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 Florian Longueteau July 9, 2007 06:21

Math. nature of incompressible Euler equations

Hi all !

I'm a French Ph.D student. I'm developping a stability code for curvilinear geometries. The equations used in this code are the Incompresible Linearized Euler Equations.

I'm wondering if these equations are hyperbolic, parabolic or elliptic. Can you advise me a way to state on this nature ?

Thank you.

Florian.

 Tom July 9, 2007 07:00

Re: Math. nature of incompressible Euler equations

The unsteady euler equations are hyperbolic (although the incompressibility condition gives them a "elliptic/parabolic" character - the acoustic waves are effectively travelling at infinite velocity giving rise to action at a distance).

 Florian Longueteau July 9, 2007 07:50

Re: Math. nature of incompressible Euler equations

I don't understand how the unsteady euler equations can be at the same time hyperbolic and elliptic/parabolic. Furthermore how can there be acoustic waves under the incompressibility condition ?

On the other hand, the steady euler equations are of "hybrid" nature (i.e. elliptic/parabolic), aren't tey ?

 Tom July 9, 2007 08:46

Re: Math. nature of incompressible Euler equations

"On the other hand, the steady euler equations are of "hybrid" nature"

No- the steady incompressible equations are elliptic - its the steady compressible equations that change type.

The usteady equations are hyperbolic as I said (they are neither parabolic or elliptic). However this tells you very little by itself and you need to look carefully at the characteristics of the system. Basically in order to satisfy continuity a "global constraint" on the flow must be satisfied; i.e. the pressure satisfies an elliptic equation (the reason for this is that the filtering out of the acoustic modes in the incompressible limit is achieved my making them move at infinite velocity). This global constraint gives the underlying equations an "elliptic" character (parabolic equations also have this behaviour) even though the equations are hyperbolic!

 Florian Longueteau July 9, 2007 09:17

Re: Math. nature of incompressible Euler equations

"No- the steady incompressible equations are elliptic - its the steady compressible equations that change type."

Ok. But when I compute the characteristics of these equations, I find three eigenvalues : the complex number "i" and his conjugate, and the real number v0/u0, where the flow field (u0,v0) is the basic state around which the linearisation is made. I thought that finding these three eigenvalues (2 complex and on real) give a "hybrid" nature to the equations. Perhaps I'm wrong in the way I compute the characteristic directions...

" you need to look carefully at the characteristics of the system"

I think I don't have the good way to exhibit the characteristics in case of the unsteady incompressible euler equations. I'm trying to put the PDE equations in the form dU/dt+A1dU/dx+A2dU/dy (where U is the vector of unknowns, A1 et A2 two matrices, et x and y the spatial variables). But I can't do this because the incompressibility suppress the time dependency for the pressure.

Consequently, I'm only able to exhibit the characteristics in the compressible case (by computing the determinant of A1*n1+A2*n2, for n1 and n2 real numbers). And I don't know how to do this in the incompressible case...

 Tom July 9, 2007 10:14

Re: Math. nature of incompressible Euler equations

The incompressible Euler equations are hyerbolic in time and elliptical in space. Therefore if you are solving the steady state equations they will be purely elliptical. If you are solving the transient equations they will be hyperbolic in time. (time marching) The equations are elliptical in space in both the compressible and incompressible form as long as the mach number is less than 1.

 Florian Longueteau July 9, 2007 10:21

Re: Math. nature of incompressible Euler equations

Ok ! Thanks !

Now let's go a litlle further. What does this mean physically to be "hyperbolic in time" and "elliptical in space" ?

In fact, I do solve the steady state equations. This is because the stability approach postulates an exlicit dependance in time (in exp(iwt)) for the perturbations. Therefore all temporal derivatives vanishes in an "iw" product.

So I guess my equations will be elliptic. What does this imply on the solutions or on the boundary conditions ?

Thanks for helping.

 Tom July 9, 2007 10:37

Re: Math. nature of incompressible Euler equations

"Now let's go a litlle further. What does this mean physically to be "hyperbolic in time" and "elliptical in space" ?"

It doesn't mean anything - I guess what you mean is that the problem is parabolic. But saying hyperbolic in time is rather strange!

"So I guess my equations will be elliptic. What does this imply on the solutions or on the boundary conditions ?"

It means you have to write down boundary conditions on all the boundaries and then solve the whole (rather large) eigenvalue problem for w.

 Florian Longueteau July 9, 2007 10:47

Re: Math. nature of incompressible Euler equations

"It doesn't mean anything - I guess what you mean is that the problem is parabolic. But saying hyperbolic in time is rather strange!"

Uh ! but you said that before : "The incompressible Euler equations are hyerbolic in time and elliptical in space." I was just wondering what you meant by that...

"It means you have to write down boundary conditions on all the boundaries and then solve the whole (rather large) eigenvalue problem for w."

Yes ! That's what I'm doing, and in fact it works quite well. Except for another problem due to the incompressibility hypothesis : the pressure is determined within an additive constant. Therefore I have to set-up an additional condition (zero mean pressure for instance) on the pressure in order to close the problem.

 Tom July 9, 2007 10:57

Re: Math. nature of incompressible Euler equations

"Uh ! but you said that before : "The incompressible Euler equations are hyerbolic in time and elliptical in space." I was just wondering what you meant by that..."

No I didn't say that - that's a different Tom! Someones trying to confuse you.

 Florian Longueteau July 9, 2007 11:05

Re: Math. nature of incompressible Euler equations

Damn ! But what's the point of doing that ?!

In which Tom can I trust now ? :)

 Tom July 9, 2007 11:10

Re: Math. nature of incompressible Euler equations

yes this is a different Tom...but not trying to confuse you.

The equations are hyperbolic in time in that you can march the soulution in time. An analogy is the space marching parabolized navier stokes codes. For mach numbers less than unitity, the euations are elliptic in space. What this means is that you must set the appropriate boundary conditions on all sides of the domain. (Of course they must be well posed) A good suggestion is to get Tannehills CFD book. He did a very good job in explaining this.

 anon July 9, 2007 15:14

Re: Math. nature of incompressible Euler equations

A clear case of a 'peeping Tom' :)

Sorry, but this last bit was classic.

 jojo July 10, 2007 16:12

Re: Math. nature of incompressible Euler equations

From a pure formal point of view, the transient Euler equations are hyperbolic but the steady-state ones are elliptic.

 Florian Longueteau July 10, 2007 16:55

Re: Math. nature of incompressible Euler equations

Thanks Tom for helping.

I'll try to see the book of Tannehills.

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