Scaling/non-dimensional navier-stokes

In summary, the conversation discusses the process of non-dimensionalizing the Navier-Stokes equation and the use of reference lengths in this process. The participants question the possibility of using multiple reference lengths and provide resources where this has been done in climate science. There is also a mention of coordinate invariance and its relation to the non-dimensionalization process.
  • #1
VooDoo
59
0
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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  • #2
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?
 
  • #3
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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  • #4
John Creighto said:
Yes, you can. It is done in climate science.
I don't believe this would be coordinate invarant, would it?
 
  • #5
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.
 

1. What is scaling in the context of Navier-Stokes equations?

Scaling in the context of Navier-Stokes equations refers to the process of converting the equations from their original form to a non-dimensional form. This involves removing all units and expressing the variables in terms of dimensionless parameters. This allows for easier analysis and comparison of results across different scales.

2. Why is non-dimensionalization important in Navier-Stokes equations?

Non-dimensionalization is important in Navier-Stokes equations because it simplifies the equations and makes them easier to solve. It also allows for the comparison of results across different scales, which is important in many scientific and engineering applications.

3. How is non-dimensionalization achieved in Navier-Stokes equations?

Non-dimensionalization in Navier-Stokes equations is achieved by choosing suitable reference scales for length, velocity, and pressure. These scales are then used to non-dimensionalize the variables in the equations, resulting in a set of dimensionless parameters.

4. What are some common dimensionless parameters used in Navier-Stokes equations?

Some common dimensionless parameters used in Navier-Stokes equations include the Reynolds number, which represents the ratio of inertial forces to viscous forces, and the Mach number, which represents the ratio of flow speed to the speed of sound. Other parameters may also be used depending on the specific application or problem being studied.

5. How does scaling affect the solutions of Navier-Stokes equations?

Scaling can greatly affect the solutions of Navier-Stokes equations. By removing units and expressing variables in terms of dimensionless parameters, the equations become more general and can be applied to a wider range of situations. Scaling also allows for the identification of important physical phenomena and simplifies the analysis of results.

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