tke(f)=E(f) ???

lucamirtanini · June 23, 2019, 15:29

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

bhigahAshish · June 25, 2019, 09:56

I think that (1) and (2) are correct.
But I notice some problem with the third. Firt of all you need to do a Fourier transform and not a fft which are very different things.
Unfortunately, I am not sure that tke(f) and E(f) are the same thing.

June 23, 2019, 15:29	tke(f)=E(f) ???	#1
lucamirtanini Senior Member luca mirtanini Join Date: Apr 2018 Posts: 165 Rep Power: 6	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

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June 25, 2019, 09:56		#2
bhigahAshish Member Ashish bhigah Join Date: Jan 2019 Posts: 31 Rep Power: 6	I think that (1) and (2) are correct. But I notice some problem with the third. Firt of all you need to do a Fourier transform and not a fft which are very different things. Unfortunately, I am not sure that tke(f) and E(f) are the same thing.