Dimensional analysis means analysis of the dimensions of physical quantities. Dimensional analysis lowers the number of variables in a fluid phenomenon by mixing the some variables to form parameters which have no dimensions.
All physical phenomena are expressed in terms of a set of elemental 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 called MLT system and FLT system. These systems of dimensions are somewhat related to Newton’s second law of motion i.e. Force = mass x acceleration or
F = M x L/T2
Other physical quantities are expressed by these quantities.
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 could be lowered into a smaller number of dimensionless parameters by various methods. Commonly used are two types of methods:
i ) Rayleigh’s Method
ii ) Buckingham Pi Method
This method expresses a functional relationship in an exponential form which is homogeneous, dimensionally. For instance, if A1 is a dependent variable and A2, A3 A4 …………… An are independent, in a phenomenon, the functional equation could be written as
A1 = f (A2, A3 A4 …………… An)
This equation is written in the exponential form using powers a,b,c ………n as shown below:
A1 = K[ A2a A3b A4c …………… Ann ]
Where K, is a dimensionless constant.
Now dimensions of the quantities A1, A2, A3 , A4 …………… An are written and equated with sum of the exponents of fundamental quantities on both sides. After solution of the equations the values of a , b, c, …. Are found out and these values are substituted to the main equation.
From the new equation, after simplification produces dimensionless groups that control the phenomenon. To mention, with involvement of large number of parameters, this method becomes complicated.
Buckingham Pi Theorem
This theory says that if there are n dimensional variables in a dimensional equation described by m fundamental dimensions they might be grouped in (n-m) dimensionless groups.
This dimensionless group is referred to as Pi groups. The advantage of this theorem is that one can predict the number of dimensionless groups that could be expected. For the application of this method, m number of repeating variables is selected and dimensionless groups are obtained by each one of the remaining variables one at a time. Usually, a geometric property (like length), a fluid property (like mass density) and flow characteristics (like velocity) are most suitable as repeating variables.
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