Dimensional Analysis

Dimensional analysis is a mathematical technique that makes use of the study of dimensions as an aid to the solution of several engineering problems. It deals with the dimensions of the physical quantities is measured by comparison, which is made with respect to an arbitrarily fixed value.

Length L, mass M and Time T are three fixed dimensions that are of importance in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the temperature is also taken as fixed dimension. These fixed dimensions are called fundamental dimensions or fundamental quantity.

Secondary or Derived quantities are those quantities that possess more than one fundamental dimension. For example, velocity is defined by distance per unit time (L/T), density by mass per unit volume (M/L³) and acceleration by distance per second square (L/T²). Then the velocity, density and acceleration become as secondary or derived quantities. The expressions (L/T), (M/L³) and (L/T²) are called the dimensions of velocity, density and acceleration respectively.

Dimensional Analysis

Dimensional homogeneity

Dimensional homogeneity means the dimensions of each term in an equation on both sides are equal. Thus if the dimensions of each term on both sides of an equation are the same the equation is known as the dimensionally homogeneous equation.

The powers of fundamental dimensions i.e., L, M, T on both sides of the equation will be identical for a dimensionally homogeneous equation. Such equations are independent of the system of units.
Let us consider the equation V = u + at
Dimensions of L.H.S = V= L/T = LT-1
Dimensions of R.H.S = LT^-1 + (LT^-2) (T)
= LT^-1 + LT^-1
= LT^-1
Dimensions of L.H.S = Dimensions of R.H.S = LT^-1

Therefore, equation V = u + at is dimensionally homogeneous

Uses of Dimensional Analysis

It is used to test the dimensional homogeneity of any derived equation.
It is used to derive the equation.
Dimensional analysis helps in planning model tests.

Methods of Dimensional Analysis

If the number of variables involved in a physical phenomenon is known, then the relationship among the variables can be determined by the following two methods.

Rayleigh's Method

Rayleigh‟s method of analysis is adopted when a number of parameters or variables is less (3 or 4 or 5).
If the number of independent variables becomes more than four, then it is very difficult to find the expression for the dependent variable

Buckingham's (Π– theorem) Method

If there are n – variables in a physical phenomenon and those n-variables contains 'm' dimensions, then the variables can be arranged into (n-m) dimensionless groups called Π terms.
If f (X₁, X₂, X₃, ……… X_n) = 0 and variables can be expressed using m dimensions then. f (Π1, Π2,Π3, ……… Πn - m) = 0 Where, Π1, Π2, Π3, ……… are dimensionless groups.
Each Π term contains (m + 1) variables out of which m are of repeating type and one is of non-repeating type.
Each Π term being dimensionless, the dimensional homogeneity can be used to get each Π term.

Method of Selecting Repeating Variables

Avoid taking the quantity required as the repeating variable.
Repeating variables put together should not form a dimensionless group.
No two repeating variables should have same dimensions.
Repeating variables can be selected from each of the following properties
- Geometric property - Length, Height, Width, Area
- Flow property - Velocity, Acceleration, Discharge
- Fluid property – Mass Density, Viscosity, Surface Tension

Model Studies

Before constructing or manufacturing hydraulics structures or hydraulics machines tests are performed on their models to obtain desired information about their performance.
Models are a small scale replica of actual structure or machine.
The actual structure is called prototype.

Similitude

It is defined as the similarity between the prototype and its model. It is also known as similarity. There three types of similarities and they are as follows. 

Geometric similarity

Geometric similarity is said to exist between the model and prototype if the ratio of corresponding linear dimensions between model and prototype are equal. i.e.

where Lr is known as scale ratio or linear ratio.

V_m/V_p= √(Lr) .... (Velocity Scale Ratio)

t_m/t_p= √(Lr) .....(Time Scale Ratio)

Hence, a_m/a_p= 1 .....(Acceleration Scale Ratio)

Kinematic Similarity

Kinematic similarity exists between prototype and model if quantities such at velocity and acceleration at corresponding points on model and prototype are same.

Where Vr is known as velocity ratio

Dynamic Similarity

Dynamic similarity is said to exist between model and prototype if the ratio of forces at corresponding points of model and prototype is constant.

Where Fr is known as force ratio.

Dimensionless Numbers

Following dimensionless numbers are used in fluid mechanics.

Reynolds's number
Froude's number
Euler's number
Weber's number
Mach number

Reynolds’s number

It is defined as the ratio of inertia force of the fluid to viscous force.

N_Re= F_i/F_v

Froude’s Number (Fr)

It is defined as the ratio of square root of inertia force to gravity force.

F_r=√F_i/F_g

Model Laws (Similarity laws)

Reynolds’s Model Law

For the flows where in addition to inertia force, the similarity of flow in the model and predominant force, the similarity of flow in model and prototype can be established if Re is same for both the system.
This is known as Reynolds's Model Law.
R_e for model = R_e for prototype
(NR_e)m = (NR_e)p

Applications

In the flow of in-compressible fluids in closed pipes.
The motion of submarine completely under water.
Motion of airplanes.

Froude’s Model Law

When the force of gravity is predominant in addition to inertia force then similarity can be established by Froude's number.
This is known as Froude's model law.
(Fr)m = (Fr)p