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I often see that dimensionless numbers are determined by using their definitions, by the method as shown in the figure:

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However, I'm having trouble with what is really happening? What does that symbol '~' stand for? Why did they take the length simply $L$, velocity $w$?

Is there something to do with order of magnitudes?

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2 Answers 2

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In fluid dynamics, exact numerical relationships and equalities are often too restrictive to be of broad utility. Particularly when working with dimensional analysis, small numerical factors ($\pi,\sqrt{2},$ etc) aren't important when trying to understand how various quantities scale with one another.

Take for instance the "characteristic size" $\ell$. For a cube, it seems reasonable to take that to be the length of one side, but what about for a less regular shape like, say, an American football?

enter image description here

What do you choose here? Do you choose the length, or the diameter through the center, or the average of the two? The point is that it doesn't really matter - the characteristic size of the ball, which is somewhere in the neighborhood of 20 cm, is a rough estimate. If it's flying through the air at 20 m/s, then the characteristic time $\tau \sim \ell/w \sim 0.01$ seconds. This sets the time scale for an air parcel to flow around the ball.

The value of this number is not as a numerically accurate statement about reality but rather as a scaling factor. It gives us an order of magnitude estimate for characteristic phenomena, and it tells us how those orders of magnitude scale with the fundamental parameters of the model ($\ell$ and $w$ in this case).

So to answer your question, you should interpret $\sim$ basically like an $=$ sign, but with the added understanding that specific numerical values are not actually important, and that the corresponding relationships are understood in the sense of scaling and order of magnitude.

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  • $\begingroup$ That was very helpful to me. I was getting an intuition it had something to do with order of magnitudes, but wasn't able to find anything which could confirm that. Thank you again. $\endgroup$ Commented Jul 27, 2021 at 15:43
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You should think of the '~' symbol as meaning 'has dimensions the same as'.

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