Passive scalar spectrum in isotropic turbulence: Prediction by the Lagrangian direct-interaction approximation

1999 ◽  
Vol 11 (7) ◽  
pp. 1936-1952 ◽  
Author(s):  
Susumu Goto ◽  
Shigeo Kida
Keyword(s):  
Isotropic Turbulence ◽  
Direct Interaction ◽  
Passive Scalar ◽  
Direct Interaction Approximation ◽  
Interaction Approximation

Eulerian and Lagrangian renormalization in turbulence theory

1977 ◽  
Vol 83 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Direct Interaction ◽  
Exact Result ◽  
Passive Scalar ◽  
Turbulence Theory ◽  
Perturbation Expansions ◽  
Gaussian Form ◽  
Lagrangian Velocity ◽  
Direct Interaction Approximation ◽  
Gaussian Statistics ◽  
Interaction Approximation

Systematic renormalized perturbation expansions for turbulence and turbulent convection are constructed which are invariant at each order under random Galilean transformations. Two types of expansion are developed whose lowest truncations give, respectively, the Lagrangian-history direct-interaction approximation and the abridged Lagrangian-history direct-interaction approximation. These approximations previously were derived as heuristic modifications of the Eulerian direct-interaction approximation (Kraichnan 1965). The techniques used involve reversion of primitive perturbation expansions for the generalized velocity field u(x, t/s), defined as the velocity measured at time s in the fluid element which passes through x at time t. The new expansions are illustrated by application to a random linear oscillator, to passive-scalar convection by a random velocity and to the Lagrangian velocity covariance. The lowest term of the expansion for the passive scalar gives Taylor's (1921) exact result for dispersion of fluid elements, and higher terms describe the deviations of the particle-displacement distribution from Gaussian form. In all the applications the assumed underlying statistics are more general than Gaussian statistics, which appear as a special case.

Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory

1989 ◽  
Vol 208 ◽  
pp. 91-114 ◽  
Author(s):  
W. D. Mccomb ◽  
V. Shanmugasundaram ◽  
P. Hutchinson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Direct Interaction ◽  
Reynolds Numbers ◽  
Time Correlation ◽  
General Belief ◽  
Direct Interaction Approximation ◽  
The Difference ◽  
Decaying Turbulence ◽  
Interaction Approximation

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

Self-consistent effective-medium parameters for oceanic internal waves

1984 ◽  
Vol 146 ◽  
pp. 253-270 ◽  
Author(s):  
R. J. Dewitt ◽  
Jon Wright
Keyword(s):  
Internal Waves ◽  
Direct Interaction ◽  
Decay Rates ◽  
The Self ◽  
Time Behaviour ◽  
Equilibrium Time ◽  
Self Consistent ◽  
Direct Interaction Approximation ◽  
Interaction Approximation ◽  
Extra Parameter

In this paper we apply a formalism introduced in a previous paper to write down a self-consistent set of equations for the functions that describe the near-equilibrium time behaviour of random oceanic internal waves. These equations are based on the direct-interaction approximation. The self-consistent equations are solved numerically (using the Garrett-Munk spectrum as input) and the results are compared to parameters obtained in the weak-interaction approximation (WIA). The formalism points out that an extra parameter that is implicitly vanishingly small in the WIA has a significant effect on decay rates when computed self-consistently. We end by mentioning possible future self-consistent calculations that would improve upon our own.

KOLMOGOROV CONSTANT IN LARGE SPACE DIMENSIONS

2002 ◽  
Vol 16 (32) ◽  
pp. 4839-4845 ◽  
Author(s):  
MALAY K. NANDY
Keyword(s):  
Eddy Viscosity ◽  
Energy Equation ◽  
Space Dimension ◽  
Direct Interaction ◽  
Large Space ◽  
Distant Interaction ◽  
Direct Interaction Approximation ◽  
Dimension Expansion ◽  
Kolmogorov Constant ◽  
Interaction Approximation

A large d (space dimension) expansion together with the ∊-expansion is implemented to calculate the Kolmogorov constant from the energy equation of Kraichnan's direct-interaction approximation using the Heisenberg's eddy-viscosity approximation and Kraichnan's distant-interaction algorithm. The Kolmogorov constant C is found to be C = C0 d1/3 in the leading order of a 1/d expansion. This is consistent with Fournier, Frisch, and Rose. The constant C0 evaluated in the above scheme, is found to be C0 = (16/27)1/3.

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