Before examining the process of dimensional analysis, it is helpful to review the subject of units and measurement systems. A unit is a standardized measurement of a parameter or quantity that has been adopted by consensus and defined by convention, e.g., the SI system. The expression of quantities that are measured in terms of standardized units allows their magnitude or numerical value to be evaluated and compared. Adopting a units system for the measurement of a broad range of physical parameters allows engineers and scientists across the world to communicate and work with each other using a common standard.

The value of a particular physical quantity is always given the product of a number and a unit (unless unitless), e.g., 12.4 meters per second (m/s) in SI units is equal to 40.68 feet per second (SI) in USC or British Imperial units, where the conversion factor from meters to feet is 3.281. This means the numerical value of a quantity depends on the system of units in which it is expressed. Engineers in the U.S. typically use both SI and USC units. However, there is a preference to use SI units because of their widespread adoption and use in scientific research and compliance with international standards. Conversion factors are available so that numerical values in one unit system can be expressed as equivalent values in the other.

SI Units

Definitions for common SI units.

Quantity	Quantity symbol	SI unit name	Unit symbol
Length	L	meter	m
Mass	M	kilogram	kg
Time	T	second	s
Temperature		Kelvin	K or oK
Velocity	V	meter per second	m/s
Acceleration	a	meter per second2	m/s2
Force	F	Newton	N
Energy	E	Joule	J
Power	P	Joule per second = Watt	W
Pressure	p	Newtons per m2 = Pascal	Pa
Viscosity		Pascal-second	Pa s

USC Units

Definitions for USC units. *Power is commonly measured in horsepower (hp) where 1 hp = 550 ft-lb/s.

Quantity	Quantity symbol	USC unit name	Unit symbol
Length	L	foot	ft
Mass	M	slug	slug or sl
Time	T	second	s
Temperature		Rankine	R or oR
Velocity	V	feet per second	ft/s
Acceleration	a	feet per second2	ft/s2
Force	F	pound	lb
Energy	E	foot-pound	ft-lb
Power*	P	foot-pounds per second	ft-lb/s
Pressure	p	pounds per foot2	lb/ft2
Viscosity		slug/(ft s) or pound seconds per foot	slug/(ft s) or lb s/ft

Notice that the name “slug” as a unit of mass originates from the concept of mass as an inertia or “sluggish” and has been referred to as the “engineer’s mass unit.” The unit of  a “slug” has also been associated with the mass of a lead projectile, but because a slug as a unit mass has a corresponding weight of 32.17 lb, then this is a more unlikely association.

The use of “pounds-mass” (lb_m) as a measure of mass is generally considered to be a source of confusion in the engineering and scientific fields. Therefore, the unit of “pounds-mass” is is best avoided in preference of using the “slug” as the base mass unit. While USC units are largely similar to British Imperial units, there are differences in regard to volumetric measurements.

In 1975, the U.S. government declared the international SI system to be the “preferred” unit system but did not require it. Indeed, the USC system still prevails in the U.S. as the preferred measurement system, and the U.S. is one of the few countries in the world that does not exclusively use the SI system. In Britain, the use of mixed SI and Imperial units is also common, but not as much as in the U.S. The SI units system is preferred in all engineering and scientific work, and many situations it may be required by specification, international code, or regulation.  However, in today’s global community, engineers and scientists from all countries need to be conversant in the use of both SI and USC units.

NOTE: In aviation, distances are measured in terms of nautical miles, which is neither Imperial or SI, airspeed is measured in terms of knots (nautical miles per hour), and altitude is measured in feet, the use of such units being in accordance with ICAO aviation and aeronautical standards.

CGS Units

The centimeter–gram–second or CGS system of units is an increasingly obsolete variant of the metric system. CGS units are based on the centimeter as the base unit of length, the gram as the base unit of mass, and the second as the base unit of time. The CGS system has been largely supplanted by the use of SI units, but sometimes quantities measured in CGS units are still encountered in engineering and scientific practice. Converting between CGS and SI units is easy because the conversion factors are all powers of 10, as reviewed in the table blow, For example, 100 cm = 1 m and 1,000 g = 1 kg. The CGS unit of force is called the dyne, which is 1 g cm/s, so a Newton (N) in SI units is equal to 100,000 dynes in CGS units.

Definitions of CGS units and conversion factors from CGS to SI units

Quantity	Quantity symbol	CGS name	Unit symbol	Unit definition	SI units
Length	L	centimeter	cm	1/100 of meter	10−2 m
Mass	M	gram	g	1/1000 of kilogram	10−3 kg
Time	T	second	s	1 second	1 s
Velocity	V	centimeter per second	cm/s	cm/s	10−2 m/s
Acceleration	a	gal (Galileo)	Gal	cm/s2	10−2 m/s2
Force	F	dyne	dyn	g cm/s2	10−5 N
Energy	E	erg	erg	g cm2/s2	10−7 J
Power	P	erg per second	erg/s	g cm2/s3	10−7 W
Pressure	p	barye	Ba	g/(cm s2)	10−1 Pa
Dynamic viscosity		poise	P	g/(cm s)	10−1 Pa s
Kinematic viscosity		Stokes	St	cm2/s	10−4 m2/s

Physical Quantities

Science and engineering involve observations of the physical world; engineers aim to express observations in the most general and the simplest type of formulation. The role of mathematics is essential here, and the resulting formulation will stem directly from the properties or quantities considered in the quantitative analysis. The allowed types of properties are called physical quantities.

Physical quantities are of two types: base quantities and derived quantities. Base quantities such as mass, length, time, and temperature can be readily measured. Derived quantities are things like velocity, pressure, energy, and power, all of which are derived from these base quantities. In dimensional analysis, only base quantities are used; those mostly encountered in mechanics and engineering are listed in the table below. Other base quantities that may arise in problem-solving are electrical current (Amp), the intensity of light (Candela), and concentration (Mole).

List of key base quantities used in engineering mechanics.

Base quantity	Symbol	SI unit	USC unit
Mass	M	kilogram (kg)	slug (or slugs or sl)
Length	L	meter (m)	foot (ft)
Time	T	second (s)	second (s)
Temperature		Kelvin (K or K)	Rankine (R or R)

Finding Base Dimensions

Many of the parameters routinely used in engineering are derived parameters, in that their units are derived from combinations of base units,  e.g., volume, density, force, work, energy, power, etc. From now on, notice that the parameter inside a set of square brackets, i.e., “[parameter],” is used to mean the dimensions of that parameter. Notice also that scientific notation must be used throughout to avoid ambiguity in the final units of the derived parameter, e.g., use and not . Consider the following examples:

• Volume, , is a derived quantity because to calculate the volume of something such as a cuboid, then its width (units of ) must be multiplied by its length (units of ) by its height (units of ). Therefore, the derived units for volume are or [] =  .
• Density, , is also a derived parameter because the density of something is its mass () per unit volume (), so density will have units of , i..e., [] = .
• Force, , is a derived parameter, although in some cases, “force” could be carried through the problem as a base unit. A force is the product of a mass and acceleration (Newton’s second law), so in terms of its base units, then [] = .
• Torque, , is a derived parameter because a torque is a moment, so the product of a force times a distance or “arm.” Therefore, [] = .

Obtaining the correct base units of any given parameter takes some practice, but it is essential for correctly using dimensional analysis Again, the scientific notation should be used when expressing the units of a parameter to avoid ambiguity, e.g., always use the form and never the form .

Aerodynamic Force on a Wing

To introduce the subject and methods of dimensional analysis, consider the forces produced on a wing in the wind tunnel, as shown in the photo below. The resultant aerodynamic force on the wing is . For convenience, it can be assumed that this force acts at a center of pressure, which, by definition, is a point where there are no pitching moments.

On a physical intuitive basis, then it would be expected that the resultant aerodynamic force on the wing for a given orientation to the flow will depend on:

1. Free-stream velocity, .
2. Free-stream density, .
3. The viscosity of the air, i.e., on the coefficient of viscosity .
4. The size of the wing, as described by its planform area , which for a rectangular wing is the product of its chord and span , i.e., , as shown in the figure below.
5. The sonic velocity , i.e., the potential for compressibility effects.

The subscript or “infinity” is used to mean undisturbed conditions, i.e., the “free-stream” or reference values of these quantities that are constant well upstream of the wing. This approach makes sense because as the flow approaches the wing, these values are all likely to change. Notice that the angle of attack of the wing, , is the angle between the chordline of the wing (i.e., a line running from the nose to the tail of the wing) and the direction of the free-stream velocity .

The relationship between and the air properties may now be written in a general functional form as

(1)  

where is referred to as the dependent variable and , , and are called the independent variables. The functional dependence, i.e., the relationship between the variables, which in this case is denoted by “” is still to be determined. Although this latter equation expresses a very general relationship, its use in this form is mostly impractical for the direct measurement or calculation of .

To see why consider how could be measured in the wind tunnel by a student of aerodynamics. The procedure might start with the student placing a wing at a given angle of attack, , and then (somehow) systematically varying each of the dependencies , , and (i.e., a total of five parameters) in turn, and in each case measuring the corresponding values of . Varying the wind speed is easy because it can be controlled easily in the wind tunnel, and different wings could be tested to change the wing area . The other parameters, however, are more challenging to vary directly or systematically. However, if it could be done, then the number of possible experimental runs required in the wind tunnel would be (!).

Then that process would need to be repeated by the student for different values of the wing origination to the flow, . Eventually, the needed relationships between and all dependent quantities at each angle of attack would be determined. Therefore, it does not take much to figure out that this process of systematically varying each of the dependent parameters, in turn, could take the student a very long time, even by doing parallel calculations using aerodynamic theory.

This approach would also result in a large quantity of data, from which cross-plotting and perhaps curve-fitting could be used to find the needed functional relationships as to how , , etc., would each affect . If just one dependent variable was used, then just a single column of numbers would be obtained, and the functional relationship would be easy to establish. However, a book of tables would be needed for four or more dependent variables or many pages of charts and graphs. Hence, it soon becomes apparent to the student that managing an engineering problem with three, four, or more independent variables is challenging. Systematically varying each independent parameter to establish the “cause and effect” on the dependent parameter would be impractical from a measurement perspective, nor would it be attractive from a computational (modeling) approach.

Basis of Dimensional Analysis

Fortunately, the approach to the preceding problem can be simplified by employing dimensional analysis, which can be used to determine sets of non-dimensional (or dimensionless) groups of parameters that affect the problem of interest. Such an approach will also reduce the number of independent variables, so it becomes an essential technique for understanding many engineering problems, not just aerodynamics.

Dimensional analysis is based on the fact that if an equation of parameters such as

(2)  

expresses some physical relation, then the parameters , , , and must all have the same dimensions for the equation to be mathematically valid, e.g., , , , and could all be values of pressures or velocity components. Furthermore, to make sense then each parameter must also be expressed numerically in the same units system.

This previous equation can be made non-dimensional or dimensionless, or what is called dimensionally homogeneous by dividing through by one of the terms, i.e., the same result can be expressed as

(3)  

Another advantage of this form is that all of the quantities are now expressed as ratios so the system of units is irrelevant.

Lord Rayleigh and Joseph Bertrand originally set down these simple but profoundly important ideas in what is called the “Method of Dimensions.” However, they are now formally embodied in what is known as the “Buckingham Theorem” after Edgar Buckingham, who developed it into a more structured form suitable for general use. The Buckingham Theorem (usually just called the Buckingham Method) is a generalized formalization of the method of dimensions, which can now be examined.

The Buckingham Π Method

Let be equal to the number of fundamental dimensions required to describe the physical variables. In engineering, most physical variables can be expressed a combinations of mass, length, time, and temperature, so in general . In mechanics, then mass, length, and time are involved, so . Let represent a list of physical variables in the general physical relationship

(4)  

where is the dependent parameter and are the independent parameters. The function represents a general functional relationship, as yet unknown. The previous equation can also be written is implicit form as

(5)  

where represents another functional relationship between the parameters yet to be determined. The Buckingham method then states that the above relationship may be expressed as a relationship of fewer dimensionless products called products, where each product is a dimensionless product of a selected set of repeating variables plus one other variable. One key to the method, therefore, is how to go about the process of selecting the most suitable repeating variables.

To see how this method works, let be the selected repeating variables, i.e.,

(6)  

where now there are new functional relationships, i.e., , …., but now they are expressed in terms of the non-dimensional products. Therefore, it can be written that

(7)  

where is the functional relationship (still to be determined) but now it is expressed in terms of each of the products. In explicit form, this latter equation can also be written as

(8)  

where is some other functional relationship.

The upshot of the dimensional analysis process means that the functional relationship has been reduced from the initial dimensional variables to non-dimensional or dimensionless variables. The reduction the number of dependent parameters, therefore, is a significant simplification in the analysis of the original problem. For example, if originally and , i.e., the dependent variable is a function of five other parameters as in the wing problem discussed above, the dimensional analysis will reduce the problem to three non-dimensional groupings. If and , then the dimensional analysis will reduce the problem to two non-dimensional groupings in that one grouping becomes a sole function of the other grouping.

Π Method Applied to the Drag on a Sphere

Consider a sphere to be tested in a low-speed wind tunnel to measure its drag, as shown in the photograph below. Perhaps the drag of a sphere sounds boring compared to a wing, but this problem helps first to introduce the process of dimensional analysis without making the process too lengthy.

Furthermore, the aerodynamic characteristics of a sphere is particularly relevant to all sorts of ball games. The motion of balls through air is a complex fluid dynamic problem that is influenced by many factors, including the density, velocity, viscosity, the size of the ball, and its surface finish. For example, the aerodynamics of a golf ball affect the distance, spin, and overall trajectory of the ball. Ever wonder why a golf ball has dimples in its surface? In baseball, the aerodynamics of a baseball can influence the speed and trajectory of a pitch.

Functional Dependency

The objective is to find the non-dimensional groupings that will describe this aerodynamic problem. On a physical intuitive basis, the aerodynamic drag force, , on the sphere would be expected to depend on:

1. Free-stream velocity, .
2. Free-stream density, .
3. Free-stream viscosity, i.e., the coefficient of viscosity .
4. The size of the sphere, as described by its diameter .

The relationship between and the air properties may be written in a general functional form as

The drag is the dependent variable. , , and are the independent variables. The functional dependence in this case is denoted by

The functional dependence in implicit form is

where is some other function. In this case, , , , so there are two products to be determined.

Understanding explicit versus implicit functional forms

What is the difference between writing an equation in the form versus the form ? In mathematics, the dependent function say , for example, can be written in terms of other parameters, e.g., where is some function, or more generally as , which are called explicit functional dependencies. However, the relationship can also be written in implicit form, i.e., is equivalent to . Notice that the function is NOT the same as the function . If , for example, then , which is an explicit dependency of on . Alternatively, it can be written that , which is in the form , which is the implicit form of the same relationship. Likewise, as . In dimensional analysis, the process starts off by writing the functional dependency of the problem parameters in implicit form.

Dimensions of the Variables

Now the dimensions of the variables in this problem must be determined. For each dependency, then

Recall that the units of force are obtained from the product of mass and acceleration, i.e., .

Dimensional Matrix

The solution to the dimensional analysis problem proceeds by setting up a dimensional matrix with all of the parameters being given in terms of their base units. Setting up the dimensional matrix in this case gives

One purpose of the dimensional matrix is to lay the problem out in a clear manner and test for linear independence of the repeating variables in terms of the chosen primary dimensions.

Repeating Variables

Although the selection of the repeating variables can be viewed as somewhat arbitrary, in general, they must always be selected based on four essential rules:

1. The repeating variables must all have a primary dependency on the dependent variable. In this case, the variables and can all be expected to have a strong effects on the dependent variable, .
2. The dependent variable, , cannot be used as a repeating variable, so it must appear only in one group, by default. No exceptions.
3. The repeating variables must include all of the physical based dimensions of the problem, i.e., they must collectively include mass and/or length and/or time, as appropriate, for the particular problem in question. In this case, , and collectively include all of the base dimensions of mass, length, and time.
4. The repeating variables must be linearly independent of one another, i.e., repeating variables must not be chosen that will have the same units or products of the same units, e.g., repeating variables with two different length scales or two different reference velocities cannot be used. In this case, , and are all linearly independent of each other.

If these prior rules are followed, the dimensional analysis process will likely succeed. If not, then it will fail, often catastrophically. For complex problems, of which there are many in engineering, the choice of repeating variables may not be obvious but should still be selected to be quantities that are most likely to have substantial effects on the dependent variable.  For example, the velocity and density of the flow in an aerodynamic problem are excellent examples of two quantities that will have significant effects on the aerodynamic forces.

Formal Check for Linear Independence

To formally check for linear independence (although, in most cases, this will be obvious), the determinant of the sub-matrix formed by the selected repeating variables must be determined, which in this case is

(9)  

If , and are non-zero constants in the following equation, for linear independence then

(10)  

Reassembling this in matrix form gives

(11)  

For this matrix equation to have a nontrivial solution then the determinant must be zero, i.e., the test is whether

(12)  

In this case, the determinant is 1.

Therefore, it has now been formally verified that the selected repeating variables and are indeed linearly independent of each other. In most cases, however, the formal test is not needed because if the variables are chosen such that they do not have the same units or products of the same units, then they will be linearly independent.

Development of the Π Products

Following the Buckingham method, then the products in this case will be

First Π Product

For then

The values of the coefficients , and must now be obtained to make the equation dimensionally homogeneous and non-dimensional.  In terms of the dimensions of the problem then

Because the left-hand side of the above equation is dimensionless, it means that the powers or exponents of , and must add to zero to make the equation dimensionally homogeneous. Therefore, for to be dimensionless, then

so that

By inspection, , , and . Therefore, the first product is

or

which is a force coefficient. Normally, aerodynamic force coefficients are defined in terms of the dynamic pressure, i.e., . Also, in this case a good reference is to use projected area . Therefore, the force coefficient as would be written as

Second Π Product

For then

so in terms of dimensions then

Equating the powers gives

Therefore, in this case , , and .

i.e.,

or by inverting the quantity a non-dimensional grouping is still obtained, i.e.,

Recall from previously that this non-dimensional parameter is called the Reynolds number.

Final Result for Functional Dependency

As a result of the dimensional analysis on the sphere then

or

or in explicit form

The functional dependency to be measured is now . The force can be measured with a balance, and with measurements of the density and viscosity (via measurements of pressure and temperature) the calculations are made of:

1. The drag coefficient, .

2. The Reynolds number, .

The results are shown in the figure below for the drag coefficient on the sphere as measured in the wind tunnel as a function of Reynolds number. Notice the rapid decrease in the drag coefficient between and , which suggests that something very profound is happening to the flow about the sphere. This behavior is tied to the characteristics of the flow on the surface of the sphere, which change from being dominated by laminar flow events to turbulent flow events.

Π Method Applied to a Wing

Returning to the wing problem previously discussed, the explicit dependency of the resultant aerodynamic force (the dependent quantity) in terms of the independent variables is

(13)  

The functional dependence  can also be written in implicit form, i.e.,

(14)  

where is some functional dependency to be determined. Following the method then , , , and so three products must be found in this case.

Dimensions of the Variables

Now, the dimensions of the variables must be determined. Remember that the square brackets [-] around a variable mean “the dimensions of” that variable. So, for each dependency in this case then

Notice that the units of force are obtained from the product of mass and acceleration, i.e., [] = . It is critically important that each of the variables is assigned the correct base units. If not, the Buckingham method will fail.

Forming the Dimensional Matrix

The solution to the dimensional analysis problem proceeds by setting up a dimensional matrix with all of the parameters being given in terms of their base units, i.e.,

(15)  

Choice of Repeating Variables

In this problem, a good choice of the repeating variables is , and , which is a common choice for aerodynamic problems. Remember not to choose the dependent variable as a repeating variable. Remember also that the repeating variables must be linearly independent, i.e., the units of any one repeating variable cannot be linearly obtained from the units of any other. In a practical sense, choosing repeating variables with different dimensions or products of dimensions is necessary. For example, two repeating variables comprising a length and an area, or two repeating variables with units, cannot be used, or the method will fail. Furthermore, the repeating variables must collectively include all of the base dimensions of the problem. In this case, the choice of the repeating variables , and satisfy all of these requirements.

Development of Π Products

Continuing to follow the Buckingham procedure, then the products in this problem are

First Π Product

Take the first product, , then

(16)  

where the coefficients , and are unknowns and their values must be obtained appropriately to make the equation dimensionally homogeneous, i.e., finding their values to make the equation non-dimensional with units of 1. In terms of the dimensions, then

(17)  

where is used to mean “the dimensions of variable .” Collecting the terms gives

(18)  

Because the left-hand side of the above equation is dimensionless, it means that the powers or exponents of , and must add to zero to make the equation dimensionally homogeneous, i.e.,

Therefore, in this case by inspection then , , and and so the first product is

(19)  

or

(20)  

Hence, , is a form of force coefficient. Aerodynamic force coefficients are nearly always defined in terms of the dynamic pressure, i.e., , which is referred to as , so more usually the force coefficient would be written as

(21)  

Notice that multiplying a similarity parameter by a constant value does not change the units of that parameter.Paralleling the above approach used to find , the two remaining products, namely and , can now be derived.

Second Π Product

For then

(22)  

so

(23)  

and

Therefore, in this case , , and and so this product becomes

(24)  

Notice that this derived product is a non-dimensional parameter that will become infinite as the velocity tends to zero. This behavior is not very convenient in practice, but the product can simply be inverted to prevent that from happening. The outcomes is still a non-dimensional parameter, but now one that is more convenient, i.e., the product becomes

(25)  

Because, in this case, there is a square-root of an area, it can be replaced by a linear dimension, such as the chord of the wing , i.e., the area of a rectangular wing will be its chord times the span of the wing or , so now

(26)  

The choice of the appropriate length scale is somewhat arbitrary (e.g., the wing span could have been used) but generally in an aerodynamic problem a parameter describing a downstream distance should be used, e.g., wing chord in this case.

This latter non-dimensional parameter is known as the Reynolds number, which is denoted by the symbol . The Reynolds number is one of the most significant parameters in aerodynamics because it governs the relative magnitude of viscous effects to inertia effects in the flow.

Third Π Product

Similarly, for the third product then

(27)  

and so in terms of dimensions, then

(28)  

giving

Therefore, in this case , , and , i.e.,

(29)  

Again, the product can be inverted for convenience (to put on the numerator) and still have a non-dimensional quantity so

(30)  

This particular non-dimensional parameter is called the Mach number (the free-stream Mach number in this case), which is the ratio of the free-stream velocity to the free-stream speed of sound. The Mach number is an important quantity in aerodynamics that not only affects the magnitude of the forces on a body but also the compressible nature of the flow, in general.

Final Result for Functional Dependency

Therefore, as a result of the dimensional analysis on the wing problem then

(31)  

or

(32)  

or in explicit form

(33)  

The previous result is a significant outcome because it means the original aerodynamic problem in the wind tunnel has been reduced from one that involves five independent variables (for which is practically impossible to determine the functional relationships to ) to one that now only has only two variables! Therefore, with a little effort, it has been shown that to determine the force on a given wing at a given angle of attack , then only the dependence of and on need to be established. The non-dimensional parameters (the Reynolds number) and (the Mach number) are known as similarity parameters.

Both the Reynolds number and the Mach number are known to have substantial effects on the aerodynamic forces produced on a wing. In the figure below, the wing lift coefficient (non-dimensional lift) is plotted versus the angle of attack for a series of free-stream Mach numbers from 0.3 (subsonic) up to 0.75 (transonic). Notice that there are two primary effects. The first effect is that increasing the Mach number increases the lift coefficient for a given angle of attack, at least in the lower angle of attack regime. The second effect is that the maximum value of the attainable lift coefficient is reduced with increasing Mach number, which is a consequence of the flow detaching or separating from the wing and is called stall. In other words, the angle of attack at which stall occurs decreases with increasing Mach number. The effects of Reynolds number on the lift on a wing will be discussed later.

Adding another dependency to the problem

Think about what might happen if span was added as a seventh variable to the parameter list in this previous problem, i.e., in this case assume that

   

Therefore, , , , and so four products will result from the dimensional analysis. The first first three have already been determined so it should be fairly obvious that the forth non-dimensional grouping will be

   

which is a span to chord ratio and is known as the aspect ratio of the wing, . Notice in the figure below, the profound influence of aspect ratio on the lift coefficient. Wings with high aspect ratio are more efficient in that for a given angle of attack and all other parameters held constant then a wing with high aspect will produce more lift and less drag.

Other Worked Examples

The best way of learning the method of dimensional analysis is by doing examples. After working through two or three more examples, the student will quickly realize the power of dimensional analysis and  how this technique can help in engineering problem solving. Naturally, the method is not restricted to aerodynamics, and the process of dimensional analysis is useful in all branches of engineering and the physical sciences.

Worked Example #2 – The Bobbing Booster

A solid rocket booster parachutes back to Earth and falls into the sea. The booster bobs around in the sea in an upright position at a frequency of . Using dimensional analysis, show that the non-dimensional frequency of this motion, , is given by

   

It may be assumed that for this problem the frequency of motion depends on the diameter of the booster , the mass of the booster, , the density of the water , and acceleration under gravity, .

Functional Dependency

By using the information provided then the frequency of motion can be written in explicit functional form as

   

where “” is some function to be determined, or in implicit form as

   

where “” is some other function. Therefore, = 5, = 3 and so there are two products.

Repeating Variables

The repeating variables must collectively include all the units of mass , length and time , so the best choice here is , and . Notice that if were to be chosen instead of as a repeating variable, then collectively the repeating variables would not include time and so the Buckingham method will fail in this case. The dependent variable, , cannot be used as a repeating variable.

Products

Therefore, the two products are

   

and

   

Dimensional Matrix

The dimensional matrix is

   

Derivation of the Products

The first product is

   

where the values of the coefficients , and must be obtained to make the equation dimensionally homogeneous.  In terms of the dimensions of the problem then

   

For to be dimensionless, then the powers or exponents of , and must add to zero, i.e., in this case

   

Therefore, = 1/2, = 0, and = -1/2, so the product is

   

For the second product then

   

In terms of the dimensions then

   

For to be dimensionless, then

   

Therefore, = -1, = 0, and , so the product is

   

which is a buoyancy similarity parameter, i.e., the ratio of the mass (or weight ) of the water displaced to the mass (or weight ) of the rocket booster.

Final Non-Dimensional Functionality

Finally, in non-dimensional form then

(34)  

or that

(35)  

Other Dimensionless Quantities

Dimensional analysis can be a powerful tool for engineering analysis because it reduces and simplifies the size and scope of the problem in terms of non-dimensional groupings called similarity parameters. The strength of dimensional analysis as a tool usually becomes apparent when it is applied to increasingly more complicated engineering problems, such as where the effects of all the dependencies cannot be established explicitly and/or the governing equations for the solution of the problem can become formidably complicated. For example, it is common for engineering problems to have five, six, seven or even more dependencies. However, sorting out the problem parameters into non-dimensional groupings helps understand what parameters are essential as they will affect the physical behavior and what parameters are not.

The dimensional analysis must be used in a systematic and structured approach, which is formally embodied in the Buckingham method. A critical first step is to carefully lay out the problem in terms of the dependent and independent quantities, which must all be assigned the correct base units. While the choice of the repeating quantities is somewhat arbitrary, specific rules must be adhered to for the method to work. For example, it is critical that the dependent quantity NOT be used as a repeating variable. The repeating variables must also have substantial effects on the dependent quantity and be linearly independent of each other, i.e., two repeating parameters with the same dimensions cannot be used such as two reference length scales or two reference velocities. If the basic rules of the Buckingham method are followed, then the outcomes will be groups of non-dimensional (unitless) parameters that should not only help in the understanding of the problem but will simplify its scope.