In my YouTube video called “Multi-Airfoil Source/Vortex Panel Method“, we added the capability to analyze multiple element (airfoil) systems, as opposed to just single airfoil systems. The derivations used in this code are the same as those in this derivation YouTube video. To see my single airfoil implementation, you can watch this video.
Below are the files needed to run the code. Note that I only have this code written in MATLAB. If you feel so inclined, you can update my Python code for the single airfoil implementation. You will need all of these files in the same directory. You will also need the “xfoil.exe” program in this directory (download it here). If you want to be able to load airfoils, you’ll also need a directory called “Airfoil_DAT_Selig” in this directory, which includes all the “.dat” files that can be downloaded from the UIUC airfoil database. The main file that you will run is called SPVP_Airfoil_N.m. All these files can also be found on my GitHub site.
To run this program without errors, you need to also download COMPUTE_IJ_SPM_N.m, COMPUTE_KL_VPM_N.m, STREAMLINE_SPM_N.m, STREAMLINE_VPM_N.m, XFOIL_N.m, and COMPUTE_CIRCULATION.m files. All files must be placed in the same directory/folder. You will also need the “xfoil.exe” program and a directory called “Airfoil_DAT_Selig” that includes all the airfoil .dat files.
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, 
). Depending on the geometry we are trying to define, more points will make a better defined shape. For instance, to specify a line, we only need two points. To specify a triangle, we only need three points, and a rectangle needs four. To define a true circle, we need infinite coordinate pairs. That’s a lot of points, so instead we can approximate its shape using a finite number of points. Let us first look at the points shown in Fig. 1. It looks like these eight points are arranged to define a circular-ish polygonal shape. It’s defined by (
![\begin{equation*} \begin{aligned} V_x &= V_\infty\cos\left(\alpha\right) + \frac{\Gamma dy}{2\pi r^2} \\[5pt] V_y &= V_\infty\sin\left(\alpha\right) + \frac{-\Gamma dx}{2\pi r^2} \end{aligned} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-f71a62df75a505069f2f4ae222de04bd_l3.png "Rendered by QuickLaTeX.com")
, whereas Fig. 2 is for the combined uniform and vortex flow when the vortex strength is 
. In both plots, the red arrows are the velocity vectors, the black lines are streamlines, and the blue and magenta curves are used to compute the line integral in the circulation equation.
. The term (capital gamma) 
is called the vortex strength, and is the same thing as the circulation defined earlier in my 
and 
velocity components by taking the appropriate derivatives of the velocity potential.![\begin{equation*} \begin{aligned} V_x &= \frac{\partial \varphi_v}{\partial x} = -\frac{\Gamma}{2\pi}\frac{\partial \theta}{\partial x} \\[5pt] V_y &= \frac{\partial \varphi_v}{\partial y} = -\frac{\Gamma}{2\pi} \frac{\partial \theta}{\partial y} \end{aligned} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-9d689c1b66860ec2b0024e3019ebf66a_l3.png "Rendered by QuickLaTeX.com")
and 
) to the point on the circle can be found knowing the vortex center point (
, 
).
. Now we can perform the partial derivative of ![\begin{equation*} \begin{aligned} \frac{\partial\theta}{\partial x} &= \frac{\partial}{\partial x}\tan^{-1}\left(\frac{dy}{dx}\right) = \left[\frac{1}{1+\left(dy/dx\right)^2}\right]\left(\frac{-dy}{dx^2}\right) \\[2pt] \frac{\partial\theta}{\partial y} &= \frac{\partial}{\partial y}\tan^{-1}\left(\frac{dy}{dx}\right) = \left[\frac{1}{1+\left(dy/dx\right)^2}\right]\left(\frac{1}{dx}\right) \end{aligned} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-e0cb719b2ad13bb842d5868bd79bd4db_l3.png "Rendered by QuickLaTeX.com")
, and noting that 
.![\begin{equation*} \left[\frac{1}{1+\left(\frac{dy}{dx}\right)^2}\right]\left(\frac{dx^2}{dx^2}\right) = \frac{dx^2}{dx^2 + dy^2} = \frac{dx^2}{r^2} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-04ce8a5edea28ebe4ce3c232bae69995_l3.png "Rendered by QuickLaTeX.com")
![\begin{equation*} \begin{aligned} V_x &= \frac{-\Gamma}{2\pi}\frac{\partial\theta}{\partial x} = \frac{-\Gamma}{2\pi}\left(\frac{dx^2}{r^2}\right)\left(\frac{-dy}{dx^2}\right) = \frac{-\Gamma}{2\pi}\left(\frac{-dy}{r^2}\right) \\[5pt] V_x &= \frac{\Gamma dy}{2\pi r^2} \\[5pt] V_y &= \frac{-\Gamma}{2\pi}\frac{\partial\theta}{\partial y} = \frac{-\Gamma}{2\pi}\left(\frac{dx^2}{r^2}\right)\left(\frac{1}{dx}\right) = \frac{-\Gamma}{2\pi}\left(\frac{dx}{r^2}\right) \\[5pt] V_y &= \frac{-\Gamma dx}{2\pi r^2} \end{aligned} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-a5028b4da6b05da22e31e0d57c27d0d2_l3.png "Rendered by QuickLaTeX.com")
. Blue line is the curve used for circulation calculation.
.
, and the computed circulation is 
. For the negative vortex, we specified a strength of 
, and the compute circulation is 
. Note that for these calculations, we have broken the curve into 
segments. If less segments are used, the results are less accurate (
using 
segments).
) with a 
). For brevity, only the loop where the velocity components are computed for this combined flow are shown in the code. The Cartesian velocity components are shown below.![\begin{equation*} \begin{aligned} V_x &= V_\infty\cos\left(\alpha\right) +\frac{\Lambda\left(X_\text{P}-X_0\right)}{2\pi r_\text{P}^2} \\[2pt] V_y &= V_\infty\sin\left(\alpha\right) +\frac{\Lambda\left(Y_\text{P}-Y_0\right)}{2\pi r_\text{P}^2} \end{aligned} \end{equation*}](https://www.joshtheengineer.com/wp-content/ql-cache/quicklatex.com-92a5402c3b6f6a233a6db21f0881a157_l3.png "Rendered by QuickLaTeX.com")
and 
are equal to the sum of the uniform flow and source flow solutions. The resulting flow field can be seen in Fig. 1. The circulation is computed the same way as before, and results again in a value nearly zero (
), which makes sense because we still have no vortex in this flow (no source of circulation).