On the evolution of material lines and vorticity in homogeneous turbulence

Abstract

The evolution of material lines, $l$, and vorticity, $\omega$, is investigated experimentally through three-dimensional particle-tracking velocimetry (3D-PTV) in quasi-homogeneous isotropic turbulence at $Re_{\lambda }\,{=}\,50$. Through 3D-PTV data the full set of velocity derivatives, $\partial u_{i}/\partial x_{j}$, is accessible. This allows us to monitor the evolution of various turbulent quantities along fluid particle trajectories. The main emphasis of the present work is on the physical mechanisms that govern the Lagrangian evolution of $l$ and $\omega$ and the essential differences inherent in these two processes. For example, we show that vortex stretching is smaller than material lines stretching, i.e. $\langle\omega_{i}\omega_{j}s_{ij}/\omega^{2}\rangle \,{<}\,\langle l_{i}l_{j}s_{ij}/l^{2}\rangle$, and expand on how this issue is closely related to the predominant alignment of $\omega$ and the intermediate principal strain eigenvector $\lambda_{2}$ of the rate of strain tensor, $s_{ij}$. By focusing on Lagrangian quantities we discern whether these alignments are driven and maintained mainly by vorticity or by strain. In this context, the tilting of $\omega$ and the rotation of the eigenframe $\lambda_{i}$ of the rate of strain tensor $s_{ij}$ are investigated systematically conditioned on different magnitudes of strain, $s^{2}$, and enstrophy, $\omega^{2}$. Further, we infer that viscosity contributes through the term $\nu\omega_{i}\nabla^2\omega_{i}$ to ${\rm D}\omega^{2}/{\rm D}t$, whereas ${\rm D}l^{2}/{\rm D}t$ has no diffusive term. This difference plays a key role in defining the mutual orientation between $\omega$ and $\lambda_{i}$. Viscosity thus contributes significantly to the difference in growth rates of $\langle\omega_{i}\omega_{j}s_{ij}\rangle$ and $\langle l_{i}l_{j}s_{ij}\rangle$.