A fluid flowing through a pipe invariably enters the pipe at some locations.
The boundary layer on the walls quickly grows until it reaches the center line of the pipe
and the flow becomes fully developed.
The region over which the velocity profile of the fluid changes is called, the entrance
region.
Past the entrance length, the flow becomes fully developed meaning that the rate of change
of all the main quantities along the floor stream direction is zero, except for the pressure.
The length of the entrance region can vary significantly depending if the flow is laminar
or turbulent.
From the analysis of extensive data on laminar and turbulent flows through round pipes these
two empirical relations had been found to give a reasonable estimate of the entrance
length based on the Reynolds number.
Reported here in this plot we can see how the entrance length varies with Reynolds number
for the laminar and turbulent flows.
For large Reynolds numbers we can have entrance lengths that are several hundreds of pipe
diameters log.
Any relations that describe fully developed flows cannot be applied to analyze the flow
in the entrance region, so it's very important to know where the flow becomes fully developed.
Now let's consider the flow through two pipes.
They have the same length and diameter, and we want to analyze a laminar flow and a turbulent
flow passing through them.
Let’s see what goes on inside the pipes.
Shown here is the contour plot of the flow velocity magnitude on a vertical section of
the pipes.
The black lines show us the velocity vectors at different locations.
The length of the lines is proportional to the intensity of the velocity.
As you can see, the velocity profile for the laminar case and turbulent case are completely
different in shape due to the different characteristics of the two flows.
Let’s plot the centerline velocity along the ducts.
The laminar and the turbulent entrance length are roughly reported here.
The laminar case reaches fully developed condition much earlier than the turbulent case due to
the low Reynolds number.
We analyzed how to empirically determine the entrance region.
Now we're going to study the fully developed turbulent flows in pipes and channels.
The first case we introduce is the flow in a circular pipe of radius R with certain velocity
profile forming in it.
Recall that a laminar flow becomes unstable around Reynolds number of 2000 and becomes
fully turbulent for Reynolds number of 4000.
For laminar flows is possible to obtain an exact solution to this problem through the
Poiseuille solution.
However, for turbulent flows we cannot obtain an exact solution, but only approximate expressions,
that can give us a reasonable velocity profile.
For fully developed turbulent flows in pipes we can assume that the entire flow is affected
by viscosity, and that the velocity should follow the universal law of the wall.
Hence, we would have a laminar sublayer really close to the wall, then a transition to the
log layer and at specific profile near the centerline region.
Let's recall that in these expressions the V_star term represents the friction velocity,
and the Y coordinate represents the distance from the wall.
If we assume that the logarithmic distribution comprehends the largest part of the boundary
layer we can obtain an approximate estimate of the average pipe velocity.
This can be done integrating the log law along the pipe radius.
Blasius derived an empirical form of the friction factor, or pipe friction factor, based on
the average velocity.
Using this expression we can also derive this other three relations that show us that the
average velocity is essentially a skin friction relation.
We can then introduce the Darcy friction factor, that is equal to 4 times the pipe friction
factor, and rewrite the relation into this form, that was originally derived by Prandtl
in 1935, using the constants K and Beta equal to 0.41 and 5 respectively.
This expression is a good first approximation of the friction factor, however, the formula
does not take into account the viscous sublayer and the centerline region.
Prandtl improved this relation changing the constants so that the solution would be a
better fit with experimental pipe friction data.
This is valid for any Reynolds number larger than 4000.
Reported here is the variation of the friction factor with Reynolds number for laminar and
turbulent flows.
In the transition region of Reynolds number between two and four thousand we have bursts
of turbulence appearing in the flow and is a region to avoid when designing a pipe system.
Another approximation that can be used for a turbulent profile in circular pipes is the
power law relation, where the N parameter is a function of Reynolds number.
This is a reasonable approximation for practical turbulent pipe flows and a value of N equal
to 7 reflects a wide range of many practical flows.
Hence it is common to find this relation used in its one-7th power form.
This velocity profile has as some limitations, is not valid near the wall and near the center
line.
This because it provides an infinite gradient of velocity at the wall, and a non-zero gradient
at the center line.
The profile of a fully developed laminar flow and a fully developed turbulent flow are significantly
different.
The laminar profile is much more pointy compared to the turbulent one that presents a much
larger velocity near the wall, this because the turbulent flow has much more momentum
than the laminar flow.
Let's consider now a turbulent channel flow between two parallel plates.
If we take a look at the velocity contours and velocity vectors along the channel we
can see that it looks similar to what we have seen in the round pipe.
Indeed using a similar analysis we can obtain an expression for the Darcy friction factor
for the channel flow.
This expression is very close to the circular pipe expression we derived before and predicts
a friction factor about 4 to 7% higher than in the circular pipe.
This similarity in the expressions and results leads us to the unified expression for not
circular ducts.
Indeed, the Darcy friction correlation derived for the circular pipe can be extended for
analyzing non circular ducts, simply introducing the concept of effective diameter originally
proposed in 1976.
The expression uses the effective Reynolds number that is based on the effective diameter.
This parameter is calculated using the skin friction coefficient and the Reynolds number
calculated using the laminar theory.
This relation is purely an empirical correlation that does not have any theoretical background
and was only verified through experimental tests on ducts with different shapes.