Panel Methods

This page is the launching point for all my content related to panel methods, including both source panel and vortex panel methods. There are links below to both blog posts (if you want to read the information) and videos (if you want to watch a video about the subject). Just click on the link and it will open in a new tab. My GitHub page has the same information as below, and includes all my code as well.

UIUC Airfoil Database File Download

This section is for downloading all the airfoil coordinate files from the UIUC Airfoil Database. The files are all of type dat, which can be opened in any text editor. This code is run in Python, since it’s free to download.

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Load Airfoil Coordinates in MATLAB

With the airfoils downloaded from the UIUC airfoil database in the previous section, we now need to read them into our MATLAB code in order to use them in the panel code. The video and accompanying blog post with code can be found below.

Running XFOIL from MATLAB

This section is for running XFOIL from a script in MATLAB. This is convenient if you want to be able to quickly change a parameter (or multiple parameters) and output the results, but you don’t want to go into the command line of XFOIL every time. This is useful for getting airfoil data in a MATLAB GUI for instance. If you have an edit text box for changing angle of attack, then every time it’s changed by the user, the MATLAB program will call XFOIL using the updated AoA value and return the airfoil coordinates and pressure coefficient data (for example).

Running XFOIL from Python

This is the same code that was used in MATLAB, but now written in Python for those of you that want an open-source option.

Compute Circulation

Circulation is a fundamental concept in aerodynamics (and more generally, in multivariable calculus). In order to find the lift of an airfoil using the vortex panel method, the circulation around the airfoil needs to be computed. My YouTube video explaining how to compute circulation numerically, along with a blog post that contains the code can be found below.

Incompressible Potential Flow

When assuming that a flow is both irrotational and incompressible, we can simplify down some complicated equations into a simpler equation (Laplace’s equation), and the resulting flow is called incompressible potential flow. My video and blog post on this subject can be found below.

Elementary Potential Flows

In order to build more complex flows, we need to understand the simplest types of potential flows. Here, we go through three elementary flows (uniform, source/sink, and vortex), along with two combinations of these flows (uniform + source/sink and uniform + vortex). Both MATLAB and Python code is included.

Panel Method Geometry

The first step in writing your own panel method code is to understand the geometry and its associated variables. Here, we go through this in detail for an arbitrary shape (circle approximated by eight panels). In the MATLAB and Python codes, we also show how this works for an airfoil.

Building More Complex Potential Flows

In the Elementary Potential Flows section, we went over all the relevant elementary incompressible potential flows. We can now use these to build up a more complex flow step-by-step. The end result of this section is to obtain an expression for the velocity potential induced at an arbitrary point P in the flow due to uniform flow and N source panels (that approximate the airfoil geometry).

Flow Around an Airfoil

How do we use the velocity potential equation derived in the previous section to compute the flow around an airfoil? This section explains the necessary conditions that we can use to compute the unknown source panel strengths. We use the boundary condition for an impermeable (solid) object to stipulate that the normal velocity at each panel control point should be zero.

Source Panel Method: Geometric Integrals

When we take the appropriate partial derivative (normal, tangential, X, or Y) of the velocity potential equation, we end up with a complicated integral expression inside the source panel term. In order to code up our system of equations in the next section, we need to find an explicit expression for this integral (also called the geometric integral since it only depends on the airfoil’s geometry). The videos below have the full derivations for the normal, tangential, X, and Y geometric integrals. You should at least watch the I(ij) video since it includes the entire derivation, whereas the other videos don’t repeat the portions that are the same as the I(ij) derivation video (to keep them shorter).

Source Panel Method System of Equations

Now that we have the normal velocity expression for each panel on the airfoil, we can set up a system of equations that can be easily solved.

Source Panel Method: Circular Cylinder

We have finally finished the derivations needed to code up a working version of the source panel method. Recall that this implementation uses constant source panel strengths (which can vary from panel to panel). This first implementation of the method is for the simplest case: flow over a circular cylinder. The reason to use this simple case is that it has an analytical solution for the pressure coefficient that we can compare to (and it is also generally used as a source panel method comparison test case). The code is provided for both MATLAB and Python. The main code is the SP_Circle file, which needs both functions to run properly (COMPUTE_IJ_SPM and STREAMLINE_SPM).

Source Panel Method: Airfoil

After making sure the simple validation case of the circular cylinder worked properly, we updated the code to be able to run with airfoils. There are a couple more functions, programs, and directories needed when running this code. You can run my code in either MATLAB or Python. Whichever you choose, make sure to download all the files with that extension (.m or .py) from the list below. You will also need the directory with the airfoils (Airfoil_DAT_Selig, make sure to extract after downloading the zip file) and the XFOIL executable (xfoil.exe) in the directory with all the code. To run the code, open the SP_Airfoil script and press Run or F5.

Vortex Panel Method: Geometric Integrals

For the source panel method, we derived the geometric integral expressions for the normal, tangential, X, and Y directions. Here, we go through similar derivations, but this time for the vortex panel method.

Vortex Panel Method System of Equations

In the same way that we wrote the system of equations for the source panel method, we can write the system of equations for the vortex panel method. To include the Kutta condition equation in the matrix, we must remove one of the normal velocity equations for one of the control points.

Vortex Panel Method: Airfoil

The source panel method code was updated to be able to solve for the vortex panel strengths instead of the source panel strengths. You can run my code in either MATLAB or Python. Whichever you choose, make sure to download all the files with that extension (.m or .py) from the list below. You will also need the directory with the airfoils (Airfoil_DAT_Selig, make sure to extract after downloading the zip file) and the XFOIL executable (xfoil.exe) in the directory with all the code. To run the code, open the VP_Airfoil script and press Run or F5. The limitations of this VPM implementation are shown in my YouTube video, and motivate the need for the combined source/vortex panel method.

Source/Vortex Panel Method System of Equations

The previous video on the constant strength vortex panel method showed that there are still some limitations with this simplified method, and to improve it we can combine the source and vortex panel methods into one source panel/vortex panel (SPVP) method. This video covers how to build the system of equations for the SPVP method and how to derive and apply a different version of the Kutta condition.

Source/Vortex Panel Method: Airfoil

We can now implement the matrix system defined in the previous section in our code. Note that to run the code, you will need both the SPM and the VPM functions, since this method uses a combination of both of them. In the YouTube video, we will look at how to implement the changes in the code, and go through some examples of different airfoils.

Multi-Element Source/Vortex Panel Method

This will be the final video of the series, where I adapt the code from the previous section to be able to load multiple airfoils. I’ll show how to load airfoils, manipulate their locations relative to each other, and how to implement the necessary changes to the code to handle multiple airfoils. We will go over a couple simple examples to test the validity, and then hopefully I’ll be able to show a validation case to an airfoil system from a paper.

  • YouTube Video
  • Blog Post (including code)