Transition to chaos in lid–driven square cavity flow* *Project supported by theNational Natural Science Foundation of China (Grant No. 12162001), the Natural Science Foundation of Ningxia (Grant No. 2019AAC03129), and the Construction Project of First-ClassDisciplines in Ningxia Higher Education (Grant No. NXYLXK2017B09).

Abstract

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid–driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.