Ways to compute the turbulent velocity kinetic energy spectrum

The implicit definition of the kinetic energy spectrum $\hat{E}_k$ of the turbulent flow is simply:

$$E = \int \hat{E}(k) \mathrm{d}k$$

I have found the definition using Fourier transform of a two-point correlation tensor (compute the two-point correlation tensor $R_{ij}$, take the Fourier transform, calculate the trace of the transformed tensor, integrate the trace in the wave-vector domain to obtain just $\hat{E}(|k|)$).

Then I have read that one could use simply:

$$\hat{E}(|k|) = 4\pi k^2|\hat{v}(k)|^2$$

where $\hat{v}$ is the transformed velocity and $k$ could be obtained from appropriate FFT theorems for numerical purposes.

• Is it really that simple?
• How would possible weak inhomogenity and/or anisotropy harm the results or their interpretation?

Why do you think the path you have described is "simple"? Think about how you could obtain the information required to follow your prescription. For example, where does that two-point correlation tensor come from?

Now, for your "simple" approach, how do you find $\vert\hat{v}(k)\vert$?

So, yeah, if you have the information required to evaluate your expression, then it is that simple. Trouble is, finding such data is anything but simple. It could be done experimentally/numerically, or you could do an asymptotic analysis for, say, homogeneous isotropic turbulence, in which case you'd eventually recover Kolmogorov's results. If you relax your assumptions to include non-homogeneity/anisotropy, things become pretty much unmanageable for a first-principles approach, and you'll be forced to introduce modeling assumptions.

I like to recommend Pope's book on turbulence for a more detailed expansion on these topics.

• +1 for the Pope reference. The appendices are of use for this question. Commented Nov 27, 2016 at 17:55

It is not that simple. The steps you cite are the correct way to find it. To be more precise, the first relation you wrote should be using the wave vector instead of the wave number: $$E(t)=\int_{\mathbb{R}^3}\hat{E}(\vec{k},t)\,\mathrm{d}\vec{k}$$ (We should not use the same letter, $\mathcal{K}(t)$ could be more appropriate.) The integration is made over the whole real space ($\mathbb{R}^3$). We can rewrite this in spherical coordinates using : $$\mathrm{d}\vec{k} = k^2\sin\theta\, \mathrm{d}k \, \mathrm{d}\theta \, \mathrm{d}\phi$$ with $\theta$ the polar angle and $\phi$ the azimuthal angle. This leads to \begin{align}E(t)=\mathcal{K}(t)&=\int_{\mathbb{R}^3}\hat{E}(\vec{k},t)\,\mathrm{d}\vec{k}\\&=\color{blue}{\int_{-\infty}^{+\infty}}\color{red}{\int_{0}^{\pi}}\color{green}{\int_{0}^{2\pi}}k^2\sin\theta\,\hat{E}(k,\theta,\phi,t) \, \color{blue}{\mathrm{d}k}\, \color{red}{\mathrm{d}\theta}\,\color{green}{\mathrm{d}\phi}. \end{align}

Now if the flow is isotropic, the direction of the wave vector has no influence on the results, i.e. $\hat{E}(k,\theta,\phi,t)=\hat{E}(k,t)$. So the integration over $\theta$ and $\phi$ leads you to your simple definition : $$\mathcal{K}(t)=\int_{-\infty}^{+\infty}4\pi k^2 \hat{E}(k,t) \, \mathrm{d}k$$ with only a dependency over the wave number. Also depending on the characteristic of your flow, your definition is not enough to take all the effects into account. If you have an axisymmetric flow for instance, you may be able to integrate over $\phi$ but not $\theta$ (with $\theta$ the angle between the wave vector and the preferred direction). Using only the wave number, you will have a global and mean value which cannot describes the complexity of the flow. It will overestimate or underestimate the turbulent kinetic energy depending on the case.

And as Pirx stated, what is $|\hat{v}(k)|$? In an anisotropic flow, the component of the fluctuating velocity field are not equivalent. Also using one direction over another will biased the value of the kinetic energy. If $\hat{v}$ is a simple mean of the three components, you will lost information and then again over or underestimate the turbulent kinetic energy.

In addition to Pope's book, you could take a look at the chapter 5 of Lesieur's book Turbulence in Fluids on the Fourier Analysis of Homogeneous Turbulence. You could also find useful (and a bit complex) tools on anisotropic decomposition of turbulent field in the chapter 2.6 of the book Homogeneous Turbulence Dynamics by Sagaut & Cambon. (I will edit this message with more references if I find any.)

• McCombs Homogeneous, Isotropic Turbulence might also be relevant. Commented Jan 30, 2023 at 13:21