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

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 $k_T \le CU/ν= (CU/\norm{\u})^{1/2}(ε/ν^3)^{1/4}$. Here $U$ and $\norm{\u}$ are, respectively, a ``microscale'' velocity and the root mean square velocity, and $C\le 1$ 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 $\Re$.

6--7 journal pages, to appear in Physics of Fluids

References in corpus (4)

• The multifractal nature of turbulent energy dissipation
• Matrix Exponential-Based Closures for the Turbulent Subgrid-Scale Stress Tensor
• Number of degrees of freedom of two-dimensional turbulence
• Estimating intermittency in three-dimensional Navier-Stokes turbulence

Cited by in corpus (2)

• Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence
• Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows