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four fundamental dimensions)
F
P
Cr
D
gc
V
F M L t
1 0 0 0
0 1 - 3 0
0 1 -1-1
0 0 1 0
0 0 1-1
-1 1 1 -2
(5.2.1-2)
It can be shown that this matrix is of rank 4, so in a complete set we should be
able to obtain 6 - 4 = 2 dimensionless products.
Carrying out the dimensional analysis yields
228
W1
Chapter 5: Application of Dimensional Analysis
[q.),i&quot;)] p v2 D2 = 0 (5.2.1-3)
In principle, we can solve W I explicitly for (p:fb2) - to obtain a new
function
(*) = W2[(.?)]
This relation may be rewritten as
(5.2.1 -4)
(5.2.1-5)
where (p v2) is proportional to the kinetic energy per unit volume and D2 is
proportional to a characteristic area.
For internal flows in circular conduits this may be expressed a..
(5.2.1-6)
where we have used the bulk velocity as the characteristic velocity, the product
(R L) in place of the square of the characteristic length, and the conduit diameter
as the characteristic length in the Reynolds number (for geometrically similar
systems, defined below, these lengths remain in constant ratio to one another).
The Fanning friction factor, f, is then defined by the relation
F = (2xRL)(\$p(v)*)f [(D(v)p,l .p (5.2.1-7)
where the first factor on the right-hand side is the lateral area of the conduit and
the second factor on the right-hand side is the kinetic energy per unit volume.
The function f is just the function ~2 divided by (ng,). Consistent with our
practice throughout the text, we do not incorporate g, in the formula, because
Chapter 5: Application of Dimensional Analysis 229
the definition of f is the same for all systems of units, and when g, is needed it
is self-evident by dimensional homogeneity. The friction factor is a
dimensionless function of Reynolds number.
A friction factor (unfortunately also designated by f') four times the
magnitude of the Fanning friction factor is sometimes also defined in the
literature. This friction factor is called the Darcy or BIasius friction factor.
Two different friction factors, both designated by f, cause no real difficulty as
long as one is consistent in which friction factor one uses and observes the factor
of four. The biggest danger of mistake is usually in using a value from a chart or
table which incorporates one friction factor in a formula using the other.
For external flows, Equation (5.2.1-5) may be written as
F = ( q q w &quot; ( y ) ] (5.2.1-8)
where we now have a different function because we have a different flow
situation.
For flow around objects, we usually convert the square of the characteristic
length to the cross-section of the object normal to flow. The reference
velocity usually used in the kinetic energy term and in the Reynolds number is
the freestream velocity (the velocity far enough away from the surface that
there is no perturbation by the object). The characteristic length, if the cross-
sectional area is circular, is the diameter of the circle - if not, a specific
dimension must be specified. Then
(5.2.1 -9)
where CD, called the drag coefficient, is a dimensionless function of
Reynolds number (CD and ~3 are different functional forms because of the
constants introduced). Again, we do not incorporate g, because the definition is
valid for any consistent system of units.
5.2.2 Shape factors
Dimensional analysis can seldom be used to predict ways in which
phenomena are affected by different shapes of object. Therefore, when applying
dimensional analysis to a problem it is usually convenient to eliminate the
230 Chapter 5: Application of Dimensional Analysis
consideration of shape by considering bodies of similar shape, that is, bodies that
are geometrically similar.
D r u force on ship hull
Tbe drag force exerted on the hull of a ship depends on the shape of the hull.
Hulls of similar shape may be specified by a single characteristic dimension such
as keel17 length, L, and hulls of other lengths but similar shapes have their
dimensions in the same ratio to this characteristic length - for example, the ratio
of beam width (width across the hull) to length will be the same for all hulls of
similar shape. If the hulls are considered to have the same relative water line (for
example, with respect to freeboard, the distance of the top of the hull above the
waterline), keel length will also be proportional to the draft (displacement).
The drag force will then depend on the keel length, L, the speed of the ship,
v, the viscosity of the liquid, p, the density of the liquid, p, and the acceleration
of gravity (which affects the drag produced by waves). What are the
dimensionless groups that describe this situation?
So 1 u t ion
Our function is
(5.2.2- 1)
The rank of the dimensional matrix may be shown to be 3 for a system with
three fundamental dimensions. We will therefore have (6 - 3) = 3 dimensionless
groups.
If we carry out the dimensional analysis, we obtain
(5.2.2-2)
(the dimensionless group F/[pv2L2] is sometimes called the pressure
coefficient, P) . l*
l7 The keel is the longitudinal member reaching from stem to stern to which the ribs
are attached.
I * Langhaar, H. L. (1951). Dimensional Analysis and Theory of Models. New York,
NY, John Wiley and Sons, p. 21.
Chapter 5: Application of Dimensional Analysis 231
To run experiments, we would fearrange (5.2.2-1) as
and would rearrange (5.2.2-2) as
F= P = w2(Re, Fr)
p v2 L2
(5.2.2-3)
(5.2.2-4)
In each case the experiments would consist of measuring the variable on the left-
hand side of the equation as a function of the variables on the right-hand side of
the equation, which would be varied from experiment to experiment.
Notice the numerous advantages of (5.2.2-4) over (5.2.2-3).
Using (5.2.2-3), to run three levels of each variable would
require 35 = 243 experiments. Using (5.2.2-4), to run three
levels of each variable would require only 32 = 9 experiments.
. To use (5.2.2-4), we could easily vary both Reynolds and
Froude numbers by varying velocity (which is easily
accomplished in a towing tank) and hull length (which could
be accomplished by using models of the same shape but
different sizes).
To use (5.2.2-3), we would also have to find a way to vary p
(perhaps by vaqing temperature or fluid or adding a thickening
agent to water), p (perhaps by varying fluid; or dissolving
something in water to vary density; or varying temperature to
vary density, even though this is not a very big effect), and g
(perhaps by building two laboratories, one on a very high
mountain and one in a mine, or building the experiment in a
centrifuge). None of these alternatives is particularly
appealing.
To present the results of (5.2.2-3) in a compact form is
difficult, as is extrapolating or interpolating the results. The
results of (5.2.2-4), however, can be concisely presented on a
two-dimensional graph, where the approximate results of
interpolation or extrapolation can be easily visualized, as is
illustrated in the following figure (the data is artificial and
must not be used for design purposes). The figure
shows the data from nine hypothetical experiments arranged in
232 Chapter 5: Application of Dimensional Analysis
a factorial design (each of three levels of Re combined with
each of three levels of Fr, and hypothetical curve fits which
could be used for interpolation (or extrapolation, if the curves
were to be extended beyond the range of the data - which would
be dangerous).
Fr = 6
F r = 4
Fr = 2
Reynolds number
vle 5.2.2-2 Deceleration o f corUpressible -fh&i
The important variables that determine the maximum pressure, pmax,
resulting when flow of a compressible fluid is stopped instantly (for example, by
shutting a valve