arXiv subjects

Shijie Qin

Publications and source records attributed to Shijie Qin.

14 recordsLinked to original sources

arXiv2026

A paradox of the Navier-Stokes turbulence

The Navier-Stokes (NS) equations as a turbulence model have been widely applied in lots of fields. The NS equations contain such a fundamental assumption that all small physical/artificial disturbances could be neglected. Is this assumption correct? In this paper a two-dimensional Rayleigh-Bénard convection governed by the NS equations is predicted by traditional direct numerical simulation (DNS) using double precision arithmetic and a range of different time-steps. It is found that the final flow type tends either to vortical flow or zonal flow, whose statistics are completely different. Notably, these two flow types frequently alternate as the time-step is reduced to a very small value, suggesting that the time-step corresponding to each turbulent flow type should be densely distributed. Thus, stochastic numerical noise exerts a huge influence on the final flow type and statistics of numerically simulated NS turbulence because the time-step has a close relationship with numerical noise. This clearly indicates that small disturbances have significant influences on the NS turbulence, which therefore should not be neglected. This leads to a logical paradox for the NS turbulence, which is a great challenge for us, although a paradox often leads to some significant breakthroughs.

physics.flu-dyn↗

arXiv2026

Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

Using clean numerical simulation (CNS) which can give very accurate spatiotemporal trajectory of Navier-Stokes turbulence in a finite but long enough interval of time, we give some numerical evidences that the Navier-Stokes equations admit distinct global solutions from almost the same initial conditions whose difference is very small, i.e. even at the order $10^{-40}$ of magnitude. Hopefully these examples could provide some enlightenments for the uniqueness and existence of Navier-Stokes equations, which are related to one Millennium Prize Problem of Clay Institute.

math.AP↗

arXiv2025

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.

physics.flu-dyn↗

arXiv2025

Reply to the comments of McMullen et al. (arXiv:2510.04828)

McMullen et al. [1] comment that the numerical simulations that explicitly include random velocity fluctuations ``should exhibit a thermal-fluctuation-dominated range'' consistent with the literature, so that our results (J. Fluid Mech. 1008, R2, 2025) [2] ``contradict other results in the literature''. First of all, we would give an opposite example against this viewpoint: DNS results (that are badly polluted by numerical noises quickly, as mention in Section 2) implicitly include random numerical noises, but they also DO NOT exhibit a thermal-fluctuation-dominated range. In other words, DNS results in the literature qualitatively agree with ours at this point. In addition, we highly suggest that influences of numerical noises on statistics of turbulent flows given by ALL numerical approaches should be carefully checked, since numerical noises might have huge influences on statistics of chaotic systems (including turbulence), as pointed by Lorenz [3] in 2006. Detailed replies are given below.

physics.flu-dyn↗

arXiv2025

Ultra-chaotic property of Navier-Stokes turbulence

A chaotic system is called ultra-chaos when its statistics have sensitivity dependence on initial condition and/or other small disturbances. In this paper, using two-dimensional turbulent Kolmogorov flow as an example, we illustrate that tiny variation of initial condition of Navier-Stokes equations can lead to huge differences not only in spatiotemporal trajectory but also in flow symmetry and its statistics. Here, in order to avoid the influence of artificial numerical noise, we apply ``clean numerical simulation'' (CNS) which can guarantee that the numerical noise can be reduced to such a desired low level that they are negligible in a time interval long enough for calculating statistics. This discovery highly suggests that the Navier-Stokes turbulence (i.e. turbulence governed by the Navier-Stokes equations) might be an ultra-chaos, say, small disturbances must be considered even from viewpoint of statistics. This however leads to a paradox in logic, since small disturbances, which are unavoidable in practice, are unfortunately neglected by the Navier-Stokes turbulence. Some fundamental characteristics of turbulence model are discussed and suggested in general meanings.

nlin.CD↗

arXiv2025

Physical significance of artificial numerical noise in direct numerical simulation of turbulence

Using clean numerical simulation (CNS) in which artificial numerical noise is negligible over a finite, sufficiently long interval of time, we provide evidence, for the first time, that artificial numerical noise in direct numerical simulation (DNS) of turbulence is approximately equivalent to thermal fluctuation and/or stochastic environmental noise. This confers physical significance on the artificial numerical noise of DNS of the Navier-Stokes equations. As a result, DNS on a fine mesh should correspond to turbulence under small internal/external physical disturbance, whereas DNS on a sparse mesh corresponds to turbulent flow under large physical disturbance, respectively. The key point is that: all of them have physical meanings and so are correct in terms of their deterministic physics, even if their statistics are quite different. This is illustrated herein. Our paper provides a positive viewpoint regarding the presence of artificial numerical noise in DNS.

physics.flu-dyn↗

arXiv2025

Noise-expansion cascade: an origin of randomness of turbulence

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ``thought experiment'' via several clean numerical experiments based on the Navier-Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ``noise-expansion cascade'' whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier-Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro-level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. Besides, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. Hopefully, the ``noise-expansion cascade'' as a fundamental property of the NS equations could greatly deepen our understandings about turbulence, and besides is helpful for attacking the fourth millennium problem posed by Clay Mathematics Institute in 2000.

physics.flu-dyn↗

arXiv2023

A kind of Lagrangian chaotic property of the Arnold-Beltrami-Childress flow

Three-dimensional steady-state Arnold-Beltrami-Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier-Stokes (NS) equations with an external force per unit mass. It is well-known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ``butterfly-effect'', their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when $Re=50$ with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.

nlin.CD↗

arXiv2023

A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos

The background numerical noise $\varepsilon_{0} $ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $\varepsilon(t)$ grows exponentially, say, $\varepsilon(t) = \varepsilon_{0} \exp(κ\,t)$, where $κ>0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in double precision, since the background numerical noise $\varepsilon_{0}$ might stop decreasing because of the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $\varepsilon_{0}$ to any required tiny level. A lot of successful applications show the novelty and validity of the CNS. In this paper, we further propose some strategies to greatly increase the computational efficiency of the CNS algorithms for chaotic dynamical systems. It is highly suggested to keep a balance between truncation error and round-off error and besides to progressively enlarge the background numerical noise $\varepsilon_{0}$, since the exponentially increasing numerical noise $\varepsilon(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.

physics.comp-ph↗

arXiv2022

Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using the two-dimensional (2D) turbulent Rayleigh-Bénard (RB) convection as an example, we numerically solve the NS equations, separately, by means of a traditional algorithm with double precision (marked by RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by the RKwD is a mixture of the "true" physical solution and the "false" numerical noises that is random and can be regarded as a kind of artificial stochastic disturbances: unfortunately, the "true" physical solution is mostly at the same level as the "false" numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the "false" numerical noises are negligible compared with the "true" physical solution and thus the CNS solution can be used as a "clean" benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. Especially, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows: the shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter, however the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus, we provide a rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2D turbulent RB convection considered in this paper.

physics.flu-dyn↗

arXiv2021

Ultra-chaos: an insurmountable objective obstacle of reproducibility and replicability

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.

physics.gen-ph↗

arXiv2021

Influence of database noises to machine learning for spatiotemporal chaos

A new strategy, namely the "clean numerical simulation" (CNS), was proposed (J. Computational Physics, 418:109629, 2020) to gain reliable/convergent simulations (with negligible numerical noises) of spatiotemporal chaotic systems in a long enough interval of time, which provide us benchmark solution for comparison. Here we illustrate that machine learning (ML) can always give good enough fitting predictions of a spatiotemporal chaos by using, separately, two quite different training sets: one is the "clean database" given by the CNS with negligible numerical noises, the other is the "polluted database" given by the traditional algorithms in single/double precision with considerably large numerical noises. However, even in statistics, the ML predictions based on the "polluted database" are quite different from those based on the "clean database". It illustrates that the database noises have huge influences on ML predictions of some spatiotemporal chaos, even in statistics. Thus, we must use a "clean" database for machine learning of some spatiotemporal chaos. This surprising result might open a new door and possibility to study machine learning.

physics.comp-ph↗

arXiv2019

Internal flows of ventilated partial cavitation

Our study provides the first experimental investigation of the internal flows of ventilated partial cavitation (VPC) formed by air injection behind a backward-facing step. The experiments are conducted using flow visualization and planar particle image velocimetry (PIV) with fog particles for two different cavity regimes of VPC, i.e., open cavity (OC) and two-branch cavity (TBC), under various range of free stream velocity (U) and ventilation rates (Q). Our experiments reveal similar flow patterns for both OC and TBC, including forward flow region near the air-water interface, reverse flow region, near-cavitator vortex, and internal flow circulation vortex. However, OC internal flow exhibits highly unsteady internal flow features, while TBC internal flow shows laminar-like flow patterns with a Kelvin-Helmholtz instability developed at the interface between forward and reverse flow regions within the cavity. Internal flow patterns and the unsteadiness of OC resemble those of turbulent flow separation past a backward-facing step (BFS flow), suggesting a strong coupling of internal flow and turbulent external recirculation region for OC. Likewise, internal flow patterns of TBC resemble those of laminar BFS flow, with the presence of unsteadiness due to the strong velocity gradient across the forward-reverse flow interface. The variation of the internal flow upon changing U or Q is further employed to explain the cavity regime transition and the corresponding change of cavity geometry. Our study suggests that ventilation control can potentially stabilize the cavity in the TBC regime by delaying its internal flow regime transition from laminar-like to highly unsteady.

physics.flu-dyn↗

arXiv2017

Influence of round-off errors on the reliability of numerical simulations of chaotic dynamic systems

We illustrate that, like the truncation error, the round-off error has a significant influence on the reliability of numerical simulations of chaotic dynamic systems. Due to the butterfly-effect, all numerical approaches in double precision cannot give a reliable long-term simulation of chaotic dynamic systems. So, in order to avoid man-made uncertainty of numerical simulations of chaos, we had to greatly decrease both of the truncation and round-off error to a small enough level, plus a verification of solution reliability by means of an additional computation using even smaller truncation and round-off errors.

nlin.CD↗