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How to develop a Parabolic Velocity Profile in 3D tube/channel flow

  1. Sep 18, 2014 #1
    i am trying to develop a Parabolic Velocity Profile in 3D tube/channel flow. for the 2D case i use

    u = 1.5*Um *(1-(2y/H)^2)
    where

    Um= fluid velocity
    y = position of solid on "y" axis (x,y)
    H = width of channel
    The above equation is not possible to used for 3D case (x,y,z). i try to search for 3D case not i am not successful.

    Any body suggest me a equation/formula for developing Parabolic Velocity Profile in 3D tube/channel flow.
     
  2. jcsd
  3. Sep 18, 2014 #2
    Your post is a little confusing. By 2D are you referring to flow between parallel plates, and by 3D are you referring to axial flow in a duct of circular cross section?

    Chet
     
  4. Sep 18, 2014 #3

    boneh3ad

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    Gold Member

    Are you familiar with how the equation you used was derived? If so, you can easily derive the equation for the velocity profile in a circular tube for a steady, fully-developed, laminar flow with constant pressure gradient. It is still parabolic. The flow is called Poiseuille flow. I won't go through the derivation, but here is the velocity profile for a circular tube:
    [tex]u_z = \dfrac{1}{4\mu}\dfrac{dp}{dz}(r^2 -R^2).[/tex]

    Here, ##u_z## is the flow velocity, ##\mu## is the dynamic viscosity, ##r## is the distance from the centerline, ##R## is the inner radius of the tube, and ##dp/dz## is the pressure gradient through the tube.
     
  5. Sep 18, 2014 #4
    To expand on what boneh3ad has said, that axial velocity can also be expressed in the same form as your "2D" equation (in terms of the mean velocity) by writing:

    ##u_z=2u_m\left(1-(\frac{r}{R})^2\right)##

    Chet
     
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