Ch4 Basics of Fluid Flow
Objectives
• Identify the basic categories of fluid flow.
• Understand the definition of pathline, streamline and
streakline.
• Understand how flow rates are defined and how to relate
volume, mass, and weight based flow rates.
• Understand the Equation of Continuity and know how to
apply it to various types of flow.
Outline
•
•
•
•
•
•
•
•
4.1 Two ways to describe fluid motion
4.2 Types of flows
4.3 Steady Flow and Uniform Flow
4.4 Pathline, Streamline And Streak line
4.5 Flow Rate and Mean Velocity
4.6 Fluid System and Control Volume
4.7 Equation of Continuity
4.8 One-, Two-, and Three-Dimensional Flow
4.1 Two ways to describe fluid motion
Two ways to describe motion are Lagrangian and
Eulerian description.
Lagrangian Description
•
•
•
•
Lagrangian description of fluid flow tracks the
position and velocity of individual particles. (eg.
Track the location of a migrating bird.)
Motion is described based upon Newton's laws.
Difficult to use for practical flow analysis.
• Fluids are composed of billions of molecules.
• Interaction between molecules hard to
describe/model.
Named after Italian mathematician Joseph Louis
Lagrange (1736-1813).
Eulerian Description
•Eulerian
¾ Describes the flow field (velocity,acceleration,
pressure, temperature, etc.) as functions of
position and time
¾ Count the birds passing a particular location
Velocity field:
r r
V = V ( x, y , z , t )
If you were going to study water flowing in a pipeline,
which approach would you use? Eulerian
Flow descriptions:
Lagrangian vs. Eulerian
4.2 Types of flow
z
Flow of an ideal fluid
No viscosity
Flow of a real fluid
Viscosity effects included
Flow of Ideal fluid in straight conduit
All fluid particles move
at constant speed
z
Flow of real fluid in straight conduit
particles @wall
u=0
non-uniform velocity
profile
4.2 Types of flow
Incompressible flow: liquids
Compressible : gases mainly
Pressure flow-----liquids /gases in pipes
Gravity flow ------free surface flow in open channels
4.3 Steady flow and Uniform flow
¾
Steady - unsteady
Changing in time
¾Uniform
- nonuniform
Changing in space
4.3 Steady flow and Uniform flow
Steady flow is constant with time,
while unsteady flow changes with time.
Uniform flow is constant with space, while
nonuniform flow changes over space.
4.3 Steady flow and Uniform flow
1. Steady uniform flow. Conditions do not change with position in the
stream or with time. An example is the flow of water in a pipe of
constant diameter at constant velocity.
2. Steady non-uniform flow. Conditions change from point to point in
the stream but do not change with time. An example is flow in a
tapering pipe with constant velocity at the inlet - velocity will change as
you move along the length of the pipe toward the exit.
3. Unsteady uniform flow. At a given instant in time the conditions at
every point are the same, but will change with time.
4. Unsteady non-uniform flow. Every condition of the flow may change
from point to point and with time at every point.
Flow Patterns
•
•
Uniform flow
Non-uniform flow
•
•
∂V
=0
∂s
∂V
≠0
∂s
Steady flow
∂V
=0
∂t
Unsteady flow
∂V
≠0
∂t
Uniform flow
y
u
u
x
z
The velocity is constant across any section normal to
the flow.
The flow model is one-dimensional
Example
•
•
•
Valve at C is opened
slowly
Classify the flow at B
while valve is opened
Classify the flow at A
4.4 Pathline, Streamline
The flow velocity is the basic description of
how a fluid moves in time and space, but in
order to visualize the flow pattern it is useful to
define some other properties of the flow. These
definitions correspond to various experimental
methods of visualizing fluid flow.
Streamlines
•
A Streamline is a curve that is
everywhere tangent to the
instantaneous local velocity vector.
•
Geometric arguments results in the
equation for a streamline
dr dx dy dz
=
=
=
V
u
v
w
Streamlines
V2, b2
V1, b1
Ideal flow
Character of Streamline
1 Streamlines can not cross each other.
(otherwise, the cross point will have two
tangential lines.)
2 Streamline can't be a folding line, but a smooth
curve.
3 Streamline cluster density reflects the
magnitude of velocity. （Dense streamlines
mean large velocity; while sparse streamlines
mean small velocity.）
Example
CAR surface
pressure contours
and streamlines
Streamtube
• Streamtube
is an
imaginary tube whose
boundary consists of
streamlines.
• The volume flow rate
must be the same for
all cross sections of
the streamtube.
Pathlines
Streaklines (脉线 )
•
•
•
A Streakline is the locus of fluid
particles that have passed
sequentially through a prescribed
point in the flow.
Easy to generate in experiments:
dye in a water flow, or smoke in an
airflow.
same as pathline and streamline
for steady flow
相继通过某空间点的质点连线
Comparisons
„ For steady flows: streamlines, pathlines and
streaklines are identical.
„ For unsteady flows: they can be very different.
¾Streamlines are an instantaneous picture of the
flow field
¾Pathlines and Streaklines are flow patterns that
have a time history associated with them.
¾Streakline: instantaneous snapshot of a timeintegrated flow pattern.
¾Pathline: time-exposed flow path of an individual
particle.
4.5 FLOW RATE
Volume flow rate - Discharge.
Mass flow rate
Discharge and mean velocity.
Volume flow rate (discharge)
The volume of fluid passing a
given cross-section in unit time
is called the discharge. It is
measured in cubic metres per
second, or similar units, and
denoted by Q.
- Constant velocity over
Q = VA
cross-section
- Variable velocity
Q = ∫ VdA
A
Mass flow rate
The mass of fluid passing a given cross-section in
unit time is called the mass flow rate. It is
measured in kilogrammes per second or similar
units.
m& = ∫ ρVdA = ρ ∫ VdA = ρQ
A
A
Discharge and mean velocity
•
If we know the
size of a pipe,
and we know the
discharge, we
can deduce the
mean velocity
•
Volume flow rate Q
is given by
Q＝∫ Vn dAc = Vavg Ac = VAc
A
Examples
•
Discharge in a 25cm pipe is 0.03 m3/s. What
is the average velocity?
Q = VA
V=
Q
Q
0.03
=
=
= 0.611 m / s
π
π
A
d2
(0.25) 2
4
4
Exercise
How many dimensions do most fluid flows have?
A.2
B.3
C. None. They are dimensionless
What is a pathline?
A. A distance along a streakline.
B. The line traced out by a given particle over time.
C. Pathline is just another term for streakline
Multiple Choice Quiz
If a flow is unsteady, its ____ may
change with time at a given location.
A. Velocity
B. Temperature
C. Density
D. All of the above
If the flow is steady, there can be changes in
velocity at different locations in the flow field,
True or False.
A. True
B. False
4.6 Fluid System and Control Volume
z
z
System method
¾
In mechanics courses.
¾
Dealing with an easily identifiable rigid body.
Control volume method
¾
¾
¾
In fluid mechanics course.
Difficult to focus attention on a fixed
identifiable quantity of mass
Dealing with the flow of fluids
System
●
●
●
A system is defined as a fixed, identifiable quantity of mass.
The boundaries separate the system from the surrounding.
The boundaries of the system may be fixed or movable.
No mass crosses the system boundaries.
Figure : Piston cylinder arrangement
Control volumes
● Is an arbitrary volume in space through which the fluid flows.
● The geometric boundary of the control volume (CV) is called the
“Control Surface (CS).”
● The CS may be real or imaginary.
● The CV may be at rest or in motion.
Control Volumes
•Solid Mechanics
–Follow the system, determine what
happens to it
•Fluid Mechanics
–Consider the behavior in a specific
region or Control Volume
•Convert System approach to CV approach
–Look at specific regions, rather than
specific masses
CV Inflow & Outflow
4.7 Equation of continuity
Mass flow entering = Mass flow leaving
An arbitrarily shaped control volume
The continuity equation
Mass flow entering = (Density) x (volume of fluid entering per second)
Mass flow leaving = (Density) x (volume of fluid exiting per second)
Assumption: Flow is incompressible (density is constant)
ρ x Q(entering) = ρ x Q(leaving)
Q(entering) = u1A1
Thus
u1A1 = u2A2
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
For steady flow
Mass entering per unit time = Mass leaving per unit time
The continuity equation
Eq. of Continuity
ρ1 A1v1 = ρ 2 A2 v2
Example
Water flows through a 4.0 cm diameter pipe at 5 cm/s.
The pipe then narrows downstream and has a diameter
of 2.0 cm. What is the velocity of the water through the
smaller pipe?
Solution
A1v1 = A2 v2
r12
v2 = 2 v1 = 4v1 = 20 cm/s
r2
Two streams join to become one river.
The dimensions are:
w1 = 8.2m, d1 = 3.4m, v1 = 2.3m/s
w2 = 6.8m, d2 = 3.2m, v2 = 2.6m/s
wf = 10.5m, df = ?, vf = 2.9m/s
Continuity Equation
A1v1 + A2v2 = Afvf
8.2×3.4×2.3+6.8×3.2×2.6=10.5×df×2.9
df = 3.964m
4.8 One-, Two-, and ThreeDimensional Flow
•Flow dimension is depending on the
number of space coordinates required to
describe a flow
– Most flow field are inherently threedimensional, analysis based on a fewer
dimensions is frequently meaningful.
One-dimensional flow
One-dimensional Flow: Fluid flows mainly in one
direction, and other two directions flow is negligible.
r
R
x
⎡ ⎛ r ⎞2 ⎤
u = umax ⎢1 − ⎜ ⎟ ⎥
⎢⎣ ⎝ R ⎠ ⎥⎦
θ
r
One-dimensional flow
Two-dimensional flow
Fluid flows mainly in two directions, and the third
direction's flow is negligible, that is, fluid motion
factors are functions of two space coordinates.
y
u
x
z
u
Two-dimensional flow
Three-dimensional Flow
Fluid flow's motion factors are functions of three
space coordinates.