An enhanced catastrophe theory that explains turbulence

Turbulence is a non-equilibrium phase transition involving dividing vortices that, as of recently, has not received satisfactory quantitative explanation. In 1941, Andre Kolmogorov quantitatively explained turbulence for fluid systems with a large Reynolds number but not for those with a smaller one. Moreover, Kolmogorov’s theory did not explain the system’s process that led to turbulence. Recently, researchers from two labs in China combined catastrophe theory with dimensionless analysis to produce a theory that quantitatively describes turbulence across phase conditions. They report on their research in Physics of Fluids.

Within catastrophe theory the occurrence of discontinuities or singularities is determined by the theory’s equation control variables, explains lead author Xiao Liang. Xiao and colleagues expanded the applicability of these control variables for the phase states of turbulence by introducing a novel dimensionless analysis. In particular, the dimensionless analysis incorporates key physical relationships and tracks the dimensional features undergoing change. The dimensionless analysis addition yields control variables that represent the consecutively dividing vortices characteristic of turbulence. The authors thereby obtained equations that demonstrated the energy spectral densities for the full dynamics of turbulence.

After deriving the quantitative theory, the researchers applied their equations to two classic phase transitions: the turbulent phase transition and the bottleneck particle flow. They found that their combination of catastrophe and dimensionless analyses produced results that comported with published experimental data and also for incidents of turbulence that Kolmogorov’s theory already explained.

Xiao says this method quantitatively demonstrates how a turbulent system transitions from phase to phase. He adds this method can be used in many areas of physics that have phase transitions, and with engineering focuses, such as aerodynamics, in which turbulence is a central issue.

Source: “Quantitative analysis of non-equilibrium phase transition process by the catastrophe theory,” by Xiao Liang, Jiu Hui Wu, and H. B. Zhong, Physics of Fluids (2017). The article can be accessed at https://doi.org/10.1063/1.4998429 .