Eulerian-Lagrangian aspects of a steady multiscale laminar flow

One key feature for the understanding and control of turbulent flows is the relation between Eulerian and Lagrangian statistics. This Brief Communication investigates such a relation for a laminar quasi-two-dimensional multiscale flow generated by a multiscale (fractal) forcing, which reproduces some aspects of turbulent flows in the laboratory, e.g., broadband power-law energy spectrum and Richardson's diffusion. We show that these multiscale flows abide with Corrsin's estimation of the Lagrangian integral time scale, T-L, as proportional to the Eulerian (integral) time scale, L-E/u(rms), even though Corrsin's approach was originally constructed for high Reynolds number turbulence. We check and explain why this relation is verified in our flows. The Lagrangian energy spectrum, Phi(w), presents a plateau at low frequencies followed by a power-law energy spectrum Phi(w)similar to w(-alpha) at higher ones, similarly to turbulent flows. Furthermore, Phi(omega) scales with L-E and u(rms) with alpha>1. These are the key elements to obtain such a relation [Phi(w)similar to epsilon w(-2) is not necessary] as in our flows the dissipation rate varies as epsilon similar to u(rms)(3)/LERe lambda-1. To complete our analysis, we investigate a recently proposed relation [M. A. I. Khan and J. C. Vassilicos, Phys. Fluids 16, 216 (2004)] between Eulerian and Lagrangian structure functions, which uses pair-diffusion statistics and the implications of this relation on Phi(omega). Our results support this relation, <[u(L)(t)-u(L)(t+tau)](2)>=<[u(E)(x)-u(E)(x+Delta(2)(tau)e)](2)>, which leads to alpha=gamma/2(p-1)+1. This Eulerian-Lagrangian relation is striking as in the present flows it is imposed by the multiscale distribution of stagnation points, which are an Eulerian property. (C) 2007 American Institute of Physics.