Can we even measure the Magnus effect?
I found a webpage which describes the mathematical foundation for the Magnus effect. It is located at the following URL, and describes the effect for rotating cylinders of infinite length:
I think it could be applied to our paintball, at least as a first cut approximation. After staring blearily at the equations, we come up with
L = - (rho) * v0 * (gamma), where
rho is the density of the fluid,
v0 is the velocity of the fluid (relative to the object in question), and
gamma is something called the circulation, defined as the line integral of the flow velocity around a closed loop.
What gamma basically means for us is "take the surface speed of the object and multiply it by the object's circumference". Thus,
(gamma) = vc * 2 * (pi) * r, where
vc is the circumferential (surface) velocity of the object,
pi is 3.14159 or thereabouts, and
r is the object's radius.
Let's plug in some of the data derived from the test 101 picture.
rho = 1.293 kg/m^3 (reasonable value for the density of air)
v0 = 85.34 m/s (280 fps)
vc = 1.25 m/s (4.1 fps surface speed)
r = 0.0086 m (radius of a paintball 0.68 inches in diameter)
So from this we get
L = -1.293 * 85.34 * 1.25 * 2 * 3.14159 * 0.0086
L = -7.45 (units?)
I'm not sure what the units should be. Doing a bit of unit analysis on the equation for L, we have the following:
L-units = (kg/m^3) * (m/s) * (m/s) * (unitless) * (unitless) * (m)
L-units = kg/s^2 ???
That sure isn't a unit of force I'm familiar with. We're missing a distance unit in the numerator somewhere, and darn if I can find it. Any help, anyone? :)
I also have happened across the following webpage, which gives a different equation (of similar form) for the Magnus effect:
Here they describe the effect as follows:
M = cM * (rho) * D^3 * f * v, where
cM is the Magnus force coefficient (1.23 works pretty well, according to the webpage),
rho is the density of the fluid,
D is the diameter of the object,
f is the object's rotational frequency (rotations per second), and
v is the velocity of the fluid (relative to the object).
If the Magnus force coefficient is unitless, then a unit analysis of this equation actually ends up with units of force coming out of it. Let's plug in some numbers derived from test 101:
cM = 1.23 (unitless?)
rho = 1.293 kg/m^3 (reasonable value for the density of air)
D = 0.0173 m (0.68 inch diameter paintball)
f = 23.1 rotations/s (15 degrees per strobe flash, one strobe flash every 1.8 milliseconds)
v = 85.34 m/s (280 fps)
M = 1.23 * 1.293 * 0.0173^3 * 23.1 * 85.34
M = 0.016 N
So for that amount of spin we end up with a Magnus effect force of 0.016 newtons. For a 3 gram paintball, this force results in an acceleration of 5.3 m/s^2, or right around 0.54 g's. Let's assume this equation has given us a correct answer, and see what the picture can tell us based on what we've calculated.
I am going to define the "first" strobe as the first image of the ball after it has exited the barrel, and incrementally name each successive ball image to the right of the first strobe (second, third, etc.).
An acceleration of 5.3 m/s should deflect the ball approximately 8.6 microns between the first and second strobes. This deflection is nearly two orders of magnitude smaller than the spatial sampling in the image (around 0.6 millimeters per pixel for the "bottom view" portion of test image 101). So even if the Magnus effect is at work, we simply can't see it from strobe to strobe. So let's look at the first and last strobes in the (strobes 1 and 5).
In this situation, the time interval is four times as large. Assuming the acceleration resulting from the Magnus effect is constant throughout the measured time, we should expect a deflection 16 times greater than that predicted between strobes 1 and 2, or around 140 microns (0.14 millimeters). This deflection is still smaller than the spatial sampling in the image.
If the ball is spinning at 15 degrees per strobe flash (and based on my other spin calculations and Tom's additional comments, I think we can safely assume the ball is not spinning at 195 or -165 degrees per strobe flash), and if the formula I used above is reasonable and accurate, then the high resolution pictures we have of the test are simply not sufficient to detect the resultant Magnus effect.
I looked at the "bottom view" portion of the test 101 picture to see if I could see any horizontal deviation in the ball's path, and I noticed something interesting... the laser aligned string is not straight. It curves slightly in the image. This leads me to believe we either have a camer/lens perspective effect going on here, or the string is vibrating during firing. The blast of air that escapes the barrel could be moving the string around.
I attempted to correct the slightly curved string by using Photoshop's transformation tools, but had little success. It seems that the transformation I'm looking for just isn't available in my version of Photoshop. Instead of measuring each ball's position from a single reference line, I generated local references corresponding to the string's location at each ball's position. I measured the following offsets:
Strobe 1: -4 pixels (-2.48 mm)
Strobe 2: -2 pixels (-1.24 mm)
Strobe 3: -2 pixels (-1.24 mm)
Strobe 4: -1 pixel (-0.62 mm)
Strobe 5: 0 pixels (0 mm)
These offsets are greater than the Magnus effect alone would suggest, unless the Magnus equation I used was incorrect. They may be the result of the escaping gas buffetting the ball around during the first few milliseconds of flight.
Well, that's about all I've got for now. Time to go do some work that actually fills up a paycheck. :)
Re: Paintball Spin Physics - Getting to the final Answer
Well, I've stared at the pictures until I'm cross-eyed and there seems to be very little that I can reach any value conclusions on that would fit into the ongoing discussion.
I cannot see anything here that addresses either, the issue of closed bolt versus open bolt or what effect internal ballistics has on the flight of the ball. However, I did do a bunch of shooting through powdered barrels as Tom suggested and arrived at the same conclusions that I reached through reading the scuff marks on paintballs shot through an un-powdered barrel. Results do seem to indicate that the ball does in fact distort or compress lineally from the forces of acceleration; causing it to tighten against the bore of the barrel for a period of time when launched.
My tests were done with a Blazer (closed bolt) operating with 400 to 450 psi input to the gun and firing Pro-ball paint at velocities of between 305 fps and 220 fps.
Only balls that demonstrated a consistent, loose fit in the barrel were used and each was blown through the unpowdered barrel with breath alone to relativly ensure consistent fit in the barrels before moving to higher pressures for launch.
In addition, each ball was chambered manually to ensure consistent positioning of the ball in the chamber prior to launch. Positioning of the ball was only made relative to the face of the bolt and did not address the position of the seem in relation to the axis of the bore.
Two barrels used: One with a straight bore of .690 and the other with an eliptically honed barrel, also with .690 base bore size but the center section of the barrel tapers out to .694 at 6" from the chamber and back down to .690 at 10" from the chamber. I did not have any Desenex brand powder on hand but Gold Bond, medicated powder seems close in consistency to Desenex.
The results: Almost every shot fired wiped the powder from the complete perimeter of the bore during the acceleration in the first part of the bore and then showed only a two-point track in the front half of the barrel. The length of time that the ball made full perimeter contact with the bore decreased as velocity was lowered. Also noted that the transition from full perimeter contact to a two-point track was more abrupt in the eliptically shaped barrel.
To further verify my results, the target was a bed sheet setup to catch the paint so I could read the balls as well as the barrel bore. Looking at the balls showed that the powder that was wiped from the bore was built up on the ball well forward of the center line of the ball with a wide "scuff" mark of imbeded powder. Thus indicating a significantly wide contact of the surface of the ball with the surface of the barrel bore.
Also, I did not see anything to indicate that any "spin" was happening inside the bore but I was not looking real close in that regard. My focus was on "data" that would indicate whether or not the ball would upset under acceleration. All indications to me are that the ball does in fact change its shape to a somewhat cylindrical form when pressure is applied and acceleration begins and the amount of distortion is relative to the velocity achieved. Your results may vary with different types of valving that might generate different rates of acceleration or different blast impact characteristics.
Now, to address some of the issues "on the table":
"Spin is the only major factor accounting for paintball inaccuracy. Promoted by Pbjosh"
In essence, a true statement IMHO. In this regard, I can only go by what I've seen which indicates to me that the less spin seen on a ball in flight, the more likely it is to go where it is intended. Less spin = tighter shot groups.
"Closed bolt operation has an effect on overall accuracy. Promoted by Glen Palmer." (two N's for this Glenn please :p
Actually, my contention is that closed bolt firing gives me the best opportunity to tune my gun (the whole gun) to maximize the effectiveness of the shot. Without appropriate setup and tuning, closed bolt firing is not likely to be any more effective than any other mode of operation.
"The paintball flight is subject to "knuckleball effect"."
It certainly is and I'd bet that we have all seen it at one time or another.
"Spin may or may not be possible because of the liquid in the paintball."
Spin is certainly possible but in relation to the effect on the flight of a paintball, is this discussion based on the lateral spin as would be imparted by rifling a barrel or the random spin generated by numerous other factors ??
"Barrels have something to do with accuracy."
This is one of the few points that I can see as being addressed in the "data" posted. The data in both pictures of "shot patterns" indicates that different barrels will achieve different shot groups. In both "shotpattern" jpgs, the Smart Parts barrel shows a tighter shot group than either, the Crown Point or Rail ? barrel. However, the tightest grouping seems to be shown in the lower left of the "shotpatterns2" jpg and I cannot make out what that barrel is.
"Seams have something to do with accuracy."
I believe that the seems themselves have less to do with overall accuracy than the size/condition and position of the seem on the ball which can and does effect the consistency of the flight path.
"Balls distort with the impact of the air blast."
A certain amount of distortion seems to be a fact. However, either as a result of impact of the air blast or the "G" forces of acceleration?
"Balls distort when leaving the barrel."
Lets hope not. Hopefully, the forces that cause distortion are relieved before the ball leaves the barrel. Although, the "smoke" tests seem to indicate a very interesting shape to the ball after it leaves the barrel.
The "smoke" shots also seem to show that there is not a collum of air being forced out ahead of the ball but that there is a small cushion of air around the ball. This brings up a question or three about the barrel and valving used for the smoke shots. Automag valving? Length of barrel? Venting in the barrel ?
The biggest problem that I run into with all of this is that I don't/can't use scientific calculations to prove out what I see and can measure. When you guys start spouting all sorts of scientific jargon and presenting formulas that often seem to me to be lacking in variables and such, you put the conversation well out of reach for me. I learned the old style math where 2+2=4 so I am relagated to believing what I see. When the things that I do can demonstrate consistent velocity readings and a tight shot group, I'm pretty well convinced that I'm doing something right. Then when something is changed and the results change too, it is quite easy to see whether or not the change was an improvement or not. Common sense and K.I.S.S. principle engineering has served me quite well for a long time. Unfortunately those things are just not accepted here.
Originally posted by AGD
Currently on the table:
Spin is the only major factor accounting for paintball inaccuracy. Promoted by Pbjosh
Closed bolt operation has an effect on overall accuracy. Promoted by Glen Palmer.
The paintball flight is subject to "knuckleball effect".
Spin may or may not be possible because of the liquid in the paintball.
Barrels have something to do with accuracy.
Seams have something to do with accuracy.
Balls distort with the impact of the air blast.
Balls distort when leaving the barrel.
Ok lets have a pointed discussion on the subject. For those of you just reading this, this thread is a continuation of the "closed bolt" thread found here.
Speaking of the Flatline barrel...
Tom, do you know if anyone other than Tippmann has performed tests to measure the amount of spin imparted to a paintball fired from the Flatline barrel system? It would give us another set of datapoints to work with for a barrel that is supposed to put spin on the ball.
Got sidetracked last night and didn't get a chance to look at test 114. I'll see if I can work on it this evening instead. :)