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NASA NTRS1992

Spurious Solutions Of Nonlinear Differential Equations

Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.

Yee, H. C.↗

NASA NTRS1991

Direct computation of turbulence and noise

Jet exhaust turbulence noise is computed using a time dependent solution of the three dimensional Navier-Stokes equations to supply the source terms for an acoustic computation based on the Phillips convected wave equation. An extrapolation procedure is then used to determine the far field noise spectrum in terms of the near field sound. This will lay the groundwork for studies of more complex flows typical of noise suppression nozzles.

Berman, C.↗

NASA NTRS1991

Turbulence modeling in supersonic combusting flows

To support the National Aerospace Plane project, the RPLUS3D CFD code has been developed at NASA Lewis. The code has the ability to solve three-dimensional flowfields with finite rate combustion of hydrogen and air. The combustion processes of the hydrogen-air system are simulated by an 18-reaction path, 8-species chemical kinetic mechanism. The code uses a Lower-Upper (LU) decomposition numerical algorithm as its basis, making it a very efficient and robust code. Except for the Jacobian matrix for the implicit chemistry source terms, there is no inversion of a matrix even though it uses a fully implicit numerical algorithm. A k-epsilon (two equation) turbulence model is incorporated into the RPLUS3D code.

Chitsomboon, Tawit↗

NASA NTRS1992

A compressible Navier-Stokes solver with two-equation and Reynolds stress turbulence closure models

This report outlines the development of a general purpose aerodynamic solver for compressible turbulent flows. Turbulent closure is achieved using either two equation or Reynolds stress transportation equations. The applicable equation set consists of Favre-averaged conservation equations for the mass, momentum and total energy, and transport equations for the turbulent stresses and turbulent dissipation rate. In order to develop a scheme with good shock capturing capabilities, good accuracy and general geometric capabilities, a multi-block cell centered finite volume approach is used. Viscous fluxes are discretized using a finite volume representation of a central difference operator and the source terms are treated as an integral over the control volume. The methodology is validated by testing the algorithm on both two and three dimensional flows. Both the two equation and Reynolds stress models are used on a two dimensional 10 degree compression ramp at Mach 3, and the two equation model is used on the three dimensional flow over a cone at angle of attack at Mach 3.5. With the development of this algorithm, it is now possible to compute complex, compressible high speed flow fields using both two equation and Reynolds stress turbulent closure models, with the capability of eventually evaluating their predictive performance.

Navier-Stoke solver↗

NASA NTRS1992

Upwind schemes and bifurcating solutions in real gas computations

The area of high speed flow is seeing a renewed interest due to advanced propulsion concepts such as the National Aerospace Plane (NASP), Space Shuttle, and future civil transport concepts. Upwind schemes to solve such flows have become increasingly popular in the last decade due to their excellent shock capturing properties. In the first part of this paper the authors present the extension of the Osher scheme to equilibrium and non-equilibrium gases. For simplicity, the source terms are treated explicitly. Computations based on the above scheme are presented to demonstrate the feasibility, accuracy and efficiency of the proposed scheme. One of the test problems is a Chapman-Jouguet detonation problem for which numerical solutions have been known to bifurcate into spurious weak detonation solutions on coarse grids. Results indicate that the numerical solution obtained depends both on the upwinding scheme used and the limiter employed to obtain second order accuracy. For example, the Osher scheme gives the correct CJ solution when the super-bee limiter is used, but gives the spurious solution when the Van Leer limiter is used. With the Roe scheme the spurious solution is obtained for all limiters.

Suresh, Ambady↗

NASA NTRS1992

On recontamination and directional-bias problems in Monte Carlo simulation of PDF turbulence models

Turbulent combustion can not be simulated adequately by conventional moment closure turbulent models. The probability density function (PDF) method offers an attractive alternative: in a PDF model, the chemical source terms are closed and do not require additional models. Because the number of computational operations grows only linearly in the Monte Carlo scheme, it is chosen over finite differencing schemes. A grid dependent Monte Carlo scheme following J.Y. Chen and W. Kollmann has been studied in the present work. It was found that in order to conserve the mass fractions absolutely, one needs to add further restrictions to the scheme, namely alpha(sub j) + gamma(sub j) = alpha(sub j - 1) + gamma(sub j + 1). A new algorithm was devised that satisfied this restriction in the case of pure diffusion or uniform flow problems. Using examples, it is shown that absolute conservation can be achieved. Although for non-uniform flows absolute conservation seems impossible, the present scheme has reduced the error considerably.

Hsu, Andrew T.↗

NASA NTRS1992

Implementation of a kappa-epsilon turbulence model to RPLUS3D code

The RPLUS3D code has been developed at the NASA Lewis Research Center to support the National Aerospace Plane (NASP) project. The code has the ability to solve three dimensional flowfields with finite rate combustion of hydrogen and air. The combustion process of the hydrogen-air system are simulated by an 18 reaction path, 8 species chemical kinetic mechanism. The code uses a Lower-Upper (LU) decomposition numerical algorithm as its basis, making it a very efficient and robust code. Except for the Jacobian matrix for the implicit chemistry source terms, there is no inversion of a matrix even though a fully implicit numerical algorithm is used. A k-epsilon turbulence model has recently been incorporated into the code. Initial validations have been conducted for a flow over a flat plate. Results of the validation studies are shown. Some difficulties in implementing the k-epsilon equations to the code are also discussed.

Chitsomboon, Tawit↗

NASA NTRS1991

Transient cooling of a square region of radiating medium

A time-accurate numerical solution was carried out for transient radiative cooling of a gray emitting and absorbing medium in a square two-dimensional region. The integro-differential energy equation for transient temperature distributions was solved in two stages. At each time increment, the local radiative source term was obtained by numerical integration of the temperature field using two-dimensional Gaussian integration over rectangular subregions. Then the differential portion of the equation was integrated forward in time by use of the local first and second time derivatives. The results were compared with available limiting case, and excellent agreement was obtained. Transient results are given for a wide range of optical thicknesses of the region. Optimum transient cooling is obtained when the optical side length is about 4.

Siegel, Robert↗

NASA NTRS1992

Finite difference solution for transient cooling of a radiating-conducting semitransparent layer

Transient solutions were obtained for cooling a semitransparent material by radiation and conduction. The layer is in a vacuum environment so the only means for heat dissipation is by radiation from within the medium leaving through the boundaries. Heat conduction serves only to partially equalize temperatures across the layer. As the optical thickness is increased, steep temperature gradients exist near the boundaries when conduction is relatively small. A solution procedure is required that will provide accurate temperature distributions adjacent to the boundaries, or radiative heat losses will be in error. The approach utilized numerical Gaussian integration to obtain the local radiative source term, and a finite difference procedure with variable space and time increments to solve the transient energy equation.

Siegel, Robert↗

NASA NTRS1992

An assumed pdf approach for the calculation of supersonic mixing layers

In an effort to predict the effect that turbulent mixing has on the extent of combustion, a one-equation turbulence model is added to an existing Navier-Stokes solver with finite-rate chemistry. To average the chemical-source terms appearing in the species-continuity equations, an assumed pdf approach is also used. This code was used to analyze the mixing and combustion caused by the mixing layer formed by supersonic coaxial H2-air streams. The chemistry model employed allows for the formation of H2O2 and HO2. Comparisons are made with recent measurements using laser Raman diagnostics. Comparisons include temperature and its rms, and concentrations of H2, O2, N2, H2O, and OH. In general, good agreement with experiment was noted.

Baurle, R. A.↗

NASA NTRS1992

Thermochemical nonequilibrium and radiative interactions in supersonic hydrogen-air combustion

The two-dimensional, elliptic Navier-Stokes equations are used to investigate supersonic flows with nonequilibrium chemistry and thermodynamics, coupled with radiation, for hydrogen-air systems. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The explicit, unsplit MacCormack finite-difference scheme is used to advance the governing equations in time, until convergence is achieved. The specific problem considered is the premixed, expanding flow in a supersonic nozzle. The reacting flow consists of seven species, one of which is the inert N2 molecule. The thermal state of the gas is modeled with one translational-rotational temperature and five vibrational temperatures. The harmonic oscillator model is used in the formulation for vibrational relaxation. The tangent slab approximation is used in the radiative flux formulation. A pseudo-gray model is used to represent the absorption-emission characteristics of the participating species. Results obtained for specific conditions indicate the presence of nonequilibrium in the expansion region. This reduces the radiative interactions and can have a significant influence on the flowfield.

Chandrasekhar, R.↗

NASA NTRS1992

Direct computation of the sound from a compressible co-rotating vortex pair

The far-field sound from corotating vortices is computed by direct computation of the unsteady, compressible Navier-Stokes equations on a computational mesh that extends to two acoustic wavelengths in all directions. The vortices undergo a period of corotation followed by a sudden merger. A 2D version of Moehring's equation is developed and used in conjunction with source terms computed in the simulation to predict the far-field sound. The prediction agrees with the simulation to within 3 percent. Results of far-field pressure fluctuations for an acoustically noncompact case are also presented for which the prediction is 66 percent too high. Results also indicate that the monopole contribution of 'viscous sound' is negligible for this flow.

Mitchell, Brian E.↗

NASA NTRS1992

Turbulence modeling for impinging jet flows

The objective of the present work is to improve the accuracy of the k-epsilon turbulence model for flows involving one or more jets impinging on a plate in a crossflow which generate a horseshoe vortex. The k-epsilon model is modified by adding source terms to the epsilon equation, which enables it to more accurately predict the shear stress in flows subject to streamline curvature and vortex stretching (or lateral divergence). Calculations with the modified model predict the ground vortex core to be about 15 percent upstream of its experimental location. This is a significant improvement over the standard model which yields higher errors for calculation of the vortex-core location.

Childs, Robert E.↗

NASA NTRS1992

Numerical modeling of fluid and electromagnetic phenomena in an arcjet

An explicit numerical technique is used to solve the axisymmetric reduced electromagnetic field equation. The effect of an electrical arc on a viscous, axisymmetric flow is approximated using an implicit thin layer Navier-Stokes solver with additional electromagnetic source terms in conjunction with the explicit finite difference code.

Flowe, Anita C.↗

NASA NTRS1992

Liquid rocket spray combustion stability analysis

A computational approach to the analysis of spray combustion stability in liquid rocket combustors is proposed which is based on the unsteady quasi-two-dimensional Euler equations with interphase source terms derived from a Lagrangian treatment of the combusting spray. Based on a preliminary evaluation, the computational methodology presented here is a promising research tool and a potential design/development aid for investigating the stability characteristics of liquid rocket engines. The method is characterized by low numerical noise; the Lagrangian treatment of the spray offers improved flexibility for the direct modeling of spray combustion.

Litchford, Ron J.↗

NASA NTRS1992

A k-omega-multivariate beta PDF for supersonic combustion

In an attempt to study the interaction between combustion and turbulence in supersonic flows, an assumed PDF has been employed. This makes it possible to calculate the time average of the chemical source terms that appear in the species conservation equations. In order to determine the averages indicated in an equation, two transport equations, one for the temperature (enthalpy) variance and one for Q, are required. Model equations are formulated for such quantities. The turbulent time scale controls the evolution. An algebraic model similar to that used by Eklund et al was used in an attempt to predict the recent measurements of Cheng et al. Predictions were satisfactory before ignition but were less satisfactory after ignition. One of the reasons for this behavior is the inadequacy of the algebraic turbulence model employed. Because of this, the objective of this work is to develop a k-omega model to remedy the situation.

Alexopoulos, G. A.↗

NASA NTRS1993

A nonequilibrium model for a moderate pressure hydrogen microwave discharge plasma

This document describes a simple nonequilibrium energy exchange and chemical reaction model to be used in a computational fluid dynamics calculation for a hydrogen plasma excited by microwaves. The model takes into account the exchange between the electrons and excited states of molecular and atomic hydrogen. Specifically, electron-translation, electron-vibration, translation-vibration, ionization, and dissociation are included. The model assumes three temperatures, translational/rotational, vibrational, and electron, each describing a Boltzmann distribution for its respective energy mode. The energy from the microwave source is coupled to the energy equation via a source term that depends on an effective electric field which must be calculated outside the present model. This electric field must be found by coupling the results of the fluid dynamics and kinetics solution with a solution to Maxwell's equations that includes the effects of the plasma permittivity. The solution to Maxwell's equations is not within the scope of this present paper.

Scott, Carl D.↗

NASA NTRS1991

Potential health risks from postulated accidents involving the Pu-238 RTG on the Ulysses solar exploration mission

Potential radiation impacts from launch of the Ulysses solar exploration experiment were evaluated using eight postulated accident scenarios. Lifetime individual dose estimates rarely exceeded 1 mrem. Most of the potential health effects would come from inhalation exposures immediately after an accident, rather than from ingestion of contaminated food or water, or from inhalation of resuspended plutonium from contaminated ground. For local Florida accidents (that is, during the first minute after launch), an average source term accident was estimated to cause a total added cancer risk of up to 0.2 deaths. For accidents at later time after launch, a worldwide cancer risk of up to three cases was calculated (with a four in a million probability). Upper bound estimates were calculated to be about 10 times higher.

Goldman, Marvin↗