Similarity Criterion

a dimensionless (abstract) number formed from dimensional physical parameters that determine the physical phenomenon under consideration. The equality of all similarity criteria of the same type for two physical phenomena or systems is a necessary and sufficient condition for the physical similarity of the systems. Trivial similarity criteria, which are ratios of physical parameters of a system that are of the same kind—for example, ratios of lengths—are usually disregarded when controlling similarity criteria are established. By definition, two systems are physically similar if these ratios are equal. Nontrivial dimensionless combinations formed from the defining physical parameters are also similarity criteria. Since every combination of similarity criteria is itself a similarity criterion, the most convenient and characteristic criteria can be selected in each actual case. The number of controlling nontrivial similarity criteria is equal to the difference between the number of the different dimensions of defining physical parameters and the number of independent dimensions.

If the equations that describe a given physical phenomenon are known, the similarity criteria for the phenomenon can be obtained by reducing the equations to dimensionless form by introducing certain characteristic values for each of the defining physical parameters in the system of equations. The similarity criteria are then defined as the dimensionless coefficients of certain terms of the new dimensionless system of equations. If the equations of a physical phenomenon are not known, the similarity criteria are found by analyzing the dimensions that define the physical parameters.

The similarity criterion of mechanical motion is obtained from the equation expressing Newton’s second law and is called the Newton number Ne = Ft²/ml, where F is the force acting on the body, m is the mass of the body, t is time, and l is a characteristic linear dimension.

In the study of elastic deformations of a structure under the action of external forces, the chief similarity criteria are Pois-son’s ratio for the material of the structure v = ǀ∊₁/∊ǀ and the criteria ρgl/E and F/El², where ∊ = ΔL/L. is the longitudinal strain, ∊1 = Δd/d is the transverse strain, E is Young’s modulus, ρ is the density of the material of the structure, F is the characteristic external force, and g is the acceleration of gravity.

In hydromechanics, the most important similarity criteria are the Reynolds number Re = ρ vl/μ = vl/ν, the Mach number M = v/a*, and the Froude number Fr = v²/gl, where ρ is liquid or gas density, v is the velocity of flow, μ is the coefficient of dynamic viscosity, v = μ/ρ is the coefficient of kinematic viscosity, and a* is the local propagation velocity of sound in the moving medium. Each similarity criterion has a definite physical meaning: each is proportional to the ratio of physical quantities of the same type. Thus, the number Re characterizes the ratio of inertial forces in the motion of a liquid or gas to viscous forces, and the number Fr is the ratio of inertial forces to gravitational forces.

The fundamental similarity criteria for heat transfer processes between a liquid or gas and a body around which the fluid is flowing are the Prandtl number Pr = ν/a = μc_ρ/λ, the Nusselt number Nu = al/λ, the Grashof number Gr = βgl³ΔT/v², the Péclet number Pe = vl/a, and the Stanton number St = α/pc _ρv. Here, α is the coefficient of heat transfer, λ is the coefficient of thermal conductivity, c_ρ is the specific heat of the liquid or gas at constant pressure, a = λ/ρc_ρ is the coefficient of thermal diffusivity, β is the coefficient of bulk expansion, and ΔT is the temperature difference between the surface of the body and the liquid or gas. The last two numbers are related to the preceding ones by the equations

Pe = Pr X Re St = Nu/Pe

The Fourier number Fo= at/l² and Biot number Bi= αl/λ are the characteristic similarity criteria for heat transfer in a body. The number Bi defines the nature of the correspondence between the thermal conditions of the surrounding medium and the temperature distribution in the body.

For processes that vary over time t, the basic similarity criterion characterizing the similarity of the courses of the processes is the homochronicity criterion Ho = vt/l. In the hydroaerome-chanics of unsteady flows, this criterion is usually called the Strouhal number Sh. In the case of similarity of electrodynamic phenomena, the homochronicity criterion is written in the form Ho= ωt, where ω is a characteristic frequency.

Examples of similarity criteria for electromagnetic fields include μγl²/t and ∊/γt, where μ is the permeability, γ the conductivity, and ∊ the dielectric constant of the medium. In the case of similar electrical circuits with distributed parameters, the similarity criteria L/Rt and C/Gt are used, where L is inductance, R resistance, C capacitance, and G conductivity.