摘要
Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to investigate the parameter range within which the axisymmetric flow becomes unstable due to an external flow bifurcation.The hydrodynamics is governed by three dimensionless parameters:the viscosity ratioμ*,the external Reynolds numbers Re^(e),and internal Reynolds numbers Re^(i),respectively.The drop-to-fluid density ratio is related to these parameters asρ*=μ*Re^(i)/Re^(e).This study focuses on highly viscous droplets withμ*≥5,where wake instability is driven by the vorticity flux transferred from the droplet surface into the surrounding fluid.By analysing the wake structure,we confirm that the onset of the external bifurcation is linked to the tilting of the azimuthal vorticityωϕ,in the wake and that the bifurcation occurs once the isocontours ofωϕalign nearly perpendicular to the symmetry axis.We propose an empirical criterion for predicting the onset of the external bifurcation,formulated in terms of the maximum vorticity on the external side of the droplet surface.This criterion is applicable for sufficiently high Re^(i) and holds over a wide range ofμ*and Re^(e).Additionally,we examine the bifurcation sequence for two specific external Reynolds numbers,Re^(e)=300 and Re^(e)=500,and show that,beyond a critical viscosity ratio,the axisymmetric wake first transitions to a steady planar-symmetric state before undergoing a secondary Hopf bifurcation.Finally,we highlight the influence of Re^(i) on external bifurcation and show that,at moderate Re^(i),wake instability may set in at a lower vorticity threshold than predicted by our criterion.These findings provide new insights into the external flow bifurcation of viscous droplets.
本文通过直接数值模拟研究了均匀流作用下固定球形液滴尾流的不稳定性, 旨在确定流动因外部分叉而导致轴对称性失稳的参数范围. 该问题由三个无量纲参数控制: 黏度比μ* 、外部雷诺数Ree 和内部雷诺数Rei, 其中液滴与外部流体的密度比可表示为ρ* = μ*Rei/Ree. 本研究聚焦于高黏度液滴(μ* ≥ 5), 在此类液滴中, 尾流不稳定性由液滴表面向周围流体输运的涡量通量所驱动. 通过对尾流结构的分析, 我们确认外部流动分叉的触发与尾部环向涡量 ωϕ 的倾斜密切相关: 当 ωϕ 的等值线趋于垂直于对称轴排列时, 流动即发生失稳. 基于此, 我们提出了一个经验性判据, 用于预测外部分叉的临界条件, 该判据以液滴外侧表面上的最大涡量为基础. 该判据适用于足够大的Rei, 且在较广的 μ* 与 Ree 范围内成立. 此外, 我们考察了在两个特定外部雷诺数(Ree = 300和500)条件下的分叉序列, 结果显示: 当黏度比超过临界值后, 轴对称尾流首先转变为稳定的平面对称状态, 随后经历次级Hopf分叉而产生非定常性. 最后, 我 们强调了Rei 对外部流动分叉的影响, 并指出在中等 Rei 下, 尾流不稳定可能在低于该判据预测的涡量阈值下发生. 上述结果为高黏性液滴外部流动分叉问题提供了新的物理认识.
基金
supported by the Deutsche Forschungsgemeinschaft(DFG)(Grant No.501298479)。
