Introduction of Dimensional Analysis
- Dimensional analysis is the study of the relation between physical quantities with the help of dimensions and units of measurement. It helps in performing mathematical calculations smoothly without changing the units.
- The dimensional analysis combines dimensional variables, non-dimensional variables, and dimensional constants into non-dimensional parameters, which reduces the number of necessary independent parameters to solve a problem.
A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension.
Primary dimensions (Fundamental or Basic dimensions):
- All physical quantities are measured by comparison with length (L), Mass(M), Time(T).
- There are seven basic dimensions named mass (m), length (L), time (t), temperature (T), electric current (i), amount of light and amount of matter. These dimensions are called fundamental dimensions.
They possess more than one fundamental dimension, i.e. Velocity is LT-1
Methods of Dimensional Analysis
The relation among know variables can be determined by these two methods:
- Rayleigh method
- Buckingham -theorem
It is used to establish the relation for a variable that depends on not more than three or four variables.
Let X be a variable that depends on X1, X2, X3, then, according to the Rayleigh method.
X=f (X1, X2, X3)
Buckingham π- theorem:
It defines that “Any problem having n variables and out of which m variables have primary dimensions (such as M, L, T) then equation establishing a relation between all the variables will have (n-m) dimensionless groups”.
These dimensionless groups represent as π groups, and they can be written in the form of:
πl = f (π2, π3,……. πn-m)
There are some conditions that must be considered in this method for solving it:
- Each fundamental dimension must be present in at least one of the m variables.
- Number of repeated variables is equal to the number of fundamental quantities.
- A recurring/repeating variables don't have the same dimensions and any two variables or more don't form a dimensionless quantity.
Method of selecting repeating variables:
The repeating variables is chosen with the consideration of the following points:
- Repeating variable should not include dependent variable.
- The selection of repeating variable should be in such a manner that one of the variables should represent the geometric property, the second should be the flow property, and the third variable should be the fluid property.
- The repeating variables selected should not form a dimensionless group.
- The repeating variables together must have the same number of fundamental dimensions.
- No two repeating variables should have the same dimensions.
Various Forces in Fluid Mechanics
1. Inertia force (Fi): It is the product of the mass and acceleration of the flowing fluid and acts in the opposite direction of acceleration.
Fi = ρL2V2
2. Surface tension force (Fs): It is defined as the product of surface tension and surface length of the flowing fluid.
Fs =σ L
3. Gravity force (Fg): It is defined as the product of mass and acceleration resulting from the gravity of the flowing fluid.
Fg = ρl3g
4. Pressure force (FP): It is equal to the product of pressure and normal cross-sectional area of the flowing fluid.
FP = P × A
⟹ FP = PL2
5. Viscous force (Fv): It is defined as the product of shear stress (τ) resulting from viscosity and surface area (As) of the flow.
Fv = μLv
6. Elastic force (Fe): It is equal to the product of elastic stress and area (A) of the flowing fluid.
Fe = KL2
Introduction of Similitude & Modelling
A similitude is a similarity between the model and prototype in every respect, which means that the model and prototype has similar properties. Three types of similarities must exist between the model and prototype are as follows:
- Geometric similarity: If the ratio of corresponding dimensions in model & Prototype are same.
- Kinematic similarity: The ratio of velocities or accelerations at corresponding points in model and prototype are the same for kinematic similarity. For kinematic similarity, Geometric Similarity is mandatory.
- Dynamic similarity: The ratio of forces at the corresponding points in model and prototype are the same. For dynamic similarity, geometric and kinematic similarities are necessary.
Reynold’s Model Law:
According to Reynold’s model law, the Reynold number for the model must be equal to the Reynold number for the prototype for dynamic similarity.
Froude’s Model Law:
According to the Froude model law, the Froude number for both the model and the prototype is equal for the dynamic similarity. Froude model law applies when the gravity force is the only predominant force.
Euler’s model law:
According to Euler’s model law, the Euler number for model and prototype should be equal to the dynamic similarity. Euler’s model law is applicable when the pressure forces are alone predominant in addition to the inertia force.
Weber model law:
Weber model law is applicable where surface tension force effects dominant to inertia force. Weber number of the model and its prototype should be the same for the dynamic similarity.
Mach Model Law:
This law is applicable where the elastic compression forces predominate over inertia force. The Mach number of the model and its prototype should be equal for the dynamic similarity.
Undistorted models are those models that are geometrically similar to their prototypes. Thus, the scale ratio for the linear dimensions of the model and its prototype is the same.
A model is said to be distorted if it is not geometrically similar to its prototype. For a distorted model, different scale ratios for the linear dimensions are adopted.
In the case of rivers, harbours, reservoirs, etc., two different scale ratios are considered. One scale ratio for horizontal and while the other for vertical dimensions are taken.
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