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 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...

 Ashley July 15, 2009 15:13

Hi,
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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