Ch7 Dim Analysis

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Fluid Mechanics

II : Chapter 7 11

Chapter 7

Dimensional Analysis&

Modeling

Fluid Mechanics - II


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Fluid Mechanics

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Introduction Although many practical engineering problems involving fluid mechanics can be

solved by using the equations and analytical procedures described in the preceding

chapters The solution to many problems is achieved through the use of a combination of

analysis and experimental data. Thus, engineers working on fluid mechanicsproblems should be familiar with the experimental approach to these problems sothat they can interpret and make use of data obtained by others

In this chapter we consider some techniques and ideas that are important in the

planning and execution of experiments, as well as in understanding and correlatingdata that may have been obtained by other experimenters

The laboratory systems are usually thought of as modelsand are used to study thephenomenon of interest under carefully controlled conditions

From these model studies, empirical formulations can be developed, or specificpredictions of one or more characteristics of some other similar system can be

made To do this, it is necessary to establish the relationship between the laboratory

model and the other system,Actual Full Scale Product

In the following sections, we find out how this can be accomplished in a systematicmanner


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Dimensional Analysis Experimental analysis of a problem is generally done using Dimensionless Groups

of properties so that a large number of variable are treated in groups and facilitatingthe following:

Behavior of 2 variables on a 2-D graphs while keeping the other variable constant

Simulation through using lab models on a much smaller scale

Correlation of experimental data using smaller model to actual big scale geometry

It is clear that the experiment would be much simpler, easier to do, and lessexpensive, when small scaled models are used instead of real big products

The basis for this simplification lies in a consideration of the dimensions of thevariables involved

As was discussed in Chapter 1, a qualitative description of physical quantities can

be given in terms of basic dimensions such as mass (M), length (L), and time (T) Alternatively, we could use force (F), L, and Tas basic dimensions : from Newtons

second law


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Dimensional Analysis /(Contd.) Consider, steady flow of an incompressible Newtonian fluid through a long, smooth-

walled, horizontal, circular pipe

An important characteristic of this system, which would be of interest to an engineerdesigning a pipeline, is the pressure drop per unit length that develops along thepipe as a result of friction

This would appear to be a relatively simple flow problem, it cannot generally besolved analytically (even with the aid of large computers) without given data aboutPipe and Fluid

Such data (which we use in analytical solutions of problems) is attained from seriesof experiments

The first step in the planning of an experiment to study this problem would be todecide on the factors, or variables, that will have an effect on the pressure drop perunit length DPl

We expect the list to include the pipe diameterD, the fluid density r , fluid viscosity

m , and the mean velocity V, at which the fluid is flowing through the pipe Thus, we can express this relationship as

DPl = f(D, r, m, V) 7.1

A complete picture of flow characteristics can be plotted by four 2-D graphs,Fig 7.1


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Dimensional Analysis /(Contd.)

Fortunately, there is a much simpler approach can be adopted in which the four graphs above

can be represented by only ONE graph. See Fig 7.2

Note that the Properties plotted at X and Y axisare dimensionless.

The original 5 variablesare presented in two ND Groups

In the following sections we will study that how to reduce original list of variables, as described

in Eq. 7.1, into fewer non-dimensional combinations of variables (called dimensionless

products ordimensionless groups) like in this case 2 groups; that are :


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Buckingham Pi Theorem If an equation involving k variables is dimensionally homogeneous, it can be reduced to a

relationship among independent dimensionless products, where r is the minimum number

of reference dimensions required to describe the variables The min. required number of DL groups to describe the phenomena, would then be (kr)

The dimensionless products are frequently referred to as pi terms, and the theorem is calledthe Buckingham pi Theorem

Buckingham used the symbolP to represent a dimensionless product, and this notation iscommonly used by all

The pi theorem is based on the idea of dimensional homogeneity which was introduced inChapter 1

Essentially we assume that for any physically equation involving kvariables, such as

u1 = f (u2, u3, u4 uk)

The dimensions of the variable on the left side of the equal sign must be equal to thedimensions of any term that stands by itself on the right side of the equal sign. It then followsthat we can rearrange the equation into a set of dimensionless products (pi terms) so that

P1 = f (P2 ,, P3 , P4 ......... Pk-r) The required number of pi terms is fewer than the number of original variables by r, where ris

determined by the minimum number of reference dimensions required to describe the originallist of variables

Usually the reference dimensions required to describe the variables will be the basicdimensions M, L, and TorF, L, and T


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Determination of Pi Terms

Following the steps to be followed in performing a dimensional analysis

using the method of repeating variables are as follows: Step 1. List all the variables that are involved in the problem.

Step 2. Express each of the variables in terms of basic dimensions.

Step 3. Determine the required number of pi terms.

Step 4. Select a number of repeating variables, where the number required is

equal to the number of reference dimensions (usually the same as the number ofbasic dimensions)

Step 5. Form a pi term by multiplying one of the non-repeating variables by theproduct of repeating variables each raised to an exponent that will make thecombination dimensionless

Step 6. Repeat Step 5 for each of the remaining repeating variables. Step 7. Check all the resulting pi terms to make sure they are dimensionless.

Step 8. Express the final form as a relationship among the pi terms and thinkabout what it means.


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To illustrate these various steps we will again consider the problem discussed

earlier in this chapter which was concerned with the steady flow of anincompressible Newtonian fluid through a long, smooth-walled, horizontal circularpipe

We are interested in the pressure drop per unit length, DPl along the pipe

According to Step 1 we must list all of the pertinent variables that are involvedbased on the experimenters knowledge of the problem

In this problem we assume that DPl = f(D, r, m, V)

Next (Step 2) we express all the variables in terms of basic dimensions. Using F,L, and Tas basic dimensions it follows that

DPl =FL-3, D=L, r= F L-4 T2, m= F L-2T, V = LT-1

We could also use M, L, and Tas basic dimensions if desired the final result will

be the same

We can now apply the pi theorem to determine the required number of pi terms(Step 3)


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Determination of Pi Terms An inspection of the dimensions of the variables from Step 2 reveals that all three

basic dimensions are required to describe the variables

Since there are five variables (k=5), (do not forget to count the dependent variableDPl ) and three required reference dimensions ( r = 3 ) then according to the pitheorem there will be (5 3), two pi terms required

The repeating variables to be used to form the pi terms (Step 4) need to beselected from the list D, r , m and V

Since three reference dimensions are required, we will need to select threerepeating variables

Generally, we would try to select from repeating variables those that are thesimplest, dimensionally

For example, if one of the variables has the dimension of a length, choose it asone of the repeating variables. In this example we will use D, V, and r asrepeating variables

Note that these are dimensionally independent, since D is a length, Vinvolvesboth length and time, and r involves force, length, and time

This means that we cannot form a dimensionless product from this set


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We are now ready to form the two pi terms (Step 5). Typically, we would start with

the dependent variable and combine it with the repeating variables to form the firstpi term; that is,

P1 = DPl DaVb rc

Since this combination is to be dimensionless, it follows that

(FL -3) (L)a(LT-1)b (FL -4T2)c = F0 L0 T0

Or (ML -2T-2 ) (L)a(LT-1)b (ML -3)c = M0 L0 T0 where F= ML T-2

The exponents, a, b, and cmust be determined such that the resulting exponentfor each of the basic dimensions; F, L, and Tmust be zero (so that the resultingcombination is dimensionless). Thus, we can write

For F 1 + c = 0

For L -3 + a + b - 4c = 0

For T -b + 2c = 0

The solution of this system of algebraic equations gives the desired values fora, b,and c,that a = 1, b = -2 and c = -1

Therefore , Similarly,21

V

DPl

r

D=

VDr

m=P

2


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Common Dimensionless Groups in Fluid Mechanics

Ex 7.1 & 7.2


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Correlation with Experimental Data

Dimensional analysis is as an aid in the efficient handling,

interpretation, and correlation of experimental data. It cannot provide a complete answer to any given problem, since the

analysis only provides the dimensionless groups describing thephenomenon, and not the specific relationship among the groups

To determine this relationship, suitable experimental data must beobtained

The degree of difficulty involved in this process depends on the Number of pi terms

Nature of the experiments (How hard is it to obtain the measurements?)

The simplest problems are obviously those involving the fewest piterms, we will now see how the complexity of the analysis increaseswith the increasing number of pi terms.


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Phenomena's with one Pi termP1 = Constant

This is one situation in which a dimensional analysis reveals the

specific form of the relationship and shows how the individual

variables are related (See Example 7.3)

The value of the constant, however, must still be determined by

experiment

Phenomena's with 2 Pi termsP1 = f(P2)

The functional relationship among the variables can then be

determined by varying P2 and measuring the corresponding values

of P1

For this case the results can be conveniently presented in graphical

form by plotting P1 versus P2 and is illustrated in Fig. 7.4


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Figure 7.4

Note that there is only a single relationship between P2 and P1 and is valid over

the range ofP2 covered by the experiments

It would be unwise to extrapolate beyond this range, since as illustrated with the

dashed lines in the figure

In actual case, the nature of the phenomenon could dramatically change as the

range P2 of is extended

In addition to presenting the data graphically, it may be possible (and desirable) to

obtain an empirical equation relating P1 and P2 by using a standard curve-fitting

technique, see example 7.5


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Phenomena's with more than 2 Pi termsP1 = f(P2, P3) (7.7)

It becomes more difficult to display the results in a convenient graphical form andto determine a specific empirical equation that describes the phenomenon

For problems involving three pi termsP1 = f(P2, P3)

It is still possible to show data correlations on simple graphs by plotting families of

curves as illustrated in Fig. 7.5

This is an informative and useful way of representing the data in a general way

It may also be possible to determine a suitable empirical equation relating the

three pi terms


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Modeling & Simulation However, as the number of pi terms continues to increase, corresponding to an

increase in the general complexity of the problem of interest, both the graphical

presentation and the determination of a suitable empirical equation becomeintractable

For these more complicated problems, it is often more feasible to use models topredict specific characteristics of the system rather than to try to develop generalcorrelations

Models are widely used in fluid mechanics. Major engineering projects involvingstructures, aircraft, ships, rivers, harbors, dams, air and water pollution, and so on,frequently involve the use of model

The physical system for which the predictions are to be made is called theprototype. Although mathematicalorcomputermodels may also conform to thisdefinition, our interest will be in physical models, that is, models that resemble the

prototype but are generally of a different size, may involve different fluids, andoften operate under different conditions (pressures, velocities, etc.)

Usually a model is smaller than the prototype. Therefore, it is more easily handledin the laboratory and less expensive to construct and operate than a largeprototype


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Modeling & Simulation/(Contd.) It is imperative that the model be properly designed and tested and that the results

be interpreted correctly

Theory of Models

The theory of models can be readily developed by using the principles ofdimensional analysis

It has been shown that any given problem can be described in terms of a set of piterms as P1 = f(P2, P3) (7.7)

In formulating this relationship, only a knowledge of the general nature of thephysical phenomenon, and the variables involved, is required. Specific values forvariables (size of components, fluid properties, and so on) are not needed toperform the dimensional analysis

Thus, Eq. 7.7 applies to any system that is governed by the same variables. If Eq.7.7 describes the behavior of a particular prototype, a similar relationship can bewritten for a model of this prototype; that is, P1m = f(P2m, P3m)

where the form of the function will be the same as long as the samephenomenon is involved in both the prototype and the model. Variables, or piterms, without a subscript will refer to the prototype, whereas the subscript m willbe used to designate the model variables or pi terms (ex. 7.5)


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Fluid Mechanics

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Modeling & Simulation/(Contd.)

Model Scales

It is clear from the preceding section that the ratio of like quantities for the modeland prototype naturally arises from the similarity requirements. For example, if in a

given problem there are two length variables l1 and l2 the resulting similarity

requirement based on a pi term obtained from these two variables is

Such ratio is called as Scale Length ; ( ll )

For true models there will be only one length scale, and all lengths are fixed in

accordance with this scale

There are, however, other scales such as the velocity scale, density scale,

viscosity scale, and so on ( lv , lr, lm etc.)

In fact, we can define a scale for each of the variables in the problem. Thus, it is

actually meaningless to talk about a scale of a model without specifying which

scale

mmm

m

l

l

l

lor

l

l

l

l

2

2

1

1

2

1

2

1==


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Fluid Mechanics

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Some Typical Model Studies

Flow through Closed Conduits

Flow around Immersed Bodies

Flow with a Free Surface


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Fluid Mechanics

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Assignments / Self Study

Examples; Solve yourself

Do at least 15 from first 60 problems from

Chapter 7 of text book

Ch7 Dim Analysis

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