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This work describes an inviscid solution for a cyclonic flowfield evolving in a hemispherical chamber configuration. In this context, a vigorously swirling motion is triggered by a gaseous stream that is introduced tangentially to the inner circumference of the chamber's equatorial plane. The resulting updraft spirals around while sweeping the chamber wall, reverses direction while approaching the headend, and then tunnels itself out through the inner core portion of the chamber. Our analysis proceeds from the Bragg–Hawthorne formulation, which proves effective in the treatment of steady, incompressible, and axisymmetric motions. Then using appropriate boundary conditions, we are able to obtain a closed-form expression for the Stokes streamfunction in both spherical and cylindrical coordinates. Other properties of interest are subsequently deduced, and these include the principal velocities and pressure distributions, vorticity, swirl intensity, helicity density, crossflow velocity, and the location of the axial mantle; the latter separates the outer annular updraft from the inner, centralized downdraft. Due in large part to the overarching spherical curvature, the cyclonic motion also exhibits a polar mantle across which the flow becomes entirely radial inward. Along this lower and shorter spherical interface, the polar velocity vanishes while switching angular direction. Interestingly, both polar and axial mantles coincide in the exit plane where the ideal outlet size, prescribed by the mantle position, is found to be approximately 70.7% of the chamber radius. We thus recover the same mantle fraction of the cyclonic flow analog in a right-cylindrical chamber where an essentially complex-lamellar motion is established.
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A model for the constant-density boundary layer surrounding fire whirls
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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- Award ID(s):
- 1916979
- PAR ID:
- 10238803
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 900
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.513
