Dimensional analysis - Methods of Dimensional analysis

Dimensional analysis :

Dimensional analysis deals with the dimensions of physical quantities. Dimensional analysis reduces the number of variables in a fluid phenomenon by combining the some variables to form non dimensional parameters. Instead of observing the effect of individual parameters the effect of non-dimensional parameters are studied. All physical phenomena is expressed in terms of a set of basic or fundamental dimensions. In fluid mechanics mass (m), length (L), and time (T) or force (F), length (L) and time (T) are considered as fundamental quantities. These two systems are known as MLT system and FLT system. These systems of dimensions are related to Newton's second law of motion i.e. Force = mass x acceleration or

F = M x L/T²  

^{Other physical quantities can be expressed by this quantities.}  

Advantages of  dimensional analysis

There are a lot of advantages of Dimensional analysis and similitude.

• By dimensional analysis number of experiments can be reduced. 
• Dimensional analysis help us to do experiments in air or water and then applying the results to a fluid which is less convenient to work with. Such as gas, steam or oil. 
• Cost can be reduced by doing experiments with the models of full size operations. Performance of the prototype can be determined from the test models. 
• Models can be used for the design of ships, Airplanes, pumps , turbines, dams, river channels, rockets  and missiles etc.  Model can bigger, smaller or of the same size of the prototypes. 

Methods of Dimensional analysis

The number of dimensional variables can be reduced into a smaller number of dimensionless parameters by several methods. Commonly used two types of methods are  

i ) Rayleigh's Method 

ii ) Buckingham Pi Method

Rayleigh's method 

This method was proposed by Lord Rayleigh in 1889. In this method a functional relationship is expressed in an exponential form which is dimensionally homogeneous. For exmaple , if A_{1 is a dependent variable and} A_2, A₃ A₄  …………… A_{n are independent variables in a phenomenon, the functional equation can be written as} 

A_{1  = f (}A_2, A₃ A₄  …………… A_n)

This equalition can be written in the exponential form using powers a,b,c .........n as 

A₁ = K[ A₂^a A₃^b A₄^c  …………… Anⁿ ]

where K is a dimensionless constant. Now the dimensions of the each of the quantities A_1, A_2, A_{3 ,} A₄  …………… A_{n are written and the sum of the exponents of fundamental quantities on both sides are equated. After solution of the equations the values ofa , b, c, .... are found out and these values are substituted in the main equation. From the new main equation, after simplification yields dimensionless groups controlling the phenomenon. However, when a large number of parameters are involved this method becomes complicated.} 

_{Buckingham Pi Theorem} 

<sub>According to this theorem if there are n dimensional variables in a dimensionally homogeneous equation described by m fundamental dimensions they may be grouped in (n-m) dimensionless groups. Buckingham referred to this dimensionless groups as Pi groups. The advantage of this theorem  is that one can predict the number of dimensionless groups are to be expected. For the application of this method, m number of repeating variables are selected and dimensionless groups are obtained by each one of the remaining variables one at a time. Generally, a geometric property (such as length), a fluid property (such as mass density) and a flow characterstics (such as velocity) are generally most suitable as repeating variables.   

Read :</sub> 

• How to read a moody diagram 
• How to solve fluid mechanics problems by moody chart 
• Newton's law of viscosity 
• Importance of fluid Mechanics for engineers and physicists 
• Fluid and some of its properties