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NASA NTRS1989

New approximate boundary conditions for large eddy simulations of wall-bounded flows

Two new approximate boundary conditions have been applied to the large eddy simulation of channel flow with and without transpiration. These new boundary conditions give more accurate results than those previously in use, and allow significant reduction of the required CPU time over simulations in which no-slip conditions are applied. Mean velocity profiles and turbulence intensities compare well both with experimental data and with the results of resolved simulations. The influence of the approximate boundary conditions remains confined near the point of application and does not affect the turbulence statistics in the core of the flow.

Piomelli, Ugo↗

NASA NTRS1987

Acoustic gravity waves: A computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.

Hariharan, S. I.↗

NASA NTRS1988

Acoustic gravity waves - A computational approach

Hariharan, S. I.↗

NASA NTRS1992

Electromagnetic scattering by coated convex surfaces and wedges simulated by approximate boundary conditions

Asymptotic/high-frequency solutions are developed for analyzing the non-specular scattering mechanisms associated with coated convex surfaces and edges simulated by approximate boundary conditions. In particular, the standard impedance boundary conditions (SIBC's) and the second order generalized impedance boundary conditions (GIBC's) are employed for a characterization of the edge diffraction, creeping wave, and surface diffracted wave contributions. To study the creeping wave and surface diffracted wave mechanisms, rigorous UTD (uniform geometrical theory of diffraction) diffraction coefficients are developed for a convex coated cylinder simulated with SIBC's and GIBC's. The ray solutions obtained remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. The diffraction coefficient for a convex cylinder are obtained via a generalization of the corresponding ones of the circular cylinder. To validate the asymptotic/high-frequency solution, integral equations are derived for both E and H-polarization and solved numerically using the method of moments. Results are presented for a single and three layered coated convex cylinder. Some insights are also provided on the accuracy of the employed GIBC's versus SIBC's for application to curved surfaces. To characterize the scattering by impedance wedges illuminated at skew incidence, diffraction coefficients are derived from an approximate solution of the governing functional difference equations. This solution exactly recovers the known ones for an impedance half plane or an arbitrary wedge at normal incidence, and to validate it for other wedge angles, a moment method code was used. Finally, to test the usefulness of the approximate skew incidence impedance wedge diffraction coefficient for three dimensional structures, equivalent currents are derived in the context of the physical theory of diffraction (PTD) for a finite length impedance wedge of arbitrary internal angles. These are incorporated in a standard general purpose PTD code and results are presented for a number of different impedance structures.

Syed, H. H.↗

NASA NTRS1982

Approximate boundary condition procedure for the two-dimensional numerical solution of vortex wakes

Research on efficient computational methods for general vorticity fields has been conducted in connection with a need for basic research on vortex-dominated flows. The present investigation is concerned with the evolution of vortex wakes behind aircraft wings. An efficient procedure is presented for the calculation of the boundary values used in the numerical solution of the unsteady, incompressible, two-dimensional Navier-Stokes equations for an unbounded flow field. The extent of the computational grid can be reduced compared to methods utilizing standard boundary conditions, without loss of accuracy. The efficiencies realized make it feasible to calculate the vortex wake development for realistic wing configurations, including the merging of multiple vortices, for Reynolds numbers of about 10,000 based on wing chord.

Weston, R. P.↗

NASA NTRS1989

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and nonzero interblade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the periodic boundary conditions. Approximate nonreflecting boundary conditions were coded for the inlet and exit boundary conditions. Sample unsteady solutions were performed for an oscillating cascade and compared to experimental data. Also, test cases were run for a flat plate cascade to compare with the unsteady, small-perturbation, subsonic analyis. The predictions for oscillating cascades with nonzero interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. The zero degree interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus nonreflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distributions for certain flow conditions.

Huff, Dennis L.↗

NASA NTRS1989

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and non-zero inter-blade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the periodic boundary conditions. Approximate nonreflecting boundary conditions have been coded for the inlet and exit boundary conditions. Sample unsteady solutions have been performed for an oscillating cascade and compared to experimental data. Also, test cases were fun for a flat plate cascade to compare with an unsteady, small-perturbation, subsonic analysis. The predictions for oscillating cascades with non-zero inter-blade phase angles are in good agreement with experimental data and small-perturbation theory. The zero degree inter-blade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus non-reflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distributions for certain flow conditions. This code is expected to be used as a tool for reviewing simpler models that do not include the full nonlinear aerodynamics or as a final check for designs against flutter in turbomachinery.

Huff, Dennis L.↗

NASA NTRS1981

Influence of boundary approximations and conditions on finite difference solutions

Numerical representations of boundary approximations and conditions for three problems are investigated to determine the resulting global accuracy of the steady state solution. Numerical accuracy with various boundary approximations is determined for quasi-one-dimensional inviscid flow in a duct with the interior grid points evaluated using the MacCormack scheme. When an extrapolation approximation with first order local truncation error is used, the global second order accuracy of the difference scheme can be destroyed. For one dimensional flow in a porous medium, an implicit midpoint difference scheme which is consistent with the boundary conditions is developed without the need of boundary approximations. A dissipative model problem is solved with the boundary conditions discretized with first and second order accuracy. The overall second order accuracy of the difference scheme is destroyed if first order numerical representation of one of the boundary conditions is used. With a boundary approximation, the second order global accuracy of the model problem is retained if either second order extrapolation or first order representation of the governing equation is used.

Blottner, F. G.↗

NASA NTRS1989

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and nonzero interblade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the perioidic boundary conditions. Approximate nonreflecting boundary conditons were coded for the inlet and exit boundary conditions. Sample unsteady solutions were performed for an oscillating cascade and compared to experimental data. Also, test cases were run for a flat plate cascade to compare with the unsteady, small-perturbation, subsonic analysis. The predictions for oscillating cascades with nonzero interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. The zero degree interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus nonreflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distribution for certain flow conditions.

Huff, Dennis L.↗

NASA NTRS1983

Stability analysis of intermediate boundary conditions in approximate factorization schemes

In many cases, approximate factorization schemes have provided a significant increase in efficiency over previously used solution methods in certain problems. The present investigation is concerned with the importance of intermediate boundary conditions in approximate factorization schemes, taking into account a specific example regarding a boundary-induced stability restriction in a scheme for the transonic full-potential equation. The considered scheme has been discussed by Holst (1979). Holst's scheme is a variation of the AF2 schemes described by Ballhaus and Steger (1975). The application of the AF2 scheme to the two-dimensional Laplace equation in a rectangle is studied giving attention to the stability of the scheme in connection with various boundary conditions for the intermediate variable.

South, J. C., Jr.↗

NASA NTRS1986

Stability analysis of intermediate boundary conditions in approximate factorization schemes

The paper discusses the role of the intermediate boundary condition in the AF2 scheme used by Holst for simulation of the transonic full potential equation. It is shown that the treatment suggested by Holst led to a restriction on the time step and ways to overcome this restriction are suggested. The discussion is based on the theory developed by Gustafsson, Kreiss, and Sundstrom and also on the von Neumann method.

South, J. C., Jr.↗

NASA NTRS2002

Analytical Modal Analysis for Thin-Film Flat Lenses

Due to strong potential applications and more demanding requirements imposed upon thin-film structures for space deployable, there has been increasing research and development activities during recent years in the field of vibration analysis of these types of structures. Moreover, interests in employing these structural components have received renewed emphasis in recent years within NASA and the Air Force. This is due to their inherent lightweight, low packaging and launch volume, and relative simplicity of deployment. Among the potential mission concepts for which these structural elements are included, one can mention solar sails, space solar power generation systems, solar thermal propulsion vehicles, large space telescopes, and inflatable communication antennas. This paper presents analytical procedures to determine vibration and physical characteristics of thin film lenses with circular and elliptical shapes membranes considered in design of a solar concentrator. In general, three methods are used to obtain approximate solutions of Helmholtz boundary value problems. One method requires that solution satisfy the differential equation exactly and the boundary condition approximately. Another method demands a solution that satisfies the boundary conditions exactly and the governing equations approximately. The third method sees a solution that satisfies both the governing equation and boundary conditions approximately. Extensive reviews of vibrations of membrane and plates are provided by Leissa and Mazumdar.

Hamid R. Hamidzadeh↗

NASA NTRS1993

On the accurate long-time solution of the wave equation in exterior domains: Asymptotic expansions and corrected boundary conditions

We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.

Hagstrom, Thomas↗

NASA NTRS1979

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.↗

NASA NTRS1981

Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

Yee, H. C.↗

NASA NTRS1988

Interaction and merging of vortex filaments

The asymptotic solutions of Navier-Stokes equations for vortex filaments of finite strength with small effective vortical cores are summarized with special emphasis placed on the physical meaning and the practical limit to the applicability of the asymptotic solution. Finite-difference solutions of Navier-Stokes equations for the marging of the filament(s) are described with a focus on the development of the approximate boundary conditions for the computational domain. An efficiency study employing a model problem is used to assess the advantages of the present approximate boundary condition method over previously used techniques. Applications of the present method are presented for the motion and decay of a 3:1 elliptic vortex ring, and for the merging process of a pair of coaxial vortex rings. A numerical procedure for the problem of local merging of vortex filaments, which requires the asymptotic analysis as well as the numerical Navier-Stokes solver, is also presented.

Liu, C. H.↗

NASA NTRS1988

A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

Funaro, D.↗

NASA NTRS1983

Numerical solution to the glancing sidewall oblique shock wave/turbulent boundary layer interaction in three dimension

A supersonic three-dimensional viscous forward-marching computer design code called PEPSIS is used to obtain a numerical solution of the three-dimensional problem of the interaction of a glancing sidewall oblique shock wave and a turbulent boundary layer. Very good results are obtained for a test case that was run to investigate the use of the wall-function boundary-condition approximation for a highly complex three-dimensional shock-boundary layer interaction. Two additional test cases (coarse mesh and medium mesh) are run to examine the question of near-wall resolution when no-slip boundary conditions are applied. A comparison with experimental data shows that the PEPSIS code gives excellent results in general and is practical for three-dimensional supersonic inlet calculations.

Anderson, B. H.↗