Download Introduction - The Purposes and Usefulness of

Dimensional analysis is the analysis of the relationships between different physical
quantities by identifying their fundamental dimensions (such as length, mass, time,
and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs.
kilograms vs. grams) and tracking these dimensions as calculations or comparisons are
performed. Converting from one dimensional unit to another is often somewhat complex.
Dimensional analysis, or more specifically the factor-label method, also known as the unitfactor method, is a widely used technique for such conversions using the rules of algebra.
1. Nondimensionalization
Scale-invariance implies that we can reduce the number of quantities appearing in
a problem by introducing dimensionless quantities.
Suppose that (a1, . . . , ar) are a set of quantities whose dimensions form a
fun- damental system of units. We denote the remaining quantities in the
model by (b1, . . . , bm), where r + m = n. Then, for suitable exponents (β1,i, . . . ,
βr,i) deter- mined by the dimensions of (a1 , . . . , ar ) and bi , the quantity
bi
Πi = aβ1,i
βr,i
1
. . . ar
is dimensionless, meaning that it is invariant under the scaling transformations induced
by changes in units.
A dimensionless parameter Πi can typically be interpreted as the ratio of two
quantities of the same dimension appearing in the problem (such as a ratio of lengths,
times, diffusivities, and so on). In studying a problem, it is crucial to know the magnitude
of the dimensionless parameters on which it depends, and whether they are small, large,
or roughly of the order one.
Any dimensional equation
a = f (a1, . . . , ar , b1, . . . , bm)
is, after rescaling, equivalent to the dimensionless equation
Π = f (1, . . . , 1, Π1, . . . , Πm).
Thus, the introduction of dimensionless quantities reduces the number of variables in the
problem by the number of fundamental units. This fact is called the ‘Bucking- ham Pitheorem.’ Moreover, any two systems with the same values of dimensionless parameters
behave in the same way, up to a rescaling.
Introduction - The Purposes and Usefulness of Dimensional Analysis

Dimensional analysis is a very powerful tool, not just in fluid mechanics, but in
many disciplines. It provides a way to plan and carry out experiments, and
enables one to scale up results from model to prototype. Consider, for example,
the design of an airplane wing.
The full-size wing, or prototype, has some chord length, cp, operates at speed
Vp, and generates a lift force, Lp, which varies with angle of attack. In addition,
the fluid properties of importance to this flow are the density and viscosity.
After the preliminary design, it is usually necessary to perform experiments to
verify and fine-tune the design. To save both time and money, these tests are
usually conducted with a smaller scale model in a wind tunnel or water tunnel.
In the sketch above, a geometrically similar model is constructed. In this case,
the model is smaller than the prototype. In some cases the opposite is true; i.e.
it may be prudent to build a large model of some small prototype in order to
perform more accurate experimental analysis.

The goal of the experimental tests is to find a relationship between the
dependent variable (in this case the wing's lift) and the independent variables in
the problem (in this case the velocity, the wing's angle of attack, chord length,
and the density and viscosity of the fluid). Note that here we are neglecting the
speed of sound, which is only important at very high speeds. The functional
relationship can be stated as follows:

There is a wrong way and a right way to conduct the experiments. The wrong
way is to try to analyze the dependence of lift on each of the five independent
variables separately. In other words, run the tests at many velocities (to see the
effect of velocity on lift), and many angles of attack (to see the effect of angle
of attack on lift), with many different model sizes (to see the effect of chord
length on lift), and in many different fluids (to see the effect of viscosity and
density on lift). This would take an enormous amount of time and resources,
and it would be very difficult to summarize the results succinctly.
The right way to do the experiments is to first perform a dimensional analysis
of the above functional relationship, which leads to a revised form of the
relationship in terms of nondimensional parameters or nondimensional groups.
In this particular problem, dimensional analysis yields

which is much simpler than the original functional relationship. In particular,
instead of a dependent variable as a function of five independent variables, the
problem has been reduced to one dependent parameter as a function of only
two independent parameters. Furthermore, each of these three parameters is
dimensionless, which makes them completely independent of the unit system
used in the measurements.

The parameter on the left is a kind of lift coefficient (the actual lift coefficient
has a factor of 2 thrown in for convenience), while the first independent
parameter on the right is called the Reynolds number. Angle of attack is
already dimensionless, so it is a dimensionless group by itself.

One plot is enough to completely describe the above functional relationship. In
particular, lift coefficient is plotted versus angle of attack, and several curves
are plotted at constant Reynolds number. This single plot is then valid for any
size wing, in any Newtonian incompressible fluid, and at any speed. When
experiments are conducted after performing the dimensional analysis, it is
realized that only one wind tunnel model needs to be made, and only one fluid
needs to be used (that fluid can be air or water or any other Newtonian
incompressible fluid)! The wind tunnel or water tunnel test needs to consist of
simply measuring lift as a function of velocity and angle of attack. Results of
the experiment are plotted nondimensionally as indicated above.
Dynamic Similarity

The principle of dynamic similarity can be stated as follows:
If the model and prototype are geometrically similar (i.e. the model is a perfect
scale replica of the prototype), and if each independent dimensionless
parameter for the model is equal to the corresponding independent
dimensionless parameter of the prototype, then the dependent dimensionless
parameter for the prototype will be equal to the
corresponding dependent dimensionless parameter for the model.

Consider the airplane wing example above. In this case, the two independent
dimensionless parameters (those on the right hand side) are Reynolds number
and angle of attack. The dependent parameter is the lift coefficient. The model
wing in the wind tunnel must obviously be set at the same angle of attack as the
desired angle of attack of the prototype. In order to achieve dynamic similarity,
the Reynolds number of the model must also equal that of the prototype. Then,
dynamic similarity assures us that the lift coefficient of the prototype will equal
that of the model. Mathematically, we can solve for the wind tunnel velocity,
Vm, required to match Reynolds number, and we can scale up the lift
measurement from the wind tunnel tests to the full scale prototype as follows:
In this manner, we can set the wind tunnel speed properly to match Reynolds
number. Then, after measuring the lift on the model wing, Lm, we can properly
scale (using the last equation above) to predict the lift, Lp, on the prototype.
Non dimensionlized temperature difference
The law of dimensional homogeneity guarantees that every additive term in an equation has the
same dimensions. It follows that if we divide each term in the equation by a collection of
variables and constants whose product has those same dimensions, the equation is rendered
nondimensional . If, in addition, the nondimensional terms in the equation are of order unity, the
equation is called normalized. Normalization is thus more restrictive than nondimensionalization,
even though the two terms are sometimes (incorrectly) used interchangeably. Each term in a
nondimensional equation is dimensionless. In the process of nondimensionalizing an equation of
motion, nondimensional parameters often appear—most of which are named after a notable
scientist or engineer (e.g., the Reynolds number and the Froude number). This process is referred
to by some authors as inspectional analysis.
{C} em t2Lf
{rgz} e Mass Volume
Length Time2 LengthfeMass Length2 Length3 Time2f em t2Lf
e1 2
rV2f e Mass Volume aLength Time b2feMass Length2 Length3 Time2fem t2Lf {P}
{Pressure} eForce AreafeMass Length Time2 1 Length2f em t2Lf
P
12
rV2 rgz C
Equation of the Day The Bernoulli equation P rV2 rgz = C 1 2
FIGURE 7–6 The Bernoulli equation is a good example of a dimensionally homogeneous
equation. All additive terms, including the constant, have the same dimensions, namely that of
pressure. In terms of primary dimensions, each term has dimensions {m/(t2L)}.
The nondimensionalized Bernoulli equation P
rV2
rgz
C P2PP
P
{1} {1} {1} {1}
++=
FIGURE 7–7 A nondimensionalized form of the Bernoulli equation is formed by dividing each
additive term by a pressure (here we use P). Each resulting term is dimensionless (dimensions of
{1}).
As a simple example, consider the equation of motion describing the elevation z of an object
falling by gravity through a vacuum (no air drag), as in Fig. 7–8. The initial location of the object
is z0 and its initial velocity is w0 in the z-direction. From high school physics,
Equation of motion: (7–4)
Dimensional variables are defined as dimensional quantities that change or vary in the problem.
For the simple differential equation given in Eq. 7–4, there are two dimensional variables: z
(dimension of length) and t (dimension of time). Nondimensional (or dimensionless) variables
are defined as quantities that change or vary in the problem, but have no dimensions; an example
is angle of rotation, measured in degrees or radians which are dimensionless units. Gravitational
constant g, while dimensional, remains constant and is called a dimensional constant. Two
additional dimensional constants are relevant to this particular problem, initial location z0 and
initial vertical speed w0. While dimensional constants may change from problem to problem,
they are fixed for a particular problem and are thus distinguished from dimensional variables.
We use the term parameters for the combined set of dimensional variables, nondimensional
variables, and dimensional constants in the problem. Equation 7–4 is easily solved by integrating
twice and applying the initial conditions. The result is an expression for elevation z at any time t:
Dimensional result: (7–5)
The constant and the exponent 2 in Eq. 7–5 are dimensionless results of the integration. Such
constants are called pure constants. Other common examples of pure constants are p and e. To
nondimensionalize Eq. 7–4, we need to select scaling parameters, based on the primary
dimensions contained in the original equation. In fluid flow problems there are typically at least
three scaling parameters, e.g., L, V, and P0 P (Fig. 7–9), since there are at least three primary
dimensions in the general problem (e.g., mass, length, and time). In the case of the falling object
being discussed here, there are only two primary dimensions, length and time, and thus we are
limited to selecting only two scaling parameters. We have some options in the selection of the
scaling parameters since we have three available dimensional constants g, z0, and w0. We
choose z0 and w0. You are invited to repeat the analysis with g and z0 and/or with g and w0.
With these two chosen scaling parameters we nondimensionalize the dimensional variables z and
t. The first step is to list the primary dimensions of all dimensional variables and dimensional
constants in the problem, Primary dimensions of all parameters:
The second step is to use our two scaling parameters to nondimensionalize z and t (by
inspection) into nondimensional variables z* and t*,
Nondimensionalized variables: (7–6) z* z z0 t* w0t z0
{z} {L} {t} {t} {z0} {L} {w0} {L/t} {g} {L/t2}
We now know thath`2ρc κ i= T. Define the new, dimensionless variables ξ = x ` τ = κt `2ρc so
that x = ξ`, dx = `dξ t = τ`2ρc/κ, dt = `2ρc/κdτ. Then the heat equation is κ ρc`2×ρc∂u ∂τ =1
`2×κ∂2u ∂ξ2 , or after cancellation, ∂u ∂τ = ∂2u ∂ξ2 . We’ve eliminated all parameters from the
problem; if we write w = u/u0 where u0 is some reference temperature, then the equation is
completely dimensionless, ∂w ∂τ = ∂2w ∂ξ2 . If we solve this problem, then we can use that
solution for any body (having the same shape) of any length and material properties, simply by
rescaling the time, length, and temperature.