How to compute turbulent energy cascade

Question

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?

Answer 1

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).