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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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At least 127 records · Page 7
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NASA NTRS2009

A Systematic Methodology for Constructing High-Order Energy Stable WENO Schemes

A third-order Energy Stable Weighted Essentially Non{Oscillatory (ESWENO) finite difference scheme developed by Yamaleev and Carpenter [1] was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables "energy stable" modifications for existing WENO schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO schemes; ESWENO schemes up to eighth order are presented in the appendix. New weight functions are also developed that provide (1) formal consistency, (2) much faster convergence for smooth solutions with an arbitrary number of vanishing derivatives, and (3) improved resolution near strong discontinuities.

Yamaleev, Nail K.↗
NASA NTRS1983

Application of the implicit MacCormack scheme to the PNS equations

The two-dimensional parabolized Navier-Stokes equations are solved using MacCormack's (1981) implicit finite-difference scheme. It is shown that this method for solving the parabolized Navier-Stokes equations does not require the inversion of block tridiagonal systems of algebraic equations and allows the original explicit scheme to be employed in those regions where implicit treatment is not needed. The finite-difference algorithm is discussed and the computational results for two laminar test cases are presented. Results obtained using this method for the case of a flat plate boundary layer are compared with those obtained using the conventional Beam-Warming scheme, as well as those obtained from a boundary layer code. The computed results for a more severe test of the method, the hypersonic flow past a 15 deg compression corner, are found to compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.

Lawrence, S. L.↗
NASA NTRS1982

High-resolution schemes for hyperbolic conservation laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurae scheme to an appropriately modified flux function. The so derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme.

Harten, A.↗
NASA NTRS1983

High resolution schemes for hyperbolic conservation laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

Harten, A.↗
NASA NTRS1985

Computing with high-resolution upwind schemes for hyperbolic equations

Computational aspects of modern high-resolution upwind finite-difference schemes for hyperbolic systems of conservation laws are examined. An operational unification is demonstrated for constructing a wide class of flux-difference-split and flux-split schemes based on the design principles underlying total variation diminishing (TVD) schemes. Consideration is also given to TVD scheme design by preprocessing, the extension of preprocessing and postprocessing approaches to general control volumes, the removal of expansion shocks and 'glitches', relaxation methods for implicit TVD schemes, and a new family of high-accuracy TVD schemes.

Chakravarthy, S. R.↗
NASA NTRS2008

A Systematic Methodology for Constructing High-Order Energy-Stable WENO Schemes

A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme developed by Yamaleev and Carpenter (AIAA 2008-2876, 2008) was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables \energy stable" modifications for existing WENO schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO schemes; ESWENO schemes up to eighth order are presented in the appendix. New weight functions are also developed that provide (1) formal consistency, (2) much faster convergence for smooth solutions with an arbitrary number of vanishing derivatives, and (3) improved resolution near strong discontinuities.

Yamaleev, Nail K.↗
NASA NTRS2009

High-Order Energy Stable WENO Schemes

A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme developed by Yamaleev and Carpenter was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables 'energy stable' modifications for existing WENO schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO schemes; ESWENO schemes up to eighth order are presented in the appendix. New weight functions are also developed that provide (1) formal consistency, (2) much faster convergence for smooth solutions with an arbitrary number of vanishing derivatives, and (3) improved resolution near strong discontinuities.

Yamaleev, Nail K.↗
NASA NTRS1975

Forward marching procedure for separated boundary-layer flows

A forward-marching procedure for separated boundary-layer flows which permits the rapid and accurate solution of flows of limited extent is presented. The streamwise convection of vorticity in the reversed flow region is neglected, and this approximation is incorporated into a previously developed (Carter, 1974) inverse boundary-layer procedure. The equations are solved by the Crank-Nicolson finite-difference scheme in which column iteration is carried out at each streamwise station. Instabilities encountered in the column iterations are removed by introducing timelike terms in the finite-difference equations. This provides both unconditional diagonal dominance and a column iterative scheme, found to be stable using the von Neumann stability analysis.

Carter, J. E.↗
NASA NTRS1995

A compact solution to computational acoustics

This paper demonstrates that the linearized, dimensional Euler equations for acoustic computation can be accurately solved as a set of decoupled first-order wave equations, and that if ordered properly, this system of simple waves has unambiguous, easily implemented boundary conditions, allowing waves of same group speeds to pass through numerical boundaries or comply with wall conditions. Thus, the task of designing a complex multi-dimensional scheme with approximate far-field boundary conditions reduces to the design of higher order schemes for the one-dimensional simple wave equation. A compact finite-difference scheme and a characteristically exact but numerically n(th) order accurate boundary condition are introduced for solving the first order wave equation. Spanning a three-point two-level stencil, this low-dispersion implicit scheme has a third order spatial accuracy when used on nonuniform meshes, fourth order accurate on uniform meshes, and a temporal accuracy of second order due to the choice of trapezoidal integration for algorithmic simplicity. The robustness and accuracy of the scheme are demonstrated through a series of numerical experiments and comparisons with published results. When tested on the one-dimensional wave equation on a uniform grid, this scheme allows a Gaussian wave packet to pass through any finite domain with low numerical dispersion characteristic of a spatially fourth-order scheme and reflections at numerical boundaries maintained below truncation error. On highly stretched and irregular grids, only mild dispersions are found in the solution while solutions by other methods fail or are severely distorted. Yet, this scheme is no more sophisticated to solve or implement than the Crank-Nicolson scheme. This scheme has been tested on four categories of the ICASE/LaRC benchmark problems, which include propagation of acoustic and convective waves in Cartesian and cylindrical domains, reflection of acoustic wave at stationary/moving boundaries, and sound generation by gust-blade interaction.

Fung, K.-Y.↗
NASA NTRS1978

An old integration scheme for compressible flows revisited, refurbished and put to work

A scheme for integrating the Euler equations of compressible flow in any hyperbolic case is presented. The scheme relies on the concept of characteristics but is strictly a finite difference scheme. Improvements in accuracy and physical consistence due to the scheme are discussed and results of its application to complex flows are shown.

Moretti, G.↗
NASA NTRS1998

Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.

Wilson, Robert V.↗
NASA NTRS1995

Second-order numerical solution of time-dependent, first-order hyperbolic equations

A finite difference scheme is developed to find an approximate solution of two similar hyperbolic equations, namely a first-order plane wave and spherical wave problem. Finite difference approximations are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.

Shah, Patricia L.↗
NASA NTRS1989

Low gravity transfer line chilldown

A code has been developed that solves for the transfer line chilldown time and flow and heat transfer characteristics in one-g environment. The code solves the transient, one dimensional, space averaged mass, momentum, and energy conservation equations for liquid-vapor two-phase flow in tubes. The physical configuration solved is that appropriate for bottom coolant injection in a vertically supported heated tube. Four distinct regions are considered consecutively: fully liquid, inverted annular, dispersed, and fully vapor flow. The conservation equations for both the liquid and the vapor are solved in each region separately. Also, in each region the mass and energy transport between each phase as well as the energy and momentum transport between the tube wall and the fluid are accounted for. A finite wall thickness is also considered. The model described above was solved numerically through a mixed finite difference scheme with forward time marching. The inverted annular regime was resolved using a semi-implicit finite differencing while the dispersed regime was solved explicitly. Also, a staggered mesh was used in which the velocity was resolved at mesh boundaries while all other field variables were resolved at the mesh centroids. Different mesh sizes were used depending on the region of solution. A coarse mesh was used in the dispersed flow region while a much finer mesh was used in both the inverted annular flow region and the tube walls.

B. N. Antar↗
NASA NTRS1985

Assessment of discretization schemes to reduce numerical diffusion in the calculation of complex flows

Comprehensive criteria for evaluating finite-difference schemes to reduce numerical diffusion in the solution of the Reynolds averaged Navier-Stokes equations have been established in this study. Several candidate schemes have been examined, based on these criteria. Taylor series analyses, model problem studies and examination of the difference equations have been used to assess the schemes. Two schemes, which showed the most promise, and satisfied a majority of the criteria, are recommended for incorporation into two dimensional and three dimensional computer codes for further evaluation.

Syed, S. A.↗
NASA NTRS1985

Application of TVD schemes for the Euler equations of gas dynamics

Highly accurate and yet stable shock-capturing finite difference schemes have been designed for the computation of the Euler equations of gas dynamics. Four different principles for the construction of high resolution total variation diminishing (TVD) schemes are available, including hybrid schemes, a second-order extension of Godunov's scheme by van Leer (1979), the modified flux approach of Harten (1983, 1984), and the numerical fluctuation approach of Roe (1985). The present paper has the objective to review the class of second-order TVD schemes via the modified flux approach. Attention is given to first-order TVD schemes, a second-order accurate explicit TVD scheme, the global order of accuracy of the second-order TVD scheme, extensions to systems and two-dimensional conservation laws, numerical experiments with a second-order explicit TVD scheme, implicit TVD schemes, and second-order implicit TVD schemes.

Yee, H. C.↗
NASA NTRS1991

High-order ENO schemes applied to two- and three-dimensional compressible flow

High order essentially non-oscillatory (ENO) finite difference schemes are applied to the 2-D and 3-D compressible Euler and Navier-Stokes equations. Practical issues, such as vectorization, efficiency of coding, cost comparison with other numerical methods, and accuracy degeneracy effects, are discussed. Numerical examples are provided which are representative of computational problems of current interest in transition and turbulence physics. These require both nonoscillatory shock capturing and high resolution for detailed structures in the smooth regions and demonstrate the advantage of ENO schemes.

Shu, Chi-Wang↗
NASA NTRS1984

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it was previously proposed to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which moves all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping is potentially effective in accelerating convergence to steady state.

Jespersen, D. C.↗
NASA NTRS1985

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it was previously proposed to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which moves all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping is potentially effective in accelerating convergence to steady state.

Jespersen, D. C.↗