Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Sport ball — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The cliff a rough ball reaches sooner. The drag coefficient of a sphere against Reynolds number, on log axes. The smooth curve is Morrison's correlation, which is a fit to measurements and is drawn in the colour this site reserves for a borrowed claim. The other is the same curve shifted along the Reynolds axis by a factor of 6 — a stated model of what roughness does, which is to trip the boundary layer early, and not a measurement of any real ball.

    The drag that falls as it speeds up

    There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.

    part 1 · applied
  2. Two sides of one ball, at different pressures. The surface pressure coefficient round a ball, measured from the front stagnation point, on the side the seam trips and on the side it does not. Up to separation both follow the exact potential-flow distribution 1 − (9/4)sin²θ. After it both take the same wake pressure, which is what a manometer measures rather than what the ideal theory predicts. The asymmetry is the shaded area between them, and integrating it gives a side force of 0.2949 towards the later-separating side.

    A ball that swings without spinning

    A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.

    part 2 · applied
  3. Spin against distance flown, not against time. The spin of a struck golf ball as a fraction of its launch spin, against how far it has flown. The integrated flight sits exactly on an exponential in distance, because the spin-down torque is proportional to speed times spin and the speed cancels. The constant is 1,202 metres and the drive is 200, so the ball arrives with five sixths of the spin it left with.

    The ball that never forgets its spin

    Commentary explains a late-swerving ball by saying the spin is dying away. The spin-down torque goes as speed times spin, so spin decays over a distance rather than a time — 1,202 metres for a golf ball against a 200-metre drive — and what dies away is the speed, which makes the swerve stronger.

    part 3 · applied
  4. Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°.

    The terminal speed sets the best launch angle

    In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

    part 4 · applied
  5. Lift takes the launch angle down far faster than drag does. The best launch angle against β for a ball whose lift is a fixed fraction ℓ of its drag, from ℓ = 0 — the drag-only curve — to ℓ = 1. At β = 10 drag alone puts the best angle at 32.6°; a lift-to-drag ratio of 0.4 takes it to 15.9° and 0.8 makes a level launch best. Once the curve reaches zero it stays there: the lift lifts the ball, and any angle given at launch is height the lift would have bought for less drag.

    A spinning ball should leave low

    Drag alone takes a ball's best launch angle from 45° to the high thirties at the speeds sport is played at. Backspin takes it into the teens, because a ball with lift is a glider and a glider buys its height with lift rather than with angle. Every tenth of lift-to-drag ratio takes a fixed slice off the angle, and above a line that depends on the launch speed the best flight leaves the ground level.

    part 5 · applied
  6. With lift in proportion to spin, a drive gains from spin until its launch is level; the measured curves give it a best spin where drives are hit. The longest carry of a 70 m/s drive at each spin rate, its launch angle chosen for that spin, for four pairs of coefficients: lift in proportion to the spin ratio with fixed drag; drag rising with spin; lift growing as its 0.4 power; both measured forms together. The fixed pair peaks near 4,800 rpm, where its best launch reaches level, and drag rising alone near 4,400 rpm; lift saturating alone is still rising at 6,000; with both measured forms the carry peaks at 234.6 m at 2273 rpm and 16.8°.

    A drive's best spin falls where players hit it only because lift tires and drag grows

    With lift in proportion to spin and drag fixed, a golf drive keeps gaining from backspin until its best launch has fallen to level, near 4,800 rpm, and past a few thousand more the model flies a ball the length of eight football pitches. Give lift and drag the curves wind tunnels measure — lift growing ever more slowly with the spin ratio, drag rising with it — and the best spin moves to near 2,300 rpm at 70 m/s, with a launch well above level, inside the range launch monitors recommend. Either curve alone leaves it at twice that or more. And along a ridge of best launches the carry hardly moves while the angle falls by ten degrees, which is why the launch angle is the softer of the two numbers.

    part 6 · applied
  7. Along the carry's ridge the run hardly moves the best spin, because every drive on it lands alike. The total distance of a 70 m/s drive, carry and run, at each spin rate launched at the angle that carries furthest for that spin, on a firm, a medium and a soft fairway, with the carry alone. The carry peaks at 234.6 m at 2273 rpm; the total along the ridge peaks at 1728 rpm on the firm turf, 274.3 m, 2052 rpm on the medium turf, 249.3 m, 2375 rpm on the soft turf, 239.9 m. Every drive on the ridge lands between 45.2° and 49.2°.

    The run moves the best drive below the ridge that carries furthest

    A drive's carry is greatest along a ridge of launches on which more spin trades for a lower angle at almost no cost. Every drive on that ridge lands at about the same steep angle, so the run after landing hardly chooses between them. What the run rewards is leaving the ridge: less spin and a launch several degrees lower, a lower flight that lands flatter, gives up thirteen metres of carry and gets back twenty in the run. On an ordinary fairway the best drive in all lands near thirty-five degrees, at every ball speed; on a firm one the model's ground stops being a turf anybody has measured before the run stops paying.

    part 7 · applied
  8. A strike cannot choose its spin and launch apart: each angle of attack is one line across the plane, and only an upward one reaches low spin with height. Backspin and launch angle of the ball as the dynamic loft is changed from 1° to 24° above the angle of attack, for a 45 m/s head striking at angles of attack of −4°, 0°, +4° and +8°; on each line, the loft that gives the longest total. At −4° the best is 2413 rpm at 4.43°; at 0° the best is 2129 rpm at 7.43°; at +4° the best is 1636 rpm at 9.71°; at +8° the best is 1221 rpm at 12.3°. The free plane's best for a 66.6 m/s ball is 1349 rpm at 12.5°.

    A driver reaches the best drive only by striking upward

    The longest drive asks for a launch near twelve degrees with little more than a thousand revolutions a minute of backspin, and a club cannot ask for those separately. Its strike sets the spin by the angle between the head's path and its face, and the launch by the face's loft, so a player who swings level can trade spin for launch only along one line — and that line misses the best drive. Striking upward moves the line. At a 45 m/s head speed the longest total comes at an attack of nine degrees upward with fourteen of loft, 4.6 metres past the best level strike, and it still lands near thirty-five degrees.

    part 8 · applied

All series