Article

Dual approach to averaged values of functions: Advanced formulas

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Abstract

Averaged values play major roles in the study of dynamic processes. The useof those values allows transforming varying processes to some constant characteristicsthat are much easier to be investigated. In order to extend the use of averaged valuesone may apply the dual approach which suggests a consideration of two different aspectsof a problem in question. This short communication proposes new advanced formulas foraveraged values of functions based on the dual conception.

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Chapter
The solute transport equation of one-dimensional (1D) or two-dimensional vertical (2DV) flow is normally constructed by the classic average method. These solute transport equations are integrated from the right to the left river bank; the average values received by this approach therefore do not generalize by means of dual approach. This paper presents the application of a dual approach to establish the 2DV solute transport equation. In particular, the concentration in a 2DV flow is obtained by twice integrals: (i) integration from the right river bank to the intermediate vertical surface layer between the right bank and the left bank, and then (ii) integration from the right bank to the left bank. From the two-dimensional horizontal (2DH) [7] and 2DV flow constructed by a dual approach, the researcher receives the 1D flow equation. The average concentration obtained from the dual approach is better than the classical approach, particularly, in the case of mixed solute transport, stratification, and etc. The basic equation obtained is based on the dual approach that describes the solute transport is more accurate than the classical method. In other words, it provides some flexible parameters to adjust based on the field or experimental data. A case study of solute transport (salinity transport) in Huong river system is illustrated.
Chapter
The two-dimensional horizontal flow model in the classical integration approach is integrated from the three-dimensional Navier-Stokes system of equations. Using the classical theory, the integral is taken directly from the bed to the free water surfaces. Consequently, the effects between the channel bed and free water surface, in the process of integration, was disappeared. However, with the proposed dual-process approach, the integral can be performed locally several times. The receiving equations thus allow to contain many physical phenomena which may be lost in the classical integral process. As a result, the derived model based on the proposed dual approach will be more complex and accurate than the classical one. In this paper, the authors perform twice integrals. The improved two-dimensional horizontal flow model was received from the dual approach which allows the calculation of flow parameters, which, having the unusual phenomena in the channel as solid objects, liquids containing other added ingredients, external forces, reversals, and so on.
Chapter
The classic average method is usually applied to describe the solute transport equation of one-dimensional horizontal flow or two-dimensional horizontal flow. The solute transport equation is totally integrated one time from the bed to the water surface; the average values received by classic average method do not generalize by means of dual approach. So, in this paper, a dual approach is applied to solve the solute transport equation of two-dimensional horizontal flow. The equation describing the depth average concentration is obtained by two times integration: The first time integral is from the bed to the intermediate surface lays between bed and water surface, and the second time integral is from the bed to the water surface. With the dual approach, the received depth average concentration is better, particularly, in the case of stratification, mixed solute, and so on. The received governing equation based on the dual approach describes more accurately the physical characteristic of the transport phenomena in nature. Moreover, it provides flexible parameter adjustment based on the experimental data. A case study of salinity transport in Huong river is illustrated.
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