PERSISTENT MULTIPLE-SCALE STAGNATION POINT STRUCTURE

摘要

In isotropic turbulence, stagnation points form a fractal multiple-scale network in space such that their number density n_s = C_sL~(-d) (L/η)~(D_s) where C_s is a dimensionless constant, L/q is the inner to outer length-scale ratio and the fractal dimension Dn is given by p + 2D*/d = 3; d is the dimensionality of the flow and p is the exponent of the energy spectrum. On the other hand, the statistical persistence of stagnation points is defined in terms of the statistics of stagnation point velocities, and we show that, on average, stagnation points stop moving as the Reynolds number tends to infinity in the frame where the mean flow is zero. In that same limit, stagnation points tend to become zero-acceleration points on average, and to be persistent. Turbulent-like velocity fields obtained by Kinematic Simulations (KS) can be made to reproduce some persistence properties of the multiple-scale stagnation point network by appropriately choosing the KS time-dependence. Studies of turbulent pair diffusion in such KS lead to (△~2)≈G_△L~2(u't/L)~γ (where A is the pair separation at time I, u' is the r.m.s. turbulent velocity and G& is the Richardson dimensionless constant) with 7 = 2d/DH. A simple argument based on the time between successive encounters of particle-pairs with stagnation points and on a re-interpretation of the locality-in-scale hypothesis in terms of a multiplicative pair-separation process confirms this relation between the Lagrangian exponent 7 and the Eulerian exponent DH. This model also leads to GA ~ C.,2/D" thus suggesting that the Richardson constant might not be universal. Simulations confirm that GA is an increasing function of C.,. Finally, we seek to corroborate these ideas and results wi th a low Reynolds number laboratory simulation of high Reynolds number two-dimensional turbulence. We complement this laboratory simulation with DNS of the same and similar flows. In this laboratory simulation we reproduce the cat's eyes within cat's eyes topological streamline structure of two-dimensional turbulence by appropriate multiple-scale electromagnetic forcing of a quasi-two-dimensional brine flow. In particular, we are able to impose the value of DH. PIV measurements of the energy spectrum corroborate the formula p + 2Ds/d, = 3.