Temporal dynamics of the alignment of the turbulent stress and strain rate

The flux of energy between scales in turbulence can be cast as the result of the interaction between a turbulent stress and a rate of strain. Efficient energy transfer requires that these two tensors be oriented properly relative to each other, but previous work has shown that the instantaneous alignment between them is poor. Here, we consider the temporal dynamics of this alignment in a direct numerical simulation of isotropic turbulence. We show that the orientation of the stress lags behind that of the strain rate, both at a single location and along trajectories. However, the timescale of the reorientation of the stress in these cases does not follow the expected dynamical scaling with length scale. To capture the proper dynamical scaling, we reformulate the energy flux between scales using the right Cauchy-Green strain tensor and the second Piola-Kirchhoff stress tensor. Our results highlight the key role played by the deformation of fluid elements in the physics of the energy cascade, and suggest that their irreversible deformation is a Lagrangian manifestation of the cascade's broken time-reversal symmetry.

Figure 1

The energy spectrum $E\left(k\right)$, where $k$ is a wave number, plotted both (a) raw and (b) premultiplied, for the direct numerical simulation used for this analysis, with dashed lines showing the inertial range spectrum $E\left(k\right)=1.6{\epsilon }^{2/3}{k}^{-5/3}$.

Figure 2

(a) In-place, time-lagged mean efficiency, computed using the energy flux defined in Eq. (5), as a function of the time lag $\mathrm{\Delta }t$. Data are shown for filter scales $r=4.4\eta$, $8.8\eta$, $17.6\eta$, $35.2\eta$, $70.4\eta$, and $140.8\eta$, and the peak efficiency monotonically increases with $r$. (b) The value of $\mathrm{\Delta }t$ at which the efficiency peaks as a function of filter scale. The black dashed line is a reference $r^{2/3}$ power law; the red line is a best-fit power law with an exponent of 1.5.

Figure 3

(a) Time-lagged mean efficiency computed along Lagrangian trajectories using the energy flux defined in Eq. (6), as a function of the time lag $\mathrm{\Delta }t$. Data are again shown for filter scales $r=4.4\eta$, $8.8\eta$, $17.6\eta$, $35.2\eta$, $70.4\eta$, and $140.8\eta$, and the peak efficiency monotonically increases with $r$. (b) The value of $\mathrm{\Delta }t$ at which the efficiency peaks as a function of filter scale. The black dashed line is a reference $r^{2/3}$ power law; the red line is a best-fit power law with an exponent of 0.4.

Figure 4

(a) Mean Lagrangian efficiency computed using the energy flux defined in Eq. (10) as a function of the evolution time $\mathrm{\Delta }t$. Data are again shown for filter scales $r=4.4\eta$, $8.8\eta$, $17.6\eta$, $35.2\eta$, $70.4\eta$, and $140.8\eta$, and the peak efficiency monotonically increases with $r$. (b) The value of $\mathrm{\Delta }t$ at which the efficiency peaks as a function of filter scale. The black dashed line is an $r^{2/3}$ power law.