Self-similar two-particle separation model

Abstract

We present a new stochastic model for relative two-particle separation in turbulence. Inspired by material line stretching, we suggest that a similar process also occurs beyond the viscous range, with time scaling according to the longitudinal second-order structure function S 2 ͑r͒, e.g.; in the inertial range as −1/3 r 2/3 . Particle separation is modeled as a Gaussian process without invoking information of Eulerian acceleration statistics or of precise shapes of Eulerian velocity distribution functions. The time scale is a function of S 2 ͑r͒ and thus of the Lagrangian evolving separation. The model predictions agree with numerical and experimental results for various initial particle separations. We present model results for fixed time and fixed scale statistics. We find that for the Richardson-Obukhov law, i.e., ͗r͑t͒ 2 ͘ = g t 3 , to hold and to also be observed in experiments, high Reynolds numbers are necessary, i.e., Re Ͼ O͑1000͒, and the integral scale needs to be large compared to initial separation, i.e., L / r 0 Ͼ 30 and d / L Ͼ 3 need to be fulfilled, where d is the size of the field of view. Removing the constraint of finite inertial range, the model is used to explore separation dynamics in the asymptotic regime. As Re → ϱ, the distance neighbor function takes on a constant shape, almost as predicted by the Richardson diffusion equation. For the Richardson constant we obtain that g → 0.95 as Re → ϱ. This asymptotic limit is reached at Re Ͼ 1000. For the Richardson constant g, the model predicts a ratio of g b / g f Ϸ 1.9 between backwards and forwards dispersion.

Turbulent pair separation due to multiscale stagnation point structure and its time asymmetry in two-dimensional turbulence

“…While the effect of time asymmetry of pair separation has been reported for 3D experiments and DNS, 8,9 there are no published results about this asymmetry in 2D turbulence.…”

“…8. ͑Color online͒ Best fit of numerically integrated mean square separation ͗⌬ 2 ͘ to DNS data with initial separations ⌬ 0 = f / 4 ͑a͒ and ⌬ 0 = f / 2 ͑b͒.…”

Time-reversal-symmetry Breaking in Turbulence

“…The larger the relative velocity between two particles, |u(0)|, the shorter they reside in the measurement volume [8,31]. The experimentally measured mean squared displacement, δR 2 (t) m , determined by particle pairs which could be tracked up to time t, is smaller than the true value δR 2 (t) (see Fig.…”

Probability density function of small separation between two inertial particles in homogenous isotropic turbulence

“…In fact, a similar random differential equation, dr / dt = S f ͑t͒r, has been used to model material line stretching in turbulence. [15][16][17][18][19] In that model, S f ͑t͒ is considered as the rate of the projected strain sensor of a velocity field of fluid particles rather than that of a velocity field of inertial particles.…”