Flow of Spatiotemporal Turbulentlike Random Fields

Abstract: We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

• The paper constructs a divergence-free Eulerian velocity field with multifractal properties to emulate high-Reynolds-number turbulence.
• It uses numerical simulations to reveal rough Lagrangian trajectories consistent with the characteristics of a fractional Gaussian field.
• Statistical analyses uncover additional intermittency in Lagrangian dynamics, shedding light on scaling laws in turbulent transport.

Flow of Spatiotemporal Turbulentlike Random Fields

The paper "Flow of Spatiotemporal Turbulentlike Random Fields" by Jason Reneuve and Laurent Chevillard investigates the complex behavior of fluid dynamics through the lens of spatiotemporal random vector fields, specifically focusing on the properties of Lagrangian trajectories within these fields. The study centers on the statistical characterization of an advecting Eulerian velocity field designed to emulate rough, multifractal turbulence as observed at infinite Reynolds numbers, a critical regime in fluid mechanics.

Summary of Key Contributions

1. Construction of the Velocity Field: The authors construct a divergence-free spatiotemporal random vector field to model turbulence. This field is statistically isotropic, homogeneous, and stationary. The velocity field is formulated to become asymptotically rough and exhibit multifractal properties both spatially and temporally. This construction aims to reflect the characteristics of turbulent flows observed in high-Reynolds-number regimes.
2. Numerical Solutions and Observations: Numerical simulations are employed to solve the flow equations for the field. The simulations reveal that the particle trajectories within this field develop rough characteristics, denoted by the Hurst exponent, consistent with the roughness imposed on the advecting field. Intriguingly, even for a simplified scenario of advection by a fractional Gaussian field, the analysis uncovers additional intermittent corrections in Lagrangian dynamics.
3. Statistical Analysis: The study provides an intricate analysis of trajectory statistics, particularly focusing on structure functions and Lagrangian intermittency. The paper systematically approaches the classic turbulence problems by leveraging fractional Gaussian fields and multiplicative chaos theory to model intermittency effects.
4. Intermittency and Scaling Laws: A significant finding is the evidence of additional intermittency in Lagrangian trajectories, independent of the Gaussian nature of the underlying velocity field. This multifractality of trajectories opens more profound questions about the scaling laws governing the turbulent transport of fluid particles.

Implications and Future Directions

The implications of this work extend across theoretical and applied domains within turbulence research. By providing a structured model to examine the Lagrangian dynamics in rough, turbulent fields, this research supports a deeper understanding of how turbulent processes affect particle transport and mixing. The demonstration of intermittent corrections, even in Gaussian settings, enhances the established understanding of turbulence phenomenology.

Future research could explore several avenues, including:

• Developing theoretical frameworks to derive the observed trajectory behaviors from first principles.
• Extending the analysis to three-dimensional fields, particularly focusing on their applications in complex geophysical and industrial fluid systems.
• Investigating the influence of additional factors such as anisotropy and compressibility on Lagrangian trajectories.
• Linking the observed statistical characteristics with experimental data and direct numerical simulations to validate the theoretical findings further.

The study's exploration of the sophisticated statistical mechanics of turbulent flows contributes valuably to the ongoing pursuit of a deeper grasp of turbulence, a pivotal challenge in fluid dynamics.