combination of the fictitious domain method and the sharp interface method for direct numerical...

International Journal of Heat and Mass Transfer 55 (2012) 6775–6785

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International Journal of Heat and Mass Transfer

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Combination of the fictitious domain method and the sharp interface methodfor direct numerical simulation of particulate flows with heat transfer

Xueming Shao, Yang Shi, Zhaosheng Yu ⇑State Key Laboratory of Fluid Power Transmission and Control, Department of Mechanics, Zhejiang University, 310027 Hangzhou, China

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

Article history:Received 6 May 2011Received in revised form 9 February 2012Accepted 15 June 2012Available online 21 July 2012

Keywords:Direct numerical simulationFictitious domain methodSharp interface methodHeat transferNon-isothermal

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⇑ Corresponding author. Tel.: +86 571 87952698; faE-mail address: [email protected] (Z. Yu).

In this paper, the fictitious domain (FD) method and the sharp interface (SI) method are combined for thedirect numerical simulations of particulate flows with heat transfer in three dimensions. The flow fieldand the motion of particles are solved with the FD method. The temperature field is solved in both fluidand solid media with the SI method. The accuracy of the proposed FD/SI method is validated via twoproblems: the natural convection in a twodimensional cavity with fixed solid particles, and the flowover a cold sphere. The method is then applied to the natural convection in a three dimensional cavitywith a fixed sphere, the motion of a spherical particle in a non-isothermal fluid, and the rising of sphericalcatalyst particles in an enclosure. The effects of the thermal conductivity ratio are examined in the firstand third problems, respectively, and the significant effects of the thermal expansion coefficient ratio onthe particle motion are demonstrated in the second problem.

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1. Introduction

The heat transfer between solid particles and fluid mediumexists widely in both nature and industrial settings. Besides thetraditional two-fluids and point-particle models [1], the inter-face-resolved method (or so-called direct numerical simulationmethod) has been developed to deal with the particulate flowswith heat transfer. The boundary-fitted methods have been widelyused for the direct numerical simulation of the heat transfer be-tween the fluids and the fixed particles [2–9]. There are limitedworks on the application of the non-boundary-fitted methods tothe heat transfer problems. Kim and Choi [10], Pacheco et al.[11], Pan [12] and Wang et al. [13] developed the immersed-boundary method for the heat transfer in complex geometries,respectively, where no particle motion was considered. Ha andhis coworkers [14,15] applied the immersed-boundary method tothe natural convection in an enclosure with a fixed particle.Regarding the works on the heat transfer between the fluid andmoving particles, Gan et al. [16,17] first numerically simulatedthe sedimentation of solid particles with thermal convection usingthe ALE finite-element method, in which a fixed temperature onthe particle boundary was assumed. Yu et al. [18] extended the dis-tributed-Lagrange-multiplier based fictitious-domain (DLM/FD)method [19–21] to handle the particulate flows with heat transferwhere the thermal conduction inside the particle boundary andthermal convection in the fluids are coupled with the constraint

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of continuous temperature and heat flux across the particle bound-ary. Feng and Michaelides [22,23] extended the immersed bound-ary (IB) method for the particulate flows with heat transfer wherethe isothermal condition was assumed in the particle domain,therefore, the method is particularly suited to the case of large par-ticle thermal diffusivity; it was shown that good agreement be-tween their IB results [23] and the DLM/FD results of Yu et al.[18] can be achieved at the particle-fluid thermal conduction coef-ficient ratio down to 5. Mandujano and Rechtman [24] extendedthe lattice Boltzmann method to simulate the particle motion ina non-isothermal fluid. The aforementioned methods for the freelymoving particles were implemented in two dimensions, althoughthe extension to the three-dimensional case is straightforward.Dan and Wachs [25] extended the DLM/FD method to the three-dimensional heat transfer problems, but only presented thenumerical examples in which the temperature on the particleboundary was fixed. Wachs [26] recently implemented a three-dimensional DLM/FD method in which the temperature on the par-ticle boundary was varied according to the heat transfer betweenthe particles and the fluids, and the isothermal condition was as-sumed over the particle domain.

The primary aim of the present study is to propose a new directnumerical simulation (DNS) method for the particulate flows withheat transfer by combining the direct-forcing fictitious domain(DF/FD) method for the fluid-particle motion and the sharp inter-face (SI) method for the heat transfer. The DF/FD method is animproved version of our previous DLM/FD code, and is more effi-cient than the DLM/FD method since the Lagrange multiplier andthe particle velocities are obtained without iteration [27]. The

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P

P∂

Ω

Γ

Fig. 1. Schematic diagram of the fictitious domain method.

6776 X. Shao et al. / International Journal of Heat and Mass Transfer 55 (2012) 6775–6785

DF/FD method has been demonstrated to be a stable, efficient andaccurate method for the solution of the fluid-solid motion [27,28].The sharp interface (SI) method (also called ghost-fluid method)[29] was developed to handle the Poisson equation with jumpcoefficients across the interface. Like the FD method, the SI methodis one type of fixed Cartesian grid methods for the problem withinner interfaces. Its main feature is that the jump condition onthe interface is used to modify the discretization of the differentialoperators on the Cartesian grids in the immediate vicinity of theinterface, and its advantage is that the interface is kept sharp,and the jump condition on the interface is accurately captured[30]. The SI method has been applied to the droplet [30,31] anddielectrophoresis problems [32,33]. We here apply it to the conju-gate heat transfer problem. The advantage of the SI method overthe FD method is that the former appeared more accurate in caseof large coefficient jump across the interface [32], presumablydue to the fact that the strong discontinuity on the interface is bet-ter captured by the SI method. The SI method is also computation-ally efficient as a result of the use of the multi-grid solver; ournumerical tests showed that only 2–3 iterations are required ifthe thermal convection predominates over the thermal diffusion,irrespective of the problem scale. The FD method proposed by Yuet al. [18] is normally more efficient for the unsteady thermal prob-lem since no iteration was required, and is easier to implement forthe non-spherical particles. Overall, we think that the SI method isbetter suited to the case of large thermal diffusion coefficient ratioor steady thermal conduction problem, and the FD method to theunsteady heat transfer problem with a relatively small thermalconductivity ratio. In the present study, we choose the SI schemefor the solution of the temperature field. The secondary aim of thiswork is to provide some new insights into the mechanisms in theheat transfer between the fluid and particles via the applications ofthe proposed method.

The rest of this paper is organized as follows. In next section, theDF/FD and SI methods are described, respectively. In Section 3, wefirst verify the code of the FD/SI method via two problems: the nat-ural convection in a two dimensional cavity with solid particles,and the flow over a cold sphere. Then, the method is applied toother three problems: the natural convection in a three-dimen-sional cavity with a fix sphere, the motion of a spherical particlein a box filled with a non-isothermal fluid, and the rising of spher-ical catalyst particles in an enclosure. Concluding remarks are gi-ven in the final section.

2. Numerical model

2.1. The fictitious domain method for the flow field

The spirit of the fictitious domain method is that the interior ofparticles is filled with the fluid and the inner fictitious fluid is en-forced to satisfy the rigid body motion constraint via a pseudo-body force [19]. We let P represent the particle domain, oP itsboundary, and X the entire computational domain, as shown inFig. 1.

To simplify reasonably the computation, the Boussinesqapproximation is used to deal with the effect of temperature vari-ation on the flow field. The change of temperature is assumed notto influence the properties of fluid medium, except for the densityin the gravitational term which has the form:

qf ¼ qf0½1� bf ðTf � T0Þ� ð1Þ

where qf0represents the reference density of the fluid at the refer-

ence temperature T0, and bf is the fluid thermal expansion coeffi-cient. The same approximation is also applied to the solid particle,

considering that the particle density and the buoyant force are ex-pected to be affected by the temperature variation, like the fluid.

The characteristic scales used for the non-dimensionlizationscheme are: Lc for length, Uc for velocity, Lc/Uc for time, qf 0U2

c forpressure p, and qf 0U2

c=Lc for the Lagrange multiplier k. Then, thedimensionless governing equations for the flow field and the par-ticle motion can be written as follows [18]:

@u@tþ u � ru ¼ r

2uRe�rp� Gr

Re2 Tfggþ k in X ð2Þ

u ¼ Uþxs � r in P ð3Þ

r � u ¼ 0 in X ð4Þ

ðqr � 1ÞV�pdUdt� Fr

gg

� �¼ �

Zp

kdx�Z

pðqrbr � 1Þ Gr

Re2 Tfgg

dx ð5Þ

ðqr � 1ÞdðJ� �xsÞdt

¼ �Z

pr� kdx�

Zp

r� ðqrbr � 1Þ Gr

Re2 Tfgg

� �dx

ð6Þ

Here u is the fluid velocity. U and xs are the particle translationaland angular velocities, respectively. qr represents the solid-fluiddensity ratio defined by qr = qs0/qf0, qs0 being the solid density atthe reference temperature. br denotes the solid-fluid thermal expan-sion coefficient ratio. r is the position vector with respect to the par-ticle centre. V�p is the dimensionless particle volume defined byV�p ¼ M=ðqsL

3c Þ, here M being the particle mass. J� is the dimension-

less moment of inertia tensor defined by J� ¼ J=ðqsL5c Þ. Re denotes

the Reynolds number, defined by Re = qf0UcLc/l,l being the fluid vis-cosity, Fr represents the Froude number, defined by Fr ¼ gLc=U2

c . Gr isthe Grashof number, defined by Gr ¼ q2

f 0bf L3c gðT2 � T1Þ=l2, where T1

and T2 are two reference temperatures. The dimensionless tempera-ture is normally defined by T ¼ ðT � T1Þ=ðT2 � T1Þ .

The reader is referred to [27] for the detailed description of thenumerical schemes for the solution of Eqs. (2)–(6). The tempera-ture term in Eq. (2) is discretized in time with the second-orderAdam–Bashforth scheme, as in [18].

2.2. The sharp interface method for the temperature field

The dimensionless governing equation for the temperaturefield is

X. Shao et al. / International Journal of Heat and Mass Transfer 55 (2012) 6775–6785 6777

q�c�p@T@tþ u � rT

!¼ r � k�

PerT

� �þ Q ; ð7Þ

with the following jump conditions on the solid–fluid interface:

½T�I ¼ 0 on @P; ð8Þ

k�@T@n

" #I

¼ 0 on @P: ð9Þ

Here q⁄, c�p and k⁄ are the dimensionless density, heat capacity andheat conductivity in the individual medium, with the reference ofqf0, cpf and kf, respectively. The sub-symbol ‘f’ represents the fluidmedium. From our definition, k⁄ is unity in the fluid medium, andequals the solid-fluid heat conductivity ratio kr (defined by kr = ks/kf) in the particle domain. [ ]I represents the jump across the inter-face. Pe denotes the Peclet number defined by Pe = qf0cpfUcLc/kf. Qrepresents the dimensionless heat source and is set to be zero, un-less otherwise specified. Note that Pe = RePr, where Pr is the Prandtlnumber defined by Pr = lcpf/kf .

A second-order semi-implicit Adam–Bashforth/Crank–Nicolsonscheme is employed to discretize Eq. (7) in time:

q�c�pTnþ1 � Tn

Dt

!¼ 1

2r � k�

PerTnþ1

� �þr � k�

PerTn

� ��

� 12

3ðu � rTÞn � u � rT� �n�1

h iþ Q ð10Þ

The diffusion term in the above equation involves the second-ordertemperature derivative and its singularity arising from the coeffi-cient jump across the interface is expected to be more severe thanthe convection term. Therefore, we discretize the diffusion termwith the sharp interface method, and for simplicity, the convectionterm is discretized with the normal central difference scheme, as inthe previous FD method [18]. For the sharp interface method, thespatial discretization can be performed dimension-by-dimension.Hence, we describe the principle of the sharp interface method inthe one-dimensional case in the following, as in [29,30].

2.2.1. One dimensional caseBy defining b = k⁄/Pe, the one dimensional form of the Laplace

term on the right hand of Eq. (7) can be written as (bTx)x. As shownin Fig. 2, we assume that an interior interface C lies between thegrid points of i and i+1, which divides the computational domaininto two parts X� and X+.

We consider general jump conditions:

½T�I ¼ TþI � T�I ¼ aI; ð11Þ

½bTx�I ¼ ðbTxÞþI � ðbTxÞ�I ¼ bI; ð12Þ

where T�I and TþI are the temperatures on two sides of the interface.

i-1 i

-

xθΔ 1(

T

T

I-

Fig. 2. Illustration of the interface lying between the gri

When the Laplace operator (bTx)x is discretized for grid point i,the value on the interface is used instead of that at grid point i+1,which is on the other side of the interface. T�I can be obtained fromthe jump conditions Eq. (11) and (12):

T�I ¼bþðTiþ1 � aIÞhþ b�Tið1� hÞ � bIhð1� hÞDx

bþhþ b�ð1� hÞð13Þ

We can finally write the discretization of the Laplace operator (bTx)x

at grid point i as

ðbTxÞx ¼ b�T�I � ThDx

� b�Ti � Ti�1

Dx

� �� Dx

¼ bTiþ1 � Ti

Dx

� �� b�

Ti � Ti�1

Dx

� �� Dx� baI

ðDxÞ2� bbIð1� hÞ

bþDx;

ð14Þ

where b ¼ bþb�=½bþhþ b�ð1� hÞ�. The expression of the discretiza-tion at grid point i+1 can be obtained similarly. It should be notedthat the above discretization of the Laplace operator is first-orderaccurate for the points in the immediate vicinity of the interface[30].

2.2.2. Three dimensional caseFor the three dimensional case, as long as the jump condition in

each Cartesian coordinate direction is given, the spatial discretiza-tion of the Laplace operator can be performed dimension-by-dimension, in the form of Eq. (14). The jump condition involvingthe derivative on the interface is normally given in the normaland tangential directions rather than the Cartesian coordinatedirections. Thus, we need to transform the jump values from theformer frame to the Cartesian frame. In the present study, onlyspherical particles are considered. The normal and tangentialdirections for a point on the spherical surface can be determinedfrom the spherical coordinates (a, u), here a denoting the anglefor the latitude ranging from ( - p/2, p/2), and u for the longituderanging from (0, 2p). The spherical coordinate system is relatedto the Cartesian system via

ðx; y; zÞ ¼ ðr cos a cos u; r cos a sinu; r sinaÞ: ð15Þ

One then can find the following formulas to compute the gradient ofT in the x-, y- and z- directions Tx, Ty and Tz from Tn, Tt1 and Tt2 ,

Tx ¼ Tn cos a cos uþ Tt1 sina cos uþ Tt2 sinu ð16Þ

Ty ¼ Tn cos a sinuþ Tt1 sina sin u� Tt2 cos u ð17Þ

Tz ¼ Tn sina� Tt1 cos a ð18Þ

Here Tn represents the gradient of T in the direction of outward nor-mal vector. Tt1 ½bTx� ¼ ½bTn� cosa cos uþ ½bTt1 � sin a cosuþ ½bTt2 �sinu and Tt2 are the gradient values in two tangential directions,

i+1 i+2

+

) xθ− Δ

I+

d points of i and i+1 in the sharp interface method.


with t1 pointing to the direction in which a reduces, and t2 thedirection in which u reduces.

Multiplying (16)–(18) by the coefficient b, one can obtain:

½bTx� ¼ ½bTn� cos a sinuþ ½bTt1 � sin a sin u� ½bTt2 � sin u ð19Þ

½bTy� ¼ ½bTn� cos a sinuþ ½bTt1 � sin a sin u� ½bTt2 � cos u ð20Þ

½bTz� ¼ ½bTn� sina� ½bTt1 � cos a ð21Þ

For the problem studied, [T]C =0, [bTn]C =0, and [Tt]C = 0.[bTt]C = [b]CTtwhich is not zero since the coefficient is discontinu-ous on the interface. Because the tangential derivative Tt is notknown a priori, we use the value at the previous time level. To com-pute Tt at one point on the spherical surface, we first calculate T ontwo adjacent points along the longitude or latitude direction, andthen determine Tt on this point with the central-difference scheme.The resulting algebraic equation is solved with the multi-grid meth-od [34], and excellent convergence rate has been observed in ournumerical tests.

3. Numerical examples

3.1. Natural convection in two dimensions

The first validation problem is the natural convection in a twodimensional cavity with four solid particles. The schematic dia-gram is depicted in Fig. 3(a). The adiabatic boundary condition isimposed on the bottom and top walls of the cavity. On the leftand right boundaries, the dimensionless temperatures are fixedto be T1 = 1 and T2 = 0, respectively. In our computations, kr = 1,Pr=1 and Gr=1221,300. The steady temperature field obtained withthe grid number of 128� 128 is shown in Fig. 3(b).

The Nusselt number is defined as Nu =rCT/rbT, in whichrbT = (T2 - T1)/(x2 - x1), and rCT ¼ 1

y2�y1

R y2y1ð@T@x Þx¼0dy. Our values of

Nu are 9.28, 9.35 and 9.36 for the grid number of 128 � 128,256 � 256 and 512 � 512, respectively, and compare favorablywith that of Braga and de Lemos [9] who obtained Nu = 9.4945.

3.2. Flow over a cold sphere

The second test problem is the thermal evolution of a fixed coldsphere in a uniform flow, which was studied by Balachandar andHa [4]. Fig. 4 shows the schematic diagram of the problem. Theno-slip condition is imposed on the sphere surface and the uniform

0 0.5 1

1

0.5T1 T2

adiabatic

adiabatic

X

Y

(a)

Fig. 3. (a) Schematic diagram of the natural convection in a two dimensio

flow conditions are imposed on all outer boundaries. Natural con-vection is not considered in the problem. We choose the particlediameter d as the characteristic length and the velocity of themainstream as the characteristic velocity. The fluid temperatureat inlet and the sphere temperature at the initial time are takenas two reference temperatures. The adiabatic conditions are im-posed on the other boundaries. The size of the computational do-main is 12 � 6 � 6, and the sphere center is fixed at (4, 3, 3). Thenumber of grids is 256 � 128 � 128, and the time step is 0.01.The Reynolds number is 50. We set c = (qcp)s/(qcp)f = 10, andPr = 0.7.

When the diffusion time scale inside the sphere is much shorterthan the convective time scale of the flow outside, the sphere canbe considered to be isothermal. This assumption was used in [4] toavoid solving the temperature equation in the interior of thesphere. By employing the FD/SI method, we solve the temperatureequation in both particle exterior and interior domains. To ensurethe isothermal condition inside the sphere, a large thermal diffu-sion coefficient ratio kr is required in our simulations. We fistexamine the effects of kr in order to choose a reasonable value.

The variations of the average temperatures inside the spherewith time for different kr are shown in Fig. 5. The sphere is heatedby the ambient hot fluid and its temperature rises till it reaches theone of the mainstream. Fig. 5 shows that the sphere’s temperaturerises faster for a larger kr. The rates are roughly the same forkr = 100 and kr = 1000, indicating that the sphere can be consideredisothermal for kr P 100. We compare the results at kr = 1000 tothose of Balachandar and Ha [4] in Fig. 6. The agreement is favor-ably good. The discrepancy may be due to the following two rea-sons. First, the computational domain of Balachandar and Ha ismuch larger. Because we employ homogeneous mesh and conse-quently it is computationally prohibitive to adopt a very large size.We therefore conduct the simulation with a smaller size of6 � 3 � 3 to examine the effect of the computational domain size.Fig. 5 shows that our results for a larger domain size become closerto those of Balachandar and Ha. Second, the heat transfer betweenthe sphere and the fluid is directly computed in our simulation, butwas treated with a model by Balachandar and Ha, which mightcause some discrepancy.

3.3. Natural convection in a three-dimensional cavity

We consider the natural convection in a three-dimensional cu-bic cavity with a fixed sphere, whose schematic diagram is shown

(b)

nal cavity with four solid particles, and (b) isotherms at steady-state.

0 12

6

4 8

1T∞ =0 0pT =

xo

z

y

Fig. 4. Schematic diagram for the flow over a cold sphere.

Fig. 5. Variation of the average temperature inside the sphere in a uniform flowwith time for different thermal conductivity ratios.

Fig. 6. Comparison of the results on the temperature variation of the sphere in auniform flow.

0 1

1

T1 T2

xo

z

y

0.5

1

0.5

0.5

Fig. 7. Schematic diagram of the problem on the natural convection in a cubiccavity with a spherical particle.

Fig. 8. Comparison of the Nusselt number as a function of the thermal conductivityratio in the absence of flow obtained from our computations and the analyticalsolution.


in Fig. 7. The edge length of the cavity is taken as the characteristiclength. The sphere is fixed at the center of the cavity. The temper-atures on the left and right boundaries are fixed to be T1 and T2,respectively. The adiabatic conditions are imposed on the remain-

ing four boundaries. The dimensionless temperature is defined byT ¼ ðT � T2Þ=ðT1 � T2Þ. The number of grids is 64 � 64 � 64.

The average Nusselt number is defined as

Nu ¼ 1ðz2 � z1Þ � ðy2 � y1Þ

Z z2

z1

Z y2

y1

@T@x

� �x¼0

dydz

" #,T2 � T1

x2 � x1

� �:

ð22Þ

The average Nusselt number is actually the relative effective ther-mal conductivity of the mixture in the cavity. If there is no flow,it has an analytical solution in the dilute limit [35]:

Nu ¼ 1þ 3ðkr � 1Þ=ðkr þ 2Þ/; ð23Þ

where / is the solid volume fraction. We first make a validation testby comparing our results on the Nusselt number without flow tothe analytical solution. The comparison is shown in Fig. 8. We seethat our results for both particle radii of a = 0.125 and a = 0.2 agreewell with the analytical solutions. The small discrepancy at a = 0.2and larger kr can be explained by the effect of the finite size ofthe computational domain. The analytical solution is valid for thedilute limit, whereas the edge length of our cavity is only 2.5 parti-cle diameters and the size effect is expected more pronounced forlarger kr.

Fig. 9. Nusselt number as a function of Grashaf number for a 3D cavity with aspherical particle.


We now investigate the natural convection problem for the caseof a = 0.2. We set Pr = 0.7. The gravity points to the negative z-axisdirection. The results on the Nusselt number for Gr 6 8 � 104 arepresented in Fig. 9. It is not surprising that the Nusselt number in-creases with Gr.

The results on the Nusselt number as a function of kr forGr = 10,000 and 80,000 are plotted in Fig. 10. We have seen thatkr has a positive effect on the Nusselt number without the flowin Fig. 8. By contrast, it is interesting to observe that the Nusseltnumber decreases with increasing kr in the presence of the naturalconvection for both Grashaf numbers, although the effect of kr isinsignificant; the Nusselt number changes only around 3% forGr = 10,000 and 2% for Gr = 80,000, as kr increases from 0.01 to100. In the following, we try to elucidate the mechanism why kr

plays such a negative role in the heat transfer.From the temperature governing equation, one can derive [18]:

Nu ¼ 1VrbTkf

ZX�qxdV þ

ZX

xqcpdTdt

dV� �

; ð24Þ

where V is the volume of the box, qx denotes the heat flux in the x-axis direction (i.e. the direction of the bulk temperature gradient),and rbT represents the bulk temperature gradient, namely,rbT ¼ T2�T1

x2�x1.From Eq. (23), the Nusselt number depends on the ther-

mal diffusion (the first term on the right-hand side) and convection(the second term) in the cavity. As mentioned earlier, the Nusseltnumber is the relative effective thermal conductivity of the mixture.Yu et al. [18] defined the first term as the diffusion-related thermal

Fig. 10. Nusselt number as a function of the thermal cond

conductivity kd and the second term as the convection-inducedthermal conductivity kc:

kd ¼1

VrbTkf

ZX�qxdV ¼ 1þ c/; ð25Þ

kc ¼1

VrbTkf

ZX

xqcpdTdt

dV : ð26Þ

In Eq. (24),

c ¼ ðkr � 1ÞV/rbT

Z@p

nxTds; ð27Þ

where nx denotes the x-component of the outward unit normal vec-tor on the particle surface.

We now examine whether the thermal conductivity ratio kr af-fects the Nusselt number mainly through the thermal diffusion orconvection. Figs. 11 and 12 show respectively the diffusion-relatedand convection-related thermal conductivities as a function of kr

for Gr = 10,000 and Gr = 80,000. From Fig. 11, kd increases with in-creasing kr for Gr = 10,000, whereas it decreases with increasing kr

for Gr = 80,000. Fig. 12 shows that kc decreases with kr for bothGr = 10,000 and Gr = 80,000. For both cases, the effects of kr on kc

are much more significant than those of kr on kd, indicating thatkr affects adversely the Nusselt number mainly through the ther-mal convection effect.

The isotherms at steady-state in the symmetric plane forGr = 10,000 and Gr = 80,000 at kr = 0.1 are compared in Fig. 13.The direction of temperature gradient inside the particle boundarywould be exactly the same as the bulk one if the natural convectionis absent. The natural convection leads to the deviation of the di-rection of the temperature gradient inside the particle boundary.The deviation angle is less than 90� for Gr = 10,000, and by contrastlarger than 90� for Gr = 80,000, implying that the direction of thetemperature gradient inside the particle boundary becomes oppo-site to the bulk one for Gr = 80,000. This explains the opposite ef-fects of kr on kd for Gr = 10,000 and Gr = 80,000, as observed inFig. 11.

The isotherms at steady-state in the symmetric plane forkr = 0.01 and kr = 100 at Gr = 80,000 are compared in Fig. 14. Asmaller kr leads to larger temperature gradient and thereby moreisotherms in the particle domain. As a result, the temperature atthe same position below the particle is lower for a smaller kr,and consequently the isotherm of the same value is closer to thehot wall for a smaller kr (see the isotherm of 0.3 in Fig. 14, for ex-ample), which is expected to result in more crowded isothermsnear the lower half of the hot wall and thereby a larger Nusseltnumber.

uctivity ratio for (a) Gr = 10,000 and (b) Gr = 80,000.

Fig. 11. Diffusion-related thermal conductivity as a function of the thermal conductivity ratio for (a) Gr = 10,000 and (b) Gr = 80,000.

Fig. 12. Covection-related thermal conductivity as a function of the thermal conductivity ratio for (a) Gr = 10,000 and (b) Gr = 80,000.

Fig. 13. Isotherms at steady-state in the symmetric plane for kr = 0.1, and (a) Gr = 10,000; (b) Gr = 80,000.


3.4. Motion of a spherical particle in a box: effect of the thermalexpansion coefficient ratio

The motion of a spherical particle in a box filled with a non-iso-thermal fluid is examined, whose schematic diagram is depicted inFig. 15. The temperatures on the top and bottom boundaries arefixed, whose dimensionless values are 0 and 1, respectively. Allside walls are insulated. We take the particle diameter as the char-acteristic length, and Uc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8a3 ðqr � 1Þg

qas the characteristic veloc-

ity, following Yu et al. [27]. The dimensionless size of the box is 4� 4� 8, and the number of grids is 64 � 64 � 128. The time step is

0.005. In our simulations, qr = 1.1 , Re = 40, cprqr = 0.1, kr = 5,Pr = 0.7, Gr = 4000. The gravity points to the negative z-axis direc-tion. The particle is released at the center of the box.

Fig. 16 shows the evolution of the height of the particle for dif-ferent solid-fluid thermal expansion coefficient ratios br.

Zero value of br means that the particle density does not changewith the temperature. In this case, the particle settles under grav-ity, since its density is larger than the fluid one and the naturalconvection does not occur at the given parameters. A bigger br

means more reduction in the solid density and thereby a largerbuoyant force on the particle. The settling velocity of the particle

Fig. 14. Isotherms at steady-state in the symmetric plane for Gr = 80,000, and (a) kr = 0.01; (b) kr = 100.

-2 2-4

4

X

YZ

T2=0

T1=12

Fig. 15. Schematic diagram of the problem on the motion of a sphere in a box.

Fig. 16. Evolution of the height of the particle for different thermal expansioncoefficient ratios.

Fig. 17. Evolution of the vertical velocity of a single spherical catalyst particle in anenclosure initially released at the center for different thermal conduction ratios.(Re, qr, Gr, br, cpr, Pr, Qs) = (40,1.1,1000,0, 1,0.7,1).


becomes smaller, as br increases. When br reaches a critical value,the particle suspends in the fluid medium. For the parametersstudied, the critical value is about 1.45, as seen in Fig. 16. As br in-creases beyond the critical value, the particle rises upwards insteadof sedimentation. This problem demonstrates that the effects of thetemperature on the particle density need to be taken account for inorder to accurately simulate the motion of particles in a non-iso-thermal fluid.

3.5. Rising of spherical catalyst particles in an enclosure

Finally we apply our method to simulate the rising of a singlespherical catalyst particle from the center of an enclosure and500 particles from the bottom, respectively, following Wachs[26]. The temperature on the boundaries of the enclosure is fixed,and the heat source is assumed to homogeneous over the particledomain and time. For the single particle case, the dimensionlesssize of the enclosure is 8 � 8 � 16, and the parameters are set tobe (Re, qr, Gr, br, cpr, Pr, Qs) =(40,1.1,1000,0, 1,0.7,1), with kr rang-ing from 0.1 to 50. The mesh size is h = d/16, leading to the gridnumber of 128 � 128 � 256. The time step is 0.001. Fig. 17 showsthe time evolution of the vertical velocity of the spherical catalystparticle at different kr. We see that the particle with a larger ther-mal conductivity rises faster. The reason is that the enhancementin kr causes more rapid heat transfer from the solid bulk regionto the boundary and then to the fluids, resulting stronger naturalconvection which is responsible for the rising of the particle, aspreviously explained by Yu et al. [18]. From Fig. 17, the effect ofthe thermal conductivity becomes insignificant for kr > 10, andour result for kr = 50 is in remarkably good agreement with that

of Wachs [26] who employed a finite-element-based fictitious do-main method and assumed a homogeneous temperature over theparticle domain (a nice approximation for large kr).

Fig. 18. Iso-therms in the center plane and particle distribution at different times during the rising of 500 spherical catalyst particles from the bottom of an enclosure at(kr, Re, qr, Gr, br, cpr, Pr, Qs) = (1,40,1.1,1000,0, 1,0.7,4.64). The temperature increment for the contours is unity.

Fig. 19. Iso-therms in the center plane and particle distribution at different times during the rising of 500 spherical catalyst particles from the bottom of an enclosure at(kr, Re, qr, Gr, br, cpr, Pr, Qs) = (50,40,1.1,1000,0,1,0.7,4.64). The temperature increment for the contours is unity.


For the simulation of the rising of 500 particles from the bot-tom, we use the following simple soft-sphere collision model,which was found more robust than the lubrication model [18]for the challenging problem from our experience:

Fij ¼ F0ð1� dij=dcÞnij; ð28Þ

where Fij, dij, and nij are the repulsive force, the gap distance andthe unit normal vector between the particles i and j, respectively. dc

represents a cut-off distance and the repulsive force is activated atdij < dc. F0 is the magnitude of the force at contact. We set dc = 0.1a(a being the particle radius), and F0 = 103. The motions of the par-ticles due to the collision force (28) and due to the hydrodynamicforce (5), (6) are handled separately with a fractional step scheme.The time step for the collision model is set to be one tenth of thelatter (i.e. 4t /10) to circumvent the stiffness problem rising fromthe explicit integration scheme with a large value of F0, as sug-

gested by Glowinski et al. [19]. The collision between a particleand a wall is treated similarly as two particles with the coefficientF0 in (28) doubled.

For the case of 500 particles, the enclosure dimension is8� 8� 32. We use the same mesh size (h = d/16) and the time step(Dt = 0.001), as in the single particle case. A packed bed of 500 par-ticles at the bottom forms from the settling of randomly distributedparticles in the enclosure under gravity alone with the above colli-sion model (28). We let (Re, qr, Gr, br, cpr, Pr, Qs) = (40,1.1,1000,0,1,0.7,4.64), same as those of Wachs [26]. Figs. 18 and 19 showthe iso-therms in the center plane and particle distributions at dif-ferent times during the rising of 500 particles for kr = 1 and kr = 50,respectively. The evolution of the average vertical position of 500particles is shown in Fig. 20. From these figures, one can see thatthe effects of thermal conductivity on the rising of 500 particlesfrom the bottom are different from the case of a single particle re-leased at the center: at the early stage of the rising (t < 12) the effect

Fig. 20. Evolution of the average vertical position of 500 spherical catalyst particlesrising from the bottom of an enclosure at (Re, qr, Gr, br, cpr, Pr, Qs) =(40,1.1,1000,0,1,0.7,4.64).


of kr on the average particle vertical position is small and at the latestage (t > 25), the particles even rise faster for smaller kr. The possi-ble reasons are the following. Firstly, for the packed bed at the bot-tom, the thermal energy loss through the wall is much largercompared to the case of a particle in the center, since substantiallylarge temperature gradients form in the gap between the particlesand the wall whose temperature is fixed. The energy loss is largerfor a larger kr. In the limiting case of infinite kr, the particle in com-pletely contact with the wall cannot rise because its temperature isretained the same as the wall temperature, from the theoreticalpoint of view; in our simulations, the particles do not complete con-tact each other and the wall, due to the repulsive-force collisionmodel. On the other hand, the thermal flux into the fluids is largerfor a larger kr, if the thermal energy loss through the wall is not con-sidered. As a consequence, the effect of the thermal conductivity ra-tio on the rising process of the particles from the bottom becomescomplex, unlike the case of a single particle released in the center.Secondly, at the late stage when more particles are located in theupper half box than in the lower half box, the temperatures insidethe particles with a lower thermal conductivity are generally muchhigher than those for a higher thermal conductivity, resulting inhigher average fluid temperatures in the upper half box and therebyfaster rising of the particles, as indicated from the comparison of thetemperature and velocity fields in the center plane for kr = 1 and 50(not shown here).

Our results on the rising of 500 particles from the bottom do notagree well with those of Wachs [26]; our particles rise much faster.The primary reason may be attributed to different collision modelsused in two works. In the model of Wachs [26], the particlestouched each other and the wall, which caused more energy lossand thereby the retarded rising of the particles, compared to oursystem in which there exists a gap between the particles and thewall. In addition, a tangential Coulomb-like friction force was usedby Wachs, which also impeded the rising motion of the particles.Accurate prediction of the rising instability for the packed bed isvery difficult due to the difficulty in accurately computing the tem-perature gradient in the particle-particle and particle-wall gap.

4. Conclusions

We have presented a new method for the direct numerical sim-ulations of particulate flows with heat transfer in three dimensionsby combining the fictitious domain (FD) method for the fluid-particle motion and the sharp interface (SI) method for the temper-

ature field in both fluid and solid mediums. The accuracy of theproposed method is validated via two problems: the natural con-vection in a two dimensional cavity with fixed solid particles,and the flow around a sphere. The method is then applied to thenatural convection in a three-dimensional cavity with a fixedsphere, the motion of a spherical particle in a non-isothermal fluid,and the rising of spherical catalyst particles in an enclosure. Our re-sults indicate that the thermal conductivity of the particle insidethe cavity has a slightly negative effect on the heat transfer acrossthe cavity wall through modifying mainly the thermal convectionrather than the thermal diffusion. A smaller particle thermal con-ductivity leads to larger temperature gradient inside the particle,a lower temperature below the particle, more crowded isothermsnear the lower half of the hot wall, and then a larger Nusselt num-ber. The results on the motion of a particle in a box filled with anon-isothermal fluid demonstrate that the particle-fluid thermalexpansion coefficient ratio has a significant effect on the particlemotion.

Acknowledgments

The work was supported by the National Natural Science Foun-dation of China (Nos. 11072217, 10872181 and 11132008), theFundamental Research Funds for the Central Universities, and theProgram for New Century Excellent Talents in University. Theauthors wish to thank A. Wachs for helpful discussions.

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