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Section B Fire and Explosion - A Study of Flame Spread in Engineered Cardboard Fuel Beds Part II: Scaling Law Approach


In this second part of a two-part exploration into the dynamic behavior observed in wildland fires, time scales differentiating convective and radiative heat transfer are further explored. Scaling laws for the two different types of heat transfer were considered: radiation-driven fire spread and convection-driven fire spread, which can both occur during wildland fires. A new interpretation of the inertial forces introduced a downstream, time-dependent frequency ω, which captures the dynamic, vortex shedding behavior of flames due to the unstable nature of the turbulent flow created in the wake of the fire. Excelsior and paper strip experiments suggest many wildland fire scenarios fall into the convection-driven spread regime.


  • Inertial Force
  • Strouhal Number
  • Radiative Heat Transfer
  • Diffusion Flame
  • Flame Spread

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Δθ 1 :

Temperature rise of air and gas

Δθ 2 :

Temperature rise of fuel

Δρ 1 :

Density change of air and gas

c 2 :

Specific heat of fuel

c p :

Specific heat of a gas at atmospheric pressure

E :

Irradiance received by radiometer

F b :

Buoyancy force of air and gas

F i :

Inertial force

F i :

Inertial force of air and gas

Fr :

Froude number

g :

Acceleration due to gravity

H :

Fuel height

H f :

Flame height

I :

Fire intensity

l 2 :

Fuel bed width

L a :

Height of flame and fire plume

L e :

Effective length over which the majority of heat transfer occurs (also effective length for vortex shedding)

l H :

Characteristic length, height of fuel

L w :

Flame zone thickness


Ratio of consumed fuel to the total available fuel

Q :

Heat generated

Q c1 :

Heat stored in the air and gas associated with the temperature rise

Q c2 :

Heat stored in the unburned fuel

q f :

Heat value per unit mass of fuel

Q r :

Radiant heat received by the unburned fuel

Q λ :

Latent heat of fuel (heat value per unit mass of fuel)

R :

Spread velocity of flame front

St :

Strouhal number

t :

Characteristic time

u :

Horizontal wind velocity

α :

Fuel bed angle


pi number

ρ 1 :

Gas density

ρ f :

Fuel density


Downstream, shedding frequency


  1. Pyne, S., Andrews, P., Laven, R.: Introduction to Wildland Fires, 2nd edn. Wiley, New York (2000)

    Google Scholar 

  2. Finney, M., Cohen, J., McAllister, S., Jolly, W.: On the need for a theory of wildland fire spread. Int. J. Wildl. Fire 22, 25–36 (2013)

    CrossRef  Google Scholar 

  3. Finney, M., Forthofer, J., Adam, B., Akafuah, N., Saito, K.: A study of flame spread in engineered cardboard fuelbeds, part I: correlations and observations. In: The Seventh ISSM, Submitted, Hirosaki, Japan (2013)

    Google Scholar 

  4. Williams, F.: Scaling mass fires. Fire Res. Abst. Rev. 11, 1 (1969)

    Google Scholar 

  5. Emori, R., Saito, K.: A study of scaling laws in pool and crib fires. Combust. Sci. Technol. 31(5–6), 217–231 (1983)

    CrossRef  Google Scholar 

  6. Emori, R., Iguchi, Y, Saito, K., Wichman, I.: Simplified scale modeling of turbulent flame spread with implication to wildland fires. Fire Safety Science–Proceedings of the Second International Symposium, pp. 263–273 (1988)

    Google Scholar 

  7. Schlichting, H.: Boundary Layer Theory, 7th edn. McGraw-Hill, New York, NY (1979)

    MATH  Google Scholar 

  8. Cetegen, B.M., Dong, Y.: Experiments on the instability modes of buoyant diffusion flames and effects of ambient atmosphere on the instabilities. Exp. Fluids 28(6), 546–558 (2000)

    CrossRef  Google Scholar 

  9. Jiang, X., Luo, K.H.: Dynamics and structure of transitional buoyant jet diffusion flames with side-wall effects. Combust. Flame 133(1), 29–45 (2003)

    CrossRef  Google Scholar 

  10. Gotoda, H., Asano, Y., Chuah, K.H., Kushida, G.: Nonlinear analysis on dynamic behavior of buoyancy-induced flame oscillation under swirling flow. Int. J. Heat Mass Transf. 52(23), 5423–5432 (2009)

    CrossRef  Google Scholar 

  11. Banta, R., Olivier, L., Holloway, E., Kropfli, R., Bartram, B., Cupp, R., Post, M.: Smoke-column observations from two forest fires using Doppler lidar and Doppler radar. J. Appl. Meteorol. 31(11), 1328–1349 (1992)

    CrossRef  Google Scholar 

  12. Clark, T.L., Radke, L., Coen, J., Middleton, D.: Analysis of small-scale convective dynamics in a crown fire using infrared video camera imagery. J. Appl. Meteorol. 38(10), 1401–1420 (1999)

    CrossRef  Google Scholar 

  13. Coen, J., Mahalingam, S., Daily, J.: Infrared imagery of crown-fire dynamics during FROSTFIRE. J. Appl. Meteorol. 43(9), 1241–1259 (2004)

    CrossRef  Google Scholar 

  14. Bejan, A.: Plumes. In: Convection Heat Transfer, 3rd edn, pp. 430–439. Wiley, Hoboken, NJ (2004)

    Google Scholar 

  15. Kottke, V.: Taylor–Görtler vortices and their effect on heat and mass transfer. In: Proceedings of the Eighth International Heat Transfer Conference, San Francisco, CA (1986)

    Google Scholar 

  16. McCormack, P.D., Welker, H., Kelleher, M.: Taylor-Görtler vortices and their effect on heat transfer. J. Heat Transf. 92, 101–112 (1970)

    CrossRef  Google Scholar 

  17. Emori, R., Saito, K., Sekimoto, K.: Scale models in engineering (Mokei Jikken no Riron to Ohyou), 3rd edn. Gihodo, Tokyo (2000)

    Google Scholar 

  18. Kuwana, K., Sekimoto, K., Saito, K., Williams, F.A.: Scaling fire whirls. Fire Saf. J. 43(4), 252–257 (2008)

    CrossRef  Google Scholar 

  19. Kuwana, K., Morishita, S., Dobashi, R., Chuah, K.H., Saito, K.: The burning rate’s effect on the flame length of weak fire whirls. Proc. Combust. Inst. 33(2), 2425–2432 (2011)

    CrossRef  Google Scholar 

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We acknowledge the late Professor Ichiro Emori and his former student Yasuo Iguchi for their pioneering work on scaling laws on flame spread. We also thank Professor Forman Williams for his invaluable discussions on the StFr correlation on wildland fires. This study was supported by USDA Forest Service under Collaboration Forest Service agreement: 12-CS-11221637-133.

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Correspondence to Nelson K. Akafuah .

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Adam, B.A., Akafuah, N.K., Finney, M.A., Forthofer, J., Saito, K. (2015). Section B Fire and Explosion - A Study of Flame Spread in Engineered Cardboard Fuel Beds Part II: Scaling Law Approach. In: Saito, K., Ito, A., Nakamura, Y., Kuwana, K. (eds) Progress in Scale Modeling, Volume II. Springer, Cham.

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