[cfdmms]

In the case of explicit filtering in LES, most of the literature suggest that if the numerical scheme is fourth-order then the filter width should be at least twice of the cell size and for a second-order numerical scheme the filter width should be at least four times of the cell size. I would like to know what are the reasons for different filter size for different numerical schemes? How it effects the LES solution?

[FMDenaro]

Originally Posted by cfdmms

In the case of explicit filtering in LES, most of the literature suggest that if the numerical scheme is fourth-order then the filter width should be at least twice of the cell size and for a second-order numerical scheme the filter width should be at least four times of the cell size. I would like to know what are the reasons for different filter size for different numerical schemes? How it effects the LES solution?

Consider that the goal of an explicit filtering is to remove high (but resolved) frequencies before the Nyquist one. Indeed a numerical discretization causes errors that are prevalent in that portion of the spectrum, which is very important as the SGS model exploits information from that.
The lower is the order of the scheme, the higher is the portion of the spectrum contaminated by the error (e.g. the local truncation error).
Therefore, a second order scheme is supposed to require a greater filter width than a second order one

[cfdmms]

Originally Posted by FMDenaro

Consider that the goal of an explicit filtering is to remove high (but resolved) frequencies before the Nyquist one. Indeed a numerical discretization causes errors that are prevalent in that portion of the spectrum, which is very important as the SGS model exploits information from that.
The lower is the order of the scheme, the higher is the portion of the spectrum contaminated by the error (e.g. the local truncation error).
Therefore, a second order scheme is supposed to require a greater filter width than a second order one

Thanks Denaro for your reply.
To be more clear, say I have filter size DELTA= 4*dx=8cm as grid cell size dx=2cm in second-order scheme and DELTA=2*dx=8cm with grid cell dx=4cm for fourth-order scheme, from that above condition can we conclude which numerical scheme will give us better solution and why?

[FMDenaro]

Originally Posted by cfdmms

Thanks Denaro for your reply.
To be more clear, say I have filter size DELTA= 4*dx=8cm as grid cell size dx=2cm in second-order scheme and DELTA=2*dx=8cm with grid cell dx=4cm for fourth-order scheme, from that above condition can we conclude which numerical scheme will give us better solution and why?

no, you need to compare second and fourth order discretizations on one grid having the same dx measure.
The example you are talking leads to the same width of the explicit filter, that is you resolve for both discretization until the pi/DELTA frequency.

[cfdmms]

Originally Posted by FMDenaro

The example you are talking leads to the same width of the explicit filter, that is you resolve for both discretization until the pi/DELTA frequency.

Hi Denaro, I dont understand this, if you kindly explain this portion. Thanks.

[FMDenaro]

for both discretization you fixed the same width of your explicit filter, that is DELTA=8 cm, therefore the range of resolved frequency in the spectrum is the same.
The only difference is in the effect of the lower truncation error for the fourth order scheme that, however, is used on a grid size twice than the second order scheme. This way, the improvement you can obtain with the fourth order discretization will be almost disregardable

[cfdmms]

Originally Posted by FMDenaro

for both discretization you fixed the same width of your explicit filter, that is DELTA=8 cm, therefore the range of resolved frequency in the spectrum is the same.
The only difference is in the effect of the lower truncation error for the fourth order scheme that, however, is used on a grid size twice than the second order scheme. This way, the improvement you can obtain with the fourth order discretization will be almost disregardable

Hi Denaro, I have another question about the filter function in explicit LES.

If the filter width is taken twice of the cell size, most of the literature recommend to use three point filter function to obtain the filtered velocity like

phi_bar_(i)= 1/4 phi_(i-1)+1/2 phi_(i)+1/4 phi_(i+1)

Now in the case of explicit filtering for fixed filter size DELTA, if the fixed filter (DELTA) is taken four or eight times of the cell size, what will be the filter functions for these conditions?

Or we can still use the above filter function to refine the grid resolutions for the fixed DELTA?

[FMDenaro]

Originally Posted by cfdmms

Hi Denaro, I have another question about the filter function in explicit LES.

If the filter width is taken twice of the cell size, most of the literature recommend to use three point filter function to obtain the filtered velocity like

phi_bar_(i)= 1/4 phi_(i-1)+1/2 phi_(i)+1/4 phi_(i+1)

Now in the case of explicit filtering for fixed filter size DELTA, if the fixed filter (DELTA) is taken four or eight times of the cell size, what will be the filter functions for these conditions?

Or we can still use the above filter function to refine the grid resolutions for the fixed DELTA?

This issue is quite uncorrect...assume h is the mesh size, so that kc=pi/h is the Nyquist cut-off.
Now, if you use a spectral method then the component are resolved unaffected by the numerical errors until kc.

If you adopt FD/FV methods, the resolved frequencies are affected by the numerical errors before kc (less than 1/2 of the components are exactly resolved). The consequence is that the real filter width (for implicit filter, that is the one induced by the discretization) is quite larger than h, says Delta_f = q*h (q is between 1 and 2, depending on the order of discretization).
As a consequence, the EXPLICIT filter width must be Delta_ex= N*Delta_f=N*q*h.

Try to compute the numerical transfer function for phi_bar_(i)= 1/4 phi_(i-1)+1/2 phi_(i)+1/4 phi_(i+1) and estimate the real filter width it induces.

In practice, on a fixed mesh size h, you can construct an explicit top-hat filter function, that is a volume integral over more cells. The more are the cells, the more is increased Delta_ex