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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 NTRS2010

RANS Modeling of Benchmark Shockwave / Boundary Layer Interaction Experiments

This presentation summarizes the computations of a set of shock wave / turbulent boundary layer interaction (SWTBLI) test cases using the Wind-US code, as part of the 2010 American Institute of Aeronautics and Astronautics (AIAA) shock / boundary layer interaction workshop. The experiments involve supersonic flows in wind tunnels with a shock generator that directs an oblique shock wave toward the boundary layer along one of the walls of the wind tunnel. The Wind-US calculations utilized structured grid computations performed in Reynolds-averaged Navier-Stokes mode. Three turbulence models were investigated: the Spalart-Allmaras one-equation model, the Menter Shear Stress Transport wavenumber-angular frequency two-equation model, and an explicit algebraic stress wavenumber-angular frequency formulation. Effects of grid resolution and upwinding scheme were also considered. The results from the CFD calculations are compared to particle image velocimetry (PIV) data from the experiments. As expected, turbulence model effects dominated the accuracy of the solutions with upwinding scheme selection indicating minimal effects.!

Georgiadis, Nick↗
NASA NTRS2003

Analysis and Improvement of Upwind and Centered Schemes on Quadrilateral and Triangular Meshes

Second-order accurate upwind and centered schemes are presented in a framework that facilitates their analysis and comparison. The upwind scheme employed consists of a reconstruction step (MUSCL approach) followed by an upwind step (Roe's flux-difference splitting). The two centered schemes are of Lax-Friedrichs (L-F) type. They are the nonstaggered versions of the Nessyahu-Tadmor (N-T) scheme and the CE/SE method with epilson = 1/2. The upwind scheme is extended to the case of two spatial dimensions (2D) in a straightforward manner. The N-T and CE/SE schemes are extended in a manner similar to the 2D extensions of the CE/SE scheme by Wang and Chang for a triangular mesh and by Zhang, Yu, and Chang for a quadrilateral mesh. The slope estimates, however, are simplified. Fourier stability and accuracy analyses are carried out for these schemes for the standard 1D and the 2D quadrilateral mesh cases. In the nonstandard case of a triangular mesh, the triangles must be paired up when analyzing the upwind and N-T schemes. An observation resulting in an extended N-T scheme which is faster and uses only one-third of the storage for flow data compared with the CE/SE method is presented. Numerical results are shown. Other improvements to the schemes are discussed.

Huynh, H. T.↗
NASA NTRS1989

Computational studies of hard-body and 3-D effects in plume flows

Axisymmetric and three-dimensional, multi-nozzle plume flows around generic rocket geometries are investigated with a three-dimensional Navier-Stokes solver to study the interactive effects between hard body and the plume. Time-asymptotic, laminar, ideal-gas solutions obtained with a two-factor, flux-split scheme and a diagonal, upwind scheme are presented. Computed solutions to three-dimensional, multi-nozzle problems and single-nozzle, axisymmetric problems demonstrate flow field features including three-dimensionality and hard-body effects. Geometry and three-dimensional effects are shown to be significant in multi-nozzle flows.

Venkatapathy, Ethiraj↗
NASA NTRS1989

Numerical solution of a two-dimensional jet in a supersonic crossflow using an upwind relaxation scheme

A numerical scheme for predicting complex two-dimensional viscous flow fields is presented. The algorithm uses Roe's method for defining the inviscid fluxes and a line relaxation scheme to solve the resulting discrete approximation. The effects of turbulence are modelled using an algebraic eddy viscosity model. An adaptive grid scheme is employed to improve the resolution of complex flow field phenomena. The algorithm is applied to the solution of a sonic jet in a supersonic crossflow. Numerical results are compared with experimental data to validate the numerical approach.

Thompson, D. S.↗
NASA NTRS1990

Numerical study of unsteady shockwave reflections using an upwind TVD scheme

An unsteady TVD Navier-Stokes solver was developed and applied to the problem of shock reflection on a circular cylinder. The obtained numerical results were compared with the Schlieren photos from an experimental study. These results show that the present computer code has the ability of capturing moving shocks.

Hsu, Andrew T.↗
NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗
NASA NTRS1989

A nonoscillatory, characteristically convected, finite volume scheme for multidimensional convection problems

A new, nonoscillatory upwind scheme is developed for the multidimensional convection equation. The scheme consists of an upwind, nonoscillatory interpolation of data to the surfaces of an intermediate finite volume; a characteristic convection of surface data to a midpoint time level; and a conservative time integration based on the midpoint rule. This procedure results in a convection scheme capable of resolving discontinuities neither aligned with, nor convected along, grid lines.

Yokota, Jeffrey W.↗
NASA NTRS1990

A nonoscillatory, characteristically convected, finite volume scheme for multidimensional convection problems

A new, nonoscillatory upwind scheme is developed for the multidimensional convection equation. The scheme consists of an upwind, nonoscillatory interpolation of data to the surfaces of an intermediate finite volume; a characteristic convection of surface data to a midpoint time level; and a conservative time integration based on the midpoint rule. This procedure results in a convection scheme capable of resolving discontinuities neither aligned with, nor convected along, grid lines.

Yokota, Jeffrey W.↗
NASA NTRS1992

Central difference TVD and TVB schemes for time dependent and steady state problems

We use central differences to solve the time dependent Euler equations. The schemes are all advanced using a Runge-Kutta formula in time. Near shocks, a second difference is added as an artificial viscosity. This reduces the scheme to a first order upwind scheme at shocks. The switch that is used guarantees that the scheme is locally total variation diminishing (TVD). For steady state problems it is usually advantageous to relax this condition. Then small oscillations do not activate the switches and the convergence to a steady state is improved. To sharpen the shocks, different coefficients are needed for different equations and so a matrix valued dissipation is introduced and compared with the scalar viscosity. The connection between this artificial viscosity and flux limiters is shown. Any flux limiter can be used as the basis of a shock detector for an artificial viscosity. We compare the use of the van Leer, van Albada, mimmod, superbee, and the 'average' flux limiters for this central difference scheme. For time dependent problems, we need to use a small enough time step so that the CFL was less than one even though the scheme was linearly stable for larger time steps. Using a total variation bounded (TVB) Runge-Kutta scheme yields minor improvements in the accuracy.

Jorgenson, P.↗
NASA NTRS1992

Central difference TVD and TVB schemes for time dependent and steady state problems

We use central differences to solve the time dependent Euler equations. The schemes are all advanced using a Runge-Kutta formula in time. Near shocks, a second difference is added as an artificial viscosity. This reduces the scheme to a first order upwind scheme at shocks. The switch that is used guarantees that the scheme is locally total variation diminishing (TVD). For steady state problems it is usually advantageous to relax this condition. Then small oscillations do not activate the switches and the convergence to a steady state is improved. To sharpen the shocks, different coefficients are needed for different equations and so a matrix valued dissipation is introduced and compared with the scalar viscosity. The connection between this artificial viscosity and flux limiters is shown. Any flux limiter can be used as the basis of a shock detector for an artificial viscosity. We compare the use of the van Leer, van Albada, mimmod, superbee, and the 'average' flux limiters for this central difference scheme. For time dependent problems, we need to use a small enough time step so that the CFL was less than one even though the scheme was linearly stable for larger time steps. Using a total variation bounded (TVB) Runge-Kutta scheme yields minor improvements in the accuracy.

Jorgenson, P.↗
NASA NTRS1993

Fourier analysis of numerical algorithms for the Maxwell equations

The Fourier method is used to analyze the dispersive, dissipative, and isotropy errors of various spatial and time discretizations applied to the Maxwell equations on multi-dimensional grids. Both Cartesian grids and non-Cartesian grids based on hexagons and tetradecahedra are studied and compared. The numerical errors are quantitatively determined in terms of phase speed, wave number, propagation direction, gridspacings, and CFL number. The study shows that centered schemes are more efficient than upwind schemes. The non-Cartesian grids yield superior isotropy and higher accuracy than the Cartesian ones. For the centered schemes, the staggered grids produce less errors than the unstaggered ones. A new unstaggered scheme which has all the best properties is introduced. The study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization.

Liu, Yen↗
NASA NTRS1988

Universal limiter for transient interpolation modeling of the advective transport equations: The ULTIMATE conservative difference scheme

A fresh approach is taken to the embarrassingly difficult problem of adequately modeling simple pure advection. An explicit conservative control-volume formation makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ULTIMATE conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three critical test profiles: an isolated sine-squared wave, a discontinuous step, and a semi-ellipse. The goal, of course, is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservation schemes of any order of accuracy. Second-order schemes are unsatisfactory, showing steepening and clipping typical of currently popular so-called high resolution shock-capturing of TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. Little is to be gained above ULTIMATE fifth-order upwinding which gives results close to the ultimate for which one might hope.

Leonard, B. P.↗
NASA NTRS1985

Large-scale computations in fluid mechanics; Proceedings of the Fifteenth Summer Seminar on Applied Mathematics, University of California, La Jolla, CA, June 27-July 8, 1983. Parts 1 & 2

Papers are presented on such topics as the use of semi-Lagrangian advective schemes in meteorological modeling; computation with high-resolution upwind schemes for hyperbolic equations; dynamics of flame propagation in a turbulent field; a modified finite element method for solving the incompressible Navier-Stokes equations; computational fusion magnetohydrodynamics; and a nonoscillatory shock capturing scheme using flux-limited dissipation. Consideration is also given to the use of spectral techniques in numerical weather prediction; numerical methods for the incorporation of mountains in atmospheric models; techniques for the numerical simulation of large-scale eddies in geophysical fluid dynamics; high-resolution TVD schemes using flux limiters; upwind-difference methods for aerodynamic problems governed by the Euler equations; and an MHD model of the earth's magnetosphere.

Engquist, B. E.↗
NASA NTRS1990

Vortical flows over delta wings and numerical prediction of vortex breakdown

Navier-Stokes solutions of subsonic vortical flow over a 75 deg sweep delta wing with a sharp leading edge are presented. The sensitivity of the solution to the numerical scheme is examined using both a partially upwind scheme and a scheme with central differencing in all directions. At moderate angles of attack, no vortex breakdown is observed, whereas the higher angle-of-attack cases exhibit breakdown. The effect of numerical grid density is investigated, and solutions that are obtained with various grid densities are compared with experimental data. An embedded grid approach is implemented to enable higher resolution in selected isolated flow regions, such as the leeward-side surface, the leading-edge vortical flow, and the vortex breakdown region.

Ekaterinaris, J. A.↗
NASA NTRS1992

The Osher scheme for non-equilibrium reacting flows

An extension of the Osher upwind scheme to nonequilibrium reacting flows is presented. Owing to the presence of source terms, the Riemann problem is no longer self-similar and therefore its approximate solution becomes tedious. With simplicity in mind, a linearized approach which avoids an iterative solution is used to define the intermediate states and sonic points. The source terms are treated explicitly. Numerical computations are presented to demonstrate the feasibility, efficiency and accuracy of the proposed method. The test problems include a ZND (Zeldovich-Neumann-Doring) detonation problem for which spurious numerical solutions which propagate at mesh speed have been observed on coarse grids. With the present method, a change of limiter causes the solution to change from the physically correct CJ detonation solution to the spurious weak detonation solution.

Suresh, Ambady↗
NASA NTRS2002

Multi-dimensional Upwind Fluctuation Splitting Scheme with Mesh Adaption for Hypersonic Viscous Flow

A multi-dimensional upwind fluctuation splitting scheme is developed and implemented for two-dimensional and axisymmetric formulations of the Navier-Stokes equations on unstructured meshes. Key features of the scheme are the compact stencil, full upwinding, and non-linear discretization which allow for second-order accuracy with enforced positivity. Throughout, the fluctuation splitting scheme is compared to a current state-of-the-art finite volume approach, a second-order, dual mesh upwind flux difference splitting scheme (DMFDSFV), and is shown to produce more accurate results using fewer computer resources for a wide range of test cases. A Blasius flat plate viscous validation case reveals a more accurate upsilon-velocity profile for fluctuation splitting, and the reduced artificial dissipation production is shown relative to DMFDSFV. Remarkably, the fluctuation splitting scheme shows grid converged skin friction coefficients with only five points in the boundary layer for this case. The second half of the report develops a local, compact, anisotropic unstructured mesh adaptation scheme in conjunction with the multi-dimensional upwind solver, exhibiting a characteristic alignment behavior for scalar problems. The adaptation strategy is extended to the two-dimensional and axisymmetric Navier-Stokes equations of motion through the concept of fluctuation minimization.

Wood, William A., III↗
NASA NTRS2004

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge-Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.

Bryson, Steve↗