CFD Online Discussion Forums

Hi,
In a lot of book about turbulence (Pope, Reynolds, Tenneke and Lumley...) there is written this formula that define the spectrum:
$\frac{1}{2}$ ( $\overline{u^{2}}$ + $\overline{v^{2}}$ + $\overline{w^{2}}$ )= $\int$ E(k)dk (1)

Of course, it could be written as well:
$\frac{1}{2}$ ( $\overline{u^{2}}$ + $\overline{v^{2}}$ + $\overline{w^{2}}$ )= $\int$ E(f)df (2)

Now, if the (2) is true, starting from the time history of the specific turbulent kinetic energy (let's call it tke(t)), I can do the fft of it

fft(tke(t))= tke(f) (3)

and the tke(f) should be equal to E(f).

Unfortunately, when I apply the formula (2) doing
$\frac{1}{2}$ ( $\overline{u^{2}}$ + $\overline{v^{2}}$ + $\overline{w^{2}}$ )= $\int$ tke(f)df
this formula is not verified (there are a lot of magnitude order of differences).

So I am wandering if I am doing mistake in the theory and which are my mistakes. I hope you can help me