Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMILITUDE

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Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMILITUDE

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Title: Chapter 4 DIMENSIONAL ANALYSIS AND DYNAMIC SIMILITUDE

1
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.

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
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 ?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.

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/?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.

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.

11
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.

12
Table 4.1 Dimensions of physical quantities used
in fluid mechanics
13
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)

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.

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 ? 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 p/128 Eq. (5.4.10a).

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.

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 ? 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 ?,
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.

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.

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.,

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.

25
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.

26
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.

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.

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.

29
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.

30
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.

31
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.

32
Figure 4.1 Wave speed vs. wavelength for surface
waves
33
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.

34
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.

35
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.

36
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.

38
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

39
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.

41
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.

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