chapter 7_part 2
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jjTRANSCRIPT
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Dr. Mohammad Shakir Nasif
Email: [email protected]
Tel: 05-3687026
Office Location: 17-03-11
3/10/2015 1:46 PM 1
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Chapter 7 Assignment
• Question: 7.3, 7.7, 7.29, 7.43, 7.45, 7.52, 7.80
Quiz: Monday or Tuesday 16 or 17 March
2015.
• Quiz will start 5 minutes from the start of the
lecture) so make sure you arrive before the
time).
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Some comments about
dimensional analysis• There are also other methods in dimensional
analysis but the method of repeating variables
is the easiest.
• There is not a unique set of pi terms which
arises from a dimensional analysis. However,
the required number of pi terms is fixed.
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Common Dimensionless Groups in
Fluid MechanicsFrom dimensionless terms, many variables arise which are commonly used
in fluid mechanics problems.
Fortunately not all of these variables are used in one problem, however if
combination of some of these variables are present, it is a standard practice
to combine them into dimensionless groups (PI terms).
These combinations appear so frequent that special names are associated
with them.
It is also possible to provide physical interpretation to the dimensionless
groups which can be helpful in assessing their influence in particular
application.
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Common Dimensionless Groups in
Fluid MechanicsFor example:
This term is dimensionless, which is very well known in fluid mechanics, heat
transfer….etc. It is called Reynolds number (Re).
VL
VLRe
The physical interpretation to the
dimensionless term (Reynolds number ) isforceViscous
forceInertiaVLRe
•When flow enters any region there will be some
disturbances. If the flow velocity is not too fast, these
disturbances get damped out by the fluid viscosity.
•We can see that the velocity in Reynolds number is on top
and the viscosity is on the bottom. For a large Reynolds
number this means that the velocity X length are large
compared to the viscosity.
•At a certain large Reynolds number, the flow is moving too
fast for the viscosity to damp out the disturbance, hence it is
called turbulent flow.
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Common Dimensionless Groups in Fluid Mechanics
These groups have
be developed by
using dimensional
analysis, then it
has been realised
that they have
been widely
repeated in
research. Hence
they were given
names.
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Common Dimensionless Groups in
Fluid Mechanics (cont.)
• Froude number (Fr no): It is the ratio of the inertia force on an element of fluid to the weight of the element.
• It is important in problems involving the study of flow of water around ships, or flow through rivers or open conduits.
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Common Dimensionless Groups in
Fluid Mechanics (cont.)
• Euler number (Eu no): Represents the ratio of the pressure force to the inertia force.
• It is normally used where pressure or pressure difference between two points is an important variables.
• For problems where cavitation is of concern, this number is commonly used.
• It can be also called as Cavitation number
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Common Dimensionless Groups in
Fluid Mechanics (cont.)
• Weber number (We no). It is the ratio of inertia force to surface tension force.
• It is important in problems in which there is an interface between two fluids, in this situation the surface tension will play an important role.
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Common Dimensionless Groups in
Fluid Mechanics (cont.)
• Mach number (Ma number): Represents the ratio of the fluid speed to the sonic speed.
• For jet fighters or airplanes fly with a speed higher than sonic speed, the Ma >1.
• If less than sonic speed Ma<1.
• If equal to sonic speed Ma=1
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Flow Similarity and Model Studies• A question comes to our mind what’s the importance of all these
numbers and how we can use them.
• These numbers applies when we want to study certain phenomena and it is not possible to construct full scale model. Therefore we can construct small scale model and use the same numbers in our study.
• We call it Similarity between full scale and small scale model.
• Geometric Similarity
– Model and prototype have same shape
– Linear dimensions on model and prototype correspond within constant scale factor
• Kinematic Similarity
– Velocities at corresponding points on model and prototype differ only by a constant scale factor
• Dynamic Similarity
– Forces on model and prototype differ only by a constant scale factor
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Flow Similarity and Model Studies
• Example: Drag on a Sphere
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Flow Similarity and Model Studies
• Example: Drag on a Sphere
For dynamic similarity …
… then …
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What is “Concept of Similitude”?
• Similitude is the study of predicting prototype
conditions (flow, pressure distribution….etc.)
from model (small model) test observation.
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Concept of Similitude
• The concept of similitude is used so that
measurements made on one system (for
example, in the laboratory) can be used to
describe the behavior of other similar systems
(outside laboratory)
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Modeling and Similitude
• A model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect.
• The physical system for which the predictions are to be made is called the prototype.
• Usually a model is smaller than the prototype and therefore, easier to handle in the lab.
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Modeling and Similitude (cont.)
Model Design Conditions (or Similarity Requirements
or Modeling Laws)
• To achieve similarity between model and prototype
behavior, all the corresponding pi terms must be
equated between model and prototype
– Geometric Similarity
– Dynamic Similarity
– Kinematic Similarity
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Modeling and Similitude (cont.)
Geometric Similarity
A model and prototype are geometrically
similar if and only if all body dimensions in
all three coordinates have the same linear-
scale ratio. All angles are preserved in
geometric similarity. All flow directions are
preserved. The orientations of model and
prototype w.r.t. the surroundings must be
identical.
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Modeling and Similitude (cont.)
Kinematic Similarity
Velocities are related to the full scale by a
constant scale factor. They also have the
same directions as in the full scale.
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Modeling and Similitude (cont.)
Dynamic Similarity
Forces are related to full scale by a constant
factor. Also requires geometric and kinematic
similarity.
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Similitude Summary
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Example 3
The drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be
determined through a wind tunnel test using a geometrically similar 0.4-m-
diameter model dish. Assume standard air for both model and prototype.
(a) At what air speed should the model test be run?
(b) With all similarity conditions satisfied, the measured drag on the model
was determined to be 170 N. What is the predicted drag on the prototype
dish?
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Example 3 SolutionThe drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be
determined through a wind tunnel test using a geometrically similar 0.4-
m-diameter model dish. Assume standard air for both model and
prototype.
(a) At what air speed should the model test be run?
(b) With all similarity conditions satisfied, the measured drag on the
model was determined to be 170 N. What is the predicted drag on the
prototype dish?
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Example 3 SolutionThe drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be
determined through a wind tunnel test using a geometrically similar 0.4-
m-diameter model dish. Assume standard air for both model and
prototype.
(a) At what air speed should the model test be run?
(b) With all similarity conditions satisfied, the measured drag on the
model was determined to be 170 N. What is the predicted drag on the
prototype dish? Drag force
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Example 4 The drag on an airplane shown in Fig.E7.7 cruising at 386
km/h in standard air is to be determined from tests on a
1:10 scale model placed in a pressurized wind tunnel. To
minimize compressibility effects, the air speed in the wind
tunnel is also to be 386 km/h.
Determine:
a)The required air pressure in the tunnel (assuming the
same air temperature for model and prototype), and
b)The drag on the prototype corresponding to a measured
force of 4 N on the model.
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Example 4
a) Drag can be predicted from a geometrically similar model if the Reynolds numbers of the
prototype and the model are the same. Thus:
For this example, Vm = V and lm/l = 1/10 so that:
And therefore
The result shows that the same fluid with ρm = ρ and μm = μ can’t be used if Reynolds number
similarity to be maintained. Instead, we can pressurize the wind tunnel to increase the density
of the air (with assumption that increase in density doesn’t significantly change the viscosity).
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Example 4
Therefore, if the viscosity is the same, the equation becomes:
For an ideal gas, p = ρRT so that:
And therefore the wind tunnel need to be pressurized so that:
Since the prototype is at standard atmospheric pressure, the required pressure in the wind
tunnel is:
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Example 4
b) The drag could be obtained from:
or
Thus, for a drag of 4 N on the model the corresponding drag on the prototype is: