A paradox of the Navier-Stokes turbulence

The Navier-Stokes (NS) equations as a turbulence model have been widely applied in lots of fields. The NS equations contain such a fundamental assumption that all small physical/artificial disturbances could be neglected. Is this assumption correct? In this paper a two-dimensional Rayleigh-B\'{e}nard convection governed by the NS equations is predicted by traditional direct numerical simulation (DNS) using double precision arithmetic and a range of different time-steps. It is found that the final flow type tends either to vortical flow or zonal flow, whose statistics are completely different. Notably, these two flow types frequently alternate as the time-step is reduced to a very small value, suggesting that the time-step corresponding to each turbulent flow type should be densely distributed. Thus, stochastic numerical noise exerts a huge influence on the final flow type and statistics of numerically simulated NS turbulence because the time-step has a close relationship with numerical noise. This clearly indicates that small disturbances have significant influences on the NS turbulence, which therefore should not be neglected. This leads to a logical paradox for the NS turbulence, which is a great challenge for us, although a paradox often leads to some significant breakthroughs.