Application of Dynamic Similarity | Fluid Mechanics for Mechanical Engineering PDF Download

The Application of Dynamic Similarity - The Dimensional Analysis

The concept:

A physical problem may be characterised by a group of dimensionless similarity parameters or variables rather than by the original dimensional variables.

 This gives a clue to the reduction in the number of parameters requiring separate consideration in an experimental investigation.

For an example, if the Reynolds number Re = ρV D_h /µ is considered as the independent variable, in case of a flow of fluid through a closed duct of hydraulic diameter D_h, then a change in Re may be caused through a change in flow velocity V only. Thus a range of Re can be covered simply by the variation in V without varying other independent dimensional variables ρ,D_h and µ.

In fact, the variation in the Reynolds number physically implies the variation in any of the dimensional parameters defining it, though the change in Re, may be obtained through the variation in anyone parameter, say the velocity V.

A number of such dimensionless parameters in relation to dynamic similarity are shown in Table 5.1. Sometimes it becomes diffcult to derive these parameters straight forward from an estimation of the representative order of magnitudes of the forces involved. An alternative method of determining these dimensionless parameters by a mathematical technique is known as dimensional analysis .

The Technique:

The requirement of dimensional homogeneity imposes conditions on the quantities involved in a physical problem, and these restrictions, placed in the form of an algebraic function by the requirement of dimensional homogeneity, play the central role in dimensional analysis.

There are two existing approaches;

• one due to Buckingham known as Buckingham's pi theorem

•  other due to Rayleigh known as Rayleigh's Indicial method

In our next slides we'll see few examples of the dimensions of physical quantities

 

Dimensions of Physical Quantities

All physical quantities are expressed by magnitudes and units.

For example , the velocity and acceleration of a fluid particle are 8m/s and 10m/s² respectively. Here the dimensions of velocity and acceleration are ms^-1 and ms^-2 respectively.

In SI (System International) units, the primary physical quantities which are assigned base dimensions are the mass, length, time, temperature, current and luminous intensity. Of these, the first four are used in fluid mechanics and they are symbolized as M (mass), L (length), T (time), and θ (temperature).

• Any physical quantity can be expressed in terms of these primary quantities by using the basic mathematical definition of the quantity.

• The resulting expression is known as the dimension of the quantity.

Let us take some  examples:

1. Dimension of Stress

Shear stress τ is defined as force/area. Again, force = mass × acceleration

Dimensions of acceleration = Dimensions of velocity/Dimension of time.

 

Dimension of area = (Length)² =L²

Hence, dimension of shear stress
   (19.1)


2.Dimension of Viscosity

Consider Newton's law for the definition of viscosity as
or 
 The dimension of velocity gradient du/dy can be written as

                 dimension of du/dy= dimension of u/dimension of y = (L / T)/L = T ^-1 

The dimension of shear stress τ is given in Eq. (19.1).

Hence dimension of


Dimensions of Various Physical Quantities in Tabular Format


 

Buckingham's Pi Theorem
Assume, a physical phenomenon is described by m number of independent variables like x₁ , x₂ , x₃ , ..., x_m

The phenomenon may be expressed analytically by an implicit functional relationship of the controlling variables as
f(x₁,x₂.x₃.....,x_m) = 0     (19.2)

Now if n be the number of fundamental dimensions like mass, length, time, temperature etc ., involved in these m variables, then according to Buckingham's p theorem -

The phenomenon can be described in terms of (m - n) independent dimensionless groups like π₁ ,π₂ , ..., π_m-n , where p terms, represent the dimensionless parameters and consist of different combinations of a number of dimensional variables out of the m independent variables defining the problem.

Therefore. the analytical version of the phenomenon given by Eq. (19.2) can be reduced to
     (19.3)

according to Buckingham's pi theorem

• This physically implies that the phenomenon which is basically described by m independent dimensional variables, 
  is ultimately controlled by (m-n) independent dimensionless parameters known as π terms.

Alternative Mathematical Description of (π) Pi Theorem
A physical problem described by m number of variables involving n number of fundamental dimensions (n < m) leads to a system of n linear algebraic equations with m variables of the form
                 (19.4)

or in a matrix form 

where,  




Determination of π terms

• A group of n (n = number of fundamental dimensions) variables out of m (m = total number of independent variables defining the problem) variables is first chosen to form a basis so that all n dimensions are represented . These n variables are referred to as repeating variables.

• Then the p terms are formed by the product of these repeating variables raised to arbitrary unknown integer exponents and anyone of the excluded (m -n) variables.

For example , if x₁ x₂ ...x_n are taken as the repeating variables. Then

• The sets of integer exponents a₁, a₂ . . . a_n are different for each p term.

• Since p terms are dimensionless, it requires that when  all the variables in any p term are expressed in terms of their fundamental dimensions, the exponent of all the fundamental dimensions must be zero.

•  This leads to a system of n linear equations in a, a₂ . . . a_n which gives a unique solution for the exponents. This gives  the values of a₁ a₂ . . . a_n for each p term  and hence the p terms are uniquely defined.

In selecting the repeating variables, the following points have to be considered:

1. The repeating variables must include among them all the n fundamental dimensions, not necessarily in each one but collectively.
2. The dependent variable or the output parameter of the physical phenomenon should not be included in the repeating variables.

No physical phenomena is represented when -

• m < n    because there is no solution   and
•  m = n   because there is a unique solution of the variables involved and hence all the parameters have fixed values.

. Therefore all feasible phenomena are defined with m > n .

• When m = n + 1, then, according to the Pi theorem, the number of pi term is one and the phenomenon can be expressed as
  f(π₁) = 0

where, the non-dimensional term π₁ is some specific combination of n + 1 variables involved in the problem.

When m > n+ 1 ,

1.  the number of π terms are more than one.

2.  A number of choices regarding the repeating variables arise in this case.

Again, it is true that if one of the repeating variables is changed, it results in a different set of π terms. Therefore the interesting question is which set of repeating variables is to be chosen , to arrive at the correct set of π terms to describe the problem. The answer to this question lies in the fact that different sets of π terms resulting from the use of different sets of repeating variables are not independent. Thus, anyone of such interdependent sets is meaningful in describing the same physical phenomenon.

From any set of such π terms, one can obtain the other meaningful sets from some combination of the π terms of the existing set without altering their total numbers (m-n) as fixed by the Pi theorem.

 

Rayleigh's Indicial Method
This alternative method is also based on the fundamental principle of dimensional homogeneity of physical variables involved in a problem.

Procedure-

1. The dependent variable is identified and expressed as a product of all the independent variables raised to an unknown integer exponent.

2. Equating the indices of n fundamental dimensions of the variables involved, n independent equations are obtained .

3. These n equations are solved to  obtain the dimensionless groups.

Example

Let us illustrate this method by solving the pipe flow problem

. Step 1 - ----- Here, the dependent variable Δp/l can be written as
                (where, A is a dimensionless constant.)

Step 2 -----Inserting the dimensions of each variable in the above equation, we obtain,
    
 

Equating the indices of M, L, and T on both sides, we get ,

   c + d = 1                             
 a + b - 3c - d = -2
 -a - d = -2  

 Step 3 -----There are three equations and four unknowns. Solving these equations in terms of the unknown d, we have

a = 2- d 
 b = -d - 1 
 c = 1- d

Hence , we can be written

 

Therefore we see that there are two independent dimensionless terms of the problem, namely,

• Both Buckingham's method and Rayleigh's method of dimensional analysis determine only the relevant independent dimensionless parameters of a problem, but not the exact relationship between them.

 For example, the numerical values of A and d   can never be known from dimensional analysis. They are found out from experiments.

 If the system of equations is solved for the unknown c, it results,


Therefore different interdependent sets of dimensionless terms are obtained with the change of unknown indices in terms of which the set of indicial equations are solved. This is similar to the situations arising with different possible choices of repeating variables in Buckingham's Pi theorem.