Unit 1: Steady Unaccelerated Flight

I. Flight Condition — Foundations of Performance Analysis

A. Introduction

Steady unaccelerated flight is a condition in which an aircraft’s velocity vector remains constant, so the resultant external force and acceleration are zero.

• Steady condition: Flight variables such as speed (V), altitude (h), and flight-path angle (\gamma) do not change with time.
• Unaccelerated condition: Newton’s second law gives (\sum \mathbf{F}=m\mathbf{a}=0); therefore, forces balance along every axis.
• Reference case: In straight, level, wings-level flight, (\gamma=0), bank angle (\phi=0), and load factor (n=L/W=1).
• Standard assumptions:
  • The aircraft is treated as a rigid body.
  • The atmosphere is locally uniform and stationary.
  • Aerodynamic forces act at representative points.
  • Propulsive effects on the airflow are neglected unless specified.
• Performance objective: Force balance is combined with aerodynamic relations to determine required thrust, required power, attainable speed, and efficiency.

II. Force Balance and Motion — Equilibrium of the Aircraft

A. Four forces of flight

An aircraft in steady flight is governed primarily by lift, weight, thrust, and drag.

• Lift (L): The aerodynamic force perpendicular to the relative wind, measured in newtons (N).
• Weight (W): The gravitational force acting vertically downward through the centre of gravity:
  
  TEXT

    W = mg

  
  where (m) is aircraft mass in kilograms and (g) is gravitational acceleration in (\text{m/s}^2).
• Thrust (T): The propulsive force produced by an engine-propeller, turbojet, turbofan, or other propulsion system; it generally acts approximately along the aircraft’s longitudinal axis.
• Drag (D): The aerodynamic force parallel and opposite to the relative wind.
• Straight-and-level balance:
  
  TEXT

    L = W
    T = D

  
  These equalities indicate equilibrium, not equality between all four forces.
• Physical interpretation: Lift balances gravity, while thrust supplies the force needed to overcome aerodynamic resistance.

B. General equation of motion

The general equations of translational motion resolve Newton’s second law along and normal to the flight path.

• Flight-path equations: If thrust is inclined by angle (\epsilon) above the flight path,
  
  TEXT

    m(dV/dt) = T cos ε - D - W sin γ
    mV(dγ/dt) = L + T sin ε - W cos γ

  
  where (V) is flight speed, (\gamma) is flight-path angle, and (\epsilon) is thrust inclination.
• Steady flight reduction: With (dV/dt=0) and (d\gamma/dt=0),
  
  TEXT

    T cos ε = D + W sin γ
    L + T sin ε = W cos γ

• Level flight: For (\gamma=0) and approximately aligned thrust (\epsilon=0), the equations reduce to (T=D) and (L=W).
• Rotational equilibrium: Steady attitude additionally requires zero resultant moments:
  
  TEXT

    ΣM = 0

  
  Thus pitching, rolling, and yawing moments must balance through control-surface deflection and aircraft trim.

III. Performance Curves — Propulsion versus Aerodynamic Demand

A. Power available and power required curves

Power curves compare the engine’s usable output with the aerodynamic power needed to maintain steady flight.

• Power required (P_R): Required power is the rate at which thrust must perform work:
  
  TEXT

    P_R = T_R V = DV

  
  where (T_R) is thrust required and (V) is true airspeed.
• Level-flight form: For a parabolic drag polar, required power has the form
  
  TEXT

    P_R = A/V + BV³

  
  where (A) represents the induced-drag contribution and (B) represents the parasite-drag contribution.
• Curve shape: At low speed, induced drag produces a large (A/V) term; at high speed, parasite drag makes the (BV^3) term dominant. The result is a U-shaped curve.
• Power available (P_A):
  1. Propeller aircraft: Shaft power available is approximately constant with speed over a limited operating range, after accounting for propeller efficiency.
  2. Jet aircraft: Nearly constant thrust gives (P_A=T_AV), so power available rises approximately linearly with speed.
• Performance margin: Excess power determines climb capability:
  
  TEXT

    Rate of climb = (P_A - P_R)/W

  
  where (W) is aircraft weight.
• Minimum-power condition: For (CD=C{D0}+kC_L^2), minimum power occurs at (CL=\sqrt{3C{D0}/k}).

B. Thrust available and thrust required curves

Thrust curves identify the speeds at which propulsion can exactly meet or exceed aerodynamic drag.

• Thrust required (T_R): In steady level flight, (T_R=D), giving
  
  TEXT

    T_R = A/V² + BV²

  
  where the first term represents induced drag and the second represents parasite drag.
• Minimum thrust: The bottom of the U-shaped thrust-required curve corresponds to minimum drag and maximum lift-to-drag ratio.
• Thrust available (T_A):
  1. Turbojet approximation: (T_A) is nearly constant with speed at a fixed altitude.
  2. Propeller approximation: Since (T_A=P_A/V), available thrust generally decreases as speed increases.
• Steady-speed points: Intersections satisfying (T_A=T_R) define possible steady level-flight speeds.
• Excess thrust: The difference (T_A-T_R) supplies acceleration or a climb component; the maximum excess-thrust condition is associated with the steepest climb angle.

IV. Aircraft Parameters — Propulsive and Geometric Loading

A. Thrust-to-weight ratio

The thrust-to-weight ratio measures propulsive force relative to aircraft weight and indicates the ability to overcome drag and change the flight path.

• Definition:
  
  TEXT

    T/W = available thrust / aircraft weight

  
  Both quantities are forces, so (T/W) is dimensionless.
• Level-flight requirement:
  
  TEXT

    T_R/W = D/W = 1/(L/D)

  
  because (L=W) in level flight.
• Maximum aerodynamic efficiency: The minimum ratio required for level flight is
  
  TEXT

    (T/W)min = 1/(L/D)max

• Performance effect: A larger available (T/W) generally improves acceleration, climb angle, and maximum speed.
• Operational variation: (T/W) increases as fuel burn reduces weight, but available thrust usually falls with increasing altitude.

B. Wing loading

Wing loading expresses aircraft weight supported per unit wing planform area.

• Definition:
  
  TEXT

    W/S = aircraft weight / wing area

  
  where (S) is wing area; SI units are (\text{N/m}^2).
• Lift relation:
  
  TEXT

    W/S = ½ρV²C_L

  
  where (\rho) is air density and (C_L) is lift coefficient.
• High wing loading: Produces higher stall and take-off speeds, but usually reduces sensitivity to atmospheric disturbances and permits a smaller wing.
• Low wing loading: Produces lower minimum speed and better low-speed manoeuvrability, but requires greater wing area.
• Performance connection: At fixed (C_L) and density, characteristic flight speeds vary approximately as (\sqrt{W/S}).

V. Aerodynamic Efficiency — Drag and Lift Relationships

A. Drag polar

The drag polar relates drag coefficient to lift coefficient and separates parasite and lift-dependent drag.

• Parabolic polar:
  
  TEXT

    C_D = C_D0 + kC_L²

  
  where (CD) is total drag coefficient, (C{D0}) is zero-lift drag coefficient, and (k) is the induced-drag factor.
• Dimensional drag:
  
  TEXT

    D = ½ρV²SC_D

• Parasite drag: The (C_{D0}) term includes skin-friction, form, and interference drag; dimensional parasite drag rises approximately with (V^2).
• Induced drag: The (kC_L^2) term arises from finite-wing lift and trailing vortices.
• Induced-drag factor:
  
  TEXT

    k = 1/(πeAR)

  
  where (e) is Oswald efficiency factor and (AR=b^2/S) is aspect ratio, with (b) denoting wingspan.
• Level-flight consequence: Because (C_L=2W/(\rho V^2S)), induced drag decreases approximately as (1/V^2).

B. Lift-to-drag ratio

The lift-to-drag ratio measures aerodynamic efficiency by comparing useful lift with resisting drag.

• Coefficient form:
  
  TEXT

    L/D = C_L/C_D = C_L/(C_D0 + kC_L²)

• Maximum condition: Differentiating (L/D) with respect to (C_L) gives
  
  TEXT

    C_L,(L/D)max = √(C_D0/k)

• Equal-drag condition: At maximum (L/D),
  
  TEXT

    C_D0 = kC_L²

  
  so parasite and induced drag coefficients are equal.
• Maximum value:
  
  TEXT

    (L/D)max = 1/[2√(C_D0k)]

• Significance: Maximum (L/D) corresponds to minimum drag, minimum thrust required, and the shallowest power-off glide angle in still air.

C. Aerodynamic relations associated with lift-to-drag ratio

Lift-to-drag ratio links aerodynamic coefficients directly to speed, glide performance, thrust requirement, and range-related conditions.

• Dynamic-pressure relation:
  
  TEXT

    q = ½ρV²
    L = qSC_L
    D = qSC_D

  
  where (q) is dynamic pressure in pascals.
• Speed for maximum ratio:
  
  TEXT

    V_(L/D)max = √[(2W)/(ρS C_L,(L/D)max)]

  
  Thus this speed increases with weight and wing loading and decreases with air density.
• Gliding flight: For a steady power-off glide,
  
  TEXT

    tan |γ| = D/L = 1/(L/D)

  
  so maximum (L/D) gives the minimum magnitude of glide angle and maximum still-air glide distance per unit altitude.
• Jet range connection: Under standard simplified assumptions, jet range is strongly favoured by a high (L/D).
• Propeller endurance connection: Minimum power rather than maximum (L/D) controls maximum endurance; the relevant aerodynamic quantity is (C_L^{3/2}/C_D).
• Speed relationship: For the parabolic polar,
  
  TEXT

    V_min-power = V_(L/D)max / 3^(1/4)

  
  so minimum-power speed is about (0.76) times minimum-drag speed.

VI. Low-Speed Boundary — Lift Limitation

A. Minimum velocity

Minimum steady level-flight velocity is set by the greatest lift coefficient the wing can produce before stall.

• Lift equilibrium:
  
  TEXT

    W = ½ρV²SC_L

  
  Reducing speed requires an increase in (C_L), normally achieved by increasing angle of attack.
• Stall-speed equation:
  
  TEXT

    V_min = V_s = √[2W/(ρS C_Lmax)]

  
  where (Vs) is stall speed and (C{L\max}) is maximum lift coefficient.
• Controlling factors: Minimum velocity rises with weight and wing loading, and falls with density, wing area, or increased (C_{L\max}).
• High-lift devices: Flaps and slats increase (C_{L\max}), thereby reducing take-off and landing speeds.
• Load-factor effect:
  
  TEXT

    V_s,n = V_s,1 √n

  
  where (n=L/W); consequently, a level turn with (n>1) has a higher stall speed than wings-level flight.
• Practical boundary: Sustainable flight also requires (T_A\geq T_R); therefore, propulsion limits may prevent operation near the aerodynamic stall boundary.