Why is the turbulent energy cascade described as function of a wavenumber?

Question

In all the literature I've seen the turbulent energy spectrum described as $E(k)$ instead of $E(L)$, i.e. as a function of a wave number not eddy size. The connection via $k=2\pi/\lambda$ is clear, but exactly what wave process is meant here. Is the idea that turbulent flow can be viewed as a superposition of waves? Waves of what? Or is this just a common notation used for energy spectra?

Answer 1

I think this is just straightforward from linear wave problems. For example, a single wave with fix wavelength $\lambda$ has the form $E(x) \sim exp(-ik_0x)$ in position space can be very simple in spectral space as $E(k) \sim \delta(k-k_0)$. For nonlinear wave (turbulence and so on) problems, many information can also be more clearly in spectral space, e.g., some waves are very turbulent in position space can be decoupled to several single k.

And, the energy and momentum can be also very simple in spectral space.

The most famous work of turbulence should be Kolmogorov1941 paper (see a modern description in "Frisch, U., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995"), which had already discussed in k space, $E(k)\sim\epsilon^{2/3}k^{-2/5}$.

However, I wonder whether people can develop other mathematical treatments to this problem. I think the best tool for turbulence is yet to be born.

Answer 2

A primary reason for this terminology is the nature of the nonlinear term in most fluid equations. In the 1st order Navier-Stokes equation, \begin{equation} \frac{\partial \mathbf{u}}{\partial \mathbf t} + (\mathbf{u}\cdot\nabla) \mathbf{u} = -\frac{1}{\rho}\nabla\overleftrightarrow{P}+\frac{1}{\rho}\nabla\overleftrightarrow{\pi} + \mathbf{F} \end{equation} the nonlinear term (the one that interacts the velocity field $\mathbf{u}$) with itself) is \begin{equation} (\mathbf{u} \cdot \nabla) \mathbf{u} \end{equation} Invoking a plane wave expansion, we can replace the $\nabla$ (which is $\partial/\partial\mathbf{x}$) with $\mathbf{k}$, so that the nonlinear part of the expression is \begin{equation} (\mathbf{u}\cdot\mathbf{k})\mathbf{u}. \end{equation} Expressing the waves in terms of their wavenumber, rather than their length, directly communicates the way in which they nonlinearly interact with other waves.