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NASA NTRS1986

An embedding method for the steady Euler equations

Certain difficulties arise in connection with the numerical solution of a direct finite difference representation of the steady Euler equations. Johnson (1979, 1981, 1982) has, therefore, proposed a surrogate-equation technique, in which the first-order steady Euler equations are embedded in a certain second-order system of equations. The present paper is concerned with the theoretical justification for such an embedding approach. For the numerical solution of the two-dimensional steady Euler equations, it is shown that, under a continuity restriction, it is possible to solve a second-order embedded system together with appropriate additional boundary conditions. The result indicates that a more direct and potentially more efficient approach to the steady solutions exists than the alternative of solving the unsteady equations.

Chang, S.-H.↗

NASA NTRS1987

Development of upwind schemes for the Euler equations

Described are many algorithmic and computational aspects of upwind schemes and their second-order accurate formulations based on Total-Variation-Diminishing (TVD) approaches. An operational unification of the underlying first-order scheme is first presented encompassing Godunov's, Roe's, Osher's, and Split-Flux methods. For higher order versions, the preprocessing and postprocessing approaches to constructing TVD discretizations are considered. TVD formulations can be used to construct relaxation methods for unfactored implicit upwind schemes, which in turn can be exploited to construct space-marching procedures for even the unsteady Euler equations. A major part of the report describes time- and space-marching procedures for solving the Euler equations in 2-D, 3-D, Cartesian, and curvilinear coordinates. Along with many illustrative examples, several results of efficient computations on 3-D supersonic flows with subsonic pockets are presented.

Chakravarthy, Sukumar R.↗

NASA NTRS1985

Numerical solution of the unsteady Navier-Stokes equation

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

Osher, Stanley J.↗

NASA NTRS1987

Exposure time considerations in high temperature low cycle fatigue

The Conventional Strainrange Partitioning (CSRP) method for High-Temperature, Low-Cycle Fatigue (HTLCF) life prediction has its origins in the modeling of first-order, creep-fatigue waveform effects while treating as second-order effects, the influence of metallurgical or environmental time dependencies. Procedures are proposed to include the latter explicitly in the inelastic strainrange--life relations. For brevity, only the CP life relation will be presented in detail. The exposure-time effect within the CP inelastic strainrange (tensile creep reversed by compressive plasticity) was determined by tensile stresshold-time experiments for 316 SS at 816 C. Reductions in CP cyclic life of a factor of about two were observed with an increase in exposure time or a corresponding decrease in creep rate by a factor of about 100. The CP life relation has been modified to be expressed in terms of either Steady State Creep Rate (SSCR) or Exposure Time (ET). The applicability and accuracy of the time-dependent CP life relations is demonstrated by conducting verification experiments involving complex hysteresis loops. Metallographic examination revealed time-dependent degradation attributable to oxide formation and precipitation of carbides along grain boundaries.

Kalluri, S.↗

NASA NTRS1987

Study of instability and active control of boundary layers using Navier-Stokes equations

Spatial stability of a three-dimensional growing boundary layer is studied using Navier-Stokes equations. The finite difference method used is an explicit, predictor-corrector type which is fourth-order accurate in space and second-order in time. Active control of two-dimensional disturbances is simulated by localized surface heating. The computations show features of the flow relevant to early stages of transition such as lifting of vorticity away from the wall and localized regions of intense shear. Active control with single heating strip is only marginally effective, but may be improved by multiple strips.

Parikh, Paresh↗

NASA NTRS1987

High resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations

Based on flux-difference splitting, implicit high resolution schemes are constructed for efficient computations of steady-state solutions to the three-dimensional, incompressible Navier-Stokes equations in curvilinear coordinates. These schemes use first-order accurate Euler backward-time differencing and second-order central differencing for the viscous shear fluxes. Up to third-order accurate upwind differencing is achieved through a reconstruction of the solution from its cell averages. The reconstruction is accomplished by linear interpolation, where the node stencils are selected such that in regions of smooth solution the flow is highly resolved while spurious oscillations in regions of rapid changes in gradient are still suppressed. Fairly rapid convergence to steady-state solutions is attained with a completely vectorizable hybrid time-marching method. Flows around a sharp-edged delta wing are computed with the maximum accuracy of the upwind-differencing restricted to first-, second-, and third-order, to illustrate the effect of accuracy on the global and on the local vortical flow fields. The results are validated with experimental data.

Hartwich, PETER-M.↗

NASA NTRS1988

Third-order multi-dimensional Euler/Navier-Stokes solver

A nonoscillatory multidimensional advection code is presented which is applicable to the study of very high Reynolds number steady flow, and results are obtained for the oblique advection of a step profile. The scheme is based on an exponential upwinding or linear extrapolation refinement of a previous QUICK scheme, allowing sharp monotonic resolution of the step while retaining third-order accuracy. The resulting simple flux-limiting criteria have been applied to a higher order method and a formally third-order supercompressive algorithm. Results are presented for first-order, second-order, and third-order upwinding.

Leonard, B. P.↗

NASA NTRS1988

Universal limiter for transient interpolation modeling of the advective transport equations: The ULTIMATE conservative difference scheme

A fresh approach is taken to the embarrassingly difficult problem of adequately modeling simple pure advection. An explicit conservative control-volume formation makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ULTIMATE conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three critical test profiles: an isolated sine-squared wave, a discontinuous step, and a semi-ellipse. The goal, of course, is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservation schemes of any order of accuracy. Second-order schemes are unsatisfactory, showing steepening and clipping typical of currently popular so-called high resolution shock-capturing of TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. Little is to be gained above ULTIMATE fifth-order upwinding which gives results close to the ultimate for which one might hope.

Leonard, B. P.↗

NASA NTRS1989

A defect stream function, law of the wall/wake method for compressible turbulent boundary layers

The application of the defect stream function to the solution of the two-dimensional, compressible boundary layer is examined. A law of the wall/law of the wake formulation for the inner part of the boundary layer is presented which greatly simplifies the computational task near the wall and eliminates the need for an eddy viscosity model in this region. The eddy viscosity model in the outer region is arbitrary. The modified Crocco temperature-velocity relationship is used as a simplification of the differential energy equation. Formulations for both equilibrium and nonequilibrium boundary layers are presented including a constrained zero-order form which significantly reduces the computational workload while retaining the significant physics of the flow. A formulation for primitive variables is also presented. Results are given for the constrained zero-order and second-order equilibrium formulations and are compared with experimental data. A compressible wake function valid near the wall has been developed from the present results.

Barnwell, Richard W.↗

NASA NTRS1989

Numerical solution of the incompressible Navier-Stokes equations for steady-state and time-dependent problems

An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a higher-order flux-difference splitting technique for the convective terms and a second-order central difference for the viscous terms. The steady-state solution of flow through a square duct with a 90 deg bend is computed and the results are compared with experimental data. Good agreement is observed. A comparison with an analytically known exact solution is then performed to verify the time accuracy of the algorithm. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Rogers, Stuart E.↗

NASA NTRS1988

Exposure time considerations in high temperature low cycle fatigue

The Conventional Strainrange Partitioning (CSRP) method for High-Temperature, Low-Cycle Fatigue (HTLCF) life prediction has its origins in the modeling of first-order, creep-fatigue waveform effects while treating as second-order effects, the influence of metallurgical or environmental time dependencies. Procedures are proposed to include the latter explicitly in the inelastic strainrange-life relations. For brevity, only the CP life relation will be presented in detail. The exposure-time effect within the CP inelastic strainrange (tensile creep reversed by compressive plasticity) was determined by tensile stresshold-time experiments for 316 SS at 816 C. Reductions in CP cyclic life of a factor of about two were observed with an increase in exposure time or a corresponding decrease in creep rate by a factor of about 100. The CP life relation has been modified to be expressed in terms of either Steady State Creep Rate (SSCR) or Exposure Time (ET). The applicability and accuracy of the time-dependent CP life relations is demonstrated by conducting verification experiments involving complex hysteresis loops. Metallographic examination revealed time-dependent degradation attributable to oxide formation and precipitation of carbides along grain boundaries.

Kalluri, S.↗

NASA NTRS1989

First and second order convex approximation strategies in structural optimization

In this paper, various methods based on convex approximation schemes are discussed that have demonstrated strong potential for efficient solution of structural optimization problems. First, the convex linearization method (Conlin) is briefly described, as well as one of its recent generalizations, the method of moving asymptotes (MMA). Both Conlin and MMA can be interpreted as first-order convex approximation methods that attempt to estimate the curvature of the problem functions on the basis of semiempirical rules. Attention is next directed toward methods that use diagonal second derivatives in order to provide a sound basis for building up high-quality explicit approximations of the behavior constraints. In particular, it is shown how second-order information can be effectively used without demanding a prohibitive computational cost. Various first-order and second-order approaches are compared by applying them to simple problems that have a closed form solution.

Fleury, C.↗

NASA NTRS1989

Numerical Study of Three-dimensional Spatial Instability of a Supersonic Flat Plate Boundary Layer

The behavior of spatially growing three-dimensional waves in a supersonic boundary layer was studied numerically by solving the complete Navier-Stokes equations. Satisfactory comparison with linear parallel and non-parallel stability theories, and experiment are obtained when a small amplitude inflow disturbance is used. The three-dimensional unsteady Navier-Stokes equations are solved by a finite difference method which is fourth-order and second-order accurate in the convection and viscous terms respectively, and second-order accurate in time. Spanwise periodicity is assumed. The inflow disturbance is composed of eigenfunctions from linear stability theory. By increasing the amplitude of the inflow disturbance, nonlinear effects in the form of a relaxation type oscillation of the time signal of rho(u) are observed.

Maestrello, Lucio↗

NASA NTRS1990

A fully Sinc-Galerkin method for Euler-Bernoulli beam models

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The Sinc discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given which prove to be especially useful when applying the forward techniques to parameter recovery problems. The discrete system which corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and a robust and efficient algorithm for solving the resulting matrix system is outlined. Numerical results which highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.

Smith, R. C.↗

NASA NTRS1991

Steady and unsteady solutions of the incompressible Navier-Stokes equations

An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Rogers, Stuart E.↗

NASA NTRS1991

Design of algorithms for a dispersive hyperbolic problem

In order to develop numerical schemes for stiff problems, a model of relaxing heat flow is studied. To isolate those errors unavoidably associated with discretization, a method of characteristics is developed, containing three free parameters depending on the stiffness ratio. It is shown that such 'decoupled' schemes do not take into account the interaction between the wave families, and hence result in incorrect wavespeeds. Schemes can differ by up to two orders of magnitude in their rms errors, even while maintaining second-order accuracy. 'Coupled' schemes which account for the interactions are developed to obtain two additional free parameters. Numerical results are given for several decoupled and coupled schemes.

Roe, Philip L.↗

NASA NTRS1951

On the particular integrals of the Prandtl-Busemann iteration equations for the flow of a compressible fluid

The particular integrals of the second-order and third-order Prandtl-Busemann iteration equations for the flow of a compressible fluid are obtained by means of the method in which the complex conjugate variables are utilized as the independent variables of the analysis. The assumption is made that the Prandtl-Glauert solution of the linearized or first-order iteration equation for the two-dimensional flow of a compressible fluid is known. The forms of the particular integrals, derived for subsonic flow, are readily adapted to supersonic flows with only a change in sign of one of the parameters of the problem.

Kaplan, Carl↗

NASA NTRS1990

Computation of three-dimensional compressible boundary layers to fourth-order accuracy on wings and fuselages

A solution method, fourth-order accurate in the body-normal direction and second-order accurate in the stream surface directions, to solve the compressible 3-D boundary layer equations is presented. The transformation used, the discretization details, and the solution procedure are described. Ten validation cases of varying complexity are presented and results of calculation given. The results range from subsonic flow to supersonic flow and involve 2-D or 3-D geometries. Applications to laminar flow past wing and fuselage-type bodies are discussed. An interface procedure is used to solve the surface Euler equations with the inviscid flow pressure field as the input to assure accurate boundary conditions at the boundary layer edge. Complete details of the computer program used and information necessary to run each of the test cases are given in the Appendix.

Iyer, Venkit↗