Wake instability past a sphere settling in a strongly stratified flow

TL;DR

This study uses fully resolved 3D DNS (JADIM) to explore the wake of a sphere settling through a strongly stratified fluid across a wide range of $Fr$, $Re$, and $Pr$. It reveals two primary instability pathways: a near-axisymmetric varicose mode leading to vortex-ring formation, and a non-axisymmetric sinuous mode driven by baroclinic vorticity generation and vortex tilting, with a detailed stage-by-stage development and a parameter-space regime map. The work connects the wake dynamics to internal-wave generation and density-transport processes, providing mechanistic insight into how stratification modulates drag and wake stability. These findings advance understanding of particle and organism transport in geophysical flows and have potential implications for predicting settling trajectories under strong stratification.

Abstract

The wake of a body moving across the isopycnals of a strongly stratified fluid is characterized by the presence of an intense jet which, under certain circumstances, may become unstable. To get insight into the phenomenology of this instability and the underlying mechanisms, we conduct fully-resolved three-dimensional time-dependent simulations of the flow past a rigid sphere settling steadily through a linearly stratified fluid over a wide range of flow parameters which include the case of salt-stratified water. Results reveal a rich dynamics characterized by distinct wake symmetries, vortical structures and transverse force signatures. Simulations evidence the existence of varicose and sinuous instability modes that arise from distinct physical mechanisms. Thanks to several metrics, we build a phase map delineating the bounds of each instability regime as a function of the three control parameters of the problem, namely the Froude, Reynolds and Prandtl numbers. We show that the instability mechanism results from a subtle interplay between baroclinic vorticity generation, vortex tilting and radial transport of the mean density gradient within the jet.

Figures (25)

• Figure 1: $(a)$: Sketch of the flow configuration and definition of some quantities; $(b)$: light fluid dragged down by the sphere at $(Fr,Re, Pr) = (1,100, 70)$, with the iso-contours and colors highlighting isopycnals and density variations; $(c)$: stable jet observed in a 2D axisymmetric simulation at $(Fr,Re, Pr) = (0.1,100, 700)$; $(d)$: unstable jet observed with the same set of parameters in a 3D simulation. In $(c,d)$, colors refer to the absolute vertical velocity $\boldsymbol{u} \cdot \boldsymbol{e}_x - 1$; the inset provides an enlarged view of the upper part of the jet.
• Figure 2: Parameter range $(Re, Fr)$ investigated in simulations, compared with experiments by akiyama2019unstable at $Pr=700$. Triangles and circles correspond to experimental and numerical observations, respectively. Open and closed symbols refer to stable and unstable jets, respectively. The solid blue line is the experimentally determined stability threshold in the range $5\leq Re \leq 50$, given, with present normalisation, by $Fr/Re \approx 3.14 \times 10^{-3}$.
• Figure 3: Examples of the jet evolution and instability characteristics observed by varying $Fr$, with $Re=100$ and $Pr=700$ in all cases. $(a-d)$: contours of the vertical velocity projected onto the vertical cross-sectional $(x,y)$ plane, with $(a)$: $Fr = 0.3$ (stable); $(b)$$Fr = 0.1$ (chaotic instability); $(c)$: $Fr = 0.05$ (spiral instability); $(d)$: $Fr = 0.02$ (standing wave instability). $(e-g)$: time histories of the $y$- and $z$-components of $C_L$ during the saturated oscillation stage, with $(e)$: $Fr = 0.1$; $(f)$: $Fr = 0.05$; $(g)$: $Fr = 0.02$.
• Figure 4: Wake evolution tracked in the $(C_{L,y},C_{L,z})$ phase plane in three distinct unstable regimes, with $Re=100$ and $Pr=700$ in all cases. $(a)$: $Fr = 0.1$ (chaotic regime); $(b)$: $Fr = 0.05$ (spiral regime); $(c)$: $Fr = 0.02$ (standing wave regime); $(d)$: Fourier transforms (FFT) of the lift coefficient $C_{L,y}$ for different stratification levels. In $(a-c)$, the color along the path transitions from grey to black following time progression.
• Figure 5: Variation with the Froude number of the maximum amplitude of the transverse force, $C_L^{\mathrm{max}}$ (blue line, right axis), and the axial location $h_x^{\mathrm{max}}$ of the peak vertical velocity on the jet axis in the axisymmetric configuration (red line, left axis).