Part A | 7.1
128 Part A Fundamentals
force felt on the sphere comes from the change in momentum which occurs when the flowing fluid impacts
the spherical body. As the momentum is related to both
the specific mass and speed of the fluid, we can add the
density of the fluid as another important characteristic
parameter that should be taken into account (we already
have speed). The viscosity of the fluid might also be
expected to set up shear stresses in the flow, which creates additional drag on the spherical body. Thus, the
dynamic (molecular) viscosity of the flow might also
be expected to be an important parameter that should
be included in our analysis. Thus, we get some sort of
functional relationship of the form
D D f .A; U; ;; ;/ :
(7.1)
In order to determine this functional relationship, one
could propose performing an experiment in which the
drag is measured for 10 different values of A, 10 different values of U, 10 different values of , and 10
different values of . This process would require 10
4
experiments and would be infeasible in terms of cost
and time for most! How can this process be made more
tractable?
If one thinks of the functional relation in (7.1) as
being an equation, D is the dependent variable and A,
U, , and are the independent variables. If we could
somehow reduce the number of independent variables,
we can reduce the number of experiments that would
need to be performed to characterize the functional dependence in (7.1). Dimensional analysis is the process
by which we will do this. However, before we embark
on a description of how to use dimensional analysis, we
will need to consider two basic concepts: dimensional
homogeneity and the Buckingham-˘ theorem.
Dimensional Homogeneity
Any equation that describes a basic law of physics must
have the same dimensions (units) on every term and on
both the right hand side and the left-hand side of the
equation. The units of all additive terms in an equation must be the same, and the dimensions will remain
homogeneous after any mathematical operation on the
equations, such as differentiation or integration. The basic dimensions are mass M, length L, and time T – all
physical quantities can be quantified by combinations
of these units. In practice, it may be convenient to use
nonfundamental units in dimensional analysis. For example in engineering, the temperature is sometimes
taken as a basic unit and in physics the luminosity L is
often used as a basic unit of measure, although strictly
speaking, temperature is a measure of energy and luminosity is a measure of power flux, both of which are
composed of the more fundamental M, L, and T units.
Here, we will use square brackets to denote dimensions
of, e.g.,
ŒU D
L
T
:
Buckingham-˘ Theorem
The Buckingham-˘ theorem can be used to determine
the number of dimensionless groups that can be found
in a functional relation, such as (7.1). If we let n be
the number of variables we believe it is involved in our
functional dependence and m be the number of basic
physical dimensions involved in those n variables, the
Buckingham-˘ theorem states that .n m/ dimensionless groups can be found by combining variables. The
combinations of variables are referred to as ˘ groups
and are used to re-write the sought after functional dependence as a function of ˘ groups.
˘ 1 D 2 ; ˘ 3 : : : ; ˘ nm /
Getting back to the spherical body, we see that the
number of variables involved in (7.1) is n D 5 and the
number of basic dimensions can be found by looking at
the units of the five variables.
ŒD D
ML
T 2 ŒU D
L
T
ŒŒ D
M
LT
;
ŒA D L ŒŒ D
M
L 3 ;
(7.2)
where we see that m D 3. Thus, we expect to be able
to reduce (7.1) to a functional dependence between n
m D 2 ˘ groups
˘ 1 D 2 / ;
(7.3)
which, as one might expect, will vastly reduce the number of experiments required to characterize the functional relation. There are several different approaches
that can be used to identify the ˘ groups in (7.3). Here
two approaches will be discussed.
The Step-by-Step Elimination Method [7.1]
Each variable in a functional dependence such as (7.1)
is combined with other variables by multiplying or
dividing it into the other variables. One starts by eliminating the simplest or purest variables first, meaning
those with the fewest number of basic units. In general,
one should not eliminate the dependent variable. Each
elimination can be thought of as being the same as removing one of the basic dimensions, length L, mass M,
or time T. Going back to our example, from (7.2) we
can see that the variable with the simplest dimensions
is the diameter A, so we begin by using A to eliminate
the length dimension from our problem:
128 Part A Fundamentals
force felt on the sphere comes from the change in momentum which occurs when the flowing fluid impacts
the spherical body. As the momentum is related to both
the specific mass and speed of the fluid, we can add the
density of the fluid as another important characteristic
parameter that should be taken into account (we already
have speed). The viscosity of the fluid might also be
expected to set up shear stresses in the flow, which creates additional drag on the spherical body. Thus, the
dynamic (molecular) viscosity of the flow might also
be expected to be an important parameter that should
be included in our analysis. Thus, we get some sort of
functional relationship of the form
D D f .A; U; ;; ;/ :
(7.1)
In order to determine this functional relationship, one
could propose performing an experiment in which the
drag is measured for 10 different values of A, 10 different values of U, 10 different values of , and 10
different values of . This process would require 10
4
experiments and would be infeasible in terms of cost
and time for most! How can this process be made more
tractable?
If one thinks of the functional relation in (7.1) as
being an equation, D is the dependent variable and A,
U, , and are the independent variables. If we could
somehow reduce the number of independent variables,
we can reduce the number of experiments that would
need to be performed to characterize the functional dependence in (7.1). Dimensional analysis is the process
by which we will do this. However, before we embark
on a description of how to use dimensional analysis, we
will need to consider two basic concepts: dimensional
homogeneity and the Buckingham-˘ theorem.
Dimensional Homogeneity
Any equation that describes a basic law of physics must
have the same dimensions (units) on every term and on
both the right hand side and the left-hand side of the
equation. The units of all additive terms in an equation must be the same, and the dimensions will remain
homogeneous after any mathematical operation on the
equations, such as differentiation or integration. The basic dimensions are mass M, length L, and time T – all
physical quantities can be quantified by combinations
of these units. In practice, it may be convenient to use
nonfundamental units in dimensional analysis. For example in engineering, the temperature is sometimes
taken as a basic unit and in physics the luminosity L is
often used as a basic unit of measure, although strictly
speaking, temperature is a measure of energy and luminosity is a measure of power flux, both of which are
composed of the more fundamental M, L, and T units.
Here, we will use square brackets to denote dimensions
of, e.g.,
ŒU D
L
T
:
Buckingham-˘ Theorem
The Buckingham-˘ theorem can be used to determine
the number of dimensionless groups that can be found
in a functional relation, such as (7.1). If we let n be
the number of variables we believe it is involved in our
functional dependence and m be the number of basic
physical dimensions involved in those n variables, the
Buckingham-˘ theorem states that .n m/ dimensionless groups can be found by combining variables. The
combinations of variables are referred to as ˘ groups
and are used to re-write the sought after functional dependence as a function of ˘ groups.
˘ 1 D 2 ; ˘ 3 : : : ; ˘ nm /
Getting back to the spherical body, we see that the
number of variables involved in (7.1) is n D 5 and the
number of basic dimensions can be found by looking at
the units of the five variables.
ŒD D
ML
T 2 ŒU D
L
T
ŒŒ D
M
LT
;
ŒA D L ŒŒ D
M
L 3 ;
(7.2)
where we see that m D 3. Thus, we expect to be able
to reduce (7.1) to a functional dependence between n
m D 2 ˘ groups
˘ 1 D 2 / ;
(7.3)
which, as one might expect, will vastly reduce the number of experiments required to characterize the functional relation. There are several different approaches
that can be used to identify the ˘ groups in (7.3). Here
two approaches will be discussed.
The Step-by-Step Elimination Method [7.1]
Each variable in a functional dependence such as (7.1)
is combined with other variables by multiplying or
dividing it into the other variables. One starts by eliminating the simplest or purest variables first, meaning
those with the fewest number of basic units. In general,
one should not eliminate the dependent variable. Each
elimination can be thought of as being the same as removing one of the basic dimensions, length L, mass M,
or time T. Going back to our example, from (7.2) we
can see that the variable with the simplest dimensions
is the diameter A, so we begin by using A to eliminate
the length dimension from our problem: