# Linear-Strength Vortex Unsteady Panel Method (2D)

• Sonik_87
In summary: I don't think you are calculating the potential PHI correctly. You should be using the result of the Euler equation, which is:
Sonik_87
Hi everyone,

I am working on a project for the development of a 2D Unsteady Panel Method for Airfoils. As often already suggested in this forum I have been using the book "Low Speed Aerodynamics" which has helped me a lot to produce the Steady State solver (I am writing the code in Matlab at the moment), which works very well.

However I am having great problems in validating my unsteady code.
I have followed the book, mainly the steps are:

1) compute the steady solution at first time step
2) move the airfoil via a prescribed motion path (in my case pitching A*sin(wt))
3) position a concentrated vortex at a certain distance from the T.E., which has unknown intensity GAMMA_W and which is therefore part of the solution
4) use the Kelvin theorem for completing the coefficients matrix A which will be N+2 times N+2 (N+1 gammas on the nodes of the panels) + 1 (latter) unknown shed concentrated vortex
5) compute the RHS (right-hand-side) term considering free stream velocity and wake induced velocity
6) solve the system

The problem seems that the Cp variations are not big enough. I am using a NACA report on wind tunnel experimental data (pitching motion 5 + 5sin(Wt) motion) to check the code. But also the comparison with results contained in the Katz and Plotkin (plot of CL-ALPHA of a 0012) isn't satisfactory.

I compute the Cp at each collocation point via the formula:

Cp_i = 1 - Vel_i^2

where Vel_i is the velocity at the i-th collocation point obtained from the different contributes: free stream (similar to steady state) + kinematic movement (sinusoidal motion) + induced velocities (by panels and by wake).
I consider the free stream V to be equal to 1.Honestly I am not quite sure that all of this is right (talking about Cp calculation).

Moreover I have great doubt about the Kutta condition: in the steady state case the last line of matrix A is set equal to: [1 0 0 0 . . . 0 1] so that vorticity at T.E. is 0:

gamma_1 + gamma_(N+1) = 0

I did not change this in the unsteady code only because I couldn't find any indication on how to modify it (the book says that for small oscillations the steady case condition should still work fine).

I would be really thankful if any of you could help me out, as it is more than a month that I am trying to solve the problem. I have also read many articles like Hancock and Mook's one, or similar, but they don't really say explicitly how to do things.

Thank you very much.

PS: do ask questions if you don't understand some of the procedures I have used, I will be happy to explain more thoroughly.

It looks to me like the problem is how you are actually calculating the Cp. You are using the result of the steady Bernoulli equation, but your flow is obviously unsteady. You should have a term that contains the time derivative of the velocity potential in you Cp calculation. Take a look through the book Low Speed Aero and you will find it.

Well the formula would be:

Cp = 1 - (V/U)^2 - (2/U^2)*dPHI/dt

but I do not know exactly how to calculate my potential PHI from my Gammas. Do you have any idea? I have tried in a few ways but the results are not correct.

## 1. What is a Linear-Strength Vortex Unsteady Panel Method (2D)?

A Linear-Strength Vortex Unsteady Panel Method (2D) is a numerical method used to simulate the flow of a fluid around a two-dimensional object. It uses a combination of vortex panels and linear strength sources to model the effects of circulation and lift on the object.

## 2. How does the Linear-Strength Vortex Unsteady Panel Method (2D) work?

The method works by dividing the surface of the object into small panels and calculating the strength and position of vortices and sources on each panel. These strengths and positions are then used to calculate the flow field around the object and the resulting lift and drag forces.

## 3. What are the advantages of using the Linear-Strength Vortex Unsteady Panel Method (2D)?

One advantage is that it is a relatively simple and efficient method for simulating the flow around two-dimensional objects. It also allows for the calculation of unsteady flows, making it useful for studying dynamic effects such as vortex shedding.

## 4. What are the limitations of the Linear-Strength Vortex Unsteady Panel Method (2D)?

One limitation is that it is only applicable for two-dimensional flows, so it cannot be used for three-dimensional objects. Additionally, the method assumes inviscid flow, so it does not account for the effects of viscous forces.

## 5. What are some applications of the Linear-Strength Vortex Unsteady Panel Method (2D)?

The method is commonly used in the design and analysis of airfoils, wings, and other aerodynamic shapes. It is also used in the study of vortex dynamics and the prediction of flow separation on surfaces.

• Aerospace Engineering
Replies
8
Views
2K
• Aerospace Engineering
Replies
4
Views
6K
• Differential Equations
Replies
4
Views
1K
• Differential Equations
Replies
1
Views
1K
Replies
1
Views
2K
• Special and General Relativity
Replies
6
Views
1K
• Differential Equations
Replies
6
Views
2K
• Engineering and Comp Sci Homework Help
Replies
13
Views
3K
• Engineering and Comp Sci Homework Help
Replies
4
Views
1K
• Differential Equations
Replies
1
Views
1K