English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In Kolmogorov's phenomenological theory of turbulence, the energy spectrum inthe inertial range scales with the wave number $k$ as $k^{-5/3}$ and extends upto a dissipation wave number $k_\nu$, which is given in terms of the energydissipation rate $\epsilon$ and viscosity $\nu$ by$k_\nu\propto(\epsilon/\nu^3)^{1/4}$. This result leads to Landau's heuristicestimate 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 establishinga quantitative basis for these results from first principles. In particular, weexamine the extent to which they can be derived from the three-dimensionalNavier--Stokes system, making use of Kolmogorov's hypothesis of finite andviscosity-independent energy dissipation only. It is found that the Taylormicroscale wave number $k_T$ (a close cousin of $k_\nu$) can be expressed inthe 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 rootmean square velocity, and $C\le1$ is a dynamical parameter. This result can beseen to be in line with Kolmogorov's prediction for $k_\nu$. Furthermore, it isshown that the minimum number of greatest Lyapunov exponents whose sum becomesnegative does not exceed $\Re^{9/4}$, where $\Re$ is defined in terms of anaverage energy dissipation rate, the system length scale, and $\nu$. Thisresult is in a remarkable agreement with the Landau estimate, up to apresumably slight discrepancy between the conventional and the present energydissipation rates used in the definition of $\Re$.

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