We were
discussing the basic concept of streamline
and equipotential line, dimensional
homogeneity, Buckingham
pi theorem, difference
between model and prototype, basic
principle of similitude i.e. types of similarity and various forces acting on moving fluid in the subject of fluid mechanics, in our recent posts.
Now we
will go ahead to understand the basic concept of model laws or similarity laws in
the subject of fluid mechanics with the help of this post.
Model laws or similarity laws
For the
dynamic similarity between the model and the prototype, ratio of corresponding
forces acting on corresponding points in the model and the prototype should be
same.
Ratios of the forces are dimensionless numbers. Therefore we can say that for the dynamic
similarity between the model and the prototype, dimensionless numbers should be
equal for the model and the prototype.
However,
it is quite difficult to satisfy the condition that all the dimensionless
numbers should be equal for the model and the prototype.
However for
practical problems, it is observed that one force will be most significant as
compared to others and that force is considered as predominant force. Therefore
for dynamic similarity, predominant force will be considered in practical
problems.
Therefore,
models are designed on the basis of ratio of force which is dominating in the phenomenon.
Hence, we can
define the model laws or similarity laws as the law on which models are designed
for the dynamic similarity.
There are following types of model laws
Reynold’s
Model law
Froude Model
law
Euler
Model law
Weber
Model law
Mach Model
law
Reynold’s Model law
Reynold’s
model law could be defined as a model law or similarity law where models are
designed on the basis of Reynold’s numbers.
According
to the Reynold’s model law, for the dynamic similarity between the model and the
prototype, Reynold’s number should be equal for the model and the prototype.
In simple,
we can say that Reynold’s number for the model must be equal to the Reynold’s number
for the prototype.
As we know
that Reynold’s number is basically the ratio of inertia force and viscous
force, therefore a fluid flow situation where viscous forces are alone
predominant, models will be designed on the basis of Reynold’s model law for the
dynamic similarity between the model and the prototype.
Image: Reynold’s
model law
Where,
Vm
= Velocity of the fluid in the model
Lm
= Length of the model
νm
= Kinematic viscosity of the fluid in the model
VP
= Velocity of the fluid in the prototype
LP
= Length of the prototype
νP
= Kinematic viscosity of the fluid in the prototype
Models based on the Reynold’s model law
Pipe flow
Resistance
experienced by submarines, airplanes etc.
Froude Model law
Froude model
law could be defined as a model law or similarity law where models are designed
on the basis of Froude numbers.
According
to the Froude model law, for the dynamic similarity between the model and the prototype,
Froude number should be equal for the model and the prototype.
In simple,
we can say that Froude number for the model must be equal to the Froude number for
the prototype.
As we know that Froude number is basically the ratio of inertia force and gravity force,
therefore a fluid flow situation where gravity forces are alone predominant,
models will be designed on the basis of Froude model law for the dynamic
similarity between the model and the prototype.
Image: Froude
model law
Where,
Vm
= Velocity of the fluid in the model
Lm
= Length of the model
gm
= Acceleration due to gravity at a place where model is tested
VP
= Velocity of the fluid in the prototype
LP
= Length of the prototype
gP
= Acceleration due to gravity at a place where prototype is tested
Models based on the Froude model law
Free
surface flows such as flow over spillways, weirs, sluices, channels etc,
Flow of
jet from an orifice or from a nozzle,
Where
waves are likely to be formed on surface
Where
fluids of different densities flow over one another
Euler’s Model law
Euler’s model
law could be defined as a model law or similarity law where models are designed
on the basis of Euler’s numbers.
According
to the Euler’s model law, for the dynamic similarity between the model and the prototype,
Euler’s number should be equal for the model and the prototype.
In simple,
we can say that Euler’s number for the model must be equal to the Euler’s number
for the prototype.
As we know
that Euler’s number is basically the ratio of pressure force and inertia force,
therefore a fluid flow situation where pressure forces are alone predominant,
models will be designed on the basis of Euler’s model law for the dynamic
similarity between the model and the prototype.
Image: Euler’s
model law
Where,
Vm
= Velocity of the fluid in the model
Pm
= Pressure of fluid in the model
ρm
= Density of the fluid in the model
VP
= Velocity of the fluid in the prototype
PP
= Pressure of fluid in the prototype
ρP
= Density of the fluid in the prototype
Models based on the Euler’s model law
Euler’s model
law will be applicable for a fluid flow situation where flow is taking place in
a closed pipe, in which case turbulence will be fully developed so that viscous
forces will be negligible and gravity force and surface tension force will be
absent.
Weber Model law
Weber model
law could be defined as a model law or similarity law where models are designed
on the basis of Weber numbers.
According
to the Weber model law, for the dynamic similarity between the model and the prototype,
Weber number should be equal for the model and the prototype.
In simple,
we can say that Weber number for the model must be equal to the Weber number for
the prototype.
As we know
that Weber number is basically the ratio of inertia force and surface tension
force, therefore a fluid flow situation where surface tension forces are alone
predominant, models will be designed on the basis of Weber model law for the
dynamic similarity between the model and the prototype.
Image: Weber
model law
Where,
Vm
= Velocity of the fluid in the model
σm
= Surface tension force in the model
ρm
= Density of the fluid in the model
Lm
= Length of surface in the model
VP
= Velocity of the fluid in the prototype
σP
= Surface tension force in the prototype
ρP
= Density of the fluid in the prototype
LP
= Length of surface in the prototype
Models based on the Weber model law
Capillary
rise in narrow passage
Capillary
movement of water in soil
Capillary waves
in channels
Flow over
weirs for small heads
Mach Model law
Mach model
law could be defined as a model law or similarity law where models are designed
on the basis of Mach numbers.
According
to the Mach model law, for the dynamic similarity between the model and the prototype,
Mach number should be equal for the model and the prototype.
In simple,
we can say that Mach number for the model must be equal to the Mach number for
the prototype.
As we know
that Mach number is basically the ratio of inertia force and Elastic force,
therefore a fluid flow situation where elastic forces are alone predominant,
models will be designed on the basis of Mach model law for the dynamic
similarity between the model and the prototype.
Image: Mach
model law
Where,
Vm
= Velocity of the fluid in the model
Km
= Elastic stress for model
ρm
= Density of the fluid in the model
VP
= Velocity of the fluid in the prototype
KP
= Elastic stress for prototype
ρP
= Density of the fluid in the prototype
Models based on the Mach model law
Water
hammer problems
Under
water testing of torpedoes
Aerodynamic
testing
Flow of
aeroplane and projectile through air at supersonic speed
We will discuss another important topic i.e. Euler's Equation of motions in the subject of fluid mechanics in our next post.
Do you
have any suggestions? Please write in comment box.
Reference:
Fluid
mechanics, By R. K. Bansal
Image Courtesy: Google
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