[Paper Review] Short term unpredictability of high Reynolds number turbulence --- rough dependence on initial data

Z. C. Feng, Y. Charles Li|arXiv (Cornell University)|Feb 9, 2017

Fluid Dynamics and Turbulent Flows17 references3 citations

TL;DR

This paper demonstrates through numerical simulations that high Reynolds number turbulence exhibits super-fast, short-term unpredictability due to rough dependence on initial data, where perturbations grow as $ e^{σ\sqrt{Re}\sqrt{t}} $, dominating before classical chaos takes over. The phenomenon is generic, abundant across modes and base solutions, and underpins the persistence of fully developed turbulence.

ABSTRACT

Short term unpredictability is discovered numerically for high Reynolds number fluid flows under periodic boundary conditions. Furthermore, the abundance of the short term unpredictability is also discovered. These discoveries support our theory that fully developed turbulence is constantly driven by such short term unpredictability.

Motivation & Objective

• To numerically demonstrate short-term unpredictability in high Reynolds number fluid flows under periodic boundary conditions.
• To investigate the ubiquity and abundance of super-fast perturbation growth across different initial data and base solutions.
• To support the theoretical claim that fully developed turbulence is sustained by rough dependence on initial data, not long-term chaos.
• To show that perturbation amplification at high Re is dominated by a $ \sqrt{t} $-dependent term before the classical Liapunov exponent takes over.
• To establish that such rapid growth is generic and independent of spatial dimension, observed in both 2D and 3D simulations.

Proposed method

• Numerical simulation of the 2D and 3D Navier-Stokes equations with periodic boundary conditions at high Reynolds number (Re = 1000).
• Use of Fourier spectral methods to represent base flows and perturbations in the form of random Fourier modes with Gaussian amplitudes and uniform phase distributions.
• Definition of the perturbation norm $ \Lambda(t) = \|du(t)\|_{H^3} $ to quantify growth over time.
• Systematic variation of initial perturbation modes (wave numbers) and base solution modes to assess growth dependence.
• Comparison of perturbation growth across different Reynolds numbers and spatial dimensions to isolate the $ \sqrt{Re}\sqrt{t} $-dominated regime.
• Analysis of time evolution of perturbations to identify the temporal separation point $ t \sim Re $, where classical chaos begins to dominate.

Experimental results

Research questions

• RQ1 Does high Reynolds number turbulence exhibit super-fast, short-term perturbation growth due to rough dependence on initial data?
• RQ2 Is such rapid amplification abundant across different perturbation modes and base solution configurations?
• RQ3 How does the $ \sqrt{Re}\sqrt{t} $-dependent growth term compare to the classical $ \sigma_1 t $-dependent Liapunov term in determining early-time dynamics?
• RQ4 Is the super-fast growth phenomenon generic and independent of spatial dimension (2D vs 3D)?
• RQ5 Can such rapid amplification explain the persistence of fully developed turbulence, distinct from transient chaos at moderate Re?

Key findings

• Perturbations in high Reynolds number flows grow super fast as $ e^{\sigma\sqrt{Re}\sqrt{t}} $, dominating for $ t \ll Re $, confirming short-term unpredictability.
• Lower wave number perturbations exhibit faster super-fast growth, indicating a hierarchy in amplification rates.
• The phenomenon is abundant: super-fast growth occurs across a wide range of perturbation and base solution modes.
• In 3D simulations, the same super-fast growth is observed for different base solutions and perturbation modes, confirming generality.
• Even with complex initial conditions modeling the turbulence regime (e.g., random Fourier modes), super-fast growth persists for lower modes.
• The time $ t \sim Re $ marks the transition from short-term rough dependence to long-term chaos, with the former dominating in fully developed turbulence.