Gaussian Random Velocity Field Research Articles

A quenched local limit theorem for stochastic flows

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particle after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field U. If the velocity field is additionally assumed to be incompressible, then U≡1 almost surely and we obtain a local central limit theorem.

Suppression of thermal noise in a non-Markovian random velocity field

We study the diffusion of Brownian particles in a Gaussian random velocity field with short memory. By extending the derivation of an effective Fokker–Planck equation for the Lanvegin equation with weakly colored noise to a random velocity-field problem, we find that the effect of thermal noise on particles is suppressed by the existence of memory. We also find that the renormalization effect for the relative diffusion of two particles is stronger than that for single-particle diffusion. The results are compared with those of molecular dynamics simulations.

Influence of velocity spatiotemporal correlations on the anomalous scaling exponents of passive scalars

In this paper, we consider spatial-temporal correlation functions of the turbulent velocities. With numerical simulations on the Gledzer–Ohkitani–Yamada (GOY) shell model, we show that the correlation function decays exponentially. The advecting velocity field is regarded as a colored noise field, which is spatially and temporally correlative. For comparison, we are also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field, the multi-dimensional normal velocity field and the She–Leveque velocity field, introduced by She, et al. We observe that extended self-similarity scaling exponents H(p)/H(2) of passive scalar obtained by the colored noise field are more anomalous than those obtained by the other three velocity fields.

Influence of Velocity Shell Correlations on Anomalous Scaling Exponents of Passive Scalars in Shell Models

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence. Different to the original problem, the distribution function of the prescribed random velocity field is multi-dimensional normal and delta-correlated in time. Here, our random velocity field is spatially correlative. For comparison, we also give the result obtained by the Gaussian random velocity field without spatial correlation. The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p = 15 by numerical simulations of the random shell model with Runge–Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimensional normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.

Anomalous scaling in a non-Gaussian random shell model for passive scalars

In this paper, we have introduced a shell-model of Kraichnan's passive scalar problem. Different from the original problem, the prescribed random velocity field is non-Gaussian and δ correlated in time, and its introduction is inspired by She and Leveque (Phys. Rev. Lett. 72, 336 (1994)). For comparison, we also give the passive scalar advected by the Gaussian random velocity field. The anomalous scaling exponents H(p) of passive scalar advected by these two kinds of random velocities above are determined for structure function with values of p up to 15 by Monte Carlo simulations of the random shell model, with Gear methods used to solve the stochastic differential equations. We find that the H(p) advected by the non-Gaussian random velocity is not more anomalous than that advected by the Gaussian random velocity. Whether the advecting velocity is non-Gaussian or Gaussian, similar scaling exponents of passive scalar are obtained with the same molecular diffusivity.

Heat equilibrium distribution in a turbulent flow

We consider a shear flow of a scale invariant Gaussian random velocity field that does not depend on the coordinates in the direction of the flow. We investigate a heat advection coming from a Gaussian random homogeneous source. We discuss a relaxation at large time of a temperature distribution determined by the forced advection-diffusion equation. We represent the temperature correlation functions by means of the Feynman-Kac formula. Jensen inequalities are applied for lower and upper bounds on the correlation functions. We show that at finite time there is no velocity dependence of long range temperature correlations (low momentum asymptotics) in the direction of the flow but the equilibrium heat distribution has large distance correlations (low momentum behavior) with an index depending on the scaling index of the random flow and of the index of the random forcing. If the velocity has correlations growing with the distance (a turbulent flow), then the large distance correlations depend in a crucial way on the scaling index of the turbulent flow. In such a case the correlations increase in the direction of the flow and decrease in the direction perpendicular to the flow, making the stream of heat more coherent.

Numerical studies of the transport behavior of a passive solute in a two-dimensional incompressible random flow field

We study the transport behavior of a passive scalar in a two-dimensional (2D) time-independent Gaussian random velocity field by efficient and highly accurate numerical simulations. The model under consideration has been used in order to gain basic understanding of transport processes in incompressible flow through heterogeneous porous media. The velocity field is derived from the linearized solution of the Darcy equation with a Gauss-distributed log-hydraulic conductivity. The transport of a passive scalar is studied by a high precision random-walk method, which allows for a systematic nonperturbative study of the ensemble and effective dispersion coefficients. The conclusive numerical results validate the range of applicability of the perturbation theory and the consistency of nonperturbative approaches to the transport problem in a random medium. Furthermore, we observe closed streamlines in incompressible 2D Gaussian random fields, which restricts the direct applicability of the simulation method for transport in heterogeneous porous media, and questions the results of similar studies that do not observe this phenomenon.

Self-Similar Decay in the Kraichnan Model of a Passive Scalar

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0<ζ<2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a≤(d/γ)+1 are statistically realizable, where d is space dimension and γ=2−ζ. An infinite sequence of invariants Jp, p=0, 1, 2,..., is pointed out, where J0 is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J0 or J1 must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being Jp—converges at long times to a scaling solution Φp with a=(d/γ)+p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a “two-scale” decay with breakdown of self-similarity for a range of exponents (d+γ)/γ<a<(d+2)/γ, analogous to what has recently been found in the decay of Burgers turbulence.

Transport by Vector Fields with Kolmogorov Spectrum

We study a model of turbulent transport described by the motion in a Gaussian random velocity field with Kolmogorov spectrum. The field is assumed to be divergence-free, homogeneous in time and space, and Markovian in time. The molecular viscosity defines the cutoff in the Fourier space, thus regularizing the vector field of the pure infinite-Reynolds-number Kolmogorov spectrum by vector fields with smooth realizations. We provide an asymptotic bound on the effective diffusivity of the finite-Reynolds number fields as R→∞. Namely, with macroscopic parameters of the system fixed and the viscosity tending to zero, the effective diffusivity is bounded above by a constant which does not depend on the Reynolds number.

Effect of nonlinear drag on the settling velocity of particles in homogeneous isotropic turbulence

We study the average settling velocity of small spherical particles under gravity through a Gaussian random velocity field generated by Fourier modes with a von Kárman energy spectrum. The particles are subjected to the effects of a linear (Stokes) or nonlinear drag force, inertia and gravity. It is shown that the effect of drag nonlinearity is a function of the particle to fluid density ratio ρp/ρf and a function of the ratio τp/τk of the particle time constant to the Kolmogorov timescale of the fluid. Simulations show that as ρp/ρf decreases from 877 to 2.65 or as τp/τk increases from 1 to 2.74, the drag nonlinearity increases as a result of the increase in particle Reynolds numbers. Hence the settling velocity changes from larger to smaller as compared with the still fluid settling velocity, showing that one of the major mechanisms governing the fall velocity reduction in a turbulent flow is the drag nonlinearity. The maximum increase in settling rate occurs at VT/υk ≈ 2 for ρp/ρf ≥ 87 (where VT is the terminal velocity of the particle and υk is the Kolmogorov velocity); this is consistent with the results of Wang and Maxey [1993]. The maximum decrease in settling rate occurs at VT/VK ≈ 1 for ρp/ρf ≈ 2.65, consistent with the results of Fung [1993]. In addition, the role of the spatial and temporal variations of the flow field on the settling rate is investigated. Finally, the Gaussian velocity field is also simulated with an exponential energy spectrum, and similar results are observed.

Multi-length-scale theories for scale-up problem and renormalized perturbation expansion

Randomness is an essential feature for groundwater ecology and fluid turbulence. The scale-up problem refers to the situation in which the randomness occurs on all scales. Classical dispersion theories are not applicable in this situation due to the coupling between adjacent length scales. The difficulty in theoretical formulations lies in the closure problem — namely, the higher order random quantities emerge in the derivations for the dynamical equations of the lower order random quantities. In this paper, we consider tracer particles driven by a Gaussian random velocity field. We develop a renormalized perturbation expansion method for this problem. Our theory shows that the effective dispersivity has very complicated structures. We introduce a pairwise time ordering approximation to derive an approximate expression for a local diffusion operator. Under this approximation, we develop a multi-length-scale theory which gives analytical predictions for the effective diffusion coefficient and for the size of the mixing region along with its scaling exponent. Our theory has no adjustable parameter. The quantitative theoretical predictions are in excellent agreement with the results from full numerical simulations. © 1997 Elsevier Science Ltd. All rights reserved

Dissipation fluctuations of a passive scalar advected by a random velocity field.

The dynamics of fluctuations of the dissipation rate of a passive scalar advected by a rapidly changing in time Gaussian random velocity field with the variance [${\mathit{v}}_{\mathit{i}}$(x)-${\mathit{v}}_{\mathit{i}}$(x+r)${]}^{2}$\ensuremath{\propto}${\mathit{r}}^{\ensuremath{\xi}}$ is considered. It is shown that when \ensuremath{\xi}/d\ensuremath{\rightarrow}0 the dissipation correlation function 〈E(x)E(x+r)〉\ensuremath{\propto}${\mathit{r}}^{\ensuremath{\gamma}}$ with \ensuremath{\gamma}=-4\ensuremath{\xi}/(d+2) in agreement with the recent works of Gawedzkii and Kupianen [Phys. Rev. Lett. 75, 3608 (1995)], and Chertkov et al. [Phys. Rev. E 52, 4924 (1995)]. It is shown that in this limit the fourth-order moment of the scalar difference is completely described in terms of the second-order moment and the scalar dissipation rate correlation function. \textcopyright{} 1996 The American Physical Society.

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With e the spectral parameter, for −∞<e<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for e<2 to anomalous behavior for e with 2<e<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for e with e<2 to an explicit family with a non-Gaussian covariance for e with 2<e<4. These non-Gaussian distributions have tails that are broader than Gaussian as e varies with 2<e<4 and behave for large values like exp(−C c |x|4−e), withC c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.

Large-scale radial velocity correlations as a test of Gaussian initial conditions

view Abstract Citations (13) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Large-Scale Radial Velocity Correlations as a Test of Gaussian Initial Conditions Landy, Stephen D. ; Szalay, Alexander S. Abstract Here we test for the Gaussian character of the large-scale mass fluctuation spectrum as a function of scale using a relation between higher order velocity correlation functions of Gaussian random fields. An underlying Gaussian distribution should satisfy the following relationship between the two-point second- and fourth-order radial velocity correlation functions <u^2^_1_u^2^_2_>-2<u_1_u_2_>^2^-<u^2^(0)>^2^=0, where u_1_, u_2_ are the radial velocities of mass points in the cosmic microwave background frame separated by a distance r = |r_1_ - r_2_|. In addition, the two-point third-order radial velocity correlation function should equal zero. Our analysis checks the consistency of the hypothesis that the initial fluctuation spectrum can be considered an isotropic Gaussian random field with random phase and that the present distribution has grown linearly as a result of gravitational instabilities. Utilizing a simulated Gaussian random velocity field, we investigate the effects of distance errors, thermal motion, selection functions, Malmquist corrections, and different weighting schemes on the correlation functions. We show a superior fit to the underlying correlation functions using a Malmquist correction derived from the actual distribution of estimated distances and an even weighting scheme. These methods are then applied to a sample of elliptical galaxies. The results show a consistency with the assumption of a Gaussian field; however, the inherent limitations of the data set preclude a definitive answer. Publication: The Astrophysical Journal Pub Date: July 1992 DOI: 10.1086/171556 Bibcode: 1992ApJ...394...25L Keywords: Computational Astrophysics; Elliptical Galaxies; Radial Velocity; Astronomical Models; Correlation Coefficients; Mass Distribution; Monte Carlo Method; Relic Radiation; Velocity Distribution; Astrophysics; COSMOLOGY: THEORY; GALAXIES: DISTANCES AND REDSHIFTS; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; METHODS: NUMERICAL full text sources ADS |

Averaged-equation and diagrammatic approximations to the average concentration of a tracer dispersed by a Gaussian random velocity field

Approximate methods for predicting the spread of a passive tracer due to a Gaussian random velocity field are examined. Previous methods such as the direct-interaction approximation [J. Fluid Mech. 11, 257 (1961)] and Phythian’s short-time expansion [J. Fluid Mech. 67, 145 (1975)] make predictions for the second moment of the concentration that are qualitatively consistent with numerical simulations [Phys. Fluids 13, 22 (1970)]. However, it is shown that the higher moments obtained from these approximations are incorrect. The derivations of the direct-interaction approximation using a diagrammatic expansion and the method of averaged equations are reviewed [Phys. Fluids A 1, 47 (1989)]. A higher-order (two-body) approximation is developed which makes predictions for the fourth moment that are qualitatively correct in both the short- and long-time limits.

The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, is shown to depend on the particle inertia and the free-fall terminal velocity in still fluid. With no inertia the particle settles on average at the same rate as in still fluid, assuming there is no mean flow. Particle inertia produces a bias in each trajectory towards regions of high strain rate or low vorticity, which affects the mean settling velocity. Results from a Gaussian random velocity field show that this produces an increased settling velocity.