The number of degrees of freedom of three-dimensional Navier--Stokes turbulence

TL;DR

This paper investigates the fundamental limits of degrees of freedom in 3D Navier-Stokes turbulence, deriving bounds consistent with Kolmogorov's theory and Landau's heuristic estimates using first principles.

Contribution

It provides a quantitative derivation of turbulence scales and degrees of freedom bounds directly from the Navier-Stokes equations, confirming classical turbulence predictions.

Findings

Derived an upper bound for the Taylor microscale wave number consistent with Kolmogorov's theory.

Established that the number of dominant Lyapunov exponents scales as Re^{9/4}.

Confirmed the Landau estimate for degrees of freedom from first principles.

Abstract

In Kolmogorov's phenomenological theory of turbulence, the energy spectrum in the inertial range scales with the wave number $k$ as $k^{-5/3}$ and extends up to a dissipation wave number $k_\nu$, which is given in terms of the energy dissipation rate $\epsilon$ and viscosity $\nu$ by $k_\nu\propto(\epsilon/\nu^3)^{1/4}$. This result leads to Landau's heuristic estimate for the number of degrees of freedom that scales as $\Re^{9/4}$, where $\Re$ is the Reynolds number. Here we consider the possibility of establishing a quantitative basis for these results from first principles. In particular, we examine the extent to which they can be derived from the three-dimensional Navier--Stokes system, making use of Kolmogorov's hypothesis of finite and viscosity-independent energy dissipation only. It is found that the Taylor microscale wave number $k_T$ (a close cousin of $k_\nu$) can be expressed in the form $k_T \le CU/\nu = (CU/\norm{\u})^{1/2}(\epsilon/\nu^3)^{1/4}$. Here $U$ and $\norm{\u}$ are, respectively, a ``microscale'' velocity and the root mean square velocity, and $C\le1$ is a dynamical parameter. This result can be seen to be in line with Kolmogorov's prediction for $k_\nu$. Furthermore, it is shown that the minimum number of greatest Lyapunov exponents whose sum becomes negative does not exceed $\Re^{9/4}$, where $\Re$ is defined in terms of an average energy dissipation rate, the system length scale, and $\nu$. This result is in a remarkable agreement with the Landau estimate, up to a presumably slight discrepancy between the conventional and the present energy dissipation rates used in the definition of $\Re$.