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Abstract: The property of the velocity field and the cascade process of the fluid flow are key problems in turbulence research. This study presents the scaling property of the turbulent velocity field and a mathematical description of the cascade process, using the following methods: (1) a discussion of the general self-similarity and scaling invariance of fluid flow from the viewpoint of the physical mechanism of turbulent flow; (2) the development of the relationship between the scaling indices and the key parameters of the She and Leveque (SL) model in the inertial range; (3) an investigation of the basis of the fractal model and the multi-fractal model of turbulence; (4) a demonstration of the physical meaning of the flowing field scaling that is related to the real flowing vortex. The results illustrate that the SL model could be regarded as an approximate mathematical solution of Navier-Stokes (N-S) equations, and that the phenomena of normal scaling and anomalous scaling is the result of the mutual interactions among the physical factors of nonlinearity, dissipation, and dispersion. Finally, a simple turbulent movement conceptional description model is developed to show the local properties and the instantaneous properties of turbulence.

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