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All sourcesNASA NTRSDOE OSTI
NASA NTRS2001

On the Definition of Surface Potentials for Finite-Difference Operators

For a class of linear constant-coefficient finite-difference operators of the second order, we introduce the concepts similar to those of conventional single- and double-layer potentials for differential operators. The discrete potentials are defined completely independently of any notion related to the approximation of the continuous potentials on the grid. We rather use all approach based on differentiating, and then inverting the differentiation of a function with surface discontinuity of a particular kind, which is the most general way of introducing surface potentials in the theory of distributions. The resulting finite-difference "surface" potentials appear to be solutions of the corresponding continuous potentials. Primarily, this pertains to the possibility of representing a given solution to the homogeneous equation on the domain as a variety of surface potentials, with the density defined on the domain's boundary. At the same time the discrete surface potentials can be interpreted as one specific realization of the generalized potentials of Calderon's type, and consequently, their approximation properties can be studied independently in the framework of the difference potentials method by Ryaben'kii. The motivation for introducing and analyzing the discrete surface potentials was provided by the problems of active shielding and control of sound, in which the aforementioned source terms that drive the potentials are interpreted as the acoustic control sources that cancel out the unwanted noise on a predetermined region of interest.

Tsynkov, S. V.↗
NASA NTRS1980

On One-Dimensional Stretching Functions for Finite-Difference Calculations

The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids.

Vinokur, M.↗
NASA NTRS1981

Computation of wing-vortex interaction in transonic flow using implicit finite difference algorithm

An implicit delta form finite difference algorithm for Euler equations in conservation law form was used in preliminary calculations of three dimensional wing vortex interaction. Both steady and unsteady transonic flow wing vortex interactions are computed. The computations themselves are meant to guide upcoming wind tunnel experiments of the same flow field. Various modifications to the numerical method that are intended to improve computational efficiency are also described and tested in both two and three dimensions. Combination of these methods can reduce the overall computational time by a factor of 4.

Srinivasan, G.↗
NASA NTRS1990

Prediction of blade-vortex interaction noise using airloads generated by a finite-difference technique

The present numerical finite-difference scheme for helicopter blade-load prediction during realistic, self-generated three-dimensional blade-vortex interactions (BVI) derives the velocity field through a nonlinear superposition of the rotor flow-field yielded by the full potential rotor flow solver RFS2 for BVI, on the one hand, over the rotational vortex flow field computed with the Biot-Savart law. Despite the accurate prediction of the acoustic waveforms, peak amplitudes are found to have been persistently underpredicted. The inclusion of BVI noise source in the acoustic analysis significantly improved the perceived noise level-corrected tone prediction.

Tadghighi, Hormoz↗
DOE OSTI2022

Application of linear prolongation to coarse mesh finite difference acceleration in CASMO5

The Coarse-Mesh Finite Difference (CMFD) method has been used for over a decade to accelerate the convergence of the Method of Characteristics (MOC) solution to the two- dimensional particle transport equation in CASMO5. Numerical testing, along with widespread use in production-level calculations, have shown that the current CMFD implementation provides stability and robustness for a wide range of realistic reactor physics problems. However, the recent development of linear prolongation has attracted attention from the community as a way to further improve the performance and stability of CMFD. Two interpolation methods for linear prolongation are presented in this work and implemented into a test version of CASMO5. The performance of the proposed interpolations, referred to as the bilinear and linear directional schemes, is evaluated in terms of runtime relative to the default constant or uniform scaling update. Numerical results indicate that the use of linear prolongation can reduce the transport solver runtime on average by approximately 10% when tested with two hundred randomly selected cases. The new directional linear interpolation, combined with default constant boundary updates, is found to provide the highest reduction in runtime for the cases analyzed. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1978

The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations

A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.

Hannah, S. R.↗
NASA NTRS1985

Selecting step sizes in sensitivity analysis by finite differences

This paper deals with methods for obtaining near-optimum step sizes for finite difference approximations to first derivatives with particular application to sensitivity analysis. A technique denoted the finite difference (FD) algorithm, previously described in the literature and applicable to one derivative at a time, is extended to the calculation of several simultaneously. Both the original and extended FD algorithms are applied to sensitivity analysis for a data-fitting problem in which derivatives of the coefficients of an interpolation polynomial are calculated with respect to uncertainties in the data. The methods are also applied to sensitivity analysis of the structural response of a finite-element-modeled swept wing. In a previous study, this sensitivity analysis of the swept wing required a time-consuming trial-and-error effort to obtain a suitable step size, but it proved to be a routine application for the extended FD algorithm herein.

Iott, J.↗
NASA NTRS1959

Some Finite Difference Solutions of the Laminar Compressible Boundary Layer Showing the Effects of Upstream Transpiration Cooling

Three numerical solutions of the partial differential equations describing the compressible laminar boundary layer are obtained by the finite difference method described in reports by I. Flugge-Lotz, D.C. Baxter, and this author. The solutions apply to steady-state supersonic flow without pressure gradient, over a cold wall and over an adiabatic wall, both having transpiration cooling upstream, and over an adiabatic wall with upstream cooling but without upstream transpiration. It is shown that for a given upstream wall temperature, upstream transpiration cooling affords much better protection to the adiabatic solid wall than does upstream cooling without transpiration. The results of the numerical solutions are compared with those of approximate solutions. The thermal results of the finite difference solution lie between the results of Rubesin and Inouye, and those of Libby and Pallone. When the skin-friction results of one finite difference solution are used in the thermal analysis of Rubesin and Inouye, improved agreement between the thermal results of the two methods of solution is obtained.

Howe, John T.↗
NASA NTRS1991

Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations

Finite-difference approximations for steady-state compressible Navier-Stokes equations, whose two spatial dimensions are written in generalized curvilinear coordinates and strong conservation-law form, are presently solved by means of Newton's method in order to obtain a lifting-airfoil flow field under subsonic and transonnic conditions. In addition to ascertaining the computational requirements of an initial guess ensuring convergence and the degree of computational efficiency obtainable via the approximate Newton method's freezing of the Jacobian matrices, attention is given to the need for auxiliary methods assessing the temporal stability of steady-state solutions. It is demonstrated that nonunique solutions of the finite-difference equations are obtainable by Newton's method in conjunction with a continuation method.

Bailey, Harry E.↗
NASA NTRS1983

Nonlinear truncation error analysis of finite difference schemes for the Euler equations

It is pointed out that, in general, dissipative finite difference integration schemes have been found to be quite robust when applied to the Euler equations of gas dynamics. The present investigation considers a modified equation analysis of both implicit and explicit finite difference techniques as applied to the Euler equations. The analysis is used to identify those error terms which contribute most to the observed solution errors. A technique for analytically removing the dominant error terms is demonstrated, resulting in a greatly improved solution for the explicit Lax-Wendroff schemes. It is shown that the nonlinear truncation errors are quite large and distributed quite differently for each of the three conservation equations as applied to a one-dimensional shock tube problem.

Klopfer, G. H.↗
NASA NTRS1985

Finite-difference computations of rotor loads

This paper demonstrates the current and future potential of finite-difference methods for solving real rotor problems which now rely largely on empiricism. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advance-ratio flight. Comparisons are made with experimental pressure data.

Caradonna, F. X.↗
NASA NTRS1987

A finite-difference outer layer and integral inner layer method for the solution of the turbulent boundary layer equations

A new turbulent boundary-layer method is developed which models the inner region with the law of the wall while the outer region uses Clauser's eddy viscosity in Matsuno's finite-difference method. The match point between the inner and outer regions as well as the wall shear stress are determined at each marching step during the computation. Results obtained for incompressible, two-dimensional flow over flat plates and ellipses are compared with solutions from a baseline method which uses a finite-difference method for the entire boundary layer. Since the present method used the finite-difference method in the outer region only, the number of grid points required was about half that needed for the baseline method. Accurate displacement and momentum thicknesses were predicted for all cases. Skin friction was predicted well for the flat plate, but the accuracy decreased significantly for the ellipses. Adding a wake functions to the law of the wall allows some of the pressure gradient effect to be taken into account thereby increasing the accuracy of the method.

Barnwell, R. W.↗
NASA NTRS1981

Numerical stability of an explicit finite difference scheme for the solution of transient conduction in composite media

A theoretical evaluation of the stability of an explicit finite difference solution of the transient temperature field in a composite medium is presented. The grid points of the field are assumed uniformly spaced, and media interfaces are either vertical or horizontal and pass through grid points. In addition, perfect contact between different media (infinite interfacial conductance) is assumed. A finite difference form of the conduction equation is not valid at media interfaces; therefore, heat balance forms are derived. These equations were subjected to stability analysis, and a computer graphics code was developed that permitted determination of a maximum time step for a given grid spacing.

Campbell, W.↗
NASA NTRS1993

Multigrid methods and high order finite difference for flow in transition - Effects of isolated and distributed roughness elements

The high order finite difference and multigrid methods have been successfully applied to direct numerical simulation (DNS) for flow transition in 3D channels and 3D boundary layers with 2D and 3D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semicoarsening multigrid method associated with line distributive relaxation scheme, and a new treatment of the outflow boundary condition, which needs only a very short buffer domain to damp all wave reflection, are developed. These approaches make the multigrid DNS code very accurate and efficient. This makes us not only able to do spatial DNS for the 3D channel and flat plate at low computational costs, but also able to do spatial DNS for transition in the 3D boundary layer with 3D single and multiple roughness elements. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments.

Liu, C.↗
NASA NTRS1972

A finite difference method for predicting supersonic turbulent boundary layer flows with tangential slot injection

An implicit finite difference method has been applied to tangential slot injection into supersonic turbulent boundary layer flows. In addition, the effects induced by the interaction between the boundary layer displacement thickness and the external pressure field are considered. In the present method, three different eddy viscosity models have been used to specify the turbulent momentum exchange. One model depends on the species concentration profile and the species conservation equation has been included in the system of governing partial differential equations. Results are compared with experimental data at stream Mach numbers of 2.4 and 6.0 and with results of another finite difference method. Good agreement was generally obtained for the reduction of wall skin friction with slot injection and with experimental Mach number and pitot pressure profiles. Calculations with the effects of pressure interaction included showed these effects to be smaller than effects of changing eddy viscosity models.

Miner, E. W.↗
NASA NTRS1985

An implicit finite-difference code for a two-equation turbulence model for three-dimensional flows

An implicit finite difference code was developed which solves the transport equations for the turbulence kinetic energy and its dissipation rate in generalized coordinates in three dimensions. The finite difference equations are solved using the Beam-Warming algorithm. The kinetic energy-dissipation code, KEM, provides the closure; i.e., the turbulent viscosity for calculation of either compressible or incompressible flows. Turbulent internal flow over a backward-facing step has been calculated using the present code in conjunction with the Incompressible Navier-Stokes Code, INS3D. The results are in good agreement with experiments and two dimensional computations of other researchers.

Kaul, U. K.↗