred with axial distance along the ramp surface. The surface pressure coefficient varied as a complex function of the vane oscillation frequency and its trailing edge displacement amplitude; that is, it varied as a function of the vane oscillation frequency throughout the entire range of frequencies covered during the test, but it varied over only a limited range of the trailing edge displacement amplitudes covered.The complexity of these findings prompted a detailed investigation, the results of which revealed a combination of phenomena that explain qualitatively how the mechanically generated, periodic, sinusoidal perturbing signal produced by the oscillating vane reacts with the fluid flow to delay the detachment of a turbulent boundary layer experiencing transitory detachment.

McKinzie, Daniel J., Jr.↗

A Chebyshev Collocation Method for Moving Boundaries, Heat Transfer, and Convection During Directional Solidification

Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid-fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and name propagation. The directional solidification of semi-conductor crystals by the Bridgman-Stockbarger method is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt-solid interface. In this work, a Chebyshev pseudospectra collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, finite-difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.

Zhang, Yiqiang↗

Optical Properties of Thin Film Molecular Mixtures

Thin films composed of molecular mixtures of metal and dielectric are being considered for use as solar selective coatings for a variety of space power applications. By controlling the degree of molecular mixing, the solar selective coatings can be tailored to have the combined properties of high solar absorptance, alpha, and low infrared emittance, epsilon. On orbit, these combined properties would simultaneously maximize the amount of solar energy captured by the coating and minimize the amount of thermal energy radiated. Mini-satellites equipped with solar collectors coated with these cermet coatings may utilize the captured heat energy to power a heat engine to generate electricity, or to power a thermal bus that directs heat to remote regions of the spacecraft. Early work in this area identified the theoretical boundary conditions needed to operate a Carnot cycle in space, including the need for a solar concentrator, a solar selective coating at the heat inlet of the engine, and a radiator. A solar concentrator that can concentrate sunlight by a factor of 100 is ideal. At lower values, the temperature of the solar absorbing surface becomes too low for efficient heat engine operation, and at higher values, cavity type heat receivers become attractive. In designing the solar selective coating, the wavelength region yielding high solar absorptance must be separated from the wavelength region yielding low infrared emittance by establishing a sharp transition in optical properties. In particular, a sharp transition in reflectance is desired in the infrared to achieve the desired optical performance. For a heat engine operating at 450C, a sharp transition at 1.8 micrometers is desired. The radiator completes the heat flow through the Carnot cycle. Additional work has been done supporting the use of molecular mixtures for terrestrial applications. Sputter deposition provides a means to apply coatings to the tubes that carry a working fluid at the focus of trough-style collectors. Sputtering offers considerable flexibility in coating conditions, including a wide variety of metal and dielectric targets. Coating designs range from simple two or three layer coatings to complex coatings that are purposely graded to be metal-rich at their base and oxide-rich at their surface in order to yield the desired solar selective properties. In these cermet coatings, molecular islands of metal are thought to be embedded in a three dimensional matrix of dielectric. Recent work has identified the use of custom made ion beam sputter deposition targets to produce coatings containing molecular mixtures of metal and dielectric. The targets are cylindrical and the surface consists of a gradually changing composition of metal and dielectric. Rotating the cylinder under the beam during ion beam sputter deposition yields a coating that is a molecular mixture of metal and dielectric, with the composition changing through the thickness of the coating. The optical properties of these coatings are not only dependent on their thickness and chemical composition, but are also dependent on the extent of the through thickness gradient established during deposition. This paper presents a summary of the optical properties of several thin film molecular mixtures designed as solar selective coatings. Optical performance is first identified as a function of wavelength, from the ultraviolet to the visible and infrared. Coating composition, thickness, and gradient from metal to dielectric also play an important role. Additional work for future activities is also identified.

Jaworske, Donald A.↗

Inertial Motion-Tracking Technology for Virtual 3-D

In the 1990s, NASA pioneered virtual reality research. The concept was present long before, but, prior to this, the technology did not exist to make a viable virtual reality system. Scientists had theories and ideas they knew that the concept had potential, but the computers of the 1970s and 1980s were not fast enough, sensors were heavy and cumbersome, and people had difficulty blending fluidly with the machines. Scientists at Ames Research Center built upon the research of previous decades and put the necessary technology behind them, making the theories of virtual reality a reality. Virtual reality systems depend on complex motion-tracking sensors to convey information between the user and the computer to give the user the feeling that he is operating in the real world. These motion-tracking sensors measure and report an object s position and orientation as it changes. A simple example of motion tracking would be the cursor on a computer screen moving in correspondence to the shifting of the mouse. Tracking in 3-D, necessary to create virtual reality, however, is much more complex. To be successful, the perspective of the virtual image seen on the computer must be an accurate representation of what is seen in the real world. As the user s head or camera moves, turns, or tilts, the computer-generated environment must change accordingly with no noticeable lag, jitter, or distortion. Historically, the lack of smooth and rapid tracking of the user s motion has thwarted the widespread use of immersive 3-D computer graphics. NASA uses virtual reality technology for a variety of purposes, mostly training of astronauts. The actual missions are costly and dangerous, so any opportunity the crews have to practice their maneuvering in accurate situations before the mission is valuable and instructive. For that purpose, NASA has funded a great deal of virtual reality research, and benefited from the results.

Source record↗

NASA-USRP Summer 2013 Internship Final Report

Three major projects were undertaken during the Summer 2013 USRP Internship: (A) assisting the cTAPS group with component and pressure vessel system analyses and documentation, (B) designing a hoisting fixture for a solid rocket motor, (C) finding an alternative to removing the DOT rated gaseous nitrogen tank from the roof for hydrostatic testing. Hypergolic Material Assessments (HMAs) and pressure calculations were performed on components of pressure systems. Additionally, component information was logged in the Standard Parts Database to provide a location where system designers can find information regarding components, including their specifications and compatibility with fluids. A hoisting fixture was designed to hoist a solid rocket motor and meets the specifications related to stress and size. However, there are issues with the fixtures bolt head allotment, the bolt spacing, and the complexity of the part. Finally, calculations were performed on an expiring DOT rated gaseous nitrogen tank in an attempt to re-rate it per ASME standards. This was unsuccessful so other options are being explored for the tank. While much progress was made on all three projects, there is still work to be performed on each project to achieve the desired results.

Gurganus, S. Christine↗

Future of the Search for Life: Workshop Report

The 2-week, virtual Future of the Search for Life science and engineering workshop brought together more than 100 scientists, engineers, and technologists in March and April 2022 to provide their expert opinion on the interconnections between life-detection science and technology. Participants identified the advances in measurement and sampling technologies they believed to be necessary to perform in situ searches for life elsewhere in our Solar System, 20 years or more in the future. Among suggested measurements for these searches, those pertaining to three potential indicators of life termed “dynamic disequilibrium,” “catalysis,” and “informational polymers” were identified as particularly promising avenues for further exploration. For these three indicators, small breakout groups of participants identified measurement needs and knowledge gaps, along with corresponding constraints on sample handling (acquisition and processing) approaches for a variety of environments on Enceladus, Europa, Mars, and Titan. Despite the diversity of these environments, sample processing approaches all tend to be more complex than those that have been implemented on missions or envisioned for mission concepts to date. The approaches considered by workshop breakout groups progress from nondestructive to destructive measurement techniques, and most involve the need for fluid (especially liquid) sample processing. Sample processing needs were identified as technology gaps. These gaps include technology and associated sampling strategies that allow the preservation of the thermal, mechanical, and chemical integrity of the samples upon acquisition; and to optimize the sample information obtained by operating suites of instruments on common samples. Crucially, the interplay between science-driven life-detection strategies and their technological implementation highlights the need for an unprecedented level of payload integration and extensive collaboration between scientists and engineers, starting from concept formulation through mission deployment of life-detection instruments and sample processing systems.

Marc Neveu↗

Experimental, Theoretical, and Computational Investigation of Separated Nozzle Flows

A detailed experimental, theoretical, and computational study of separated nozzle flows has been conducted. Experimental testing was performed at the NASA Langley 16-Foot Transonic Tunnel Complex. As part of a comprehensive static performance investigation, force, moment, and pressure measurements were made and schlieren flow visualization was obtained for a sub-scale, non-axisymmetric, two-dimensional, convergent- divergent nozzle. In addition, two-dimensional numerical simulations were run using the computational fluid dynamics code PAB3D with two-equation turbulence closure and algebraic Reynolds stress modeling. For reference, experimental and computational results were compared with theoretical predictions based on one-dimensional gas dynamics and an approximate integral momentum boundary layer method. Experimental results from this study indicate that off-design overexpanded nozzle flow was dominated by shock induced boundary layer separation, which was divided into two distinct flow regimes; three- dimensional separation with partial reattachment, and fully detached two-dimensional separation. The test nozzle was observed to go through a marked transition in passing from one regime to the other. In all cases, separation provided a significant increase in static thrust efficiency compared to the ideal prediction. Results indicate that with controlled separation, the entire overexpanded range of nozzle performance would be within 10% of the peak thrust efficiency. By offering savings in weight and complexity over a conventional mechanical exhaust system, this may allow a fixed geometry nozzle to cover an entire flight envelope. The computational simulation was in excellent agreement with experimental data over most of the test range, and did a good job of modeling internal flow and thrust performance. An exception occurred at low nozzle pressure ratios, where the two-dimensional computational model was inconsistent with the three-dimensional separation observed in the experiment. In general, the computation captured the physics of the shock boundary layer interaction and shock induced boundary layer separation in the nozzle, though there were some differences in shock structure compared to experiment. Though minor, these differences could be important for studies involving flow control or thrust vectoring of separated nozzles. Combined with other observations, this indicates that more detailed, three-dimensional computational modeling needs to be conducted to more realistically simulate shock-separated nozzle flows.

Hunter, Craig A.↗

End-to-end modelling of He II flow systems

A practical computer code has been developed which uses the accepted two-fluid model to simulate He II flow in complicated systems. The full set of equations are used, retaining the coupling between the pressure, temperature and velocity fields. This permits modeling He II flow over the full range of conditions, from strongly or weakly driven flow through large pipes, narrow channels and porous media. The system may include most of the components used in modern superfluid flow systems: non-ideal thermomechanical pumps, tapered sections, constrictions, lines with heated side walls and heat exchangers. The model is validated by comparison with published experimental data. It is applied to a complex system to show some of the non-intuitive feedback effects that can occur. This code is ready to be used as a design tool for practical applications of He II. It can also be used for the design of He II experiments and as a tool for comparison of experimental data with the standard two-fluid model.

Mord, A. J.↗

Mean Flow Augmented Acoustics in Rocket Systems

Oscillatory motion in solid rocket motors and liquid engines has long been a subject of concern. Many rockets display violent fluctuations in pressure, velocity, and temperature originating from the complex interactions between the combustion process and gas dynamics. The customary approach to modeling acoustic waves inside a rocket chamber is to apply the classical inhomogeneous wave equation to the combustion gas. The assumption of a linear, non-dissipative wave in a quiescent fluid remains valid while the acoustic amplitudes are small and local gas velocities stay below Mach 0.2. The converging section of a rocket nozzle, where gradients in pressure, density, and velocity become large, is a notable region where this approach is not applicable. The expulsion of unsteady energy through the nozzle of a rocket is identified as the predominate source of acoustic damping for most rocket systems. An accurate model of the acoustic behavior within this region where acoustic modes are influenced by the presence of a steady mean flow is required for reliable stability predictions. Recently, an approach to address nozzle damping with mean flow effects was implemented by French [1]. This new approach extends the work originated by Sigman and Zinn [2] by solving the acoustic velocity potential equation (AVPE) formulated by perturbing the Euler equations [3]. The acoustic velocity potential (psi) describing the acoustic wave motion in the presence of an inhomogeneous steady high-speed flow is defined by, (del squared)(psi) − (lambda/c)(exp 2)(psi) − M(dot)[M(dot)(del)(del(psi))] − 2(lambda(M/c) + (M(dot)del(M))(dot)del(psi)−2(lambda)(psi)[M(dot)del(1/c)]=0 (1) with M as the Mach vector, c as the speed of sound, and lambda as the complex eigenvalue. French apply the finite volume method to solve the steady flow field within the combustion chamber and nozzle with inviscid walls. The complex eigenvalues and eigenvector are determined with the use of the ARPACK eigensolver. The present study employs the COMSOL Multphysics framework to solve the coupled eigenvalue problem using the finite element approach. The study requires one way coupling of the CFD High Mach Number Flow (HMNF) and mathematics module. The HMNF module evaluated the gas flow inside of a solid rocket motor using St. Robert's law modeling solid propellant burn rate, slip boundary conditions, and the supersonic outflow condition. Results from the HMNF model are used by the coefficient form of the mathematics module to determine the eigenvalues of the AVPE. The mathematics model is truncated at the nozzle sonic line, where a zero flux boundary condition is self-satisfying. The remaining boundaries are modeled with a zero flux boundary condition, assuming zero acoustic absorption on all surfaces. Pertinent results from these analyses are the complex valued eigenvalue and eigenvectors. Comparisons are made to the French results to evaluate the modeling approach. A comparison of the French results with that of the present analysis is displayed in figures 1 and 2, respectively. The graphic shows the first tangential eigenvector's real (a) and imaginary (b) values.

Fischbach, Sean R.↗

Adjoint-based Sensitivities of Flutter Predictions based on the Linearized Frequency-domain Approach

Flutter is a critical factor in designing and certifying aircraft. The linearized frequency-domain method offers a lower cost alternative to time-marching computational fluid dynamics for high-fidelity flutter analysis. In this work, adjoint-based sensitivities are added to a flutter analysis based on the linearized frequency-domain method to efficiently compute derivatives of flutter cost functions with respect to design variables or uncertain parameters. The derivation of the adjoint equations, which involve complications such as derivatives of a nonlinear generalized eigenvalue problem with complex-valued inputs and derivatives of the linearized Navier-Stokes equations, is provided. The implemented adjoint terms and derivatives are verified before demonstrating the approach for derivatives of flutter dynamic pressure with respect to Mach number for the AGARD 445.6 wing.

Aeroelasticity↗

Long-life, space-maintainable nuclear stage regulators and shutoff valves

The six most promising valve, regulator, and remote coupling concepts, representing the more radical designs from twenty concepts generated, were investigated. Of the three valves, one has no moving parts because shutoff sealing is accomplished by an electromagnetic field which ionized the flowing fluid. Another valve uses liquid metal to obtain sealing. In the third valve, high sealing forces are generated by heating and expanding trapped hydrogen. The pressure regulator is an electronically controlled, electromechanically operated, single state valve. Its complexity is in electronic circuitry, and the design results in less weight, increased reliability and performance flexibility, and multipurpose application. The two remote couplings feature the minimization of weight and mechanical complexity. One concept uses a low melting temperature metal alloy which is injected into the joint cavity; upon solidification, the alloy provides a seal and a structural joint. The second concept is based on the differential thermal expansion of the coupling mating parts. At thermal equilibrium there is a predetermined interference between the parts, and sealing is achieved by interference loading.

Source record↗

Visualization of fluid dynamics at NASA Ames

Some of the hardware and software tools and techniques in use at NASA's numerical aerodynamic simulation facility for the analysis of computational fluid dynamics are described. The visualization process can be illustrated by video tapes and stereo pictures. Although these visualization tools have dramatically improved the ability to conduct research in fluid dynamics, a comparison of the current environment for analysis with an 'ideal' environment illustrates that there are still major improvements that should be made. The most time-consuming task in future analyses of the increasingly complex computer simulations will be the extraction and clear display of the key features. In addition, the interface between the workstation and the scientist should be improved significantly. Current research on techniques for creating these improvements is described.

Watson, V.↗

Computational analysis of Variable Thrust Engine (VTE) performance

The Variable Thrust Engine (VTE) of the Orbital Maneuvering Vehicle (OMV) uses a hypergolic propellant combination of Monomethyl Hydrazine (MMH) and Nitrogen Tetroxide (NTO) as fuel and oxidizer, respectively. The performance of the VTE depends on a number of complex interacting phenomena such as atomization, spray dynamics, vaporization, turbulent mixing, convective/radiative heat transfer, and hypergolic combustion. This study involved the development of a comprehensive numerical methodology to facilitate detailed analysis of the VTE. An existing Computational Fluid Dynamics (CFD) code was extensively modified to include the following models: a two-liquid, two-phase Eulerian-Lagrangian spray model; a chemical equilibrium model; and a discrete ordinate radiation heat transfer model. The modified code was used to conduct a series of simulations to assess the effects of various physical phenomena and boundary conditions on the VTE performance. The details of the models and the results of the simulations are presented.

Giridharan, M. G.↗

Workshop on Computational Turbulence Modeling

The purpose of this meeting was to discuss the current status and future development of turbulence modeling in computational fluid dynamics for aerospace propulsion systems. Various turbulence models have been developed and applied to different turbulent flows over the past several decades and it is becoming more and more urgent to assess their performance in various complex situations. In order to help users in selecting and implementing appropriate models in their engineering calculations, it is important to identify the capabilities as well as the deficiencies of these models. This also benefits turbulence modelers by permitting them to further improve upon the existing models. This workshop was designed for exchanging ideas and enhancing collaboration between different groups in the Lewis community who are using turbulence models in propulsion related CFD. In this respect this workshop will help the Lewis goal of excelling in propulsion related research. This meeting had seven sessions for presentations and one panel discussion over a period of two days. Each presentation session was assigned to one or two branches (or groups) to present their turbulence related research work. Each group was asked to address at least the following points: current status of turbulence model applications and developments in the research; progress and existing problems; and requests about turbulence modeling. The panel discussion session was designed for organizing committee members to answer management and technical questions from the audience and to make concluding remarks.

Shabbir, A.↗

Magnetohydrodynamic Augmented Propulsion Experiment

A fundamental obstacle to routine space access is the specific energy limitations associated with chemical fuels. In the case of vertical take-off, the high thrust needed for vertical liftoff and acceleration to orbit translates into power levels in the 10 GW range. Furthermore, useful payload mass fractions are possible only if the exhaust particle energy (i.e., exhaust velocity) is much greater than that available with traditional chemical propulsion. The electronic binding energy released by the best chemical reactions (e.g., LOX/LH2 for example, is less than 2 eV per product molecule (approx. 1.8 eV per H2O molecule), which translates into particle velocities less than 5 km/s. Useful payload fractions, however, will require exhaust velocities exceeding 15 km/s (i.e., particle energies greater than 20 eV). As an added challenge, the envisioned hypothetical RLV (reusable launch vehicle) should accomplish these amazing performance feats while providing relatively low acceleration levels to orbit (2-3g maximum). From such fundamental considerations, it is painfully obvious that planned and current RLV solutions based on chemical fuels alone represent only a temporary solution and can only result in minor gains, at best. What is truly needed is a revolutionary approach that will dramatically reduce the amount of fuel and size of the launch vehicle. This implies the need for new compact high-power energy sources as well as advanced accelerator technologies for increasing engine exhaust velocity. Electromagnetic acceleration techniques are of immense interest since they can be used to circumvent the thermal limits associated with conventional propulsion systems. This paper describes the Magnetohydrodynamic Augmented Propulsion Experiment (MAPX) being undertaken at NASA Marshall Space Flight Center (MSFC). In this experiment, a 1-MW arc heater is being used as a feeder for a 1-MW magnetohydrodynamic (MHD) accelerator. The purpose of the experiment is to demonstrate that an MHD accelerator can be an effective augmentation system for increasing engine exhaust velocity. More specifically, the experiment is intended to show that electromagnetic effects are effective at producing flow acceleration whereas electrothermal effects do not cause unacceptable heating of the working fluid. The MHD accelerator was designed as an externally diagonalized segmented Faraday channel, which will be inserted into an existing 2-tesla electromagnet. This allows the external power to be connected through two terminals thereby minimizing the complexity and cost associated with powering each segment independently. The design of the accelerator and other components in the flow path has been completed and fabrication activities are underway. This paper provides a full description of MAPX including performance analysis, design, and test plans, and current status.

Litchford, Ron J.↗

Devescovinid trichomonad with axostyle-based rotary motor ("Rubberneckia"): taxonomic assignment as Caduceia versatilis sp. nov

An amitochondriate trichomonad cell of the family Devescovinidae (Class Parabasalia), helped demonstrate the fluid model of lipoprotein cell membranes. This wood-ingesting symbiont in the hindgut of the dry wood-eating termite Cryptotermes cavifrons is informally known to cell biologists as "Rubberneckia". As the microtubular axo-style complex generates force causing clockwise movement of the entire anterior portion of the cell at the shear zone the protist displays "head" rotation. Studies by phase contrast and videomicroscopy of live cells, of whole mounts by scanning, and thin sections by transmission electron microscopy extend the observations of Tamm and Tamm [24-26] and Tamm [19-23]. Habitat, cell shape, size, nuclear features, parabasal apparatus and other morphological details permit the assignment of "Rubberneckia" to Kirby's cosmopolitan genus Caduceia. This large-sized devescovinid has distinctive parabasal gyres, an axostylar rotary, motor, and regularly-associated nonflagellated, fusiform and flagellated rod epibiotic surface bacteria. In addition to regularly aligned epibionts intranuclear and endocytoplasmic bacteria are abundant and hydrogenosomes are Present. "Rubberneckia" is compared here to the other seven species of Caduceia. Since it is clearly sufficiently distinctive to warrant new species status, we named it C. versatilis.

Non-NASA Center↗

Controls/CFD Interdisciplinary Research Software Generates Low-Order Linear Models for Control Design From Steady-State CFD Results

The NASA Lewis Research Center is developing analytical methods and software tools to create a bridge between the controls and computational fluid dynamics (CFD) disciplines. Traditionally, control design engineers have used coarse nonlinear simulations to generate information for the design of new propulsion system controls. However, such traditional methods are not adequate for modeling the propulsion systems of complex, high-speed vehicles like the High Speed Civil Transport. To properly model the relevant flow physics of high-speed propulsion systems, one must use simulations based on CFD methods. Such CFD simulations have become useful tools for engineers that are designing propulsion system components. The analysis techniques and software being developed as part of this effort are an attempt to evolve CFD into a useful tool for control design as well. One major aspect of this research is the generation of linear models from steady-state CFD results. CFD simulations, often used during the design of high-speed inlets, yield high resolution operating point data. Under a NASA grant, the University of Akron has developed analytical techniques and software tools that use these data to generate linear models for control design. The resulting linear models have the same number of states as the original CFD simulation, so they are still very large and computationally cumbersome. Model reduction techniques have been successfully applied to reduce these large linear models by several orders of magnitude without significantly changing the dynamic response. The result is an accurate, easy to use, low-order linear model that takes less time to generate than those generated by traditional means. The development of methods for generating low-order linear models from steady-state CFD is most complete at the one-dimensional level, where software is available to generate models with different kinds of input and output variables. One-dimensional methods have been extended somewhat so that linear models can also be generated from two- and three-dimensional steady-state results. Standard techniques are adequate for reducing the order of one-dimensional CFD-based linear models. However, reduction of linear models based on two- and three-dimensional CFD results is complicated by very sparse, ill-conditioned matrices. Some novel approaches are being investigated to solve this problem.

Melcher, Kevin J.↗