Topological entropy of stationary three-dimensional turbulence

Abstract

Topological entropy serves as a viable candidate for quantifying mixing and complexity of a highly chaotic system. Particularly in turbulence, this is determined as the exponential stretching rate of a fluid material line that typically necessitates a Lagrangian description. We extend our recent work [A. Manoharan, S. Subramanian, and A. Joy, Phys. Rev. E 112, 015106] to three dimensions, and present an exact Eulerian framework to compute the topological entropy of stationary turbulent flows. The only prerequisite is a distribution of eigenvalues of the local strain-rate tensor and their decorrelation times. This can be easily obtained from a single wire probe at a fixed location, thereby eliminating the need for Lagrangian particle tracking which is formidable due to the chaotic nature of the flow. We believe that our results lend great utility in experiments targeting transport and mixing in many industrial and natural flows.

Figures (7)

• Figure 1: Top: Time evolution of the material curve initialized as a straight line in a typical numerical simulation of a turbulent flow at Re $=10^3$. The domain of the flow is a periodic box of size $(2\pi)^3$. For the details of the numerical simulation, refer Section \ref{['sec:level3']}. Bottom: The exponential growth rate at late time ($\mathcal{ST} > 10$) provides the topological entropy of the flow that is plotted in Fig. \ref{['fig:top_ent']} as the Lagrangian measure.
• Figure 2: A typical tracer pair separated by a distance $d_0$ deforms to $d_\tau$ over a small time $\tau$
• Figure 3: Auto-correlation functions of the eigenvalues $\lambda$ and $\zeta$ as defined in Eq. \ref{['eqn:acf']} for a representative Reynold's number Re $=10^3$. Solid lines are fits to the exponential decays. The e-folding time of the eigenvalue $\lambda$ is taken as $\tau \approx 1.5$ as it exhibits the slower decay. This is the working protocol at all Re. Inset: Decorrelation time as a function of Re. One must typically wait for a time $\mathcal{T} \gg \tau$ for the material curve to reach exponential growth.
• Figure 4: Cumulative distribution functions of the top two eigenvalues $\lambda$ and $\zeta$ at a representative Re = $10^3$. They are stationary as confirmed by the corresponding insets. By independently sampling the data in both space and time we get an identical distribution that clearly demonstrates ergodicity in our flow. Dashed line in each figure shows a fit with the log-normal distribution. Note: The distribution of $\zeta$ is shifted on the right by 0.46 as it can sample negative values too. The mean however, is positive, with $\langle \zeta \rangle \approx 0.14$ at this Re.
• Figure 5: Convergence in the estimate of the topological entropy ($\approx$ 0.376) as we increase the sample size of eigenvalues $N$ at a representative Re $= 10^3$. Inset shows that the relative error in the estimation drops below 3% as $N \sim \mathcal{O}(10^3)$ which is easily attainable in experiments.