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/sci/ - Science & Math

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11181909 No.11181909 [Reply] [Original]

I don't understand dimensionless parameters.

I'm taking fluid mechanics right now, and I know how to USE the Reynolds Number and how to non-dimensionalize parameters.

But I can't wrap my head around the whole concept of dimensionless parameters. Like what do they represent, why do they work, and how come characteristic lengths are arbitrary.

And don't even fucking tell me that the Reynolds number is the ratio of inertial to viscous effects.

Is there something I can read for this?

>> No.11181932

>>11181909
1 meter is 1/2 the size of 2 meters. What is 1/2? Apples? Grapes? No, 1/2 is just 1/2. Dimensionless numbers are scaling factors which are used to show that certain quantites correlate linearly with variation. If you divide 1 meter by 2 meters, you have just created a ratio of 1/2, and certainly this ratio shouldn't have a unit. You don't say that 1 meter is 1/2 meters of 2 meters, you say it is a half of 2 meters. This is exactly what a dimensionless parameter means.

>> No.11181958

>>11181909
Do you have another example of a dimensionless parameter you don't understand besides the Reynolds number?

It seems to me your question is "I don't understand the Reynolds number." I don't see why the lack of units is something difficult to understand.

>> No.11181960
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11181960

>>11181909
Let's say I'm building an airplane and I need to design a wing. The problem is that I don't fully understand the physics of a wing because it turns out fluid mech is very complicated, but still I need to design one. So I think: what parameters influence the lifting power of a wing? I do a little tinkering and thinking and I conclude that the lift of a wing depends on 1) it's span, 2) its camber, 3) its angle of attack 4) airspeed, 4) viscosity of air, and so on (I'm just using lift as an example).
So, I have LIFT as a function of N variables. Note that lift is a kind of force and that a force is measured in pounds and that a pound is a derived quantity composed of M=3 fundamental quantities: length (feet), time (seconds), mass (slugs). You still with me?
So it turns out there is something called the Buckingham Pi theorem. This theorem basically says that if I have a quantity involving M physical dimensions that is a function of N variables, then I can find ANOTHER function that involves P=N-M terms called pi terms. There are a couple methods to find pi terms, see https://en.wikipedia.org/wiki/Rayleigh%27s_method_of_dimensional_analysis for example. Once you have pi terms, it is much, much easier to perform experiments and find empirical, quantitative relationships between them. Reynold's number is an example of a pi term. So is the Froude number, Mach number, etc. Any decent fluid mech text should have a section on dimensional analysis.

>> No.11181964

>>11181932
so then how come there are so many different dimensionless parameters like Reynold's number, mach number etc...

they're all calculated differently even though all the dimensions cancel out in the end

like for the Reynolds number, what does density, velocity, diameter, and viscosity have to do with anything?

>> No.11181969

>>11181909
Dimensionless numbers are useful because it allows you to use scale models.
The reason they work is because fluids even with different individual properties all follow the same physical laws.

>> No.11181970

All scientific constructs exist only to the extent they are useful in making accurate predictions
Dimensions are as arbitrary as dimensionless in this sense

>> No.11181976

>>11181964
>what does density, velocity, diameter, and viscosity have to do with anything?
What does mass have to do with acceleration?

>> No.11181980

>>11181976
it don't? idk

>> No.11181982

>>11181964
Consider the fact that all of those quantities will be in SI units, i.e. for any value of those numbers in SI units, you KNOW the ratio of those dimensioned units will equal Reynolds number.

Consider V, D and u as fixed, and you're looking for p. Then you know via this equation that the density of the fluid will be fixed by the dimensionless Reynolds number.
It's just a ratio anon

>> No.11181990

>>11181980
>what's f=ma?
So you're either a brainlet or you are a troll.

>> No.11181989

>>11181980
It doesn't matter "why" viscosity influences the behavior of a fluid in a pipe. Yes, there is well understood physical reasons that viscosity influences velocity profiles and sure, you are totally allowed to solve navier-stokes wherever you go, but Reynold's number simplifies this.

The point is, an increase in Reynold's number has consistent and experimentally shown to predict where the transition from laminar to turbulent flow is, as well as a bunch of other useful stuff. Did you read my post on Buckingham Pi theorem, like a good boy?

>> No.11181994

>>11181989
Sorry it was too long. Can't you spoonfeed me the answer in bite-sized chunks?

>> No.11181995

>>11181990
dont be mean to someone trying to learn

>> No.11181999

>>11181994
1) look up buckingham pi theorem and rayleigh method of dim analysis
2) reynold's number is a pi term that pops out when you apply this theorem

>> No.11182003

A clock will loose accuracy in units of second per second

>> No.11182005

>>11181989
Yeah, my understanding of it is that dimensionless parameters are just a "condition" that affects something.

I don't understand why you can apparently choose different characteristic lengths, and it still works.

>> No.11182011

>>11182005
>I don't understand why you can apparently choose different characteristic lengths
Can you be more specific? I think I know what you are asking, though.

The Reynold's number that appears when you are working with the flow in a pipe is not the same Reynold's number that appears when you are calculating drag coefficients. They are mathematically identical, yes, but they are PI TERMS that pop out in the analysis of totally different objects. So yeah, it makes perfect sense that characteristic length would represent something different for each.

>> No.11182017

>>11182011
My professor would be like

"oh wow this problem would be much easier if I non-dimensionless it"

and then he proceeds to do L* = diameter/length or some other arbitrary stuff.

So there seems to be multiple ways to non-dimensionalize a problem, and they all seem to be correct. And that's what I'm confused about.

>> No.11182021

>>11181995
You shouldn't be studying fluid mechanics without having done basic physics.

>> No.11182034

>>11182017
What other ways can you think of to non-dimensionalize the problem?
>they all seem to be correct
What does "correct" mean? Do they predict natural phenomena?

I'm not trying to be rude, I'm trying to wrap my head around what exactly your problem is. IMO they should teach this shit to all freshman STEM majors, dimensional analysis is fundamental and very powerful.

>> No.11182039

Correct as in if I chose to do L* = diameter/length, and you can solve the problem.

But if you do L* = length/diameter, you can apparently also solve the problem.

Even though the two L* would be different value

>> No.11182052

>>11182039
The thing is, applying one and only one of these ratios gives us predictive power. You cannot deduce which one, you have to do an experiment to find out.
But yes, you are correct in saying they both non-dimensionalize the problem. The difference is that if you use the first definition of L* you will find that L* appears in the Pi function in a different place than if you used the second definition.

>> No.11182065

>>11182039
I'm not who your replying to, but I think you're not being very clear about what "solve the problem" means. That's probably where your confusion lies.

diameter/length and length/diameter are equally good. The Reynolds number and 1 over the Reynolds number have equal predictive power. You just have to be clear how they are actually being used.

>> No.11182085

honestly I don't even know specifically what I'm confused about, but I'm confused nevertheless.

I'll do some thinking, don't want to bother you guys too long.

>> No.11182119

>>11182085
it's no bother, i like helping people. just figure out what precisely your question is lol

>> No.11182324

>>11181909
>And don't even fucking tell me that the Reynolds number is the ratio of inertial to viscous effects.
But it is... and it predicts whether a flow is turbulent or laminar by giving you a quantification of how one set of forces is dominating the other.

>> No.11182392

>>11181909
The absolute fucking state of engineers, holy shit. This is why, beyond all the memes and shitposts, engineers are actually the laughing stock of the world.

>> No.11182533

>>11181960
>force is measured in pounds
Not in the developed world, where we don't want satellites to explode.

>> No.11182540

>>11182533
>t. no freedom

>> No.11182575

>>11182392
Don't worry, OP is going to fail or switch majors soon enough.

>> No.11182709

>>11181909
>And don't even fucking tell me that the Reynolds number is the ratio of inertial to viscous effects.
well you're retarded because that's exactly why dimensionless numbers exist.

When a physical system can have different behaviors depending on the dominating effect, you can study and classify those behaviors using dimensionless numbers and give immediate results based on a simple calculation.

mach number<1? zoom
mach number>1? boom
Re<1000? woosh
Re>1000? slosh
etc
It also tells you which effects you can ignore for simplification.

>> No.11182850
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11182850

>>11182709
Not OP, reynold's number make sense, but lets say you were to look at some other arbitrary system, there's plenty of unit ratios you could come up with that would be dimensionless, but how do you determine whether a given parameter will be meaningful at all? Educated guesswork and experimentation based on what you're looking for?

>> No.11182911

>>11182540
t. shart-in-mart

>> No.11183222

>>11181964
om you are retarded

>> No.11184792

>>11182034
That's exactly what they do, well at least for engineering majors

>> No.11185646

>>11181964
>like for the Reynolds number, what does density, velocity, diameter, and viscosity have to do with anything?
What does the speed of light defined as traveling 299,792,458 unit distances in 5.91246(60)^44 Planck seconds of the unit that light travels in 1/299,792,458 X 5.91246(60)^44 Planck seconds have to do with anything?