[bhigahAshish]

Hi all,
I am reading the book "Turbulence: The Legacy of A. N. Kolmogorov" and I am facing a doubt. (you can find it here https://epdf.pub/turbulence-the-lega...olmogorov.html)

In this book Energy Spectrum E(f) is introduced as:

E(k)=dε(f)/df

where ε(f) is called "cumulative energy spectrum".

I did not understand why ε(f) is called "cumulative" and how ε(f) can be calculated. At page 53 (see image) of this book there is a definition of this parameter. Is this definition equal to the Fourier transform of the turbulent kinetic specific energy? If not how can I calculate ε(f)??

Thanks in advance.

[LuckyTran]

The velocity has been filtered using a low-pass filter. Therefore, when you take the average of this filterered velocity, it contains only the cumulative energy of the frequency scales up to the cut-off frequency and isn't the infinite-time average kinetic energy at 0 Hz of the unfiltered velocity. Taking this average using a bunch of different cutoff frequecies you then build a distribution at different frequencies which they call a spectrum.

[bhigahAshish]

But, for istance, if we take the value of the cumulative energy spectrum at the frequency of 50 Hz, the value contains the cumulative energy from 0 up to 50 Hz or just the value at 50 Hz?

[LuckyTran]

ε(50 Hz) contains the cumulative energy from 0 to 50 Hz.


E(50 Hz) is the energy density at 50 Hz.

[bhigahAshish]

Ok thank you