Accelerations in isotropic and homogeneous turbulence and Taylor’s hypothesis

Abstract

The validity of Taylor’s hypothesis is analyzed by comparing the root mean square (rms) values of full (Lagrangian) and inertial accelerations in an isotropic and homogeneous turbulent flow. Full, local, and inertial accelerations in turbulence were decomposed into solenoidal and potential components, which made it possible to avoid dealing, at least directly, with the pressure-gradient term in the Navier–Stokes equation. The evaluations of the correlation functions and spectra of the accelerations are presented. These evaluations have been obtained using the Batchelor [Proc. Cambridge Philos. Soc. 47, 359 (1951)] longitudinal structure function that describes statistical properties of the turbulent velocity field. This function is equally valid for both inertial and dissipative subranges. It was shown that the ratio of the rms values of the full and inertial accelerations depends on the Reynolds number Rλ only and decreases at large Rλ as Rλ−1/2. At Rλ of about 20 this ratio is close to 0.72. At Rλ of 1000 the ratio is less than 0.1. The validity of Taylor’s hypothesis depends on the ratio of the rms values of the accelerations. The results indicate that Taylor’s hypothesis is valid for large Rλ (exceeding about 1000) and becomes questionable at Rλ below 100. At large Rλ the full acceleration in homogeneous and isotropic turbulence turned out to be independent of the Reynolds number.