Atmospheric Flight: Performance, Stability and Control

The International Standard Atmosphere (ISA) is a reference, not a forecast: a dry, still, perfect gas (R = 287 J/(kg·K), γ = 1.4) obeying the hydrostatic equation dp = −ρ g dh with a prescribed temperature profile. At sea level T0 = 288.15 K, p0 = 101 325 Pa and ρ0 = 1.225 kg/m³. In the troposphere temperature falls linearly at 6.5 K per km, so T = 288.15 − 0.0065 h, reaching 216.65 K at 11 km; from 11 km to 20 km the temperature is constant at 216.65 K. Combining the hydrostatic equation with p = ρRT gives the two working laws: in a gradient layer p/p1 = (T/T1)^(−g0/(aR)), with −g0/(aR) = 5.256 for a = −0.0065 K/m, and ρ/ρ1 = (T/T1)^4.256; in an isothermal layer p/p1 = ρ/ρ1 = exp[−g0(h − h1)/(RT)].

Altitude has several meanings. Geometric altitude is the tape-measure height; geopotential altitude is the height that would give the same potential energy with g held at g0, and it is the altitude the ISA tables are written in. Pressure altitude is the ISA altitude at which the standard pressure equals the pressure the aircraft is actually in, and density altitude is the same idea for density — a hot day raises density altitude and lengthens the take-off run. The density ratio σ = ρ/ρ0 links the airspeeds: indicated airspeed (IAS) is what the airspeed indicator shows; calibrated airspeed (CAS) is IAS corrected for instrument and position (installation) errors; equivalent airspeed (EAS) is CAS corrected for compressibility; and true airspeed (TAS) is the actual speed through the air, with EAS = TAS·√σ. EAS is the speed that gives the true dynamic pressure at sea-level density: q = ½ρV_TAS² = ½ρ0V_EAS².

Aircraft are classified first as lighter-than-air (balloons and airships, supported by buoyancy) or heavier-than-air (supported by aerodynamic force); the latter are fixed-wing (aeroplanes and gliders), rotary-wing (helicopters and autogyros) or flapping-wing. A fixed-wing aeroplane has a fuselage carrying payload and crew; a wing producing lift, fitted with ailerons, flaps and often slats and spoilers; an empennage — the horizontal stabiliser with its elevator and the vertical fin with its rudder; landing gear, tricycle or tail-wheel; and a powerplant. The resultant aerodynamic force is resolved into lift (normal to the free stream), drag (along it) and side force, and the resultant moment into rolling (about the longitudinal x-axis), pitching (about the lateral y-axis) and yawing (about the normal z-axis) moments. The angle of attack α is the angle between the relative wind and the body x-axis (or chord) in the plane of symmetry; the sideslip angle β is the angle between the relative wind and that plane, β = sin⁻¹(v/V).

• Altimeter — an aneroid capsule senses static pressure and the dial is graduated in ISA altitude; with the subscale set to 1013.25 hPa it reads pressure altitude.
• Airspeed indicator — measures pitot (total) minus static pressure and is calibrated with sea-level ISA density, so it reads IAS; only at sea level in the standard atmosphere does a perfect instrument read true airspeed.
• Vertical speed indicator — static pressure goes straight into a capsule and into the case through a calibrated leak; while the aircraft climbs or descends the two pressures differ in proportion to the rate of change of static pressure, i.e. the rate of climb. It lags a sudden change.
• Turn-and-bank (turn-and-slip) indicator — a rate gyroscope whose precession shows the rate of turn, plus a ball in a curved tube (an inclinometer) that shows whether the turn is balanced or the aircraft is slipping or skidding.

High-lift devices raise C_L,max so that the aeroplane can fly slowly for take-off and landing. Trailing-edge flaps — plain, split, slotted and Fowler (which also extends the chord and so the area) — increase effective camber: the C_L-α curve shifts up and to the left, α_L0 becomes more negative and the stall comes at a slightly lower α. Leading-edge devices — slats, slots, Krueger flaps and droop — delay leading-edge separation, so the curve simply extends to a higher stalling angle. The primary controls are the elevator (pitch), the ailerons (roll) and the rudder (yaw). Aileron deflection produces adverse yaw: the wing with the down-going aileron makes more lift and therefore more induced drag, yawing the nose away from the intended turn — the reason for differential and Frise ailerons and for co-ordinated rudder.

Below the stall the lift curve is linear, C_L = a(α − α_L0). The drag polar writes total drag as parasite plus lift-dependent drag: C_D = C_D0 + K·C_L², with K = 1/(π e AR) where AR is the aspect ratio and e the Oswald efficiency. In steady level flight L = W and T = D, so the thrust required is T_R = W/(L/D) and is least where L/D is greatest. Differentiating C_L/C_D gives the three conditions that the whole of performance rests on.

The maximum lift-to-drag ratio is (L/D)max = 1/(2√(K·C_D0)). In a steady climb at flight-path angle γ, T − D − W sin γ = 0, so the rate of climb is R/C = V sin γ = (TV − DV)/W — excess power per unit weight. In a power-off glide tan γ = D/L, so the flattest glide is at (L/D)max and the glide range from height h is h·(L/D); the smallest sink rate is at the minimum-power condition, which is a different, slower speed. As altitude rises the excess power shrinks: the absolute ceiling is where the maximum rate of climb falls to zero, the service ceiling where it falls to 100 ft/min (0.5 m/s). Stall speed is V_s = √(2W/(ρ S C_L,max)). For take-off, lift-off is taken at about 1.2 V_s and the ground roll is approximately s_LO ≈ 1.44 W²/(g ρ S C_L,max T) — it grows with the square of weight and falls with density, which is why a hot, high airfield is the hardest case. Landing approach is at about 1.3 V_s.

Range and endurance come from integrating the fuel burn as weight falls from W0 (take-off) to W1 (empty tanks). For a jet the thrust-specific fuel consumption c_t is the fuel weight burned per unit thrust per unit time, so dW/dt = −c_t T = −c_t W/(L/D). Endurance is then E = (1/c_t)(L/D) ln(W0/W1), largest at (L/D)max. Range, flying at constant V and L/D (a cruise-climb), is R = (V/c_t)(L/D) ln(W0/W1); at constant altitude, where V must fall as W falls, the optimum is at maximum C_L^(1/2)/C_D. For a propeller aircraft the engine meters fuel per unit power, so with specific fuel consumption c (fuel weight per unit power per unit time) and propeller efficiency η_p, R = (η_p/c)(L/D) ln(W0/W1) — best at (L/D)max, independent of speed — while endurance is best at maximum C_L^(3/2)/C_D, i.e. minimum power.

The load factor is n = L/W. In a level, co-ordinated turn at bank angle φ the vertical component of lift supports the weight, L cos φ = W, so n = 1/cos φ — 2 at 60° of bank. The horizontal component supplies the centripetal force, giving the turn radius R = V²/(g√(n² − 1)) = V²/(g tan φ) and the turn rate ω = V/R = g√(n² − 1)/V. A tight, fast turn therefore needs a high load factor, and the stall speed in the turn rises to V_s√n. In a pull-up from level flight n = 1 + V²/(gR).

The V-n (manoeuvre) diagram is the flight envelope the structure is designed for, plotted against equivalent airspeed. On the left it is bounded by the stall curves n = ½ρ0V_E²S C_L,max/W, parabolas that reach the positive and negative limit load factors; the positive curve meets the limit load factor at the corner or manoeuvring speed V_A = V_s√n_max, the lowest speed at which the limit load can be pulled and the speed at which the smallest radius and highest rate of turn are obtained. The top and bottom are the limit load factors, the right edge is the design dive speed. Gust lines are superimposed on it: a sharp-edged vertical gust raises the angle of attack and so the load factor, the increment growing with speed and lift-curve slope and falling with wing loading W/S.

Static stability asks only which way the first moment acts after a disturbance. Longitudinally, the pitching-moment coefficient about the centre of gravity is C_m = C_m0 + C_mα·α. The aeroplane is statically stable if a nose-up disturbance produces a nose-down moment, C_mα < 0, and it can be trimmed (C_m = 0) at a positive angle of attack only if C_m0 > 0 as well. With the centre of gravity at h and the wing aerodynamic centre at h_ac (both as fractions of the mean chord, measured aft from the leading edge), the wing contributes C_L,w(h − h_ac) — destabilising when the c.g. is behind the a.c. — and the tail contributes −η V_H C_L,t, where V_H = l_t S_t/(S c̄) is the horizontal tail volume ratio. With tail lift slope a_t, wing-body slope a and downwash gradient dε/dα, C_mα = a(h − h_ac) − η V_H a_t (1 − dε/dα).

The neutral point h_n is the c.g. position that makes C_mα zero: h_n = h_ac + η V_H (a_t/a)(1 − dε/dα). The static margin is h_n − h, and C_mα = −a(h_n − h), so the c.g. must lie ahead of the neutral point. A bigger tail, a longer tail arm or a smaller downwash gradient moves h_n aft and widens the allowable c.g. range; this is the whole reasoning behind horizontal tail position and size. If the pilot lets go of the stick, a free elevator floats with the local flow and the tail loses part of its restoring power. With the hinge-moment coefficient C_h = b0 + b1 α_t + b2 δ_e (+ b3 δ_tab) set to zero, the floating angle is δ_float = −(b1/b2)α_t, and the tail lift slope is multiplied by (1 − τ b1/b2), where τ = dα_t/dδ_e is the elevator effectiveness. Since b1/b2 is normally positive, the stick-free neutral point lies ahead of the stick-fixed one and the stick-free static margin is the smaller.

The hinge moment H = C_h q S_e c_e is what the pilot must hold, and through the control gearing G the stick force is F = G·H. A trim tab (b3 term) is used to drive the hinge moment, and so the stick force, to zero at the chosen speed; away from trim the stick-force gradient dF/dV is proportional to the stick-free static margin, which is what gives the pilot a feel for speed. Aerodynamic balance — a set-back hinge, a horn balance or an internal seal — reduces b1 and b2 and so the forces, but over-balancing makes them reverse. The elevator’s control power is C_mδe = −η V_H a_t τ.

Directional (weathercock) stability requires that a sideslip produce a yawing moment that turns the nose back into the relative wind: C_nβ > 0 in the usual sign convention. The fuselage ahead of the c.g. is destabilising; the vertical tail provides the restoring moment, its contribution C_nβ,v ≈ η_v V_v a_v (corrected for the sidewash the fuselage and wing induce) growing with the vertical tail volume ratio V_v = l_v S_v/(S b) — so the fin is sized by its area and its moment arm behind the c.g. The rudder’s control power C_nδr must be enough to hold the aircraft straight with an engine out and in a crosswind landing.

Lateral stability (the dihedral effect) requires that a sideslip produce a rolling moment that lifts the leading, into-wind wing away from the slip: C_lβ < 0. Geometric dihedral Γ raises the angle of attack of the leading wing and is stabilising; wing sweepback makes the leading wing more nearly perpendicular to the flow, and its effect grows with C_L; a high wing on a fuselage is stabilising and a low wing destabilising, from the cross-flow round the fuselage; and a vertical tail above the c.g. adds a stabilising rolling moment. Too much dihedral effect is as bad as too little, which is why many high-wing swept transports carry anhedral. The aileron’s control power is C_lδa.

Dynamic stability asks what the motion does over time. The aeroplane is a rigid body with six degrees of freedom; in body axes the force equations are X − mg sin θ = m(u̇ + qw − rv) and its two companions, and the moment equations are Euler’s equations with the products of inertia that the plane of symmetry leaves (only I_xz). Attitude is described by Euler angles in the 3-2-1 sequence — yaw ψ, then pitch θ, then roll φ — whose rates are not the body rates: φ̇ = p + (q sin φ + r cos φ) tan θ, θ̇ = q cos φ − r sin φ, ψ̇ = (q sin φ + r cos φ) sec θ. The tan θ and sec θ terms blow up at θ = ±90°, the singularity (gimbal lock) of this description.

Linearising about steady, wings-level, symmetric flight splits the equations in two. Because the aeroplane is symmetric about its xz-plane, small symmetric perturbations (u, w or α, q, θ) produce no asymmetric forces to first order and vice versa, so the longitudinal set decouples from the lateral-directional set (v or β, p, r, φ, ψ). Each set has characteristic roots whose pattern is remarkably consistent across aircraft.

• Short period (longitudinal) — a fast, heavily damped oscillation in α and pitch rate at almost constant speed; period of a few seconds.
• Phugoid (longitudinal) — a slow, lightly damped exchange of kinetic and potential energy, speed and height oscillating at nearly constant α. Lanchester’s approximation gives ω_n ≈ √2·g/U0, a period T ≈ π√2·U0/g, and a damping ratio ζ ≈ 1/(√2·L/D).
• Roll subsidence (lateral) — a fast, non-oscillatory, heavily damped convergence of roll rate, set by the roll damping derivative.
• Spiral (lateral) — a very slow, non-oscillatory mode that may converge or diverge; strong directional stability with weak dihedral effect tends to make it divergent.
• Dutch roll (lateral-directional) — a lightly damped oscillation coupling yaw and roll; strong dihedral effect with weak directional stability worsens it, and a yaw damper is the usual cure.