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 m...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 a necessary condition but not sufficient.All of these findings might challenge some of our general beliefs in turbulence:for example,it might be wrong in physics to neglect the influences of small disturbances to NS equations.Our results suggest that,from physical point of view,it should be better to use the Landau-Lifshitz-Navier-Stokes(LLNS)equations,which consider the influence of unavoidable thermal fluctuations,instead of the NS equations,to model turbulent flows.展开更多
In this paper,a new concept,i.e.,ultra-chaos,is proposed for the first time.Unlike a normal-chaos,statistical properties such as the probability density functions(PDF)of an ultra-chaos are sensitive to tiny disturbanc...In this paper,a new concept,i.e.,ultra-chaos,is proposed for the first time.Unlike a normal-chaos,statistical properties such as the probability density functions(PDF)of an ultra-chaos are sensitive to tiny disturbances.We illustrate that ultra-chaos is widely existed and thus has general scientific meanings.It is found that statistical non-reproducibility is an inherent property of an ultra-chaos so that an ultra-chaos is at a higher-level of disorder than a normal-chaos.Thus,it is impossible in practice to replicate experimental/numerical results of an ultra-chaos even in statistical meanings,since random environmental noises always exist and are out of control.Thus,the ultra-chaos should be an insurmountable obstacle of reproducibility and replicability.Similar to Gödel’s incompleteness theorem,such kind of“incompleteness of reproducibility”reveals a limitation of our traditional scientific paradigm based on reproducible experiments,which can be traced back to Galileo.The ultra-chaos opens a new door and possibility to study chaos theory,turbulence theory,computational fluid dynamics(CFD),the statistical significance,reproducibility crisis,and so on.展开更多
文摘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 a necessary condition but not sufficient.All of these findings might challenge some of our general beliefs in turbulence:for example,it might be wrong in physics to neglect the influences of small disturbances to NS equations.Our results suggest that,from physical point of view,it should be better to use the Landau-Lifshitz-Navier-Stokes(LLNS)equations,which consider the influence of unavoidable thermal fluctuations,instead of the NS equations,to model turbulent flows.
基金supported by the National Natural Science Foundation of China(No.91752104).
文摘In this paper,a new concept,i.e.,ultra-chaos,is proposed for the first time.Unlike a normal-chaos,statistical properties such as the probability density functions(PDF)of an ultra-chaos are sensitive to tiny disturbances.We illustrate that ultra-chaos is widely existed and thus has general scientific meanings.It is found that statistical non-reproducibility is an inherent property of an ultra-chaos so that an ultra-chaos is at a higher-level of disorder than a normal-chaos.Thus,it is impossible in practice to replicate experimental/numerical results of an ultra-chaos even in statistical meanings,since random environmental noises always exist and are out of control.Thus,the ultra-chaos should be an insurmountable obstacle of reproducibility and replicability.Similar to Gödel’s incompleteness theorem,such kind of“incompleteness of reproducibility”reveals a limitation of our traditional scientific paradigm based on reproducible experiments,which can be traced back to Galileo.The ultra-chaos opens a new door and possibility to study chaos theory,turbulence theory,computational fluid dynamics(CFD),the statistical significance,reproducibility crisis,and so on.
