Senthil Seliyan Elango
Title: Fluid Dynamics
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INTRODUCTION
Real fluids
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 viscosity:

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
fluids.
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.
DESCRIPTION
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?
This

---

this

or this

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 laminar flow.

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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 this expression

where = density, u = mean velocity, d = diameter and = viscosity

---

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
zone.
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 non-dimensional number.
We can go through an example to discover at what velocity the flow in a pipe stops being
laminar.
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.

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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.
In summary:
Laminar flow
Re < 2000
'low' velocity
Dye does not mix with water
Fluid particles move in straight lines
Simple mathematical analysis possible
Rare in practice in water systems.

---

Transitional flow
2000 > Re < 4000
'medium' velocity
Dye stream wavers in water - mixes slightly.
Turbulent flow
Re > 4000
'high' velocity
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 used
Most common type of flow.
GENERAL ANALYSIS
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
the fluid.
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 written
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. Many experiments
have been done with various fluids measuring the pressure loss at various Reynolds
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
a pipe.

---

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 the centre

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
laminar.
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.
GENERAL RECOMMENDATION
Boundary Layers
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 the flow.

Looking at this two-dimensionally we get the above velocity profile from the wall to the
centre of the flow.

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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 = 0.99 umainstream
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
developed flow.
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
momentum equation.
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
increases.
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 types:
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 gradient become
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 the friction
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
layer.
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 high).
From earlier we saw how we could calculate whether a particular flow in a pipe is
laminar or turbulent using the Reynolds number.

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= 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
friction.)

---

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 thinner.
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 shown below.

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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.
Tee-Junctions

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.

---

Y-Junctions
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.
Bends

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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
here.
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 boundary layer
separation.
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
below.

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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 whistling of
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
bridge collapsed.
Aerofoil
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 dive.
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 boundary layer
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 separation take
place.
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. Pressure will
decrease on the top so the adverse pressure gradient which would cause the
boundary layer separation reduces.

3. Putting a flap on the end of the wing and tilting it before separation occurs
increases the velocity over the top of the wing, again reducing the pressure and
chance of separation occurring.

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ACTUALIZATION
Pipe Network Calculation
Applications
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 method.
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 single-valued. This
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 criteria.
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 temperatures typical
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 seconds units:

Darcy Weisbach equation (Mays, 1999; Munson et al., 1998; Streeter et al., 1998):

Where "log" is base 10 logarithm and "ln" is natural logarithm.
Pressure computation
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:

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Pj = S(Zi - Zj - Hpipe) + Pi where node j is down-gradient from node i. S = fluid weight
density [F/L3].
Minor Losses
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).
Example:
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 nodes. Usually,
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 required.
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

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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.
CONCLUSION
The discussion thus far has been rather general and has introduced many important ideas
and principles. Fluid flow behavior has been demonstrated. Numerous references to
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 two-dimensional
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
subsonic flow.

Reference:
Fluid Mechanics by Dr.Andrew Sleigh
Fluid Mechanics with engineering Application-
J.Franzini/E.Finnemore, McGraw ­Hill