Three-dimensional temporal instabilities, leading to spray formation of a round liquid jet segment with an outer, coaxial high-density gas flow, are studied with Navier-Stokes and level-set computations. These computations predict the liquid surface shape showing the smaller structures on the conical wave crests, i.e. lobes, holes, bridges and ligaments, which are the precursors to droplet and spray formations. These structures and their time scales affect droplet size and velocity distributions as well as spray cone angles. The gas-to-liquid density ratio, liquid Reynolds number (Re) and liquid Weber number (We) range between 0.02-0.9, 320-16 000 and 2000-230 000, respectively, which cover three distinct physical domains. (1) At higher Re and We, ligaments and then drops develop following hole and liquid bridge formations. (2) At higher gas densities throughout the Re range, several holes merge forming two bridges per lobe before breaking to form ligaments; this hole merging is explained by slower development of hairpin vortices and lobe shape. (3) In cases where both gas density and Re or We are lower, the well-ordered lobes are replaced by more irregular, smaller-scale corrugations along the conical wave crest edge; ligaments form differently by stretching from the lobes before holes form. Thicker ligaments and larger droplets form in the low Re, low gas density range. The surface wave dynamics, vortex dynamics and their interactions are explained. Understandings of liquid stream break up and concurrent smaller structure formation are built upon an examination of both translation and rotation of the fluid. In all cases, hole formation is correlated with hairpin and helical vortices; fluid motion through a perforation in the thin sheet near the wave crest corresponds to these vortices. The hole formation process is dominated by inertial forces rather than capillary action, which differs from mechanisms suggested previously for other configurations. Circulation due to streamwise vorticity increases while the lobes thin and holes form. For larger surface tension, cavities in the jet core rather than perforations in a sheet occur. The more rapid radial extension of the two-phase mixture with increasing gas density is explained by greater circulation in the ring (i.e. wave crest) region. Experimental descriptions of the smaller structures are available only at lower Re and lower density, † Email address for correspondence: sirignan@uci.edu Hole, ligament and bridge formations 187 agreeing with the computations. Computed scales of bridges, ligaments, early droplets and emerging spray radii agree qualitatively with experimental evidence through the high Re and We domains.

The liquid jet from a round orifice during the transient start-up and steady mass flux periods of a high pressure injector is studied via Navier-Stokes and level-set computations. Via post-processing, the role of vorticity dynamics is examined and shown to reveal crucial new insights. A brief review of relevant literature is made. An unsteady, axisymmetric full-jet case is solved. Then, a less computationally intensive case is studied with a segment of the jet core undergoing temporal instability; agreement with the full-jet calculation is satisfactory justifying the segment analysis for three-dimensional computation. The results for surface-shape development are in agreement with experimental observations and other three-dimensional computations; the initial, axisymmetric waves at the jet surface created by Kelvin-Helmholtz (KH) instability distort to cone shapes; next, three-dimensional character develops through an azimuthal instability that leads to the creation of streamwise vorticity, lobe shapes on the cones, and formation of liquid ligaments which extend from lobes on the cones. The cause of this azimuthal instability has been widely described as a Rayleigh-Taylor instability. However, additional and sometimes more important causes are identified here. Counter-rotating, streamwise vortices within and around the ligaments show a relationship in the instability behavior for jets flowing into like-density fluid; thus, density difference cannot explain fully the three-dimensional instability as previously suggested. Furthermore, the formation of ligaments that eventually break into droplets and the formation of streamwise vorticity are caused by the same vortical dynamics. Waviness is identified on the ligaments which should result in droplet formation. The nonlinear development of the shorter azimuthal waves and ligament waves explains the experimental results that droplet sizes are usually smaller than KH wavelengths. The higher the relative velocity and/or the lower the surface tension the shorter are the values of the most unstable wavelengths.

SUMMARYThe main objective of this work is to develop a novel moving-mesh finite-volume method capable of solving the seepage problem in domains with arbitrary geometries. One major difficulty in analysing the seepage problem is the position of phreatic boundary which is unknown at the beginning of solution. In the current algorithm, we first choose an arbitrary solution domain with a hypothetical phreatic boundary and distribute the finite volumes therein. Then, we derive the conservative statement on a curvilinear co-ordinate system for each cell and implement the known boundary conditions all over the solution domain. Defining a consistency factor, the inconsistency between the hypothesis boundary and the known boundary conditions is measured at the phreatic boundary. Subsequently, the preceding mesh is suitably deformed so that its upper boundary matches the new location of the phreatic surface. This tactic results in a moving-mesh procedure which is continued until the nonlinear boundary conditions are fully satisfied at the phreatic boundary. To validate the developed algorithm, a number of seepage models, which have been previously targeted by the other investigators, are solved. Comparisons between the current results and those of other numerical methods as well as the experimental data show that the current moving-grid finite-volume method is highly robust and it provides sufficient accuracy and reliability.

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