19th Australasian Fluid Mechanics ConferenceMelbourne, Australia8-11 December 2014

A Numerical and Experimental Study on the Unsteady Conjugate N atural ConvectionBoundary Layers in a Water Filled Rectangular Cavity with a C onducting Partition Wall at

Varied Locations

M. Khatamifar 1, R. D’Urso 1, W. Lin 1,2, D. Holmes 1, S. W. Armfield 3 and M. P. Kirkpatrick 3

1College of Science, Technology & EngineeringJames Cook University, Townsville, QLD 4811, Australia

2Solar Energy Research InstituteYunnan Normal University, Kunming, Yunnan 650092, China

2School of Aerospace, Mechanical and Mechatronic EngineeringThe University of Sydney, NSW 2006, Australia

Abstract

This paper presents some preliminary results on the behav-ior of unsteady conjugate natural convection boundary layers(CNCBLs) in a water filled rectangular cavity with a partitionwall at varied locations through a series of experiments and two-dimensional direct numerical simulations (DNS). The experi-ments were carried out over 2.428×109 ≤ Ra ≤ 2.458×1010

and 5.3701≤ Pr ≤ 6.1697, with the partition wall placed atthe center of the cavity, whereRa andPr are the Rayleigh andPrandtl numbers, respectively. The temperatures at eight loca-tions in the cavity were measured by Resistance TemperatureDetectors (RTDs). The DNS runs were carried out using a codedeveloped in house based on the finite-difference method andwritten in Visual C#.NET programming language. The DNSruns complement and expand the experiments by providing ad-ditional information about the behavior of unsteady CNCBLsand the effects of the variation of the partition position.

Introduction

Heat transfer between two fluid reservoirs at different tem-peratures through a vertical separating conducting wall oc-curs widely in nature and engineering applications. The phe-nomenon is particularly important in power plants, polymerproduction, chemical reactors, food processing, computer andelectronic instruments design, building design, and geophysi-cal flows, in addition to its fundamental significance. The heattransfer in such a configuration is via a mechanism called conju-gate natural convection boundary layers (CNCBLs). On the sideof the partition wall contacting with the fluid at a higher tem-perature, a natural convection boundary layer forms and movesdown along the wall whereas on the other side of the wall whichis in contact with the fluid at a lower temperature, a similarboundary layer forms and moves up along the wall, both due tothe buoyancy caused by the temperature difference between thewall and the fluid in contact. Heat are transferred through theconducting partition wall, from the fluid at a higher temperatureto the fluid at a lower temperature. As the partition wall temper-ature changes with the development of the CNCBLs, it becomespart of the solution of the flow and heat transfer, which makesthe prediction of the behaviour of the flow and heat transfer insuch a configuration complicated and hard.

There have been many studies on CNCBLs, with the majoritydealing only with their steady behavior (see, e.g., [1, 2, 3]). Thebehavior of unsteady CNCBLs is still poorly understood, whichmotivates this study, where both physical experiments and two-dimensional DNS simulations are carried out to examine the be-havior of unsteady CNCBLs in a water filled rectangular cavitywith a conducting partition wall at varied locations.

Methodologies

Experimental Methods

Figure 1. Schematic of the experimental setup: 1. OTS heater (240V5000W), 2. Davey D40VA sump pump, 3. 205 L galvanized steel drumwith water at temperature higher than the ambient, 4. Experimental tank(400×400×600 mm, Perspex front and rear, PVC top and bottom, 3mm aluminum alloy 5083 partition plate and side heat exchangers), 5.8×Resistance temperature detectors (RTDs), 6. 1000 L plastic tankwith water at ambient temperature, 7. Davey D15VA sump pump.

The experimental setup is sketched in figure 1. The rectangu-lar experimental tank was of the dimensions 400 mm× 400mm × 600 mm (L ×W × H). It had Perspex front and rearsides, and PVC top and bottom sides. Its left and right verti-cal sides were two aluminium heat exchangers, which, by cir-culating hot/cold water through them from the hot/cold waterreservoirs respectively, produced and maintained a constant butdifferent temperature (Th andTc for hot and cold water) on eachside during the course of each experiment run. The tank wasinsulated by 50 mm thick polystyrene foam sheets at the top,bottom, rear and front sides, which made it appropriate to as-sume that the heat transferred across these four sides is neg-ligible. A galvanized 205 L drum and a 1000 L plastic tankwere used as the hot/cold water reservoirs, respectively. In eachof these tanks, a submersible sump pump was placed to pumphot/cold water through the aluminium heat exchangers. TheDavey D40VA/D15VA sump pump used for the hot/cold waterreservoir pumped hot/cold water at a flow rate of approximately189/131 L/min. The use of such large flow rates was to ensurethat the inside wall of each heat exchanger was maintained ata constant temperature. A 3 mm aluminum alloy 5083 parti-tion wall with a thermal conductivity ofκp ≈ 121 W/m-K wascentrally positioned in the tank (i.e., xp=0.5, wherexp is thepartition position ratio with respect to the cavity length), whichdivides the tank into two separate but equal-sized domains.

To examine the evolution of the temperatures in the cavity, eight

calibrated PT100 RTD probes were inserted through the holesat predetermined locations on the top surface of the experimen-tal tank. The RTDs were in the central plane of the tank in thespanwise direction and 75 mm below the top surface. Their lo-cations were 20 mm, 75 mm, 130 mm, 185 mm, 215 mm, 270mm, 325 mm, and 380 mm away from the left vertical side-wall, respectively, for the RTDs numbered from 1 to 8. TheRTDs were connected to a DT85 dataTaker Data Logger thatrecorded the temperatures measured by the probes at 1-secondintervals. This frequency of logging was found to give an idealresolution on the temperature evolution. The temperature of thehot water tank was controlled by a 5000W digital temperaturecontrolled immersion heater. This contained a built-in AC1-5PLAE electronic controller and a PT100 RTD that allowed foraccurate (±0.3 oC) PID heater control. The temperature of thecold water tank was at the room temperature and a PT100 RTDwas inserted into the tank to measure it.

The flow is governed mainly by the dimensionless parametersRa, Pr, A, xp, andκr, whereRa andPr are the Rayleigh andPrandtl numbers,A is the aspect ratio of the tank, andκr is theratio of the thermal conductivity of the partition wall (κp) andthat of water (κw), respectively, which are defined as,

Ra =gβw(Th −Tc)H3

νwαw, Pr =

νw

αw, A =

LH, κr =

κp

κw, (1)

in which g is the acceleration due to gravity,βw, νw and αware the thermal expansion coefficient, kinematic viscosity andthermal diffusivity of the fluid (water), respectively.αw =κw/(ρwCp), whereρw andCp(=4180 kJ/kg-K) are the densityand specific heat of water, andκw ≈ 0.6 W/m-K.

Run Th (oC) Tc (oC) Ra Pr1 25.7 23.7 2.428× 109 6.16972 26.1 23.1 3.621× 109 6.18553 26.9 22.9 4.913× 109 6.13814 28.6 23.6 6.582× 109 5.94855 30.5 23.5 9.689× 109 5.81116 32.5 23.5 1.308× 1010 5.68517 34.5 23.5 1.678× 1010 5.55918 37.0 24.0 2.130× 1010 5.37019 38.0 23.0 2.458× 1010 5.3701

Table 1. Details of the experimental runs.

Totally nine experiment runs were carried out, over 2.428×109 ≤ Ra ≤ 2.458× 1010 and 5.3701≤ Pr ≤ 6.1697, all atA = 1, xp = 0.5 andκr ≈ 202. All these experiments were con-ducted with the same tank and partition configuration and thevariations ofRa andPr were achieved by changing the hot wa-ter temperature through the OTS heater. The details about theseexperimental runs are listed in table 1.

Numerical Methods

With the experiments, the scope of this study is quite limitedas only measured temperatures at eight fixed locations in thecavity can be provided by the experiments due to the experi-mental setup. In addition, only one specific water tank and par-tition configuration is used in the experiments. A series of two-dimensional DNS runs are therefore carried out to complementand expand the experiments, which provide additional detailedinformation about the behavior of unsteady CNCBLs, such asthe temperature and velocity fields, thermal and viscous bound-ary layer thicknesses, Nusselt number, etc. Furthermore, withthe DNS runs, the behavior of unsteady CNCBLs at varied par-tition positions and different configurations can be examined.

The governing equations for DNS are the continuity, Navier-Stokes, and temperature equations for the fluid in the cavityand the heat conduction equation in the partition wall. With theOberbeck-Boussinesq approximation for buoyancy, these equa-tions can be written in dimensionless forms as follows,

∂u∂x

+∂v∂y

= 0, (2)

∂u∂τ

+u∂u∂x

+ v∂u∂y

=−∂p∂x

+

√

PrRa

(

∂2u

∂x2 +∂2u

∂y2

)

, (3)

∂v∂τ

+u∂v∂x

+ v∂v∂y

=−∂p∂y

+

√

PrRa

(

∂2v

∂x2 +∂2v

∂y2

)

+θ, (4)

∂θ∂τ

+u∂θ∂x

+ v∂θ∂y

=

√

1PrRa

(

∂2θ∂x2 +

∂2θ∂y2

)

, (5)

∂θ∂τ

=

√

κr

PrRa

(

∂2θ∂x2 +

∂2θ∂y2

)

, (6)

whereu andv are velocity components in thex andy directions,p is pressure,θ is temperature, andτ is time, which are madedimensionless by their respective characteristic scales,i.e.,

x =XH, y =

YH, u =

UV0

, v =VV0

,

p =P

ρV 20

, τ =t

H/V0, θ =

T −T0

Th −Tc,

(7)

in which U , V , X , Y , P, T , andt are the dimensional counter-parts ofu, v, x, y, p, θ, andτ, respectively.V0 =

√

gβ(Th −Tc)is used as the characteristic velocity scale.T0 = (Th +Tc)/2 isused as the reference temperature.

The initial conditions (whenτ = 0) are,

u = v = 0, θ =−0.5 at all x andy, (8)

and whenτ > 0 the boundary conditions for the fluid are,

u = v = 0, θ = 0.5 at x = 0, 0≤ y ≤ 1,

u = v = 0, θ =−0.5 at x = A, 0≤ y ≤ 1,

u = v = 0,∂θ∂y

= 0 at y = 0, 0≤ x ≤ A,

u = v = 0,∂θ∂y

= 0 at y = 1, 0≤ x ≤ A,

(9)

and for the partition wall the boundary conditions are,

u = v = 0, κw∂θ∂y

∣

∣

∣

∣

x−1

= κp∂θ∂y

∣

∣

∣

∣

x+1

at x = x1, 0≤ y ≤ 1,

u = v = 0, κw∂θ∂y

∣

∣

∣

∣

x−2

= κp∂θ∂y

∣

∣

∣

∣

x+2

at x = x2, 0≤ y ≤ 1,

(10)wherex1 andx2 are the locations of the left and right sides of thepartition wall and the associated superscripts ‘-’ and ‘+’ denotetheir nearest left and right cells, respectively.

A DNS code based on the finite difference method and writtenin Visual C# .NET programming language was developed inhouse. The governing equations were discretised with a com-bination of explicit and implicit schemes. The SIMPLE algo-rithm was used to solve the coupled pressure and velocity. Themethod introduced by [4], in which ghost nodes at the fluid-solid interface are employed and the cavity flow and partitionconduction are solved simultaneously, was used to deal withthe conjugate flow and heat transfer at the partition wall. As

a staggered non-uniform mesh with 500×500 grids was used,some modifications were made to the method. The code has theexplicit and implicit sections of calculation, with the explicitsection calculating velocities and temperature for all cells andthe implicit section consisting of a loop that calculates the con-tinuity equation until convergence is attained. The velocitiesand temperatures are then updated and stored in the cells for thenext time-step calculations.

Results

Time Series of Temperatures

0

1

2

3

Experimental

Numerical

0

1

2

0

1

2

3

0

1

2

0

1

2

3

0

0.02

0.04

0.06

0

1

2

3

0 200 400 6000

0.02

0.04

0.06

0

1

2

0

1

2

3

0

1

2

0

1

2

3

0

0.02

0.04

0.06

0

1

2

3

0

0.02

0.04

0.06

0 5000 10000 150000

1

2

3

t (s)t (s)

RTD 1RTD 1

RTD 2

RTD 3

RTD 4

RTD 5

RTD 6

RTD 7

RTD 8

RTD 2

RTD 3

RTD 4

RTD 5

RTD 6

RTD 7

RTD 8

T -

T (

C)

T -

T (

C)

0 0oo

Figure 2. Time series of temperature(T −T0) at the eight points mea-sured by the RTDs and obtained by the DNS runs, with the left and rightcolumns for the whole durations and early stages of the measurementsand simulations, respectively.

The time series of the temperatures at the eight points measuredby the RTDs and obtained by the DNS runs for the representa-tive case corresponding to the experiment run 2 are presentedin figure 2. The results show that in the left domain the tem-peratures rise quickly at the early development stage but thenapproach toTh very slowly and asymptotically, whereas thetemperatures in the right domain rise in a significantly slowerrate but also approach a final temperature slowly and asymptot-ically. For the case presented in the figure, the final temperatureattained in the right domain is about 2/3 ofTh at the end of theexperiment and DNS run, apparently due to the conjugate heattransfer on the partition wall. Theoretically it is expected thatthe temperatures in both domains will attainTh if the heatingapplied on the left vertical side lasts infinitely.

The DNS results obtained so far only cover the early develop-ment stage due to the extensive computation requirements. The

comparison with the experimental results shows that the DNSresults are in general in good agreement with the experimen-tal results, although noticeable deviations are observed duringthe course of the passage of first intrusion through these mea-surement points. The deviations are most likely caused by themesh used which may not be fine enough to resolve the smallestscales as the flow, atRa = 3.621×109, is at the transition to tur-bulent regimes. A finer mesh is expected to significantly reducethe deviations and the results will be reported at the conference.Another possible cause may come from the two-dimensionalflow assumption in the DNS as at such a highRa, the flowshould be three-dimensional.

20

30

40

Run 1Run 2Run 3

0

0.5

1

20

30

40

0

0.5

1

20

25

30

0

0.5

1

20

25

30

0 1000 2000 30000

0.5

1

0

0.5

1

Run 7Run 8Run 9

20

30

40

Run 4Run 6Run 6

0

0.5

1

20

30

40

0

0.5

1

20

25

30

0

0.5

1

0 5000 10000 1500020

25

30

τt (s)

RTD 1RTD 1

RTD 2

RTD 3

RTD 4

RTD 5

RTD 6

RTD 7

RTD 8

RTD 2

RTD 3

RTD 4

RTD 5

RTD 6

RTD 7

RTD 8

T (

C)

θo

Figure 3. Time series of dimensional temperaturesT (left column) anddimensionless temperaturesθ (right column) at the eight points mea-sured by the RTDs for all 9 experimental runs.

The time series of the measured temperatures at the 8 RTDSfor all 9 experiment runs are presented in figure 3. The gen-eral behavior observed from figure 2 for the experiment run 2 isalso observed for all other 8 experiment runs. It is interestingto note that whenTh − Tc increases, the temperatures at the 4RTDs in the right domain approach an eventual dimensionlesstemperature, again asymptotically, which is smaller than thatfor a smallerTh−Tc. This may be caused by the increasing heatloss through the insulation whenTh − Tc increases. But somefurther studies should be carried out to examine this.

Evolution of Temperature Fields

The evolution of temperature fields obtained by DNS is pre-sented, as an example, in figure 4 forRa = 3.621×109, Pr =6.1855,A = 1, andκr = 202, but atxp=0.3, 0.5, and 0.7, re-spectively. The effect ofxp on the stratification of fluids in theleft and right domains is obvious; the stratification in the leftdomain develops quicker for smallerxp but that in the right do-

t=20s t=50s t=500s t=2000s

Figure 4. Temperature contours at four times forxp = 0.3 (first row), 0.5 (second row), and 0.7 (third row), respectively.

main is just the opposite, due to smaller heat transfer rate acrossthe partition wall. It is expected that the fluid in the left domainwill attain Th faster for smallerxp but that in the right domainwill take much longer time to attain its asymptotic temperature.

Horizontal Profiles of θ and v

Figure 5 presents the evolution of the horizontal profiles ofθ and v at y = 0.5 obtained by DNS for the case ofRa =3.621×109, Pr = 6.1855,A = 1, κr = 202, andxp=0.5. Theformation and development of boundary layers on the left andright vertical sides and the CNCBLs on the both sides of thepartition wall are clearly shown and in the middle regions ofboth domains, stratification is well developed at the later stage.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t=10 st=20 st=50 st=100 st=200 st=500 st=1000 st=2000 s

x

x

(a)

(b)

θ

v

Par

titio

n w

all

Par

titio

n w

all

Figure 5. Evolution of the horizontal profiles ofθ andv.

Conclusions

In this paper, some preliminary results obtained experimentallyand numerically are presented to show the behavior of unsteadyCNCBLs with varied partition locations. The two-dimensionalDNS results, which were obtained by using a code specificallydeveloped in house based on the finite-difference method andwritten in Visual C#.NET programming language, are found ingeneral to be in good agreement with the experimental results,but they complement and expand the experiments by providingadditional information about the behavior of unsteady CNCBLsand the effects of the variation of the partition position. Moredetailed results will be presented at the conference.

Acknowledgements

The support from the Australian Research Council (ARC),the National Natural Science Foundation of China (51469035,11072211), and the Yunnan Natural Science Foundation(2011FA017) is gratefully acknowledged. M. Khatamifar alsothanks James Cook University for the JCUPRS scholarship.

References

[1] Lock, G.S.H. and Ko, R.S., Coupling Throught a Wall Be-tween Two Free Convective Systems,Int. J. Heat MassTransfer, 16, 1973, 2087–2096.

[2] Xu, F., Patterson, J.C. and Lei, C., Heat Transfer ThroughCoupled Thermal Boundary Layers Induced by a Sud-denly Generated Temperature Difference,Int. J. HeatMass Transfer, 52, 2009, 4966–4975.

[3] Williamson, N., Armfield, S.W. and Kirkpatrick, M.P.,Transition to Oscillatory Flow in a Differentially HeatedCavity with a Conducting Partition,J. Fluid Mech., 693,2012, 93–114.

[4] Carlson, K.D., Lin, W. and Chen, C.-J., Pressure Bound-ary Conditions of Incompressible Flows with ConjugateHeat Transfer on Nonstaggered Grids. Part ii: Applica-tions,Numerical Heat Transfer, Part A: Applications, 32,1997, 481–501.