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 S.S. July 27, 2003 05:00

fully developed velocity profile in 3D

I have a laminar, viscous 3D flow in a duct (brick type geometry) with rectangular cross-section. I want to use velocity inlet and pressure outlet boundary conditions. There is a nice expression for fully-developed 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.

 Jackie July 27, 2003 15:12

BC at the inlet of the tube

Hi all,

I've had a problem of specifying the boundary condition at the inlet of an opened-end 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

 xueying July 27, 2003 16:00

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).

 xueying July 27, 2003 16:05

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 $u_n(y) = \frac{a_n}{\lambda_n} \left( -1+\frac{sinh\sqrt{\lambda_n}y+sinh\sqrt{\lambda_n} (b-y)}{sinh\sqrt{\lambda_n}b}\right)$

$\phi_n(x) = \sqrt{\frac{2}{a}}sin(\frac{n \pi x}{a}), \;\; n=1,2,...$

where $a$ is the length in the $x$ direction, $b$ is the length in the $y$ direction, $\lambda_n = \frac{n^2\pi^2}{a^2}$, $a_n = \sqrt{\frac{2}{a}}\alpha \frac{a}{n\pi}[(-1)^n-1]$ and $\alpha = - \Delta p/(L\mu)$, $\Delta p$ is pressure drop in the $z$ direction with length $L$, and flow viscosity $\mu$.

 S.S. July 27, 2003 16:47

Re: fully developed velocity profile in 3D

Yes, but can it be expressed in terms of u_max (maximum velocity)? In 2-D, 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 y-coordinates. 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.

 Hall July 27, 2003 23:07

Re: fully developed velocity profile in 3D

Hi,

If u have a nice expression for fully-developed 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 3-D, 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 2-d velocity field to 3-d

 xueying July 29, 2003 12:02

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.

 mugurg July 6, 2013 15:36