# Sheared Parabolic Grid Generator

1. Aug 29, 2006

### minger

Hi,

I have created a program to generate grid points for a Joukowski airfoil. I was wondering if anyone knew of a 2D Sheared Parabolic Grid Generator that I could use to get a grid. If you are unfamiliar with this grid, it's basically a long rectangle that is "wrapped" around the airfoil so that the incoming part of the grid (towards the leading edge) is curved. The grid points are spaced radially outward from the airfoil.

My professor said there might be one out there somewhere. I just really don't want to spend the next week learning GRIDPRO (even though I know it would be useful, I just want to get past this part).

Thanks,

2. Aug 29, 2006

### Clausius2

I think you mean an elliptic generator, don't you?

I did that one year ago for my research when I was undergrad. I made an elliptic grid generator, I remember I meshed a NACA 0012 airfoil (or something like that) for doing potential flow around it, and later I meshed the nose cone of the fairing of a russian rocket launcher. But I coded the generator in Matlab. It's not such a difficult thing, you only have to solve a couple of elliptic partial non linear equations.

Last edited: Aug 30, 2006
3. Aug 30, 2006

### minger

eh...my advisor told me explicitly that it was a sheared parabolic grid that we were going for. I'm pretty new to the CAA field, so I honestly just know what it looks like.

I also think I might have found some fortran70 code that I can change a little bit to work...hopefully.

4. Aug 30, 2006

### Clausius2

Well, I don't know. I have never heard about parabolic meshing. But let me know if you have troubles with the code.

5. Sep 5, 2006

### minger

Alright, after some research, I have an answer. There are three "classes" of mesh-generating schemes based on shearing transformations.
1) Straight through meshes in which one set of mesh lines is roughly parallel to the streamlines of the flow.
2) Meshes wrapping around the leading edge of the wing and passing smoothly off the trailing edge (Parabolic)
3) Meshes wrapping around both the leading and the trailing edge (elliptical)

There are advantages and disadvantages to all three. The parabolic scheme leads to a natural bunching of cells near the trailing edge which lends to a good resolution of the Kutta condition. However, they require the use of comparatively complicated mapping procedures, and the small mesh-width near the trailing edge can lead to a low rate of convergence of the iterative scheme.

Here’s basically how to parabolic mesh (if anyone is interested).

1. Generate a series of gridpoints which lie on the airfoil.

2. Include a “wake” off the trailing edge (otherwise the grid will stop at the end of the airfoil). Note there need to be two sets of points on the wake (so it can “unwrap”).

3. Use the transformation:

(X’ + iY’)² = {x-xo + i(y-yo)}/t

where t is a scaling factor, xo and yo are coordinates of a point just inside the leading edge which will define the origin of the parabolic coordinates; x and y are the gridpoints of the airfoil.

The effect of this transformation “unwraps” the airfoil to form a shallow bump.

4. Then apply a shearing transformation where:

X = X’; Y = Y’ – S(X’,Y’)

Where S(X’,Y’) is the bump we produced. The effect of this transformation basically just flattens the bump into a straight line.

5. Place your grid on top of the straight line.

6. Then go backwards, first finding the inverse of the shearing transformation, then the initial transformation.

ref: Remarks on the calculation of a transonic potential flow by a finite volume method, Jameson