Derivation of the similarity indicators and numerical integration of the similarity criterions, which are interval-based in system center-radius
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Abstract
In frame of this work only physically realizable systems will be considered, namely such systems, which may be presented as unity of physical elements, structurally adjusted each to another and interacted with external medium. The physical modeling for such systems, that it is based on theory of similarities on starting stage of projecting, works as useful source of knowledge about their properties. It is known that theory of similarities studies the conditions for similarities of physical processes. Two physical processes are called similar, when they obey the same physical laws. Each quantitative characteristic for one of them is obtained from another by means of multiplication on the constant value. This value is called the constant of similarity, and it is the same for all uniform values, which are involved in process under investigation. It is a rule in theory of similarities, that two phenomena are similar if and only if, when they are qualitatively similar and have equal values for some dimensionless parameters, which are called criterions of similarities. It is also rule in theory of similarities, that dimension of any physical value may be only the multiplication of values, which are in powers, and taken as basis values. Dimensions for both parts of equality, that presents same physical law, must be the same. Dimensionless multiplications of different powers are called as criteria of similarity. In this work some features of calculation processes, connected with definitions of numerical values for criterions of similarities, will be presented. Due to the fact, that numerical values of the similarity criterions are obtained from the experimental results, these results are defined with some error, which influences the decision about similarities of the systems under comparison. To define this influence the procedure of calculation for criterions and indicators of similarities with the interval numbers is used, defined in the system center-radius. In this work the algorithm of consecutive multiplication in the interval-based view of factors, each of those presents a power function, and the algorithm for derivation of the similarity indicator are presented. The interval evaluation for the similarity indicators is obtained, depending on value of the definition intervals for corresponding values. Algorithms for numerical integration with trapezoidal rule with regular and arbitrary distribution of nodes are presented. It is shown, that for relatively simple, in respect of calculation, criteria of similarities, the neglect by experimental errors may lead to misleading conclusions about quality of the models proposed. Numerical example with evaluation of the quality for proposed physical model is considered, comparison was done for Reynolds criterion.
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