Dimensional analysis

A method for finding relations between physical quantities that are significant for a phenomenon under study, based on considering the dimensions of these quantities. In dimensional analysis one considers the problem of establishing various systems of units of measurement, questions of the choice of the primary quantities and the corresponding experimental units of measurement and, associated with the choice of the primary units of measurement, the formation of secondary units of measurement for quantities defined in terms of the primary ones.

There are various possibilities for the quantities to be chosen as primary units of measurement. Various areas of application have found it advantageous and convenient to choose their own local units as the primary units of measurement. In this connection, and because of established usage, in practice various systems of units of measurement have been created, and the problem of transition (recalibration) from one system of measurement to another has arisen. Numerous systems of units of measurement are in wide use; among the main ones are — the CGS system, in which the primary units are the centimetre, the gram-mass, and the second; the MKS system, in which the primary units are the metre, the kilogram-force, and the second; and the SI (Système Internationale) system, in which the primary units are the metre, the kilogram-mass, the second, the Ampère, the Kelvin, and the candela. The number of primary units of measurement in the existing and potentially admissible systems can vary: less than three, equal to three, as in the CGS and MKS systems, and others. An expression of a derived unit of measurement in terms of the basic units is called a dimension formula; it can be written in terms of the symbols of the primary units of measurement and has the form of a monomial. For example, in the CGS system the dimension formulas contain three arguments: the length unit $L$, the time unit $T$ and the mass unit $M$; on the basis of the definition of force in terms of mass and acceleration, the dimension formula for the force $K$ has the form

$$K=\frac{ML}{T^2}=MLT^{-2}.\label{1}\tag{1}$$

In the CGS system the formula for any quantity $N$ of a mechanical, thermal or electromagnetic nature has the form

$$N=L^lT^tM^m,\label{2}\tag{2}$$

where the exponents $l,t,m$ are some integers or fractions, called the dimension exponents, or dimensionalities, of $N$. It is understood that the dimensionality of a primary quantity with respect to itself is one, and with respect to any other primary quantity — zero.

Dimension formulas make it possible to determine the numerical scale factors for the computation of corresponding characteristics when changing the primary units of measurement. If instead of given units of length $L$, time $T$ and mass $M$, one passes to new units, $\alpha$ times less in length, $\beta$ times less in time and $\gamma$ times less in mass, then the new unit of measurement for the quantity $N$ with the dimension formula \eqref{2} will be smaller than the original one by the factor $a^l\beta^t\gamma^m$.

If $l=t=m=0$, then the numerical value of the quantity is independent of the scale of the primary units, and such a quantity is dimensionless, or abstract. Examples of dimensionless quantities are: the Reynolds number $\mathrm{Re}=\rho vl/\mu$; the Froude number $\mathrm{Fr}=v^2/\sqrt{gl}$, the Mach number $\mathrm M=v/a$, and the cavitation number $\kappa=2\Delta P/\rho v^2l^2$, where for the dimensional quantities typical in some phenomenon the following notations have been adopted: $\rho$ — density, $v$ — velocity, $l$ — linear size, $\mu$ — dynamic viscosity coefficient, $\Delta P$ — characteristic pressure difference, and $g$ — gravitational acceleration.

The number of primary units of measurement can be increased if one agrees to choose independent units of measurements for some additional quantities, e.g. one can take independently the units: for thermal energy — the calorie, and for mechanical energy — the kilogram-metre, adding the relation: $I$ is thermal energy in calories or mechanical energy in kilogram-metres, where $I$ is a dimensional "physical" constant, called the mechanical equivalent of heat.

Physical laws containing dimensional constants can be used to reduce the number of primary units of measurement. E.g. the constant of velocity of light can be assumed to be absolute, that is, a dimensionless quantity, and so one can express the dimension and unit of measurement for length in terms of the dimension and unit of measurement for time; when considering the gravitational constant as an absolute dimensionless number, one can express the dimension and unit of measurement for mass in terms of the dimension and unit of measurement for length $L$ and time $T$.

In some questions there is an obvious tendency to standardization by introducing a unified universal system of units of measurement. However, it is artificial to link a universal system of units of measurement to particular physically fixed scales or to the corresponding physical constants. On the contrary, the possibility of using arbitrary units of measurement and the independence of observed laws on the choice of the system of units can be the source of valuable conclusions.

Physical laws, in general, are independent of the system of units of measurement. This circumstance determines a special structure of the functions and functionals which express physical relations independent of the system of units in terms of dimensional quantities. This special structure of the functional relations is established by the $\pi$-theorem: Every relation between dimensional characteristics which has physical significance is essentially a relation between the abstract dimensionless combinations which can be composed of the dimensional quantities that have been or are to be determined. Among these one should include the dimensional physical constants having significance for the phenomena under consideration.

Additional data on the absence of some physical constants in the relationship in question lead to the suppression, in these relations, of significant arguments. For example, if in the thermal or mechanical phenomena under consideration there is no transformation from thermal energy, measured in calories, to mechanical energy, measured in ergs, then the constant of the mechanical equivalent of heat is absent from the arguments of the function describing the corresponding law.

To obtain useful conclusions using dimensional analysis it is necessary to schematize the problem, and in the first place to fix a general model for the phenomena and properties of the objects considered. In many cases, such a schematization can be associated with several working hypotheses. In the framework of some models, one sets up a system of characteristics which are interconnected by physical relations, which by the $\pi$-theorem must be represented as relations between dimensionless parameters. Thus one must introduce a system of defining parameters for the constants or variables following from the statement of the distinguished class of problems and characterizing completely, in general, the particular problem for a given domain.

When isolating the minimal number of defining parameters, one should take into account symmetry conditions and the choice of advantageous coordinates. Independent variables (coordinates of points in space and time) and physical parameters such as the coefficients of thermal conductivity, viscosity, moduli of elasticity, etc., should be included in the table of defining parameters. Such constants as the gravitational acceleration or the mechanical equivalent of heat should also appear in the list of defining parameters when gravitation or transformation of thermal energy, measured in calories, into mechanical energy, measured in kilogram-metres (or in ergs), are significant.

For a particular class of problems it may be necessary to include among the defining parameters the dimensional or dimensionless characteristics of the boundaries of the bodies under consideration and of the functions that figure in the formulation of the initial, boundary and other conditions.

If the problem in question was formulated as a mathematical one, then one can write down a full table of arguments for relations of the form

$$a=f(a_1,\dots,a_n),\label{3}\tag{3}$$

where $a$ is the unknown quantity and $a_1,\dots,a_n$ are the defining parameters. Depending on the formulation of the problem, the number $n$ can be finite or infinite. Usually, by a suitable specialization of the class of problems under consideration, one can restrict oneself to the case when $n$ is finite and not large. The parameters $a_1,\dots,a_n$ are composed of the given values appearing in the basic equations, the boundary conditions and the initial data. The defining parameters are all the starting data which one must know in advance from the statement of the mathematical problem for the calculation of the desired function by various methods, including also computations by special machines or using analogue systems.

In the case of obtaining the necessary answers experimentally, the defining parameters are the quantities characterizing each separate test and the quantities that are necessary and sufficient for the comparison of different experiments.

One can write out a system of defining parameters also in those cases when detailed properties of the model and the equations describing the phenomena being studied are in general unknown: It suffices to rely on preliminary data or hypotheses on the form of the given functions and constants which enter or may enter into the definition of the model, and on other conditions distinguishing particular solutions of the problem.

The conclusions of a dimensional analysis are obtained on the basis of the $\pi$-theorem about the existence of relations of the form \eqref{3}, which hold for the quantity to be determined, and the determining quantities $a_1,\dots,a_n$ in any system of units of measurement.

The system of arguments in \eqref{3} must be complete in the sense of the dimensional theory. This means that if the quantity $a$ is neither zero nor infinite, then there exist numbers $\alpha_1,\dots,\alpha_n$ for which the dimensions of $a$ and the combination $a_1^{\alpha_1},\dots,a_n^{\alpha_n}$ are the same. Using this and the $\pi$-theorem, relation \eqref{3} can be rewritten in the form

$$\pi=\frac{a}{a_1^{\alpha_1}\dots a_n^{\alpha_n}}=g(\pi_1,\dots,\pi_{n-k}),\label{4}\tag{4}$$

where $\pi_1,\dots,\pi_{n-k}$ are dimensionless monomials (analogous to $\pi=3.14\dots$), formed from $a_1,\dots,a_n$, and $k\leq n$. The number $k$ is equal to the number of parameters among the $a_1,\dots,a_n$ with independent dimensions.

After explicitly writing down physical laws in the form \eqref{4}, the possibilities for using dimensional analysis are essentially exhausted. The corresponding results so obtained have a restricted character. If one modifies the equations of motion in any way that does not introduce new parameters, then the physical laws \eqref{4} may be drastically changed, but the main conclusion contained in \eqref{4} remains valid.

There exists a whole series of examples of obtaining fruitful conclusions and of establishing expedient and convenient methods for the processing of experimental results using dimensional analysis and the $\pi$-theorem. A special virtue is the reduction of the number of arguments in the desired functions; this reduction is ensured by the $\pi$-theorem and in many cases is achieved just by stating the problem using experimental data or some hypotheses about the mechanisms of the important effects in the problem being studied.

The main practical advantage consists in establishing the possibility of transferring the results of an experiment under certain conditions to other conditions under which the experiment was not performed. For example, from experiments with water flowing in a tube of fixed diameter one can automatically state the results required on the flow of water in tubes of the same form but of different dimensions, and on the motion of oil, air, etc. in tubes. However, for such a transfer of the results of one experiment to other experiments it is necessary to ensure that in corresponding cases one has the same values of the dimensionless parameters. The values of some of these quantities are easily afforded by the geometrical conditions under which the experiment is performed. The values of other parameters are associated with physico-mechanical conditions of the performance of the experiments.

Examples of the use of dimensional analysis are examples of establishing the self-similarity of the desired solutions from the statement of problems. The self-similarity property consists in the following. In the first approach to a mathematical solution of the given problem, one can note the independent arguments of functions; these are usually the three coordinates of points in some region of space and the time. As well as these variable quantities, among the arguments of the desired functions there may occur some given constant parameters arising in the description of the properties of the model under study and in singling out a particular schematic problem. Every problem can be formulated in dimensionless variables both for the desired functions and for their independently varying arguments. Self-similarity manifests itself in the fact that the number of independent dimensionless variables is reduced in comparison to the number of dimensional ones.

Effective results associated with self-similarity have been obtained in the theory of certain one-dimensional unsteady flow problems with spherical, cylindrical and flat waves, when there are only two independent variables — a coordinate $r$ with the dimension of length and the time $t$. If from the defining dimensional constants present in the formulation of the problem it is impossible to form two combinations — one with the dimension of length and the other with the dimension of time, then the only variable dimensionless argument in the desired dimensionless functions is a parameter of the form $\lambda=Cr/t^\alpha$, where $\alpha$ is some exponent and $C$ is a dimensional constant expressible in terms of the given constants from the formulation of the problem. In particular, if a mathematical problem is reduced to the solution of a system of partial differential equations in $r$ and $t$, then in the presence of only one variable parameter $\lambda$, the partial differential equations in $r$ and $t$ can be transformed into ordinary differential equations in the single independent variable $\lambda$, as a consequence of which the mathematical problem is significantly simplified. In this way, the solution of many practically important problems has been obtained, for example the problem of the perturbed motion of air caused by an atomic explosion in the atmosphere.

The considerations of dimensional analysis for self-similar processes can serve not only to reduce partial differential equations to ordinary ones, but also to obtain finite relations between the desired functions (the integrals of these equations), which makes it possible to find the solution of some self-similar problems by a closed formula. These methods have been demonstrated in the solution of problems about strong explosion in the atmosphere and in problems on the propagation of explosive waves in gas-like media in which the density in the initial state is variable, changing along a radius by a power law. These problems are presented as strongly schematized models for the explosion of stars.

Dimensional analysis is immediately connected with the concept of physical similarity, which is studied in similarity theory.

References

[1]	P.V. Bridgeman, "Dimensional analysis" , Yale Univ. Press (1937)
[2]	L.I. Sedov, "Similarity and dimensional methods in mechanics" , Infosearch (1959) (Translated from Russian)

References

[a1]	G. Birhoff, "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press (1960)