Abstract
The spreading of passive particles immersed in a two-dimensional turbulent flow confined within a closed domain is studied analytically and numerically. The primary goal is to investigate the effect of the confinement and the geometry of the container on one and two-particle Lagrangian statistics (absolute dispersion from point sources and relative dispersion of pairs of particles, respectively). The influence of the flow confinement is analysed by performing numerical experiments with numerous particles in square boxes with different sizes. The results examine the modification of the time-dependent, dispersion curves as the particles spread out (in comparison to the turbulent regimes for unbounded flows). At long times, such curves asymptotically reach a constant value of saturation as the particles fill the container. Theoretical saturation values are calculated, and the obtained formulae are tested with the numerical results. To study the influence of the domain shape, saturation values are computed analytically for different geometries (rectangles, triangles, and ellipses). To our knowledge, the obtained expressions are new. The saturation values depend on the characteristic lengths of the domain for both regular and irregular shapes. Ranges of saturated values for the different geometries are provided. The results are compared with well-known asymptotic values for unbounded flows, thus determining the influence of the closed boundaries on particle dispersion.
Highlights
Particle dispersion, and the Lagrangian statistics, can be modified by the effects of boundaries
We present the Lagrangian statistics of a forced, 2D turbulent flow bounded by solid walls measured from numerical simulations and analytical considerations
The saturation kurtosis is smaller than the asymptotic kurtosis in the standard dispersion regime for an unbounded domain; Ks can be regarded as a quantitative measure of the influence of the confinement on the particle statistics
Summary
We consider a 2D flow in a Cartesian coordinate system (x, y). The flow dynamics is described by the vorticity-stream function formulation of the 2D Navier-Stokes equation:. The corresponding fluctuations over the interval 500 ≤ t ≤ 5000 are small (about 3 and 6%, respectively) During this period the flow is considered to be in a statistically stationary state, adequate to perform experiments on particle dispersion. Two inertial ranges are observed, as predicted by homogeneous, 2D turbulence theory for unbounded flows.4 These ranges are separated at the cut-off wave number kf = 131.946 (vertical solid line) determined by the external forcing. To compare the turbulent flow developed in the simulations with medium (L = 0.5) and small (L = 0.25) domains, it is convenient to define the Reynolds number based on the forcing scale rf = 2π/kf = 0.0476. The turbulent layer near the boundaries, which is of the order of the forcing scale, contains approximately rf /∆x ∼ 26 grid points. There are at least ten grid points to solve these scales
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