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HYDRAULICS-2HYDRAULICS-2 3. DIMENSIONAL ANALYSIS AND SIMILITUDE

3. Dimensional Analysis and Similitude

Dimensional analysis is a method of simplifying physical problems of dimensions. It is a mathematical technique used in research work for design and for conducting model tests. It deals with the dimensions of the physical quantities involved in the phenomenon. All physical quantities are measured by comparison, which is made with respect to an arbitrarily fixed value. Length L, mass M and time T are three fixed dimensions which are of importance in Fluid Mechanics. If in any problem of fluid mechanics, heat is involved then temperature is also taken as fixed dimension. These fixed dimensions are called fundamental dimensions or fundamental quantity.When a hydraulic structure is build it undergoes some analysis in the design stage. Often the structures are too complex for simple mathematical analysis and a hydraulic model is build. Usually the model is less than full size but it may be greater. The real structure is known as the prototype. The model is usually built to an exact geometric scale of the prototype but in some cases - notably river model - this is not possible. Measurements can be taken from the model and a suitable scaling law applied to predict the values in the prototype.

3.1 Dimensional Analysis

It enables to formulate equations by deriving the relation between variables. Each physical phenomenon can be expressed by an equation, composed of variables which may be dimensional and non dimensional quantities. Dimensional analysis helps in determining a systematic arrangement of variables in the physical relationship and combining dimensional variables to form non-dimensional parameters. In the study of fluid mechanics dimensional analysis has been found to be useful in both analytical and experimental investigation. Some of the uses are:

i) Testing dimensional homogeneity of any equation of fluid motion.

ii) Deriving equations expressed in terms of non dimensional parameters to show the relative significance of each parameter.

iii) Planning model tests

iv) Attacking problems that are not amenable to direct theoretical solutionsIn engineering, the application of fluid mechanics in designs makes much of the use of empirical results from a lot of experiments. This data is often difficult to present in a readable form. Even from graphs it may be difficult to interpret. Dimensional analysis provides a strategy for choosing relevant data and how it should be presented. This is a useful technique in all experimentally based areas of engineering. If it is possible to identify the factors involved in a physical situation, dimensional analysis can form a relationship between them.

The resulting expressions may not at first sight appear rigorous but these qualitative results converted to quantitative forms can be used to obtain any unknown factors from experimental analysis. In general, the result of performing dimensional analysis on a physical problem is a single equation. This equation relates all of the physical factors involved to one another.Dimensional Homogeneity Dimensional homogeneity means the dimensions of each terms in an equation on both sides are equal. Thus if the dimensions of each term on both sides of an equation are the same the equation is known as dimensionally homogeneous equation. The powers of fundamental dimensions (i.e., L, M, T ) on both sides of the equation will be identical for a dimensionally homogeneous equation. Such equations are independent of the system of units.

Fourier principle of dimensional homogeneity states that an equation which expresses physical phenomena of fluid flow must be algebraically correct and dimensionally homogenous.

Q =( 2/ 3) Cd LgH3/2 (flow over rectangular weir)

The above equation is dimensionally homogenous equation. Where as the next is dimensionally non homogenous equation.

V = 1/nR2/3 S1/2 (Manning equation)

Dimensions and units

Any physical situation can be described by certain familiar properties e.g. length, velocity, area, volume, acceleration etc. These are all known as dimensions.

Of course dimensions are of no use without a magnitude being attached. We must know more than that something has a length. It must also have a standardized unit - such as a meter, a foot, a yard etc.

Dimensions are properties which can be measured. Units are the standard elements we use to quantify these dimensions. In dimensional analysis we are only concerned with the nature of the dimension i.e. its quality not its quantity. The following are the fundamental quantities and their abbreviations used in hydraulics: length = L , mass = M , time = TThe following table lists dimensions of some common derived physical quantities:

QuantitySI Unit.Dimension

Velocitym/sms-1LT-1

Accelerationm/s2ms-2LT-2

ForceN

kg m/s2kg ms-2M LT-2

energy (or work)Joule J

N m,

kg m2/s2kg m2s-2ML2T-2

PowerWatt W

N m/s

kg m2/s3Nms-1kg m2s-3ML2T-3

pressure ( or stress)Pascal P,

N/m2,

kg/m/s2Nm-2kg m-1s-2ML-1T-2

Densitykg/m3kg m-3ML-3

specific weightN/m3kg/m2/s2kg m-2s-2ML-2T-2

Relative densitya ratio

no units.1

no dimension

ViscosityN s/m2kg/m sN sm-2kg m-1s-1M L-1T-1

Surface tensionN/m

kg /s2Nm-1kg s-2MT-2

3.2 Methods of Dimensional AnalysisIf the number of variable involved in a physical phenomenon is known, then the relation among the variables can be determined by the following two methods:

i. Rayleighs method, and

ii. Buckinghams ( -theorem.

3.2.1 Raleigh MethodIn this method of dimensional analysis a functional relationship of some variables is expressed in the form of an exponential equation which must be dimensionally homogenous. If the number of independent variables becomes more than four, then it is difficult to find expressions for the dependent variable.

Let X be a function of independent variables x1, x2, x3. Then according to Raleighs method: X = f [x1, x2, x3],

This can be written as : X = kx1a. x2b . x3c

Where, k = constant

a, b ,c = arbitrary powers

The values of a, b , c are obtained by comparing the powers of fundamental dimension on both sides.

3.2.2 Buckinghams ( Theorem

The Raleighs method of dimensional analysis becomes more laborious if the variables are more than the number of fundamental dimensions (M, L, and T). This difficulty is over comed by Buckinghams ( Theorem, Which states that, If there are n-variables (independent and dependant) in a physical phenomena and if these variables contain m-fundamental dimension (M, L, and T), then the variables are arranged into (n - m) dimensionless terms , called (-terms.

Let x1, x2,x3......xn are variables involved in the phenomena.

x1 = f(x2,x3....xn)

This can be written as: f (x1, x2, x3......xn) = 0 The above equation is a dimensionally homogeneous equation. It contains n variables, If there are m fundamental dimensions then according to Buckinghams (-theorem, the above equation can be written in terms of number of dimensionless groups or (-terms in which number of (-terms is equal to (n - m). Hence the above equation becomes: f((1, (2, (3,... (n-m) = 0Each of (-terms is dimensionless and is independent of the system. Division or multiplication by a constant does not change the character of the (-term. Each (-term contains m + 1 variables, where m is the number of fundamental dimensions and is also called repeating variables. Let in the above case X2 , X3 , and X4 are repeating variables. If the fundamental dimensions (M,L, T) =3. Then each (-term is written as:

Hence, the arbitrary powers can be obtained using the same procedure used for Raleighs method.

Method of Selecting Repeating Variables

The number of repeating variables is equal to the number of fundamental dimensions of the problem. The choice of repeating variables is governed by the following considerations:

As far as possible, the dependent variable should not be selected as repeating variable.

The repeating variables should be chosen in such a way that one variable contains geometric property (height, diameter, length), other variable contains flow property (velocity, acceleration, pressure.), and third variable fluid property (viscosity, density, unit weight etc.).

The repeating variables selected should not form a dimensionless group.

The repeating variables together must have the same number of fundamental

dimensions.

No two repeating variables should have the same dimensions.

Steps in Buckingham ( - Theoremi. List all the n physical quantities involved in the phenomena.

ii. Select m variables out of these which are to serve as repeating variables (none of them is dimensionless, the dependant variable should not be selected, usually in fluid flow problem linear geometric dimension fluid property and flow behavior are chosen)

iii. Write general equation for equation of (-terms.

iv. Write the dimensional equations for equations of (-terms (Obtain the unknown exponents)

v. Write the general equation of the phenomena in terms of (-terms.

Wrong choice of physical properties.

If, when defining the problem, extra - unimportant - variables are introduced then extra groups will be formed. They will play very little role influencing the physical behaviors of the problem concerned and should be identified during experimental work. If an important / influential variable was missed then a group would be missing. Experimental analysis based on these results may miss significant behavioral changes. It is therefore, very important that the initial choice of variables is carried out with great care.

3.3 Model Analysis and SimilitudeFor predicting the performance of the hydraulic structures (such as dams, spill ways etc.) or hydraulic machines (such as turbines, pumps etc.), before actually constructing or manufacturing, models of the structures or machines are made and tests are performed on them to obtain the desired information.The model is the small scale replica of the actual structure or machine. The actual structure or machine is called Prototype. It is not necessary that the models should be smaller than the prototypes (though in most of cases it is), they may be larger than the prototype. The study of models of actual machines is called model analysis. Model analysis is actually an experimental method of finding solutions of complex flow problems. Exact analytical solutions are possible only for a limited number of flow problems. The followings are the advantages of the dimensional and model analysis:1. The performance of the hydraulic structure or hydraulic machine can be easily predicted, in advance, from its model.

2. With the help of dimensional analysis, a relationship between the variables influencing a flow problem in terms of dimensionless parameters is obtained. This relationship helps in conducting tests on the model.

3. The merits of alternative designs can be predicted with the help of model testing. The most economical and safe design may be, finally, adopted.

4. The tests performed on the models can be utilized for obtaining, in advance, useful information about the performance of the prototypes only if a complete similarity exists between the model and the prototype.

3.3.1 Similitude and types of SimilaritiesSimilitude is defined as the similarity between the model and its prototype in every respect, which means that the model and prototype have similar properties or model and prototype are completely similar. Three types of similarities must exist between the model and prototype.

1. Geometric similarity:- The geometric similarity is said to exist between the model and the prototype if the ratio of all corresponding linear dimension in the model and prototype are equal. Lp / Lm = Dp / Dm = Hp / Hm = Lr = Length ratio

Where, Lp and Lm = Length of the prototype and Length of the model

Dp and Dm = Diameter/Depth of the prototype and the model Hp and Hm = Height of the prototype and the height of the model

Ap / Am = (LpDp) / ( LmDm) = Lr2

Vp / Vm = ( Lp Dp Hp) / ( Lm Dm Hm) = Lr3Where, Ap & Vp = Area and Volume of the prototype

Am & Vm = Area and Volume of the model

2. Kinematic Similarity:

Kinematic similarity means the similarity of motion between model and prototype. Thus kinematic similarity is said to exist between the model and the prototype if the ratios of the velocity and acceleration at the corresponding points in the model and at the corresponding points in the prototype are the same. Since velocity and acceleration are vector quantities, not only the ratio of magnitude of velocity and acceleration at the corresponding points in the model and prototype should be same; but the directions of velocity and accelerations at the corresponding points in the model and prototype also should be parallel.

For kinematic similarity, we must have:

and Where Vr is the velocity ratio, ar is the acceleration ratio.Vp1 = Velocity of fluid at point 1 in prototype,

Vp2 = Velocity of fluid at point 2 in prototype,

ap1 = Acceleration of fluid at 1 in prototype,

ap2 = Acceleration of fluid at 2 in prototype, and

Vm1, Vm2 , am1, am2 = Corresponding values at the corresponding points of fluid

Velocity and acceleration in the model.

Also the directions of the velocities in the model and prototype should be same. 3. Dynamic Similarity.

Dynamic similarity means the similarity of forces between the model and prototype. Thus dynamic similarity is said to exist between the model and the prototype if the ratios of the corresponding forces acting at the corresponding points ate equal. Also the directions of the corresponding forces at the corresponding points should be same.

Then for dynamic similarity, we have

Fr = the force ratio (Fi)p = Inertia force at a point in prototype,

(Fv)p = Viscous force at the point in prototype,

(Fg)p = Gravity force at the point in prototype,

(Fi)m (Fv)m, (Fq)m = Corresponding values of forces at the corresponding point in model. Also the directions of the corresponding forces at the corresponding points in the model and prototype should be same. Types of forces acting in moving fluid

For the fluid flow problems, the forces acting on a fluid mass may be any one, or a combination of the several of the following forces:

1. Inertia Force (Fi ).

It is equal to the product of mass and acceleration of the flowing fluid and acts in the direction opposite to the direction of acceleration. It is always existing in the fluid flow problems.

2. Viscous Force (Fv). It is equal to the product of shear stress (t) due to viscosity and surface area of the flow. It is present in fluid flow problems where viscosity is having an important role to pJay.

3. Gravity Force (Fg).It is equal to the product of mass and acceleration due to gravity of the flowing fluid. It is present in case of open surface flow.

4. Pressure Force (Fp) It is equal to the product of pressure intensity and cross-sectional area of the flowing fluid. It is present in case of pipe-flow.

5. Surface Tension Force (Fs). It is equal to the product of surface tension and length of surface of the flowing fluid.

6. Elastic Force (Fe)It is equal to the product of elastic stress and area of the flowing fluid.

For a flowing fluid, the above-mentioned forces may not always be present. And also the forces, which are present in a fluid flow problem, are not of equal magnitude. There are always one or two forces which dominate the other forces. These dominating forces govern the flow of fluid. 3.3.2 Dimensionless numbers

Dimensionless numbers are those numbers which are obtained by dividing the inertia force by viscous force or gravity force by pressure force or surface tension force by elastic force. As this is a ratio of one force to the other force, it will be a dimensionless number. These dimensionless numbers are also called non-dimensional parameters. The followings are the important dimensionless numbers:

1. Reynolds number( Re) Reynolds Number (Re). It is defined as the ratio of inertia force of a flowing fluid and the viscous force of the fluid. The expression for Reynolds number is:

2. Froudes number (Fe)The Froudes number is defined as the square root of the ratio of inertia force of a flowing fluid to the gravity force. Mathematically, it is expressed as 3. Eulers number

It is defined as the square root of the ratio of the inertia force of a flowing fluid to the pressure force. Mathematically, it is expressed as:

4. Webers number (We):

It is defined as the square root of the ratio of the inertia force of a flowing fluid to the surface tension force. Mathematically, it is expressed as:

5. Machs number (M):

Machs number is defined as the square root of the ratio of the inertia force of a flowing fluid to the elastic force. Mathematically, it is defined as:

3.3.3 Model laws or Similarity laws

For the dynamic similarity between the model and the prototype the ratio of the corresponding forces acting at the corresponding points in the model and prototype should be equal. The ratio of the forces is dimensionless numbers. It means for dynamic similarity between the model and prototype, the dimensionless numbers should be same for model and the prototype. But it is quite difficult to satisfy the condition that all the dimensionless numbers (i.e., Re, Fe, We, E and M) are the same for the model and prototype. Hence models are designed on the basis of ratio of the force, which is dominating in the phenomenon. The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. The followings are the model laws:

A) Reynolds model law: Reynolds model law is the law in which models are based on Reynoldss number. Models based on Reynolds number include:(i) Pipe flow

(ii) Resistance experienced by sub-marines, airplanes, fully immersed bodies etc.

As defined earlier that Reynolds number is the ratio of inertia force and viscous force, and hence fluid flow problems where viscous forces alone are predominant, the models are designed for dynamic similarity on Reynolds law, which states that the Reynold number for the model must be equal to the Reynolds number for the prototype. Then according to Reynolds model law,

are the corresponding values of velocity, density, linear dimension and viscosity of fluid in prototype. And also are called the scale ratios for density, velocity, linear dimension and viscosity. The scale ratios for time, acceleration, force and discharge for Reynolds model law are obtained as

B) Froude model law:

Froude model law is the law in which the models are based on Froude number which means for dynamic similarity between the model and prototype, the Froude number for both of them should be equal. Froude model law is applicable when the gravity force is only predominant force which controls the flow in addition to the force of inertia. Froude model law is applied in the following fluid flow problems:

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

2. Flow of jet from an orifice or nozzle

3. Where waves are likely to be formed on surface,

4. Where fluids of different densities flow over one another.Then according to Froude model law:

are the corresponding values of the velocity, length and acceleration due to gravity for the prototype. Scale ratios for various physical quantities based on Froude model law are:

(i) Scale ratio for time.

(ii) Scale ratio for acceleration:

Acceleration is given by

(iii) Scale ratio for discharge

(iv) Scale ratio for force

If the fluid used in model and prototype is same, then

(v) Scale ratio for pressure intensity

Pressure ratio: (vi) Scale ratio for work, energy, torque, moment etc.

(vii) Scale ratio for Power

C) Euler model law

Eulers model law is the law in which the models are designed on Eulers number which means for dynamic similarity between the model and prototype, the Euler number for model and prototype should be equal. Eulers model law is applicable when the pressure forces are alone predominant in addition to the inertia force. According to this law: ,

If fluid is same in model and prototype, then the above equation becomes:

Where,

Eulers model law is applied for fluid flow problems where flow is taking place in a closed pipe in which case turbulence is fully developed so that viscous forces are negligible and gravity force and surface tension force are absent. This law is also used where the phenomenon of cavitations takes place.

D) Weber model law. Weber model law is the law in which models are based on Webers number, which is the ratio of the square root of inertia force to surface tension force. Hence where surface tension effects predominate in addition to inertia force, the dynamic similarity between the model and prototype is obtained by equating the Weber number of the model and its prototype. Hence, according to this law:

Weber model law is applied in following cases:

1. Capillary rise in narrow passages,

2. Capillary movement of water in soil,

3. Capillary waves in channels,

4. Flow over weirs for small heads. E ) Mach model law.

Mach model law is the law in which models are designed on Mach number, which is the ratio of the square root of inertia force to elastic force of a fluid. Hence where the forces due to elastic compression predominate in addition to inertia force, the dynamic similarity between the model and its prototype is obtained by equating the Mach number of the model and its prototype. Hence according to this law: ,

Mach model law is applied in the following cases

1. Flow of aero plane and projectile through air at supersonic speed, i.e., at a velocity more than the velocity of sound,

2. Aerodynamic testing,

3. Under water testing of torpedoes,

4. Water-hammer problems.

3.3.4 Classification of models

The hydraulic models are classified as:

1. Undistorted models Undistorted models are those models which are geometrically similar to their prototypes or in other words if the scale ratio for the linear dimensions of the model and its prototype is same, the model is called undistorted model. The behavior of the prototype can be easily predicted from the results of undistorted model.

2. Distorted models.

A model is said to be distorted if it is not geometrically similar to its prototype. For a distorted model different scale ratios for the linear dimensions are adopted. For example, in case of rivers, harbors, reservoirs etc. two different scale ratios, one for horizontal dimensions and other for vertical dimensions are taken. Thus the models of rivers, harbors and reservoirs will become as distorted models. If for the river, the horizontal and vertical scale ratios are taken to be same so that the model is undistorted, then the depth of water in the model of the river will be very-very small which may not be measured accurately. The followings are the advantages of distorted models:

I) The vertical dimensions of the model can be measured accurately.

II) The cost of the model can be reduced.

III) Turbulent flow in the model can be maintained.

Though there are some advantages of the distorted model, yet the results of the distorted model cannot be directly transferred to its prototype. But sometimes from the distorted models very useful information can be obtained.

Models with free surfaces - rivers, estuaries etc.

When modeling rivers and other fluid with free surfaces the effect of gravity becomes important and the major governing non-dimensional number becomes the Froude (Fr) number. The resistance to motion formula above would then be derived with g as an extra dependent variables to give an extra group. So the defining equation is:

(R, , u, l, , g ) = 0

From which dimensional analysis gives:

Generally the prototype will have a very large Reynolds number, in which case slight variation in Re causes little effect on the behavior of the problem. Unfortunately models are sometimes so small and the Reynolds numbers are large and the viscous effects take effect. This situation should be avoided to achieve correct results. Solutions to this problem would be to increase the size of the model - or more difficult - to change the fluid (i.e. change the viscosity of the fluid) to reduce the Re.

Geometric distortion in river models

When river and estuary models are to be built, considerable problems must be addressed. It is very difficult to choose a suitable scale for the model and to keep geometric similarity. A model which has a suitable depth of flow will often be far to big - take up too much floor space. Reducing the size and retaining geometric similarity can give tiny depth where viscous force comes into play. These result in the following problems:

accurate depths and depth changes become very difficult to measure;

the bed roughness of the channel becomes impracticably small;

Laminar flow may result - (turbulent flow is normal in river hydraulics.)

The solution often adopted to overcome these problems is to abandon strict geometric similarity by having different scales in the horizontal and the vertical. Typical scales are 1/100 in the vertical and between 1/200 and 1/500 in the horizontal. Good overall flow patterns and discharge characteristics can be produced by this technique; however local detail of flow is not well modeled.

In this model the Froude number (u2/d) is used as the dominant non-dimensional number. Equivalence in Froude numbers can be achieved between model and prototype even for distorted models. To address the roughness problem artificially high surface roughness of wire mesh or small blocks is usually used. Examples:

3.1 Find the expression for the power (P) required by a pump which depends up

on the head (H), the discharge (Q) and specific weight (w) using Raleigh Method.

P = f( H,Q,W)

P = k. Ha. Qb, Wc

Where k = non dimensional constant. Substituting the dimensions on both sides of the equation:

[ ML2T-3 ] = K .[ L ]a [ L3T-1.]b. [ M L-2T-2.]c Equating the powers of dimensions on both sides

Equate the power of M , 1 = c

c = 1

Equate the Power of L ,2 = a + 3b 2c

a = -3b + 4

Equate the Power of T , -3 = -b - 2c , b = 1

Then, a = 4-3b , a = -3 (1) + 4, a = 1

P = KH1Q1W1

P = H Q W

Where is the efficiency of the pump.

3.2 The physical quantities involved in the phenomena are Q, d, H, g, ( and (. Obtain the functional equation for discharge Q using Buckinghams ( -theorem..

Q = f (d, H, g, (, ()

f1 (Q, d, H, g, (, () = C

m = 3 , n = 6

f2 ((1, (2, (3) = C1 Let us choose three repeating variables: d, (, g

(1 = (a1 db1 gc1 Q

(2 = (a2 db2 gc2 ((3 = (a3 db3 gc3 H

Substituting dimensions on both sides for each (-terms, determine the arbitrary powers.

i) For (1 = M0L0T0 = [M/L3]a1 [L]b1 [L/T2]c1 [L3/T]

Equate the power of M, 0 = a1Equate the power of L, 0 = -3a1 + b1 + c1 + 3

Equate the power of T,0 = -2c1 1

c1 = -1/2

b1 = -5/2

(1 = Q

g1/2 d5/2

For (2 = (a2 db2 gc2 (M0L0T0 = [M L-3]a2 [L]b2 [LT-2]c2 [ML-1T-1]

Equate the power of M, 0 = a2 + 1

a2 = -1

Equate the power of L,0 = -3 a2 + b2 + c2 - 1

Equate the power of T ,0 = -2c2 1

c2 = -1/2

b2 = -3/2

(2 = ( (g1/2 d3/2

iii) For (3 = (a3 db3 gc3 H

M0L0T0 = (ML-3)a3 (L)b3 (LT-2)c3 (L)

Equate the power of M, 0 = a3Equate the power of L,0 = -3a3 + b3 + c3 + 1

Equate the power of T,0 = -2c3

c3 = 0

b3 = -1

(3 = H / d

Therefore, f2 ((1, (2, (3) = C1

196Lecture notes AU, CIVIL ENGINEERING23