Initial conditions in physical space from energy spectrum

arunsmec · February 19, 2016, 00:51

Hi,
For validation purpose, I want to simulate homogeneous isotropic turbulence using my code. Following literature, my initial conditions have a known one dimensional energy spectrum

. I need to give initial conditions in physical space, with

grid points.
I have learnt that I have to proceed as follows.

Generate a random velocity field in the physical domain.
Transform to wavenumber space.
Impose incompressibility condition $\hat{u}.\vec{k}=0$ , by projecting velocity vectors on plane normal to $\vec{k}$ .
Scale to the given spectrum $\hat{u}\rightarrow\frac{\hat{u}}{|\hat{u}|}E(k)$ .
Transform back to physical space

I am using Matlab to generate the initial conditions, and steps 1 and 2 looks traightforward to me.
However in step 3, how will I do the orthogonalization process $\hat{u}=\hat{u}-(\hat{u}.\vec{k})\vec{k}/k^2$ with my FFT data? What is the wavenumber vector $\vec{k}$ , when FFT data has wavenumbers in the range

in all 3 directions?

FMDenaro · February 19, 2016, 04:06

have a look here

http://users.ictp.it/~krs/pdf/1993_008.pdf

arunsmec · February 19, 2016, 05:58

Thank you for the response.
However, the document does not have any additional information than what I understand now. My problem is specific. I have the amplitude-wavenumber information (data) after transformation to Fourier space. With my limited knowledge, I don't know how to get $\hat{u}.\vec{k}$ from this available data.

FMDenaro · February 19, 2016, 15:32

using search function, you can find many posts in this forum about the initial conditions, for example:

http://www.cfd-online.com/Forums/mai...procedure.html

February 19, 2016, 00:51	Initial conditions in physical space from energy spectrum	#1
arunsmec Member Aru Join Date: Feb 2012 Location: Chennai, India Posts: 40 Rep Power: 14	Hi, For validation purpose, I want to simulate homogeneous isotropic turbulence using my code. Following literature, my initial conditions have a known one dimensional energy spectrum . I need to give initial conditions in physical space, with grid points. I have learnt that I have to proceed as follows. Generate a random velocity field in the physical domain. Transform to wavenumber space. Impose incompressibility condition $\hat{u}.\vec{k}=0$ , by projecting velocity vectors on plane normal to $\vec{k}$ . Scale to the given spectrum $\hat{u}\rightarrow\frac{\hat{u}}{\|\hat{u}\|}E(k)$ . Transform back to physical space I am using Matlab to generate the initial conditions, and steps 1 and 2 looks traightforward to me. However in step 3, how will I do the orthogonalization process $\hat{u}=\hat{u}-(\hat{u}.\vec{k})\vec{k}/k^2$ with my FFT data? What is the wavenumber vector $\vec{k}$ , when FFT data has wavenumbers in the range in all 3 directions?

February 19, 2016, 05:58	Thank you	#3
arunsmec Member Aru Join Date: Feb 2012 Location: Chennai, India Posts: 40 Rep Power: 14	Thank you for the response. However, the document does not have any additional information than what I understand now. My problem is specific. I have the amplitude-wavenumber information (data) after transformation to Fourier space. With my limited knowledge, I don't know how to get $\hat{u}.\vec{k}$ from this available data.

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February 19, 2016, 04:06		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	have a look here http://users.ictp.it/~krs/pdf/1993_008.pdf

February 19, 2016, 15:32		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	using search function, you can find many posts in this forum about the initial conditions, for example: http://www.cfd-online.com/Forums/mai...procedure.html