# How to compute turbulent energy cascade

I need to compute the kinetic energy cascade using a finite volume solution in an equally spaced grid. I wonder if it is more correct to first compute the kinetic energy in the space (or time) domain, $$E(x,t)=0.5(u(x,t)^2+v(x,t)^2)$$, and then apply the FFT to obtain $$E(k)$$, or applying first the FFT to $$u$$ and $$v$$, obtaining $$u(k)$$ and $$v(k)$$, to finally compute $$E(k)=(u(k)^2+v(k)^2)$$. Is there a correct option?

Of course, the Fourier transform is a linear operator. So, you have the kinetic energy defined as: $$E(\mathbf{r}) = \frac{1}{2} \mathbf{u}(\mathbf{r}) \cdot \mathbf{u}(\mathbf{r})$$. The Fourier transform of $$\mathbf{u}$$ and $$E$$ are:

$$\tilde{E}(\mathbf{k}) = \int_{\Omega} E(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^{3} \mathbf{r}$$

$$\tilde{\mathbf{u}}(\mathbf{k}) = \int_{\Omega} \mathbf{u}(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^{3} \mathbf{r}$$

But you see obviously: $$\tilde{E}(\mathbf{k}) \neq \frac{1}{2} \tilde{\mathbf{u}} (\mathbf{k}) \cdot \tilde{\mathbf{u}}(\mathbf{k})$$ because:

$$\frac{1}{2} \tilde{\mathbf{u}} (\mathbf{k}) \cdot \tilde{\mathbf{u}}(\mathbf{k}) = \frac{1}{2} \int_{\Omega} \int_{\Omega} \mathbf{u}(\mathbf{r}) \cdot \mathbf{u}(\mathbf{r}^{'}) e^{-i\mathbf{k}\cdot (\mathbf{r}+\mathbf{r}^{'})} d^{3} \mathbf{r} d^{3} \mathbf{r}^{'} \neq \int_{\Omega} E(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^{3} \mathbf{r} = \tilde{E}(\mathbf{k})$$

So, just calculate $$E$$ everywhere in your domain ($$\Omega$$) and then take the Fourier transform of it and not taking Fourier transform of velocity ($$\mathbf{u}$$) and then take its Euclidean norm.

Update:

If you insist to calculate Fourier transform ($$\mathscr{F}$$) of $$E$$ from $$\mathbf{u}$$, you have this relation based on convolution theorem:

$$\mathscr{F}\{\mathbf{u}(\mathbf{r})\cdot\mathbf{u}(\mathbf{r})\}=\mathscr{F}\{\mathbf{u}(\mathbf{r})\}*\mathscr{F}\{\mathbf{u}(\mathbf{r})\}$$

or

$$\tilde{E}(\mathbf{k}) = \mathscr{F} \{ E(\mathbf{r}) \} = \frac{1}{2} \mathscr{F} \{ \mathbf{u}(\mathbf{r}) \} * \mathscr{F} \{ \mathbf{u}(\mathbf{r}) \} = \frac{1}{2} \int_{\Omega_{\mathbf{k}}} \tilde{\mathbf{u}}(\mathbf{k}^{'}) \cdot \tilde{\mathbf{u}}(\mathbf{k}-\mathbf{k}^{'}) d^{3} \mathbf{k}^{'}$$

Where $$\Omega_{\mathbf{k}}$$ is the computational domain in reciprocal space (Fourier space).