CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Fluid Dynamics
School of Civil Engineering
FACULTY OF ENGINEERING
• Analysis of fluids in motion
• Liquid and Gasses
An Introduction to Fluid Mechanics
Lecture 8: Fluid Dynamics Introduction
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•
•
•
Dr Andrew Sleigh
Dr Cath Noakes
Use fundamental laws of physics
Conservation of energy
Conservation of momentum
Conservation of mass (Continuity)
www.efm.leeds.ac.uk/CIVE/FluidsLevel1
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Most Fluid Flow is Complex!
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Unit 3
Fluid Mechanics: Lecture 8
1
CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Complex fluid flow
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•
•
•
•
Wake behind vehicles
Wakes on beaches
Hurricanes & Tornadoes
Weather prediction
… and many many more
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
All can be analysed
With greater or lesser success
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Fluid Mechanics: Lecture 8
2
CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Flow classification
Uniform flow
• Many common situations can be simplified
• Yet still give very accurate predictions
• Flow is uniform if

 Same velocity
 Same depth
 Same pressure
 Same width
 Same everything
• To help simplify analysis we classify flow




Uniform
Non-uniform
Steady
Unsteady
At all positions it is the same …

(note: spatial definition)
• Else it is Non-Uniform
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Steady
Most flow
• Flow is steady if
• Most flow is Non-uniform and unsteady

At all time conditions do not change
 velocity constant
 depth constant
 pressure constant
 width constant
 Everything constant

(note: temporal definition)
• Else it is Unsteady
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics

A river for example
 Non-uniform as depth and width not constant
 Unsteady as flow not constant e.g. rain, flow from
land a tributaries
• Can combine all classifications


Steady uniform / Steady non-uniform
Unsteady uniform / Unsteady non-uniform
• This course is all Steady-Uniform
Fluid Mechanics: Lecture 8
3
CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Unit 3
Simplify analysis
Channel flow
• Usually possible to simplify analysis
• Judge the problem and look at dominant
behaviour
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•
•
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Rivers/Channels Clearly non-uniform
Depth changes
velocity not constant
Even across section
• Often analysis can be steady-uniform

(it is easiest!)
• Steady Uniform flow
is commonly used
• Gives good results
• Usually give good starting point
• May be sufficient
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Compressible - Incompressible
Unit 3
Three dimensional flow
• Gasses are compressible
• So are liquids (a little)
• Under steady conditions

With small pressure change

Can assume constant density
And get very good results

Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Three dimensional flow
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Unit 3
Three dimensional flow
Fluid Mechanics: Lecture 8
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CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Unit 3
One dimensional flow - Pipe
One dimensional flow - Channel
• Velocity will vary across pipe
• Zero at wall, max in centre
• Channel flow is clearly non-uniform
• Clearly three-dimensional
• Clearly one-dimensional?
• Clearly one-dimensional?
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Two-dimensional
Two-dimensional
• May be predominantly in two dimensions
• Rivers often predominantly in two
dimensions – in plan view
• Simplify analysis?
• Simplify analysis
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Two-dimensional
Analysis on this course
• Rivers occasionally analysed in two
dimensions – in cross section view
• We will consider

Steady
 Often uniform
Incompressible
 One dimensional

 occasionally two-dimensional
• Simplify analysis?
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Fluid Mechanics: Lecture 8
5
CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Streamlines
Unit 3
Three dimensional flow
• Contours of equal velocity
• Help visualise the flow pattern
• Known as Streamlines
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Features of Streamlines
Streamline change at different velocities
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• Low velocity
Parallel to solid surfaces
Direction in direction of fluid
Fluid (mass) cannot cross a streamline
A streamline cannot cross another
streamline
• Particles stay on the same stramline
• In unsteady flow they can change position
• In steady flow they do not move position
Fluid Mechanics: Lecture 8
• High velocity
Fluid Mechanics: Lecture 8
Unit 3
Streamline - Particles
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Unit 3
Flow Patterns
Fluid Mechanics: Lecture 8
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CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Unit 3
Features of Streamtubes
Features of Streamtubes
• Take a circle of points in 3-D space
• Streamlines from these form a Streamtube
• “walls” of streamtubes are streamlines
• Consequently fluid cannot cross a
streamtube “wall”
• It is not like a pipe – as it “walls” move with
the fluid
• In unsteady flow they can change position
• In steady flow they do not move position
Sometimes “flattened”
to 2-D for analysis
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Continuity and Streamlines/Streamtubes
Continuity and Streamlines/Streamtubes
• Continuity
• In a real pipe (or any other vessel)
• Use mean velocity
 1 A1 u m 1   2 A 2 u m 2  C onstant  m
Mass entering =
per unit time
Mass leaving
per unit time
1 A1u1   2  A2 u 2
• For steady flow
• Incompressible: 1 = 2 = 
• Continuity equation:
1A1u1  2A2 u2  Constant  m
 
A1u1  A2 u2  Q
dm
dt
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Unit 3
Flow and Continuity: Example 1
Continuity Example 2
• Q = Area  Mean Velocity = Au
• Qin = Qout
• What is Q?
1.5 m3/s
1.0 m3/s
1.2 m3/s
1.5 m3/s
1.5 m3/s
2.8 m3/s
Q
• What is Q?

Q = 3 m3/s
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
Q
• Q + 1.5 + 1 = 1.2 + 2.8
• Q = 1.5 m3/s
Fluid Mechanics: Lecture 8
7
CIVE1400 Fluid Mechanics: Intro to Fluid
Dynamics
07/04/2009
Unit 3
Unit 3
Continuity Example 3
Continuity Example 4
• What is Q?
•
u = 1.5 m/s
A = 0.5 m2
•
u = 1.0 m/s
A = 0.7 m2
Water flows in a circular pipe which increases in diameter from
400mm at point A to 500mm at point B. Then pipe then splits into
two branches of diameters 0.3m and 0.2m discharging at C and
D respectively.
If the velocity at A is 1.0m/s and at D is 0.8m/s, what are the
discharges at C and D and the velocities at B and C?
U = 0.2 m/s
A = 1.3 m2
Q = 2.8 m3/s
C
A
Q
dB=0.5m
B
dC=0.3m
dA=0.4m
vA=1.0m/s
• Q + 1.50.5 + 10.7 = 0.21.3 + 2.8
• Q = 3.72 m3/s
D
dD=0.2m
vD=0.8m/s
Fluid Mechanics: Lecture 8
Fluid Mechanics: Lecture 8
Unit 3
Today’s lecture:
Continuity Example 4
•
If the velocity at A is 1.0m/s and at D is 0.8m/s, what are the
discharges at C and D and the velocities at B and C?
Point
Velocity m/s
Diameter m
Area m²
Q m³/s
A
1.00
0.4
0.126
0.126
B
0.64
0.5
0.196
0.126
C
1.42
0.3
0.071
0.101
D
0.80
0.2
0.031
0.025
C
A
dB=0.5m
B
• Classification of Flow


Steady / Unsteady
Uniform / Non-uniform
• Streamlines / Streamtubes
dC=0.3m
• One, Two and Three Dimensions
dA=0.4m
vA=1.0m/s
D
dD=0.2m
Fluid Mechanics: Lecture 8
CIVE1400 Fluid Mechanics
• Continuity
vD=0.8m/s
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