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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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NASA NTRS1991

Modeling flow at the nozzle of a solid rocket motor

The mechanical behavior of a rocket motor internal flow field results in a system of nonlinear partial differential equations which can be solved numerically. The accuracy and the convergence of the solution of the system of equations depends largely on how precisely the sharp gradients can be resolved. An adaptive grid generation scheme is incorporated into the computer algorithm to enhance the capability of numerical modeling. With this scheme, the grid is refined as the solution evolves. This scheme significantly improves the methodology of solving flow problems in rocket nozzle by putting the refinement part of grid generation into the computer algorithm.

Chow, Alan S.↗
NASA NTRS1990

Modelling crystal growth: Convection in an asymmetrically heated ampoule

The objective was to develop and implement a numerical method capable of solving the nonlinear partial differential equations governing heat, mass, and momentum transfer in a 3-D cylindrical geometry in order to examine the character of convection in an asymmetrically heated cylindrical ampoule. The details of the numerical method, including verification tests involving comparison with results obtained from other methods, are presented. The results of the study of 3-D convection in an asymmetrically heated cylinder are described.

Alexander, J. Iwan D.↗
NASA NTRS1992

Essentially nonoscillatory postprocessing filtering methods

High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

Lafon, F.↗
NASA NTRS1992

Using adaptive grid in modeling rocket nozzle flow

The mechanical behavior of a rocket motor internal flow field results in a system of nonlinear partial differential equations which cannot be solved analytically. However, this system of equations called the Navier-Stokes equations can be solved numerically. The accuracy and the convergence of the solution of the system of equations will depend largely on how precisely the sharp gradients in the domain of interest can be resolved. With the advances in computer technology, more sophisticated algorithms are available to improve the accuracy and convergence of the solutions. An adaptive grid generation is one of the schemes which can be incorporated into the algorithm to enhance the capability of numerical modeling. It is equivalent to putting intelligence into the algorithm to optimize the use of computer memory. With this scheme, the finite difference domain of the flow field called the grid does neither have to be very fine nor strategically placed at the location of sharp gradients. The grid is self adapting as the solution evolves. This scheme significantly improves the methodology of solving flow problems in rocket nozzles by taking the refinement part of grid generation out of the hands of computational fluid dynamics (CFD) specialists and place it into the computer algorithm itself.

Chow, Alan S.↗
NASA NTRS1992

Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design

Various computational methodologies relevant to large-scale multidisciplinary gradient-based optimization for engineering systems design problems are examined with emphasis on the situation where one or more discipline responses required by the optimized design procedure involve the solution of a system of nonlinear partial differential equations. Such situations occur when advanced CFD codes are applied in a multidisciplinary procedure for optimizing an aerospace vehicle design. A technique for satisfying the multidisciplinary design requirements for gradient information is presented. The technique is shown to permit some leeway in the CFD algorithms which can be used, an expansion to 3D problems, and straightforward use of other computational methodologies.

Newman, P. A.↗
NASA NTRS1992

Invariant two-dimensional structures of vorticity field in nonviscous fluid

It is shown that a fluid flow which are invariant under translations along their vorticity field directions can be described in terms of a formally 2D system. Its nonviscous behavior of the scalar vorticity is governed by Hamiltonian dynamics, similar to the 2D line vortex case. Applying a statistical mechanical method, the equilibrium distribution of the vorticity field for such a general 2D system is shown to obey a nonlinear partial differential equation that is a generalization of the sinh-Poisson equation derived by Montgomery and Joyce (1973, 1974). Thus, if the domain containing the fluid is finite, similar negative temperature states may occur and the distribution of the vorticity intensity may become localized in space. The results are believed to have applications in studying large-scale partial and temporal properties of flows having small but nonzero viscosity. The analysis can probably be extended to other fluid systems with curved magnetic field lines.

Chen, Hudong↗
NASA NTRS1993

Theory of Metastable State Relaxation in a Gravitational Field for Non-Critical Binary Systems with Non-Conserved Order Parameter

A new mathematical ansatz is developed for solution of the time-dependent Ginzburg-Landau nonlinear partial differential equation describing metastable state relaxation in binary (solute+solvent) non-critical solutions with non-conserved scalar order parameter in presence of a gravitational field. It has been demonstrated analytically that in such systems metastability initiates heterogeneous solute redistribution which results in the formation of a non-equilibrium singly-periodic spatial solute structure in the new solute-rich phase. The critical radius of nucleation and the induction time in these systems are gravity-dependent. It has also been proved that metastable state relaxation in vertical columns of supersaturated non-critical binary solutions leads to formation of the solute concentration gradient. Analytical expression for this concentration gradient is found and analysed. It is concluded that gravity can initiate phase separation (nucleation or spinodal decomposition).

Izmailov, Alexander F.↗
NASA NTRS1947

The Elasto-Plastic Stability of Plates

This article explains results developed from the following research: 'The Stability of Plates and Shells beyond the Elastic Limit.' A significant improvement is found in the derivation of the relations between the stress factors and the strains resulting from the instability of plates and shells. In a strict analysis, the problem reduces to the solution of two simultaneous nonlinear partial differential equations of the fourth order in the deflection and stress function, and in the approximate analysis to a single linear equation of the Bryan type. Solutions are given for the special cases of a rectangular plate buckling into a cylindrical form, and of an arbitrarily shaped plate under uniform compression. These solutions indicate that the accuracy obtained by the approximate method is satisfactory.

Ilyushin, A. A.↗
NASA NTRS1989

Hypervelocity atmospheric flight: Real gas flow fields

Flight in the atmosphere is examined from the viewpoint of including real gas phenomena in the flow field about a vehicle flying at hypervelocity. That is to say, the flow field is subject not only to compressible phenomena, but is dominated by energetic phenomena. There are several significant features of such a flow field. Spatially, its composition can vary by both chemical and elemental species. The equations which describe the flow field include equations of state and mass, species, elemental, and electric charge continuity; momentum; and energy equations. These are nonlinear, coupled, partial differential equations that have been reduced to a relatively compact set of equations in a self-consistent manner (which allows mass addition at the surface at a rate comparable to the free-stream mass flux). The equations and their inputs allow for transport of these quantities relative to the mass-average behavior of the flow field. Thus transport of mass by chemical, thermal, pressure, and forced diffusion; transport of momentum by viscosity; and transport of energy by conduction, chemical considerations, viscosity, and radiative transfer are included. The last of these complicate the set of equations by making the energy equations a partial integrodifferential equation. Each phenomenon is considered and represented mathematically by one or more developments. The coefficients which pertain are both thermodynamically and chemically dependent. Solutions of the equations are presented and discussed in considerable detail, with emphasis on severe energetic flow fields. Hypervelocity flight in low-density environments where gaseous reactions proceed at finite rates chemical nonequilibrium is considered, and some illustrations are presented. Finally, flight where the flow field may be out of equilibrium, both chemically and thermodynamically, is presented briefly.

Howe, John T.↗
NASA NTRS1990

Hypervelocity atmospheric flight: Real gas flow fields

Flight in the atmosphere is examined from the viewpoint of including real gas phenomena in the flow field about a vehicle flying at hypervelocity. That is to say, the flow field is subject not only to compressible phenomena, but is dominated by energetic phenomena. There are several significant features of such a flow field. Spatially, its composition can vary by both chemical and elemental species. The equations which describe the flow field include equations of state and mass, species, elemental, and electric charge continuity; momentum; and energy equations. These are nonlinear, coupled, partial differential equations that were reduced to a relatively compact set of equations of a self-consistent manner (which allows mass addition at the surface at a rate comparable to the free-stream mass flux). The equations and their inputs allow for transport of these quantities relative to the mass-averaged behavior of the flow field. Thus transport of mass by chemical, thermal, pressure, and forced diffusion; transport of momentum by viscosity; and transport of energy by conduction, chemical considerations, viscosity, and radiative transfer are included. The last of these complicate the set of equations by making the energy equation a partial integrodifferential equation. Each phenomenon is considered and represented mathematically by one or more developments. The coefficients which pertain are both thermodynamically and chemically dependent. Solutions of the equations are presented and discussed in considerable detail, with emphasis on severe energetic flow fields. For hypervelocity flight in low-density environments where gaseous reactions proceed at finite rates, chemical nonequilibrium is considered and some illustrations are presented. Finally, flight where the flow field may be out of equilibrium, both chemically and thermodynamically, is presented briefly.

Howe, John T.↗
NASA NTRS2004

Rapid Switching and Modulation by use of Coupled VCSELs

Devices incorporating coupled vertical-cavity surface-emitting lasers (VCSELs) have been proposed as means to effect unprecedentedly rapid modulation and/or switching of light beams in advanced optical communication and information-processing systems. A basic device according to the proposal would comprise two or more VCSELs (1) positioned so close to each other that they are coupled and (2) connected to a current source that biases them at a steady current above the threshold current for lasing. The operation of VCSELs under the conditions stated above has been simulated computationally with an algorithm that solves an approximation of the Maxwell-Bloch equations. (These are nonlinear coupled partial differential equations that model the relevant aspects of the physics of semiconductor lasers.) The results of the simulations show that the intensities of the beams oscillate, out of phase relative to each other, at a frequency that is typically of the order of tens of gigahertz. In particular, one simulation was performed for two coupled VCSELs, each having a circular aperture 5.6 microns in diameter, operating at a wavelength of 980 nm. The results of the simulation showed that the intensities of the two beams oscillated in opposite phase (see Figure 1) at a frequency of 42 GHz. In the far field, the beams were found to be separated by an angle of approx.=8deg. The intensities of the beams were also found to oscillate in the near field, 90deg out of phase with the oscillation in the far field. Figure 2 depicts a simple near-field switching device in which an assembly of microlenses would couple the two oscillating beams to two receivers.

Goorjian, Peter M.↗
NASA NTRS1990

A parallel iterative solution method for systems of nonlinear hyperbolic equations

An iterative algorithm suitable for the solution of a system of nonlinear hyperbolic partial differentiation equations in multiple dimensions is discussed. Current numerical methods for systems of nonlinear PDEs have limited parallelism due to strong coupling between the equations. This method decouples the PDEs by linearizing the convention coefficient for a space-time domain. This provides large grain parallelism. The linearization also allows the treatment of some terms in the equations as source terms, providing more freedom to choose from a wider variety of numerical methods. Smaller grain parallelism may be exploited within the solves for each equation. Thus, the method has potential for parallelism at several levels.

Scroggs, Jeffrey S.↗
NASA NTRS1972

A numerical investigation of film cooling as an alternate thermal protection system for spacecraft during atmospheric reentry

Research was undertaken to determine the mass required to protect a surface during atmospheric reentry by injecting a cold gas tangentially along the surface. A consideration of this problem revealed that the boundary layer equations could be employed to approximate the most important flow configurations which occur in film cooling. However, even with this simplification, the equations to be solved are complex, nonlinear, coupled partial differential equations. Due to the relative weakness of classical methods of boundary layer analysis, finite difference methods were chosen as the best approach to the formidable problem under consideration. Instead of blindly adopting a proposed technique and trying to force it to the problem, a more systematic study was deemed to be not only desirable but necessary. A relatively basic study of the application of finite difference methods was undertaken with the objective of developing an algorithm capable of obtaining accurate solutions to film cooling problems.

Clausing, A. M.↗
NASA NTRS1988

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Achtemeier, Gary L.↗
NASA NTRS1993

On the dynamics of approximating schemes for dissipative nonlinear equations

Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

Jones, Donald A.↗
NASA NTRS1979

Stability analysis of a liquid fuel annular combustion chamber

The problems of combustion instability in an annular combustion chamber are investigated. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations. By evaluating these effects, parameters which cause instabilities to occur in the combustion chamber can be ascertained. It is assumed that in the annular combustion chamber, the liquid propellants are injected uniformly across the injector face, the combustion processes are distributed throughout the combustion chamber, and that no time delay occurs in the combustion processes.

Mcdonald, G. H.↗
NASA NTRS1978

Stability analysis of a liquid fuel annular combustion chamber

High frequency combustion instability problems in a liquid fuel annular combustion chamber are examined. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation in order to analyze the problem of instability. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations.

Mcdonald, G. H.↗