Exploring a Parabolic Velocity Profile: Basics & Calculations

[sifr]

I know it must relate to the velocity profile being a parabola shape and the max velocity being at the peak of the parabola -

I wanted to know whether there are actual calculations I can do to show this?

I only know the basics so as much details as possible would be great help

http://blogs.tlt.psu.edu/projects/accessibilitydemo/examples/VelocityProfileLaminar.png

 

[Chestermiller]

I know it must relate to the velocity profile being a parabola shape and the max velocity being at the peak of the parabola -

I wanted to know whether there are actual calculations I can do to show this?

I only know the basics so as much details as possible would be great help

http://blogs.tlt.psu.edu/projects/accessibilitydemo/examples/VelocityProfileLaminar.png

The volumetric throughput rate is equal to the velocity integrated over the cross section area of the pipe. The average velocity is equal to the volumetric throughput rate divided by the total cross sectional area. See Bird, Steward, and Lightfoot, Transport Phenomena.

A parabolic profile only applies to laminar flow. In turbulent flow, the factor is much less than 2.

 

[Aero51]

You are looking for the derivation of laminar flow through a pipe. Most undergraduate fluid mechanics books derive this velocity profile through the Navier-Stokes equations. My favorite is Granger's "Fluid Mechanics"; its a cheap dover book more detailed than many modern texts. Assuming its parabolic (that is to say, without justifying it) the sum of the velocities along the diameter of the pipe divide by the diameter will equal the average velocity. This is analgous to finding the vertex in high school precalx.

 

[Chestermiller]

You are looking for the derivation of laminar flow through a pipe. Most undergraduate fluid mechanics books derive this velocity profile through the Navier-Stokes equations. My favorite is Granger's "Fluid Mechanics"; its a cheap dover book more detailed than many modern texts. Assuming its parabolic (that is to say, without justifying it) the sum of the velocities along the diameter of the pipe divide by the diameter will equal the average velocity. This is analgous to finding the vertex in high school precalx.

The line in bold above is not correct. You have to weight the velocities in terms of the differential areas.

dQ = 2∏r v dr

You then divide, not by the total radius or the diameter, but by the total cross sectional area ∏R²

 

[Aero51]

Oh sorry i was thinking for a strictly 2d profile. I probably should have checked first.