Unconfined deflagrations occur at nearly constant pressure. Constant pressure conditions can be obtained using the soap bubble technique.1 A soap bubble can be filled with a reactive mixture, which is then ignited. The present study addresses the dynamics of such weakly confined deflagrations, focusing on the effect of buoyant forces. We wish to verify experimentally the scaling laws suggested by Babkin et al.2 and Zingale and Dursi3 for determining the flame size at which buoyant effects become important.

For relatively fast burning flames, the flame takes on a spherically symmetric structure, as shown in Fig. 1. Buoyancy forces do not have time to operate. For weaker mixtures, i.e., mixtures characterized by relatively low flame speeds, we find that the flames remain spherical initially, but begin to rise, as shown in Figs. 2 and 3. This phenomenon is due to the gravitational acceleration (buoyancy forces) preferentially accelerating the light combustion products contained inside the flame kernel.

The characteristic flame size for which buoyancy is expected to play a role can be obtained by equating the characteristic time for flame growth by diffusive-convective effects (as in the absence of gravity) and the time scale for buoyancy effects.3 The time scale for the flame to grow to a characteristic dimension R is

$t_{burn}=R/\dot{R}=\left(R/S\right) \left(\rho _u / \rho _b \right)^{-1}$
tburn=R/Ṙ=R/Sρu/ρb1⁠, where S is the laminar flame speed, ρub is gas expansion ratio, and the subscripts b and u refer, respectively, to the burned and unburned gases. The time scale for buoyancy forces to displace the flame bubble the same characteristic distance R is trise = R/Vrise, where Vrise is the characteristic rise speed of the bubble due to buoyant forces. The latter can be estimated as being the terminal velocity of a non-reactive buoyant bubble, where the inertia dominated drag force balances the buoyant force.3,4 Adapting the result of Davies and Taylor4 to a bubble with non-negligible density, we obtain
$V_{rise}=2/3 \sqrt{Rg (1- (\rho _u / \rho _b)^{-1})}$
Vrise=2/3Rg(1(ρu/ρb)1)
.

The ratio of the time scales for burning and flame kernel advection becomes

\begin{equation}\frac{t_{burn}}{t_{rise}}=\frac{2}{3}\frac{1}{S} (\rho _u / \rho _b)^{-1}\sqrt{Rg (1-(\rho _u / \rho _b)^{-1})}.\end{equation}
tburntrise=231S(ρu/ρb)1Rg(1(ρu/ρb)1).
(1)

Without the 2/3 factor, it is the square root of the Richardson number.

Setting this time scale ratio to unity provides an estimate of the critical flame dimension in which buoyant forces will begin to play a substantial role:3 

\begin{equation}R_*= \frac{9}{4} \frac{S^2}{g} \left( \frac{\rho _b}{\rho _u} \right) ^{-2} \left( 1-\frac{\rho _b}{\rho _u} \right) ^{-1} .\end{equation}
R*=94S2gρbρu21ρbρu1.
(2)

We can compare the above criterion with the experiments shown in Figs. 1–3. For a 12%H2 flame, we predict a detaching flame radius of 70 cm, larger than the field of view (Fig. 1). On the scale of the visualization, the flame does not rise noticeably. For the 9%H2 and 7%H2 flames, the predicted critical radii are 10 cm and 3 cm, respectively. This is in good agreement with our experiments, where flame kernels of characteristic dimensions comparable to these estimates detach from the bottom wall (Figs. 2 and 3).

The authors are grateful for financial support on this project from Atomic Energy Canada Limited (AECL) - Chalk River under Contract No. 189-513710-021-000/RSD-12-11 and from the National Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant to M.I.R. and through the NSERC H2CAN Strategic Network of Excellence.

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