Ultra-chaotic property of Navier-Stokes turbulence

TL;DR

The paper proposes that Navier–Stokes turbulence may be ultra-chaotic, with statistics that remain sensitive to tiny initial disturbances. It tests this idea using a 2D Kolmogorov flow at $Re=2000$ and $n_K=16$, employing clean numerical simulation (CNS) to suppress numerical noise and compare three nearly identical initial fields with distinct symmetries. The results show that perturbations of order $10^{-10}$ can yield macroscopically different symmetry outcomes and statistical measures, including $\langle D_\Omega\rangle_A$, PDFs of $E$, $\Omega$, $D$, $D_\Omega$, and the energy spectra, while the flows still obey the $-5/3$ law. This challenges the notion that NS turbulence statistics are robust to small disturbances and motivates incorporating stochastic disturbances or non-differentiable dynamics (e.g., LLNS) into turbulence models; it also points toward alternative approaches like wave-particle turbulence simulations for more faithful representations of turbulent flows.

Abstract

A chaotic system is called ultra-chaos when its statistics have sensitivity dependence on initial condition and/or other small disturbances. In this paper, using two-dimensional turbulent Kolmogorov flow as an example, we illustrate that tiny variation of initial condition of Navier-Stokes equations can lead to huge differences not only in spatiotemporal trajectory but also in flow symmetry and its statistics. Here, in order to avoid the influence of artificial numerical noise, we apply ``clean numerical simulation'' (CNS) which can guarantee that the numerical noise can be reduced to such a desired low level that they are negligible in a time interval long enough for calculating statistics. This discovery highly suggests that the Navier-Stokes turbulence (i.e. turbulence governed by the Navier-Stokes equations) might be an ultra-chaos, say, small disturbances must be considered even from viewpoint of statistics. This however leads to a paradox in logic, since small disturbances, which are unavoidable in practice, are unfortunately neglected by the Navier-Stokes turbulence. Some fundamental characteristics of turbulence model are discussed and suggested in general meanings.

Figures (6)

• Figure 1: Vorticity fields $\omega$ at $t=100$ of the 2D turbulent Kolmogorov flow governed by (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) in the case of $n_K=16$ and $Re=2000$, given by CNS subject to the initial conditions (\ref{['initial_condition-1']}) (left, marked by Flow CNS-1), (\ref{['initial_condition-2']}) (middle, marked by Flow CNS-2), and (\ref{['initial_condition-3']}) (right, marked by Flow CNS-3), respectively.
• Figure 2: Comparison of time histories of the spatially averaged enstrophy dissipation rate $\langle D_\Omega\rangle_A$ of the 2D turbulent Kolmogorov flow governed by (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) in the case of $n_K=16$ and $Re=2000$, given by Flow CNS-1 (red line), Flow CNS-2 (green line), Flow CNS-3 (blue line), respectively.
• Figure 3: Comparison of probability density functions (PDFs) of (top-left) kinetic energy $E(x,y,t)$, (top-right) enstrophy $\Omega(x,y,t)$, (bottom-left) kinetic energy dissipation $D(x,y,t)$, and (bottom-right) enstrophy dissipation rate $D_\Omega(x,y,t)$ of the 2D turbulent Kolmogorov flow, governed by (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) in the case of $n_K=16$ and $Re=2000$, given by Flow CNS-1 (red line), Flow CNS-2 (green line), Flow CNS-3 (blue line), respectively. Their definitions are given in Appendix.
• Figure 4: Comparison of the spatiotemporally averaged (left) kinetic energy dissipation rate $\langle D\rangle_{x,t}(y)$ and (right) enstrophy dissipation rate $\langle D_\Omega\rangle_{x,t}(y)$ of the 2D turbulent Kolmogorov flow governed by Eqs. (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) in the case of $n_K=16$ and $Re=2000$, given by Flow CNS-1 (red line), Flow CNS-2 (green line), Flow CNS-3 (blue line), respectively.
• Figure 5: Comparison of the spatiotemporally averaged (top-left) horizontal velocity $\langle u\rangle_{x,t}(y)$, (top-right) normal stresses $\langle u'u'\rangle_{x,t}(y)$, (bottom-left) normal stresses$\langle v'v'\rangle_{x,t}(y)$, and (bottom-right) shear stress $\langle u'v'\rangle_{x,t}(y)$ of the 2D turbulent Kolmogorov flow, governed by (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) in the case of $n_K=16$ and $Re=2000$, given by Flow CNS-1 (red line), Flow CNS-2 (green line), Flow CNS-3 (blue line), respectively. Their definitions are given in Appendix.