Upward and downward annular liquid jets: Conservation properties, singularities, and numerical errors

Abstract

A one-dimensional hydraulic model for inviscid incompressible axisymmetric anular liquid jets is derived by assuming that the presure is uniform throughout the jet and that the velocity components are uniform on each cross-section. This model can be derived from that of Boussinesq if the slope of the annular jet is small. Both models indicate that the liquid jet's acceleration and curvature may become singular for Weber numbers less than or equal to one. The singularity does not depend on the Froude number and pressure difference between the gases enclosed by and surrounding the annular liquid jet, and it is in good accord with available experimental data. Since jets are observed experimentally for Weber numbers less than one, the analysis presented here indicates that annular jets leave the nozzle exit with an angle that differs from that of the nozzle. An asymptotic analysis of the governing equations for small Weber numbers indicates that the shape of annular liquid jets may be a circular arc, and this is in accordance with available experimental and theoretical data. Numerical experiments and comparisons with analytical solutions for long annular liquid jets indicate that the convergence length is nearly independent of the accuracy of the numerical method and computer precision, while the local and global energy errors increase as the computer precision is decreased. It is also shown that upwind finite difference methods which conserve linear momentum do not conserve mechanical energy and yield larger errors than explicit fourth-order accurate Runge-Kutta techniques. The shape of annular liquid jets exhibits a rounded form for Weber numbers much less than unity; however, the long wave approximation employed in the derivation of the hydraulic model may not be valid for small Weber numbers.