Fully developed 3d velocity profile: Square inlet!
Hi
I need to simulate fully developed flow in a square channel, for which I need a UDF for obtaining the 3d velocity profile. The code itself is simple, however I'm unable to find a relation for velocity profile in a square channel. For a circular channel, it is simply the paraboloid U=2*Uavg*(1(r/R)^2) But for a square channel (laminar flow) I do not know what would be the right equation to use. Has anyone ever simulated this in a 3d square duct? I'd be really grateful if you could help me this. I also tried solving analytically using the Navier Stokes equation but kinda got stuck at the PDE: Uxx+Uyy=K Where U=0 at x=+/ a and at y= +/a (a = 0.5* square channel side) Would really appreciate a formula or code regarding this.. Thanks. 
Re: Fully developed 3d velocity profile: Square in
Check White, Viscous Fluid Flow, p 120

Re: Fully developed 3d velocity profile: Square in
Alternatively, just run a case where you let the flow fully develop, then record a profile of the fully developed profile and use that as a boundary condition for your primary case.

Re: Fully developed 3d velocity profile: Square in
Alternatively use periodic/cyclic boundaries to get fully developed flow. If your flow is compressible then unfortunatley there is no short cut. You will have to do what paolo suggested. If you do come up with an equation, please let me know I have been looking for one for long time.
Regards 
Re: Fully developed 3d velocity profile: Square in
Sorry I meant what William suggested.
Regards 
Re: Fully developed 3d velocity profile: Square in
Thank Paolo, William and cfd.newbie.
For the moment I tried using the profile for circular channels as an approximation at the inlet (and letting the corners be zero) and since I had an entrance length of about 2.5*Width where flow could develop, I've got fairly good results. I had previously observed that using a uniform velocity at inlet wasn't sufficient, at least with this entrance length, but it was computationally expensive to use a larger grid for a 3d simulation. Thanks William, the reference looks extremely useful, (also found it here: Threedimensional simulation of square jets in crossflow Sau et al (2004) PHYSICAL REVIEW E 69, 066302) though the equation is slightly complex. 
Fully developed flow in Square and Rectangular Duct
You can find this relation in "Advanced Engineering Fluid Mechanics" by Muralidhar and Biswas.
I 
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