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

摘要

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 alpha to quantitative study the whole process of the laminar flow to turbulence, in which alpha varies from -2 to -6/5 corresponding to energy containing range and alpha = -9/5 to energy containing scale where -10/9 power law is deduced, and at alpha = -6/5 the -5/3 law of Kolmogorov turbulence theory is obtained, and fully developed turbulence phase starts at alpha = -2/3 giving -3 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.