Is a direct numerical simulation (DNS) of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct?
arXiv:2404.10007 · doi:10.1016/j.joes.2024.04.002
Abstract
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.
27 pages, 18 figures
References in corpus (23)
- On the mathematically reliable long-term simulation of chaos of Lorenz equation in the interval [0,10000]
- On the reliability of computed chaotic solutions of nonlinear differential equations
- Chaos and predictability of homogeneous-isotropic turbulence
- More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits
- The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum
- On the risks of using double precision in numerical simulations of spatio-temporal chaos
- Chaotic motion of three-body problem : an origin of macroscopic randomness of the universe
- Physical limit of prediction for chaotic motion of three-body problem
- On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems
- Emergent Bistability and Switching in a Nonequilibrium Crystal
- Three-body problem -- from Newton to supercomputer plus machine learning
- Predictability of the energy cascade in 2D turbulence
- Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence
- Ultra-chaos: an insurmountable objective obstacle of reproducibility and replicability
- On the origin of intrinsic randomness of Rayleigh-Benard turbulence
- Accurate predictions of chaotic motion of a free fall disk
- A kind of Lagrangian chaotic property of the Arnold-Beltrami-Childress flow
- On the inherent self-excited macroscopic randomness of chaotic three-body system
- Superfast amplification and superfast nonlinear saturation of perturbations as the mechanism of turbulence
- Intrinsic flow structure and multifractality in two-dimensional bacterial turbulence
- Energy-Enstrophy Stability of beta-plane Kolmogorov Flow with Drag
- Intermittent "Turbulence" in a Many-body System
- A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos
Cited by in corpus (4)
- Noise-expansion cascade: an origin of randomness of turbulence
- Cascades and Kolmogorov's lognormal scaling in two-dimensional bacterial turbulence
- Clean numerical simulation (CNS) of three-dimensional turbulent Kolmogorov flow
- Physical significance of artificial numerical noise in direct numerical simulation of turbulence