Search NASA⌕ Search

SEARCH · Search NASA

Results for “explicit solution”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

A finite volume Euler calculation of the aerodynamics of transonic airfoil-vortex interaction

Unsteady inviscid transonic airfoil-vortex interaction is numerically analyzed by solving the two-dimensional unsteady Euler equations in integral form using a finite volume scheme. The solution procedure is based on an explicit Runge-Kutta time-stepping scheme wherein the spatial terms are central-differenced and a combination of second- and fourth-differences in the flow variables is used to form the numerical dissipation terms to stabilize the scheme. A velocity decomposition technique is applied to alleviate the problem of vortex diffusion by the numerical dissipation terms and to treat the interaction of a Rankine vortex with an airfoil accurately. Results obtained are compared with available numerical data.

Damodaran, Murali↗
NASA NTRS1988

Self-similar dynamics of polytropic gaseous spheres

The fundamental equations governing the self-similar dynamics of 'polytropic' gaseous spheres are derived, and the asymptotic solutions are given. The solutions divide into cases with and without 'critical points' in closed analogy with the solar wind solutions of Holzer and Axford (1970). Properties for solutions with critical points are discussed, and their behavior around the critical point is derived explicitly for n = 1. Numerical examples of self-similar solutions for n = 1 and n = 2 - gamma are presented, and the properties of the solutions are discussed.

Suto, Yasushi↗
NASA NTRS1994

Energy and maximum norm estimates for nonlinear conservation laws

We have devised a technique that makes it possible to obtain energy estimates for initial-boundary value problems for nonlinear conservation laws. The two major tools to achieve the energy estimates are a certain splitting of the flux vector derivative f(u)(sub x), and a structural hypothesis, referred to as a cone condition, on the flux vector f(u). These hypotheses are fulfilled for many equations that occur in practice, such as the Euler equations of gas dynamics. It should be noted that the energy estimates are obtained without any assumptions on the gradient of the solution u. The results extend to weak solutions that are obtained as point wise limits of vanishing viscosity solutions. As a byproduct we obtain explicit expressions for the entropy function and the entropy flux of symmetrizable systems of conservation laws. Under certain circumstances the proposed technique can be applied repeatedly so as to yield estimates in the maximum norm.

Olsson, Pelle↗
NASA NTRS1993

Time-dependent modulation of galactic cosmic rays by merged interaction regions

Models that solve the one-dimensional, solar modulation equation have reproduced the 11-year galactic cosmic ray using functional representations of global merged interaction regions (MIRs). This study extends those results to the solution of the modulation equation with explicit time dependence. The magnetometers on Voyagers 1 and 2 provide local magnetic field intensities at regular intervals, from which one calculates the ratio of the field intensity to the average local field. These ratios in turn are inverted to form diffusion coefficients. Strung together in radius and time, these coefficents then fall and rise with the strength of the interplanetary magnetic field, becoming representations of MIRs. These diffusion coefficients, calculated locally, propagate unchanged from approx. 10 AU to the outer boundary (120 AU). Inside 10 AU, all parameters, including the diffusion coefficient are assumed constant in time and space. The model reproduces the time-intensity profiles of Voyager 2 and Pioneer 10. Radial gradient data from 1982-1990 between Pioneer 10 and Voyager 2 are about the same magnitude as those calculated in the model. It is also shows agreement in rough magnitude with the radial gradient between Pioneer 10 and 1 AU. When coupled with enhanced, time-dependent solar wind speed at the probe's high latitude, as measured by independent observers, the model also follows Voyager 1's time-intensity profile reasonably well, providing a natural source the model also follows Voyager 1's time-intensity profile reasonably well, providing a natural source for the observed negative latitudinal gradients. The model exhibits the 11-year cyclical cosmic ray intensity behavior at all radii, including 1 AU, not just at the location of the spacecraft where the magnetic fields are measured. In addition, the model's point of cosmic ray maximum correctly travels at the solar wind speed, illustrating the well-known propagation of modulation. Finally, at least in the inner heliosphere this model accounts for the delay experienced by lower-rigidity protons in reaching their time-intensity peak. The actual delays in this model, however, are somewhat smaller than the data. In the outer heliosphere the models sees no delays, and the data are ambiguous as to their existence. It appears that strong magnetic field compression regions (merged interaction regions) that are 3-4 times the average field strength can, at least in a helioequatorial band, disrupt effects, such as drifts, that could dominate in quieter magnetic fields. The question remains: Is the heliosphere ever quiet enough to allow such effects to be unambiguously measured, at least in the midlatitudes?

Perko, J. S.↗
NASA NTRS1992

Vertical landing on an asteroid

This work is concerned with the final approach phase and vertical landing on an asteroid with a power-limited, electrically propelled spacecraft. With gravitational effects taken into account, a new solution to the fuel optimal vertical landing on an asteroid was obtained. In this solution, the spacecraft commanded acceleration is explicitly expressed as a function of vehicle velocity and time to go. Based on qualitative methods of analysis, the guidance strategy and the resulting trajectories were studied. It is shown that these fuel-optimal trajectories effectively assure a vertical soft landing on the asteroid. Results of numerical simulations for the vertical landing, starting from an elliptical orbit are presented.

Harel, D.↗
NASA NTRS1998

Prediction of Transitional Flows in the Low Pressure Turbine

Current turbulence models tend to give too early and too short a length of flow transition to turbulence, and hence fail to predict flow separation induced by the adverse pressure gradients and streamline flow curvatures. Our discussion will focus on the development and validation of transition models. The baseline data for model comparisons are the T3 series, which include a range of free-stream turbulence intensity and cover zero-pressure gradient to aft-loaded turbine pressure gradient flows. The method will be based on the conditioned N-S equations and a transport equation for the intermittency factor. First, several of the most popular 2-equation models in predicting flow transition are examined: k-e [Launder-Sharina], k-w [Wilcox], Lien-Leschiziner and SST [Menter] models. All models fail to predict the onset and the length of transition, even for the simplest flat plate with zero-pressure gradient(T3A). Although the predicted onset position of transition can be varied by providing different inlet turbulent energy dissipation rates, the appropriate inlet conditions for turbulence quantities should be adjusted to match the decay of the free-stream turbulence. Arguably, one may adjust the low-Reynolds-number part of the model to predict transition. This approach has so far not been very successful. However, we have found that the low-Reynolds-number model of Launder and Sharma [1974], which is an improved version of Jones and Launder [1972] gave the best overall performance. The Launder and Sharma model was designed to capture flow re-laminarization (a reverse of flow transition), but tends to give rise to a too early and too fast transition in comparison with the physical transition. The three test cases were for flows with zero pressure gradient but with different free-stream turbulent intensities. The same can be said about the model when considering flows subject to pressure gradient(T3C1). To capture the effects of transition using existing turbulence models, one approach is to make use of the concept of the intermittency to predict the flow transition. It was originally based on the intermittency distribution of Narasimha [1957], and then gradually evolved into a transport equation for the intermittency factor. Gostelow and associates [1994,1995] have made some improvements to Narasimha's method in an attempt to account for both favorable and adverse pressure gradients. Their approach is based on a linear, explicit combination of laminar and turbulent solutions. This approach fails to predict the overshoot of the skin friction on a flat plate near the end of transition zone, even though the length of transition is well predicted. The major flaw of Gostelow's approach is that it assumes the non-turbulent part being the laminar solution and the turbulent part being the turbulent solution and they do not interact across the transitional region. The technique in condition averaging the flow equations in intermittent flows was first introduced by Libby [1975] and Dopazo [1977] and further refined by Dick and associates [1988, 1996]. This approach employs two sets of transport equations for the non-turbulent part and the other for the turbulent part. The advantage of this approach is that it allows the interaction of non-turbulent and turbulent velocities through the introduction of additional source terms in the continuity and momentum equations for the non-turbulent and turbulent velocities. However, the strong coupling of the two sets of equations has caused some numerical difficulties, which requires special attention. The prediction of the skin friction can be improved by this approach via the implicit coupling of non-turbulent and turbulent velocity flelds. Another improvement of the interrmittency model can be further made by allowing the intermittency to vary in the cross-stream direction. This is one step prior to testing any proposal for the transport equation for the intermittency factor. Instead of solving the transport equation for the intermittency factor, the distribution for the intermittency factor is prescribed by Klebanoff's empirical formula [1955]. The skin friction is very well predicted by this new modification, including the overshoot of the profile near the end of the transition zone. The outcome of this study is very encouraging since it indicates that the proper description of the intermittency distribution is the key to the success of the model prediction. This study will be used to guide us on the modelling of the intermittency transport equation.

Huang, George↗
NASA NTRS1997

Heart Fibrillation and Parallel Supercomputers

The Luo and Rudy 3 cardiac cell mathematical model is implemented on the parallel supercomputer CRAY - T3D. The splitting algorithm combined with variable time step and an explicit method of integration provide reasonable solution times and almost perfect scaling for rectilinear wave propagation. The computer simulation makes it possible to observe new phenomena: the break-up of spiral waves caused by intracellular calcium and dynamics and the non-uniformity of the calcium distribution in space during the onset of the spiral wave.

Kogan, B. Y.↗
NASA NTRS1993

Progressive wave equations and algorithms for sonic boom propagation

Raspet, Bass, and others have carried out a number of studies of sonic boom propagation using algorithms that can be traced back to a 1973 doctoral thesis written by Pestorius at the University of Texas. The algorithm was originally set forth without a formal derivation from a governing set of partial differential equations. Perusal of Pestorius's report and of subsequent reports from the Texas nonlinear accoustics group suggests that such a derivation has not yet appeared in the archival literature. The author and his colleagues, on the other hand, have been working with an explicit set of approximate partial differential equations analogous to Burgers' equation. The present paper traces through the pertinent statements in the Pestorius reports and shows that the derived algorithm, at least in the limit of sufficiently short steps, is equivalent to the numerical solution of a similar (and nearly equivalent) explicit set of partial differential equations. The tying-together of the two approaches is important because it is possible that some of the basic ideas embodied in the Pestorius algorithm may be incorporated in other contexts where more nearly general models of sonic boom propagation through realistic atmospheres are to be solved numerically.

Pierce, Allan D.↗
NASA NTRS1993

Near-field supersonic flow predictions by an adaptive unstructured tetrahedral grid solver

Applicability of a three-dimensional solution adaptive unstructured tetrahedral Euler flow solver about generic models for near-field sonic boom pressure signature predictions is evaluated. Comparisons of computational and experimental data demonstrates the capability of the method for predicting inviscid solutions useful for high speed calculations about simple 3-D geometries. The approach has promising features and results indicate potential for application to more complex configurations. The mesh generation is based on the advancing front technique, and steady state solutions of the Euler equations are achieved by explicit time integration. Spatial discretization uses the Taylor-Galerkin approach; an alternate time integration, based on the Runge-Kutta method, is also included. The solution-adaptive grid procedure is based on either remeshing or mesh refinement techniques. An alternative geometry-adaptive grid procedure has also been incorporated.

Djomehri, M. J.↗
NASA NTRS1987

Guidance and control strategies for aerospace vehicles

A simplified method of matched asymptotic expansions was developed where the common part in composite solution is generated as a polynomial in stretched variable instead of actually evaluating the same from the outer solution. This methodology was applied to the solution of the exact equations for three dimensional atmospheric entry problems. Compared to previous works, the present simplified methodology yields explicit analytical expressions for various components of the composite solution without resorting to any type of transcendental equations to be solved only by numerical methods. The optimal control problem arising in the noncoplanar orbital transfer employing aeroassist was also addressed.

Hibey, Joseph L.↗
NASA NTRS1988

Three-dimensional atmospheric entry problem using method of matched asymptotic expansions

The analysis of a three-dimensional atmospheric entry problem using the method of matched asymptotic expansions is considered. A composite solution is formed in terms of an outer solution, an inner solution and a common solution. The outer solution is obtained from the gravitationally dominant region, whereas the aerodynamically dominant region contributes to the inner solution. The common solution accounts for the overlap between the outer and inner regions. In comparison to previous work, this simplified methodology yields explicit analytical expressions for various components of the composite solution without resorting to any type of transcendental equations to be solved only by numerical methods.

Naidu, D. S.↗
NASA NTRS2016

Dynamic Beam Solutions for Real-Time Simulation and Control Development of Flexible Rockets

In this study, flexible rockets are structurally represented by linear beams. Both direct and indirect solutions of beam dynamic equations are sought to facilitate real-time simulation and control development for flexible rockets. The direct solution is completed by numerically integrate the beam structural dynamic equation using an explicit Newmark-based scheme, which allows for stable and fast transient solutions to the dynamics of flexile rockets. Furthermore, in the real-time operation, the bending strain of the beam is measured by fiber optical sensors (FOS) at intermittent locations along the span, while both angular velocity and translational acceleration are measured at a single point by the inertial measurement unit (IMU). Another study in this paper is to find the analytical and numerical solutions of the beam dynamics based on the limited measurement data to facilitate the real-time control development. Numerical studies demonstrate the accuracy of these real-time solutions to the beam dynamics. Such analytical and numerical solutions, when integrated with data processing and control algorithms and mechanisms, have the potential to increase launch availability by processing flight data into the flexible launch vehicle's control system.

flexible body dyanmics fiber optic strain sensor p↗
NASA NTRS1985

Optimal aeroassisted transfer between coplanar elliptical orbits

An attempt is made to solve the problem of orbital transfer between coplanar elliptical orbits. Pure propulsive transfer is analyzed with the restriction of two impulses for the transfer. The optimal switching conditions are reviewed, and it is shown that the solution is obtained by solving a set of three nonlinear equations for three unknowns. A semianalytical solution is obtained to the problem of planar rotation of an orbit for the pure propulsive maneuver, and it is shown that, for high eccentricity and rotation angle, aeroassisted transfer is a fuel-saving maneuver. It is demonstrated that complete circularization of the intermediate orbit is not necessary in the optimal aeroassisted transfer. An analytical proof is presented, giving an explicit condition for noncircularization. A complete numerical solution is presented for a case of optimal aeroassisted transfer from a low-energy orbit to a high-energy orbit.

Vinh, N. X.↗
NASA NTRS1994

Application of a distributed network in computational fluid dynamic simulations

A general-purpose 3-D, incompressible Navier-Stokes algorithm is implemented on a network of concurrently operating workstations using parallel virtual machine (PVM) and compared with its performance on a CRAY Y-MP and on an Intel iPSC/860. The problem is relatively computationally intensive, and has a communication structure based primarily on nearest-neighbor communication, making it ideally suited to message passing. Such problems are frequently encountered in computational fluid dynamics (CDF), and their solution is increasingly in demand. The communication structure is explicitly coded in the implementation to fully exploit the regularity in message passing in order to produce a near-optimal solution. Results are presented for various grid sizes using up to eight processors.

Deshpande, Manish↗
NASA NTRS1990

Solution techniques for the vorticity-streamfunction formulation of two-dimensional unsteady incompressible flows

A review of the solution techniques of Tezduyar et al. (1988 and 1989) for the vorticity-streamfunction formulation of two-dimensional incompressible flows is presented. While both the viscous and inviscid cases are considered, the derivation of the proper finite-element formulations for multiply connected domains is emphasized. In all formulations associated with the vorticity transport equation, the streamline upwind/Petrov-Galerkin method is used. The adaptive implicit-explicit and grouped element-by-element solution strategies are employed to maximize the computational efficiency. The solutions obtained in all test cases compare well with solutions from previously published investigations. The convergence and benchmark studies performed in this paper show that the solution techniques presented are accurate, reliable, and efficient.

Tezduyar, T. E.↗
NASA NTRS1995

An outflow boundary condition for aeroacoustic computations

A formulation of boundary condition for flows with small disturbances is presented. The authors test their methodology in an axisymmetric jet flow calculation, using both the Navier-Stokes and Euler equations. Solutions in the far field are assumed to be oscillatory. If the oscillatory disturbances are small, the growth of the solution variables can be predicted by linear theory. Eigenfunctions of the linear theory are used explicitly in the formulation of the boundary conditions. This guarantees correct solutions at the boundary in the limit where the predictions of linear theory are valid.

Hayder, M. Ehtesham↗
NASA NTRS1991

Derivative interface conditions for multiblock grids

Methods are developed for computing numerical solutions along block boundaries, even when there is a discontinuity in the grid lines or slopes. The technique is based on matching derivatives and does not require overlapping and interpolation of solution values at block boundaries. The comparison of block boundary values is implicit and has proven to be stable for both implicit and explicit numerical algorithms. Examples are included for the numerical solution of the Euler equations for compressible flow on grids with both grid line discontinuities and discontinuous slopes at block boundaries.

Mastin, C. W.↗