Modelling and synthesizing turbulence with multi-scale coherent vortices

Abstract

Turbulence is a complex system exhibiting both universal statistical features and prominent coherent structures. We model turbulence using coherent vortices distributed within a multi-scale statistical framework, termed `woven turbulence'. These entangled vortices are generated based on fractional Brownian bridges, with scale-dependent parameters set by dimensional analysis and geometric similarity. By integrating statistical and structural modeling, our approach naturally captures both the universal statistical features of turbulence and its coherent vortex structures. The spatial filling fraction of vortices in woven turbulence, termed `vortex density', is tunable, enabling us to investigate the statistical-structural interaction and uncover two concise physical insights of turbulence. First, the invariance of the hierarchical vortex density across scales corresponds to Kolmogorov's $-5/3$ law in the inertial range. Second, there exists a critical total vortex density at which the intermittency of woven turbulence closely matches that of real turbulence, and this critical density converges to a finite value in the inviscid limit. Deviating from this critical density reveals a negative correlation between intermittency and total vortex density. In addition, woven turbulence also serves as a fast turbulence synthesis method, requiring only the Taylor-Reynolds number as input and exhibiting an extremely low computational cost proportional to the grid size. It generates instantaneous turbulent fields at Taylor-Reynolds numbers of order $10^3$ on $4096^3$ grid points, with computational cost over five orders of magnitude lower than that of direct numerical simulation.