Adaptive block-implicit methods for annular liquid jets

Abstract

A domain-adaptive technique is used to analyze the dynamic response of inviscid, isothermal, annular liquid jets to velocity fluctuations at the nozzel exit. The adaptive technique maps the unknown, time-dependent, curvilinear geometry of the annular jet into a unit interval and yields a system of integro-differential equations for the mass per unit length, radius, and radial velocity components of the liquid jet. The convergence length is governed by an ordinary differential equation that depends on the values of the dependent variables at the convergence point. The governing equations are solved by means of two methods of lines, which discretize the spatial coordinate but keep continuous the time, and a Newton method. A block-bidiagonal technique is also used to solve the fluid dynamics equations. It is shown that the dynamic response of liquid jets to sinusoidal velocity fluctuations at the nozzle exit is periodic and nearly sinusoidal for small amplitudes of the velocity oscillations at the nozzle exit. For large amplitudes the fluctuations of both the convergence length and the pressure of the gases enclosed by the annular jet are periodic but not sinusoidal. It is also shown that there is a delay time between the velocity fluctuations at the nozzle exit and those of the convergence length. This delay time is nearly independent of the amplitude of the velocity fluctuations at the nozzle exit and decreases as the Weber and Froude numbers and the nozzle exit angle are decreased.