International Journal of Heat and Mass Transfer 54 (2011) 717–721

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International Journal of Heat and Mass Transfer

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

Technical Note

Unsteady numerical computation of combined thermally andelectromagnetically driven convection in a rectangular cavity

Xiaohui Zhang a,⇑, Mo Yang b

a Department of Thermal Energy Engineering, School of Energy, Soochow University, Suzhou, Chinab Thermal Engineering Institute, University of Shanghai for Science and Technology, Shanghai, China

a r t i c l e i n f o

Article history:Received 23 April 2010Received in revised form 6 October 2010Accepted 12 October 2010Available online 2 November 2010

Keywords:Joule heatingLow Prandtl number fluidUnsteady modelMagnetohydrodynamicNatural convection

0017-9310/$ - see front matter Crown Copyright � 2doi:10.1016/j.ijheatmasstransfer.2010.10.006

⇑ Corresponding author. Address: Department of TSchool of Energy, Soochow University, No. 1, Shizi StrChina. Tel.: +86 512 61080967; fax: +86 512 6511190

E-mail address: xhzhang@suda.edu.cn (X. Zhang).

a b s t r a c t

A series of numerical simulations of fluid flow and heat transfer based on two-dimensional unsteadymodel of MHD thermal convection have been performed. The computational domain is a rectangularcavity with an aspect ratio of 2, filled with electrically conductive fluids at different Prandtl numbers.The process medium is assumed to be subjected to DC heating by a pair of plate electrodes located atthe cavity sidewalls. The top and bottom walls are assumed to be electrically insulated. The upper bound-ary of the cavity is cooled by the atmosphere and all the other walls are kept thermally insulated. ForPr = 1 and Pr = 0.1 fluid, the simulation results show that the fluid flow and heat transfer rate becometime independent and reach steady-state conditions. On the contrary, for Pr = 0.01 fluid, it is found thatphysically realizable periodic oscillation flow evolves, significantly affecting the heat transfer. Thesetransient characteristics of velocity and temperature fields are presented graphically.

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1. Introduction

Application of Joule heating in engineering can be found invarious industrial processing, such as electric glass melting, orthe heating molten slag in electro-slag remelters. Comprehensivereview of this application is given by Sugilal et al. [1].

Physical phenomena taking place inside Joule-heated liquidpools are quite complex. Two types of body forces are present:the buoyancy force resulted from thermal gradients due to volu-metric Joule heating, and the Lorentz force given by the interactionbetween self-induced magnetic field and moving charge carriers inthe liquid. The convective behavior in the liquid pool under Jouleheating depends strongly on the interaction between these forces.

The magnetic fields produce important effects on the buoyancy-induced flow and temperature distributions of electrically conduc-tive fluids in various systems or processes. Ozoe and his co-workers[2–5] presented a series of experiment and numerical investigationson this topic, leading to some interesting results. Sarris et al. [6],employing a numerical method, studied the unsteady two-dimen-sional natural convection of an electrically conducted fluid in alaterally and volumetrically heated square cavity under the influ-ence of a magnetic field. Three-dimensional numerical simulationin cubic enclosures with both internal heat generation and an

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hermal Energy Engineering,eet, Suzhou, 215006 Jiang Su,7.

imposed uniform magnetic field was performed by Piazza and Ciofa-lo [7] to investigate the MHD free convection. In that case, the exter-nal magnetic force applied to the model, was different from the self-induced magnetic field due to the current.

Sugilal et al. [1] performed a numerical study to identify theconditions under which each of the two body forces prevailing ina Joule-heated liquid pool dominates the other. Results show thatthe combined flow results are almost identical to those of gravita-tionally driven flows when Ha2 Pr=

ffiffiffiffiffiffiRap

< 120 and the combinedflow results are almost identical to those of electromagneticallydriven flows when Ha2 Pr=

ffiffiffiffiffiffiRap

> 100. The region betweenHa2 Pr=

ffiffiffiffiffiffiRap

> 120 and Ha2 Pr=ffiffiffiffiffiffiRap

< 100 is characterized by theeffect of both electromagnetic and gravitational body forces. Thecorrelations of the heat transfer under these three types of flow re-gimes are also presented. This study is based on the steady model.

These previous experimental [3,4] and numerical [1,2,5–7]works presented useful results, but few works have been reportedabout the study of the simultaneously low Prandtl number fluidwith non-steady state model for the non-linear coupling of ther-mally and electromagnetically driven convection. The purpose ofthis paper is to clarify the differences between steady-state andunsteady computational results for Pr = 1, Pr = 0.1 and Pr = 0.01fluid.

2. Physical model and problem formulation

Consider the same system shown in Ref. [1], except that in thepresent case the height of the cavity L is aligned with the y-axis

ights reserved.

Nomenclature

E electric field intensityg gravitational accelerationHa Hartmann numberJ electric current densityk thermal conductivityL characteristic length of cavity (Cartesian coordinate in y

direction)p pressureP non-dimensional pressurePr Prandtl numberRa Rayleigh numberT temperatureu;v velocity componentsU;V dimensionless velocityW width of cavity (Cartesian coordinate in x direction)x, y Cartesian coordinatesX;Y dimensionless coordinates

Greek symbolsa thermal diffusivityb coefficient of thermal expansionh dimensionless temperaturem kinematic viscosity of the fluidq density of the fluidr electrical conductivitys dimensionless timeu electric scalar potential

Superscript0 dummy variable, dimension physical quantity

Subscript0 reference value

718 X. Zhang, M. Yang / International Journal of Heat and Mass Transfer 54 (2011) 717–721

direction and the width of the cavity W is aligned the x-axis direc-tion; here, we set W : L ¼ 2 : 1. The fluid in the 2D orthogonal cav-ity is heated by an electrode pair, which is assumed to be suitablyrepresented by two isopotential surfaces with an externally ap-plied potential difference of u0 across them. The rest of the bound-aries are assumed to be electrically insulated. In the present study,low frequency alternating current sources are considered for Jouleheating.

In the present model, the flow is simulated as a two dimen-sional phenomenon with the following assumptions: (a) the fluidis Newtonian, incompressible and the flow is laminar; (b) the effectof temperature on fluid density is expressed adequately by theBoussinesq approximation; (c) the local electrical conductivity isindependent of the thermal field.

The following equations are used to describe electric charge andelectric scalar potential, respectively [1]:r � J ¼ 0; E ¼ �ru. FromOhm’s law: J ¼ rE and consider the above assumptions, we can get

d2udX2 ¼ 0 ð1Þ

J ¼ �2dudX

ð2Þ

The self-induced magnetic flux density induced by a currentdensity J at position ðX 0;Y 0Þ can be obtained from Biot–Savart’slaw [1]:

B½X;Y� ¼ 12p

Z 1

0

Z 2

0

�ðY � Y 0Þ � J

ðX � X 0Þ2 þ ðY � Y 0Þ2dX 0dY 0 ð3Þ

ðX 0;Y 0Þ is indicated as dummy variable as it is identical to (X, Y)in the sense that both represent position in the space. Eq. (1) issolved to obtain the scalar potential distribution. Subsequently Jcan be evaluated using Eq. (2) and B½X;Y� can be evaluated usingEq. (3). After non-dimensionalization, the unsteady fluid flow andheat transfer equations developed from steady model [1] read asfollows:

@U@Xþ @V@Y¼ 0 ð4Þ

@U@sþ U

@U@Xþ V

@U@Y¼ � @P

@Xþ @

2U

@X2 þ@2U

@Y2

@V@sþ U

@V@Xþ V

@V@Y¼ � @P

@Yþ @

2V

@X2 þ@2V

@Y2 þHa2 � J � Bþ RaPr� h

ð5Þ

@h@sþ U

@h@Xþ V

@h@Y¼ 1

Pr@2h

@X2 þ@2h

@Y2

" #þ 2

PrJ � J ð6Þ

The following dimensionless variables are employed:

ðX;YÞ ¼ ðx; yÞ=L; ðU;VÞ ¼ ðu;vÞL=m; s ¼ tðL2=mÞ;h ¼ 2ðT � T0ÞkW2

=ru20L2; J ¼ J0W=ru0L;

B ¼ B0W=rlu0L; P ¼ pL2=qm2; u ¼ u0=u0

The non-dimensional parameters that appear in the equations are:

Ha ¼ J0L2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil=q0m2

q; Ra ¼ L3gbDT0=va

The Rayleigh number is defined in the conventional manner ex-cept that the T0 is re-defined in terms of the internal heat genera-tion: ru2

0. Here, DT0 ¼ ðru20Þ=½2k W

L

� �2�. J0 is a reference currentdensity for potential u0, i.e., J0 ¼ ru0. q0 is the fluid density corre-sponding to the reference temperature T0. The Hartmann numbergoverns the magnetohydrodynamic flows which are determinedfrom the additional electric current induction.

The electrical boundary conductions for the isopotential sur-faces are as follows: X ¼ 0; 0 6 Y 6 1;u ¼ 1; X ¼ 2; 0 6Y 6 1;u ¼ 0. A no-slip condition is imposed on all the boundariesfor the velocities: U ¼ 0;V ¼ 0. Conditions for the dimensionlesstemperature are h ¼ 0 for the upper isothermal surface and@h=@X ¼ 0; @h=@Y ¼ 0 for the rest of the boundaries. The expres-sion for the Nusselt number in Ref. [1] is adopted.

Eqs. (4)–(6) are solved by using a finite volume method (FVM)on a staggered grid system [8]. The QUICK discretization scheme[9] is adopted to deal with convection and diffusion terms. TheSIMPLE algorithm [8] is used to treat the coupling of the momen-tum and mass continuity equations. The chosen convergence crite-rion is such that the maximum residual of all the governingequations shall be less than 10�6.

We performed a grid sensitivity analysis by setting Rayleighnumber to Ra = 104 and Ra = 105 and testing six different uniformgrids of 30 � 15, 60 � 30, 90 � 45, 120 � 60, 180 � 90 and240 � 120 elements, respectively, with a fixed Hartmann numberof Ha = 1.414 � 103. The average Nusselt number is chosen as thereference quantity for the analysis. The results are summarizedin Table 1, reporting relative errors between solutions at successivegrid refinement steps. Overall, the grid 120 � 60 is chosen as thebest compromise between spatial grid convergence and computa-tional weight.

Table 1Effect of grid numbers on the average Nusselt number for Pr = 1.

Grid 30 � 15 60 � 30 90 � 45 120 � 60 180 � 90 240 � 120

Ra = 105

Ha = 1.414 � 10310.785 9.6637 9.4033 9.2894 9.1855 9.1372

Relative error 0.11603 0.02769 0.01226 0.01131 0.00528

Ra = 104

Ha = 1.414 � 1039.9539 7.9471 7.2258 7.0115 6.8708 6.7929

Relative error 0.25252 0.09982 0.03056 0.02477 0.01149

X. Zhang, M. Yang / International Journal of Heat and Mass Transfer 54 (2011) 717–721 719

3. Results and discussion

All the computations are started from a zero-field initializationand the time step is chosen to be uniform, Ds ¼ 10�3. Throughoutthe simulations, time histories of the dimensionless temperatureand velocity components are recorded at a monitoring pointX;Y ¼ ð0:25; 0:483Þ, together with the time evolution of the aver-age Nusselt number. Snapshots of the entire velocity and temper-ature fields are also collected at selected time steps.

Depending on the interaction between buoyancy force and self-induced electromagnetic force, for Pr = 1 and 0.1, thermally drivenand electromagnetically-driven stable flows can be observed. Allthe results are steady-state and are in good agreement with previousresults [1]. For the sake of brevity, we do not discuss them here.

Fig. 1. Periodic oscillations for Ha = 10, Ra = 10000. (a) Dimensionless temperature at mopoint ðX;Y ¼ 0:25;0:483Þ, (c) y-component of velocity (i.e., V) at monitoring point ðX;Y

Extensive computations are then carried out by varying theHartmann number and Rayleigh number for low Prandtl fluid,Pr = 0.01, In this case, the results differ from those predicted witha steady-state model [1]. In this section, we will deal with the mostrepresentative case, Ha = 10 and Ra = 10000.

The initial transient for the case investigated for Ha = 10 andRa = 10000 is illustrated in more detail in Fig. 1a–c reports the timedependent behavior of dimensionless temperature, velocity at themonitoring point X;Y ¼ 0:25;0:483 of the cavity. The time of theaverage Nusselt number is depicted in Fig. 1d. The first noteworthyfeature of Fig. 1 is that, after a few (�0.5) time units, a transition tooscillatory flow and heat transfer rate occurs. The quantities atlocation X;Y ¼ 0:25;0:483 exhibit a clearly periodic behavior, theperiod is about 0.024 time unit.

nitoring point ðX;Y ¼ 0:25;0:483Þ, (b) x-component of velocity (i.e., U) at monitoring¼ 0:25;0:483Þ and (d) the average Nusselt number Nu.

Fig. 2. A series views of instantaneous streamlines (left panel) and isotherms (right panel) for Pr = 0.01, Ha = 10 and Ra = 10000. (a) s ¼ 0:73953, (b) s ¼ 0:751724,(c) s ¼ 0:76372 and (d) s ¼ 0:77622.

720 X. Zhang, M. Yang / International Journal of Heat and Mass Transfer 54 (2011) 717–721

Stream lines and isotherms are plotted in Fig. 2 for approxi-mately equal time interval in one periodic circle, in order toanalyze the macroscopic nature of the velocity fields and temper-ature fields. We observe that the stream lines and isotherms arenot centro-symmetric at any point in time. The oscillation consistsof counter-rotating vortex near the left of the cavity and clockwiserotating vertex near the right of cavity, where it is shown that thevortices are squeezed each other as time goes on. As can seen fromFig. 2 (left column), for the moment of s ¼ 0:73953, a small coun-ter-rotating vortex remains on the left-hand side of the cavitywhile a large clockwise rotating vertex locates on the right-handof the cavity, then, around s ¼ 0:751724, the counter-rotating vor-tex remained on the left-hand side of the cavity become largewhile the clockwise rotating vertex located on the right-hand ofthe cavity is small. The flow patterns demonstrate the oscillatorymotion of the convective rolls from the left to the right of the cav-ity, and backwards. The corresponding temperature fields are sim-ilar to that shown in Fig. 2 (right column).

It can be seen that the natural convection is governed by grav-itational body force, while the effect of electromagnetic body forcecan be neglected. A periodic oscillation flow has not been predictedso far in previous works [1] for the present case. The oscillationreason is that if the Pr is low, the thermal gradients are dampedmore quickly than the velocities, this disequilibrium is source ofinstabilities. Fundamental studies on oscillatory convection are ex-pected to clarify the general mechanism of oscillatory convectionand suggest effective way for its control. These predictions stillneed be tested with experimental measurements.

4. Conclusion

Numerical simulations of unsteady combined thermal and MHDconvection have been performed.

For Pr = 1 and 0.1 fluid, the predicted results demonstrate thatthe fluid flow and heat transfer rate become time independent,in agreement with previous steady-state results [1]. It is shownthat the solutions for Pr = 0.01 are instead time dependent. A peri-odic oscillation of an asymmetric 2-cell flow is found for Ha = 10and Ra = 10000. The convection rolls oscillate by the effect of thebuoyancy force, showing a swaying motion across the xy plane.The isotherms are not centro-symmetric at any point in time,and the Nusselt number oscillates accordingly. When the computa-tion is based on the steady state model, this intrinsic oscillationcould not be depicted. It is concluded that the unsteady modeladopted here provides reasonable results in the low Prandtl num-ber range.

Acknowledgments

This work was supported by the Jiangsu Planned Projects forPostdoctoral Research Funds of China (0901017B). This work wascarried out as part of research project awarded by National NaturalScience Foundation of China (50876067).

The authors are indebted to Dr. Sugilal,Bhabha Atomic ResearchCentre, India, for his discussion. The authors would like to thankDr. Angeli, University of Modena and Reggio Emilia, Italy, for hiscontributions.

X. Zhang, M. Yang / International Journal of Heat and Mass Transfer 54 (2011) 717–721 721

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