22-Mec-B7 Aero and Space Flight · May 2016

Question 6 of 7: Stability, Turning Flight and Propulsion

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 07-Mec-B7 Aero and Space Flight, National Examinations, May 2016. Three hours, open book; any non-communicating calculator permitted. Seven questions of equal value; any six constitute a complete paper and only the first six answered are marked, so full marks are 120 and the percentage grade is (mark obtained / 120) × 100. Several questions call for an essay-format answer, where clarity and organisation carry marks. All seven questions are solved here.

Reference texts. Solutions follow the conventions of the texts recommended for this examination code:

J. D. Anderson Jr., Introduction to Flight, 9th ed. — the standard atmosphere and the four altitudes (Ch. 3), incompressible and compressible flow with the Pitot-static tube (§3.4, §4.11, §8.7), airplane performance (Ch. 6), stability and control (Ch. 7), propulsion (Ch. 9), space flight and atmospheric entry (Ch. 8). This is the primary reference throughout.
J. D. Anderson Jr., Fundamentals of Aerodynamics, 6th ed. — finite-wing theory and induced drag (Ch. 5), transonic flow, drag divergence and the area rule (Ch. 11).
B. N. Pamadi, Performance, Stability, Dynamics and Control of Airplanes, 3rd ed. — take-off and landing distances and gust load factors (Ch. 2, Ch. 5).
G. P. Sutton and O. Biblarz, Rocket Propulsion Elements, 9th ed. — the ideal rocket equation and propellant-system comparison (Ch. 4, Ch. 11–12).
H. D. Curtis, Orbital Mechanics for Engineering Students, 4th ed. — the orbit equation, vis-viva and orbital elements (Ch. 2–3).

Check — assumptions carried through the performance questions. The paper's page-1 note invites the candidate to "submit with their answer paper a clear statement of any assumptions made", and three quantities the performance questions need are never stated:

Standard atmosphere. ISA sea-level values $T_0 = 288.15\ \text{K}$, $p_0 = 101\,325\ \text{Pa}$, $\rho_0 = 1.225\ \text{kg}\,\text{m}^{-3}$, troposphere lapse rate $L = 0.0065\ \text{K}\,\text{m}^{-1}$ to 11 km, then isothermal at $216.65\ \text{K}$; $R = 287.05\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$, $\gamma = 1.4$. Inside the atmosphere model $g = 9.80665\ \text{m}\,\text{s}^{-2}$, giving the exponents $g/(LR) = 5.2559$ for pressure and $4.2559$ for density; aircraft weights use $g = 9.81\ \text{m}\,\text{s}^{-2}$.
Thrust lapse (Questions 4 and 5). For a fixed-geometry turbojet the thrust is taken proportional to density, $T = T_{SL}\,(\rho/\rho_0)$. Nothing in the paper states a lapse law, and this is the conventional first approximation.
Ground-run averaging (Question 5b). The net accelerating force is evaluated once at $V_{LO}/\sqrt{2}$ — the speed at which $V^2$ equals its mean over the run — and the run is then taken as uniformly accelerated.

Compressibility corrections to lift and drag are ignored wherever the question says so (Question 4a) and elsewhere in Questions 4 and 5, consistent with the parabolic drag polar supplied.

Question 6: Stability, Turning Flight and Propulsion (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

This question is descriptive throughout, apart from the governing relations quoted in part (c); each part is answered in turn.

(a) Static versus dynamic stability. Static stability concerns the initial tendency of the aircraft after a disturbance from a trimmed equilibrium. The aircraft is statically stable if the forces and moments generated by the disturbance act to return it towards equilibrium: in pitch this requires $\mathrm{d}C_M/\mathrm{d}\alpha \lt 0$ together with a positive trim moment $C_{M,0} \gt 0$, which is what a tailplane set at a suitable incidence, with the centre of gravity ahead of the neutral point, provides. Static stability says nothing about what happens subsequently. Dynamic stability concerns the time history of the motion: the aircraft is dynamically stable if the disturbance eventually dies away. A statically stable aircraft can be dynamically unstable — it will initially head back towards trim, overshoot, and oscillate with growing amplitude — but a statically unstable aircraft cannot be dynamically stable, so static stability is a necessary but not sufficient condition. The distinction matters in practice because the characteristic modes of an aeroplane (short-period and phugoid in the longitudinal axis; roll subsidence, spiral and Dutch roll in the lateral-directional axis) differ enormously in damping and period: the phugoid is typically very lightly damped and the spiral mode is frequently mildly divergent, both being acceptable because the pilot or autopilot can correct on that timescale, whereas an undamped short-period or Dutch-roll mode is unflyable.

(b) Achieving static lateral stability. Lateral (roll) static stability means that a disturbance in bank, which produces a sideslip, generates a restoring rolling moment — formally, that the rolling-moment derivative with sideslip is negative, $C_{l\beta} \lt 0$, the "dihedral effect". The devices commonly used on a subsonic aircraft are:

Geometric dihedral. Raising the wing tips above the root means that in a sideslip the leading wing meets the relative wind at a larger effective angle of attack than the trailing wing, and the lift difference rolls the aircraft back to level. This is the primary and most direct means, typically 3–7° on a low-wing aeroplane.
Wing sweep-back. In a sideslip the leading wing presents a smaller effective sweep angle, so the velocity component normal to its leading edge is larger and it generates more lift — the same restoring sense. The effect is proportional to $C_L$, so it is strongest at low speed, and a swept-wing jet often needs anhedral to avoid excessive dihedral effect and the resulting Dutch-roll tendency.
Wing vertical position. A high wing contributes a substantial positive dihedral effect through the sidewash the fuselage induces around it; this is why high-wing transports typically need little or no geometric dihedral, and low-wing aircraft need more.
Vertical fin and dorsal fin. Because the fin's side force acts above the roll axis, the sideslip-induced side force produces a rolling moment in the stabilising sense as well as the directional (weathercock) moment it is chiefly there for.

Lateral and directional stability cannot be designed independently: too much dihedral effect relative to directional stability produces a poorly damped Dutch roll, while too little produces spiral divergence, so the ratio is a design compromise rather than a maximisation.

(c) Factors setting the minimum turn radius. In a steady, level, coordinated turn at bank angle $\phi$ the lift must both carry the weight and supply the centripetal force, so with load factor $n = L/W = 1/\cos\phi$, $$R = \frac{V^{2}}{g\sqrt{n^{2}-1}}$$ The radius is therefore reduced by flying slower and by pulling more $g$. Three limits bound how far each can be pushed at a given altitude:

The structural limit. The airframe's design limit load factor $n_{max}$ (2.5 for a transport, 7–9 for a fighter) caps $n$ outright; above the corner speed, the turn is structurally limited and $R$ grows as $V^2$.
The aerodynamic (stall) limit. The wing can only produce $L = \tfrac12\rho V^{2}SC_{L,max}$, so $n \le \rho V^{2}C_{L,max}/(2W/S)$. Below the corner speed the turn is lift-limited, and the tightest possible radius, $R_{min} = 2(W/S)/(\rho g C_{L,max})$ in the limit of large $n$, depends on wing loading, maximum lift coefficient and density — hence directly on altitude.
The thrust limit. Turning at load factor $n$ multiplies the induced drag by $n^{2}$, so a sustained turn requires $T \ge qSC_{D0} + n^{2}KW^{2}/(qS)$. If the available thrust cannot meet this, the turn can only be flown by trading speed or height, and the sustained turn radius is larger than the instantaneous one. Since thrust lapses with density while the lift limit also worsens, turn performance degrades markedly with altitude.

The tightest turn is achieved at the corner speed, where the stall and structural limits intersect, and it is generally thrust-limited if it is to be sustained.

(d) Propulsive efficiency of a jet engine. Propulsive (or Froude) efficiency measures how much of the mechanical power the engine imparts to the working fluid actually appears as useful propulsive power on the airframe. With a mass flow $\dot m$ taken in at flight speed $V_\infty$ and expelled at $V_j$, the thrust is $F = \dot m (V_j - V_\infty)$, the useful power is $FV_\infty$, and the power added to the stream is $\tfrac12\dot m (V_j^{2} - V_\infty^{2})$, so $$\eta_{p} = \frac{F V_\infty}{\tfrac12 \dot m (V_j^{2}-V_\infty^{2})} = \frac{2}{1 + V_j/V_\infty}$$ The shortfall is the kinetic energy left behind in the jet wake, which is wasted. The expression shows the central design tension: a given thrust can be produced either by giving a small mass flow a large velocity increment or by giving a large mass flow a small one, and only the second is efficient, since $\eta_p \to 1$ as $V_j \to V_\infty$. It also shows that propulsive efficiency improves as flight speed rises towards the jet speed, which is why the pure turbojet is well suited to supersonic flight and poor at low subsonic speeds. Overall efficiency is the product of the thermal efficiency of the cycle and this propulsive efficiency, $\eta_o = \eta_{th}\eta_p$.

(e) Why by-pass engines and afterburning are used. The two are opposite responses to the propulsive-efficiency relation above. A by-pass (turbofan) engine uses the core turbine to drive a large fan that accelerates a much greater mass of air through a modest velocity increment, bypassing the combustor. For the same thrust this raises $\dot m$ and lowers $V_j$, so $\eta_p$ improves sharply; specific fuel consumption falls by 30–50 per cent relative to a turbojet at subsonic cruise, and the lower jet velocity also reduces jet noise, which scales with roughly the eighth power of exhaust velocity. High-bypass turbofans (bypass ratios of 5–12, and above) are therefore universal on subsonic transports, at the cost of frontal area, nacelle drag and weight. Afterburning (reheat) does the reverse: additional fuel is burned in the exhaust duct downstream of the turbine, where oxygen remains, raising the exhaust temperature and hence $V_j$ and giving a thrust augmentation of 50 per cent or more from an engine of unchanged frontal area. Because it raises $V_j$ it degrades propulsive efficiency badly, and the specific fuel consumption may double or triple, so it is used only in short bursts — take-off from short runways, transonic acceleration through the drag-rise peak of Question 3(a), and combat manoeuvre. The two are complementary rather than contradictory: a modern military engine is a low-bypass turbofan with reheat, cruising efficiently and augmenting when required.

(f) Solid versus liquid propellant rocket engines. In a solid motor the fuel and oxidiser are pre-mixed into a rubbery grain cast directly into the motor case, which is also the combustion chamber. Ignition burns the exposed surface of the grain, and the thrust-time history is fixed at manufacture by the grain geometry (star, tubular, end-burning). The advantages are simplicity, ruggedness, very high propellant density (hence compact stages), long storage life without maintenance, and instant readiness — which is why solids dominate military and tactical applications and are standard for launch-vehicle boosters. The disadvantages are that the burn cannot ordinarily be throttled, stopped or restarted; specific impulse is lower (roughly 240–270 s against 300–460 s for liquids); and a cracked or debonded grain exposes extra burning area and can burst the case. In a liquid engine fuel and oxidiser are stored separately in tanks and delivered to a small combustion chamber by pressure feed or turbopumps. The propellant combinations reach far higher specific impulse (liquid oxygen with kerosene, or with liquid hydrogen at over 450 s in vacuum), the engine can be throttled, shut down and restarted, and the propellants can regeneratively cool the chamber and nozzle. The costs are mechanical complexity (pumps, valves, injectors, plumbing), the handling difficulties of cryogenic or hypergolic and toxic propellants, low bulk density requiring large tanks, and lengthy launch preparation. Hybrid motors, with a solid fuel grain and a liquid or gaseous oxidiser, seek the throttleability of the liquid with much of the simplicity and safety of the solid.

Topic. Aircraft stability concepts, the turn performance envelope, and the efficiency logic behind air-breathing and rocket propulsion choices.

Key relations. Longitudinal static stability $\mathrm{d}C_M/\mathrm{d}\alpha \lt 0$; lateral $C_{l\beta} \lt 0$; turn $n = 1/\cos\phi$, $R = V^{2}/(g\sqrt{n^{2}-1})$; Froude efficiency $\eta_p = 2/(1 + V_j/V_\infty)$; overall $\eta_o = \eta_{th}\eta_p$; rocket specific impulse $I_{sp} = c/g_0$.

Why this works. Parts (d) and (e) are one idea: thrust is a momentum transaction, $F = \dot m \Delta V$, but the energy cost is quadratic in $\Delta V$, so spreading a given thrust over more mass at less velocity increment is always thermodynamically cheaper. Every propulsion decision in the question follows from that asymmetry — the propeller is its extreme (enormous $\dot m$, tiny $\Delta V$, high efficiency but limited to low Mach numbers), the turbofan is the practical compromise, the turbojet suits high $V_\infty$ where the ratio $V_j/V_\infty$ is naturally close to one, and the afterburner deliberately sacrifices efficiency for thrust density. The stability discussion in (a) and (b) rests on the distinction between the sign of a restoring derivative and the roots of the characteristic equation of the resulting motion, and the turn discussion in (c) is the intersection of three independent constraint curves on the $V$–$n$ plane.

Common pitfall. Treating static and dynamic stability as synonyms, or asserting that dynamic stability implies static stability without noting the one-way logical relation. In (b), listing dihedral alone and omitting the roles of sweep and wing vertical position, or confusing lateral with directional stability. In (c), quoting a turn radius without stating which of the three limits is active. In (d), defining propulsive efficiency as thrust per unit fuel flow, which is specific fuel consumption, a different quantity altogether.

Source. Anderson, Introduction to Flight, 9th ed., Ch. 7 (stability and control), §6.16–6.17 (turning flight and the V-n diagram), Ch. 9 (propulsion, propulsive efficiency, turbofans and afterburning), §9.9 (rocket engines); Sutton and Biblarz, Rocket Propulsion Elements, 9th ed., Ch. 11–12.

← Question 5 Next: Question 7 →