Multi-dimensional complex phase transition model by catastrophe theory developing Kolmogorov turbulence theory

Abstract

In this paper, a novel multi-dimensional complex non-equilibrium phase transition model is put forward to describe quantitatively the physical development process of turbulence and develop the Kolmogorov turbulence theory from the catastrophe theory, in which the well-known −5/3 power law is only a special case in this paper proving the accuracy of our methods. Catastrophe theory is a highly generalized mathematical tool that summarizes the laws of non-equilibrium phase transition. Every control variable in catastrophe theory could be skillfully expanded into multi-parameter multiplication with different indices and the relationship among these characteristic indices can be determined by dimensionless analysis. Thus, the state variables can be expressed quantitatively with multiple parameters, and the multi-dimensional non-equilibrium phase transition theory is established. As an example, by adopting the folding catastrophe model, we strictly derive out the quantitative relationship between energy and wave number with respect to a new scale index [Formula: see text] to quantitative study the whole process of the laminar flow to turbulence, in which [Formula: see text] varies from [Formula: see text] to [Formula: see text] corresponding to energy containing range and [Formula: see text] to energy containing scale where [Formula: see text] power law is deduced, and at [Formula: see text] the [Formula: see text] law of Kolmogorov turbulence theory is obtained, and fully developed turbulence phase starts at [Formula: see text] giving [Formula: see text] law. Furthermore, this theory presented is verified by our wind tunnel experiments. This novel non-equilibrium phase transition methods cannot only provide a new insight into the turbulence model, but also be applied to other non-equilibrium phase transitions.

Quantitative analysis method for laminar-turbulence transition process of mechanical seal by a catastrophic model

“…When a ¼ �2/3, E ¼ 0, indicating the turbulent kinetic energy is completely dissipated into viscous heat, and the flow regime is fully developed turbulence. 24 Turbulence transition factor based on the quantitative analysis method…”

“…According to Thom's theory, 21 any system jumps from one phase to another, and the whole process can be described by a smooth surface (catastrophe manifold). Liang and Zhou [22][23][24][25] proposed a multi-dimensional complex phase transition model to analyze the process of turbulence transition, and the theoretical results are verified by experiments. In general, the phase transition model provides a new understanding of turbulence, which has the potential to deal with turbulent lubrication problems.…”

Quantitative analysis method for extraordinary optical transmission of thin microcavity by the catastrophic theory

“…Any system from one phase to another can be described by 7 types of elementary catastrophe models with the corresponding different potential functions based on Thom's theory. In some previous papers, Liang et al [13][14][15][16] proposed a general non-equilibrium phase transition process to analyze the turbulence transition, which has been applied to explain the sound absorption of porous materials. Wu [17] introduced a novel kind of equations linking the quantum dynamics and the classical wave motions.…”