The Kutta Condition Applied to Aerofoils

When a smooth symmetric body, such as a cylinder with oval cross-section, moves with zero angle of attack through a fluid it generates no lift. There are two stagnation points on the body – one at the front and the other at the back. If the oval cylinder moves with a nonzero angle of attack through the fluid there are still two stagnation points on the body – one on the underside of the cylinder, near the front edge; and the other on the topside of the cylinder, near the back edge. The circulation around this smooth cylinder is zero and no lift is generated, despite the positive angle of attack.

If an aerofoil with a sharp trailing edge begins to move with a positive angle of attack through air, the two stagnation points are initially located on the underside near the leading edge and on the topside near the trailing edge, just as with the cylinder. As the air passing the underside of the aerofoil reaches the trailing edge it must flow around the trailing edge and along the topside of the aerofoil toward the stagnation point on the topside of the aerofoil. Vortex flow occurs at the trailing edge and, because the radius of the sharp trailing edge is zero, the speed of the air around the trailing edge should be infinitely fast! Real fluids cannot move at infinite speed but they can move very fast. The very fast airspeed around the trailing edge causes strong viscous forces to act on the air adjacent to the trailing edge of the aerofoil and the result is that a strong vortex accumulates on the topside of the aerofoil, near the trailing edge. As the aerofoil begins to move, it carries this vortex, known as the starting vortex, along with it. Pioneering aerodynamicists were able to photograph starting vortices in liquids to confirm their existence.

The vorticity in the starting vortex is matched by the vorticity in the bound vortex in the aerofoil, in accordance with Kelvin’s circulation theorem. As the vorticity in the starting vortex progressively increases, the vorticity in the bound vortex also progressively increases, and causes the flow over the topside of the aerofoil to increase in speed. The stagnation point on the topside of the aerofoil moves progressively towards the trailing edge. After the aerofoil has moved only a short distance through the air, the stagnation point on the topside reaches the trailing edge and the starting vortex is cast off the aerofoil and is left behind, spinning in the air where the aerofoil left it. The starting vortex quickly dissipates due to viscous forces.

As the aerofoil continues on its way, there is a stagnation point at the trailing edge. The flow over the topside conforms to the upper surface of the aerofoil. The flow over both the topside and the underside join up at the trailing edge and leave the aerofoil traveling parallel to one another. This is known as the Kutta condition.

When an aerofoil is moving with a positive angle of attack, the starting vortex will be cast off, and the Kutta condition will be established. There will be a finite circulation of the air around the aerofoil and the aerofoil will generate lift, with magnitude equal to that given by the Kutta—Joukowski theorem.

One of the consequences of the Kutta condition is that the airflow over the upper surface of the aerofoil travels much faster than the airflow over the bottom surface. A portion of air flow which approaches the aerofoil along the stagnation streamline is split into two parts at the stagnation point, one half traveling over the upper surface and the other half traveling along the bottom surface. The flow over the topside is so much faster than the flow along the bottom that these two halves never meet again. They do not even re-join in the wake long after the aerofoil has passed. This is sometimes known as “cleavage.” There is a popular fallacy called the equal transit-time fallacy that claims the two halves rejoin at the trailing edge of the aerofoil. This fallacy is in conflict with the phenomenon of cleavage that has been understood since Martin Kutta’s discovery.

Whenever the speed or angle of attack of an aerofoil changes there is a weak starting vortex which begins to form, either above or below the trailing edge. This weak starting vortex causes the Kutta condition to be re-established for the new speed or angle of attack. As a result, the circulation around the aerofoil changes and so too does the lift in response to the changed speed or angle of attack.

The Kutta condition gives some insight into why aerofoils always have sharp trailing edges, even though this is undesirable from structural and manufacturing viewpoints. An aircraft wing with a smoothly rounded trailing edge would generate little or no lift.