Modelling and synthesizing turbulence with multi-scale coherent vortices

TL;DR

This work introduces woven turbulence, a framework that explicitly constructs multi-scale coherent vortices along centerlines generated by fractional Brownian bridges and nests them inside a self-similar fractal hierarchy. By tying scale-dependent vortex parameters to a fractal skeleton, the model reproduces the three-regime energy spectrum, inertial-range Kolmogorov scaling, and tunable intermittency through a critical vortex density varrho_c, while enabling ultra-fast turbulence synthesis with cost scaling as $O(N^3)$. The approach achieves DNS-like statistics for $Re_\lambda$ up to around $10^3$ at a fraction of the computational expense, and reveals physical insights such as the invariance of hierarchical vortex density across scales corresponding to the -5/3 law. Limitations include absence of cross-scale vortex interactions and non-tubular structures, which the authors plan to address through cross-scale coupling, richer vortex geometries, and multifractal extensions, broadening applicability to turbulence modeling and data generation.

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.

Figures (12)

• Figure 1: Construction of woven turbulence. (a) Schematic of multi-scale vortex tubes constituting woven turbulence. Each vortex tube is constructed around its curved centerline, and this centerline is generated from spline interpolation of the FBB discrete points. At the first scale level, we annotate the three fundamental elements of each vortex tube: the core size $\sigma$, the circulation $\Gamma$, and the centerline $\mathcal{C}$, with the underlying FBB discrete points $\boldsymbol{B}({J})$ indicated as purple points. For vortex tubes across different scales, both the centerline length and circulation are scale-dependent but uniform within each scale, whereas centerline shapes vary stochastically. Note that the vortex tubes are sketched as segments, but the actual ones in woven turbulence are closed. (b) Visualization of the woven turbulence case WT6 (with $4096^3$ grid points and $Re_{\lambda}=1237$, see table \ref{['tab:set-up of cases']}). Flow fields in a $1/2^3$ spatial portion are visualized by isosurfaces of the vorticity magnitude $|\boldsymbol{\omega}|=3\sqrt{\langle\Omega\rangle}$, where $\Omega = |\boldsymbol{\omega}|^2/2$ denotes the enstrophy and $\langle\cdot\rangle$ represents the volume average over the computational domain ${V}$. These isosurfaces are color-coded by the normalized helicity density $h/\sqrt{\langle h^2\rangle}$ with helicity density $h = \boldsymbol{u}\cdot\boldsymbol{\omega}$, where $\boldsymbol{u}$ and $\boldsymbol{\omega}$ denote the velocity and the vorticity, respectively.
• Figure 2: Effect of Hurst exponent $H$ on FBB and spatial arrangement of vortex tubes in woven turbulence. These cases use the same parameter values $\mathcal{N}=1$, $\sigma_{\mathcal{N}}=0.01$, $\varrho=1.98 \times 10^{-4}$, and $\lambda_{\sigma}=0$, with $H = 0.3$, $0.5$, and $0.9$. (a) Sample path of the FBB $x$-component $B_{x}(J)=\boldsymbol{B}(J)\cdot\boldsymbol{e}_{x}$, normalized by its standard deviation $\sqrt{\langle B_{x}^2 \rangle_{\mathscr{N}}}$. (b) Scaling of mean-squared distance $\langle (\boldsymbol{B}({J}) - \boldsymbol{B}({J}^{\prime}))^2 \rangle_{\mathscr{N}}$ in generated FBB (solid lines) and corresponding theoretical scaling law in \ref{['eq:FBB_scaling']} (symbols). (c) Energy-containing range spectra of woven turbulence cases (solid lines) and corresponding model spectra $E(k) \propto k^{r_{E}}$ (symbols). (d) Vorticity magnitude isosurfaces $|\boldsymbol{\omega}|=3\sqrt{\langle\Omega\rangle}$ of woven turbulence cases with $H=0.3$, $0.5$, and $0.9$ from left to right. These isosurfaces are color-coded by the normalized helicity density $h/\sqrt{\langle h^2\rangle}$ with the colorbar same as that in figure \ref{['fig:multi-scale']}(b).
• Figure 3: Energy spectrum $E(k)$ of case WT6 (see table \ref{['tab:set-up of cases']}) with $Re_{\lambda}=1237$ and the corresponding model spectrum in \ref{['eq:model_spectrum']}. The inset plots the hierarchical energy spectra $E_{i}(k)$ in \ref{['eq:hierarchical energy spectrum']} of case WT6 and the corresponding characteristic energy $E_{i}(k_{i})$. Each hierarchical energy spectrum is shown around its characteristic wavenumber $k_i$, within the range $[k_i/10, 10k_i]$.
• Figure 4: Effect of the population ratio $r_{n}$ on woven turbulence. Cases WT4-S, WT4, and WT4-D (see table \ref{['tab:set-up of cases']}) vary $r_{n}$ with values of 4, 8, and 16, respectively, while keeping the vortices at the first scale level fixed. (a) Schematic of monofractal vortex tubes for different $r_{n}$. For $r_{n} = 8$, the "total volume" of vortex tubes at each scale is scale-invariant, while it decreases and increases with decreasing scale for $r_{n} < 8$ and $r_{n} > 8$, respectively. (b) Energy spectra with varying $r_{n}$. The variation of the hierarchical vortex density $\varrho_i$ with scale level $i$ is shown in the inset for various $r_{n}$. (c) Vorticity magnitude isosurfaces $|\boldsymbol{\omega}|=\Gamma_{i}/(8\pi\sigma_{i}^2)$ of the vortex tubes at the first (blue) and second (cyan) scale levels for various $r_{n}$.
• Figure 5: Effect of $\lambda_{\sigma}$ on vortex structures and dissipation range energy spectrum in woven turbulence with varying $\lambda_{\sigma} = 0$, $0.5$, and $1.5$, along with the same parameter values $\mathcal{N}=1$, $\sigma_{\mathcal{N}}=0.04$, $\varrho=1.59 \times 10^{-3}$, and $H=5/6$. (a) Vorticity magnitude isosurfaces $|\boldsymbol{\omega}|=10$ for $\lambda_{\sigma}=0$ (light blue) and $\lambda_{\sigma}=3/2$ (translucent dark blue). A $1/2^3$ subdomain of the flow field is shown to highlight the structure in the dissipation range. (b) Kernel function $G_{1}(s,\rho)$ given by \ref{['eq:kernel']} for $\lambda_{\sigma}=0$ (left) and $\lambda_{\sigma}=3/2$ (right). (c) Dissipation range spectra of cases (solid lines) with varying $\lambda_{\sigma}$ and exponential decay model (symbols).