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NASA NTRS1983

Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm

A multiple-grid algorithm for use in efficiently obtaining steady solutions to the Euler and Navier-Stokes equations is presented. The convergence of the explicit MacCormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids. Both the fine and coarse grid schemes are readily vectorizable. The combination of multiple-gridding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems. Results are presented for subsonic, transonic, and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows. Work reduction factors over a scalar, single-grid algorithm range as high as 76.8.

Navier Stokes simulation↗

NASA NTRS1983

Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm

Chima, R. V.↗

NASA NTRS1992

Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations

A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.

Soh, Woo Y.↗

NASA NTRS1997

SYNTHESIS OF NOVEL ALL-DIELECTRIC GRATING FILTERS USING GENETIC ALGORITHMS

We are concerned with the design of inhomogeneous, all dielectric (lossless) periodic structures which act as filters. Dielectric filters made as stacks of inhomogeneous gratings and layers of materials are being used in optical technology, but are not common at microwave frequencies. The problem is then finding the periodic cell's geometric configuration and permittivity values which correspond to a specified reflectivity/transmittivity response as a function of frequency/illumination angle. This type of design can be thought of as an inverse-source problem, since it entails finding a distribution of sources which produce fields (or quantities derived from them) of given characteristics. Electromagnetic sources (electric and magnetic current densities) in a volume are related to the outside fields by a well known linear integral equation. Additionally, the sources are related to the fields inside the volume by a constitutive equation, involving the material properties. Then, the relationship linking the fields outside the source region to those inside is non-linear, in terms of material properties such as permittivity, permeability and conductivity. The solution of the non-linear inverse problem is cast here as a combination of two linear steps, by explicitly introducing the electromagnetic sources in the computational volume as a set of unknowns in addition to the material unknowns. This allows to solve for material parameters and related electric fields in the source volume which are consistent with Maxwell's equations. Solutions are obtained iteratively by decoupling the two steps. First, we invert for the permittivity only in the minimization of a cost function and second, given the materials, we find the corresponding electric fields through direct solution of the integral equation in the source volume. The sources thus computed are used to generate the far fields and the synthesized triter response. The cost function is obtained by calculating the deviation between the synthesized value of reflectivity/transmittivity and the desired one. Solution geometries for the periodic cell are sought as gratings (ensembles of columns of different heights and widths), or combinations of homogeneous layers of different dielectric materials and gratings. Hence the explicit unknowns of the inversion step are the material permittivities and the relative boundaries separating homogeneous parcels of the periodic cell.

Zuffada, Cinzia↗

NASA NTRS1975

Higher order accurate partial implicitization: An unconditionally stable fourth-order-accurate explicit numerical technique

The previously obtained second-order-accurate partial implicitization numerical technique used in the solution of fluid dynamic problems was modified with little complication to achieve fourth-order accuracy. The Von Neumann stability analysis demonstrated the unconditional linear stability of the technique. The order of the truncation error was deduced from the Taylor series expansions of the linearized difference equations and was verified by numerical solutions to Burger's equation. For comparison, results were also obtained for Burger's equation using a second-order-accurate partial-implicitization scheme, as well as the fourth-order scheme of Kreiss.

Graves, R. A., Jr.↗

NASA NTRS1982

Stability of mean flows over an infinite flat plate

The present investigation is mainly concerned with a stability analysis for the linearized Navier-Stokes equations for parallel and nonparallel mean flows over an infinite flat plate. The system of equations for parallel flows is presented. The system is viewed as a generalized Orr-Sommerfeld equation. Attention is given to an explicit criterion characterizing the case when the stability of all physically reasonable solutions is determined by the eigenvalues. The proof given in the investigation is applicable to both the generalized Orr-Somerfeld equations and the modified equations for nonparallel flow. The fact that the criterion is independent of the completeness or incompleteness of eigenfunctions is contrary to some expectations.

Miklavcic, M.↗

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 NTRS1983

Efficient three-dimensional global models for climate studies - Models I and II

Climate modeling based on numerical solution of the fundamental equations for atmospheric structure and motion permits the explicit modeling of physical processes in the climate system and the natural treatment of interactions and feedbacks among parts of the system. The main difficulty concerning this approach is related to the computational requirements. The present investigation is concerned with the development of a grid-point model which is programmed so that both horizontal and vertical resolutions can easily be changed. Attention is given to a description of Model I, the performance of sensitivity experiments by varying parameters, the definition of an improved Model II, and a study of the dependence of climate simulation on resolution with Model II. It is shown that the major features of global climate can be simulated reasonably well with a horizontal resolution as coarse as 1000 km. Such a resolution allows the possibility of long-range climate studies with moderate computer resources.

Russel, G.↗

NASA NTRS1984

Investigation of the three-dimensional flow field within a transonic fan rotor - Experiment and analysis

An experimental investigation of the three-dimensional flow field through a low aspect ratio, transonic, axial flow fan rotor has been conducted, using an advanced laser anemometer (LA) system. Laser velocimeter measurements of the rotor flow field at the design operating speed and over a range of throughflow conditions are compared to analytical solutions. The numerical technique used herein yields the solution to the full, three-dimensional, unsteady Euler equations using an explicit time-marching, finite volume approach. The numerical analysis, when coupled with a simplified boundary layer calculation, generally yields good agreement with the experimental data. The test rotor has an aspect ratio of 1.56, a design total pressure ratio of 1.629 and a tip relative Mach number of 1.38. The high spatial resolution of the LA data matrix (9 radial x 30 axial x 50 blade-to-blade) permits details of the transonic flow field such as shock location, turning distribution, and blade loading levels to be investigated and compared to analytical results.

Pierzga, M. J.↗

NASA NTRS1986

Smoothing and the second law

The technique of obtaining second order, oscillation free, total variation diminishing (TVD), scalar difference schemes by adding a limited diffusion flux (smoothing) to a second order centered scheme is explored. It is shown that such schemes do not always converge to the correct physical answer. The approach presented here is to construct schemes that numerically satisfy the second law of thermodynamics on a cell by cell basis. Such schemes can only converge to the correct physical solution and in some cases can be shown to be TVD. An explicit scheme with this property and second order spatial accuracy was found to have an extremely restrictive time step limitation (Delta t less than Delta x squared). Switching to an implicit scheme removed the time step limitation.

Merriam, Marshal L.↗

NASA NTRS1988

Inflation in Einstein-Cartan theory with energy-momentum tensor with spin

Generalized, or power-law, inflation is shown to necessarily exist for a simple, anisotropic (Bianchi Type I) cosmology in the Einstein-Cartan gravitational theory with the Ray-Smalley (RS) improved energy-momentum tensor with spin. Formal solution of the EC field equations with the fluid equations of motion explicitly shows inflation caused by the RS spin angular kinetic energy density.

Fennelly, A. J.↗

NASA NTRS1988

Inevitable inflation in Einstein-Cartan theory with improved energy-momentum tensor with spin

Generalized, or power-law, inflation is shown to necessarily exist for a simple, anisotropic, (Bianchi Type-1) cosmology in the Einstein-Cartan gravitational theory with the Ray-Smalley improved energy momentum tensor with spin. Formal solution of the EC field equations with the fluid equations of motion explicitly shows inflation caused by the RS spin angular kinetic energy density. Shear is not effective in preventing inflation in the ECRS model. The relation between fluid vorticity, torsion, reference axis rotation, and shear ellipsoid precession shows through clearly.

Fennelly, A. J.↗

NASA NTRS1988

Navier-Stokes calculations of transonic flows past cavities

A computational investigation of subsonic and transonic flows past three dimensional deep and transitional cavities is presented. Computational simulations of these self induced oscillatory flows were generated through time accurate solutions of the Reynolds averaged full Navier-Stokes equations, using the explicit MacCormack scheme. The Reynolds stresses were included through the Baldwin-Lomax algebraic turbulence model with certain modifications. Two cases were computed to demonstrate the capability of the numerical scheme in modeling the complex three dimensional flow features inside a cavity. The results from an experimental investigation were used not only to benchmark the computations, but also to widen the database used for the discussions and conclusions. The computational results include instantaneous and time averaged flow properties everywhere in the computational zone. Time series analyses were performed for the instantaneous pressure values on the cavity floor. The features of deep and transitional cavity flows, and the effects of the sidewall on the cavity flow flowfield are illustrated through computational graphics.

Baysal, Oktay↗

NASA NTRS1987

Smoothing and the second law

The technique of obtaining second-order oscillation-free total -variation-diminishing (TVD), scalar difference schemes by adding a limited diffusive flux ('smoothing') to a second-order centered scheme is explored. It is shown that such schemes do not always converge to the correct physical answer. The approach presented here is to construct schemes that numerically satisfy the second law of thermodynamics on a cell-by-cell basis. Such schemes can only converge to the correct physical solution and in some cases can be shown to be TVD. An explicit scheme with this property and second-order spatial accuracy was found to have extremely restrictive time-step limitation. Switching to an implicit scheme removed the time-step limitation.

Merriam, Marshal L.↗

NASA NTRS1989

Navier-Stokes calculations of transonic flows past cavities

A computational investigation of subsonic and transonic flows past 3-D deep transitional cavities is presented. Computational simulations of these self-induced oscillatory flows were generated through time-accurate solutions of the Reynolds averaged full Navier-Stokes equations, using the explicit MacCormack scheme. The Reynolds stresses were included through the Baldwin-Lomax algebraic turbulence model with certain modifications. Two cases were computed to demonstrate the capability of the numerical scheme in modeling the complex 3-D flow features inside a cavity. The results from an experimental investigation were used not only to benchmark the computations, but also to widen the database used for the discussions and conclusions. The computational results include instantaneous and time averaged flow properties everywhere in the computational zone. Time series analyses were performed for the instantaneous pressure values on the cavity floor. The features of deep and transitional cavity flows, and the effect of the sidewall on the cavity flow flowfield are illustrated through computational graphics.

Baysal, Oktay↗

NASA NTRS1988

Convergence speeding up in the calculation of the viscous flow about an airfoil

A finite volume method to solve the three dimensional Navier-Stokes equations was developed. It is based on a cell-vertex scheme with central differences and explicit Runge-Kutta time steps. A good convergence for a stationary solution was obtained by the use of local time steps, implicit smoothing of the residues, a multigrid algorithm, and a carefully controlled artificial dissipative term. The method is illustrated by results for transonic profiles and airfoils. The method allows a routine solution of the Navier-Stokes equations.

Radespiel, R.↗

NASA NTRS1989

Recent advances in Runge-Kutta schemes for solving 3-D Navier-Stokes equations

A thin-layer Navier-Stokes has been developed for solving high Reynolds number, turbulent flows past aircraft components under transonic flow conditions. The computer code has been validated through data comparisons for flow past isolated wings, wing-body configurations, prolate spheroids and wings mounted inside wind-tunnels. The basic code employs an explicit Runge-Kutta time-stepping scheme to obtain steady state solution to the unsteady governing equations. Significant gain in the efficiency of the code has been obtained by implementing a multigrid acceleration technique to achieve steady-state solutions. The improved efficiency of the code has made it feasible to conduct grid-refinement and turbulence model studies in a reasonable amount of computer time. The non-equilibrium turbulence model of Johnson and King has been extended to three-dimensional flows and excellent agreement with pressure data has been obtained for transonic separated flow over a transport type of wing.

Vatsa, Veer N.↗