For a body with a sharp trailing edge, the no-penetration condition alone does not determine one unique potential flow. Different amounts of circulation can be superposed while still keeping the body surface impermeable.

Those mathematical solutions generally predict different velocity distributions and, through Kutta-Joukowski, different lift forces.

The Kutta condition supplies the additional physical selection rule:

$$\boxed{\text{the flow leaves a sharp trailing edge smoothly with finite velocity.}}$$

It is not another force law. Its role is to select the circulation of the physically relevant outer-flow solution.

Why potential flow needs an extra condition

In an inviscid potential-flow model, an impermeable body imposes only a normal-velocity condition:

$$\mathbf v\cdot\mathbf n=0$$

for a stationary surface.

A circulation field can be added without changing this normal condition. Consequently, many potential flows can satisfy the same body geometry and far-field velocity.

For a streamlined body with a sharp trailing edge, however, most of these solutions require the flow to turn sharply around the edge or produce an unbounded velocity there. Such behavior is not observed in the corresponding attached physical flow.

The Kutta condition rejects those solutions.

Finite-angle trailing edge

Suppose the upper and lower surfaces meet at a finite included angle at the trailing edge.

The velocity immediately adjacent to each surface must be tangent to that surface. Because the two surface tangents point in different directions at the corner, a single finite velocity vector cannot be tangent to both unless its magnitude tends to zero at the meeting point.

Thus, for an idealized finite-angle sharp trailing edge, the Kutta condition places a stagnation point at the edge:

$$\boxed{\mathbf v_{TE}=\mathbf0}.$$

The selected circulation is the one that moves the relevant stagnation point to the trailing edge.

Cusped trailing edge

If the upper and lower surfaces meet tangentially in a cusp, their limiting tangent directions coincide. The trailing-edge velocity need not be zero.

Instead, smooth finite departure requires the limiting upper- and lower-surface velocities to agree:

$$\boxed{V_{upper,TE}=V_{lower,TE}}$$

with a finite common direction downstream.

Thus “the trailing edge is always a stagnation point” is too strong. It is true for the finite-angle idealization, while a cusped trailing edge can have a finite nonzero departure speed.

Circulation selection

Imagine a family of potential-flow solutions around the same airfoil, each with a different circulation $\Gamma$.

The Kutta condition selects the member of that family for which the trailing-edge behavior is regular and the flow leaves smoothly. Once that circulation has been determined, the Kutta-Joukowski theorem gives the corresponding lift per unit span.

The logical sequence is therefore

1. body geometry and far-field flow define a family of potential solutions;
2. the Kutta condition selects the relevant circulation;
3. Kutta-Joukowski converts that circulation into lift.

Why viscosity matters even though the selected outer flow may be inviscid

A real fluid satisfies no slip at the solid surface and develops thin viscous layers. During startup, viscous stresses generate and redistribute vorticity near the body and into the wake.

This viscous process allows the outer flow to evolve toward a circulation state that an everywhere-inviscid model could not select from geometry alone.

Once a high-Reynolds-number flow is established and attached, much of the region outside the thin viscous layers may be modeled approximately by inviscid potential flow. The Kutta condition is then used as an effective boundary condition that carries information from the small viscous trailing-edge region into the outer solution.

This is another example of a singular-perturbation structure: a thin region where viscosity matters can determine an order-one quantity—the circulation and therefore the lift—of a much larger nearly inviscid region.

Circulation during startup

Kelvin's theorem says that circulation around a material loop is conserved only under inviscid, barotropic flow with conservative body forces. The no-slip viscous region formed during startup violates the inviscid assumption, so circulation can be generated and rearranged there.

A starting wing typically sheds vorticity into the wake while an oppositely signed bound circulation develops around the wing. This provides the dynamical bridge between an initially noncirculating state and the later lifting outer flow.

The Kutta condition does not replace that viscous startup physics. It summarizes its effect when constructing the later idealized outer-flow solution.

Worked selection example: circulating cylinder analogy

For a circulating cylinder, the surface tangential velocity is

$$v_\theta(a,\theta) =-2U\sin\theta+rac{\Gamma}{2\pi a}.$$

Suppose, as an analogy, we require a chosen surface point $\theta=\theta_s$ to be a stagnation point. Then

$$v_\theta(a,\theta_s)=0,$$

which selects

$$\boxed{\Gamma=4\pi aU\sin\theta_s}.$$

For example, if

$$a=0.25,\mathrm m,$$

$$U=20,\mathrm{m/s},$$

and the required stagnation point is at

$$\theta_s=-30^\circ,$$

then

$$\Gamma =4\pi(0.25)(20)\sin(-30^\circ) =-10\pi,\mathrm{m^2/s}.$$

Thus

$$\boxed{\Gamma\approx-31.4,\mathrm{m^2/s}}.$$

The cylinder is not an airfoil with a sharp trailing edge, so this is not itself a Kutta-condition calculation. It demonstrates the mathematical mechanism: an additional physical condition on the location or behavior of a stagnation point selects one circulation from an otherwise continuous family.

Scope

The Kutta condition is most useful for attached flow around a body with a sharp trailing edge. If the flow is massively separated or stalled, the ideal attached potential-flow picture is no longer sufficient and a simple Kutta-condition model cannot determine the real aerodynamic force by itself.

The condition should therefore be understood as a physically motivated closure for an ideal outer-flow model, not as a universal law that overrides viscous separation.