A numerical investigation of the stability of expanding liquid sheets and the influence of boundary conditions

Abstract

Incompressible flow solutions are found numerically for a radially expanding liquid sheet in order to confirm analytical results for inviscid flow and to investigate viscous and non-linear effects. An hp -finite element method is used to perform the numerical simulations. In our unsteady simulations, we observe that forced sinuous pulses cause two different speed waves to travel downstream for Weber numbers greater than one. We also witness wave deceleration for Weber numbers approaching one, confirming the predictions of inviscid linear stability analysis. Comparisons are also made to theoretical predictions of the radius where the sheet becomes unstable. To determine the critical radius, the inlet Weber number is reduced until the theoretical critical radius is within the simulated domain. Surprisingly, instead of leading to breakup, this causes the sheet to change from a stable symmetric shape to a stable asymmetric shape. The transition between these shapes occurs by both supercritical and subcritical bifurcations when the Weber number based on the sheet thickness approaches one, in agreement with the theoretical work of Taylor. The absence of breakup in our simulations appears to be a direct result of allowing the interface to span the entire domain. To verify this, we examine the dependence of the solutions on domain aspect ratio, shape, and exit boundary conditions. No boundary conditions are found that allows the sheet to break-up.