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April 24, 2008, 13:48 
Airfoil conditions for unsteady flow

#1 
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Hi Guys, I was talking to a colleage today about a code I've been working on. I'm solving the transonic flow around a moving airfoil. I have taken the boundary condition on the airfoil to be that of flow tangency  my colleage informs me that in unsteady flow the flow is not really tangential to the moving airfoil surface.
Could someone please let me know why? I'm pretty confused over this issue. Dave 

April 24, 2008, 15:44 
Re: Airfoil conditions for unsteady flow

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A general motion of the airfoil would mean that the airfoil surface is moving but not necessarily in a direction parallel to itself. In general, there will be a component of motion of the surface perpendicular to itself. Thus, if you pick any element ds of the airfoil surface, at any given instant of time, it will be moving with a velocity vector vs = (vsx,vsy,vsz). This vs will in general not be perpendicular to the normal vector ns = (nsx,nsy,nsz) of that surface element at the same instant of time. Thus, the instantaneous velocity vs need not lie in the instantaneous tangent plane.
The condition that the fluid not penetrate the moving surface or separate from it implies that the fluid particle adjacent to that surface element ds and moving with a velocity vf = (vfx,vfy,vfz) must satisfy a certain condition on the velocity. That condition is most easily understood by viewing the motion in a frame F' which is moving at that instant with the velocity vs relative to the original frame F. In the frame F', the element ds appears stationary at the given time, and the fluid motion (i.e., its relative velocity) must be tangential to it so as not to penetrate it or separate from it. Thus, the condition on the fluid velocity is that, at that instant, the relative velocity vf' = (vf  vs) must lie in the tangent plane to ds and be perpendicular to ns. Alternatively, you can think of it in the original frame F as : the rate of motion of the fluid in the direction normal to the surface (vf .dot. ns) must equal the rate of motion of the surface element itself in the normal direction (vs .dot. ns). In particular, a noslip viscous condition would imply that vf = vs. 

April 24, 2008, 17:43 
Re: Airfoil conditions for unsteady flow

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Thanks Ananda your explanation helped clear things up. From what you've said I think I have formulated my boundary condition correctly but was sloppy in my language when referring to the condition as the "tangency condition".
I derived my boundary condition by assuming that a fluid element on the surface of the aifoil would remain on the surface until it leaves the trailing edge. This seems to agree with what you said but I understand now that this doesn't mean that the flow is tangential to the airfoil. Many thanks, Dave 

April 24, 2008, 18:11 
Re: Airfoil conditions for unsteady flow

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You are welcome. Yes, I believe you had correctly derived your boundary condition. My long explanation could have been summarized as "in a reference frame attached to the airfoil, the fluid slides along the airfoil surface", which means that the flow is tangential to the airfoil surface in the moving frame (which sees the relative flow), but not in the original frame (which sees the absolute flow). The airfoil surface attitude is the same in both frames.


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