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.

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

B.C. at inlet might be fully developed, i.e. \vec{n} \cdot \nabla \vec{v} = \vec{0} (in latex form).

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 $\mu$.

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.

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

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]

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 335-338 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.

[vidyadhar]

Originally Posted by mugurg

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 335-338 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.

Hello Ugur,

I have seen this thread more than a decade after you have posted it.

I want to have fully developed velocity profile at the inlet of a duct with rectangular cross section.

I request you to help me in executing the same.

Thank you,
vidyadhar

[lcarasik]

Originally Posted by vidyadhar

Hello Ugur,

I have seen this thread more than a decade after you have posted it.

I want to have fully developed velocity profile at the inlet of a duct with rectangular cross section.

I request you to help me in executing the same.

Thank you,
vidyadhar

The matlab script in question: https://www.mathworks.com/matlabcent...tangular-ducts

[vidyadhar]

Originally Posted by lcarasik

The matlab script in question: https://www.mathworks.com/matlabcent...tangular-ducts

Dear Lane Carasik,

Thank you for the quick reply.

I have seen the code provided in the link.

May I request you to please let me know the equation used in the code.Also, what is um in the equation given. Though the reference mentioned in the code is a book by Shah and London, I am unable to access it.

Thanks & Regards,
Vidyadhar

[FMDenaro]

Originally Posted by vidyadhar

Dear Lane Carasik,

Thank you for the quick reply.

I have seen the code provided in the link.

May I request you to please let me know the equation used in the code.Also, what is um in the equation given. Though the reference mentioned in the code is a book by Shah and London, I am unable to access it.

Thanks & Regards,
Vidyadhar

the solution for a laminar steady developed flow in a rectangular duct satisfies the Poisson equation Lap u = Re*dp/dx with u=0 on the boundaries.

[vidyadhar]

Originally Posted by FMDenaro

the solution for a laminar steady developed flow in a rectangular duct satisfies the Poisson equation Lap u = Re*dp/dx with u=0 on the boundaries.

Hello Filippo Maria Denaro,

Thank you for the reply.

Since there are many analytical expressions available in various sources for the Poisson equation, I would like to know which form was used by the coder to develop the code.

I request you to please let me know the same.

Thanks & Regards,
vidyadhar

[FMDenaro]

https://books.google.it/books?hl=it&...20duct&f=false

[vidyadhar]

Originally Posted by FMDenaro

https://books.google.it/books?hl=it&...20duct&f=false

Thank you.

I will try to find the equation in the book, if it is not hidden.


Regards,
vidyadhar