Dynamic Similarity of Flows

Dynamic Similarity of Flows governed by Viscous, Pressure and Inertia Forces

The criterion of dynamic similarity for the flows controlled by viscous, pressure and inertia forces are derived from the ratios of the representative magnitudes of these forces with the help of Eq. (1a) to (1c) as follows:
 


The term   is known as Reynolds number, R_e after the name of the scientist who first developed it and is thus proportional to the magnitude ratio of inertia force to viscous force .(Reynolds number plays a vital role in the analysis of fluid flow)

The term Δp/pV²  is known as Euler number, E_u after the name of the scientist who first derived it. The dimensionless terms, R_e and E_u, represent the criteria of dynamic similarity for the flows which are affected only by viscous, pressure and inertia forces. Such instances, for example, are
    
1. the full flow of fluid in a completely closed conduit,
    
2. flow of air past a low-speed aircraft and
    
3. The flow of water past a submarine deeply submerged to produce no waves on the surface.

Hence, for a complete dynamic similarity to exist between the prototype and the model for this class of flows, the Reynolds number, R_e and Euler number, E_u has to be same for the two (prototype and model). Thus

where, the suffix p and suffix m refer to the parameters for prototype and model respectively.

In practice, the pressure drop is the dependent variable, and hence it is compared for the two systems with the help of Eq. (1d), while the equality of Reynolds number (Eq. (1c)) along with the equalities of other parameters in relation to kinematic and geometric similarities are maintained.

1. The characteristic geometrical dimension l and the reference velocity V in the expression of the Reynolds number may be any geometrical dimension and any velocity which are significant in determining the pattern of flow.
    
2. For internal flows through a closed duct, the hydraulic diameter of the duct D_h and the average flow velocity at a section are invariably used for l and V respectively.
    
3. The hydraulic diameter D_h is defined as D_h= 4A/P where A and P are the cross-sectional area and wetted perimeter respectively.

Dynamic Similarity of Flows with Gravity, Pressure and Inertia Forces

A flow of the type in which significant forces are gravity force, pressure force and inertia force, is found when a free surface is present.
 
Examples can be

1. The flow of a liquid in an open channel.
    
2. The wave motion caused by the passage of a ship through water.
      
3. The flows over weirs and spillways.

The condition for dynamic similarity of such flows requires
    
(i) the equality of the Euler number E_u (the magnitude ratio of pressure to inertia force),

(ii) The equality of the magnitude ratio of gravity to inertia force at corresponding points in the systems being compared.
       

(iii) In practice, it is often convenient to use the square root of this ratio  so to deal with the first power of the velocity.
    
From a physical point of view, equality of (1g)^1/2/V   implies equality of 1g/V²  as regard to the concept of dynamic similarity.

The reciprocal of the term (1g)^1/2/V  is known as Froude number ( after William Froude who first suggested the use of this number in the study of naval architecture. )

Hence Froude number,

F_r = V/(1g)^1/2

Therefore, the primary requirement for dynamic similarity between the prototype and the model involving flow of fluid with gravity as the significant force, is the equality of Froude number, F_r, i.e.,



Dynamic Similarity of Flows with Surface Tension as the Dominant Force

Surface tension forces are important in certain classes of practical problems such as ,
    
1. flows in which capillary waves appear
    
2. flows of small jets and thin sheets of liquid injected by a nozzle in air
    
3. flow of a thin sheet of liquid over a solid surface.

Here the significant parameter for dynamic similarity is the magnitude ratio of the surface tension force to the inertia force.
 
This can be written as  



The term  σ/pV²l  is usually known as Weber number, W_b (after the German naval architect Moritz Weber who first suggested the use of this term as a relevant parameter. )

Thus for dynamically similar flows (Wb)_m =(Wb)_p



Dynamic Similarity of Flows with Elastic Force

When the compressibility of fluid in the course of its flow becomes significant, the elastic force along with the pressure and inertia forces has to be considered.

Therefore, the magnitude ratio of inertia to elastic force becomes a relevant parameter for dynamic similarity under this situation.

Thus we can write,


 
The parameter pV²/E  is known as Cauchy number, (after the French mathematician A.L. Cauchy)

If we consider the flow to be isentropic, then it can be written



(where E_s is the isentropic bulk modulus of elasticity)

Thus for dynamically similar flows (cauchy)_m=(cauchy)_p


    
The velocity with which a sound wave propagates through a fluid medium equals to .
    
Hence, the term pV²/E_s  can be written as V²/a²  where a is the acoustic velocity in the fluid medium.

The ratio V/a is known as Mach number, Ma ( after an Austrian physicist Earnst Mach)

The situation arises in the flow of air past high-speed aircraft, missiles, propellers and rotory compressors. In these cases equality of Mach number is a condition for dynamic similarity.

Therefore,

(Ma)_p=(Ma)_m    

i.e.      

V_p/a_p = V_m/a_m     (1j)

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