Transcript Slide 1

Chapter 4
DIMENSIONAL 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.
4.1 DIMENSIONAL HOMOGENEITY AND
DIMENSIONLESS RATIOS
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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 ϒy12/2 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 lefthand 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/ϒy12, 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/ϒy12, 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/ϒy12, 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.
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For if one can create in a model and force ratios that occur on the fullscale 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
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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.
Figure 4.1
Dimensions of
physical
quantities used
in fluid
mechanics
4.3 THE Π THEOREM
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The Buckingham Π 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 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)
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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
- 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  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.
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The steps in a dimensional analysis may be summarized as follows:
1.
Select the pertinent variables (requires some knowledge of the
process)
Write the functional relations, e.g.,
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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 or unknown exponents, e.g.,
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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:
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Recombine, if desired, to alter the forms of the Π parameters,
keeping the same number or independent parameters.
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/(ρV2/2)
 the ratio of pressure to dynamic pressure, when multiplied by area
 the ratio of pressure force to inertial force, as (ρV2/2)A would be the
force needed to reduce the velocity to zero
 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
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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 ε'; hence,
(4.4.1)
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If compressibility is important,
(4.4.2)
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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)
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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)
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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.
The Reynolds Number
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VDρ/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.
The Froude Number
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when squared and then multiplied and divided by ρA, is a ratio or
dynamic (or inertial) force to weight
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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
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V2lρ/σ
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.
Figure 4.2 Wave speed vs. wavelength for surface waves
The Mach Number
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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 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
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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.
Wind- and Water-Tunnel Tests
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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.
Figure 4.2
Wind tunnel
tests on an
aircraft carrier
superstructur
e. Model is
inverted and
suspended
from ceiling.
Pipe Flow
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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.
Open Hydraulic Structures
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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
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Since gravity is the same, the velocity ratio varies as thc square root of
the scale ratio λ = lp/lm
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The corresponding times for events to take place (as time for passage
of a particle through a transition) are related; thus
Figure 4.3
Model test on
a harbor to
determine the
effect of a
breakwater
Ship’s Resistance
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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.
Hydraulic Machinery
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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.
Figure 4.4
Model tests
showing the
influence of a
bulbous bow
on bow wave