An experimental test of Corrsin's conjecture and some related ideas

Abstract

We present an experimental test of Corrsin's conjecture against experimental data obtained by a particle tracking technique in approximately homogeneous and isotropic turbulent flow at Reynolds numbers Rλ ≈ 100. The conjecture states that where RL(t − t′) = ⟨v(t)·v(t′)⟩ is the Lagrangian velocity covariance function, G is the single particle mean Green's function, and RE(x − x′, t − t′) ≡ ⟨u(x, t)·u(x′, t′)⟩ is the Eulerian two-point, two-time velocity covariance function. All terms in the relation have been measured in the experiment. The equation is exact if a conditional Lagrangian velocity function RL(t|x) is inserted in place of RE(x, t) on the right-hand side. RL(t|x) is obtained by restricting sampling of the two velocities to situations where both belong to the same fluid particle trajectory. The experimental data show that the RE(x, t) and RL(t|x) behave fundamentally differently, thereby seriously questioning the rationale of the conjecture. The estimate of RL(t), based on Corrsin's conjecture and the experimentally determined RE and G, is found to decrease too fast compared to the directly measured RL, thus underestimating the Lagrangian timescale by about 40%. Even asymptotically (t → ∞) the estimate is considerably lower than the measured Lagrangian correlation function. The simpler relation RL(t) = RE(0, t), which has also been attributed to Corrsin, appears to agree much better with data. Various simple physical models of the spatiotemporal Eulerian correlation function have been compared with data, and models inspired by eddy sweeping seem to perform well.