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Chapter 9 Dimensional Analysis
This chapter is dedicated to my adviser, Dr.~E.R.G. Eckert.
Genick BarMeir

9.1 Introductory Remarks
Dimensional analysis refers to techniques dealing with units or conversion to a unitless system.
The definition of dimensional analysis is not consistent in the literature which span over various fields and times.
Possible topics that dimensional analysis deals with are
consistency of the units, change order of magnitude, applying from the old
and known to unknown (see the Book of Ecclesiastes), and creation of group
parameters without any dimensions.
In this chapter, the focus is on the applying the old to unknown as
different scales and the creation of dimensionless groups.
These techniques gave birth to dimensional parameters which have a great scientific importance.
Since the 1940s
the dimensional analysis is taught and written in all fluid mechanics textbooks.
The approach or the technique used in these books is referred to as Buckingham–$\pi$–theory.
The $\pi$–theory was coined by Buckingham.
However, there is another technique which is referred to in the literature as the Nusselt's method.
Both these methods attempt to reduce the number of parameters which affect the problem
and reduce the labor in solving the problem.
The key in these techniques lays in the fact of consistency of the dimensions of any
possible governing equation(s) and the fact that some dimensions are reoccurring.
The Buckingham–$\pi$ goes further and no equations are solved
and even no knowledge about these equations is required.
In Buckingham's technique only the dimensions or the properties of the problem at hand are analyzed.
This author is aware of only a single class of cases were Buckingham's methods is useful and or
can solve the problem namely the pendulum class problem (and similar).
The dimensional analysis was independently developed by Nusselt and improved by his students/co–workers
(Schmidt, Eckert) in which the governing equations are used as well.
Thus, more information is put into the problem and thus a better understanding on the
dimensionless parameters is extracted.
The advantage or disadvantage of these similar methods depend on the point of view.
The Buckingham–$\pi$ technique is simpler while Nusselt's technique produces
a better result.
Sometime, the simplicity of Buckingham's technique yields insufficient knowledge or
simply becomes useless.
When no governing equations are found, Buckingham's method has usefulness.
It can be argued that these situations really do not exist in the Thermo–Fluid field.
Nusselt's technique is more cumbersome but more precise and provide more useful information.
Both techniques are discussed in this book.
The advantage of the Nusselt's technique are:
a) compact presentation, b)knowledge what parameters affect the problem,
c) easier to extent the solution to more general situations.
In very complex problems both techniques suffer from in inability to provide a significant
information on the effective parameters such multi–phase flow etc.
It has to be recognized that the dimensional analysis
provides answer to what group of parameters affecting the problem
and not the answer to the problem.
In fact, there are fields in thermo–fluid where dimensional analysis, is recognized as
useless.
For example, the area of multiphase flows there is no solution based
on dimensionless parameters (with the exception of the rough solution of
In the Buckingham's approach it merely suggests the number of dimensional parameters
based on a guess of all parameters affecting the problem.
Nusselt's technique provides the form of these dimensionless parameters,
and the relative relationship of these parameters.
9.1.1 Brief History
The idea of experimentation with a different, rather than the actual, dimension was suggested
by several individuals independently.
Some attribute it to Newton (1686) who coined the phrase of ``great Principle of Similitude.''
basic units of mass, length, and time as building blocks of all other units.
Another example, John Smeaton (8 June 1724–28 October 1792) was an English civil and
mechanical engineer who study relation between propeller/wind mill and similar
devices to the pressure and velocity of the driving forces.
dimensional analysis theory.
open channel flow and actual body but more importantly the relationship between drag of
models to actual ships.
While the majority of the contributions were done by thermo–fluid guys the concept of the equivalent
or similar propagated to other fields.
Aim\'{e}em Vaschy, a German Mathematical Physicist (1857–1899), suggested using similarity
in electrical engineering and suggested the Norton circuit equivalence
Rayleigh probably was the first one who used dimensional analysis
the relationships between the physical quantities (see the question why the sky is blue story).
Osborne Reynolds (1842–1912) was the first to derive and use dimensionless parameters
temperature by molecules velocity and thus creating dimensionless group with the byproduct of
compact solution (solution presented in a compact and simple form).
In the about the same time (1915, Wilhelm Nusselt (November 25, 1882 – September 1, 1957),
a German engineer, developed the dimensional analysis (proposed
of heat transfer without knowledge about previous work of Buckingham.
9.1.2 Theory Behind Dimensional Analysis
In chemistry it was recognized that there are fundamental elements that
all the material is made from (the atoms).
That is, all the molecules are made from a combination of different atoms.
Similarly to this concept, it was recognized that in many physical systems
there are basic fundamental units which can describe all the other
dimensions or units in the system.
For example, isothermal single component systems (which does not undergo phase change,
temperature change and observed no magnetic or electrical effect) can be described by just
The units or dimensions are, time, length, mass, quantity of substance (mole).
For example, the dimension or the units of force can be constructed utilizing
Newton's second law i.e.~mass times acceleration $\longrightarrow m\,a = M\,L/t^2$.
Increase of degree of freedom, allowing this system to be non–isothermal
will increase only by one additional dimension of temperature, $\theta$.
These five fundamental units are commonly the building blocks for most
of the discussion in fluid mechanics (see Table of basic units 9.1).
Basic Units of Common System 
Standard System 
Old System 
Name 
Latter 
Units 
Name 
Latter 
Units 
Mass 
$M$ 
$kg$ 
Force 
$F$ 
$N$ 
Length 
$L$ 
$m$ 
Length 
$L$ 
$m$ 
Time 
$t$ 
$sec$ 
Time 
$t$ 
$sec$ 
Temperature 
$\theta$ 
$\,^{\circ}C$ 
Temperature 
$T$ 
$\,^{\circ}C$ 
Additional Basic Units for Magnetohydrodynamics 
Electric Current 
$A$ 
$[A]$mpere 
Electric Current 
$A$ 
$[A]$mpere 
Luminous Intensity 
$cd$ 
$[cd]$ candle 
Luminous Intensity 
$cd$ 
$[cd]$ candle 
Chemical Reactions 
Quantity of substance 
$\mathfrak{M}$ 
$mol$ 
Quantity of substance 
$\mathfrak{M}$ 
$mol$ 
The choice of these basic units is not unique and several books and researchers
suggest a different choice of fundamental units.
One common selection is substituting the mass with the force in the previous
selection (F, t, L, mol, Temperature).
This author is not aware of any discussion on the benefits of one method over
the other method.
Yet, there are situations in which first method is better than the second one
while in other situations, it can be the reverse.
In this book, these two selections are presented.
Other selections are possible but not common and, at the moment, will not be discussed here.
What are the units of force when the basic units are: mass, length, time, temperature
(M, L, t, $\theta$)?
What are the units of mass when the basic units are: force, length, time, temperature
(F, L, t, T)?
Notice the different notation for the temperature in the two systems of basic units.
This notation has no significance but for historical reasons remained in use.
Solution
These two systems are related as the questions are the reversed of each other.
The connection between the mass and force can be obtained from the simplified
Newton's second law $F = m\, a$ where $F$ is the force, $m$ is the mass, and
$a$ is the acceleration.
Thus, the units of force are
\begin{align}
\label{basicMass:force}
F = \dfrac{ M\, L}{ t^2}
\end{align}
For the second method the unit of mass are obtain from Equation \eqref{basicMass:force} as
\begin{align}
\label{basicMass:mass}
M = \dfrac{ F\,t^2}{L}
\end{align}
The number of fundamental or basic dimensions determines the number of the combinations
which affect the physical
situations.
The dimensions or units which affect the problem at hand can be reduced because
these dimensions are repeating or reoccurring.
The Buckingham method is based on the fact that all equations must be consistent with their units.
That is the left hand side and the right hand side have to have the same units.
Because they have the same units the equations can be divided to create unitless equations.
This idea alludes to the fact that these unitless parameters can be found without any knowledge
of the governing equations.
Thus, the arrangement of the effecting parameters in unitless groups yields the affecting parameters.
These unitless parameters are the dimensional parameters.
The following trivial example demonstrates the consistency of units
Newton's equation has two terms that related to force $F= m\,a +\dot{m}\, U$.
Where $F$ is force, $m$ is the mass, $a$ is the acceleration and dot above $\dot{m}$
indicating the mass derivative with respect to time.
In particular case, this equation get a form of
\begin{align}
\label{unitsInEq:gov}
F = m \, a + 7
\end{align}
where $7$ represent the second term.
What are the requirement on equation \eqref{unitsInEq:gov}?
Solution
Clearly, the units of [$F$], $m\,a$ and $7$ have to be same.
The units of force are [$N$] which is defined by first term of the right hand side.
The same units force has to be applied to $7$ thus it must be in [$N$].
Suppose that there is a relationship between a quantity a under the question and several others parameters
which either determined from experiments or theoretical consideration which is of the form
\begin{align}
\label{dim:eq:generalRelationship}
D = f (a_1, a_2, \cdots, a_i, \cdots, a_n)
\end{align}
where $D$ is dependent parameters and $a_1, a_2, \cdots, a_i, \cdots, a_n$ are have independent dimensions.
From these independent parameters $a_1, a_2, \cdots, a_i$ have independent dimensions (have basic dimensions).
This mean that all the dimensions of the parameters $a_{i+1}, \cdots, a_n$ can be written as combination of the
the independent parameters $a_1, a_2, \cdots, a_i$.
In that case it is possible to write that every parameter in the later set can written as dimensionless
\begin{align}
\label{dim:eq:laterDimless}
\dfrac{a_{i+1}}{{a_1}^{p_1},{a_2}^{p_2},\cdots,{a_i}^{p_i}} = \text{dimensionless }
\end{align}
The ``non–basic'' parameter would be dimensionless when divided by appropriately and selectively
chosen set of constants ${p_1}, {p_2}, \cdots, p_i$.
In a experiment, the clamping force is measured.
It was found that the clamping force depends on the length of experimental setup, velocity of the
upper part, mass of the part, height of the experimental setup, and leverage the force is applied.
Chose the basic units and dependent parameters.
Show that one of the dependent parameters can be normalized.
Solution
The example suggest that the following relationship can be written.
\begin{align}
\label{dim:eq:appropriatelySelectively}
F = f ( L, U, H, \tau, m)
\end{align}
The basic units in this case are in this case or length, mass, and time.
No other basic unit is need to represent the problem.
Either $L$, $H$, or $\tau$ can represent the length.
The mass will be represented by mass while the velocity has to be represented by the velocity (or some combination
of the velocity).
Hence a one possible choice for the basic dimension is $L$, $m$, and $U$.
Any of the other Lengths can be reprsented by simple division by the $L$.
For example
\begin{align}
\label{dim:eq:appropriatelySelectivelyRatio}
\text{Normalize parameter} = \dfrac{H}{L}
\end{align}
Or the force also can be normalized as
\begin{align}
\label{dim:eq:appropriatelySelectivelyR2}
\text{Another Normalize parameter} = \dfrac{F}{m\,U^2\,L^{1} }
\end{align}
The acceleration can be any part of acceleration component such as centrifugal acceleration.
Hence, the force is mass times the acceleration.
The relationship \eqref{dim:eq:generalRelationship} can be written in the light of the above explanation as
\begin{multline}
\label{dim:eq:gRD}
\dfrac{D}{ {a_1}^{p_1}, {a_2}^{p_2}, \cdots, {a_i}^{p_1}} = \
F \left( \dfrac{a_{i+1}} {{a_{1}}^{p_{i+1,1}},\,{a_2}^{p_{i+1, 2} } ,\cdots, \,{a_i}^{p_{i+1, i} } } ,
\cdots,
\dfrac{a_{n}} {{a_{n}}^{p_{n,1}},\,{a_n}^{p_{n, 2} } ,\cdots, \,{a_n}^{p_{n, i} } }
\right)
\end{multline}
where the indexes of the power $p$ on the right hand side are single digit and the double digits on the
on the right hand side.
While this ``proof'' shows the basic of the Buckingham's method it actually provides merely the minimum
number of the dimension parameters.
In fact, this method entrenched into the field while in most cases provides incomplete results.
The fundamental reason for the erroneous results is because the fundamental assumption of
equation \eqref{dim:eq:generalRelationship}.
This method provides a crude tool of understanding.
9.1.3 Dimensional Parameters Application for Experimental Study
The solutions for any situations which are controlled by the same governing equations with same
boundary conditions regardless of the origin the equation.
The solutions are similar or identical regardless to the origin
of the field no matter if the field is physical, or economical, or biological.
The Buckingham's technique implicitly suggested that since the governing equations
(in fluid mechanics) are essentially are the same, just knowing the parameters
is enough the identify the problem.
This idea alludes to connections between similar parameters to similar solution.
The non–dimensionalization i.e. operation of reducing the number affecting parameters, has
a useful by–product, the analogy in other words, the solution by experiments or other cases.
The analogy or similitude refers to understanding one phenomenon from the study of another phenomenon.
This technique is employed in many fluid mechanics situations.
For example, study of compressible flow (a flow where the density change plays a significant
part) can be achieved by study of surface of open channel flow.
The compressible flow is also similar to traffic on the highway.
Thus for similar governing equations if the solution exists for one case
it is a solution to both cases.
The analogy can be used to conduct experiment in a cheaper way and/or a safer way.
Experiments in different scale than actual dimensions can be conducted for
cases where the actual dimensions are difficult to handle.
For example, study of large air planes can done on small models.
On the other situations, larger models are used to study small or fast situations.
This author believes that at the present the Buckingham method has extremely
limited use for the real world and yet this method is presented in the classes on fluid mechanics.
Thus, many examples on the use of this method will be presented in this book.
On the other hand, Nusselt's method has a larger practical use in the real world
and therefore will be presented for those who need dimensional analysis for the real world.
Dimensional analysis is useful also for those who are dealing with the numerical research/calculation.
This method supplement knowledge when some parameters should be taken into account and why.
Fig. 9.1 Fitting rod into a hole.
Fitting a rod into a circular hole (see Figure 9.1) is an example
how dimensional analysis can be used.
To solve this problem, it is required to know two parameters;
1) the rode diameter and 2) the diameter of the hole.
Actually, it is required to have only one parameter, the ratio
of the rode diameter to the hole diameter.
The ratio is a dimensionless number and with this number one can
tell that for a ratio larger than one, the rode will not enter the hole; and a ratio
smaller than one, the rod is too small.
Only when the ratio is equal to one, the rode is said to be fit.
This presentation allows one to draw or present the situation by using only one coordinate,
the radius ratio.
Furthermore, if one wants to deal with tolerances,
the dimensional analysis can easily be extended to say that
when the ratio is equal from 0.99 to 1.0 the rode is fitting, and etc.
If one were to use the two diameters description, further significant information will
be needed.
In the preceding simplistic example, the advantages are minimal.
In many real problems this approach can remove clattered views and put the problem into focus.
Throughout this book the reader will notice that the systems/equations in many cases are
converted to a dimensionless form to augment understanding.
9.1.4 The Pendulum Class Problem
The only known problem that dimensional analysis can solve (to some degree) is the pendulum class problem.
In this section several examples of the pendulum type problem are presented.
The first example is the classic Pendulum problem.
Fig. 9.2 Figure for example
length of pendulum [$\ell$].
Assume that no other parameter including the mass affects the problem.
That is, the relationship can be expressed as
\begin{align}
\label{pendulum:form}
\omega = f \left(\ell, g \right)
\end{align}
Notice in this problem, the real knowledge is provided,
however in the real world, this knowledge is not necessarily given or known.
Here it is provided because the real solution is already known from standard
Solution
The solution technique is based on the assumption that the indexical form
The Indexical form
\begin{align}
\label{pendulum:basic}
\omega = C_1\times \ell^{a} g^{b}
\end{align}
The solution functional complexity is limited to the basic combination which has to be
in some form of multiplication of $\ell$ and $g$ in some power.
In other words, the multiplication of $\ell\,g$ have to be in the same units of the frequency units.
Furthermore, assuming, for example, that a trigonometric function relates
$\ell$ and $g$ and frequency.
For example, if a $\sin$ function is used, then the functionality looks like
$\omega = \sin (\ell\,g)$.
From the units point of view, the result of operation not match i.e. ($sec \neq \sin\,(sec)$).
For that reason the form in equation \eqref{pendulum:basic} is selected.
To satisfy equation \eqref{pendulum:basic} the units of every term are examined
and summarized the following table.
Table 9.2 Units of the Pendulum Parameters
Parameter 
Units 
Parameter 
Units 
Parameter 
Units 
$\omega$  $t^{1}$  $\ell$  $L^{1}$  $g$  $L^{1}t^{2}$ 
Thus substituting of the Table 9.2 in equation \eqref{pendulum:basic}
results in
\begin{align}
\label{pendulum:govIndalign}
t^{1} = C_1\left( L^1 \right)^a \, \left(L^1 \,t^{2}\right)^b
\Longrightarrow L ^{a +b} t^{2\,b}
\end{align}
after further rearrangement by multiply the left hand side by $L^0$ results in
\begin{align}
\label{pendulum:govRe}
L^{0}t^{1} = C\,L ^{a +b} t^{2\,b}
\end{align}
In order to satisfy equation \eqref{pendulum:govRe}, the following must exist
\begin{align}
\begin{array}{ccc}
\label{pendulum:reEq}
0 = a + b & and & 1 = \dfrac{2}{b}
\end{array}
\end{align}
The solution of the equations \eqref{pendulum:reEq} is $a= 1/2 $ and $b= 1/2$.
Thus, the solution is in the form of
\begin{align}
\label{pendulum:solution}
\omega = C_1\, \ell^{1/2} \, g^{1/2} = C_1 \, \sqrt{\dfrac{g}{\ell}}
\end{align}
It can be observed that the value of $C_1$ is unknown.
The pendulum frequency is known to be
\begin{align}
\label{pendulum:frequency}
\omega = \dfrac{1}{2\pi} \sqrt{\dfrac{g}{\ell}}
\end{align}
What was found in this example is the form of the solution's equation and frequency.
Yet, the functionality e.g. the value of the constant was not found.
The constant can be obtained from experiment for plotting $\omega$ as the abscissa
and $\sqrt{\ell/g}$ as ordinate.
According to some books and researchers, this part is the importance of the dimensional analysis.
It can be noticed that the initial guess merely and actually determine the results.
If, however, the mass is added to considerations, a different result will be obtained.
If the guess is relevant and correct then the functional relationship can be obtained
by experiments.
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