Drag Polar

Module 5

5.1 Drag Force
5.2 Drag Polar Equation
5.3 Range & Endurance
5.4 Winds Effects
5.5 Aerodynamic efficiency
5.1 Drag Force
  • TOTAL DRAG
    • PARASITE DRAG
      • INTERFERENCE
      • PROFILE DRAG
        • SKIN FRICTION
        • FORM DRAG
    • INDUCED DRAG
    • WAVE DRAG

TOTAL DRAG CLASSIFICATION

WING DRAG
Induced Drag

Depends on aspect ratio. Greatest at low speeds.

Form Drag

Depends on shape. Goes up with square of speed.

Skin Friction

Depends on surface. Goes up with square of speed.

PARASITE DRAG
Form Drag

Depends on shape. Goes up with square of speed.

Skin Friction

Depends on surface. Goes up with square of speed.

SHOCK DRAG
Wave Drag

Only occurs at transonic and supersonic speeds.

Shock-Turbulence Drag

Only occurs at transonic and supersonic speeds.

Drag Equation
The Total Drag Equation
The total drag coefficient CD is expressed using the parabolic drag polar equation:
CD = CD,0 + kCL2
Where CD,0 is the parasite drag coefficient and kCL2 represents the induced drag. The factor k is defined as:
k = 1 / (π · AR · e)
Induced Drag (CD,i)
Mathematically, induced drag is tied to the Lift Coefficient (CL). As the aircraft slows down, it must fly at a higher angle of attack to maintain lift, which increases CL 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.
Di = ½ ρ V2 S [ CL2 / (π · AR · e) ]
As velocity (V) increases, induced drag decreases (Di ∝ 1/V2).
Parasite Drag (Dp)
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.
Dp = ½ ρ V2 S CD,0
Parasite drag increases with the square of velocity (Dp ∝ 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: Cf = 1.328/Re1/2

    For Turbulent flow: Cf = 0.074/Re1/5

  • 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.
CD,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.
ΔCD,trim = ktail CL,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).
Dcooling = ṁ (V∞ - Vexit)
5.2 Drag Polar Equation
ISA Deviation
ISA Deviation

It is essential to present performance data at temperatures other than the ISA temperature for all flight levels within the performance-spectrum envelope. If this were to be attempted for the actual or forecast temperatures, it would usually be impracticable and in some instances impossible.

To overcome the presentation difficulty and retain the coverage or range required, it is necessary to use ISA deviation. This is simply the algebraic difference between the actual (or forecast) temperature and the ISA temperature for the flight level under consideration. It is calculated by subtracting the ISA temperature from the actual (or forecast) temperature for that particular altitude.

ISA Deviation = Ambient temperature − Standard Temperature
JSA Deviation
As an alternative to ISA deviation some aircraft manuals use the Jet Standard Atmosphere (JSA) Deviation that assumes a temperature lapse rate of 2◦/1000 ft and that the atmosphere has no tropopause, the temperature is, therefore, assumed to continue decreasing at this rate beyond 36 090 ft.
Height and Altitude
Three parameters are used for vertical referencing of position in aviation. They are the airfield surface level, mean sea level (MSL) and the standard pressure level of 1013.2 hPa. It would be convenient if the performance data could be related to the aerodrome elevation because this is fixed and published in the Aeronautical Information Publication. However, this is impractical because of the vast range that would have to be covered. Mean sea level and pressure altitude are the only permissible references for assessing altitude for the purposes of aircraft performance calculations, provided that the one selected by the manufacturers for the Flight Manual is used consistently throughout the manual.
Pressure Altitude
In Aeroplane Flight Manuals (AFMs) the word altitude refers strictly to pressure altitude, which can be defined as the vertical distance from the 1013.2 hPa pressure level. Therefore, aerodrome and obstacle elevations must be converted to pressure altitude before they can be used in performance graphs. Many large aerodromes provide the aerodrome pressure altitude as part of their hourly weather reports.

A/F Pressure Altitude = Aerodrome elevation in ft + [(1013.2 hPa − QNH) × 27 ft] Aerodrome Pressure Altitude = (1013.2 hPa − QFE) × 27 ft

To correct an altitude for the temperature errors of the altimeter use the following formula:

Altitude Correction = 4 × ISA Deviation × Indicated Altitude ÷ 1000

Density Altitude
The performance data for small piston/propeller-driven aeroplanes is calculated using density altitude, which is pressure altitude corrected for nonstandard temperature. It is the altitude in the standard atmosphere at which the prevailing density occurs and can be calculated by using the formula:

Density Altitude = Pressure Altitude + (118.8 × ISA Deviation)

ISA Calculator (Table-Based)
ISA Calculator (0–20 km, Table-Based)
PropertySymbolValue
Drag Equation
The Total Drag Equation
The total drag coefficient CD is expressed using the parabolic drag polar equation:
CD = CD,0 + kCL2
Where CD,0 is the parasite drag coefficient and kCL2 represents the induced drag. The factor k is defined as:
k = 1 / (π · AR · e)
Induced Drag (CD,i)
Mathematically, induced drag is tied to the Lift Coefficient (CL). As the aircraft slows down, it must fly at a higher angle of attack to maintain lift, which increases CL 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.
Di = ½ ρ V2 S [ CL2 / (π · AR · e) ]
As velocity (V) increases, induced drag decreases (Di ∝ 1/V2).
Parasite Drag (Dp)
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.
Dp = ½ ρ V2 S CD,0
Parasite drag increases with the square of velocity (Dp ∝ 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: Cf = 1.328/Re1/2

    For Turbulent flow: Cf = 0.074/Re1/5

  • 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.
CD,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.
ΔCD,trim = ktail CL,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).
Dcooling = ṁ (V∞ - Vexit)
5.3 Range & Endurance

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5.4 Winds Effects

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5.5 Aerodynamic efficiency

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