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At least 19 records

Pseudospectral collocation methods for fourth order differential equations

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

Malek, Alaeddin↗

Flight Test Results from the Rake Airflow Gage Experiment on the F-15B

The results are described of the Rake Airflow Gage Experiment (RAGE), which was designed and fabricated to support the flight test of a new supersonic inlet design using Dryden's Propulsion Flight Test Fixture (PFTF) and F-15B testbed airplane (see figure). The PFTF is a unique pylon that was developed for flight-testing propulsion-related experiments such as inlets, nozzles, and combustors over a range of subsonic and supersonic flight conditions. The objective of the RAGE program was to quantify the local flowfield at the aerodynamic interface plane of the Channeled Centerbody Inlet Experiment (CCIE). The CCIE is a fixed representation of a conceptual mixed-compression supersonic inlet with a translating biconic centerbody. The primary goal of RAGE was to identify the relationship between free-stream and local Mach number in the low supersonic regime, with emphasis on the identification of the particular free-stream Mach number that produced a local Mach number of 1.5. Measurements of the local flow angularity, total pressure distortion, and dynamic pressure over the interface plane were also desired. The experimental data for the RAGE program were obtained during two separate research flights. During both flights, local flowfield data were obtained during straight and level acceleration segments out to steady-state test points. The data obtained from the two flights showed small variations in Mach number, flow angularity, and dynamic pressure across the interface plane at all flight conditions. The data show that a free-stream Mach number of 1.65 will produce the desired local Mach number of 1.5 for CCIE. The local total pressure distortion over the interface plane at this condition was approximately 1.5%. At this condition, there was an average of nearly 2 of downwash over the interface plane. This small amount of downwash is not expected to adversely affect the performance of the CCIE inlet.

Frederick, Michael↗

On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach

The coupling of hyperbolic and parabolic systems is discussed in a domain Omega divided into two distinct subdomains omega(+) and omega(-). The main concern is to find the proper interface conditions to be fulfilled at the surface separating the two domains. Next, they are used in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, i.e., the hyperbolic system is rendered parabolic by adding a small artifical viscosity. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These are taken as interface conditions for the hyperbolic-parabolic problem. Actually, two alternative sets of interface conditions are discussed according to whether the regularization procedure is variational or nonvariational. It is shown how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, a method of resolution is discussed which alternates the resolution of the hyperbolic problem within omega(-) and of the parabolic one within omega(+). The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows.

Gastaldi, Fabio↗

Stability of interfaces with mesh refinement

A study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, a root condition which implies stability for the initial-boundary value problem for this type of interface is derived. From the root condition, the stability of several interface equations is proved, using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, stability for Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved.

Berger, M. J.↗

Bubble departure radii at solidification interfaces

A model has been developed for the prediction of bubble departure radii from flat solidification interfaces, including the effect of thermocapillary forces. Under normal gravity conditions, the necessary gas bubble radius required for departure from a CBr4 solidification interface is predicted to be approximately 1/2 mm in agreement with measured values. Under microgravity conditions, however, where surface forces predominate, the model predicts a seemingly prohibitive value of 40 mm. This result is at least in agreement with the microgravity tests conducted on the NASA SPAR I and SPAR III sounding rockets (1978) where the bubbles were not larger than 2 mm in radius and no bubble detachment was observed.

Cole, R.↗

Stresses and deformations in cross-ply composite tubes subjected to a uniform temperature change: Elasticity and Approximate Solutions

The effects of a uniform temperature change on the stresses and deformations of composite tubes are investigated. The accuracy of an approximate solution based on the principle of complementary virtual work is determined. Interest centers on tube response away from the ends and so a planar elasticity approach is used. For the approximate solution a piecewise linear variation of stresses with the radial coordinate is assumed. The results from the approximate solution are compared with the elasticity solution. The stress predictions agree well, particularly peak interlaminar stresses. Surprisingly, the axial deformations also agree well. This, despite the fact that the deformations predicted by the approximate solution do not satisfy the interface displacement continuity conditions required by the elasticity solution. The study shows that the axial thermal expansion coefficient of tubes with a specific number of axial and circumferential layers depends on the stacking sequence. This is in contrast to classical lamination theory which predicts the expansion to be independent of the stacking arrangement. As expected, the sign and magnitude of the peak interlaminar stresses depends on stacking sequence.

Hyer, M. W.↗

Stresses and deformations in cross-ply composite tubes subjected to a uniform temperature change

This study investigates the effects of a uniform temperature change on the stresses and deformations of composite tubes and determines the accuracy of an approximate solution based on the principle of complementary virtual work. Interest centers on tube response away from the ends and so a planar elasticity approach is used. For the approximate solution a piecewise linear variation of stresses with the radial coordinate is assumed. The results from the approximate solution are compared with the elasticity solution. The stress predictions agree well, particularly peak interlaminar stresses. Surprisingly, the axial deformations also agree well, despite the fact that the deformations predicted by the approximate solution do not satisfy the interface displacement continuity conditions required by the elasticity solution. The study shows that the axial thermal expansion coefficient of tubes with a specific number of axial and circumferential layers depends on the stacking sequence. This is in contrast to classical lamination theory, which predicts that the expansion will be independent of the stacking arrangement. As expected, the sign and magnitude of the peak interlaminar stresses depend on stacking sequence. For tubes with a specific number of axial and circumferential layers, thermally induced interlaminar stresses can be controlled by altering stacking arrangement.

Hyer, M. W.↗

Development and Validation of a Slosh-Based Ullage Collapse Model

Ullage collapse modeling is an essential tool for the success of spacecraft powered by cryogenic propellants. The degree of ullage collapse can have a major impact on the design of cryogenic propellant tanks as well as the mission’s concept of operations. Additional commodities (e.g., helium gas for pressure control) and their quantities are dependent on the amount of heat transfer and phase change of propellants expected during flight. Currently, high-fidelity computational fluid dynamics (CFD) with heat transfer and phase change is the most accurate method for predicting ullage collapse. CFD simulations of this kind, however, can become very costly for problems requiring large domains, long simulation times, or significant liquid motion. Consequently, there is a need for a reduced order model to provide fast responses which can be used in engineering design decisions. The proposed numerical model is a nodal approach, augmented by outputs from CFD analyses that are generated rapidly and reliably. These outputs include transient values for liquid-vapor interface area, interface velocity, and liquid position using a volume of fluid (VoF) method. There are several benefits of combining a nodal approach with slosh predictions from CFD simulations: leveraging of the Loci/STREAM-VoF solver that has been validated in high-g and low-g environments, rapid modeling of VoF with isothermal propellant simplifications and relatively coarse meshes, and improved model predictions from the nodal tool. The nodal approach, informed by the slosh predictions from CFD, assumes uniform conditions in the liquid and gas domains. Modifications to account for temperature stratification in the liquid and gas may be implemented as a future improvement. Thermodynamic changes within the tank are solved at user-defined timesteps. First, impacts due to mass entering or leaving the control volume are captured. Second, heat transfer impacts are accounted for by tracking the volume of the gas which has been contacted by liquid in that timestep and rebalancing the internal energy. Finally, phase change is approximated using the area and velocity of the interface. The tank conditions are re-computed by conserving mass and energy. Calibration of the model to several test flights and experiments are required to baseline the predictive power and uncertainty. The final product will enable designers and engineers to create quick-turnaround predictions for ullage collapse and inform design feasibility.

Computational Fluid Dynamics↗

Theoretical and Experimental Investigations on Droplet Evaporation and Droplet Ignition at High Pressures

The evaporation of fuel droplets under high ambient pressure and temperature in normal gravity and microgravity has been investigated experimentally. For subcritical ambient conditions, droplet evaporation after a heat-up period follows the d(exp 2)-law. For all data the evaporation constant increases as the ambient temperature increases. At identical ambient conditions the evaporation constant under microgravity is smaller compared to normal gravity. This effect can first be observed at 1 bar and increases with ambient pressure. Preliminary experiments on ignition delay for self-igniting fuel droplets have been performed. Above a 1 s delay time, at identical ambient conditions, significant differences in the results of the normal and microgravity data are observed. Self-ignition occurs within different temperature ranges due to the influence of gravity. The time dependent behavior of the droplet is examined theoretically. In the calculations two different approaches for the gas phase are applied. In the first approach the conditions at the interface are given using a quasi steady theory approximation. The second approach uses a set of time dependent governing equations for the gas phase which are then evaluated. In comparison, the second model shows a better agreement with the drop tower experiments. In both cases a time dependent gasification rate is observed.

Ristau, R.↗

The optical depth of the 158 micrometer (C-12 II) line: Detection of the F=1 yields 0 (C-13 III) hyperfine-structure component

The first detection of the F = 1 yields 0 hyperfine component of the 158 micrometer (C-13 II) fine structure line in the interstellar medium is reported. A twelve point intensity map was obtained of the (C-13 II) distribution over the inner 190 inch (right ascension) by 190 inch (declination) regions of the Orion nebula using an imaging Fabry-Perot interferometer. The (C-12 II)/(C-13 II) line intensity ratio varied significantly over the region mapped. It is highest (86 plus or minus 9) in the core of the Orion H II region and significantly lower (62 plus or minus 7) in the outer regions of the map, reflecting higher optical depth in the (C-12 II) line here. It is suggested that this enhanced optical depth is the result of limb brightening of the optically thin (C-13 II) line at the edges of the bowl-shaped H II region blister. If the C-12/C-13 abundance ratio is 43, the (C-12 II) line in the inner regions of the Orion nebula, has a low optical depth: tau sub 12 approximately = 0.75 plus or minus 0.25. The optical depth together with the large brightness temperature of the (C-12 II) line (approximately 160 K) requires that the excitation temperature of the P-2 sub 3/2 level be approximately 310 K, in very good agreement with the previous analysis of the physical conditions of the Orion interface region based on fine structure line intensity ratios and photodissociation region models. If the C-12/C-13 abundance ratio is 67, the line optical depth is somewhat larger (tau sub 12 approximately = 1.85), and the transition excitation temperature is somewhat smaller (approximately 190 K) than that predicted by these models. The present results therefore support values approximately = 43 for the C-12/C-13 abundance ratio in the Orion nebula.

Stacey, G. J.↗

Radiative Transfer in a Discrete Random Medium Adjacent to a Half-Space with a Rough Interface

For a macroscopically plane-parallel discrete random medium, the boundary conditions for the specific coherency dyadic at a rough interface are derived. The derivation is based on a modification of the Twersky approximation for a scattering system consisting of a group of particles and the rough surface, and reduces to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave propagating in a discrete random medium with non-scattering boundaries. In a matrix-form setting, the boundary conditions for the specific coherency dyadic imply the boundary conditions for specific intensity column vectors which in turn, yield the expressions for the reflection and transmission matrices. The derived expressions are shown to be identical to those obtained by applying a phenomenological approach based on a facet model to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave.

Adrian Doicu↗

Application of the MHD energy principle to magnetostatic atmospheres

The MHD energy principle is applied to the stability of a magnetized atmosphere which is bounded below by much denser fluid, as is the solar corona. The two fluids are treated as ideal; the approximation is consistent with the energy principle, and the dynamical conditions that must hold at a fluid-fluid interface are used to show that if vertical displacements of the lower boundary are premitted, then the lower atmosphere must be perturbed as well. However, displacements which do not perturb the coronal boundary can be properly treated as isolated perturbations of the corona alone.

Zweibel, E. G.↗

Application of the MHD energy principle to magnetostatic atmospheres

The MHD energy principle is applied to the stability of a magnetized atmosphere which is bounded below by much denser fluid, as is the solar corona. The two fluids are treated as ideal; the approximation is consistent with the energy principle, and the dynamical conditions that must hold at a fluid-fluid interface are used to show that if vertical displacements of the lower boundary are permitted, then the lower atmosphere must be perturbed as well. However, displacements which do not perturb the coronal boundary can be properly treated as isolated perturbations of the corona alone.

Zweibel, E. G.↗

Analysis of Hydrodynamic (Landau) Instability in Liquid-Propellant Combustion at Normal and Reduced Gravity

The burning of liquid propellants is a fundamental combustion problem that is applicable to various types of propulsion and energetic systems. The deflagration process is often rather complex, with vaporization and pyrolysis occurring at the liquid/gas interface and distributed combustion occurring either in the gas phase or in a spray. Nonetheless, there are realistic limiting cases in which combustion may be approximated by an overall reaction at the liquid/gas interface. In one such limit, the gas flame occurs under near-breakaway conditions, exerting little thermal or hydrodynamic influence on the burning propellant. In another such limit, distributed combustion occurs in an intrusive regime, the reaction zone lying closer to the liquid/gas interface than the length scale of any disturbance of interest. Finally, the liquid propellant may simply undergo exothermic decomposition at the surface without any significant distributed combustion, such as appears to occur in some types of HydroxylAmmonium Nitrate (HAN)-based liquid propellants at low pressures. Such limiting models have recently been formulated,thereby significantly generalizing earlier classical models that were originally introduced to study the hydrodynamic stability of a reactive liquid/gas interface. In all of these investigations, gravity appears explicitly and plays a significant role, along with surface tension, viscosity, and, in the more recent models, certain reaction-rate parameters associated with the pressure and temperature sensitivities of the reaction itself. In particular, these parameters determine the stability of the deflagration with respect to not only classical hydrodynamic disturbances, but also with respect to reactive/diffusive influences as well. Indeed, the inverse Froude number, representing the ratio of buoyant to inertial forces, appears explicitly in all of these models, and consequently, in the dispersion relation that determines the neutral stability boundaries beyond which steady, planar burning is unstable to nonsteady, and/or nonplanar (cellular) modes of burning. These instabilities thus lead to a number of interesting phenomena, such as the sloshing type of waves that have been observed in mixtures of HAN and TriEthanolAmmonium Nitrate (TEAN) with water. Although the Froude number was treated as an O(1) quantity in these studies, the limit of small inverse Froude number corresponding to the microgravity regime is increasingly of interest and can be treated explicitly, leading to various limiting forms of the models, the neutral stability boundaries, and, ultimately, the evolution equations that govern the nonlinear dynamics of the propagating reaction front. In the present work, we formally exploit this limiting parameter regime to compare some of the features of hydrodynamic instability of liquid-propellant combustion at reduced gravity with the same phenomenon at normal gravity.

Margolis, Stephen B.↗

Gas Diffusion in Fluids Containing Bubbles

Mathematical model describes movement of gases in fluid containing many bubbles. Model makes it possible to predict growth and shrink age of bubbles as function of time. New model overcomes complexities involved in analysis of varying conditions by making two simplifying assumptions. It treats bubbles as point sources, and it employs approximate expression for gas concentration gradient at liquid/bubble interface. In particular, it is expected to help in developing processes for production of high-quality optical glasses in space.

Zak, M.↗

Coupling finite element and spectral methods: First results

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved.

Bernardi, Christine↗

Coupling finite element and spectral methods - First results

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved.

Bernardi, Christine↗

Interactive Educational Tool for Turbofan and Afterburning Turbojet Engines

A workstation-based, interactive educational computer program has been developed at the NASA Lewis Research Center to aid in the teaching and understanding of turbine engine design and analysis. This tool has recently been extended to model the performance of two-spool turbofans and afterburning turbojets. The program solves for the flow conditions through the engine by using classical one-dimensional thermodynamic analysis found in various propulsion textbooks. Either an approximately thermally perfect or calorically perfect gas can be used in the thermodynamic analysis. Students can vary the design conditions through a graphical user interface; engine performance is calculated immediately. A variety of graphical formats are used to present results, including numerical results, moving bar charts, and student-generated temperature versus entropy (Ts), pressure versus specific volume (pv), and engine performance plots. The package includes user-controlled printed output, restart capability, online help screens, and a browser that displays teacher-prepared lessons in turbomachinery. The program runs on a variety of workstations or a personal computer using the UNIX operating system and X-based graphics. It is being tested at several universities in the midwestern United States; the source and executables are available free from the author.

Benson, Thomas J.↗