Gerard Westendorp
May17-08, 05:02 AM
I have just a free N-dimensional fluid simulator (CFD code) on the web:
http://www.xs4all.nl/~westy31/CellFlow/CellFlow.html
I include a picture of a 4-dimensional flow.
One thing I am curious about is the amount of turbulent kinetic energy
as you go from 1D to 4D.
In 1D, there is no turbulence, because there is only one mass flux, at
least for incompressible flow.
The 2D case appears to be more "ordered" than the 3D case. So what about 4D?
According to Kolmogorov's theory, the energy spectrum should scale
with k^-5/3, where k is the wave number. There is no dimension
mentioned in this scaling law. Obviously this cannot be correct for 1D.
I suspect that for 4D, there may be an infinite amount of turbulent
kinetic energy, because the number of wave-modes between |k| and |k| +
dk increases with a power of k compared to the 3D case. My program does
not show that, because the number of cells is finite.
Gerard
http://www.xs4all.nl/~westy31/CellFlow/CellFlow.html
I include a picture of a 4-dimensional flow.
One thing I am curious about is the amount of turbulent kinetic energy
as you go from 1D to 4D.
In 1D, there is no turbulence, because there is only one mass flux, at
least for incompressible flow.
The 2D case appears to be more "ordered" than the 3D case. So what about 4D?
According to Kolmogorov's theory, the energy spectrum should scale
with k^-5/3, where k is the wave number. There is no dimension
mentioned in this scaling law. Obviously this cannot be correct for 1D.
I suspect that for 4D, there may be an infinite amount of turbulent
kinetic energy, because the number of wave-modes between |k| and |k| +
dk increases with a power of k compared to the 3D case. My program does
not show that, because the number of cells is finite.
Gerard