Chapter 4
Fluid Kinematics: Steady and unsteady flow, laminar
and turbulent flow, uniform and non-uniform flow.
Path-line, streamlines and stream tubes. Velocity and
discharge. Control volume, Equation of continuity for
compressible and incompressible fluids.
Dr. Muhammad Ashraf Javid
Assistant Professor
1
Fluid Engineering Mechanics
Fluid Kinematics
2
Branch of fluid mechanics which deals with response of fluids in motion without considering forces and energies in them.
The study of kinematics is often referred to as the geometry of motion.
Car surface pressure contours
and streamlines
Flow around cylindrical object
Fluid Flow
3
Rate of flow: Quantity of fluid passing through any
section in a unit time.
Type:
1. Volume flow rate:
2. Mass flow rate
3. Weigh flow rate
time
fluid ofQuantity flow of Rate
time
fluid of volume
time
fluid of mass
time
fluid ofweight
Fluid Flow
4
Let’s consider a pipe in which a fluid is flowing with mean velocity, V (=L/t).
Let, in unit time, t, volume of fluid (AL) passes through section X-X,
1. Volume flow rate:
2. Mass flow rate:
3. Weigh flow rate:
V
L
A
Longitudinal Section Cross Section
AVt
ALQ
time
fluid of volume
QAV
t
ALM
time
fluid of mass
QAVG
t
AL
t
ALgG
time
fluid ofweight
Units
X
X
m3/sec
Kg/sec
N/sec
Reynolds Number (NO.)
5
It is the ratio between inertial forces and viscous
forces.
v
ieN
F
F
forceViscous
forceInertialRorR
1)(....... 2222
2
3 VLT
LL
T
LLavaMFi
2)()().(.2
2 VLT
LL
T
LL
LT
LA
dy
duAFv
VLVL
VL
VL
F
FRorR
v
ieN
22
L = characteristics length
L = D in case of pipe
L = R = hydraulic mean depth in case of open channel
It is used to differentiate between laminar and turbulent flow (nature of flow).
Types of Flow
6
Depending upon fluid properties
Ideal and Real flow
Incompressible and compressible flow
Depending upon properties of flow
Laminar and turbulent flows
Steady and unsteady flow
Uniform and Non-uniform flow
Rotational flow and Ir-rotational flow
Critical flow, Sub critical flow, and Super critical flow
Ideal and Real flow, and Velocity Profile
7
Real fluid flows implies friction effects. Ideal fluid flow is hypothetical;
it assumes no friction.
Velocity distribution of pipe flow
Compressible and incompressible flows
8
Incompressible fluid flows assumes the fluid have constant density
while in compressible fluid flows density is variable and becomes
function of temperature and pressure.
P1 P2
v1
v2
v2
P1
P2
v1
v2
Incompressible fluid Compressible fluid
Laminar and turbulent flow
9
The flow in laminations (layers) is
termed as laminar flow while the
case when fluid flow layers
intermix with each other is termed
as turbulent flow.
Transition of flow from Laminar to turbulent
Laminar flow
Turbulent flow
Laminar and turbulent flow
10
• Re ≤ 2000, flow is laminar in pipe and Re > 4000, flow in turbulent
• Re ≤ 500, flow is laminar in open channel, Re > 1000, flow is turbulent
• In-between of above values, the flow is in transition stage.
Steady and Unsteady flows
11
Steady flow: It is the flow in
which conditions of flow remains
constant w.r.t. time at a particular
section but the condition may be
different at different sections.
Flow conditions: velocity, pressure,
density or cross-sectional area etc.
e.g., constant discharge through a
pipe
Unsteady flow: It is the flow in
which conditions of flow changes
w.r.t. time at a particular section.
e.g., variable discharge through a
pipe
V
Longitudinal Section
X
X
conttVdt
d
dt
dp
dt
dv ;0,0,0
variable;0,0,0 Vdt
d
dt
dp
dt
dv
Uniform and Non-uniform flow
12
Uniform flow: It is the flow in which conditions of flow remains constant from section to section at any instant of time.
e.g., Constant discharge though a constant diameter pipe
Non-uniform flow: It is the flow in which conditions of flow does not remain constant from section to section.
e.g., Constant discharge through variable diameter pipe
V
Longitudinal Section
X’
X’
V
Longitudinal Section X’
conttVdt
d
dt
dp
dt
dv ;0,0,0
variable0,0,0 Vdt
d
dt
dp
dt
dv
X’
X
X
X
X
Describe flow condition
13
Constant discharge though non
variable diameter pipe
V
Longitudinal Section X
Steady flow !!
Non-uniform flow !!
X
Steady-non-uniform flow
13
Variable discharge though non
variable diameter pipe
V
Longitudinal Section X
Unsteady flow !!
Non-uniform flow !!
X
unsteady-non-uniform flow
Flow Combinations
14
Type Example
1. Steady Uniform flow Flow at constant rate through a duct
of uniform cross-section
2. Steady non-uniform flow
Flow at constant rate through a duct
of non-uniform cross-section (tapering
pipe)
3. Unsteady Uniform flow Flow at varying rates through a long
straight pipe of uniform cross-section.
4. Unsteady non-uniform flow Flow at varying rates through a duct
of non-uniform cross-section.
Other Flow Types
Rotational Flow In this case the fluid particles are moving around a curved path about a
fixed axis of rotation.
Ir-rotational Flow Fluid particles are not moving around a curved path about a fixed axis of
rotation.
Critical Flow Velocity of flow becomes equal to critical velocity.
Critical velocity is the velocity below which all the turbulence will be damaged out by the viscosity of fluid. V = Vcr , D = Dcr
Sub critical Flow V < Vcr and Depth of flow > critical depth of flow
Super critical Flow V > Vcr and Depth of flow < critical depth of flow
Froude Number and Flow States
16
The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow. The Froude number is a ratio of inertial and gravitational forces.
Gravity (numerator) - moves water downhill
Inertia (denominator) - reflects its willingness to do so.
Where:
V = Water velocity
D = Hydraulic depth (cross sectional area of flow / top width)
g = Gravity
When:
Fr = 1, critical flow,
Fr > 1, supercritical flow (fast rapid flow),
Fr < 1, subcritical flow (slow / tranquil flow)
gD
VFr
One, Two and Three Dimensional Flows
17
Although in general all fluids flow three-dimensionally, with
pressures and velocities and other flow properties varying in all
directions, in many cases the greatest changes only occur in two
directions or even only in one. In these cases changes in the other
direction can be effectively ignored making analysis much more
simple.
Flow is one dimensional if the flow parameters (such as velocity,
pressure, depth etc.) at a given instant in time only vary in the
direction of flow and not across the cross-section
Longitudinal section of rectangular channel Cross-section Velocity profile
Mean
velocity Water surface
One, Two and Three Dimensional Flows
18
Flow is two-dimensional if it can be
assumed that the flow parameters
vary in the direction of flow and in
one direction at right angles to this
direction
Flow is three-dimensional if the flow
parameters vary in all three directions
of flow
Two-dimensional flow over a weir
Three-dimensional flow in stilling basin
Visualization of flow Pattern
19
Streamlines around a wing shaped body Flow around a skiing athlete
Path line and stream line
20
Path-line: It is a trace made by
single particle over a period of time.
Streamline is a imaginary line
shows the mean direction of a
number of particles at the same
instance of time.
Character of Streamline
Streamlines can not cross each other.
Streamline can't be a folding line, but
a smooth curve.
Streamline cluster density reflects
the magnitude of velocity.
(Dense streamlines mean large velocity;
while sparse streamlines mean small
velocity.) Flow around cylindrical object
Mean Velocity and Discharge
21
Let’s consider a fluid flowing with mean velocity, V, in a pipe of uniform cross-
section, A,. Thus volume of fluid that passes through section XX in unit time ,
Δt, becomes;
Volume flow rate:
V
V Δt
A
Longitudinal Section Cross Section
AV
t
AtVQ
time
fluid of volume
X
X
AtVfluid of Volume
VAG
VAM
Similarly
Velocity
Distribution
Vmax
V=Vavg
Continuity
22
In a continuous flowing fluid, quantity of fluid flowing per unit
time is same for each section.
Assumptions:
Fluid is ideal (viscosity is zero).
Stream surface is ideal (frictionless).
Flow is steady.
Law of conservation of mass is applicable.
Continuity
23
Matter cannot be created or destroyed
- (it is simply changed in to a different
form of matter).
This principle is know as the
conservation of mass and we use it in the
analysis of flowing fluids.
The principle is applied to fixed
volumes, known as control volumes
shown in figure:
An arbitrarily shaped control volume.
For any control volume the principle of conservation of mass says
Mass entering per unit time - Mass leaving per unit time
= Increase of mass in the control volume per unit time
Continuity Equation
24
For steady flow there is no increase in the mass within the control
volume, so
Mass entering per unit time = Mass leaving per unit time
A stream tube
Derivation:
Lets consider a stream tube.
ρ1, v1 and A1 are mass density, velocity and cross-sectional area at section 1. Similarly, ρ2, v2 and A2 are mass density, velocity and cross-sectional area at section 2.
According to mass conservation
2222
1111
VAM
VAM
dt
MdVAVA
dt
MdMM
CV
CV
222111
21
Continuity Equation
25
For steady flow condition
Hence, for stead flow condition, mass flow rate at section 1= mass flow rate at section 2. i.e., mass flow rate is constant.
Similarly
Assuming incompressible fluid,
Therefore, according to mass conservation for steady flow of incompressible fluids volume flow rate remains same from section to section.
0/ dtMd CV
222111222111 0 VAVAVAVA
222111 VAVAM
222111 VgAVgAG
21
2211 VAVA 21 QQ 4321 QQQQ
Continuity Equation
26
For incompressible fluids:
Q=A1V1=A2v2=A3v3= constant
For compressible fluids:
M=ρ1A1v1=ρ2A2v2=ρ3A3v3= constant
or
W=γ1A1v1=γ2A2v2=γ3A3v3= constant
NUMERICALS
27
If the diameter at section 1 is d1 = 30mm and at section 2 d2 = 40mm and
the mean velocity at section 2 is u2 = 3.0 m/s, determine velocity at Section
1.
NUMERICALS
28
If pipe 1 diameter = 50mm, mean velocity 2m/s,
pipe 2 diameter 40mm takes 30% of total
discharge and pipe 3 diameter 60mm. What are
the values of discharge and mean velocity in
each pipe?
29
Fluid Dynamics: Hydrodynamics: Different forms of energy in a
flowing liquid, head, Bernoulli's equation and its
application, Energy line and Hydraulic Gradient Line,
and Energy Equation, Free and forced vortex, Forces
on pressure conduits, reducers and bends
Hydrodynamics
It is the study of liquids in motion
considering the forces and energies
exerted by or upon the liquid.
Forms of Energy
Liquid has three form of energy.
Kinetic Energy
If a liquid mass “m” is flowing with velocity “V” and all the liquid
particles are moving with same velocity then the K.E. = (1/2)mV2
Pressure energy: If a liquid exist under a pressure “p” then the Pr.E. of that liquid will
be.
Pr.E = p.dv
dv = small volume of liquid
V
Z
Q
Datum line
Forms of Energy
Potential Energy:
If a liquid particles of weight “w” is situated at a elevation “z” above datum line then P.E
P.E = WZ = mg z
The P.E of a particle of the liquid depends on its elevation above any arbitrary datum.
We are only interested in difference of elevation and therefore the location of datum is determined solely by consideration of convenience.
1
2
Z1
Z2
W Z1 W Z2
Types of head
p
W
dvpheadPressure
g
V
mg
Vm
W
VmheadKinetic
zW
ZWheadPotential
.
222
222
All the three heads have the unit of length (m).
Head: energy per unit weight is called as head. = E/W • For all three forms of energy we have three different head;
• Kinetic head, • potential head or elevation head, • pressure head.
• When elevation head is zero, the pressure head is also called as static head, otherwise static head is equal to pressure head plus elevation head.
Bernoulli's Theorem
For a perfect, frictionless,
incompressible liquid the total
head of the particles of liquid
moving in a continuous stream
remains constant.
It is another form of law of
conservation of energy.
P2
D
P1
Z1
Z2
Q
A
B
C
A’
B’ C’
D’
1
2
ds1
ds2
Assumptions
Fluid is ideal
Fluid is incompressible
Flow is steady
There is no loss of
energy due to friction
Bernoulli's Theorem
Consider a perfect, incompressible, liquid is flowing through a pipe.
Consider a controlled volume of liquid between section-1 and section-2.
Original position of the volume of liquid can be defined by position A, B, C, D.
Let V1 is velocity of flow, Z1 elevation, p1 pressure at section-1 respectively.
Similarly V2 is velocity of flow, Z2 elevation, p2 pressure at section-2
respectively.
Let us assume this volume displaces small distance ds1 at section-1 due to
pressure p1.
Then as the fluid is incompressible, so it will also move by a distance ds2 at
section-2.
Such that the new position of the liquid is defined as A`, B`, C`, D`.
Bernoulli's Theorem
)(2
)( 21
2
1
2
221 ZZWVVg
Wppdv
)1(.. EPinlossEKingainpressurebydonework
Wdvandppdvpressurebydonework )2()( 21
)3()(2
. 2
1
2
2 VVg
WEKinGain
)4()(. 21 ZZWEPinLoss
)5()(2
)( 21
2
1
2
221 ZZWVVg
Wpp
W
Put equation 2, 3 and 4 in equation 1
Bernoulli's Theorem
Dividing the previous equation (5) by W
22
2
2
2
11
1
22Z
p
g
V
g
VZ
p
)(2
1)(
121
2
1
2
221 ZZVVg
pp
21
2
1
2
221
22ZZ
g
V
g
Vpp
)6(22
2
2
22
1
2
11
p
g
VZ
p
g
VZ
Energy Equation
This equation (5) is applicable only for 1st three
assumptions.
Considering the last assumption of Bernoulli’s Theorem
hL is the loss of head due to friction.
The above equation is called as Energy Equation:
Lhp
g
VZ
p
g
VZ
2
2
22
1
2
11
22
Application of Bernoulli's Principle
Operation of an airplane wing
Design of Venturimeter •The shape of the wing forces the
air moving above the wing to move
faster than the air moving below
the wing.
•This cause the pressure above the
wing to be lower than the pressure
below the wing.
•The pressure difference pushes up
on the wing.
•This is called as lift.
Terminologies
Static head
Sum of pressure and elevation head.
Static head = Z + (p/γ)
If datum is selected in such a way that elevation of the fluid
particles is zero then static head will be equal to pressure
head only.
Terminologies
Hydraulic grade line (H.G.L)
It is the graphical representation of the static head in the form of a line above any datum along the flow path.
Static head = Z + (p/γ)
The height of piezometer tube indicates the static head.
Every ordinate of H.G.L. gives us the value of static head w.r.t. the relative datum.
If we install piezometer at different section then the line joining the piezometeric liquid level will be the H.G.L. that is why H.G.L. is also called as piezometer head line.
Q
H.G.L.
Terminologies
Energy Line
Total energy head
Total energy head = elevation head + pressure head + velocity head
If there is no head loss then the total energy head will be equal to total head.
Energy Grade Line (E.G.L.)
Graphical representation of the total energy head in the form of line above any datum along the flow path.
Every ordinate of E.L gives us the value of total energy head with respect to the datum.
If we install pitot tube at a section then the level above datum to which liquid will rise in the tube reflect the value of total energy head and the line joining these levels will be the energy line.
As liquid entering into tube will comes into rest and transfer all kinetic energy into potential energy.
p
Datum line
Z Z
p
g
V
2
2
Examples to H.G.L. and E.G.L.
p
g
V
2
2
Datum line Z Z
Datum line Z Z
•hL= o
•Velocity and
pressure head is
same
•hL≠ o
•Velocity head is
same and pressure
head will change
E.G.L.
H.G.L
hL
g
V
2
2
E.G.L.
H.G.L
p
Examples to H.G.L. and E.G.L.
•hL= o
•Velocity and
pressure head is
changing.
p
g
V
2
2
E.G.L
H.G.L
p
p
g
V
2
2
E.G.L
H.G.L
p
hL •hL≠ o
•Velocity and
pressure head is
changing.
Examples to H.G.L. and E.G.L.
Examples to H.G.L. and E.G.L.
Examples to H.G.L. and E.G.L.
Examples to H.G.L. and E.G.L.
Problem
49
5.2.3
Apply Bernoulli's Equation
Problem
50
Power
51
Rate of work done is termed as power
Power=Energy/time
Power=(Energy/weight)(weight/time)
If H is total head=total energy/weight and γQ is the weight flow rate
then above equation can be written as
Power=(H)(γQ)= γQH
In BG:
Power in (horsepower)=(H)(γQ)/550
In SI:
Power in (Kilowatts)=(H)(γQ)/1000
1 horsepower=550ft.lb/s
Problem
52
53
Free and Forced Vortex Flow Momentum and Forces in Fluid Flow
Free and Forced Vortex Flow
54
Vortex flow is defined as flow along curved path. Or It is a type of
flow in which the liquid particles are continuously moving
around a curved path about any axis of rotation.
It is of two types namely; (1). Free vortex flow and (2) forced
vortex flow
If the fluid particles are moving around a curved path with the help
of some external torque the flow is called forced vortex flow. And if
no external force is acquired to rotate the fluid particle, the flow is
called free vortex flow.
Forced Vortex Flow (Rotational Flow)
55
It is defined as that type of flow, in which some external torque is
required to rotate the fluid mass.
The fluid mass in this type of flow rotate at constant angular
velocity, ω. The tangential velocity, V, of any fluid particle is given by
V= ω r,
Where, r is radius of fluid particle from the axis of rotation
Examples of forced vortex flow are;
1. A vertical cylinder containing liquid which is rotated about its central axis
with a constant angular velocity ω,
2. Flow of liquid inside impeller of a centrifugal pump
3. Flow of water through runner
Forced Vortex Flow (Rotational Flow)
56
Free Vortex Flow (Irrotational flow)
57
When no external torque is required to rotate the fluid mass, that
type of flow is called free vortex flow.
Thus the liquid in case of free vortex flow is rotating due to the
rotation which is imparted to the fluid previously.
Example of free vortex flow are
1. Flow of liquid through a hole provided at the bottom of container
2. Flow of liquid around a circular bend in pipe
3. Flow of fluid in a centrifugal pump casing
Forced vortex
58
It is a type of flow in which
particles of liquid are forced to
move in a curved path about any
axis of rotation.
In this case fluid is subjected to
external torque.
We are only interested to profile
of water surface and level (y).
Consider a cylinder rotated at ω
(rad/sec). ω(rad/sec).
Forced vortex
59
The free surface of water will
not remain level but get a curve
shape as shown in the figure.
Consider a point “P” on the
free surface of forced vortex.
Let
W = weight of particle
F = centrifugal force
R = reaction of F and W
making “θ” with vertical.
Forced vortex
60
x = radius of the point with
respect to axis of rotation.
r = radius of cylinder
V = Tangential velocity
xVxrFor
rV
,
r
VmF
2
xg
W
x
x
g
WF 2
22
tan
sin
)2()1(
)2(
)1(sin
sin
2
2
2
g
x
CosR
R
W
xg
W
byEqdividing
CosRW
oFy
Rxg
W
RF
oFx
Forced vortex
61
)4(2
,
)3(2
xw.r.t.equationabovegintegratin
tan
22
22
2
g
rY
Yyrxwhen
g
xy
dx
dy
g
x
dx
dy
Equation (3) states that
free surface of liquid
subjected to forced
vortex will by paraboloid.
Height of paraboloid w.r.t. original surface
62
Height of paraboloid w.r.t. original surface
63
Consider “AA” level surface when
there is no rotation.
“BB” is the lowest level when
cylinder is subjected to rotation
and “CC” is maximum level of
paraboloid.
Volume of liquid between “BB”
and “CC” will remain same as the
volume between “AA” and “BB”
when there is no rotation.
Volume of liquid before rotation =
volume of liquid after rotation
)5(2
)2
(
2
22
222
Yh
YYrhr
YrYrhr
•Equation (5) states that
depression at the axis of
rotation (or at center) w.r.t
AA is same as elevation at
radius r.
Free vortex
64
It is a type of flow in
which particles of a
liquid move in a curve
path about any axis of
rotation without any
external force.
Consider a particle of
the liquid at the surface
of free vortex.
Free vortex
65
Let
m = mass of particle
v = tangential velocity
r = radius of circular path
Angular momentum of this particle is
L = M.r = mvr
As torque is defined as rate of change of angular momentum
In case of free vortex the product of tangential velocity and corresponding radius is constant.
2211
1
1
timew.r.titgintegratin
)(0
0
)(
rvrvC
vrm
C
mvrC
dt
mvrd
T
dt
mvrdT
Numerical: Forced vortex flow
66
Y
Y
Numerical: Forced Vortex flow
67
Y
Y
= h
= h
Numerical: Free Vortex Flow
68
Numerical: Free Vortex Flow
69
Momentum and Forces in Fluid Flow
70
We have all seen moving fluids exerting forces. The lift force on an aircraft
is exerted by the air moving over the wing. A jet of water from a hose
exerts a force on whatever it hits.
In fluid mechanics the analysis of motion is performed in the same way as in
solid mechanics - by use of Newton’s laws of motion.
i.e., F = ma, which is used in the analysis of solid mechanics to relate applied
force to acceleration.
In fluid mechanics it is not clear what mass of moving fluid we should use,
so, we use a different form of the equation.
dt
mdma sV
F
Momentum and Forces in Fluid Flow
71
Newton’s 2nd Law can be written:
The Rate of change of momentum of a body is equal to the resultant force acting
on the body, and takes place in the direction of the force.
The symbols F and V represent vectors and so the change in momentum must be
in the same direction as force.
It is also termed as impulse momentum principle
dt
md sVF
mV
F Sum of all external forces on a body of fluid or systems
Momentum of fluid body in direction s
smddt VF
Momentum and Forces in Fluid Flow
72
Let’s start by assuming that we
have steady flow which is non-uniform
flowing in a stream tube.
In time δt a volume of the fluid
moves from the inlet a distance u1 δt ,
so the volume entering the stream-
tube in the time δt is A stream-tube in three and two-dimensions
volume entering the stream tube = area x distance
mass entering stream tube = volume x density
momentum of fluid entering stream tube = mass x velocity
tuA 11
tuA 111
1111 utuA
momentum of fluid leaving stream tube 2222 utuA
Momentum and Forces in Fluid Flow
73
Now, according to Newton’s 2nd Law the force exerted by the fluid
is equal to the rate of change of momentum. So
Force=rate of change of momentum
We know from continuity of incompressible flow, ρ=ρ1= ρ2 &
Q=Q1=Q2
111222
11112222
1111222211112222
F
F
uQuQt
utuA
t
utuA
t
tuuA
t
tuuA
t
tuuAtuuA
1212 uumuuQF
This analysis assumed that the inlet and outlet velocities were in the
same direction - i.e. a one dimensional system. What happens when
this is not the case?
Momentum and Forces in Fluid Flow
74
Consider the two dimensional
system in the figure below:
At the inlet the velocity vector, u1 ,
makes an angle, θ1 , with the x-axis,
while at the outlet u2 make an
angle θ 2.
In this case we consider the forces
by resolving in the directions of the
co-ordinate axes.
The force in the x-direction
Two dimensional flow in a streamtube
Momentum and Forces in Fluid Flow
75
The force in the y-direction
The resultant force can be determined by combining Fx and Fy
vectorially as
And the angle at which F acts is given by
Momentum and Forces in Fluid Flow
76
For a three-dimensional (x, y, z) system we then have an extra force
to calculate and resolve in the z direction.
This is considered in exactly the same way.
In summary we can say:
The total force of the fluid = rate of change of momentum through the control volume