Theoretical Pipe Flow: Understanding Poiseuille's Law

[ATKrank]

Hey guys, I am doing an internship and I have had some thoughts about fluid flow that have come up and I am having trouble fully grasping some concepts due to no one being able to thoroughly explain any answer that they might come up with.

So I have a crude understanding of some fluid dynamics already, but here is my dilemma and I would like any answers or clues to help me fully understand these principles.

According to Poiseuille's Law, the volumetric flow rate is a function of differential pressure, pipe radius, and fluid viscosity. However this is only applicable to laminar flow situations. So with this calculable flow rate, it is possible to evaluate/design large piping systems as long as the flow stays laminar.

What precisely makes Poiseuille's Law ineffective at calculating flow rates in turbulent flow even though the fluid is still incompressible? I understand that there are eddies and unpredictable flow patterns associated with turbulent flow. But the way I am thinking about it is that since the fluid is still incompressible, the differential pressure would still drive the same flow rate of the fluid. Basically saying that the flow rate should be independent of the flow pattern. Why is this wrong?

 

[Chestermiller]

The thing that determines the pressure drop is the shear stress at the wall. Although the Newtonian flow equations apply to turbulent flow as well as laminar flow, there are fluctuations in the velocity components in turbulent flow, and there is a fluctuating radial velocity component that is not present in laminar flow. The radial velocity fluctuations are somewhat coupled with the axial velocity oscillations. The momentum effect of this coupling translates into a radial transport of momentum, equivalent to a shear stress. So the average shear stress distribution in turbulent flow is different from that in laminar flow, and, more importantly, the shear stress at the wall is higher in turbulent flow than in laminar flow (for the same volumertic flow rate).

 

[ATKrank]

I think I get what you are saying. It helps for me to think of fluids as molecules in instances like this so I am going to re-cap it like that. I believe what you are saying is that if we look at a molecule of water in turbulent flow, we know that not all the velocity of the molecule is in the axial direction of the pipe. I am picturing the molecule carrying momentum into the wall and that is what is causing there to be a larger shear stress than it would with laminar flow.

So basically another way to say it is that in laminar flow, the shear stress in the pipe wall is purely caused by static pressure of the fluid since all the momentum is in the axial direction. While with turbulent flow the shear stress is caused by a summation of the static pressure, and a portion of dynamic pressure of the fluid since a portion of the momentum is not in the axial direction. Is this a correct analysis?

 

[Vedward]

I always like to use the example of 50 rabbits into the pipe mean 50 rabbits out.

 

[Chestermiller]

I think I get what you are saying. It helps for me to think of fluids as molecules in instances like this so I am going to re-cap it like that. I believe what you are saying is that if we look at a molecule of water in turbulent flow, we know that not all the velocity of the molecule is in the axial direction of the pipe. I am picturing the molecule carrying momentum into the wall and that is what is causing there to be a larger shear stress than it would with laminar flow.

So basically another way to say it is that in laminar flow, the shear stress in the pipe wall is purely caused by static pressure of the fluid since all the momentum is in the axial direction. While with turbulent flow the shear stress is caused by a summation of the static pressure, and a portion of dynamic pressure of the fluid since a portion of the momentum is not in the axial direction. Is this a correct analysis?

No. It is the coupling (correlation) between the radial and axial velocity fluctuations that give rise to the higher shear stress in turbulent flow. It has nothing to do with dynamic pressure. If u'(t) is the radial velocity fluctuation at time t in turbulent flow, and w'(t) is the axial velocity fluctuation at time t, then the time average of the product u'w' is not equal to zero. Multiplying this time average by the density gives the rate of axial momentum transfer radially (per unit volume), and is also the shear stress.

If you want to learn more about this, see Transport Phenomena by Bird, Stewart, and Lightfoot.