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IMBSIR 84-2841

Buoyancy-Induced Wall Flow

Due to Fire in A Room

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

May 1984

Final Report

U.S. DEPARTMENT OF COMMERCE

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NBSIR 84-2841

BUOYANCY-INDUCED WALL FLOW

DUE TO FIRE IN A ROOM

Yogesh Jaluria

Mechanical and Aerospace

Engineering Department

Rutgers University

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

May 1 984

Final Report

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. EmMt Amblar. Director
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TABLE OF CONTENTS

Page

LIST OF FIGURES Iv

NOMENCLATURE vi

Abstract 1

1. INTRODUCTION 3

2. THE PROBLEM 5

2.1 Background 5 2.2 Objectives of the Study 7

3. EXPERIMENTAL RESULTS 8

4. RELATIVE MAGNITUDE OF WALL FLOWS 12

5. WALL FLOW PENETRATION ACROSS THE INTERFACE 21

5.1 Flow in Stratified and Negative ly-Buoyant Situations 21

5.2 Penetration in a Three-Layer System 24

5.3 Penetration in a Two-Layer System 28

5.4 Concluding Remarks on Penetration 30

6. FLOW OF FIRE PLUME ACROSS THE INTERFACE 30

7. ESTIMATION OF WALL FLOW EFFECTS 37

8. SUMMARY AND CONCLUSIONS 45

9. FUTURE WORK 48

10. ACKNOWLEDGEMENTS 51

11. REFERENCES 52

-lii-
LIST OF FIGURES

Page

Figure 1. Flow configuration considered 37

Figure 2. Sketch of the temperature distribution in a two-layer

zone mode 1 58

Figure 3. Sketch of the velocity and temperature profiles in the

wall flow 58

Figure 4. Flow in the negatively-buoyant wall flow and in a corresponding jet 59

Figure 5. Interaction of opposing wall flows.... 59

Figure 6. Interaction of a fire plume with the stably stratified

upper layer 60

Figure 7. Experimental arrangement 61

Figure 8. Measured gas and wall temperatures in the room for a circular burner and Q = 6 3.2 kW 62

Figure 9. Measured gas and wall temperatures in the room for a circular burner and Q = 6 3.2 kW, at a different

location... 63

Figure 10. ^feasured gas and wall tenperatures for a single line

burner and Q

= 31.6 kW 64

Figure 11. ^feasured gas and wall temperatures for the line burner

at various door openings and heat input values.... 65

Figure 12. Transient wall and gas tenperatures for a line burner

and Q

= 31.6 kW, with a 1.83 m x 0.24 m door opening..... 66

Figure 13. Actual and idealized tenperature profiles 67

Figure 14. Actual and idealized boundary layer wall flow 67

Figure 15. Mass flow rate in the wall flow, as a function of the

height Z 68

Figure 16. A conparison between the wall flow rate and the plume

flow rate, for a circular burner at Q

= 63.2 kW and a 1.83 m X 0.74 m door opening 69

Figure 17. Mass flow rate and the momentum flow rate in the

laminar wall flow, for various models considered in

the negatively-buoyant region, employing values from

Fig. 8 70

-Iv-
Figure 18. Momentum flow rate for various heights Li of the

isothermal lower region, considering laminar flow

in both the zones 71

Figure 19. Momentum flow rate for turbulent flow in the two zones... 72

Figure 20. Mass flow for laminar and turbulent flow, as a function

of the height of the lower isothermal zone 73

Figure 21. Dependence of the location of the separation point on

Lj for turbulent flow 74

Figure 22. Momentum flow at various values of and L

2 for

turbulent flow 75

Figure 23. Dependence of the height of the separation point on the

height L

2 of the stratified intermediate layer 76

Figure 24. Variaton of the mass and momentum flow across the

stratified layer with L

2

* 77

Figure 25. Mass and momentum flow for a two-layer system in laminar

flow, considering various models for the negativelybuoyant circumstance...... 78

Figure 26. Mass and momentum flow for a two-layer system in

turbulent flow 79

Figure 27. Penetration depth as a function of interface height

for turbulent flow in a two-zone system 80

Figure 28. Sketch of the tenperature and velocity profiles as the

fire plume flows across the interface 81

Figure 29. Mass flow rate as the wall flow penetrates the

stratified intermediate layer 82

Figure 30. Velocity profiles, for the inviscid and the viscous

models, as the wall flow penetrates the stratified

layer 83

Figure 31. Mass flow rate in the plume as it flows across the

stratified layer for various values of Lj and maximum

temperature defect D^, at Q

= 63.2 kW 84

Figure 32. Mass flow rate in the plume for various values of the

temperature defect D^ for Q

= 63.2 kW..... 85

Figure 33. Besults for plume flow across the interface, from the

Integral model of Ref. [38] for Q

= 63.2 kW 86

Figure 34. Variation of the upward penetrating wall flow rate with

the interface height for a two-zone model 8"

-V-
NOMENCLATURE

2 horizontal cross-sectional area of the room, m

specific heat at constant pressure, J/kgK

width of the fire plume, defined in Eqn, (25), m

heat input parameter, defined in Eqn. (24)

magnitude of gravitational acceleration, m/s

Grashof number, defined in Eqn, (17)

Grashof number at transition from laminar to turbulent flow

convective heat transfer coefficient at the floor, W/m K

height of the room, m

height of lower isothermal layer, m

height of intermediate stratified layer, m

mass flow rate in the wall flow per unit flow perimeter, kg/ms

total mass flow rate in the wall flow, kg/s

mass flow rates defined in Fig. 1, kg/s

mass flow rate of the fuel, kg/s

outflow mass flow rate at the opening, kg/s

mass flow rate in the plume entering the upper layer, kg/s

mass flow peeled off from the plume at the interface, kg/s

momentum flow rate in the wall flow per unit perimeter, kg/s

O total momentum flow rate in the wall flow, kg m/s^

coefficient which gives the variation of flow region thickness

with height, 6 a Z*'

flow perimeter for the wall flow, m

Prandtl number, Pr = v/a

heat input into the fire, kW

convective heat input into the plume, kW
w

T

AT

y

z

z

a

a

3

6

X

V

P

T

convective energy flux in wall flow per unit perimeter, W/m

total convective energy flux in wall flow, W

radial distance from fire plume axis, m

temperature, K

difference between wall and gas temperatures, K

floor temperature, K

outside ambient tenperatur e , K

vertical velocity, m/s

vertical velocity parameter in Eqns. (4) and (12), m/s

centerline velocity in the pliime

vertical distance above the top of the lower isothermal layer, ro

horizontal coordinate distance from the walls, m

vertical coordinate distance from room floor, m

interface height, m

entrainment coefficient, Eqn. (21)

O thermal diffusivity, m /s

coefficient of thermal expansion, 3 = 1/T for perfect gas,

boundary layer thickness, m

constant, defined in Eqn. (25)

kinematic viscosity, m /s

density, kg/m

t ime , s

-vii-
Subscripts

u

1

g

w

00

O

C

upper layer

lower layer

gas

wall

local conditions outside the flow region

conditions at the level of the heat source

centerline conditions in the plume

-vill-
BUOYANCY- INDUCED WALL FLOW DUE TO FIRE IN A ROOM

Yogesh Jaluria

Abstract

A study of the buoyancy-driven flow generated adjacent to the vertical

walls of a room due to fire in the room has been carried out. Considering a

room with a single opening, experimental measurements of the wall and the gas

temperatures in the room were made. Though the emphasis was on the steady and

quasi-steady state situations, some measurements were also taken at the very

early stages of the fire to consider the transient variation of the temperature levels. The boundary layer flow that arises over the vertical walls due

to the resulting difference between the wall and gas temperatures was

analyzed, employing the integral analysis method. The flow rate, momentum and

convected energy in the downward flow that arises in the heated upper layer

and those in the upward flow that is generated in the cooler lower layer were

determined. As the fluid flows from one layer into the other, through an

intermediate stratified region with large temperature gradients, it becomes

negatively buoyant, penetrates to a finite distance and then falls back due to

negative buoyancy. These circumstances are also analyzed by the use of an

integral analysis and calculations are carried out for selected experimental

results. The flow is predicted to separate from the wall at the location

where the downward and the upward momenta of the two opposing flows are equal.

The separation point and the region near it where the flow starts separating

from the wall are studied. Results are obtained for a three-layer model,

which incorporates an intermediate stratified layer lying between two isothermal regions, and are eventually considered in terms of the two-layer approxi-

- 1 -
mation used in zone models. The penetration of the fire plume into the upper

layer is also considered in terms of the Integral approach and a simple model

based on the dynamics of the fluid elements in the flow.

For the range of experimental data considered, it is found from this

analysis that the wall flow is quite significant as compared to the flow in

the plume and should, in general, be taken into account in the mass balance

considerations of the two layers. The energy transfer across the interface

due to the wall flow is relatively small and may often be neglected.

Depending on the temperature levels and the height of the interface, the

upward flow in the lower region may penetrate into the upper layer or the

downward flow in the upper layer may be injected into the lower layer. The

study indicates the method for determining the direction and magnitude of the

penetrating flow. Though more detailed and more accurate computations are

included, an approximate method for evaluating the wall effects for a twolayer model and for an experimental study is outlined. The penetration of the

plume into the upper layer is studied and it is found that only a small fraction of the flow is unable to penetrate the Intermediate stratified layer. As

a first approximation, the mass flow rate may be taken as unaltered across the

interface. Employing this approximation, results are presented on the plume

flow in the upper layer, as obtained from the integral analysis. Finally, the

study indicates future experimental and analytical work that may be carried

out to answer some of the basic questions raised here and to provide further

Information on the wall and plume flow penetration across the interface.

Key words: wall flows, plume penetration, enclosure fires, zonemodeling, stratified media, density Interface.

- 2 -
1 . INTRODUCTION

In recent years, there has been a considerable interest and research

activity in the buoyancy-driven flow of air and gases in room fires, with a

view to determine the changing environment within the enclosure and to provide

inputs for the eventual spread of combustion products into other connected

enclosures [1-4]. Much of the present information on the development of the

fire, with respect to the temperatures in the room and the movement of gases,

has been obtained from zone modeling, which divides the room into distinct

homogeneous regions, generally the hot upper gas zone and the cooler lower

zone [5-7]. The zone models have received corroboration from detailed experimental measurements of temperature and velocity distributions in enclosures

with ventilation from single openings [8-11]. The zone models predict the

average gas and wall temperatures in the various regions and do not provide

Information on local conditions, which may be obtained from field model

studies which have also been carried out in recent years [12,13].

The basic features of the zone model for fire in a room with an opening,

such as a door or a window, are shown in Fig. 1. As a fire starts in a room,

buoyant hot fluid rises from the fire, entrains fresh air from the room and

moves towards the ceiling in the form of a buoyant plume. This flow generates

a distinct layer of hot and, thus, lighter gas below the ceiling. The layer

increases in thickness as the plume brings more hot fluid into the layer. The

hot gases eventually start flowing out of the room, while the cold air from

outside flows into the room due to the resulting pressure difference. As

estimated by zone modeling studies and also observed in several experiments,

fires of any thermal significance (> 10 kW) in typical residential rooms

- 3 -
display a rapid increase in the upper layer thickness, its interface with the

lower layer reaching the opening in a matter of seconds. However, conventional walls have a much slower time response than the gas layers [2] and,

following the rapid increase in the tenperatures in the two layers, the walls

continue to increase in temperature at a very gradual rate. Therefore, an

initially rapid increase in the gas tenperatures is followed by slowly varying

wall and gas temperatures. This has led to the modeling of many practical

circumstances as quasi-steady, with the flow and tenperature distributions

progressing from one essentially steady situation to another, until rapid

pyrolysis of combustible material in the room, followed by room flashover,

occurs.

The air entrainment by the plume and the flow rate through room openings

are important inputs into the modeling of fires in enclosures. Several

studies of entrainment in fire plumes have been carried out [14,15] and the

results have been compared with those for an idealized point source plume. A

larger entrainment has been obtained in the actual case and the entrainment

has been found to be influenced by the size of the source, by disturbances in

the room, by the incoming air flow through the openings, by the proximity to

walls and by several other effects [16,17]. It is, therefore, important to

develop analytical procedures for an accurate determination of plume entrainment under various operating conditions. In addition, the flow of the plume

across the Interface, between the upper and lower layers, is important for

modeling of the flow.

Due to the difference between the temperatures of the gas and the

adjacent walls that are being heated up due to convective and radiative heat

-4-
transfer, a significant buoyancy-induced flow may arise adjacent to the walls

and may lead to additional mixing across the interface. At the initial stages

of the fire in a room, the upper layer generates a downward flow adjacent to

the cooler upper region of the wall. The flow emerges from the upper layer,

becomes negatively buoyant in the lower layer and rises towards the interface

to cause greater mixing in the lower zone, near the interface. At later

stages, the lower region of the wall, in contact with the lower layer, heats

up and an upward flow is also generated in the cooler lower layer. These

opposing streams of wall flows may be important in the mass and energy

balances of the zones for modeling. The present study was undertaken to study

the wall flows thus generated in enclosure fires. Also of interest were the

nature of the interface between the two layers, with respect to temperature

distributions in an actual full-scale experimental system, and the penetration

of the buoyancy-driven wall flows and of the fire plume across the interface.

2. THE PROBLEM

2.1 Background

Let us consider some of the important physical aspects relevant to a

study of tenperatures and flows due to fire in an enclosure. In terms of a

two-layer zone model, the upper layer is characterized by a uniform average

temperature T and the lower layer by T_ i , as shown in Fig. 2. Similarly,

6 » ^ 6 »-*- the wall temperatures may be considered in terms of an upper region value Tw , u and a lower region value T ,. Thus, a step change in the temperatures is W ,

J. assumed to occur at the interface height In actual practice, of course, a

step change would be replaced by a more gradual variation, as shown In Fig. 1

- 5 -
and discussed later in terras of experiraental results. Now, as is evident from

Fig. 2, the wall temperature is higher than the adjacent gas temperature in

the lower layer and lower in the upper layer. Therefore, a buoyancy-induced

wall flow rises in the lower layer and descends in the upper layer. The

boundary layer flows that arise adjacent to heated or cooled vertical surfaces

have been studied extensively [18]. A qualitative sketch of these flows is

shown in Fig. 3, along with the corresponding velocity and temperature distributions.

In the absence of the upper opposing flow, a buoyant lower flow arises in

the lower layer, penetrates the interface, becomes negatively buoyant as it

encounters the high temperature in the upper layer, stagnates at a certain

height and drops towards the Interface due to the negative buoyancy. This is

similar to the case of a negatively buoyant jet, as shown in Fig. 4 [19-22].

In the presence of an opposing stream, the flow stagnates and separates from

the wall, either above or below the interface, as shown in Fig. 5. The fire

plume is, on the other hand, generally, buoyant with respect to the upper

layer and rises in the upper layer towards the celling. It may, however, shed

some flow at the interface due to negative buoyancy in the outer regions of

the plume. This "peeling off" phenomenon can manifest Itself as a multi-layer

counter current stratified flow as has been observed In several experimental

studies [1,23] Also, If the heat Input Into the fire Is reduced after the

layers are established, the plume may be unable to penetrate Into the upper

layer, as shown In Fig. 6 [24,25].

There are several Important questions that must be answered In a consideration of wall flows. The most significant one relates to the magnitude of

- 6-
the flow generated in an actual room fire situation. The wall flow rate may

be compared with the flow in the fire plume and also with that entering

through the opening to determine the importance of this effect. The second

question is whether this flow is able to enter the hot upper layer, despite

the opposing descending wall flow and the stable thermal stratification that

exists in the room. The third question relates to how these considerations

modify the present two-layer zone models and experimental studies of enclosure

fires. Besides these three crucial questions, there are several related

considerations. It is important to consider the nature of the interface in an

actual fire situation and to determine the mixing across it due to wall flows.

Also of interest is the penetration of the fire plume across the interface and

the determination of the magnitude of the flow that is unable to penetrate

into the upper layer.

2.2 Objectives of the Study

This study was undertaken to determine the nature and magnitude of the

buoyancy-driven wall flows generated by fire in an enclosure. It is to be

ascertained whether the effects due to these flows are important in the

consideration of the mass and energy balances in the present two-layer models,

which are employed to predict temperature levels and flows in the room. To

the author's knowledge, all existing zone models neglect the wall flow.

In addition to the above consideration of the wall flow effects, it Is

also desirable to determine the magnitude of the fire plume flow which does

not penetrate into the upper layer, since this flow should also be Included In

the overall energy and mass balances. A consideration of the experimentally

- 7 -
observed interface region in terms of the step change assumed by a two-layer

model is also needed. Such discrimination bears on the whole concept of

penetration across the interface.

In summary, the basic objectives of the study are to develop a methodology to evaluate the wall flow effects, in terms of their magnitude and

penetration across the interface, and to employ the methodology in an analysis

of recent full-scale enclosure fire experiments and in incorporating these

effects in existing two-layer zone models for enclosure fires.

3. EXPERIMENTAL RESULTS

The study considers steady and quasi-steady situations that arise in an

enclosure fire. This will be based on fires of 30 to 130 kW energy release in

a typical residential room with a single opening. The analysis and the

results obtained are not expected to apply for the very early stages of the

fire, following ignition, since transient effects are important and may have

to be included for a realistic analysis of the flow. For the experimental

arrangement considered in this study, it can be shown, employing the results

presented in Ref. [18], that the boundary layer flow takes about 2 seconds to

be established. This is of the same order as the time taken for the interface

to descend to the door soffit. For the experimental arrangement considered in

Refs. [10,11] and employed here, a steady state circumstance is attained, for

a constant heat release fire, after the transient heating of the walls is

complete. Therefore, this study examines wall flows for such steady or quasisteady cases.

-8-
Experimental work was carried out to determine the wall and gas temperature profiles generated by fire in a room. The experiments were conducted in

a 2.8 m x 2.8 ra x 2.18 m room shown in Fig. 7. Essentially steady state

conditions are attained in this arrangement within about 30 minutes following

ignition of the burner. The experiments employed circular and line burners,

the latter being a long slot supplied by an array of circular holes. The gas

flow rate and the length of the line burner could be adjusted. Velocity and

temperature measurements were taken by means of bidirectional velocity probes

and thermocouples. Bare thermocouples were Imbedded in the walls and the

ceiling to obtain wall temperature data. Further details on the experimental

arrangement and the measurement techniques may be obtained from Refs. [10,11].

Several experiments were carried out with various burner locations and

configurations, door openings and fire heat Inputs. Some of the typical

results obtained are shown in Figs. 8-11 (the room symbol on the figures

indicates the burner location and door size [10,11]) and Table 1 for the two

burner configurations. It is apparent that the results depend on the heat

input, the door opening, the dimensions of which are given in terms of the

notation included in Table 1, and the burner location. However, the basic

form of the profiles is similar in all the cases shown. The wall temperature

increases gradually from the floor and then more sharply as the upper layer is

approached, attaining essentially uniform temperature in the upper layer in

most cases. The gas tenperature shows greater uniformity in the two layers

and a much sharper temperature rise from the lower to the upper layer. The

gas temperature starts increasing gradually as the upper layer is approached

and then rises very rapidly, over relatively a very short distance. It then

gradually attains the upper layer temperature. Even the region of rapid

-9 -
temperature variation spreads over a distance of about 20 cm, being smaller in

some cases and much larger in others, particularly at small door openings

which give rise to much greater mixing in the room [11]. Only a few profiles

are shown here for brevity and similar trends were observed at other operating

conditions.

It is evident from the results presented that a two-layer zone model can

be employed to obtain average temperatures in the two zones. Gas temperatures

are fairly uniform in the two layers and quite a sharp change in the temperature occurs in going from one layer to the other. However, it is not clear

where one might locate the interface for a two-layer model. Following a

suggestion by Emmons [52]

,

Steckler [10] has considered this question in

detail. In replacing the actual measured profile with an idealized two-layer

profile, the two profiles are made equivalent by requiring equal mass and

equal values of the integral / Tdz. The second condition is, therefore,

arbitrary, though for a constant density fluid it would imply equal thermal

energy in the two profiles. For gases, since the product of density and

temperature is essentially a constant, a similar constraint on energy does not

lead to any useful constraint on the choice of the idealized profile. The

method discussed by Steckler [10] was shown to yield satisfactory results with

respect to flow rates and other physical variables measured experimentally.

Some experimental observations were also made in order to determine the

position of the interface as indicated by the occurrence of a stable stratification. Employing smoke, introduced into the upper layer, a fairly distinct

interface was observed. The upper region was seen to be fairly disturbed and

the lower region, away from the plume, relatively quiescent. A comparison was

- 10 -
made between the interface height observed by means of the smoke and the

measured temperature profile. Ihe interface observed with smoke was found to

be quite close to that obtained by the method outlined above [10] . The height

in the former case versus that in the latter for four experiments yielded

1.27, 1.09, 1.38, l.OA m as compared to the corresponding values of 1.4, 1.1,

1.38, 1.15 m. Obviously, more work is needed on the relationship between the

interface as defined by a two-layer temperature profile and observations of

stable stratification by some visualization technique is needed. Some visualization of the wall flows, by means of a mineral oil smoke generator, was also

attempted in this study. A downward flow adjacent to the wall was observed in

the upper region and an upward flow in the lower region. The region near the

interface was found to be very disturbed and a distinct flow direction could

not be discerned.

The experimental study also measured opening flow rates, as outlined in

Ref. [11]. Ihble 1 shows the results for a few configurations and heat

inputs. Time-dependent temperature distributions were also measured. It was

observed that the upper layer descends to the door soffit within a few

seconds, generally 2-5 seconds. This is followed by a transient process while

the walls, floor and the ceiling are heated. Within about 30 minutes, the

variation in the temperature profiles was found to become quite small. During

a tjrplcal experiment, such as that shown in Fig. 8, wall temperatures in the

upper and lower regions varied by only about 5% over a period of more than 30

minutes following the initial transient. Therefore, measurements were taken

in an essentially steady-state situation. Following the initial transient of

a few minutes, the interface is more or less established and a quasi-steady

approach would be quite appropriate while the walls are heating up. After

- 11 -
about 30 minutes, the conditions may be taken as steady. Fig. 12 shows the

measured variation with time of the gas and wall tenperatures in the upper and

lower layers for a single line burner, with 31.6 kW input and 1.83 m x 0.24 m

door opening. It is seen that the system may be taken as quasi-steady after a

few minutes. These experimental results will serve as the basis for analysis.

The next section describes the computational methods employed for estimating

the wall flow rate and the degree of its penetration across the hot-cold

Interface and for considering the nature of plume flow across the interface.

Wall and gas temperature profiles representative of Fig. 8 will be used as the

conditions of stratification. For this experiment, the values of Tg T^

T T 0 and Z. are obtained as 241°C, 184°C, 110°C, 50°C and 1.42 m, w,* g,Jl 1

» 1 » »

respectively. Variations from these experimental data will also be considered.

4. RELATIVE MAGNITUDE OF WALL FLOWS

With information now available on the tenperature profiles, we may

proceed to a consideration of the buoyancy-induced flows generated adjacent to

the walls. The actual wall and gas temperature distributions are of the form

shown in Fig. 13(a). The two-layer zone model requires an idealized distribution of the form shown in Fig. 13(b). Though our interest lies mainly in this

distribution for zone modeling purposes, the nature of flow penetration and

various other aspects outlined earlier can be best considered in terms of a

three-layer approximation, which considers uniform temperatures in the upper

and lower regions and an intermediate region where sharp changes in the

temperature occur from one region to the other. Though the actual profiles

may be employed for such a consideration, a more convenient approach would be

- 12-
to consider a profile of the form shown in Fig. 13(c), where a linear temperature variation exists between the two layers. This approximation has been

used by several researchers in the study of stratified water bodies, where

zone models have yielded results which closely approximate the experimental

results [26-29]

.

The wall flows generated in the two isothermal regions may be analyzed as

boundary layer flows adjacent to isothermal surfaces immersed in uniform

temperature media. In the upper region the wall is colder than the ambient

fluid and a downward flow arises, whereas in the lower region an upward flow

is generated. These flows encounter a stably-stratified region as they go

from one layer to the other, through the step change of Fig. 13(b) or the more

gradual variation of Fig. 13(c). Therefore, we need to consider the flow in

the two Isothermal regions, followed by a consideration of flow in the negative ly-buoyant or stably-stratified regions.

A considerable amount of work has been done on laminar and turbulent

natural convection flows adjacent to isothermal vertical surfaces in isothermal ambient media [18]. For the coordinate system of Fig. 14, the vertical

velocity u and the temperature distribution may be determined for laminar flow

from similarity analysis [30,31]. One could then calculate the flow rate

m, convected energy q and momentum M in the boundary layer. These are

obtained, per unit flow perimeter, from the following expressions:

6 m = / pudy ( 1) o

6 q = / PC u (t-T„) dy (2)

o

^

- 13-
( 3 )

where Cp is the fluid specific heat at constant pressure, is the local gas

tenperature, p is the fluid density, which varies as 1/T for the assumed

perfect gas, and 6 is the boundary layer thickness. For the exact boundary

layer similarity analysis, 6 is replaced by infinity and the velocity and

temperature distributions from the numerical solution of the governing equations are employed. An approximate integral analysis has also been carried

out [32] . It yields results quite close to the exact analysis for laminar

flow. The integral analysis is probably more appropriate here, since there is

no satisfactory exact analysis for turbulent flow, which is of interest in

many cases, and since the numerical values of the integrals required for

determining the above physical quantities are not readily available.

The integral analysis assumes the velocity and temperature distributions,

substitutes these in the governing integral equations and solves for the

unknown quantities like 6 , maximum velocity, etc. The distributions employed

for laminar flow are:

where U and 6 are functions of the height Z. The governing equations are

obtained as ordinary differential equations for U and 6. These are:

T - T

(4) w

VU

6 (5)

3

|^(SU/30) =|5i-

( 6 )

- 14-
where the Bousslnesq approximation, - p = 3p(T-T^), has been employed.

Here, g is the acceleration due to gravity, v the fluid kinematic viscosity,

a its thermal diffusivlty and 0 the coefficient of thermal expansion. For the

temperature differences, AT = observed in the experimental study, the

Bousslnesq approximation is quite satisfactory. These equations yield U

and 6 as:

U = 5.17 V (Pr.ff)

20^- 1 /2

V

1/2 „l/2

(7)

6 3.93 Pr“^^^ (Pr + —)20>i 1/4 |- g0AT> ~l/4

( 8 )

where, Pr is the Prandtl number and is given by Pr = v/a. Similarly, m, M and

q may be determined. From the above integral expressions, these are obtained

for the constant density approximation as:

m = p6u/12 (9)

M = p6u^/105 (10)

q = pC 6mT/30 (11)

P

In this study, p was taken as constant in each of the two regions, with

different values, evaluated at the average of the wall and gas temperatures,

being employed for each layer.

Similarly, for turbulent flow, various analyses have been carried out

[33-35], supported by various experimental studies [18,36]. The profiles

employed by several workers are [33]

- 15 -
( 12)

Hie governing ordinary differential equations are obtained as:

0.0523 (6U^) = 0.125 gpATS - 0.0225 (13)

0.0366 -^ (6U) = 0.0225 U Pr”^^^ (14)

If the boundary layer is assumed to be fully turbulent from the leading edge,

Z = 0, the expressions for U and 6 are obtained as:

U = 1«185|- (Gr)^^^[l + 0.494 (Pr)^^^]”^^^

6 = 0.565 Z (Gr)“^^^® Pr”®^^^ [1 + 0.494 (Pr) 2/3jl/10

where Gr is the Grashof nimber and is given by

(15)

(16)

Gr = g3AT Z^/v^ (17)

Again the net flow rate, momentum and converted energy in the boundary layer

may be obtained, per unit perimeter of the flow region, as:

m = 0.1463 p6U (18)

M = 0.052 3 p6u2

(19)

- 16-
( 20 ) q = 0^366 pc 61BT

P

The maximuni vertical velocity obtained as 0.537 U.

Therefore, the physical quantities of interest may be obtained from the

above expressions for heat, mass and momentum. More accurate results may be

obtained from the other analytical and experimental studies mentioned above.

A question of accuracy may be raised regarding the application of the above

boundary layer analysis since considerable turbulent flow and mixing occurs in

the region away from the walls. To some extent, this effect can be taken into

consideration by determining the location where the boundary layer flow undergoes transition to turbulence. Transition occurs earlier if the flow away

from the walls is highly turbtilent. In addition to this effect, the turbulent

flow outside the boundary layer gives rise to a non-zero velocity in the

ambient region. Consequently, a pure natural convection flow is not obtained

and the mixed convection effects may have to be considered. As reviewed by

Jaluria [18], this effect may be quantized and the overall variation in the

flow rate, momentum flow rate and the convected thermal energy may be determined. For the range of velocities generally encountered in these flows, as

estimated from the flow in the fire plume, it can be shown from current

knowledge that the effect is not very large. However, a detailed numerical

solution would be needed to account for the complete three-dimensional characteristics of this problem. A word may also be said about property variations.

In the above expressions, fluid properties are taken as constant. The values

were determined at the average of the local wall and gas temperatures. For

the range of AT observed experimentally, this approach is quite accurate

[18], If fluid property variations are to be incorporated in the analysis,

the governing equations and the results obtained would need to be suitably

- 17 -
modified. However, it was numerically found to be a small effect for the

experimental range of tenperature difference considered for the wall flow.

In actual practice, the boundary layer is laminar near the leading edge,

undergoes transition downstream and eventually attains fully-developed turbulence, as shown in Fig. 14. This problem is obviously a complicated one and a

detailed study becomes quite involved [37]. One may assume laminar flow up to

a certain point, followed by turbulent flow downstream. This implies a step

change from laminar to turbulent flow and is an approximation of the actual

circumstance. This requires a solution of the governing equations for turbulent flow, Eqns. (13) and (14), with inputs from the laminar flow which

applies up to transition. A numerical solution, employing the Runge-Kutta

method, was obtained. The values at the first step are obtained from the

expressions for 6 and U given above and numerical solution of the equations

yields the downstream values. For various values chosen for the Grashof

number Gr^ at which transition occurs, the numerical solution was obtained by

matching the flow rate and the momentum across the transition location. Fig.

15 shows the flow rate in kg/s per m of the flow perimeter for various conditions. The results are shown mainly for AT = 60°C, a typical value obtained

from Fig. 8. One case for AT = 30^C is also shown.

It is obvious from Fig. 15 that the boundary layer mass flow rate depends

strongly on the transitional Grashof nunber. Many earlier studies have

Q indicated the value of Gr^ to be around 5 x 10 . However, it can be as large

as 10 and as low as 10 , depending on the background disturbance. For a

relatively quiescent compartment fire problem, the former value is expected to

apply and for a disturbed one, the latter. For the upper zone, the flow is

- 18-
generally very disturbed and a f ully-turbulent flow may be expected over much

of the region. It is, therefore, clear that there is uncertainty regarding

the onset of turbulence. The two extreme cases of laminar and completely

turbulent flow may be considered and an average taken to represent the

resulting values. This approach appears reasonable in terms of the results

shown in Fig. 15. One needs to know the position of the interface so that the

expressions for laminar and turbulent flow, given above, may be employed to

obtain the flow rate, momentum and convected energy due to the wall flow. Let

us consider the data of Fig. 8 and determine the upward lower region flow at

the interface. From Fig. 8, the Interface for a two-layer model may be taken

at a height of 1.42 m, for which the laminar solution gives a flow rate in the

lower region at the interface height of 0.0121 kg/ms and the turbulent one of

0.0225 kg/ms. The opening flow measured from this case was 0.424 kg/s. For a

flow perimeter of about 10 m, which applies for the experimental arrangement,

the upward wall flow in the lower region ranges from 28.5% to 53% of the

opening flow rate. An average of the two gives a value of 41%, which

Indicates the level of importance of the wall flow in this case. This average

value of the upward flow at the interface was obtained as 42.5, 39.4 and 21.6%

for test numbers 169, 300 and 303 of Table 1.

A similar comparison of the estimated wall flows may be made with the

corresponding fire plume flow rate. Following the model of Morton, Taylor and

Turner [38] for a point heat source, the plume width d, centerline velocity

and the centerline density difference Ap

^ may be obtained in terms of the heat

input into the convective flow. Ihe relevant expressions are:

d =

5

Z ( 21 )

- 19-
( 22)

-1/3 2-5/3 (23)

where

g(X^ + 1)

(24)

Here, the subscript o refers to the ambient conditions at the level of the

heat source, a is the entrainment coefficient and X a constant that conpares

the spread of the thermal field with that of the velocity field in the plume.

The radial velocity and density distributions at any height Z have been

assumed to the Gaussian, i.e. ,

The constants a and X may be taken as 0.1042 and 1.15, as based on the work of

Yokoi [39] and Zukoski et al. [15] and derived by Evans [40]. The numerical

results from this model were also conpared with the measurements of McCaffrey

[14] , indicating a fairly good agreement far from the source.

As measured by McCaffrey [41], about 20% of the input energy is lost as

radiation for a methane flame. Employing this result to determine the

convected energy Q^, the flow rate in the fire plume was determined. Fig. 16

shows a comparison between the wall flow rate and the plume flow rate, as a

function of the height Z. For the experimental interface location at 1.42 m,

corresponding to the data of Fig. 8, the fractional strength of the wall flow

(25)

-20 -
varies from 27.5% in laminar flow to 51% in fully turbulent flow, giving an

average value of 39.3%. At two values of Gr^, the corresponding results are

also shown in Fig. 16. The relative significance of the wall flow increases

as the interface height decreases, indicating the dominant effect of a

decrease in entrainment height on the plume flow rate. These results are

shown for Q

= 63.2 kW, at the conditions of Fig. 8, and the interface height

obtained is around 1.42 m. It is, therefore, seen that a significant flow

rate, as compared to the flow in the fire plume and to that through the

opening, is generated adjacent to the walls. If this flow penetrates into the

upper layer from below or the lower layer from the upper region, it conveys

air and gases across the interface. This implies a transport of mass and

energy from one layer to the other and should be included with other mechanisms of transport across the interface. Hie next step in our consideration

of the wall flows is a study of the penetration of these flows across the

Interface. We need to determine if the flows do penetrate, under what conditions they do so and what the magnitude of the penetrating flow is. These

questions are considered in detail in the next section.

5. WALL FLOW PENETRATION ACROSS THE INTERFACE

5.1 Flow in Stratified and Negatively-Buoyant Situations

As mentioned in the preceding section and as seen in Fig. 13(c), the wall

flow rising in the lower region, or descending in the upper region, encounters

a stably stratified region before it emerges in another Isothermal region

where it is negatively buoyant. In fact, the flow is buoyant as it initially

enters the stratified region, between the upper and lower Isothermal regions,

- 21 -
and as It moves downstream, its buoyancy decreases, becomes zero and then

negative. Some flow may be shed in this region and the flow that does penetrate into the isothermal region would turn back after attaining a certain

height of penetration. For the idealized two-layer system of Fig. 13(b), the

stratified region is of zero thickness and a step change in temperature is

assumed. This implies a negatively buoyant flow as the flow moves from one

layer to the other. Both these circumstances are considered here, the actual

or the three-layer profile being of interest when experimental data are available on the profiles and the two-layer system being applicable to the zonemodeling studies. The two are obviously interrelated and the ultimate focus

for modeling purposes is on the two-layer approximation.

Let us first consider the three-zone system of Fig. 13(c), which involves

a linearly stratified region of height L

2 lying between a lower Isothermal

region of height , and an upper isothermal region of height L-Lj-L2

, where L

is the total height of the enclosure. Interest lies in studying the flow

through this stratified region and then considering the situation as L

2 0,

the limiting case being the two-layer zone model of Fig. 13(b). This is

clearly a very complex flow situation and boundary layer approximations will

not apply when reverse flow or shedding occurs. A proper approach would be to

consider the full governing partial differential equations and obtain the

velocity and tenperature fields numerically, employing various finite difference or finite element methods available. However, that would require a

considerable effort. Approximate results may be obtained by employing integral methods, as outlined earlier. The relevant information on the profiles

and other boundary layer variables may be obtained from earlier exact analysis

of the boundary layer flow. This is the approach considered here.

-22-
The natural convection flow over a heated surface Immersed in a stablystratified medium has been considered by several investigators [42-45]. The

problem has been treated in these studies as a boundary layer flow, with the

ambient and surface tenperatures varying as specified. The linear ambient

temperature variation for an Isothermal surface is of considerable practical

interest and has received much of the attention in literature. It can be seen

from these studies that the velocity and the temperature distributions are

similar in form to those for an isothermal medium over the considered parametric range. The temperature in the outer region of the flow has been found

to be less than the local ambient temperature and this tenperature defect has

been observed to be on the order of 5% of the overall temperature difference

across the flow. If this effect is neglected, which is justified when the

parametric range of condition in the present work is compared with that of

Ref. [43], the velocity and temperature distributions may be taken as:

for laminar flow and, similarly, Eqn. (12) for turbulent flow. The above

assumption clearly constrains the tenperature variation in the flow, though

the profiles have been found to be satisfactory for an unstratlfled medium.

The variation of the gas tenperature with height is numerically treated as

steps of isothermal regions as is customary in a finite difference approximation. The solution of the governing momentum and energy equations, Eqns. (5)

and (6) or Eqns. (13) and (14), is obtained numerically with these step

changes. This procedure is often employed, with good success, with the moving

boundary heat transfer problems.

(26)

-23 -
5.2 Penetration In a Three-Layer System

The laminar results for the mass flow rate per unit perimeter, m, and for

the momentum flow rate, M, are shown in Fig. 17. Ifere, is taken at a

typical value of 1.2 m and L

2 at 0.4 m. Results for the unstratifled, or

isothermal, ambient medium case are shown at T -i . Ihe tenperature difference, T i ~ T , is taken as 60°C. The governing equations are first solved

w >

J- a > until the momentum and, hence, the flow becomes zero. This is shown as the

boundary layer solution in Fig. 17 and is similar to the treatment for

negatively buoyant plumes by Iforton [21]. The flow continues to increase

because of entrainment and the momentum decreases because of buoyancy, after

an initial increase, since T > T up to the point of crossover, beyond which

w g

T < T . Beyond a height of (Li + L^), (T - T ) is constant and taken as

57°C. All the values employed here are typical values taken from the experimental results of Fig. 8. Therefore, the flow increases due to entrainment,

while the momentum decreases sharply. Ultimately, the momentum becomes zero

and the flow turns horizontally.

Obviously, the boundary layer treatment of the flow is not physically

realistic. As pointed out in Ref. [46], shedding of the flow from the

boundary layer must start before the stagnation point (beyond which no flow

penetrates) is reached. The tenperature of this shed flow is not known. But

it will be cooler than the local ambient and would fall downwards. A

vertically rising flow adjacent to the wall and a descending flow outside the

rising flow region are expected to arise [20,22] . This is a complex nonboundary layer phenomenon, which requires a much more elaborate treatment than

that outlined above. However, in order to approximate the flow and obtain

-24 -
estimates of the flow and momentum in this region, some further assumptions

and approximations may be made. The validity of these assumptions can be

ascertained by detailed experimentation or by an exact solution, analytical or

numerical, of the governing elliptic partial differential equations.

An assumption is made regarding the variation of the boundary layer

thickness 6 in the region of negative buoyancy where becomes less than

T . It is assumed that 6 varies as z’^, where n is a constant. For the

olaminar boundary layer flow, it can be shown analytically that n = 0.25 and

for the turbulent flow, n = 0.7 [18]. A Froude nunber dependence, based on

local velocity and temperature levels, may also be employed [46], though that

would also be arbitrary. With this assunptlon, the momentum decrease beyond

the point where T = T may be determined. The Integral equation is: W g

This gives Eqn. (5) with the assuned profiles, which obviously do not take the

shear due to the reversing shed flow into account. However, as seen for

recirculating flow in enclosures [18], the reverse flow and the wall flow are

generally separated by a relatively stagnant fluid. The flow shed from the

wall flow may then be obtained from the assumed 6 variation and the computed

upward momentum. The tenperature of the outflowing fluid, if needed, may be

obtained from the integral energy equation. While several objections may be

raised against this assunptlon, it is probably consistent with the approximate

nature of this analysis and a much more elaborate analysis will be needed to

consider the shed flow, its descent towards the Interface and its consequent

effect on the wall flow [47].

(27)

-25 -
Fig. 17 shows the flow rate and the momentum flow rate at three values of

n. As n increases, the results approach that of the boundary layer analysis

that gives an increasing flow which is finally turned horizontally when

momentum becomes zero. The basic trends are similar and the n = 0.25 case is

simply the upstream boundary layer variation. It is this case which is

considered in greater detail later because it is a more realistic variation

[46]. For n = 0, the boundary layer thickness remains constant after

becomes less than T . For all these values of n, the flow decreases along

O

with the momentum, as may be realistically expected in a stratified flow.

Fig. 18 shows the momentum flow rate at various values of Lj, the other

variables being kept unaltered from those of Fig. 17. The downward flow

momentum is also shown. The point where the two momenta are equal is

predicted to be the separation point from the wall. Therefore, if the upward

flow is stronger at the interface, the flow penetrates into the upper layer.

Otherwise the downward flow mixes with the lower layer. The turning of the

flow away from the wall would obviously occur upstream of the separation

point. This distance may be expected to be of the order of the boundary layer

thickness [48] and an estimate of the temperature and flow rate of the turned

flow may be determined by considering a small control volume in the neighborhood of the separation point. It can be easily shown that the shed flow wotild

be negatively buoyant and will flow towards the interface. In fact, if its

temperature, which lies between T and T , is assumed, the governing momentum

w g

and energy equations may be solved to yield results quite similar to those

obtained with the above treatment.

The results for a fully turbulent wall flow were also obtained and Fig.

19 shows the variation of momentum for several values of Lj and also for the

- 26-
downflow, thereby Indicating the separation point. Fig. 20 shows th'e corresponding flow rates. The turbulent results are shown for n = 0.7, which

applies for a turbulent boundary layer. Fig. 21 shows the dependence of the

height of the separation point on Lj. The results are quite similar for

laminar and turbulent flow and the height of penetration, beyond Lj , may be

determined from these results.

Fig. 22 shows the dependence of penetration on both Lj and L

2

, the height

of the stratified region. The temperature differences of Fig. 8 are employed.

The results are shown at Lj = 0.8, 1.0 and 1.2 m. For the first value of Lj

,

curves for L

2 = 0 .2 and 0.3 m are shown. For the second value of Lj, results

are shown for L

2 = 0.2 m. For the last case, results are shown at L

2 = 0.1,

0.2, 0.4, 0.5 and 0.6 m. This is an important consideration since L

2 0 for

the two-layer model and it is desirable to determine the results that apply

for that idealized case. In Fig. 22, the upward momentum is shown in solid

lines and the downward momentum in broken lines. It is seen that, in all

cases, the flow initially increases and then decreases as the flow becomes

negatively buoyant. The computed momentum at the top of the stratified region

is found to decrease as L2 Increases. A similar trend is found for the flow

rate, though the Inaccuracy of the numerical results increases as L

2 decreases. However, the results Indicate that as L

2 0, the flow momentum

change across the stratified region becomes smaller. Fig. 23 shows the height

of the separation point at two values of Lj for various values of L

2

* A

stronger effect is seen at the smaller Lj , since the flow in the lower layer

is weaker and the downward flow may penetrate into the lower layer Instead.

Fig. 24 shows m and M at the top of the stratified layer for various values of

L

2 at Lj = 1.2 m. It is interesting to note that M decreases gradually with

-27 -
L

2

» whereas ni changes very little. It may be mentioned that the points at

L

2 = 0 obtained by extrapolation give an increase in m and M of 5 and 4%,

respectively, over the corresponding values just below the stratified region.

The above result is an interesting one since it implies a small change

in m and M across a stratified region whose thickness goes to zero, l.e. , for

a step change. However, it must be remembered that the numerical results are

not very accurate at small values of L

2

and such an extrapolation may not be

valid. Still, it points out a few Interesting features which may be employed

for the simulation of a two-layer model, as discussed below.

5.3 Penetration in a Two-Layer System

We may now consider flow penetration in a two-zone system, such as the

one shown in Fig. 13(b). As the wall flow from below crosses the interface,

it is subjected to a step change in the ambient temperature. Under the action

of the strong buoyancy forces and the large temperature gradients that arise

across the flow, the tenperature profile will adjust in a very short distance

to attain the wall temperature, at y

= 0 and the ambient temperature,

Tci „> at y

= 6. In actual practice, of course, there is a finite region over

which this adjustment occurs and a three-layer model, as discussed above, may

be employed to consider the adjustment. As suggested by Fig. 24, m and M may

be taken as unchanged while T changes from T . to T . This temperature

change Implies a considerable energy flux into the flow due to convective flow

and diffusion. This can be achieved by a substantial amount of mixing at the

interface, as observed in the experiments, which raises the temperature in the

flow to the resulting profile. Such a mixing process will obviously lead to a

- 28-
finite amount of entrainment Into the flow, which being negatively-buoyant is

expected to shed flow further downstream. In view of the disturbed nature of

this penetration, an analytical or experimental study of the flow in the

region of the interface is very desirable. If the entire increase in

converted energy q is assumed to come from entrainment, a step change in m

occurs along with that in q, contrary to what is observed for a finite stratified region. Since there are shear and buoyancy forces, M will change downstream, though right across the interface it may be taken as unchanged. For

the results presented here for the step change in T , m and M are taken as

O

unchanged across the interface and q obviously changes. A more realistic

approximation would be a change in m along with q. But more information on

the mixing process is needed for a satisfactory model. Due to large temperature gradients across the flow, a substantial amount of energy transfer also

occurs due to convective transport mechanisms.

Fig. 25 shows the mass flow rate, m, and the momentum flow rate, M, per

unit perimeter for laminar flow. Two values of n, along with the boundary

layer analysis results are shown. The separation point may be determined by

employing the upward flow momentum, as before. Fig, 26 shows the results for

turbulent flow. Fig. 27 shows the penetration distance 6 as a function of

P

the interface height with and without opposing flow for the turbulent

case. A considerable difference between the two is seen. Without the

opposing stream, the flow always penetrates, with 6^ becoming smaller as the

interface height becomes smaller. But with opposing flow, the penetration

is restricted. For Z^ < 0.9 m, the flow from the upper layer penetrates into

the lower layer. Also note that at large Z^ (> 1.6 m), the flow from below

does not stagnate at any height, but ultimately hits the celling whose height

is 2.18 m in this case.

-29 -
5.4 Concluding Remarks on Penetration

It is seen that the penetration of the wall flow from one isothermal

layer into another at a different temperature is a very involved process. The

penetration depends on the tenperature profile and the opposing flow that

arises in the other layer. The flow separates from the wall when the two

opposing streams have equal momentum. The flow turns away from the wall and

descends or rises towards the interface since it is negatively buoyant. A

considerable amount of mixing occurs in this region and flow stability

considerations are Important. The actual profiles may be considered for

experimental results and a two-zone idealized profile for zone modeling. In

two-layer consideration, a computation of the momentum for the two streams

indicates which zone the flow enters and gets mixed with the fluid there,

conveying mass and energy. For the actual profiles, a three-layer system may

be considered and the location of the separation point determined. The corresponding interface relevant to a two-zone idealization may be located as

outlined by Steckler [10] and also confirmed from some of the results in this

study. The flow rates and the energy transported across the interface may be

determined in order to provide inputs into the zone models.

6. FLOW OF FIRE PLUME ACROSS THE INTERFACE

In order to further study the basic mechanisms underlying flow penetration from one Isothermal layer into the other, some consideration is given to

the flow of the fire plume across the interface. The plume is generally

buoyant with respect to the upper layer, unless the heat input is reduced

after the layers are established by a higher energy input source. The wall

- 30-
flow, on the other hand, Is negatively buoyant as it crosses the interface.

The temperature and velocity levels in the wall flow are also much lower than

those encountered for the corresponding fire plume, though the much larger

flow perimeter makes the flow rate in the wall boundary layer conqiarable to

that in the plume, as seen earlier. These differences in characteristics may

be expected to lead to differences in the penetration across the interface.

The wall flow generally penetrates a given distance and then turns back,

whereas the plume would penetrate and flow in the upper layer at a reduced

buoyancy level because of the higher gas temperature there.

Some of the relevant aspects may be considered in terms of Fig. 28. As

the plume enters the upper layer, the gas temperature rises sharply. As a

result, some portion of the flow is buoyant and some negatively buoyant, with

respect to the upper layer temperature. Tlie buoyancy flux in the flow is the

convected thermal energy q, given by Eqn. (2). Therefore, the net buoyancy of

the flow with respect to the upper layer temperature T is given by:

g » u

5 q = / PC u(T-T )dy (28)

o

P

This quantity may be negative, zero or positive. As pointed out by Cooper

[49], a negative buoyancy leads to a finite penetration height, zero buoyancy

to a nonbuoyant jet and a positive buoyancy to a flow which penetrates and

rises as a buoyant plume. Note that any entrainment into the flow in the

upper layer does not alter q, since the entering fluid is at temperature

Tg The wall flows considered earlier are clearly negatively buoyant and

have a finite penetration distance, whereas the plumes, in the context of the

present work, would be positively buoyant.

- 31 -
As the fire plume enters the upper layer from below, its buoyancy, with

respect to the local environment, is reduced, since T_ „ is larger than T

Consequently, the outer regions of the plume become negatively buoyant,

causing flow there to stagnate and reverse, as shown qualitatively in Fig.

28(b). This implies that some flow may be shed at the Interface and further

entrainment of the upper layer gas occurs as the plume moves downstream.

Since in actual practice, there is always a finite region over which such

changes occur, one may think of a region of adjustment over which m, M and q

change, while the temperature profile becomes consistent with the changed

value of as shown in Fig. 28(c). However, the changes in temperature do

occur over a very short distance and an idealization of a two-zone model is

appropriate in most cases. This implies a step change in , which results in

a step change in temperature in the outer region of the plume. These large

temperature gradients lead to large energy transport rates, due to conduction

as well as ambient fluid entrainment, leading to a smoothenlng of the

profiles, so that an esentially Gaussian profile is obtained further downstream. The nature and height of this region of adjustment are not known in

detail. For a three-layer model, the intermediate layer allows a consideration of these adjustments, whereas in a two-layer model, one assumes an

immediate adjustment, followed by the plume flow in the isothermal upper

region.

A simple model, based on the dynamics of the fluid elements that

constitute the flow, is first considered. If one considers the fluid at any

given location in the flow, just below the Interface, one could determine its

vertical velocity and tenperatur e. This fluid element is exposed to an

increasing temperature in the stratified zone of height L

2

, where the two-

-32-
layer model is obtained for 0, until it emerges in the upper zone at

temperature T„ . Neglecting viscous effects, a balance between the kinetic

and potential energy leads to;

(29)

or,

+ / 2gs(l^T„)dx (30)

o

where x is the vertical distance above the isothermal lower region of height

Lj, Uq is the local velocity just below this height, u the velocity at height

X for the given fluid element, T the temperature and the local ambient

ten^jerature. Thus, if T is assumed unchanged over the stratified layer, or a

chosen region of adjustment, a step change in temperature, from T -i to T ,

exists at the edge of the flow. This is the maximum temperature defect that

could arise, where the temperature defect D.ji is the amount by which the

smallest temperature in the flow is less than the gas temperature at the same

height, as outlined earlier in Section 5.1. One may also consider a defect

which varies from zero at Z = Lj + L

2

> with a parabolic variation and a

maximum value at mid-height. For the wall flow, D.j. was taken as zero,

because of analytical work done earlier on the flow that indicated that the

temperature in the outer region of the boundary layer flow was only slightly

less than the local ambient temperature, giving a temperature defect of less

than 5% of the total temperature difference in the flow. For a plume, it has

been seen to be higher [45], though still only a few percent of the total

temperature difference. We shall consider a range of values for IXp and deter-

- 33 -
mine its effect on penetration. This is, obviously, an approximate model for

penetration and is used only to study the effects qualitatively.

The above model was first used for the wall flow rising from below and

penetrating into the upper layer, with no temperature defect, that is, the

assumed profiles of Eqn. (4) or Eqn. (12), for laminar or turbulent flow,

respectively. Fig. 29 shows the results obtained. The trends observed are

similar to those obtained from the integral analysis given earlier. A viscous

3^u model was also considered by Including the viscous force v along with the

buoyancy force. The corresponding results are shown in Fig. 29 and it is seen

that the integral results lie within the invlscid and viscous models, lending

support to the basic features obtained by such a simplistic model. Fig. 30

shows the velocity profiles as the flow penetrates and the lowering of the

maximum velocity and stagnation at the two ends propagating Inwards is seen.

The viscous model constrains the outer region and an outward shift in the

profiles arises. This figure indicates the physical nature of the penetration

process. It must be noted that, though not shown here, the flow in the outer

region of the wall flow reverses under the action of the negative buoyancy.

Figs. 31 and 32 show the results for a fire plume, employing the inviscid

model, which is more appropriate due to the absence of the constraining wall

of the earlier case. The three layer profile from Fig. 8 is employed and the

observed results indicate the physical processes Involved in peeling off of

the flow as it penetrates into the upper zone. It is seen that the flow rate

initially Increases and then decreases in the stratified region, as seen

earlier for wall flows. The flow rate Increases as the plume emerges into the

upper layer. These effects are more significant at larger interface heights

- 34-
where the flows are weaker. Fig. 31 Is drawn for the maximum temperature

defect, so that the negative buoyancy effect is the strongest. Fig. 32 shows

the results for various D.j, and it is seen that a decrease in flow rate occurs

only for large D.p. It is, therefore, seen that, depending on the interface

height, the flow rate decreases in the stratified layer due to negative

buoyancy in the outer region of the flow. The model gives an order of magnitude value of this flow. For high interfaces, which arise for large door

openings [11], this effect is large and needs to be accounted for. For low

interfaces, under the same conditions, the effect may be neglected. The

results presented here give a qualitative picture of the processes involved.

Though Dq, is not known and can only be determined by a detailed analysis, a

defect of 5-10% of the total change in T„ is a reasonable approximation. At

O

this level of D.J., the amount of fluid peeled off is small and may, therefore,

be neglected, as a first-order approximation.

In view of the above qualitative picture and the earlier results on wall

flows, it is now possible to consider the plume flow across a two layer interface. The flow momentum changes because of the forces acting over a finite

distance. Since a step change is considered, momentum remains unaltered

immediately above the Interface. The temperature profile is assumed to adjust

to the new ambient tenperature immediately. This implies a considerable

Inflow of energy and a lowering of the buoyancy with respect to the upper

layer temperature, from that obtained with respect to the lower layer temperature. This inflow of energy occurs due to conduction and mass Inflow, arising

from large tenperature differences. For a step change in the profile, a mass

inflow may be Included. But as a first approximation, in view of the results

presented, it may be taken as unaltered. The governing equations from which

the results presented in Eqns. (21-24) are obtained are:

-35 -
(d^u) = 2dau (31)

d 2v 2 ,2 , . , (d u ) = 2X d g (p^ - p^)/p^ (32)

Iz “ (33)

These equations may be solved analytically or numerically.

Fig. 33 presents the centerline temperature and velocity, with the above

model. At the Interface, only the temperature profiles change to accommodate

the change In . llie rest of the variables remain unchanged, so that m and M

are unaltered. The above equations are then solved numerically beyond the

Interface with the new temperature distribution. As displayed In Fig. 33, the

centerline temperature T^ varies as and the velocity as up to

the Interface. Beyond the Interface, the reduced buoyancy leads to a more

gradual change In T^ and a more rapid decay In U^, as expected. Hie observed

trends are quite similar to those shown In Ref. [40]. There Is no step change

In and T^,, as physically expected. For a lower Interface, the downstream

temperature decrease Is more pronounced, since the buoyancy In the upper

region Is higher than that for higher Interfaces, with the other conditions

kept the same. The flow rate continues to Increase beyond the Interface due

to continued entrainment and the momentum also Increases at a decreased rate,

due to lowered buoyancy.

It Is evident from the above discussion that the penetration of the wall

flows, as well as of the plume flow, across a sharp ambient tenperature change

Is a fairly complex phenomenon. It Is not a boundary layer effect, since

-36 -
stagnation and flow reversals may arise. Some simple analytical models are

considered in order to study the basic features of the flow. Ihe results

obtained are physically realistic and some quantitative Information is also

obtained. Though several aspects need detailed further effort, some of the

major considerations and conclusions may be outlined. Let us first consider

the relevance of these effects in zone-modeling and in experimental studies of

enclosure fires.

7. ESTIMATION OF WALL FLOW EFFECTS

Let us consider the two-layer zone model for the circumstance shown in

Fig. 1. In addition to the flow rates shown, let us denote the total wall

flow rate by m^, the plume flow rate that crosses the interface by m^, and the

plume flow that is peeled off and is unable to penetrate by m^. For steady

state conditions in a lower layer control volume that excludes the fire plume,

a mass balance gives;

m-+m +m. = m +m (34)

I a j p w

where m^ is the mass flow rate of the fuel. Therefore,

m =m +m. +m£-m (35)

P a j f w

^

Als o.

m = m + m'

e p p

( 36 )

- 37 -
This gives the pltime flow entering the upper layer, normalized by the flow

through the opening, as:

m m. m_ m

-E-= ;1 + -j- +

f w

. * • * m m m m

a a a a

(37)

and the rate of entrainment over height as:

•m• • in . TH- m m*

e

1 _ ^ • • • • • m m m m m

a a a a a

(38)

A comparison between the theoretical models of Refs* [6,15,39] and experimentation is made in terms of the fire plume entrainment. For a wind-blown

plume, there would be an Increase in the entrainment and that additional

effect must also be considered when a couparison with the results from a

theoretical model for a free standing plume is being made [17].

It has been shown that, in a three layer system, some of the plume flow

is shed in the intermediate stratified region. Ifowever, the effect is not

large for the range of physical variables considered here and, for the twolayer approximation, the flow rate may be assumed to remain essentially

unaltered. The fuel flow rate is generally much smaller than the opening flow

rate. Neglecting these two quantities, the rate of entrainment becomes

m

1 mw

(39) m m

•m

a a a

air flow rate may be measured experimentally with good accuracy [10,11] .

If we consider an energy balance for the lower layer, the energy input into

the plume flow at the source and into the wall flow is conveyed to the upper

-38 -
layer, unless a significant amount of these flows peel off and get mixed with

the lower layer, this being a negligible effect according to the results

obtained here. Thus, the energy balance gives:

iTi (T , - T ) = m. (T - T , ) + 7T- a g,l o" j g,u g,l Cp

(AO)

where is the outside air tenperature and is the heat transfer between

the floor and the gas. It may be estimated as

Qf

= hA(Tj- ( 41 )

where h is the convective heat transfer coefficient, A the floor area, and

the floor temperature, which is greater than the gas temperature in the lower

layer. If is neglected, an upper bound for m^ is obtained as:

- T°

(4 2 )

If Tf is measured, available heat transfer correlations [18] may be employed

to determine m. more accurately. As discussed in Refs. [10,11], m,/m

j

* j a increases as the door opening is decreased. It nust be noted that if the wall

flow from the upper layer penetrates into the lower layer, the energy added to

the lower layer due to this flow and that due to heat transfer from the lower

wall should be included in Eqn. (40) for greater accuracy.

Therefore, the wall flow rate enters into the calculation for ifl . As has

e been shown earlier, this flow is quite significant as compared to the flow

through the opening m . A comparison between the two may be made by consid-

- 39 -
ering the results shown in Fig. 15 for T_. -i - T • = 60°C and 30°C, along with w ,

J- e > those presented in Ihble 1. It is therefore, evident that the estimated value

of m , obtained from the measured value of m , is reduced by 20-40% by

considering the wall flow. This additional consideration could, thus, lead to

a better agreement between the theoretical and experimental predictions of the

plume entrainment.

The wall flows not only transport mass across the interface, they also

lead to an energy transfer. The energy transfer for a two-layer system is

given by:

6 Hp6UC

q = Pq = P / p C u ( T-T , ) dy =

( T , -T .

)

^w ^ P g»l 30 ^ w,l g,!''

for laminar flow.

and.

= 0.0366 Pp6uc (t t -T ,) , for turbulent flow (43)

P w,l g,l^*

where the physical variables 6 and U are determined in the lower layer just

below the Interface and P is the perimeter of the wall flow. The perimeter P

is the horizontal perimeter of the room minus the door width. Similarly the

mass flow rate due to the wall flow is:

6

ra = Pm = P / pudy = Pp6U/12, for laminar flow, and

o

= 0.1463 pp6U, for turbulent flow (44)

-40 -
The variables 6 and U are determined from Eqns. (7) and (8) for laminar flow

and Eqns. (15) and (16) for fully turbulent flow. As explained in Section 4,

the governing equations may be solved with the transition Grashof nimber

9

Gr " 5 X 10 to obtain these variables, or an average from the laminar and

c

f ully-turbulent analyses may be employed as an approximation. The above mass

and energy transported across the interface nust, therefore, be included in

the mass and energy balance equations for the two layers. The energy transport is relatively small conpared to the energy input into the upper layer due

to the plume because of the much larger temperature levels in the latter

case. The energy transfer across the Interface due to the fire plume is Q^,

which is the total energy input minus the radiative loss from the fire.

It must be noted that the wall flow m^ may be positive or negative,

depending on the wall and gas temperature levels and the location of the

interface. If the downward flow momentum is larger than the upward flow

momentum, the flow penetrates from the upper layer into the lower layer and

mixes with the fluid there. If the upward flow is stronger, it penetrates

into the upper layer, as considered above. The flow momentum is given by:

5 2 2 M = PM = P / pu dy = Pp6u /105, for laminar flow, and

o

= 0J)523 Pp6u^, for turbulent flow (45)

If the downward flow momentim in the upper zone, (M^)^* is much larger than

the upward flow momentum in the lower zone, (M^)]^» both being calculated at

the interface height Z^, i.e. , (M^)^ >> ^^^1» flow penetrates into the

lower zone from the upper one. Similarly, the flow penetrates into the upper

- 41-
zone if (M^)^ << is negative in the first case and positive

in the other. It is difficult to predict penetration when (M ) » (M ),. The

penetrative flow under consideration is a stability problem and a considerable

disturbance is expected in the region where the flow turns away from the

wall. When the difference between the two flow momenta is large, this

disturbed region is well within one of the zones. But when the two are close

in magnitude, it is very difficult to predict the penetrative flows for the

two zones. However, some conjectures may be made, a detailed analysis being

beyond the scope of the present work.

As mentioned earlier, the interaction region in which the flow turns is

expected to be of order 6 . If the separation point, as determined from the

analysis of Section 5 is farther than distance 6 from the interface, a

complete penetration into that zone may be assumed, since the turning region

is beyond the interface. If it is less than 6, a proportional distribution of

the flow in the two zones may be considered, assuming a uniform flow away from

the wall over a flow height 6. If the two momenta are exactly equal, a rather

unlikely situation, the wall flows do not penetrate. Similar considerations

are outlined in Refs. [43,48] and the experimental observations in these

studies support these conjectures. In many cases, information on the separation point, as a function of the physical variables of the problem, may not be

available. In zone modeling, for instance, the temperature levels and the

interface height at a given time is all that is known. For such cases, ra may w

be determined for various interface heights, going from a large positive value

to a large negative value as decreases. The value of at which the two

momenta are equal may be taken as zero ra^ and an interpolation employed to

determine m^ at other values of Z^. Depending on the desired accuracy in

- 42-
considering the wall flow effects, one may construct the scheme to determine

the penetrative flow. The simplest one is the one that assumes complete

penetration when the two momenta are different, as shown in Fig. 34,

If experimental results are available on the wall and gas tenperatures ,

one may approximate the profiles as a two-layer system [10,11] and apply the

above considerations. For greater accuracy, the exact profiles may be

employed to determine mass flow rate, momentim and energy at the estimated

interface by solving the governing equations, as outlined in Section 5. This

would then indicate the location of the point where the two momenta are equal

and allow one to consider the case when the two are close to each other at the

Interface, as outlined above.

In summary, the approximate method discussed above for incorporating the

wall flow effects in the present zone models may be outlined. Let us assume

that the upper layer temperatures, Tg ^ and the lower layer temperatures, T_ 1 and T„ -i , and the Interface height are available from the zonemodel analysis or from experimentation. For the experimental results, as

pointed out earlier, the model of Bef. [10] may be employed to obtain the

characteristic quantities for a two-layer system. The first step is to determine the momentum flow rates, for the downward flow in the upper zone and for

the upward flow in the lower zone, at the Interface height, from the boundary

layer analysis. For laminar and turbulent flows, respectively, M is given by:

M = pv2 (pr + (46)

-43 -
and,

M = 0.04149 pv^ [1 + 0.494 (47)

The corresponding mass flow rates are:

m

20>|-l/4 „ -1/2 rg3AT^ 1/4 ^3/4

= 1.6932 pv (pr Pr“^'" (^) (48)

and

m = 0.09795 pv

rgBAT>|0.4 „ -8/15

Pr [1 + 0.494 (Pr)^^^]"^**^ (49)

These values are given per unit flow perimeter. The laminar as well as the

turbulent values are computed and their average taken to approximately give

the flow momentum and mass flow rate. For more accurate results, the

considerations outlined in Section 5 may be followed.

The downward flow momentum in the upper layer (M^)^ and that in the lower

layer (M^)^ are conpared. If (M^^)^ > upper region flow penetrates

into the lower layer and the corresponding value of m is determined. If

^^^u ^ ^^^1* lower region flow penetrates into the upper layer. This

approximation amounts to the replacement of a sharp variation of m with Z^

,

near the location where the two momenta are equal, by a step change, as shown

in Fig. 34. For this figure, the temperature data of Fig. 8 are employed and

the effect of a variation in the interface height is Indicated. The solid

curve refers to a three layer model and the broken lines to the two-layer

approximation. The interpolated curve around = 0.97 is also obtained

- 44-
approximately as outlined earlier and the step change assimption is reasonable. The determined wall flow rate may then be employed in the determination

of m from Eqn, (39). Thus an Insltu plume entrainment rate could be compared

e with Idealized free plume models.

8. SUMMARY AND CONCLUSIONS

A study of the buoyancy-induced flow generated adjacent to the vertical

walls of a room with an opening due to fire in the room has been carried

out. Experimental results on wall and gas temperatures are obtained for a

full-scale fire in a room. Though some data are taken at the very early

stages of the fire, most of the measurements are taken for the quasi-steady

and the essentially steady state circumstance attained after the initial

transient. The boundary layer flows generated adjacent to the walls due to

the temperature differences that arisa are studied for laminar as well as

turbulent flow. The corresponding physical quantities, such as mass flow

rate, momentum flow rate and the convected thermal energy, are computed for

the experimentally-determined temperatures. The study is considered largely

in terms of the two-layer zone model, which assumes an isothermal hot upper

layer lying above a colder isothermal lower layer, and a three-layer model,

which considers a stratified intermediate region between the two isothermal

zones. In actual practice, a step change in temperature from one zone to the

other does not exist and a more gradual variation is observed. However,

considerable effort has been directed at the two-zone idealization because of

the resulting simplicity. The three-zone model is considered largely to

provide insight into the basic fluid flow processes and to provide suitable

inputs for the two-layer model.

- 45-
The penetration of the wall flow across the Interface between the two

isothermal regions is studied. The momentum of the generated wall flows is

determined for the upward flow in the lower layer and for the downward flow in

the upper layer. Ihe variation of the momentum with height is determined in

the boundary layer flow as well as in the negatively-buoyant circumstance that

arises following penetration. The point of separation of the flow from the

wall is obtained from the condition of net zero momentum for the two opposing

gas flows. The penetration of the fire plume across the interface is also

considered. The study considers the results obtained in terms of the twolayer zone model and indicates the method that may be adopted to incorporate

the wall flow effects in the model.

The main conclusions of the study may now be outlined for the range of

experimental conditions considered. Experimentation was carried out over a

wide range of heat release and opening slze» as related to enclosure fires.

The important conclusions obtained from this study are given below, the

particular values obtained for specific experiments being given in the various

figures and discussed earlier. Of course, it should be realized that all of

the computations were based on boundary layer or plume models, and did not

address the separated flows. Since the separated flows can redistribute

themselves across the interface region, the quantitative results presented

here may be over-estimated. Also the fact that the interface is not an ideal

plane, but an ill-defined transition region, contributes to uncertainties in

these conclusions on transport rates across the interface. Nevertheless the

following can be concluded from this analysis subject to these qualifications.

-46 -
1. The wall flow effects are found to be quite significant

subject to the limitations of the quiescent external flow

field picture discussed above* The flow rates were found to

be in the range 20-40% of the flow rate in the plume.

2. The energy transported across the interface by the wall flow

is small compared to that due to the plume, being less than

about 5% of the energy input into the fire. Therefore, the

energy transfer due to the wall flow may be neglected but the

mass transfer across the Interface must be included in the

analysis.

3. The fire plume penetration into the upper layer was also

studied and a small amount of flow, 5-10%, was found to be

shed in the stratified zone. For the two-layer model, the

mass flow may be taken as unaltered across the Interface.

The general features of the study may be sunmarized as:

1. A downward flow arises in the upper layer and an upward flow

in the lower layer. Depending on which flow has a larger

momentum at the interface, the flow penetrates into the other

layer. When the two momenta are close to each other, the flow

gets distributed over the two layers and interpolation may be

employed to determine the flow that penetrates.

-47 -
2. Employing the Integral formulation for plume flow, its penetration across the interface is studied and physically

realistic results are obtained.

3. An approximate method for taking the wall flow effects into

account is outlined for zone-models as well as for experimental studies, on the basis of the results obtained in this

study

.

9. FUTURE WORK

This study has considered approximate integral models for the evaluation

of the wall flow effects due to fire in a room. Though the general trends and

approximate values of the relevant physical quantities have been obtained, a

more detailed analytical and experimental study is needed to justify some of

the assunptions made and to obtain more accurate results. In particular,

future analytical effort may be directed at solving the governing flow equations for two-dimensional wall flows and the axlsjmmetrlc plume flow to

provide further information and insight into the following aspects:

1. Temperature field in the wall flow as it penetrates a

stratified intermediate layer, particularly any temperature

defect or reverse flow that might arise, effect of a variation

in the height L

2 of this layer on the flow, particularly

as L

2 0 which corresponds to a two-layer system, and the

effect of the shed flow on the flow adjacent to the wall. The

problem does not allow boundary layer simplification and the

full equations need to be considered.

-48 -
2. Interaction of opposing wall flows needs further study with

respect to the dependence of penetration on the difference

between the momenta of the two streams » particularly when the

two are close to each other* It is a nonboundary layer

problem, which may possibly be studied in terms of available

results on flows in corners*

3* Plume penetration across the stratified layer, followed by a

consideration of the two-layer model* The boundary layer

results would indicate regions of reverse flow and separation,

but a nonboundary layer analysis would give the desired quantitative information on flow rates, momentum flow and buoyancy

flux*

A. A parametric study of the general problem is needed to determine the governing dimensionless parameters* This would allow

a generalization of the physical processes* Though the

present results may be presented in dimensionless terms, a

more detailed study is needed to study the effect of the

various physical variables in the problem*

Further experimental work is also desirable in order to provide inputs

for the analytical work* Though some visualization of the interface region

was done in this study, clearly more work is needed to obtain a better understanding of the interaction of the wall flows with the Interface and with each

other* Temperature measurements in the flow near the Interface would also

help in the analysis* The mineral oil smoke generator may be employed to

-49-
visualize the flow adjacent to the walls at various operating conditions and

recorded on video. Of particular Interest would be an estimation of the

penetration distances, height and location of the region where flow separates

from the wall and the nature of the flow. Also of interest would be further

results on the transient effects, both at start up and at shut down of the

fire. This would permit a consideration of the very early stages of the fire,

in terms of the boundary layer transients versus the interface movement

[50,51].

When the fire is turned off, the walls and the gases cool down gradually

due to energy loss to the environment. Since the walls remain hot for a

fairly long time following the shutting off of the fire, it is possible to

obtain quantitative information on the wall flows penetrating into the upper

layer by measuring the outflow at the door and the temperatures in the room.

In the absence of wall flow effects, mass balance of the upper layer gives:

= m^ (50)

where A is the cross-sectional area of the room, t is time and m is the mass

o outflow rate at the opening. If a flow rate of m^ penetrates into the upper

layer, the governing equation is:

|^(a>Zi) + (51)

Therefore, if ) and Z^(t) are determined experimentally, the wall flow

rate may be determined as a function of the conditions at a given time. If

the process can be approximated as quasi-steady, which appears reasonable

- 50-
given the slow rate of cooling of the walls, these results may be employed to

yield wall flow rates when the fire is present, by considering the corresponding temperature profiles. Though further work is needed on ascertaining the

validity of this approach, the results obtained are expected to be valuable in

the verification of the analytical model predictions.

Experimental work on the wall flows may also be carried out on a laboratory scale. The work would need to be directed at the magnitude of the wall

flow and its penetration as a function of the physical variables of the

problem. Detailed velocity and temperature measurements in the flow would

answer most of the questions raised in the present study and would permit the

development of more accurate models for determining the wall flow effects.

Similarly, the penetration of the fire plume into the upper layer may be

studied to provide suitable inputs into the analysis of the flow as it goes

from one zone to the other, across a sharp interface.

10. ACKNOWLEDGEMENTS

This study was carried out during the sunmer of 1982, while the author

was a visiting scientist with the Fire (Jrowth Processes Research Group at the

(inter for Fire Research, Ifetional Bureau of Standards, (Jalthersbur g,

Maryland. The financial support for this work was provided by the (inter.

The author acknowledges the several discussions he had with J. G. (Julntlere

and L. Y. Cooper on the mathematical modeling of the basic processes and the

extensive help he had from K. D. Steckler and W. J. Rtnklnen on the experimentation. He also had several interesting and stimulating discussions with

B. J. McCaffrey and H. R. Baum.

- 51-
11.

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30. Ostrach, S. , An Analysis of Laminar Free Oanvectlon Flow and Heat

Transfer About a Flat Plate Parallel to the Direction of the Generating

Body Force, NACA TR-1111 (1953).

31. Sparrow, E.M. and Gregg, J.L. , Similar Solutions for Free Convection from

a Nonisothermal Vertical Plate, J. Heat Ttansfer, 80 379-386 (1958).

- 53-
32. Squire, H.B., Referenced in Modem Developments in Fluid Dynamics by

S. Goldstein, Oxford Univ. Press, N.Y. (1938).

33. Eckert, E.R.G. and Jackson, T.W. , Analysis of Turbulent Free Convection

Boundary Layer, NACA TN-2207 (1950).

34. Kato, H. , Nichiwaki, N. and Hirata, M. , On the Turbulent Heat Transfer by

Free Convection from a Vertical Plate, Int. J. Heat ^fass Transfer, 11,

1117-1125 (1968).

35. Nbto, K. and Matsumoto, R., Turbulent Heat Transfer by Natural Convection

Along an Isothermal Vertical flat Surface, J. Heat Transfer, 97, 621-624

(1975).

36. Cheesewright, R. , Turbulent Natural Convection from a Vertical Plane

Surface, J. Heat Transfer, 90, 1-8 (1968).

37. Jaluria, Y. and Gebhart, B., On Transition Mechanisms in Vertical Natural

Convection Flow, J. Fluid Ifech. , 66, 309-337 (1974).

38. Morton, B.R., Taylor, G.I. and Turner, J.S., Turbulent Gravitational

Convection from Maintained and Instantaneous Sources, Proc. Roy. Soc.

,

A234, 1-23 (1956).

39. Yokoi, S., Plume Rising from Heat Source, Bull. J. Assoc. Fire Sci. Eng.,

5, 53-59 (1956).

40. Evans, D.D. , Determining Sprinkler Actuation Time Phase I - Approximating

Fire Plume Temperature and Velocity in a Two Layer Eiivlronment , NBS Int.

Rep. to appear.

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

Diffusion Flames, Oomb. Inst. West. States Sect. >feeting, Pullman, WA

(1981).

42. Elchhorn, R., Natural Convection in a Thermally Stratified Fluid, Prog.

Heat Mass Transfer, Pergamon Press, U.K. , 2, 41-53 (1969).

43. Chen, C.C. and Eichhorn, R. , Natural Convection from a Vertical Surface

to a Thermally Stratified Fluid, J. Ifeat Ttansfer, 98, 446-451 (1976).

44. Jaluria, Y. and Gebhart, B., Stability and Transition of Buoyancy-Induced

Flows in a Stratified Medium, J. Fluid Mech. , 66, 593-612 (1974).

45. Hlmasekhar, K. and Jaluria, Y. , Laminar Buoyancy-Induced Axlsymmetric

Free Boundary Flows in a Thermally Stratified Ifedlum, Int. J. Heat Mass

Transfer, 25, 213-221 (1982).

46. Fox, D.G. , Forced Plume in a Stratified Fluid, J. Geophys. Res., 75,

6818-6835 (1970).

47. Nforton, B.R., Coaxial Turbulent Jets, Int. J. Heat Mass Transfer, 5,

955-965 (1962).

- 54-
48. Schetz, J.A. and Elchhorn, R. , Natural Convection with Discontinuous

Wall-Tenperature Variations, J. Fluid Mach., 18, 167-176 (1964).

49. Cooper, L.Y., Personal Comm. (1982).

50. Siegel, R. , Transient Free Obnvection from a Vertical Flat Plate, Trans.

ASME, 80, 347-359 (1958).

51. Ede, A.J., Advances in Free Convection, Advances in Heat Transfer,

Academic Press, 4, 1-64 (1968).

52. Emmons, H.W., Personal Communication with K. Steckler (1979).

-55 -
Table 1

^ Heat Input, Q Location and

Test No. Door Opening (kW) No. of Burners Opening Flow (kg/s)

168 6/6 63.2 1, Ftont Left Garner 0.424

169 6/6 63.2 1, Back Right Corner 0.426

300 6/6 31.6 1. Back Wall 0.37 2

301 6/6 126.4 4, Back Wall 0.622

302 8/6 31.6 1. Back Wall 0.436

303 8/6 126.4 4. Back Wall 0.741

304 4/6 31.6 1. Back Whll 0.290

305 4/6 126.4 4 . Back Wall 0.484

306 2/6 126.4 4 . Back Wall 0.267

307 4/6 126.4 4 , Back Wall 0.478

308 2/6 31.6 1, Back Wall 0.171

All door openings are 1. 83 m high. The width Is 0.74 m for the 6/6 door.

0.99 m for the 8/6 door. 0.49 m for the 4/6 door and 0 .24 m for the 2/6 door.

-56-
TEMPERATURE PROFILE

ii

h

Fig. 1. Flow configuration considered

-57 -
HOT UPPER LAYER

IHTERF^E

I INTERFME

COLDER LOWER LAYER

I OPENING

i

Fig. 2. Sketch of the temperature distribution in a two-layer zone mode 1.

Fig. 3. Sketch of the velocity and temperature profiles in

the wall flow.
Fig. 4. Flow in the negatively-buoyant wall flow

and in a corresponding jet.

-59-
Fig. 6. Interaction of a fire plume with the

stably stratified upper layer.

-60-
ASPIRATED THERMOCOUPLES

CEILING THERMOCOUPLE

MOVABLE BIDIRECTIONAL VELOCITY

PROBES AND THERMOCOUPLES

WALL THERMOCOUPLES

Fig. 7. Experimental arrangement.
HEIGHT,

Z

|m)

0 100 200 300

TEMPERATURE |°C|

Fig. 8. Measured gas and wall temperatures in the room for

a circular burner and Q

= 63.2 kW.

-62-
HEIGHT,

Z

(m|

Q=63.2 KW I *1

6/6 DOOR

CEILING

I I 0 100 200 300

TEMPERATURE rC|

Fig. 9. Measured gas and wall temperatures in the room for

a circular burner and Q

= 63.2 kW, at a different location.

- 63-
HEIGHT,

Z

|m|

2.0

1.0

CEILING

1 LME BURNER

Q=31£ KW

6/6 DOOR

aOOR

50 100

TEMPERATURE |°C)

150

Fig. 10. Measured gas and wall temperatures for a single line

burner and Q = 63.2 KW, at a different location.

-6A-
HEIGHT,

Z

(mj

CEILM6

T

LINE BURNER

n

2.0

1.0

0

0 100 200 300

TEMPERATURE |°C|

Fig. 11. Measured gas and wall temperatures for the line burner

at various door openings and heat input values.

-65-
=

1.67m -66-

Fig.

12.

Transient

wall

and

gas

temperatures

fora

line

burner

and 0=

31.6

kW,

with

1.83 mX

0.24 m

door

opening.
z z

z

Actual and idealized temperature profiles.

Fig. 14. Actual and idealized boundary layer wall flow.

-67 -
CO c4

|iu s/3i() 001 X ui

-68-

Fig.

15.

Mass

flow

rate

in

the

wall

flow,

as

a

function

of

the

height
- 69-

HEIGHT,

Z

|m|

16.

A

comparison

between

the

wall

flow

rate

and

the

plume

flow

rate,

for

a

circular

burner

at Q

-

63.2

kW

and

a

1.83

m

x

0.74

m

door

opening.
m

CM

lO

(zS/3)|) 001 X (M) 31Vd M01J NfllNSNON

(sui/3i|j 001 X (lu) 31Vd MOIJ SSVW

ISJ

00

- 70-

17.

Mass

flow

rate

and

the

momentum

flow

rate

in

the

laminar

wall

flow,

for

various

models,

considered

in

the

negatlvely-buoyant

region,

employing

values

from

Fig.
CM

in

lO

INI I—

C9

CO

cc

- 71-

Momentum

flow

rate

for

various

heights

of

the

isothermal

lower

region,

considering

laminar

flow

in

both

the

zones.
(;S/3)|| 001 X 31VH M01J MIUN3NON

- 72 -

Fig.

19.

Momentum

flow

rate

for

turbulent

flow

in

the

two

zones.
CO c\i

(sui/3)|) 001 X (ui| BlVdMOlJ

- 73-

Mass

flow

for

laminar

and

turbulent

flow,

as

a

function

of

the

height

of

of

the

of

the

lower

isothermal

zone.
CM

(ui) INIOd N0liVliVd3S JO 1H9I3H

-74-

Fig.

21.

Dependence

of

the

location

of

the

separation

point

on L

for

turbulent

flow.
|2S/S)|) 001 X |M)31Vd M01i NniN3N0N

'^4

- 75-

Momentum

flow

at

various

values

of

L,

and

L„

for

turbulent

flow.
TJ

(U

(ui) INIOd NOliVdVdBS JO 1H9IJH

COu

4-) CO

(1) xiu

iwo

r>-l

<U (U 43U cO CMO (U 4J

<U CO O «H C T3

(U 0) TO B

13 ft

(U (1) O. 4-) (U 13 Q -H

ro

cvl

•H

-76-

HEIGHT

OF

STRATIFIED

LAYER,

L

2

(m)
m

X

100|kg/msJ

M

X

100(kg/s2j

HEIGHT OF STRATIFIED LAYER, L2(m|

Fig. 24. Variation of the mass and momentum flow across

the stratified layer v;ith

- 77 -
lzS/3)|) 001 x(w| im M01J NniNBNOW

(sui/3)|) 001 X (ui| 31VII M01J SSVW

- 78 -

Fig.

25.

Mass

and

momentum

flow

for

a

two-layer

system

in

laminar

flow,

considering

various

models

for

the

negatively-buoyant

circumstance.
CO CNJ

(zS/ 3)|) OOlxiMl 3iVd M01J NniN3M0N

(sui/3)|) 001 X (lu) 31Vd M01J SSVN

- 79-

FiR,

26.

Mass

and

rnomentum

flow

for

a

two-layer

system

in

turbulent

flow.
PENETRATION

DISTANCE,

dp(m)

Fig. 27. Penetration depth as a function of interface height

for turbulent flow in a two-zone system.

-80-
Region of adjustment

Am

AM

AQ

(r)

Fig. 28. Sketch of the temperature and velocity profiles as the

fire plume flows across the interface.

-81 -
(zs/3)|)00l X M

(sui/3)|| 001 X lu

- 82-

Fig.

29.

Mass

flow

rate

as

the

wall

flow

penetrates

the

stratified

Intermediate

layer.
(s/uiln A1I0013A IVOlidBA

om

oc

-83-

Velocity

profiles,

for

the

Invlscid

and

the

viscous

models,

as

the

wall

flow

penetrates

the

stratified

layer.
|sui/3)|| 001 X |iu| 31VH M01i SSVM

-84-

Fig.

31.

Mass

flow

rate

in

the

plume

as

it

flows

across

the

stratified

layer

for

various

values

of

Land

maximum

temperature

defect

D

,

at Q

=

63.2

kW.
- 85-

Mass

flow

rate

in

the

plume

for

various

values

of

the

temperature

defect

for Q

=

63.2

kW.
(s/ui) 001 x*n

(0o)*l

-86-

Fig.

33.

Results

for

plume

flow

across

the

Interface,

from

the

integral

model

of

Ref.

[38]

for Q

=

63.2

kW.
|SUI/8)|| ‘001 X (iji) 3iv)i M01i 11VM 9Minil3IGd OUV/MdO

-87 -

Variation

of

the

upward

penetrating

wall

flow

rate

with

the

interface

height

Z

for

a

two-zone

model.
NBS-114A (REV. 2-eo) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No, 3. Publication Date I BIBLIOGRAPHIC DATA

REPORT NO.

SHEET (See instructions) NBSIR - 84-2841 May 1984

4. TITLE AND SUBTITLE

Buoyancy-Induced Wall Flow due to Fire in a Room

5. AUTHOR(S)

Yogesh Jaluria

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State, ZIP)

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) A Study of buoyancy-drlven flow generated adjacent

to the vertical walls of a room due to fire in the room has been carried out. The

boundary layer flow that arises over the vertical walls due to the resulting difference between the wall and gas temperatures was analyzed, employing the integral

analysis method. The flow rate, momentum and convected energy in the downward flow

that arises in the heated upper layer and those in the upward flow that is generated

in the cooler lower layer were determined. The separation point and the region near

it where the flow starts separating from the wall are studied. Though more detailed

and more accurate computations are included, an approximate method for evaluating the

wall effects for a two-layer model and for an experimental study is outlined.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Density interface; enclosure fires; plume penetration; stratified media;

wall flows; zone-modeling

13, AVAILABILITY

[IQ Unlimited

1 I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402. 02 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

96

15. Price

$11.50

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.7 '