Gyroscopic effect is minimal in a properly designed shell, which represents about 99.5% of all cases. The range of proper stabilization is quite large, and shells only tumble or exhibit overstabilization phenomena under rather extreme conditions. The aerodynamic couple causing the projectile to try to turn sideways is rather small. This is for two reasons; first, the projectile hardly ever gets too far out of line before gyroscopic forces drive it back, and second because the center of pressure - i.e. the point at which the air pushes back is really not that far ahead of the center of gravity - i.e. the point where the projectile in effect presses forward. If the center of pressure is aft of the center of gravity, the aerodynamic forces on a canted projectile will tend to push the projectile back into line again. This why we put (tail) fins on arrows (and airplanes); the presence of the fins moves the center of pressure aft.
Usually the center of pressure of a projectile is forward of the center of gravity. So if it were not spun, it would, indeed, tend to turn sideways as you suggest. Spinning it allows it to overcome the overturning forces, and, at the same time - in a sort of serendipity - also causes it to follow the trajectory.
Normal trajectories don't curve that rapidly. A shot fired from a battleship's gun at 45 degrees of elevation does in fact turn through about 90 degrees on its way from the gun to the target, but it also has about 90 seconds to make the turn, so it's not turning all that fast - about one degree per second. And this is a fairly extreme case - usually the 'turning rate,' if I may call it that, is much less.
Because the trajectory does curve in an arch, due to gravity, the nose of the projectile always tends to be slightly above the tangent to the trajectory, i.e. it tends to fly very slightly 'nose high.' Because the projectile is spinning, the rotational force upward on the nose causes the nose to precess to the right, and it's this rightward cant which causes what gunners usually call 'drift.' The aerodynamic forces pushing on the right-canted nose now cause the nose to precess downward, and this (more or less) steady downward push on the nose is what causes the projectile to follow the tangent to the trajectory almost exactly, i.e. to 'trail' properly. Actually, the projectile is in a sort of imbalance most of this time as the various forces readjust each other and the nose actually describes a small arc in the air, known as 'nutation' from the Greek word for 'nodding.' Very descriptive of what the shell really does.
The designer works things out - or rather attempts to work things out - so that the precessional forces due to the spin of the projectile just balances the average overturning forces due to aerodynamics. For normal shells this seems to work best if the spin rate, i.e., the rifling twist, is somewhere between 20 and 35 calibers of projectile travel for one complete turn.
If the projectile is spinning too rapidly, then the bullet is said to be "overspun" or overstabilized, and in such cases one does indeed find that the so-called 'summital yaw' - i.e., the cant at the top of the trajectory (where air resistance is least) is too large. It can be - in extreme cases - too large for the projectile to 'turn over' at the summit, and it will then fall base down. This actually happens with rockets at high altitude as well, in spite of the fact that they have fins. No air, and a slight residual spin, means that the fins don't have any effect and the rocket then re-enters the atmosphere base first. The fins then cause it to 'snap' through to a nose-first attitude again, which often causes it to break up.
If the projectile is spinning too slowly, then it is said to be underspun. In this type of situation the aerodynamic forces overpower the spin and the projectile wobbles and rotates violently, just like a top when it has run out of spin when set on your floor.
In fact, if you can, get a small top or gyroscope and spin it up on your floor or countertop and see what happens. In this case, the force of gravity acts in the same way that the aerodynamic forces act on a projectile. You will see the nose, i.e., the top of the top, first remain steady, then slowly start to precess in a circle as gravity takes the upper hand, i.e., as the spin rate decreases due to friction. The top begins by being overspun, and then, as friction dampens the spin, moves to being underspun. The big precessional circle tends to get a bit jerky in its motion, with little circles superimposed upon the bigger one. The little circles are the nutation.
Basically, curvature of trajectory means that gyroscopically stabilized projectile is always slightly nose up. Air pressure on bottom of nose causes nose to precess to the right (causing drift). Air pressure on the side of the nose causes the nose to precess downward, so if everything is balanced properly, the projectile ends up following the trajectory nose-first. It's the air that does it, all right. Problems occur when there is not enough air around to cause the nose to tip down rapidly enough, either because the spin is too great, the projectile velocity is too small, the altitude is so high that air density is too low or the trajectory is so steeply curved (e.g., as in mortar fire) that the bullet simply doesn't have time to turn around properly. That's why one doesn't spin mortar rounds, so fins are used to move the center of pressure back, and spin - which then becomes superfluous, is abandoned.
It's hard to find written sources that describe these phenomena in "regular" libraries, though if you have access to a large university engineering library, you will likely be able to find out at least a little bit more about this. "Classic" work, if you can find it - and make your way through the rather ponderous math - is Fowler's "The Aerodynamics of a Spinning Shell," Philosophical Transactions of the Royal Society (London) (A) Vol 221 pp. 295-387, printed in 1920 or so.
- 6 October 1999