Numerical Simulation and Comparison With Experiment of Natural Convection Between Two Floors of a Building Model via a Stairwell

International Journal of Heat and Mass Transfer 54 (2011) 19–33

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Numerical simulation and comparison with experiment of natural convectionbetween two floors of a building model via a stairwell

M.R. Mokhtarzadeh-DehghanSchool of Engineering and Design, Brunel University, Uxbridge, Middlesex UB8 3PH, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 May 2010Accepted 8 September 2010Available online 2 November 2010

Keywords:StairwellNatural convectionModellingLES

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.09.067

E-mail address: [email protected]

The paper presents a numerical study of three-dimensional buoyancy-driven flow in a half-scale model ofa two-floor building model. The model consists of an upper compartment and a lower compartment witha stairway connecting the two floors. The model forms a closed system, with no inlet or outlet. The flow isdriven by a single heat source placed in the lower compartment. The study is linked closely to a previ-ously published experimental study by the present author, which provided the details of the geometryand the boundary conditions as well as data for comparison with the present numerical results. Thenumerical method is large eddy simulation with the dynamic kinetic energy transport subgrid model.Radiation exchange is modelled using the discrete ordinates (DO) radiation model. The thermal boundaryconditions on the model walls are set as heat flux. It is shown that the air temperature level is sensitive tothe initial conditions for temperature, but air velocity is unaffected. In order to study this effect further,with the aid of the k–e model, the measured wall temperatures are set as boundary conditions, whichremoves the dependency on initial temperature. For the cases studied, comparisons are made betweenthe measured and computed wall temperatures, wall heat fluxes, air temperature and air velocity. Thereis a general agreement between the two results.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

There are numerous practical situations involving natural con-vection in a fluid e.g. [1–4]. Fundamentals of natural convectionflows may be studied by defining very simple problems; a basicexample being a fluid-field cavity of certain height-to-width ratiowith two vertical walls kept at different temperatures. Another ap-proach to study is to investigate the flows in the real environment,or defining experimental models, full-scale or small-scale, whichrepresent the real situation closely. A further approach is by math-ematical modelling using numerical techniques. The progress in ra-pid generation of complex geometries and meshing of the flowdomain, together with numerous modelling techniques, has meantthat the use of numerical modelling is now widespread. However,the question of validity of the results produced by a numericalmodel remains. The term numerical or mathematical model re-ferred to here embodies the geometry representing the actualproblem, a set of differential equations for the fluid flow and heattransfer, the boundary conditions, plus all other aspects such as thegrid, modelling of turbulence and details of discretization and solu-tion techniques. It is usual practice to validate a numerical modelin a particular computer code using simple test cases. These test

ll rights reserved.

cases are usually more aligned to the simplifying assumptions em-ployed within the numerical model. As an example, a typical gen-eral purpose computer model may rely on using relationshipsderived from studying two-dimensional zero pressure gradientturbulent boundary layer flows in order to model the near-wallflow, which may depart significantly from the flows which occurin practice.

The flow between different zones of a building has been thesubject of study by numerous investigators. Among these are asmall number of studies on buoyancy-driven flows in stairwellsconnecting compartments at different levels of a building. The flowis induced by density differences caused by, for example, a sourceof heat such as fire. The previous experimental and numerical stud-ies of buoyancy-driven flows, more closely related to the presentwork, in terms of practical applications and the numerical tech-niques adapted, are those of Peppes et al. [5,6], Jiang and Chen[7], Qin et al. [8], Sun et al. [9] and Ergin [10]. Peppes et al. [5,6]studied buoyancy-driven flows in real buildings using large eddysimulation (LES). They studied a naturally ventilated building of6.3 m height comprising two floors connected via a stairwell. Fur-ther studies were carried out in a full-scale real building of height13.0 m, comprising three floors and connecting stairwells. Theseworks also included numerical simulation based on Reynolds Aver-aged Navier–Stokes (RANS) of the transient flow using RNG k–emodel. Jiang and Chen [7] studied natural convection between

Nomenclature

A area of horizontal opening (m2)CP specific heat at constant pressure (kJ/kg K)DT differential temperature (K)g gravitational acceleration (m/s2)Gr Grashof number, Gr = gb DT A h/m2

h height of the model (m)I radiation intensity (W/m2)k turbulence energy (m2/s2)ksgs subgrid-scale kinetic energy (m2/s2)Nh, Nu control angles in radiation model (rad)p filtered pressure (N/m2)q heat transfer rate through wall by conduction (W)qc heat transfer rate by convection (W)qr heat transfer rate by radiation (W)qin incoming radiation heat flux (W/m2)qout outgoing radiation heat flux (W/m2)Q total nominal heat input (W)_Ql wall heat loss (W/m2)r, s position and direction vectors (m)Pr Prandtl numberRe Reynolds number, Re = VA0.5/mSt Stanton number, St = Q/qCP TavA(gh)0.5

Sij rate of strain tensor (1/s)t time (s)T air temperature (�C)

T filtered temperature (�C)Tav average temperature in horizontal opening (�C)Tw local wall temperature (�C)Twav wall average temperature (�C)u, v, w velocity components (m/s)�u filtered velocity component (m/s)V air velocity (m/s), volume of computational cell (m3)xi coordinate systemy+ non-dimensional distance from wallX1 Distance along the length of horizontal opening (m)

Greek symbolsb coefficient of thermal expansion (1/K)e energy dissipation rate (m2/s3); emissivityh, u polar and azimuthal angle (rad)l dynamic viscosity (kg/m s)m kinematic viscosity (m2/s)mt kinematic eddy viscosity (m2/s)q density (kg/m3)rt turbulent Prantdl numberr Stefan-Boltzmann constant (W/m2 K4)

AbbreviationsTBC temperature boundary conditionHFBC heat flux boundary condition

20 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

two compartments connected via a simulated window. One com-partment was a test chamber and the other compartment wasthe outer laboratory environment, both located on the same level.Both measurements and numerical simulations were reported. Forthe numerical simulation, they used the standard k–e model, andLES. Their results showed better agreement between the predictedand measured air velocities and temperatures for LES comparedwith RANS. Qin et al. [8] carried out a LES of the flow in a two-sto-rey building model with two stairways. The flow was induced by afire source placed on the lower floor. They validated their tech-niques by simulating the experimental study of Ergin-Özkanet al. [11] and compared their predicted results with the experi-mental data. Their simulations indicated the formation of distinctlayers of smoke and air flow in the stairway. Sun et al. [9] usedCFD to predict smoke movement in a six-storey stairwell, inducedas a result of a fire in an adjacent compartment. They also used LES,indicating an increasing trend in the use of this technique in large-scale three-dimensional buoyancy-driven flows. The experimentaldata of Ergin-Özkan et al. [11] was also used [12] for numericalsimulation using the standard k–e model. The case presented inRef. [12] formed an open case, having two openings, one in thelower compartment allowing for air to enter at a velocity of0.7 m/s, and the other in the upper compartment used as an outlet.The results showed correct flow patterns as well as reasonableagreement between the measured and computed velocity profilesat the mid-section of the opening between the two floors. The airtemperature, however, were higher than the measured values. Er-gin [10] presented a further study of the flow in a one-half scalestairwell model described previously in Ergin-Özkan et al. [11].The study of Ergin [10] completed a series of works e.g. [11,13]which were conducted on a model which had a rather simplifiedgeometry of a two-floor building with a connecting stairwell anda heat source placed on the lower compartment. Because of thespecific design of this model, the induced flow had a two-dimen-sional character. The experimental model was then extended toform a one-half scale model of more realistic geometry [14].

The experimental model presented in Ref. [14] consisted of alower compartment and an upper compartment connected via astairway. A heater placed in the lower compartment induced abuoyancy-driven three-dimensional airflow within the model.The model was a closed system without an inlet or outlet. In theabsence of flow visualisation, the pattern of the flow was deducedon the basis of similar previous work [12] and variations in themeasured temperature distributions on the walls of the model. Itwas suggested that the experimental model could be used as a testcase for numerical simulation, with the measured heat losses fromthe walls of the model as the boundary conditions. Having sug-gested such a possible test case, provided the motivation for theauthor to carry out a numerical simulation of the flow and refinethe model. The contribution of the present paper is in the numer-ical investigation of the flow and direct comparison with theexperimental data. In doing so, the mathematical model is definedin such a way that it can be readily used by other investigators.Previously reported wall heat fluxes [14] are supplemented byadditional data for walls temperatures so that both heat flux andtemperature boundary conditions may be implemented, or be usedfor comparison with the predicted results. It should be noted thatthe case investigated in this paper is different from the case pre-sented in Ref. [12], which, as was noted earlier, was a simpler mod-el having a two-dimensional character, and from the studiesreported in Refs. [5–9], because of the very different geometryand flow conditions involved in these studies.

2. Problem under investigation

The set up of the numerical simulation is selected in accordancewith the stairwell model as described in Ref. [14]. A brief descrip-tion of the experiment is therefore first provided. Fig. 1 togetherwith Tables 1 and 2 provide the geometry, dimensions and defini-tion of each wall of the stairwell model, which is the same as theone described in Ref. [14], except for the omission of the space

B S

E D

ED

F C

A

C

U

J K J

T

N M

O L

B

X1

W A1W1A1

T1

t

A2

D

B A

J

B

D

E

F

G H I

N

L

O

S R

P Q

M

C

Floor

Outlet

Inlet

Front

Back Ceiling

Partition

K

Z Z1

Q1

R1

Fig. 1. Schematic diagram of the stairwell model.

Table 1Dimensions of the walls (mm).

AB 1220 RS, HI 612AC 2386 TU 580BS 3460 UW 650AK, BJ 1344 AA1, BW1 100AP, KL 940 BT1 180PQ, LM 608 AA2 50MN, LO 1320 t 18OG 1200

Table 2Definition of the walls. Walls shown in italic are omitted from the numerical model.

AKLP Front lower 1 QRSQ1 Floor 2PLO Front lower 2 PQRR1 Floor 3POGR1 Front lower 3 AKJB Inlet lowerKGFC Front upper KCDJ Inlet upperBSIJ Back lower RHIS Outlet lower 1JIED Back upper RHGR1 Outlet lower 2CFED Ceiling GFEI Outlet upperAPQ1B Floor 1 ONHG Landing

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 21

below the stairway and landing, as shown in Table 2. There is ahorizontal opening between the two floors. The heat source wasa typical panel heater (radiator) used to heat rooms in buildings.There is no inlet or outlet. The experimental stairwell model wasplaced in a large room so that heat losses took place freely fromall the walls of the model. The model was raised from the floor le-vel and was placed on a platform structure. Heat inputs of 300, 450,600 and 750 W by the heater produced three-dimensional recircu-lating flows with measured velocities less than 0.2 m/s in the hor-izontal opening. The measured temperature of air was in the rangeof 28–42 �C. The characteristics dimensionless numbers, based onthe dimensions of the horizontal opening and the measured airvelocity and temperature in the horizontal opening were5997 < Re < 7363, 3.1 � 10�4 < St < 7.7 � 10�4, 2 � 10+8 < Gr < 3.68� 10+8. The flow is considered to have a laminar nature in mostpart of the space, but transitional or turbulent in the areas in thevicinity and above the heater. The problem also involves heattransfer by radiation. The inclusion of the Reynolds number followsthe conclusion of Reynolds [15] and Reynolds et al. [16] that thestairwell flows, characterised by circulating air flow between twofloors via a throat area, fall near the regime of transition and theReynolds number plays a role, for example in relation to the losscoefficient. Reynolds [15] defined Re based on the reticulating vol-

ume flow rate and the throat area. It was also stated that, fordesigning a stairwell model, it was necessary to match the Rey-nolds number for the model and the prototype. Reynolds et al.[16] argued that, while Re in the range of 3800–7000 were ob-tained in a particular experiment, the Re for the resistance-gener-ating processes is lower, about 1000, due to smaller-scaleprocesses adjacent to the walls.

The measured heat fluxes from the walls of the stairwell, or walltemperatures, and the heat input from the heater can be used asthe boundary conditions, whereas the measured air temperatureand velocity in the horizontal opening can be used for comparisonwith the predicted results. Furthermore, if heat fluxes are used asthe boundary conditions, the measured wall temperatures can alsobe used for comparison, and vice versa, if temperatures are set asboundary conditions. The results presented here are for the nomi-nal heat inputs of 300 and 600 W, referred to as Case 1 and Case 2,respectively. It should be noted that the actual measured total heatloss was less than these nominal values [14] and therefore the ac-tual rather than the nominal values were used. In doing so, thespace below the stairway and landing (Fig. 1) was omitted in thepresent work, because this space formed a separate zone fromthe main compartments of the model and had a small contributionof less than 2.5% to the total heat loss. The exclusion of this spacemeant that the associated heat loss had to be subtracted from thetotal heat input, in order to have a balance between the heat inputset on the heater and heat output set on the walls. Having the mea-sured heat losses on all surfaces, the heat fluxes were calculatedbased on the surface areas of each wall. In order to clarify themethod adopted, an example is provided here. In Ref. [14] for anominal heat input of 300 W from the heater, the actual measuredtotal heat loss reported was 275.2 W from which 7.4 W was asso-ciated with the space below the stairway. The difference of267.8 W for the heater surface area of 0.816 m2 gives a heat fluxof 328.2 W/m2. Table 3 provides the values of heat flux set as theboundary conditions. This table also gives the areas as calculatedby the CFD code. There are small differences with the actual valuesused and also with the values given in Ref. [14] because of round-ing the figures. An assumption implicit in the values given in Table3 is the uniformity of heat flux per unit area over the walls as de-fined in Fig. 1. In the experiment, the measured wall temperaturescorresponded to smaller cells on each wall, but incorporating theheat fluxes from each cell as boundary condition was considerednot useful from a practical point of view, as in practice such de-tailed information is not normally available. It was also possibleto set the measured wall temperatures as the boundary conditions.

Table 3Heat flux set on the stairwell walls.

Wall Case 1 Case 2 Area (m2)_Ql ð½W=m2Þ _Ql ðW=m2Þ

Inlet Lower wall �17.81 �34.48 1.66Upper wall �6.16 �13.82 1.25

Outlet Lower wall �8.88 �18.38 0.83Upper wall �6.54 �12.91 1.25

Back Lower wall �12.9 �26.93 4.71Upper wall �8.16 �15.72 3.54

Front Lower wall 1 �26.95 �57.35 1.28Lower wall 2 �9.77 �21.87 0.96Upper wall �6.18 �15.01 3.54

Floor Floor 1 �20.78 �38.36 1.14Floor 2 �10.05 �14.34 1.54

Ceiling �4.79 �12.96 4.22Heater 328.51 683.13 0.82

22 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

These are shown in Table 4. It should be noted that these temper-atures, as in the case of heat fluxes, are the averaged values overthe walls. As will be discussed later, although some results wereobtained using temperature boundary conditions, this approachwas not adopted in the LES simulations. The reason is that, itwas thought that setting the heat flux as the boundary conditionat least ensured the correct heat loss from each wall.

Ideally, it is desirable to obtain the wall temperatures or heatfluxes as part of the solution. Depending on the problem, it maybe possible to set the boundary conditions on the external walls[17,18]. The resulting conjugate heat transfer problem would pro-vide the internal wall temperatures. In the present study, the thick-ness of the walls were such that the same level of variations weremeasured on the external walls and therefore the method wouldsimply add further calculations for heat transfer through the wallsand for radiation from the external walls to the outside atmo-sphere. A better solution would be to place the boundaries of thecomputational domain in the surrounding atmosphere, where theroom temperature can be set. In this case, both internal and exter-nal wall temperatures would be calculated as part of the solution.This method would be more difficult to implement for the presentcase due to the complexity of the geometry and extra computingresources required.

3. Mathematical model

The fluid flow within the stairwell model is three dimensional,incompressible and non-isothermal. The mathematical model con-sists of the governing equations of continuity, momentum and en-ergy. The fluid flow is driven solely by buoyancy forces, generated

Table 4Average measured wall temperatures at two different heat inputs.

Wall Case 1 Case 2Twav (�C) Twav (�C)

Inlet Lower wall 30.1 35.6Upper wall 28.0 31.4

Outlet Lower wall 27.7 31.9Upper wall 27.7 31.1

Back Lower wall 28.2 32.6Upper wall 27.7 30.9

Front Lower wall 1 31.4 38.6Lower wall 2 28.1 32.1Upper wall 27.8 30.9

Floor Floor 1 30.4 37.1Floor 2 26.8 30.7

Ceiling 28.1 31.9

as a result of the heat transfer from the heater to the air. The calcu-lation of buoyancy forces was achieved by making the air density tobe variable as a function of temperature, using the ideal gas lawwith the pressure set at 101325 N/m2. Part of the flow rising fromthe immediate vicinity of the heater is expected to reach the parti-tion just above the heater and is diverted towards the interior of thelower compartment. Other part of the rising flow finds its way tothe opening area on top of the stairway and enters the upper com-partment. The flow is expected to be transitional or turbulent in thisearly stage of the development, but becomes more laminar when itslows down, as it reaches the regions farther away from the heatsource. The flow therefore cannot be modelled as laminar or fullyturbulent. The complex nature of the geometry with regard to theposition and orientation of the walls relative to each other and tothe flow is expected to make the modelling of the near-wall flowdifficult. For example, the flow adjacent to the floor is a slowly mov-ing flow and tends to rise, whereas on the ceiling the flow is partlyimpinging on the surface and partly moving strongly in parallel tothe wall with a tendency to flow downward in the regions closeto the side walls. The nature of the flow on the front and inlet wallsis also complex. Additionally, it may be said that the large-scaleeddy structures in the early development of the unsteady flowabove the heater, and also in the shear layer in the horizontal open-ing have an important role in determining the overall velocity andtemperature distributions. The prevailing complex flow conditionstherefore suggest an approach based on large eddy simulation. Inlarge eddy simulation, large eddies of size greater than a filter sizeare resolved whereas the contribution of the smaller eddies aremodelled. This approach is based on the derivation of a set of ‘‘fil-tered” Navier-Stokes equations which govern the dynamics of largeeddies. Such derivation leads to the appearance of new terms, thecalculation of which requires modelling in order to achieve a closedsystem of equations. Through a so-called subgrid-scale model, theeffects of the smaller subgrid-scale eddies on the larger, resolvededdies are taken into account.

The results presented in this paper were obtained using largeeddy simulation as implemented in the ANSYS12 Fluent CFD code[19]. The governing equations for the resolved quantities are writ-ten as

oui

oxi¼ 0; ð1Þ

oui

otþ oðuiujÞ

oxj¼ � 1

qopoxiþ o

oxjm

oui

oxj

� �� osij

oxjþ gi; ð2Þ

oTotþ oðuiTÞ

oxi¼ o

oxi

mPr

oToxi

!� oqi

oxi; ð3Þ

where sij and qi are unknown terms, given by

sij ¼ uiuj � ui uj; ð4Þ

qi ¼ uiT � uiT: ð5Þ

The subgrid-scale models are introduced to calculate these un-known terms as part of the solution of the governing equations.The subgrid-scale stress sij is obtained using the dynamic kineticenergy transport (DKET) model [19]. This is an eddy viscosity mod-el, where the kinematic eddy viscosity, mt, is obtained from:

mt ¼ Ckk1=2sgs Df ; ð6Þ

where Df ¼ffiffiffiffiV3p

and V is the volume the computational cell. Thesubgrid-scale kinetic energy ksfg is obtained from:

oksgs

otþ oujksgs

oxj¼ �sij

oui

oxj� Ce

ksgs

Df

� �32

þ o

oxj

mt

rk

oksgs

oxj

� �ð7Þ

Fig. 2. Stairwell walls and measurement positions: (a) front, (b) back, (c) ceiling, (d)floor, (e) inlet, (f) outlet.

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 23

where

ksgs ¼12

u2k � u2

k

� �: ð8Þ

The parameters Ck and Ce are obtained dynamically and re is setto 1.0. The term sij is then obtained from the rate-of-strain tensorfor the resolved scale Sij using:

sij �23

ksgsdij ¼ �2mtSij; ð9Þ

where

Sij ¼12

oui

oxjþ ouj

oxi

� �: ð10Þ

The subgrid-scale heat flux qi is obtained from:

qi ¼ �mt

rt

oToxi

; ð11Þ

where the turbulent Prandtl number rt is obtained dynamically.The boundary conditions for the governing equations are the

zero velocity on the wall and heat flux on each wall as defined inTable 3. The calculation of these heat fluxes were based on themeasurements of internal and external wall temperatures [14]. Inthe experimental investigation [14] the wall temperatures weremeasured after a long period of time, allowed for the establishmentof a steady state flow condition within the experimental model.After achieving this condition, the variations in the measured walltemperatures were small, typically <1%. Therefore, constant valueswere set at the walls as boundary conditions.

The positions at which these temperatures were measured areshown in Fig. 2(a)–(f). In the experiments [14], the values of theheat flux (or rate of heat loss per unit area, thus the negative signin Table 3) were obtained by calculating the heat transfer by con-duction through the wall. Designating this heat flux as q, it can beset equal to the sum of the heat flux to the air by convection, qc,and the radiative heat flux, qr. The value of qr was obtained as partof the numerical simulation using the discrete ordinates (DO) radi-ation model as implemented in the FLUENT code [19]. This modeluses the radiative transfer equation for the total radiation inten-sity, I(r, s), where r and s are the position vector and direction vec-tor, respectively, over the three-dimensional space. Each octant ofthe angular space is discretized into Nh � NU control angles, whereN is the number for discretization, and the angles h and U are thepolar and azimuthal angles, respectively. For the 3D space, theradiative transfer equation is then solved 8 Nh � NU times. In thepresent study Nh and NU were set equal to 6.0. The radiation calcu-lation was updated every 10 flow iterations. The walls were takenas grey diffuse walls. The rate of incoming radiative heat flux, qin isobtained from [19]:

qin ¼Z~s:~n>0

Iin~s �~ndX ð12Þ

where Iin is the radiation intensity,~s is the direction vector, ~n is thenormal vector, and X is the hemispherical solid angle. The outgoingradiative heat flux is given by

qout ¼ 1� ewð Þqin þ ewrT4W ; ð13Þ

where ew is the wall emissivity. For the medium air, the absorptionand scattering coefficients were set to zero, the refractive index wasset to 1.0. In the experiments [14] the front and back walls weremade of Perspex and the other walls were made of wood, paintedblack. The surface of the heater was painted cream. The emissivitiesof the wooden, Perspex and heater walls were set as 0.9, 0.9 and 0.7[10], respectively.

For LES, applying the no-slip condition (zero velocity) at thewall and, in the absence of any near-wall flow modelling, the wall

shear stress is obtained from the laminar stress–strain relation-ship. Similarly, the Fourier’s law of conduction is used at the wall.

24 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

For the RANS based on the k–e model, a so-called enhanced walltreatment was adopted [19]. This method resolves the flow inthe near-wall region affected by viscosity but it requires a fine gridresolution.

The partition between the lower and upper compartment wastreated as a solid wall allowing heat transfer by conductionthrough it.

4. Other computational details

The pressure has no equation of its own. The method to derivethe pressure was SIMPLEC. This iterative method makes use of aso-called pressure correction equation, the derivation of which isbased on the continuity equation. The main parameter obtainedby solving this equation is the pressure–correction, which whenadded to a guessed value of pressure gives an improved value forthe pressure. Starting from an initial guess, the pressure is there-fore improved step by step as the iteration process progresses.

In LES, the discretization scheme for the convection terms of themomentum equations was the bounded central differencing. Thediscretization scheme for the thermal energy equation and subgridkinetic energy equation was second order upwind scheme. In thek–e model, a second order upwind scheme was used for the equa-tions of momentum, k, and thermal energy, whereas 1st order

Fig. 3. Computational grid: (a) front, floor, ou

scheme was used for e. A first order upwind was used for the DOmodel in both LES and the k–e model. As part of the discretizationof the momentum equation and implementation of a pressureinterpolation scheme at a cell face, the body force weighted meth-od was invoked. This method provides improvements over the so-called standard pressure interpolation schemes in buoyancy-dri-ven flows with large body forces.

The LES calculations are transient in nature for which a solutionbased on second order implicit formulation was adopted. The timestep was kept constant, equal to 0.02 s. Suitability of the time stepmay be checked with reference to the Courant number, which isthe ratio of the time step to the time a particle takes to travel a dis-tance equal to a cell length. Inspection of the converged solutionshowed that 94% of all cells had a Courant number less than 1.0,and 99% less than 3.0. The results presented here were obtainedafter 420 s, i.e. 21000 time steps. Taking an air velocity of0.05 m/s over a circulation path of 7.2 m, the total computationtime corresponds to about 3 flow-through times. In a real life situ-ation, as in experiments [14], for a given heat input on the heaterand initial wall temperatures being at, say, ambient temperature,it takes many hours to achieve an established flow and certain walltemperatures. But in the present case, it is expected that, sinceexperimental wall thermal conditions are already set on the walls,a much shorter time is needed to achieve an established flow

tlet; (b) stairs and stairwell wall, heater.

Fig. 4. Histograms of air temperature and air velocity over the computationaldomain. The vertical axis shows the percentage of total elements (Case 2).

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 25

pattern. The local values of flow parameters such as velocity andtemperature may still vary about an average value. This is due tothe dynamic unsteady nature of the flow in the rising and spread-ing plume rather than any numerical instability. A further 20 s wasused to obtain statistical results.

The convergence was checked using the criterion that, for eachequation, the scaled sum of all the cells residuals was reduced tovalues below a set limit. In the present simulations, the scaledresiduals fell below 3.0 � 10�5 for mass conservation, 6.0 � 10�7

for the velocity components, 4 � 10�8 for the energy equation,1.5 � 10�6 for subgrid turbulence energy and 1.1 � 10�6 for theradiative transfer equation. For the k–e solutions, higher valuesfor the limits were accepted, e.g. about 2.0 � 10�4 for mass conser-vation, 5.0 � 10�5 for velocity and 2.0 � 10�4 for k and e. The over-all balance for total heat transfer rate and radiation heat transferrate for LES and k–e simulations with heat flux boundary condi-tions were achieved to a high degree. The worst cases were forthe k–e simulations with temperature boundary conditions, butthe overall balance for total heat transfer rate was still less than0.5%.

Apart from the air density referred to above, other physicalproperties involved in the calculations are laminar dynamic viscos-ity, specific heat at constant pressure and laminar heat conductiv-ity. These properties were allowed to change as a function of airtemperature by entering five values of the property at five temper-atures over the range of 20–100 �C, and allowing the computercode to interpolate.

The initialization of the problem at the start of the iteration pro-cess was as follows. The u, v and w-components of the velocitywere set to 0.0, 0.0 and 0.1 m/s, respectively. The initial tempera-ture was chosen based on the range of average temperatures ob-tained on the walls as seen in Table 4. For the lower heat input(Case 1) the initial value of temperature was set to 29.85 �C(303 K), which is close to the highest temperature achieved onthe inlet wall. Similarly, for the higher input (Case 2), the initialtemperature was set to 34.85 �C. As will be discussed later, the set-ting of the initial temperature was found to affect the final temper-ature level within the model when heat flux boundary conditionswere set.

The strategy in choosing the grid was to adopt a fine grid so thatimportant regions of the flow are well resolved. In a large eddysimulation model, the size of the grid cells determine the rangeof scales which are resolved and the range of scales which aremodelled. The areas of the model which were considered moreimportant and had a greater influence on the flow were the regionsnext to the heater and the stairwell walls, and the opening be-tween the two floors. It was aimed to place the first grid line nextto the wall at about 1–2 mm from the wall. However, due to thecomplexity of the geometry, this was not achieved for all the walls,such as the stairs. The grid had a total of 1,640,030 cells. The grid ismuch finer than the ones used by other investigators in similarwork. Qin et al. [8], in their study of fire-induced flow through afull-scale two-storey building model using large eddy simulation,had a grid of about 158,000 cells and placed the first grid lines ata nominal distance of y+ = 5. Jiang and Chen [7] in their study ofbuoyancy-driven flow in a large-scale model of a building used agrid of 700,000 cells. Because of large dimensions involved in thesestudies, the distances between the nodes should have been muchgreater than the distances used in the present study. Details ofthe present grid are as follows.

The grid was structured comprising rectangular cuboid cells.Fig. 3(a) shows the cells distributions on the stairwell walls withthe back, ceiling and stairwell wall removed. The distribution ofgrid cells on the heater and stairway are shown in Fig. 3(b) and(c). The nature of the grid with respect to the cell aspect ratiomay be noted from Fig. 3. The areas which appear darker indicate

high concentration of the grid lines, causing high aspect ratios.These were caused by the necessity to use fine grids near the walls,in particular for the heater surfaces, and due to a relatively smallheater thickness and a small gap between the heater and the frontwall. It should be noted that in most areas concerned, air flows par-allel to the surface. It may be said, however, that a more advancedgrid could be set up or a larger number of cells could be used to im-prove the aspect ratio in these regions, but the results shown laterindicate that the present grid has produced acceptable results com-parable with experiment.

The total volume of the computational domain was 8.52 m3.From 1,640,030 cells, 1,594,590 cells were located in the fluidand 45,440 cells were placed in the solid partition. The total num-ber of grid divisions was 174 � 95 � 109 in the x, y and z-direc-tions, respectively. The grid divisions on the heater were40 � 65 � 5. The throat area had a distribution of 60 � 55. Fourgrid cells were placed within the thickness of the partition. Thedistribution on a typical stair was 5 � 55 � 5. The first grid linenext to the heater front wall, floor, inlet, back and stairwell wallswere located at a distance of less than 1 mm from the wall. Forthe front and ceiling walls and the heater back wall, this distancewas between 1 and 3 mm. For the outlet, stairs and the patricianwalls the distance was about 10 mm and on the stairs about20 mm. The calculated values of the non-dimensional parametery+ on the inlet, back, floor were <1, ceiling <2, front <4, heater <3,outlet <8, stairs <15, landing <10 and stairwell wall <2. The cellarea ranged from a minimum of 1.11 � 10�7 m2 to a maximumof 6.31 � 10�3 m 2. The cell volume was in the range of 9.89� 10�11 to 3.4 � 10�4

It was necessary to use parallel processing for the LES simula-tions. Here an example of the processing time needed is provided.

26 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

Starting the simulation from time t = 0.0, the air motion startedmainly from the heater and then gradually spread to other partsof the stairwell model as t increased. Convergence was achievedat each time step. The time, t, taken to achieve convergence at each

Fig. 5. Histograms of air temperature on the stairwell walls. The vertical axis shows theoutlet, (c) front, (d) back, (e) ceiling, (f) floor, (g) heater.

time step varied, being much greater at the initial stages of compu-tation than towards the final time of 420 s. Using a Streamline par-allel processing cluster in the Linux 64-bit environment, thecomputational grid was partitioned into 16 parts which ran on 4

percentage of the total elements in the computational domain (Case 2). (a) Inlet, (b)

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 27

nodes, each with 2 processors and 2 cores per processor, running at1.8 GHz. The processing time required for one time step (0.02 s)around the total time of 420 s for Case 1 was about 50 s.

5. Results and discussion

The results were obtained for two heat inputs set on the heater,referred to as Case 1 and Case 2 (Table 3). These correspond to thenominal heat inputs of 300 and 600 W in the experiments [14],respectively.

5.1. Overall assessment of the results

The main parameters of interest are the air velocity and temper-ature in the horizontal opening between the two floors, and themain stairwell wall temperatures, because they can be directlycompared with the experimental data. However, before these re-sults are presented in detail, an overall scrutiny of the results isfirst presented for typical results obtained for LES of Case 2 withheat flux boundary conditions.

Fig. 4(a) shows the histograms of the air temperature within thecomputational domain. The air temperature is in the range of 21–57 �C. It will become clear from the results which will be presentedlater that the highest temperatures, as expected, relate to the re-gions near the heater, and the lowest ones to the regions in thefar corners of the stairwell model. Since in Case 2 the initial tem-perature was set at 34.85 �C over the entire flow domain, the re-sults indicate that the simulation allows a readjustment of thetemperature, such that in some areas the air temperature has de-creased and in some areas it has increased, compared with the ini-tial temperature. However, the room air temperature (outside ofthe stairwell model) in the experiments for this heat input wasabout 27 �C. The air temperature is expected to be higher than thisvalue everywhere within the stairwell model. Therefore, the impo-sition of heat flux on the walls has forced the air temperature to gobelow what was attained experimentally in the remote part of themodel.

Fig. 4(b) shows the histogram of predicted air velocity over thefull domain. The maximum calculated air velocity is about0.88 m/s, but for the vast majority of the flow the velocity is below0.1 m/s. As will be shown later the predicted values are realistic. Astatistical analysis of velocity fluctuations for the period of 420–440 s showed that 75% of all the elements had RMS (root meansquare) values below 0.05 m/s, less than 1% had values greater than0.15 m/s, and the higher values were about 0.2 m/s.

Table 5Heat transfer rate as boundary condition (q) and calculated (qc, qr) heat transfer rates usin

Wall Case 1

qr (W) Cal. qc (W) Cal.

Inlet Lower wall �21.8 �7.8Upper wall �3.4 �4.3

Outlet Lower wall �1.2 �6.2Upper wall �2.0 �6.2

Back Lower wall �34.6 �26.1Upper wall �14.0 �14.9

Front Lower wall 1 �46.3 11.8Lower wall 2 �3.7 �5.1Upper wall �6.7 �15.2

Floor Floor 1 �26.0 2.2Floor 2 �15.4 �0.1

Ceiling �2.6 �17.6Heater 136.6 130.9

The histograms of the temperatures on the stairwell walls andthe heater surface are shown in Fig. 5(a)–(g), for Case 2. Only asmall number of elements have a temperature below 27 �C, whichas was noted earlier was the room temperature outside of thestairwell model. These occur at far ends on the walls. The figuresare also useful in indicating the degree of variability of tempera-ture (or heat flux) on these walls, which is an indication of whethersetting a uniform boundary condition over a large section of a wall,in a typical application such as this, is justified. It may be said thatit is only on the ceiling where the degree of uniformity allows thiscondition to be reasonably set. On the heater the temperature ran-ged from about 67 to 127 �C. On the front of the heater, high tem-peratures were located at the top of the heater reaching amaximum temperature of about 107 �C, but most parts of the sur-face are between 92 and 107 �C. In the experiments, the tempera-ture on the centre of the heater front surface was measured atabout 100 �C. In the study of Ergin [10] the calculated temperaturewas also about 100 �C. On the back of the heater, the temperatureshowed greater variations, with the upper half being in the rangeof 114–124 �C. It should be noted that this level of agreementhas been obtained with a simplified model of the heater. Althoughthe total heat flux set on the heater as the boundary condition wasbased on the experimental value, the surface area differed from theactual one. This is because the heater used in the experiments hada corrugated surface with a round base, compared with the heaterwith flat surfaces and a uniform width assumed in the simulation.Reproducing the actual geometry or a similar one would have ma-jor implications from the point of view of geometry constructionand gridding. The simplification made is therefore based on takinga practical approach. It was not envisaged that the flow patternwould be significantly affected. The results shown later indicatethat this was in fact the case as the predicted flow pattern agreeswith the pattern obtained experimentally.

The above scrutiny of the range and order of magnitude of theresults indicate that acceptable values are obtained for both airvelocity and wall temperatures, which are generally in comparablerange with the experiment and independent calculations of Ergin[10].

As was stated above, the total rate of heat transfer per unit area,q, set on each wall as the boundary condition (Table 3) is the sumof qc, by convection, and qr, by radiation. Table 5 shows the calcu-lated values of qc and qr. A negative value of qc indicates heat trans-fer from the air to the wall. Therefore, apart from the front lowerwall 1 and floor 1 which show positive values, all other wallsshown in Table 5 receive heat by convection. The differing resultsfor these two walls can be attributed to their positions, that is,

g LES.

Case 2

q (W) HFBC qr (W) Cal. qc (W) Cal. q (W) HFBC

�29.6 �44.5 �12.8 �57.3�7.7 �6.6 �10.7 �17.3

�7.4 �3.6 �11.7 �15.3�8.2 �3.2 �12.9 �16.1

�60.7 �76.4 �50.5 �126.9�28.9 �24.3 �31.4 �55.7

�34.5 �99.9 26.5 �73.4�8.8 �9.2 �10.5 �19.7�21.9 �16.3 �36.9 �53.2

�23.8 �49.8 5.9 �43.9�15.5 �22.7 0.6 �22.1

�20.2 �12.7 �42.0 �54.7267.5 264.4 291.8 556.2

Fig. 6. Schematic diagram of flow pattern.

28 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

being close to the heater and being affected by radiation from it.This can also be noted from the high values of qr for these wallsand also the wall temperatures on the cells 1, 2 (front wall) and11, 12 (floor) in Fig. 7a and f, respectively. For front lower 1 andfloor 1, inspection of the local temperature differences, between lo-cal wall temperatures and local air temperatures at 1 mm distancefrom the wall, showed variations in both values and sign. However,on the whole, the effect of positive temperature differences wasgreater, leading to the positive values of qc as given in Table 5.

For example, comparing the wall temperature on the front lower1 which is located behind the heater with the air temperatureshowed temperature differences in the range of �3.3 and 6.6 �C,with a cumulative average of 1.16 �C.

In the arrangement of Ergin [10], the wall behind the heater hada similar position to the front lower wall 1 in the present study,and showed the same behaviour. But this is not the case for thewall in Ergin’s model which has a similar position to floor 1 inthe present model. In Ergin’s case, the results showed that heatwas transferred to this part of the floor by convection. Negativevalues of qr indicates that all walls, except the heater, also receiveheat transfer by radiation. The results indicate that radiation ex-change between the walls plays a major role. The results of Erginshowed that the main mode of heat transfer to the enclosure isby convection. The present results show smaller contribution byconvection, which could be attributed to generally lower air veloc-ities achieved here, caused by the larger domain of the presentmodel. From the total heat input set on the heater, about 51% istransferred by radiation. This differs from the calculations of Erginwhere a value of 36.3% was reported. The results for the horizontalwalls, namely, floor and ceiling shows that the main mode of heattransfer from the ceiling is by convection. This may be attributed to

the impingement of the rising plume against the wall and thendiversion of a strong flow along the wall. In the opposite, convec-tion plays only a small role on the horizontal floor because oflow velocities involved. Except for the outlet wall, radiation ismore important in the lower compartment of the stairwell thanthe upper compartment. The opposite is true in the upper compart-ment, where convection is more important.

In the discussion of the experimental results [14] the expectedflow pattern in the stairwell model was speculated, based on thedistribution of the wall temperatures and the air velocity and tem-perature profiles in the opening between the two floors. Fig. 6shows a sketch of the flow pattern based on the present predictedresults, which is in general agreement with the one proposed inRef. [14]. The predicted results, however, indicated a more complexflow especially in the opening area. The air rising from the maintwo surfaces of the heater tends to be diverted more towards theback wall and the proportion of the air which enters the openingarea is concentrated in the region close to the front wall. Thedownflow is clearly from the landing area on top of the stairway,and the downward flow along the stairs is maintained down tothe midway along the stairs, beyond which air tends to rise. Be-cause of the opening next to the stairs, part of the downflow inthe stairway spills over into the lower compartment. There is alsoflow of air entering the opening between the two floors from theunderneath of the partition. The rising plume of air, upon reachingthe ceiling is diverted towards the inlet and outlet walls formingtwo large recirculation zones in the upper compartment. Inspec-tion of the results showed that very slow flow, generally recirculat-ing, was established in the region between the back wall and thestairway side wall. Complex flow was noticed in the region be-tween the foot of the stairway and the back wall, with air tends

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Fig. 7. Wall temperatures on the walls of stairwell model. s Experiment, D 300 s, e 360 s, h 420 s LES, x k–e.

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 29

to flow towards the inlet wall. Downward flow of air along thewalls was observed on the outlet, inlet and back walls due to therecirculating nature of the flow. In the opening between the two

floors, although the rising air was mainly on the heater side andthe downflow was mainly on the landing side on top of the stair-way, there were small pockets of rising or falling air elsewhere in

30 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

this region. On the whole, the flow pattern appears to be accept-able and realistic.

5.2. Comparison with experimental data

Having assessed the general validity of the results, direct com-parisons with the experimental data is now presented. Fig. 7 pro-vides comparisons of the stairwell walls temperatures with thepredicted values for both heat inputs, at the same positions wherethe temperatures were measured. The cell numbers referred to inFig. 7 correspond to the numbers used in Fig. 2. The predicted val-ues are given at 300, 360 and 420 s time interval. Full agreementbetween the results is not sought here because of the simplifica-tions made in the boundary conditions. By setting heat flux asboundary condition, it is possible to obtain a different temperaturedistribution, yet the overall heat balance is satisfied. Here, there-fore, we are contented with achieving agreements between overalltrends and orders of magnitude. Fig. 8 provides comparisons for airtemperature and air velocity. The discussion first focuses on theLES results with heat flux boundary conditions. The results ob-tained using the k–e model (also shown in Fig. 7) will be discussedlater.

5.3. Wall temperatures

On the front wall (Fig. 7a,g), significant over-prediction isnoticeable for cells 1 and 2, where the former being opposite tothe heater and the latter just above it. The differences betweenthe values for these two cells are 70% and 33% (for Case 2, higherheat input) and 50% and 15% (for Case 1, lower heat input), respec-tively. The percentage difference is much lower for all other pointsand fall below 3% for cell 11 located just above the opening be-tween the two floors, where the rising plume is located. The vari-ation between two consecutive time steps, i.e. 360 s compared to300 and 420 s compared to 360 s, is in the range of �1.5–2.5% forCase 1 and �4.5–6% for Case 2.

Fig. 7(b) and (h) shows the results for the back wall. Two dis-tinct levels of agreement are evident. For cells 1–10, where they

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Fig. 8. Air velocity and temperature in the opening between the two floors. (a)Velocity, (b) temperature; d, - - -, Case 1; s, —, Case 2; d, s Experiment.

are located far from the heater and the air flow is relatively slowand mixing is weak, higher disagreement has resulted, reachingmaximum values of 3.5% and 10% for Case 1 and Case 2, respec-tively. For cells 11–20, which are located in the region where thereis greater air movement and activity, the percentage difference islower and reaches a maximum of 2.5%. The variation of tempera-ture between time steps as stated above is in the range of ±2%.

Fig. 7(c) and (i) shows the results for the inlet wall. Better agree-ment is obtained on the part of the wall which is located in theupper compartment where the maximum difference is 2%. In thelower compartment the differences reach a maximum of 6% and11% for Case 1 and Case 2, respectively. The highest percentage dif-ference correspond to cells 7 and 8 which are located in the lowerparts of the wall adjacent to the back wall. The variations in thewall temperature between time steps are ±2% and ±4% for Case 1and Case 2, respectively.

Fig. 7(d) and (j) shows the results for the outlet wall. Greaterdifferences are noticeable in the lower compartment, in particularfor positions 1 and 4, which are closest to the far end of the floor.The percentage differences fall between �3.5% to 5% and �9% to1.5% for the two cases, respectively. In the upper compartmentthe values are in the range of �1–2.5% for both cases. In the lowercompartment, the predicted values tend to be lower than the mea-sured values, whereas the opposite is mostly true for the uppercompartment. The changes in wall temperatures between differenttime steps are in the range of ±3%.

Fig. 7(e) and (k) shows the results for the ceiling. For Case 1, thepredicted values are lower than the measured values, reaching amaximum of about 5% for position 8, which is just above the mid-dle of the horizontal opening and is in the region where the risingplume impinges on the ceiling. For the higher heat input, the vari-ations are between �2.5% and 4.5%, where the latter percentagevalue occurs at position 9 which is also located in the same regionas referred to above, but is closer to the front wall. The maximumvariations in the predicted temperatures between two consecutivetime steps are in the range of �2–5%, which also occur in the sameregion.

The results for the floor are shown in Fig. 7(f) and (l). The pre-dicted results are always lower than the measured values. It ap-pears that greater mixing due to a stronger flow takes place inthe experiment than in the simulation. By imposing the experi-mental values of heat flux as the boundary condition, the real heatloss has been achieved, but with lower temperature levels in thefluid and on the wall. The variations between time steps are small,being in the range of �0.7–1.3%.

On the whole the general trends for the wall temperatures arewell predicted, but as expected quantitative differences exist.However, except for the front wall, the differences between pre-dicted and measured values are less than 11%. Greater differencesrelate to the areas close to the heater, where the effects of bothconvection and radiation are strong, as well as areas far from themain activity, such as the lower end of the outlet wall, where verylow velocities are predicted. Another area where larger differencesare noticeable is on the ceiling above the opening between the twofloors. This is where the rising plume impinges on the ceiling.

5.4. Air temperature and velocity

Fig. 8 shows a comparison between the measured and predictedair temperature and velocity in the opening between the twofloors. The experimental profiles were obtained in the mid-sectionof the opening (along ZZ1 in Fig. 1). For the predicted results, it wasconsidered likely that the experimental conditions in the mid-sec-tion are achieved in the vicinity of the mid-line rather than at theexact location where the experimental results were obtained.Therefore the following procedure was followed. First statistical

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M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 31

averages over 20 s from 420 to 440 s were obtained. Next, the pro-files of velocity and temperature were obtained along five equally-spaced lines within a band of 0.2 m in the middle of the openingand then averaged. In other words, the profiles shown in Fig. 8are an average of five different profiles.

For the lower heat input the velocity is mainly underpredicted(Fig. 8(a)). In the experiments, changes in the air velocity by about0.05 m/s was possible; this amounts to 30% for a velocity of0.15 m/s. On the whole, the magnitude and trends obtained by thesimulation can be considered as satisfactory. Fig. 8(b) shows the cor-responding air temperature. The main features of the experimentalprofiles are (i) the region of relatively higher temperatures in the up-flow region (X1 < 0.6 m), (ii) the region of lower temperatures in thedownflow (X1 > 0.9 m), and a region around X1 = 0.75 m where apeak is noticeable. In the interpretation of the experimental data[14] the peak was associated with the rising hot plume. In the sim-ulation, the peak value is closer to the start of the opening area.Fig. 6 shows that, in the simulation, the rising plume was placedcloser to the front wall than the middle of the opening where theseresults are plotted. In this respect, therefore, there is a disagreementwith experiment in relation to the location of the plume.

Further realization of what is achievable in this type of flow interms of agreement between measured and computed values maybe obtained with reference to the previous work by other investi-gators. The experimental data in the half-scale stairwell model ofErgin-Özkan et al. [11] was used for comparison with the numeri-cal simulation using the standard k–e model by Mokhtarzadeh-Dehghan et al. [12]. The computed velocities along a line drawnperpendicular, and half-way along the stairway, reached maximumvalues of about 0.3 m/s. The computed values were generally high-er showing differences of about 0.05–0.1 m/s with the measure-ment. Jiang and Chen [7] used LES to study natural convection ina building model with a large opening simulating a window. Theair velocities were mostly less than 0.1 m/s. In their comparisonsof computed and measured velocity profiles similar differences tothe present study were reported. Better agreement was obtainedfor the temperature distribution. As part of the validation of theirwork on fire-induced flow using large eddy simulation, Qin et al.[8] simulated the half-scale stairwell model of Ergin et al. [11].They compared the simulated velocity distribution with those fromthe experiments along two lines, one along a vertical line at thestart of the stairway and another along a horizontal line at theend of the stairway. The measured values ranged from 0 to0.25 m/s and the differences of 0.05 to 0.1 m/s between the pre-dicted and measured values were common. No values for temper-ature distribution were reported. Sun et al. [9] studied smokemovement in a real six-storey stairwell, driven by fire on theground floor. They reported average measured velocities at differ-ent heights, reaching a maximum value of about 1.0 m/s. Compar-ison with the predicted results obtained using LES showed correcttrends with differences of mostly 0.1–0.2 m/s.

Considering the present LES results as whole, it may be con-cluded that the simulation has been reasonably successful inobtaining the correct picture of the overall flow pattern, the overalltrends and orders of magnitudes of the wall temperatures, and airvelocity and temperature. The degree of agreement between pre-diction and experiment has been consistent with those reportedby other investigators in similar works.

5.5. Effect of initial temperature and the case of k–e model

In the preliminary present LES calculations, for the case of heatflux boundary condition (HFBC), it was noted that there was adependency of the temperature field to the initial air temperatureset within the stairwell model, whereas the velocity field was notaffected. For temperature boundary condition (TBC), however, both

the temperature and velocity fields were independent of the initialtemperature. The initial temperature which is set uniformly every-where at the start of computation serves as an initial guess and thetemperature field is free to change and take the correct distributionin the final converged solution. With HFBC, however, a convergedsolution can be obtained with different level of temperature, setby the initial air temperature. With TBC, since the walls tempera-tures are fixed, the initial air temperature has not such an influenceon the final solution. It should be noted that in both cases, for anyconverged solution and at any time step, the heat input from theheater is balanced by the heat output from the walls, thereforethe internal energy of the system remains constant. Also, the pres-sure differentials determining the velocity field are not affected. Inorder to demonstrate the effect of initial temperature, additionalsimulations were carried out using the k–e model.

The wall temperatures were referred to in relation to Fig. 7. Thisfigure also shows the results obtained using the k–e model withthe same initial temperatures, for Case 1 and Case 2, respectively.The results for the k–e model show the same general pattern butoverall they have deviated from the experimental values morethan the LES results, except for the floor (Fig. 7(f) and (l)). Otherexceptions are in the regions far from the centre of flow activitywhere LES showed difficulty, i.e. far corners in the lower compart-ment, near the outlet wall (Fig. 7(d) and (j)) and in the region adja-cent to the lower inlet and lower back walls (Fig. 7(h) and (i)).These differences indicate that although the general flow patternof flow moving between the two floors are similar, different detailsin the flow pattern have been established. This deduction may bealso noticed from the air velocity and air temperature in the hori-zontal opening as shown in Figs. 9 and 10. These figures also showthe results obtained using HFBC with different initial tempera-tures. Fig. 9 shows smooth and mostly linear variations of temper-ature predicted in the case of the k–e model. The profiles of airvelocity (Fig. 10) show more distinct peaks and troughs, placedat different locations in the horizontal opening. The effect of initialtemperature is clearly seen in these figures. Changes in the initialtemperature by ±2.0 �C have produced shifting of the air tempera-

(a) 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 0.3 0.6 0.9 1.2 1.5X1 (m)

V (m

/s)

(b)0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 0.3 0.6 0.9 1.2 1.5X1 (m)

V (m

/s)

Fig. 10. Air velocity in the opening between the two floors. (a) Case 1, (b) Case 2; —,LES, HFBC; , k–e, HFBC, initial temperature; , k–e, TBC, initial tempera-ture, (a) 27.85 �C, 29.85 �C, 31.85 �C, (b) 32.85 �C, 34.85 �C, 36.85 �C.

Fig. 11. Heat transfer rate on stairewell walls. (a) Case 1, (b) Case 2; ,Calculated using temperature boundary condition; , Measured; On x-axis: 1:inlet lower; 2: inlet upper; 3: outlet lower; 4: outlet upper; 5: back lower; 6: backupper; 7: front lower 1; 8: front lower 2; 9: front upper 1; 10: floor 1; 11: floor 2;12: ceiling.

32 M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33

ture profile by the same amount, whereas the velocity profiles re-main unaffected (Fig. 10). The results for the stairwell wall temper-atures (not presented here) also showed the same shift. Inspectingthe flow pattern showed more orderly flow behaviour than thatobtained using LES. Upon reaching the opening area, the air rosemuch closer to the entrance to the horizontal opening and movedstraight up to the ceiling, which was then diverted to form largeclearly established recirculation zones. Such zones were also moredistinguishable and orderly in the lower compartment. The upflowand downflow in the horizontal opening were also more distin-guishable causing the variation in the profile seen in Fig. 10. Onthe whole, it was concluded that the LES flow field was more real-istic than the one obtained using the k–e model. The profiles withTBC are also shown in Figs. 9 and 10. As can be seen the profiles are

Table 6Calculated heat transfer rate using temperature boundary conditions with the k–e model.

Wall Case 1TBC

qr (W) qc (W)

Inlet Lower wall �12.6 �5.2Upper wall �2.4 �4.1

Outlet Lower wall �2.1 �4.1Upper wall �2.3 �4.4

Back Lower wall �37.4 �23.8Upper wall �10.0 �13.2

Front Lower wall 1 �59.3 �3.5Lower wall 2 �4.6 �5.8Upper wall �7.5 �16.1

Floor Floor 1 �11.6 2.7Floor 2 �14.8 �4.5

Ceiling �3.9 �15.4Heater 146.2 121.2

similar to those with HFBC but noticeable differences exist. In thiscase, there is no dependency on initial temperature. This findingindicates that for natural convection in a closed system, settingthe wall boundary conditions based solely on heat flux has impli-cations for the prediction of the temperature field. The problemis that for a practical flow application none of these conditionsare usually known in advance of the solution. When heat fluxboundary conditions were set, measured wall temperatures werecompared with the predicted ones in Fig. 7. Similarly, when tem-perature boundary conditions were set the measured wall heatfluxes (Table 5) can be compared with the predicted heat fluxes gi-ven in Table 6 and shown in Fig. 11. The main differences relate tothe walls close to the heater, namely, inlet lower, floor 1 and frontlower 1, arising from the differences in the proportion of radiativeand convection heat transfer rates.

Case 2TBC

q (W) qr (W) qc (W) q (W)

�17.8 �32.8 �10.5 �43.3�6.5 �6.3 �9.0 �15.3

�6.2 �3.4 �8.3 �11.7�6.7 �3.9 �12.0 �15.9

�61.2 �76.7 �44.7 �121.4�23.2 �22.5 �31.5 �54.0

�62.8 �124.0 �2.1 �126.1�10.4 �9.2 �10.7 �19.9�23.6 �22.7 �40.0 �62.7

�8.9 �21.4 8.0 �13.4�19.3 �22.5 �7.5 �30.0

�19.3 �5.1 �34.9 �40.0267.4 308.3 247.9 556.2

M.R. Mokhtarzadeh-Dehghan / International Journal of Heat and Mass Transfer 54 (2011) 19–33 33

6. Conclusions

The study aimed to investigate a previously published experi-mental data by the present author as a possible test case fornumerical simulation of a highly three-dimensional natural con-vection flow. The results showed a good level of consistency be-tween different simulation results and the experimental data,which gave supported to the validity of the experimental data forthis purpose, as well as the numerical methods used. The resultsindicate that the case under investigation may provide a challeng-ing test case for CFD codes under development. For such a closedsystem, the solution may be sensitive to factors such as type ofthermal boundary conditions, near-wall treatment and turbulencemodelling, and therefore different solutions may be obtained. Theinclusion of radiation exchange between the walls and ability tocalculate the proportion of heat transfer by convection for such acomplex geometry may also prove challenging.

Because of the nature of the flow LES provided a better optionthan the mostly common approach utilizing the k–e model, butat the expense of significant demand on computing resources.The results obtained using LES were more realistic, but the resultsobtained using the k–e model still proved to be useful and,although displaying somewhat different flow details, it producedthe overall features and predicted correct order of magnitudes.This was in spite of the use of near-wall modelling.

It was found that the use of heat flux boundary conditions onthe walls resulted in dependency of the temperature field on theinitial air temperature, but the velocity field was independent ofthis parameter. This aspect of the simulation requires furtherinvestigation in order to reduce this effect, as it has practical impli-cations. Setting temperature boundary conditions on the walls pro-duced the same temperature field and thus such a dependency wasnot present.

Acknowledgement

The simulations reported in this paper were carried out usingANSYS FLUENT flow modelling software under an academic license.

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