Drag Polar

Module 5

5.1 Drag Polar

5.2 Range & Endurance

5.3 Aerodynamic efficiency

5.4 Winds Effects

5.1 Drag Polar

TOTAL DRAG
- PARASITE DRAG
  - INTERFERENCE
  - PROFILE DRAG
    - SKIN FRICTION
    - FORM DRAG
- INDUCED DRAG
- WAVE DRAG

TOTAL DRAG CLASSIFICATION

WING DRAG

PARASITE DRAG

SHOCK DRAG

Drag Equation

The Total Drag Equation

Drag force D = ½ ρ V² S C_D

Where, ρ = density of air at certain altitute, V = Velocity of flight, S= Wing Area, C_D = Drag coefficient. The total drag coefficient C_D is expressed using the parabolic drag polar equation:

C_D = C_D,0 + kC_L²

Where C_D,0 is the parasite drag coefficient and kC_L² represents the induced drag (C_D,i). The factor k is defined as:

k = 1 / (π · AR · e)

Induced Drag (C_D,i)

Mathematically, induced drag is tied to the Lift Coefficient (C_L). As the aircraft slows down, it must fly at a higher angle of attack to maintain lift, which increases C_L and, consequently, induced drag. This type of drag is induced by the lift force. It results from the generation of a trailing vortex system downstream of a lifting surface with a finite aspect ratio.

D_i = ½ ρ V² S [ C_L² / (π · AR · e) ]

As velocity (V) increases, induced drag decreases (D_i ∝ 1/V²).

Parasite Drag (D_p)

Parasite drag is the total drag of an airplane minus the induced drag. It represents the resistance not directly associated with the production of lift. It is composed of skin friction, form drag, and interference drag.

D_p = ½ ρ V² S C_D,0

Parasite drag increases with the square of velocity (D_p ∝ V²). It consists of:

Skin Friction Drag: Results from viscous shearing stresses over the aircraft's skin. It depends on whether the boundary layer is laminar or turbulent, which is determined by the Reynolds Number:
Re = (ρ V L) / μ
For Laminar flow: C_f = 1.328/Re^1/2
For Turbulent flow: C_f = 0.074/Re^1/5
Form Drag (Pressure Drag): Results from the distribution of pressure normal to the body's surface. It is based on the projected frontal area of the object.

Wave Drag

Limited to supersonic flow, this results from non-canceling static pressure components on either side of a shock wave.

C_D,w = 4α² / √(M² - 1)

Trim & Cooling Drag

Trim Drag: The increment in drag resulting from the aerodynamic forces required to trim the aircraft about its center of gravity. It is usually a form of induced and form drag acting on the horizontal tail.

ΔC_D,trim = k_tail C_L,tail²

Cooling Drag: This drag results from the momentum lost by the air that passes through the power plant installation for the purpose of cooling the engine. It is calculated by the change in momentum mass flow rate (ṁ).

D_cooling = ṁ (V_∞ - V_exit)

Overview influence and its drag recovery

Drag Type	Influence	Recovery / Reduction
Induced Drag	High at low speed due to high CL	Winglets, high AR wings, optimal speed
Parasite Drag	Increases with V², surface roughness	Streamlining, smooth finish
Skin Friction	Depends on Reynolds number	Laminar flow control, polished skin
Form Drag	Flow separation, bluff bodies	Aerodynamic shaping
Wave Drag	Shock waves in transonic/supersonic flow	Swept wings, area ruling
Trim Drag	Tail forces for stability	Proper CG, FBW systems
Cooling Drag	Momentum loss in cooling airflow	Optimized inlet/exhaust design

Drag Polar Numerical Formulae

Maximum Lift-to-Drag Ratio:
(L / D)_max = 1 / (2 √(C_D,0 K))

Lift Coefficient for Maximum L/D:
C_L,opt = √(C_D,0 / K)

Minimum Drag Velocity:
V_minD = √( (2 W) / (ρ S C_L,opt) )

Responsive Flight Drag Calculator

Aerodynamic Drag Polar Analysis

Min Velocity (m/s)

Max Velocity (m/s)

Density (kg/m³)

Wing Area S (m²)

Cdo

k factor

Weight W (N)

Enter values and click calculate to see the Minimum Drag Velocity.

Sample Numerical Example: Maximum Lift-to-Drag Ratio

Problem Statement:
A light twin-engine airplane has a drag polar given by:
C_D = 0.0358 + 0.0405 C_L²
Calculate the maximum lift-to-drag ratio.

C_D,0 = 0.0358 (zero-lift drag coefficient)
K = 0.0405 (induced drag factor)

C_L,opt = √(C_D,0 / K)
= √(0.0358 / 0.0405)
= √(0.884)
≈ 0.94

(L / D)_max = 1 / (2 √(C_D,0 K))
= 1 / (2 √(0.0358 × 0.0405))
= 1 / (2 √0.00145)
= 1 / (2 × 0.038)
≈ 13.1

The maximum lift-to-drag ratio is approximately 13.1,
occurring at a lift coefficient of C_L = 0.94.

5.2 Range & Endurance

Jet Aircraft Performance: Range & Endurance

Specific Fuel Consumption

In the case of jet powered aircraft, specific fuel consumption is given as Thrust Specific Fuel Consumption (TSFC). It is defined as the weight of fuel consumed per unit thrust per unit time. Here, thrust is used in contrast to power, which is typical for propeller-driven aircraft. The relationship between fuel consumption and thrust available (T_A) is expressed as:

N(fuel) / s = (TSFC) T_A

Maximum Endurance

For level, un-accelerated flight, the pilot adjusts the throttle such that thrust available (T_A) equals the thrust required (T_R). Therefore, the weight of fuel consumed per hour will be minimum when thrust required is minimum. For minimum thrust required, the ratio C_L / C_D should be maximum. Maximum endurance occurs when the airplane is flying at a velocity such that T_R is minimum. Endurance is derived by considering the rate of fuel weight change over time. The weight of fuel consumed per unit time is proportional to the thrust available and the specific fuel consumption coefficient (c_t):

dW = -c_t T_A dt

Rearranging to solve for the time increment (dt):

dt = -dW / (c_t T_A)

For steady, level flight, the pilot maintains T_A = T_R. Substituting T_A = W / (C_L/C_D):

dt = (1 / c_t) (C_L / C_D) (dW / W)

Integrating from the initial weight (W_o) to the final weight (W₁) gives the overall endurance:

E = (1 / c_t) (C_L / C_D) ln(W_o / W₁)

Maximum endurance is achieved when the aircraft flies at a velocity where the thrust required is minimum. The overall endurance (E) is computed as:

E = (1 / c_t) (C_L / C_D) ln(W₀ / W₁)

Maximum Range

Range is the integral of the distance covered (ds) over the change in aircraft weight. To obtain the expression for range, the aircraft must consume the minimum weight of fuel per unit distance. For a jet aircraft, minimum fuel per unit distance corresponds to a minimum T_A / V ratio. Since T_A = T_R in steady flight, range is maximum when T_R / V is minimum. This corresponds to the tangent point on the thrust required curve. The distance increment is defined as:

ds = V dt = -V dW / (c_t T_A)

In steady level flight, T_A = D and L = W. Substituting these into the range equation:

R = ∫ (V / c_t) (C_L / C_D) (dW / W)

For a jet aircraft, maximum range is achieved when flying at a velocity such that C_L^1/2 / C_D is maximized, which corresponds to C_L = (3C_Do / K)^1/2. Maximum range occurs when C_L^1/2 / C_D is maximum. The requirement for this corresponds to:

C_L = (3C_Do / K)^1/2

Evaluating the integral from W₁ to W_o yields the final range expression:

R = 2 (2 / (ρ S))^1/2 (1 / c_t) (C_L^1/2 / C_D) (W_o^1/2 - W₁^1/2)

Mathematical Derivations: Propeller Performance

Derivation of Endurance (E)

For a propeller aircraft, endurance is the total time spent in the air. We begin with the relationship where the change in fuel weight (dW) is proportional to the engine power (P), the specific fuel consumption (c), and the time increment (dt):

dW = -c P dt ⇒ dt = -dW / (c P)

The engine power (P) is related to the power required (P_R) and propeller efficiency (η) by P = P_R / η. Since P_R = D V, we substitute:

dt = -(η / c) (1 / (D V)) dW

Using steady level flight conditions (L = W) and substituting Velocity V = (2W / (ρ S C_L))^1/2 and Drag D = W (C_D / C_L):

dt = (η / c) (C_L^3/2 / C_D) (ρ S / 2)^1/2 (W^-3/2) dW

Integrating from W₁ to W_o yields the maximum endurance formula:

E = (η / c) (C_L^3/2 / C_D) (2 ρ S)^1/2 (W₁^-1/2 - W_o^-1/2)

Endurance is maximized when the aircraft flies at a velocity where C_L^3/2 / C_D is maximum.

Derivation of Range (R)

Range is the total distance covered on a full tank of fuel. The distance increment (ds) is defined as velocity (V) multiplied by the time increment (dt):

ds = V dt = -V dW / (c P)

Substituting the propeller engine power relationship P = D V / η:

ds = -V dW / (c (D V / η)) = -(η / c) (1 / D) dW

By multiplying by (W / W) and noting that for steady level flight L = W, we get:

ds = (η / c) (L / D) (dW / W) = (η / c) (C_L / C_D) (dW / W)

Integrating from the final weight (W₁) to the initial weight (W_o) results in the Breguet Range Equation for propeller aircraft:

R = (η / c) (C_L / C_D) ln(W_o / W₁)

This derivation shows that maximum range for a propeller-driven aircraft occurs when the aircraft is flying at a velocity where the lift-to-drag ratio (C_L / C_D) is maximum.

5.3 Aerodynamic efficiency

Coming soon

5.4 Winds Effects

Coming soon

Main course

Course

Back