Vortex Panel Method: Solving for \theta_{ij}

[sid_galt]

In the vortex panel method the following equation is used
$$V_{freestream}sin \beta_i - \sum_{j=1}^n \displaystyle\frac{\lambda_i}{2\pi} \int \displaystyle\frac{d\theta_{ij}}{dn_i} ds_j = 0$$

where n is the panel number, i is the control point at which the vortex strength is being calculated and j is the panel which is inducing some vortex at i, $$\lambda_i$$ is the vortex strength at i and

$$\theta_{ij} = \arctan{\displaystyle\frac{y_i-y_j}{x_i-x_j}}$$

My question is what is the value of $$\theta_{ij}$$ when i = j?

 

[sid_galt]

Does anyone know? What is the value of $$\theta_{ij}$$ when i = j?

For source panels I have read it is $$\frac{\lambda_i}{2}$$. But I don't know what it is for vortex panels.

 

[InfernoSun]

According to my reference, your original equation is incorrect. It should have cosBeta to the right of Vfreestream. Also, the subscript on lambda should be [j], not what your equation shows.

Both source panel and vortex panel methods are numerical methods for flows over arbitrary shapes. But because the source panel method has zero circulation it only applies to non-lifting cases. Vortex panel method can be used for lifting cases. The idea behind both methods is to treat the surface as if it were a streamline, i.e. a physical surface does not exist, it is replaced by a flow line. A source panel achieves this by balancing the dynamic pressure of the onrushing flow with an infinitesimal flow source on a surface, which exerts an outward pressure against it. Like air hockey. The table is a set of source panels balancing the weight per unit area of the puck. But try to imagine this system without the table (the table doesn't exist), just the sources. And the puck is the freestream airflow impinging on the source panels.

To answer your question, at least attempt to, these methods are numerical. To obtain a solution for i = j is the whole point of the method. The solution is physically the contribution of all the panels to the potential at the control point of the ith panel.

Just out of curiosity, what are you applying this method to?

 

[DuncanM]

You may want to check out the resources listed here:

http://www.aae.uiuc.edu/m-selig/ads.html

In particular, see the online utility posted by Professor Kevin Jones of the Naval Postgraduate School:

http://www.aa.nps.navy.mil/~jones/online_tools/panel/about/

He even posts his source code (in FORTRAN). If all else fails, perhaps running your question by him might provide an answer.

Regards,


DuncanM
http://www.rocketscientists.ca/

 

[sid_galt]

According to my reference, your original equation is incorrect. It should have cosBeta to the right of Vfreestream. Also, the subscript on lambda should be [j], not what your equation shows.

Ya, I realized that a few days after posting the question

Just out of curiosity, what are you applying this method to?

To airfoils. To be more precise, I am trying to create a Java program to implement this method.
Thank you for the reply.

 

[sid_galt]

You may want to check out the resources listed here:

http://www.aae.uiuc.edu/m-selig/ads.html

In particular, see the online utility posted by Professor Kevin Jones of the Naval Postgraduate School:

http://www.aa.nps.navy.mil/~jones/online_tools/panel/about/

He even posts his source code (in FORTRAN). If all else fails, perhaps running your question by him might provide an answer.

Regards,


DuncanM
http://www.rocketscientists.ca/

Thank you for the links. The second one is very helpful.