Chapter 7DIMENSIONAL ANALYSIS

AND MODELING

Lecture slides by

Adam, KV Sharma

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DIMENSIONS AND UNITSDimension: A measure of a physical quantity (without numerical values).

Unit: A way to assign a number to that dimension.

There are seven primary dimensions : 1. Mass m (kg)2. Length L (m)3. Time t (sec)4. Temperature T (K)5. Current I (A)6. Amount of Light C (cd)7. Amount of matter N (mol)

All non-primary dimensions can be formed by some combination of the seven primary dimensions.

{Velocity} = {Length/Time} = {L/t}{Force} = {Mass Length/Time} = {mL/t2}

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The water strider is an insect that can walk on water due to surface tension.

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7–2 ■ DIMENSIONAL HOMOGENEITY

You can’t add apples and oranges!

The law of dimensional homogeneity: Every additive term in an equation must have the same dimensions.

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7–2 ■ DIMENSIONAL HOMOGENEITY

Bernoulli equation

The law of dimensional homogeneity: Every additive term in an equation must have the same dimensions.

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Nondimensionalization of Equations

Nondimensional equation: If we divide each term in the equation by a collection of variables and constants whose product has those same dimensions, the equation is rendered nondimensional.

Most of which are named after a notable scientist or engineer (e.g., the Reynolds number and the Froude number).

A nondimensionalized form of the Bernoulli equation is formed by dividing each additive term by a pressure (here we use P). Each resulting term is dimensionless (dimensions of {1}).

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In a general unsteady fluid flow problem with a free surface, the scaling parameters include a characteristic length L, a characteristic velocity V, a characteristic frequency f, and a reference pressure difference P0 P. Nondimensionalization of the differential equations of fluid flow produces four dimensionless parameters: the Reynolds number, Froude number, Strouhal number, and Euler number.

In Fluid Mechanics,

•the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces.

•The Froude number (Fr) is a dimensionless number defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of a partially submerged object moving through water, and permits the comparison of objects of different sizes.

•The Strouhal number (St) is a dimensionless number describing oscillating flow mechanisms.

•The Euler number (Eu) is a dimensionless number used in fluid flow calculations. It expresses the relationship between a local pressure drop over a restriction and the kinetic energy per volume, and is used to characterize losses in the flow, where a perfect frictionless flow corresponds to an Euler number of 1.

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DIMENSIONAL ANALYSIS AND SIMILARITY

In most experiments, to save time and money, tests are performed on a geometrically scaled model, not on the full-scale prototype.

In such cases, care must be taken to properly scale the results. Thus, powerful technique called dimensional analysis is needed.

The three primary purposes of dimensional analysis are

• To generate non-dimensional parameters that help in the design of experiments and in the reporting of experimental results

• To obtain scaling laws so that prototype performance can be predicted from model performance

• To predict trends in the relationship between parameters

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Greek letter Pi () denote a non-dimensional parameter.

In a general dimensional analysis problem, there is one that we call the dependent , giving it the notation 1.

The parameter 1 is in general a function of several other ’s, which we call independent ’s.

DIMENSIONAL ANALYSIS AND SIMILARITY

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To achieve similarity

The principle of similarity

Three necessary conditions for complete similarity between a model and a prototype.

(1) Geometric similarity—the model must be the same shape as the prototype, but may be scaled by some constant scale factor.

(2) Kinematic similarity—the velocity at any point in the model flow must be proportional (by a constant scale factor) to the velocity at the corresponding point in the prototype flow.

(3) dynamic similarity—When all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow (force-scale equivalence).

DIMENSIONAL ANALYSIS AND SIMILARITY

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In a general flow field, complete similarity between a model and prototype is achieved only when there is geometric, kinematic, and dynamic similarity.

Kinematic similarity is achieved when, at all locations, the speed in the model flow is proportional to that at corresponding locations in the prototype flow, and points in the same direction.

To ensure complete similarity, the model and prototype must be geometrically similar, and all independent groups must match between model and prototype.

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A 1 : 46.6 scale model of an Arleigh Burke class U.S. Navy fleet destroyer being tested in the 100-m long towing tank at the University of Iowa. The model is 3.048 m long. In tests like this, the Froude number is the most importantnondimensional parameter.

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Geometric similarity between a prototype car of length Lp and a model car of length Lm. In the case of aerodynamic drag on the automobile, there are only two ’s in the problem.

FD is the magnitude of the aerodynamic drag on the car, and so on forming drag coefficient equation.The Reynolds number is the most well known and useful dimensionless parameter in all of fluid mechanics.

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A drag balance is a device usedin a wind tunnel to measure theaerodynamic drag of a body. When testing automobile models, a moving belt is often added to the floor of the wind tunnel to simulate the moving ground (from the car’s frame of reference).

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Similarity can be achieved even when the model fluid is different than the prototype fluid. Here a submarine model is tested in a wind tunnel.

If a water tunnel is used instead of a wind tunnel to test their one-fifth scale model, the water tunnel speed required to achieve similarity is

One advantage of a water tunnel is that the required water tunnel speed is much lower than that required for a wind tunnel using the same size model (221 mi/h for air and 16.1 mi/h for water) .

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A drag balance is a device usedin a wind tunnel to measure theaerodynamic drag of a body. When testing automobile models, a moving belt is often added to the floor of the wind tunnel to simulate the moving ground (from the car’s frame of reference).

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THE METHOD OF REPEATING VARIABLESAND THE BUCKINGHAM PI THEOREM

How to generate the nondimensional analysis?

There are several method but the most popular was introduced by Edgar Buckingham called the method of repeating variables.

Step must be taken to generate the non-dimensional parameters, i.e., the ’s?

A concise summary of the six steps that comprise the method of repeating variables.

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Setup for dimensional analysis of a ball falling in a vacuum.

Pretend that we do not know the equation related but only know the relation of elevation z is a function of time t, initial vertical speed w0, initial elevation z0, and gravitational constant g. (Step 1)

Step 1

THE METHOD OF REPEATING VARIABLES

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A concise summary of the six steps that comprise the method of repeating variables.

Step 2

n = 5

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A concise summary of the six steps that comprise the method of repeating variables.

Step 3

The primary dimensions are [M], [L] and [t]. The number of primary dimensions in the problem are (L and t).

Then the number of ’s predicted by the Buckingham Pi theorem is

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A concise summary of the six steps that comprise the method of repeating variables.

Step 4

Need to choose two repeating parameters since j=2. Therefore

Caution1. Never choose dependent variable2. Do not choose variables that can form dimensionless group3. If there are three primary dimension available , must choose repeating variables which include all three primary dimensions. 4. Don’t pick dimensionless variables. For example, radian or degree.5. Never pick two variables with same dimensions or dimensions that differ by only an exponent. For example, w0 and g.6. Pick common variables such as length, velocity, mass or density. Don’t pick less common like viscosity or surface tension.7. Always pick simple variables instead of complex variables such as energy or pressure.

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Step 5: Construct the k ’s , and manipulate as necessary

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Step 6

Need modification for commonly used nondimensional parameters.

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The pressure inside a soap bubble is greater than that surrounding the soap bubble due to surface tension in the soap film.

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If the method of repeating variables indicates zero ’s, we have either made an error, or we need to reduce j by one and start over.

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Although the Darcy friction factor for pipe flows is most common, you should be aware of an alternative, less common friction factor called the Fanning friction factor. The relationship between the two is f = 4Cf .

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DIMENSIONLESS PARAMETER

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension.

It is thus a "pure" number, and as such always has a dimension of 1.

Other examples of dimensionless quantities:

- Weber number (We),

- Mach (M),

- Darcy friction factor (Cf or f),

- Drag coefficient (Cd) etc.

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RAYLEIGH METHOD

Rayleigh's method of dimensional analysis is a conceptual tool used in physics, chemistry, and engineering.

This form of dimensional analysis expresses a functional relationship of some variables in the form of an exponential equation.

It was named after Lord Rayleigh.

The method involves the following steps:

•Gather all the independent variables that are likely to influence the dependent variable.

•If X is a variable that depends upon independent variables X1, X2, X3, ..., Xn, then the functional equation can be written as X = F(X1, X2, X3, ..., Xn).

•Write the above equation in the form where C is a dimensionless constant and a, b, c, ..., m are arbitrary exponents.

•Express each of the quantities in the equation in some fundamental units in which the solution is required.

•By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents a, b, c, ..., m.

•Solve these equations to obtain the value of exponents a, b, c, ..., m.

•Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents. 41