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Sunday, 20 May 2018

MODEL LAWS OR SIMILARITY LAWS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference 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.

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