Generalized Langevin equation for relative turbulent dispersion

1998 ◽  
Vol 357 ◽  
pp. 167-198 ◽  
Author(s):  
BURKHARD M. O. HEPPE
Keyword(s):  
Langevin Equation ◽  
Stokes Equations ◽  
Gaussian White Noise ◽  
Random Force ◽  
Turbulent Dispersion ◽  
Relative Dispersion ◽  
Generalized Langevin Equation ◽  
Exact Equation ◽  
Velocity Statistics ◽  
Moderate Reynolds Number

The relative velocity of two fluid particles in homogeneous and stationary turbulence is considered. Looking for reduced dynamics of turbulent dispersion, we apply the nonlinear Mori–Zwanzig projector method to the Navier–Stokes equations. The projector method decomposes the Lagrangian acceleration into a conditionally averaged part and a random force. The result is an exact generalized Langevin equation for the Lagrangian velocity differences accounting for the exact equation of the Eulerian probability density. From the generalized Langevin equation, we obtain a stochastic model of relative dispersion by stochastic estimation of conditional averages and by assuming the random force to be Gaussian white noise. This new approach to dispersion modelling generalizes and unifies stochastic models based on the well-mixed condition and the moments approximation. Furthermore, we incorporate viscous effects in a systematic way. At a moderate Reynolds number, the model agrees qualitatively with direct numerical simulations showing highly non-Gaussian separation and velocity statistics for particle pairs initially close together. At very large Reynolds numbers, the mean-square separation obeys a Richardson law with coefficient of the order of 0.1.

scholarly journals Turbulent pair dispersion of inertial particles

2010 ◽  
Vol 645 ◽  
pp. 497-528 ◽  
Author(s):  
J. BEC ◽  
L. BIFERALE ◽  
A. S. LANOTTE ◽  
A. SCAGLIARINI ◽  
F. TOSCHI
Keyword(s):  
Particle Density ◽  
Heavy Particle ◽  
Density Ratio ◽  
Turbulent Dispersion ◽  
Light Particle ◽  
Relative Dispersion ◽  
Small Scale ◽  
Heavy Particles ◽  
Velocity Statistics ◽  
Light Particles

The relative dispersion of pairs of inertial point particles in incompressible, homogeneous and isotropic three-dimensional turbulence is studied by means of direct numerical simulations at two values of the Taylor-scale Reynolds number Reλ ~ 200 and Reλ ~ 400, corresponding to resolutions of 5123 and 20483 grid points, respectively. The evolution of both heavy and light particle pairs is analysed by varying the particle Stokes number and the fluid-to-particle density ratio. For particles much heavier than the fluid, the range of available Stokes numbers is St ∈ [0.1 : 70], while for light particles the Stokes numbers span the range St ∈ [0.1 : 3] and the density ratio is varied up to the limit of vanishing particle density. For heavy particles, it is found that turbulent dispersion is schematically governed by two temporal regimes. The first is dominated by the presence, at large Stokes numbers, of small-scale caustics in the particle velocity statistics, and it lasts until heavy particle velocities have relaxed towards the underlying flow velocities. At such large scales, a second regime starts where heavy particles separate as tracers' particles would do. As a consequence, at increasing inertia, a larger transient stage is observed, and the Richardson diffusion of simple tracers is recovered only at large times and large scales. These features also arise from a statistical closure of the equation of motion for heavy particle separation that is proposed and is supported by the numerical results. In the case of light particles with high density ratio, strong small-scale clustering leads to a considerable fraction of pairs that do not separate at all, although the mean separation increases with time. This effect strongly alters the shape of the probability density function of light particle separations.

Nonlinear drift interactions between fluctuating colloidal particles: oscillatory and stochastic motions

1993 ◽  
Vol 256 ◽  
pp. 343-401 ◽  
Author(s):  
E. J. Hinch ◽  
Ludwig C. Nitsche
Keyword(s):  
Brownian Motion ◽  
Langevin Equation ◽  
Colloidal Particles ◽  
Stokes Equations ◽  
Thermal Fluctuations ◽  
Random Force ◽  
Stochastic Motion ◽  
Singularity Method ◽  
Nonlinear Force ◽  
Uniform Equilibrium

In this work we consider how nonlinear hydrodynamic effects can lead to a mean force of interaction between two spheres of equal radius a undergoing translational fluctuations parallel or perpendicular to their line of centres. Motivated by amplitudes and Reynolds numbers characteristic of Brownian motion in colloidal systems, nonlinearities due to motion of the boundaries and to inertia throughout the fluid are treated as regular perturbations of the time-dependent Stokes equations. This formulation ultimately leads to a prescription for computing, at leading order, the time-average nonlinear force for the case of pure oscillatory modes – which represents the Fourier decomposition of more general motions. The associated hydrodynamic problems are solved numerically using a least-squares boundary singularity method. Frequency-dependent results over the whole spectrum are presented for a sphere-sphere gap equal to one radius; illustrative calculations are also carried out at other separations. Subsequently we extend the analysis of nonlinear drift to a Langevin equation formulation of the more complex problem of stochastic motion due to thermal fluctuations in the suspending fluid, i.e. Brownian motion. By integrating (numerically) over the spectrum of frequencies, we quantify how the mutual interactions of all translational disturbance modes give rise, on ensemble average, to a stochastic nonlinear force of interaction between the particles. It is particularly interesting that this net interaction – arising from a zero-mean random force – is of O(1) on the Brownian scale kT/a, even though it represents a small O(Re) correction at each frequency of pure oscillations. Finally, we discuss how the presence of stochastic nonlinear drift would lead to non-uniform equilibrium distributions of dilute colloidal suspensions, unless one adds to the random force in the Langevin equation a cancelling non-zero mean component.

scholarly journals The study of dynamic singularities of seismic signals by the generalized Langevin equation

2009 ◽  
Vol 388 (17) ◽  
pp. 3629-3635 ◽  
Author(s):  
Renat Yulmetyev ◽  
Ramil Khusnutdinoff ◽  
Timur Tezel ◽  
Yildiz Iravul ◽  
Bekir Tuzel ◽  
...  
Keyword(s):  
Langevin Equation ◽  
Generalized Langevin Equation ◽  
Seismic Signals

Turbulent dispersion in a non-homogeneous field

1999 ◽  
Vol 392 ◽  
pp. 45-71 ◽  
Author(s):  
ILIAS ILIOPOULOS ◽  
THOMAS J. HANRATTY
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Stochastic Model ◽  
Point Source ◽  
Time Scales ◽  
Langevin Equation ◽  
Turbulent Dispersion ◽  
Homogeneous Field ◽  
Fully Developed Flow ◽  
Good Agreement

Dispersion of fluid particles in non-homogeneous turbulence was studied for fully developed flow in a channel. A point source at a distance of 40 wall units from the wall is considered. Data obtained by carrying out experiments in a direct numerical simulation (DNS) are used to test a stochastic model which utilized a modified Langevin equation. All of the parameters, with the exception of the time scales, are obtained from Eulerian statistics. Good agreement is obtained by making simple assumptions about the spatial variation of the time scales.

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