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Abstract:
The presence of mean velocity gradients induces anisotropies in turbulent flows, which affect even the smallest scales of motion at finite Reynolds numbers. By performing direct numerical simulations of the Navier-Stokes equations, we study the return to isotropy of a homogeneous turbulent flow initially in a statistically stationary state under a uniform shear, S = partial derivative U-1/partial derivative x(2), in the conceptually simple situation where the mean shear is suddenly released. In particular, we characterize the timescales involved in the dynamics. We observe that the Reynolds stress tensor, which measures the large-scale flow anisotropy, relaxes towards an isotropic form over a timescale of the order of the large-eddy turnover time of turbulence, in qualitative agreement with previous studies with different types of initially imposed mean velocity gradient. We also investigate how the correlations of the velocity gradient tensor relax to isotropy with time. In particular, we focus on the properties of the one-point vorticity correlations <omega(i)omega(j)> and <omega(i)omega(j)omega(k)>. The nonzero off-diagonal term of the second-order correlation tensor, i.e., the correlation between the streamwise and the transverse components of vorticity, <omega(1)omega(2)>, decreases towards 0 over a time of the order of the Kolmogorov timescale. In comparison, the anisotropies in the diagonal components <omega(2)(i)> (i = 1, 2, or 3) relax over a time significantly longer than the Kolmogorov timescale. This difference can be explained by an elementary theoretical analysis of the dynamics of the anisotropy tensor b(ij)(omega) <omega(i)omega(j)>/<omega(k)omega(k)> - 1/3 delta(ij) at the instant when the shear is released. We also observe that the skewness of the spanwise component of vorticity, S-omega 3, relaxes slowly towards zero. The relaxation of a small-scale quantity over a time much longer than the Kolmogorov timescale, as surprising as it may seem, is in fact consistent with a known relation between velocity-gradient correlations and the pressure-rate-of-strain correlation, and raises the important question of separation between the timescales characterizing the return to isotropy at large and small scales.
