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P. Liu
Similarity theory, from the phenomenon of occurrence and development
of internal regularity (mathematical equations) and the external conditions
(definite condition), based on the main premise that the mathematical equations to inherent homogeneity on the dimension and the correctness of the
mathematical equations is not affected by the influence of the measurement
system of units, comes to a conclusion through the mathematical deduction methods such as linear transformation. Similarity theory is characterized
by a combination of high abstraction and application. The similarity theory
provides guidance for simulation test, determines the reduction or enlargement of the model scale, increases or decreases the parameters, and changes
the medium properties, etc. The purpose is to find out the internal regularity
of the model with the lowest cost and the shortest operating cycle. Although
the similarity theory itself is a more rigorous mathematical logic system, once
it is applied in practice, it cannot be very accurate in many cases, because
the problems dealt with by similarity theory are usually extremely complex.
Three theorems in similarity theory are based on the following:
(1) Definition of similar phenomena;
(2) The relationship between the physical quantities involved in any physical phenomenon is subject to various objective laws, and they cannot be
changed arbitrarily;
(3) The size of each physical quantity involved in the physical phenomenon
exists objectively, regardless of the measurement unit used.
If the prototype and the model are proportional to each physical quantity
corresponding to each point and time, the two systems are similar. The similarity number (called similarity scale, similarity coefficient, etc.) is the ratio of
the physical quantity of the prototype to the physical quantity of the corresponding model. There are mainly geometric similarities, similar motions,
and similar dynamics. Traditionally, length, time, and mass are taken as basic
physical quantities, and the similarity number relationship between prototypes and models is called the similarity index. If the two are similar, the
similarity indicator is 1. The dimensionless quantity group derived from
similar indicators is called the similarity criterion.
If two flows are similar, they are similar as a single-valued condition, and
the inertial force acting on the two systems corresponds to the proportion of
other forces. In a fluid mechanics problem, if the forces acting on a particle
satisfy the dynamic similarity, the ratios between the following forces must be
equalized, including the ratio of inertial force to pressure (or pressure difference), the ratio of inertial force to gravity, the ratio of inertial force to viscous
P. Liu
Similarity theory, from the phenomenon of occurrence and development
of internal regularity (mathematical equations) and the external conditions
(definite condition), based on the main premise that the mathematical equations to inherent homogeneity on the dimension and the correctness of the
mathematical equations is not affected by the influence of the measurement
system of units, comes to a conclusion through the mathematical deduction methods such as linear transformation. Similarity theory is characterized
by a combination of high abstraction and application. The similarity theory
provides guidance for simulation test, determines the reduction or enlargement of the model scale, increases or decreases the parameters, and changes
the medium properties, etc. The purpose is to find out the internal regularity
of the model with the lowest cost and the shortest operating cycle. Although
the similarity theory itself is a more rigorous mathematical logic system, once
it is applied in practice, it cannot be very accurate in many cases, because
the problems dealt with by similarity theory are usually extremely complex.
Three theorems in similarity theory are based on the following:
(1) Definition of similar phenomena;
(2) The relationship between the physical quantities involved in any physical phenomenon is subject to various objective laws, and they cannot be
changed arbitrarily;
(3) The size of each physical quantity involved in the physical phenomenon
exists objectively, regardless of the measurement unit used.
If the prototype and the model are proportional to each physical quantity
corresponding to each point and time, the two systems are similar. The similarity number (called similarity scale, similarity coefficient, etc.) is the ratio of
the physical quantity of the prototype to the physical quantity of the corresponding model. There are mainly geometric similarities, similar motions,
and similar dynamics. Traditionally, length, time, and mass are taken as basic
physical quantities, and the similarity number relationship between prototypes and models is called the similarity index. If the two are similar, the
similarity indicator is 1. The dimensionless quantity group derived from
similar indicators is called the similarity criterion.
If two flows are similar, they are similar as a single-valued condition, and
the inertial force acting on the two systems corresponds to the proportion of
other forces. In a fluid mechanics problem, if the forces acting on a particle
satisfy the dynamic similarity, the ratios between the following forces must be
equalized, including the ratio of inertial force to pressure (or pressure difference), the ratio of inertial force to gravity, the ratio of inertial force to viscous