Analysis of instabilities in liquid sheets

Numerical and analytical solutions are presented for liquid sheets in order to confirm linearized predictions for inviscid flow and to investigate the transition to instability. Two sheet configurations are investigated. The first is a radially expanding sheet formed by impinging jets. Incompressible flow solutions are found for these sheets using an hp-finite element method. 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 when the Weber number based on the sheet thickness approaches one, in agreement with the theoretical work of G.I. Taylor. The absence of breakup in our simulations appears to be a direct result of allowing the interface to span the entire domain.;The second configuration examined is a thin inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. A series solution method is developed to solve the boundary value problem. We show that solutions with non-real wave-numbers can be interpreted as a superposition of eigenmodes with jump-periodic boundary conditions. These solutions are used to validate asymptotic stability predictions for sinuous and varicose perturbations. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate (or reduce the order of) singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow like t 1/3. This agrees with the prediction of deLuca and Costa (1997). If only the position is perturbed, we find that the sheet is stable and decays like t-2/3 at the origin. This agrees with the prediction of Luchini (2004). Furthermore, if both the position and velocity of a sheet are perturbed in the absence of ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and convectively unstable varicose waves grow like t1/2 .