The Kutta-Joukowski Condition

With the Joukowski transformation, the mapping of the cylinder to the profile is singular at B, X = a, Z = 0 (the other singular point is at X = – a, Z = 0). This point corresponds to the cusped trailing edge B of the Joukowski profile, see Fig.2.12. It has been found experimentally that the flow will leave the profile at a sharp trailing edge (cusp or small angle). This is due to viscosity. As the boundary layer grows from the leading edge to the trailing edge, the fluid particles do not have enough momentum to come around the trailing edge, and hence separate from the profile there. The sharp trailing edge of the profile plays the same role as the flap for the cylinder. This condition makes the inviscid flow solution past a profile unique by fixing the circulation.

The Kutta-Joukowski (K-J) condition states the the flow must leave the profile at the sharp trailing edge “smoothly”. See Fig.2.15.

Revisiting the Joukowski transformation of Sect. 2.5.1, one can see that with an incoming flow making an angle a with the X-axis and with an arbitrary circulation around the cylinder, the rear stagnation point will be located at an arbitrary point such as P in Fig.2.12, therefore the stagnation streamline will leave the profile at point p and the K-J condition will not be satisfied. The velocity will be infinite at the trailing edge. In order to enforce the K-J condition, the circulation must be adjusted such that the rear stagnation point is located at point B. This uniquely determines the value of the circulation Г as a function of incidence a.

1.5 Definitions

The chord c of a profile is the radius of the largest circle centered at the trailing edge and touching the leading edge.

The incidence angle or angle of attack is defined as the angle between the profile chord and the incoming flow velocity vector.

It is convenient to use dimensionless coefficients to represent pressure, forces and moments. The pressure coefficient is defined as

where the denominator, 2 p y° is called the dynamic pressure (has dimension of a pressure, unit Pa) and use has been made of the Bernoulli equation.

The lift, drag and moment coefficients per unit span are made dimensionless, in 2-D flow, with the dynamic pressure and a reference length or length squared as

where the prime indicates that the force or moment is per unit span and the chord is the reference length in all cases in 2-D. Note that lower case subscripts will be used for two-dimensional coefficients to leave upper-case subscripts notation for wing, tail and complete configurations. Here, the moment coefficient is taken about point O, which in general will represent the leading edge of a profile or the nose of the airplane.

Other important quantities are:

• the lift slope d1

• the zero incidence lift coefficient Ci, o

• the moment slope d Jf0

• the zero incidence moment coefficient Cm,0,0.