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gtheler committed Jun 15, 2024
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14 changes: 8 additions & 6 deletions examples/laplace.markdown
Original file line number Diff line number Diff line change
Expand Up @@ -195,22 +195,22 @@ wing has to satisfy the Kutta condition.

This example

1. Creates a symmetric airfoil-like Joukowsky profile using the the
Gmsh Python API.
1. Creates a symmetric airfoil-like [Joukowsky profile] using the the
Gmsh Python API
2. Solves the steady-state 2D Laplace equation with a different
Dirichlet value at the airfoil until the solution $\phi$ evaluated
at the continuation of the wing tip matches the boundary value $c$.
It then computes the circulation integral of the velocities
1. over the profile itself
2. over a circle around the airfoil, computing the unitary
tangential vector
1. from the internal normal variables `nx` and `ny`
2. from two functions `tx` and `ty` using the circle’s equation
1. from the internal normal variables `nx` and `ny`
2. from two functions `tx` and `ty` using the circle’s equation
3. around the original rectangular domain

It also computes the drag and the lift scalars as the integral of
the pressure (computed from Bernoulli’s principle) times `nx` and
`nz` over the circle, respectively.
the pressure (computed from Bernoulli’s principle) times `nx`
and `nz` over the circle, respectively.

<figure>
<img src="airfoil-msh1.png" style="width:100.0%" alt="Full domain" />
Expand Down Expand Up @@ -482,3 +482,5 @@ alt="Potential and velocities zoomed over the airfoil" />
<figcaption aria-hidden="true">Potential and velocities zoomed over the
airfoil</figcaption>
</figure>

[Joukowsky profile]: https://en.wikipedia.org/wiki/Joukowsky_transform
13 changes: 7 additions & 6 deletions examples/laplace.md
Original file line number Diff line number Diff line change
Expand Up @@ -179,22 +179,23 @@ wing has to satisfy the Kutta condition.

This example

1. Creates a symmetric airfoil-like Joukowsky profile using the the
Gmsh Python API.
1. Creates a symmetric airfoil-like [Joukowsky
profile](https://en.wikipedia.org/wiki/Joukowsky_transform) using
the the Gmsh Python API
2. Solves the steady-state 2D Laplace equation with a different
Dirichlet value at the airfoil until the solution $\phi$ evaluated
at the continuation of the wing tip matches the boundary value $c$.
It then computes the circulation integral of the velocities
a. over the profile itself
b. over a circle around the airfoil, computing the unitary
tangential vector
i. from the internal normal variables `nx` and `ny`
ii. from two functions `tx` and `ty` using the circle's equation
i. from the internal normal variables `nx` and `ny`
ii. from two functions `tx` and `ty` using the circle's equation
c. around the original rectangular domain

It also computes the drag and the lift scalars as the integral of
the pressure (computed from Bernoulli's principle) times `nx` and
`nz` over the circle, respectively.
the pressure (computed from Bernoulli's principle) times `nx`
and `nz` over the circle, respectively.

![Full domain](airfoil-msh1.png){width="100%"}

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