Superfast amplification and superfast nonlinear saturation of perturbations as a mechanism of turbulence

Self Cite

We study the properties of homogeneous and isotropic turbulence in higher spatial dimensions through the lens of chaos and predictability using numerical simulations. We employ both direct numerical simulations and numerical calculations of the eddy damped quasi-normal Markovian closure approximation. Our closure results show a remarkable transition to a non-chaotic regime above the critical dimension, $d_c$ , which is found to be approximately 5.88. We relate these results to the properties of the energy cascade as a function of spatial dimension in the context of the idea of a critical dimension for turbulence where Kolmogorov's 1941 theory becomes exact.

Section: Error Decay and Critical Dimension In D >mentioning

confidence: 72%

Section: Chaos and Predictability In Turbulencementioning

confidence: 99%

Critical transition to a non-chaotic regime in isotropic turbulence

Clark

Armua

et al. 2021

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“…Moreover, Lorenz (1989Lorenz ( , 2006 further discovered that the trajectories of chaotic dynamical systems have sensitive dependence not only on initial conditions (SDIC) but also on numerical algorithms (SDNA), because numerical noise, arising from truncation error and round-off error, is unavoidable for all numerical algorithms. All of these phenomena are based on the exponential increase of noise (or small disturbances), especially for the long-duration numerical simulation of a chaotic dynamical system (Ruelle & Takens 1971;Li et al 2020). Naturally, the non-replicability/unreliability of chaotic trajectories has certainly led to heated debate about the credibility of numerical simulations of chaotic systems, with Teixeira, Reynolds & Judd (2007) reaching the pessimistic conclusion that 'for chaotic systems, numerical convergence cannot be guaranteed forever'.…”

Section: Introductionmentioning

confidence: 99%

“…All of these phenomena are based on the exponential increase of noise (or small disturbances), especially for the long-duration numerical simulation of a chaotic dynamical system (Ruelle & Takens 1971; Li et al. 2020). Naturally, the non-replicability/unreliability of chaotic trajectories has certainly led to heated debate about the credibility of numerical simulations of chaotic systems, with Teixeira, Reynolds & Judd (2007) reaching the pessimistic conclusion that ‘for chaotic systems, numerical convergence cannot be guaranteed forever’.…”

Section: Introductionmentioning

confidence: 99%

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

Qin

Liao

2023

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.

“…In fact, the question may be even more widely open as a superfast uncertainty growth may have been observed at very early times in some DNS results (Li et al. 2020). Such superfast growth is not ruled out by the rigorous constraint on the uncertainty growth derived from the Navier–Stokes equation by Li (2014): where and are the coefficients depending on the perturbations.…”

Section: Introductionmentioning

confidence: 99%

The production of uncertainty in three-dimensional Navier–Stokes turbulence

Ge,

Rolland,

Vassilicos

2023

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier–Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is, in turn, followed by a linear growth in the one DNS case where the Navier–Stokes forcing also produces uncertainty.