Scaling of the N-S equations

by hanson
Tags: equations, scaling
hanson is offline
Oct3-05, 01:20 PM
P: 320
Hi all!
I am having problems with understanding the scaling process of the N-S equations in fluid dynamics.
From textbooks, I see that each quantity say velocity, time, length...etc are all divided some some reference values in order to obtain some dimensionaless quantity V*, t*, p*, g* etc..
And the N-S equations are then rewrite into a dimensionless form, the coefficients beceome the Reynolds number, Froude number etc...
And the writer says after having this dimensionless equation, we can know the importance of the terms by just looking at the coefficients.

That's what the textbook said, and I don't really understand. I can't catch the reason for making it in a dimensionless form. Can't I still judge the importance of the terms by looking at the coefficients of the terms when the equation have dimensions? Why must we transform it to be dimensionless?

Can anyone help me out?
Phys.Org News Partner Science news on
Lemurs match scent of a friend to sound of her voice
Repeated self-healing now possible in composite materials
'Heartbleed' fix may slow Web performance
hanson is offline
Oct4-05, 06:14 AM
P: 320
could anyone please help?
robphy is offline
Oct4-05, 07:09 AM
Sci Advisor
HW Helper
PF Gold
robphy's Avatar
P: 4,107
Certainly, you don't have to use dimensionless variables.

However, using dimensionless variables can simplify the analysis of a problem.
First, it cuts down the number of symbols in the equation. (Less writing!) This often highlights the "[standard] form" of an equation. It may be easier to recognize the mathematics, and possibly draw analogies between different physical systems. (For example, it may help you recognize that the mass-spring system is analogous to an inductor-capacitor system.)

Additionally, one can't compare A with B (i.e. one can't say A>B) if they carry different dimensions. However, if [tex]\alpha[/tex] and [tex]\beta[/tex] are dimensionless, then one can compare them. In particular, it may useful to know that [tex]\alpha\gg\beta[/tex] so that the [tex]\beta[/tex]-term in an expression like [tex]\alpha \blacksquare + \beta \blacksquare +\ldots [/tex] may be neglected.

Implicit in the above is the idea of "scaling". For example, if you know how a problem scales, you can experimentally model it less expensively. (For example, wind tunnel tests for airplanes. Another example: special effects using miniatures and slow-motion.)

Register to reply

Related Discussions
Scaling a PDE Differential Equations 1
un-scaling the matrix - how? Precalculus Mathematics Homework 5
Scaling Laws General Physics 0
question about RG and scaling in qft Quantum Physics 0
Scaling of functions General Math 1