Lift generation by Kutta Joukowski Theorem

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Kutta Joukowski Theorem


Lift generation by Kutta Joukowski Theorem


Background and Historical Note:

When asked how lift is generated by the wings, we usually hear arguments about velocity being higher on the upper surface of the wing relative to the lower surface and then applying Bernoulli’s principle, the pressure is higher on the lower surface of the wing than the upper, resulting in a net upward force called a lift. But now the question arises what is the reason for this pressure difference? We already know from our previous article on Coanda effect that how airflow remains attached to the airfoil surface. The generation of lift by the wings has a bit complex foothold. The “Bernoulli” explanation was established in the mid-18th century and has few assumptions. More curious about Bernoulli's equation? Check out this article here.

One more popular explanation of lift takes circulations into consideration. It was during the time of the first powered flights (1903) in the early 20th century, that investigators such as Martin Kutta and Nikolai Y. Joukowski (Zhukouski) were connecting lift with the circulation of air on the contours of the wings and lift was no longer looked simply as an interplay between pressure and velocity. The “Kutta-Joukowski” (KJ) theorem, which is well-established now, had its origin in Great Britain (by Frederick W. Lanchester) in 1894 but was fully explored in the early 20th century. The developments in KJ theorem has allowed us to calculate lift for any type of two-dimensional shapes and helped in improving our understanding of the wing aerodynamics. In the following text, we shall further explore the theorem.

What is Circulation?

The frictional force which negatively affects the efficiency of most of the mechanical devices turns out to be very important for the production of the lift if this theory is considered. In further reading, we will see how the lift cannot be produced without friction. 

To understand lift production, let us visualize an airfoil (cut section of a wing) flying through the air. With this picture let us now zoom closely into what is happening on the surface of the wing. The air close to the surface of the airfoil has zero relative velocity due to surface friction (due to Van der Waals forces). Due to the viscous effect, this zero-velocity fluid layer slows down the layer of the air just above it. Similarly, the air layer with reduced velocity tries to slow down the air layer above it and so on. This happens till air velocity reaches almost the same as free stream velocity. These layers of air where the effect of viscosity is significant near the airfoil surface altogether are called a 'Boundary Layer'. This boundary layer is instrumental in the wing’s ability to produce lift.


Boundary-Layer

Let's begin with the rotating cylinder in the uniform flow field. It is easy to visualize lift produced due to circulation around the rotating cylinder. Assume clockwise rotating cylinder. Clockwise circulation sets around the cylinder due to its rotation. The upper part of the cylinder which moves toward right boosts the free stream velocity just above it in the same direction. On the other hand, the lower part of the cylinder which moves toward the left slows down the free velocity. This velocity difference causes static pressure difference above and below the cylinder which results in a lift in a rotating cylinder.

Lift over rotating cylinder
Lift over a rotating cylinder


In a case of an airfoil, at the very initial stage of the flow two stagnation points set in (Stagnation point is the highest static pressure location on the airfoil where flow separates or rejoins). One near the leading edge of the airfoil and other near the trailing edge on the upper surface of the airfoil (As shown in figure a). As the angle of attack of an airfoil is increased stagnation point moves toward the lower surface of the airfoil. It was observed that stagnation point near the trailing edge which is on the upper surface of the airfoil initially, always moves exactly at the trailing edge (Shown in figure b). This condition is known as the Kutta condition. Which hints the presence of circulation around an airfoil which displaces the stagnation point. To maintain conservation of forces, circulation of the equal and opposite strength sets and leaves from the trailing edge of the airfoil (Kelvin's circulation theorem). This vortex formed is called as starting vortex. 






Relation Between Circulation and Lift

Having a picture of what “circulation on the wing” means, we now can proceed to link how this circulation produces lift. In the figure below, the diagram in the left describes airflow around the wing and the middle diagram describes the circulation due to the vortex as we earlier described. The addition (Vector) of the two flows gives the resultant diagram. The length of the arrows corresponds to the magnitude of the velocity of the airflow. The flow on the upper surface adds up whereas the flow on the lower surface subtracts, leading to higher pressure on the lower surface as compared to the upper surface. Therefore, Bernoulli’s principle comes into the picture again, resulting in a net upward force which is called Lift.


 Resultant of circulation and flow over the wing
Therefore, the “Kutta-Joukowski” theorem completes the Bernoulli’s high-low pressure argument for lift production by deepening our understanding of this high and low-pressure generation.
Mathematical Formulation of Kutta-Joukowski Theorem:

The theorem relates the lift produced by a two-dimensional object to the velocity of the flow field, the density of flow field, and circulation on the contours of the wing.



The rightmost term in the equation represents circulation mathematically and is evaluated using vector integrals. It is important that Kutta condition is satisfied to use this equation.
Concluding Points:

The Kutta–Joukowski theorem has improved our understanding as to how lift is generated, allowing us to craft better, faster, and more efficient lift producing aircraft. These days, with superfast computers, the computational value is no longer significant, but the theorem is still very instructive and marks the foundation for students of aerodynamics. 

Thanks for reading!

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