The flow field due to a sphere moving in a viscous, density stratified fluid

TL;DR

This paper analyzes the flow field caused by a sphere moving in a viscous, density-stratified fluid at low Reynolds and Richardson numbers, revealing complex columnar structures and multiple length scales influenced by diffusion effects.

Contribution

It extends previous work by characterizing the detailed structure of the flow, including multiple annular cells and the impact of diffusion on the flow decay and structure.

Findings

Identification of a columnar flow structure with multiple annular cells.

Derivation of boundary and cell count expressions as functions of downstream distance.

Discovery of secondary and tertiary screening lengths influenced by diffusion.

Abstract

We study the flow field induced by a sphere translating in a viscous density-stratified ambient, specifically, in the limit of small Reynolds $(Re = \rho U a/\mu \ll 1)$, and viscous Richardson numbers $(Ri_v = \gamma a^3 g/\mu U\ll 1)$, and large Peclet number $(Pe = Ua/D\gg 1)$. Here, $a$ is the sphere radius, $U$ its translational velocity, $\rho$ an appropriate reference density within the Boussinesq framework, $\mu$ the ambient viscosity, $\gamma$ the absolute value of the background density gradient, and $D$ the diffusivity of the stratifying agent. For the scenario where buoyancy forces first become comparable to viscous forces at large distances, corresponding to the Stokes-stratification regime defined by $Re \ll Ri_v^{1/3} \ll 1$ for $Pe \gg 1$, important flow features such as a vertical reverse jet and a horizontal wake, on scales larger than the primary screening length of $\mathcal{O}(aRi_v^{-1/3})$, have been identified by Varanasi and Subramanian (2022). Here, we show that the reverse jet is only the central portion of a columnar structure with multiple annular cells. In the absence of diffusion this columnar structure extends to downstream infinity with the number of annular cells diverging in this limit. We provide expressions for the boundary of the structure, and the number of cells within, as a function of the downstream distance. For small but finite diffusion, two additional length scales emerge - a secondary screening length of $O(aRi_v^{-1/2}Pe^{1/2})$, where diffusion starts to smear out density variations across cells, leading to exponentially decaying flow field; and a tertiary screening length, of $O(aRi_v^{-1/2}Pe^{1/2}\ln(Ri_v^{-1}Pe^3))$, beyond which the columnar structure ceases to exist and the downstream disturbance field reverts from an exponential to eventual algebraic decay, analogous to that prevalent at large distances upstream.