Chapter 7 Similitude and Dimensional Analysis

7/31/2019 Chapter 7 Similitude and Dimensional Analysis

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SIMILITUDE AND DIMENSIONAL ANALYSIS

DEFINITION AND USES OF SIMILITUDE

Similitude means similarity

it impossible to determine all the essential facts for a givenfluid flow by pure theory alone

we must often depend on experimental investigations.

we can greatly reduce the number of tests needed bysystematically using dimensional analysis and the laws of

similitude or similarity. Because, these enable us to apply test data to other cases

than those observed.

we can obtain valuable results at a minimum cost fromtests made with small-scale models of the full-sizeapparatus. The laws of similitude enable us to predict theperformance of the prototype, which means the full-sizedevice, from tests made with the model. for example, wemight study the flow in a carburetor in a very large model.


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A few examples of where we have used models are

Ships in towing basins, Airplanes in wind tunnels,

Hydraulic turbines,

Centrifugal pumps,

Spillways of dams, River channels and the study of such

phenomena as the action of waves and tides on

beaches,

Soil erosion and Transport of sediment.


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GEOMETRIC SIMILARITY

geometric similarity means that the model andits prototype have identical shapes but differ only

in size. the flow patterns must be geometrically similar. If

subscripts p and m denote prototype and model,respectively, we define the length scale ratios as

the ratio of the linear dimensions of theprototype to the corresponding dimensions in themodel.

Area ratio Lr2 and volume ratio Lr

3=


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Complete geometric similarity is not always easy toattain.

For example,

we may not be able to reduce the surface roughnessof a small model in proportion unless we can make

its surface very much smoother than that of theprototype.

Similarly, in the study of sediment transport, we maynot be able to scale down the bed materials withouthaving material so fine as to be impractical. Finepowder, because of cohesive forces between theparticles, does not simulate the behavior of sand.



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Again, in the case of a river the horizontal scale is

usually limited by the available floor space, and thissame scale used for the vertical dimensions may

produce a stream so shallow that capillarity has an

appreciable effect and also the bed slope may be so

small that the flow is laminar. In such cases we need touse a distorted model, which means that the vertical

scale is larger than the horizontal scale. Then, if the

horizontal scale ratio is denoted by Lr

and the vertical

scale ratio by Lr, the cross section area ratio is LrLr



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KINEMATIC SIMILITARY

Kinematic similarity implies that, in addition to

geometric similarity, the ratio of the velocities at all

corresponding points in the flows are the same. The

velocity scale ratio is

and this is a constant for kinematic similarity. Its valuein terms ofL, is determined by dynamic considerations.

As time Tis dimensionally L/V, the time scale ratio is

and in a similar manner the acceleration scale ratio is

=

=

=

2=

2


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DYNAMIC SIMILARITY

Two systems have dynamic similarityif, in addition

to kinematic similarity, corresponding forces are inthe same ratio in both. Theforce scale ratio is

which must be constant for dynamic similarity.

Forces that may act on a fluid element include

those due to gravity (FG), pressure (FP), viscosity(Fv), and elasticity (FE). Also, if the element of fluid

is at a liquid-gas interface, there are forces due to

surface tension (FT).

=


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DYNAMIC SIMILARITY(Cont..)

If the sum of forces on a fluid element does not add up

to zero, the element will accelerate in accordance withNewton's law. We can transform such an unbalanced

force system into a balanced system by adding an inertia

force F, that is equal and opposite to the resultant of the

acting forces. Thus, generally,

F = FG+ FP + FV + FE+ FT= Resultant

and FI = - Resultant

Thus FG +FP+FV+FE+FT+FI = 0


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Dynamic Similarity

These forces can be expressed in the simplest

terms as:

Gravity: FG = mg = pL3g

Pressure: FP = (p)A = (p)L2

Viscosity: FV= du/dyA = (V/L)L2 = VL

Elasticity: FE= EvA = EvL2

Surface tension: FT

= L

Inertia: FI = ma = L3 L/T2 = L4T-2 = V2L2


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In many flow problems some of these forces are eitherabsent or insignificant. In Fig we see two geometricallysimilar flow systems. Let us assume that they also possesskinematic similarity, and that the forces acting on any fluidelement are FG , FP ,FV, and FI. Then we will have dynamicsimilarity if

=

=

=

=


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Reynolds Number

In the flow of a fluid through a completelyfilled conduit, gravity does not affect the flowpattern. Also, since there are no free liquidsurfaces, capillarity is obviously of no practical

importance. Therefore the significant forcesare inertia and fluid friction due to viscosity.

For the ratio ofinertia forces to viscous forces,we call the resulting parameter the Reynoldsnumber

This is actually a theory of dynamic similarity


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Reynolds Number(Cont)

The ratio of these two forces is

For any consistent system of units, R is a dimensionlessnumber, which turns out to be useful for comparing

different flows. The linear dimension L may be any

length that is significant in the flow pattern. Thus, for a

pipe completely filled, commonly we use the pipe

diameter for L.

= =22 =

=


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Thus, for a pipe flowing full,

where D is the diameter of the pipe.

If two systems, such as a model and itsprototype, or two pipelines with differentfluids, are dynamically equivalent so far asinertia and viscous forces are concerned, they

must both have the same value of R. Thus, forsuch cases, we will have dynamic similaritywhen

= =

=

=

=

Reynolds Number(Cont.)


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Froude Number

When we consider inertia and gravity forces

alone, we obtain a ratio called a Froudenumber, or F. The ratio of inertia forces to

gravity forces is

Although this is sometimes defined as a

Froude number, it is more common to use the

square root so that V is in the first power, as inthe Reynolds number. Thus a Froude number

is

223

=2

=


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Froude Number (Cont.) Systems involving gravity and inertia forces include

the wave action set up by a ship, the flow of water in

open channels, the forces of a stream on a bridgepier, the flow over a spillway, the flow of a jet from

an orifice, and other cases where gravity is the

dominant factor.

To compute F, the length L must be some lineardimension that is significant in the flow pattern. For

a ship, we commonly take this as the length at the

waterline. For an open channel, we take it as the

depth of flow. For situations where inertia and

gravity forces predominate, dynamic similarity

occurs when

=

=

=


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Dynamic Similarity

Mach Number

When compressibility is important, we need to consider theratio of the inertia to the elastic forces. The Mach numberM is defined as the square root of this ratio.

where c is the sonic velocity (or celerity) in the medium inquestion (see Sec. 13.3). So the Mach number is the ratio ofthe fluid velocity (or the velocity of the body through astationary fluid) to that of a sound wave in the samemedium. If M is less than 1, the flow is subsonicif it is equalto 1, it is sonic; if it is greater than 1, the flow is calledsupersonic.

D i Si il it


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Dynamic SimilarityWeber Number

Surface tension may be important in a few

cases of flow, but usually it is negligible. Theratio of inertia forces to surface tension is

pV2L2/L, the square root of which is known as

the Weber number:Euler Number

A dimensionless quantity related to the ratio

of the inertia forces to the pressure forces isknown as the Euler number.

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Chapter 7 Similitude and Dimensional Analysis