Random Velocity Field Research Articles

Turbulence and Cloud Angular Momentum

Numerical models of molecular cloud cores are obtained, with assumed Gaussian random velocity fields that are consistent with moderate turbulence. If the velocity power spectrum is taken to be P(k) ∼ kn, with n = − 4 or −3, then the constructed line-of-sight velocity maps on the plane of the sky show gradients that can be interpreted as rotation. Deduced values of angular velocity, angular momentum, and dimensionless rotational parameter β are consistent with observations. The velocity gradient Ω, of an individual core is not a good indicator of its intrinsic angular momentum J/M. However, the distribution of deduced angular momenta from a large sample of cores with different random velocity fields is close to the distribution of actual angular momenta of these model cores if one assumes J/M = pΩR2 where R is the core radius and p must be determined from a Monte-Carlo study. For centrally condensed cores the standard value of p = 0.4 overestimates the mean intrinsic angular momentum by a factor of 3.

Fractional Brownian motions in a limit of turbulent transport

W show that the motion of a particle advected by a random Gaussian velocity field with long-range correlations converges to a fractional Brownian motion in the long time limit.

Probability density function of a passive scalar gradient

Fractal and multi-fractal models are used to derive the probability density function (PDF) for the gradient of a passive scalar diffusing in a random velocity field. Although the PDF is symmetrical, it is non-Gaussian and compares reasonably well with the measured PDF. In the process, we also provide a unified systematic framework to derive the appropriate scalar microscales for the following regimes: inertial-diffusive range; viscous-convective range; viscous-diffusive range.

Simple example of the development of cluster structure of a passive tracer field in random flows

As a follow-up to Ref. [1], the formation of particle clusters (Lagrangian description) and cluster structure of the field of a passive tracer (Eulerian description) in a random velocity field is analyzed for a simple problem amenable to an analytical solution.

Escape probability and mean residence time in random flows withunsteady drift

We investigate fluid transport in random velocity fields with unsteady drift. First, we propose to quantify fluid transport between flow regimes of different characteristic motion, by escape probability and mean residence time. We then develop numerical algorithms to solve for escape probability and mean residence time, which are described by backward Fokker‐Planck type partial differential equations. A few computational issues are also discussed. Finally, we apply these ideas and numerical algorithms to a tidal flow model.

Generalized flows, intrinsic stochasticity, and turbulent transport.

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows, which are families of probability distributions on the space of solutions to the associated ordinary differential equations which no longer satisfy the uniqueness theorem for ordinary differential equations. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field, are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes, characterized by their degrees of compressibility, are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behaviors for the structure functions of the passive scalar. Physically, this means that the scaling depends on Prandtl number. In the other two regimes, the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The "one force, one solution" principle is established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime, which is the regime of inverse cascade. Existence and uniqueness of an invariant measure are also proved in these regimes when the transport equation is suitably forced. Finally incomplete self similarity in the sense of Barenblatt and Chorin is established.

Simulation of stochastic metal-forming process for rigid-viscoplastic material

Numerical solution of stochastic metal-forming process is analysed. The theoretical formulation is described which presents probabilistic distributions for velocities and strains taking into account random initial and boundary conditions. Finite element equations for rigid-viscoplastic materials are solved for the first two probabilistic moments of the random velocity field. The example of ring compression with random parameters is presented.

One‐Point Probability Distribution Functions of Supersonic Turbulent Flows in Self‐gravitating Media

Turbulence is essential for understanding the structure and dynamics of molecular clouds and star-forming regions. There is a need for adequate tools to describe and characterize the properties of turbulent flows. One-point probability distribution functions (PDFs) of dynamical variables have been suggested as appropriate statistical measures and applied to several observed molecular clouds. However, the interpretation of these data requires comparison with numerical simulations. To address this issue, smoothed particle hydrodynamics (SPH) simulations of driven and decaying, supersonic, turbulent flows with and without self-gravity are presented. In addition, random Gaussian velocity fields are analyzed to estimate the influence of variance effects. To characterize the flow properties, the PDFs of the density, of the line-of-sight velocity centroids, and of the line centroid increments are studied. This is supplemented by a discussion of the dispersion and the kurtosis of the increment PDFs, as well as the spatial distribution of velocity increments for small spatial lags. From the comparison between different models of interstellar turbulence, it follows that the inclusion of self-gravity leads to better agreement with the observed PDFs in molecular clouds. The increment PDFs for small spatial lags become exponential for all considered velocities. However, all the processes considered here lead to non-Gaussian signatures, differences are only gradual, and the analyzed PDFs are in addition projection dependent. It appears therefore very difficult to distinguish between different physical processes on the basis of PDFs only, which limits their applicability for adequately characterizing interstellar turbulence.

Relative molecular diffusion

We study both analytically and numerically the motion of a passive floating admixture subject to molecular collisions in a random velocity field. Within the framework of the one-dimensional model, we study the phenomena of relative molecular diffusion and the closely related fluctuations of the density of a cluster of particles that were initially located in the same physically infinitesimal volume.

Different transport regimes in a spatially-extended recirculating background

Passive scalar transport in a spatially-extended background of roll convection is considered in the time-periodic regime. The latter arises due to the even oscillatory instability of the cell lateral boundary, here accounted for by sinusoidal oscillations of frequency ω. We show that the strength of anticorrelated regions of the velocity field can be controlled by varying the frequency ω and the conditions under which a strong enhancement/reduction of transport takes place can be created. Such two ubiquitous regimes are triggered by a small-scale (random) velocity field superimposed to the recirculating background. The crucial point for the emergence of transport enhancement/reduction is played by the dependence of Lagrangian trajectories on the statistical properties of the small-scale velocity field, e.g. its correlation time or its energy.

Velocity-fluctuation-induced anomalous kinetics of the A+A--> reaction

The effect of a random velocity field on the kinetics of the single-species annihilation reaction A+A--> is analyzed near two dimensions with the aid of the perturbative renormalization group. The previously found asymptotic behavior induced by density fluctuations only in the diffusion-limited reaction is shown to be unstable to any velocity fluctuations (including thermal fluctuations near equilibrium) in spatial dimensions d</=d(c)=2. Four different stable long-time asymptotic regimes induced by the combined effect of velocity and density fluctuations are identified and the corresponding decay rates calculated in the leading order.

Diffusion coefficients as function of Kubo number in random fields

Particle standard diffusion D (K ) in 2D homogeneous, isotropic, stationary, random velocity fields is studied numerically as a function of the dimensionless Kubo number K defined as the ratio between the correlation time of the velocity field and its sweeping time. The three cases corresponding to the energy spectra proportional, in the inertial range, respectively to k -3 , k -(5/3) and k 2 are considered and two different power law regimes are found for large but finite K .

Passive scalar transport in a random flow with a finite renewal time: Mean-field equations

A mean-field equation for a passive scalar (e.g., for a mean number density of particles) in a random velocity field (incompressible and compressible) with a finite constant renewal time is derived. The finite renewal time of a random velocity field results in the appearance of high-order spatial derivatives in the mean-field equation for a passive scalar. We considered three models of a random velocity field: (i) a velocity field with a small renewal time; (ii) the Gaussian approximation for Lagrangian trajectories; and (iii) a small inhomogeneity of the velocity and mean passive scalar fields. For a small renewal time we recovered results obtained using the $\ensuremath{\delta}$-function-correlated in time random velocity field. The finite renewal time and compressibility of the velocity field can cause a depletion of turbulent diffusion and a modification of an effective drift velocity. For a compressible velocity field the form of the mean-field equation for a passive scalar depends on the details of the velocity field, i.e., the universality is lost. For an incompressible velocity field the universality exists in spite of the finite renewal time. Results by Saffman [J. Fluid Mech. 8, 273 (1960)] for the effect of molecular diffusivity in turbulent diffusion are generalized for the case of a compressible and anisotropic random velocity field. The obtained results may be of relevance in some atmospheric phenomena (e.g., atmospheric aerosols and smog formation).

Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbitrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution α and β=τD, where τ is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity\(\bar \lambda \) and the dispersion Δ. We use the method of higher space dimensions d to estimate\(\bar \lambda \) and Δ for different values of α and β. It was shown previously that\(\bar \lambda \sim D\) for β≪ 1 and\(\bar \lambda \sim \sqrt {D/\tau } \) for β≫ 1. The estimate of Δ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to Δ estimated as αD2τ for β≪1 and αβ/τ for β≫1. For α above some critical value σcr, the values of\(\bar \lambda \) and Δ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as\(\alpha _{cr} \sim \beta ^{ - 1} \) for β≪1 and\(\alpha _{cr} \sim \beta ^{ - 1/2} \) for β≫1.

Linear and nonlinear interactions between a columnar vortex and external turbulence

The structure of initially isotropic homogeneous turbulence interacting with a columnar vortex (with circulation Γ and radius σ), idealized both as a solid cylinder and a hollow core model is analysed using the inhomogeneous form of linear rapid distortion theory (RDT), for flows where the r.m.s. turbulence velocity u0 is small compared with Γ/σ. The turbulent eddies with scale Γ are distorted by the mean velocity gradient and also, over a distance Γ from the surface of the vortex, by their direct impingement onto it, whether it is solid or hollow. The distortion of the azimuthal component of turbulent vorticity by the differential rotation in the mean flow around the columnar vortex causes the mean-square radial velocity away from the cylinder to increase as (Γt/2πr2)2 (Γx/r)u20, when (r − σ) &gt; Γx, but on the surface of the vortices ((r − σ) &lt; Γx) where 〈u2r〉 is reduced, 〈u2z〉 increases to the same order, while the other components do not grow. Statistically, while the vorticity field remains asymmetric, the velocity field of small-scale eddies near the vortex core rapidly becomes axisymmetric, within a period of two or three revolutions of the columnar vortex. Calculation of the distortion of small-scale initially random velocity fields shows how the turbulent eddies, as they are wrapped around the columnar vortex, become like vortex rings, with similar properties to those computed by Melander &amp; Hussain (1993) using a fully nonlinear direct numerical simulation. A mechanism is proposed for how interactions between the external turbulence and the columnar vortex can lead to non-axisymmetric vortex waves being excited on the vortex and damped fluctuations in its interior. If the columnar vortex is not significantly distorted by these linear effects, estimates are made of how nonlinear effects lead to the formation of axisymmetric turbulent vortices which move as result of their image vorticity (in addition to the self-induction velocity) at a velocity of order u0tΓ/σ2 parallel to the vortex. Even when the circulation (γ) of the turbulent vortices is a small fraction of Γ, they can excite self-destructive displacements through resonance on a time scale σ/u0.

Turbulent transport of atmospheric aerosols and formation of large-scale structures

Turbulent transport of aerosols and droplets in a random velocity field with a finite correlation time is studied. We derived a mean-field equation and an equation for the second moment for a number density of aerosols. The finite correlation time of random velocity field results in the appearance of the high-order spatial derivatives in these equations. The finite correlation time and compressibility of the velocity field can cause a depletion of turbulent diffusion and a modification of an effective mean drift velocity. The coefficient of turbulent diffusion in the vertical direction can be depleted by 25 % due to the finite correlation time of a turbulent velocity field. The latter result is in compliance with the known anisotropy of the coefficient of turbulent diffusion in the atmosphere. The effective mean drift velocity is caused by a compressibility of particles velocity field and results in formation of large-scale inhomogeneities in spatial distribution of aerosols in the vicinity of the atmospheric temperature inversion. Results obtained by Saffman (1960) for the effect of molecular diffusivity in turbulent diffusion are generalized for the case of compressible and anisotropic random velocity field. A mechanism of formation of small-scale inhomogeneities in particles spatial distribution is also discussed. This mechanism is associated with an excitation of a small-scale instability of the second moment of number density of particles. The obtained results are important in the analysis of various atmospheric phenomena, e.g., atmospheric aerosols, droplets and smog formation.

The Passive Polymer Problem

In this article, we introduce a generalization of the diffusive motion of point-particles in a turbulent convective flow with given correlations to a polymer or membrane. In analogy to the passive scalar problem we call this the passive polymer or membrane problem. We shall focus on the expansion about the marginal limit of velocity–velocity correlations which are uncorrelated in time and grow with the distance x as |x|e, and e small. This relation gets modified in the case of polymers and membranes (the marginal advecting flow has correlations which are shorter ringed.) The construction is done in three steps: First, we reconsider the treatment of the passive scalar problem using the most convenient treatment via field theory and renormalization group. We explicitly show why IR-divergences and thus the system-size appear in physical observables, which is rather unusual in the context of ordinary field-theories, like the φ4-model. We also discuss, why the renormalization group can nevertheless be used to sum these divergences and leads to anomalous scaling of 2n-point correlation functions as e.g., S2n(x)≔〈[Θ(x, t)−Θ(0, t)]2n〉. In a second step, we reformulate the problem in terms of a Langevin equation. This is interesting in its own, since it allows for a distinction between single-particle and multi-particle contributions, which is not obvious in the Focker–Planck treatment. It also gives an efficient algorithm to determine S2n numerically, by measuring the diffusion of particles in a random velocity field. In a third and final step, we generalize the Langevin treatment of a particle to polymers and membranes, or more generally to an elastic object of inner dimension D with 0≤D≤2. These objects can intersect each other. We also analyze what happens when self-intersections are no longer allowed.

Fractional Brownian Motions and Enhanced Diffusion in a Unidirectional Wave-Like Turbulence

We study transport in random undirectional wave-like velocity fields with nonlinear dispersion relations. For this simple model, we have several interesting findings: (1) In the absence of molecular diffusion the entire family of fractional Brownian motions (FBMs), persistent or anti-persistent, can arise in the scaling limit. (2) The infrared cutoff may alter the scaling limit depending on whether the cutoff exceeds certain critical value or not. (3) Small, but nonzero, molecular diffusion can drastically change the scaling limit. As a result, some regimes stay intact; some (persistent) FBM regimes become non-Gaussian and some other FBM regimes become Brownian motions with enhanced diffusion coefficients. Moreover, in the particular regime where the scaling limit is a Brownian motion in the absence of molecular diffusion, the vanishing molecular diffusion limit of the enhanced diffusion coefficient is strictly larger than the diffusion coefficient with zero molecular diffusion. This is the first such example that we are aware of to demonstrate rigorously a nonperturbative effect of vanishing molecular diffusion on turbulent diffusion coefficient.

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.

Spectrum of the fokker-planck operator representing diffusion in a random velocity field

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. 79, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time dependence of the mean-square displacement, <r(2)> approximately t(2/d) in dimension d>2, associated with the imaginary parts of eigenvalues.