The word "criterion" of Greek origin means a sign that is the basis for the formation of an assessment of an object or phenomenon. Over the past years, it has been widely used both in the scientific community and in education, management, the economy, the service sector, and sociology. If the criteria of science (these are certain conditions and requirements that are mandatory) are presented in an abstract form for the entire scientific community, the similarity criteria affect only those areas of science that deal with physical phenomena and their parameters: aerodynamics, heat transfer and mass transfer. In order to understand the practical value of applying the criteria, it is necessary to study some concepts from the categorical apparatus of the theory. It is worth noting that similarity criteria were used in technical specialties long before they got their name. The most trivial criterion of similarity can be called finding a percentage of the whole. A similar operation was done without any special problems and difficulties. And the coefficient of performance, which reflects the dependence of the power consumed by the machine and the power supplied, has always been a criterion of similarity and from this did not become perceived as something foggy-sky-high.
The foundation of the theory
The physical similarity of phenomena, whether it be nature or a man-made technical world, is used by man in research on aerodynamics, mass and heat transfer. In the scientific community, the method of studying processes and mechanisms by means of modeling has established itself well. Naturally, when planning and conducting the experiment, the support is the energy-dynamic system of quantities and concepts (ESWP). A reservation should be made that the system of quantities and the system of units (SI) are not equivalent. In practice, ESWP exists objectively in the outside world, and research only reveals them, so the basic quantities (or criteria of physical similarity) do not have to coincide with the basic units. But the basic units (systematized in SI), meeting the requirements of practice, are approved (conditionally) through international conferences.
The conceptual apparatus of similarities
Similarity theory - concepts and rules whose purpose is to determine the similarity of processes and phenomena and to ensure the possibility of transferring the studied phenomena from a prototype to a real object. The basis of the terminological dictionary is constituted by such concepts as homogeneous, homonymous and dimensionless quantities, the similarity constant. To facilitate understanding of the essence of the theory should consider the meaning of these terms.
- Homogeneous - quantities that have equal physical meaning and dimension (an expression showing how the unit of measurement of a given quantity is made up of units of basic quantities; speed has the dimension of length divided by time).
- The same name - processes that differ in value, but have the same dimension (induction and mutual induction).
- Dimensionless - quantities in the dimension of which the basic physical quantities are included in the degree equal to zero.
A constant is a dimensionless quantity in which the base is a quantity with a fixed size (for example, an elementary electric charge). It allows you to make the transition from model to natural system.
The main types of similarity
Any physical quantities may be similar. It is customary to distinguish four types:
- geometric (observed when the relations of similar linear dimensions of the sample and model are equal);
- temporary (observed on similar particles of similar systems moving along similar paths for a certain period of time);
- physical quantities (can be observed at two similar points of the model and the sample, for which the ratio of physical quantities will be constant);
- initial and boundary conditions (can be observed if the previous three similarities are observed).
The similarity invariant (usually denoted by idem in calculations and means invariantly or “the same”) is an expression of quantities in relative units (that is, the ratio of similar values within the same system).
In the event that the invariant contains relations of homogeneous quantities, it is called a simplex, and if it is heterogeneous, then the similarity criterion (all the properties of invariants are inherent in them).
Laws and rules of the theory of similarity
In science, all processes are governed by axioms and theorems. The axiomatic component of the theory includes three rules:
- the value h of the quantity H is the same as the ratio of the quantity to the unit of measurement [H];
- the physical quantity is independent of the choice of unit of measurement;
- The mathematical description of the phenomenon is not subject to a specific choice of units of measure.
Basic postulates
Using theorems, the following theory rules are described:
- Newton-Bertrand theorem: for all similar processes, all the similarity criteria under study are pairwise equal to each other (π 1 * = π 1 **; π 2 * = π 2 **, etc.). The ratio of the criteria of the two systems (model and sample) is always 1.
- Buckingham-Federman theorem: similarity criteria are related using the similarity equation, which is represented by a dimensionless solution (integral) and is called a criterion equation.
- Kirinchen-Guchman theorem: for the similarity of two processes, their qualitative equivalence and pairwise equivalence of the determining similarity criteria are necessary.
- Theorem π (sometimes referred to as Buckingham or yours): the relationship between h quantities, which are measured using m units, is represented as the ratio h - m by dimensionless combinations of π 1 , ..., π hm of these h quantities.
The similarity criterion is complexes combined using the π – theorem. The type of criterion can be established by compiling a list of quantities (A 1 , ..., A n ) describing the process, and apply the theorem in question to the dependence F (a 1 , ..., a n ) = 0, which is a solution to the problem.
Similarity criteria and research methods
There is an opinion that the most accurately describing name of the theory of similarity should sound like a method of generalized variables, since it is one of the methods of generalization in science and experimental research. The main areas of influence of the theory are modeling methods and analogies. The use of the main criteria of similarity as a particular theory existed long before the introduction of this term (previously called coefficients or degrees). As an example, you can cite the trigonometric functions of all the angles of such triangles - they are dimensionless. They provide an example of geometric similarity. In mathematics, the most famous criterion is the Pi number (the ratio of the size of a circle and the diameter of a circle). Today, the theory of similarity is a widespread tool of scientific research, which is being qualitatively transformed.
Physical phenomena studied through similarity theory
In the modern world, it is difficult to imagine the study of the processes of hydrodynamics, heat transfer, mass transfer, aerodynamics, bypassing the theory of similarities. Criteria are derived for any phenomena. The main thing is that between their variables there was a dependence. The physical meaning of the similarity criteria is reflected in the record (formula) and its preceding calculations. Typically, criteria, like some laws, are named after famous scientists.
Heat transfer study
The criteria for thermal similarity consist of quantities that are able to describe the process of heat transfer and heat transfer. Four best-known criteria are:
- Reynolds similarity criterion (Re).
The following values are presented in the formula:
- c is the velocity of the heat carrier;
- l is a geometric parameter (size);
- v is the kinematic viscosity coefficient
Using the criterion, the dependence of inertial forces and viscosity is established.
It includes such components:
- α - is the heat transfer coefficient;
- l is a geometric parameter (size);
- λ - is the coefficient of thermal conductivity.
This criterion describes the relationship between the intensity of heat transfer and the conductivity of the coolant.
The following values are presented in the formula:
- v is the kinematic viscosity coefficient;
- α - is the coefficient of thermal diffusivity.
This criterion describes the ratio of temperature and velocity fields in a stream.
The formula is composed using the following variables:
- g - indicates the acceleration of gravity;
- β - is the coefficient of volume expansion of the coolant;
- ∆T - indicates the temperature difference between the coolant and the conductor.
This criterion describes the ratio of two molecular friction forces and lifting force (occurs due to different fluid densities).
Criteria for the similarity of heat transfer in a free convention are usually called the Nusselt, Grashof and Prandtl criteria, and in a forced convention, Peclet, Nusselt, Reynolds and Prandtl.
The study of hydrodynamics
The criteria for hydrodynamic similarity are presented by the following examples.
- Froud similarity criterion (Fr).
The following values are presented in the formula:
- υ - denotes the speed of matter at a distance from the object streamlined by it;
- l - describes the geometric (linear) parameters of the subject;
- g - indicates the acceleration of gravity.
This criterion describes the ratio of inertia and gravity in the flow of matter.
- Criterion for the similarity of Strouhal (St).
The formula contains the following variables:
- υ - means speed;
- l - denotes geometric (linear) parameters;
- T - indicates the time interval.
This criterion describes the transient motion of a substance.
- Mach similarity criterion (M).
The following values are presented in the formula:
- υ - denotes the speed of a substance at a particular point;
- c - denotes the speed of sound (in liquid) at a specific point.
This criterion of hydrodynamic similarity describes the dependence of the motion of a substance on its compressibility.
Briefly about the remaining criteria
The most common criteria for physical similarity are listed. Equally important are such as:
- Weber (We) - describes the dependence of surface tension forces.
- Archimedes (Ar) - describes the dependence of the lifting forces and inertia.
- Fourier (Fo) - describes the dependence of the rate of change of the temperature field, physical properties and body size.
- Pomerantseva (Po) - describes the ratio of the intensity of internal sources of heat and temperature field.
- Peclet (Pe) - describes the ratio of convective and molecular heat transfers in a stream.
- Hydrodynamic homochronism (Ho) - describes the dependence of the portable (convective) acceleration and acceleration at a given point.
- Euler (Eu) - describes the dependence of the forces of pressure and inertia in the flow.
- Galilee (Ga) - describes the relationship between the forces of viscosity and gravity in a stream.
Conclusion
Similarity criteria may consist of certain values, but may also be deduced from other criteria. And such a combination will also be a criterion. From the above examples it can be seen that the principle of similarity is indispensable in hydrodynamics, geometry, mechanics, significantly simplifying the research process in some cases. Achievements of modern science became possible largely due to the ability to simulate complex processes with great accuracy. Thanks to the theory of similarity, more than one scientific discovery was made, then awarded the Nobel Prize.