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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2404.10007 |
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| _version_ | 1866917679067037696 |
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| author | Qin, Shejie Yang, Yu Huang, Yongxiang Mei, Xinyu Wang, Lipo Liao, Shijun |
| author_facet | Qin, Shejie Yang, Yu Huang, Yongxiang Mei, Xinyu Wang, Lipo Liao, Shijun |
| contents | Traditionally, results given by the direct numerical simulation (DNS) of Navier-Stokes equations 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. Is this really true? In this paper a two-dimensional sustained turbulent Kolmogorov flow is investigated numerically by the two numerical methods with detailed comparisons: one is the traditional `direct numerical simulation' (DNS), the other is the `clean numerical simulation' (CNS). The results given by DNS are a kind of mixture of the false numerical noise and the true physical solution, which however are mostly at the same order of magnitude due to the butterfly-effect of chaos. On the contrary, the false numerical noise of the results given by CNS is much smaller than the true physical solution of turbulence in a long enough interval of time so that a CNS result is very close to the true physical solution and thus can be used as a benchmark solution. 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, not only quantitatively (even in statistics) but also qualitatively (such as symmetry of flow). Thus, fine enough spatial grid spacing with small enough time-step alone cannot guarantee the validity of the DNS: it is only a necessary condition but not sufficient. This finding might challenge some assumptions in investigation of turbulence. So, DNS results of a few sustained turbulent flows might have huge deviations on both of small and large scales from the true solution of Navier-Stokes equations even in statistics. Hopefully, CNS as a new tool to investigate turbulent flows more accurately than DNS could bring us some new discoveries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Is a direct numerical simulation (DNS) of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct? Qin, Shejie Yang, Yu Huang, Yongxiang Mei, Xinyu Wang, Lipo Liao, Shijun Fluid Dynamics Traditionally, results given by the direct numerical simulation (DNS) of Navier-Stokes equations 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. Is this really true? In this paper a two-dimensional sustained turbulent Kolmogorov flow is investigated numerically by the two numerical methods with detailed comparisons: one is the traditional `direct numerical simulation' (DNS), the other is the `clean numerical simulation' (CNS). The results given by DNS are a kind of mixture of the false numerical noise and the true physical solution, which however are mostly at the same order of magnitude due to the butterfly-effect of chaos. On the contrary, the false numerical noise of the results given by CNS is much smaller than the true physical solution of turbulence in a long enough interval of time so that a CNS result is very close to the true physical solution and thus can be used as a benchmark solution. 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, not only quantitatively (even in statistics) but also qualitatively (such as symmetry of flow). Thus, fine enough spatial grid spacing with small enough time-step alone cannot guarantee the validity of the DNS: it is only a necessary condition but not sufficient. This finding might challenge some assumptions in investigation of turbulence. So, DNS results of a few sustained turbulent flows might have huge deviations on both of small and large scales from the true solution of Navier-Stokes equations even in statistics. Hopefully, CNS as a new tool to investigate turbulent flows more accurately than DNS could bring us some new discoveries. |
| title | Is a direct numerical simulation (DNS) of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct? |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2404.10007 |