Aircraft Performance, Propulsion and Wing – Key Vocabulary

Aims and Objectives

Aim: Introduce the basics of flight within Aircraft Performance, Propulsion and Wing (AE2111).
Objectives
• Establish deeper understanding of aircraft performance.
• Further introduce aircraft propulsion knowledge.
• Develop broader understanding of wings and airfoils.

Learning Outcomes

Apply fundamental principles of fluid flow to solve simple problems.
Apply flight-mechanic & aerodynamic principles to estimate aircraft performance.
Describe boundary-layer flow characteristics over a wing.
Describe & perform calculations in transonic and supersonic flows.

Teaching Arrangement & Weekly Topics

30 Jan Introduction & Cruise Performance
06 Feb Range Calculation
13 Feb Drag Estimation
20 Feb Take-Off Performance
27 Feb Aircraft Propulsion
05 Mar Propellers
12 Mar Airfoils
19 Mar Wing Aerodynamic Performance
26 Mar Longitudinal Stability & Control
16 Apr Reading Week (incl. Flying)
23 Apr Airworthiness
30 Apr Revision

Assessment Overview

Coursework (30 %) – shared with other module parts; within this part: 15 % Drag-calculation assignment.
Exam (70 %) – explanation, discussion & calculation questions.

Introduction: What Is Aircraft Performance?

Concerned with extreme quantities of translational motion of the aircraft CG that govern operational & economic use.
Typical outputs:
• Flight envelope (min/max $V,\ h,\ n$ ).
• Range & endurance.
• Climb/Descent, Cruise, Turning, Take-off & Landing performance.

Aircraft Performance Analysis – Sub-Domains

Mathematical modelling (3-DOF point-mass EoM, environment, aero, engine, systems).
Analytical calculations.
Non-linear desktop simulation for complex/dangerous manoeuvres.
Flight-test validation.

Key Performance Parameters

Speeds: $V<em>{min},\ V</em>{max}$ .
Range (Bréguet).
Endurance (Loiter).
$L/D$ ratio.
Loading factor $n$ .
Thrust-to-weight $T/W$ .
Wing loading $W/S$ .

Equations of Motion (Point-Mass 1-D Example)

Newton’s 2nd law: $F = m a$ .
• Forces: $F = T - D$ (steady level).
• If $F$ constant → $a = \tfrac{F}{m}$ .
• Velocity: $V = a t + V<em>0$ . • Position: $X = \tfrac{1}{2} a t^2 + V</em>0 t + X_0$ .

Thrust–Velocity Curves

Derived from drag polar $C<em>D(C</em>L)$ by equating $T= D = \tfrac{1}{2}\rho V^2 S C_D$ in steady level flight.
Curve indicates thrust required & excess thrust across $V$ range.

Phase Mission Profile (Range & Endurance)

0–1 Engine start → 1–2 Taxi → 2–3 Take-off → 3–4 Climb → 4–5 Cruise → 5–6 Loiter → 6–7 Descent → 7–8 Divert → 8–9 Land/Shutdown.

V–n & Gust Diagram Basics

Defines structural/operational envelope (limit load factors vs. airspeed).
Includes design-gust lines, flap-down limits, dive speed $V<em>D$ , manoeuvre speed $V</em>B$ .

Cruise Performance Topics

Recap aerodynamics (drag polar).
Specific range & fuel flow for turbofan vs. propeller.
Optimisation with compressibility & wind effects.

Aerodynamic Fundamentals Recap

Lift: $L = \tfrac{1}{2}\rho V^2 S C_L$ .
Drag: $D = \tfrac{1}{2}\rho V^2 S C_D$ .
Drag polar: $C<em>D = C</em>{D0} + k C_L^2$ with $k = \tfrac{1}{\pi e AR}$ .
Minimum $C<em>D$ occurs at $C</em>L = 0$ but flight impossible → examine $D/L$ instead.

Zero-Lift Drag $C_{D0}$

Dominated by skin-friction & form drag.
Typical values: $0.003 \text{ (streamlined)}$ to $0.005 \text{ (poorly streamlined)}$ .
Depends on wetted-area/wing-area ratio (≈2 for flying wing, 5–6 for transport).
Reduction methods: better streamlining or lower wetted-area ratio.

Induced Drag Coefficient & $D/L$ Minimum

$C<em>{Di} = k C</em>L^2$ .
To reduce $k$ : raise aspect ratio or achieve elliptic lift distribution ( $e\to1$ ).
Ratio $\tfrac{D}{L} = C<em>D/C</em>L = C<em>{D0}/C</em>L + k C_L$ .
Minimum occurs at $C<em>{L,\min (D/L)} = \sqrt{\tfrac{C</em>{D0}}{k}}$ giving $(D/L)<em>{\min} = 2\sqrt{C</em>{D0}k}$ .
Same $C<em>L$ independent of altitude ⇒ same minimum $D/L$ , but corresponding speed increases with altitude because $V \propto \sqrt{W/(\rho S C</em>L)}$ .

Example: Drag vs. Airspeed at Multiple Altitudes

Given: $m = 100000\,\text{kg},\ S=200\,\text{m}^2,\ \rho<em>{SL}=1.225,\ \rho</em>{5km}=0.736,\ \rho_{10km}=0.414$ .
Procedure:
• Choose speed grid, compute $C<em>L = \tfrac{2W}{\rho V^2 S}$ . • Evaluate $C</em>D$ via polar, then $D$ .
Result: curves shift rightward with altitude; minimum-drag speed higher aloft.

Fuel Flow Models

Turbofan: $\dot m<em>f = \text{sfc}</em>T \; T$ .
Propeller: $\dot m<em>f = \text{sfc}</em>P \; P$ , with prop efficiency $\eta<em>p = \tfrac{T V}{P}$ ⇒ $P = \tfrac{T V}{\eta</em>p}$ .

Minimum Fuel Flow (Maximum Endurance) Speed

Turbofan:

Occurs at minimum drag speed.
$V<em>{\min FF}=V</em>{\min D}$ , $\dot m<em>{f,\min}=\text{sfc}</em>T D_{\min}$ .
Slightly lower at high altitude since sfc drops.
Propeller:
Fuel proportional to power ⇒ graph power vs. $V$ .
Minimum power occurs at $C<em>{L,\min P}=\sqrt{\tfrac{3 C</em>{D0}}{k}}$ .
$V<em>{\min P}=0.76 V</em>{\min D}$ ; rises with altitude.
Aircraft is speed-unstable near $V_{\min P}$ (drag decreases when speeding up).

Specific Range (SR)

Definition (instantaneous): $\text{SR}=\tfrac{\text{distance}}{\text{fuel mass}}=\tfrac{V}{\dot m_f}$ .
Turbofan analytic form in cruise (steady): $\text{SR}=\tfrac{1}{\text{sfc}_T}\;\tfrac{L}{D}\;\tfrac{1}{W}$ (for unit weight loss).
Classic Bréguet Range: $R=\tfrac{V}{\text{sfc}<em>T}\;\tfrac{L}{D}\;\ln!\left(\tfrac{W</em>i}{W_f}\right)$ .

Graphical Optimisation of SR

Plot thrust-required (drag) curve.
Draw rays from origin; slope $=\tfrac{\dot m_f}{V}$ .
Tangency point gives max SR.
Increasing altitude: curve shifts; optimal $V$ grows, SR improves due to lower sfc & higher $L/D$ product $V/\rho$ .

Spreadsheet Optimisation Example

Given: $S=200\,\text{m}^2,\ C_{D0}=0.02,\ k=0.04,\ W=100000g,\ \text{sfc}=0.6\,\text{h}^{-1}$ .
Sea-level optimal $V≈140\,\text{m/s}$ ; at 10 km optimal $V≈222\,\text{m/s}$ with higher SR.

Analytical Optimisation Result

Substitute $V=\sqrt{\tfrac{2W}{\rho S C_L}}$ and $T=D$ into SR; differentiate.
Maximum SR for turbofan occurs at $C<em>{L,opt}=\sqrt{\tfrac{C</em>{D0}}{3k}}$ (same as max $M(L/D)$ ).
Speed follows from lift equation.

Propeller Aircraft – SR Characteristics

$\text{SR}=\tfrac{\eta<em>p}{\text{sfc}</em>P}\;\tfrac{L}{D}\;\tfrac{1}{W}$ .
Max SR independent of altitude; governed by same $C_L$ as minimum drag (not power).
Graphical: draw rays from origin on power-required curve.

Compressibility Effects (High-Subsonic & Transonic)

Above $M≈0.7$ : drag polar depends on Mach: $C<em>D(M)=C</em>{D0}+k C<em>L^2 + \Delta C</em>D(M)$ .
Optimisation now seeks max $M(L/D)$ rather than $L/D$ .
Plot $M(L/D)$ vs. $C<em>L$ to pick optimal Mach/ $C</em>L$ combination.

M(L/D) Optimisation & Step-Climb Logic

In stratosphere, speed of sound nearly constant with altitude ⇒ fixed Mach implies fixed TAS.
To keep optimal $C_L$ while weight drops, aircraft climbs (step climbs) maintaining Mach; may be ATC-constrained.
Available thrust limits high-altitude operation especially post-engine failure.

Wind Effects

Distinguish airspeed $V<em>a$ and ground speed $V</em>g = V<em>a + V</em>{wind}$ (tailwind positive).
Specific range with wind: $\text{SR}=\tfrac{V<em>g}{\dot m</em>f}=\tfrac{V<em>a+V</em>{wind}}{\dot m_f}$ .
Tailwind ⇒ higher SR & lower optimum $V_a$ ; headwind opposite.

Drag-Characteristics “Three Speeds” (Level Flight)

Minimum power speed $V<em>{\min P}=0.76 V</em>{\min D}$ (prop focus).
Minimum drag speed $V_{\min D}$ (turbofan endurance).
Max $L/D$ speed (gives max range for given sfc).
Designers balance these according to mission.

Recap – Core Formulae

Drag polar: $C<em>D=C</em>{D0}+kC_L^2$ .
Induced-drag factor: $k=\tfrac{1}{\pi e AR}$ .
Lift & Drag: $L=\tfrac12\rho V^2 S C<em>L,\; D=\tfrac12\rho V^2 S C</em>D$ .
Min $D/L$ : $C<em>{L}=\sqrt{C</em>{D0}/k}$ .
Min power $C<em>L$ : $\sqrt{3C</em>{D0}/k}$ .
Prop efficiency: $\eta_p=\tfrac{TV}{P}$ .
Bréguet range: $R=\tfrac{V}{\text{sfc}}\tfrac{L}{D}\ln!\bigl(\tfrac{W<em>i}{W</em>f}\bigr)$ .
Endurance: $E=\tfrac{1}{\text{sfc}}\tfrac{L}{D}\ln!\bigl(\tfrac{W<em>i}{W</em>f}\bigr)$ .

Ethical & Practical Considerations

Efficient cruise reduces emissions & operating cost.
Range/endurance predictions critical for safety (fuel reserves, diversion).
High-altitude cruise must respect structural loads (V-n diagram) & thrust availability.
Step-climb vs. ATC constraints illustrate trade-off between optimal performance and operational rules.