Clean numerical simulation (CNS) of three-dimensional turbulent Kolmogorov flow

Turbulence holds immense importance across various scientific and engineering disciplines. The direct numerical simulation (DNS) of turbulence proposed by Orszag in 1970 is a milestone in fluid mechanics, which began an era of numerical experiment for turbulence. Many researchers have reported that turbulence should be chaotic, since spatiotemporal trajectories are very sensitive to small disturbance. Thus, due to the famous butterfly-effect of chaos, unavoidable numerical noises of DNS might have great influence on spatiotemporal trajectories of turbulence. This is indeed true for a two-dimensional (2D) Kolmogorov turbulent flow, as currently revealed by a much more accurate algorithm than DNS, namely the ``clean numerical simulation'' (CNS). Different from DNS, CNS can greatly reduce both of truncation error and round-off error to any required small level so that numerical noise can be rigorously negligible throughout a time interval long enough for calculating statistics. However, In physics, 3D turbulent flow is more important than 2D turbulence. Thus, for the first time, we solve a 3D turbulent Kolmogorov flow by means of CNS in this paper, and compare our CNS result with that given by DNS in details. It is found that the spatial-temporal trajectories of the 3D Kolmogorov turbulent flow given by DNS are indeed badly polluted by numerical noise rather quickly, and besides the DNS result has significant deviations from the CNS benchmark solution not only in the spatial symmetry of flow field and the energy cascade but also even in statistics.