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Building and Environment 43 (2008) 17191733

The influence of stacks on flow patterns and stratification associated

with natural ventilation

Shaun D. Fitzgerald, Andrew W. Woods

BP Institute for Multiphase Flow, Madingley Rise, Madingley Road, Cambridge CB3 OEZ, UK

Received 16 September 2007; received in revised form 24 October 2007; accepted 26 October 2007

Abstract

We investigate the steady state natural ventilation of an enclosed space in which vent A, located at height hA above the floor, is

connected to a vertical stack with a termination at height H, while the second vent, B, at height hB above the floor, connects directly to

the exterior. We first examine the flow regimes which develop with a distributed source of heating at the base of the space. If hBohA, then

the unique flow solution involves inflow through vent Band outflow through vent A up the stack. IfH4hB4hA, then two different flow

regimes may develop. Either (i) there is inflow through vent Band outflow through vent A, or (ii) the flow reverses, with inflow down the

stack into vent A and outflow through vent B. With inflow through vent A, the internal temperature and ventilation rate depend on the

relative height of the two vents, A and B, while with inflow through vent B, they depend on the height of vent Brelative to the height of

the termination of the stack H. With a point source of heating, a similar transition occurs, with a unique flow regime when vent Bis lower

than vent A, and two possible regimes with vent Bhigher than vent A. In general, with a point source of buoyancy, each steady state is

characterised by a two-layer density stratification. Depending on the relative heights of the two vents, in the case of outflow through vent

A connected to the stack, the interface between these layers may lie above, at the same level as or below vent A, leading to discharge of

either pure upper layer, a mixture of upper and lower layer, or pure lower layer fluid. In the case of inflow through vent A connected to

the stack, the interface always lies below the outflow vent B. Also, in this case, if the inflow vent A lies above the interface, then the lower

layer becomes of intermediate density between the upper layer and the external fluid, whereas if the interface lies above the inflow vent A,

then the lower layer is composed purely of external fluid. We develop expressions to predict the transitions between these flow regimes, in

terms of the heights and areas of the two vents and the stack, and we successfully test these with new laboratory experiments. We

conclude with a discussion of the implications of our results for real buildings.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Natural ventilation; Stacks; Stratification; Flow regimes

1. Introduction

The growing interest in reducing energy demand from

buildings has stimulated much research in natural ventila-tion. Many of the key principles of natural ventilation have

been identified using simplified analogue laboratory

experiments, and supporting theoretical models [15]. The

majority of these studies focus on buoyancy driven

displacement ventilation in which relatively cool air enters

the base of the building, is heated, and then discharges

from a vent at the top of the building.

As an approximation, the vent and stack geometry are

often simplified as being openings at the base and top of

the building. However, in many real naturally ventilated

buildings, the vents (e.g. windows) may be located atintermediate levels in a room and there may be substantial

stack structures which draw air from the building and

channel this upwards prior to venting to the exterior. In

buildings with vaulted ceilings and sloping roofs, there may

be vents or stacks connected to both the lower and the

upper part of the roof (Fig. 1a). The present study was

inspired by a new classroom block at the Hagley School in

Worcestershire in which the top floor classrooms are

ventilated using a stackvent configuration analogous to

Fig. 1a. A critical question in the design phase of the

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building concerned the height of the stack which would

optimise the outflow from the classroom. As we identify in

this work, with the geometry of Fig. 1a, it is possible that

the stack may involve inflow to the room or outflow from

the room. We will show that a rich spectrum of natural

ventilation flows may arise from such a configuration of

openings, depending on the relative heights of the vents

and termination of the stack. We also show that with alocalised source of buoyancy, the nature of the temperature

stratification which develops also depends on the locations

of the vents and stack.

In order to develop a systematic understanding of the

different flow regimes, we explore the natural ventilation in

a simplified model building with two vents, and in which

one vent is connected to a stack. We allow the level of both

vents to vary from the floor to the level of the top of the

stack (Fig. 1bd). By comparison with Fig. 1a, one can

recognise that the design of the stack and vents at Hagley

School, with a sloping roof, is captured by this model

geometry. In this paper, we explore the flow regimes and

patterns of internal stratification which develop with such a

configuration of the vents. To our knowledge, many of the

flow regimes which we identify have not been described in

the literature on natural ventilation (eg. [3]).

Earlier work on the inclusion of intermediate level vents

[5] has been restricted to the case in which vents are also

placed at both the top and bottom of a building. In this

paper we focus on a simplified open-plan type building, in

which there is a low level heat load and only two vents, one

at the floor or at an intermediate level, and one which

accesses a stack venting from the side of the building. The

case of a distributed heat load, in which the air becomes

well-mixed within the space, builds on the work of

Gladstone and Woods [4] who considered the case of a

room with vents at high and low level in the space. In that

case, there is a unique upward displacement ventilation

flow. If in contrast, there are two vents, A and B, at

intermediate level in the space, with vent A connected to a

stack which extends to the top of the building, then the

flow regime depends on the relative height of vents A and

B. If vent A lies above vent B, then we expect an upwarddisplacement flow in the stack (Fig. 1b). However, if vent B

lies above vent A, then there may be two different regimes.

Firstly, the flow may continue to enter the space through

vent Band exit through vent A rising up the stack (Fig. 1c).

In this case, the flow is driven by the buoyancy force

associated with the column of buoyant air in the stack

above the level of the inflow, vent B. In the second regime,

the external air enters the stack and descends into the room

through vent A, while air vents from the space through

vent B (Fig. 1d). In this case, the flow is driven by the

buoyancy force associated with the column of air between

the level of vent A and vent B. The two different flow

regimes arise from the non-linearity associated with upflow

and downflow in the stack. In Section 2, we develop a

model of these multiple states, and, in Section 3, we

successfully compare our predictions of the different flow

regimes with some new laboratory experiments. Such

multiple flow regimes which arise from the non-linearity

of flow in a stack have been recognised in the different

context of mixing ventilation through roof mounted stacks

[6].

The case in which the room is heated by a localised

source of buoyancy is more complex and forms the subject

of the remainder of this paper. In broad-brush terms, as

with a distributed heat source, there is a transition from a

ARTICLE IN PRESS

A

BhA

hB

H

A A

BB

hB

hBhA

B

A

stack

window

Sloping roof

Internal heat load

H

Fig. 1. Schematic of the steady ventilation regimes in a room heated by a distributed source at the base and ventilated by two intermediate level openings,

one of which is connected to a stack which extends to the top of the room. In (a) we show a schematic of the configuration of the vent and stack at Hagley

School Worcestershire, in which there is a sloping roof, a high level window on the vertical wall adjacent to the highest point of the roof, and a stack risingabove the vent which connects to the lower side of the roof. In (b)(d) we show a generic building, used for the modelling, which can accommodate the

stackvent configuration of (a) by suitable choice the vent and stack elevations, but which also allows for the opening of a low level vent/window (Fig. b).

In (b)(c) the room ventilates in simple upward displacement mode, while in (d), which has the same geometrical configuration as (c), the flow reverses,

now entering the room through the stack.

S.D. Fitzgerald, A.W. Woods / Building and Environment 43 (2008) 171917331720


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unique upflow displacement regime, in the case that vent

A connected to the stack lies above the other vent B, to two

complementary flow regimes when vent A which is

connected to the stack lies below the other vent B.

However, with a localised source of heat, a two-layer

stratification typically becomes established in the space

[1,2]. In the original work of Linden et al. [2], in whichthere were vents at the top and base of the space, it was

shown that the interface lies between the inflow and

outflow vents, that the lower layer is composed purely of

external fluid and that the outflow fluid is derived purely

from the upper layer. We show here that with a stack

connected to one of the vents, there are a number of

different flow regimes which can develop depending on

whether the vent connected to the stack lies above or below

the vent connected directly to the exterior. In each of these

flow regimes, the interior fluid develops a two-layer

stratification, but depending on the vent sizes and

elevations, the outflow may issue from either the lower or

the upper layer. Similarly, the inflow may enter either the

upper or lower layer, and in the case in which the fluid

enters the upper layer, the lower layer becomes of

intermediate density between the exterior and upper layer.

In Section 4 we describe a theoretical model which

categorises these different flow regimes, and we present

some new analogue laboratory experiments in Section 5 in

which we demonstrate each of the regimes. We also test our

predictions, by developing an analogue theoretical model

for the experiments and comparing the transitions in flow

regime with that model. In Section 6, we discuss the

implications of these results for the design of naturally

ventilated buildings, and consider some avenues for furtherresearch.

2. Distributed heat loads

We consider a room in which there is a distributed heat

load QH at the base of the room which leads to vigorous

convection and a well-mixed interior [4]. It is assumed that

there are openings A and B on the sides of the room, of

area aA and aB, at heights hA and hB above the floor, with

opening A connected to a stack which rises to a termination

at elevation Habove the floor, where HXmaxhA; hB. It isimportant to recognise that in this model, H does not

correspond to the height of the building, but the height of

the top of the stack. However, with a sloping roof

configuration, one can imagine that the elevation of vent

B, on the upper end of the roof, could coincide with the

elevation of a stack above vent A, located on the lower end

of the roof.

We investigate first the flow regime which develops when

the outflow is through vent A and rises up the stack while

the inflow is through vent B(Fig. 1b and c). The buoyancy

driving the flow in this case is associated with a column of

buoyant room air extending from the level of vent Bto the

top of the stack, g0H hB, where g0 is the reduced gravity

of the air in the room, defined as g0 gre rr=re, where

re and rr are the density of the exterior and interior fluid,

and g is the acceleration due to gravity.

If the effective opening area of the two vents is A (cf. [2];

also see Section 4 herein), then the flow rate V is given by

V Ag0H hB1=2, (2.1)

while the heat flux QH is given by the balanceQH rCpDTV, (2.2)

where DT is the temperature elevation in the room, r is the

density of air and Cp is the specific heat capacity of air. For

small changes in temperature,

g0$gaDT, (2.3)

where the coefficient of expansion for air a 1=T, T isabsolute temperature expressed in Kelvin, and so the

temperature elevation of the room is

D

T

Q2H

ar2C2pA2gH hB !

1=3

(2.4)

and the ventilation rate is given by combining (2.2) and

(2.4),

VA2H hBgaQH

rCp

1=3. (2.5)

If the height of vent A, which connects to the stack, lies

below vent B, then it is possible that the reverse flow regime

is established in which there is inflow through the stack,

and outflow through vent B (Fig. 1d). In this case, the

buoyancy driving the flow is associated with the buoyancy

of the air in a column between vent A and vent B, given byg0hB hA, and the temperature elevation of the room is

DTQ2H

r2C2pA2gahB hA

!1=3, (2.6)

while the ventilation is now given by

VA2hB hAgaQH

rCp

1=3. (2.7)

Fig. 2 illustrates the variation of dimensionless internal

temperature Dy DTr2C2pA2gaH=Q2H

1=3 as a function

of the height of vent B hB hB=H in the case that vent A,which connects to the stack, is located at the points hA 13and 2

3above the floor of the space, where hA hA=H. The

two complementary flow regimes, which lead to two

different temperatures, develop when hA4hB, as expected.

3. Experimental investigation of multiple flow regimes with

distributed heating

Following a similar approach to Gladstone and Woods

[4], we carried out a series of analogue laboratory

experiments to test whether the multiple flow regimes

predicted in Section 2 do indeed develop in an analogue

experimental system. Using water as the working fluid, we

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S.D. Fitzgerald, A.W. Woods / Building and Environment 43 (2008) 17191733 1721


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immersed a perspex tank of height 28.6cm and area

17:8 cm 17:8 cm, with walls of thickness 0.8 cm, in a largereservoir of water. The experimental tank included a

number of openings at different levels on the side of the

tank, and there was also a central vent, which connectedthe room to the exterior through a stack. To model a

distributed source of heating, we used a coiled high

resistance wire placed just above the floor of the tank.

The heat flux produced could then be calculated from

measurement of the current through and voltage loss

across the wire [6]. Note that this is somewhat different to

the technique used by Gladstone and Woods [4] in which

the base plate was maintained at a constant temperature

and the associated heat flux was inferred from empirical

laws for turbulent thermal convection. The temperature

distribution in the tank was recorded by type K thermo-

couples, of accuracy 0:1 C, which were spaced at regularvertical intervals of 3 cm within the tank. With sufficient

heat load, in the range 200500 W, this experimental

system leads to turbulent thermal convection in the

experimental room, with Rayleigh numbers of order

108109. At these values of the Rayleigh number the fluid

within the tank is well-mixed and isothermal, as measured

directly from thermocouples in the tank [4], thereby

providing a good analogue to the well-mixed ventilation

regime within a room heated by a distributed source of

buoyancy. The flow through the openings, of diameter

12 cm, with speed 510 cm/s, has Reynolds number of

order 10002000. In order to apply the model of Section 2

to such flows, we require an estimate of the loss coefficient

for the openings. The effective combined loss coefficients

for the vents and stacks were measured using a calibration

experiment in which the tank was filled with a saline

solution and then immersed in the reservoir of fresh water.

Rubber plugs were removed from two of the ventilation

openings, at high and low level, and the rate at which the

solution then drained from the tank was recorded (cf. [4]).As described in the Appendix, the effective loss coefficient

in the calibration experiment was found to be $0:6, withtypical Reynolds numbers in the stack of order 1000. We

therefore use this value in comparing our experimental

data with the predictions of the model of Section 2. In

principle, one can model the frictional losses through both

the pipe, which serves as the stack, and the opening from

the tank into the pipe [7], but for the range of Reynolds

numbers in the present experiments, the empirical value for

the loss coefficient of 0:6 is sufficiently accurate. By makingthese measurements, and demonstrating that the constant

value for the loss coefficient describes this experimental

data, we have established that our model of the flow

through the experimental stack system is analogous to that

for a real building; in applying the model to a real building

one would however need to determine the loss coefficient

which pertains to the actual building. Note that in this

experimental system, the heat losses from the walls of the

experimental tank are small compared to the heat flux

convected with the ventilation flow (cf. [4,7]). We infer that

the flow regimes which develop within the experimental

model are dynamically analogous to those in a real

building (cf. [3,4]).

We then conducted a systematic series of experiments to

explore the steady state flow regimes as a function of theelevation of vent B, with a fixed height of vent A which

connects to the stack of 12cm. Experiments were

conducted for a range of heat fluxes, and both the inflow

and outflow regimes in the stack were realised. In Fig. 2, we

compare the experimental measurements of the dimension-

less temperature excess in the space,

Dy DTA2r2C2pgHaw

Q2H

!1=3, (3.1)

where aw is the coefficient of expansion for water, as a

function of the dimensionless height of the vent B, hB.

According to the model of Section 2,

Dy 1

1 hB1=3

, (3.2)

for the stack outflow mode, while

Dy 1

hB hA1=3

, (3.3)

for the stack inflow mode. It is seen in Fig. 2 that there is

very good agreement between the predictions and the

experimental results. Note, however, that as the vertical

separation between the mid-points of the two vents

becomes small relative to the vertical extent of the

ARTICLE IN PRESS

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

hB

Fig. 2. Non-dimensional temperature Dy as a function of the non-

dimensional height of vent B. The thick solid line corresponds to outflow

from vent A and through the stack and the thin solid and dotted lines to

inflow through the stack for hA 13

and 23, respectively. Also, shown is

comparison of experimental data (symbols) with theory for the case

hA 0:58. The dot-dashed line and open squares correspond to inflow andblack squares to outflow through vent A.



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openings, the nature of the flow through each vent changes

from uni-directional to bi-directional and the model is then

no longer valid.

4. Localised source of buoyancy

We now turn to the more complex case in which the heatsource is localised rather than distributed over the floor of

the space. We explore the effect of the height of the two

vents, and the presence of a stack, on the ventilation

regime. A localised source of heating at the base of the

room typically generates a turbulent buoyant plume and if

the space is ventilated through vents at the top and base of

the room, then the interaction of the plume with the

ventilation leads to the formation of a two-layer stratifica-

tion within the space [2]. In this flow regime, the interface

lies between the two vents, the lower layer is typically

composed of pure external fluid and only upper layer fluid

vents from the space. We now show how variations in the

height of the two vents, and the presence of a stack

connected to one of the vents, can change the internal

stratification and flow regime substantially.

For our analysis, we assume there is a stack connected to

a vent A, located at an intermediate height, hA, while the

other vent, B, connected directly to the exterior, has height

hB (cf. Fig. 1). We assume there is a localised heat source of

magnitude QH at the base of the room. We explore the

different flow regimes which may develop as the elevation

of vent B rises from the base of the room to the top of the

room, and we focus on the case hBoH, where H is the

height of the termination of the stack, as is typically the

case in practice. Again, we emphasise that the height of the

top of the stack H does not need to correspond to theheight of the top of the room; indeed, even in the case that

vent Bhas the same elevation as the top of the stack above

vent A, one can imagine a building with a sloping roof,

with the stack connected to vent A on the lower part of the

roof, and a simple vent B connected to the upper part of

the roof (cf. Fig. 1a).

The results for the case A 0:1 are shown in the regimediagram of Fig. 3(a), where A A%=l3=2H2 with l % 0:12for a fully developed turbulent buoyant plume [8], A%

cAaAcBaB=12c2Aa

2A c

2Ba

2B

1=2 and where c is the loss

coefficient for the vent. When vent B is located at the base

of the room, two regimes, which we denote as I and V, may

develop depending on the elevation of vent A above the

base of the room relative to the theoretical height of the

fluid interface, hL, as predicted by the theory of Linden

et al. [2]. If vent A is located sufficiently high in the room,

then the interface is predicted to lie below vent A, hLohA,

and there is pure outflow of the upper layer fluid, while

the lower layer is composed of external fluid (regime I,

ARTICLE IN PRESS

I

V

II

IV

III

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

hB

hA

A

BB

A

A=0.1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

II

IV

V

III

I

hB

hA

A=0.001

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

III

IV

V

I II

hB

hA

A=1

Fig. 3. Regime diagram illustrating the different flow regimes which develop in the case of a localised heat source for the cases in which (a) A 0:1,

(b) A 0:001, (c) A 1. In each case, the stack provides the conduit for the outflow.



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state, this is not possible since there would be no loss of

buoyant fluid from the room. Instead, the interface depth

becomes fixed at hA in order that buoyant fluid can vent

from the space. The density of the upper layer then equals

that of the plume as it reaches the lower interface of this

layer. However, now the volume flux supplied to the upper

layer by the plume is smaller than the buoyancy drivenoutflow through the stack associated with fluid of that

buoyancy. As a result, to maintain a steady state, lower

layer fluid is also convected up the stack. This decreases the

buoyancy of the fluid in the stack. This regime is denoted

as V in the regime diagram (Fig. 3a).

In this equilibrium regime, the volume flux in the stack V

is given by the sum of the flux in the plume Vp at height hA,

Vp lb1=3h

5=3A , (4.5)

and the flux from the lower layer, Vl into the stack, where b

is the buoyancy flux at the base and l 0:12 as before. The

buoyancy of the fluid in the stack g

0

s, which is assumed tobe homogenous, is given by the buoyancy of the plume at

height hA, g0p b=Vp, multiplied by the factor Vp=Vp

Vl owing to the dilution by the lower layer (ambient) fluid

which enters the stack. By matching the volume flux

associated with the outflow of fluid from the stack

A%ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g0sH hAp

with the sum of the volume flux supplied

to the upper layer by the plume and Vl, we deduce that the

ratio of the flux from the lower layer to the total fluid

entering the stack, Vl=Vl Vp, is given by the relation,

Vl

Vl Vp 1

h5=3A

A2=31 hA1=3

. (4.6)

The variation of Vl=Vl Vp with hA is shown in Fig. 4for several values of A. As the ratio Vl=Vl Vpincreases, the buoyancy of the outflow decreases, since

the fluid is being diluted with progressively more lower

layer ambient fluid. As expected, it is seen that Vl=Vl Vp increases with the effective opening area, A, and with

decreasing height of the point of access to the stack hA.

Note that in practice, when there is outflow from both

layers, the interface elevation far from the outflow vent

may be a little different from that at the vent. This is

because near the outflow the fluid is moving much more

rapidly than far from the vent and this can lead to a

reduction in the fluid pressure. Indeed, in the experimentsreported in the next section, in which H 20230cm, we

find that in this mixed flow regime there is a small

difference, o0:5 cm, between the far-field height of theinterface and the elevation of vent A, hA.

4.2.2. Transition VIII

We now consider how the internal stratification evolves

as the height of the inflow vent Brises from the base of the

room in the situation in which the flow regime commences

in regime V. The discussion corresponds to moving along

line BB0 in Fig. 3a. As hB rises, the interface remains fixed

to the level of the outflow vent, hA. Eventually, hB reaches

this same level and then rises above the interface and

outflow vent A, so that the flow adjusts to regime III as

described above and shown in Fig. 3a.

Now, the incoming fluid will enter the upper layer and,

being relatively dense, will descend through this layer. As itdescends, it will entrain some of the upper layer fluid before

reaching the interface. As a result, it supplies the lower

layer with fluid of intermediate density. While the interface

remains fixed at the level of the outflow vent A, the fluid

which vents from the room is composed of a mixture of

upper and lower layer fluid. As the height of the inflow vent

B increases, the lower layer becomes progressively more

buoyant, and the fraction of lower layer fluid venting from

the space increases. Eventually, a critical point is reached at

which all the fluid which vents from vent A and through the

stack originates from the lower layer. IfhB, the elevation of

the inflow vent B increases any further then this will cause

the interface to rise above the level hA corresponding to the

height of the outflow vent A. At this critical point, the flow

regime evolves to regime IV, as shown in Fig. 3a.

4.2.3. Regime transition IIIIV

We now develop a simple model to predict the point of

transition between regime III and IV, which occurs when

the interface is located at the level of the outflow stack, hA,

but all the outflow is derived from the lower layer. To

model this transition, we note that the conservation of

buoyancy in the room may be written as

b Vg0l (4.7)

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

hA

Vl/(Vl+Vp)

Fig. 4. Variation of the proportion of lower layer fluid which vents from

the stack as a function of the height of access to the stack hA when a room

is heated by a point source at the base and ventilated by an opening in the

base and an opening on the side wall which connects into a stack. Curves

are given for dimensionless vent areas A 0:1, 0.5, 1 and 2 by the solid,dashed, dotted and dot-dashed lines, respectively.



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upper layer fluid is given by

hA

1

2 hB A

2=5

1 hB

1=5

. (4.21)The solution of this relation is also shown on the regime

diagram (Fig. 3a). Again, we find that in the limit hB ! 1

then Eq. (4.21) requires that hA again decreases, owing to

the non-linear parameterisation of the mixing in the fluid

which flows in through the inflow opening. Indeed, in the

limit hB ! 1 we find that hA ! 0:5. Again, although theoverall predictions are consistent with our experiments, we

have not been able to test this detailed prediction using our

experimental system owing to limitations of the size of our

apparatus (Section 5).

The regime diagram shown in Fig. 3a corresponds to the

case A 0:1. In Figs. 3b and c we show the regime diagramfor the cases in which A 0:001 and 1, respectively. It canbe seen that in the case A51, for which the room is

effectively very tall relative to the typical dimension of the

openings, then regimes II and IV dominate, whereas when

A is larger, the other, intermediate regimes can be observed

for a wider range of values of hA and hB. Note that in the

case of very small A, the assumption that there is no heat

loss through the fabric of the ventilated space becomes

less accurate, and, in the case of outflow through the stack

(Fig. 3b) this will lead to a reduction in the temperature of

the upper layer and hence the buoyancy force and overall

ventilation flow rate. Modelling such effects in detail is

beyond the scope of the present study, but some of theeffects of such heat losses have recently been described by

Livermore and Woods [13].

4.3. Reverse flow regime with inflow through the stack and

vent A and outflow through vent B

As mentioned in the introduction, once the elevation of

vent A, connected to the stack, lies below that of vent B

which is connected directly to the exterior, it is possible for

the flow to reverse in direction, with inflow through the

stack and vent A and outflow through vent B. In this flow

regime, the interface either lies above (regime i, Fig. 5) or

below (regime ii, Fig. 5) the inflow vent A at the base of the

stack, where the cold ambient fluid enters the room. In

regime i, the ventilation of the room is similar to the classiccase described by Linden et al. [2], but with a modified

height hB hL over which the buoyancy acts to drive the

flow, where hL is the height of the interface within the

room. Consequently, hL is given by a modified form of

(4.1)

A h5L

hB hL

1=2. (4.22)

In order that the interface lies above the inflow vent, we

require hAohL where hL is given by (4.22). If the interface

lies below the point of access to the stack then regime ii

develops whereby the incoming fluid will enter therelatively hot upper layer and descend. As the relatively

dense ambient air descends, it will form a plume and

entrain some of the hot upper layer. Consequently, the

lower layer will be warmer than the exterior fluid. The

delineation between this regime ii and regime i is shown in

the regime diagram (Fig. 6), with the transition from

regime i to ii being given by setting hA hL in Eq. (4.22).

5. Experimental observations

We have developed a series of new analogue laboratory

experiments to investigate whether the different flow

regimes that we described in Section 4 actually develop in

practice. In the experiments of Section 3, we modelled a

distributed source of heating using a heated wire. However,

in order to model the convective flow associated with a

localised heat source, it is easier to use the analogue system

of fresh and saline water as described by Baines and Turner

[10]. The focus of the experiments is to demonstrate that

the multiplicity of different flow regimes predicted in

Section 4 can indeed develop in an analogue experimental

system. We also develop the model of Section 4 for

application to the experiments, now using properties of a

saline plume in water, and we compare the experimental

observations of the conditions for transitions in the flow

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REGIME i

hA

REGIME ii

hBhA

Fig. 5. Schematic diagram showing the two layer stratification in the room when inflow through the stack occurs. Regime i occurs when the interface in

the room lies above the base of the stack and regime ii when the interface lies below the base of the stack.



10/15

regime with the predictions of this model. Similar experi-

mental models exploring the dynamics of saline turbulent

plumes within a ventilated, enclosed chamber have been

applied as an analogue model for natural ventilation flows

from localised sources of heating in a number of earlier

studies (cf. [3,11]). The turbulent buoyant plumes gener-

ated from the localised source of saline water within themodel room represent an analogue for the natural

convective flows generated from localised heat sources

within a building, provided that the source mass flux is

much smaller than the ventilation flow, and that the plume

is of sufficient Reynolds number to be fully turbulent [11].

The experimental system consisted of a small perspex

tank (the room) of height 28.6 c m and area

17:8 cm 17:8 cm. This was partially immersed in a largereservoir of height 48 cm and area 88cm 43 cm, which

acted as the exterior environment. We used fresh water in

the environment and saline solution as the source of

(negatively) buoyant fluid. The system was therefore run

upside down, with the plume source located below the free

surface of the water in the tank, and the stack termination

being located above the base of the tank (Fig. 7). The dense

saline solution injected through the plume source, which

moves downwards, corresponds to a source of hot air risingfrom a localised heat source on the base of a real room.

One side wall of the experimental tank had a large

number of circular ventilation holes, of diameter 5, 6 and

15 mm. These ventilation holes were sealed by rubber

stoppers. These could be removed to model a wide range of

effective vent areas and heights between the model room

and the exterior reservoir. A hole in the base of the tank

also enabled stacks of internal diameter d, and with the

entry point being located at a height below the roof of the

tank, ha, taking values d; hA given by (13 mm, 189 mm),(13 mm, 245 mm), (10 mm, 208 mm), (10 mm, 230 mm). The

stack entrance for the fluid was the horizontal plane of the

top of the cylindrical pipe used for the stack.

The room and exterior tank were initially filled with

fresh water. Then, for convenience a 5% salt solution was

supplied from above at a constant rate 0.3 cc/s via a twin-

feed peristaltic pump, as the source for a descending

(negatively) buoyant plume. The nozzle used for the plume

source was described by Woods et al. [11]. The room was

elevated 20 cm from the base of the exterior tank so that

the dense fluid exiting from the room could sink to the base

of the reservoir, away from the room. A systematic series

of experiments were conducted with the plume source 7.1,

10.8, 13 and 14.5 cm from the base of the tank, and with the

number of upper vents ranging from one 6 mm diametervent to, at most, 4 mm 6 mm and 3 mm 15mm

diameter vents all being open simultaneously. As in Section

3, the effective combined loss coefficients for the vents and

stacks were found to be $0:6 with typical Reynolds

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

iiii

i

hB

hA

Fig. 6. Regime diagram illustrating the different flow regimes which

develop, in the case of a localised heat source, for the case in which

A 0:1 and when the stack is the source of inflow.

Reservoir

tank

Experimental

model room

Source of saline

fluid to pump

into tank

Peristaltic Pump

Plume

source

Pipe connected

to the opening in

base represents

analogue of a stack

Ventilation

opening in

side of model

room

Free surface

Supports to

raise model

room off floor

of reservoir

Fig. 7. Schematic of the experimental apparatus, illustrating how the model room is immersed within the external reservoir tank, and the geometry of the

inflow nozzle and the experimental stack.



11/15

numbers of order 1000. This is discussed further in the

Appendix. In comparing the experimental observations of

the transition in flow regime with the predictions of the

theoretical model, we adopt the model of Section 4, but

now use physical properties of water and aqueous saline

solution. The main comparison of the experimental results

with the theoretical predictions is in testing the flow

regimes which develop with different plume source heights

and with different net sizes of ventilation opening;

however, we also compare the interface height observed

in the experiments with the theoretical predictions (Fig. 8a)

and the density of the outflowing fluid with the theoretical

predictions (Fig. 8b).

5.1. Regimes I and V

In the first series of experiments a stack was located in

the base of the tank whilst the number of ventilation holes

at the top of the tank was systematically increased. For

each configuration, we fixed (a) the distance between the

plume source nozzle and external termination of the stack,

H, which in our experiments also corresponds to the depth

of the room, and (b) the stack height hA, and we

determined the critical value of A% at which there was a

transition in the flow regime, as may be seen in themeasurements of the steady state height of the interface

(Fig. 8a). The transition point corresponds to a change in

regime from single layer outflow (regime I, Fig. 3) to mixed

outflow with the interface fixed at the level of vent A

connected to the stack (regime V, Fig. 3). The photographs

in Fig. 9 illustrate the two different regimes, with Fig. 9a

showing the case with both upper and lower layer fluid

supplying the stack and Fig. 9b the case of only fluid from

the buoyant layer in the stack. Note that in Fig. 8a we have

combined the experimental results for a range of different

stack heights by plotting hL as a function ofA. This allows

the theoretical prediction of interface height for the single

layer outflow mode (regime I, Eq. (4.1)) to be compared

with experimental observations whilst clearly separating

the transition points for different stack heights.

The proportion of upper and lower layer fluid in the

outflow from the stack can be deduced from measurements

of salt concentration in the stack and the lower saline layer.

In Fig. 8b we show how this varies as a function of effective

vent area in the mixing outflow regime. Note that in each

case, for small vent area the outflow is derived purely from

the lower layer of saline fluid and the interface lies above

the outflow stack in the experiment (regime I, Fig. 3).

However, with larger area, the interface is fixed at the level

of the access point to the outflow stack and the outflow is amixture of fluid from the upper ambient layer and lower

saline layer.

To compare the flow behaviour directly with the

predictions of our model we need to account for the effect

of a virtual origin since the plume source has finite mass

flux (cf. [11,12]). The correction method suggested by Hunt

and Kaye [12] has been used, and it may be seen in Figs. 8a

and b that the model predictions are in good accord with

the experimental observations. In particular the agreement

of interface height in the cases for which the interface lies

below the level of the outflow vent, as shown in Fig. 8a,

suggests that the experimental plumes are well modelled by

the theory of turbulent plumes (cf. [2]).

5.2. Intermediate level vent and intermediate access point to

stack

In the main series of experiments, the plume source was

located just below the free surface of the water in the room,

1317 cm above the floor of the room (Fig. 7), whilst a

series of stacks of different heights and a series of vents

connecting directly to the exterior were used in order to

map out all the different flow regimes predicted in Fig. 3.

Five percent of saline solution was injected from the plume

source at a rate of 0.3 cc/s and each experiment allowed to

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Vl/(Vl

+Vp

)

hA=0.16

hA=0.16

hA=0.36

hA=0.3

hA=0.48

hA=0.22

hA=0.36

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3

hL

A

A

Fig. 8. (a) Variation of interface height hL as a function of vent area A for

the case in which the room is ventilated by a stack and a vent at the height

of the plume source. The thick solid grey line corresponds to Eq. (4.1) cf.

Linden et al. [2]. The other lines indicate the stack heights hA used in the

experiments, results of which are represented by symbols. Open squares

correspond to hA 0:16, filled squares to hA 0:22, open triangles tohA 0:3, filled triangles to hA 0:36 open diamonds to hA 0:48 andfilled diamonds to hA 0:57. (b) Variation of proportion of ambient fluidin the stack as a function of A for mixed outflow mode. The solid and

dashed lines denote the predictions from (4.6) for hA 0:16 and 0.36,respectively, and the open squares and filled triangles represent the

corresponding experimental results. Experimental errors due to measure-

ment of interface height are estimated to be 0.5 cm, which is at most 5%.



12/15


13/15

this case, the interface always lies below the outflow vent.

However, depending on the location of the inflow vent

supplied by the stack, the interface may lie above or below

the inflow vent, and therefore the lower layer is composed

of either pure external fluid or fluid of intermediate

composition.

ARTICLE IN PRESS

Fig. 10. Photographs from seven experiments in which the room was ventilated by a stack accessed at an intermediate level and an intermediate level vent.

Cases shown for internal stack diameter 13.5 mm, intermediate level vent 15mm diameter. (a) Regime Ionly upper layer fluid vents from the stack, cold

lower layer (hA 0:46, hB 0:15); (b) Regime Vfluid from both the lower and upper layers vents from the stack, cold lower layer (hA 0:25, hB 0:1);(c) Regime IIonly upper layer fluid vents from the stack, warm lower layer (hA 0:46, hB 0:77); (d) Regime IVonly lower layer fluid vents from thestack, warm lower layer (hA 0:16, hB 0:5); (e) Regime IIIfluid from both the lower and upper layers vents from the stack, warm lower layerhA 0:33; hB 0:38; (f) Regime iinflow through stack, cold lower layer (hA 0:21; hB 0:94); (g) Regime iiinflow through stack, warm lower layer(hA 0:46; hB 0:77). All experiments are conducted upside down c.f. a heated room and in the descriptions above cold refers to fresh water in theactual experiment and warm to saline water. Dotted lines indicate the position of the interfaces, arrows the direction of flow through the vents.



14/15

The flow regimes predicted by the models have been

demonstrated in a series of new analogue laboratory

experiments, for both the distributed and the point sources

of buoyancy. Furthermore, we have developed the

theoretical models for application to the laboratory

experiments, and found reasonable agreement between

the experimental observations of transitions betweenregimes and the prediction of the model.

The work is significant in that it exposes the dominant

control that the elevation of the ventilation openings, and

the presence of stacks, may have on natural displacement

ventilation flows. The situation is of relevance, since many

naturally ventilated buildings are designed to have outflow

through stacks. However, we have shown that a range of

different flow patterns can develop with the stacks

providing a pathway for either inflow or outflow, and that

in some situations, both of these different flow regimes may

develop. As well as stacks and vents designed for natural

ventilation, the work has implications for the natural flows

which may develop when a room exchanges air through a

combination of a chimney and window.

In some buildings with flat roofs, stacks extend above the

height of the building, to height H above the base of the

room. In this case, the dimensionless height of the vent (e.g.

a window) which opens directly to the environment, scaled

relative to H, hB is always smaller than unity. As a result,

only a sub-set of the flow regimes described herein ( Figs. 3

and 6) are accessible. However, in other buildings, for

example, those with sloping roofs, as at the Hagley School

in Worcestershire, the height of the opening which connects

directly to the exterior may be intermediate to the height of

the opening to the stack from the interior space and the

height of the top of the stack. In this case, all the regimes

may be accessible.

In addition to the importance of recognising these

regimes for developing control systems for the flow, and

in particular for locating sensors to detect the different flow

regimes, these findings may also be of importance formodelling the dispersal of smoke or chemicals from a point

source release within a building, and for predicting the

concentration of contaminants issuing from the building.

Appendix

The effective loss coefficient for a vent connected to a

stack was determined experimentally. The tank with an

open top and a stack connected to the base was filled with

5% saline solution. The external reservoir was then

carefully filled such that the water level was 7 cm above

the top of the experimental tank. A loosely placed rubberbung which was used to seal the stack whilst the apparatus

was filled was then dislodged and the descending interface

recorded by video using the shadowgraph technique. The

effective area A was determined by comparing the data

with the theoretical prediction [3] and fitting the loss

coefficient. The best fit was obtained for a loss coefficient of

0.6 as shown in Fig. A1. This example corresponded to a

stack of vertical extent 9.5 cm and diameter 13 mm.

The deviation from the theoretical prediction at late time

time 130s corresponded to the time at which the

interface approached the top of the stack. At this time

the fluid exiting from the stack became a mixture of fresh

and saline fluid and the draining box model [3] was

therefore no longer appropriate to describe the ventilation

flow.

References

[1] Sandberg M, Lindstrom S. Stratified flow in ventilated roomsa

model study. In: Proceedings of roomvent 2nd international

conference on air distribution in rooms, Oslo, Norway; 1990.

[2] Linden PF, Lane-Serff GF, Smeed DA. Emptying filling boxes: the

fluid mechanics of natural ventilation. Journal of Fluid Mechanics

1990;212:30935.

[3] Linden PF. The fluid mechanics of natural ventilation. Annual

Review of Fluid Mechanics 1999;31:20138.

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0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

I

V

II

IV, i

II, ii

IV, ii

III, i III, iihB

hA

Fig. 11. Regime diagram for the laboratory experiments in which the

room is ventilated by a stack accessed at an intermediate level and anintermediate level vent. The case shown is for A 0:13. Experimentalobservations of the flow regime are indicated by symbols: Diamonds

correspond to regime I, open circles to V, filled squares to II, open squares

to IV, filled circles to III, triangles to regime i and crosses to ii.

0

5

10

15

20

25

30

35

0 50 100 150 200 250 300 350

time (s)

distance(cm)

Fig. A1. Experimental measurements of height of interface as a function

of time (diamonds) compared with the theoretical prediction (solid line)

for the case in which the loss coefficient for the stack and vent 0:6.



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[4] Gladstone C, Woods AW. On buoyancy-driven natural ventilation of

a room with a heated floor. Journal of Fluid Mechanics

2001;441:293314.

[5] Fitzgerald SD, Woods AW. Natural ventilation of a room with vents

at multiple levels. Building and Environment 2004;39:50521.

[6] Chenvidyakarn T, Woods A. Multiple steady states in stack

ventilation. Building and Environment 2005;40:399410.

[7] Livermore S, Woods AW. Natural ventilation of a building withheating at multiple levels. Building and Environment 2007;42:141730.

[8] Morton BR, Taylor GI, Turner JS. Turbulent gravitational convec-

tion from maintained and instantaneous sources. Proceedings of

Royal Society of London Series A 1956;234:123.

[9] Turner JS. Buoyancy effects in fluids. Cambridge: Cambridge

University Press; 1979.

[10] Baines WD, Turner JS. Turbulent buoyant convection from a

source in a confined region. Journal of Fluid Mechanics 1969;37:

5180.

[11] Woods AW, Caulfield CP, Phillips JC. Blocked natural ventilation:

the effect of a source mass flux. Journal of Fluid Mechanics

2003;495:11933.[12] Hunt GR, Kaye NG. Virtual origin correction for lazy turbulent

plumes. Journal of Fluid Mechanics 2001;435:37796.

[13] Livermore S, Woods AW. On the effect of distributed cooling in

natural ventilation. Journal of Fluid Mechanics 2007, in press.

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