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  1. Free, publicly-accessible full text available January 1, 2024
  2. Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number $S$ for the onset of the bubble ( $S^*_B$ ) and the cone ( $S^*_C$ ) is determined as the jet-to-ambient density ratio $\varLambda$ is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts $S^*_B$ that decreases with increasing values of $\varLambda$ for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at $S^*_C$ , found to be independent of $\varLambda$ , in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number $Re_{eff}$ , defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at $Re_{eff}=200$ for all $\varLambda$ . Similar to the theoretical predictions, numerical calculations of $S^*_B$ decrease monotonically as $\varLambda$ is increased. The critical swirl numbers for the cone, $S^*_C$ , are found to depend strongly on viscous effects; for $\varLambda =1/5$ , the low jet Reynolds number (51) at $Re_{eff}=200$ delays the transition to the cone, while for $\varLambda =5$ at $Re_{eff}=200$ , the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing $S^*_C$ . 
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    This paper investigates the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The inviscid swirling motion found outside the boundary layer, driven by the entrainment of the buoyant turbulent plume of hot combustion products that develops above the fire, is described by an irrotational solution, obtained by combining Taylor's self-similar solution for the motion in the axial plane with the azimuthal motion induced by a line vortex of circulation $2 {\rm \pi}\Gamma$ . The development of the boundary layer from a prescribed radial location is determined by numerical integration for different swirl levels, measured by the value of the radial-to-azimuthal velocity ratio $\sigma$ at the initial radial location. As in the case $\sigma =0$ , treated in the seminal boundary-layer analysis of Burggraf et al. ( Phys. Fluids , vol. 14, 1971, pp. 1821–1833), the pressure gradient associated with the centripetal acceleration of the inviscid flow is seen to generate a pronounced radial inflow. Specific attention is given to the terminal shape of the boundary-layer velocity near the axis, which displays a three-layered structure that is described by matched asymptotic expansions. The resulting composite expansion, dependent on the level of ambient swirl through the parameter $\sigma$ , is employed as boundary condition to describe the deflection of the boundary-layer flow near the axis to form a vertical swirl jet. Numerical solutions of the resulting non-slender collision region for different values of $\sigma$ are presented both for inviscid flow and for viscous flow with moderately large values of the controlling Reynolds number $\Gamma /\nu$ . The velocity description provided is useful in mathematical formulations of localized fire-whirl flows, providing consistent boundary conditions accounting for the ambient swirl level. 
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