Kutta condition & Starting vortex

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  \Gamma needed for smooth trailing-edge flow.
✅ This circulation then gives lift via the Kutta–Joukowski theorem:

 L' = \rho U \Gamma


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 =  \Gamma .
  • Starting vortex in wake =  -\Gamma .
  • 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:

 \Gamma = \text{Unique value ensuring smooth TE flow}

And lift per unit span is given by:

 L' = \rho U \Gamma

✅ 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.

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