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 Iain June 16, 2008 21:34

Transonic Far field - any advice

Hi Guys, I'm trying to incorporate non-reflecting far-field boundary conditions into my problem. I have very limited experience in this matter. I was hoping someone could shed some light on the issue?

I'm solving the low frequency transonic small disturbance equation and need to use non-reflecting bc's at the top and bottom boundaries of my computational domain. I have a reference that introduces some simple conditions; the far-field conditions at the top and bottom of the domain are stated as

phi_y +/- sqrt(B)*phi_x =0

phi is the disturbance potential, subscripts denote differentiation and B is a constant.

My problem is that I'm not sure where I should use the + or - sign at the top or bottom boundary. If anyone has some insight into this problem I'd greatly appreciate some advice. The reference I'm using is very clear on most of the details but seems to miss this key point of which sign to use at top & bottom boundaries.

many thanks, Iain

 Iain June 17, 2008 11:14

Solution

Hi Guys, I found a second reference that clears this matter up.

For those who are interested - the top BC should be phi_y + sqrt(B)*phi_x =0

and the bottom BC is phi_y - sqrt(B)*phi_x =0.

Figured I'd let you guys know in case anyone has similar problems in future. Iain

 Praveen. C June 17, 2008 16:43

Re: Solution

But why ?

 Iain June 17, 2008 18:45

Re: Solution

For the transonic small disturbance equation (low frequency) an oscillating airfoil generates disturbances that reach the far-field boundary.

Non-reflecting conditions stop some portion of these disturbances from being reflected back into the interior of the computational domain where they can degrade the quality of the result. Much of the literature features "perfecly reflecting" conditions - these send all the waves back into the domain.

The equations were originally derived by Engquist for the general case and specifically for the low frequency case by Kwak. Sorry I don't have the actual reference details with me at the moment. The details of the derivations are in the refs.

 Praveen. C June 18, 2008 15:42

Re: Solution

That still does not explain your boundary conditions choice.

If you read some book on gas dynamics, you will find that the small disturbance equation has two characteristics, each with constant slope. One of them goes out of the computational domain at top and the other one goes out from the bottom. Thats how the +/- sign in your boundary conditions is determined.

 Iain June 18, 2008 18:32

Re: Solution

I'm dealing with transonic, low frequency, small disturbance flow. The characteristics in this case are parabolic.

The derivation of the boundary conditions, in terms of characteristics, is given in Kwak's paper. The parabolic characteristics are also derived also in Cook & Cole's "Transonic Aerodynamics"; at the end of chapter 3.

The straight line characteristics would surely be useful for steady flows but the flow I'm dealing with is unsteady.

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