Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMIILITUDE

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Chapter 4DIMENSIONAL ANALYSIS AND DYNAMIC
SIMIILITUDE
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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 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 ? 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 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
  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.

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• 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.

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• An example of the use of dimensional analysis and
  its advantages by considering the hydraulic jump
  (Sec. 3.11). The momentum equation
• (4.1.1)
• right-hand side the inertial forces left-hand
  side the pressure forces due to gravity two
  forces are of equal magnitude (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 ol 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.
• This is the key to much of what follows.

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• 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 electro-magnetics 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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Figure 4.1 Dimensions of physical quantities
used in fluid mechanics
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4.3 THE ? THEOREM

• The Buckingham ? theorem proves 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
• - 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 ? 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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• 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 or 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

• pressure coefficient
• Reynolds number
• Froude number
• Weber number
• Mach number

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Pressure Coefficient

• The pressure coefficient
• ?p/(?V2/2)
• the ratio of pressure to dynamic pressure, when
  multiplied by area
• the ratio of pressure force to inertial force, as
  (?V2/2)A would be the force needed to reduce the
  velocity to zero
• 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 c in the pipe
  wall and l2 to their spacing e' hence,
• (4.4.1)
• If compressibility is important,
• (4.4.2)
• With orifice flow, studied in Chap. 8,
• (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 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 to wave drag if there is a free
  surface, for large Mach numbers CD may vary more
  markedly with M than with the other paramelers
  the length ratios may refer to shape or roughness
  of the surface.

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The Reynolds Number

• VD?/µ
• 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

• 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 ol hydraulic jump,
  in design of hydraulic structures, and in ship
  design.

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The Weber Number

• V2l?/s
• 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 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 4.2 Wave speed vs. wavelength for surface
waves
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The Mach Number

• The speed of sound in a liquid
• if K is the bulk modulus of
  elasticity
• (k is the specific heat ratio
  and T the absolute temperature for a perfect
  gas).
• V/c or is the Mach number - 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. 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 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
• ? 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 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.
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Pipe Flow

• Steady flow in a pipe viscous and inertial
  forces are the only ones of consequence
• ? 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 thc 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
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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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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
  Reinolds 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.

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