Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T18:06:46.608Z Has data issue: false hasContentIssue false
  • English
  • Français

A model for the constant-density boundary layer surrounding fire whirls

Published online by Cambridge University Press:  06 August 2020

A. D. Weiss Open the ORCID record for A. D. Weiss [Opens in a new window] ,
P. Rajamanickam ,
W. Coenen Open the ORCID record for W. Coenen [Opens in a new window] ,
A. L. Sánchez Open the ORCID record for A. L. Sánchez [Opens in a new window]  and
F. A. Williams
A. D. Weiss
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA
P. Rajamanickam
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA
W. Coenen*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Leganés (Madrid), Spain
A. L. Sánchez
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA
F. A. Williams
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Reacting Flows: Flames Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A22
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Department of Aerospace Engineering, Auburn University, Auburn, USA.

References

REFERENCES

Baker, G. L. 1981 Boundary layers in laminar vortex flows. PhD thesis, Purdue University.Google Scholar
Batchelor, G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Meteorol. Soc. 80 (345), 339–358.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14 (4), 593–629.CrossRefGoogle Scholar
Burggraf, O. R., Stewartson, K. & Belcher, R. 1971 Boundary layer induced by a potential vortex. Phys. Fluids 14 (9), 1821–1833.CrossRefGoogle Scholar
Byram, G. M. & Martin, R. E. 1962 Fire whirlwinds in the laboratory. Fire Control Notes 33 (1), 13–17.Google Scholar
Candel, S., Durox, D., Schuller, T., Bourgouin, J.-F. & Moeck, J. P. 2014 Dynamics of swirling flames. Annu. Rev. Fluid Mech. 46, 147–173.CrossRefGoogle Scholar
Carpio, J., Coenen, W., Sánchez, A. L., Oran, E. & Williams, F. A. 2020 Numerical description of axisymmetric blue whirls over liquid-fuel pools. P. Combust. Inst. (submitted).Google Scholar
Coenen, W., Kolb, E. J., Sánchez, A. L. & Williams, F. A. 2019 a Observed dependence of characteristics of liquid-pool fires on swirl magnitude. Combust. Flame 205, 1–6.CrossRefGoogle Scholar
Coenen, W., Rajamanickam, P., Weiss, A. D., Sánchez, A. L. & Williams, F. A. 2019 b Swirling flow induced by jets and plumes. Acta Mechanica 230 (6), 2221–2231.CrossRefGoogle Scholar
Emmons, H. W. & Ying, S.-J. 1967 The fire whirl. Proc. Combust. Inst. 11, 475–488.CrossRefGoogle Scholar
Fiedler, B. H. & Rotunno, R. 1986 A theory for the maximum windspeeds in tornado-like vortices. J. Atmos. Sci. 43 (21), 2328–2340.2.0.CO;2>CrossRefGoogle Scholar
Gol'dshtik, M. A. 1960 A paradoxical solution of the Navier–Stokes equations. Appl. Math. Mech. (Sov.) 24 (4), 913–929.CrossRefGoogle Scholar
Gupta, A. K., Lilley, D. G. & Syred, N. 1984 Swirl Flows. Abacus.Google Scholar
Head, M. R., Prahlad, T. S. & Phillips, W. R. C. 1977 Transition to a rising core at the centre of a vortex. Simplified analysis for laminar flow. Aeronaut. Q. 28 (3), 197–210.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Maths 20 (3-4), 1–14.Google Scholar
Hicks, W. M. 1899 II. Researches in vortex motion. Part III. On spiral or gyrostatic vortex aggregates. Philos. Trans. R. Soc. A 192, 33–99.Google Scholar
King, W. S. & Lewellen, W. S. 1964 Boundary-layer similarity solutions for rotating flows with and without magnetic interaction. Phys. Fluids 7 (10), 1674–1680.CrossRefGoogle Scholar
Lagerstrom, P. A. 1988 Matched Asymptotic Expansions: Ideas and Techniques. Springer.CrossRefGoogle Scholar
Lei, J., Liu, N., Zhang, L. & Satoh, K. 2015 Temperature, velocity and air entrainment of fire whirl plume: a comprehensive experimental investigation. Combust. Flame 162 (3), 745–758.CrossRefGoogle Scholar
Li, S., Yao, Q. & Law, C. K. 2019 The bottom boundary-layer structure of fire whirls. Proc. Combust. Inst. 37 (3), 4277–4284.CrossRefGoogle Scholar
List, E. J. 1982 Turbulent jets and plumes. Annu. Rev. Fluid Mech. 14 (1), 189–212.CrossRefGoogle Scholar
Maxworthy, T. 1973 A vorticity source for large-scale dust devils and other comments on naturally occurring columnar vortices. J. Atmos. Sci. 30 (8), 1717–1722.2.0.CO;2>CrossRefGoogle Scholar
Mills, R. H. 1935 The boundary layer for some axial symmetric flows. PhD thesis, California Institute of Technology.Google Scholar
Mullen, J. B. & Maxworthy, T. 1977 A laboratory model of dust devil vortices. Dyn. Atmos. Oceans 1 (3), 181–214.CrossRefGoogle Scholar
Phillips, W. R. C. 1985 On vortex boundary layers. Proc. R. Soc. Lond. A 400 (1819), 253–261.Google Scholar
Potter, J. M. & Riley, N. 1980 Free convection from a heated sphere at large Grashof number. J. Fluid Mech. 100 (4), 769–783.CrossRefGoogle Scholar
Rotunno, R. 1979 A study in tornado-like vortex dynamics. J. Atmos. Sci. 36 (1), 140–155.2.0.CO;2>CrossRefGoogle Scholar
Rotunno, R. 1980 Vorticity dynamics of a convective swirling boundary layer. J. Fluid Mech. 97 (3), 623–640.CrossRefGoogle Scholar
Rotunno, R. 2013 The fluid dynamics of tornadoes. Annu. Rev. Fluid Mech. 45, 59–84.CrossRefGoogle Scholar
Rotunno, R., Bryan, G. H., Nolan, D. S. & Dahl, N. A. 2016 Axisymmetric tornado simulations at high Reynolds number. J. Atmos. Sci. 73 (10), 3843–3854.CrossRefGoogle Scholar
Rouse, H., Yih, C. S. & Humphreys, H. W. 1952 Gravitational convection from a boundary source. Tellus 4 (3), 201–210.CrossRefGoogle Scholar
Schneider, W. 1981 Flow induced by jets and plumes. J. Fluid Mech. 108, 55–65.CrossRefGoogle Scholar
Taylor, G. I. 1958 Flow induced by jets. J. Aerosp. Sci. 25 (7), 464–465.Google Scholar
Tohidi, A., Gollner, M. J. & Xiao, H. 2018 Fire whirls. Annu. Rev. Fluid Mech. 50, 187–213.CrossRefGoogle Scholar
Wilson, T. & Rotunno, R. 1986 Numerical simulation of a laminar end-wall vortex and boundary layer. Phys. Fluids 29 (12), 3993–4005.CrossRefGoogle Scholar
Xiao, H., Gollner, M. J. & Oran, E. S. 2016 From fire whirls to blue whirls and combustion with reduced pollution. Proc. Natl Acad. Sci. 113 (34), 9457–9462.CrossRefGoogle ScholarPubMed