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Computational cost - Re ^3

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Old   May 27, 2013, 05:18
Default Computational cost - Re ^3
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Karthik
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Hi,
I just started reading a textbook on cfd and found this statement- The computational cost for a DNS increases as the cube of the Reynold number.
Can someone please explain how this order of magnitude estimate was arrived at.

Thanks
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Old   May 27, 2013, 06:38
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Hi. Imagine you want to compute the turbulent flow over a body of characteristic length L.

Suppose your mesh is uniform. Turbulence is unsteady so you will need to perform a computation over one characteristic time of the largest turbulence scale t_L at least. Also, turbulence is 3D so the dimensions of your domain are, at least, L \times L \times L.

The minimum number of operations are the result of multiplying the number of cells by the number of time steps: N_{ops}= N_{cells} \cdot N_{timesteps}.

The Kolmogorov scale characteristic length and turnover time scale with those of the largest scale as:
\eta/L\sim Re_L^{-3/4}.
t_\eta/t_L\sim Re_L^{-1/2}.

In a DNS you have to resolve all the turbulence scales, both in space and time. Therefore, the cell size and time step in your simulation must be \Delta x \sim \eta and \Delta t \sim t_\eta (length scale and characteristic turnover time of the smallest, i.e. Kolmogorov, scale).

The total number of cells is therefore (notice that your domain is 3D) N_{cells} \sim (L/\Delta x)^3 \sim (Re_L^{3/4})^3. Likewise, the number of time steps is N_{timesteps} \sim t_L/\Delta t \sim Re_L^{1/2}.

The total number of operations scales as N_{ops}= N_{cells} \cdot N_{timesteps} \sim (Re_L^{3/4})^3 \cdot Re_L^{1/2} = Re_L^{11/4} = Re_L^{2.75}.

Notice that these are minimum requirements, so I guess you can easily reach the scaling with the third power of the Reynolds number. A detailed explanation can be found in the book of Pope.

Cheers,
Michujo.
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Old   May 27, 2013, 07:27
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Filippo Maria Denaro
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a very quick (and brutal) estimation can be done assuming that you have to work with a cell Reynolds number = O(1). Therefore, for each direction, you have

Reh = u h /ni = ReL h/L = ReL / N = O(1)
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Old   May 28, 2013, 04:28
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From the michujo estimate, the exact Re^3 scaling is then reached when considering an unitary Courant number (as most DNS methods rely on explicit convection schemes). As a result, the time step also scales like the grid step and the final estimate is reached.
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