# About the accuracy of DNS with spectral method and FVM method

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 May 20, 2014, 21:16 About the accuracy of DNS with spectral method and FVM method #1 Senior Member   Huang Xianbei Join Date: Sep 2013 Location: CAU,China Posts: 277 Rep Power: 6 As known, the usual way to perform a DNS is the pseudo-spectral method, even though the grid is larger than Kolmogorov scale, the calculation is expected to be accurate. While when FVM is used, I'm suggested to use a finer mesh if I want to perform a DNS with FVM. So here I have a question: Is FVM DNS needs the grid to be smaller than the Kolmogorov scale?

May 21, 2014, 03:29
#2
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Filippo Maria Denaro
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Quote:
 Originally Posted by huangxianbei As known, the usual way to perform a DNS is the pseudo-spectral method, even though the grid is larger than Kolmogorov scale, the calculation is expected to be accurate. While when FVM is used, I'm suggested to use a finer mesh if I want to perform a DNS with FVM. So here I have a question: Is FVM DNS needs the grid to be smaller than the Kolmogorov scale?
1) No matter SM or FVM are used, the key is that the grid size is of the order of the Kolmogorov lenght scale to perform a DNS.

2) Using FVM implies that the resolved frequency before Nyquist are smoothed. If your grid is correctly refined, this smoothing lies over the range of physical dissipative frequencies and the simulation is accurate. A general suggestion can be to ensure that the Taylor micro-scale frequency is well resolved by ensuring that pi/h is far from it

May 21, 2014, 20:45
#3
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Huang Xianbei
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Location: CAU,China
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Quote:
 Originally Posted by FMDenaro 1) No matter SM or FVM are used, the key is that the grid size is of the order of the Kolmogorov lenght scale to perform a DNS. 2) Using FVM implies that the resolved frequency before Nyquist are smoothed. If your grid is correctly refined, this smoothing lies over the range of physical dissipative frequencies and the simulation is accurate. A general suggestion can be to ensure that the Taylor micro-scale frequency is well resolved by ensuring that pi/h is far from it
As in Moin(1998,DIRECT NUMERICAL SIMULATION:
A Tool in Turbulence Research
):
Code:
```It is straightforward to show, using
the modified wavenumber, that the second-order central difference requires a
mesh spacing equal to 0.26 to meet this requirement, while the fourth-order
central difference, sixth-order PadŽe, and Fourier spectral schemes require mesh
spacings of 0.55, 0.95, and 1.5 respectively```
Here, the 0.26 and others mean the Kolmogorov scale's multiply.
So it however shows the spectral method is somewhat more accurate in the same resolution.
What's h mean in the formula 'pi/h'?

May 21, 2014, 21:52
#4
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Huang Xianbei
Join Date: Sep 2013
Location: CAU,China
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Quote:
 Originally Posted by FMDenaro 1) No matter SM or FVM are used, the key is that the grid size is of the order of the Kolmogorov lenght scale to perform a DNS. 2) Using FVM implies that the resolved frequency before Nyquist are smoothed. If your grid is correctly refined, this smoothing lies over the range of physical dissipative frequencies and the simulation is accurate. A general suggestion can be to ensure that the Taylor micro-scale frequency is well resolved by ensuring that pi/h is far from it
Hi,Filippo:
I'm not sure the method I estimate the Kolmogorov scale is right,so I write down my process:
The channel flow I focus on, Re=2900,Re_tau=194,FVM method is used. As dissipation rate can be estimated as epsilon=u_tau^3/l, u_tau is the friction velocity, l is the half-width of channel. So the Kolmogorov scale is write as : yita=Re_tau^(-4/3)*l
The channel dimension is 4*pi*l, 2*l, 2*pi*l in three directions, l=1m. The grid is 128*128*128
Using the former formula, yita=0.0192375031
deltax=0.098125=5*yita
deltaz=0.0490625=2.55*yita
deltay(max)=0.041238518=2.14*yita
Is this calculation reasonable to show a acceptable grid resolution?

May 22, 2014, 03:42
#5
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Filippo Maria Denaro
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Quote:
 Originally Posted by huangxianbei As in Moin(1998,DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research): Code: ```It is straightforward to show, using the modified wavenumber, that the second-order central difference requires a mesh spacing equal to 0.26 to meet this requirement, while the fourth-order central difference, sixth-order PadŽe, and Fourier spectral schemes require mesh spacings of 0.55, 0.95, and 1.5 respectively``` Here, the 0.26 and others mean the Kolmogorov scale's multiply. So it however shows the spectral method is somewhat more accurate in the same resolution. What's h mean in the formula 'pi/h'?

- h is the mesh size
- SM are theoretically more accurate than FD/FV but the problem is that in practice you use SM on the non linear term that produces aliasing. The de-aliasing techniques (filtering and so on) degrades the accuracy of the resolved high frequency and you are forced to use refined grids
- this paper contains some other estimations (for LES but is useful for your aims) http://scitation.aip.org/content/aip...1063/1.3676783

May 25, 2014, 09:06
#6
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Huang Xianbei
Join Date: Sep 2013
Location: CAU,China
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Quote:
 Originally Posted by FMDenaro - h is the mesh size - SM are theoretically more accurate than FD/FV but the problem is that in practice you use SM on the non linear term that produces aliasing. The de-aliasing techniques (filtering and so on) degrades the accuracy of the resolved high frequency and you are forced to use refined grids - this paper contains some other estimations (for LES but is useful for your aims) http://scitation.aip.org/content/aip...1063/1.3676783
Thank you very much.I compared the fvm DNS with that from SM and find a considerable deviation in Reynolds stress vv, I think it's part from the numerical diffusion in the time scheme and the interpolation scheme. So now I use the Crank-Nicholson scheme which is fully second order accurate instead of backward. Also, the fourth order interpolation is used. While I find it's quite unstable when using Crank-Nicholson, the Courant number will go craze when the timestep size is not so small(while comparing with that of the former calculation, the timestep is only 1/6 of that).
So how can I make Crank-Nicholson more stable? until now, the only thing I can do is to use a smaller timestep size when the calculation crashed.

May 25, 2014, 11:25
#7
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Filippo Maria Denaro
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Quote:
 Originally Posted by huangxianbei Thank you very much.I compared the fvm DNS with that from SM and find a considerable deviation in Reynolds stress vv, I think it's part from the numerical diffusion in the time scheme and the interpolation scheme. So now I use the Crank-Nicholson scheme which is fully second order accurate instead of backward. Also, the fourth order interpolation is used. While I find it's quite unstable when using Crank-Nicholson, the Courant number will go craze when the timestep size is not so small(while comparing with that of the former calculation, the timestep is only 1/6 of that). So how can I make Crank-Nicholson more stable? until now, the only thing I can do is to use a smaller timestep size when the calculation crashed.

Generally, the codes us CN for the diffusive terms and Adams-Bashforth for convective terms, that will increase the stability. However, the stability region for small cell Reynolds numbers require very small time steps

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