The Fluid Mechanics of Natural Ventilation-Displacement Ventilation by Buoyancy-Driven Flows Assisted by Wind G.R

Home , Displacement ventilation, Numbers (season 4)

AIVC 12,435

PERGAMON Building and Em·ironment 34 ( 1999) 707-710 --

The fluid mechanics of natural ventilation-displacement ventilation by buoyancy-driven flows assisted by wind G.R. Hunt*, P.F. Linden

Depanme111 oj Applied .Hatlil'111a1ics Theol'<'li('llf Phi-sics. L"11ireni1r(}(Camhridge. Silrer Street. Ca111hriclg<'. CBJ 9EW. L'.K. mu/

Abstract

This paper describes the fluid mechanics of natural ventilation by the combined effects of buoyancy and wind. Attention is restricted to transient draining flows in a space containing buoyant fluid. when the wind and buoyancy forces rei11/(Jrc<' one another. The flows have been studied theoretically and the results compared with small-scale laboratory experiments. Connections between the enclosure and the surrounding fluid are with high-level and low-level openings on both windward and leeward faces. Dense fluid enters through \\indward openings at low levt:ls and displaces the lighter fluid within the enclosure through high-level. leeward openings. A strong. stable stratification develops in this case and a displacement flow is maintained for a range of Froude numbers. The rate at which the enclosure drains increases as the wind-induced pressure drop between the inlet and outlet is increased and as the density difference between the exterior and interior environment is increased. A major result of this work is the identification of the form of the nonlinear relationship between the buoyancy and wind effects. It is shown that there is a Pythagorean relationship between the combined buoyancy and wind-driven velocity and the velocities which are produced by buoyancy and wind forces acting in isolation. This study has particular relevance to understanding and predicting the air flow in a building which is night cooled by natural ventilation. and to the flushing of gas from a building after a leak. Elsevier Science Ltd. All rights reser\ed. ·(' 1999

Nomenclature pressure inside the space at windward opening P1 lli_, ll" area of leeward and windward openings, respec pressure inside the space at leeward opening p, tively wind pressure at the inlet and outlet, respec P";' P,", the 'effective' area of the openings tively A* B buoyancy flux of the buoyant jet Peclet number Pc Cn. C" coefficients of discharge and expansion. respec Q volume flux tively Reynolds number Re Cpi· Cr., pressure coefficients at the inlet and outlet, S cross-sectional area of the space respectively time [s] Fr Froude number re time taken for the space to empty by buoyancy forces Fr(r) Froude number at time t assisted by wind fr(O) initial Froude number time taken for the space to empty by buoyancy tc0, few Frcn1(0) critical initial Froude number forces alone and wind forces alone, respectively g acceleration due to gravity absolute temperature T reduced gravity,= g/'J.p/p Uw velocity associated with the buoyancy and wind g' U". Ii depth of the buoyant layer of fluid force. respectively '11 vertical height of the leeward opening L:1_, Uw velocity through the leeward and windward Ii* vertical distance over which the buoyancy force acts openings. respectively vertical distance between midpoints of windward and C";"" average velocity of the wind /-! leeward openings Creek sy111ho/.1· length of the intruding current L "J. empirical constant, = 1.85 momentum flux of the buoyant jet A1 /'J. T temperature difference between the external and reference surface pressure P0 internal environment /'J.p density difference between the external and internal

* Corrcsronding author environment

fl wind pressure drop tigation of natural ventilation by combined wind and "h""'' diffusivity of heat in air and salt in water. buoyancy effects has previously been published [7]. 1',., t respectively1 Wind flowaround buildings. which are typically sharp ''";,. '"'"'"' kinematic viscosity of air and water. respec edged and of rectangular cross-section. results in a pres tively sure drop tJ. across the building. The wind pressure is p density of the external environment positive at the windward face and negative in the lee of p' density of the internal environment the building due to separation and turbulent effects. As a consequence of this pressure drop the air in a heated room, with ventilation openings on both windward and Subscripts leeward faces. is subject to both a buoyancy force and a model refers to quantities used at small-scale m the pressure force associated with the wind. The processes are laboratory nonlinear and so the combined effectscannot be obtained full-scale refers to quantities at full-scale simply by adding the results of the two different processes acting in isolation. The nonlinearity arises because the flows through openings are a nonlinear function of the Introduction pressure drops across them. 1. The extent to which the wind hinders or enhances the The use of natural ventilation has become an increas ventilation of the enclosure depends upon its strength ingly important and prominent aspect of building design and direction, as well as upon the size and location of and is re-emerging as a desirable alternative to mech the openings and the temperature difference between the anical systems. Architects, building services engineers internal and external environment. Air speeds induced and their clients are attracted to natural ventilation as by a temperature difference of I 0 C in a building 5 m a way of providing a comfortable environment in the high are of the order of 1.3 ms-1• Average wind speeds workplace because of the potential savings in energy costs for the U .K. are of the order of 4 and therefore the ms-1 and for wider environmental reasons. This increased effects of the wind on the buoyancy-driven flow might be interest has exposed the limitations of existing knowledge expected to be considerable. There are clearly situations to predict how naturally ventilated buildings v.ill perform where the wind and buoyancy forces reinforce one under the wide range of atmospheric conditions typically another to ventilate a space. as in Fig. I (a). and situations experienced over the year. where the buoyancy and wind forces oppose one another. A number of earlier cxperimen ts and theoretical studies as in Fig. l(b). The two situations depicted in Fig. I will have examined the natural ventilation of an enclosure. result in very different air flows within the enclosure. The majority of these studies have concentrated on ven Transient ventilation flows, such as those which may tilation flows which are solely driven by either (i) the occur at night in naturally ventilated buildings in which buoyancy force associated with the temperature differ there are no internal sources of buoyancy, are considered. ence between the fluid inside the enclosure and that of its Night-time ventilation allows far higher ventilation rates surroundings (stack effect) [I]. or (ii) the wind. Stack than would be acceptable during the hours of occupancy. driven flowsare well understood and mathematical mod The aim of night cooling by natural ventilation is to els have been developed to predict the thermal strati remove the warm internalair and to replace it with cooler fication within the enclosure [2. 3]. Work on wind-driven external air in order to promote heat transfer from the ventilation flows has received attention over many years, building fabric to the air. In this way the fabric of the particularly with the use of wind tunnel models for the building is cooled which then reduces peak daytime tem determination of pressure coefficients for model build peratures. In this paper the fluid mechanics of the flow ings. Pressure coefficients have been determined for a that occurs when wind and buoyancy forces reinforce one wide range of building geometrics. wind directions and another to ventilate an enclosure are examined. Exper- exposure levels and are well documented [4. 5]. In contrast to the numerous studies on stack and wind effects in isolation. ventilation by the combined effects of stack and wind has received little attention in the open literature. Wind Wind Davies and Linden (6] examined the combined effect of buoyancy and wind on the ventilation of a corridor (the corridor was of rectangular cross-section with a doorway

at one end that occupied the full cross-section of the cool cool corridor). However. for an enclosure with high and low level openings the nonlinear interaction between wind (a) (h) and buoyancy is not well understood and provides the Fig. I. A schematic diagram illustratin\!.:onditions for wind und which motivation for the present study. A preliminary inves- fo rces (a) one a11otl1er oppose buoyancy reinforce and (b) G.R. H11111, Linden Building and E11riro11me111 1999) 707-720 P.F. I 34 ( 709 imental techniques, together with simple mathematical The fluid pressures inside the enclosure at z = H and modelling have been used to determine the temperature z = 0 are denoted by P1 and P�, respectively. Using and stratification within the enclosure as a function of Bernoulli's theorem we have the wind speed and the temperature or other buoyancy difference. ( a) 1

2. A mathematical model for wind assisted buoyancy and driven displacement flows (I b) A mathematical model for combined bu yanc and o wind-dri en di'placement ventilution is now proroscd. The fluid velocity is assumed to be constant over the nsider the empt ing ofa pac initially containing nuid entire depth of the opening. Jn reality, there will be a of uniform density p', which b heavier is ·mrounch::d variati n in the fluid velocity across the vc rtic;tl depth of fluid of density p' +/)..p. e make the Bou· ·incsq p = \ the openi ng. However. thi variation in pres ure is likely approximati n in which the Jensity differenc� ().p o > 0 to be small compared with the variation in pres-ure between the fluid in the pace and the urroundin fluid. g between the two openings due to the (large) vertical dis i urned t be mall ompa d p. We write a· c n..: 1 tance between them. The flow will contract on flowing g' = gf..p/p for the reduced gravity. If openings, whose through an orifice and so eqn (1) should only be applied centres are separated by a vertical distance H. are now to the velocities and areas after contraction. Also, the made at low-level and at high-level, then there will be a flows are not dissipationless unless the contractions are buoyancy-driven displacement flow through the open perfectly smooth. For this reason eqn ( 1) is replaced with ing . The dense tluid nter· through the lower opening and light fluid leave' through the upper, producing a (2a) table interface at height H-'1(1). If. in ddi t i n to tht: a o buoyancy force, there is a steady external wind then the and fluid inside the enclosure will also be subject to a force associated with the pressure drop L1 between the wind ward and leeward openings. In the case where the wind (2b) ward opening is at low-level and the leeward opening is at high-level the wind adds to this flow maintaining the where the expansion coefficient C0 is a constant lying same structure. It v. ill be assumed. as is observed to be between one half. fo r a sharp expansion at the inlet, and the case. that the interface is stable. at least for a range unity. for a perfectly smooth expansion (8]. CJ is related of so that no mixing occurs. The cross-sectional area to the discharge coefficient Cn ( � 0.6) by C n = CJ 1, see L1. of the enclosure S (independent of height), is assumed to (9]. Assuming a hydrostatic variation in pressure within be much larger than the area of either opening so that the enclosure we write the velocity of the ascending interface is small. Under = P +pg(H-h)+p'gh. (3) these assumptions we may assume that the variation in P1 1 pressure within the enclosure is hydrostatic. The notation If the fluid is incompressible then the volume flux, Q, used and the geometry of the enclosure is illustrated in which enters the enclosure equals the volume flux which Fig. 2. For convenience. a reference pressure P0 has been leaves. so that chosen to denote the total fluid pressure at the centre of (4) the windward opening on the outside of the. enclosure. Eliminating P11 and P1 from eqns (2) and (3) and using eqn (4) gives the volume flux through the openings, namely. wind ...... ··-·· ·-···-·· · ··· ···· (P. pcrh-{).) P ... 0 - 0 ' .J Q q'h+ (5) �p =A* ( pL1)� . h where H

- (6) p ...... Pii· p,

Fig.�. sc.:hc.:nwtic.: diagram illustrating the.: gc.:nm.:try ol' the.: c.:nc.:losurc.: A and the.: notation usc.:d. is an 'effective' area of the openings. The respective fluid

r ' H11111, P.F. Li11de11 , B11ildi11g and 1999; 707-720 710 G.R. E11riro11111e11134 ( velocities through the inlet and outlet of the enclosure were considered. In the limit of purely wind-driven flow are now we have

h A* -=!---.l (15) u" (7) H te" =-(g'h+ -�)+- (/\\ p where re". the total emptying time under the influence of and wind forces alone. is given by

A* SH -p UL= . (8) " =- I . ( 16) (/L (g'h+ -�)+- f(' A* ( ) - p

Equation ( 15) that a di plac ment mode of The fluid velocities given in eqns (7) and (8) can be seen n "umc e ven to be composed of the sum of the qua re or the buoyancy tilation prevai l' within the enclo·ure in the ab cnce or velocity. i,1/h. and of tht: square or the lo it associ buoyancy. In thi limit mixing now will develop within c c ct ated with the pre· ·urc drop due to thi;; wind.,' The th enclosure as a density differencei needed maintain p. e 10 total fluid velocity U through an opening is of the form the stratification. However, eqn (15) serves to describe the fluid behaviour as the wind-driven limit is approached . ,.ii + v;-;1 1 ' ., (9) U - ..J (see Section 5). From eqns (13) and (15) we note that as the wind strength increases, from initially being zero to where and u\\ denote the velocity associated with the vb finally dominating the fluid motion, the descent of the buoyancy and wind force, respectively. interface changes from a quadratic to a linear dependence The rate of ascent of the interface between the dense with time. and light fluid is given by The relative sizes of the wind and the buoyancy pro dh Q duced velocities are measured as a Froude number, (I 0) namely, the ratio of the �erage wind velocity to dt s U";"d the buoyancy velocity Considering the wind pres and so. if Hat time = 0. Jg'h. h = / sures at the inlet and outlet we obtain

' I pl = - - (./2 - • h(t) I (11) � \\llld � ( 17) = ((g H, +--�)2 ---f/A"'t)�- --� ( - I'<') H g'H p 2 pg'H p \\'here Cl';= P,,,1( � pC2 ;11<1)andC11" = P"'' ('.pl'2111.i)de1101e and the total time te for the space to empty is given by - 1he pressurt: coefticients at the inlet und outlet respec tively. and P,,, and P"'' demHc the wind pres ·ur,e at !he re = 2S . ( 12) respective openings. The Froude number may therefore --.. [(g' H + -�)i --(�)i] .fl, I " p p be written in the form From the general expressions (11) and (12) the fluid behaviour within the enclosure can be examined in two Fr(t) = ( 18) limiting cases of interest. The first caseof interest is when the buoyancy force drives the motion in the absence of The non-dimensional evolution of the interface height, wind. The second case is when there is no density differ as a function of time. becomes ence between the fluid in the enclosure and the sur roundings and the ventilation flowis purely wind-driven. ' l -h = I +Fr(O)-- - -Fr(O)-. ( 19) In the absence of wind, eqns (1 1) and (12) reduce to H (J t�h)2 '

where -- I l ; h - - ( 13) H ( - teh) • Fr(O)= �. (20) where teh. the time for the enclosure to empty under the �P?H influence of buoyancy forces alone, is given by denotes the initial Froude number. The total time for the enclosure to empty under the combined forces of 2S buoyancy and wind. as a fraction of the time to empty ten= * H i . ( 14) A (g' ) under buoyancy alone. is then

These equations are identical to those derived by Linden !!!__ = +Fr(O)--Fr(O). (21) al. [I] in which flowsdriven solely by buoyancy forces j1 et f('b Hunt. Linden. Bui/di11g and Enriron111l!nt34 f 1999) 707-720 G.R. P.F. 711

From eqns (18) and (20) the Froude number. at time t, allowed a number of different generic ventilation flows may be expressed as to be modelled. Note that the geometry of the box and the positioning_of the box openings is not intended to (22) represent any particular room or building. The box was suspended in the flume with its windward face normal to and hence, for a constant wind velocity and density the oncoming flow. The cross-sectional area of the box difference, the Froude number Fr(t) increases as the was approximately 40% of the total cross-sectional area enclosure empties. In other words. wind effects become of the flume and therefore represented a considerable blockage to the flow. Hmvever, the effect of the wind on increasingly more dominant as the thickness h of the lighter layer decreases. In fact, the wind force begins to the ventilation flow within the box is governed only by dominate the buoyancy force once the pressure drop between the windward and leeward openings, which has been measured in the present experi � ments, and hence the structure of the fluid flow around h< (23) pg'' the box is not important. Density differencesin the apparatus, necessary to simu late the stack effect. were achieved by adding salt solution 3. The experiments to the box. The density of the fluid in the box was then greater than that of the fresh water in the flume, and 3.1. E.rperimentaf se1-11p hence dense fluid drained from low-level openings in the box. The large volume of fresh water contained in the In the experiments, the enclosure was a tran·parent flume tank ensured that the fluid around the outside of box suspended in the test section of a large llum1:: tank, the box remained at an approximately constant density see Fig. 3. The flume had the dimensions of throughout the experiments. The case of salt water in the 2.65 x 0.30 0. 57 m (length x width x height) and was x box represents a space cooler than the environment; for filled with fresh water. A four horse-power. electric pump the (more common) case of a warm sp

0 2 0.15 x 0.25 m (length width x height). Multiple . 95 x x circular opening·. each of area n � on both wind•\· a rd cm . and leeward faces of the box at high and low levels.

reflector honeycomb section I box I 00000

<:-- ""'-- 0. 5m 2

,' o-6 o o o

control valve pump 0.15m Fig The tlume lank Fig. The ventilation box. J. 4. Hunt. P.F. Linden. Building and Enrironment 1999; 707-720 712 G.R. 34 (

the fluid contained in the box using a food dye. A bank and brine to create density differences in the system. the offtuorescent tubes was used to light the experiments and rc:duction in �he magnitude of the dimensionless groups

a sheet of tracing paper diffused the light. A video camera due to the change in scale is counteracted as l'uir » r ·'"k' was used to record the experiments. Each frame of the and Khcat » Ksah• where 1,·hcat and 11.·,.111 denote the diffus recording was then digitised in a false-colour format i\ ities of heat in air and salt in water, respectiYely. In using the image analysis system Diglmage [1 0] and cor addition, significantly larger density differences can be rected for background variations in the lighting. From ac:hieved in the laboratory using fresh water and brine the resulting images the position of the interface, which than occur in air due to temperature differences. hence

corresponded to a particular concentration of dye. was _(/ooud » g;·uu-sc

was then switched on and the required 'wind' velocity where Umodel denotes the speed of the fluid driven by

selected. Each experiment was initiated by opening sim = the flume and Urull·SC'11« u\\ind) denotes the wind speed. ultaneously both windward and leeward vents. This Dynamical similarity is achieved at small-scale in the resulted in dense fluid leaving the box through the low laboratory as 9�10ue1 » g;·u11-sca1c and Umodet « Uru11-scak· leveL leeward opening and ambient fluid entering the box By varying the density difference between the fluid through the high-level, windward opening, i.e. a dis c:ontained in the box and its surroundings and the 'wind' placement flow was established. vdocity, the ventilation of the enclosure observed during experiments can be used to predict how naturally ven 3 .2. Scaling considerations tilated buildings will perform under a wide range of con ditions. Friction and diffusioneffects are small for typical ven tilation flows as is evident fr om the Reynolds, and Re, Peclet. Pl!, numbers which are both of the order of IO' 4. Results and discussion (domestic)-! 06 (industrial). Dynamical similarity between buoyancy-driven flows at small-scale and those Experime111a/ obserrntions 4. I. at fu ll-scale is achieved when Images showing the development of a typical experi ment, in which the box empties under buoyancy forces alone, are shown in Fig. 5; the density differenceg' 0. I = m s-2 and the flow is shown at = 45, 105, 165 and 225 t and s. The displacement mode of ventilation is made clearly visible by the stable descending interface. The devel opment of a typical experiment in which the buoyancy forces are reinforced by wind fo rces, is shown in Fig. 6. The density difference is identical to that of the no-wind

where r is the kinematic viscosity of the fluid. K is the case and the wind flow is from left to right. A dis diffusivity and H is a characteristic length scale. If air placement flow was maintained for a range of initial was used as the working fluid and density differences Froude numbers when the effects of wind and buoyancy were created using heat then the Reynolds and Peclct reinforce one another and the ventilation rate was sig numbers at small-scale would be reduced by a factor niflcantlyenhanced over the corresponding no-wind case. of (H111.,,1c1/Hru11-scalY 2• Laboratory models are typically as is clearly evident on comparing Figs 5 and 6.

I : I scale and hence the Reynolds and Peclet In the initial stages of the experiment. turbulent mixing 20--1 : 00 numbers would be reduced by two or three orders of was observed as a jet of fluid entered through the wind magnitude. However. by using water as the working fluid " ard opening and mixed with the denser fluid contained H11111. P.F. Li11d<'11 B11i/di11g and Enriro111111'11I 3411999) 707-720 G.R. 713

Fig. Buoyancy-driven displacement ventilation. was initial I� tilled with a dense fluid (q' = __ and placed in a large 5. Thebo.� O. l ms ,) reservoir tank. experime nt was sturteci by removing a plug upper left and I0\1cr right The digitised shado11graph images are at (a) The rro111 the races of the box. I= and id) s. 45. (b) 105. (c) 165 225

in the box. The incoming jet or fluid impinged on the layered fluid formed during this initial mixing stage, see downstream face of the box and intruded downwards Figs 5 and 6. was typically 4 cm thick. i.e. approximately into and vigorously mixed with the dense layer of th1id one sixth of the height of the enclosure. A stable strati below. The downward momentum of the intrusion was fication within the box was then clearly visible. The strati opposed by the buoyancy forces and the intrusion fication in the layered band was very strong, so that the increased in length until its momentum \\as reduced to overall stratification in the box was essentially two-layer zero. The intruding current was then retkcted back across with the narrow intermediate layered band effectively a the box and a displacement flow established (see Section smearing of the interface. Jn both the buoyancy-driven 5). Numerous secondary intrusions and retlections result and the wind-assisted buoyancy-driven displacement ing from collisions between the current and both leeward flows a small amount of mixing was observed in the and windward internal faces of the box \\ere observed region above the fluid interface. The amount of mixing during all experiments. During each collision the current increased as the initial Froude number. and hence the dissipated energy. the depth of the subsequent intrusion momentum or the incoming buoyant jet, increased. The decreased and the reflections became progressively incoming buoyant jct scoured the interface, lifting dense weaker. Turbulent mixing between the intruding current fluid from the lower layer which was then entrained into and the dense fluid layer below increased the density the jet. The sharp curved interface formed by entrainment of the current and thin fluid layers v,ere formed as the into the jct is visible in Figs 5(a) and (b) and 6(a) and (b). intrusion travelled horizontally across the enclosure: each new layer the result or a single rellection. density or Co111pari.rn11s he11i-ce11 da1i1 and lheory The 4.:!. these l<1yers was between that or the ambient fluid and the dense layer of th1id. Each new layer was denser than. The mathematical model developed in Section 2 does and hence formed below. the previous layer. The hand or not incorporate the initial mixing stage of the flow. How-

r ! H11111. Li11de11 : Buildi11g a11d £iwiro11111e111 34 1999) 707-720 714 G.R. P.F. (

- . � .

(d)

Fig. 6. Buoyancy-driven displacement ventilation assisted by wind The box 11as initially filled 11 ith a dense fluid(g' 0. 1 ms__ and placed in a = ,) Aume tank. The experiment was started by removing a plug from the upper left (wind11 arcl) and lo11er right (ke11 ard) faces ol'the box: The pressure drop across the box was 10.7 kg 111-1 s _, gi1ing an initial Froude number of 0.99. The digitised shado11graph images are at (a) r (b) 105. ti.= = 45. (c) 165 and (d) 215 s.

ever, a correction has been made for the mixing stage average of the mixing lengths determined fo r each value by introducing a virtual origin. When the interface has ofg' at that particular ·wind' velocity. descended sufficiently far, tosay h = H-h*. the mixing Figure 7(a) depicts the position of the interfaceas a fu nc due to the incoming flow is negligible and a true dis tion of time for the experiments in which only the buoy placement flow then begins. The height over which the ancy force drives the ventilation flow (� = 0 kg m-1 s-2) buoyancy force acts for the displacement flow is then and fo r the density differences corresponding to H-h*, where the 'mixing length' h* will vary according g' = 0.02, 0.05, 0. 1 and 0.2 m s-2• Also shown in Fig. to the amount of mixing which occurs. The virtual origin 7(a) are the predicted interface positions deduced from is then given by h = H-h* at = 0. Equation ( 11) sug eqn (11). The expansions and contractions at the open t gests that a quadratic relationship governs the time evol ings are sharp and therefore we adopt the codl1cien ts ution of the interface and hence the mixing length, h*, C = 0.5 and C

(a) g'=0.02 (b) 25 25 g'=0.02 [!] [!] g'=0.05 g'=0.05 (!) (!) g'=0.1 l> g'=0.1 g'=0.2 g'=0.2 20 + 20 + Model, g'=0.02 Model, g'=0.02 Model, g'=0.05 -- -- Model, g'=0.05 ----- Model, 1 - g'=0. ----- Model, g'=0.1 ...... 15 - - - Model, g'=0.2 ...... 15 E E -- - Model, g'=0.2 � ._.0 ..c ..c 10 10

5 5

0 200 400 600 800 1000 1200 0 100 200 300 400 500 600 t (seconds) t (seconds)

-- Model, g'=0.05 g'=0.2 + ------Model, g'=0.1 ------Model, g'=0.1 15 - --- Model, g'=0.2 - 15 --- Model, E E g'=0.2 ._.0 ._.(..) ..c ..c 10 10

5 5

0 50 100 150 200 250 300 0 30 60 90 120 150 180 t (seconds) t (seconds) ' 7. (a)-(d). The rnsitinn ol" interl"a <:c. as a 1·un..:1inn ol" ti111c. 1. ro r .c/ = 0.02. 0.05. 0. 1and 0.2 111 s = 0 kg 111 s ': = 4.0 kg 111 · 1 Fig ' Ir. ' ' ': (al i'. (b) i'. s ': = 20. 7 kg 111 s 'and (d) = 49.5 kg 111 s . (..:)i'. i'.

Tahk I For a 1: 100 scale model of a building and assuming The tknsil) dilkr..:nces used in the e\pcrimenls and Lhc •1ppro.\imatc that the difference between the pressure coelticients at the l..:mpcralure dilkrenees the) represent inlet and outlet of the building is unity. i.e. (Cp; - Cp.,)ru11."'"k

.<( S : = Lhen the wind pressure drop � used in the experi (Ill ) i'.T(CJ I. menls scales to give the fo llowing approximate wind 0 02 0.6 speeds at fu ll-scale, see Table 2. These equivalent full 0.05 1.5 scale wind speeds are in the range or low to average 0. 10 3.0 0.:20 6.0 United Kingd om wind speeds. IL can be seen from Fig. 7 that. foi· a fixed density t

716 G.R. and 34 I J f/11111. P.F. Li11dr11 . B11ildi11g E11riro11111l'11l 1999 707-720

Table 2 been non-dimensionalised \Vith respect to h* and time The 11 ind pressure drops used in the experiments and the approximate with respect the total emptying time le. It can be seen ll"ind speeds they represent for a building m tall to 25 from Fig. 8 that the theoretical results have collapsed

Wind pressLtre drop in experiment Wind speed onto a single line thus confirming the scalings in eqns ( 11) i'ikgm· 1s-' m s-1 and (l2). The initial rate of descent of the interface. deduced 00 0.0 from eqn (19). namely. 4.0 0.9 20.7 2.0 d(/1 h* 5 -- = -2"'/ 1 +Fr(O)-. (25) 49 3.2 � d(I te0))I , '"' 0 , It has been assumed that g:'"'"" = g;."'"""'' and. for comen ience. 11 e ha1 e taken = is illustrated in Fig. 9 as a function of the initial Froude (Cr, - C",, )'"''-·""' 1. number, Fr(O). Also shown in Fig. 9 are the experimental observations and the theoretical wind-driven limit, i.e. difference, the total time for the box to empty decreases d(h h*) -·- as the wind pressure drop between the openings increases. = -?- 'F r(O) Fr(O) » 1, (26) d(t/teb) In other words, fo r a given density difference the rate at ' rc,�o which the enclosure drains increases as the initial Froude I which is shown as the dashed line. For Fr(O) < 1 the number increases. The descent of the interface can also initial rate of descent of the interface is a weak function be seen to evolve from a quadratic to an approximately of the initial Froude number and tends to a constant linear dependence with time. As the pressure force exerted value of - 2 , while for fr(O) I the rate of descent of > by the wind becomes increasingly dominant so the effect the interface increases monotonically and tends to the of the buoyancy force plays a less significant role in the \V ind-driven limit of -2Fr(O). Agreement between the emptying of the box. In Figs 7(c) and (d) the displacement theoretical results and the experimental observations is flowis approaching the purely wind-driven limit as only good for Froude numbers up to the order of 2.5. Ven small changes in the total emptying time result from tilation flows examined in the par ameter range variations in g'. Fr(O) 1.5 were subject to considerable mixing between > At the higher 'wind' velocities considered, disturbances the incoming, wind-driven fluid and the fl uid contained within the box. caused mainly by the incoming jet of air in the box. Despite these additional effects the data still leading to three-dimensional effects. resulted in in ter agrees with the theory at high fr(O). The displacement facial fluctuations of approximately 5 mm. An error bar mode of ventilation broke down when the ventilation indicating the magnitude of the interracial Auctuations is flow ·short circuited". which occurs when the initial shown in Fig. 7(d). The accuracy with which the position incoming current of light fluid impinges on the down of the interface could be resolved resulted in the exper stream wa ll of the box and plunges through the dense imental data poi nts falling in a narrow ba nd. see Figs layer of fl uid to pass out through the low-level, leeward 7(c) and (d). Displacement flows in the box were not opening. This si tuation is discussed in Section 5. observed at the flow rate corresponding to �=49 .5 kg m-1 s-2 and with a density difference ofg' = 0.02 m s-2• these conditions correspond to an initial Froude number Conditions for the transition to a mixing flow 5. of approximately 3.43. In this case. the high speed of the incoming jet, driven by the wind. caused almost total The experiments have shown that a displacement t1ow mixing within the box (see Section 5). persists in the enclosure for a range of initial Froude In these transient. draining. ventilation fl ows. the rate numbers from 0 to about 2.5. The initial stages of a of descent of the interface is the important quantity to typical displacement flow. are depicted schematically in predict accurately. Agreement between the experimental Fig. 10. The incoming current impinges on the down observations and the pred ictions of the present theory stream wall of the enclosure and intrudes downwards a are good for each 'wind' velocity and density difference distance L into the layer of dense Auid below. The current considered. The majority of the theoretical predictions is reflected back across the enclosure and a displacement are within 10% of the observed values. The theoretical flow isestabl ished. predictions of interface descent rate tend to overestimate On increasing the ini tial Froude number above those observed during experiment. The main cause of this approximately 2.5 there was a flow transition from a discrepancy is believed to be the initial mixing stage which displacement to a mixing mode of ventilation. During acts to dissipate energy through turbulence. thereby hin this transitional regime. which occurred in the parameter dering the rate at which the interface descends. range 2. 5 < Fr( O) < 3.3. the Aow was observed to initially The non-dimensional fo rm of the results presented in short circuit and then. as the intrusion was fo rced back Fig. 7 is shown in Fig. 8 where the interface height has upwards by the buoyancy forces. a displacement t1owwas G.R. Hun!. Linden 1 Building and Enl'iron11um1 34 1999) 707-720 P.F. ( 717 (a) 1.0 (b) 1.0

g'=0.02 g'=0.02 ['] ['] g'=0.05 g'=0.05 C) C) 0.8 0.8 g'=0. 1 � g'=0.1 A A g'=0.2 g'=0.2 + + Model Model -- -- 0.6 0.6

iC iC .c .c - - .c .c 0.4 0.4

0.2 0.2 ['] �[']

0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 tlte t/te

g'=0.02 ['] g'=0.05 0.8 C) 0.8 g',,,0.1 g'=0.1 + A A g'=0.2 g'=0.2 + I!> + -- Model Model +& -- 0.6 0.6 +& iC iC + t:.. .c .c + - - A .c .c + A+ 0.4 @> 0.4 � � -t A + A 0.2 �!) 0.2 + '� +A 4'

0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t/te tlte Fig. (a)--(d). The position or interface. as ;1 !'u nction or tim�. = 0.02. 0.05. 0. I and 0.2 m s-': (a) = 0 kg 111 - Is ': (b) kg 8. h.h*. /, /('. for,11' l'J. - l'J. =4.0 111 . I S -l; kg 111 I S 'and = 49.5 kg 111-' s -'. (C) l'J. = :w. 7 (

re-established. For initial Froude numbers greater than For a fixed enclosure geometry. the length of the 3.3 a short circuiting cross-Aow was maintained and a intrusion increases as the initial Froude number. and mixing mode of ventilation was observed. Dense ·fluid hence the kinetic energy of the intrusion, is increased. A was rapidly entrained by the short circuiting jct of Auid short circuiting flow. and hence a transition to a mixing whose momentum caused an overturning of the dense mode of ventilation. occurs when the intrusion extends fluid within the enclosure. A single reci rculating cell. of as far as the leeward opening, i.e. when L = H - 11 1• 2. denser than ambient fluid.was then established. The mix where '11 denotes the vertical height of the leeward open ing mode of ventilation observed provided a very ing. see Fig. 0. The minimum initial Froude number I ineflieient means of flushing the dense fluid from the req uired to bring about short circuiting will be referred enclosure as the fluid inside the box was gradually diluted, to as the critical initial Froudc number, Frcrii(O). For ra ther than displaced . Fr(O) ;:;:: Fr,,.i1(0) a displacement flow is not established H11111. Li11de11 B11ildi11g 11111/ £11riro11111<'111 34 I1999 707-7:!0 718 G.R. P.F. J

6 · ducing a constant of proportionality '.Y., eqn (27) may be .. .. ··· Q) written as (.) {g 5 .· Q) Q - ,.. (28) c L='.Y.Ui/' .cJ'1 -" (54 and hence substituting for Q at time t = 0 see eqn (5), Q) . gives the length of the intrusion as a function of the initial "§ 3 - Froude number. namely, c �Q) 2 _..._-t:i , , L '.)'. -A*-,- . (I +Fr(Q)-)1 -. (29) Q) - = "C H H' -a{v � co :E Experiments performed by Turner[12 ], in which a source c of fluid heavier than its environment and with initial momentum vertically upwards (so that the forces of 0.0 0.5 1.0 1.5 2.0 2.5 3.0 momentum and buoyancy oppose), found that the Fr(O) negatively buoyant jet reached a vertical height of initial rate of descent of the as a fu nction of 1.85 1 before the momentum decreased to zero. Fig. 9. The interface the 4 s- � initial theoretical wind-driven limit. gi\·en M' Froude number. Fr (O). The Similarly, the intrusion is driven by the opposing forces by eqn (26), is shown as the dashed line. of momentum and buoyancy and hence for simplicity we assume that '.Y. = .85. However, in contrast to the nega J tively buoyant jet studied by Turner[12 ]. the momentum of the buoyant jet which enters the enclosure will be reduced on impact with the leeward wall. The predictions

of eqn (29) with '.Y. 1.85 are therefore expected to over = estimate the observed values of the intrusion length. The length of the intrusion L/H is shown in Fig. 11 as a - wind fu nction of the initial Froude number. The experimental a) b) c) results (shown by the square markers) and the theoretical

Fig. 0. The initial stages of a typi<:ctl Jisplacernenl tlow: (a) inlrusion: predictions, eqn (19). (shown by the continuous cu1Te) I (b) reAection and (c) displacement no\1 established. are in close agreement over the range of initial Froude numbers considered and as expected the theoretical pre dictions slightly overestimate the observed length of the intrusion. When the intrusion extends to the leeward and a transition to mixing fl ow is observed: the efficiency opening, = H-h1. 2 and Fr(O) Frc1;,(0) and hence, L = of the ventilation flow is then lower than the no-wind case. Fluid enters the windward opening as a buoyant jet whose density initially diffe rs from the interior fluid by 1.0 an amount !J.p.The buoyant jet has jet-like or plume-like characteristics depending on the relative strengths of the 0.8 initial momentum fluxM( = QU") and initial buoyancy flux B( = Qg'). On impact with the leeward face of the enclosure the fluid intrudes downwards into the dense 0.6 layer due to the momentum of the jet. The length of the oC!l :::JI 0 intrusion is inhibited by the buoyancy fo rces. If we [CJ experiment 0.4 assume that impact takes place sufficiently far from the -- (29) equation jet origin so that the initial volume Q is not important. analysis indicates that the only characteristic dimensional 02 length is given by 1

·' 4 O.O -t-��Tj��.....,.,��-..,��--.1--- ,1�---,1 M - (27) 3.0 Lx- . �0.0 0.5 1.0 2.5 B' 2 1.5 2.0 Fr(O) The length scale thus determines the distance over L Fig. I I. The experimentall] o e ve intrusion length as a fu nction or bs r d , which the buoyant jet is momentum driven and hence the initial Froude n m e . The theoretically prcdicted intrusion length. u b r provides the scaling for the length of the intrusion. Intro- eqn is show n as the continuous line. (29). H11111. Li11de11 / B11i/di11g and £11 l"iro11111e111 34 1999) 707-720 C.R. P.F. ( 719 from eqn (29), the critical initial Froude number is defined 25 11\ (11)�] = -.- -) solely in terms of the enclosure geometry: g'H+ � - - � 215s. te g A* [( - p p

(30) It is interesting to note that the total time taken for the room to empty by the buoyancy forcealone, is teb. By increasing the ceiling height H and the area of the '?S(H)�- windward openings aw and reducing the height of the = � 671 s. teb .::__A* -; leeward opening hL it is therefore possible to maintain a g displacement flowover a wider range of wind speeds and This example illustrates that when the forces of buoy temperature differences. For the geometry of the box ancy and wind combine to reinforce one another, the used in the experimen ts eqn (30) predicts a critical initial total emptying time of an enclosure is significantly less Froude number of Fr",;1(0) = 2.18 which is in good agree than that for buoyancy-driven ventilation. For the con ment with the observed value of approximately 2.5. ditions prescribed in this particular example, the time taken to empty the space by buoyancy forces alone (671 s) is considerably greater than the emptying time Example 6. if the space were ventilated by the combined effects of buoyancy and wind (215 s). Here, the buoyancy force As an example of combined wind and stack effects. acts merely to keep the fl uid stratified andprovi des little ·ider a low-rise biuildingon ·rn expo ed "ile expelling force. con a room in . whi hi· be ventilated b nuwral \'e ntil ion Suppo e lO a1 . Cht' occupies the entire ·ectional a re ay room cro. s- a 100 m�. of the building. and ha · pening on both the 7. Conclusions

w i nd wa rd and leeward face· who centre, are eparated ·e An experimental and theoretical investigation of natu a rtical distance of 6 m. Th1:: indward opening by � ral ventilation by the combined effects of buoyancy and 6 m ) i at kn -level and the leeward opening (area ('1 .rea wind has been conducted. We have restricted our atten 1.5 m2 and verti al height m) i at high-le el. Let the I internal and external air temperatures be 25 and 15 'C, tion to transient ventilation flows in which the effects of buoyancy and wind one another to ventilate respectively, and the wind speed be 3 m s-1• reinj(Jrce the enclosure. In this case. a displacement mode of From Orme et al. [5]. the pressure coefficients at the or a range of windward and leeward faces of an exposed. low-rise ve ntilation is maintained in the enclosure f wind speeds and the spacc_..£.mpties in a time building. for a normally incident wind. are Cl'; = 0.7 and = (2S g'A*)(, c/ H+l1p-v 11;p). Increasing the Cl'" = -0.2. respectively. From eqn ( 17) the square of te . - wind pressure drop 11 between wind\\ ard and leeward the wind-induced velocity is then 11,p = 4.05 s 2 • For me a perfect gas. density and temperature are related by the openings decreases the total emptying time as does increasing the density difference between the space and equation of state = pRT and hence P the exterior. For a given geometry. initial density differ ence and wind speed. the emptying time is controlled by T p the effective area of the openings A*. The effective area is dependent mainly upon the smaller of the ventilation where 11 Tdeno tes the temperature difference and the openings and so the flushing rate is determined by the T absolute temperature. A 10 C temperature difference ven t with the smaller total area. This has implications 1 results in a reduced gravity ofg' = 0.34 1 m s- and hence for providing occupants of naturally ventilated buildings the square of the initial buo ancy-induced velocity is with a sense of control over their environment: they may e e g· H = 2.05 m s- . The e 'pansions and contractions to open a local vent. providing the controlling vent is the flow at window or door openings arc normally sharp unaltered. without disrupting the overall ventilation of e and. therefore. we take C = 0.5 and ("_1 = 0.6 and the the workplace. 2 effective area of the openings m . The initial Another important result deduced from the theory A* = 1.25 Froude number of the flow Fr( O) = 1.41 and eqn indicates the form or the nonl i near relationship between from (30) the critical initial Froude numhcr Fr"111(0) = 3.59. the natural forces or wind and buoyancy. It has been

The initial Froudc number is considerahly smaller than shown that the wind and buoyanc 1 · combine to · ti.m:e the critical initial Froude number and hence we expect a drive a total volume fl ux_Q_= A*,./ U� U� through the + displacement flow lo be established within the enclosure. openings. where uh = g'h and u\\ = \ !:J./p are the air y The total lime tc for the room to e pt or the warm air speeds induced by the effects or buoyancy and wind, m y by a wind-assisted buoyancy-driven displacement flowis respectively. A convenient aidc-111e1110ire and graphical then representation of this relationship is a "natural ventilation 720 Hun t. Linde11 811ilding and Enrironment 707-720 C.R. P.F. 34 ( 1999)

stratification within an enclosure to be predicted as a function of time and is easily extended to consider single sided ventilation flows by combined buoyancy and wind.

Acknowledgements Ub The authors would like to thank the Building Research Establishment for their financial support of this project through the Department of the Environment's Energy Related Environmental Issues (En REI) in Buildings pro gramme. We would also like to thank D. Cheesely; B. Uw Dean and D. Lipman for constructing the experimental Fig. 12. The ·natural ventilation triangle' for buoyancy-dri\en dis apparatus and for their technical support. placement flows assisted by wind. The base and vertical sides of the triangle are set by the magnitudes of the wind and buoyancy produced velocities. and Ub. respectively. The length of the hypotenuse deter U" References mines the total fluid velocity produced by buoyancy forces reinforced L' by wind. [I] Linden PF, Lane-Serif GF, Smeed DA. Emptying filling boxes: the fluid mechanics of natural ventilation. J Fluid Mech 1990:1 1 2:300-35. [1] Cooper P. Linden PF. Natural ventilation of enclosures containing triangle', see Fig. 12. The lengths of the base and vertical two buoyancy sources. J Fluid Mech 1996:3 11:155-76. (3] Linden PF. Cooper P. Multiple sources of buoyancy in a naturally side of the right angled triangle are set by the magnitudes ventilated enclosure. J Fluid Mech 1996:3 11: 177-92. of the wind and buoyancy induced velocities, respectively. [4] Allen C. Wind pressure data requirements for air infiltration cal The magnitude of the velocity produced by buoyancy culations. Technical Note 13. The Air Infiltration and Ventilation forces reinforced by wind is then given by the length of Centre. Coventry, U.K .. 1986. the hypotenuse. [5] Orme M, Liddament M. Wilson A. An analysis and data summary of the AIVC's numerical database. Technical Note 44. The Air On increasing the initial Froude number a transition Infiltration and Ventilation Centre. Coventry. U.K • . 1994. from a displacement to a mixing mode of ventilation [6] Davies GMJ. Linden PF. The effects of a head11 ind on buoyancy occurs in which the efficiency of the ventilation tlow is driven flow through a door-w ay. In: Proceedings of ROOM lower than for the no-wind case. The value of the initial VENT'92. Third International Conference on Air Distribution in Froude nmnber at which this transition occurs depends Rooms. Vol 3. 1992. p. 419·-33. [7] Hunt GR. Linden PF. The natural ventilation of an enclosure by solely upon the geometry of the enclosure and is given the combined effects of buoyancy and wind. In: Proceedings of Frcr;,(0) = by .,/(ai� 4(H-hL!2)/(A*H1 �:x))!-1. Hence ROOMVENT'96. Fifth Internal Conference on Air Distribution Frc,;,(O) also determines the conditions required to drain in Rooms. Vol 3. 1996. p. 139-46. an enclosure by displacement ventilation at the maximum (8] Batchelor GK. An introduction to fluid dynamics. Cambridge possible rate. By increasing the ceiling height Hand the University Press, 1967. p. 615. [9] Ward-Smith AJ. Internal fluid flow-the fluid dynamics of flow in area of the windward openings a and reducing the height w pipes and ducts. Oxford: Oxford Science Publications. Clarcndcn of the leeward opening hL it is possible to maintain a Press. 1980. p. 566. displacement flowover a wider range of wind speeds and [IO] Dalziel SB. Rayleigh-Taylor instability: experiments with image temperature differences. analysis. Dyn Atmos Oceans 1993:20: 117--53. The simple.mathematical model developed in Section (1 1] Baker N. Linden PF. Physical models of air !lows-a new design tool. In: Mills F. editor. Atrium Buildings Architecture and Engin 2 shows good quantitative agreement with the exper eering. 1991. p. 13-22. imental observations over a wide range of density differ (12] Turner JS. Jets and plumes with negative or reversing buoyancy. ences and wind speeds. It allows the temperature and J Fluid Mech 1966:26:779-92.