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April 21, 1999, 17:06 
Far field conditions

#1 
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Hi,
I have two questions: 1. Does anyone have an opinion/experience on mapping an infinite domain onto a finite domain in order to invoke a far field condition? 2. Does anyone have experience with 2d stagnation flow as a test case and if so what is your opinion of it as a test case? Thanks, Anay LuketaHanlin 

April 23, 1999, 09:50 
Re: Far field conditions

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Hi. I will only consider point 1, since I am not quite sure what is the definition of a stagnation flow.
First I'll give you my opinion. You can handle the problem in several ways. You can make a change of variables for the infinite dimension. For example if x is supposed to go to infinity, you can define z as varying between 0 and 1 and then write simply x=1/(1+z). And you will work with z now where z=1/x  1. Then of course you have to change all your equations accordingly (derivatives, etc..). Another change of variable would be z=log(x) for example, etc... While this can be handled quite easily from the point of view of the equations and the algebra, it is more difficult from the point of view of the numerics. From my experience the simplest way to do this is to take a domain large enough so that at the outer boundary the solution is very close to the solution at infinity (far field condition). Then you might have to take a lot of grid points, and the spacing between them might be increasing. Then you solve the problem a first time. Then check the solution by solving the same problem but now you put the outer boundary a bit further away. In this manner you check that the location of the outer boundary is far enough and does not alter the solution (you should get the same solution). The important point here is to treat the outer boundary (the 'far' boundary) correctly, in that sens that no oscillations are emerging from it and waves propagating outwards are not reflected inwards. THe best way to do that is to implement the boundary conditions on the characteristics of the flow, or even to solve for the characteristics of the flow at the outer (far) boundary. I have solved in this manner twodimensional rotating compressible viscous flow, including external forcing and where the flow is known far from the center (cylindrical coordinates integrated in z), and it worked well. So I never needed to go to infinity, but just far enough. I hope you can do the same. PG. 

April 23, 1999, 14:02 
Re: Far field conditions

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Patrick,
The method you state I know has been used. But, why do you state that a transformation is more difficult from a numerics point of view? Thank you for your response. Anay 

April 23, 1999, 14:26 
Re: Far field conditions

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Hi Anay,
I stated that the transformation is more difficult from the point of view of the numerics, mainly because you completely change your equations and all you stability criteria can be affected (these are not anymore the NavierStokes equations). And in addition the transformation includes a singularity point (x goes to infinity when z goes to 1). Patrick. 

April 27, 1999, 20:27 
Re: Far field conditions

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I am investigating a jet discharging into an "open" atmosphere and therefore have the same problem. I have not come up with the solution, but I have just found a paper about setting up 'free boundary conditions' in a truncated domain.
The paper is T.C. Papanastasiou, N. Malamataris and K. Ellwood, 'A New Outflow Boundary Condition', Int. J. Numer. Methods Fluids, 14, 587608 (1992). It may be of use. James Hart 

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