Search NASA⌕ Search

SEARCH · Search NASA

Results for “HYPERBOLIC EQUATION”

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 NTRS1973

Resonant oscillations of intermediate frequency in a stratified atmosphere.

A class of solutions to a model of forced oscillations in a rotating stratified atmospheric layer is derived and analyzed. The basic model is found to reduce to a boundary value problem with a second-order linear partial differential equation of the hyperbolic type for this range of forcing frequencies. The forced solutions are shown to exhibit resonances with the normal modes of oscillation of the layer. The characteristics of the resonant modes are analyzed in terms of mean tropospheric values of temperature, temperature lapse, wind speed, horizontal and vertical wind shears, latitude, and the frequency and horizontal wavelength of the forcing mechanism. These solutions are compared with solutions to the model for a different (subinertial) range of forcing frequencies. This comparison leads to an elliptic boundary value problem. The solutions in that case do not exhibit the same type of resonance and generally decay away from the region of forcing.

Thompson, O. E.↗
NASA NTRS1973

The starting transient of solid propellant rocket motors with high internal gas velocities

A comprehensive analytical model which considers time and space development of the flow field in solid propellant rocket motors with high volumetric loading density is described. The gas dynamics in the motor chamber is governed by a set of hyperbolic partial differential equations, that are coupled with the ignition and flame spreading events, and with the axial variation of mass addition. The flame spreading rate is calculated by successive heating-to-ignition along the propellant surface. Experimental diagnostic studies have been performed with a rectangular window motor (50 cm grain length, 5 cm burning perimeter and 1 cm hydraulic port diameter), using a controllable head-end gaseous igniter. Tests were conducted with AP composite propellant at port-to-throat area ratios of 2.0, 1.5, 1.2, and 1.06, and head-end pressures from 35 to 70 atm. Calculated pressure transients and flame spreading rates are in very good agreement with those measured in the experimental system.

Peretz, A.↗
NASA NTRS1976

Characteristics procedure for supersonic flows including consideration of viscous contributions to flow rotationality

A characteristics procedure has been developed which includes viscous and conductive transport terms normal to streamlines. By introducing the transport terms as corrections, the equations retain their hyperbolic character. The solutions have been developed using an inverse grid scheme in a streamline-normal network. Inclusion of the transport terms enables tracing of shock waves through the supersonic portion of the boundary layer so that shock reflections are well-located. Excellent agreement is obtained with the results of other characteristics procedures and with the experimental pressure distributions for the waisted body of Winter, Smith and Rotta.

Anderson, B. H.↗
NASA NTRS1976

Diffraction of a shock wave by a compression corner; regular and single Mach reflection

The two dimensional, time dependent Euler equations which govern the flow field resulting from the injection of a planar shock with a compression corner are solved with initial conditions that result in either regular reflection or single Mach reflection of the incident planar shock. The Euler equations which are hyperbolic are transformed to include the self similarity of the problem. A normalization procedure is employed to align the reflected shock and the Mach stem as computational boundaries to implement the shock fitting procedure. A special floating fitting scheme is developed in conjunction with the method of characteristics to fit the slip surface. The reflected shock, the Mach stem, and the slip surface are all treated as harp discontinuities, thus, resulting in a more accurate description of the inviscid flow field. The resulting numerical solutions are compared with available experimental data and existing first-order, shock-capturing numerical solutions.

Vijayashankar, V. S.↗
NASA NTRS1976

An efficient numerical method for solving the time-dependent compressible Navier-Stokes equations at high Reynolds number

A fine-mesh method incorporating two new operators, which drastically reduces the computation time, has been developed for solving the time-dependent Navier-Stokes equations at flight Reynolds numbers. The approach time-splits the equations into a hyperbolic part and a parabolic part, solves the hyperbolic part by a new explicit numerical method based on characteristics theory, and solves the parabolic part by a new efficient implicit parabolic method. The method has reduced the computation time by one and two orders of magnitude from that required previously to solve for the interaction of a shock wave with a boundary layer on a flat plate.

Maccormack, R. W.↗
NASA NTRS1978

A high order accurate finite element algorithm for high Reynolds number flow prediction

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Baker, A. J.↗
NASA NTRS1979

Recent developments in finite element analysis for transonic airfoils

The prediction of aerodynamic forces in the transonic regime generally requires a flow field calculation to solve the governing non-linear mixed elliptic-hyperbolic partial differential equations. Finite difference techniques were developed to the point that design and analysis application are routine, and continual improvements are being made by various research groups. The principal limitation in extending finite difference methods to complex three-dimensional geometries is the construction of a suitable mesh system. Finite element techniques are attractive since their application to other problems have permitted irregular mesh elements to be employed. The purpose of this paper is to review the recent developments in the application of finite element methods to transonic flow problems and to report some recent results.

Hafez, M. M.↗
NASA NTRS1984

Mathematical Instability Criteria for Elastic Structures

Theoretical paper discusses physical significance of vanishing of hyperbolic coefficients in equations of elastodynamics. Paper presents generalized approach to structural elastodynamics as part of continuing effort to develop mathematical stability criteria for structures and simulate postinstabilty behavior of elastics in general.

Zak, M.↗
NASA NTRS1983

Second-order explicit finite-difference methods for transient-flow analysis

Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.

Chaudhry, M. H.↗
NASA NTRS1982

A minimal residual method for transonic potential flows

For transonic flow calculations, a combination of the successive line over-relaxation (SLOR) and the preconditioned conjugate gradient (CG) method has been suggested by Wong and Hafez (1981). This paper studies the method of minimal residual (MR) which avoids a combined iteration. This method is closely related to the CG method, may be regarded as a first-order gradient method, and is applicable to symmetric and nonsymmetric matrices. The problem is formulated as a nonlinear mixed elliptic-hyperbolic partial differential equation which includes an artificial viscosity and a switching function which is zero in subsonic regions and nonzero in supersonic regions. Alternatives to the SLOR method which provide faster convergence rates are introduced. The preconditioned MR algorithm is developed, and transonic potential flows around NACA 0012 airfoil are calculated for different Mach numbers and angles of attack. Preliminary results are presented, demonstrating that the MR algorithm requires no parameter estimation and rapidly converges for subsonic flows.

Wong, Y. S.↗
NASA NTRS1985

Shock capturing

Recent developments which have improved the understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations are discussed. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. A conservative difference scheme is defined. Conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, the reason for this excellent shock resolution can be determined. This result is used to provide a mechanism for improving the resolution of two dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. Total variation diminishing (TVD) finite difference schemes and flux limiters are introduced to deal with money shocks and contact discontinuities.

Davis, S. F.↗
NASA NTRS1983

Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.

Durgun, K.↗
NASA NTRS1986

Preconditioned methods for solving the incompressible and low speed compressible equations

Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. The compressible equations in conservation form with slow flow are also considered. Two arbitrary functions, alpha and beta, are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for beta is determined given a constant, alpha. It is further shown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which generalize previous results.

Turkel, E.↗
NASA NTRS1986

A spectral multidomain method for the solution of hyperbolic systems

A multidomain Chebyshev spectral collocation method for solving hyperbolic partial differential equations were developed. Though spectral methods are global methods, an attractive idea is to break a computational domain into several domains, and a way to handle the interfaces is described. The multidomain approach offers advantages over the use of a single Chebyshev grid. It allows complex geometries to be covered, and local refinement can be used to resolve important features. For steady state problems it reduces the stiffness associated with the use of explicit time integration as a relaxation scheme. Furthermore, the proposed method remains spectrally accurate. Results showing performance of the method on one dimensional linear models and one and two dimensional nonlinear gas dynamics problems are presented.

Kopriva, D.↗
NASA NTRS1987

Steady state computations for wave propagation problems

The behavior of difference approximations of hyperbolic partial differential equations as time t goes to infinity is studied. The rate of convergence to steady state is analyzed theoretically and experimentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady-state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties.

Engquist, Bjorn↗
NASA NTRS1988

Non-oscillatory spectral Fourier methods for shock wave calculations

A non-oscillatory spectral Fourier method is presented for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.

Cai, Wei↗
NASA NTRS1988

Far field boundary conditions for computation over long time

A new class of computational far field boundary conditions for hyperbolic partial differential equations is developed. These boundary conditions combine properties of absorbing boundary conditions for transient solutions and properties of far field boundary conditions for steady-state problems. The conditions can be used to limit the computational domain when both traveling waves and evanescent waves are present. Boundary conditions for scalar wave equations are derived and analyzed. Extensions to systems of equations are discussed and results from numerical experiments are presented.

Engquist, Bjorn↗