Dimensional Analysis and Dynamic Similarity

Introduction

When we were in school, you must have noticed the fact that while solving problems on physical quantities, we often were asked to make the Units the same. This is because this was an easy way to derive a relation between two identities- which have the same unit. Similarly, physical parameters having dimensions can be added or subtracted, or related to another quantity that will have the same dimension. Therefore, it is important to have a clear and concise concept of dimensions and units- to make the concept of Dimensional analysis clear and rigid in your mind. 

Dimensions and Units are an important part of measurements in the world of Physics and Mathematics. Dimensions measure the physical parameters- with no numerical value in them. Units are the numerical values assigned to the dimensions. For example, Length and Breadth are Dimensions; while Units are the numerical values that will help in measuring these dimensions. Units can be a meter (m), centimetre (cm), feet (ft), etc. Often, different symbols and Greek letters are used as a Unit. For example, ampere, ohm, etc. 

Since we have a basic idea about the concept of Units and Dimensions, let us proceed to our brief discussion on Dimensional Analysis. 

Dimensional Analysis

In the most basic sense, Dimensional Analysis is practised to conclude a relationship between different physical parameters of a Dimension. Since dimensions do not need numerical values; they are independent of numerical values, constants, and multiples. In the field of Engineering, the use of dimensional analysis is to highlight the association between physical parameters based on their fundamental aspects- mass, time, electric current, etc.

This analysis can be used to recognize the equivalent dimensional groups to procure Dynamic Similarity. Dynamic Similarity points that the fluids flow identically when there is a presence of two geometric vessels that have the same boundary levels and are similar. An important factor here is that the geometric vessels must have the same Reynolds number, too. 

We must keep in mind that in the world of Physics, there are two kinds of physical parameters. These are Derived and Fundamental. The Fundamental parameters are- length, time, mass, electric current, intensity, and substance quantity. The derived parameter is when we add two fundamental parameters together. Let us look at an example for a better understanding. 

Let us denote time with “s” and length with “m”. Length and Time are the Fundamental parameters in Physics. We know that speed is calculated by finding the ratio between the distance (or, length) and the time. Therefore, the unit for speed will be “m/s” or metres per second. This is a derived parameter in Physics. Because it is the combination of two fundamental parameters. These units and conversions solely depend on the requirements of the question or the desirable answer to the question. 

Another name of Dimensional Analysis is Unit Factor Method- because it helps in the evaluation of the units. Suppose, we want to convert 6 km into a metre. How will do that? Indeed, we will multiply the kilometre value by 1000 because 1 km = 1000 m. Therefore, in this case, the conversion factor is 1000. We must also remember that both sides of any equation must have the same dimensions, to make an association or relation. 

Now, let us have a glance at the Dimensional Analysis of Force. As we know Force is the product of Acceleration and Mass. Therefore, the dimension of mass is mass and acceleration can be velocity per time. Hence, the Dimensional analysis of Force is Mass, Velocity, time, and distance. Velocity can be measured as the product of distance and time.

Let us look at a Dimensional Analysis example and the uses of Dimensional Analysis. 

Example and Uses

• Suppose you have packed 12 pieces of clothes in a tiny bag. Now, let us consider that you have a total of 10 such bags. The total pieces of cloths can be calculated as-

12 * 10 = 120

Here, 10 is the conversion factor. Because it helps us to calculate the total pieces of clothes in all the tiny bags. 

• Let us convert 100 centimetres into feet. Here, we must know that-

1 ft = 12 inches and 1 inch = 2.5 cm. 

Therefore, to convert cm into inches, we must convert cm into inches and then convert the resulting inches in feet. Hence, 100 cm will be 3.3 inches. 

The above-mentioned Dimensional analysis examples can be used in our daily lives. Therefore, some of the uses of Dimensional analysis includes-

• Helps in making an association between different physical quantities
• Helps in conversion of one unit into another
• Helps in calculating the dimensions of various dimension constants
• Helps in checking the accuracy and correctness of an equation
• Helps in attaining Dynamic Similarity

Conclusion

Dimensions and Units are an important part of measurements in the world of Physics and Mathematics. Dimensions measure the physical parameters. Units are the numerical values assigned to the dimensions. The field of study of the association between physical parameters based on their dimensions and units is widely known as Dimensional Analysis. The use of dimensional analysis is to highlight the association between physical parameters based on their fundamental aspects of parameters. This analysis can be used to recognize the similar dimensional groups to procure Dynamic Similarity. The Dimensional analysis of Force includes Mass, Velocity, time, and distance.