Similitude in Fluid Mechanics - Similarity Laws

Similitude in Fluid Mechanics 

There are a lot of advantages of dimensional analysis and similitude in fluid mechanics. With the application of dimensional analysis and similitude or laws of similarity, number of experiments can be reduced. Costs can be reduced by doing experiments with models of the full size apparatus. Performance of the prototype can be predicted from tests made with model. With the help of similarity laws the results obtained from experiments done with air or water can be applied to a fluid, which is less convenient to work with, such as gas, steam or oil.  

Description of Similarity Law for Fluid Mechanics

The similarity relation between a prototype and its model is known as similitude. Types of similarities must exist for complete similitude between a model and its prototype. These are - 

j) Geometric similarity
ii) Kinematic similarity
iii) Dynamic similarity 

Geometric Similarity laws in fluid mechanics 

A model and its prototype are said to be in geometric similarity, if the ratios of their corresponding linear dimensions are equal (such as length, breadth, width etc.) For geometric similarity, the corresponding areas are related by the square of ratio and the corresponding volumes by the cube of the length scale ratio.

 Length scale ratio = l_m/l_p = b_m/b_p = d_m/d_p  

(Length scale ratio)2 = Area-model / Area-prototype = (l_m/l_p )²  = ( b_m/b_p )² = ( d_m/d_p)²

(Length scale ratio)3 = Volume-model / Volume-prototype = (l_m/l_p )³ = ( b_m/b_p )³ = (d_m/d_p )³


Here, l_m,b_m,d_m  are the linear dimensions of the model and l_p,b_p,d_p are the linear dimensions of the prototype.  

Kinematic Similarity/ similitude in fluid mechanics 

A model and its prototype are said to be kinematically similar if the flow patterns in the model and the prototype for any fluid motion has geometric similarity and if the ratios of the velocities as well as accelerations at all corresponding  points in the flow is the same.

Let,

V1 and V2 — velocities of fluid in prototype at points 1 and 2

v1 and v2 — velocities of fluid in model at corresponding points 1 and 2

A1 and A2 — acceleration of fluid in prototype at points I and 2

a1 and a2 — acceleration of fluid in model at corresponding points 1 and 2

Velocity ratio = V1/v1 = V2/v2

Acceleration ratio = A1/a1 = A2/a2

Dynamic Similarity law in fluid mechanics 

A model and its prototype are said to be dynamically similar if the ratio of the forces acting at the corresponding points are equal. Geometric and kinematic similarities exist for dynamically similar systems. 

1onding points are equal. Geometric and kinem

F1 and F2 — forces acting in prototype at points I and 2

 f1 and f2 — forces acting in the model at the corresponding points 1 and 2

Now F1/f1 = F2/f2  = constant

If the dynamic, kinematic and geometric similarities can be obtained then experiments with models can give very accurate results. That's why similitude in fluid mechanics is very important. 

Please read:

• Principles of similitude fir engine design. 
• Methods of Dimensional analysis 
• An Introduction to fluid mechanics 
• A Complete Guide to Fluid Mechanics