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Scaling analyses based on subsurface layer instability were performed to explore the role of three independent (surface tension, gravity, and viscosity) influences on the mechanism of pulsating flame spread under normal and microgravity conditions. These three influences form two independent pi-numbers: the Marangoni (Ma) number and Grashof (Gr) number, which include the characteristic length scale ratio (depth of subsurface circulation)/(horizontal length of preheated liquid surface). The Prandtl (Pr) number was introduced to compensate for the different thermal diffusivity and kinematic viscosity of different liquids. Also a nondimensional flame spread rate,

Flame
spread over liquids is one of our current interests, because of both its relevance to fire safety and our basic
curiosity about its complex mechanism. A series of studies was performed to
understand the flame spread mechanism on alcohols [

Over the past ten years, a series of experimental studies
of flame spread over liquids has been conducted [

Step in a single
cycle of pulsating flame spread and surface wave in step (b) [

To
benefit from all these experimental and numerical studies both under normal
gravity and microgravity and to update the theory of flame spread over liquids
formulated by Williams [

A shadowgraph
and a particle laser-sheet image defining thermal characteristic length,

William
used the liquid pool depth as an approximation for the characteristic length in
his flame spread theory and pointed out that it is both possible and desirable
to pursue more careful analysis to obtain improved estimates of spread rate
using an appropriate characteristic length [

Our
objective is to study the instability of laminar subsurface layer flow ahead of
a spreading flame. The temperature coefficient of surface tension and other
physical properties are treated as constants. Our instability analysis is based
on the previous study [

Subsurface layer flow model and symbols.

If the surface perturbation
is expressed as

The neutral stable line obtained from (_{i}

The neutral stable line.

The area above the neutral stable
line in Figure

In experiments, uniform flame spread changes to pulsation at a certain temperature below the flash point. The Marangoni force is proportional to the temperature difference between the hot zone underneath the flame leading edge and the bulk liquid temperature. When the liquid bulk temperature is lower, the Marangoni force is greater. This may lead to a subsurface flow rate increase and the generation of a surface wave.

An
overall energy balance and an overall momentum balance for the subsurface layer,
respectively, produce the following:

A schematic of the
experimental apparatus is shown in Figure

Experimental apparatus.

The fuel container is 480 mm long

Temperature structures created in the liquid were visualized by
our specially designed high sensitivity shadowgraph (HSSG) [

The microgravity
experiments were conducted using the 4.5-second drop tower facility at Microgravity
Laboratory of Japan (MGLAB). A 4.5-second period of microgravity is obtained by
allowing the experiment to free-fall in an evacuated tube through a distance of
about 150 m. A schematic of the test rack for microgravity tests is shown in
Figure

Experimental apparatus for a shallow liquid pool in microgravity.

The
package housed the test rack, containing a wind tunnel, a fuel tray, a fuel
delivery syringe, three video cameras, and a gas control system. A fuel tray (

Figure

Marangoni number versus characteristic depth of subsurface layer circulation.

The
critical Ma number under microgravity
is much larger than that under normal gravity, suggesting that when the liquid surface
becomes unstable and a surface wave is formed under microgravity, the value

Figure

Ratio of characteristic length and characteristic depth of subsurface layer circulation versus initial fuel layer depth (fuel is n-propanol and tray width = 20 mm).

The quotient

The thermal field is targeted in the experiment, while the
theoretical analysis targets the flow field. Therefore, the shadowgraph and
PTLS methods were used to measure the thermal scale and the flow scale at the
same time. Figure

Thermal and flow characteristic depth of subsurface layer circulation versus initial fuel layer depth.

The characteristic depth in n-butanol was larger than the
depth in n-propanol by a factor of about 1.4. This value is close to the 1.3
ratio of the viscosity coefficients of n-propanol and n-butanol. As a result, when
a dimensionless number is used, it corrects it by the Prandtl (Pr)
number. The Pr number is defined as
follows:

The nondimensional flame spread rate may be
influenced by the quenching distance

In a previous scaling analysis [

By submitting (

The time averaged flame spread rates under subflash conditions
measured in this study (for four different fuels: methanol, ethanol,
n-propanol, and n-butanol) and experimental data from [

Flame spread rate as a function
of for four different alcohols (

Instability and scaling analyses were conducted on experimental data quantifying flame spread over liquid fuels obtained both under normal and microgravity by other researchers and by our group. We arrived at the following conclusions.

We found
four important (Ma, Gr, Fr, We) pi-numbers and a characteristic length
scale ratio,

The flow and
thermal characteristic depths of subsurface circulation were measured by our
specially designed high sensitivity shadowgraph (HSSG) combined with a
particle-track laser-sheet (PTLS) technique. The flow characteristic depth for
n-butanol was twice as deep as the thermal characteristic depth. Based on these
experimental results, the characteristic depth scale ratio:

We
correlated the nondimensional flame spread rate,

Amplitude of surface wave, m

Thermal diffusivity, m^{2}/s

Forward and reverse propagation velocities of dynamic wave, m/s

Propagation velocity of kinematic wave, m/s

Wave velocity, m/s

Diffusion coefficient of fuel vapor, m^{2}/s

Froude number

Acceleration due to
gravity, m/s^{2}

Initial fuel layer depth, m

Sub-surface layer depth, m

Flow characteristic depth of sub-surface layer circulation, m

Thermal characteristic depth of the sub-surface layer circulation, m

Characteristic length scale ratio

Wave number

Thermal characteristic length of the sub-surface layer circulation, m

Marangoni number

Prandtl number

Heat flux from the liquid surface, W/m^{2}

Temperature, K

Liquid bulk temperature, K

Flash point temperature, K

Local
liquid velocity

Liquid surface velocity, m/s

Average
liquid velocity

Flame spread rate (time averaged), m/s

Vapor diffusion rate, m/s

Weber number

Flow direction coordinate, m

Normal direction to the wall, m

Heat release factor

Momentum displacement thickness

Characteristic depth scale ratio

Temperature
difference between flash point,

Quenching distance, m

Liquid volumetric flow rate per unit width of the tray ^{2}/s

Critical
flow rate required for surface wave to occur, m^{2}/s

Wavelength, m

Thermal conductivity of the liquid, W/(m

Kinematic
viscosity, m^{2}/s

Density
of the liquid, kg/m^{3}

Surface tension force, N/m

Temperature derivative of surface tension coefficient, N/m

Shear stress in the liquid, Pa

Surface shear stress due to Marangoni effect, Pa

Liquid phase (bulk)

Interface between gas and liquid phase

Partial derivatives in time

Partial
derivatives in the

Time-averaged quantity

Perturbation quantity.

This study was supported in part by NASA under Grant NAG3-2567 and in part by the Japan Space Forum on “Ground-based Research Program for Space Utilization.” We would like to thank Dr. K. Kuwana and Dr. F. Miller for valuable technical discussions.