Similarity theory

The study of physical phenomena based on the concept of physical similarity.

Two physical phenomena are similar if the numerical values for the characteristics of one phenomenon enable one to derive the numerical values for the characteristics of the other by a simple conversion which is analogous to transferring from one system of units of measurement to another. For any set of similar phenomena, all the corresponding dimensionless characteristics (dimensionless combinations of dimensional quantities) have the same numerical values (see Dimensional analysis). The converse conclusion is also correct, i.e. if all the corresponding dimensionless characteristics are identical for two phenomena, then these phenomena are physically similar.

Dimensional analysis and similarity theory are closely related and are used in experiments with models. In such experiments, one replaces the investigation of a phenomenon in nature by the investigation of an analogous phenomenon in a model of smaller or larger scale (usually under special laboratory conditions).

After one has established the system of parameters defining a relevant class of phenomena, one establishes the similarity conditions for two phenomena. For example, a phenomenon may be determined by $n$ independent parameters, some of which may be dimensionless. Also, let the dimensions of certain variables and physical constants be expressible by the dimensions of $k$ parameters with independent dimensions ($k \le n$). Then from the $n$ quantities one can form only $(n-k)$ independent dimensionless combinations. All the desired dimensionless characteristics of the phenomenon may be considered as functions of these $(n-k)$ independent dimensionless combinations, consisting of the defining parameters. In the set of dimensionless quantities composed of the defining characteristics of a phenomenon one can always indicate a certain basis, i.e. a system of dimensionless quantities that determines all the others.

The class of phenomena defined by the corresponding substitution contains phenomena that, in general, are not similar to one another. The following conditions are used to isolate a subclass of similar phenomena from this.

For two phenomena to be similar, it is necessary and sufficient for the numerical values of the dimensionless combinations constructed from the complete list of defining parameters forming the basis to be the same for these two phenomena. The name similarity criteria is given to conditions on the constancy of the bases for the relevant parameters composed of the given quantities determining the phenomenon.

In hydrodynamics, the major similarity criteria are based on the Reynolds number, which characterizes the relation between the inertial and the viscous forces, the Mach number, which incorporates the compressibility of a gas, and the Froude number, which characterizes the relation between the inertial forces and the gravitational ones. The basic similarity criteria for heat transfer between a liquid (gas) and a body are: the Prandtl number, which characterizes the thermodynamic state of the medium, the Nusselt number, which characterizes the rate of convective heat transfer between the surface of the body and the bulk of the liquid (gas), the Péclet number, which characterizes the relation between the convective and the molecular heat transfer processes in a fluid, and the Stanton number, which characterizes the energy dissipation rate in a flow of liquid or gas. In the case of heat distribution in a solid, the similarity criteria are based on the Fourier number, which characterizes the rate of change in the thermal conditions in the environment and the rate of adaptation of the temperature field within the body, and the Biot number, which characterizes the rate of conductive heat transfer between two solid bodies and the temperature distribution within the bodies. In time-varying processes, the basic similarity criteria characterizing an identical course of processes in time are the homochronicity criteria. In aerohydrodynamics, this criterion is called the Strouhal number. The criterion for similarity in mechanical motion is the Newton number. The Poisson ratio is the similarity criterion for elastic deformation.

If the similarity conditions are fulfilled, it is necessary to know the scale factors for all the corresponding quantities in order to calculate all the characteristics in nature from data on the dimensional characteristics in the model. If a phenomenon is determined by $n$ parameters, of which $k$ have independent dimensions, these scale factors can take any values for the quantities with independent dimensions, and the values must be specified on the basis of the conditions of the problem, including the conditions of a trial in an experiment. The conversion factors for all the other dimensional quantities are derived from formulas expressing the dimension of each dimensional quantity via the dimensions of the $k$ quantities with independent dimensions, for which the scale factors are indicated by the experimental conditions and the formulation.

For example, in the case of steady-state flow of an incompressible viscous liquid around a body, all the dimensionless quantities characterizing the motion as a whole are determined by three parameters: the angles $\alpha$ and $\beta$ (the direction of the translational velocity of the body relative to the surface) and the Reynolds number $R$. The conditions for physical similarity (the similarity criteria) are: $$ \alpha = \text{const.}\,,\ \ \ \beta= \text{const.}\,,\ \ \ R = \frac{Pvd}{\mu} = \text{const.} $$

Here it is understood that in simulating the phenomenon, the results from experiments with the model can be transferred to nature only if $\alpha$,$\beta$ and $R$ are the same. The first two conditions are always readily met in practice, but the third is more difficult, particularly when the model is smaller than the natural body, which may be of large dimensions, for example, the wing of an aircraft. When the dimensions are reduced, one can maintain the value of the Reynolds number either by increasing the speed of the flow, which in practice is usually not feasible, or by substantially altering the density and viscosity of the liquid. In practice, these features lead to considerable difficulties in researching aerodynamic resistance (for example, air flow around an aircraft of natural size in a wind tunnel, and also in tunnels of closed type, in which compressed, i.e. more dense, air circulates with a high velocity).

Special theoretical and experimental studies have shown that in some cases of bodies of streamlined form, the Reynolds number has a noticeable effect only on the dimensionless coefficient of heat resistance and sometimes has very little effect on the dimensionless lift coefficient and on certain other quantities playing very important parts in practical applications. Differences in the Reynolds number in the model and in nature are unimportant for certain aspects.

Analogously, in simulating the motion of a body in a gas at high speed, it is necessary to have identical values for the Mach number in the model and in nature.

In simulating the passage of a ship through water, it is necessary to provide equality of the Froude and Reynolds numbers for the natural object and the model. However, if the linear dimensions are reduced in experiments with water in the laboratory, the condition for constancy of the Reynolds number implies that the speed of the model should be increased, whereas constancy of the Froude number implies that the speed should be reduced, and therefore an exact simulation is in general impossible when testing ship models in a laboratory. Sometimes such difficulties can be avoided by using different liquids or by artificially altering the acceleration due to gravity by "centrifugal simulation" , where the object is placed in a rotating system of large diameter.

Detailed consideration of the essence of hydrodynamic phenomena shows that in many cases one can take the effects of the Reynolds number into account by means of additional calculations or via simple experiments and by using data on the bending of flat plates. In the hydrodynamics of an ordinary water-displacing vessel, the Froude number is of primary importance, and therefore the simulation is based on constancy of the Froude number.

Research on models is often the only possible way of experimenting to solve major practical problems. This is the situation in research on natural phenomena occurring over scales of tens, hundreds or even thousands of years; under the conditions of model experiments, the similar phenomenon may last for a few hours or days (for example, in simulating oil extraction). There are also converse cases, where instead of examining extremely fast processes in nature one examines some similar phenomenon occurring much more slowly in a model.

Simulation is the basis also for the determination of laws of nature and for defining general features and characteristics in various classes of phenomena, while it also is involved in devising experimental and theoretical methods for researching various problems and deriving systematic information, techniques, rules, and recommendations for handling particular practical problems.

References

[1]	P.W. Bridgman, "Dimensional analysis" , Yale Univ. Press (1937)
[2]	L.I. Sedov, "Similarity and dimensional methods in mechanics" , Infosearch (1959) (Translated from Russian)
[a1]	H.E. Huntley, "Dimensional analysis" , Dover, reprint (1967)
[a2]	G. Birkhoff, "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press (1960) pp. Chapt. IV Zbl 0095.20303