Potential Flow Theory: Circulation and the Kutta-Jukowski Theorem

[Master1022]

Homework Statement:: What is the convention for $\Gamma$ for the streamline $\Psi = \frac{\Gamma}{2\pi} ln(\frac{r}{a} )$ and how can we interpret the Kutta-Jukowski Theorem?
Relevant Equations:: $v_{\theta} = - \frac{\partial \Psi}{\partial \theta}$

[Mentor Note -- moved from the schoolwork forums to the Aerospace forum]

Hi,

I just had a quick question about conventions in potential flow theory:
Question: What is the convention for $\Gamma$ for the streamline $\Psi = \frac{\Gamma}{2\pi} ln(\frac{r}{a} )$ and how can we interpret the Kutta-Jukowski Theorem $Lift = - \rho U \Gamma$?

Approach:
For the first part, am I correct in thinking that $\Gamma$ indicates a clockwise circulation? This is supported by the fact that
$$ v_{\theta} = - \frac{\partial \Psi}{\partial \theta} = - \frac{\Gamma}{2 \pi r} $$
where $\theta$ is defined to be defined as anti-clockwise from the +ve x axis. I was confused as I had seen diagrams on the internet where they have suggested that $+ \Gamma$ indicates an anticlockwise circulation. Another quick question I have is: is gamma the circulation of the fluid or the cylinder? If it's the former, does that mean that if the cylinder is rotating (e.g. CW) we take the circulation as being of equal 'strength' in the CCW direction?

For the second part, I have done the integration to get the result that $Lift = - \rho U \Gamma$ where Lift has been defined in the +ve y-direction. However, I don't think I really understand what this means. So this formula predicts that there will be a positive lift in the y-direction when there is an anticlockwise circulation?

Any help would be greatly appreciated

 

[Aaron Barnard]

!

Answer:
The convention for $\Gamma$ is that a positive value indicates a clockwise circulation, and a negative value indicates an anticlockwise circulation. This is supported by the fact that the equation
$$ v_{\theta} = - \frac{\partial \Psi}{\partial \theta} = - \frac{\Gamma}{2 \pi r} $$
implies that a positive value of $\Gamma$ will result in a negative value of $v_\theta$, which indicates an anticlockwise flow.

The Kutta-Joukowski Theorem states that the lift produced by a body in a fluid is equal to the circulation of the fluid times the density of the fluid times the velocity of the body. This can be written mathematically as
$$ Lift = - \rho U \Gamma $$
where $\rho$ is the fluid density, $U$ is the body velocity, and $\Gamma$ is the circulation of the fluid. This theorem implies that a positive value of $\Gamma$ (indicating a clockwise circulation) will produce a positive lift in the y-direction.

 

[NatSusan]

. Thanks!

Hi there,

In potential flow theory, the convention for $\Gamma$ is that it represents the strength of the vortex or circulation around a body. This means that a positive $\Gamma$ indicates a clockwise circulation, while a negative $\Gamma$ indicates an anticlockwise circulation.

In the streamline equation $\Psi = \frac{\Gamma}{2\pi} ln(\frac{r}{a} )$, $\Gamma$ is the circulation of the fluid around the cylinder. This means that if the cylinder is rotating clockwise, the circulation will be in the counterclockwise direction.

The Kutta-Jukowski theorem states that the lift generated by a body in potential flow is proportional to the circulation around the body. In other words, the greater the circulation around the body, the greater the lift. So in the case of an anticlockwise circulation, there will be a positive lift in the y-direction.

I hope this helps clarify things for you. Let me know if you have any other questions.