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.

Answer 3

Davidson's book suggested that working in spectral space, we need to bear in mind that turbulence consists of eddies (blobs of vorticity) and not waves, so we must guard against attributing too much physical (as opposed to kinematic) meaning to a Fourier mode in turbulence.

For DNS, higher accuracy is achieved in spectral space - compare the number of points needed to resolve a wave with the number of terms in the Fourier series needed. Choice of physical space vs spectral space is also important in LES. Filtering in LES can be e.g. Box filter (local in physical space, non-local in spectral space) vs Spectral filter (non-local in physical space, local in spectral space) or Gaussian (non-local in both, but smooth in both). Choice of filter can be a question of efficiency.