FROUDE NUMBER (Fr)

Froude Number

The Froude number represents the ratio between inertia force and gravity force.  Froude number is expressed as:

The Froude number is applied where gravity forces are predominant.  The number is used in the analysis of:

§  Wave action.  Examples include breakwaters and ships

§  Free surface flow in open-channels

§  Hydraulic structures, such as spillways, stilling basins, weirs, and notches

§  Forces on bridge piers and offshore structures

The equation for dynamic similarity where gravity forces is predominant is given below.

EXAMPLE 1:

Flow in wide river 4.0 meters deep is to be simulated in a laboratory with a model whose channel is 1.0 meter deep. If the average speed of flow in the river is 1.0 meter per second, the speed of flow at which the model should be studied is?

Solution :

Lp = 4m            Vp = 1m/s

Lm= 1m            Vm = ?

     

REYNOLDS NUMBER (R)

Reynolds number is a non-dimensional parameter that is used when viscous force is dominant.  Reynold’s number represents the ratio between inertia force F_I and viscosity force F_V. 

Reynolds number is used for the following types of flow:

§  Completely submerged flow e.g. ship

§  Completely enclosed e.g. flow through pipes and plates

§  Viscous flow e.g settling of particles in fluids

§  Flow in flow meter in pipes, venturi meter, or orifice meter

The Reynolds number can be expressed as:

  

     

Below is the equation for dynamic similarity where viscous forces are predominant.

Example 1
An underwater missile, diameter 2m and length 10m is tested in a water tunnel to determine the forces acting on the real prototype. A 1/20^th scale model is to be used. If the maximum allowable speed of the prototype missile is 10 m/s, what should be the speed of the water in the tunnel to achieve dynamic similarity?

For dynamic similarity the Reynolds number of the model and prototype must be equal:

So the model velocity should be

As both the model and prototype are in water then, m_m = m_p and r_m = r_p so


Note that this is a very high velocity. This is one reason why model tests are not always done at exactly equal Reynolds numbers. Some relaxation of the equivalence requirement is often acceptable when the Reynolds number is high. Using a wind tunnel may have been possible in this example. If this were the case then the appropriate values of the r and m ratios need to be used in the above equation.

SIMILARITY

In hydraulic and aeronautical engineering valuable results are obtained at relatively small cost by performing tests on small scale models of full size systems (prototypes). Similarity laws help us to interpret the results of model studies. The relation between model and prototype is classified into three:

1. Geometry Similarity
2. Kinematics Similarity
3. Dynamic Similarity

M o d e l 

Model is similar with object/structure required in certain scale ratio. It is need to be tested in laboratory with similar condition in real phenomenon. The size of model is not necessary smaller than prototype.

1. Not necessarily functional (don’t need to work).
2. Can be to any scale (usually smaller but can also be of the original size or bigger).
3. Used for Display or/and [Visual] Demonstration of product.
4. May consist of only the exterior of the object/product it replicates.
5. Relatively cheap to manufacture.

 P r o t o t y p e 

Prototype is an object/actual structure in full size. It is need properly tested in actual phenomenon.

1. Is fully functional, but not fault-proof.
2. Is an actual version of the intended product.
3. Used for performance evaluation and further improvement of product.
4. Contains complete interior and exterior.
5. Is relatively expensive to produce.
6. Often used as a technology demonstrator

An Example: Flow Past a Sphere

Flow Past a Sphere: Geometric and

Kinematic Similitude

https://ecourses.ou.edu/cgi-bin/view_anime.cgi?file=m06221.swf&course=fl&chap_sec=06.2

Flow Past a Sphere: Dynamic Similitude

https://ecourses.ou.edu/cgi-bin/view_anime.cgi?file=m06222.swf&course=fl&chap_sec=06.2

Types of Similarities

Geometry similarity

The prototype and model have identical shapes but differ only in size. Ratio of corresponding length in prototype and model show as,

Kinematic Similarity

In addition to geometric similarity, ratio of velocities at all corresponding points in flow are the same.

Dynamic Similarity

Two systems have dynamic similarity if, in addition to dynamic similarity, corresponding forces are in the same ratio in both. The force scale ratio is 

Basically, if the geometric and kinematics similarities exist, it shows two system are dynamically similar. The ratios of these systems of all corresponding forces are the same. The respective forces includes:

1. Gravity
2. Viscosity
3. Elasticity
4. Surface tension
5. Inertia

Advantages of Using Similarities

1. Performances of object can be predicted.
2. Economy and easy to build, where design of model can be done many times until reach a certain values.
3. Non-functional structure also can be measured such as dam.

 NON-DIMENSIONAL PARAMETER

By using Raleigh’s Method or the Buckingham Phi Theorem, the number of dimensional variables such as mass, length and time used in an analysis of flow is reduced to a few non-dimensional variables.

Listed below are the five non-dimensional parameters that represent the ratio of forces per unit volume.

1.    Reynolds Number

2.    Froude Number

3.    Mach Number

4.    Euler Number

5.    Weber Number

THEOREM PI BUCKINGHAM

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 homogenous equation as:
                                             

 

          

where, the dimension at each section is the same.

The Buckingham Phi Theorem can also be expressed in terms of Õ as shown in on the right.

                                     
                                 Π1 = function (Π2, Π3, …, Πn-k)

      
      where, m = the primary dimensions

                          n = dimensional variables such as velocity, discharge and density.

                          k = reduction

                                          

THE STEP-BY-STEP METHOD

QUESTION 1

Find the dimensionless form of the solution for the thrust force, FT of a propeller if it depends upon the fluid density ρ, the diameter D, the rotational speed ω, and the relative fluid velocity, V.

Solution :

STEP 1 :

FT = f (ρ, D, ω, V )

                      

FT = Dependent variable

ρ, D, ω, V = Independent variable

STEP 2 :

STEP 3

                                                                                        

Thus, the value of k is equal to 3. Find n-k to find the number dimensionless π groups needed.

5-3=2 , so we can write f (π1,π2)= 0

STEP 4

                                                                             

We need to choose three repeating variables since m=3. These variables must contain all the m.

               Contain all m

STEP 5

                                                                                                                                                      

                                                                                                                                          

STEP 6                                                                                                               

STEP 7                                                                                                    

                                                                                               

STEP 8

STEP 9 

Example 2 :

Solution : http://youtu.be/zxJ4R9qe3Yc