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Vortex Panel Method: System of Equations

In this video, we will build up the system of equations we need to solve in order to get the vortex panel strengths. First, we get N equations by setting the normal velocity at every control point equal to zero. Next, we get an equation for the Kutta condition. If we simply add this equation into the system of equations, we end up with N+1 equations for N unknowns, which is over-constrained. Instead, we will replace the last normal velocity equation with the Kutta condition equation. We can now solve the system of equations to get the vortex panel strengths. In the next video, we will adapt my SPM code into the VPM code, look at some results, compare to the Kuethe and Chow implementation, and motivate the need for my combined SP/VP implementation. ===== RELEVANT VIDEOS ===== ► Panel Methods Playlist    • How To: Run XFoil from MATLAB   ► Source Panel Method: System of Equations    • Source Panel Method: System of Equations   ► K(ij) Geometric Integral Derivation    • Vortex Panel Method: Normal Velocity ...   ► L(ij) Geometric Integral Derivation    • Vortex Panel Method: Tangential Veloc...   ► Nx(pj) and Ny(pj) Geometric Integral Derivation    • Streamline Geometric Integral VPM [Nx...   ===== NOTES ===== At 2:56, I show a screenshot from the Kuethe and Chow book (1976), and this expression does not have a negative sign. In their book, they generally take the negative sign out of the summation and move it to the uniform flow term immediately, hence the missing negative sign. The focus of that screenshot is actually the text below the equation. ===== ERRORS ===== If you see an error in the video, please let me know and I will include it here. ===== REFERENCES ===== Note: the links are Amazon affiliate links. If you do happen to want to buy the book and use the link below, it helps me out a little. ► Fundamentals of Aerodynamics, Anderson https://amzn.to/3emVuXU ► Foundations of Aerodynamics, Kuethe and Chow https://amzn.to/2yMg1Vi ► Theory of Wing Sections, Abbott and Doenhoff https://amzn.to/2wvZyUt