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Vorticity dynamics in a spatially developing liquid jet inside a co-flowing gas
- A. Zandian, W. A. Sirignano, F. Hussain
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- Journal:
- Journal of Fluid Mechanics / Volume 877 / 25 October 2019
- Published online by Cambridge University Press:
- 27 August 2019, pp. 429-470
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A three-dimensional transient round liquid jet within a low-speed coaxial outer gas flow is numerically simulated and analysed via vortex dynamics ($\unicode[STIX]{x1D706}_{2}$ analysis). Two types of surface deformations are distinguished, which are separated by a large indentation on the jet stem. First, there are those inside the recirculation zone behind the leading cap – directly affecting the cap dynamics and well explained by the local vortices. Second, deformations upstream of the cap are mainly driven by the Kelvin–Helmholtz (KH) instability, unaffected by the vortices in the behind-the-cap region (BCR), and are important in the eventual atomization process. Different atomization mechanisms are identified and are delineated on a gas Weber number ($We_{g}$) versus liquid Reynolds number ($Re_{l}$) map based on the relative gas–liquid velocity. In a frame moving with the liquid velocity, this result is consistent with prior temporal studies. A simpler and clearer portrait of similarity of the atomization domains is shown by using the relative gas–liquid axial velocity, i.e. $We_{r}$ and $Re_{r}$, and avoiding the widely used velocity ratio as a third key parameter. A detailed comparison of vorticity along the axis in an Eulerian frame versus a frame fixed to a surface wave reveals that the vortex development and surface deformations are periodic in the upstream region, but this periodicity is lost closer to the BCR. In the practical range of the density ratio and for early times in the process, axial vorticity is mainly generated by baroclinicity while streamwise vortex stretching becomes more important at later times and only at lower relative velocities when pressure gradients are reduced. The inertia, vortex, pressure, viscous and surface tension forces are analysed to delineate the dominant causes of the three-dimensional instability of the axisymmetric KH structure due to surface acceleration in the axial, radial and azimuthal directions. The inertia force related to the axial gradient of kinetic energy is the main cause of the axial acceleration of the waves, while the azimuthal acceleration is mainly caused by the pressure and viscous forces. The viscous forces are negligible in the radial direction and away from the nozzle exit in the axial direction. It is interesting to note that azimuthal viscous forces are important even at high $Re_{l}$, indicating that inertia is not totally dominant in this instability occurring early in the atomization cascade.
Understanding liquid-jet atomization cascades via vortex dynamics
- A. Zandian, W. A. Sirignano, F. Hussain
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- Journal:
- Journal of Fluid Mechanics / Volume 843 / 25 May 2018
- Published online by Cambridge University Press:
- 21 March 2018, pp. 293-354
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Temporal instabilities of a planar liquid jet are studied using direct numerical simulation (DNS) of the incompressible Navier–Stokes equations with level-set (LS) and volume-of-fluid (VoF) surface tracking methods. $\unicode[STIX]{x1D706}_{2}$ contours are used to relate the vortex dynamics to the surface dynamics at different stages of the jet breakup – namely, lobe formation, lobe perforation, ligament formation, stretching and tearing. Three distinct breakup mechanisms are identified in the primary breakup, which are well categorized on the parameter space of gas Weber number ($We_{g}$) versus liquid Reynolds number ($Re_{l}$). These mechanisms are analysed here from a vortex dynamics perspective. Vortex dynamics explains the hairpin formation, and the interaction between the hairpins and the Kelvin–Helmholtz (KH) roller explains the perforation of the lobes, which is attributed to the streamwise overlapping of two oppositely oriented hairpin vortices on top and bottom of the lobe. The formation of corrugations on the lobe front edge at high $Re_{l}$ is also related to the location and structure of the hairpins with respect to the KH vortex. The lobe perforation and corrugation formation are inhibited at low $Re_{l}$ and low $We_{g}$ due to the high surface tension and viscous forces, which damp the small-scale corrugations and resist hole formation. Streamwise vorticity generation – resulting in three-dimensional instabilities – is mainly caused by vortex stretching and baroclinic torque at high and low density ratios, respectively. Generation of streamwise vortices and their interaction with spanwise vortices produce the liquid structures seen at various flow conditions. Understanding the liquid sheet breakup and the related vortex dynamics are crucial for controlling the droplet-size distribution in primary atomization.
Early spray development at high gas density: hole, ligament and bridge formations
- D. Jarrahbashi, W. A. Sirignano, P. P. Popov, F. Hussain
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- Journal:
- Journal of Fluid Mechanics / Volume 792 / 10 April 2016
- Published online by Cambridge University Press:
- 01 March 2016, pp. 186-231
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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, 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.
Stress-induced cavitation for the streaming motion of a viscous liquid past a sphere
- J. C. PADRINO, D. D. JOSEPH, T. FUNADA, J. WANG, W. A. SIRIGNANO
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- Journal:
- Journal of Fluid Mechanics / Volume 578 / 10 May 2007
- Published online by Cambridge University Press:
- 26 April 2007, pp. 381-411
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The theory of stress-induced cavitation is applied here to the problem of cavitation of a viscous liquid in the streaming flow past a stationary sphere. This theory is a revision of the pressure theory which states that a flowing liquid will cavitate when and where the pressure drops below a cavitation threshold, or breaking strength, of the liquid. In the theory of stress-induced cavitation the liquid will cavitate when and where the maximum tensile stress exceeds the breaking strength of the liquid. For example, liquids at atmospheric pressure which cannot withstand tension will cavitate when and where additive tensile stresses due to motion exceed one atmosphere. A cavity will open in the direction of the maximum tensile stress, which is 45° from the plane of shearing in pure shear of a Newtonian fluid. This maximum tension criterion is applied here to analyse the onset of cavitation for the irrotational motion of a viscous fluid, the special case imposed by the limit of very low Reynolds numbers and the fluid flow obtained from the numerical solution of the Navier–Stokes equations. The analysis leads to a dimensionless expression for the maximum tensile stress as a function of position which depends on the cavitation and Reynolds numbers. The main conclusion is that at a fixed cavitation number the extent of the region of flow at risk to cavitation increases as the Reynolds number decreases. This prediction that more viscous liquids at a fixed cavitation number are at greater risk of cavitation seems not to be addressed, affirmed nor denied, in the cavitation literature known to us.
Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets
- C. MEHRING, W. A. SIRIGNANO
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- Journal:
- Journal of Fluid Mechanics / Volume 388 / 10 June 1999
- Published online by Cambridge University Press:
- 10 June 1999, pp. 69-113
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Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimensional simulations using a discrete-vortex method. For semi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appearance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a short-wavelength oscillation. For semi-infinite sheets with sinuous waves, qualitative agreement between the dimensionally reduced analysis and experimental results is found. For example, a half-wave thinning and a sawtooth wave shape is found for the nonlinear sinuous mode. For the semi-infinite dilational case, a critical frequency-dependent Weber number is found below which one component of the disturbances decays with downstream distance. For the semi-infinite sinuous case, a critical Weber number equal to 2 is found; below this value, only one characteristic is emitted in the positive time direction from the nozzle exit.