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Re: Turbulence and fluid mechanics

One of the most bizarre real-world things I ever saw was in my Fluid Dynamics class for Chemical Engineering, and the name of this freak of nature is the Reynolds number.

For those who haven't heard of this value, it's a quantity that describes when the flow of a stream changes from laminar to turbulent. So why is that weird?

Consider that most numbers that describe real-world things have some sort of dimensionality to them, which is expressed using some sort of units. Distances (one dimension) may be measured in cubits or feet or parsecs or ångstroms; areas (two dimensions) in square feet or hectares; volumes (three dimensions) in cubic centimeters; time in seconds; speed in meters per second (or furlongs/fortnight); acceleration in meters per second squared; force in newtons; frequencies in hertz; and and so on.

The Reynolds number has no dimension.

It's a real number, describing a real quantity: the fluid's speed with respect to an object in the stream (m/s), times a linear dimension (m), divided by the fluid's kinematic viscosity (m2/s), yielding a number that tells you when turbulent flow will happen. But the units cancel out! The number has no dimensionality of any kind. It's just a number. And yet it describes a real-world property that has real engineering utility.

There are some others, such as the Mach number. But I always found the Reynolds number particularly fascinating.

Yes, I probably should get out more.

Re: Turbulence and fluid mechanics

In a way it makes sense though right? There's a text by Brian Cantwell that introduces the dimensional analysis chapter with,
Any physical relationship must be expressible in dimensionless form... This is because the variables are subject to measurement by an observer in terms of units that are selected at the arbitrary discretion of the observer. It is clear that a physical event ... cannot depend on the particular ruler used to measure space, the clock used for time, the scale used to measure mass... depending on the dimensions that appear in the problem.
If you and I run a race on a straight track that is 30 antehiros long, I run at an average pace of vScytale = 3 kilosquattes per hemtistool, and you run at an average pace of vFlatfingers = 49 terribads per naysec, who's going to win?

We don't know because we have no idea of the relative size of the measures we're using here, but we do know the winner depends on only these two quantities. The underlying principle of dimensional analysis is that, in this case, these two quantities can be collapsed into one because they have the same dimension: length per time. That means we can form a dimensionless number, call it the Flatfingers number,

Fa = vScytale / vFlatfingers

And so we can see that the winner of the race depends not on *two* quantities, but on one: the Flatfingers number. I don't know how big a kilosquatte is, and I don't know how long a naysec takes, but I know there are three solutions: if Fa > 1, I win: if Fa < 1, you win. If Fa = 1, it's a tie. In fact, Fa=1 is degenerate ("we both win").

Of course, to determine the Flatfingers number I need to know the conversion factors between the various units cited, so this all seems rather trivial. We've just shifted the problem into different language. But the language makes very clear what it is we should care about: it doesn't matter how fast I run, it only matters if I run faster than you.
If Fa is huge, it means I'm a speed demon compared to you, and if Fa is tiny, it means I'm a dismal failure compared to you.

We could say the Flatfingers number describes the "relative effects" of Scytale vs Flatfingers in the race. This is exactly the same language people use to talk about things like the Reynolds number: it describes the relative effects of inertia vs viscosity. And we know it works because it's dimensionless. If it weren't dimensionless, we couldn't know in general if inertia and viscosity were being compared on the same terms.

This is what makes these numbers so useful: Bond number is relative effects of surface tension vs gravity. Froude number, inertia/gravity, and so on.

Thank goodness for dimensionless numbers! If it weren't for Reynolds number, we couldn't describe this lab photo in exactly the same terms as this satellite view . Same relative effects, even though our measuring sticks are different.

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