Non-dimensionalizing the N-S equations

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In summary: However, it may be more convenient to use the first approach for computational purposes. In summary, the choice of characteristic variable in non-dimensionalizing the N-S equations is important and can affect the resulting equations and their interpretation. The two versions of Stokes equations differ mainly in their scaling of time, with the second approach being more suitable for transient cases. It may be more convenient to use the first approach for computational purposes. There is no clear consensus on which approach is better, and it ultimately depends on the specific problem being studied.
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hanson
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In non-dimensionalizing the N-S equations, are the choices of the characteristic variable important? I am currently reading the Stokes approximation to the N-S equation and find two versions of Stokes equations. They have used different scaling in their derivation and I find no clue in understanding the importance of scaling and am wondering if someone here can help me out.

Please kindly read the figure attached.

While the version 1 has characterise the time by U and L, version2 uses the kinemtic viscosity to scale the time. Are there any difference between these two approaches? I don't know what kind of difference will be produced and what effect would different choice of scales bring out. Could anyone explain clearly?

And are the two versions of Stokes equation the same? If not, what are the differences?

I find no clue in non-dimensionalization...
 

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hanson said:
In non-dimensionalizing the N-S equations, are the choices of the characteristic variable important? I am currently reading the Stokes approximation to the N-S equation and find two versions of Stokes equations. They have used different scaling in their derivation and I find no clue in understanding the importance of scaling and am wondering if someone here can help me out.

Please kindly read the figure attached.

While the version 1 has characterise the time by U and L, version2 uses the kinemtic viscosity to scale the time. Are there any difference between these two approaches? I don't know what kind of difference will be produced and what effect would different choice of scales bring out. Could anyone explain clearly?

And are the two versions of Stokes equation the same? If not, what are the differences?

I find no clue in non-dimensionalization...
The only real difference between these two approaches is in the definition of the dimensionless time. The second approach looks more appropriate for considering the transient case.
 

1. What is the purpose of non-dimensionalizing the Navier-Stokes equations?

The purpose of non-dimensionalizing the Navier-Stokes equations is to simplify and generalize the equations so they can be applied to a wide range of fluid flows. By removing specific units and scaling the variables, the equations become more universal and easier to solve.

2. How do you non-dimensionalize the Navier-Stokes equations?

The Navier-Stokes equations can be non-dimensionalized by dividing each variable by a characteristic value, such as the characteristic length, velocity, or time of the flow. This transforms the equations into dimensionless form, where all variables are expressed relative to the characteristic values.

3. What are the benefits of non-dimensionalizing the Navier-Stokes equations?

Non-dimensionalizing the Navier-Stokes equations allows for easier comparison between different fluid flows and simplifies the equations for analytical and numerical solutions. It also helps in identifying the key parameters that influence the flow behavior and allows for more efficient modeling and experimentation.

4. Are there any limitations to non-dimensionalizing the Navier-Stokes equations?

While non-dimensionalizing the Navier-Stokes equations can greatly simplify the equations, it may also lead to loss of physical insight and accuracy. This is because the characteristic values used for non-dimensionalization may not fully capture all the complex features of the flow, and certain assumptions may need to be made in the process.

5. Can non-dimensionalizing the Navier-Stokes equations be applied to all types of fluid flows?

Non-dimensionalizing the Navier-Stokes equations can be applied to a wide range of fluid flows, including both laminar and turbulent flows. However, the choice of characteristic values and assumptions made during the process may vary depending on the type of flow and the specific problem being solved.

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