Download Fluid Dynamics

By C.K.Cheung
Fluid Dynamics
Fluid Flow
1/
Steady flow:
The fluid velocity, v, at a fixed point of a section is constant in time.
v
2/
Non-steady flow:
The fluid velocity at a fixed point depends on time.
3/
Streamline
Path taken by fluid particles.
V
streamline
V
1
By C.K.Cheung
V2
P2A2
Density = 
L2
V1
Y2
P1A1
L1
Y1
Reference level
Assume:
Nonviscous, steady, incompressible flow of fluid through the pipeline.
Consider the shaded portion:
1/
net W.D. on fluid = P1A1L1 – P2A2L2
2/
 (K.E.) =
1
1
mv22  mv12
2
2
3/  ( P.E. ) = mg(y2 – y1 )
Since, (1) = (2) + (3)
1
1
mv22  mv12 + mg(y2 – y1 )
2
2
 P1A1L1 – P2A2L2 =
For incompressible fluid:
m
A1L1 = A2L2 =

 P1(
m

) + mgy1 +
m
1
1
mv12 = P2( ) + mgy2 + mv 22
2
2

Multiply both side by (

):
m
2
By C.K.Cheung
 P1 +  gy1 +
1 2
1
v1 = P2 +  gy2 + v 22
2
2
Since, “1” & “2” refers to any two sections
 At any section along the pipeline:
P +  gy +
1 2
v = constant
2
( Bernoulli’s equation )
Note :
1/ P called static pressure
2/
1 2
v called the dynamic pressure
2
94’ MC
12.
B
0.4 m
A
The figure above shows part of a pipe having circular cross-sections.
The area of the
cross-section at B is double that at A and the centre of the cross-section at B is 0.4 m higher
than that at A. If an ideal liquid flows steadily through the pipe with speed 4 m/s at A, what is
the difference in static pressure between A and B? (Given: density of the liquid is 1200
kg/m³)
A.
B.
C.
D.
2 400 N/m²
4 200 N/m²
4 800 N/m²
7 200 N/m²
E.
12 000 N/m²
A
 AaVa = AbVb  Aa(4) = (2Aa)Vb  Vb = 2 ms-1
Also,
Pa + (1/2) (1200)(4)2 = Pb + (1200)(10)(0.4) + (1/2) (1200)(2)2
Static pressure difference = Pb - Pa = 2400 Nm-1
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By C.K.Cheung
Application of Bernoulli’s Equation
1/
Spinning ball (moving leftwards)
V+V '
V’
V
1
w
w=0
V-V '
2
V’
V
Consider points ‘ 1 ‘ & ‘ 2 ‘
 P +  gy +
1 2
v = constant
2
 Assume small ball  y ~ 0
P+
1 2
v = constant
2
Since V1 > V2  P1 < P2  net upward force.
2/
Air foil
V1
V1 > V2
V2
 P2 > P1  upward lift
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By C.K.Cheung
3/
Against the wind:
Force on sail
Fast moving air
Yacht
4/
Filter pump
Water from water- tape
air
Air from apparatus
(Water + air)
The high velocity of water at constriction produces a drop in air pressure  air flows in from
the side tube, and together with water is expelled through the lower part of the pump.
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By C.K.Cheung
5/
Bunsen burner
jet
air
gas
6
By C.K.Cheung
6/
Carburettor
Petrol
+ air
mixture
e
air
float
Petrol
level
A float mechanism maintain the petrol level just below the top of a fine jet. When the engine
is running, air is swept past the jet swiftly  drop in air pressure, the greater atmospheric pressure
on the petrol in the reservoir then forces petrol out of the jet in the form of fine spray. This occurs
when the accelerator pedal is pressed down.
7/
Spray
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By C.K.Cheung
Pitot – static tube
To measure gases flow speed.
1
2
Density of
P1
gas = ’
P2
h
Density of liquid = 
Assume the probe is small ( not to disturb the fluid flow )
Apply Bernoulli’s eqt., to “1 “
 P2 = P1 +
1
'v2
2
& “2”
(1)
Also:
P2 = P1 + gh
(2)
(1) = (2):

1
 ' v 2  gh
2
v=
2 gh
'
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By C.K.Cheung
More e.g. about Bernoulli’s principle:
1/
2/
The 2 boats will draw together & collide
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By C.K.Cheung
3/
Umbrella in windy day
4/
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By C.K.Cheung
4/
Wharfs are made with pilings that permit the free passage of water.
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By C.K.Cheung
e.g.
P1
P2
v
h
H
X=?
Given:
1/
A large open tank
2/
A small hole is punched at a depth h below water surface.
Find:
1) x = ?
2)
Could a hole be punched at another depth so that this second stream would have the same
range? If so, at what depth?
1)
 P1 + gh = P2 + (1/2)v2
P1 = P2
gh =
2/
(1/2)v2
v = (2gh)1/2  ~ free fall
Since (H-h) = (1/2)gt2  t = (2(H-h)/g)1/2
X = vt = (4h(H-h))1/2
For another hole at a depth h’
X’ = (4h’(H-h’))1/2
 h’ = h (reject)
or
h’ = (H-h)
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By C.K.Cheung
93’ MC
15.
fluid flow
speed v
h
The above figure shows a Pitot-static tube situated in a moving fluid.
Which of the following graphs best shows
the relation between the speed v of the fluid and the difference in manometer levels h?
A.
v
h
B.
v
h
C.
v
h
D.
v
h
E.
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By C.K.Cheung
v
h
A
90’ MC
11. In fluid dynamics, which of the following assumptions is/are used in deriving Bernoulli’s equation?
(1)
The fluid undergoes streamline flow.
(2)
The fluid is compressible.
(3)
Viscous forces act on the fluid.
A.
(1), (2) and (3)
B.
(1) and (2) only
C.
(2) and (3) only
D.
(1) only
E.
(3) only
D
84’MC
42.
The above figure shows the stream lines for air flowing past the wing of an aeroplane. Which
of the following is/are correct?
(1) The pressure above the wing is greater than that below the wing.
(2) The speed of the air flow above the wing is greater than that below the wing.
(3) The pressure difference between locations above and below the wing increases when the
density of air increases.
(Assume the same speed of air flow.)
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By C.K.Cheung
A.
(1), (2) and (3)
B.
C.
D.
E.
(1) and (2) only
(2) and (3) only
(1) only
(3) only
C
87’MC
12.
fluid
flow
S
T
The above diagram shows a Pitot-static tube situated in a moving fluid.
shows a difference h in the liquid levels.
A manometer connected to S and T
If
v = the velocity of the moving fluid,
d = the density of the moving fluid,
 = the density of the liquid in the manometer,
then v² is equal to
A.
2gh/d.
B.
2dgh/.
C.
gh/d.
D.
dgh/.
E.
dgh.
94’ essay
3. (a) (i)
(ii)
A
Solids can be thought of as networks of atoms connected by ‘small springs’.
Explain how this method can be deduced from solids’ observed resistance to
deformation. ( out of syllabus )
Sketch the curve of potential energy against interatomic separation and use it to
explain the phenomenon of thermal expansion of solids.
(6 marks)
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By C.K.Cheung
(b) Glass is a strong, stiff and brittle material.
Sketch the stress-strain graph for glass and
briefly explain why it is so described.
(3 marks)
(c) With suitable diagrams, use the Bernoulli principle to explain
(i) how a yacht can sail against the wind;
(ii) the curved flight of a spinning ball. Also suggest one design feature which increases
the curvature of the ball’s flight.
(7 marks)
91’ IIB
8. Figure 8.1 shows a petrol pump in a garage. The pump delivers petrol with a density of 750
kg/m3 at a rate of 1.2  10-2 m3/s. The input to the pump is from a pipe with a cross-sectional
area A1 of 4  10-3 m2 at a suction pressure P1 of 1  104 Pa. The discharge of the pump is at a
gauge pressure P2 of 2.8  105 Pa into a pipe with a cross-sectional area A2 of 8  10-4 m2.
The pipes at the entrance and exit are at the same horizontal level and the temperature of the
petrol remains constant throughout the flow.
Suction pressure
Gauge pressure
P1 = 1 x 10 4 Pa
P2 = 2.8 x 10 5 Pa
Input
Output
v1
v2
Pump
rotor
Figure 8.1
(a) Find the average flow speeds v1 and v2 of the petrol into and out of the pump.
(2 marks)
(b) Find the change in kinetic energy per unit mass of petrol.
(2 marks)
(c) By considering the work done per unit mass of petrol at the entrance and exit of the pump,
find the work done by the pump in delivering a unit mass of petrol.
(4
marks)
(d) Find the mechanical power developed in the pump in order to maintain the above flow
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By C.K.Cheung
conditions.
(2 marks)
(e) Explain why, in reality, the mechanical power of the pump required is higher than that
calculated in (d).
(2 marks)
89” Essay
2. (a) Discuss qualitatively the motion of a small metal ball allowed to drop downwards into a
long vertical tube of liquid so that it moves along the axis of the tube (radius of tube >>
radius of ball).
(3 marks)
(b) Derive Bernoulli’s equation,
P + hg + ½v² = a constant
for the flow of a fluid through a tube of varying cross-sectional area. Clearly explain all
the physical parameters used in this equation.
(6 marks)
(c) In practice, discuss the likely sources of error in applying Bernoulli’s equation to the flow
of
(i)
liquids, and
Due to viscosity the velocity of the liquid at any particular cross-section of the tube will vary
from a maximum at the center to zero on the sides of the tube.
Even if the cross-section and the height remained constant the pressure would drop due to
energy dissipation against viscous
(ii)
gases
in tubes.
(3 marks)
The fluid is compressible so that the density would vary with the pressure P, affecting the ‘hg’ term in
the equation.
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By C.K.Cheung
88’ IIB
8. (a)
direction of ball
Figure 8.1
(i)
Figure 8.1 shows the streamlines around a tennis ball when it is projected in a straight
line through still air. In Figure 8.2, sketch the streamlines in the vicinity of the ball
if; apart from the forward motion, it is also spinning about an axis, through its centre,
perpendicular to the plane of paper in an anti-clockwise direction.(2 marks)
forward motion
Pattern correct:
lines crowded on L.H.S.-----1M
Arrows inserted---------1M
Figure 8.2
(ii)
Describe, with reasons, the subsequent motion of the ball.
(4 marks)
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By C.K.Cheung
Ball rotates; drags surrounding air
Air on LHS flows faster than that on RHS
Pressure on LHS less
Ball deflected towards the left
(b) (i)
One end of an open tube is put vertically into water. By blowing strongly across the
open end, water can be drawn up the tube. Suppose a few centimetres of the tube is
above the water surface. What should be the air velocity at the open end for water in
the tube to rise up by 1 cm? Explain your working. (Surface tension effects may
be ignored.)
(3 marks)
( Density of air
Density of water
Acceleration due to gravity
= 1.29 kg/m3,
= 1 000 kg/m3,
= 10 m/s2 )
Pressure at upper end reduced due to air-flow
1
 ' v 2  gh
2
(1/2)(1.29)v2=1000x10x0.01
v = 12.5 ms-1
(ii)
Mention one daily application making use of the principle described in (b)(i).
mark)
(1
84’Essay
1. (a) (i)
Using a practical example, demonstrate what is meant by ‘the conservation of
mechanical energy’.
(ii) By means of a further practical example, show that in ‘real-life’ situations mechanical
energy is often not conserved.
(b) Derive Bernoulli’s equation for fluid flow:
1 2
v  a constant .
2
(c) Explain why Bernoulli’s equation is not strictly applicable to
(i) a gas, and
(ii) a viscous liquid flowing through a narrow tube.
(d) With the aid of diagrams and Bernoulli’s equation, explain the observed effects of
(i) the motion of a spinning ball, and
(ii) the mixing of coal gas and air in a bunsen burner.
P  hg 
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