Joukowsky Transform - Velocity Field and Circulation For The Joukowsky Airfoil

Velocity Field and Circulation For The Joukowsky Airfoil

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.

The complex velocity around the circle in the plane is

where

• is the complex coordinate of the centre of the circle
• is the freestream velocity of the fluid
• is the angle of attack of the airfoil with respect to the freestream flow
• R is the radius of the circle, calculated using
• is the circulation, found using the Kutta condition, which reduces in this case to

The complex velocity W around the airfoil in the z plane is, according to the rules of conformal mapping and using the Joukowsky transformation:

Here with and the velocity components in the and directions, respectively ( with and real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure or lift can be calculated.

A Joukowsky airfoil has a cusp at the trailing edge.

The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically been romanized in a number of ways, thus the variation in spelling of the transform.