Return to isotropy of homogeneous shear-released turbulence

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, $S=\frac{\partial U_1}{\partial x_2}$, 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 $\langle {\omega }_i{\omega }_j\rangle$ and $\langle {\omega }_i{\omega }_j{\omega }_k\rangle$. The nonzero off-diagonal term of the second-order correlation tensor, i.e., the correlation between the streamwise and the transverse components of vorticity, $\langle {\omega }_1{\omega }_2\rangle$, decreases towards 0 over a time of the order of the Kolmogorov timescale. In comparison, the anisotropies in the diagonal components $\langle {\omega }_i^2\rangle$ ($i=1$, 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 $b_{ij}^{\omega }\equiv \frac{\langle {\omega }_i{\omega }_j\rangle }{\langle {\omega }_k{\omega }_k\rangle }-\frac{1}{3}{\delta }_{ij}$ 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.

Figure 1

(a) Evolution of components of the Reynolds stress anisotropy tensor $b_{ij}$. (b) Trajectories of the return to isotropy of HSRT on the $\xi \text{−}\eta$ invariant map. The green lines delineate the Lumley triangle. In both plots, the full lines correspond to $R_{\lambda }=155$ and the dashed lines to $R_{\lambda }=113$. The trajectories of the two Reynolds numbers on the Lumley triangle essentially superpose.

Figure 2

Time dependence of the rates of return to isotropy of the invariants $\mathrm{II}$ and $\mathrm{III}$. The blue lines correspond to ${\rho }_{\mathrm{II}}=-\frac{E/\varepsilon }{\mathrm{II}}\frac{d\mathrm{II}}{dt}$ and the magenta lines to ${\rho }_{\mathrm{III}}=-\frac{E/\varepsilon }{\mathrm{III}}\frac{d\mathrm{III}}{dt}$. The solid and dashed lines correspond to $R_{\lambda }=155$ and $R_{\lambda }=113$, respectively.

Figure 3

Evolution of the components of the anisotropy tensor $b_{ij}^{\omega }$, as a function of $t/{\tau }_{\eta }$. (a) diagonal components $b_{ii}^{\omega }$ for $i=1$ (blue), 2 (red), and 3 (magenta); (b) the nonzero off-diagonal component $b_{12}^{\omega }$. The full lines represent the solution at $R_{\lambda }=155$ and the dashed lines at $R_{\lambda }=113$.

Figure 4

Evolution of the skewness of ${\partial }_2{u}_1, S_{{\partial }_2{u}_1}$ (blue) and of ${\omega }_3, S_{{\omega }_3}$ (magenta). The full lines correspond to $R_{\lambda }=155$ and the dashed lines to $R_{\lambda }=113$.

Figure 5

Evolution of norms of odd and even parts of deviation ${\mathit{\Theta }}_{od,ev}^n$ normalized by $\left|\right|{\mathbf{T}}^{n,HIT}\left|\right|$ ($n=2,3$), (a) $\left|\right|{\mathit{\Theta }}_{ev}^2\left|\right|/\left|\right|{\mathbf{T}}^{2,HIT}\left|\right|$, (b) $\left|\right|{\mathit{\Theta }}_{od}^2\left|\right|/\left|\right|{\mathbf{T}}^{2,HIT}\left|\right|$, (c) $\left|\right|{\mathit{\Theta }}_{ev}^3\left|\right|/\left|\right|{\mathbf{T}}^{3,HIT}\left|\right|$, and (d) $\left|\right|{\mathit{\Theta }}_{od}^3\left|\right|/\left|\right|{\mathbf{T}}^{3,HIT}\left|\right|$. Time has been scaled on all graphs by $T_E$, except in the inset of panel (b), where it was scaled by ${\tau }_{\eta }$.

Figure 6

Evolution of the normalized third moment of the spanwise component of vorticity, ${\omega }_3$. The relaxation of this moment occurs over a timescale of the order $T_E$, as predicted by the elementary considerations developed in this text. As shown in the inset, however, the evolution of the third moment is nonmonotonic. The full lines correspond to $R_{\lambda }=155$ and the dashed lines to $R_{\lambda }=113$.

Figure 7

Evolution of the components of the dissipation anisotropy tensor $b_{ij}^{\varepsilon }$, defined by Eq. (7), as a function of $t/{\tau }_{\eta }$ (a); and as a function of $t/T_E$, (b). Full lines correspond to the solution at $R_{\lambda }=155$ and dashed lines to $R_{\lambda }=113$.

Figure 8

Evolution of norms of odd and even parts of deviation ${\mathit{\Theta }}_{od,ev}^n$ normalized by its values at $t=0$. $\left|\right|{\mathit{\Theta }}_{ev}^2\left|\right|\left(t\right)/\left|\right|{\mathit{\Theta }}_{ev}^2\left|\right|\left(0\right)$, time $t$ normalized by (a) ${\tau }_{\eta }$ and (b) $T_E, \left|\right|{\mathit{\Theta }}_{od}^2\left|\right|\left(t\right)/\left|\right|{\mathit{\Theta }}_{od}^2\left|\right|\left(0\right)$, time $t$ normalized by (c) ${\tau }_{\eta }$, and (d) $T_E, \left|\right|{\mathit{\Theta }}_{ev}^3\left|\right|\left(t\right)/\left|\right|{\mathit{\Theta }}_{ev}^3\left|\right|\left(0\right)$, time $t$ normalized by (e) ${\tau }_{\eta }$ and (f) $T_E, \left|\right|{\mathit{\Theta }}_{od}^3\left|\right|\left(t\right)/\left|\right|{\mathit{\Theta }}_{od}^3\left|\right|\left(0\right)$, time $t$ normalized by (g) ${\tau }_{\eta }$ and (h) $T_E$.