Ultra-chaotic property of Navier–Stokes turbulence

A chaotic system is called ultra-chaos when its statistics have sensitivity dependence on initial condition and/or other small disturbances. In this paper, using two-dimensional turbulent Kolmogorov flow as an example, we illustrate that tiny variation of initial condition of Navier–Stokes equations can lead to huge differences not only in spatiotemporal trajectory but also in flow symmetry and its statistics. Here, to avoid the influence of artificial numerical noise, we apply “clean numerical simulation” (CNS) proposed by Liao in 2009. Different from direct numerical simulation (DNS), CNS can guarantee that the numerical noise is reduced to such a desired low level that it is negligible in a time interval that is long enough for calculating statistics. This discovery highly suggests that the Navier–Stokes turbulence (i.e. turbulence governed by the Navier–Stokes equations) should be an ultra-chaos, say, small disturbances must be considered even from viewpoint of statistics. Some fundamental characteristics of turbulence model are discussed and suggested.