TBA

1. KV
FLUID MECHANICS
{for energy conversion}

2. KV
Identify the unique vocabulary associated with fluid mechanics
with a focus on energy conservation
Explain the physical properties of fluids and derive the
conservation laws of mass and energy for an ideal fluid (i.e.
ignoring the viscous effects)
Recognise the various types of fluid flow problems
encountered in practice.
Understand the vapour pressure and its role in the occurrence
of cavitation.
Develop a comprehensive understanding of the effect of
viscosity on the motion of a fluid flowing around an immersed
body
Determine the forces acting on an immersed body arising from
the flow of fluid around it
OBJECTIVE
S

3. KV
BULK
PROPERTIES OF
FLUIDS
Viscosity - Force per unit area due to
internal friction (we will discuss this further in this lecture)
Density (ρ) - Mass per unit volume
𝜌 =
𝑚
𝑉
𝑃 =
𝐹
𝐴
Pressure (P) - Force per unit area in a fluid

4. KV
NO-SLIP
CONDITION
The image shows the evolution of a velocity profile gradient as a
result of the fluid sticking to the surface of a blunt nose. The layer
that sticks to the surface slows the adjacent fluid layer because of
viscous forces between the layers, which slows the next layer, and
so on. A consequence of the no-slip condition is that all velocity
profiles must have zero values with respect to the surface at the
points of contact between a fluid and a solid surface. Therefore the
no-slip condition is responsible for the development of the velocity
profile. (Cengel, 2008)
The development of a velocity profile
due to the no-slip conditions as a fluid
flows over a blunt nose.

5. KV
A fluid flowing over a stationery surface because
of the no-slip condition
Bell Profile
A fluid flowing over in a pipe between two surfaces - no-
slip condition at each surface interface
Flow separation during flow over a curved surface
Boundary layer: The flow region adjacent to
the wall in which the viscous effects (and
thus the velocity gradients) are significant.

6. KV
The flow of an originally uniform fluid
stream over a flat plate, and the regions
of viscous flow (next to the plate on both
sides) and inviscid flow (away from the
plate).
CLASSIFICATIO
N OF FLUID
FLOWS
Viscous vs Inviscid
regions of flow
Viscous flows: Flows in which the frictional effects are significant.
Inviscid flow regions: In many flows of practical interest, there are
regions (typically regions not close to solid surfaces) where viscous
forces are negligibly small compared to inertial or pressure forces.

7. KV
External flow over a tennis ball, and the
turbulent wake region behind
CLASSIFICATIO
N OF FLUID
FLOWS
Internal vs. External flow
External flow: The flow of an unbounded fluid over a surface such
as a plate, a wire, or a pipe.
Internal flow: The flow in a pipe or duct if the fluid is completely
bounded by solid surfaces.
• Water flow in a pipe is internal flow, and airflow over a
ball is external flow.
• The flow of liquids in a duct is called open-channel flow
if the duct is only partially filled with the liquid and there
is a free surface.

8. KV
Schlieren image of a small model of
the space shuttle orbiter being tested
at Mach 3 in the supersonic wind
tunnel of the Penn State Gas
Dynamics Lab. Several oblique
shocks are seen in the air
surrounding the spacecraft.
CLASSIFICATIO
N OF FLUID
FLOWS
Laminar vs Turbulent Flow
Incompressible flow: If the density of flowing fluid remains nearly
constant throughout (e.g., liquid flow).
Compressible flow: If the density of fluid changes during flow (e.g.,
high-speed gas flow)
When analysing rockets, spacecraft, and other systems that
involve high-speed gas flows, the flow speed is often expressed
by Mach number
Ma = 1 Sonic flow
Ma < 1 Subsonic flow
Ma > 1 Supersonic flow
Ma >> 1 Hypersonic flow
𝑀𝑎 =
𝑉
𝑐
=
𝑆𝑝𝑒𝑒𝑑𝑜𝑓𝑓𝑙𝑜𝑤
𝑆𝑝𝑒𝑒𝑑𝑜𝑓𝑠𝑜𝑢𝑛𝑑

9. KV
CLASSIFICATIO
N OF FLUID
FLOWS
Laminar flow: The highly ordered fluid motion characterised by
smooth layers of fluid. The flow of high-viscosity fluids such as
oils at low velocities is typically laminar.
Turbulent flow: The highly disordered fluid motion that typically
occurs at high velocities and is characterised by velocity
fluctuations. The flow of low-viscosity fluids such as air at high
velocities is typically turbulent.
Transitional flow: A flow that alternates between being laminar
and turbulent.
Laminar vs Turbulent Flow
Laminar, transitional, and turbulent
flows.

10. KV
CLASSIFICATIO
N OF FLUID
FLOWS
Forced flow: A fluid is forced to flow over a surface or in a pipe by
external means such as a pump or a fan.
Natural flow: Fluid motion is due to natural means such as the
buoyancy effect, which manifests itself as the rise of warmer (and
thus lighter) fluid and the fall of cooler (and thus denser) fluid.
Natural (or Unforced) vs
Forced Flow
In this schlieren image of a girl in a
swimming suit, the rise of lighter,
warmer air adjacent to her body
indicates that humans and warm-
blooded animals are surrounded by
thermal plumes of rising warm air.

11. KV
CLASSIFICATIO
N OF FLUID
FLOWS
• The term steady implies no change at a point with time.
• The opposite of steady is unsteady.
• The term uniform implies no change with location over a specified region.
• The term periodic refers to the kind of unsteady flow in which the flow
oscillates about a steady mean.
• Many devices such as turbines, compressors, boilers, condensers, and
heat exchangers operate for long periods of time under the same
conditions, and they are classified as steady-flow devices.
Steady vs Unsteady Flow
Oscillating wake of a blunt-based airfoil at Mach
number 0.6. Photo (a) is an instantaneous image, while
photo (b) is a long-exposure (time-averaged) image.

12. KV
CLASSIFICATIO
N OF FLUID
FLOWS
• A flow field is best characterised by its velocity distribution.
• A flow is said to be one-, two-, or three-dimensional if the flow
velocity varies in one, two, or three dimensions, respectively.
• However, the variation of velocity in certain directions can be small
relative to the variation in other directions and can be ignored.
One-, Two-, and Three-Dimensional Flows
Flow over a car antenna is
approximately two-dimensional
except near the top and bottom of
the antenna.

13. KV
CLASSIFICATIO
N OF FLUID
FLOWS
One-, Two-, and Three-Dimensional Flows
The development of the velocity profile in a circular pipe. V = V(r, z) and thus the flow is two-
dimensional in the entrance region, and becomes one-dimensional downstream when the
velocity profile fully develops and remains unchanged in the flow direction, V = V(r).

14. KV
STREAMLINES
& STREAM-
TUBES

15. KV
Fluid stream
velocity (u)
Stream tube
streamlin
es
MASS
CONTINUITY
𝑚
·
= 𝜌 × 𝑈 × 𝐴
for continuity
𝐴1𝑈1 = 𝐴2𝑈2 = 𝐴3𝑈3. . 𝐴∞𝑈∞

16. KEITH VAUGH
MASS & VOLUME
FLOW RATES
Mass flow rate
The average velocity Vavg is defined as the
average speed through a cross section.
Definition of average
velocity
Volumetric flow rate
The volume flow rate is the
volume of fluid flowing through
a cross section per unit time.
Recall Lecture 8 - Semester 1 - Slide 4

17. KV
An incompressible ideal fluid flows at a speed of 2 ms-1 through a
1.2 m2 sectional area in which a constriction of 0.25 m2 sectional
area has been inserted. What is the speed of the fluid inside the
constriction?
Putting ρ1 = ρ2, we have U1A1 = U2A2
EXAMPLE 1
𝑈2 = 𝑈1
𝐴1
𝐴2
= 2𝑚/𝑠
1.2𝑚2
0.25𝑚2
= 9.6𝑚/𝑠

18. KV
In many practical situations, viscous forces are
much smaller that those due to gravity and
pressure gradients over large regions of the flow
field. We can ignore viscosity to a good
approximation and derive an equation for energy
conservation in a fluid known as Bernoulli’s
equation. For steady flow, Bernoulli’s equation is
of the form;
For a stationary fluid, u = 0
ENERGY
CONSERVATION
IDEAL FLUID
𝑃
𝜌
+ 𝑔𝑧 +
1
2
𝑈2
= 𝑐𝑜𝑛𝑠𝑡
𝑃
𝜌
+ 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡 Hydrostatic Pressure

19. KV
BERNOULLI’s
EQUATION FOR
STEADY FLOW
(Andrews & Jelley 2007)

20. KV
Work done in pushing elemental mass
δm a small distance δs
Similarly at P2
The net work done is
By energy conservation, this is equal to
the increase in Potential Energy plus the
increase in Kinetic Energy, therefore;
(Andrews & Jelley 2007)

21. KV
and kinetic energy (KE) of liquid per unit mass
total energy of liquid per unit mass z
ENERGY OF A
MOVING LIQUID
𝑔
𝑃
𝜌𝑔
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡
Hydrostatic Pressure:
𝐸 = 𝑔
𝑃
𝜌𝑔
+
𝑈2
2𝑔
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡
𝐾𝐸 =
𝑈2
𝜌𝑔

22. KV
No losses between stations ① and ②
Losses between stations ① and ②
Energy additions or extractions
(2)
①
②
z1
z2
p1
p2
u1
u2
𝑧1 +
𝑃1
𝜌𝑔
+
𝑈1
2
2𝑔
= 𝑧2 +
𝑃2
𝜌𝑔
+
𝑈2
2
2𝑔
𝑧1 +
𝑃1
𝜌𝑔
+
𝑈1
2
2𝑔
= 𝑧2 +
𝑃2
𝜌𝑔
+
𝑈2
2
2𝑔
+ ℎ𝐿
𝑧1 +
𝑃1
𝜌𝑔
+
𝑈1
2
2𝑔
+
𝑊𝑖𝑛
𝑔
= 𝑧2 +
𝑃2
𝜌𝑔
+
𝑈2
2
2𝑔
+
𝑊𝑜𝑢𝑡
𝑔
+ ℎ𝐿 (3)
(1)

23. KV
(a) The atmospheric pressure on a surface is 105 N-m-2. If it is
assumed that the water is stationary, what is the pressure at a
depth of 10 m?
(assume ρwater = 103 kgm-3 & g = 9.81 ms-2)
(b) What is the significance of Bernoulli’s equation?
(c) Assuming the pressure of stationary air is 105 Nm-2, calculate
the percentage change due to a wind of 20 ms-1.
(assume ρair ≈ 1.2 kgm-3)
EXAMPLE 2

24. KV
A Pitot tube is a device for measuring the velocity in a fluid.
Essentially, it consists of two tubes, (a) and (b). Each tube has one
end open to the fluid and one end connected to a pressure gauge.
Tube (a) has the open end facing the flow and tube (b) has the
open end normal to the flow. In the case when the gauge (a) reads
p = po+½pU2 and gauge (b) reads p = po, derive an expression for
the velocity of the fluid in terms of the difference in pressure
between the gauges.
EXAMPLE 3

25. KV
Applying Bernoulli’s equation
Bernoulli’s equation
Rearranging
(Andrews & Jelley 2007)
𝑃
𝜌
+ 𝑔𝑧 +
1
2
𝑈2
= 𝑐𝑜𝑛𝑠𝑡
𝑃𝑜
𝜌
+
1
2
𝑈2
=
𝑃𝑠
𝜌
𝑈 =
2 𝑃𝑆 − 𝑃𝑜
𝜌
1
2

26. KV
The velocity in a flow stream varies considerably
over the cross-section. A pitot tube only measures
at one particular point, therefore, in order to
determine the velocity distribution over the entire
cross section a number of readings are required.
VELOCITY
DISTRIBUTION &
FLOW RATE
(Bacon, D., Stephens, R. 1990)

27. KV
The flow rate can be determine;
If the total cross-sectional area if the section is A;
The velocity profile for a circular pipe is the same across any diameter
where ur is the velocity at radius r
The velocity at the boundary of any duct or pipe is zero.

28. KV
In a Venturi meter an ideal fluid flows with a volume flow rate and a
pressure p1 through a horizontal pipe of cross sectional area A1. A
constriction of cross-sectional area A2 is inserted in the pipe and
the pressure is p2 inside the constriction. Derive an expression for
the volume flow rate in terms of p1, p2, A1 and A2.
EXAMPLE 4

29. KV
From Bernoulli’s equation
Also by mass continuity
Eliminating u2 we obtain the volume flow rate as
(Andrews & Jelley 2007)

30. KV
Applying Bernoulli’s equation to stations ① and ②
Mass continuity
ORIFICE
PLATE
(Bacon, D., Stephens, R. 1990)

31. KV
• Saturation temperature Tsat: The temperature at which a pure
substance changes phase at a given pressure.
• Saturation pressure Psat: The pressure at which a pure substance
changes phase at a given temperature.
• Vapor pressure (Pv): The pressure exerted by its vapour in phase
equilibrium with its liquid at a given temperature. It is identical to the
saturation pressure Psat of the liquid (Pv = Psat).
• Partial pressure: The pressure of a gas or vapor in a mixture with
other gases. For example, atmospheric air is a mixture of dry air
and water vapour, and atmospheric pressure is the sum of the
partial pressure of dry air and the partial pressure of water vapour.
VAPOUR
PRESSURE AND
CAVITATION

32. KV
• There is a possibility of the liquid pressure in liquid-flow
systems dropping below the vapour pressure at some
locations, and the resulting unplanned vaporisation.
• The vapour bubbles (called cavitation bubbles since they
form “cavities” in the liquid) collapse as they are swept
away from the low-pressure regions, generating highly
destructive, extremely high-pressure waves.
• This phenomenon, which is a common cause for drop in
performance and even the erosion of impeller blades, is
called cavitation, and it is an important consideration in
the design of hydraulic turbines and pumps.
Cavitation damage on a 16-mm by 23-mm
aluminium sample tested at 60 m/s for 2.5 h.
The sample was located at the cavity collapse
region downstream of a cavity generator
specifically designed to produce high damage
potential.

33. KV
Viscosity: A property that represents the
internal resistance of a fluid to motion or the
“fluidity”.
Drag force: The force a flowing fluid exerts on a
body in the flow direction. The magnitude of this
force depends, in part, on viscosity
VISCOSITY
A fluid moving relative to a body exerts a drag
force on the body, partly because of friction
caused by viscosity.
The motion of a viscous fluid is more complicated
than that of an inviscid fluid.

34. KV
Fluids for which the rate of deformation is proportional
to the shear stress are called Newtonian fluids.
DYNAMICS OF A
VISCOUS FLUID
The viscous shear force per unit
area in the fluid is proportional to
the velocity gradient
μ - coefficient of dynamic viscosity
kg/m  s or N  s/m2 or Pa  s
1 poise = 0.1 Pa  s
The behaviour of a fluid in laminar flow between two
parallel plates when the upper plate moves with a
constant velocity.
𝜏 ∝
𝑑(𝑑𝛽)
𝑑𝑡
𝑜𝑟𝜏 ∝
𝑑𝑢
𝑑𝑦
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
𝑁/𝑚2
𝐹 = 𝜏𝐴 = 𝜇𝐴
𝑑𝑢
𝑑𝑦
𝑁
Shear Stress
Shear Force

35. KV
The rate of deformation (velocity gradient) of a
Newtonian fluid is proportional to shear stress, and the
constant of proportionality is the viscosity.
Variation of shear stress with the rate of deformation
for Newtonian and non-Newtonian fluids (the slope of
a curve at a point is the apparent viscosity of the fluid
at that point).

36. KV
The viscosity of liquids decreases and the viscosity of gases increases
with temperature.
Kinematic Viscosity, 𝜈 = 𝜇/𝜌
m2/s or stoke (1 stoke = 1 cm2/s)

37. KV
L length of the cylinder
𝑛
·
number of revolutions per unit time
This equation can be used to calculate the viscosity of a fluid
by measuring torque at a specified angular velocity.
Therefore, two concentric cylinders can be used as a
viscometer, a device that measures viscosity.
𝑇 = 𝐹𝑅 = 𝜇
2𝜋𝑅3
𝜔𝐿
𝑙
= 𝜇
4𝜋2
𝑅3
𝑛
·
𝐿
𝑙

38. KV
Many methods have been devised for the measurement of fluid viscosity.
One method - the Saybolt viscometer uses a tank fitted with a small orifice. The time taken for a s standard
quantity of oil (60 mL) to flow may be used to determine the viscosity since the higher the viscosity of the fluid the
longer the time taken.
A second method us to enclose a thin flip of fluid between two concentric cylinders the inner one of which is held
by a torsion wire or spring. The outer cylinder is rotated at a given velocity and the torque on the inner cylinder
measured, form which the fluid viscosity may be determined since the great the torque the higher the viscosity.
A third mentor is to drop a ball of know size and weight a given distance through the fluid medium. The higher the
viscosity of the fluid the grease the time taken and hence the time taken may be used to calculate the viscosity.

39. KV
Laminar flow in a pipe Turbulent flow in a pipe
Typical flow regimes
(Andrews & Jelley 2007)

40. KV
Flow around a cylinder
for an inviscid fluid
Flow around a cylinder for a
viscous fluid
(Andrews & Jelley 2007)

41. KV
Lift and Drag
(Andrews & Jelley 2007)

42. KV
Variation of the lift and drag coefficients CL and
CD with angle of attack, (Andrews & Jelley 2007)

43. KV
Bulk properties of fluids
Streamlines & stream-tubes
Mass continuity
Energy conservation in an ideal fluid
Bernoulli’s equation for steady flow
Applied example’s (pitot tube, Venturi meter & questions
Dynamics of a viscous fluid
Flow regimes
Laminar and turbulent flow
Vortices
Lift & Drag

44. KV
Andrews, J., Jelley, N. 2007 Energy science: principles, technologies and impacts, Oxford
University Press
Bacon, D., Stephens, R. 1990 Mechanical Technology, second edition, Butterworth Heinemann
Boyle, G. 2004 Renewable Energy: Power for a sustainable future, second edition, Oxford
University Press
Çengel, Y., Turner, R., Cimbala, J. 2008 Fundamentals of thermal fluid sciences, Third edition,
McGraw Hill
Turns, S. 2006 Thermal fluid sciences: An integrated approach, Cambridge University Press
Twidell, J. and Weir, T. 2006 Renewable energy resources, second edition, Oxon: Taylor and
Francis
Young, D., Munson, B., Okiishi, T., Huebsch, W., 2011 Introduction to Fluid Mechanics, Fifth
edition, John Wiley & Sons, Inc.
Illustrations taken from Energy science: principles, technologies and impacts, Mechanical Technology and Fundamentals of Thermal Fluid Sciences