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 Farid Moussaoui March 25, 2001 15:48

Multiple solutions for Euler equations

I am searching the definition of airfoils admitting multiple solutions for the Euler equations.

Thanks.

F++

 T. Matsuzawa March 26, 2001 23:26

Re: Multiple solutions for Euler equations

Hello

The nonunique solutions of the Euler equations were closely investigated by Prof. Jameson in AIAA Paper 91-1625.

First you should try to find this report.

Best Regards.

T. Matsuzawa

 Selina Tracy March 27, 2001 02:21

Re: Multiple solutions for Euler equations

Interesting. I heard about the bifurcation but could you explain of the mean 'multiple solutions' for airfoil? What factor decides the flow? Just briefly.

Selina.

 kalyan March 27, 2001 13:28

Re: Multiple solutions for Euler equations

You get a unique solution for lift around airfoils only if they have a sharp trailing edge and the Kutta condition is imposed there. If there are no sharp edges, it is not clear where the Kutta condition should be imposed. A good example is an ellipse at an angle of attack. So the lift prediction can not be made using invicid flow computations. The reason for this is quite simple (though it is often not explained along side the Kutta condition).

Inviscid flows can not produce vorticity in a flow that is initially irrotational except in the presence of baroclinic torque. Baroclinic torque can often be small or absolutely zero if the fluid has constant density. In that case, Euler equations can not produce any body force (lift or drag). If you solved the Euler equations around an ellipse, you would end up with zero lift (the flow pattern would look similar to a Hele-Shaw flow). For airfoils with sharp leading edges, you impose the Kutta condition the justification for which is the fact that no real fluid (i.e., viscous fluid) can fully turn around a sharp corner. This constraint generates vorticity around the airfoil and the lift. The condition itself has nothing to do with Euler equations. It is a physical observation based on real fluid behavior.

You would however end with non-zero drag if you the flow around ellipse shaped airfoils numerically. The trailing edge separation point would depend on the extent of truncation error (artifial dissipation) and so would the lift. So, two different Euler schemes would give two different lift coefficients.

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