Fluid Dynamics Notes | EduRev

: Fluid Dynamics Notes | EduRev

 Page 1


CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
44
3. Fluid Dynamics
Objectives
• Introduce concepts necessary to analyse fluids in motion
• Identify differences between Steady/unsteady uniform/non-uniform compressible/incompressible flow
• Demonstrate streamlines and stream tubes
• Introduce the Continuity principle through conservation of mass and control volumes
• Derive the Bernoulli (energy) equation
• Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow
• Introduce the momentum equation for a fluid
• Demonstrate how the momentum equation and principle of conservation of momentum is used to
predict forces induced by flowing fluids
This section discusses the analysis of fluid in motion - fluid dynamics. The motion of fluids can be
predicted in the same way as the motion of solids are predicted using the fundamental laws of physics
together with the physical properties of the fluid.
It is not difficult to envisage a very complex fluid flow. Spray behind a car; waves on beaches; hurricanes
and tornadoes or any other atmospheric phenomenon are all example of highly complex fluid flows which
can be analysed with varying degrees of success (in some cases hardly at all!). There are many common
situations which are easily analysed.
3.1 Uniform Flow, Steady Flow
It is possible - and useful - to classify the type of flow which is being examined into small number of
groups.
If we look at a fluid flowing under normal circumstances - a river for example - the conditions at one
point will vary from those at another point (e.g. different velocity) we have non-uniform flow. If the
conditions at one point vary as time passes then we have unsteady flow.
Under some circumstances the flow will not be as changeable as this. He following terms describe the
states which are used to classify fluid flow:
• uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is
said to be uniform.
• non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
(In practice, by this definition, every fluid that flows near a solid boundary will be non-uniform - as
the fluid at the boundary must take the speed of the boundary, usually zero. However if the size and
shape of the of the cross-section of the stream of fluid is constant the flow is considered uniform.)
• steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ
from point to point but DO NOT change with time.
• unsteady: If at any point in the fluid, the conditions change with time, the flow is described as
unsteady. (In practise there is always slight variations in velocity and pressure, but if the average
values are constant, the flow is considered steady.
Combining the above we can classify any flow in to one of four type:
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.
Page 2


CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
44
3. Fluid Dynamics
Objectives
• Introduce concepts necessary to analyse fluids in motion
• Identify differences between Steady/unsteady uniform/non-uniform compressible/incompressible flow
• Demonstrate streamlines and stream tubes
• Introduce the Continuity principle through conservation of mass and control volumes
• Derive the Bernoulli (energy) equation
• Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow
• Introduce the momentum equation for a fluid
• Demonstrate how the momentum equation and principle of conservation of momentum is used to
predict forces induced by flowing fluids
This section discusses the analysis of fluid in motion - fluid dynamics. The motion of fluids can be
predicted in the same way as the motion of solids are predicted using the fundamental laws of physics
together with the physical properties of the fluid.
It is not difficult to envisage a very complex fluid flow. Spray behind a car; waves on beaches; hurricanes
and tornadoes or any other atmospheric phenomenon are all example of highly complex fluid flows which
can be analysed with varying degrees of success (in some cases hardly at all!). There are many common
situations which are easily analysed.
3.1 Uniform Flow, Steady Flow
It is possible - and useful - to classify the type of flow which is being examined into small number of
groups.
If we look at a fluid flowing under normal circumstances - a river for example - the conditions at one
point will vary from those at another point (e.g. different velocity) we have non-uniform flow. If the
conditions at one point vary as time passes then we have unsteady flow.
Under some circumstances the flow will not be as changeable as this. He following terms describe the
states which are used to classify fluid flow:
• uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is
said to be uniform.
• non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
(In practice, by this definition, every fluid that flows near a solid boundary will be non-uniform - as
the fluid at the boundary must take the speed of the boundary, usually zero. However if the size and
shape of the of the cross-section of the stream of fluid is constant the flow is considered uniform.)
• steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ
from point to point but DO NOT change with time.
• unsteady: If at any point in the fluid, the conditions change with time, the flow is described as
unsteady. (In practise there is always slight variations in velocity and pressure, but if the average
values are constant, the flow is considered steady.
Combining the above we can classify any flow in to one of four type:
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.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
45
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. An example is a pipe of constant diameter connected to a pump pumping at a
constant rate which is then switched off.
4. Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time
at every point. For example waves in a channel.
If you imaging the flow in each of the above classes you may imagine that one class is more complex than
another. And this is the case - steady uniform flow is by far the most simple of the four. You will then be
pleased to hear that this course is restricted to only this class of flow. We will not be encountering any
non-uniform or unsteady effects in any of the examples (except for one or two quasi-time dependent
problems which can be treated at steady).
3.1.1 Compressible or Incompressible
All fluids are compressible - even water - their density will change as pressure changes. Under steady
conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis
of the flow by assuming it is incompressible and has constant density. As you will appreciate, liquids are
quite difficult to compress - so under most steady conditions they are treated as incompressible. In some
unsteady conditions very high pressure differences can occur and it is necessary to take these into account
- even for liquids. Gasses, on the contrary, are very easily compressed, it is essential in most cases to treat
these as compressible, taking changes in pressure into account.
3.1.2 Three-dimensional flow
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. The flow may be unsteady, in this
case the parameter vary in time but still not across the cross-section. An example of one-dimensional flow
is the flow in a pipe. Note that since flow must be zero at the pipe wall - yet non-zero in the centre - there
is a difference of parameters across the cross-section. Should this be treated as two-dimensional flow?
Possibly - but it is only necessary if very high accuracy is required. A correction factor is then usually
applied.
Pipe Ideal flow Real flow
One dimensional flow in a pipe.
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. Streamlines in two-dimensional flow are curved lines on a
plane and are the same on all parallel planes. An example is flow over a weir foe which typical
Page 3


CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
44
3. Fluid Dynamics
Objectives
• Introduce concepts necessary to analyse fluids in motion
• Identify differences between Steady/unsteady uniform/non-uniform compressible/incompressible flow
• Demonstrate streamlines and stream tubes
• Introduce the Continuity principle through conservation of mass and control volumes
• Derive the Bernoulli (energy) equation
• Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow
• Introduce the momentum equation for a fluid
• Demonstrate how the momentum equation and principle of conservation of momentum is used to
predict forces induced by flowing fluids
This section discusses the analysis of fluid in motion - fluid dynamics. The motion of fluids can be
predicted in the same way as the motion of solids are predicted using the fundamental laws of physics
together with the physical properties of the fluid.
It is not difficult to envisage a very complex fluid flow. Spray behind a car; waves on beaches; hurricanes
and tornadoes or any other atmospheric phenomenon are all example of highly complex fluid flows which
can be analysed with varying degrees of success (in some cases hardly at all!). There are many common
situations which are easily analysed.
3.1 Uniform Flow, Steady Flow
It is possible - and useful - to classify the type of flow which is being examined into small number of
groups.
If we look at a fluid flowing under normal circumstances - a river for example - the conditions at one
point will vary from those at another point (e.g. different velocity) we have non-uniform flow. If the
conditions at one point vary as time passes then we have unsteady flow.
Under some circumstances the flow will not be as changeable as this. He following terms describe the
states which are used to classify fluid flow:
• uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is
said to be uniform.
• non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
(In practice, by this definition, every fluid that flows near a solid boundary will be non-uniform - as
the fluid at the boundary must take the speed of the boundary, usually zero. However if the size and
shape of the of the cross-section of the stream of fluid is constant the flow is considered uniform.)
• steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ
from point to point but DO NOT change with time.
• unsteady: If at any point in the fluid, the conditions change with time, the flow is described as
unsteady. (In practise there is always slight variations in velocity and pressure, but if the average
values are constant, the flow is considered steady.
Combining the above we can classify any flow in to one of four type:
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.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
45
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. An example is a pipe of constant diameter connected to a pump pumping at a
constant rate which is then switched off.
4. Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time
at every point. For example waves in a channel.
If you imaging the flow in each of the above classes you may imagine that one class is more complex than
another. And this is the case - steady uniform flow is by far the most simple of the four. You will then be
pleased to hear that this course is restricted to only this class of flow. We will not be encountering any
non-uniform or unsteady effects in any of the examples (except for one or two quasi-time dependent
problems which can be treated at steady).
3.1.1 Compressible or Incompressible
All fluids are compressible - even water - their density will change as pressure changes. Under steady
conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis
of the flow by assuming it is incompressible and has constant density. As you will appreciate, liquids are
quite difficult to compress - so under most steady conditions they are treated as incompressible. In some
unsteady conditions very high pressure differences can occur and it is necessary to take these into account
- even for liquids. Gasses, on the contrary, are very easily compressed, it is essential in most cases to treat
these as compressible, taking changes in pressure into account.
3.1.2 Three-dimensional flow
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. The flow may be unsteady, in this
case the parameter vary in time but still not across the cross-section. An example of one-dimensional flow
is the flow in a pipe. Note that since flow must be zero at the pipe wall - yet non-zero in the centre - there
is a difference of parameters across the cross-section. Should this be treated as two-dimensional flow?
Possibly - but it is only necessary if very high accuracy is required. A correction factor is then usually
applied.
Pipe Ideal flow Real flow
One dimensional flow in a pipe.
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. Streamlines in two-dimensional flow are curved lines on a
plane and are the same on all parallel planes. An example is flow over a weir foe which typical
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
46
streamlines can be seen in the figure below. Over the majority of the length of the weir the flow is the
same - only at the two ends does it change slightly. Here correction factors may be applied.
Two-dimensional flow over a weir.
In this course we will only be considering steady, incompressible one and two-dimensional flow.
3.1.3 Streamlines and streamtubes
In analysing fluid flow it is useful to visualise the flow pattern. This can be done by drawing lines joining
points of equal velocity - velocity contours. These lines are know as streamlines. Here is a simple
example of the streamlines around a cross-section of an aircraft wing shaped body:
Streamlines around a wing shaped body
When fluid is flowing past a solid boundary, e.g. the surface of an aerofoil or the wall of a pipe, fluid
obviously does not flow into or out of the surface. So very close to a boundary wall the flow direction
must be parallel to the boundary.
• Close to a solid boundary streamlines are parallel to that boundary
At all points the direction of the streamline is the direction of the fluid velocity: this is how they are
defined. Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall.
It is also important to recognise that the position of streamlines can change with  time - this is the case in
unsteady flow. In steady flow, the position of streamlines does not change.
Some things to know about streamlines
• Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline.
• Streamlines can not cross each other. If they were to cross this would indicate two different velocities
at the same point. This is not physically possible.
• The above point implies that any particles of fluid starting on one streamline will stay on that same
streamline throughout the fluid.
A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the
rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows.
This tubular surface is known as a streamtube.
Page 4


CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
44
3. Fluid Dynamics
Objectives
• Introduce concepts necessary to analyse fluids in motion
• Identify differences between Steady/unsteady uniform/non-uniform compressible/incompressible flow
• Demonstrate streamlines and stream tubes
• Introduce the Continuity principle through conservation of mass and control volumes
• Derive the Bernoulli (energy) equation
• Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow
• Introduce the momentum equation for a fluid
• Demonstrate how the momentum equation and principle of conservation of momentum is used to
predict forces induced by flowing fluids
This section discusses the analysis of fluid in motion - fluid dynamics. The motion of fluids can be
predicted in the same way as the motion of solids are predicted using the fundamental laws of physics
together with the physical properties of the fluid.
It is not difficult to envisage a very complex fluid flow. Spray behind a car; waves on beaches; hurricanes
and tornadoes or any other atmospheric phenomenon are all example of highly complex fluid flows which
can be analysed with varying degrees of success (in some cases hardly at all!). There are many common
situations which are easily analysed.
3.1 Uniform Flow, Steady Flow
It is possible - and useful - to classify the type of flow which is being examined into small number of
groups.
If we look at a fluid flowing under normal circumstances - a river for example - the conditions at one
point will vary from those at another point (e.g. different velocity) we have non-uniform flow. If the
conditions at one point vary as time passes then we have unsteady flow.
Under some circumstances the flow will not be as changeable as this. He following terms describe the
states which are used to classify fluid flow:
• uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is
said to be uniform.
• non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
(In practice, by this definition, every fluid that flows near a solid boundary will be non-uniform - as
the fluid at the boundary must take the speed of the boundary, usually zero. However if the size and
shape of the of the cross-section of the stream of fluid is constant the flow is considered uniform.)
• steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ
from point to point but DO NOT change with time.
• unsteady: If at any point in the fluid, the conditions change with time, the flow is described as
unsteady. (In practise there is always slight variations in velocity and pressure, but if the average
values are constant, the flow is considered steady.
Combining the above we can classify any flow in to one of four type:
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.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
45
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. An example is a pipe of constant diameter connected to a pump pumping at a
constant rate which is then switched off.
4. Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time
at every point. For example waves in a channel.
If you imaging the flow in each of the above classes you may imagine that one class is more complex than
another. And this is the case - steady uniform flow is by far the most simple of the four. You will then be
pleased to hear that this course is restricted to only this class of flow. We will not be encountering any
non-uniform or unsteady effects in any of the examples (except for one or two quasi-time dependent
problems which can be treated at steady).
3.1.1 Compressible or Incompressible
All fluids are compressible - even water - their density will change as pressure changes. Under steady
conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis
of the flow by assuming it is incompressible and has constant density. As you will appreciate, liquids are
quite difficult to compress - so under most steady conditions they are treated as incompressible. In some
unsteady conditions very high pressure differences can occur and it is necessary to take these into account
- even for liquids. Gasses, on the contrary, are very easily compressed, it is essential in most cases to treat
these as compressible, taking changes in pressure into account.
3.1.2 Three-dimensional flow
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. The flow may be unsteady, in this
case the parameter vary in time but still not across the cross-section. An example of one-dimensional flow
is the flow in a pipe. Note that since flow must be zero at the pipe wall - yet non-zero in the centre - there
is a difference of parameters across the cross-section. Should this be treated as two-dimensional flow?
Possibly - but it is only necessary if very high accuracy is required. A correction factor is then usually
applied.
Pipe Ideal flow Real flow
One dimensional flow in a pipe.
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. Streamlines in two-dimensional flow are curved lines on a
plane and are the same on all parallel planes. An example is flow over a weir foe which typical
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
46
streamlines can be seen in the figure below. Over the majority of the length of the weir the flow is the
same - only at the two ends does it change slightly. Here correction factors may be applied.
Two-dimensional flow over a weir.
In this course we will only be considering steady, incompressible one and two-dimensional flow.
3.1.3 Streamlines and streamtubes
In analysing fluid flow it is useful to visualise the flow pattern. This can be done by drawing lines joining
points of equal velocity - velocity contours. These lines are know as streamlines. Here is a simple
example of the streamlines around a cross-section of an aircraft wing shaped body:
Streamlines around a wing shaped body
When fluid is flowing past a solid boundary, e.g. the surface of an aerofoil or the wall of a pipe, fluid
obviously does not flow into or out of the surface. So very close to a boundary wall the flow direction
must be parallel to the boundary.
• Close to a solid boundary streamlines are parallel to that boundary
At all points the direction of the streamline is the direction of the fluid velocity: this is how they are
defined. Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall.
It is also important to recognise that the position of streamlines can change with  time - this is the case in
unsteady flow. In steady flow, the position of streamlines does not change.
Some things to know about streamlines
• Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline.
• Streamlines can not cross each other. If they were to cross this would indicate two different velocities
at the same point. This is not physically possible.
• The above point implies that any particles of fluid starting on one streamline will stay on that same
streamline throughout the fluid.
A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the
rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows.
This tubular surface is known as a streamtube.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
47
A Streamtube
And in a two-dimensional flow we have a streamtube which is flat (in the plane of the paper):
A two dimensional version of the streamtube
The ?walls? of a streamtube are made of streamlines. As we have seen above, fluid cannot flow across a
streamline, so fluid cannot cross a streamtube wall. The streamtube can often be viewed as a solid walled
pipe. A streamtube is not a pipe - it differs in unsteady flow as the walls will move with time. And it
differs because the ?wall? is moving with the fluid
3.2 Flow rate.
3.2.1 Mass flow rate
If we want to measure the rate at which water is flowing along a pipe. A very simple way of doing this is
to catch all the water coming out of the pipe in a bucket over a fixed time period. Measuring the weight of
the water in the bucket and dividing this by the time taken to collect this water gives a rate of
accumulation of mass. This is know as the mass flow rate.
For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg,
then:
mass flow rate m =
mass of fluid in bucket
time taken to collect the fluid
=
=
- =
- &
. .
. / ( )
8 0 2 0
7
0857
1
kg s kg s
Performing a similar calculation, if  we know the mass flow is 1.7kg/s, how long will it take to fill a
container with 8kg of fluid?
Page 5


CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
44
3. Fluid Dynamics
Objectives
• Introduce concepts necessary to analyse fluids in motion
• Identify differences between Steady/unsteady uniform/non-uniform compressible/incompressible flow
• Demonstrate streamlines and stream tubes
• Introduce the Continuity principle through conservation of mass and control volumes
• Derive the Bernoulli (energy) equation
• Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow
• Introduce the momentum equation for a fluid
• Demonstrate how the momentum equation and principle of conservation of momentum is used to
predict forces induced by flowing fluids
This section discusses the analysis of fluid in motion - fluid dynamics. The motion of fluids can be
predicted in the same way as the motion of solids are predicted using the fundamental laws of physics
together with the physical properties of the fluid.
It is not difficult to envisage a very complex fluid flow. Spray behind a car; waves on beaches; hurricanes
and tornadoes or any other atmospheric phenomenon are all example of highly complex fluid flows which
can be analysed with varying degrees of success (in some cases hardly at all!). There are many common
situations which are easily analysed.
3.1 Uniform Flow, Steady Flow
It is possible - and useful - to classify the type of flow which is being examined into small number of
groups.
If we look at a fluid flowing under normal circumstances - a river for example - the conditions at one
point will vary from those at another point (e.g. different velocity) we have non-uniform flow. If the
conditions at one point vary as time passes then we have unsteady flow.
Under some circumstances the flow will not be as changeable as this. He following terms describe the
states which are used to classify fluid flow:
• uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is
said to be uniform.
• non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
(In practice, by this definition, every fluid that flows near a solid boundary will be non-uniform - as
the fluid at the boundary must take the speed of the boundary, usually zero. However if the size and
shape of the of the cross-section of the stream of fluid is constant the flow is considered uniform.)
• steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ
from point to point but DO NOT change with time.
• unsteady: If at any point in the fluid, the conditions change with time, the flow is described as
unsteady. (In practise there is always slight variations in velocity and pressure, but if the average
values are constant, the flow is considered steady.
Combining the above we can classify any flow in to one of four type:
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.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
45
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. An example is a pipe of constant diameter connected to a pump pumping at a
constant rate which is then switched off.
4. Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time
at every point. For example waves in a channel.
If you imaging the flow in each of the above classes you may imagine that one class is more complex than
another. And this is the case - steady uniform flow is by far the most simple of the four. You will then be
pleased to hear that this course is restricted to only this class of flow. We will not be encountering any
non-uniform or unsteady effects in any of the examples (except for one or two quasi-time dependent
problems which can be treated at steady).
3.1.1 Compressible or Incompressible
All fluids are compressible - even water - their density will change as pressure changes. Under steady
conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis
of the flow by assuming it is incompressible and has constant density. As you will appreciate, liquids are
quite difficult to compress - so under most steady conditions they are treated as incompressible. In some
unsteady conditions very high pressure differences can occur and it is necessary to take these into account
- even for liquids. Gasses, on the contrary, are very easily compressed, it is essential in most cases to treat
these as compressible, taking changes in pressure into account.
3.1.2 Three-dimensional flow
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. The flow may be unsteady, in this
case the parameter vary in time but still not across the cross-section. An example of one-dimensional flow
is the flow in a pipe. Note that since flow must be zero at the pipe wall - yet non-zero in the centre - there
is a difference of parameters across the cross-section. Should this be treated as two-dimensional flow?
Possibly - but it is only necessary if very high accuracy is required. A correction factor is then usually
applied.
Pipe Ideal flow Real flow
One dimensional flow in a pipe.
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. Streamlines in two-dimensional flow are curved lines on a
plane and are the same on all parallel planes. An example is flow over a weir foe which typical
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
46
streamlines can be seen in the figure below. Over the majority of the length of the weir the flow is the
same - only at the two ends does it change slightly. Here correction factors may be applied.
Two-dimensional flow over a weir.
In this course we will only be considering steady, incompressible one and two-dimensional flow.
3.1.3 Streamlines and streamtubes
In analysing fluid flow it is useful to visualise the flow pattern. This can be done by drawing lines joining
points of equal velocity - velocity contours. These lines are know as streamlines. Here is a simple
example of the streamlines around a cross-section of an aircraft wing shaped body:
Streamlines around a wing shaped body
When fluid is flowing past a solid boundary, e.g. the surface of an aerofoil or the wall of a pipe, fluid
obviously does not flow into or out of the surface. So very close to a boundary wall the flow direction
must be parallel to the boundary.
• Close to a solid boundary streamlines are parallel to that boundary
At all points the direction of the streamline is the direction of the fluid velocity: this is how they are
defined. Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall.
It is also important to recognise that the position of streamlines can change with  time - this is the case in
unsteady flow. In steady flow, the position of streamlines does not change.
Some things to know about streamlines
• Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline.
• Streamlines can not cross each other. If they were to cross this would indicate two different velocities
at the same point. This is not physically possible.
• The above point implies that any particles of fluid starting on one streamline will stay on that same
streamline throughout the fluid.
A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the
rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows.
This tubular surface is known as a streamtube.
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
47
A Streamtube
And in a two-dimensional flow we have a streamtube which is flat (in the plane of the paper):
A two dimensional version of the streamtube
The ?walls? of a streamtube are made of streamlines. As we have seen above, fluid cannot flow across a
streamline, so fluid cannot cross a streamtube wall. The streamtube can often be viewed as a solid walled
pipe. A streamtube is not a pipe - it differs in unsteady flow as the walls will move with time. And it
differs because the ?wall? is moving with the fluid
3.2 Flow rate.
3.2.1 Mass flow rate
If we want to measure the rate at which water is flowing along a pipe. A very simple way of doing this is
to catch all the water coming out of the pipe in a bucket over a fixed time period. Measuring the weight of
the water in the bucket and dividing this by the time taken to collect this water gives a rate of
accumulation of mass. This is know as the mass flow rate.
For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg,
then:
mass flow rate m =
mass of fluid in bucket
time taken to collect the fluid
=
=
- =
- &
. .
. / ( )
8 0 2 0
7
0857
1
kg s kg s
Performing a similar calculation, if  we know the mass flow is 1.7kg/s, how long will it take to fill a
container with 8kg of fluid?
CIVE 1400: Fluid Mechanics Fluid Dynamics: The Momentum and Bernoulli Equations
48
time 
mass
mass flow rate
=
=
=
8
17
4 7
.
. s
3.2.2 Volume flow rate - Discharge.
More commonly we need to know the volume flow rate - this is more commonly know as discharge. (It is
also commonly, but inaccurately, simply called flow rate). The symbol normally used for discharge is Q.
The discharge is the volume of fluid flowing per unit time. Multiplying this by the density of the fluid
gives us the mass flow rate. Consequently, if the density of the fluid in the above example is
850 kg m
3
then:
discharge, Q
volume of fluid
time 
mass of fluid
density  time 
mass flow rate
density 
0.857
850 
=
=
×
=
=
=
= ×
=
- - 0 001008
1008 10
1008
3 3 1
3 3
. / ( )
. /
. /
m s m s
m s
l s
An important aside about units should be made here:
As has already been stressed, we must always use a consistent set of units when applying values to
equations. It would make sense therefore to always quote the values in this consistent set. This set of units
will be the SI units. Unfortunately, and this is the case above, these actual practical values are very small
or very large (0.001008m
3
/s is very small). These numbers are difficult to imagine physically. In these
cases it is useful to use derived units, and in the case above the useful derived unit is the litre.
(1 litre = 1.0 × 10
-3
m
3
). So the solution becomes 1008 . / l s. It is far easier to imagine 1 litre than  1.0 × 10
-
3
m
3
. Units must always be checked, and converted if necessary to a consistent set before using in an
equation.
3.2.3 Discharge and mean velocity.
If we know the size of a pipe, and we know the discharge, we can deduce the mean velocity
Discharge in a pipe
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