Author: Senthil Seliyan Elango
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The flow of real fluids exhibits viscous effect, which are they tend
to "stick" to solid
surfaces and have stresses within their body.
You might remember from earlier in the course Newton's law of
This tells us that the shear stress, , in a fluid is proportional to
the velocity gradient - the
rate of change of velocity across the fluid path. For a "Newtonian"
fluid we can write:
where the constant of proportionality, is known as the coefficient
of viscosity (or
simply viscosity). We saw that for some fluids - sometimes known as
exotic fluids - the
value of changes with stress or velocity gradient. We shall only
deal with Newtonian
In his lecture we shall look at how the forces due to momentum
changes on the fluid and
viscous forces compare and what changes take place.
Laminar and turbulent flow
If we were to take a pipe of free flowing water and inject a dye
into the middle of the
stream, what would we expect to happen?
Actually both would happen - but for different flow rates. The top
occurs when the fluid
is flowing fast and the lower when it is flowing slowly.
The top situation is known as turbulent flow and the lower as
In laminar flow the motion of the particles of
fluid is very orderly with all particles
moving in straight lines parallel to the pipe walls.
But what is fast or slow? And at what speed does the flow pattern
change? And why
might we want to know this?
The phenomenon was first investigated in the 1880s by Osbourne
Reynolds in an
experiment which has become a classic in fluid mechanics.
He used a tank arranged as above with a pipe taking water from the
centre into which he
injected a dye through a needle. After many experiments he saw that
where = density, u = mean velocity, d = diameter and =
would help predict the change in flow type. If
the value is less than about 2000 then flow
is laminar, if greater than 4000 then turbulent and in between these
then in the transition
This value is known as the Reynolds number, Re:
Laminar flow: Re < 2000
Transitional flow: 2000 < Re < 4000
Turbulent flow: Re > 4000
What are the units of this Reynolds number? We can fill in the
equation with SI units:
i.e. it has no units. A quantity that has no units is
known as a non-dimensional (or
dimensionless) quantity. Thus the Reynolds number, Re, is a
We can go through an example to discover at what velocity the flow
in a pipe stops being
If the pipe and the fluid have the following properties:
water density = 1000 kg/m3
pipe diameter d = 0.5m
(dynamic) viscosity, = 0.55x103 Ns/m2
We want to know the maximum velocity when the Re is 2000.
If this were a pipe in a house central heating system, where the
pipe diameter is typically
0.015m, the limiting velocity for laminar flow would be, 0.0733 m/s.
Both of these are very slow. In practice it very rarely occurs in a
piped water system - the
velocities of flow are much greater. Laminar flow does occur in
situations with fluids of
greater viscosity - e.g. in bearing with oil as the lubricant.
At small values of Re above 2000 the flow exhibits small
instabilities. At values of about
4000 we can say that the flow is truly turbulent. Over the past 100
years since this
experiment, numerous more experiments have shown this phenomenon of
limits of Re for
many different Newtonian fluids - including gasses.
What does this abstract number mean?
We can say that the number has a physical meaning, by doing so it
helps to understand
some of the reasons for the changes from laminar to turbulent flow.
It can be interpreted that when the inertial forces dominate over
the viscous forces (when
the fluid is flowing faster and Re is larger) then the flow is
turbulent. When the viscous
forces are dominant (slow flow, low Re) they are sufficient enough
to keep all the fluid
particles in line, then the flow is laminar.
Re < 2000
Dye does not mix with water
Fluid particles move in straight lines
Simple mathematical analysis possible
Rare in practice in water systems.
2000 > Re < 4000
Dye stream wavers in water - mixes slightly.
Re > 4000
Dye mixes rapidly and completely
Particle paths completely irregular
Average motion is in the direction of the flow
Cannot be seen by the naked eye
Changes/fluctuations are very difficult to detect. Must use laser.
Mathematical analysis very difficult - so experimental measures are
Most common type of flow.
Pressure loss due to friction in a pipeline
Up to this point on the course we have considered ideal fluids where
there have been no
losses due to friction or any other factors. In reality, because
fluids are viscous, energy is
lost by flowing fluids due to friction which must be taken into
account. The effect of the
friction shows itself as a pressure (or head) loss.
In a pipe with a real fluid flowing, at the wall there is a shearing
stress retarding the flow,
as shown below.
If a manometer is attached as the pressure (head) difference due to
the energy lost by the
fluid overcoming the shear stress can be easily seen.
The pressure at 1 (upstream) is higher than the pressure at 2.
We can do some analysis to express this loss in pressure in terms of
the forces acting on
Consider a cylindrical element of incompressible fluid flowing in
the pipe, as shown
The pressure at the upstream end is p, and at the downstream
end the pressure has fallen
by p to (p-p).
The driving force due to pressure (F = Pressure x Area) can then be
driving force = Pressure force at 1 - pressure force at 2
The retarding force is that due to the shear stress by the walls
As the flow is in equilibrium,
driving force = retarding force
Giving an expression for pressure loss in a pipe in terms of the
pipe diameter and the
shear stress at the wall on the pipe.
The shear stress will vary with velocity of flow and hence with Re.
have been done with various fluids measuring the pressure loss at
numbers. These results plotted to show a graph of the relationship
between pressure loss
and Re look similar to the figure below:
This graph shows that the relationship between pressure loss and Re
can be expressed as
As these are empirical relationships, they help in determining the
pressure loss but not in
finding the magnitude of the shear stress at the wall w on a
particular fluid. If we knew
w we could then use it to give a general equation to predict
the pressure loss.
Pressure loss during laminar flow in a pipe
In general the shear stress w. is almost impossible to
measure. But for laminar flow it is
possible to calculate a theoretical value for a given velocity,
fluid and pipe dimension.
In laminar flow the paths of individual particles of fluid do not
cross, so the flow may be
considered as a series of concentric cylinders sliding over each
other - rather like the
cylinders of a collapsible pocket telescope.
As before, consider a cylinder of fluid, length L, radius r,
flowing steadily in the centre of
We are in equilibrium, so the shearing forces on the cylinder equal
the pressure forces.
By Newtons law of viscosity we have
, where y is the distance from the wall.
As we are measuring from the pipe centre then we change the sign and
replace y with r
distance from the centre, giving
Which can be combined with the equation above to give
In an integral form this gives an expression for velocity,
Integrating gives the value of velocity at a point distance r from
At r = 0, (the centre of the pipe), u = umax, at r
= R (the pipe wall) u = 0, giving
so, an expression for velocity at a point r from the pipe centre
when the flow is laminar is
Note how this is a parabolic profile (of the form y = ax2 + b ) so
the velocity profile in the
pipe looks similar to the figure below
What is the discharge in the pipe?
So the discharge can be written
This is the Hagen-Poiseuille equation for
laminar flow in a pipe. It expresses the
discharge Q in terms of the pressure gradient (
), diameter of the pipe and the
viscosity of the fluid.
We are interested in the pressure loss (head loss) and want to
relate this to the velocity of
the flow. Writing pressure loss in terms of head loss hf,
i.e. p = ghf
This shows that pressure loss is directly proportional to the
velocity when flow is
It has been validated many time by experiment.
It justifies two assumptions:
1. fluid does not slip past a solid boundary
2. Newton's hypothesis.
When a fluid flows over a stationary surface, e.g. the bed of a
river or the wall of a pipe,
the fluid touching the surface is brought to rest by the shear
stress o at the wall. The
velocity increases from the wall to a maximum in the main stream of
Looking at this two-dimensionally we get the above velocity profile
from the wall to the
centre of the flow.
This profile doesn't just exit, it must build
up gradually from the point where the fluid
starts to flow past the surface - e.g. when it enters a pipe.
If we consider a flat plate in the middle of a fluid, we will look
at the build up of the
velocity profile as the fluid moves over the plate.
Upstream the velocity profile is uniform, (free stream flow) a long
way downstream we
have the velocity profile we have talked about above. This is the
known as fully
developed flow. But how do we get to that state?
This region, where there is a velocity profile in the flow due to
the shear stress at the
wall, we call the boundary layer. The stages of the formation of the
boundary layer are
shown in the figure below:
We define the thickness of this boundary layer as the distance from
the wall to the point
where the velocity is 99% of the "free stream" velocity, the
velocity in the middle of the
pipe or river.
boundary layer thickness, = distance from wall to point where u =
The value of will increase with distance from the point where the
fluid first starts to
pass over the boundary - the flat plate in our example. It increases
to a maximum in fully
Correspondingly, the drag force D on the fluid due to shear stress
oat the wall increases
from zero at the start of the plate to a maximum in the fully
developed flow region where
it remains constant. We can calculate the
magnitude of the drag force by using the
Our interest in the boundary layer is that its presence greatly
affects the flow through or
round an object. So here we will examine some of the phenomena
associated with the
boundary layer and discuss why these occur.
Formation of the boundary layer
Above we noted that the boundary layer grows from zero when a fluid
starts to flow over
a solid surface. As is passes over a greater length more fluid is
slowed by friction
between the fluid layers close to the boundary. Hence the thickness
of the slower layer
The fluid near the top of the boundary layer is dragging the fluid
nearer to the solid
surface along. The mechanism for this dragging may be one of two
The first type occurs when the normal viscous forces (the forces
which hold the fluid
together) are large enough to exert drag effects on the slower
moving fluid close to the
solid boundary. If the boundary layer is thin then the velocity
gradient normal to the
surface, (du/dy), is large so by Newton's law of viscosity
the shear stress, = (du/dy), is
also large. The corresponding force may then be large enough to
exert drag on the fluid
close to the surface.
As the boundary layer thickness becomes greater, so the velocity
smaller and the shear stress decreases until it is no longer enough
to drag the slow fluid
near the surface along. If this viscous force was the only action
then the fluid would come
to a rest.
It, of course, does not come to rest but the second mechanism comes
into play. Up to this
point the flow has been laminar and Newton's law of viscosity has
applied. This part of
the boundary layer is known as the laminar boundary layer
The viscous shear stresses have held the fluid particles in a
constant motion within layers.
They become small as the boundary layer increases in thickness and
the velocity gradient
gets smaller. Eventually they are no longer able to hold the flow in
layers and the fluid
starts to rotate.
This causes the fluid motion to rapidly become turbulent. Fluid from
the fast moving
region moves to the slower zone transferring momentum and thus
maintaining the fluid
by the wall in motion. Conversely, slow moving fluid moves to the
faster moving region
slowing it down. The net effect is an increase in momentum in the
boundary layer. We
call the part of the boundary layer the turbulent boundary layer.
At points very close to the boundary the velocity gradients become
very large and the
velocity gradients become very large with the viscous shear forces
again becoming large
enough to maintain the fluid in laminar motion. This region is known
as the laminar sub-
layer. This layer occurs within the turbulent zone and is next to
the wall and very thin - a
few hundredths of a mm.
Surface roughness effect
Despite its thinness, the laminar sub-layer can play a vital role in
characteristics of the surface.
This is particularly relevant when defining pipe friction - as will
be seen in more detail in
the level 2 module. In turbulent flow if the height of the
roughness of a pipe is greater
than the thickness of the laminar sub-layer then this increases the
amount of turbulence
and energy losses in the flow. If the height of roughness is less
than the thickness of the
laminar sub-layer the pipe is said to be smooth and it has little
effect on the boundary
In laminar flow the height of roughness has very little effect
Boundary layers in pipes
As flow enters a pipe the boundary layer will initially be of the
laminar form. This will
change depending on the ration of inertial and viscous forces; i.e.
whether we have
laminar (viscous forces high) or turbulent flow (inertial forces
From earlier we saw how we could calculate whether a particular flow
in a pipe is
laminar or turbulent using the Reynolds number.
= density u = velocity = viscosity d = pipe diameter)
Laminar flow: Re < 2000
Transitional flow: 2000 < Re < 4000
Turbulent flow: Re > 4000
If we only have laminar flow the profile is parabolic - as proved in
earlier lectures - as
only the first part of the boundary layer growth diagram is used. So
we get the top
diagram in the above figure.
If turbulent (or transitional), both the laminar and the turbulent
(transitional) zones of the
boundary layer growth diagram are used. The growth of the velocity
profile is thus like
the bottom diagram in the above figure.
Once the boundary layer has reached the centre of the pipe the flow
is said to be fully
developed. (Note that at this point the whole of the fluid is now
affected by the boundary
The length of pipe before fully developed flow
is achieved is different for the two types
of flow. The length is known as the entry length.
Laminar flow entry length 120 diameter
Turbulent flow entry length 60 diameter
Boundary layer separation
Convergent flows: Negative pressure gradients
If flow over a boundary occurs when there is a pressure decrease in
the direction of flow,
the fluid will accelerate and the boundary layer will become
This is the case for convergent flows.
The accelerating fluid maintains the fluid close to the wall in
motion. Hence the flow
remains stable and turbulence reduces. Boundary layer separation
does not occur.
Divergent flows: Positive pressure gradients
When the pressure increases in the direction of flow the situation
is very different. Fluid
outside the boundary layer has enough momentum to overcome this
pressure which is
trying to push it backwards. The fluid within the boundary layer has
so little momentum
that it will very quickly be brought to rest, and possibly reversed
in direction. If this
reversal occurs it lifts the boundary layer away from the surface as
This phenomenon is known as boundary layer separation.
At the edge of the separated boundary layer, where the velocities
change direction, a line
of vortices occur (known as a vortex sheet). This happens because
fluid to either side is
moving in the opposite direction.
This boundary layer separation and increase in the turbulence
because of the vortices
results in very large energy losses in the flow.
These separating / divergent flows are inherently unstable and far
more energy is lost
than in parallel or convergent flow.
Examples of boundary layer separation :A
divergent duct or diffuser
The increasing area of flow causes a velocity drop (according to
continuity) and hence a
pressure rises (according to the Bernoulli equation).
Increasing the angle of the diffuser increases the probability of
boundary layer separation.
In a Venturi meter it has been found that an angle of about 6
provides the optimum
balance between length of meter and danger of boundary layer
separation which would
cause unacceptable pressure energy losses.
Assuming equal sized pipes, as fluid is removed, the velocities at 2
and 3 are smaller than
at 1, the entrance to the tee. Thus the pressure at 2 and 3 are
higher than at 1. These two
adverse pressure gradients can cause the two separations shown in
the diagram above.
Tee junctions are special cases of the Y-junction with similar
separation zones occurring.
See the diagram below.
Downstream, away from the junction, the boundary layer reattaches
and normal flow
occurs i.e. the effect of the boundary layer separation is only
local. Nevertheless fluid
downstream of the junction will have lost energy.
Two separation zones occur in bends as shown
above. The pressure at b must be greater
than at a as it must provide the required radial acceleration for
the fluid to get round the
bend. There is thus an adverse pressure gradient between a and b so
separation may occur
Pressure at c is less than at the entrance to the bend but pressure
at d has returned to near
the entrance value - again this adverse pressure gradient may cause
Flow past a cylinder
The pattern of flow around a cylinder varies with the velocity of
flow. If flow is very
slow with the Reynolds number ( v diameter/ less than 0.5, then
there is no separation
of the boundary layers as the pressure difference around the
cylinder is very small. The
pattern is something like that in the figure below.
If 2 < Re < 70 then the boundary layers separate symmetrically on
either side of the
cylinder. The ends of these separated zones remain attached to the
cylinder, as shown
Above a Re of 70 the ends of the separated
zones curl up into vortices and detach
alternately from each side forming a trail of vortices on the down
stream side of the
cylinder. This trial in known as a Karman vortex trail or street.
This vortex trail can easily
be seen in a river by looking over a bridge where there is a pier to
see the line of vortices
flowing away from the bridge. The phenomenon is responsible for the
hanging telephone or power cables. A more significant event was the
famous failure of
the Tacoma narrows bridge. Here the frequency of the alternate
vortex shedding matched
the natural frequency of the bridge deck and resonance amplified the
vibrations until the
Normal flow over a aerofoil (a wing cross-section) is shown in the
figure below with the
boundary layers greatly exaggerated.
The velocity increases as air it flows over the wing. The pressure
distribution is similar to
that shown below so transverse lift force occurs.
If the angle of the wing becomes too great and
boundary layer separation occurs on the
top of the aerofoil the pressure pattern will change dramatically.
This phenomenon is
known as stalling.
When stalling occurs, all, or most, of the 'suction' pressure is
lost, and the plane will
suddenly drop from the sky! The only solution to this is to put the
plane into a dive to
regain the boundary layer. A transverse lift force is then exerted
on the wing which gives
the pilot some control and allows the plane to be pulled out of the
Fortunately there are some mechanisms for preventing stalling. They
all rely on
preventing the boundary layer from separating in the first place.
1. Arranging the engine intakes so that they draw slow air from the
at the rear of the wing though small holes helps to keep the
boundary layer close
to the wing. Greater pressure gradients can be maintained before
2. Slower moving air on the upper surface can be increased in speed
by bringing air
from the high pressure area on the bottom of the wing through slots.
decrease on the top so the adverse pressure gradient which would
boundary layer separation reduces.
3. Putting a flap on the end of the wing and tilting it before
increases the velocity over the top of the wing, again reducing the
chance of separation occurring.
Pipe Network Calculation
Pipe Network simulates steady flow of liquids or gases under
pressure. It can simulate
city water systems, car exhaust manifolds, long pipelines with
different diameter pipes in
series, parallel pipes, groundwater flow into a slotted well screen,
soil vapor extraction
well design, and more. Enter flows at nodes as positive for inflows
and negative for
outflows. Inflows plus outflows must sum to 0. Enter one pressure in
the system and all
other pressures are computed. All fields must have a number, but the
number can be 0.
You do not need to use all the pipes or nodes. Enter a diameter of
0.0 if a pipe does not
exist. If a node is surrounded on all sides by non-existent pipes,
the node's flow must be
entered as 0.0. The program allows a wide variety of units. After
clicking Calculate, the
arrows "<--, -->, v, ^" indicate the direction of flow through each
pipe (to the left, right,
down, or up).
Losses can be computed by either the
Darcy-Weisbach or Hazen-Williams (HW) method,
selectable by clicking on the "Roughness, e" drop-down menu. If HW
is used, then the
fluid must be selected as "Water, 20C (68F)".
The H, V, Re output field is scrollable using the left and right
arrow keys on your
keyboard. Velocity is in m/s if metric units are selected for flow
rate Q, and ft/s if
English units are selected for Q.
Equations and Methodology
The pipe network calculation uses the steady state energy equation,
Darcy Weisbach or
Hazen Williams friction losses, and the Hardy Cross method to
determine the flow rate in
each pipe, loss in each pipe, and node pressures. Minor losses (due
to valves, pipe bends,
etc.) can be accounted for by using the equivalent length of pipe
Hardy Cross Method (Cross, 1936; Viessman and Hammer, 1993)
The Hardy Cross method is also known as the single path adjustment
method and is a
relaxation method. The flow rate in each pipe is adjusted
iteratively until all equations
are satisfied. The method is based on two primary physical laws:
1. The sum of pipe flows into and out of a node equals the flow
entering or leaving the
system through the node.
2. Hydraulic head (i.e. elevation head + pressure head, Z+P/S) is
means that the hydraulic head at a node is the same whether it is
computed from upstream
or downstream directions.
Pipe flows are adjusted iteratively using the following equation,
until the change in flow in each pipe is less than the convergence
n=2.0 for Darcy Weisbach losses or 1.85 for Hazen Williams losses.
Friction Losses, H
Our calculation gives you a choice of computing friction losses H
using the Darcy-
Weisbach (DW) or the Hazen-Williams (HW) method. The DW method can
be used for
any liquid or gas while the HW method can only be used for water at
of municipal water supply systems. HW losses can be selected with
the menu that says
"Roughness, e (m):". The following equations are used:
Hazen Williams's equation (Mays, 1999; Streeter et al., 1998;
Viessman and Hammer,
1993) where k=0.85 for meter and seconds units or 1.318 for feet and
Darcy Weisbach equation (Mays, 1999; Munson et al., 1998;
Streeter et al., 1998):
Where "log" is base 10 logarithm and "ln" is natural logarithm.
After computing flow rate Q in each pipe and loss H in each pipe and
using the input
node elevations Z and known pressure at one node, pressure P at each
node is computed
around the network:
Pj = S(Zi - Zj - Hpipe) + Pi where node
j is down-gradient from node i. S = fluid weight
Minor losses such as pipe elbows, bends, and valves may be included
by using the
equivalent length of pipe method (Mays, 1999). Equivalent length
(Leq) may be
computed using the following calculator which uses the formula
Leq=KD/f. f is the
Darcy-Weisbach friction factor for the pipe containing the fitting,
and cannot be known
with certainty until after the pipe network program is run.
However, since you need to
know f ahead of time, a reasonable value to use is f=0.02, which is
the default value. We
also recommend using f=0.02 even if you select Hazen-Williams losses
in the pipe
network calculation. K values are from Mays (1999).
For example, there is a 100-m long 10-cm diameter (inside diameter)
pipe with one fully
open gate valve and three regular 90o elbows. Using the minor loss
calculator, Leq is 1.0
m and 1.25 m for the fully open gate valve and each elbow,
respectively. The pipe length
you should enter into the pipe network calculator is 100 + 1.0 +
3(1.25) = 104.75 m. The
calculator allows a variety of units such as m, cm, inch, and ft for
diameter; and m, km,
ft, and miles for equivalent length. If a fitting is not listed,
select "User enters K" and
enter the K value for the fitting.
The pipe network calculation has many applications. Two examples
will be provided.
1. Municipal water supply system. A water tower is located at node
D. The other nodes
could represent industries or homes. Enter the water withdrawals at
all the nodes as
negative numbers, then enter the inflow to the network from the
water tower at node D as
a positive number equal to the sum of the withdrawals from the other
cities require a certain minimum pressure everywhere in the system,
often 40 psi. Use
the drop-down menu to select the node that you expect will have the
lowest pressure -
possibly the node furthest from D or the one at the highest
elevation; we'll use node I.
Enter the pressure at node I as 40 psi. Enter all the pipe lengths,
diameters and node
elevations. Then click "Calculate". You can use your right and left
arrow keys to scroll
to the left and right to see the velocity in each pipe. Typically,
you want pipe velocities
to be around 2 ft/s. If you are designing a system (as opposed to
analyzing a system that
is already in place), vary the pipe diameters until the pipe
velocities are reasonable and
pressure at node D is as low as possible to minimize the height of
the water tower. There
will be a trade-off between pressure at D and pipe diameters.
Smaller diameter pipes
will save money on pipes but will require a taller water tower. The
water tower height is
proportional to the pressure at D according to h=P/S, where P is the
pressure at D. S is
the weight density of the water, and h is the water tower height
2. Manifold. A manifold has multiple inflows at various positions
along the same
pipeline, and one outflow. Let node I be the outflow and use all
other nodes A-H as
inflow locations; so flow is from node A through pipes 1, 2, 5, 7,
6, 8, 11, and 12 and out
node I. Enter the diameters and lengths of
these pipes and the desired inflows at nodes
A-H. Enter the outflow at node I as a positive number equal to the
sum of the inflows at
nodes A-H. Enter the diameters of pipes 3, 4, 9, and 10 as 0.0 since
they are non-existent
pipes. Enter the elevations of all nodes. For a horizontal pipe, set
all the elevations to
the same value or just to 0.0 to keep it simple. From the drop-down
menu, select the
node where you know the pressure and enter its pressure. Clicking
"Calculate" will give
the flow rate in all pipes and the pressure at all the nodes.
The discussion thus far has been rather general and has introduced
many important ideas
and principles. Fluid flow behavior has been demonstrated. Numerous
airfoil or streamline shapes have been made. Viscous flow of the
boundary layer and
unsteady flow in the turbulent wake have been examined. The flow is
since velocity and other flow parameters vary normal to the
free-stream direction as well
as parallel to it. With these ideas in mind, one may now study
aircraft operating in a
Fluid Mechanics by Dr.Andrew Sleigh
Fluid Mechanics with engineering Application-
J.Franzini/E.Finnemore, McGraw Hill