Uploaded April 2020 | Updated September 2026, 2 weeks ago
The vortex panel method code in this video is an adaptation of the source panel method code from a few videos ago. The only change we've made between the codes is the formulation of the matrix system of equations (including the addition of the Kutta condition equation).
We'll look at a few examples of the code working like you would expect, and compare resulting lift and moment coefficients to the XFOIL results. Then we'll look at a few cases where the code seems to fall apart, which is the motivation for my next two videos, the combined source/vortex panel method.
===== WHERE ARE WE GOING? =====
→ The limitations in this video motivate the need for a more robust implementation of the VPM.
→ I will derive the combined SPM/VPM formulation and code it to show how good we can get the results for a pretty simple implementation.
→ We can finally extend the SPM/VPM formulation to multiple separate airfoil elements. This will be the last video in the series.
===== CODE =====
► My website
joshtheengineer.com/2020/06/21/vortex-panel-method-airfoil
► GitHub
github.com/jte0419/Panel_Methods
===== RELEVANT VIDEOS =====
► Panel Methods Playlist
youtube.com/watch?v=bWjo3N9COz4&list=PLxT-itJ3HGuUDVMuWKBxyoY8Dm9O9qstP
► Panel Method Geometry
youtube.com/watch?v=kIqxbd937PI
► Building More Complex Flows
youtube.com/watch?v=EKzbwJvKcmw
► Flow Around an Airfoil
youtube.com/watch?v=cLdv1UfX1g8
► Normal Velocity Geometric Integral [K(ij)]
youtube.com/watch?v=5lmIv2CUpoc
► Tangential Velocity Geometric Integral [L(ij)]
youtube.com/watch?v=IxWJzwIG_gY
► Streamline Geometric Integral VPM [Nx(ij) and Ny(ij)]
youtube.com/watch?v=TBwBnW87hso
► Solving the System of Equations: VPM
youtube.com/watch?v=j3ETHFBiYOg
► Source Panel Method: Airfoil
youtube.com/watch?v=fdNOYdwY9Bw
===== NOTES =====
→ I'll add notes here if I need to.
===== 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
amzn.to/3emVuXU
► Foundations of Aerodynamics, Kuethe and Chow
amzn.to/2yMg1Vi
► Theory of Wing Sections, Abbott and Doenhoff
amzn.to/2wvZyUt
The vortex panel method code in this video is an adaptation of the source panel method code from a few videos ago. The only change we've made between the codes is the formulation of the matrix system of equations (including the addition of the Kutta condition equation).
We'll look at a few examples of the code working like you would expect, and compare resulting lift and moment coefficients to the XFOIL results. Then we'll look at a few cases where the code seems to fall apart, which is the motivation for my next two videos, the combined source/vortex panel method.
===== WHERE ARE WE GOING? =====
→ The limitations in this video motivate the need for a more robust implementation of the VPM.
→ I will derive the combined SPM/VPM formulation and code it to show how good we can get the results for a pretty simple implementation.
→ We can finally extend the SPM/VPM formulation to multiple separate airfoil elements. This will be the last video in the series.
===== CODE =====
► My website
joshtheengineer.com/2020/06/21/vortex-panel-method-airfoil
► GitHub
github.com/jte0419/Panel_Methods
===== RELEVANT VIDEOS =====
► Panel Methods Playlist
youtube.com/watch?v=bWjo3N9COz4&list=PLxT-itJ3HGuUDVMuWKBxyoY8Dm9O9qstP
► Panel Method Geometry
youtube.com/watch?v=kIqxbd937PI
► Building More Complex Flows
youtube.com/watch?v=EKzbwJvKcmw
► Flow Around an Airfoil
youtube.com/watch?v=cLdv1UfX1g8
► Normal Velocity Geometric Integral [K(ij)]
youtube.com/watch?v=5lmIv2CUpoc
► Tangential Velocity Geometric Integral [L(ij)]
youtube.com/watch?v=IxWJzwIG_gY
► Streamline Geometric Integral VPM [Nx(ij) and Ny(ij)]
youtube.com/watch?v=TBwBnW87hso
► Solving the System of Equations: VPM
youtube.com/watch?v=j3ETHFBiYOg
► Source Panel Method: Airfoil
youtube.com/watch?v=fdNOYdwY9Bw
===== NOTES =====
→ I'll add notes here if I need to.
===== 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
amzn.to/3emVuXU
► Foundations of Aerodynamics, Kuethe and Chow
amzn.to/2yMg1Vi
► Theory of Wing Sections, Abbott and Doenhoff
amzn.to/2wvZyUt
![Explained: Ghost Nodes [CFD]](https://i.ytimg.com/vi/JMP9aanQ5o0/mqdefault.jpg "Explained: Ghost Nodes [CFD]
In some finite difference/volume equations, you might need to use points that are outside of the actual grid domain. In these cases, ghost nodes can be calculated from the interior points and used for the differences (such as a central difference on a boundary node).")
![Source Panel Method: Tangential Velocity Geometric Integral [J(ij)]](https://i.ytimg.com/vi/JRHnOsueic8/mqdefault.jpg "Source Panel Method: Tangential Velocity Geometric Integral [J(ij)]
In the previous video (Geometric Integral Iij), we went through the full derivation of the geometric integral for the normal partial derivative, which was needed to solve for the source panel strengths. In this video, we will go through the (very similar) derivation of the tangential partial derivative geometric integral, which is needed to solve for the panel velocities, and thus the panel pressure coefficients.
This derivation is almost exactly the same as the normal geometric integral derivation, but there are some slight differences. Where it is exactly the same, I will refer you back to my other video (Iij) so we dont make this video longer than it needs to be.
RELEVANT VIDEOS
► Panel Methods Playlist
https://www.youtube.com/watch?v=bWjo3N9COz4&list=PLxT-itJ3HGuUDVMuWKBxyoY8Dm9O9qstP
► Panel Method Geometry
https://www.youtube.com/watch?v=kIqxbd937PI
► Building More Complex Flows
https://www.youtube.com/watch?v=EKzbwJvKcmw
► Flow Around an Airfoil
https://www.youtube.com/watch?v=cLdv1UfX1g8
► Normal Velocity Geometric Integral, Iij
https://www.youtube.com/watch?v=76vPudNET6U
NOTES
- Ill add notes here if I need to.
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")
![Explained: TI-83 Linear Interpolation Program [Math]](https://i.ytimg.com/vi/KQzK5z9jkHE/mqdefault.jpg "Explained: TI-83 Linear Interpolation Program [Math]
In this video I show you how to program a linear interpolation equation into your TI-83 graphing calculator.
If you would like to see how the equation is derived, click the link below to my other video.
https://goo.gl/7OA0UU
Here is the code if you already know how to input it into your calculator:
PROGRAM:INTERP
Input X1=,A
Input X2=,D
Input X= ,I
Input Y1=,N
Input Y2=,S
Disp (((S-N)/(D-A))(I-A)+N)")
![Explained: Croccos Theorem [Taylor-Maccoll]](https://i.ytimg.com/vi/KZYHjujYpFk/mqdefault.jpg "Explained: Croccos Theorem [Taylor-Maccoll]
Croccos theorem is needed to obtain the irrotationality condition used later in the overall derivation. The equation is essentially a combination of the momentum and energy equations. The final resulting boxed equation states that the curl of the velocity field is zero, indication an irrotational flow.
The extra videos I mention in this video can be found below.
Momentum Equation: http://www.youtube.com/watch?v=ZH0fiaYGTYQ
Assumptions: http://www.youtube.com/watch?v=HIDeIZO0z-8
Total Derivative: http://www.youtube.com/watch?v=bgUDbntoXBY
TdS Equation: http://www.youtube.com/watch?v=N126SaEGuJA
Vector Identity: http://www.youtube.com/watch?v=0LO-ZcESYdE
Enthalpy: http://www.youtube.com/watch?v=yy5HQ9syrrI")
![Explained: TdS Equation [Taylor-Maccoll]](https://i.ytimg.com/vi/N126SaEGuJA/mqdefault.jpg "Explained: TdS Equation [Taylor-Maccoll]
The TdS equation is a combined statement of the first and second law of thermodynamics. It relates the entropy, enthalpy, and pressure changes in a system (for the form that we need for Croccos theorem). To keep the big picture in mind, recall that I mentioned that Croccos theorem is a combination of the thermodynamics and kinematics of a flow. This equation gives us the thermodynamics part.")
