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LAGRANGIAN SINGLE-PARTICLE STATISTICS OF FLUID TURBULENCE
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⟨ {[u(x+r) − u(x)]}.(r/r)^3 ⟩ = −4/5 εr (1)
where r is the distance between the two points separated by vector r at which the velocities u(x + r) and u(x) are measured simultaneously. This exact “4/5-law” has been the foundations of nearly all the subsequent theoretical and phenomenological work on Eulerian statistics of turbulence. Lagrangian properties of turbulence, i.e., how turbulence looks like when traveling with fluid particles in the flow, have attracted increasing interest in the last two decades, due to a combination of theoretical breakthroughs and the rapid advances in experimental and numerical techniques. However, unlike in the cases of Eulerian statistics, there is no known exact result on inertial range Lagrangian single-particle statistics. It is commonly assumed that in the inertial range the dimensional scaling relation of the second-order Lagrangian velocity structure function holds:
D2(τ)=⟨δτv^2⟩=⟨[v(t+τ)−v(t)]^2⟩=C0ετ, (τη ≪τ ≪TL) (2)
where v(t) is the velocity following a fluid particle in turbulence, C0 is assumed to be a universal dimensionless constant, τη and TL are the Kolmogorov and integral time scales of the flow, respectively. The reason that the second-order structure function D2(t) is treated specially is that there is no apparent intermittency correction on the scaling of τ in this dimensional argument. Considerable effort has been devoted to verify Eq. (2) or to identify the numerical values of C0. However, no convincing inertial scaling relation as predicted by Eq. (2) has been observed in state-of-the-art experimental and numerical simulation data.
Based on theoretical considerations and observations of experimental and numerical data, we argue that the dimensional scaling Eq. (2) might be fundamentally flawed. For example, if Eq. (2) holds, then the following will be true:
dD2/dτ=2⟨aδτv⟩=C0ε (3)
in the inertial range. However, as shown in Figure 1, dD2/dτ observed from experiments and numerical simulations reaches maximum at approximately 2τη and then decreases nearly exponentially, without appreciable plateau range. Further investigation of the acceleration spectra obtained from numerical simulations indicate that a small correction on the scaling exponent given in Eq. (2) better fits the data: D2 ~ τ 1−μ with μ ≈ 0.1.
This small correction, if it holds true at larger Reynolds numbers, implies that the choice of available parameters in the dimensional argument leading to Eq. (2) is incorrect. If ε is not the right parameter, then what is the alternative? A deeply related question is: How are Lagrangian single-particle statistics related to ε, the energy cascade through spatial scales? We will discuss these in the presentation.
Author(s):
Haitao Xu
Max Planck Institute for Dynamics and Self-Organization (MPIDS), Gottingen
Germany