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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 NTRS1984

Spectral methods for solution of the boundary-layer equations

The basic principles of spectral methods are reviewed, and their application to the two-dimensional incompressible boundary-layer equations is discussed. It is shown that spectral collocation methods provide an accuracy sufficient for engineering applications on extremely coarse grids, and the machine time requirements are small. Although for a given stream-wise discretization, the fully spectral scheme requires more computational effort than the marching scheme, the higher accuracy of the former yields a large net gain in efficiency when considered at equivalent error levels.

Streett, C. L.↗
NASA NTRS1974

Centaur feedline dynamics study using power spectral methods

Tests were conducted to determine the dynamic characteristics of the Centaur/RL-10 oxygen and hydrogen feedlines. The fundamental-mode resonant frequencies were determined by applying power spectral methods to noise-generated data from hot firings of the RL-10 engine. The effect of net positive suction pressure of the main feed pumps on resonant frequency characteristics was determined to be a straight-line relation. Power spectral methods were also used to determine the dynamic characteristics of the boost pumps.

Lorenzo, C. F.↗
NASA NTRS1980

Spectral methods for problems in complex geometries

The properties of spectral methods are surveyed and their extension to solve problems in complex geometries is developed. A new iteration procedure is introduced to solve efficiently the full matrix equations resulting from spectral approximations to nonconstant coefficient boundary-value problems in complex geometries. It is shown that the work required to solve these spectral equations exceeds that of solving the lowest-order finite-difference approximation to the same problem by only O(N log N).

Orszag, S. A.↗
NASA NTRS1993

A multidomain spectral method for viscous compressible flows

We present a new multidomain spectral method for the solution of the compressible Navier-Stokes equations. In the subdomain interiors, Chebyshev spectral collocation is used. At interfaces, the advective terms are upwinded and the viscous terms are treated by a penalty method. The method is applied to the solution of a viscous hypersonic flow over a blunt body.

Kopriva, David A.↗
NASA NTRS2013

An Extension of the Time-Spectral Method to Overset Solvers

Relative motion in the Cartesian or overset framework causes certain spatial nodes to move in and out of the physical domain as they are dynamically blanked by moving solid bodies. This poses a problem for the conventional Time-Spectral approach, which expands the solution at every spatial node into a Fourier series spanning the period of motion. The proposed extension to the Time-Spectral method treats unblanked nodes in the conventional manner but expands the solution at dynamically blanked nodes in a basis of barycentric rational polynomials spanning partitions of contiguously defined temporal intervals. Rational polynomials avoid Runge's phenomenon on the equidistant time samples of these sub-periodic intervals. Fourier- and rational polynomial-based differentiation operators are used in tandem to provide a consistent hybrid Time-Spectral overset scheme capable of handling relative motion. The hybrid scheme is tested with a linear model problem and implemented within NASA's OVERFLOW Reynolds-averaged Navier- Stokes (RANS) solver. The hybrid Time-Spectral solver is then applied to inviscid and turbulent RANS cases of plunging and pitching airfoils and compared to time-accurate and experimental data. A limiter was applied in the turbulent case to avoid undershoots in the undamped turbulent eddy viscosity while maintaining accuracy. The hybrid scheme matches the performance of the conventional Time-Spectral method and converges to the time-accurate results with increased temporal resolution.

Leffell, Joshua Isaac↗
NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗
NASA NTRS1985

Spectral methods for modeling supersonic chemically reacting flow fields

A numerical algorithm was developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two-stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the technique were assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the procedure presented yields equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency.

Drummond, J. P.↗
NASA NTRS1985

Spectral methods for modeling supersonic chemically reacting flow fields

A partial implicit numerical algorithm has been developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two-stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the new technique have been assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the new procedure yielded equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency.

Drummond, J. P.↗
NASA NTRS1989

Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries

The numerical analysis of spectral methods when non-constant coefficients appear in the equation, either due to the original statement of the equations or to take into account the deformed geometry, is presented. Particular attention is devoted to the optimality of the discretization even for low values of the discretization parameter. The effect of some overintegration is also addressed, in order to possibly improve the accuracy of the discretization.

Maday, Yvon↗
NASA NTRS1987

Removal of spurious modes encountered in solving stability problems by spectral methods

A technique based on the Galerkin approximation is developed to remove spurious roots arising when Chebyshev spectral methods are used to solve eigenvalue problems in hydrodynamic stability. The derivation of Galerkin-Chebyshev approximations is explained, and numerical results for the Orr-Sommerfeld equations of plane Poiseuille flow and a Blasius profile are presented in tables and compared with those obtained by the method of Zebib (1984). It is pointed out that the present method does not increase the size of the algebraic system to be solved.

Zebib, Abdelfattah↗
NASA NTRS1983

A spectral method for the solution of transonic potential flow about an arbitrary two-dimensional airfoil

The application of a spectral method to the computation of transonic potential flow about an arbitrary lifting airfoil is discussed. An iterative solution algorithm is described, which was found to be robust, fairly efficient, and fit well in a multigrid coontext. Details of the application to the transonic potential flow problem are given. Results from the spectral technique for both subcritical flow and supercritical flow with shocks are given and compared with results from state-of-the-art finite-difference airfoil codes.

Streett, C. L.↗
NASA NTRS1974

Determination of rare-earth elements in Luna 16 regolith sample by chemical spectral method

An analysis was made of regolith from layer A of the Luna 16 sample for rare earth elements, by a chemical spectral method. Chemical and ion exchange concentrations were used to determine the content of 12 elements and Y at the level 0.001 to 0.0001 percent with 10 to 15 percent reproducibility of the emission determination. Results within the limits of reproducibility agree with data obtained by mass spectra, activation, and X-ray fluorescent methods.

Stroganova, N. S.↗
NASA NTRS1984

Spectral methods for exterior elliptic problems

Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases.

Canuto, C.↗
NASA NTRS1990

Spectral methods for time dependent problems

Spectral approximations are reviewed for time dependent problems. Some basic ingredients from the spectral Fourier and Chebyshev approximations theory are discussed. A brief survey was made of hyperbolic and parabolic time dependent problems which are dealt with by both the energy method and the related Fourier analysis. The ideas presented above are combined in the study of accuracy stability and convergence of the spectral Fourier approximation to time dependent problems.

Tadmor, Eitan↗
NASA NTRS1986

Stability analysis of spectral methods for hyperbolic initial-boundary value systems

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations.

Gottlieb, D.↗