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Ground Effect Theory for the Internal Ballistics of Airsoft

posted Dec 17, 2012, 12:35 PM by Danny Park   [ updated Dec 17, 2012, 4:13 PM ]
So I've been doing a bit of work on (Wing in) Ground Effect, and the more I learned, the more it seemed to apply to how a bb behaves in a barrel. 

So we'll start off with the Biot-Savart law, where we can calculate the induced velocity by approximating the spinning sphere as a vortex. Using this approximation, we can approximate the barrel walls as an opposing vortex mirrored across an imaginary plane (for the top and bottom). This vortex will cancel the induced flow normal to the plane, and magnify all the induced flow tangential. 

In the case of a plane, there are two subcategories of ground effect. Span-dominated, which "blocks" wingtip vortices when the wing is close enough to the ground, which can increase lift and decrease induced drag (so you can choose to either go faster, lift more, or some combination). The second case, which is more important to us, is the chord-dominated ground effect. What happens here is that, when near enough to the ground, the vortex will slow down the flow underneath the wing, and speed up the flow above the wing. Using Bernoulli's equation, we know that, without an external source of energy, the stagnation pressure is constant, so when the velocity/dynamic pressure increases, the static pressure will decrease. If the wing is very close to the ground, it creates a nozzle-type effect, where the flow is accelerated underneath the wing, and the plane is "sucked" down due to the venturi effect. 

Here is the tricky part, where I begin to make assumptions. Because we are shooting undersized projectile (relative to the barrel), the upper surface of the bb is not flush with the barrel, creating a sort of nozzle. My assumption might or might not be correct, because modeling turbulent flow around a cylinder (which is where our bb lives), is very difficult, and inviscid flow is not a good approximation, and I cannot estimate the boundary layer thickness; however, if the assumption holds, there is a tiny gap above the bb where some of the flow is forced through. Since there is an incoming flow provided from the piston/cylinder, the flow which already entered the area between the upper surface of the bb and the barrel cannot simply reverse and go around the "easy" way, so it is forced through a small area. Since mass must be conserved (the continuity equation), the fluid behind forces the fluid in front through the tiny area, which accelerates it greatly, and once again, reduces the pressure above the bb.

Once the bb enters this region, we are golden, because the "nozzle effect" will "suck" the bb to the top, but how does it get there? This is the most difficult question, since, traditionally, the flow is simply going in the wrong direction for magnus effect, and the assumption here might possibly be the worst approximation in the bunch, but it is not completely made up.

When the circumferential velocity is much much greater than the translational velocity, the boundary layer becomes self-enclosed, which means that it creates a more or less uniform layer around the entire bb (depending on the ratio between the circumferential and translational velocities). I have not simulated this yet, but I'm pretty sure that in this flow regime, we get the reverse magnus effect, which will slam the bb into the top of the barrel, or because the "normal" Magnus Effect is so strong initially, that it initially slams the bb into the bottom of the barrel, and subsequently slams the bb into the top, creating the initial conditions for the theory to work. This is because the relative velocity of the flow to the bb is highest just upon leaving the hopup, resulting in a large amount of lift. As the bb accelerates down the barrel, its radial velocity to the axial velocity ratio (relative to the barrel) goes down, the bouncing is damped, so every bounce the bb's trajectory is more and more tangential to the barrel, and eventually is "sucked" enough to enter a stable path down the rest of the barrel. 

The argument could be made that my theory would more likely cause the bb to stabilize on the bottom of the barrel, considering the flow is in the wrong direction, but considering the surface friction in the problem, the bb would "skip" along the bottom surface of the barrel, causing it to jump up (since the circumferential velocity of the bb is in the opposite direction of the translational velocity of its center of mass). This might be another explanation of what causes the bb to initially get up to the top of the barrel, or might contribute to the effect in the previous paragraph. 

Here is an example of the "skipping":

Example of Chord-Dominated Ground Effect:

I believe that this is the most comprehensive theory for the bb dynamics in the barrel, but it is currently very hand-wavy, and might not hold water in some areas. Modeling turbulent boundary layers is beyond my fourth year ability, so currently, I can only explain things conceptually.