Scaling/non-dimensional navier-stokes

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The discussion focuses on the non-dimensionalization of the Navier-Stokes equations using different reference lengths for the x, y, and z coordinates. One participant questions the feasibility and rationale behind using three distinct reference lengths, suggesting that traditional non-dimensionalization typically involves a single reference length, L. Another contributor points out that such an approach is indeed utilized in climate science, providing references to support this claim. Concerns about the coordinate invariance of this method are raised, with some uncertainty about the implications of the transformations on unit elimination. The conversation highlights the complexity and nuances of non-dimensionalization in fluid dynamics.
VooDoo
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Hey guys,

Just trying to non-dimensionalise the navier stokes equation. We were taught how to do it when you scale x,y,z with one reference length L...just wondering how to do it if I scale x,y,z with a,b,c respectively.

Edit - this is what I already know http://www.mme.tcd.ie/~craigm/Cnotes/4b6/nodes/week1.php#SECTION00080000000000000000 they use one reference length L to scale as well
 
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Why would you want to non-dimensionalize with 3 different reference lengths? In fact, I don't even think that you 'can' do this. When you non-dimensionalize, you only get a 1/Re because there is a 1/L in every term.

Maybe I am misunderstanding your problem, perhaps you can elaborate a little more on what you mean by scaling with a,b, and c?
 
minger said:
Why would you want to non-dimensionalize with 3 different reference lengths? In fact, I don't even think that you 'can' do this. When you non-dimensionalize, you only get a 1/Re because there is a 1/L in every term.

Maybe I am misunderstanding your problem, perhaps you can elaborate a little more on what you mean by scaling with a,b, and c?

Yes, you can. It is done in climate science.

http://arxiv.org/ftp/physics/papers/0407/0407002.pdf
http://www.mne.psu.edu/cimbala/me33web_Fall_2005/Lectures/Nondimensionalization_of_NS_equation.pdf
 
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John Creighto said:
Yes, you can. It is done in climate science.
I don't believe this would be coordinate invarant, would it?
 
Phrak said:
I don't believe this would be coordinate invarant, would it?

I'm sorry, I'm not sure what that means. The transformations do seem to get rid of the units though.
 

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