Eulerian and Lagrangian time microscales in isotropic turbulence

In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order (/) ^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 (³/) ^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_^1{2} times as large as the r. m. s. value of du / dt (R Λ 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 ^-5{3} behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.