PHYSICAL REVIEW E 67, 066711 ~2003!

Lattice Boltzmann method for simulations of liquid-vapor thermal flows

Raoyang Zhang and Hudong ChenExa Corporation, 450 Bedford Street, Lexington, Massachusetts 02420, USA

~Received 10 January 2003; published 27 June 2003!

We present a lattice Boltzmann method that has the capability of simulating thermodynamic multiphaseflows. This approach is fully thermodynamically consistent at the macroscopic level. Using this method, theliquid-vapor boiling process, including liquid-vapor formation and coalescence together with a full coupling oftemperature, is simulated.

DOI: 10.1103/PhysRevE.67.066711 PACS number~s!: 44.05.1e, 44.25.1f, 44.35.1c, 47.11.1j

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I. INTRODUCTION

After years of research, the lattice Boltzmann meth~LBM ! has become an established numerical approaccomputational fluid dynamics~CFD!. Many models and ex-tensions have been formulated that cover a wide rangcomplex fluids and flows@1#. Furthermore, the LBM hasbeen extended to include turbulence models that haveready had a direct and substantial impact on engineeringplications@2–5#.

Among many desirable LBM features such as simplicparallelizability, and robustness in dealing with complboundary conditions, one recognized advantage is its cbility of simulating fluid flows with multiple phases@6,7,1#.The core mechanism in LBM modeling of multiphase flowis its microscopic level realization of nonideal gas equatioof state. As a result, at sufficiently low temperature aproper pressure, liquid-vapor-like first-order phase trantions are spontaneously generated. There is no need toplicitly track the interfaces between immiscible phases. Fthermore, unlike static statistical physical models@8#, theLBM also contains momentum conservation, so that bubband liquid droplets are formed along with fluid hydrodnamic processes. The success and simplicity of the LBMmultiphase flows has led to various applications that inclusimulations of oil-water mixtures through porous media@9#,Rayleigh-Taylor problems@10,7#, and many more@1#. On theother hand, there is a crucial missing piece. That is, so fathe existing multiphase LBM models are limited to regimin which either the temperature dynamics is negligible oreffect on flow is unimportant. This limitation, along with thoverall unavailability in CFD, has prevented us from dealiwith an important class of flows, namely, multiphase floinvolving strong coupling with thermodynamics. Specific eamples of such types of flow range from the common waboiling processes to thermal nuclear reactor applicatioThus, from both fundamental and practical points of vieextensions of the existing CFD and LBM methods to simlation of thermal multiphase flows is extremely important

The LBM originally evolved from lattice gas modeobeying fundamental conservation laws and symmet@11–14#. Now it has also been shown to be systematicaderivable from the continuum Boltzmann equation@15#. Themost commonly known lattice Boltzmann equation~LBE!has the following form~adopting the lattice units conventioin which Dt5Dx51):

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f i~x1 ci ,t11!2 f i~x,t !5Ci , ~1!

where the timet takes on only positive integer values, anthe particle velocity takes on a finite set of discrete vecvalues ~speeds! $ci ; i 50,...,b%. These speeds form linkamong nodes on a given lattice@11,16#. The collision termon the right hand side of Eq.~1! now often uses the so calleBhatnagar-Gross-Krook~BGK! approximation@17,16#

Ci52f i2 f i

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having a single relaxation time parametert. Here, f ieq is the

local equilibrium distribution function, which has an apprpriately prescribed functional dependence on the local hyddynamic properties. The basic hydrodynamic quantities, sas fluid densityr and velocityu, are obtained through simplmoment summations,

r~x,t !5(i

f i~x,t !,

ru~x,t !5(i

ci f i~x,t !. ~3!

In addition, one can also define a fluid temperatureT by

rD

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where D is the dimension of the momentum space of tdiscrete lattice velocities@11#. It has been shown theoretcally that the hydrodynamic behavior produced from tLBE obeys the Navier-Stokes fluid dynamics in the lowavelength and low frequency limit@16#. The resultingequation of state is that of an ideal gas fluid, namely,pressurep obeys a linear relation with density and tempeture,

p5rT. ~5!

The kinematic viscosity of the fluid is related to the relaation parameter by@11,3,16#

n5S t21

2DT. ~6!

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R. ZHANG AND H. CHEN PHYSICAL REVIEW E67, 066711 ~2003!

The LBM has been extended to simulations of multiphaflows @1,6,7#. The key step is to introduce an additional terD f i(x,t) on the right hand side of Eq.~1!, to represent abody force. This force term is self-consistently generatedthe neighboring distribution functions around each lattsite, and it does not violate either the local mass conservaor the global momentum conservation. However, the lomomentum is altered by an amount

F~x,t !5(i

ciD f i~x,t !. ~7!

The appearance of the body force term can physically,mean-field sense, be attributed to a nonlocal interactiontential U among the particles@18#. The existence of such ainteraction potential is the essential mechanism in the nideal gas type of fluid. Hence, with a suitable choice ofU,spontaneous phase separations can be produced, and onuse it conveniently to study multiphase flow phenomenamerically. Through the years, there has been much progin LBM models for multiphase flows@1#. On the other handas pointed out at the beginning, all the existing attempts hbeen limited to isothermal~or ‘‘athermal’’! situations inwhich the dynamics of temperature in the fluid is suppressThat is, T is assumed either a constant or, at best, a pscribed function of space~or time!.

II. MULTIPHASE FLOWS WITH INCORPORATION OFTHERMODYNAMICS

In this paper, we present an extension of the multiphLBM to include the full thermodynamics. We focus on thmethod of solution as the first step and will present quatative studies@19# in future work.

The most natural extension of the LBM for thermodynaics has been to introduce a conserved energy degree ofdom @20#. This is relatively straightforward for the ideal gatype of model in which only pointwise collisions are involved and only kinetic energy is considered. When a sucient number of particle speeds is used, one can theoreticshow that the LBM leads to the correct full set of thermohdrodynamic equations of an ideal gas fluid@3,20,21#. Unfor-tunately, in addition to being considerably more expenscomputationally than the isothermal LB models, such anproach cannot be easily generalized to multiphase thermnamic flows. The most obvious obstacle is the difficultytracking the energy evolution while maintaining total enerconservation. For a nonideal gas system, the total energycontains an interaction energy part that is a function ofrelative positions among the particles. Without total eneconservation, a temperature variable cannot be definedself-consistently at the microscopic level. In addition, it hbeen shown that, unlike the isothermal models, a LBE wan energy degree of freedom does not guarantee a globHtheorem@22#. As a consequence, the system can exhibit snificantly less stability. Other undesirable features in thisrect approach include~1! difficulty in changing the Prandtnumber value from unity, unless a substantial generalizato the BGK collision term is made; and~2! a rather limited

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temperature range~with the maximal allowable value onlyabout twice the minimal value!, unless significantly morespeeds are added@3,21#. For these reasons, the progressthe LBM for thermal multiphase flows has been rather slo

Here we present a LBM approach that can essentiavoid all of the above mentioned drawbacks. The fundamtal idea can be briefly summarized. First of all, the fludynamics part~i.e., the density and momentum evolution! isrepresented by a modifiedisothermalLBE, while the energyevolution part is determined by an additional scalar enetransport equation@23#. The latter can be solved via eitherfinite difference scheme or an auxiliary LBE. Second, tcoupling of the two parts is through a properly defined boforce term in the LBE~and the compression and dissipatioterms in the energy equation!. As we shall realize belowalthough conceptually rather simple, this model producescorrect full thermohydrodynamic equations together withnonideal gas equation of state.

We choose a common isothermal LBE~e.g., D3Q19@16#!as a starting basis. As discussed earlier, an isothermalmodel for fluid density and velocity evolution is consideably simpler compared to its energy conserving counterpThis is certainly desirable for doing efficient fluid flow simulations. Furthermore, the equilibrium distribution in an isthermal LB model is a function of only fluid density anvelocity. The lack of temperature dynamics in the equilrium distribution is the key for achieving a higher stabilitythe LBE @22#. Having these facts in mind, it is very desirabto introduce a macroscopic mechanism to recover the tmodynamics. Specifically, instead of letting the temperatinfluence the equilibrium distributions in a LB system, ththermodynamic effect is obtained via a temperatudependent body force@6#. Because of the ‘‘external’’ natureof the coupling, the LB system and its equilibrium propertiremain microscopically isothermal. Nonetheless, asplained below, this alternative way of coupling achievesdesired thermodynamics at the macroscopic level.

Ignoring the higher order contributions, the body forterm can be simply expressed as@24#

D f i~x,t !5wi

T0ci•F~x,t !, ~8!

where the constant weightswi and T0 are directly deter-mined by the LBE model~e.g., D3Q19, in whichT051/3).One can easily verify that this gives rise to Eq.~7!. Theglobal momentum conservation is preserved as longF(x,t) is expressed as a spatial gradient of a scalar func@25#,

F~x,t !52“U~x,t !. ~9!

It is straightforward to implement this condition in a discrespace by proper finite difference procedures. Based onconsideration for higher order isometry in surface tensiwe choose~for D3Q19! the following specific form:

“U~x,t !'(i

D

bci2 ciU~x1 ci ,t !. ~10!

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LATTICE BOLTZMANN METHOD FOR SIMULATIONS OF . . . PHYSICAL REVIEW E 67, 066711 ~2003!

With the additional body force term, one can easily reconize that the overall effective pressure in the resulting flmomentum equation has become

p5rT01U, ~11!

where the first term is a result of the isothermal LBM. FroEq. ~11!, one can obtain any form of equation of statep5p(r,T), simply by making a corresponding choice forU,

U~x,t !5p„r~x,t !,T~x,t !…2r~x,t !T0 . ~12!

The above quantity is determined once the local valuesr(x,t) andT(x,t) are provided. Obviously the resulting fluiis no longer isothermal if the temperatureT(x,t) varies. Inother words, because of the macroscopic method of cpling, the resulting fluid dynamics is no longer isothermal.addition, with the great flexibility in choosing the equationstate, this approach can be applied to simulation of nonidgas fluids and multiphase flows. Indeed, we confirmedbasic feature through a set of spinodal decomposition tbased on a van der Waals gas model~Carnahan-Starlingequation of state!. As in other multiphase LBMs, a spontaneous phase separation process is well observed at sciently low temperature values.

The evolution of the temperatureT(x,t) in this approachis obtained by solving a supplemental scalar energy transequation,

r~] t1u•“ !e52p“•u1“•k“T1C, ~13!

wheree5cvT is the internal energy, andcv is the specificheat at constant volume of the fluid. The overall pressurep isdefined by the equation of state~11!, andk is the heat con-ductivity, which can be specified flexibly. The termC repre-sents the viscous dissipation of flow and the contributionsurface tension. The energy evolution equation~13! is a stan-dard macroscopic description for thermal fluids@18#. Thecomputation of an isothermal LB model along with a scaenergy equation is considerably less expensive than anycroscopic attempts, for it requires neither many partispeeds nor complicated tracking of the energy evolutiMoreover, the difficulties in stability and Prandtl number asociated with the original thermal LBM are not issues in tapproach. Solving a scalar transport equation is rastraightforward. There are many finite difference schemesaccurately and efficiently solving the scalar transport eqtion. In our particular simulations, we used an extended LWendroff scheme@4#. The combination of Eqs.~1!–~3!, ~8!–~10!, and ~12! and ~13! forms our LBM approach formodeling multiphase thermodynamic fluid flows. The thmal boundary condition can be realized via standard numcal procedures so that

kn•“Tuw5q, ~14!

with a prescribed heat fluxq, that can be either fixed orfunction of local properties in order to achieve a fixed wtemperature. The unit vectorn denotes the surface normdirection.

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There is one more important feature in this model wopointing out. That is, the approach avoids the fundamelimitation on the temperature range that has constraiother thermal LB models. Notice that the temperaturepears only in the body force term in a gradient function forHence, unlike that for equilibrium distribution functionthere is no absolute upper or lower bound on the temperavalues except that it should not change too rapidly acrosgiven resolution scale. Moreover, there is obviously nosolute bound on temperature in the energy equation.

Based on the above discussion, one can see that thismodel generates a fully macroscopically consistent desction for thermodynamic flows involving generalized equtions of state. Therefore, this approach offers a convenand efficient numerical tool for studying thermal multiphaflow problems.

III. SIMULATION OF THE LIQUID-VAPORBOILING PROCESS

In this section we present computational results of a tycal multiphase thermodynamic flow simulation with our Lapproach. In particular, the liquid-vapor boiling processvolving Rayleigh-Be´nard-like convection and phase changetogether with a complex temperature dynamics, is simulasuccessfully, albeit qualitatively. Although representative owide range of important applications, boiling flow problemhave not been very successfully treated in CFD in geneConsequently, it is very important that this approach cdemonstrate such a fundamental capability.

The Rayleigh-Be´nard convection process has been wideused as a benchmark for many fluid computations. It issimplest representation of a boiling phenomenon in whiccomplex buoyancy-driven convection process occurs at vous values of the Rayleigh number@26#. On the other handmost of the boiling processes occurring in nature alsovolve evolutions of multiple thermodynamic phases. Thatin addition to thermal convection, the fluid undergoesphase transition process in which liquid droplets and vabubbles are generated. The most obvious practical examinclude the common one of water boiling in a pot.

We choose a standard Rayleigh-Be´nard setup, in whichboth the upper and the lower solid plates obey no-slip bouary conditions, while the horizontal boundary conditionperiodic. To achieve more stable and second-order accunumerical results, we have also applied the scheme tomodified LB discretization formulation of Heet al. @7#. Asdiscussed above, the Carnahan-Starling equation of staused here for convenience. The mean density valuer51.36rc . The temperature on the upper wall is fixed atTu50.795Tc while on the lower wallTl50.954Tc . Here Tc~50.55 in lattice units for the choice of the model! is thecritical temperature for formation of two phases@7#. Theinitial temperature is set to be linearly distributed betwethe two plates and is consistent with their temperatboundary conditions. The simulation volume isL3H52563128 grid points. The gravity valueg5531026 ~latticeunits! is used. In order to avoid unnecessary complicationweak surface tension effect@10,7# is also applied in the LBM

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R. ZHANG AND H. CHEN PHYSICAL REVIEW E67, 066711 ~2003!

flow simulation, so that the surface tension contributionthe energy evolution can be neglected. Specifically,choose the surface tension coefficients to be 0.01 for thesesimulations. The kinematic viscosity and the Prandtl numare set atn50.02 and Pr510, respectively. For simplicitythe heat capacitycv in our multiphase flow is chosen asconstant~51!. All the other fluid parameters are the samein @7#. Based on the choice of these parameter values,resulting Rayleigh number is Ra;3.03105, which is muchhigher than the first threshold (Rac51708) for onset of con-vection in the conventional single-phase Rayleigh-Be´nardsystem@26#.

The simulation starts from a uniform density distributiowith 1% random fluctuations. To enhance bubble formatismall temperature fluctuations are added to the equatiostate in the first grid point layer near the bottom solid plaDue to the higher temperature, a lower density fluid is pduced that subsequently leads to small vapor bubbles abottom plate, and then these merge with each other to flarger sized ones. The phase formation process in the slation is combined with the convection process in whichhotter and lighter vapor phase rises while the colderheavier liquid phase descends due to gravity. With the abchoice of parameter values, such a full thermohydrodynacycle is seen to be able to sustain itself indefinitely. Figurshows the density distributions at some representative ndimensionalized~by AH/g) times. One can observe the fomation of two streams of bubbles near the bottom plate. O

FIG. 1. Density distributions att539.5, 41.5, 43.5, and 45.5respectively. Density ratio between liquid and vapor is 3. Light grepresents the liquid phase; dark gray represents the vapor ph

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can also observe that two pairs of counter-rotating convtion rolls are locked between the bubble streams. In additthe convection rolls are seen to pinch off the small bubbfrom the bottom plate. As bubbles rise, their sizes are seeincrease slightly. Since small bubbles move faster than labubbles, collision and coalescence often occur among thFigure 2 shows streamlines of the velocity field at time 39from which one can clearly see the two pairs of countrotating convection rolls. Figure 3 depicts the correspondtemperature deviation from a linear distribution at the satime. Interestingly, the temperature exhibits a nontrivial bhavior: Its value is seen to be relatively lower in the vapphase domains near their interfaces. As expected physicthis phenomenon is due to thep“•u term in the energyequation~13! associated with the volume expansion frowater to vapor phases. All the above simulated phenomare qualitatively correct for a realistic two-phase thermodnamic flow.

We also ran another simulation with the exact same seas the above, except that the surface tension is madetimes stronger. As seen in Fig. 4, only one pair of countrotating convection rolls is produced in its asymptotic sta

ye.

FIG. 2. Streamline and vector plots of the flow velocity fieldt539.5.

FIG. 3. Temperature deviation@Tdev5(T2Tlinear)/Ttop# at t539.5. The gray-scale range is~20.1,0.02!.

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LATTICE BOLTZMANN METHOD FOR SIMULATIONS OF . . . PHYSICAL REVIEW E 67, 066711 ~2003!

Unlike the standard single-phase Bernard convection, hwe see that other intrinsic properties such as the surfacesion can also alter the thermal convection characteristicsmultiphase flow@26#.

IV. DISCUSSION

In this paper, we present an approach that combinemultiphase lattice Boltzmann method with a scalar tempeture equation. The coupling is realized macroscopically viself-consistent body force. The basic formulation is appcable to both two- and three-dimensional flow situations. Idirectly verifiable theoretically that this LB model obeys t

FIG. 4. Density distribution and velocity vector plot with suface tension coefficients50.1.

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correct full thermohydrodynamic equations with a nonidegas equation of state. The incorporation of thermohydronamics in the approach allows for simulations of compmultiphase flows coupled with temperature dynamics. Otimportant features of the scheme include its simplicity, eciency, and robustness for simulating thermal multiphaflow processes@27#. The last has been shown to be extremedifficult with other LBM-based schemes or the more convetional numerical methods@28#. Furthermore, like otherLBMs, our approach can handle complex physical boundconditions@2#. All these features are highly desirable forpractical computational model.

We have demonstrated the capability of our approachdoing boiling flow simulations. This type of flow has so fanot been successfully handled via other methods. Thus,approach has opened a promising opportunity for numcally studying thermal multiphase flow problems. On tother hand, further improvements in the approach are nesary in order for it to become a more useful and quantitatcomputational tool. Without going into detail, the major rmaining issues include~1! incorporating more realistic equations of state instead of the van der Waals type of gas mo~2! a more physical treatment of the heat capacity and laheat;~3! further understanding and modeling of surface tesion and near interface physics;~4! further enhancement onumerical stability to achieve significantly higher density rtio and lower viscosity;~5! generalization of boundary conditions; and~6! direct comparisons with experimental resu@19#. Among all the above tasks, we consider~4! to be themost challenging for LBMs in general.

ACKNOWLEDGMENTS

The authors wish to thank Dr. Xiaoyi He, Dr. YueHonQian, Dr. Ilya Staroselsky, Dr. Adrian Tentner, Dr. StevOrszag, Dr. Sauro Succi, Dr. Rick Shock, and Dr. Nick Matys for useful discussions. This work is supported in partDOE SBIR Grant No. 65016B01-I.

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