Evolution of triangles in a two-dimensional turbulent flow

Abstract

As a turbulent flow advects a swarm of Lagrangian markers, the mutual separation between particles grows, and the shape of the swarm gets distorted. By following three points in an experimental turbulent two-dimensional flow with a k(-5/3) spectrum, we investigate the geometry of triangles, in a statistical sense. Two well-characterized shape distributions are identified. At long times when the average size of the triangles <R> is larger than the integral scale, the distribution of shapes is Gaussian. When the size of the triangle <R> is in the inertial range and grows as t(3/2) (Richardson's law), a plausibly self-similar, non-Gaussian probability distribution is observed, where very elongated triangles have a much larger probability than in the Gaussian regime. These results are discussed, and, in the latter case, compared with the predictions of a stochastic model recently introduced [A. Pumir et al., Phys. Rev. Lett. 85, 5324 (2000)].

Scale-Dependent Statistical Geometry in Two-Dimensional Flow

“…Such Lagrangian triangles have been studied previously in 2D in kinematic simulations [14] and experiments [15]. Both studies found similar results: under the action of the flow, triangles distorted from initially symmetric shapes, eventually reaching the uncorrelated random limit after long times.…”

Dispersion of tracer particles in a compressible flow

“…After a sufficiently long time, random small-scale fluctuations tend to drive I 2 towards the Gaussian equilibrium value of I 2G = (1 − π/4)/2. In the two-dimensional incompressible case, the experiments show that the Gaussian limit is achieved in roughly 20 t/τ η [18]. It was asserted in Ref.…”

“…Since a theory for their motion has yet to be developed, one must again turn to computer simulations, which have been recently carried out [16]. Secondly, first studies the competing effects of coherent shear and random motion in the surface flow are presented by analyzing the evolution of shapes formed by three particles tracked simultaneously [17,18]. By neglecting the center of mass, the relative position of three points can be conveniently expressed by the following two vectors,…”

Presence of a Richardson’s regime in kinematic simulations

“…where X 1 (t) is the position of the first particle and X 2 (t) the position of the second particle at time t. The first quantity of interest is the mean-square separation between the two particles ∆ 2 (t) as a function of time which has received much research attention since the pioneering work of [1] (see for example [2][3][4][5][6][7][8][9][10][11][12][13][14]). …”