Laminar Pipe Flow: Displacement, and Momentum Thicknesses

[Disquoveri]

Hello. I'm new to the forums. I apologize if I have posted this in the wrong topic area.

I'm new to this hydrodynamics area and I'm trying to learn the concepts of boundary layer thicknesses. My question is about laminar pipe flow and calculating displacement and momentum thicknesses. I was wondering if finding the displacement and momentum thicknesses in a pipe flow is the same as finding them in flow over a flat-plate.

I found the displacement thickness over a flat plate to be:
$\delta_1 = \int_0^\infty (1-\frac{u\rho}{u_{\infty}\rho_{\infty}})dy$

And the momentum thickness is:
$\delta_2 = \int_0^\infty \frac{\rho u}{\rho_{\infty}u_{\infty}} (1-\frac{u\rho}{u_{\infty}\rho_{\infty}})dy$

Thank you.

 

[Chestermiller]

In the region where the thicknesses are small compared to the radius of the pipe, you can treat flow in the entrance region to a pipe as flow over a flat plate. However, as the thicknesses become greater, neglect of the curvature becomes less acceptable.

Chet

 

[cabon7969]

Hello Sir,
I just want to know further in this topic that
"Can we calculate displacement thickness for fully developed laminar flow in pipes"

 

[Chestermiller]

Hello Sir,
I just want to know further in this topic that
"Can we calculate displacement thickness for fully developed laminar flow in pipes"

For fully developed laminar flow in pipes, the boundary layer has penetrated to the center of the pipe, and the displacement thickness is R.

Chet

 

[cabon7969]

Hello Sir ,
I did not understand how the displacement thickness becomes R.
According to me, displacement thickness is the distance by which streamline get deflected due to effect of boundary layer and also it can be stated as imaginary increase in thickness of wall due to boundary layer.
But for the fully developed laminar flow this thickness will be constant, but i don't know what it will be ?

 

[boneh3ad]

For a fully-developed pipe flow, why do you feel the need to define a displacement thickness in the first place? Once the flow is fully-developed, the concept of a boundary layer is somewhat meaningless. There is already a well-defined, well-predicted velocity profile.