Fire Induced Flow Field- Theory

and Experiment

HOWARD R. BAUM
Center for Fire Research
National Bureau of Standards
Gaithersburg, Maryland 20899, USA

BERNARD J. MCCAFFREY
University of Maryland
College of Engineering
Baltimore, Maryland 21228, USA

ABSTRACT

The complete flow pattern induced by unconfined fires is studied

theoretically and experimentally. The theoretical development is based
on kinematic relationships between the velocity, vorticity, and heat
release fields. The flow both inside and outside a single fire plume is
related to plume centerline velocity and temperature measurements. Very
large area fires, such as those hypothesized in the Nuclear Winter
scenario, are represented as ensembles of individual fires of differing
strengths distributed over randomly chosen sites within the burning area.
The experimental data for fire plumes over a four order of magnitude size
range is compared with these calculations and previously developed plume
velocity and temperature correlations.

INTRODUCTION

The study of flows induced by isolated unconfined fires is of interest

for two reasons. First, this represents the simplest configuration in
which to study the structure of the fire plume and its effect on the
environment, without the additional complexities introduced by enclosure
geometries and ventilation systems. Second, large fires can often be
considered to be composed of one or more such flow systems, so that it is
useful to have a methodology capable of predicting their properties.
While there is a considerable body of literature describing fire induced
flows in enclosures (for an excellent survey see Zukoski (1)), there is
much less information available describing the winds induced by uncon-
fined fires.

An idealization usually made in the study of enclosure fires is that

the fire plume pumps a stationary layer of cool gas through the heat
release region into a nearly quiescent hot layer. The only motions of
interest outside the plume are those induced by the presence of boun-
daries. However, the role of the plume as a fluid pump implies that this
picture is incomplete. The fact that the plume gases entrain air as they
rise means that a flow must be induced in the environment surrounding the
fire. While the mean velocities associated with this flow are small
compared with those in the plume for a single isolated fire, the com-
posite mean horizontal velocity induced by a large number of such fires

FIRE SAFETY SCIENCE- PROCEEDINGS OF THE SECOND INTERNATIONAL SYMPOSIUM, pp. 129-148 129

Copyright © International Association for Fire Safety Science

can considerably exceed the plume vertical velocity. Carrier, et at. (2)
have explicitly noted the need to account for such winds in their study
of urban fire storms.

The first systematic calculation of the mean flow induced outside the
fire plume was carried out by Taylor (3), who replaced the mass entrained
into a point source plume by a line of sinks distributed along the plume
centerline, and calculated the resulting potential flow. This calcula-
tion is the essential starting point for the approach to be followed
here; yet it suffers from two basic limitations. First, even though the
experimental data base available today is much better than that available
to Taylor, so that it is not necessary to assume a point source plume,
the direct measurement of entrainment which yields the sink strength is
still a very difficult one. Second, the replacement of the fire plume by
a line sink is not a good approximation near the edge of the plume
itself. Moreover, this replacement furnishes very little guidance for
the prediction of time dependent phenomena.

The basic difficulty can be traced to the fact that the velocity
field, unlike the temperature field, does not have any sharp cutoff or
change in character at the plume boundary. The flow quantity most like
the temperature is in fact the vorticity, which can only be generated by
solid boundaries and essentially horizontal temperature gradients in
plumes. This is not inconsistent with Taylor's picture, and lends itself
to a "kinematic" approach to fire induced flows. This kinematic ap-
proach, set out in the next two sections below, regards the vorticity and
rate of heat release as the fundamental sources of the flow, both inside
and outside the fire plume. This method of analysis can be applied to
either time dependent or time averaged descriptions of the flow. The
main requirement is the existence of a plume data base from which the
vorticity field can be inferred.

The data needed for the time averaged plume vorticity distribution is
obtained from McCaffrey's (4) plume correlation. This correlation was
assembled from an analysis of small scale laboratory plume centerline
velocity and temperature measurements. Since that time many larger scale
experiments have been performed, permitting the validity of this correla-
tion to be assessed over a range of four orders of magnitude in fire
strength. Time resolved data has also begun to appear. The status of
the relevant experimental data base is discussed in the fourth section.

The "Nuclear Winter" hypothesis (5) has revived interest in the flows
induced by very large fires. Here, the primary feature of interest is
the fire induced convection column covering many kilometers in the
horizontal directions and extending to the height of the atmosphere.
Existing analyses of this phenomenon, whether analytical in nature (2),
(6) or based on detailed time dependent numerical simulations (7)-(9),
have represented the fires themselves as a smooth continuous source of
heat. It is the present authors contention that a more realistic
idealization for these or any other large fires is a random distribution
of distinct fires with the same characteristics they would possess if
isolated. This is certainly true if the fires are all widely separated
from each other. While this is not necessarily always the case in a
large urban fire, the analysis in the fifth section shows that global
rates of heat release postulated in many of the simulations mentioned
above can be readily achieved with distinct fires occupying only a small
fraction of the nominally burning area.

130
 This analysis is primarily aimed at describing the low level flows in
the first few hundred meters above the firebed, before the individual
plumes above each separate fire can merge. Even with this restriction,
the calculation presents some severe computational problems. Even the
time averaged velocity field involves a three order of magnitude range of
length scales in the horizontal directions, and is inherently three
dimensional. This fact, together with the random locations and varying
strengths of the large number (hundreds to thousands) of fires used in
the simulations, requires the development of some novel means of perform-
ing the calculations. Perhaps the most interesting result to emerge is
the prediction of very large ground level horizontal inflows, without the
need to invoke an external swirling velocity field far from the burning
area.

The Kinematic Model

The flow induced by a large fire is primarily determined by three

factors: The geometrical arrangement of the burning parcels of fuel, the
rate of heat released by each individual fuel element, and the at-
mospheric environment in the neighborhood of the firebed. In any given
fire scenario, there is a considerable amount of randomness in each of
these factors. The mathematical model described below attempts to deal
with this situation by combining random distributions of location and
burning rate of the individual fuel elements with a deterministic
description of the composite flow generated by a given realization of the
fire scenario. Each fuel element is assumed to be sufficiently removed
from the others so that the buoyant plume of hot gas and smoke rising
above the burning fuel evolves independently. This assumption implies
that attention is restricted to low altitudes, before the separate plumes
merge into a large convective column. Thus, the atmospheric environment
(winds and stratification) does not playa significant role in what
follows. Under these circumstances, the composite flow field can be
decomposed into a collection of individual flows associated with each
plume. The flow pattern associated with a single plume can then be
analyzed in detail and the results compared with experimental data.

The starting point is the inviscid equations of fluid mechanics which

control the large scale fluid motion of interest. These laws expressing
conservation of mass, momentum, energy, and an equation of state, for a
low Mach number variable density flow are (10).

~ ~

Dt + pVeu ~ 0
~

Du ~

p + Vp -(p -po)g 0 (1)

Dt
DT
P~ Dt Q(;,t)
Po = pRT
~

Here P, u, and T are, the local density, velocity, and temperature in

the gas at position; and time t induced by a rate of energy relea~e
Q(;,t). The gas has a specific heat Cp and gas constant R, while g is
the gravitational acceleration. The quantities Po and Po are ground
level ambient pressure and density, while p represents the small pertur-
bation from hydrostatic pressure that drives the motion. All plumes are
assumed to evolve in a uniform atmosphere at rest far from the plumes.
The convective operator DjDt includes local time dependence:

131
D a -+
Dt at + ueV
It is certainly not feasible to contemplate solving eq. (1) directly.
Instead, we make use of two equivalent results. If the first of eqs. (1)
is multiplied by Cp and added to the third and the equation of state is
used, the divergence of the velocity field can be expressed as:

(2)

Physically, eq. (2) states that the volumetric expansion rate of any
fluid element is proportional to the net rate of heat addition. It is
important to note that although the heat is added to the atmosphere only
in the vicinity of each element of burning fuel, eq. (2) is valid
everywhere above the surface. The second result needed is Bjerknes
theorem (11). The fluid vorticity ~ is defined in terms of the fluid
-+
velocity u as:
-+ -+
Vxu '" w (3)

Now consider a closed loop of Lagrangian marked particles moving with the
local fluid velocity. The circulation r around that closed loop is
defined as:

r '" f-;;'ed~ f~e~dA (4)

.e -+
Here n is a unit vector normal to any surface A bounded by the closed
loop .e of marked particles. The momentum conservation equation, the
second of eq. (1), can then be manipulated into the form:

&
dt
r
~ ~([(p
-+ - -+
-po)/po]g -Vp/p}odr (5)
.e
Equation (5) is Bjerknes theorem. Physically, it states that vor-
ticity can be created in a fluid away from a boundary only through the
mechanism of density gradients. In the absence of such gradients the
circulation around the closed loop, and hence the vorticity contained
within the loop, cannot change.

Now consider the history of a fluid element originating from rest far
from any of the plumes generated by the fire. Initially the fluid
contained no vorticity, and in its approach to the fire it remains
irrotational. As it enters one of the plumes it encounters large density
gradients, and an intense vortex field is created. Thus, eq. (3) can be
rewritten as:

Vx-;;'=~p(~,t) (6)

Here ~p denotes the vorticity in the plumes. Just as in the case of eq.
(2), eq. (6) holds everywhere even though ~p only involves the velocity
gradients inside the plumes.

Two further points should be noted. First, since the right hand sides
of eqs. (2) and (6) involve only information inside the plumes, if the
plumes remain distinct we can write:

(7)

132
 T1ME·AVEAAQ[OCEMTERLIMEVELOCIlY
& TEMP£RATURERlSE

,,
,, 1.2
, !
PIIQTO$~APHIC
T(MH~tl\"MIl -_:,:::,::-+-------------- ----- ------ 1.1)

[XPOSUREI"'1tj
l
SalfTO£llUn,STllETCH,OOillt
flNALLYllRlAK FROM THfANCHORfO
flJl»E8rrwaN THfLOWE!l AND
UPPER INTfRSftTlONSWlfH
IWl.lJ,WflICKEllnt"Af£'/'l11l
I
I;---jlr--i--\---
TOP OFI..UfIMU.UU _ _
AIICHORlO FLAME
fl _
PULSATES CQ~ERENtl~ M
WITHINrtRMln£ltT8£HAVIOll

2 I
AR8ITRARYUNITS

Figure 1. Conceptual picture of buoyant diffusion flame.

Here the sum is taken over each individual plume. Thus, a velocity
field ~i can be associated with each plume, where the velocity field is
the solution to

(8)

Second, these formulae are valid both instantaneously and in a time

averaged sense. If time dependent information about the structure of an
individual fire plume is available, then it can be used in eq. (8). At
present, only time averaged information is available. In either case,
the kinematics of a general vector field insures that a knowledge of the
divergence and curl of the vector is sufficient to determine it every-
where uniquely.

The next step in the analysis is the representation of the vorticity

and rate of heat release fields, the right hand side of eqs. (8). There
is ample experimental support (4) for a Gaussian radial distribution for
the time averaged vertical velocity u and temperature rise T-T o in the
central region of a buoyant plume. Thus, these quantities can be
represented in the form

u = U(z) exp {-lr/R(Z)]Z}

(9)
(T-To)/T o = 6(Z)exp{-[r/AR(Z)]Z}
Here, Z denotes vertical distance above ground level and r radial
distance from the mean plume centerline. The quantity A represents' the
ratio of thermal to velocity plume widths. The time averaged convective
energy flux in the plume at any height, H(Z), is given by:

H(z) = 2~Cpfpu(T-To)rdr (10)

o
Since H(z) is closely related to the heat release distribution, a
quantity presumed known in this study, three more pieces of information
are required to completely specify the profiles given in eqs. (9). In
principle, one could proceed by adopting an integral approach, using the

133
conservation of mass and vertical momentum averaged over the plume cross
section together with an estimate of \ to obtain the necessary informa-
tion. However, that immediately leads to a requirement to estimate the
entrainment rate and the difficulties noted above. The alternative
approach adopted here is to employ the correlations developed in Ref. (4)
to estimate U, e, and \ directly.

Before examining the correlations and their consequences, it is

important to get a better idea of the plume structure. Fig. (1) shows an
idealization of a typical pool fire or low momentum burner flame. There
are three reasonably distinct regions. The lowest region is the con-
tinuous burning zone. There are always many flame sheets anchored to the
fuel bed. The flow is not steady, but pulsates fairly regularly as large
scale eddies are entrained into the plume. The second region is an
intermittent zone containing irregular patches of flame breaking off from
the anchored flame. At the top of the visible flame zone, almost all
combustion has ceased. Above this point is the plume region, charac-
terized by the classical velocity and thermal structure induced by a
weakly buoyant source. The forms chosen are not arbitrary, but make
maximum use of the classical plume theory. The vertical distance,
velocity, and temperature are made dimensionless as follows:

sa Z" z/D* (11)

U/JgD* sa U*(z*) (12)

e sa e*(z*) (13)

In these equations, it is important to note that Qo refers to the

total chemical rate of heat release. A crude accounting for radiative
losses will be made explicitly below.

Th" temperature and velocity correlations provide that U* and e* are

the following functions of z* only:

e* B(Z*)2n-l (14)

TABLE 1. Plume Correlation Parameters

Plume Range n A B

Flame 0<z*<1.32 1/2 2.18 2.91

Intermittent 1.32<z*<3.30 0 2.45 3.81

Plume 3.30<z" -1/3 3.64 8.41

The quantities n,A, and B for each region are given in Table 1. The
functions are sketched at the right of figure 1.

Substitution of eqs. (9) into eq. (10) then yields the following form
for H(z).

H(z) = ITC p p oToR2U(z)[1-I,(e*)] (15)

I (e*) =f1dt {I + e*(t)ln 2} 1

x a

134
 McCaffrey finds that A ~ 0.862 everywhere in the plume represents the
best fit to the data. This value is somewhat awkward to work with, since
the integral I A must then be evaluated numerically. Examination of the
data base used shows that a value A ~ j3/2 = .866 lies well within the
scatter. With this choice of A, the integral I can be evaluated in
closed form as:

I, (a) ~ 3(j2a) 3/4 (log K + r: - arctan L - arctan M)

(a) > (2)-1/2

3(j2a)-3/4 (log K + arctan (-L) - arctan M)

o :5 a :5 (2)-1 1 2
[(a)I/4 _(2) 1 12 ] 2 + 1/2
K (16)
[ (a) 11 4 + (2) 112] 2 + 1/2

L (2)-112[(a)I/4_(2) 1/2J- 1
M (2)-1 12[(a)114 + (2)-1 12J- 1

Eqs. (14)-(16) can now be used to determine complete structure of the

fire plume. At the top of the intermittent flame zone, the combustion
has ceased. Thus, in the plume region, the convective energy flux H(z)
must be constant and equal to the total chemical heat release Qo minus
the fraction radiated away. Denoting this fraction by ry the plume radius
can be readily obtained as:

(17)

In the intermittent flame zone itself, there is still some combustion

occurring, leading to an increase in H(z). However, this is roughly
balanced by a small amount of radiation from the burning eddies. The net
result is that the overall change in H(z) is very small and eq. (17) may
be used all the way down to the end of the continuous flame zone with
little loss in accuracy. Inside the continuous flame zone, energy is not
conserved and H(z) changes rapidly. However, by analyzing the momentum
equation, McCaffrey is able to show that the plume radius R* is ap-
proximately constant. This statement does not hold immediately adjacent
to the burner or fuel bed, where the whole idea of a plume as a slender
object breaks down. However in the upper half of the continuous flame
zone, eq. (17) can be replaced by:

R" = R~ ~ R"(z" ~ 1.32) (18)

The radiative fraction for fires using relatively smoke free hydrocar-
bon fuels is generally in the range 0.25 - 0.35. It can change dramati-
cally, however, for large fires with heavy smoke loading. This is one
reason for wanting to display the radiation loss explicitly. It is also
true that in many fire experiments it is impossible to determine the rate
of heat release directly. The quantity actually measured is the weight
loss of the burning object as a function of time. This information,
together with tests on small scale samples to yield heat release rate per
unit mass leads to an inferred value of the chemical heat release rate
rather than the convective energy flux. Hence, an explicitly correction
is required.

135
 -e-
With U and R determined, the vorticity w is then given by:

:, ~ w¢~ (19)
w¢ = [2U(Z)/R(Z)J{(r/R) exp [-(r/R)2J}

Here ~ is a unit vector in the azimuthal direction. The time averaged

vorticity is seen to be distributed in azimuthal rings about the axis of
symmetry of the plume. The time averaged rate of heat release Q is
related to the function H(z) by:
OJ

~: (z) ~ 2ffIdrrQ(r,z)(1-~) (20)

o
Equation (20) together with the additional assumption of a Gaussian
profile whose width is given by eq. (18) then uniquely determines Q.

The Single Plume Flow Field

The problem of solving for the mean induced velocity field now reduces
to the solution of eqs. (2) and (6) for the prescribed values of Q and :'.
The result is actually a kinematically and (approximately) dynamically
consistent flow field both inside and outside the plume. The calculation
is performed by decomposing the velocity field into solenoidal and
irrotational components.
-+ -+
u ~ \71>+ v (21)

Since the time averaged vorticity and heat release distributions are
axially s~nmetric, the vertical and radial velocity components U z and u r
can be obtained in terms of 1>(r,z) and a pseudo-stream function w(r,z) as
follows:

81> 1 1 8w
8z + r r 8z
(22)

The equations are made non-dimensional by introducing the velocity and

length scales defined in equations (11) and (12). The rate of heat
release is normalized as follows:

(23)

The remaining variables are scaled in the form:

(ur ' uz ) = (gO')1/2 { ~(r* ,z*), ~(r* ,z') }

(1),w) ~ (gO*)1/20" { ;(r* ,z*), >Ir(r* .z") } (24)

w¢=(g/0*)1/2 w(r*,z*); r=O*r*; z=O*z*

Now dropping all tildes and asterisks, the equations for 1> and W
become:

1 1 8w
~ rw(r,z) (25)
r r 8r
Equations (25) must be solved subject to the boundary condition u ~ 0
at the surface z = o. The additional no-slip boundary condition, v = 0,
cannot be enforced in an inviscid model. Far from the heat source in
every direction u and v vanish at least as fast as (r 2 + z2)-1/3, since

136
that is the asymptotic decay rate in the plume. In addition, the spatial
extent of the heat source is bounded. Hence, the potential flow at
infinity must be equivalent to that from a point source. Mathematically,
these conditions can be expressed in the form:

W (r , 0) = ~
8z (r,O) - 0 (26)
Lim (~,w) = 0
r,z ~ 00

Solutions to equations (25) are obtained numerically using finite

difference methods using the FISHPAK (12) separable elliptic equations
solvers. These (or indeed any) algorithms can only give accurate results
when employed on a finite domain. The solutions obtained must retain
their accuracy in the infinite domain, however, if they are to be useful
in large area fire simulations. This can be accomplished by replacing
boundary conditions at infinity by analytical asymptotic solutions to
eqs. (25) and (26). The asymptotic solutions are then assumed to apply
outside a large but finite cylindrical domain. The numerical solutions
are employed inside this domain, with the asymptotic results used as
boundary conditions.

The asymptotic solution corresponding to the potential induced by a

point source is:

(27)

Physically, the volumetric source representing the non-radiating

fraction of the heat released pushes all the fluid into the upper half
plane along straight streamlines originating from the source.

Next consider the solenoidal flow pattern. This is much more inter-
esting, due to the nature of the flow and because it dominates the motion
far from the fire. The asymptotic form of w is very simple when ex-
pressed in terms of a spherical radial coordinate p and polar angle O.

3nAB -4/3 3 nAB

n(O) = 2A [7(1 _ ~) J tanO (cos 0) exp {- y- (1 _ ~) tan 2 O} (28)
p2 = r2 + z2 ; z = p cosO ; r = p sinO
The solution to the second of eqs. (25) with w replaced by wa is obtained
by assuming W to have the form:

(29)

Then, using the variable ~ = cosO in eqs. (25), (28) and (29); the
equation and boundary conditions for F(~) are:
d2F
ct;z + [10/9(1-~2)] F = n(~)

n(~) = - 6nA2B/[7(1-~)](~)-7/3exP{-3nAB(1-~2)/7~2(1-~)} (30)

F (0) = F (1) = 0

Note that the solution to eqs. (30) together with eq. (29) constitutes
an exact solution of the second of eqs. (25) and the boundary conditions.
The solution is obtained as follows:

137
 Introduce a new variable x defined by:

x -(1 + /1-)/2 1/2 :5 x :5 1 (31)

The homogeneous terms in eq. (30) then become:

dZF 5 2
x(l - x)dXT + (~)(~)F - 0 (32)
This is the hypergeometric equation. Let one solution to this
equation which is bounded over the domain of interest be denoted u1 (x ) .
Then (13):
2 5
u1 ( x ) x ZF1 (-~, 3,2,x)

zF 1 ( - ; , ~ ,2 ,x) - [-y(~he; )
J- 1 J~ dt( (l-xt)/(l-t) )z/ 3
(33)

Here ~ denotes the Factorial function.

A second independent solution to the homogeneous equation Uz(x) can

readily be found in the form:

(34)

The solutions U1 and Uz can be used to construct two other independent

solutions Wi and Wz ' such that Wi vanishes at x ~ 1/2 and Wz vanishes at
x-l.

Denoting U1(x) and Uz(x) the solutions given in eqs. (33) and (34)
with the factor (~z(5/3) ~(1/3») 1 removed, the required solutions are:
A 1 A 1
Wi - U1(x) Uz(Z) Uz(x) U1(Z) Wz - Uz(x) (35)
From these two solutions, it is a straightforward task to determine
the solution to eq. (30) in the form:

The solution given in eq. (36) is equivalent to that obtained by

Taylor (3) outside the plume. However, unlike Taylor's result, eq. (36)
is uniformly valid, with the plume flow incorporated in a kinematically
consistent manner. (It is also dynamically consistent subject to the
inherent limitations on our ability to predict turbulent flows).
Figure (2) shows the angular dependence of the spherical polar radial
(Vp ) and angular (Ve ) components of the velocity field as determined from
eqs. (29) and (36). The (p)-1/3 radial dependence is suppressed in the
figure by evaluating the velocities at p ~ 1. The radial velocity
exhibits a strong outflow with a Gaussian profile in the plume centered
at B ~ O. This is exactly compensated by a weak inflow at all other
angles down to the ground at B ~ rr/2. The angular velocity is always
small with no significant structure, vanishing at B ~ 0 and B - rr/2. The
radiative fraction is taken to be ry ~ 0.35 in all calculations.

The asymptotic formulae were used as boundary conditions at the edge

of a right cylinder of radius r - 10 and height z ~ 20. Since eqs. (29)
and (36) are themselves exact solutions to the point source plume
equations, they were employed as a test of the accuracy of the FISHPAK
software for this problem. It was found that a discretization ~r ~ ~z
0.1 was sufficient to guarantee a numerical error in the solutions of eq.
(25) of less than 1/2 percent. Figure (3) shows a comparison of the

138
 ~ \

\\
o
.; .. ~~~-~-~~-- -----

iO.O SO.O 60.0 70.0 60.0 90.0

Figure 2. Angular dependence of radial Vp andtangential (Ve ) components

of velocity for point source plume in spherical polar coordinates.

centerline vertical velocity computed by this method with that given by

the correlation. It is a test of the internal consistency of the
calculation. The lower solid line is the correlation characterized by
table 1, with the upper solid line obtained from the numerical solutions
to eqs. (22) and (25). The worst error is less than four percent. It is
also clear that most of the flow is induced by the vortex field except

3.0

2.5

2.0

Uz
1.5

- - U z (1))
1.0 ------U z <.p)
--Uz

0.5

./\. ~

0 4 8 12 16 20
Z

Figure 3. Comparison of computed centerline velocities (upper solid

curve) with plume velocity correlation (lower solid curve).

139
 0.2
/---
0.1

o
./ /
/ /
0.5
--'------.
1.0 1.5
2.0
R ............. --
-0.1 .., . i - : "
UR

/.//'~-
-0.2

-0.3 u, ,.,
•••••• U r (t/J)
-0.4
",••,,' - - Ur

-0.5

-0.6

Figure 4. Calculated ground level radial velocity showing potential

(dashed) and solenoidal (dotted) contributions to flow.

very close to the flaming region. This is also borne out by the ground
level radial flow, plotted in Fig. (4). The vorticity induces a strong
inflow which peaks near the edge of the plume, which is partially
countered by the expansion induced by the local heat release. This
effort rapidly diminishes with increasing r; so that by r ~ 10 the
vorticity induced flow is overwhelmingly dominant.

The Experimental Database

The mathematical model developed above has been shown to be in

reasonable agreement with correlations developed from small scale
experiments. We now wish to compare the correlations with the experimen-
tal database.

Most of our knowledge of fire plumes, defined here as buoyant dif-

fusion flames from area sources, comes from observations at laboratory
scale, dimensions on the order of 1m. A variety of transducers including
thermocouples, ionization probes, and bidirectional velocity probes have
been inserted directly into these flames and recently several non
intrusive optical devices including LDV's have been used for garnering
further information. What has emerged is a qualitativae or descriptive
picture of the phenomena illustrated in Fig. 1, and a quantitative
description of the time-averaged behavior manifested in the correlations
summarized in Table 1.

Diameter or heat release rate together with fuel type ought to com-
pletely define the problem, at least for a radiatively non-participating,
quiescent, normal oxygen containing environment, i.e. D and Q are
uniquely related for a given fuel type. Hasemi (14), Heskestad (15), and
Cetegen et.al. (16) have all added refinements to the above inter-
pretation in introducing to the temperature data analysis a virtual
source, i.e., the place where one starts measuring z, which is a weak
function of Q*. For simplicity we have ignored these variations since

140
further refinements (Hasemi (14) Appendix) may be required and also it
may turn out that the effects are second order for large pool fires. In
addition, more recent measurements [Rockett, Sugawa, McCaffrey (17)]
indicate that the use of a virtual source actually makes the collapse of
velocity data worse than if one were not used.

How well the small scale-developed analysis represents the behavior of

fires can be estimated from Fig. 5. Time-averaged and scaled width,
centerline velocity, and centerline temperature rise are shown plotted
against scaled height. The lines represent the analytical predictions,
the symbols are available literature data for pool diameters between 2
and 30 m. Two meters is an arbitrary compromise between having too many
points from small fires and yet having a sufficient number of points,
especially higher in the flame plume. Recall that fire size or, im-
plicitly D, is contained in he denominator of scaled height.

In general the agreement between width and velocity data and the cor-
responding analysis seems reasonable. Recall that a Gaussian form was
chosen for the radial variation. For the width plot the line is actually
the lie profile for velocity while the three 15 m JP-4 points are lie
temperature data. Measurements made at small scale indicated that the
temperature rise profile was slightly narrower than the velocity profile.
That difference is ignored here and the velocity profile will be used as
the measure of plume width. (For the 15 m pool, data below 5.7 m
exhibited a bimodel radial distribution and were, consequently not used
for the present comparison since the analysis is no longer appropriate
very near the surface.) Surprisingly, perhaps, the velocity data appear
to scatter less and be in closer agreement with the analysis than does
the temperature data. With the temperature data there appears to be a
tendency for the temperature rise to increase with fire size for the
larger fires. One can speculate that this may be the result of a
thickening and slower moving soot layer surrounding the luminous portion
of the flame plume which might tend to block the radiation from escaping.

0.06 Continuous
:r
b 0.04
Flame I Plume

ii 0.02

01---+-----+----------;
a
2.0 o
t:
o
sw
>
D(m)
o Heptane 2 Yumolo '" K080kl
• JP~4 2x2 Hagglund'" Pernon
a Methanol 2.4 Kung 4<Stavrlanhlla
• JP~4 3.7K7.3 Gordon'" MoMlilan
0 v G080lhul 3.7x7.3 Gordon'" McMillan
0
'"
a: o Hoptane 6 Yumoto &.Kosekl
...
::> • JP~4
• JP~4
10x10 Hllgglund &. Per6llon

..'f"fi
15 NASA
600 8 .. KerOMJna 30 Japan Soc. f(H
SafelY Engr •

ill
... 400

200

oL--'_L...l_--'-_-'-_-'--,...-,-'----,-'c:---,-':-:-'
o a.OS 0.10 0.15 0.20 0.25 0.30 0.35 0.40
HEIGHT

Figure 5. Comparison of large scale data with small scale correlation

and analysis. For data sources see Refs. (28),(18),(31),(29),(32),(33).

141
 TRUPACK 1 OPEN POOL FIRE TEST 2/26/86
5.0 585 MW
I::. Gross fuel burning rate
o 535 MW Estimated total heat release rate
\l 359 MW - Convective heat release rate
(> NASA 15 meter diameter fire
- McCaffrey (10-80 kW firesl

0.5

0.1 '--_ _.L......--'_.L.......L....JL.L-'-.L-'---_ _-L-_'---L--L-LL...L.LJ

0.001 0.005 0.05 0.1

Figure 6. Comparison of fire hundred megawatt pool fire velocity

measurements with correlation.

Decreasing radiant intensity with fire size due to soot blocking has been
observed for increasing D (Hagglund and Persson (18) and for increasing Q
(McCaffrey (19). The increased temperature levels on the centerline seen
on Fig. 5 may be another manifestation of the same phenomenon.

Some very recent data has emerged for velocity in large pools of jet
fuel used to evaluate the integrity of containment of nuclear waste
material. Fig. (6) from Keltner, (20), shows the results of upward gas
velocity measurements in the lower regions of the flames in comparison to
the simple correlations presented earlier. Seen on the figure is a
dashed line of slope 1/2, the approximately value of the slope in the
continuous flame region, that is, up to about 0.08 in the units of the
figure. The solid line is a representation of the actual data near the
burner surface in the small scale result which is ignored in deriving the
simple, 1/2 slope correlation. That is, the further away from the
surface one travels the closer the data approaches the 1/2 slope. If it
turns out that this starting region with time-averaged bimodal tempera-
ture distributions is important in characterizing the phenomena than Fig.
4 is a testament to the facet that the small scale measurements could in
fact be used even to capture this behavior. The laboratory scale solid
line appears to represent the one-half gigawatt data very adequately.
Thus, the correlation is valid over a four order of magnitude range in Q.

We now turn to a discussion of time resolved data. Figure 7 showing

the characteristic pulsation frequency plotted against fire size, D,
attempts to further bridge small and large scale. Shown by open tri-
angles is the data of Byram and Nelson (30)(1970) obtained by counting
the "roaring sounds that accompanied the pulsations" of ethanol pool
fires in a given interval of time. (For the smaller diameter fires high
speed movie film was used). The value of approximately 3 Hz found
earlier (21), (22) in 0.30 m square methane gas burner would fall within
this data scatter. The filled symbols represent the large diameter
Japanese crude oil and kerosene data. The solid line has a slope of -1/2
and represents a relatively smooth transition from small to large scale.
The D to the -1/2 power follows directly from an inviscid-hydrostatic

142
 6 Ethanol
• Crude oil
• Kerosene

f (Hz)

1.0

o (m)

Figure 7. Pool fire pulsation frequency as function of diameter.

analysis with the inverse period being proportional to velocity divided

by distance. Velocity is proportional to square root of diameter and
dividing by distance which is proportional to D yields the requisite
result. A smooth transition in the data does not guarantee the absence
of any new phenomena associated with large fires. It is certainly,
however, a necessary condition for such an absence. For example, in
decreasing size, laminar-like diffusion flames of D < 0.1 m exhibit
frequencies which would not be correlated by the line shown, i.e., fJD
becomes a function of D. In this case therefore possible new phenomena
would be indicated.

To date most of our information comes from time-averaged measurements.

The fire plume however, involves many different time scales ranging from
submicrosecond molecular processes involving chemical changes through
millisecond diffusional transport processes all the way up to seconds
where large scale coherent eddy structures encompass the fluid mechanical
mixing aspects. It is quite clear that time-averaging can only give a
crude picture and will not provide the detailed information necessary for
accurate analytical characterization. The next phase in this development
is to assume the chemistry is infinitely fast and to try and resolve the
slowest (which for these pool configured diffusion flames may be the most
important) namely, to resolve the large scale mixing process. Non-
stationary studies of the structure of the flow are taking place at
several institutions including Waterloo (Wechman, et at (23)),
BorehamwoodjSouthampton (Walker & Moss (24)), Washington State (Fisher,
et al (25)) and Stuttgart (Schonbucher et al (26)). The analytical
approach needed to provide a rational framework for this data has yet to
be developed.

Large Fires

We now turn to the flows induced by fires spread over large urban or
industrial areas. Such mass fires, whether induced by industrial
accidents, earthquake, or war, could extend over many kilometers. As
mentioned above, it is our contention that such a mass fire is composed
of many hundreds or even thousands of individual fires of the type

143
described in the preceding sections. Such fires, provided they are
distinct and separated physically from each other, can be characterized
approximately by the sources (heat release rate and vorticity) that they
would have separately. Under these circumstances the velocity field can
be represented as follows:

(37)

Here, ~(~) is the composite velocity field induc~d by fires at nominal

locations ~jk and of strength Qk. The quantity w is the dimensionless
flow field calculated above for a single fire. The dependence on ambient
quantities is suppressed for clarity.

In practice, the precise locations and strengths of the individual

fires cannot be predicted with any certainty. A more reasonable approach
would be to regard these quantities as random variables, subject to
certain overall constraints. The distribution function governing the
strengths and locations of the fires could be based on the geometry and
material composition of the burning area, together with the ignition
and/or firespread scenario. Figure (8) shows an example of what can be
done. A one kilometer square area is seeded with one hundred fires
distributed uniformly in space. The fires are also distributed uniformly
over five discrete strengths starting at 1.2 x 10 9W, with each successive
size one half the previous value, for a total rate of heat release of
approximately 4.7 x 10 10W. The shaded circles denote the actual burning
areas assuming a combustible fuel density consistent with a heat release
rate of 3 x 10 5W/m2 • This is well below that obtained from burning a
variety of crude oil pool fires. Even so, the figure shows that less
than fifteen percent of the area is actually burning.

The locations and strengths are obtained using a random number

generator, and are subject only to the constraint that no two burning

Figure 8. Ground level flow induced by 100 fires burning in one kilo-
meter square. The total heat release is 4.7xl0 10W and the peak velocity
is 10m/s.

144
areas overlap. Note that the dimensionless velocity fields need only be
calculated once for a given fractional radiative loss ry. The values of
the radial and vertical dimensionless velocity components are stored for
each of the twenty thousand cells in which the numerical solutions are
obtained. Similarity, the angular dependence of the asymptotic solutions
are tabulated for one thousand values of the polar angle. The ground
level velocity components for each of the one hundred fires are then
computed on the 100 x 100 grid illustrated by the tick marks on the
boundary using eq. (37). The velocity vectors at every fourth point are
shown. Speed contours calculated from the full 100 x 100 grid are also
displayed.

The peak values for the computed velocities are approximately ten
meters per second. This may be compared with an experimental mass fire
burn reported by Adams et.al.(27). The nominal burn area was .45 km on a
side surrounded by a 300 M wide cleared area on each side. The peak
convective heat release was 2 x 10 1 0 W, with the brush fuel organized
into thirty continuous rows. The average peak inflow velocity was 4.2
m/with 10m/so total horizontal velocities at several locations reported.
Given the numerous detailed differences between the calculated and
experimental fire scenarios, the relatively good agreement between
computed and measured velocities cannot be regarded as conclusive
evidence for this approach. Still, the results are encouraging; par-
ticularly since no angular momentum induced firewhirl phenomena need be
invoked to explain such relatively large ground level velocities.

For still larger fires, the burning area is divided into one kilometer
squares. This represents a useful resolution scale for describing a
large city or industrial area. The fires within each square are laid
down as described above. Figure (9) shows the flow pattern induced by a
5 km x 1 km mass fire. The calculation of the contribution of each fire
to the flow in its own square is performed as before. However, examina-
tion of eq. (37) shows that the contribution of the fires in remote
squares can be greatly simplified. The inner sum (over j) for fires of a
given size in a single square can be replaced by J times that of a single
fire acting at the centroid of fires of strength Qk for that square.
Thus, the entire flow pattern (involving 500 fires) can be computed in
about 30 seconds of computer time per square kilometer on a CDC Cyber 205
operating in scalar mode. This computing time is quite insensitive to
the overall mass fire size, and is ideally suited for parallel computa-
tion.

Figure 9. Ground level flow induced by 500 fires burning in 5 Km strip.

The total heat release is 2.4xlO l lW and the peak velocity is 50m/s.

145
Figure 10. Detail showing central square of flow in Fig. 9.

A detail from this calculation, the center square kilometer of the

burning region, is shown in Figure (10). Note that although the overall
calculation encompasses an area of 7 Km x 3 Km, individual features with
a length scale of less than ten meters (individual plume half widths) are
preserved with no loss in accuracy. This highly granular structure of
the mass fire model leads to predictions of nearly fifty meters per
second in peak ground level velocity. This type of "subgrid model", which
is based entirely on the theory and experiments described in earlier sec
tions, is completely beyond the capability of purely numerical finite
difference calculations. As an example, the computation illustrated in
Figs. (9) and (10) would require 2-4 x 10 8 grid points to directly solve
for the velocities at a resolution comparable to that displayed. (This
scale computation is well beyond the capabilities of existing computers.)
Moreover, the methods described here have been applied to scenarios
involving many times the area and number of fires shown, without heroic
efforts. Indeed, simulations involving tens of thousands of fires spread
over hundreds of square kilometers are quite feasible with existing
computers.

Concluding Remarks

A methodology for calculating the flows induced by fires has been

presented. The approach requires a blend of analysis, experiment, and
computation that emphasizes the strengths of each of these elements.
Although the research was motivated by scenarios generated for the
Nuclear Winter hypothesis, it offers a new approach for the analysis of
both time averaged and fluctuating phenomena in fires, regardless of
origin. We believe that the insights to be gained from this approach are
not limited to fluid mechanics. The analysis of combustion processes in
particular will require an economical description of the large eddy
structure generated in fire plumes. The evolving experimental data base
together with extensions of the kinematic analysis to time dependent
flows will provide a challenging test of these ideas.

146
Acknowledgement

This research was supported in part by the Defense Nuclear Agency.

REFERENCES

1. Zukoski, E.E., "Fluid Dynamic Aspects of Room Fire", in Fire Safety

Science, Proceedin~s of the First International Symposium, ed. C.E
Grant and P.J. Pagni, pp. 1-23, Hemisphere, Washington, D.C. 1986.

2. Carrier, G.F., Fendell, F., and Feldman, P.S., Comb. and Sci. Tech.,
4, p. 135, 1984.

3. Taylor, G.I., J. Aerospace Sciences 25, p. 464, 1958.

4. McCaffrey, B.J., Combustion and Flame 52, p. 149, 1983.

5. Committee on the Atmospheric Effects of Nuclear Explosions, "The

Effects on the Atmosphere of a Major Nuclear Exchange", National
Academy Press, Washington, D.C., 1985.

6. Carrier, G.F., Fendell, F., and Feldman, P.S., J. Heat Transfer, 107,
p. 19, 1985.

7. Marcus, S., Krueger, S., Rosenblatt, M., "Numerical Simulation of

Near-Surface Environments and Particulate Clouds Generated by Large-
Area Fires", Defense Nuclear Agency Technical Report DNA-TR-87-1,
Washington, D.C. 1987.

8. Small, R.D., Larson, D.A., Remetch, D., and Brode, H.L., "Analysis
of the Burning Region and Plume of a Large Fire", Defense Nuclear
Agency Technical Report DNA-TR-86-401, Washington, D.C. 1986.

9. Cotton, W.R., American Scientist, 73, p. 275, 1985.

10. Rehm, R.G., and Baum, H.R., J. Res. Nat. Bur. Stds, 83, p. 297, 1978.

11. Turner, J.S., Buoyancy Effects in Fluids, pp. 6-9, Cambridge

University Press, Cambridge, 1973.

12. Boisvert, R.F., Howe, S.E., and Kahaner, D.K., "Guide to Available
Mathematical Software", National Bureau of Standards Report NBSIR
84-2824, Washington, D.C. 1984.

13. Oberhettinger, F., "Hypergeometric Functions", in Handbook of

Mathematical Functions with Formulas Graphs. and Mathematical
Tables, ed. M. Abramowitz and I.A. Stegun, National Bureau of
Standards Applied Mathematical Series 55, pp. 555-567, U.S. Govt.
Printing Office, Washington, D.C. 1964.

14. Hasemi, Y. and Tokunaga, T., J. Heat Transfer, 108, p. 882, 1986.

15. Heskestad, G., Fire Safety Journal, 5, p. 109, 1983.

16. Cetegen, B., Zukoski, E.E., and Kubota, T., "Entrainment and Flame
Geometry of Fire Plumes", National Bureau of Standards Report GCR
82-402, Washington, D.C. 1982.

147
17. Rockett, J., Sugawa, 0., and McCaffrey, B., in preparation.

18. Hagglund, B. and Persson, L.E., "The Heat Radiation from Petroleum
Fires", FaA Rapport C 20126-D6(A3), Stockholm, 1976.

19. McCaffrey, B.J., Some Measurements of the Radiative Power Output of

Diffusion Flames", 1981 Western States Combustion Institute Meeting,
Pullman, Paper WSS/CI 81-15, 1981.

20. Keltner, N., private communication, 1986.

21. McCaffrey, B.J., "Purely Buoyant Diffusion Flames", National Bureau

of Standards Report NBSIR 79-1910, (1979).

22. Cox, G., and Chitty, R., Fire and Materials 6, p. 127, 1982.

23. Wechman, E. J ., "The Structure of the F1owfie1d Near the Base of a

Medium-Scale Pool Fire", Ph.D. Thesis, University of Waterloo,
Waterloo, 1987.

24. Walker, N.L. and Moss, J.B., Comb. Sci. and Tech. 41, p. 43, 1984.

25. Fischer, S.J., Hardouin-Duparc, B., and Grosshandler, W.L.,

Combustion and Flame, 70, p. 291, 1987.

26. Schonbucher, A., Arnold, B., Ba11uff, C., Bieller, V., Barotz, W.,
Kasper, H., Kaufmann, M., and Schiess, N., "Simultaneous Observa-
tions of Organized Density Structures and the Visible Field in Pool
Fires", Twenty-First International Symposium on Combustion, The
Combustion Institute, Pittsburgh, in press.

27. Adams, J.S., Williams, D.W., and Tregellas-Williams, J., "Air

Velocity, Temperature, and Radiant-Heat Measurements Within and
Around a Large Free-Burning Fire", Fourteenth Symposium (Int'l) on
Combustion, p. 1045, The Combustion Institute, Pittsburgh, 1973.

28. Yumoto, T., and Loseki, H. "Effect of Tank Diameter on Combustion

Characteristics of Heptane Tank Fires", Report of Fire Research
Institute of Japan 55 (in Japanese), 1983.

29. Gordon, W., and McMillan, R.D., Fire Technology 1, p. 52, 1965.

30. Byran, G.M., and Nelson,R.M., Fire Technology 6, p. 102, 1970.

31. Kung, H.S., and Starrianidis, P., "Buoyant Plumes of Large-Scale

Pool Fires", Nineteenth Symposium (International) on Combustion,
p. 905, The Combustion Institute, Pittsburgh, 1982.

32. Raj, P.K., "Analysis of JP-4 Fire Test Data and Development of a
Simple Fire Model", ASME Paper 81-HT-17, 1981.

33. Anon, "Report of Oil Fire Experiment", Japan Society of Safety

Engineering (Kerosene) (in Japanese), 1981.

148