1

Bulletin of the JSME

Mechanical Engineering ReviewsVol.1, No.1, 2014

Paper No.13-00277© 2014 The Japan Society of Mechanical Engineers[DOI: 10.1299/mer.2014fe0003]

Multiphase Rayleigh-BenardPaolo ORESTA , Francesco FORNARELLI and Andrea PROSPERETTI

∗ Department of Mathematics, Mechanics and Management, Polytechnic of Bari, 70126 Bari, Italy, andDepartment of Engineering for Innovation, University of Salento, and INFN sez. Lecce, 73100 Lecce, Italy

E-mail: p.oresta@poliba.it∗∗ Department of Engineering for Innovation, University of Salento, and

INFN sez. Lecce, 73100 Lecce, Italy∗∗∗ Department of Mechanical Engineering, Johns Hopkins University, Baltimore MD 21218, USA, and

Physics of Fluids Group, Department of Science and Technology, J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AEEnschede, The Netherlands

AbstractNumerical simulations of two-phase Rayleigh-Benard convection in a cylindrical cell with particles or vapor bub-bles suspended in the fluid are described. The particles or bubbles are modeled as points, the Rayleigh number is2 × 106 and the fluids considered are air, for the particle case, and saturated water for bubbles. It is shown that thepresence of a second phase has a profound effect on the flow and heat transfer in the cell. The heat capacity of theparticles and the latent heat of the liquid are used, in dimensionless form, as control parameters to modulate theseeffects. It is shown that, as these parameters are varied, the nature of the flow in the cell changes substantially, insome cases with adverse and in others beneficial effects on the Nusselt number. By the analysis of several aspectsof the numerical results, a physical discussion of several mechanisms is provided.

Keywords : Two-phase Rayleigh-Benard convection,

1. Introduction

It is well known that suspended particles or bubbles have a very strong effect on turbulent flows. A considerableamount of experimental evidence has been accumulating for well over 40 years showing that small particles or bubblesdampen turbulence, while larger ones increase it (see e.g. Hetsroni & Sokolov 1971; Gore & Crowe, 1989; Sun & Faeth1986; Lance et al. 1991; Kiger & Lasheras, 1997; Hwang & Eaton 2006).

The theoretical investigation of the phenomenon has greatly benefitted from the use of the so-called point-particlemodel, in which the particles are approximated as points and are tracked in a Lagrangian fashion while the fluid is treatedwithin the usual Eulerian framework on the basis of Navier-Stokes equations augmented by the point forces due to theparticles. This model has been extensively used for gas-particle flows (see e.g. Squires & Eaton 1990; Elghobashi &Truesdell, 1992; Ferrante & Elghobashi, 2003; Eaton, 2009; Pialat et al. 2007; Soldati & Marchioli 2009), and also forbubbly flows (see, e.g., Climent & Magnaudet, 2006; van den Berg et al., 2006; Mazzitelli & Lohse 2009).

The vast majority of these studies consider isothermal systems. A recent development in this area has been theaddition of thermal effects to the point-particle model. Zonta et al. (2008, 2011) and Arcen et al. (2012) have studied theeffect of the addition of particles to the heat transfer in a channel turbulent flow. For the case of mildly turbulent Rayleigh-Benard convection, the effect of bubbles and particles has been considered in a series of recent papers by the presentauthors and co-workers (Oresta et al., 2009; Schmidt et al., 2011; Lakkaraju et al. 2011, 2013; Oresta & Prosperetti2013). The purpose of the present paper is to review the mathematical formulation of the problem and the numericalmethod of solution, and to present some new results.

The system considered is a vertical cylindrical cell with a height H, radius R and aspect ratio 2R/H = 1/2. Thehorizontal plates forming the base and top of the cell are kept at a temperature Th and Tc < Th, respectively, in such a way

convection

Received 10 November 2013

Point particles Point bubbles Natural convection Heattransfer

* ** ***

, , ,

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that, in the absence of the particles, a mildly turbulent naturally convective flow is established. We devote the first part ofthe paper to the effect of particles in a gas, and the second one to the effect of bubbles in a liquid. We start, however, witha brief derivation of the models.

2. Fluid momentum and energy balance

Since in this paper we consider particles in a gas as well as bubbles in a liquid, it is useful to present a unifiedderivation of the fluid momentum and energy equations to be used with the point particle model. We will generically usethe word “particles” to indicate true particles, drops or bubbles.

Our method is heuristic: we first write down the averaged equations and then we deduce the point-particle modelwhich, upon averaging, would give the same result. Even though we will ultimately use a point-particle model, thederivation is clearer if we start by considering the finite size of the inhomogeneities. For simplicity we adopt the languageof volume, rather than ensemble, averaging.

The general form of the volume-averaged momentum balance for the fluid-particle mixture can be written as

(1 − α)ρdudt+

1V∑k

mkdvk

dt= ∇∇∇ ·σσσ + (1 − α)ρg +

1V∑k

mkg . (1)

Here u is the average fluid velocity, σσσ the average stress in the mixture, ρ the fluid density, g the acceleration of gravityand mk and dvk/dt the mass and acceleration of the k-th particle; the summations are extended to all the particles containedin the averaging volume V . The volume fraction occupied by the particles is denoted by α and is given by

α =1V∑kvk , (2)

where vk is the volume of the k-th particle. We rewrite the previous equation as

ρdudt= ∇∇∇ · σσσ + ρg + αρ

(dudt− g)+

1V∑k

mk

(g − dvk

dt

). (3)

The point-particle formulation that, upon volume averaging, would give the same result is evident by inspection andis

ρdudt= ∇∇∇ · σσσ + ρg +

Np∑i=1vi

[ρ

(dui

dt− g)− ρp

(dvi

dt− g)]δ(x − xi) , (4)

where ρp is the particle density, Np the total number of particles in the system, and xi the position of the i-th particle.Here and in the following a particle index appended to a fluid quantity signifies that that quantity is evaluated at theparticle position; thus, dui/dt denotes the fluid acceleration evaluated at the location of the i-th particle. For simplicity weassume that the particles are equal (but allowing, as shown below, for the possibility that vapor bubbles may have differentvolume); the extension to more general cases is immediate.

Since, with this step, the particles occupy a negligible volume, the continuity equation reduces to

∇∇∇ · u = 0 . (5)

Furthermore, again because the particles occupy a negligible volume, the averaging volume can be taken arbitrarily smallso that we can simply identify du/dt and σσσ with the acceleration and stress of the fluid and write

dudt=∂u∂t+ u · ∇∇∇u , σi j = −pδi j + μ

(∂ui

∂x j+∂u j

∂xi

), (6)

in which p is the pressure and μ the fluid viscosity.Since we are interested here in flows driven by natural convection, we must allow density variations in the buoyancy

term of the momentum equation. Following the standard Boussinesq procedure (see e.g. Bergman et al., 2011; Bejan2013), we set ρ � ρc[1−β(T −Tc)], with β the isobaric thermal expansion coefficient and ρc the density at the temperatureTc of the cold (upper) plate of the cell. We also absorb the constant-density contribution ρcg into the pressure term bydefining a modified pressure according to p = p−ρcg·x and, following the standard procedure, we neglect the temperaturedependence of the viscosity. With these steps, the momentum equation (4) becomes then

ρdudt= −∇∇∇p + μ∇2u − βρ(T − Tc)g +

Np∑i=1vi

[ρ

(dui

dt− g)− ρp

(dvi

dt− g)]δ(x − xi) . (7)

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For simplicity, we have written ρ in place of ρc and p in place of p as is customary in the theory of Rayleigh-Benardconvection.

The procedure used to derive the energy equation parallels that used for the momentum equation. We write

(1 − α)ρcdTdt− 1

V∑k

Qpk = −∇∇∇ · q , (8)

where c is the constant-pressure specific heat per unit mass of the fluid, Qpk is the heat lost by the k-th particle per unittime, and q is the average heat flux in the mixture. As before, from this equation we deduce the point-particle model inthe form

ρcdTdt= −∇∇∇ · q +

Np∑i=1

[viρc

dTi

dt+ Qpi

]δ(x − xi) , (9)

in which, following the convention mentioned before, dTi/dt denotes the derivative of the fluid temperature at the particleposition. As in the step leading to (6), we use

dTdt=∂T∂t+ u · ∇∇∇T , q = −k∇∇∇T , (10)

with k the thermal conductivity of the fluid.For particles in a gas, ρ� ρp and c � cp. Thus, the momentum and energy equations can be simplified to

ρdudt= −∇∇∇p + μ∇2u − βρ(T − Tc)g −

Np∑i=1

mp

(dvi

dt− g)δ(x − xi) , (11)

and

ρcdTdt= k∇2T +

Np∑i=1

Qpiδ(x − xi) . (12)

Conversely, for bubbles in a liquid, ρ � ρp and so that the momentum equation can be simplified to

ρdudt= ∇∇∇ · σσσ + ρg +

Np∑i=1viρ

(dui

dt− g)δ(x − xi) . (13)

In the right-hand side of the energy equation the local heating of the liquid, embodied in the term viρc(dTi/dt), is muchsmaller than the latent heat exchanged with the bubbles so that the energy equation simplifies to the same form as for thecase of particles, namely

ρcdTdt= k∇2T +

Np∑i=1

Qbiδ(x − xi) , (14)

in which Qbi is the heat lost by the i-th bubble per unit time.

3. The particle model

In the Lagrangian approach that we follow, the particle position, xi, must be found by integrating the equation

dxi

dt= vi , (15)

and the particle velocity vi itself must be found by integrating an equation of motion. For particles or drops in a gas,buoyancy and the other forces introduced below in section 4 for the bubble case are very small. The only significant forcein addition to gravity is drag and we write

dvi

dt= g + fi

ui − vi

τp, (16)

where we drop the index i on the characteristic time scale

τp =mp

3πμdp=ρpd2

p

18μ, (17)

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in which dp is the particle diameter, as all the particles are assumed equal. The factor fi accounts for finite-Reynolds-number effects and is given by (see e.g. Oresta et al. 2009)

fi = 1 + 0.15 Re0.687pi , (18)

with Rep = ρdp|ui − vi|/μ the instantaneous Reynolds number of the i-th particle.Since we only consider small particles, the lumped capacitance approximation is adequate to describe their thermal

balance and we write

mpcpdTpi

dt= −Qpi , (19)

with Qpi the heat flow rate out of the particle, given by

Qpi = πd2phpi(Tpi − Ti) . (20)

Here cp is the particle specific heat per unit mass and hpi the heat transfer coefficient which we express in terms of aparticle Nusselt number:

Nupi =dphpi

k, (21)

for which we use the standard correlation (see, e.g., Bergman et al. 2011)

Nup = 2 + 0.6Re1/2p Pr1/3 , (22)

with Pr = ν/κ the fluid Prandtl number given by the ratio of the fluid kinematic viscosity ν = μ/ρ to its thermal diffusivityκ = k/ρc.

The particle energy equation (19) can be written in an alternative way which brings out an important time scaleanalogous to (16):

dTpi

dt=

Nupi

2Ti − Tpi

τt,p. (23)

The characteristic time τt,p appearing here is defined by

τt,p =ρpcpd2

p

12k, (24)

and represents the characteristic time scale for the relaxation of the particle temperature to that of the surrounding fluid.The factor 1

2 Nupi in (23) arises similarly to the factor fi in (16) to account for the deviations of Nupi from the low-velocityasymptote Nup = 2 (cf. (22)).

4. The bubble model

The mathematical model for bubbles recognizes the fundamental fact that, due to their negligible mass vis-a-vis thesurrounding fluid, many forces that are negligible in the case of particles become important for bubbles. Thus, while (15)still applies, the particle momentum equation (16) is replaced by

−18πCdρd2

b |v − u|(v − u) +12ρ

[v

(dudt− dv

dt

)+ (u − v)

dvdt

]+ ρv

dudt

+12vρ(∇∇∇ × u

) × (v − u) − ρvg = 0 , (25)

in which Cd is the drag coefficient taken as (see e.g. Mei & Klausner 1992; Mei et al. 1994)

Cd =16

Reb

⎡⎢⎢⎢⎢⎢⎣1 + Reb

8 + 12 (Reb + 3.315

√Reb)

⎤⎥⎥⎥⎥⎥⎦ , (26)

with Reb = ρdb|u − v|/μ the bubble Reynolds number defined in the same way as in the particle case. The terms in thesquare brackets in (25) are the added mass force, with the standard coefficient 1/2, followed by the ’virtual buoyancy’force and, in the next line, by the lift force (also taken with the standard coefficient 1/2) and, finally, by the buoyancyforce.

i

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Since the temperature differences are moderate, momentum effects in the radial motion of the bubbles are negligibleand the bubble diameter can be simply calculated by an energy balance between the latent heat associated to the phasechange and the heat exchanged with the liquid:

Lddt

(π

6d3

bρv

)= −Qb = πd2

bhb(T − Tsat), (27)

with Tsat the saturation temperature at the prevailing pressure. Given the small height of the cell considered here (seebelow), hydrostatic effects are negligible in the present simulations and, therefore, both Tsat and ρv can be assumed to beconstant. Thus, (27) can be simplified to

ddb

dt=

2hb

Lρv(T − Tsat) . (28)

For the heat transfer coefficient hb we use (Oresta et al. 2009)

Nub =dbhb

k= Nub0

⎡⎢⎢⎢⎢⎢⎣1 +⎛⎜⎜⎜⎜⎝ 4Peb

πNu2b0

⎞⎟⎟⎟⎟⎠n/2⎤⎥⎥⎥⎥⎥⎦

1/n

, (29)

where n � 2.65 and

Nub0 =16π

Ja[π

8Ja+

(6π2)1/3

161

Ja2/3 +34

]. (30)

The Peclet and Jakob numbers, Peb and Ja, are defined by

Peb =db|v − u|κ

, Ja =ρc(Th − Tsat)ρvL

. (31)

The relation (29) was built on the basis of results by Magnaudet et al. (1998) and Ivashnyov & Smirnov (2004) asexplained in our previous study (Oresta et al. 2009).

5. Description of the problem

For a fixed aspect ratio (cell diameter over height), which here we take equal to 1/2, the strength of single-phasenatural convection in the Boussinesq approximation is controlled by the Rayleigh number

Ra =gβ(Th − Tc)H3

νκ, (32)

where g = |g|, Th and Tc are the temperatures of the hot (bottom) and cold (top) plates respectively, and H is the height ofthe cell, and by the fluid Prandtl number Pr.

The chief objective of the numerical simulations is the determination of the effect of the particles or bubbles on theheat transfer through the cell which, in dimensionless form, is the Nusselt number. In the single-phase situation the heatflux out of the hot plate must equal that into the cold plate by conservation of energy, but the situation is different in thepresence of particles or bubbles. Thus, we define hot-plate and a cold-plate Nusselt numbers by

Nuh,c =H

Th − Tc〈|∂zT |〉A,t

∣∣∣z=0,H . (33)

Here ∂z denotes the derivative in the upward vertical direction and 〈(. . .)〉 indicates mean values with the average takenover the subscripted quantities, i.e., in this case, the cross-sectional area A and the time t. The same convention is followedconsistently in this paper to indicate averaged quantities. It is shown in Oresta et al. (2009) that the Nusselt numbers canbe expressed as

Nuh = 1 +H

κ(Th − Tc)〈uz(T − Tc)〉V,t −

1πR2k(Th − Tc)

⟨∑i

(H − zi) Qi

⟩t, (34)

and

Nuc = 1 +H

κ(Th − Tc)〈uz(T − Tc)〉V,t +

1πR2k(Th − Tc)

⟨∑i

ziQi

⟩t, (35)

which lend themselves to a more convenient numerical evaluation.

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We consider as our base case a situation in which, without particles or bubbles, Ra = 2 × 106. For moderate valuesof the fluid Prandtl number, in the single-phase case, the cell contains a single convection roll with fluid rising along oneside and descending along the opposite side with a Nusselt number Nu0 = 11.2. These values are in close agreement withthe predictions of the Grossman-Lohse theory (Stevens et al., 2013), Nu0 = 11.25, which supports the accuracy of thecalculations.

The number of particles or bubbles present in the cell at any time is held constant. When a particle deposits at thebottom of the cell, another particle is introduced at a random position at the top plate with the cold temperature, Tc, ofthe plate. Similarly, when a bubble reaches the top of the cell, it is removed and a new bubble is introduced at a randomposition at the bottom plate with an in initial diameter of 25 μm. We have shown in our earlier work that the results areindependent of this initial size provided it is reasonably small (Lakkaraju et al. 2011). Since the latent heat necessary toproduce these small bubbles is negligible, it is not included in the calculation of the heat flux at the bottom plate.

As described in Oresta & Prosperetti (2013), if the new particle is introduced imparting to it the local fluid velocity(which, near the plate, is very small), the time necessary for the particle to accelerate to its terminal velocity causes theformation of a particle-rich layer which prevents a meaningful comparison of the effect of different particle diameters.Thus, the particles are introduced at their terminal velocity found from (16) by setting dvi/dt = 0 and ui = 0, namely

vt =τp

fg . (36)

This problem does not arise with bubbles, which immediately start to grow and are quickly removed by buoyancy andconvection.

In single-phase natural convection the fluid velocity scale, the so-called free-fall velocity, is given by

U f =√gβ(Th − Tc)H , (37)

and, with the present parameters, it has the value U f = 0.204 m/s. Non-dimensional times are consistently expressed interms of H/U f .

6. Numerical methods

The fluid model consists of the continuity, momentum and energy equations (5), (4) and (9). No-slip conditions areapplied on the cell walls, the bottom and top plate maintain a fixed temperature while the lateral walls are adiabatic.

The field equations are written in cylindrical coordinates and discretized on a staggered grid using a second-order-accurate finite difference scheme. Most of the simulations were conducted with a grid having 193 × 49 × 193 nodesin the azimuthal, radial and axial directions, respectively. To assess the adequacy of the spatial discretization in thevertical direction, we can estimate the thickness λT of the thermal boundary layers as /(2λT ) ∼ 〈∂zT 〉 or,from (33), λT ∼ H/ (2 Nu). The largest Nusselt numbers encountered in this work are about 70 and the verticaldiscretization with 193 nodes places at least about 10 points inside the thermal boundary layer.

Time advancement is effected with the third-order Runge-Kutta scheme applied in a fractional step framework withthe advective terms treated explicitly and the viscous terms computed implicitly by an approximate factorization tech-nique; details can be found in Verzicco & Orlandi (1996).

The Poisson equation that enforces the flow incompressibility is solved by a direct procedure which relies on trigono-metric expansions in the azimuthal direction and the FISHPACK package (Swartztrauber, 1974) for the radial and axialdirections for which, therefore, a non-uniform mesh distribution can be used. In the radial and axial directions the grid isnon-uniform and clustered towards the boundaries to adequately resolve the viscous and thermal layers.

The position of the particles or bubbles is found from (15), the particle velocity from (16), the bubble velocity from(25), the particle temperature from (23) and the bubble radius from (28). For the parameters values used in the presentsimulations, the most stringent limitation on the time step arises from the particle/bubble momentum equation. Particularlyin the case of very small bubbles, the associated time scale becomes tiny and an accurate integration would require anextremely small time step. To address this difficulty, the momentum equation is integrated implicitly using the trapezoidalrule and a time step at least an order of magnitude smaller than the characteristic time scale of the momentum equation forthe particles and for bubbles with a diameter equal to the initial value. In this way the integration remains stable also forbubbles smaller than the initial size, even though their velocity will not be computed with great precision. This proceduredoes not negatively impact the accuracy of the calculation because the precise velocity of these tiny bubbles, which aremostly on the way to complete condensation and removal from the calculation, is inconsequential. This is all the more so

(Th Tc)−

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as these very small bubbles usually occur in the cold liquid layer near the top plate where velocities are already very small.As a check, in the course of the calculation we monitor the momentum time scale of some randomly selected bubbles tomake sure that their motion remains well resolved while their diameter is greater than the initial one. The particle timescale, given by (17), does not vary in the course of the calculation and the time step is always sufficiently small to producean accurate integration of the momentum equation.

The energy equation for particles or bubbles does not present analogous problems for the current range of values ofJ and Ja, although it might if this range was extended. For this equation and for the particle/bubble position equation (15)we use either an Adams-Bashforth or a third-order Runge-Kutta explicit scheme.

The time step for the fluid integration is set on the basis of the CFL condition and is usually larger than the time stepappropriate for the particle/bubble momentum equation. Thus, the particle/bubble problem is solved using a time step thatis an integer submultiple of the fluid time step one making sure that it remains at least one order of magnitude smallerthan the particle/bubble momentum characteristic time scale.

A significant difference with the single-phase version of the method described by Verzicco & Orlandi (1996) lies inthe new source terms introduced which are located at the position of the particle/bubble and, therefore in general not onthe grid nodes. Upon discretization of the equations it is therefore necessary to replace these sources by an equivalentsystem of forcings at the nodes. For this purpose, the force terms in the momentum equation are first decomposed intoradial, azimuthal and vertical components. Each component is then distributed by suitable weighing among the 8 verticesof the surrounding momentum cell in the same direction. For example, for a radial force component fr at a positionri + ξΔr, θ j + ηΔθ, zk + ζΔz, with Δr, Δθ and Δz the grid spacings and 0 ≤ ξ, η, ζ < 1, the portion attributed to the node(ri, θ j, zk) is

(1 − ξ)(1 − η)(1 − ζ) fr . (38)

The system of 8 forces thus obtained produces the same net resultant and couple as the original bubble force. The samestrategy has been used for the thermal forcing so that the total amount of heat that each particle or bubble exchanges withthe fluid is preserved. It can be verified that this interpolation strategy is second-order accurate and therefore consistentwith the overall spatial accuracy of the discretization.

Several validation tests of the numerical methods are described in our previous papers (Oresta et al. 2009; Lakkarajuet al. 2013).

Parameter ValueAspect ratio, 2R/H 1/2Prandtl number, Pr 0.678Rayleigh number, Ra 2 ×106

Particle number, Np 25,000Cell height, H 0.116 mCell diameter, 2R 0.058 mTemperature difference, Th − Tc 10 ◦CParticle density, ρp 917 kg/m3

Fluid density, ρ 1.29 kg/m3

Kinematic viscosity, ν 1.37 ×10−5 m2/sThermal diffusivity, κ 2.02 ×10−5 m2/sThermal expansion coefficient, β 3.67 ×10−3 K−1

Constant-pressure fluid specific heat, c 1,000 J/kgFree fall velocity, U f 2.04 ×10−1 m/s

Table 1 Summary of the fluid and particle properties used in the simulations.

7. Rayleigh-Benard convection with suspended particles

We describe here the framework and some results for Rayleigh-Benard convection with suspended particles. Thecorresponding results for bubbly flow will be given in section 8.

7.1. The situation of interestThe dimensional values of the parameters used in the simulations are shown in table 1; the particle density is close

to that of ice and the fluid physical properties are those of air at 0 ◦C and normal pressure.

K

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We keep the number of particles fixed at Np = 25,000 and focus on the effect of the particle diameter, which is giventhe values 50, 100 and 200 μm, and of the specific heat of the particles, which is varied over several orders of magnitude.The values of the volume fractions and several dimensional quantities associated with the particles are shown in table 2.

7.2. Dimensionless parametersThe physics of the phenomena investigated in this study depends on a number of dimensionless parameters which

establish the relative magnitude of several competing processes. The significance of these non-dimensional parameters isillustrated in this section and their numerical values are summarized in table 3.

In the first place, one may inquire about the extent to which the particles, present with a certain volume fractionα = Npv/V, with V = πR2H the cell volume, may be expected to affect the fluid motion. If the fluid-particle drag force isestimated according to Stokes’s law, the work that particles falling at their terminal velocity perform on the fluid is of theorder of 3πdpμvtHNp, and this may be compared with the kinetic energy of the fluid, of the order of 1

2 VρU2f . Equating

these two quantities we find an important scale for the volume fraction:

α∗ =ρ

ρpβ(Th − Tc) . (39)

One would expect a significant effect of the particles on the fluid motion only when the particle volume fraction is largerthan α∗. With the values shown in table 1, we find α∗ � 5.16 × 10−5 which, as can be seen in table 2, is close to thevolume fraction of the dp = 100 μm particles considered in this work.

After using (19) to express Qpi, the heat exchanged with the particles, a continuum approximation to the energyequation (12) is

ρcρpcp

dTdt=

kρpcp∇2T −

Np∑i=1vi

dTpi

dtδ(x − xi) �

kρpcp∇2T − αdTp

dt, (40)

where now Tp is viewed as the temperature of the “particle field”. This equation brings into prominence the ratio

J =ρcρpcp

. (41)

With the specific heat of ice, cp � 2050 J/kg, and the values shown in table 1 for the other quantities, we find J � 6.9×10−4

but actually, in this paper, we will consider J as an adjustable parameter to explore the phenomena that occur when theheat capacity of the particles is varied. Since J represents a balance between the energy that can be lost by the fluid andthat which can be absorbed by the particles, this parameter plays here a role similar to that of the Jakob number in thecase of bubbles defined in (31) and discussed in section 8.

We can now proceed to an estimate similar to (39) for the volume fraction for which the amount of heat absorbed bythe particles is comparable to the heat available from the fluid. The former is of the order of Npmcp(Th − Tc), while thelatter is of the order of Vρc(Th − Tc). Equating these two quantities leads us to

α∗∗ =ρcρpcp

= J . (42)

It is only when the actual volume fraction is larger than this quantity that one can expect a significant thermal effect of theparticles.

Regardless of its heat capacity, a particle can exchange an appreciable amount of heat with the fluid only as long asits residence time, of the order of H/vt, is longer than its thermal relaxation time defined in (24), namely provided that theparameter vtτt,p/H is smaller than 1. This quantity is shown in figure 1 as a function of the parameter J defined in (41) forparticles with diameters dp = 50, 100 and 200 μm.

The Stokes number characterizes the fluid-particle momentum exchange and is defined as the ratio of the particle tothe fluid momentum time scales. In a turbulent flow, of course, there is a range of time scales, extending from the time

dp 105〈αp〉V,t vt H/vt τp mp(μm) (mm/s) (s) (ms) (ng)50 0.534 66.6 1.72 7.14 60.0100 4.27 232 0.495 28.8 480200 34.2 658 0.175 115 3839

Table 2 Mean volume fraction 〈αp〉V,t, particle terminal velocity vt, residence time scale H/vt, characteristic par-ticle time scale τp and particle mass mp corresponding to the particle diameters used in the simulations.

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10-310-210-1100101102103104

10-6 10-5 10-4 10-3 10-2(v

t / H

) τt,p

J

Fig. 1 The thermal time constant of the particles defined in (24) for particle diameters, in ascending order, of dp= 50, 100 and 200 μm, as a function of the particle heat capacity expressed as the parameter J defined in(41).

dp 102 Mp vt/U f 102vtτp/H Ret Pet 103StK 103St f(μm)50 0.395 0.327 0.415 0.244 0.165 50.4 12.7100 3.16 1.14 5.79 1.70 1.15 201 50.8200 25.3 3.23 65.6 9.64 6.54 805 203

Table 3 Dimensionless particle parameters as functions of the particle diameter dp: mass loading Mp =π6 d3

pρpNp/(πR2Hρ); dimensionless terminal velocity (or inverse dimensionless residence time in the cell)vt/U f ; Reynolds and Peclet numbers based on the terminal velocity, Ret and Pet; Kolmogorov-scaleStokes number StK ; large-scale Stokes number St f .

scale for the large-scale circulation, H/U f , to the Kolmogorov time scale,√ν/ε, with ε the energy dissipation rate. This

latter quantity can be estimated from the well-known exact relation valid for single-phase Rayleigh-Benard convection(see, e.g., Ahlers et al. 2009)

ε =ν3

H4RaPr2 (Nu − 1) , (43)

and, with the present parameter values, for single-phase convection, it has the value√ν/ε � 0.157 s. The Stokes numbers

corresponding to these two time scales are

St f =τpU f

H, (44)

and

StK = τp

√ε

ν. (45)

If St f < 1, in their fall the particles have time to adapt to the large-scale fluid velocity while, when St f > 1, theirmomentum tends to be little affected by the fluid. Similarly, when StK < 1, the particle can follow the fluctuating fluidmotion, while it cannot when StK > 1.

Analogous quantities can be defined to characterize the fluid-particles energy exchange, namely

Stt, f =τt,pU f

H, (46)

and

Stt,K = τt,p

√ε

κ. (47)

with τt,p the thermal relaxation time defined in (24). These thermal Stokes numbers are functions of the parameter J justas τt,p.

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10

20

30

40

50

60

70

10-6

10-5

10-4

10-3

10-2

Nu

h

J

0

200

400

600

800

1000

1200

10-6

10-5

10-4

10-3

10-2

Re

J

Fig. 2 Left: Nusselt number at the hot (lower) plate for particle diameters of, in ascending order, dp = 50, 100and 200 μm, as a function of the particle heat capacity expressed as the parameter J defined in (41).Right: Fluid Reynolds number (based the volume- and time-averaged r.m.s. velocity) for particlediameters, in ascending order, of dp = 100, 50 and 200 μm.

7.3. ResultsWe start by showing in figure 2 the hot-plate Nusselt number (left) and the fluid Reynolds number based on the r.m.s.

of the velocity magnitude in the cell, both as functions of the dimensionless parameter J defined in (41). The Nusseltnumber increases with the particle size and the lines are, in ascending order, for particles with diameters dp of 50 μm(solid circles, blue line), 100 μm (diamonds, red line) and 200 μm (squares, green line); the dashed horizontal line showsthe single-phase result. Here and in the figures that follow the asterisk placed on the left vertical axis shows the resultfor the case studied in Oresta & Prosperetti (2013) in which the particles maintain a fixed temperature so that, effectively,J = 0.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

z /

H

( < Tp >A,t - Tc ) / ( Th - Tc )

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

z /

H

( < Tp >A,t - Tc ) / ( Th - Tc )

Fig. 3 Area- and time-averaged normalized particle temperature as a function of height in the cell for particlediameters of 100 μm (left) and 200 μm. In both panels the red lines (triangles) are for J = 0, i.e., a fixedparticle temperature equal to that of the cold plate. From left to right the values of the parameter J definedin (41) are 0, 10−3 and 10−2.

For all particle diameters the heat transferred from the hot plate decreases as J increases, i.e., as the particle heatcapacity decreases. This decline of Nu occurs because, as J increases, the particles have time to warm up as they falltoward the bottom plate and, therefore, they are less and less able to extract heat from it. As expected, for each particlediameter the decrease begins when the volume fraction becomes comparable to the critical value α∗∗ defined in (42).

Unlike the Nusselt number, the dependence of the Reynolds number on the particle size (figure 2, right) is notmonotonic as the dp = 100 μm result (diamonds, red line) is somewhat lower than the 50 μm one for J < 10−3. We haveencountered the same phenomenon for constant-temperature particles in our earlier paper (Oresta & Prosperetti, 2013)where we have explained it by focusing on the regions near the bases of the cell, which is the where the energy source forthe convection lies. Here the particle and fluid velocities are approximately perpendicular to each other and the particlesimpose a drag force on the fluid which increases with the particle size. For large particles, this effect is compensatedby the general mixing induced by the particle motion. The end result is that, as the particle diameter increases, Re goes

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through a minimum. The smaller strength of the convective motion for the intermediate-size particles is not sufficient toovercome their larger thermal inertia, and this is why the Nusselt number is larger for dp = 100 μm than for dp = 50 μmin spite of the larger drag on the fluid motion. Of the three particle sizes considered here, only those with dp = 200 μmare present with a volume fraction which exceeds the estimate given in (39) and, indeed, of the three this is the only casein which the particles are seen to have a major effect on the Reynolds number.

Figure 3 shows the cross-sectional and time-averaged particle temperature as a function of height for J = 0, i.e.,constant-temperature particles, J = 10−2 and 10−3 for dp = 100 μm (left) and 200 μm. For J = 10−3 the smaller particlesheat up nearly as much as the larger ones for J = 10−2. As a consequeence, the Nusselt number for the former issignificantly smaller than that for the latter as shown in figure 2.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

z /

H

(< T >A,t - Tc ) / ( Th - Tc )

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

z /

H

( < T >A,t - Tc ) / ( Th - Tc )

Fig. 4 Normalized area- and time-averaged fluid temperature at the particle location for particle diameters ofdp = 100 (left) and 200 μm. From left to right the lines correspond to J = 0 (in which case the particletemperature remains equal to that of the cold plate), J = 10−3 and J = 10−2.

Figure 4 shows the area- and time-average fluid temperature at the particle location for dp = 100 (left) and 200 μmand J = 0, 10−3 and 10−2. For particles with a fixed temperature (J = 0) the cooling caused by the particles is substantialover the entire cell height except close to the hot lower plate. As J increases and, with it, the ability of the particles towarm up, the cooling due to the particles is strongest near the top cold plate where they are introduced. In the single-phasecase, with the fluid treated in the Boussinesq approximation as here, the bulk fluid temperature is close to 1

2 (Th + Tc), i.e.,(T − Tc)/(Th − Tc) � 0.5. This is approached by the 100 μm particles for J = 10−2, while the larger particles still retain asignificant cooling ability for this value of J.

0

0.2

0.4

0.6

0.8

1

-0.5 -0.4 -0.3 -0.2 -0.1 0

z /

H

Qp*

0

0.2

0.4

0.6

0.8

1

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

z /

H

Qp*

Fig. 5 Normalized area- and time-averaged particle heat flux, defined in (48), for particle diameters of dp = 100(left) and 200 μm. In ascending order the lines correspond to J = 10−2 (squares, green lines), J = 10−3

(circles, blue lines) and J = 0 (triangles, red lines); for J = 0 the particle temperature remains equal to thatof the cold plate.

According to (20), the difference between the particle temperature and the fluid temperature at the particle positionis proportional to the heat absorbed by the particles. The average over the cross section and time of the non-dimensional

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form of this quantity, defined by

Q∗p =〈Nupi(Tpi − Ti)〉A,t

2(Th − Tc), (48)

is shown as a function of height in the cell in figure 5 for dp = 100 (left) and 200 μm. The heat absorbed is largest (mostnegative) for particles with J = 0, which maintain their initial temperature, and it becomes substantial near the lower hotplate where the temperature difference is largest. In the bulk the particles have a substantial effect on the fluid temperature,the temperature difference is much reduced and, with it, Q∗p. Near the top plate the particles are at their coldest while thefluid retains some heat and, correspondingly, the heat exchanged increases somewhat.

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

En

0

J

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

En

1

J

Fig. 6 First (left) and second Fourier modes of the kinetic energy in the angular direction for particle diameters,in ascending order, of dp = 100, 50 and 200 μm, as functions of the particle heat capacity expressed as theparameter J defined in (41).

The nature of the circulation in the cell can be investigated by considering the Fourier modes of the fluid velocitydefined by

En =π

βgH4(Th − Tc)

∫ R

0rdr

∫ H

0dz〈|un|2〉t (49)

where un is the n-th Fourier coefficient (in the angular direction) of the vector velocity field. Figure 6 shows the non-dimensional time-average energy of the first two modes. A pure n = 0 mode (left panel) would correspond to a toroidalaxisymmetric vortex, while a pure n = 1 mode would correspond to an annular vortex with a horizontal axis. We havenoticed in our earlier work that the toroidal mode is always associated with a lower energy state, and the present resultsconform with this conclusion. For dp = 100 μm, the energy of both modes (diamonds, red line) remains smaller than fordp = 50 μm, as expected from the behavior of the Reynolds number (figure 2, right). In some sense the effect of coldparticles can be described as bringing the cold temperature closer to the warm plate, which tends to decrease the “effectiveheight” of the cell and, with it, the effective Rayleigh number. Thus it is not surprising that, as J increases and the particlesget warmer, the energy of all modes increases. The energy of the n = 0 mode tends to remain smaller than that of then = 1, but the difference is larger for the dp = 200 μm particles. As explained in our earlier paper (Oresta & Prosperetti,2013), large massive particles drag fluid down with them. However, due to continuity, the cross-sectional average fluidvelocity must vanish, and this constraint “squeezes” the upward moving fluid to small regions of the cell. The upwardfluid velocity must therefore increase, which results in a marked high-velocity annular flow.

An illustration of this flow pattern is provided in figure 7 which shows, on the left, an instantaneous view of the three-dimensional trajectory of some randomly selected fluid particles in the cell and, on the right, a cross-sectional view of thevelocity distribution on the mid-plane; here dp = 200 μm and Ja = 10−2 and the color is keyed to the vertical velocity. Thegreat reduction of the cell volume occupied by the ascending fluid is quite evident here, as is the approximately annularstructure of the flow.

8. Bubbly Rayleigh-Benard convection

The large heat transfer associated with boiling is due to the absorption of latent heat to generate bubbles and tothe greatly enhanced convection due to the bubble rise. Here we address this latter aspect only, without attempting to

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Fig. 7 Left: An instantaneous three-dimensional view of the trajectory of some randomly selected fluid particlesin the cell for dp = 200 μm and Ja = 10−2; the color indicates the vertical velocity. Right: The verticalvelocity on the mid-plane cross-section of the cell for the case of the image on the left. Note in thesefigures the much smaller fraction of the cross section occupied by rising fluid.

model bubble nucleation and detachment. Another aspect in which the present study deviates from the actual boilingphenomenon is that we keep the number of bubbles Nb fixed, considering the two values Nb = 25,000 and 50,000. Whileunable to account for the full boiling heat transfer phenomenon, our study permits us to separate the two effects of bubblegeneration and enhanced convection thus helping to shed light on their relative importance. The results described in thissection are similar to those of our earlier papers (Oresta et al. 2009; Schmidt et al. 2011; Lakkaraju et al. 2011) exceptthat the Rayleigh number has been increased by an order of magnitude to 2 × 106.

The physical properties used in the simulation are shown in table 4 and correspond to those of water at 100 ◦C andnormal pressure. The bubble surface temperature is kept at saturation. The top plate is subcooled by 0.5 K and the bottomplate superheated by 0.5 K. This superheat is smaller than what would be necessary in most practical situations to causethe nucleation of bubbles, but it is sufficient to permit us to explore the role of bubbles in enhancing convection in thecell. It may also be noted that the use of a larger superheat would significantly increase the effective Rayleigh number andtherefore require a much smaller integration time step (see Lakkaraju et al. 2013).

Parameter ValueAspect ratio, 2R/H 1/2Prandtl number, Pr 1.75Rayleigh number, Ra 2 ×106

Bubble number, Nb 25,000 and 50,000Cell height, H 24 mmCell diameter, 2R 12 mmTemperature difference, Th − Tc 1 KVapor density, ρp 0.598 kg/m3

Liquid density, ρ 958 kg/m3

Liquid kinematic viscosity, ν 0.294 ×10−6 m2/sLiquid thermal diffusivity, κ 0.168 ×10−6 m2/sLiquid thermal expansion coefficient, β 0.752 ×10−3 K−1

Latent heat L 2.27×106 J/kgConstant-pressure liquid specific heat, c 4,219 J/kgFree fall velocity, U f 13.2 mm/s

Table 4 Summary of the fluid and particle properties used in the simulations.

With the superheat used in the simulations, the Jakob number defined in (31) would have the value 1.488 but, asin the particle case, we use it as a control parameter to explore the effect of buoyancy. If Ja = 0, the latent heat iseffectively infinite and the bubbles maintain the radius with which they are inserted in the system. Increasing Ja amountsto decreasing the latent heat and the bubbles grow more and more, generating a stronger and stronger convection in thecell with their buoyancy.

K

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0

20

40

60

80

0 0.1 0.2 0.3 0.4 0.5

Nu

h

Ja

0

250

500

750

1000

1250

1500

0 0.1 0.2 0.3 0.4 0.5

Re

Ja

Fig. 8 Hot plate Nusselt number (left) and Reynolds number based on the average liquid r.m.s. velocity asfunctions of the Jakob number for 25,000 (lower curves) and 50,000 bubbles.

For single-phase Rayleigh-Benard convection in the Boussinesq approximation, the mean liquid temperature in thebulk is close to the average between the cold and warm temperatures of the plates. This trend is strengthened in thepresent case by the fact that the bubbles maintain the saturation temperature, which equals this mean temperature. Ingeneral terms, therefore, the bubbles tend to grow in the warm liquid layer adjacent to the lower plate, travel upwardmaintaining essentially a constant radius, and condense (at least partially) in the top cool layer, from which they areremoved upon reaching the top plate.

Figure 8 shows the Nusselt and Reynolds numbers in the cell as the Jakob number is increased from 0 to 0.5. Inthese conditions, without bubbles, the single-phase Nusselt number has a value around Nu0 = 11.2. The addition of evenconstant-size bubbles (i.e., with Ja = 0) increases it to 14.4 and 20.1 for Nb = 25,000 and 50,000, respectively. A modestincrease of Ja to 0.01 is already sufficient to nearly double the Nusselt number, to 28.1 and 33.6, respectively, for Nb =

25,000 and 50,000, which further grows to 58.7 and 70.2 for Ja = 0.5. A similar trend is visible in the Reynolds number,also shown in figure 2: there is an initial, very rapid growth followed by a less steep, but still substantial, further growth.

0

0.002

0.004

0.006

0.008

0 0.1 0.2 0.3 0.4 0.5

α

Ja

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5

< v

b >

V,t /

vbin

j

Ja

Fig. 9 Average volume fraction (left) and bubble volume normalized by the initial value for Nb = 25,000 (circles,blue line) and 50,000 (diamonds, red line).

The origin of these trends is made clear by an analysis of the bubble volume. Figure 9 shows the time- and volume-averaged volume fraction as a function of Ja and the similarly averaged single-bubble volume. The mean volume fractionis a strongly increasing function of Ja, rising, for Ja = 0, from 1.88×10−5 for Nb = 25,000 and 3.76×10−5 for Nb =

50,000 to several tenths of 1% at Ja = 0.5. While substantial, these maximum values are still small enough to justify thepoint-bubble approximation used here.

Figure 9 also shows, on the right, the time- and volume-averaged mean bubble volume normalized by the initialvalue. It is seen here that the mean bubble size is considerably larger than the initial size at injection, and all the moreso as Ja increases. It is interesting to note that bubbles grow larger in the case with Nb = 25,000 rather than 50,000. Thereason is twofold: since the bubbles keep the saturation temperature, they are colder than the liquid near the bottom plateand absorb latent heat. Hence, their cooling effect when they are near the lower plate is stronger the larger Nb. Secondly,

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as Nb increases, the strength of the circulation also increases, which has the effect of thinning the boundary layer near thehot plate, which is the region where bubbles grow the most.

In view of the approximately linear dependence of volume fraction and bubble volume on Ja, the decline of the initialrapid growth rate of the Nusselt and Reynolds numbers of figure 8 must be the result of a process that tends to oppose thestrength of the circulation associated with larger and larger bubbles. A likely candidate for this negative feedback processresides in the fact that the bubble rise velocity is an increasing function of their volume. Since the number of bubblesis kept constant, the shorter the residence time of the bubbles in the cell, the faster the rate at which new bubbles mustbe introduced. As already noted before, growing bubbles in the hot lower liquid layer cool it and therefore reduce thestrength of the mechanism that drives the circulation.

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5

< v

b↑

>V

,t /

vbin

j

Ja

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

< N

b↑

>V

,t /

Nb

Ja

Fig. 10 Left: Time- and volume-averaged bubble volume conditional on the velocity of the surrounding liquidbeing upward. Right: Mean fraction of bubbles associated with rising liquid.

As expected, bubbles residing in the rising, warmer liquid have a larger volume than the average, as shown in the leftpanel of figure 10, which is the bubble volume averaged over time and over the cell volume conditionally on the liquidvelocity being upward at the bubble location. Here again we observe that the mean bubble volume for Nb = 25,000 islarger than for Nb = 50,000. The right panel in figure 10 shows the fraction of bubbles associated with ascending liquid.For small Ja, the bubble size is fairly small and the bubbles tend to follow the liquid motion. Thus, the very small bubblesintroduced at the bottom plate are swept up by the liquid current and tend to remain in the rising liquid stream. As Jaincreases, however, buoyancy increases and the bubbles are less dependent on the liquid motion. Asymptotically, themotion of very rapidly rising bubbles would be completely independent of the liquid motion and, since the average liquidvelocity must vanish by continuity and the bubbles are introduced randomly, one would expect that half the bubbles wouldbe associated with liquid moving up and half with liquid moving down. These considerations explain why the number ofbubbles associated with ascending liquid falls as Ja increases.

The increase of Nu with Ja can be regarded as due to the increase of an “effective” thermal expansion coefficient βe f f

of the liquid. With the present value of the Rayleigh and Prandtl numbers one expects a scaling Nu∼Ra1/4 Pr1/8 (see e.g.Ahlers et al. 2009) and we can therefore deduce βe f f by rearranging (32) as

βe f f

β=

(NuNu0

)4, (50)

with Nu0 the single-phase Nusselt number. The results given by this relation are shown in figure 11.Figure 12 permits a comparison of two instantaneous three-dimensional snapshots of the vertical fluid velocity in the

cell for Ja = 0 and Ja = 0.5 with 50,000 bubbles. For Ja = 0 there is a very clear toroidal mode, with the liquid descendingalong the axis and rising along the wall of the cell. For Ja = 0.5, the descending fluid has been pushed toward the walland a tendency toward a transition to an annluar mode becomes manifest. As stated earlier, the toroidal mode has a lowerenergy than the annular one, in keeping with the very different values of the Nusselt numbers shown in figure 8 for thesetwo cases

9. Conclusions

In the present paper we have presented a summary of the type of results that one can obtain by using an extended

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0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5

β eff

/ β

Ja

Fig. 11 The normalized effective theraml expansion coefficient βe f f defined in (50) vs Ja for 25,000 (circles andblue line) and 50,000 (diamonds and red line) bubbles.

Fig. 12 Instantaneous three-dimensional snapshots of the vertical liquid velocity in the cell for Ja = 0 (left) andJa = 0.5 with 50,000 bubbles. Note the low-energy toroidal vortex in the first case and the onset of ahigher-energy annular component in the second one.

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point-particle or point-bubble model together with the Boussinesq approximation in the study of Rayleigh-Benard con-vection. For the particle case the fluid simulated has been air while, for bubbles, we have considered water at the normalboiling temperature. We have used as a parameter the ability of the particles or bubbles to absorb heat from the fluid and,by varying this parameter, we have found a wide variety of behaviors.

The introduction of particles and bubbles has the effect of increasing considerably the number of dimensionlessparameters that characterize the system and produces a wealth of different physical effects the investigation of which hasjust begun. Some phenomena, such as the strongly increased circulation caused by growing bubbles, are expected. Butothers, such as the non-monotonic behavior of the circulation strength as the particle size is varied, appear counter-intuitiveand their explanation has required delving into the detailed physics of several distinct physical processes.

This type of phenomena is important in natural processes, such as the dynamics of clouds and rain, as well as intechnology, boiling being the primary example, but we are still quite far from a level of understanding comparable withthat reached for single-phase Rayleigh-Benard convection. The experimental study of these phenomena is also in itsinfancy (see e.g. Zhong et al. 2009). It appears therefore that this area is mature for a systematic investigation and manymore results can be expected in the near future.

Acknowledgments

The authors are grateful to Professors D. Lohse and R. Verzicco for several useful suggestions. We acknowledgethe kind support provided by the IT staff of the Centro Cultura Innovativa d’Impresa, University of Salento, where thecomputations where carried out. PO and FF gratefully acknowledge support by FIRB under grant RBFR08QIP5 001 andAP support by NSF under grant CBET 1335965.

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