Quasistationary probability density functions in the turbulent mixing of a scalar field

Flow-induced separation in wall turbulence

One of the defining characteristics of turbulence is its ability to promote mixing. We present here a case where the opposite happens-simulation results indicate that particles can separate near the wall of a turbulent channel flow, when they have sufficiently different Schmidt numbers without use of any other means. The physical mechanism of the separation is understood when the interplay between convection and diffusion, as expressed by their characteristic time scales, is considered, leading to the determination of the necessary conditions for a successful separation between particles. Practical applications of these results can be found when very small particles need to be separated or removed from a fluid.

Probability density functions of decaying passive scalars in periodic domains: an application of Sinai-Yakhot theory

Employing the formalism introduced by Phys. Rev. Lett. 63, 1962 (1989)], we study the probability density functions (pdf's) of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields. The particular regime we focus on is one where the normalized scalar pdf's attain a self-similar profile in finite time, i.e., the so-called strange or statistical eigenmode regime. In accordance with the work of Sinai and Yakhot, the central regions of the pdf's are power laws. However, the details of the pdf profiles are dependent on the physical parameters in the problem. Interestingly, for small Peclet numbers the pdf's resemble stretched or pure exponential functions, whereas in the limit of large Peclet numbers, there emerges a universal Gaussian form for the pdf. Numerical simulations are used to verify these predictions.

Regular and anomalous scaling of a randomly advected passive scalar

Extending Kolmogorov's refined similarity hypothesis to study the inertial behavior <[T(x+r,t)-T(x,t)](2n)>~r(zeta(2n)) of a passive scalar T(x,t) advected by a rapidly changing incompressible velocity field, a random variable straight theta was introduced by Ching [Phys. Rev. Lett. 79, 3644 (1997)]. In this paper, the statistical distribution of the random variable X=straight theta/sqrt[<straight theta(2)>] is investigated analytically for the scaling in two limits, n-independent scaling zeta(2n)=zeta(2) and regular scaling zeta(2n)=nzeta(2), and numerically for the scaling of the Kraichnan conjecture zeta(2n)=1 / 2[sqrt[4ndzeta(2)+(d-zeta(2))(2)]-(d-zeta(2))]. For n-independent scaling zeta(2n)=zeta(2), the statistical distribution of X tends to an exponential distribution when zeta(2)-->0 or d-->infinity and to a Gaussian distribution when zeta(2)-->2 and d=2. For regular scaling zeta(2n)=nzeta(2), the statistical distribution of X tends to a Gaussian distribution when zeta(2)-->0 or d-->infinity. In d=2, there seems to be a phase transition for the probability density function P(X) from a convex to a concave function when the value of zeta(2) is increased and the critical point is zeta(2)=4/3 where the random variable X has a uniform distribution in [-sqrt[3],sqrt[3]]. In d=3, P(X) is a convex function for all 0<zeta(2)<2 and tends to a constant on its support [-sqrt[3],sqrt[3]] when zeta(2)-->2. For the scaling of the Kraichnan conjecture, P(X) has two peaks in d=2 for zeta(2)>1.33, but, in d=3, it has only one peak for all 0<zeta(2)<2 and changes very slowly with the value of X in the neighborhood of X=0 as zeta(2)-->2.

Relation between the probability density and other properties of a stationary random process

We consider the Pope-Ching differential equation [Phys. Fluids A 5, 1529 (1993)] connecting the probability density p(x)(x) of a stationary, homogeneous stochastic process x(t) and the conditional moments of its squared velocity and acceleration. We show that the solution of the Pope-Ching equation can be expressed as n(x)</v(x)/(-1)>, where n(x) is the mean number of crossings of the x level per unit time and </v(x)/(-1)> is the mean inverse velocity of crossing. This result shows that the probability density at x is fully determined by a one-point measurement of crossing velocities, and does not imply knowledge of the x(t) behavior outside of the infinitesimally narrow window near x.