[FMDenaro]

Have a look to discretization in 3.3.3 for example

[lietoflaw]

Originally Posted by FMDenaro

Be careful, you are discussing one of the possibile FV flux reconstruction. The mean value formula is indeed not really useful as an implicit filtering, exactly as you observed. Different discretizations are more effective.

Sorry, professor. I fail to get the point.
First, why is the mean valve formula not useful as an implicit fliter. Can you explain it with more details?
Second, what do different discretizations mean? Derivative scheme or different integral scheme?

[FMDenaro]

Originally Posted by lietoflaw

Sorry, professor. I fail to get the point.
First, why is the mean valve formula not useful as an implicit fliter. Can you explain it with more details?
Second, what do different discretizations mean? Derivative scheme or different integral scheme?

The mean value formula writes the discrete integral as f(i)*h, being the node i the centroid. Use the spectral analysis and see the resulting transfer function. You will discover that there is not smoothing at all of the high wavenunber component.

[FMDenaro]

Different discretizations both for the integral and for the derivative (in the diffusive flux)

[lietoflaw]

Originally Posted by FMDenaro

The mean value formula writes the discrete integral as f(i)*h, being the node i the centroid. Use the spectral analysis and see the resulting transfer function. You will discover that there is not smoothing at all of the high wavenunber component.

I think that I've discovered the key to demonstrate the source of the puzzle.

First of all, a concept have to be illuminated.
The discreted varibles are not obtained directly and change when the discreted methods are different.

For example, in the FD method.
where means the approximation.

But it's a little more complex in the FV method
For the volume term:



For the face term:



And these approximation terms have to be displayed by some discrete schemes, which creates two levels of the whole dicrete procedure.



Due to the difference on the level of approximation, the FD method just owns the implicit filter on discrete level, leading a 1D implicit filter.

Back to the original description, the mean value formula is actually not useful as an implicit filter on the discrete level while it inherits the filter from the approximation level.

If only the implicit filter on discreted level should be considered, the filter obtained from eqn (25) is the final anwser, which works on the discreted variables.

However, if we consider the filter works on the actual variable, these filter induced by approximation effects shouldn't be overlooked

[FMDenaro]

You are in progress but be careful about the concept of averaging in wavenumber space. Only looking to the resulting transfer function induced by each scheme you can better see the effect of the implicit filtering

[lietoflaw]

Originally Posted by FMDenaro

You are in progress but be careful about the concept of averaging in wavenumber space. Only looking to the resulting transfer function induced by each scheme you can better see the effect of the implicit filtering

Thank you for your help! I had never expected that someone, rather than a professor, would like to discuss LES with me openly and deeply before this thread. Your paper and explaination really help me clarify the concept of implicit filter.

[FMDenaro]

Originally Posted by lietoflaw

Thank you for your help! I had never expected that someone, rather than a professor, would like to discuss LES with me openly and deeply before this thread. Your paper and explaination really help me clarify the concept of implicit filter.

Well, being a professor means being available to help in understanding a topic