The Kutta condition and the starting vortex are fundamental concepts in airfoil theory that explain how real wings generate circulation—and therefore lift—in a physically consistent way.
These ideas ensure that the idealized solutions of potential flow match the real behavior of viscous flows around airfoils.
1️⃣ Physical Problem: Sharp Trailing Edge
For an airfoil with a sharp trailing edge, potential flow theory (which is inviscid) predicts infinite velocity at the TE unless a unique circulation is chosen.
✅ Real flows don’t allow infinite velocities.
✅ Viscosity ensures smooth flow leaves the trailing edge at a finite, realistic speed.
This leads us to the Kutta condition.
2️⃣ Kutta Condition: Definition
The Kutta condition states:
Flow must leave the sharp trailing edge smoothly.
Mathematically, this means the velocity at the trailing edge is finite and directed along the bisector of the TE angle.
✅ For thin, sharp TE airfoils, it often simplifies to:
Equal static pressure on upper and lower surfaces at the TE.
✅ The Kutta condition selects a single, physically meaningful circulation from many mathematically possible potential flows.
3️⃣ Why Is It Needed?
Without the Kutta condition:
- Potential flow around an airfoil is ambiguous: infinite family of solutions with different circulations.
- There is no unique lift value.
✅ The Kutta condition uniquely determines the circulation needed for smooth trailing-edge flow.
✅ This circulation then gives lift via the Kutta–Joukowski theorem:
4️⃣ Starting Vortex: How Circulation Forms
The starting vortex is a real physical phenomenon that explains how an airfoil generates circulation when it begins moving.
Process:
- When the airfoil accelerates from rest, viscous forces near the sharp TE cannot turn the flow sharply around it.
- Flow rolls up into a vortex that is shed downstream.
- To conserve total vorticity (by Kelvin’s circulation theorem), an equal and opposite circulation develops around the airfoil.
✅ The starting vortex is shed into the wake.
✅ The bound circulation remains around the airfoil.
5️⃣ Conservation of Circulation (Kelvin’s Theorem)
Kelvin’s circulation theorem states:
In an inviscid, barotropic fluid with conservative body forces, the circulation around a material loop remains constant.
✅ Initially at rest: total circulation = 0.
✅ After starting:
- Airfoil circulation =
.
- Starting vortex in wake =
.
- Total remains = 0.
6️⃣ Role in Establishing Lift
- The Kutta condition forces the flow to adopt a specific circulation around the airfoil.
- The starting vortex enforces conservation of circulation when this new circulation develops.
- Once in steady flight, the starting vortex drifts away and the bound circulation around the airfoil is constant, sustaining lift.
✅ Without enforcing the Kutta condition, there would be no predictable lift behavior.
7️⃣ Mathematical Implication
For steady, inviscid, incompressible 2D flow satisfying the Kutta condition:
And lift per unit span is given by:
✅ The Kutta–Joukowski theorem relies on circulation set by the Kutta condition.
8️⃣ Example Interpretation
Example scenario:
- An airfoil at positive angle of attack.
- Flow accelerates over upper surface, decelerates below.
- To enforce smooth trailing edge flow (Kutta condition), circulation is added.
- The starting vortex with opposite circulation is shed into the wake.
- Result: sustained lift due to bound circulation.
9️⃣ Importance in Aerodynamics
✅ Explains why lift can be predicted for real airfoils with sharp trailing edges.
✅ Ensures potential flow models match physical reality.
✅ Underpins airfoil design and analysis in both classical and computational aerodynamics.
✅ Essential for understanding wake development and vortex shedding.
Summary
✅ The Kutta condition ensures realistic flow by enforcing finite velocity at the trailing edge.
✅ It selects the unique circulation needed for lift.
✅ The starting vortex is shed to conserve total circulation as the airfoil accelerates.
✅ Together, they explain the development of lift-producing circulation around an airfoil in real flows.