Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMILITUDE

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Chapter 4DIMENSIONAL ANALYSIS AND DYNAMIC
SIMILITUDE
2

• 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.

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• 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.

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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.

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• 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.

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• 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 ?y12/2 has the dimensions of force per
  unit width, and it multiplies a dimensionless
  number which is specified by the geometry or the
  hydraulic jump.

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• 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.

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• 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.

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• 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/?y12, 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/?y12, 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/?y12, 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.

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• 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.

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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.

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Table 4.1 Dimensions of physical quantities used
in fluid mechanics
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4.3 THE ? THEOREM

• The Buckingham ? 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 ?1, ?2, ..., represent dimensionless groupings
  of the quantities A1, A2, A3, ..., then with m
  dimensions involved, an equation of the following
  form exists
• (4.3.2)

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• The method of determining the ? 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 ?.
• For example, let A1, A2, A3 contain M, L and T,
  not necessarily in each one, but collectively.
  Then the ? parameters are made up as
• (4.3.3)
• - the exponents are to be determined ? each ? 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 ?
  parameter, so that the x, y, z exponents can be
  determined, and hence the ? 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 ? 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 ?
  parameter.

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• 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

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

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• 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 p/128 Eq. (5.4.10a).

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• Example 4.2
• A V-notch weir is a vertical plate with a notch
  of angle f 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. f
  is dimensionless hence, it is one ? parameter.
• Only two dimensions are used, L and T. If and H
  are the repeating variables.

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• 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.

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• If H and V0 were selected as repeating variables
  in place of g and H,
• From which , and
• Since any of the ? 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.

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• 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 ?,
  speed of advance V0, diameter D, air viscosity µ,
  density ?, 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 ?,
  ?, and D as repeating variables.

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• 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, VD?/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/??2D4 against V0/?D
  should be most informative.

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• 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,
  ?, is chosen.
• Write the ? parameters in terms of unknown
  exponents, e.g.,

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• For each of the ? 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 ? expressions of step 4
  the exponents to obtain the dimensionless ?
  parameters.
• Establish the functional relation
• or solve for one of the ?'s explicitly
• Recombine, if desired, to alter the forms of the
  ? parameters, keeping the same number or
  independent parameters.

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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.

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Pressure Coefficient

• The pressure coefficient ?p/(?V2/2) is the ratio
  of pressure to dynamic pressure
• When multiplied by area, it is the ratio of
  pressure force to inertial force, as (?V2/2)A
  would be the force needed to reduce the velocity
  to zero.
• It may also be written as ?h/(V2/2g) by division
  by ?.
• 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.

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

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• 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
  CDA?V2/2, 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.

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The Reynolds Number

• The Reynolds Number VD?/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.

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The Froude Number

• The Froude Number , when squared and then
  multiplied and divided by ?A, 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.

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The Weber Number

• The Weber Number V2l?/s 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.

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Figure 4.1 Wave speed vs. wavelength for surface
waves
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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 ?A/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.

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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.

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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.

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Figure 4.2 Wind tunnel tests on an aircraft
carrier superstructure. Model is inverted and
suspended from ceiling.
37
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.

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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 ?
  lp/lm
• The corresponding times for events to take place
  (as time for passage of a particle through a
  transition) are related thus

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Figure 4.3 Model test on a harbor to determine
the effect of a breakwater
40
Ships 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.

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Figure 4.4 Model tests showing the influence of a
bulbous bow on bow wave
42
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.

43

• Example 4.4
• The valve coefficients K ?p/(?V2/2) for a
  600-mm-diameter valve are to be determined from
  tests on a geometrically similar 300-mm-diameter
  valve using atmospheric air at 27C. The ranges
  of tests should be for flow of water at 20C 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 27C

44

• Then the ranges of air velocities are