[lucamirtanini]

Hi all,
I am trying to write the theory of the energy spectrum for a generic inhomogeneous turbulence. I used as a basis for my work the tennekes and lumley book. I attached here the page of this book and I tried to rewrite the spectrum tensor for inhomogeneous turbulence.

Can someone confirm that I did it well? My doubt is the fact that in the exp() there is no x vector and I am wandering if I am missing something.

I really hope that someone can help me

[Santiago]

Originally Posted by lucamirtanini

Hi all,
I am trying to write the theory of the energy spectrum for a generic inhomogeneous turbulence. I used as a basis for my work the tennekes and lumley book. I attached here the page of this book and I tried to rewrite the spectrum tensor for inhomogeneous turbulence.

Can someone confirm that I did it well? My doubt is the fact that in the exp() there is no x vector and I am wandering if I am missing something.

I really hope that someone can help me

Because the integration is made for a "ball", hence the radius r. That operator is quite tricky to implement, specially when you come close to a boundary. Besides, is not common to consider the cross correlations in spectral analysis... the diagonal terms have direct physical meaning in spectral space

[FMDenaro]

Originally Posted by lucamirtanini

Hi all,
I am trying to write the theory of the energy spectrum for a generic inhomogeneous turbulence. I used as a basis for my work the tennekes and lumley book. I attached here the page of this book and I tried to rewrite the spectrum tensor for inhomogeneous turbulence.

Can someone confirm that I did it well? My doubt is the fact that in the exp() there is no x vector and I am wandering if I am missing something.

I really hope that someone can help me

I am not sure of what exactly want to do. If you consider a fully 3d non-homogenous non-isotropic turbulence, it could be also questioned how do you define the Fourier analysis since it is not a periodic function.
From the general definition you wrote, there is more that the fact your function depends on the position x. The problem is also in the extension of the integrals.


I can say that in a practical case like the flow in a plane channel, where we have two homogeneous dimensions and one (normal to the wall) non-homogeneous dimension, we perform the 1D spectra along the directions of homogeneity. And these spectra are evaluated for several positions along the non-homogenous direction.

[lucamirtanini]

Originally Posted by Santiago

Because the integration is made for a "ball", hence the radius r. That operator is quite tricky to implement, specially when you come close to a boundary.

In the step of the demonstration that I have shown there is no integration on the radius.

Originally Posted by Santiago

Besides, is not common to consider the cross correlations in spectral analysis... the diagonal terms have direct physical meaning in spectral space

I will not use this formula directly in a calculation. I was trying to generalize the demonstration fora a 3d inhomogeneous turbulence.

[lucamirtanini]

Originally Posted by FMDenaro

I am not sure of what exactly want to do. If you consider a fully 3d non-homogenous non-isotropic turbulence, it could be also questioned how do you define the Fourier analysis since it is not a periodic function.

I do not see a linke between the fact of being a 3d inhomogeneous turbulnence and the use of the fourier analysis.

Originally Posted by FMDenaro

From the general definition you wrote, there is more that the fact your function depends on the position x. The problem is also in the extension of the integrals.


I can say that in a practical case like the flow in a plane channel, where we have two homogeneous dimensions and one (normal to the wall) non-homogeneous dimension, we perform the 1D spectra along the directions of homogeneity. And these spectra are evaluated for several positions along the non-homogenous direction.

In my opinion, the extension of the integral is not the problem, since it is a generic demonstration.

[FMDenaro]

https://ntrs.nasa.gov/search.jsp?R=19790022816

[lucamirtanini]

Originally Posted by FMDenaro

https://ntrs.nasa.gov/search.jsp?R=19790022816

Thank you. I have already read this one. He wrote the formula like I did, but the formula 1.7 seems to me to be wrong since there is not the (1/2pi)^3. Do you agree?

[FMDenaro]

Originally Posted by lucamirtanini

Thank you. I have already read this one. He wrote the formula like I did, but the formula 1.7 seems to me to be wrong since there is not the (1/2pi)^3. Do you agree?

If you see the 1.8, the factor appears in the correlation function

[lucamirtanini]

Yes, but it should have been placed also in the 1.7. Isn't it?

[FMDenaro]

Originally Posted by lucamirtanini

Yes, but it should have been placed also in the 1.7. Isn't it?

No, correlation and spectra are a Fourier pair transform, for example here you see the factor in the spectra but not in the correlation
https://www.google.com/url?sa=t&rct=...VEGq8lnm20jOIu

[lucamirtanini]

Originally Posted by FMDenaro

No, correlation and spectra are a Fourier pair transform, for example here you see the factor in the spectra but not in the correlation
https://www.google.com/url?sa=t&rct=...VEGq8lnm20jOIu

No, if you look at the page of the tenneke and lumley book that I have shown in the first post. You can see that the psi in the paper is the phi in the book of lumley, and the Cij in the paper is the Rij in the book of Lumley. So the (1/2pi)^3 is missing

[LuckyTran]

Originally Posted by lucamirtanini

No, if you look at the page of the tenneke and lumley book that I have shown in the first post. You can see that the psi in the paper is the phi in the book of lumley, and the Cij in the paper is the Rij in the book of Lumley. So the (1/2pi)^3 is missing

They have different definitions for the Fourier transform and that's why you see the 2pi in different places.


Or am I misunderstanding the question?

[lucamirtanini]

Originally Posted by LuckyTran

They have different definitions for the Fourier transform and that's why you see the 2pi in different places.


Or am I misunderstanding the question?

Sorry for the stupid question, what does it means that they have different definitions for the F T? Which definition of FT is used by Trevino and which one is used by Lumley and Tennekes?

[LuckyTran]

Lumley is using the one with the 1/(2pi) and Trevino is using the one with 1 as the prefactor.

[lucamirtanini]

Originally Posted by LuckyTran

Lumley is using the one with the 1/(2pi) and Trevino is using the one with 1 as the prefactor.

Ok...I am not enough expert of FT. How can they be equivalent? Is the 1/(2pi) included in the correlation coefficient in the trevino paper?

[FMDenaro]

Originally Posted by lucamirtanini

Ok...I am not enough expert of FT. How can they be equivalent? Is the 1/(2pi) included in the correlation coefficient in the trevino paper?

Have a look to pag 220 of the Pope's textbook, you see the factor (2pi)^3 in the spectrum but not in the correlation. They constitute a Fourier transform pair. The Fourier transform has the factor 2*pi, the inverse transform has not, see Appendix A (pag 678). You will read the explicit statement about the fact that this factor is sometimes interchanged between direct and inverse transform.


Let me say that I often observed similar notations that drive people to some misleading reading, for example the Fourier decomposition in components along wavenumbers k=n*2pi/L where L is the length of periodicity. When sometimes is assumed L=2*pi, the factor 2*pi disappears and k=n. But n is the wavenumber while k is the dimensional frequency.

[lucamirtanini]

Originally Posted by FMDenaro

Have a look to pag 220 of the Pope's textbook, you see the factor (2pi)^3 in the spectrum but not in the correlation. They constitute a Fourier transform pair. The Fourier transform has the factor 2*pi, the inverse transform has not, see Appendix A (pag 678). You will read the explicit statement about the fact that this factor is sometimes interchanged between direct and inverse transform.


Let me say that I often observed similar notations that drive people to some misleading reading, for example the Fourier decomposition in components along wavenumbers k=n*2pi/L where L is the length of periodicity. When sometimes is assumed L=2*pi, the factor 2*pi disappears and k=n. But n is the wavenumber while k is the dimensional frequency.

If you look carefully at the trevino paper you will see that the equations 1.7 and 1.8 are not fourier transform pairs. So the factor could not be introduced in 1.8. If there was a definition of Cij as an inverse fourier transform of psi, than I can understand your statement