Scaling/non-dimensional navier-stokes

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Discussion Overview

The discussion revolves around the non-dimensionalization of the Navier-Stokes equations, specifically exploring the implications of using different reference lengths for scaling the spatial dimensions. Participants are examining theoretical aspects of fluid dynamics and mathematical modeling.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about non-dimensionalizing the Navier-Stokes equations using three different reference lengths (a, b, c) instead of a single reference length (L).
  • Another participant questions the feasibility of this approach, suggesting that non-dimensionalization typically results in a term involving 1/Re due to the presence of 1/L in every term.
  • A different participant asserts that non-dimensionalizing with multiple reference lengths is possible, citing its application in climate science and providing links to relevant literature.
  • There is a discussion about the coordinate invariance of the proposed method, with one participant expressing uncertainty about its implications.
  • Another participant expresses confusion regarding the meaning of coordinate invariance and notes that the transformations seem to eliminate units.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of non-dimensionalizing with three different reference lengths. Some express skepticism about the approach, while others support its applicability in specific contexts.

Contextual Notes

There are unresolved questions regarding the implications of using multiple reference lengths, particularly concerning coordinate invariance and the mathematical consistency of the transformations involved.

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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