The Velocity and Pressure Distributions on the Joukowski Aerofoil

The velocity anywhere on the circle with circulation in the z – plane corresponding to the lifting cambered aerofoil in the f-plane, given by Equation (4.11), is:

Vc — 2VOT [sin 0c + sin (a + 0)].

The velocity Va, at the corresponding point on the aerofoil, is obtained directly by applying the velocity ratio between the transformed planes, given by:

We know that the transformation function is:

b2

Z = z + —,

z

Substituting this, further simplification can be made. The resulting equation is compact for obtaining the velocity distribution around the aerofoil profile. However, the velocity appears as square in the pressure coefficient, Cp, expression, which implies that, computing the pressure coefficient is a tedious process. The approximation that the eccentricity e is very small compared to unity (e ^ 1) progressively becomes unrealistic when thicker and more cambered aerofoil sections are required. Use of this approximation is justified only when they produce significant simplification at the expense of acceptable small deviations from the exact solutions of the velocity and pressure distributions around the aerofoil profile generated. In the present case, the utility of the approximate method largely ceases after the expressions for § and П, for the profile:

§ = 2b cos 0

П = 2be(1 + cos 0) sin 0 + 2be sin2 0.

have been obtained. For obtaining velocity and pressure distribution numerical solution may be employed.