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Similitude and Dimension Analysis

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SIMILITUDE AND DIMENSION ANALYSIS
DEFINITION AND USES OF SIMILITUDE • Similitude means similarity • it impossible to determine all the essential facts for a given fluid flow by pure theory alone • we must often depend on experimental investigations. • we can greatly reduce the number of tests needed by systematically using dimensional analysis and the laws of similitude or similarity. • For these enable us to apply test data to other cases than those observed. • we can obtain valuable results at a minimum cost from tests made with small-scale models of the full-size apparatus. The laws of similitude enable us to predict the performance of the prototype, which means the full-size device, from tests made with the model. for example, we might study the flow in a carburetor in a very large model. • 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.

GEOMETRIC SIMILARITY • geometric similarity means that the model and its 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 the prototype to the corresponding dimensions in the model.
=

• Area ratio Lr2 and volume ratio Lr3

GEOMETRIC SIMILARITY
• Complete geometric similarity is not always easy to attain. For example, • we may not be able to reduce the surface roughness of a small model in proportion unless we can make its surface very much smoother than that of the prototype. • Similarly, in the study of sediment transport, we may not be able to scale down the bed materials without having material so fine as to be impractical. Fine powder, because of cohesive forces between the particles, does not simulate the behavior of sand. • Again, in the case of a river the horizontal scale is usually limited by the available floor space, and this same 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 to use 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’

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 value in terms of L, is determined by dynamic considerations, as explained in the following section. As time T is dimensionally L/V, the time scale ratio is = • and in a similar manner the acceleration scale 2 ratio is
=
2

=

DYNAMIC SIMILARITY • Two systems have dynamic similarity if, in addition to kinematic similarity, corresponding forces are in the same ratio in both. The force 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). If the sum of forces on a fluid element does not add up to zero, the element will accelerate in accordance with Newton'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, ƩF = FG + FP + FV + FE + FT = Resultant and FI = - Resultant Thus FG +FP+FV+FE+FT+FI = 0

Dynamic Similarity
• These forces can be expressed in the simplest terms as: • Gravity: FG = mg = pL3g • Pressure: FP = (Δp)A = (Δp)L2 • Viscosity: FV = μdu/dyA = µ(V/L)L2 = µVL • Elasticity: FE = EvA = EvL2 • Surface tension: FT = σL • Inertia: FI = ma = ρL3 L/T2 = ρL4T-2 = ρV2L2

• In many flow problems some of these forces are either absent or insignificant. In Fig we see two geometrically similar flow systems. Let us assume that they also possess kinematic similarity, and that the forces acting on any fluid element are FG , FP ,FV , and FI. Then we will have dynamic similarity if
= = = =

Reynolds Number
• In the flow of a fluid through a completely filled conduit, gravity does not affect the flow pattern. Also, since there are no free liquid surfaces, capillarity is obviously of no practical importance. Therefore the significant forces are inertia and fluid friction due to viscosity. • For the ratio of inertia forces to viscous forces, we call the resulting parameter the Reynolds number • This is actually a theory of dynamic similarity

Reynolds Number
• The ratio of these two forces is
= =
2 2

=

=

• For any consistent system of units, R is a dimensionless number, 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. Thus, for a pipe flowing full,
= =

where D is the diameter of the pipe. • If two systems, such as a model and its prototype, or two pipelines with different fluids, are dynamically equivalent so far as inertia and viscous forces are concerned, they must both have the same value of R. Thus, for such cases, we will have dynamic similarity when
= = =

Froude Number
• When we consider inertia and gravity forces alone, we obtain a ratio called a Froude number, or F. The ratio of inertia forces to 2 2 2 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 in the Reynolds number. Thus a Froude number is
=
3

=

• 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 bridge pier, 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 linear dimension 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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