fully developed velocity profile in 3D
I have a laminar, viscous 3D flow in a duct (brick type geometry) with rectangular crosssection. I want to use velocity inlet and pressure outlet boundary conditions. There is a nice expression for fullydeveloped velocity profile (parabola) in 2D (written in terms of u_max). How is it written in 3D? Any help or reference will be highly appreciated. Thanks in anticipation.

BC at the inlet of the tube
Hi all,
I've had a problem of specifying the boundary condition at the inlet of an openedend tube. I use CFD to simulate flow inside the tube having an embedded momentum source. The momentum source is located at the center of the horizontal tube and both ends (inlet and outlet) of the tube is opened to the atmosphere. I specify zero static pressure at the outlet (reference to ambient). The flow inside the tube is driven by the given momentum source. However, I don't know what the boundary condition at the inlet of the tube should be. Could anybody please give me some suggestion? Thank you very much Jackie 
Re: BC at the inlet of the tube
B.C. at inlet might be fully developed, i.e. \vec{n} \cdot \nabla \vec{v} = \vec{0} (in latex form).

Re: fully developed velocity profile in 3D
The fully developed flow velocity profile for a rectangular channel is a serial solution; following is the analytical solution written in latex form:
\[ u(x,y) = \sum_{n=1}^{\infty} u_n(y) \phi_n(x) \] where \[ u_n(y) = \frac{a_n}{\lambda_n} \left( 1+\frac{sinh\sqrt{\lambda_n}y+sinh\sqrt{\lambda_n} (by)}{sinh\sqrt{\lambda_n}b}\right) \] \[ \phi_n(x) = \sqrt{\frac{2}{a}}sin(\frac{n \pi x}{a}), \;\; n=1,2,... \] where $a$ is the length in the $x$ direction, $b$ is the length in the $y$ direction, $\lambda_n = \frac{n^2\pi^2}{a^2}$, $a_n = \sqrt{\frac{2}{a}}\alpha \frac{a}{n\pi}[(1)^n1] $ and $\alpha =  \Delta p/(L\mu)$, $\Delta p$ is pressure drop in the $z$ direction with length $L$, and flow viscosity $\mu$. 
Re: fully developed velocity profile in 3D
Yes, but can it be expressed in terms of u_max (maximum velocity)? In 2D, there is a nice relation for u(y) which is given in terms of u_max (maximum velocity which occurs at the centerline of the parabola), and the ycoordinates. This expression is very convenient to use as velocity inlet b.c. profile in CFD codes (provided that u_max is known). Is there anything like that in 3D?
Also, can someone give me a reference where the detailed derivation of this exact 3D solution can be found? Thanks. 
Re: fully developed velocity profile in 3D
Hi,
If u have a nice expression for fullydeveloped velocity profile (parabola) in 2D, I think it is easy to expand it to 3d, just let the profile rotate around the longitude axis let's say if x is the longitude axis, and at the inlet, u = U(y) is a function of y. Now at 3D, again you set the x' as longitude axis. At inlet, you will have coordinates like (y',z'), take r = sqrt(y'^2 + z'^2), and let u = U(r). Now, u expand the 2d velocity field to 3d 
Re: fully developed velocity profile in 3D
The detailed derivation of this exact 3D solution can be found in book titled as Mathematical Methods in Chemical Engineering; the authors are: Varma, A. & Morbidelli, M.; Oxford University Press, New York Oxford, page 494.

I know it has been decade since this thread was opened, but just in case somebody googles and finds this page, I wanted to help.
You can check the equations 335338 from Shah and London. Generating parabolic velocity profile for 3D flows might not be a good idea since it does not represent fully developed flow for especially high aspect ratio channels. So, better check the equations that I have written. Actually, I have written a MATLAB script that generates fully developed velocity profile for rectangular channels and uploaded it to MATLAB file exchange forums, but it is not approved yet. If somebody replies this thread, I can supply the link for the file, or send as an attachment. 
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