skip to main content


Title: 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.  more » « less
Award ID(s):
1916979
NSF-PAR ID:
10238803
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Journal of Fluid Mechanics
Volume:
900
ISSN:
0022-1120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    This work focuses on the use of a finite-volume solver to describe the wall-bounded cyclonic flowfield that evolves in a swirl-driven thrust chamber. More specifically, a non-reactive, cold-flow simulation is carried out using an idealized chamber configuration of a square-shaped, right-cylindrical enclosure with eight tangential injectors and a variable nozzle size. For simplicity, we opt for air as the working fluid and perform our simulations under steady, incompressible, and inviscid flow conditions. First, a meticulously developed mesh that consists of tetrahedral elements is generated in a manner to minimize the overall skewness, especially near injectors; second, this mesh is converted into a polyhedral grid to improve convergence characteristics and accuracy. After achieving convergence in all variables, our three velocity components are examined and compared to an existing analytical solution obtained by Vyas and Majdalani (Vyas, A. B., and Majdalani, J., “Exact Solution of the Bidirectional Vortex,” AIAA Journal, Vol. 44, No. 10, 2006, pp. 2208-2216). We find that the numerical model is capable of predicting the expected forced vortex behavior in the inner core region as well as the free vortex tail in the inviscid region. Moreover, the results appear to be in fair agreement with the Vyas–Majdalani solution derived under similarly inviscid conditions, and thus resulting in a quasi complex-lamellar profile. In this work, we are able to ascertain the axial independence of the swirl velocity no matter the value of the outlet radius, which confirms the key assumption made in most analytical models of wall-bounded cyclonic motions. Moreover, the pressure distribution predicted numerically is found to be in fair agreement with both theoretical formulations and experimental measurements of cyclone separators. The bidirectional character of the flowfield is also corroborated by the axial and radial velocity distributions, which are found to be concurrent with theory. Then using parametric trade studies, the sensitivity of the numerical simulations to the outlet diameter of the chamber is explored to determine the influence of outlet nozzle variations on the mantle location and the number of mantles. Since none of the cases considered here promote the onset of multiple mantles, we are led to believe that more factors are involved in producing more mantles than one. Besides the exit diameter, the formation of a multiple mantle structure may be influenced by the physical boundary conditions, nozzle radius, inlet curvature, and length. In closing, we show that the latter plays a significant role in controlling the development of backflow regions inside the chamber. 
    more » « less
  2. In this work, an exact inviscid solution is developed for the incompressible Euler equations in the context of a bidirectional, cyclonic flowfield in a right-cylindrical chamber with a hollow core. The presence of a hollow core confines the flow domain to an annular swirling region that extends into a toroid in three-dimensional space. The procedure that we follow is based on the Bragg–Hawthorne framework and a judicious assortment of boundary conditions that correspond to a wall-bounded cyclonic motion with a cylindrical core. At the outset, a self-similar stream function is obtained directly from the Bragg–Hawthorne equation under the premises of steady, axisymmetric, and inviscid conditions. The resulting formulation enables us to describe the bidirectional evolution of the so-called inner and outer vortex motions, including their fundamental properties, such as the interfacial layer known as the mantle; it also unravels compact analytical expressions for the velocity, pressure, and vorticity fields, with particular attention being devoted to their peak values and spatial excursions that accompany successive expansions of the core radius. By way of confirmation, it is shown that removal of the hollow core restores the well-established solution for a fully flowing cylindrical cyclone. Immediate applications of cyclonic flows include liquid and hybrid rocket engines, swirl-driven combustion devices, as well as a multitude of heat exchangers, centrifuges, cyclone separators, and flow separation devices that offer distinct advantages over conventional, non-swirling systems. 
    more » « less
  3. This work presents an exact solution of Euler's incompressible equations in the context of a bidirectional vortex evolving inside a conically shaped cyclonic chamber. The corresponding helical flowfield is modeled under inviscid conditions assuming constant angular momentum. By leveraging the axisymmetric nature of the problem, a steady-state solution of the generalized Beltramian type is obtained directly from first principles, namely, from the Bragg–Hawthorne equation in spherical coordinates. The resulting stream function representation enables us to fully describe the ensuing swirl-dominated motion including its fundamental flow characteristics. After identifying an isolated singularity that appears at a cone divergence half-angle of 63.43°, two piecewise formulations are provided that correspond to either fluid injection or extraction at the top section of the conical cyclone. In this process, analytical expressions are readily retrieved for the three velocity components, vorticity, and pressure. Other essential flow indicators, such as the theoretically preferred mantle orientation, the empirically favored locus of zero vertical velocity, the maximum polar and axial velocities, the crossflow velocity, and other such terms, are systematically deduced. Results are validated using limiting process verifications and comparisons to both numerical and experimental measurements. The subtle differences between the present model and a strictly Beltramian flowfield are also highlighted and discussed. The conically cyclonic configuration considered here is relevant to propulsive devices, such as vortex-fired liquid rocket engines with tapered walls; meteorological phenomena, such as tornadoes, dust devils, and fire whirls; and industrial contraptions, such as cyclonic flow separators, collectors, centrifuges, boilers, vacuum cleaners, cement grinders, and so on. 
    more » « less
  4. SUMMARY

    We present a new, 3-D model of seismic velocity and anisotropy in the Pacific upper mantle, PAC13E. We invert a data set of single-station surface-wave phase-anomaly measurements sensitive only to Pacific structure for the full set of 13 anisotropic parameters that describe surface-wave anisotropy. Realistic scaling relationships for surface-wave azimuthal anisotropy are calculated from petrological information about the oceanic upper mantle and are used to help constrain the model. The strong age dependence in the oceanic velocities associated with plate cooling is also used as a priori information to constrain the model. We find strong radial anisotropy with vSH > vSV in the upper mantle; the signal peaks at depths of 100–160 km. We observe an age dependence in the depth of peak anisotropy and the thickness of the anisotropic layer, which both increase with seafloor age, but see little age dependence in the depth to the top of the radially anisotropic layer. We also find strong azimuthal anisotropy, which typically peaks in the asthenosphere. The azimuthal anisotropy at asthenospheric depths aligns better with absolute-plate-motion directions while the anisotropy within the lithosphere aligns better with palaeospreading directions. The relative strengths of radial and azimuthal anisotropy are consistent with A-type olivine fabric. Our findings are generally consistent with an explanation in which corner flow at the ridge leads to the development and freezing-in of anisotropy in the lithosphere, and shear between the lithosphere and underlying asthenosphere leads to anisotropy beneath the plate. We also observe large regions within the Pacific basin where the orientation of anisotropy and the absolute-plate-motion direction differ; this disagreement suggests the presence of shear in the asthenosphere that is not aligned with absolute-plate-motion directions. Azimuthal-anisotropy orientation rotates with depth; the depth of the maximum vertical gradient in the fast-axis orientation tends to be age dependent and agrees well with a thermally controlled lithosphere–asthenosphere boundary. We observe that azimuthal-anisotropy strength at shallow depths depends on half-spreading rate, with higher spreading rates associated with stronger anisotropy. Our model implies that corner flow is more efficient at aligning olivine to form lattice-preferred orientation anisotropy fabrics in the asthenosphere when the spreading rate at the ridge is higher.

     
    more » « less
  5. A computational study of vorticity reconnection, associated with the breaking and reconnection of vortex lines, during vortex cutting by a blade is reported. A series of Navier–Stokes simulations of vortex cutting with different values of the vortex strength are described, and the different phases in the vortex cutting process are compared to those of the more traditional vortex tube reconnection process. Each of the three phases of vortex tube reconnection described by Melander & Hussain ( Phys. Fluids  A, vol. 1(4), 1989, pp. 633–635) are found to have counterparts in the vortex cutting problem, although we also point out numerous differences in the detailed mechanics by which these phases are achieved. Of particular importance in the vortex cutting process is the presence of vorticity generation from the blade surface within the reconnection region and the presence of strong vortex stretching due to the ambient flow about the blade leading edge. A simple exact Navier–Stokes solution is presented that describes the process by which incident vorticity is stretched and carried towards the surface by the ambient flow, and then interacts with and is eventually annihilated by diffusive interaction with vorticity generated at the surface. The model combines a Hiemenz straining flow, a Burgers vortex sheet and a Stokes first problem boundary layer, resulting in a nonlinear ordinary differential equation and a partial differential equation in two scaled time and distance variables that must be solved numerically. The simple model predictions exhibit qualitative agreement with the full numerical simulation results for vorticity annihilation near the leading-edge stagnation point during vortex cutting. 
    more » « less