Unit 6 | unit 6 dimensional analysis and hydraulic similitude

Introduction to fluid mechanics

Unit - 6

Dimensional Analysis and Hydraulic Similitude

6.1 Dimensional Homogeneity

A physical equation is the relationship between two or more physical quantities.

Any correct equation expressing a physical relationship between quantities must be dimensionally homogeneous and numerically equivalent.

Dimensional homogeneity states that every term in an equation when reduced to fundamental dimensions must contain identical powers of each dimension.

A dimensionally homogeneous equation is applicable to all system of units.

In a dimensionally homogeneous equation, only quantities having the same dimensions can be added, subtracted or equated.

P = w h

Dimensions of LHS = ML-1T-2

Dimensions of RHS = ML-2T-2×L = ML-1T-2

Dimensions of LHS = Dimensions of RHS

Hence, equation P= w h is dimensionally homogeneous; so it can be used in any system of units.

Applications of Dimensional Homogeneity:

It facilitates to determine the dimensions of a physical quantity.

It helps to check whether an equation of a physical phenomenon is dimensionally homogeneous or not.

It facilitates conversion of units from one system to another.

Key takeaways

Dimensional homogeneity states that every term in an equation when reduced to fundamental dimensions must contain identical powers of each dimension.

A dimensionally homogeneous equation is applicable to all system of units.

6.2 Rayleigh Method

A basic method to dimensional analysis method and can be simplified to yield dimensionless groups controlling the phenomenon. Flow chart below shows the procedures.

Rayleigh's method - Ritz is a method of finding numbers in eigenvalue scales that are difficult to solve by analyzing, especially in the context of solving body boundary problems that can be manifested as different matrix measurements.

It is used in mechanical engineering to measure eigenmodes of the body system, such as finding resonant waves of a structure to direct proper lubrication.

The name Rayleigh - Ritz is an unfamiliar term used to describe a method that is rightly called the Ritz method, because this method was invented by Walter Ritz in 1909.

In 1911, Lord Rayleigh wrote a letter congratulating Ritz on his work, but stating that he himself had used the Ritz method in many places in his book and in another book.

This statement, although disputed later, and that a less than one vector approach leading to Rayleigh quotient makes the wrong name persist.

Procedure:

Procedure of Rayleigh method are given below

Fig no 1 Procedure of Rayleigh method

Where C = dimensionless constant

a, b, c,..m are arbitrary exponents.

1. Example: The rate at which a pressure wave propagates through a liquid can be expected to depend on the quantity of liquid represented by the bulk modulus K, and its mass established by D. A. the type of relationship that exists.

Therefore: and a possible equation is

Key takeaways

It is used in mechanical engineering to measure eigenmodes of the body system, such as finding resonant waves of a structure to direct proper lubrication.

6.3 Buckingham's Pi Method and Other Methods

When a large number of physical variables are involved Rayleigh’s method of dimensional analysis becomes increasingly laborious and cumbersome.

Buckingham’s method is an improvement over the Rayleigh’s method.

Buckingham’s designated the dimensionless group by the letter Π. It is therefore often called Buckingham Π-Method.

The Buckingham’s Π-Method states as follows:

“If there are n variables in a dimensionally homogeneous equation and if these variables contain fundamental dimensions (such as M, L, T, etc.) then the variables are arranged into (n-m) dimensionless terms.

These dimensionless terms are called Π– terms.”

Mathematically, if any variable X1, depends on independent variables, X2, X3, X4,…… Xn; the functional equation may be written as

-------- 1

Equation 1 can also be written as

-------- 2

It is dimensionally homogeneous equation and contains n variables.

If there are m fundamental dimensions, then according to Buckingham’s Π – theorem, it [equation] can be written in terms of number of Π – terms in which number of Π terms is equal to (n-m).

Hence equation 2 becomes as

--------3

Each dimensionless – term is formed by combining m variables out of the total n variables with one of the remaining (n-m) variable.

These m variables which appear repeated in each of – terms are consequently called repeating variables and are chosen from among the variables such that they together involve all the fundamental dimentions and they themselves do not form a dimensionless parameter.

Let in the above case X2, X3 and X4 are the repeating variables if the fundamental dimensions m(M, L, T) = 3, then each term is written as:

-------- 4

Where a1, b1, c1, a2, b2, c2 etc. are the constants, which are determined by considering dimensional homogeneity. These values are substituted in equation 4 and values of are obtained. These values of Π’s are substituted in equation 3.

The final general equation for the phenomenon may then be obtained by expressing anyone of the Π – terms as a function of the other as:

Key takeaways

The final general equation for the phenomenon may then be obtained by expressing anyone of the Π – terms as a function of the other as:

6.4 Dimensionless Groups

A flawless group is a combination of a limited or innumerable sizes with a general zero size. In a system of parallel units, it can therefore be represented by a pure number.

The number of flawless groups of experience details has long been known. More than a hundred years ago, G. G. Stokes pointed out the importance of a group now known as the Reynolds Number as a state of fluid similarity in liquidity under various conditions [Stokes, (1966)].

It was Helmholtz who showed the importance of the teams now known as Froude Number and Match number. Initially, flawless groups did not have specific names, and the first to enter the words was M. G. Weber in 1919, when he graduated from Froude, Reynolds and Cauchy in groups.

Valuation is an informal process, and there are several instances where the same unequal group is given more than one name, e.g., Froude and Boussinesq, Bond and Eötvös. On the other hand, some depositors have been honored with numerous numbers; Lagrange for example has three, and Damköhler, no less than five.

Yet it seems that there is one rule that, even if not written, is rarely broken; that is, that flawless degrees are awarded after death.

The following table lists the seamless groups most closely associated with heat and high transfer.

Archimedes Number
Bond number		Also called Eotvos number, Eo
Biot number, heat transfer
Biot number, mass transfer
Clausius number
Darcy number		Alternative
Deborah number		Relaxation time/ observation time, also called Weissenberg number also called Brinkam number.
Eckert number
Euler number
Fanning friction factor
Fourier number, heat transfer
Fourier number, mass transfer
Froude number		Or also called Boussinessq number, Bo.
Galileo number
Graetz number		Also called phase change number
Grash of number
Jakob number
J factor, heat transfer
J factor, mass transfer
Kutataeladze number
Lewis number
Mach number
Nussett number
Preclet number
Prandtl number
Rayleigh number		is coefft. Thermal expansion)
Reynolds number
Schmidt number
Sherwood number
Stanton number, heat transfer
Stanton number, mass transfer
Strouhal number		f’ is frequency
Weber number

Fig no 2 The seamless groups most closely associated with heat and high transfer.

Key takeaways

A flawless group is a combination of a limited or innumerable sizes with a general zero size. In a system of parallel units, it can therefore be represented by a pure number.

6.5 Similitude

Similitude are concept that applies to the testing of engineering models. The model is said to resemble a real application if it has two geometric similarities, kinematic similarities and dynamic similarities. Similarities and Similitude are the alternate in this context.

The term dynamic Similitude is often used to capture everything because it means that geometric and kinematic similarities have been met.

Similitude's main application is in hydraulic and aerospace engineering to test fluid flow conditions with rated models. And it is the main idea behind many liquid formulas for liquid formulas.

Engineering models are used to study the complex problems of fluid dynamics where computer calculations and simulations are unreliable. Models tend to be smaller than the final build, but not always. The types of scales allow for design testing before construction, and in many cases are critical step in the development process.

The design of the scale model, however, must be accompanied by analysis to determine under which conditions it was tested. While geometry can be simply measured, other parameters, such as pressure, temperature or velocity and the type of liquid may need to be adjusted. Similarities are found when experimental conditions are created in such a way that test results apply to actual construction.

Fig no 3 Three conditions are required for the model to match the system.

The following methods are needed to achieve similarity;

Geometric Similarity - the model is the same method as the system, which is usually measured.

Kinematic similarity - the fluid flow of both the model and the actual application must exceed the same levels of motion change time. (Liquid distribution is the same)

Dynamic similarity - the ratio of all forces acting on liquid-related particles and parameters in both systems remains the same.

To satisfy the above conditions the application is analyzed;

All parameters required to define a system are identified using terms from the continuation mechanics.

Quantitative analysis is used to present the system with as few independent variables and as many measurable parameters as possible.

The non-measurable parameter values are held to be the same for all scale model and usage. This can be done because it is flawless and will ensure a strong similarity between the model and use. Emerging statistics are used to obtain measurement rules governing modular test conditions.

It is often not possible to achieve strong similarities during model testing. The bigger the application, the harder it is to match. In these cases some of the features of the simulation are ignored, focusing only on the most important parameters.

The construction of ships is often more sophisticated than science for the most part because the strong similarities are very difficult to find a slightly submerged vessel: the ship is affected by air forces in the air above it, by hydrodynamic forces inside the water beneath it, and especially by wave movements in the visible connection between water and air.

The requirements for each of these scenarios vary, so the models cannot replicate what happens on a full-size vessel almost as much as a plane or submarine can - each works perfectly on the same object.

Matching is a term widely used in fragmentation techniques related to the complex lifestyle. Under loading conditions provided fatigue damage to the excluded template can be compared to that of the excluded sample. The similarities suggest that the fatigue life of two people will also be the same.

Key takeaways

The term dynamic Similitude is often used to capture everything because it means that geometric and kinematic similarities have been met.

6.6 Model Studies

In recent years hydraulic model studies have been conducted in the study and analysis of many problems in fluid mechanics. The hydraulic problem can be analyzed by analytical methods, but these analytical methods involve many limitations and assumptions and therefore their systems are often blocked. In many cases, analytical methods involve the most complex calculations that cannot be solved.

Despite the great advances made in the field of fluid mechanics, solutions of various complex flow patterns cannot be obtained by analytical methods alone. The analytical methods available need to be simplified so that their programs are not only restricted but also theoretical.

There are many cases where it is not possible or impossible to perform a balanced and complete statistical analysis of the problems. The hydraulic model can sometimes provide the only way to detect and eliminate certain undesirable conditions.

Only exemplary research and research can make progress in existing jobs, safe and economical construction and the creation of new jobs and the advancement of knowledge in many aspects of water engineering. Nowadays, model courses have a wide field of applications - dams, rivers and harbors, hydraulic machinery, buildings, ships, aircraft, water problems and more.

In short, experimenting with the model has become an important tool for the design engineer.

Model courses are designed for two purposes, namely -

(i) Obtaining information on possible performance of the species, also

(ii) Assist in the construction and avoidance of costly errors and transform the economic solution to the hydraulic problem.

Since, by comparison, it is not expensive to change the structure of the model, there is enough space to test the design of several other models in the model before using the latter. Obviously, such tests would be extremely expensive if they were performed on a complete scale. The model study is designed to provide not only quality assurance but also measurement indicators for model type features.

The hydraulic model offers itself as a powerful construction tool that establishes a fluid system derived from the potential performance of the model type. Models are the only way to find the closest way to solving many hydraulic problems.

They are very useful in studying the related relevance of different designs. They clearly show the weak points of a particular design. Today, the importance of hydraulic models, especially for large projects, is accepted in all sectors.

The idea of using models to study the features of hydraulic structures was conceived by Sir Isaac Newton who introduced the law of matching in his work Principia. The idea however remained silent until Froude adopted smaller models to learn and find satisfactory shipbuilding designs.

Today the small-scale model is the most powerful research tool in all laboratories to study hydraulic structures with the principles of hydraulic simulation. Obviously, the choice of model scales and the proper interpretation of the test results will depend largely on the experience of the hydraulic engineer, even if the rules governing hydraulic models are well established.

Key takeaways

6.7 Types of Models

The classification of models is

i. Undistorted models

ii. Distorted models

Undistorted model

If the average size of the model is the same size and type of the same condition, the model is said to be uninterrupted model.

Distorted model

The inverted model is said to be a distorted model only if it differs geometrically and prototype.

Advantages of distorted models

The advantages of distorted models are,

1. The exact size of the model can be accurately measured.

2. Model costs can be reduced.

3. Incredible flow in the model can be maintained.

Key takeaways

The classification of models is

i. Undistorted models

ii. Distorted models

6.8 Type of Models, Application of Dimensional Analysis and Model Studies to Fluid Flow Problem

Types of model laws

Types of model laws are

1. Reynolds model laws

2. Froude model laws

3. Euler model laws

4. Weber model laws

5. Mach model laws

Application of Froude model laws

The law of Froude's model is applied

Free local flow such as streaming to spill, heirs, sluices, channels etc.

Flow of the jet from the orifice or nozzle

Where waves may form more

When a flow of liquid of different strength passes one anther

Weber model laws

When only strong surface strength is the main model can be considered to be strongly similar to the prototype whereas the inertial ratio and surface tensile strength are the same in model and prototype.

(We) model = (We) prototype

Use of dimensional analysis

1. Conversion from one unit to another

2. Testing of statistical units (Size Relationship)

3. Define unparalleled relationships using

a) Rayleigh's method

b) Buckingham TT-Theorem

4. Model Analysis

Reynolds Model Law

In flow situations where in addition to inertia, viscous force is the other predominant force, the similarity of flow in the model and its prototype can be established if Reynolds number is same for both the systems.

This is known as Reynolds law and according to this law,

Where, ρm = density of fluid in model

Vm = velocity of fluid in model,

Lm = length of linear dimension of the model, and

µm = viscosity of fluid in model

And ρ P, VP, LP and µP are the corresponding values of density, velocity, linear dimension and viscosity of fluid in prototype.

[

Following are some of the phenomenon for which Reynolds model law can be a sufficient criterion for similarity of flow in the model and the prototype:

Motion of air planes,

Flow of incompressible fluid in closed pipes,

Motion of submarines completely under water, and

Flow around structures and other bodies immersed completely under moving fluids.

Froude Model Law

When the gravitational force can be considered to be the only predominant force which controls the motion in addition to the inertia force, the similarity of the flow in any two such systems can be established if the Froude’s number of both the systems is the same.

This is known as Froude’s Model Law.

Some of the phenomenon for which the Froude Model Law can be sufficient criterion for dynamic similarity to be established in the model and the prototype are:

Free surface flows such as flow over spillways, sluices, etc.

Flow of jet from an office or nozzle;

Where waves are likely to be formed on the surface;

Where fluids of different mass densities flow over one another

Let, Vm = velocity of fluid in model

Lm = length (or linear dimension) of the model

gm = acceleration due to gravity (at a place where model is tested)

and VP, LP and gP are the corresponding values of the velocity, length and acceleration due to gravity for the prototype.

Then according to Froude Law,

(Fr)m = (Fr)P

Or ------ 1

As the value of g at the site of mode testing will be practically the same as at site of the proposed prototype, therefore gm = gP and the equation 1 values as

Where Lr = scale ratio for length

Key takeaways

Types of model laws are

1. Reynolds model laws

2. Froude model laws

3. Euler model laws

4. Weber model laws

5. Mach model laws

6.9 Dynamic Similitude: Definition of Reynolds Number, Froude Number, Mach Number, Weber Number and Euler Number

Reynold’s Number (Re)

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

Reynold’s Number, Re =

For pipe flow (where the linear dimension is taken as diameter d),

Reynold’s number signifies the relative predominance of the inertia to the viscous forces occurring in the flow systems.

Fraud’s Number (Fr)

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

Mathematically,

Froude number governs the dynamics similarity of the flow situations; where gravitational force is more significant and all other forces are comparatively negligible.

Euler’s Number (Eu)

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

The Euler number is important in the flow problems / situations in which a pressure gradient exists.

Weber’s Number (We)

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

Mathematically,

This number assumes importance in the following flow situation:

i) Flow of blood in veins and arteries

ii) Liquid atomisation, and

iii) Capillary movement of water in soils

Mach Formula (M)

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

Mathematically,

The Mach number is important in compressible flow problems at high velocities, such as high velocity flow in pipes or motion of high speed projectiles and missiles.

Key takeaways

Reynold’s Number (Re)

Re =

Fraud’s Number (Fr)

Euler’s Number (Eu)

Weber’s Number (We)

Mach Formula (M)

Examples:

Q.1 An oil of specific gravity 0.9 and viscosity 0.9 poise is to be transported at the rate of 1000 l/s through a. 1.2 m diameter pipe. Tests were conducted on a 10 cm diameter pipe using water at .20°C. Viscosity of water at 20°C is 0.01 poise. Find the rate of flow in the model.

Soln.:

Prototype

D = 1.2 m.

Sp = 0.9,

Model

Dm = 10 cm = 0.1 m

Sm = 1

μp = 0.9 poise. μm = 0.01 poise = 1 x 10-² poise

To find: Velocity and flow rate.

Since it is pipe flow, Reynolds’s number must be applied.

(Re)model = (R) prototype

Rate of flow through model

Q.2 From the following data, find the scale ratio of model Velocity of water 1 m/'s through circular pipe. Prototype: Velocity of oil 0.12 ms through 50 mm diameter pipe. Assume kinematic viscosity of water 0.01 cms and that of oil 0.008 cm/s. Assume dynamic similarity. Also find the diameter of pipe used for model.

Soln.:

Prototype

Vp -012 m/s

Dp = 50mm = 0.05 m

Vp = 0.006 cm²/sec

Model

Vm= 1m/s

Dm=?

Vm= 0.01 cm^2/sec

To find:

Since it is pipe flow, Reynold's ember mast be applied

Model scale ratio is 1:5

Now, diameter of model pipe

Q. 3 Water at 15.6°C flows at 3.5 m/s in a 150 mm diameter pipe. At what velocity must a fuel oil at 32.6°C flow in a 75 mm diameter pipe for the flow to be dynamic similar? v.-1.13 x 10 m/s and 2.96 x 10 m/s.

Soln.:

Water

Dw = 150 mm = 0.15 m

Oil

To find: Velocity of oil V.

Since it is pipe flow, Reynold's number must be applied

The velocity of cl at 32.6°C is 15.33 m'k

Q.4 A ship has a length of 150 m and wetted area 3000 m² A model of this ship 5 m in length when towed in fresh water (p= 1000 kg/m) at 2 m/s produces a resistance of 40N. Calculate (1) corresponding speed of the ship. (i) the shaft power required to propel the ship at this speed through sea water (p=1030 kg/m). Take the propeller efficiency as 75%

Soln.:

Model

Lm= 5 m

Vm= 2 m/s

Fm = 40 N

Prototype

Lp= 150 m

Ap= 3000m^2

Propeller efficiency= n= 75 %

To find:

Speed of ship Vp

Since it is ship motion Reynolds number must be applied

Ratio of drag force

Drag force on prototype

Actual power input

Q. 5 A ship 300 m long moves in seawater whose density is 1030 kg/m A1.75 Model of this ship is to be tested in a wind tunnel. The velocity of air in wind tunnel is 29 m/s the resistance of the model is 60 N. Determine velocity and resistance of the ship in seawater. Air density is 1.24 kg/m Kinematic viscosity of air=0.018 stokes Kinematic viscosity of seawater 0,01 stokes.

Soln.:

Given:

Prototype

Scale 1:75

Lp = 300 m

To find:

Vp and Fp

Since it is ship motion Reynolds number must be applied

Drag force ratio

Q.6 In a geometrically similar model of weir the discharge is 0.15 m².s.. if the scale of the model is 1/50 find the discharge of the prototype.

Soln.:

Given:

Discharge per meter length = 0.15 m/sec.

Scale ratio Lr = Lp/Lm = 50

To find: qp

Since it is a spillway, Froude Number must be applicable.

The discharge ratio for spillway is given by

Q. 7 The performance of a spillway is to be studied by means of a model constructed to a scale of 1:9. Determine: (1) Rate of flow in the model for a prototype discharge of 1400 m/s. (2) Energy lost in the prototype if the energy loss in model is 0.3 kW

Soln.:

Given:

Scale of model 1:9

Lr = 9

Discharge of proto type Q = 1400 m^3 /s

Since it is a spillway, Froude number must be applicable.

Using discharge scale ratio

Energy loss

Q. 8 In the model test of a spillway the discharge per meter length is 1/6 m^3/sec if the scale of the model is 1/36 find the discharge per meter run of the prototype.

Given:

Qm = 1/6 m^3/sec

Lr = Lp/Lm = 36

To find: QP

Since it is a spillway, Froude Number must be applicable. The discharge ratio for spillway is given by

Q.9 Find the suitable scale for model of a spillway if maximum discharge available in the laboratory is 10 L.P.S. and the prototype discharge is 168.07 m/sec,

Soln.:

Given:

Q=10 lps = 10 x10³ m³/sec;

Qp = 168.07 m^3/sec

Since it is a spillway, Froude Number must be applicable. using discharge scale ratio

Model scale should be 1:49.

Q.10 A ship model of scale 1/60 is towed through sea water at a speed of 1:1 m/s A force of 2.1 N is required to tow the model. Determine the speed of ship and the propulsive force on the ship if the prototype is subjected to wave resistance only.

Solution: