Drag Coefficient of a 'flexible' object (e.g. a piece of paper)

[eddiezhang]

TL;DR Summary

Are there any analytical / mathematical methods to find the drag coefficient of an object that deforms during motion (e.g. a piece of paper travelling through air).
If not (as I suspect), how easy would it be to determine / approximate it experimentally?

Hi all and thanks for your time. I'm a little new to this site and was unsure what the prefix to this post should be, so I put it under 'intermediate'.

Imagine having a piece of paper glued to the palm of your hand. You swing your hand down and the edges of the paper bend backwards because of drag. Is it theoretically possible to determine the magnitude of this drag force, considering that parts of the paper are 'pushed' backwards, and what might be some relevant papers / physics and maths concepts to help me out here? This would be quite useful for a bit of a passion project I've got going on, so I've scoured google to some... poor results.

I guess this question could be more usefully phrased as if you can mathematically / analytically find the drag coefficient of an object that deforms during motion (in an ideal physics world of course) - let's say that a force acts on the centre of mass of a normal piece of paper, with the only other force present being drag. The answer feels like a hard 'no', or at least a 'not unless you sit through a year or two of engineering school' .

If the answer is no (as I suspect), trying to estimate the magnitude of the drag force on a piece of paper is just annoying to do experimentally, as the paper just flops around from side to side once you let go of it- unless there is some kind of obvious setup which would let me do that which I've missed. On that point, do you guys have any suggestions on how I could do that? The velocities the paper ends up going at in my particular situation are not that high - between 1 and 3 m/s.

Any answer would be really appreciated, especially if it's to do with the maths of it all. My current understanding of drag is restricted to Rayleigh's drag equation F = .5ρv^2AC_d (apologies for butchering the formatting lol), and I guess it's just the 'reference area' and 'coefficient of drag' terms that are giving me a bit of grief in terms of understanding. I've only done (finished) A-level physics, so the big guns of 'shear stress' or multivariable calculus or God forbid the Navier-Stokes equations will likely go straight over my head (I'm very willing to try understand though).

This post is long enough. I'd really appreciate some help as to the theory / potential experimental setup. Papers would be lovely, though I've found few. Go raibh maith agat!!!

 

[Arjan82]

To come up with a fully analytical solution to a fluid structure interaction (FSI) problem like this would win you a guaranteed Nobel price, so I'm not going to attempt that :). To compute the drag of any object fully analytically already is a momentous task for anything more complex than a flat plate in laminar flow, let alone if the object moves...

Note that the Rayleigh's drag equation is actually a semi-empirical equation where the Cd is determined experimentally. But we can get some notion of this value if you compare it to a flag on a flag pole. This does mean that instead of fixing the paper to your hand you need some kind of pole to make the comparison work. However, the famous work by Hoerner 'Fluid-Dynamic Drag' already provides a figure for the Cd value (I got this from an otherwise irrelevant paper, not from the book):

I suppose (hope...) that the A used in the drag equation is b*c, but I didn't check.

If you want to do the measurements yourself I would suggest attaching the paper to some kind of pole and put it into some kind of home-made wind-tunnel. Which is actually very hard to get right, but that also depends on the accuracy you're after :).

Good luck!

 

[A.T.]

If you want to do the measurements yourself I would suggest attaching the paper to some kind of pole and put it into some kind of home-made wind-tunnel.

As an alternative, one could try it on some vehicle, ideally in a large enclosed room.

 

[eddiezhang]

To come up with a fully analytical solution to a fluid structure interaction (FSI) problem like this would win you a guaranteed Nobel price, so I'm not going to attempt that :). To compute the drag of any object fully analytically already is a momentous task for anything more complex than a flat plate in laminar flow, let alone if the object moves...

Note that the Rayleigh's drag equation is actually a semi-empirical equation where the Cd is determined experimentally. But we can get some notion of this value if you compare it to a flag on a flag pole. This does mean that instead of fixing the paper to your hand you need some kind of pole to make the comparison work. However, the famous work by Hoerner 'Fluid-Dynamic Drag' already provides a figure for the Cd value (I got this from an otherwise irrelevant paper, not from the book):

View attachment 339217

I suppose (hope...) that the A used in the drag equation is b*c, but I didn't check.

If you want to do the measurements yourself I would suggest attaching the paper to some kind of pole and put it into some kind of home-made wind-tunnel. Which is actually very hard to get right, but that also depends on the accuracy you're after :).

Good luck!

Thanks! I suspected as much... approximations here I come!

 

[boneh3ad]

I'd call it a hard no even if you attends a few years of school for engineering.

Whether measuring it is "easy" depends on the fidelity you need. You could just drop it and measure it's terminal speed, but the error bars will be huge as it moves in all directions in the complex flow field. It likely will change speeds a lot as it's orientation changes. You could get a sort of average drag coefficient, I suppose.

Anything more accurate would need a more careful definition of the problem.