Constructing wall turbulence using hierarchical hairpin vortices

TL;DR

This paper introduces SWAT, a physics-driven framework to construct wall-bounded turbulence from hierarchically organized hairpin vortex packets. By incorporating height-dependent core sizes, curved centerlines, spanwise meandering, and wall-coherent superstructures, SWAT reproduces key mean-field and structural statistics, including a natural log-law and a $k_x^{-1}$ energy spectrum, across a wide range of Reynolds numbers ($\,\mathrm{Re}_\tau=1000$--$10000$). The approach provides new insights into how vortex geometry, packet organization, and VLSMs govern attached/detached dynamics and streak formation, while offering a highly efficient initialization tool for DNS/LES that significantly reduces computational cost. SWAT serves as a testable, parameter-controlled platform for mechanism testing and model validation in wall turbulence, with potential extensions to other geometries and inflow conditions.

Abstract

Wall-bounded turbulence is characterized by coherent, worm-like structures such as hairpin vortices. The attached-eddy model provides a successful statistical framework for the log-law region, yet the complex geometry and multiscale nature of wall-turbulence vortices remain challenging for physics-based modelling. Here, we model wall turbulence as an ensemble of complex vortices, introducing a systematic approach to constructing turbulence fields enriched with hierarchically organized hairpin vortex packets. The geometry and organization of the vortex packets are calibrated to match observations, enabling the model to reproduce both attached and detached motions through a height-dependent core-size variation. Our model successfully reproduces the key statistical and structural features of wall turbulence, matching direct numerical simulations of turbulent channel flow at friction Reynolds numbers from 1,000 to 10,000. More importantly, it also reveals new insights into the coherent structures, emphasizing the role of vortex geometry, packet organization, and hierarchy in setting the attached/detached balance, meandering streaks and inclination angles, superstructure alignment, and the overall partition of contributions. Moreover, the constructed channel turbulence rapidly transitions into fully developed turbulence in direct numerical simulation, demonstrating its physical self-consistency and practical utility for initializing high-fidelity simulations. This approach significantly reduces computational costs associated with turbulence development while providing a flexible framework for testing and advancing turbulence models based on vortex structures.

Figures (21)

• Figure 1: Geometry of a single hairpin vortex. (a) Front and (b) side views of the hairpin vortex centerline. (c) Vortex surface and a segment of a single hairpin vortex tube with variable thickness. The 3D hairpin vortex is visualized by VSF isosurface ($\phi_v=0.1$, blue) with embedded vortex lines (red solid). An enlarged schematic of the vortex tube segment is shown, where the vorticity is constructed in curved cylindrical coordinates $(\zeta,\rho,\theta)$. The vortex centerline $\mathcal{C}$ (green dash-dotted) is described in the Frenet--Serret frame ($\boldsymbol{T}$, $\boldsymbol{N}$, $\boldsymbol{B}$). On each cross section of the vortex tube, the vorticity follows a Gaussian distribution with a continuously varying standard deviation $\sigma$.
• Figure 2: Geometry of vortex packets and wall-coherent superstructures. (a) Alignment of sub-level hairpin vortices along the streamwise direction, exhibiting increasing heights at a growth angle of $\gamma$. (b) Top view of the vortex packet, illustrating the spanwise meandering features. (c) Vortex surface visualization $\phi_v=0.1$ of a typical wall-coherent superstructure.
• Figure 3: Construction of synthetic wall-attached turbulence. The input parameters include the prescribed friction Reynolds number $\hbox{Re}_\tau$, boundary layer thickness $\delta$, and the dimensions of the computational domain $L_x \times L_y \times L_z$. Based on the attached-eddy model, key structural properties are determined, including the hierarchical level number of vortex packets $N_p$ in \ref{['eq:Np']}, the height of individual hairpin vortices $h_i^{(j)}$ in \ref{['eq:hi']}, population density $M_i$ in \ref{['eq:Mi']}, and spanwise meandering $\Delta z_i$ in \ref{['eq:zi']}. These parameters define the centerlines of the hairpin vortices. Next, the vorticity field of the hairpin vortices is constructed based on their centerlines, circulation strengths $\Gamma_i^{(j)}$ in \ref{['eq:Gammaij']}, and core size distribution $\sigma_i^{(j)}$ in \ref{['eq:sigmaij']}. Finally, the velocity field of the synthetic wall turbulence is obtained by applying the Biot--Savart law along with wall and bulk flow corrections in \ref{['eq:damping']}.
• Figure 4: Structure and statistics of the synthetic wall-attached turbulence (SWAT) for $\hbox{Re}_\tau=1000$. (a) Visualization of vortex surfaces in SWAT, displaying hierarchically attached vortex packets with spanwise meandering features. The vortex surfaces are colour-coded according to the wall distance. (b,c) Comparison of (b) mean velocity and (c) Reynolds stress profiles between SWAT and DNS data. Symbols represent DNS data from Lee2015Direct, while solid lines in matching colours represent SWAT. The gray area marks a strictly logarithmic region Marusic2013On, defined as $3\hbox{Re}_\tau^{1/2}<y^+<0.15\hbox{Re}_\tau$.
• Figure 5: Streamwise energy spectra and higher-order statistics of SWAT and DNS at $\hbox{Re}_\tau=1000$. (a) Streamwise energy spectra at $y^+=3.9Re_\tau^{1/2}$. (b) Profiles of higher-order statistics for the streamwise velocity $\langle [(u-\langle u \rangle)/u_\tau]^{2p}\rangle^{1/p}$. Symbols represent DNS data from Lee2015Direct, while solid lines in corresponding colors denote SWAT results.