Eulerian and Lagrangian time microscales in isotropic turbulence

Abstract

In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order $(\epsilon/\nu)^{\frac{1}{2}}$ (ε is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value of du/dt of order $(\epsilon^3/\nu)^{\frac{1}{4}}$. The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of ∂u/∂t at a fixed observation point is (apart from a numerical factor) $R_{\lambda}^{\frac{1}{2}}$ times as large as the r.m.s. value of du/dt (RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a $\omega^{-\frac{5}{3}} $ behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.