Question

The Kutta-Joukowski theorem, Equation (3.140), was derived exactly for the case of the lifting cylinder. In Section 3.16 it is stated without proof that Equation (3.140) also applies in general to a two-dimensional body of arbitrary shape. Although this general result can be proven mathematically, it also can be accepted by making a physical argument as well. Make this physical argument by drawing a closed curve around the body where the closed curve is very far away from the body, so far away that in perspective the body becomes a very small speck in the middle of the domain enclosed by the closed curve.

Short answer

The Kutta-Joukowski theorem holds generally because, at a distance, any shape can be approximated as a point-like source with uniform circular flow.

Step-by-step solution

Understanding the Problem

The goal is to provide a physical argument to accept the Kutta-Joukowski theorem for any two-dimensional body by considering a closed curve far away from the body. In the theorem, the circulation around the body is associated with lift.

Concept of Circulation

The Kutta-Joukowski theorem connects the lift force experienced by a body in a fluid flow to the circulation of the flow around the body. Circulation is defined as the line integral of velocity around a closed curve enclosing the body.

Imagining the Closed Curve

Envision a closed curve that is drawn very far from the body. At this great distance, the influence of the body's precise shape becomes negligible, and the behavior around the closed curve only sees the body as a small point.

Flow Field at a Distance

At a large distance, the flow field around the closed curve can be approximated by the flow around a circular cylinder with circulation. The effect of the body is seen as causing a "uniform" distribution of circumferential velocity around this far-distanced closed curve.

Conclusion Using Symmetry and Consistency

Accepting that the body behaves similarly to a lifting cylinder at a great distance, where the details of the specific shape are irrelevant, allows the theorem to hold. The lift on the body is still determined by the circulation, which acts similarly because of the symmetrical distant influence.

Key concepts

Circulation in Fluid Mechanics

In fluid mechanics, circulation is a crucial concept that helps us understand how fluid flow creates lift in objects immersed in it.
Circulation defines the sum of the flow velocity around a closed path that encloses a body.
Think of it as walking around a loop where you sum up the velocity field vectors along the loop's perimeter.
In mathematical terms, circulation \( \Gamma \) is expressed as the line integral of velocity: \[ \Gamma = \oint_{C} \mathbf{v} \cdot d \mathbf{s} \] where \( \mathbf{v} \) is the velocity vector and \( d\mathbf{s} \) is the differential displacement along path \( C \).

• This concept becomes immensely important when analyzing lift.
• It’s tied directly to the lift that various shapes can produce.

Understanding this concept is key to connecting how fluid flow properties can influence the forces on an object.

Lift Force on Two-Dimensional Bodies

Lift is the force that acts perpendicular to the relative flow direction on a body, and understanding how this force emerges on two-dimensional bodies is fundamental in aerodynamics.
The Kutta-Joukowski theorem provides a direct link between circulation and the lift force on a two-dimensional body.
This theorem tells us that the lift per unit span \( L' \) is proportional to the product of the fluid density \( \rho \), the free stream velocity \( V \), and the circulation \( \Gamma \):\[ L' = \rho V \Gamma \]

• This implies that controlling circulation can directly control lift.
• Blowing air over wings or changing angles can modify circulation.

It’s like steering the sails of a boat to catch the wind in a way that maximizes motion in the desired direction.
Knowing this, engineers can design wings or foils to produce the needed lift efficiently.

Aerodynamics of Arbitrary Shapes

The Kutta-Joukowski theorem isn’t only applicable to simple shapes like circles or airfoils; it’s valid for any arbitrary shape immersed in fluid flow.
Even though different shapes may appear extremely complex, their aerodynamic behavior at large distances simplifies.
Imagine drawing a giant closed curve far from the shape.
To the flow field analyzing this giant loop, the complex shape impacts the loop surround like a speck.

• This allows us to treat the complex shape as if it's merely a single point influencing the flow.
• Here, the intricacies of the body's outline become negligible.

At this far distance, the lift force due to circulation can still be predicted accurately with the Kutta-Joukowski theorem, showing a beautiful consistency of aerodynamics across different shapes.
This principle aids engineers in predicting how arbitrary-shaped objects will behave in fluid flows, essential for designing anything from buildings to underwater vehicles.