Is there a limit for the solution of Navier Stokes equation?

AI Thread Summary
The discussion centers on the limits of the Navier-Stokes (N-S) equations in relation to the Kolmogorov Microscale equations. It is clarified that Kolmogorov theory pertains specifically to turbulence and may not apply to laminar flow, which can exist below certain length scales. The N-S equations can yield both laminar and turbulent solutions depending on initial conditions, with laminar flow remaining uniform unless turbulence is artificially introduced. The accuracy of Direct Numerical Simulations (DNS) is emphasized as crucial for reflecting Kolmogorov's hypothesis in turbulent flows. Overall, the conversation highlights the complexities of fluid dynamics and the conditions under which different flow types are analyzed.
robert80
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Dear all

I have 1 simple question. If the solution of Navier Stokes equation exists, its limits for infimum length, time and velocity would be the Kolmogorov Microscale equations, am I correct?

Thanks,

Robert
 
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I am not so sure about that. I think Kolmogorov Microscale apply to turbulence and eddies only. Below that scale you could have flow, but it would be laminar. I think its limits would be certain lengths below which the assumption of continuity is not satisfied (i.e. like 1000 times molecule size). That would be my guess.
 
As said above, Kolmogorov theory deals with turbulent flows at "sufficiently" high Reynolds numbers, whilst N-S equations can give either laminar-flow solutions or turbulent-flow solutions, depending on initial and boundary conditions.

Think of a free uniform rectilinear flow, for example. If you don't introduce an artificial initial instability to generate a turbulence, the solution of N-S will give a uniform rectilinear and laminar flow at all length scales.

But generally speaking, if the resulting flow is turbulent (because you've decided to make it turbulent) then DNS solution should reflect Kolmogorov's hypothesis as the length scale goes to zero. It will, of course, depend on the numerical accuracy of the solution.
 
Last edited:
robert80 said:
Dear all

I have 1 simple question. If the solution of Navier Stokes equation exists, its limits for infimum length, time and velocity would be the Kolmogorov Microscale equations, am I correct?

Thanks,

Robert

Can you provide some sort of reference for the Kolmogorov microscale equation? The reference I found:

http://www.google.com/url?sa=t&sour...sg=AFQjCNHoGK-uadD_2HuQGaqLCVA2u7EtjQ&cad=rja

indicates the equation relates to energy, not momentum- the NS equation is a momentum equation.
 
Andy Resnick said:
The reference I found
...
indicates the equation relates to energy, not momentum- the NS equation is a momentum equation.
That is correct. You can find a very good and easy-to-read short essay about Kolmogorov theory here (PowerPoint file): http://www.bakker.org/dartmouth06/engs150/09-kolm.ppt
 
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Ok thank you for all the help, this links are preety useful.
 

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