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, Φ ( w ) , presents a plateau at low frequencies followed by a power-law energy spectrum Φ ( w ) ∼ w − α at higher ones, similarly to turbulent flows. Furthermore, Φ ( ω ) scales with L E and u rms with α > 1 . These are the key elements to obtain such a relation [ Φ ( w ) ∼ ϵ w − 2 is not necessary] as in our flows the dissipation rate varies as ϵ ∼ u rms 3 ∕ L E Re λ − 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 Φ ( ω ) . Our results support this relation, ⟨ [ u L ( t ) − u L ( t + τ ) ] 2 ⟩ = ⟨ [ u E ( x ) − u E ( x + Δ 2 ¯ ( τ ) e ) ] 2 ⟩ , which leads to α = γ ∕ 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.