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Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMILITUDE.

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Dimensionless parameters significantly deepen our understanding of fluid-flow phenomena in a way which is analogous to the case of a hydraulic jack, where the ratio of piston diameters determines the mechanical advantage, a dimensionless number which is independent or the overall size of the jack.

- They permit limited experimental results to be applied to situations involving different physical dimensions and often different fluid properties.
- The concepts of dimensional analysis introduced in this chapter plus an understanding of the mechanics of the type of flow under study make possible this generalization of experimental data.
- The consequence of such generalization is manifold, since one is now able to describe the phenomenon in its entirety and is not restricted to discussing the specialized experiment that was performed. Thus,it is possible to conduct fewer (but highly selective) experiments to uncover the hidden facets of the problem and thereby achieve important savings in time and money.

Equally important is the fact that, researchers are able to discover new features and missing areas of knowledge of the problem at hand.

- This directed advancement of our understanding of a phenomenon would be impaired if the tools of dimensional analysis were not available.
- Many of the dimensionless parameters may be viewed as a ratio of a pair of fluid forces, the relative magnitude indicating the relative importance of one of the forces with respect to the other.
- If some forces in a particular flow situation are very much larger than a few others, it is often possible to neglect the effect of the smaller forces and treat the phenomenon as though it were completely determined by the major forces. This means that simpler (but not necessarily easy) mathematical and experimental procedures can be used to solve the problem.
- For situations with several forces of the same magnitude (inertial, viscous, and gravitational forces) special techniques are required.

4.1 DIMENSIONAL HOMOGENEITY AND DIMENSIONLESS RATIOS

- Solving practical design problems in fluid mechanics requires both theoretical developments and experimental results.
- By grouping significant quantities into dimensionless parameters, it is possible to reduce the number of variables appealing and to make this compact result (equations or data plots) applicable to all similar situations.
- If one were to write the equation of motion ∑F = ma for a fluid particle, including all types of force terms that could act (gravity, pressure, viscous, elastic, and surface-tension forces), an equation of the sum of these forces equated to ma (the inertial force) would result.

Each term must have the same dimensions - force.

- The division of each term of the equation by any one of the terms would make the equation dimensionless. For example, dividing through by the inertial force term would yield a sum of dimensionless parameters equated to unity.
- The relative size of any one parameter, compared with unity, would indicate its importance. If divide the force equation through by a different term, say the viscous force term, another set of dimensionless parameters would result.
- Without experience in the flow case it is difficult to determine which parameters will be most useful.

An example of the use of dimensional analysis and its advantages is given by considering the hydraulic jump. The momentum equation for this case

(4.1.1)

- The right-hand side - the inertial forces; left-hand side - the pressure forces due to gravity. These two forces are of equal magnitude, since one determines the other in this equation.
- The term γy12/2 has the dimensions of force per unit width, and it multiplies a dimensionless number which is specified by the geometry or the hydraulic jump.

If one divides this equation by the geometric term 1 - y2/y1 and a number representative of the gravity forces, one has

(4.1.2)

- The left-hand side - the ratio of the inertia and gravity forces, even though the explicit representation of the forces has been obscured through the cancellation of terms that are common in both the numerator and denominator.
- This ratio is equivalent to a dimensionless parameter, actually the square of the Froude number.
- This ratio of forces is known once the ratio y2/y1 is given, regardless or what the values y2 and y1 are.
- From this observation one can obtain an appreciation or the increased scope that Eq. (4.1.2) affords over Eq. (4.1.1) even though one is only a rearrangement of the other.

In writing the momentum equation which led to Eq. (4.1.2) only inertia and gravity forces were included in the original problem statement. But other forces, such as surface tension and viscosity, are present (were neglected as being small in comparison with gravity and inertia forces).

- However, only experience with the phenomenon, or with phenomena similar to it, would justify such an initial simplification.
- For example, if viscosity had been included because one was not sure of the magnitude of its effect, the momentum equation would become
- This statement is more complete than that given by Eq. (4.1.2). However, experiments would show that the second term on the left-hand side is usually a small fraction of the first term and could be neglected in making initial tests on a hydraulic jump.

In the last equation one can consider the ratio y2/y1 to be a dependent variable which is determined for each of the various values of the force ratios, V12/gy1 and Fviscous/γy12, which are the independent variables.

- From the previous discussion it appears that the latter variable plays only a minor role in determining the values of y2/y1. Nevertheless, if one observed that the ratios of the forces, V12/gy1 and Fviscous/γy12, had the same values in two different tests, one would expect, on the basis of the last equation, that the values of y2/y1 would be the same in the two situations.
- If the ratio of V12/gy1 was the same in the two tests but the ratio Fviscous/γy12, which has only a minor influence for this case, was not, one would conclude that the values of y2/y1 for the two cases would be almost the same.

This is the key to much of what follows. For if one can create in a model and force ratios that occur on the full-scale unit, then the dimensionless solution for the model is valid for the prototype also.

- Often it is not possible to have all the ratios equal in the model and prototype. Then one attempts to plan the experimentation in such a way that the dominant force ratios are as nearly equal as possible.
- The results obtained with such incomplete modeling are often sufficient to describe the phenomenon in the detail that is desired.
- Writing a force equation for a complex situation may not be feasible, and another process, dimensional analysis, is then used if one knows the pertinent quantities that enter into the problem.
- In a given situation several of the forces may be of little significance, leaving perhaps two or three forces of the same order or magnitude. With three forces of the same order or magnitude, two dimensionless parameters are obtained; one set of experimental data on a geometrically similar model provides the relations between parameters holding for all other similar flow cases.

4.2 DIMENSIONS AND UNITS

- The dimensions of mechanics are force, mass, length, and time; they are related by Newton's second law of motion,

F = ma (4.2.1)

- For all physical systems, it would probably be necessary to introduce two more dimensions, one dealing with electromagnetics and the other with thermal effects.
- For the compressible work in this text, it is unnecessary to include a thermal unit, because the equations or state link pressure, density, and temperature.
- Newton's second law of motion in dimensional form is

F = MLT-2 (4.2.2)

which shows that only three of the dimensions are independent. F is the force dimension, M the mass dimension, L the length dimension, and T the time dimension.

- One common system employed in dimensional analysis is the MLT system.

4.3 THE П THEOREM

- The Buckingham Π theorem proves that, in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n - m independent dimensionless parameters.
- Let A1, A2, A3.... An be the qualities involved, such as pressure, viscosity, velocity, etc. All the quantities are known to be essential to the solution, and hence some functional relation must exist

(4.3.1)

- If Π1, Π2, ..., represent dimensionless groupings of the quantities A1, A2, A3, ..., then with m dimensions involved, an equation of the following form exists

(4.3.2)

The method of determining the Π parameters is to select m of the A quantities, with different dimensions, that contain among them the m dimensions, and to use them as repeating variables together with one of the other A quantities for each Π.

- For example, let A1, A2, A3 contain M, L and T, not necessarily in each one, but collectively. Then the Π parameters are made up as

(4.3.3)

- the exponents are to be determined each Π is dimensionless. The dimensions of the A quantities are substituted, and the exponents of M, L, and T are set equal to zero respectively. These produce three equations in three unknowns for each Π parameter, so that the x, y, z exponents can be determined, and hence the Π parameter.

- If only two dimensions are involved, then two of the A quantities are selected as repeating variables, and two equations in the two unknown exponents are obtained for each Π term.
- In many cases the grouping of A terms is such that the dimensionless arrangement is evident by inspection. The simplest case is that when two quantities have the same dimensions, e.g., length, the ratio or these two terms is the Π parameter.

Example 4.1

- The discharge through a horizontal capillary tube is thought to depend upon the pressure drop per unit length, the diameter, and the viscosity. Find the form of the equation.

Solution

- The quantities are listed with their dimensions:

Then

- Three dimensions are used, and with four quantities there will be one Π parameter:
- Substituting in the dimensions gives
- The exponents of each dimension must be the same on both sides of the equation. With L first,

And similarly for M and T

- From which x1 = 1, y1 = -1, z1 = -4, and
- After solving for Q,
- From which dimensional analysis yields no information about the numerical value of the dimensionless constant C; experiment (or analysis) shows that it is π/128 [Eq. (5.4.10a)].

Example 4.2

- A V-notch weir is a vertical plate with a notch of angle φ cut into the top of it and placed across an open channel. The liquid in the channel is backed up and forced to flow through the notch. The discharge Q is some function of the elevation H of upstream liquid surface above the bottom of the notch. In addition, the discharge depends upon gravity and upon the velocity of approach V0 to the weir. Determine the form of discharge equation.

Solution

- A functional relation
- Is to be grouped into dimensionless parameters. φ is dimensionless; hence, it is one Π parameter.
- Only two dimensions are used, L and T. If and H are the repeating variables.

Then

- From which , and
- This can be written
- In which both f and f1 are unknown functions. After solving for Q,
- Either experiment or analysis is required to yield additional information about the function f1.

If H and V0 were selected as repeating variables in place of g and H,

- From which , and
- Since any of the Π parameters can be inverted or raised to any power without affecting their dimensionless status,
- The unknown function f2 has the same parameters as f1, but it could not be the same function. The last form is not very useful, in general, because frequently V0 may be neglected with V-notch weirs. This shows that a term of minor importance should not be selected as a repeating variable.

Example 4.3

- The thrust due to any one of a family of geometrically similar airplane propellers is to be determined experimentally from a wind-tunnel test on a model. Use dimensional analysis to find suitable parameters for plotting test results.

Solution

- The thrust FT depends upon speed of rotation ω, speed of advance V0, diameter D, air viscosity μ, density ρ, and speed of sound c.
- The function
- is to be arranged into four dimensionless parameters, since there are seven quantities and three dimensions. Starting first by selecting ρ, ω, and D as repeating variables.

By writing the simultaneous equations in xl, yl, zl, etc., as before and solving them gives,

- Solving for the thrust parameter leads to

..

- Since the parameters can be recombined to obtain other forms, the second term is replaced by the product of the first and second terms, VDρ/μ, and the third term is replaced by the first term divided by the third term, V0/c; thus
- Of the dimensionless parameters, the first is probably of the most importance since it relates speed of advance to speed of rotation. The second parameter is a Reynolds number and accounts for viscous effects.
- The last parameter, speed of advance divided by speed of sound, is a Mach number, which would be important for speeds near or higher than the speed of sound. Reynolds effects are usually small, so that a plot of FT/ρω2D4 against V0/ωD should be most informative.

The steps in a dimensional analysis may be summarized as follows:

- Select the pertinent variables (requires some knowledge of the process).
- Write the functional relations, e.g.,
- Select the repeating variables. (Do not make the dependent quantity a repeating variable.) These variables should contain all the m dimensions or the problem. Often one variable is chosen because it specifies the scale, another the kinematic conditions; and in the cases of major interest in this chapter one variable which is related to the forces or mass of the system, for example, D, V, ρ, is chosen.
- Write the Π parameters in terms of unknown exponents, e.g.,

For each of the Π expressions write the equations of the exponents, so that the sum of the exponents of each dimension will be zero.

- Solve the equations simultaneously.
- Substitute back into the Π expressions of step 4 the exponents to obtain the dimensionless Π parameters.
- Establish the functional relation

or solve for one of the Π's explicitly:

- Recombine, if desired, to alter the forms of the Π parameters, keeping the same number or independent parameters.

4.4 DISCUSSION OF DIMENSIONLESS PARAMETERS

- The five dimensionless parameters:
- pressure coefficient;
- Reynolds number;
- Froude number;
- Weber number;
- Mach number

- are of importance in correlating experimental data.

Pressure Coefficient

- The pressure coefficient △p/(ρV2/2) is the ratio of pressure to dynamic pressure
- When multiplied by area, it is the ratio of pressure force to inertial force, as (ρV2/2)A would be the force needed to reduce the velocity to zero.
- It may also be written as △h/(V2/2g) by division by γ.
- For pipe flow the Darcy-Weisbach equation relates losses h1 to length of pipe L, diameter D, and velocity V by a dimensionless friction factor f
- as fL/D is shown to be equal to the pressure coefficient.
- In pipe flow, gravity has no influence on losses; therefore, F may be dropped out. Similarly, surface tension has no effect, and W drops out.

For steady liquid flow, compressibility is not important, and M is dropped. l may refer to D; l1 to roughness height projection є in the pipe wall; and l2 to their spacing є'; hence,

(4.4.1)

- If compressibility is important,

(4.4.2)

- With orifice flow,

(4.4.3)

in which l may refer to orifice diameter and l1 and l2 to upstream dimensions.

Viscosity and surface tension are unimportant for large orifices and low-viscosity fluids. Mach number effects may be very important for gas flow with large pressure drops, i.e., Mach numbers approaching unity.

In steady, uniform open-channel flow, the Chezy formula relates average velocity V, slope of channel S, and hydraulic radius of cross section R (area or section divided by wetted perimeter) by

(4.4.4)

- C is a coefficient depending upon size, shape, and roughness of channel. Then

(4.4.5)

since surface tension and compressible effects are usually unimportant.

- The drag F on a body is expressed by F = CDAρV2/2, in which A is a typical area of the body, usually the projection of the body onto a plane normal to the flow. Then F/A is equivalent to △p, and

(4.4.6)

- R is related to skin friction drag due to viscous shear as well as to form, or profile, drag resulting from separation of the flow streamlines from the body; F is to wave drag if there is a free surface, for large Mach numbers CD may vary more markedly with M than with the other parameters; the length ratios may refer to shape or roughness of the surface.

The Reynolds Number

- The Reynolds Number VDρ/μ is the ratio of inertial forces to viscous forces.
- A critical Reynolds number distinguishes among flow regimes, such as laminar or turbulent flow in pipes, in the boundary layer, or around immersed objects.
- The particular value depends upon the situation.
- In compressible flow, the Mach number is generally more significant than the Reynolds number.

The Froude Number

- The Froude Number , when squared and then multiplied and divided by ρA, is a ratio or dynamic (or inertial) force to weight.
- With free liquid-surface flow the nature of the flow (rapid or tranquil) depends upon whether the Froude number is greater or less than unity.
- It is useful in calculations of hydraulic jump, in design of hydraulic structures, and in ship design.

The Weber Number

- The Weber Number V2lρ/σ is the ratio of inertial forces to surface-tension forces (evident when numerator and denominator are multiplied by l)
- It is important at gas-liquid or liquid-liquid interfaces and also where these interfaces are in contact with a boundary.
- Surface tension causes small (capillary) waves and droplet formation and has an effect on discharge of offices and weirs at very small heads.
- Fig. 4.1 shows the effect of surface tension on wave propagation.
- To the left of the curve's minimum the wave speed is controlled by surface tension (the waves are called ripples), and to the right of the curve's minimum gravity effects are dominant.

The Mach Number

- The speed of sound in a liquid is written if K is the bulk modulus of elasticity or (k is the specific heat ratio and T the absolute temperature for a perfect gas).
- V/c or is the Mach number. It is a measure of the ratio of inertial forces to elastic forces.
- By squaring V/c and multiplying by ρA/2 in numerator and denominator, the numerator is the dynamic force and the denominator is the dynamic force at sonic flow.
- It may also be shown to be a measure of the ratio or kinetic energy or the flow to internal energy of the fluid. It is the most important correlating parameter when velocities are near or above local sonic velocities.

4.5 SIMILITUDE; MODEL STUDIES

- Model studies of proposed hydraulic structures and machines: permit visual observation or the flow and make it possible to obtain certain numerical data. e.g., calibrations of weirs and gates, depths of flow, velocity distributions, forces on gates, efficiencies and capacities of pumps and turbines, pressure distributions, and losses.
- To obtain accurate quantitative data: there must be dynamic similitude between model and prototype. This similitude requires (1) that there be exact geometric similitude and (2) that the ratio of dynamic pressures at corresponding points be a constant (kinematic similitude, i.e., the streamlines must be geometrically similar).
- Geometric similitude extends to the actual surface roughness of model and prototype. For dynamic pressures to be in the same ratio at corresponding points in model and prototype, the ratios of the various types or forces must be the same at corresponding points.
- Hence, for strict dynamic similitude, the Mach, Reynolds, Froude, and Weber numbers must be the same in both model and prototype.

Wind- and Water-Tunnel Tests

- Used to examine the streamlines and the forces that are induced as the fluid flows past a fully submerged body.
- The type of test that is being conducted and the availability of the equipment determine which kind of tunnel will be used.
- Kinematic viscosity of water is about one-tenth that of air a water tunnel can be used for model studies at relatively high Reynolds numbers.
- At very high air velocities the effects of compressibility, and consequently Mach number, must be taken into consideration, and indeed may be the chief reason for undertaking an investigation.
- Figure 4.2 shows a model of an aircraft carrier being tested in a low-speed tunnel to study the flow pattern around the ship's super-structure. The model has been inverted and suspended from the ceiling so that the wool tufts can be used to give an indication of the flow direction. Behind the model there is an apparatus for sensing the air speed and direction at various locations along an aircraft's glide path.

Figure 4.2 Wind tunnel tests on an aircraft carrier superstructure. Model is inverted and suspended from ceiling.

Pipe Flow

- In steady flow in a pipe, viscous and inertial forces are the only ones of consequence.
- Hence, when geometric similitude is observed, the same Reynolds number in model and prototype provides dynamic similitude.
- The various corresponding pressure coefficients are the same
- For testing with fluids having the same kinematic viscosity in model and prototype, the product, VD, must be the same.
- Frequently this requires very high velocities in small models.

Open Hydraulic Structures

- Structures such as spillways, stilling pools, channel transitions, and weirs generally have forces due to gravity (from changes in elevation of liquid surfaces ) and inertial forces that are greater than viscous and turbulent shear forces.
- In these cases geometric similitude and the same value of Froude's number in model and prototype produce a good approximation to dynamic similitude; thus
- Since gravity is the same, the velocity ratio varies as the square root of the scale ratio λ = lp/lm
- The corresponding times for events to take place (as time for passage of a particle through a transition) are related; thus

Ship’s Resistance

- The resistance to motion of a ship through water is composed of pressure drag, skin friction, and wave resistance. Model studies are complicated by the three types of forces that are important, inertia, viscosity, and gravity. Skin friction studies should be based on equal Reynolds numbers in model and prototype, but wave resistance depends upon the Froude number. To satisfy both requirements, model and prototype must be the same size.
- The difficulty is surmounted by using a small model and measuring the total drag on it when towed. The skin friction is then computed for the model and subtracted from the total drag. The remainder is stepped up to prototype size by Froude's law, and the prototype skin friction is computed and added to yield total resistance due to the water.
- Figure 4.4 shows the dramatic change in the wave profile which resulted from a redesigned bow. From such tests it is possible to predict through Froude's law the wave formation and drag that would occur on the prototype.

Model tests showing the influence of a bulbous bow on bow wave

Hydraulic Machinery

- The moving parts in a hydraulic machine require an extra parameter to ensure that the streamline patterns are similar in model and prototype. This parameter must relate the throughflow (discharge) to the speed of moving parts.
- For geometrically similar machines, if the vector diagrams of velocity entering or leaving the moving parts are similar, the units are homologous; i.e., for practical purposes dynamic similitude exists.
- The Froude number is unimportant, but the Reynolds number effects (called scale effects because it is impossible to maintain the same Reynolds number in homologous units) may cause a discrepancy of 2 or 3 percent in efficiency between model and prototype.
- The Mach number is also of importance in axial-flow compressors and gas turbines.

Example 4.4

- The valve coefficients K = Δp/(ρV2/2) for a 600-mm-diameter valve are to be determined from tests on a geometrically similar 300-mm-diameter valve using atmospheric air at 27°C. The ranges of tests should be for flow of water at 20°C at 1 to 2.5 m/s. What ranges of airflows are needed?

Solution

- The Reynolds number range for the prototype valve is
- For testing with air at 27°C

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