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NASA NTRS1993

Calculations and experiments concerning lifting force and power in TEMPUS

A critical comparison is reported between the theoretically predicted and experimentally measured values for the electromagnetic lifting force and the heating rates which may be achieved, under simulated microgravity conditions, using the TEMPUS electromagnetic levitation device. The experiments involved the suspending of a metallic sample from one arm of a recording balance, such that it was carefully positioned between the heating and the positioning coils of the levitation device. The net force exerted by the sample was measured as a function of position, the coil currents, and the nature of the sample. Some calculations are also reported regarding the power absorption by the sample. The theoretical predictions, based on the numerical solution of Maxwell's equations using the volume integral technique, were found to be in excellent agreement with the measurements. For the idealized case of a spherical sample, analytical solutions describing the lifting force were also found to agree very well with the computed results.

Zong, J. H.↗

NASA NTRS1986

A multistage time-stepping scheme for the thin-layer Navier-Stokes equations

A finite-volume scheme for numerical integration of the Euler equations was extended to allow solution of the thin-layer Navier-Stokes equations in two and three dimensions. The extended algorithm, which is based on a class of four-stage Runge-Kutta time-stepping schemes, was made numerically efficient through the following convergence acceleration technique: (1) local time stepping, (2) enthalpy damping, and (3) residual smoothing. Also, the high degree of vectorization possible with the algorithm has yielded an efficient program for vector processors. The scheme was evaluated by solving laminar and turbulent flows. Numerical results have compared well with either theoretical or other numerical solutions and/or experimental data.

Swanson, R. C., Jr.↗

DOE OSTI2023

Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids

This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite-Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth-order-accurate reconstructions with and without Weighted Essentially Non-Oscillatory limiting. Acoustic-only upwinding for pressure and mass flux terms and advective-only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st-order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k −5/3 to smaller spatial scales by reducing dissipation either with higher-order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k −5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1988

Euler equation computations for the flow over a hovering helicopter rotor

A numerical solution technique is developed for computing the flow field around an isolated helicopter rotor in hover. The flow is governed by the compressible Euler equations which are integrated using a finite volume approach. The Euler equations are coupled to a free wake model of the rotary wing vortical wake. This wake model is incorporated into the finite volume solver using a prescribed flow, or perturbation, technique which eliminates the numerical diffusion of vorticity due to the artificial viscosity of the scheme. The work is divided into three major parts: (1) comparisons of Euler solutions to experimental data for the flow around isolated wings show good agreement with the surface pressures, but poor agreement with the vortical wake structure; (2) the perturbation method is developed and used to compute the interaction of a streamwise vortex with a semispan wing. The rapid diffusion of the vortex when only the basic Euler solver is used is illustrated, and excellent agreement with experimental section lift coefficients is demonstrated when using the perturbation approach; and (3) the free wake solution technique is described and the coupling of the wake to the Euler solver for an isolated rotor is presented. Comparisons with experimental blade load data for several cases show good agreement, with discrepancies largely attributable to the neglect of viscous effects. The computed wake geometries agree less well with experiment, the primary difference being that too rapid a wake contraction is predicted for all the cases.

Roberts, Thomas Wesley↗

DOE OSTI2023

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Here, our numerical scheme is based on the direct approximation of the nonlinear Feynman–Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman–Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1984

Comparison of free-piston Stirling engine model predictions with RE1000 engine test data

Predictions of a free-piston Stirling engine model are compared with RE1000 engine test data taken at NASA-Lewis Research Center. The model validation and the engine testing are being done under a joint interagency agreement between the Department of Energy's Oak Ridge National Laboratory and NASA-Lewis. A kinematic code developed at Lewis was upgraded to permit simulation of free-piston engine performance; it was further upgraded and modified at Lewis and is currently being validated. The model predicts engine performance by numerical integration of equations for each control volume in the working space. Piston motions are determined by numerical integration of the force balance on each piston or can be specified as Fourier series. In addition, the model Fourier analyzes the various piston forces to permit the construction of phasor force diagrams. The paper compares predicted and experimental values of power and efficiency and shows phasor force diagrams for the RE1000 engine displacer and piston. Further development plans for the model are also discussed.

Tew, R. C., Jr.↗

NASA NTRS1984

Comparison of free-piston Stirling engine model predictions with RE1000 engine test data

Tew, R. C.↗

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 NTRS1986

Finite volume solultion of the compressible boundary layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicitly to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and very simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗

NASA NTRS1986

Finite volume solution of the compressible boundary-layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗

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 NTRS1993

Accuracy of the Generalized Self-Consistent Method in Modelling the Elastic Behaviour of Periodic Composites

Local stress and strain fields in the unit cell of an infinite, two-dimensional, periodic fibrous lattice have been determined by an integral equation approach. The effect of the fibres is assimilated to an infinite two-dimensional array of fictitious body forces in the matrix constituent phase of the unit cell. By subtracting a volume averaged strain polarization term from the integral equation we effectively embed a finite number of unit cells in a homogenized medium in which the overall stress and strain correspond to the volume averaged stress and strain of the constrained unit cell. This paper demonstrates that the zeroth term in the governing integral equation expansion, which embeds one unit cell in the homogenized medium, corresponds to the generalized self-consistent approximation. By comparing the zeroth term approximation with higher order approximations to the integral equation summation, both the accuracy of the generalized self-consistent composite model and the rate of convergence of the integral summation can be assessed. Two example composites are studied. For a tungsten/copper elastic fibrous composite the generalized self-consistent model is shown to provide accurate, effective, elastic moduli and local field representations. The local elastic transverse stress field within the representative volume element of the generalized self-consistent method is shown to be in error by much larger amounts for a composite with periodically distributed voids, but homogenization leads to a cancelling of errors, and the effective transverse Young's modulus of the voided composite is shown to be in error by only 23% at a void volume fraction of 75%.

Walker, Kevin P.↗

NASA NTRS2003

Vector Radiative Transfer Equation for Arbitrarily Shaped and Arbitrarily Oriented Particles: A Microphysical Derivation from Statistical Electromagnetics

We use the concepts of statistical electromagnetics to derive the general radiative transfer equation (RTE) describing multiple scattering of polarized light by sparse discrete random media consisting of arbitrarily shaped and arbitrarily oriented particles. The derivation starts from the volume integral and Lippmann-Schwinger equations for the electric field scattered by a fixed N-particle system and proceeds to the vector form of the Foldy-Lax equations and their approximate far-field version. We then assume that particle positions are completely random and derive the vector RTE by applying the Twersky approximation to the coherent electric field and the Twersky and ladder approximations to the coherency dyad of the diffuse field in the limit N -> infinity. The concluding section discusses the physical meaning of the quantities entering the general vector RTE and the assumptions made in its derivation.

Mishchenko, Michael I.↗

NASA NTRS1987

A finite volume Euler calculation of the aerodynamics of transonic airfoil-vortex interaction

Unsteady inviscid transonic airfoil-vortex interaction is numerically analyzed by solving the two-dimensional unsteady Euler equations in integral form using a finite volume scheme. The solution procedure is based on an explicit Runge-Kutta time-stepping scheme wherein the spatial terms are central-differenced and a combination of second- and fourth-differences in the flow variables is used to form the numerical dissipation terms to stabilize the scheme. A velocity decomposition technique is applied to alleviate the problem of vortex diffusion by the numerical dissipation terms and to treat the interaction of a Rankine vortex with an airfoil accurately. Results obtained are compared with available numerical data.

Damodaran, Murali↗

NASA NTRS1995

Viscous aerodynamic design using the adjoint variable approach

The use of classical optimal control methods, in particular variational methods, to solve the airfoil optimization problem, by deriving a set of adjoint (costate) equations and boundary conditions has already been done for inviscid (potential and Euler flows) and two dimensional, steady state, incompressible flow governed by the Navier-Stokes equations. The interior and boundary terms of the volume integral have been derived (in this work) for the steady Navier-Stokes equations in three dimensions for a viscous, compressible heat conducting fluid. This can be used to derive the adjoint equations and numerical boundary conditions for general classes of problems and hence paves the way for a solution to the aerodynamic optimization problem for compressible viscous flows. The next steps to the realization of that goal are projected as below. The usual square integral pressure functional as an objective function is being replaced by a more realistic drag functional subject to a lift constraint. The feasibility of attempting the more difficult time dependent problem is being investigated. It remains to get the full system of adjoint equations and boundary conditions with the new functional. The state and adjoint equations must be discretized and coded. An appropriate optimization program must be used (steepest descents seems inadequate) and various known airfoil shapes should be recovered in test cases of the computer program.

DeRise, George↗

NASA NTRS1988

Finite volume computation of unsteady inviscid rotational transonic flows past airfoils in rigid body motion

Unsteady inviscid transonic flow over airfoils in arbitrary rigid body motion is analyzed numerically by solving the two-dimensional unsteady Euler equations in integral form using a finite volume scheme. The solution procedure is based on an explicit Runge-Kutta time-stepping scheme wherein the spatial terms are central-differenced and a combination of second- and fourth-differences in the flow variables are used to form the numerical dissipation terms to stabilize the scheme. Unsteady calculations are started from converged steady-state solutions as initial conditions. Nonreflective boundary conditions are imposed on the far-field boundaries. Results are presented and, where possible, validated against available numerical and experimental data for airfoils subjected to a step change in angle of attack, airfoils oscillating and plunging in transonic flow, and airfoils immersed in a time-varying free stream.

Damodaran, Murali↗

NASA NTRS1993

Analytical evaluation of a surface integral expressing the coupling between interior and exterior volumes in a FE-IE approach

Recently an approach which combines the finite element technique and an integral equation to determine the fields scattered by inhomogeneous bodies of complicated shape has been proposed. Basically, a mathematical surface which encloses the scatterers is introduced, thus dividing the space into an interior and an exterior volume, in which the finite element technique and an integral equation for EM scattering, respectively, are applied. The integral equation is set up for the tangential components of the fields at the surface, while the interior volume the unknowns are the total fields. Continuity of the tangential fields at the boundary, as required by Maxwell's equations, is imposed, thus coupling the two methods to obtain a consistent solution. The coupling term is expressed by a surface integral formed by the dot product of a FE basis function and an IE testing function, or viceversa. By choosing the boundary to be a surface of revolution and by making a convenient selection of IE basis (testing) functions, it is possible to evaluate the integrals analytically on surfaces such as curved triangles, curved quadrilaterals and curved pentagons. We will illustrate the salient steps involved in setting up and carrying out these integrals and discuss what class of basis (testing) functions and analytic surfaces of revolution they are applicable to. Analytic calculations offer the advantage of better accuracy than purely numerical ones, and, when combined with them, often shed light on issues of numerical convergence and limiting values. Furthermore, they may reduce computation time and storage requirements.

Zuffada, C.↗

NASA NTRS1989

Moment method with isoparametric elements for three-dimensional anisotropic scatterers

A novel method for computing the frequency-domain electromagnetic fields scattered from, and penetrating into, arbitrarily shaped, three-dimensional, lossy, inhomogeneous anisotropic scatters is presented. The method is based on a general volume integrodifferential formulation of the scattering problem and consists of the numerical solution of the coupled integral equations by the moment method and point matching. The numerical model of the scatterer is obtained by parametric volume elements, and the basis functions used to represent the field within each element are the same used in the finite-element method. Element integration problems due to the singular kernel of the integral equations are treated in some detail. Numerical results for both the isotropic and the anisotropic spherical scatterer are presented, including comparisons with results obtained by different numerical methods for the isotropic cases considered. The capability of the numerical code presented to deal with cases where the material parameters of the scatterer are given by singular matrices is discussed for two particular examples.

Graglia, Roberto D.↗