Flow in Pipes 7

[aali94]

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I'm trying to understand how fluid friction factor varies with Reynolds number. I do not understand why there is a sudden increase in friction factor at transition and why it then decreases gradually as the flow becomes turbulent?

I have read that random fluctuations in velocity in turbulent flow tend to reduce the velocity gradient. Is the decreasing thickness of the viscous sub-layer resulting in a smaller velocity gradient at the pipe wall, the reason for the decrease in friction factor in turbulent flow. But, then why does it increase when the flow is transitional?

 

[katmar]

The friction losses in pipe flow are due to three entirely separate effects. These are skin friction, eddy formation, and form drag. They all vary with the flow rate. The friction factors we use (as per the Moody chart) are the sum of the three individual effects and at different flow rates you will find that one of them will be the dominant effect.

The skin friction is proportional to the shear stress at the wall of the pipe. The shear stress is in turn proportional to the velocity gradient in the boundary layer adjacent to the pipe wall. In general the velocity gradient at the wall increases as the flow rate (indicated by the Reynolds number) increases and therefore the shear stress increases as the flow rate increases. However, the friction factor is proportional to the shear stress divided by the square of the velocity. So, even though the shear stress increases with flow rate the squared velocity below the line results in the friction factor due to skin friction decreasing with increasing Reynolds number. Note that there are several friction factors in common use (Fanning, Moody etc) but they are simply multiples of each other and this reasoning applies to them all.

The second element that contributes to the friction factor is eddy formation. In turbulent flow eddies form both perpendicular to and parallel with the pipe axis. These additional flows, and the shear introduced between them, simply absorb energy and add to the friction factor.

The third source of friction loss is due to the eddies impacting on the protrusions from the walls (i.e. the roughness of the pipe). This is known as form drag. When the boundary layer is thicker than the protrusions the eddies are isolated from them and we get what is known as "hydraulically smooth flow". In turbulent flow the boundary layer gets thinner as the Reynolds number increases. To get "smooth flow" either the Reynolds number must be low (to give a thick boundary layer) or the pipe must be smooth so that the protrusions do not go beyond the boundary layer.

With all this theory out of the way, let me try to actually answer your question. In laminar flow (sometimes called streamline flow) there are no eddies and skin friction alone applies. Even though the shear stress at the wall increases with flow rate the effect of having the squared velocity in the denominator (as described above) results in the friction factor decreasing as the Reynolds number increases in the laminar flow regime. This decrease in the contribution of skin friction to the overall friction factor continues into the turbulent zone for the same reason.

For the zone where 2300 < Re < 4000 I prefer to use the Crane terminology and call it the "Critical Zone" rather than the "Transition Zone" that some authors call it. I like to reserve the "Transition Zone" terminology for the section of turbulent flow (Re > 4000) between smooth flow and fully developed turbulent flow - again following the Crane usage.

In the Critical Zone we suddenly get pockets of turbulence and eddy formation starting. This absorbs a relatively large amount of energy and the friction factor increases abruptly. This effect continues as the Reynolds number increases into the turbulent zone but has a progressively smaller impact on the friction factor because of the same influence of the squared velocity in the denominator as described for laminar flow.

In the Turbulent Zone the elements of the friction factor due to skin friction and eddy currents continue to decrease with increasing Reynolds number. But the element of friction factor due to the impact of the eddies on the protrusions increases as the boundary layer gets thinner and thinner. Eventually, for any given e/d (roughness to pipe diameter) ratio the boundary layer becomes so thin that any further increase in Reynolds number has no further effect on the impact between the eddies and the protrusions. At this point the contributions to the friction factor from the skin friction and eddy currents are negligible because of the velocity squared factor. But the form drag is a dissipation of kinetic energy and therefore proportional to velocity squared. Since the friction factor (due to form drag) has velocity squared in the numerator and denominator it remains constant for any further increase in Reynolds number.



Katmar Software - AioFlo Pipe Hydraulics

http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

 

[zdas04]

aali94,
You are unlikely to get a better explanation than Katmar gave you.

David Simpson, PE
MuleShoe Engineering

"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
The plural of anecdote is not "data"

 

[aali94]

Thank you, that was really helpful.