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, economics, the service sector, and sociology. If the scientific criteria (these are certain conditions and requirements that must be observed) are presented in an abstract form for the entire scientific community, then 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 speci alties long before they got their name. The most trivial similarity criterion can be called a percentage of the whole. Such an operation was done by everyone without any problems and difficulties. And the efficiency factor, which reflects the dependence of the machine's power consumption and output power, has always been a similarity criterion and therefore has not been perceived as something vaguely sky-high.
Foundations of the theory
Physical similarity of phenomena, whether nature or 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 using modeling has proven itself well. Naturally, when planning and conducting an experiment, the energy-dynamic system of quantities and concepts (ESVP) is a support. It should be noted that the system of quantities and the system of units (SI) are not equivalent. In practice, ESWP exists objectively in the surrounding 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) with the help of international conferences.
Conceptual apparatus of similarities
Theory of similarity - concepts and rules, the purpose of which 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 terminological dictionary is based on such concepts as homogeneous, eponymous and dimensionless quantities, similarity constant. To facilitate understanding of the essence of the theory, the meaning of the listed terms should be considered.
- 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 basicquantities; speed has the dimension of length divided by time).
- Similar - 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.
Constant - a dimensionless quantity, in which the base value is a quantity with a fixed size (for example, an elementary electric charge). It allows the transition from a model to a natural system.
Main types of similarity
Any physical quantities can be similar. It is customary to distinguish four types:
- geometric (observed when the ratios of similar linear dimensions of the sample and model are equal);
- temporal (observed on similar particles of similar systems moving along similar paths over a certain period of time);
- physical quantities (can be observed at two similar points of the model and sample, for which the ratio of physical quantities will be constant);
- initial and boundary conditions (can be observed if the three previous similarities are observed).
A similarity invariant (usually denoted idem in calculations and means invariant or "same") is an expression of quantities in relative units (i.e. the ratio of similar quantities within one system).
If the invariant contains ratios of homogeneous quantities, it is called a simplex, and if heterogeneous quantities, then the similarity criterion (they haveall properties of invariants).
Laws and rules of similarity theory
In science, all processes are regulated by axioms and theorems. The axiomatic component of the theory includes three rules:
- the value h of the value H is the same as the ratio of the value to the unit of its measurement [H];
- a physical quantity is independent of the choice of its unit;
- mathematical description of the phenomenon is not subject to the specific choice of units.
Basic postulates
The following rules of the theory are described using theorems:
- Newton-Bertrand theorem: for all similar processes, all similarity criteria under study are pairwise equal to each other (π1=π1; π2=π2 etc.). The ratio of the criteria of two systems (model and sample) is always equal to 1.
- Buckingham-Federman theorem: similarity criteria are related using a similarity equation, which is represented by a dimensionless solution (integral) and is called a criterion equation.
- Kirinchen-Gukhman theorem: for the similarity of two processes, their qualitative equivalence and pairwise equivalence of the defining similarity criteria are necessary.
- Theorem π (sometimes called Buckingham or Vash): the relationship between h quantities, which are measured using m units of measurement, is represented as a ratio h - m by dimensionless combinations π1, …, πh-m of these h values.
The similarity criterion is the complexes united by the π-theorem. The type of criterion can be established by compiling a list of quantities (A1, …, A) describing the process, and applying the considered theorem to the dependence F(a 1, …, a )=0, which is the solution to the problem.
Similarity criteria and research methods
There is an opinion that the most accurate name of the similarity theory should sound like the method of generalized variables, since it is one of the methods of generalization in science and experimental research. The main spheres of influence of the theory are the methods of modeling and analogy. The use of basic similarity criteria as a private theory existed long before the introduction of this term (previously called coefficients or degrees). An example is the trigonometric functions of all angles of similar triangles - they are dimensionless. They represent an example of geometric similarity. In mathematics, the most famous criterion is the number Pi (the ratio of the size of a circle and the diameter of a circle). To date, the theory of similarity is a widely used 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 there was a dependence between their variables. The physical meaning of the similarity criteria is reflected in the entry (formula) and the precedingcalculations. Typically, the criteria, like some laws, are named after famous scientists.
Study of heat transfer
Thermal similarity criteria consist of quantities that are able to describe the process of heat transfer and heat transfer. The four most famous criteria are:
Reynolds similarity test (Re)
The formula contains the following quantities:
- s – heat carrier speed;
- l – geometric parameter (size);
- v – coefficient of kinematic viscosity
With the help of the criterion, the dependence of the forces of inertia and viscosity is established.
Nusselt test (Nu)
It includes the following components:
- α is the heat transfer coefficient;
- l – geometric parameter (size);
- λ is the thermal conductivity coefficient.
This criterion describes the relationship between the intensity of heat transfer and the conductivity of the coolant.
Prandtl criterion (Pr)
The formula contains the following quantities:
- v is the kinematic viscosity coefficient;
- α is the coefficient of thermal diffusivity.
This criterion describes the ratio of temperature and velocity fields in the flow.
Grashof criterion (Gr)
The formula is made using the following variables:
- g - indicates the acceleration of gravity;
- β - is the coefficient of volumetric expansion of the coolant;
- ∆T – denotes the differencetemperatures between the coolant and the conductor.
This criterion describes the ratio of the two forces of molecular friction and lift (due to the different density of the liquid).
Nusselt, Grashof and Prandtl criteria are usually called the heat transfer similarity criteria under free convention, and Peclet, Nusselt, Reynolds and Prandtl criteria under forced convention.
Study of hydrodynamics
The hydrodynamic similarity criteria are presented by the following examples.
Froude similarity test (Fr)
The formula contains the following quantities:
- υ - denotes the speed of matter at a distance from the object flowing around it;
- l - describes the geometric (linear) parameters of the subject;
- g - stands for acceleration due to gravity.
This criterion describes the ratio of the forces of inertia and gravity in the flow of matter.
Strouhal similarity test (St)
The formula contains the following variables:
- υ – denotes speed;
- l - denotes geometric (linear) parameters;
- T - indicates a time interval.
This criterion describes unsteady motions of matter.
Mach similarity criterion (M)
The formula contains the following quantities:
- υ - denotes the speed of matter at a particular point;
- s - denotes the speed of sound (in liquid) at a particular point.
This hydrodynamic similarity criterion describesthe dependence of the movement of matter on its compressibility.
Remaining criteria in brief
The most common physical similarity criteria are listed. No less important are such as:
- Weber (We) – describes the dependence of surface tension forces.
- Archimedes (Ar) - describes the relationship between lift and inertia.
- Fourier (Fo) - describes the dependence of the rate of change of the temperature field, physical properties and dimensions of the body.
- Pomerantsev (Po) - describes the ratio of the intensity of internal heat sources and the temperature field.
- Pekle (Pe) – describes the ratio of convective and molecular heat transfer in a flow.
- Hydrodynamic homochronism (Ho) – describes the dependence of the translational (convective) acceleration and acceleration at a given point.
- Euler (Eu) - describes the dependence of the forces of pressure and inertia in the flow.
- Galilean (Ga) - describes the ratio of the forces of viscosity and gravity in the flow.
Conclusion
Similarity criteria can consist of certain values, but can also be derived 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, and mechanics, greatly simplifying the research process in some cases. The achievements of modern science have become possible largely due to the ability to model complex processes with great accuracy. Thanks to the theory of similarity, more than one scientific discovery was made, then awarded the Nobel Prize.
