Pull-up and pull-down manoeuvres

Why it matters

Turning in the vertical plane — pulling up from a dive, recovering from an unusual attitude, flying a loop or a split-S — is limited by the same load factor that limits a level turn, but gravity now helps or hinders depending on the attitude. Dive recovery height, terrain-following ability and aerobatic manoeuvres all follow from a few lines of force balance, and the results feed directly into the V-n diagram.

Key ideas

Setting up the problem. Consider an aircraft at the instant it is moving horizontally at speed V, with its wings level, while its flight path curves in the vertical plane with radius R. Thrust balances drag along the path, so the speed is momentarily constant. The net force towards the centre of curvature provides the centripetal acceleration V²/R. Load factor is still n = L/W.

Pull-up from level flight. The aircraft is upright and the centre of curvature is above it. Lift acts upward (towards the centre); weight acts downward (away from the centre): L − W = (W/g)·V²/R so R = V²/(g·(n − 1)) and ω = V/R = g·(n − 1)/V. With n = 1 there is no curvature at all (straight level flight). Gravity opposes the pull-up, so one g of the load factor is "wasted" holding the aircraft up.

Pull-down from inverted flight. The aircraft is inverted and the pilot pulls (towards the canopy, which now points down) so the path curves downward — the first half of a split-S. The centre of curvature is below the aircraft, and lift and weight both act towards it: L + W = (W/g)·V²/R so R = V²/(g·(n + 1)) and ω = g·(n + 1)/V. Gravity helps, so the pull-down gives a tighter radius and a faster rate than a pull-up at the same speed and load factor.

Vertical attitude. When the path is vertical, weight is along the path and does not contribute to the curvature: R = V²/(g·n).

Large load factors. For n ≫ 1 all three cases approach R ≈ V²/(g·n) and ω ≈ g·n/V, which is also the large-n limit of the level turn. Fighters at high n therefore have nearly the same turning performance in every plane.

Limits. Exactly as in the level turn:

• the stall limit n ≤ ½ρV²C_L,max/(W/S);
• the structural limit n ≤ n_max (positive and negative limit load factors);
• a sustained manoeuvre needs thrust = drag at the higher induced drag ∝ n². Combining the stall limit with the large-n result gives the stall-limited radius R_min ≈ 2(W/S)/(ρ·g·C_L,max) — set by wing loading, density and maximum lift, not by speed. (A pull-down from inverted can be tighter still, because gravity helps.)

Practical points. A real loop is not a circle: speed and load factor change around it because of gravity along the path, which is why energy methods are used for full manoeuvres. A pull-down half-loop loses height of roughly 2R, which matters for low-altitude recovery. In a pull-up the angle of attack increases sharply at the start, so the aircraft can stall at a speed well above its 1-g stall speed (V_stall·√n — an "accelerated stall").

Formulas

n = L/W

R = V² / (g·(n − 1)), ω = g·(n − 1)/V (pull-up, at the bottom of the curve)

R = V² / (g·(n + 1)), ω = g·(n + 1)/V (pull-down from inverted, at the top)

R = V² / (g·n), ω = g·n/V (vertical flight path; also the large-n limit)

n_max,aero = ½·ρ·V²·C_L,max / (W/S)

R_min ≈ 2·(W/S) / (ρ·g·C_L,max) (stall-limited, large n)

Symbols: L lift, W weight (N); n load factor; V true airspeed (m/s); R radius of curvature of the flight path (m); ω pitch (flight-path) rate (rad/s); g = 9.81 m/s²; ρ density (kg/m³); C_L,max maximum lift coefficient; W/S wing loading (N/m²). The pull-up and pull-down formulas hold at the instant the flight path is horizontal.

Worked examples

Example 1 (standard): pull-up and pull-down at the same n. Given: V = 100 m/s, n = 4.

1. Pull-up: R = V²/(g(n − 1)) = 10 000/(9.81 × 3) = 339.8 m; ω = g(n − 1)/V = 29.43/100 = 0.294 rad/s (16.9°/s).
2. Pull-down: R = V²/(g(n + 1)) = 10 000/(9.81 × 5) = 203.9 m; ω = g(n + 1)/V = 0.491 rad/s (28.1°/s). The pull-down is 40 % tighter because gravity now adds to lift.

Example 2 (GATE level): stall-limited manoeuvre. Given: W/S = 2500 N/m², C_L,max = 1.2, sea level (ρ = 1.225 kg/m³), V = 150 m/s, structural limit n_max = 7.

1. Dynamic pressure: q = ½ × 1.225 × 150² = 13 781 Pa.
2. Stall-limited load factor: n = q·C_L,max/(W/S) = 13 781 × 1.2/2500 = 6.62, below the structural limit, so the stall governs.
3. Tightest pull-up from level: R = 22 500/(9.81 × 5.62) = 408 m.
4. Tightest pull-down from inverted: R = 22 500/(9.81 × 7.62) = 301 m. If the speed stayed constant, the half-loop would lose about 2R ≈ 602 m of height.
5. Stall-limited radius at high speed (large n): R_min ≈ 2(W/S)/(ρgC_L,max) = 5000/(1.225 × 9.81 × 1.2) = 347 m. A stall-limited pull-up approaches this from above as speed rises; a pull-down is tighter (here 301 m) because gravity adds to lift.

Common mistakes

• Using the level-turn formula R = V²/(g√(n² − 1)) for a vertical-plane manoeuvre.
• Getting the sign of gravity wrong: it subtracts in a pull-up (n − 1) and adds in a pull-down (n + 1).
• Calling a gentle push-over a "pull-down". In performance texts the pull-down is a pull from inverted flight.
• Forgetting that stall speed rises as √n, so a hard pull at moderate speed can stall the wing.
• Treating a loop as a constant-speed circle; the formulas hold only at the instant considered.

For GATE AE

Expect numericals for radius and rate of a pull-up or pull-down at a given speed and load factor, comparisons between pull-up, pull-down and level turns, the stall-limited load factor at a given speed, and minimum-radius estimates from wing loading. Practise drawing the free-body diagram for each attitude so the ± 1 terms come naturally.

Quick check

1. What is the radius of a pull-up from level flight at 120 m/s with n = 3?
2. Why is a pull-down tighter than a pull-up at the same n?
3. What is R in a vertical climb with load factor n?
4. For large n, how does R depend on n?

Answers: 1. 14 400/(9.81 × 2) = 734 m. 2. Lift and weight both point towards the centre of curvature. 3. R = V²/(g·n). 4. R ≈ V²/(g·n), inversely proportional to n.