Search NASA⌕ Search

SEARCH · Search NASA

Results for “equations”

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 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Nonparametric solution of the Euler equations for steady flow

A theory is presented for formulating well-posed boundary value problems for the Euler equations for steady rotational flow. It is shown that the Euler equations of motion are equivalent to a variational principle, which is used to define a finite difference scheme for numerically solving the Euler equations. The principle is extended to MHD problems in terms of a potential energy of a perfectly conducting plasma having a minimum number of stable configurations. The flow around a cylinder is considered, noting that time-independent solutions of the Euler equations can be used to provide limits to solutions of the Navier-Stokes equations. A sample is worked out in terms of the motion of vortices inside a circle.

Garabedian, P. R.↗
NASA NTRS1983

Numerical computation of transonic flow governed by the full-potential equation

Basic concepts associated with the numerical solution of elliptic partial differential equations are introduced, and procedures used to solve the full potential equation for transonic flow fields are discussed. Governing equations, classical relaxation schemes and concepts regarding transonic, full potential equation algorithms are covered. The equation transformation and grid generation procedures; full potential spatial differencing schemes; full potential iteration schemes, emphasizing convergence acceleration; and three dimensional applications are presented.

Holst, T. L.↗
NASA NTRS1983

A study of non-unique solutions of the two-dimensional boundary layer equations at laminar separation and reattachment points

Nonunique laminar boundary layer equation solutions in direct problem calculations are identified for the case of accelerating flow. As a separation or reattachment point is approached, the multiple solutions approach each other and become identical. The computer code used to generate these results was developed for the solution of compressible, laminar or turbulent boundary layer, and free wake problems, in either direct or inverse mode. Similarity solutions in either a primitive variable or a stream function form are possible, and the resulting equations are solved by means of a modified Keller's Box scheme in which the energy equation and turbulence modeling equations are solved simultaneously with the continuity and momentum equations. Examples illustrating the nature of the solutions at the separation and reattachment points are presented.

Drela, M.↗
NASA NTRS1983

Design and implementation of a multigrid code for the Euler equations

The steady-state equations of inviscid fluid flow, the Euler equations, are a nonlinear nonelliptic system of equations admitting solutions with discontinuities (for example, shocks). The efficient numerical solution of these equations poses a strenuous challenge to multigrid methods. A multigrid code has been developed for the numerical solution of the Euler equations. In this paper some of the factors that had to be taken into account in the design and development of the code are reviewed. These factors include the importance of choosing an appropriate difference scheme, the usefulness of local mode analysis as a design tool, and the crucial question of how to treat the nonlinearity. Sample calculations of transonic flow about airfoils will be presented. No claim is made that the particular algorithm presented is optimal.

Jespersen, D. C.↗
NASA NTRS1984

The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Bayliss, A.↗
NASA NTRS1985

Surface-slip equations for multicomponent nonequilibrium air flow

Equations are presented for the surface-slip (or jump) values of species concentration, pressure, velocity, and temperature in the low-Reynolds number, high-altitude flight regime of a space vehicle. The equations are obtained from closed form solutions of the mass, momentum, and energy flux equations using the Chapman-Enskog velocity distribution function. This function represents a solution of the Boltzmann equation in the Navier-Stokes approximation. The analysis, obtained for nonequilibrium multicomponent air flow, includes the finite-rate surface catalytic recombination and changes in the internal energy during reflection from the surface. Expressions for the various slip quantities were obtained in a form which can be employed in flowfield computations. A consistent set of equations is provided for multicomponent, binary, and single species mixtures. Expression is also provided for the finite-rate, species-concentration boundary condition for a multicomponent mixture in absence of slip.

Gupta, R. N.↗
NASA NTRS1984

Ionization Equilibrium and Equation of State in the Solar Interior

Many-body formulations of the equations of state are restated as a set of Saha-like equations. It is shown that the resulting equations are unique and convergent. These equations are similar to the usual Saha equations to the order of the Debye-Huckel theory. Higher order corrections, however, require a more general formulation. It is demonstrated that the positive free energy resulting from the interaction of unscreened particles in high orbits depletes the occupation of these states, without the introduction of shifted energy levels.

Rogers, F. J.↗
NASA NTRS1984

The Sensitivity of Solar Eigenfrequencies to the Treatment of the Equation of State

The principal uncertainties in the equation of state involve the treatment of pressure ionization, the Debye-Huckel coulomb corrections, and the treatment of many-particle interaction effects. It is found that, for the lowest degree modes (l between 0 and 3), the terms and procedures used in the equation of state which deal with these uncertainties introduce changes in the frequencies which are less than 4 micro Hz. Recently, Shibahashi, Noels and Gabriel (1983) published solar eigenfrequencies using a theory with an equation of state improved with respect to the theory used earlier by Shibahashi and Osaki (1981). Their comparison between the two sets of results suggested that uncertainties in the frequencies as large as 10 micro Hz could be caused by the equation of state. It is felt that since the entire effect of the uncertain terms is only 4 micro Hz and since the uncertainties are only a fraction of each term, the 10 micro Hz changes found by Shibahashi et al must be a consequence of differences between the earlier and later calculations in areas other than the equation of state.

Ulrich, R. K.↗
NASA NTRS1985

Finite element analysis of notch behavior using a state variable constitutive equation

The state variable constitutive equation of Bodner and Partom was used to calculate the load-strain response of Inconel 718 at 649 C in the root of a notch. The constitutive equation was used with the Bodner-Partom evolution equation and with a second evolution equation that was derived from a potential function of the stress and state variable. Data used in determining constants for the constitutive models was from one-dimensional smooth bar tests. The response was calculated for a plane stress condition at the root of the notch with a finite element code using constant strain triangular elements. Results from both evolution equations compared favorably with the observed experimental response. The accuracy and efficiency of the finite element calculations also compared favorably to existing methods.

Dame, L. T.↗
NASA NTRS1983

Transformation matrices between non-linear and linear differential equations

In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

Sartain, R. L.↗
NASA NTRS1985

A model for closing the inviscid form of the average-passage equation system

A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time average flow field through the blade passages of multistage turbomachinery. These equations referred to as the average passage equation system govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high speed counter rotating propeller and a high speed fan stage. Results from these simulations are presented.

Adamczyk, J. J.↗
NASA NTRS1985

A unique set of micromechanics equations for high temperature metal matrix composites

A unique set of micromechanic equations is presented for high temperature metal matrix composites. The set includes expressions to predict mechanical properties, thermal properties and constituent microstresses for the unidirectional fiber reinforced ply. The equations are derived based on a mechanics of materials formulation assuming a square array unit cell model of a single fiber, surrounding matrix and an interphase to account for the chemical reaction which commonly occurs between fiber and matrix. A three-dimensional finite element analysis was used to perform a preliminary validation of the equations. Excellent agreement between properties predicted using the micromechanics equations and properties simulated by the finite element analyses are demonstrated. Implementation of the micromechanics equations as part of an integrated computational capability for nonlinear structural analysis of high temperature multilayered fiber composites is illustrated.

Hopkins, D. A.↗
NASA NTRS1986

Regularity estimates up to the boundary for elliptic systems of difference equations

Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

Strikwerda, J. C.↗
NASA NTRS1986

A minimum entropy principle in the gas dynamics equations

Let u(x bar,t) be a weak solution of the Euler equations, governing the inviscid polytropic gas dynamics; in addition, u(x bar, t) is assumed to respect the usual entropy conditions connected with the conservative Euler equations. We show that such entropy solutions of the gas dynamics equations satisfy a minimum entropy principle, namely, that the spatial minimum of their specific entropy, (Ess inf s(u(x,t)))/x, is an increasing function of time. This principle equally applies to discrete approximations of the Euler equations such as the Godunov-type and Lax-Friedrichs schemes. Our derivation of this minimum principle makes use of the fact that there is a family of generalized entrophy functions connected with the conservative Euler equations.

Tadmor, E.↗
NASA NTRS1985

Approximate-factorization schemes for solving the transonic full-potential equation

The present paper provides a general discussion of approximate-factorization techniques applied to the transonic full-potential equation. Giving particular attention to the AF2 approximate-factorization scheme. This scheme was first introduced by Ballhaus and Steger (1975) for solving the low-frequency (unsteady), transonic small-disturbance equation. The full-potential equation algorithm is examined, taking into account the governing equations, grid generation, the artificial density scheme (spatial differencing), the alternating direction implicit scheme, the AF2 iteration scheme, temporal damping, and boundary conditions. Computed results are also presented. It is shown that fast, fully-implicit algorithms of the approximate-factorization variety are both efficient and reliable for solving the conservative full-potential equation.

Holst, T. L.↗
NASA NTRS1985

Accelerated solution of the steady Euler equations

The present paper is concerned with two methods for the accelerated solution of the steady Euler equations. One method makes use of a second-order embedding to facilitate the derivation of the relaxation solution of the steady equations of motion, while the other method employs a multile-gridding concept to accelerate the convergence of a simple, explicit, time-marching scheme applied to the unsteady equations. It is pointed out that the surrogate equation technique provides a means for formulating problems involving the full steady Euler equations in such a way as to allow the use of relaxation solution procedures. It is, therefore, possible to solve either irrotational or rotational flow problems spanning the entire spectrum of subsonic, transonic, and supersonic conditions. The solutions can be obtained without an employement of either derived dependent variables, semidirect methods, or an unsteady formulation.

Johnson, G. M.↗
NASA NTRS1986

Polynomial elimination theory and non-linear stability analysis for the Euler equations

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Kennon, S. R.↗
NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗