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NASA NTRS1985

Radial diffusion in magnetodiscs

The orbits of charged particles in magnetodiscs are considered. The bounce motion is assumed adiabatic except for transits of a small equatorial region of weak magnetic field strength and high field curvature. Previous theory and modeling have shown that particles scatter randomly in pitch angle with each passage through the equator. A peaked distribution thus diffuses in pitch angle on the time scale of many bounces. It is argued in this paper that spatial diffusion is a further consequence when the magnetodisc has a longitudinal asymmetry. A general expression for DLL, the diffusion of equatorial crossing radii, is derived. DLL is evaluated explicitly for ions in Jupiter's 20-35 radii magnetodisc, assumed to be represented by Connerney et al.'s (1982) Voyager model plus a small image dipole asymmetry. Rates are energy, species, and space dependent but can average as much as a few tenths of a planetary radius per bounce period.

Birmingham, T. J.↗
NASA NTRS1993

A transport equation for the scalar dissipation in reacting flows with variable density: First results

Although the different regimes of premixed combustion are not well defined, most of the recent developments in turbulent combustion modeling are led in the so-called flamelet regime. The goal of these models is to give a realistic expression to the mean reaction rate (w). Several methods can be used to estimate (w). Bray and coworkers (Libby & Bray 1980, Bray 1985, Bray & Libby 1986) express the instantaneous reaction rate by means of a flamelet library and a frequency which describes the local interaction between the laminar flamelets and the turbulent flowfield. In another way, the mean reaction rate can be directly connected to the flame surface density (Sigma). This quantity can be given by the transport equation of the coherent flame model initially proposed by Marble & Broadwell 1977 and developed elsewhere. The mean reaction rate, (w), can also be estimated thanks to the evolution of an arbitrary scalar field G(x, t) = G(sub O) which represents the flame sheet. G(x, t) is obtained from the G-equation proposed by Williams 1985, Kerstein et al. 1988 and Peters 1993. Another possibility proposed in a recent study by Mantel & Borghi 1991, where a transport equation for the mean dissipation rate (epsilon(sub c)) of the progress variable c is used to determine (w). In their model, Mantel & Borghi 1991 considered a medium with constant density and constant diffusivity in the determination of the transport equation for (epsilon(sub c)). A comparison of different flamelet models made by Duclos et al. 1993 shows the realistic behavior of this model even in the case of constant density. Our objective in this present report is to present preliminary results on the study of this equation in the case of variable density and variable diffusivity. Assumptions of constant pressure and a Lewis number equal to unity allow us to significantly simplify the equation. A systematic order of magnitude analysis based on adequate scale relations is performed on each term of the equation. As in the case of constant density and constant diffusivity, the effects of stretching of the scalar field by the turbulent strain field, of local curvature, and of chemical reactions are predominant. In this preliminary work, we suggest closure models for certain terms, which will be validated after comparisons with DNS data.

Mantel, T.↗
NASA NTRS1978

Stochastic partial differential equations in turbulence related problems

The theory of stochastic partial differential equations (PDEs) and problems relating to turbulence are discussed by employing the theories of Brownian motion and diffusion in infinite dimensions, functional differential equations, and functional integration. Relevant results in probablistic analysis, especially Gaussian measures in function spaces and the theory of stochastic PDEs of Ito type, are taken into account. Linear stochastic PDEs are analyzed through linearized Navier-Stokes equations with a random forcing. Stochastic equations for waves in random media as well as model equations in turbulent transport theory are considered. Markovian models in fully developed turbulence are discussed from a stochastic equation viewpoint.

Chow, P.-L.↗
NASA NTRS2010

Ocean Turbulence, III: New GISS Vertical Mixing Scheme

We have found a new way to express the solutions of the RSM (Reynolds Stress Model) equations that allows us to present the turbulent diffusivities for heat, salt and momentum in a way that is considerably simpler and thus easier to implement than in previous work. The RSM provides the dimensionless mixing efficiencies Gamma-alpha (alpha stands for heat, salt and momentum). However, to compute the diffusivities, one needs additional information, specifically, the dissipation Epsilon. Since a dynamic equation for the latter that includes the physical processes relevant to the ocean is still not available, one must resort to different sources of information outside the RSM to obtain a complete Mixing Scheme usable in OGCMs. As for the RSM results, we show that the Gamma-alpha s are functions of both Ri and Rq (Richardson number and density ratio representing double diffusion, DD); the Gamma-alpha are different for heat, salt and momentum; in the case of heat, the traditional value Gamma-h = 0.2 is valid only in the presence of strong shear (when DD is inoperative) while when shear subsides, NATRE data show that Gamma-h can be three times as large, a result that we reproduce. The salt Gamma-s is given in terms of Gamma-h. The momentum Gamma-m has thus far been guessed with different prescriptions while the RSM provides a well defined expression for Gamma-m(Ri,R-rho). Having tested Gamma-h, we then test the momentum Gamma-m by showing that the turbulent Prandtl number Gamma-m/Gamma-h vs. Ri reproduces the available data quite well. As for the dissipation epsilon, we use different representations, one for the mixed layer (ML), one for the thermocline and one for the ocean;s bottom. For the ML, we adopt a procedure analogous to the one successfully used in PB (planetary boundary layer) studies; for the thermocline, we employ an expression for the variable epsilon/N(exp 2) from studies of the internal gravity waves spectra which includes a latitude dependence; for the ocean bottom, we adopt the enhanced bottom diffusivity expression used by previous authors but with a state of the art internal tidal energy formulation and replace the fixed Gamma-alpha = 0.2 with the RSM result that brings into the problem the Ri, R-rho dependence of the Gamma-alpha; the unresolved bottom drag, which has thus far been either ignored or modeled with heuristic relations, is modeled using a formalism we previously developed and tested in PBL studies. We carried out several tests without an OGCM. Prandtl and flux Richardson numbers vs. Ri. The RSM model reproduces both types of data satisfactorily. DD and Mixing efficiency Gamma-h(Ri,Rq). The RSM model reproduces well the NATRE data. Bimodal epsilon-distribution. NATRE data show that epsilon (Ri < 1) approximately equals 10epsilon(Ri > 1), which our model reproduces. Heat to salt flux ratio. In the Ri much greater than 1 regime, the RSM predictions reproduce the data satisfactorily. NATRE mass diffusivity. The z-profile of the mass diffusivity reproduces well the measurements at NATRE. The local form of the mixing scheme is algebraic with one cubic equation to solve.

Canuto, V. M.↗
NASA NTRS1983

Interfacial layers in high-temperature-oxidized NiCrAl

The utility of Auger electron spectroscopy combined with ball cratering for depth analysis of oxide and diffusion layers produced in a Ni-14Cr-24Al alloy by oxidation in air at 1180 C for 25 hr is demonstrated. During postoxidation cooling, the oxide layers formed by this alloy spalled profusely. The remaining very thin oxide was primarily Cr2O3 with a trace of Ni. The underlying metal substrate exhibited gamma/gamma-prime and beta phases with a metallic interfacial layer which was similar to the bulk gamma/gamma-prime phase but slightly enriched in Cr and Al. These data are compared to electron microprobe results from a nominally identical alloy. The diffusion layer thickness is modelled with a simple mass balance equation and compared to recent results on the diffusion process in NiCrAl alloys.

Larson, L. A.↗
NASA NTRS1991

Investigations of turbulent scalar fields using probability density function approach

Scalar fields undergoing random advection have attracted much attention from researchers in both the theoretical and practical sectors. Research interest spans from the study of the small scale structures of turbulent scalar fields to the modeling and simulations of turbulent reacting flows. The probability density function (PDF) method is an effective tool in the study of turbulent scalar fields, especially for those which involve chemical reactions. It has been argued that a one-point, joint PDF approach is the one to choose from among many simulation and closure methods for turbulent combustion and chemically reacting flows based on its practical feasibility in the foreseeable future for multiple reactants. Instead of the multi-point PDF, the joint PDF of a scalar and its gradient which represents the roles of both scalar and scalar diffusion is introduced. A proper closure model for the molecular diffusion term in the PDF equation is investigated. Another direction in this research is to study the mapping closure method that has been recently proposed to deal with the PDF's in turbulent fields. This method seems to have captured the physics correctly when applied to diffusion problems. However, if the turbulent stretching is included, the amplitude mapping has to be supplemented by either adjusting the parameters representing turbulent stretching at each time step or by introducing the coordinate mapping. This technique is still under development and seems to be quite promising. The final objective of this project is to understand some fundamental properties of the turbulent scalar fields and to develop practical numerical schemes that are capable of handling turbulent reacting flows.

Gao, Feng↗
NASA NTRS1980

Nonequilibrium ionization due to thermal diffusion and mass flows

Recent calculations of diffusion coefficients are used in the continuity equation to compute ion populations of carbon in the solar transition region. Thermal diffusion causes strong departures from ionization equilibrium in the region where the temperature gradient is steepest. Mass-conserving flows are also included in our calculations. These dominate over thermal diffusion depending on the magnitude of the flows and also lead to departures from ionization equilibrium. These results have important implications for the interpretation of EUV line emission.

Roussel-Dupre, R.↗
NASA NTRS1984

Problems in simulating the stratocumulus-topped boundary layer with a third-order closure model

The Andre et al. (1976, 1978) third-order closure model, in which the time rate of change terms, the relaxation and rapid effects for pressure-related terms, and the clipping approximation are used along with the quasi-normal closure, is invoked in the study of turbulence in a cloudy layer that is radiatively cooled from above. A spurious oscillation whose greatest amplitude lies near the inversion is shown by analysis to arise from the mean gradient and buoyancy terms of the triple-moment equations. An attempt is made to damp the oscillation through the introduction of diffusion terms into the triple-moment equations. The results obtained are noted to be sensitive to the ad hoc eddy coefficient applied in the third-moment equations.

Moeng, C.-H.↗
NASA NTRS1979

The prediction of two- and three-dimensional asymmetrical turbulent wakes - A comparision of the performance of three turbulence models for the effects of streamline curvature and rotation

Two- and three-dimensional turbulent wakes which develop under the influence of streamline curvature and rotation were calculated with three turbulence closure models. The first model comprised of transport equations for the turbulent kinetic energy and the rate of energy dissipation. The second model comprised of equations for the rate of turbulent kinetic energy dissipation and Reynolds stresses, but the effects of the convection and diffusion in the Reynolds stress transport equation were handled collectively. The third model utilizes equations of turbulent kinetic energy dissipation and Reynolds stresses in nearly exact form. All of the three models were modified for the effects of streamline curvature. The second and third models include the effects of rotation through the rotation-originated redistribution term in the transport equation of Reynolds stresses. The numerical results for the wakes of airfoils, cascades and turbomachinery rotor blades demonstrate that the second and third models provide accurate predictions, but computer time and storage can be considerably saved with the second model.

Hah, C.↗
NASA NTRS1992

A finite element method for the thermochemical decomposition of polymeric materials. I - Theory

The governing differential equations are developed to model the thermomechanical behavior of chemically decomposing, polymeric materials. These equations account for thermal and gaseous diffusion through a poroelastic, transversely isotropic solid. The Bubnov-Galerkin finite element method is applied to the governing equations to cast the coupled set into a single matrix equation. A method for solving these equations simultaneously at each time step is discussed.

Sullivan, R. M.↗
NASA NTRS1980

Velocity and concentration distribution in a Stefan diffusion tube

Numerical solutions for the diffusion- and gravity-driven flow in a cylindrical Stefan tube were obtained from the coupled diffusion and Navier-Stokes equations of Peclet numbers 0.3, 1 and 5. Distributions of binary component concentrations and velocities were calculated. The mass average velocity is parabolic in nature, except at high Peclet numbers. The solvent is not stagnant but recirculates, even in the absence of gravity. Radial concentration gradients develop which act convectively destabilizing. Consequences for the deduction of diffusion coefficients from Stefan tube experiments are discussed.

Markham, B. L.↗
NASA NTRS1982

Solar reflection from a regular array of horizontally finite clouds

The reflected flux from a regular array of 2- and 3-D clouds has been computed to estimate the effect of fractional cloud cover on albedos and the solar flux available to heat the earth's surface. The broken clouds are represented by a regular array of identical cuboids for the 3-D problem and equally spaced, infinitely long, bars for the 2-D problem. A diffusion approximation to the radiative transfer equation is used to compute the fluxes leaving each face of the cloud. Interaction between clouds is simulated by assuming diffuse exitance from the cloud faces and applying angle factors to obtain modified boundary conditions on each cloud face.

Weinman, J. A.↗
NASA NTRS1986

Surface mass transport of alumina

Measurements were made of the kinetics of thermal grooving at the intersection of rhombohedral twin boundaries with the (1 0 -1 0) plane in alumina, at temperatures from 1773 to 2273 K. An analysis of the values obtained on the basis of the model of Mullins (1957) indicated that surface diffusion was the dominant mass transport mechanism. Comparisons are made with the results of other investigations of alumina, and an equation is derived for the surface diffusion value.

Gaddipati, A. R.↗
NASA NTRS1981

Numerical simulation of reentry flow around the Shuttle Orbiter including real gas effects

The flight environment of a reentry vehicle is predicted from the numerical solution of fluid dynamics equations for the region between the blunt nose and the swept wings. The inviscid portion of the shock layer is modeled by the Euler equations, but the laminar viscous flow adjacent to the wall is modeled by the approximate parabolic Navier-Stokes equations. The approximations made to the axial gradients of pressure and diffusive fluxes enable the coupled inviscid and viscous equations to be solved efficiently along the body axis. The equilibrium air aftbody code contains significant improvements over its predecessor which only considers a perfect gas model and noncircular configurations. The inclusion of numerical damping either explicitly or implicitly has extended its capabilities for predicting flow field around a winged configuration at higher Machs and greater angles of attack. The results are obtained on the cylindrical coordinates and satisfactory for the Shuttle Orbiter at a free-stream Mach number of 22 and an angle of attack of 40 deg. Also discussed are the inviscid formulation and its application for wind-tunnel conditions.

Li, C. P.↗
NASA NTRS1989

An improvement of convection fidelity in Euler calculations

A new solution procedure is presented to solve the Euler equations for steady, compressible, rotational, inviscid flows. The approach is aimed at achieving real inviscid solutions in Euler calculations by eliminating numerical diffusion. The variables in the Euler equations are divided into elliptic and convective quantities, using the Clebsch velocity decomposition. The convective quantities are then transported without numerical contamination using an efficient convection operator, while the elliptic quantities are integrated with a relaxation procedure. This approach provides a generalization of the full potential formulation to rotation Euler physics by allowing variations of convective quantities. Results are demonstrated for several transonic flows in two dimensions.

Lee, K. D.↗
NASA NTRS1984

Simulations of the stratocumulus-topped boundary layer with a third-order closure model

A third order closure model is proposed by Andre et al. (1982), in which the time rate of change terms, the relaxation and rapid effects for the pressure related terms, and the clipping approximation are included along with the quasi-normal closure, to study turbulence in a cloudy layer which is cooled radiatively from above. A spurious oscillation which is strongest near the inversion occurs. An analysis of the problem shows that the oscillation arises from the mean gradient and buoyancy terms of the triple moment equations; these terms are largest near the cloud top. The oscillation is physical, rather than computational. In nature the oscillation is effectively damped, by a mechanism which apparently is not included in our model. In the stably stratified layer just above the mixed layer top, turbulence can excite gravity waves, whose energy is radiated away. Because the closure assumption for the pressure terms does not take into account the transport of wave energy, the model generates spurious oscillations. Damping of the oscillations is possible by introducing diffusion terms into the triple moment equations. With a large enough choice for the diffusion coefficient, the oscillation is effectively eliminated. The results are quite sensitive to the ad hoc eddy coefficient.

Moeng, C. H.↗
NASA NTRS1974

The diffusive idealization of charged-particle transport in random magnetic fields

The uniqueness and accuracy of the equations which describe the transport of charged particles diffusing in a random magnetic field parallel to a relatively large guiding field is examined. With regard to uniqueness, it is found that the same coefficient of diffusion is obtained by three methods that have apparently led to discrepancies in previous work. With regard to accuracy, it is found that two corrections must be added to Fick's law in which the diffusive flux is proportional to the gradient of the density. Explicit expressions are given for a characteristic time and a characteristic length which describe the corrections.

Earl, J. A.↗