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All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that was averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. An expansion is derived for the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest nontrivial order. The validity of this expansion is seen to be the same as that of the standard quasilinear expansion.

Jones, F. C.↗
NASA NTRS1978

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that has been averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. In this note we derive an expansion of the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest non-trivial order. The validity of this expansion is seen to be the same as that of the standard quasi-linear expansion.

Jones, F. C.↗
NASA NTRS1986

Closure in turbulence theory using stabilization principle

The approach presented is based upon the concept that the velocity fluctuations driven by the instability of the original flow grow until a new stable state is approached. The application of this stabilization principle is demonstrated by describing the smoothing out of a velocity discontinuity.

Zak, Michail↗
NASA NTRS1998

Chinks in Solar Dynamo Theory: Turbulent Diffusion, Dynamo Waves and Magnetic Helicity

In this first year of our investigation we explored the role of compressibility and stratification in the dissipation of magnetic fields. The predictions of Mean Field Electrodynamics have been questioned because of the strong feedback of small scale magnetic structure on the velocity fields. In 2-D, this nonlinear feedback results in a lengthening of the turbulent decay time. In 3-D alpha-quenching is predicted. Previous studies assumed a homogeneous fluid. This first year we present recent results from 2-D compressible MHD decay simulations in a highly stratified atmosphere that more closely resembles to solar convection zone. We have applied for NCCS T3E time to assist in the performance of our 3-D calculations.

DeLuca, E. E.↗
NASA NTRS1949

Boundary Layer Theory: Turbulent Flows - Part 2

The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.

Schlichting, H.↗
NASA NTRS1955

The Theories of Turbulence

The theory of turbulence reached its full growth at the end of the 19th century as a result of the work by Boussinesq and Reynolds. It then underwent a long period of stagnation which ended under the impulse given to it by the development of wind tunnels caused by the needs of aviation. Numerous researchers, attempted to put Reynolds' elementary statistical theory into a more precise form. During the war, some isolated scientists - von Weizsacker and Heisenberg in Germany, Kolmogoroff in Russia, Onsager in the U.S.A. - started a program of research. By a system of assumptions which make it possible to approach the structure of turbulence in well-defined limiting conditions quantitatively, they obtained a certain number of laws on the correlations and the spectrum. Since the late reports have improved the mathematical language of turbulence, it was deemed advisable to start with a detailed account of the mathematical methods applicable to turbulence, inspired at first by the work of the French school, above all for the basic principles, then the work of the foreigners, above all for the theory of the spectrum.

MECHANICS, STATISTICAL↗
NASA NTRS1969

A cascade theory of turbulence

Cascade theory of turbulence, applying cascade decomposition resulting from Navier-Stokes application to hydrodynamic turbulence to Riemann equation for atmospheric and plasma turbulence

Tchen, C. M.↗
NASA NTRS1975

Wave theory of turbulence in compressible media (acoustic theory of turbulence)

The generation and the transmission of sound in turbulent flows are treated as one of the several aspects of wave propagation in turbulence. Fluid fluctuations are decomposed into orthogonal Fourier components, with five interacting modes of wave propagation: two vorticity modes, one entropy mode, and two acoustic modes. Wave interactions, governed by the inhomogeneous and nonlinear terms of the perturbed Navier-Stokes equations, are modeled by random functions which give the rates of change of wave amplitudes equal to the averaged interaction terms. The statistical framework adopted is a quantum-like formulation in terms of complex distribution functions. The spatial probability distributions are given by the squares of the absolute values of the complex characteristic functions. This formulation results in nonlinear diffusion-type transport equations for the probability densities of the five modes of wave propagation.

Kentzer, C. P.↗
NASA NTRS1993

Time dependent turbulence modeling and analytical theories of turbulence

By simplifying the direct interaction approximation (DIA) for turbulent shear flow, time dependent formulas are derived for the Reynolds stresses which can be included in two equation models. The Green's function is treated phenomenologically, however, following Smith and Yakhot, we insist on the short and long time limits required by DIA. For small strain rates, perturbative evaluation of the correlation function yields a time dependent theory which includes normal stress effects in simple shear flows. From this standpoint, the phenomenological Launder-Reece-Rodi model is obtained by replacing the Green's function by its long time limit. Eddy damping corrections to short time behavior initiate too quickly in this model; in contrast, the present theory exhibits strong suppression of eddy damping at short times. A time dependent theory for large strain rates is proposed in which large scales are governed by rapid distortion theory while small scales are governed by Kolmogorov inertial range dynamics. At short times and large strain rates, the theory closely matches rapid distortion theory, but at long times it relaxes to an eddy damping model.

Rubinstein, R.↗
NASA NTRS1942

Correlation of data on the statistical theory of turbulence

The statistical theory of turbulence affords an excellent medium for representing the kinematic conditions in turbulent flow and also serves as a valuable aid to exact experimental research. But it is still not developed enough for solving dynamic processes. Even in the simplest case of isotropic turbulence the calculation of the correlation curve or of the decrement of turbulence invariably reaches a point where clear-cut assumptions, such as omission of the inertia terms, or, earlier, mixing length assumptions or even merely general dimensional considerations, must be made.

Wieghardt, K↗
NASA NTRS1983

Theory of hydromagnetic turbulence

The present state of MHD turbulence theory as a possible solar wind research tool is surveyed. The theory is statistical, and does not make statements about individual events. The ensembles considered typically have individual realizations which differ qualitatively, unlike equilibrium statistical mechanics. Most of the theory deals with highly symmetric situations; most of these symmetries have yet to be tested in the solar wind. The applicability of MHD itself to solar wind parameters is highly questionable; yet it has no competitors, as a potentially comprehensive dynamical description. The purpose of solar wind research require sharper articulation. If they are to understand radial turbulent plasma flows from spheres, laboratory experiments and numerical solution of equations of motion may be cheap alternative to spacecraft. If "real life" information is demanded, multiple spacecraft with variable separation may be necessary to go further. The principal emphasis in the theory so far has been on spectral behavior for spatial covariances in wave number space. There is no respectable theory of these for highly anisotropic situations. A rather slow development of theory acts as a brake on justifiable measurement, at this point.

Montgomery, D.↗
NASA NTRS1996

Modeling of Inhomogeneous Compressible Turbulence Using a Two-Scale Statistical Theory

Turbulence modeling plays an important role in the study of high-speed flows in engineering and aerodynamic problems; they include flows in supersonic combustion engines and over hypersonic transport aircraft. The enhancement of the kinetic energy dissipation by the dilatational terms is one of the typical compressibility effects. Zeman (1990) and Sarkar et al. (1991) proposed that the dilatation dissipation is proportional to the solenoidal dissipation and is a function of the turbulent Mach number. Sarkar (1992) also modeled the pressure-dilatation correlation using the turbulent Mach number. Zeman (1991) related the correlation to the rate of change of the pressure variance.

Hamba, Fujihiro↗
NASA NTRS1976

Kinetic theory of turbulent compressible flows and comparison with classical theory

A complete set of turbulent correlations is given in terms of expansion coefficients in a double series of Hermite polynomials of a two-particle correlation in phase space. Only two of these coefficients, corresponding to Reynolds stress and turbulent heat flux, are shown to appear in gasdynamic equations of turbulence. This is accomplished by incorporating the phase-space correlation in the collision integral of Boltzmann's equation and by deriving generalized Navier-Stokes and Fourier transport relations. In contrast to the classical formalism, the simple expression remains invariant in form whether the flow is compressible or not. Nevertheless, a bilinear transformation of fluctuating quantities shows that the two formalisms are identical with regard to terms of double correlations. The kinetic theory justifies nonexistence of the higher-order correlations characteristic of compressible-turbulence equations in the classical regime.

Tsuge, S.↗