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Numerical Investigation of Natural Convection insideComplex EnclosuresYos S. Morsi a & Subrat Das aa Center for Modeling and Process Simulation, Swinburne University of Technology,Hawthorn, AustraliaPublished online: 30 Nov 2010.

To cite this article: Yos S. Morsi & Subrat Das (2003) Numerical Investigation of Natural Convection inside ComplexEnclosures, Heat Transfer Engineering, 24:2, 30-41, DOI: 10.1080/01457630304080

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Heat Transfer Engineering, 24(2):30–41, 2003Copyright C©© 2003 Taylor & Francis0145–7632/03 $12.00 + .00DOI: 10.1080/01457630390116266

Numerical Investigation ofNatural Convection insideComplex Enclosures

YOS S. MORSI and SUBRAT DASCenter for Modeling and Process Simulation, Swinburne University of Technology, Hawthorn,Australia

In this article, the analyses of heat transfer and free convective motion have been carried outnumerically for various structures. The solution is based on a finite element method with the frontalsolver to examine the flow parameters and heat transfer characteristics. Several domeconfigurations—such as flat, inclined, and dome shapes—are considered for the top of theenclosure. A general conic equation is considered to represent the dome as circular, elliptical,parabolic, or hyperbolic shape. The findings from this study indicate that the convectivephenomenon is greatly influenced by the shape of the top cover dome and tends to form a secondarycore even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure.In addition, the circular and elliptical shapes of the dome give higher heat transfer rate. The effectof various “offset” of the dome and inclined roof on convective heat transfer is also found to bequite significant. However, beyond 0.3 of offset of the top cover for the dome and inclined roof, thechange in overall heat transfer rate is minimal. The heat transfer coefficients of dome shaped andinclined roof enclosures are given and discussed.

There are numerous publications on the fluid flowand transport processes generated or altered by buoy-ancy force in enclosures. As a result, various numer-ical, analytical, and experimental studies related tothe analysis of fluid flow phenomena and heat trans-fer in various types of enclosures are found in litera-ture. However, even after 40 years of research, most ofthe studies are associated with uniform/regular shapes,such as square, rectangular, cylindrical, annulus, etc. Inmany practical situations including solar heating, nu-clear waste disposal, solidification, and particularly inthe area of preservation of food grains, drugs, and other

Address correspondence to Professor Yos S. Morsi, School of Engineer-ing and Science, Swinburne University of Technology, Hawthorn Victoria3122, Australia. E-mail: ymorsi@swin.edu.au

granular products, irregular-shaped enclosures are nor-mally used. In these complex geometries, the convec-tive flow and heat transfer is more complex and of in-trinsic interest due to the resulted complexity of theboundary coupled with nonlinear governing equations.Although information gained from previous studies isessential in enhancing our understanding in predict-ing the nature of the flow patterns, it fails to explainthe complications resulting from the curved walls onthe flow characteristics and heat transfer. Moreover,very few studies have addressed the question of naturalconvection in nonrectangular and noncylindrical typeenclosures.

For the sake of completeness, a brief review of regu-lar and irregular enclosures is given. The recent unifiedand the most comprehensive review of this subject is

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made by Ostrach [1] and Hoogendoorn [2]. The authorspoint out that internal flow problems are considerablymore complex than those of external ones because theinteraction between the boundary layer and core leads toseveral complexities and is inherent to all confined con-vection configurations. It is believed that the core flowis so sensitive to the geometry as well as the boundaryconditions that the flow pattern cannot be predicted apriori from the given improvised boundary conditionsand geometry. In case of an inclined/curvature bound-ary, the situation is even more intricate because theremay be more than one global core and flow subregionsare possible, such as cells and layers embedded in thecore. There has been no comprehensive model to estab-lish the effect of the curvature wall on flow and heattransfer characteristics.

Few studies have been reported on natural convectionin nonrectangular and noncylindrical enclosures. Heattransfer in a parallelogram-shaped enclosure by Asakoand Nikmura [3], triangular enclosures by Akinsate andColeman [4], and other nonrectangular enclosures byCampo et al. [5] shows that there is a significant changein shape of the core flow. However, all of these studieshave concluded that the shape has less of an effect onthe quantitative analysis than that of the dimension ofthe enclosure.

Moreover, various researchers have explored theanalysis of natural convection in different shapes of en-closures, such as a trapezoidal enclosure by Moukalledand Acharya [6] and an enclosure having inclined roofby Das and Sahoo [7]. On the other hand, Lewandowskiand Khubeiz [8] and Asfia et al. [9] have discussedthe effect of convex and concave surfaces on natu-ral convection. They point out the significant changein flow patterns such as dead space, local heating,etc., due to a curved wall. Lewandowski et al. [10]have investigated the natural convection phenomenaand its effect due to various complex surfaces. How-ever, none of these researchers has given any general-ized model to analyze the effect quantitatively; specifi-cally, the effect of a curved boundary wall on the flowpattern and the heat transfer coefficient is not clearlyoutlined.

Thus the objective of this present paper is to considerthe different shapes for the top enclosure and analyzethe flow patterns and heat transfer characteristics. Weare mainly interested in generalizing the problem byconsidering the dome as circular, elliptical, parabolic,and hyperbolic type of shapes. Further, an inclined top isalso considered in the present analyses. However, in thefirst part of the paper, the analysis is carried out for theenclosure having zero “offset” (square enclosure), andthis is primarily to build the confidence in our numericalpredictions.

PROBLEM DEFINITION AND GOVERNINGEQUATIONS

A two-dimensional steady-state analysis of an incom-pressible fluid driven by the buoyancy force is consid-ered within the enclosure. The governing equations con-sidered here are same as those of De Vahl Davis [11]and Tabbarok and Lin [12]. Using the Boussinesq ap-proximation and neglecting the dissipation effect dueto the viscous term, the governing equations in non-dimensional form are written as follows:

∂u

∂x+ ∂v

∂y= 0 (1)

u∂u

∂x+ v

∂u

∂y= −∂p

∂x+ Pr

[∂2u

∂x2+ ∂

2u

∂y2

](2)

u∂v

∂x+ v

∂v

∂y= −∂p

∂y+ Pr

[∂2v

∂x2+ ∂

2v

∂y2

]+ T ·Ra·Pr (3)

u∂T

∂x+ v

∂T

∂y= ∂

2T

∂x2+ ∂

2T

∂y2(4)

All the above Equations are normalized using thefollowing dimensionless scales:

(x, y) = (x∗, y∗)/Lref (u, v) = (u∗, v∗)Lref/α

p = p∗L2ref

/(ρα2)

T = (T∗ − Tc)/(Th − Tc)

Ra= gβ(Th − Tc)L3ref

/(να) Pr= ν/α

where asterisks denote dimensional variables.The solution is carried out for an enclosure having

two differentially heated vertical isothermal walls (Th =1, Tc = 0) and two horizontal adiabatic walls. The rangeof aspect ratioγ considered here is 1 (H = Lref) to1 + E. The no-slip condition is applied to all boundarywalls for the validation of the numerical procedure andcomparison with the dome-shaped enclosure.The boundary conditions and the flow domain areillustrated in Figure 1.

On surface of I u = v = 0; Th = 1

On surface of III u = v = 0; Tc = 0

On surface of II & IV u = v = 0;∂T

∂n= 0

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Figure 1 Coordinate system.

COMPUTATIONAL DOMAIN

The model configuration and coordinate (x, y) sys-tem used are shown in Figure 1, which represents a typi-cal dome-shaped enclosure. The general conic equationto represent the top dome shape is given by

(1− e2)y2+ 2

[(1− e2)E

2+ 1

8E

]y+ x2 = 0 (5)

whereE = E∗/Lref (“offset”) is introduced to recastEq. (5) in dimensionless form, ande is defined as ec-centricity of the curvature boundary, which decides theshape of the top adiabatic cover depending upon thevalues as given below.

e= 0 for Circular 0< e< 1 for Elliptical

e= 1 for Parabolic e> 1 for Hyperbolic

SOLUTION PROCEDURE

Equations (1) to (4) result in a set of nonlinearcoupled simultaneous equations for which an iterativescheme is adopted. The total domain is discretized into340 elements, resulting in 1083 node numbers. All theelements are isoparametric and quadrilateral, containingeight nodes. All eight nodes are associated with veloc-ities as well as temperature; only the corner nodes areassociated with pressure. This is an accepted practice ofdepicting the variation in pressure by a shape function,which has one order less than those defining velocitiesand temperature given by Taylor and Hood [13].

A suitable element numbering scheme as proposedby Sloan [14] is adopted so as to achieve an optimal frontwidth of 140 for the given geometry of the model; subse-quently, the frontal solver is used to solve the above setof equations. The advantage of using the frontal solver

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Table 1 Comparison of present results with De Vahl Davis [11]

Ra= 103 Ra= 104 Ra = 105 Ra= 106

De Vahl De Vahl De Vahl De VahlPresent Davis [11] Present Davis [11] Present Davis [11] Present Davis [11]

Nu 1.122 1.116 2.243 2.238 4.523 4.509 8.828 8.817vmax 7.86 — 19.60 19.617 68.52 68.59 220.81 219.36

is that the core memory requirement is much less thanthat of a general conventional banded solver.

PRESENTATION AND DISCUSSIONOF RESULTS

The solution is carried out for an enclosure havingtwo differentially heated vertical isothermal walls (Th =1, Tc = 0) and two horizontal adiabatic walls. The rangeof aspect ratio,γ, considered here is 1 (H = Lref) to1 + E. The no-slip condition is applied to all boundarywalls for the validation of the numerical procedure andcomparison with the dome-shaped enclosure.

At the maximum value of Rayleigh number, i.e.,Ra= 106, different grid sizes of quadrilateral eight-noded elements are used for discretization. An optimumnumber of elements is chosen when two consecutivegrid refinements yield an error of less than 1% on the av-erage steady state Nusselt number for the hot wall. Thepredicted quantities of average Nusselt number,Nu, onthe hot wall and maximum vertical velocity,vmax, havebeen compared with the benchmark numerical solutionsof De Vahl Davis [11], as illustrated in Table 1. It maybe noted that De Vahl Davis obtained the solution basedon finite difference method using a rectangular mesh.Tabbarok and Lin [12] have used finite element methodbased on stream-vorticity formulation. A comparisonbetween theNu results of Tabbarok and Lin [12] andthose obtained using the present numerical scheme areshown in Table 2. It is observed that the present resultsare in good agreement with previous published data,with the minor differences attributable to either the dif-ference in the numerical scheme or the grid used in thesestudies.

Moreover, the predicted average Nusselt number iscalculated by trapezoidal integration over the lengthconcerned. Figure 2 shows the flow and isothermal

Table 2 Comparison of present result with Tabbarokand Lin [12] with Pr= 0.733

Grashof number PresentNu Tabbarok and Lin [12]

1000 1.068 1.0735000 1.605 1.653

10000 2.029 2.108Figure 2 Isotherms and streamlines for square enclosure.

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Figure 3 Nondimensional (a)u-velocity (b) v-velocity in themid-vertical axis and mid-horizontal axis, respectively.

patterns for different Rayleigh numbers for aspect ra-tio, γ = 1(Lref = H ), which are very close to thosepredicted by De Vahl Davis. Figure 3 shows theu-vvelocities at the mid-vertical and mid-horizontal axis,

respectively. Note that the symmetric nature of the ve-locity is an additional check in favor of the present tech-nique. Based on the above argument, it can be stated thatthe proposed numerical scheme and the grid generationused are adequate to analyze buoyancy-driven flows indome-shaped enclosures.

DOME-SHAPED ENCLOSURES

The nondimensional parameters included in thisstudy are: eccentricity,e, offset,E, and Rayleigh num-ber, Ra (with Pr= 1). The eccentricity,e, indicates thenature of the dome for the adiabatic top cover as de-scribed earlier. In the present problem,e is chosen as 0.5and 1.2 for the elliptical and hyperbolic dome shapes,respectively. The offset measures the dome height fromthe isothermal walls. Note that the square enclosurewith aspect ratio of unity indicates that the height of theisothermal wall is equal to the width of the enclosure.However, a dome-shaped enclosure with offsetE has anequivalent aspect ratio of 1+ E for the correspondingrectangular enclosure having the height, 1+ E.

The isotherms and streamlines are plotted inFigure 4a. The temperature gradient is negative alongthex direction, giving rise to clockwise unicellular cir-culation at the central part of the domain. An increase inRayleigh number results in the dominance of the con-vective currents. This is evident from the large temper-ature gradient near the hot and cold isothermal walls.With a further increase in Rayleigh number, the gradientof temperature changes to zero or positive at the centralpart of the domain, resulting in a secondary circulationzone in the core region. The streamlines at the bottomof the enclosure are more densely packed than at thetop wall. This indicates that the resultant flow velocityat the bottom wall of the enclosure is more predominantthan that at the top due to the geometry of the enclosure.

The contour plots ofu-v velocities are shown inFigure 4b. The peaku-velocities are observed at thetop and bottom parts of the enclosure. With the increasein Rayleigh number, these velocities shift to the top ofthe hot wall and bottom of the cold wall. A similar sit-uation also prevails for peakv-velocities, which occursat the sides of the isothermal walls; with the increase inRayleigh number, these velocities shift similarly. Thesetwo components give the resultant velocities that havethe peaks at the top of the hot wall and at the bottomof the cold wall. It is evident from the contour plotsthat thev-velocity always remains dominant over theu-velocity at the isothermal walls due to the buoyancyforce. This results in a strong velocity gradient near thewall, as shown by the streamlines, which in turn resultsin a strong temperature gradient near the wall as well.

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Figure 4 Comparison of isotherms and streamlines andu andv velocities for different Rayleigh numbers (Pr= 1) at E = 0.3, e= 0.0.

The contour plots ofu-velocity clearly distinguish theeffect of top circular and the flat adiabatic bottom.

A comparison of velocities of each dome-shaped en-closure is made with its equivalent square, rectangu-lar, and inclined top enclosures. Figures 5 and 6 showthe u-velocity at the mid-vertical axis andv-velocityat the axis of the mid-isothermal walls, respectively. Itis observed that theu-velocity deviates a lot from its

symmetry, which may be attributed to the shape of thetop dome cover. In the case of the inclined top roof, theu-velocity at the top central part decreases significantly.It is also observed that for the same offset, differentcurvatures of the dome yield different velocities inthe upper half. However, the difference is not appre-ciable in the lower half due to the common flat bot-toms of all the enclosures. Moreover, thev-velocity in

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Figure 5 Nondimensionalu-velocity profile at the mid-vertical axis.

Figure 6 is almost symmetrical about the mid-horizontalaxis. When the dome-shaped enclosures are comparedwith the square, equivalent rectangular, and inclinedtop roof enclosure, it is observed that thev-velocityis comparatively higher at high Rayleigh numbers. Thisis mainly due to the adiabatic curved top cover, whichallows a smooth passage to the fluid near the top cor-ner of the hot wall. However, when the fluid enters thedome-shaped area, it decelerates substantially, resulting

in lower u-velocity in the top central part of the dome.This is the reason why theu-velocity is substantiallylower at the top central part of a curved top enclosurethan that of the square and the equivalent rectangularenclosures.

In the case of the horizontal or flat inclined top cover,low heat transfer rates will exist due to the severe deflec-tion of flow away from the vertical walls. The domed-shaped top cover allows the convective currents to

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Figure 6 Nondimensionalv-velocity profile at the mid-horizontal axis of the isothermal walls.

redirect more smoothly over the curved surface, whichenhances the heat transfer rate from the hot wall whencompared to that of the square enclosure. The increasein offset, E, improves the convective current; thus, anet rise in mean heat transfer rate along the hot wallis observed. But dome-shaped and inclined top enclo-sures proved to be less efficient in terms of heat transfer

rate when compared with the equivalent rectangular en-closure because of less heat transfer area along the hotisothermal wall, as shown in Figure 7. The Nusselt num-bers is also plotted for all types of conic sections, such ascircular, elliptical, parabolic, and hyperbolic top domecovers along with the inclined top enclosures at differ-ent offsets. The most interesting result noted is that an

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Figure 7 Comparison of Nusselt number for the dome, rectangular, and inclined roof enclosures with different values of offset,E.

increase in offset beyond 0.3 for all conic sections andinclined roof brings stagnation in the improvement ofconvective current near the hot wall. This indicates nosignificant increase in the mean Nusselt number. Forthe circular top cover (e = 0.0), the percentage rise inNusselt number in the lower offset (i.e., 0.1 to 0.2) isabout 5% and only 1.2% at higher offset (0.4 to 0.5; seeTable 3). Increase in the offset (E > 0.3) for all typesof conic and triangular sections increases the area at the

central dome section, resulting in a lower velocity. Thisis an important factor in the improvement of convectiveheat transfer. The effect of different curvatures of thedome-shaped cover is also shown at varying Rayleighnumbers. These curvatures have little effect on the meanNusselt number at very low values of offset, even atthe high range of Rayleigh number. However, at higheroffsets and higher Rayleigh numbers, it is found thatthe circular and the elliptical curvatures have the most

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Tabl

e3

Ave

rage

Nus

selt

num

ber

atth

eho

twal

lfor

diffe

rent

ecce

ntric

ity

Nu

(Dom

eden

clos

ures

)

E=

0.1

E=

0.2

E=

0.3

E=

0.4

E=

0.5

Ra

e=

0.0

e=

0.5

e=

1.0

e=

1.2

e=

0.0

e=

0.5

e=

1.0

e=

1.2

e=

0.0

e=

0.5

e=

1.0

e=

1.2

e=

0.0

e=

0.5

e=

1.0

e=

1.2

e=

0.0

e=

0.5

E=

1.0

e=

1.2

6×

103

2.08

02.

080

2.07

02.

070

2.19

02.

190

2.19

02.

180

2.28

52.

285

2.25

82.

255

2.32

02.

320

2.29

02.

283

2.35

12.

354

2.31

22.

292

8×

103

2.26

52.

265

2.26

52.

264

2.39

32.

393

2.38

02.

380

2.47

52.

470

2.45

32.

450

2.52

22.

522

2.49

52.

472

2.55

12.

522

2.49

010

42.

440

2.44

02.

427

2.42

22.

555

2.55

02.

547

2.54

22.

642

2.63

72.

617

2.60

02.

689

2.68

22.

662

2.63

72.

698

2.70

12.

682

2.66

02×

104

2.98

72.

987

2.98

72.

987

3.11

43.

114

3.10

43.

097

3.20

13.

182

3.16

73.

164

3.24

13.

239

3.20

73.

189

3.25

53.

261

3.23

13.

200

4×

104

3.66

13.

661

3.66

13.

661

3.79

13.

790

3.77

33.

766

3.86

93.

863

3.85

13.

838

3.91

13.

903

3.87

83.

861

3.92

73.

891

3.87

3

Nu

(Enc

losu

res

havi

ngin

clin

edto

p)E=

0.1

E=

0.2

E=

0.3

E=

0.4

E=

0.5

6×

103

1.98

12.

101

2.17

12.

238

2.24

78×

103

2.16

12.

304

2.35

62.

432.

447

104

2.37

32.

461

2.52

12.

545

2.60

12×

104

2.92

3.00

93.

071

3.11

33.

151

4×

104

3.54

03.

668

3.73

3.79

63.

81

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Figure 8 Comparison of average Nusselt number for square, dome, rectangular, and inclined roof enclosures with different Rayleighnumbers (Pr= 1).

significant effect on the convective heat transfer com-pared to other curvatures as shown in Figure 8. More-over, dome-shaped top covers will have higher heattransfer rates than any equivalent inclined roof configu-ration. The estimated values of the mean Nusselt num-ber have been tabulated in Table 3 for different Rayleighnumbers, eccentricities, and offset values.

There is a reduction in theu-velocity in the centralpart of the domed boundary. This is due to the fact thatthe flow gets a larger surface area due to the dome shape,while passing from the top hot corner to the top cen-tral part of the enclosure and gently deflecting over thecurved top cover near the hot wall. This results in aweakening of the vorticity near the wall. The weak-ening of vorticity in the case of the inclined top roof

enclosures, however, is due to severe change in flowdirection. This weaker vorticity near the wall may notbe able to sustain a weak return motion of the fluidnear the cold wall. Therefore, due to the weakening ofu-velocity, the core flow is disturbed to give rise to asecondary vorticity in the central part of the domaineven at a moderate range of Rayleigh numbers.

CONCLUSIONS

The present numerical scheme is validated for choiceof grid as well as numerical accuracy in the predic-tion of buoyancy-driven flow within complex enclo-sures having curvature and inclined top boundary. The

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results show that dome-shaped enclosures have signifi-cant effects on flow behavior and heat transfer charac-teristics. The nature of the curvature cannot be neglectedin the prediction of the heat transfer rate and flow behav-ior. Dome-shaped enclosures have higher heat transferrates than those of equivalent inclined roof and squareenclosures. Moreover, dome-shaped enclosures havingoffset in the range of 0.25 to 0.3 have the maximumheat transfer rate for all types of conic surfaces. Fur-ther rise in offset does not effect the heat transfer ratesignificantly.

NOMENCLATURE

E offset (height of dome from top of the isothermalwall)

e eccentricityg acceleration due to gravity, m/sec2

H height of the enclosure, mLref characteristic length, mn outward flux normal to boundaryNu average Nusselt numberp dimensionless fluid pressurePr Prandtl numberRa Rayleigh numberT dimensionless fluid temperatureu, v dimensionless velocity components inx and y

direction, respectivelyx, y dimensionless coordinates

Greek Symbols

α fluid thermal diffusivity, m2/secβ coefficient of volumetric expansion, K−1

ν fluid kinematic viscosity, m2/secρ fluid density, kg/m3

γ aspect ratio

Subscript

h, c hot and cold wall, respectively

REFERENCES

[1] Ostrach, S., Natural Convection in Enclosures,ASME J. HeatTransfer, vol. 110, pp. 1175–1190, 1988.

[2] Hoogendoorn, C. J., Natural Convection in Enclosures,Proc. Eighth International Heat Transfer Conference, SanFrancisco, Hemisphere Publishing Corp., Washington, DC,vol. 1, pp. 111–120, 1986.

[3] Asako, Y., and Nikamura, H., Heat Transfer in a ParallelogramShaped Enclosure,Bull. JSME, vol. 25, pp. 1419–1427, 1982.

[4] Akinsate, V. A., and Coleman, J. A., Heat Transfer by SteadyState Free Convection in Triangular Enclosures,InternationalJ. Heat Mass Transfer, vol. 25, pp. 991–998, 1982.

[5] Campo, E. M. D., Sen, M., and Ramos, E., Analysis of LaminarNatural Convection in Triangular Enclosure,Numerical HeatTransfer, vol. 13, pp. 353–372, 1988.

[6] Moulkalled, F., and Acharya, S., Buoyancy-Induced HeatTransfer in Partially Divided Trapezoidal Cavities,NumericalHeat Transfer, Part A, vol. 32, pp. 787–810, 1997.

[7] Das, S., and Sahoo, R., Velocity-Pressure Formulation for Con-vective Flow Inside Enclosure with Top Quadratic InclinedRoof,J. Energy, Heat, and Mass Transfer, vol. 20, pp. 55–64,1998.

[8] Lewandowski, W. M., and Khubeiz, M. J., Experimental Studyof Laminar Natural Convection in Cells With Various Convexand Concave Bottoms,J. Heat Transfer, vol. 114, pp. 94–98,1992.

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Yos S. Morsiobtained a Masters and Ph.D. fromLondon University in 1983. He is a professorand leader of modeling and process simulationresearch group in the School of Science and En-gineering at Swinburne University of Technology,Australia. Most of his research work is in the areaof flow-field predictive and experimental diagnos-tics techniques. He has published a number ofpapers in this area and attended and participatedin various local and international workshops and

conferences. His main research interests are physiological fluid dynamics,microfluid handling, fluid-structure interaction, ultrasonic and laser Dopplertechniques for flow measurements, lung aerosol deposition, spray cooling,particle-laden flows turbulence mixing, and chemically-reacting flows.

Subrat Das is a lecturer in the School of Sci-ence and Engineering at Swinburne University ofTechnology, Australia. He received his BE andME from South Gujarat University, India, andsubmitted his Ph.D. thesis in 2001. His researchinterests are finite element methods, complex en-closures, natural convection, porous media flowwithin enclosures, fluid-structure interaction, andultrasonic and laser Doppler techniques for flowmeasurements. Currently, he is working on top

submergence swirling flow into molten metal bath, which is an industrialproject.

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