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

[ksbiefr]

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.

 

[Chestermiller]

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

 

[boneh3ad]

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:
$$u_z = \dfrac{1}{4\mu}\dfrac{dp}{dz}(r^2 -R^2).$$

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.

 

[Chestermiller]

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:
$$u_z = \dfrac{1}{4\mu}\dfrac{dp}{dz}(r^2 -R^2).$$

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.

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

 

[PJC01]

Developing a parabolic velocity profile in 3D tube/channel flow requires understanding of the underlying fluid dynamics and the governing equations. In general, velocity profiles in 3D flow are more complex compared to 2D flow and cannot be simply described by a single equation.

To develop a parabolic velocity profile in 3D tube/channel flow, one approach is to use the Navier-Stokes equations, which describe the conservation of momentum for a fluid. These equations can be solved numerically using computational fluid dynamics (CFD) software.

Another approach is to use experimental techniques such as particle image velocimetry (PIV) to measure the velocity distribution in the 3D flow. This data can then be used to develop a mathematical model or a set of equations that can describe the velocity profile.

In either case, it is important to consider the boundary conditions, such as the geometry of the tube/channel, the flow rate, and the viscosity of the fluid. These parameters will affect the velocity profile and should be taken into account in the development of the equation or model.

In summary, developing a parabolic velocity profile in 3D tube/channel flow requires a thorough understanding of fluid dynamics and the use of advanced techniques such as CFD or experimental methods. It is not possible to provide a single equation or formula that can accurately describe the velocity profile in all cases.