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All sourcesNASA NTRSDOE OSTI
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.↗
NASA NTRS1977

Finite-difference theory for sound propagation in a lined duct with uniform flow using the wave envelope concept

Finite difference equations are derived for sound propagation in a two dimensional, straight, soft wall duct with a uniform flow by using the wave envelope concept. This concept reduces the required number of finite difference grid points by one to two orders of magnitude depending on the length of the duct and the frequency of the sound. The governing acoustic difference equations in complex notation are derived. An exit condition is developed that allows a duct of finite length to simulate the wave propagation in an infinitely long duct. Sample calculations presented for a plane wave incident upon the acoustic liner show the numerical theory to be in good agreement with closed form analytical theory. Complete pressure and velocity printouts are given to some sample problems and can be used to debug and check future computer programs.

Baumeister, K. J.↗
NASA NTRS1996

Time-Dependent Parabolic Finite Difference Formulation for Harmonic Sound Propagation in a Two-Dimensional Duct with Flow

An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Kreider, Kevin L.↗
NASA NTRS1981

Application of 'steady' state finite element and transient finite difference theory to sound propagation in a variable duct - A comparison with experiment

Experimental data are presented for sound propagation in a simulated infinite hard wall duct with a large change in duct cross sectional area. The data are conveniently tabulated for further use. The 'steady' state finite element theory of Astley and Eversman (1981) and the transient finite difference theory of White (1981) are in good agreement with the data for both the axial and transverse pressure profiles and the axial phase angle. Therefore, numerical finite difference and finite element theories appear to be ideally suited for handling duct propagation problems which encounter large axial gradients in acoustic parameters. The measured energy reflection coefficient agrees with the values from the Astley-Eversman modal coupling model.

Baumeister, K. J.↗
NASA NTRS1981

Application of steady state finite element and transient finite difference theory to sound propagation in a variable area duct: A comparison with experiment

Sound propagation without flow in a rectangular duct with a converging-diverging area variation was studied experimentally and theoretically. The area variation was of sufficient magnitude to produce large reflections and induce modal scattering. The rms (root-mean-squared) pressure and phase angle on both the flat and curved surface were measured and tabulated. The steady state finite element theory and the transient finite difference theory are in good agreement with the data. It is concluded that numerical finite difference and finite element theories appear ideally suited for handling duct propagation problems which encounter large area variations.

Baumeister, K. J.↗
NASA NTRS1975

Application of a finite difference technique to thermal wave propagation

A finite difference formulation is presented for thermal wave propagation resulting from periodic heat sources. The numerical technique can handle complex problems that might result from variable thermal diffusivity, such as heat flow in the earth with ice and snow layers. In the numerical analysis, the continuous temperature field is represented by a series of grid points at which the temperature is separated into real and imaginary terms. Computer routines previously developed for acoustic wave propagation are utilized in the solution for the temperatures. The calculation procedure is illustrated for the case of thermal wave propagation in a uniform property semi-infinite medium.

Baumeister, K. J.↗
NASA NTRS1975

Application of a finite difference technique to thermal wave propagation

A finite difference formulation is presented for thermal wave propagation resulting from periodic heat sources. The numerical technique can handle complex problems that might result from variable thermal diffusivity, such as heat flow in the earth with ice and snow layers. In the numerical analysis, the continuous temperature field is represented by a series of grid points at which the temperature is separated into real and imaginary terms. Next, computer routines previously developed for acoustic wave propagation are utilized in the solution for the temperatures. The calculation procedure is illustrated for the case of thermal wave propagation in a uniform property semi-infinite medium.

Baumeister, K. J.↗