6.5 Similitude

MODEL

Model is similar with real object required in certain scale ratio. It is needed to be tested in laboratory with similar condition which is likely to be occurring in real phenomena. The size of the model might not be smaller than prototype.

PROTOTYPE

Prototype is an actual object that is in its full scale and needed to be reacting properly in real phenomena for mankind example like: ship, spillway structure in open channel.

However, scaling effect might affect the characteristic of a real-world prototype. Factor which lead to scale effect might be air concentration, Earth characteristics.

Advantages for using similarities.

1. Performances of object can be predicted.
2. Economics and easy to build, until it’s tested to reach a value.
3. Non-functional structure can be measured. Such as dam.

Perfect model-prototype similarity: Mechanical similarity

A physical scale model satisfying mechanical similarity is completely similar to its real-world prototype and involves no scale effects.

Mechanical similarity requires three criteria:

1. Geometric similarity
2. Kinematic similarity
3. Dynamic similarity

Geometric similarity

Boundaries between 2 flows are exact dilation (expand) or shrinkages of each other.

Length, Lp/Lm  = Lr

Area,   Ap/Am  = Ar

Volume, Vp/Vm  = Vr

Kinematic similarity

Streamlines between flows are exact dilation or shrinkage of each other.

Velocity scale ratio, Vr = Vp/Vm 

Discharge scale ratio, 

Acceleration scale ratio, 

Dynamic similarity

Corresponding forces at corresponding locations mutually parallel and have the same ratio of magnitudes. This ratio is the same for all sets of corresponding forces.

∑F = Fg + Fp + Fv + Fe + Ft  = Resultant

Therefore: Fg + Fp + Fv + Fe + Ft + Fi = 0

Where Fi = -resultant force

Where:

Gravitational force

 Fg = mg = ρgL^3

Pressure

 Fp = ΔpA = ΔpL^2

Viscosity

 Fv = μ(V/L)A = μ(V/L)L^2 = μVL

Elasticity

Fe  = EvA = EvL^2

Surface Tension

Ft  = σL

Inertia force

 Fi = ma = ρL^2v^2

In a general flow field, complete similarity between a model and prototype is achieved only when there is geometric, kinematic and formed dynamic similarity to perform for real life situation.

For further description about dimensionless quantities:

http://www.ipplm.pl/bib/nrl/fluid.pdf

6.4 Methods

There are a lot of methods can be used to reduce into a smaller number of dimensionless parameters. We will discuss about the two commonly used methods.

6.4.1 Rayleigh’s Method
A basic method to dimensional analysis method and can be simplified to yield dimensionless groups controlling the phenomenon.
The Rayleigh’s method is based on the following steps:
1. First of all, write the functional relationship according to the given data.

2. Now write the equation in terms of a constant with exponents i.e. powers a, b,c,...

3. With the help of the principle of dimensional homogeneity, find out the values of a, b,
c… by obtaining simultaneous equation and simplify it.

4. Now substitute the values of these exponents in the main equation, and simplify it.


Example 1
Express dimensionless equation for the speed v with a wave pressure travels through a fluid. Consider the physical factors probably influence the speeds are compressibility, E, (or K), density,p and kinematics viscosity, v.

Solution: 
Write the fundamental dimension (used MLTθ) for all dimensions given

V= [L/T]          K= [F/L^2 ]= [M/(LT^2 )]    p = [M/L^3 ]    v = [L^2/T]

Let write the equation like this:  V = CK^a p^b v^d

Inserts fundamental dimension into the equation while C is dimensionless constant.
L/T = (M/(LT^2 ))^a (M/L^3 )^b (L^2/T)^d


To satisfy dimensional homogeneity, net power of each dimension must be identical on both sides of this equation. Thus,
For M: 0 = a + b
For L:   1 = -a + (-3b) + 2d
For T:  -1 = -2a – d
a = 1/2     b = -  1/2     d = 0

V = CK^(1/2) p^(-1/2) v^0

Or

V = C √(K/p)

Therefore, wave speed is not affected by the fluid’s kinematics viscosity, v.


6.4.2 The Buckingham pi Theorem

When a large number of variables are involved, Raleigh’s method becomes lengthy. In such circumstances, the Buckingham’s method is useful. This method expressed the variables related to a dimensional homogeneous equation as:

xr: f(xr,xr....x.)

where, the dimension at each section is the same.

Example:
The drag  D, which is a force, on a ship. What exactly is the drag a function of?

Say that we have n number of quantities (e.g. 6 quantities, which are D,l,ρ,μ,V, and g) and m number of dimensions (e.g. 3 dimensions, which are M, L, and T). These quantities can be reduced to (n - m) independent dimensionless groups.

Step 1: Setup π groups
For the MLT System, m = 3, so choose A1, A2, and A3 as the repeating variables.

Using the Buckingham π Theorem on the Drag Equation:
f(D, l, ρ, μ, V, g) = 0
Where m = 3, n = 6, so there will be n - m = 3 π groups.

We will select ρ, V, and l as the repeating variables (RV), leaving the remaining quantities as D, μ, and g. Note that if the analysis does not work out, we could always go back and repeat using new RVs. Thus,

Which are all dimensionless quantities, i.e. having units of

Step 2: Determine π groups

For the first π group,
π1
M^0 T^0 L^0=(〖M/L^3 )〗^(x_1 ) (L/T)^(y_1 ) (L)^(z_1 ) (ML/T^2 )

Expanding and collecting like units, we can solve for the exponents:
For M: 0 = x1 + 1 ⇒ x1 = -1
For T: 0 = -y1 - 2 ⇒ y1 = -2
For L: 0 = -3x1 + y1 + z1 + 1 ⇒ z1 = 3(-1) - (-2) - 1 = -2
Therefore, we find that the exponents x1, y1, and z1 are -1, -2, and -2 respectively.

This means that the first dimensionless π group, π1, is

= D/(p V^2 l^2)

For the second π group,
π2 M^0 L^0 T^0=(〖M/L^3 )〗^(x_2 ) (〖L/T)〗^(y_2 ) (〖L)〗^(z_2 ) (M/LT)

Solving for the exponents,
For M: x2 + 1 = 0 ⇒ x2 = -1
For T: -y2 - 1 = 0 ⇒ y2 = -1
For L: -3x2 + y2 + z2 - 1 = 0 ⇒ z2 = 1 - (-1) + 3(-1) = -1

Thus,
π_2=ρ^(-1) V^(-1) l^(-1)=μ/( ρVl)=v/Vl    
   
However, we will now invert π2 so that,
π_2 = (Vl/v)^-1 = Reynolds Number      
It is permissible to exponentiate any π group, e.g. π-1, π½, π2, etc., to form a new group, as this does not alter the functional form.

For the third π group,
π3 M^0 L^0 T^0=(〖M/L^3 )〗^(x_3 ) (〖L/T)〗^(y_3 ) (〖L)〗^(z_3 ) (L/T^2 )
Solving for the exponents,
For M: x3 = 0 ⇒ x3 = 0
For T: -y3 - 2 = 0 ⇒ y3 = -2
For L: -3x3 + y3 + z3 + 1 = 0 ⇒ z3 = -1 - (-2) = 1

Thus,
π_3=ρ^0 V^(-2) lg=lg/V^2            
Raising it to the power of -½,
√(1/π_3 )= V/√lg =Froude Number    
          
Thus, the three π groups can be written together as
f(D/(ρl^2 V^2 ),Re,Fr)=0

Finally,
D/(ρl^2 V^2 )=f(Re,Fr)              

Source: YouTube

6.3 Dimensional Homogeneity

         An equation is said to be dimensionally homogeneous if the dimensions of the various terms on the LEFT and RIGHT sides of the equation are IDENTICAL.
There are two system, that is MLTθ and FLTθ where,
   M = mass
   F = force
   L =length
   T = time
   θ = temperature

Example 1.0:
  Force = mass x acceleration
In MLTθ system,

Example 1.1:
Consider the equation for the distance S travelled in time t by a particle having velocity u and acceleration f,

Dimensionally,        m = m/sec  sec + 1/2  m/〖sec〗^2  〖sec〗^2      or   L = L/T .T + 1/2  L/T^2 .T^2
                             m= m + ½ m                                L = L + L  
Hence, the terms on both sides of the equation have the same unit of length and therefore the equation is dimensionally homogeneous.

Source: YouTube

Source: http://www.daviddarling.info/encyclopedia/D/dimensional_analysis.html

6.2 Units and Dimensions

Dimensions	SI	US
Mass (M)	kg	lb
Length (L)	m	ft
Time (T)	s	s
Temperature (θ)	℃	℉

All equations related to a physical phenomenon must be dimensionally homogeneous. This is known as Principle of Dimensional Homogeneity.

6.1 Dimensional Analysis

Dimensional analysis is particularly helpful in experimental work because it provides a guide to the things that significantly influence the phenomena; thus it indicates the direction in which experiment work is important.

The significant of dimensional analysis are:

     1.      Useful for research study especially in design work by reducing the number of variables.

     2.      To express in dimensionless equation to find the significant of each parameters.

     3.      To simplify the analysis of complex phenomenon in systematic order.

Source: http://physicsjourneytolife.blogspot.com/2013/01/day-10-fundamentals-of-physics-14.html