Continuum Mechanics Definition Edit

Shear pressure

Pressure force and viscous force surrounding an airfoil [1]

A body moving through a fluid is submitted to an interaction between its outer surface and the fluid.[1] This can be described as a stress  including two terms :

  • the wall shears stresses, due to viscous effects Tw
  • the normal stresses due to pressure p

The integrated or resultant aerodynamic effects of these distributions is a force. This force can be split in two terms: lift and drag.

The drag is the resultant force in the direction of the upstream velocity U relative to the airfoil. When specifically considering an aircraft,  the drag force is in the opposite direction of the freestream velocity (or flight path).[1]

Commonly used EquationEdit

In aerospace engineering, the overall drag is generally expressed as follows:

Eq drag


D is the overall drag

ρ is the air density

V is the air velocity relative to the airfoil

CD is the drag coefficient (dimensionless)

S is the aircraft wing area                            

  • To measure the change in drag coefficient CD, the drag count can be used.
One drag count is equal to an increase of CD of 10-4.
So an increase of the drag coefficient by 12 drag counts means that CD,new = CD + 0.0012
A drag count is then dimensionless.

  • The required power to overcome drag varies as the cube of  the velocity V.

Types of Drag Edit

There are essentially three types of drags that can affect a body traveling through a fluid:

  • parasite drag (including leakage and protuberance drag)
  • lift induced drag (including trim drag)
  • wave drag

Parasite drag Edit

Parasite drag (also called parasitic drag or zero-lift drag) is made up of the following components :

  • Form drag:  loss due to the shape of a wetted surface changing the flow direction of a viscous fluid.
  • Skin friction drag: loss in the boundary layer due to the roughness of a wetted surface .
  • Interference drag: the proximity of several bodies creating mutual interferences in the air flow around each different element.[3]
  • Leakage drag: it is due to  air incoming through both holes and gaps in high-pressure zones over the fuselage, wings, empennage,...This air is exiting the aircraft's skin towards low pressure zones. Loss due to incoming airflow contributes directly to drag while exiting air increases airflow separation.
  • Protuberances drag: protuberances are elements added on the "clean" body that spoil the airflow. They include antennas, door edges, control surface hinges as well as protruding rivets.

     Leakage and protuberance drag is difficult to predict but it roughly represents 2-5% of the parasite drag for jet transports and 5-10% for propeller airplane or new generation fighters.[3]

     A well designed aircraft in subsonic flight will have parasitic drag mostly due to skin friction plus small drag due to turbulent air mixing behind it. The parasite drag coefficient is denoted as CDmin or  CDo, as it describes the minimum possible drag the aircraft could see when no lift is produced at a subsonic  velocity.[3]

Lift-induced drag Edit

Drag ind lift

Lift-induced drag. [6]

Lift-induced drag is the result of lift creation on a three-dimensional lifting body, such as the wings or fuselage of an airplane. Lift-induced drag includes the creation of vortices  above the wings as well as the additional viscous drag.

The lift is perpendicular to the freestream velocity V and the induced drag is perpendicular to the lift. In most cases, the aircraft is symmetric and the induced drag is collinear to the freestream velocity V.[2]

The lift-induced drag coefficient. CDi. is approximated to vary proportionally to the square of the lift coefficient CL. [2]

Eq drag3

where AR is the aspect ratio (wingspan squared over the wing area)

So the lift-induced drag coefficient increases with the lift coefficient . Therefore, CDi will increase with the angle of attack.

Since the calculation of the lift-induced drag is based on the area of the wing S, the expression of the lift-induced drag must be completed by the trim drag. This additional drag is caused by the horizontal tail (or canard) force that is instantaneously required to balance the aircraft total pitching moment around its center of gravity.[2]

Wave drag Edit

Wave drag (also called compressibility drag) is due to the formation of shocks. A shock is the supersonic phenomenon.[3]

Though, even if the aircraft is flying at a subsonic speed some areas of its fuselage may encounter supersonic airflow . This is especially true above the wings where the airflow is strongly accelerated. Therefore wave drag starts to appear in transonic flight (Mach 0.8 to 1.2) and is still present in supersonic flights (Mach > 1,2).

As an airplane accelerates through the transonic regime, the increase of drag is due to the formation of shocks and the importance of the compressibility drag tends to dominate other forms of drag as the aircraft's speed increases towards Mach 1.0.

Two  Mach Number are then defined:

  • the critical Mach number Mcr is the dimensionless velocity at which shocks first form on the airplane. Mcr is primarily a function of airfoil thickness and sweep. A typical range for Mcr would be (0.7-0.9), depending on airfoil thickness;
  • the drag divergent Mach number MDD at which the formation of shocks begins to significantly affect the drag.

For instance, Boeing definition is MDD = 1.08 Mcr.[3]

Concorde CD CL2 drag polar

Drag coefficient versus Lift coefficient squared for Concorde. [7]

Drag polar Edit

Plots of drag coefficient CD versus lift coefficient CL are very useful. As a first approximation CD varies as the square of CL. So a linear relation between CD and CL ² can be plotted for a aircraft at a given Mach number.

To be more accurate, a linear term can be taken into account and we get the relation:

CD = K1.CL2 + K2.CL + CD0


  • CD0 stands for parasite drag.
  • K1 and K2 are dimensionless. K1.CL2 + K2.CL includes both lift-induced drag and wave drag.

Power Curve  Edit

Power curve 2

Power curve: drag VS aircraft Mach number. [6]

Depending on the aircraft speed, the importance of the different types of drag changes.

At lower speeds, induced drag is the most important because of the high angle of attack required to produce sufficientlift. [2]

As velocity increases, the induced drag decreases (as well as the angle of attack) but parasite drag increases since the airflow surrounding the aircraft is faster, generating more skin friction.  At transonic speed, the wave drag has to be taken into account.

The overall drag can be plot as a function of the aircraft velocity to obtain the power curve. This curve presents a point at which the drag is minimum: this is the velocity at which the range is maximum. This velocity is different from the velocity at which the endurance is maximized.

Influence of Mach and Reynolds numbersEdit

The drag coefficient depends largely on the velocity of the flow. Since the Mach number and the Reynolds number vary both proportionally with the airflow speed, drag can be seen as a function of both Re and M.[1]

If the Mach number, M < 0.5, CD is mostly a function of Re because most of the drag is due to friction between the aircraft and air.

For streamlined bodies like a wing, the drag coefficient increases when the boundary layer surrounding it becomes turbulent because most of the drag is due to the shear force. So the drag increases with the flow speed and the surface roughness of the wing. Hence, the more the surface roughness of the wing, the smaller the Reynolds number at which the boundary layer becomes turbulent. [1]

On the other hand, if the velocity is large enough, compressibility effects have to be taken into account. CD becomes then essentially a function of M because wave drag overcomes viscous effects. CD increases dramatically in the vicinity of M = 1 because of shocks formation. For example, a sharp wing would see its CD being maximum around M = 1, whereas a more blunt airfoil would see its maximum C right before M = 1. [1]

References Edit