Spin Decay Due to Fluid Friction for a Sphere

[nda1977]

I'm need to determine the rate of spin decay for spherical objects as they're passing through air.

Basically, I'm trying to model the Magnus Effect and the output I'm getting doesn't reflect reality. I hope that I can obtain better results if I'm ab;e tp determine the rate at which a sphere's angular velocity decreases.

Does anybody know the equations necessary to calculate spin decay?

For the situation I know the mass / size of the object and the air density. My guess is that there is some sort of friction coeffecient that needs to be estimated or determined. Beyond that, I'm at a loss as to how to calculate spin decay. Any ideas or advice would be appreciated.

Thanks,
Nathan

 

[Clausius2]

I'm need to determine the rate of spin decay for spherical objects as they're passing through air.
Basically, I'm trying to model the Magnus Effect and the output I'm getting doesn't reflect reality. I hope that I can obtain better results if I'm ab;e tp determine the rate at which a sphere's angular velocity decreases.
Does anybody know the equations necessary to calculate spin decay?
For the situation I know the mass / size of the object and the air density. My guess is that there is some sort of friction coeffecient that needs to be estimated or determined. Beyond that, I'm at a loss as to how to calculate spin decay. Any ideas or advice would be appreciated.
Thanks,
Nathan

Do you want to reduce the problem to a constant coefficient of friction (that you guess) and set up a mechanical equilibrium equation? Or are you looking for calculating the friction coefficient?? I mean, your problem may be very complicated if you want, but we don't know what level of detail do you want in your calculation. If I were you, my elegance ;) would impose me to look at Batchelor's book and calculate an Stokes unsteady flow over the sphere for calculating $$C_D=C_D(\omega)$$, but I don't know what you want nor if you are willing to talk to Mr. Batchelor.

 

[nda1977]

Thanks for the response, Clausius.

Well, unfortunately I lost all of my books on fluid mechanics during Hurricane Katrina (which has a certain irony to it). While I think that Batchelor and co. would be very useful, I don't have access to such texts at the moment. And to be honest with you, my memory of a lot of that stuff is fuzzy at best.

I'll tell you what I do know and what I'm trying to determine:
I have a solid sphere of known mass and size and a known density. I have a known initial velocity and can perform the calculations so that I'll know the velocity of the object throughout it's flight. The object has a tremendous (and consistent) spin on it, though I do not know what the frequency is. I've taken measurements that show that the deflection due to spin is faily significant, which leads me to believe that V/U is fairly high.
What I would like to determine is what the intial spin frequency is. If I can come up with an equation to calculate the rate of spin decay, I can take the trajectory data that I've observed and "fit" an initial spin frequency to satisfy the trajectory.

My initial guess is that the initial spin is very high, something around 180,000 rpm, and after 1.000 seconds of flight, the frequency has decreased to about 70,000 rpm. Again, this is just a guess made so that the program output will resemble experimental observation.

I'd like to either determine if the numbers I came up with are a very poor estimate or if it is truly possible for the objects I'm working with to experience that much spin decay.

Ultimately, what I'm looking for is information or advice about an equation that could be used to estimate spin decay. I'm honestly not sure if that would come in the form of an equation for calculating angular acceleration, friction, or the coefficient of angular drag (or, more probable, some combination of the three).

As you can see, I'm looking for a simple answer to a complex question, something that may not be possible.

Let me know if that clarifies the question somewhat. I know what I'm looking for, but I may not be asking the right question. Again, any advice or information what would point me in the right direction would be helpful.

-Nathan

 

[Clausius2]

Wow!, I am sorry about your experience with katrina.

Well, I think the easiest resolution of your problem is:

i) go to some book for acquiring data about the friction coefficient of an sphere inmersed in a rotating fluid. Surely somebody has made experiments about it.

ii) Once you have an approximate figure for this coefficient, write down the angular conservation law for the sphere:

Angular Inertia=-Friction Torque

iii) For calculating the friction torque, you may obtain an average torque integrating the friction force (you got it from the friction coefficient) over the entire sphere surface conviniently.

iv) Solve for the angular velocity. I think you are going to obtain some function decaying exponentially, so maybe it is a good first approximation.

Enjoy!

 

[Astronuc]

These might help.

http://www.grc.nasa.gov/WWW/K-12/airplane/cyl.html

http://www.mathpages.com/home/kmath258/kmath258.htm - mathematics behind effect

As Clausius2 mentioned, I think it is simply a matter of determine the shear forces acting in oppostion to the rotation. The shear forces would be a function of $\omega r$.

 

[nda1977]

Thanks, Clausius. That's really helped to get me rolling. I've found the equation for angular inertia for a sphere. I suppose the last component I need to determine angular acceleration would be to find the equation for friction torque in terms of the coefficient of friction, radius, and frequency of a rotating sphere. Once I have that, I can determine the approximate range of the friction coefficient and work from there.

Do you know of a place where I can find the torque equation in terms of the coefficient of friction, radius, and frequency, particularly a website. I'm fairly limited in terms of access to physics textbooks, though if you have good suggestions I could try to have one sent in via interlibrary loan.

Nevertheless, thanks again for the feedback.
Nathan

 

[nda1977]

Astronuc,
Thanks for the links. I was a little thrown off by the NASA page wherein it shows the relationship between spin and Cl. I always think of Cl in terms of V/U. The references I'm using all show a negative Cl for V/U between 0 and 0.3, and thereafter Cl becomes positive (low V/U leads to a Reverse Magnus Effect for smooth spheres).

Unfortunately, all of the papers I've seen deal with fairly dense spheres that do not suffer from significant spin decay for the duration of the experiments (typically for golf, tennis, and baseball).

My initial guess was that the friction force would be proportional to frequency squared, which agrees with the exponential decay that Clausius suggests. So far I've just inserted some estimated constant in front of the f^2 to determine the spin decay.

The equation I'm working on has the angular acceleration equal to the moment of inertia divided by the friction torque (apologize for writing it out, but haven't been able to get HTML equations to work in this forum).

For a solid sphere, the rotational moment of inertia should be equal to 2 / 5 * m * r ^ 2. The equation for friction torque eludes me. If I had to guess, I'd say that the friction torque part of the equation would be Tf = (coefficient of friction) * (angular velocity) * r. However that's a guess and probably not a good one.

Again, any more advice would be helpful. If I'm heading the wrong direction, please let me know.

Thanks again for the help. I'm sure I'll figure it out eventually.
Nathan