Is a direct numerical simulation(DNS)of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct?  

作　　者：Shijie Qin Yu Yang Yongxiang Huang Xinyu Mei Lipo Wang Shijun Liao 

机构地区：[1]School of Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240,China [2]State Key Laboratory of Marine Environmental Science,College of Ocean and Earth Sciences,Xiamen University,Xiamen 361102,China [3]UM-SJTU Joint Institute,Shanghai Jiao Tong University,Shanghai 200240,China [4]State Key Laboratory of Ocean Engineering,Shanghai 200240,China

出　　处：《Journal of Ocean Engineering and Science》2024年第3期293-310,共18页海洋工程与科学（英文）

摘　　要：Turbulence is strongly associated with the vast majority of fluid flows in nature and industry.Traditionally,results given by the direct numerical simulation(DNS)of Navier-Stokes(NS)equations that relate to a famous millennium problem are widely regarded as‘reliable’benchmark solutions of turbulence,as long as grid spacing is fine enough(i.e.less than the minimum Kolmogorov scale)and time-step is small enough,say,satisfying the Courant-Friedrichs-Lewy condition(Courant number<1).Is this really true?In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons:one is the traditional DNS,the other is the‘clean numerical simulation’(CNS).In theory,the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric.However,it is found that numerical noises of the DNS are quickly enlarged to the same level as the‘true’physical solution,which finally destroy the spatial symmetry of the flow field.In other words,the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly.On the contrary,the numerical noise of the CNS is much smaller than the‘true’physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the‘true’physical solution and thus can remain symmetric,which can be used as a benchmark solution for comparison.Besides,it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations,not only quantitatively(even in statistics)but also qualitatively(such as spatial symmetry of flow).This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations:it is only

关 键 词：Kolmogorov flow Statistical stability Computational reliability Clean numerical simulation Artificial numerical noises 

分 类 号：O17[理学—数学]