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February 23, 2005, 14:28 
3d parbolic inlet

#1 
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Does anyone have a good formulae for specifying 3D a parabolic (laminar) inlet velocity profile in terms of, say, x and y?


March 3, 2005, 08:14 
Re: 3d parbolic inlet

#2 
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ýf you can find please send me a copy


March 3, 2005, 08:37 
Re: 3d parbolic inlet

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Will do, but I don't think anyone knows one. I would have thought it would have been a commonly used thing.
I can't really afford the time to work my own out now, but I might do so later. 

March 3, 2005, 09:43 
Re: 3d parbolic inlet

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For what sort of cross section? If you mean pressure driven flow, then the solutions for two less common cross sections can be found in Panton, Incompressible Flow 2nd Ed, Ch 11: 1) For an ellipse with major axis a and minor axis b, defining K=(a/b)^2 and aligning the x axis with the major axis and y axis with minor axis, the equation of the wall is x^2+Ky^2=1. The steady parallel flow solution (w is dimensionless z component of velocity) is given by w=1/(2(1+K)*(1x^2Ky^2). Nondimensionalization is accomplished using the velocity scale (a^2/mu)*(dp/dx) and length scale a, where a is still the semimajor axis length, mu is the dynamic viscosity, and dp/dx it the pressure gradient. 2) For a recangular cross section the exact solution is found by separation of variables and expressed in terms of a series. In a rectangle with the origin at the center and the x and y axes parallel to sides of length 2a and 2ka respectively, then, nodimensionalizing x and y by a, and the z component of velocity as above, a "parabolic" formula must be of the form w=C(k^2y^2)(1x^2) to satisfy the no slip boundary condition. C will depend on k and on what feature of the flow you want to capture i.e. do you want the flow rate, the average wall shear stress, or the maximum velocity to match the exact solution? Unless there is a good reason not too, it would really make more sense to write your UDF using the series solution for the inlet velocity. It's too messy for me to write here clearly, but if I am on the right track with this answer and you don't have access to a source of the solution (Panton, above, is one, and the original publication is Rosenhead, Laminar Boundary Layers, 1963, p 136.) then I could send it to you.


March 4, 2005, 01:12 
Re: 3d parbolic inlet

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You can have the 3D parabolic inlet velocity profile as
u=0,v=0 and w=2U(1(x^2+y^2)/R^2) Where u, v and w are velocities in x, y and z directions respectively. R is the radius of the tube and U is the average velocity. 

March 4, 2005, 05:53 
Re: 3d parbolic inlet

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Thankyou Peter and Abdul for your comments. I sould have specified that I would like the profile for a rectangular cross section pipe. Idealy I would like to specify one parameter (I assume C) to specity the volumetric flow rate. I am trying to put the forumlea: w=C(k^2y^2)(1x^2) into Excel, but hot had much luck yet. I have access to Panton, but only the 1933 edition. I would appreciate any information you wish to send me.


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