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-   -   need the eigen vectors and their inverses for 2D Euler equations (http://www.cfd-online.com/Forums/main/66412-need-eigen-vectors-their-inverses-2d-euler-equations.html)

Ashley July 14, 2009 17:26

need the eigen vectors and their inverses for 2D Euler equations
 
Hi,
I need the eigen vectors in both x and y directions and their analytical inverses for the 2D Euler equations.Does anybody know any reference other than haffmann?

vinayender July 15, 2009 06:27

find here....
 
Hi Ashely,

for 2D flows we have 4 eigen values
u_perpendicuar+a, u_perpendicuar , u_perpendicuar, u_perpendicuar-a

eigen vectors are

for u_perpendicuar-a
( 1
u-a*nx
v-a*ny
H-u_perpendicular*a)

for u_perpendicuar
( 0
-a*ny
a*nx
u_perpendicular*a)

for u_perpendicuar
( 1
u
v
0.5*(u^2+v^2))

for u_perpendicuar+a
( 1
u+a*nx
v+a*ny
H+u_perpendicular*a)


where nx and ny are the unit normal vectors for the face across which we are computing the flux...

Please let me know if you wnat more information or clarification on this....

Ashley July 15, 2009 15:13

Hi,
Thank you so much for your answer.
I know that in x-direction the eigen values are:
u-a,u,u,u+a

and the eigen vectors are

(/1,1,0,1/)
(/(ux(i,j)-ax(i,j)),ux(i,j),0,(ux(i,j)+ax(i,j))/)
(/vx(i,j),vx(i,j),1,vx(i,j)/)
(/(Hx(i,j)-ux(i,j)*ax(i,j)),0.5*(uh(i,j)^2+vx(i,j)^2),vx(i,j) ,(Hx(i,j)+ux(i,j)*ax(i,j))/)

and in y-direction the eigen values are

v-a,v,v,v+a
Now what are the eigenvectors in y-direction?
thank you very much

vinayender July 16, 2009 10:30

I have modified my earlier post a little...
those are the eigen values and vectors normal to a given face...


If you are using these values for coding and getting flux across a face, then all you will be needed is flux perpendicuar because flux perpendicuar to a face is the only component which contributes to your residue of a cell.
Hence if you have the eigen values and eigen vector normal to a face it would be sufficient.....

gory July 16, 2009 12:09

There is a book
 
Hi Ashley,

I think that the information posted by vinayender should be enough (you can use Maple, for example, to generate the inverse) for you.

But there is a book which contains the (right-)eigenvector matrices and the corresponding inverse matrices (i.e., the left-eigenvctor matrices) for various forms of the Euler equations in 1, 2 , and 3D: conservative, primitive, symmetric forms, all interms of a face normal. Also interesting is that it contains the absolute value of the Jacobian (normal to a face), which is often needed for upwind schemes, expressed in terms of only the face normal (no tangent vectors needed). I personally find it very useful. Check out the http://www.cfdbooks.com for details if you're interested.

Good Luck!
Gory


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