Heat transport in two-phase vertical natural convection using an Euler–Lagrange approach

21st Australasian Fluid Mechanics Conference

Adelaide, Australia

10-13 December 2018

Heat transport in two-phase vertical natural convection using an Euler–Lagrange approach

C. S. Ng

1

, V. Spandan

1

, D. Lohse

1

and R. Verzicco

1,2

1

_{Physics of Fluids Group, Max Planck Center for Complex Fluid Dynamics, J. M. Burgers Center for Fluid Dynamics and}

MESA+ Research Institute, Department of Science and Technology, University of Twente, P.O. Box 217, 7500 AE

Enschede, The Netherlands

2

_{Department of Mechanical Engineering, University of Rome ‘Tor Vergata’, Italy}

Abstract

We present numerical results of dispersed droplets in vertical

natural convection (VNC) flow, which is a buoyancy driven

flow between differentially heated vertical walls. Our focus is

to study the effects of droplets on the local statistics of heat

transport in natural convection, where heat transport

enhance-ment due to bubbles has recently been reported [5]. Our

nu-merical simulations are fully-resolved and based on an Euler–

Lagrange approach with two phases: the first is the carrier phase

(liquid), which is solved by a second-order accurate

finite-difference scheme and marched in time using a fractional-step

approach; the second is the dispersed phase (droplets) that are

much larger than the Kolmogorov length scale. The interfacial

droplet boundaries and deformations are modelled by an

im-mersed boundary method and an interaction potential approach,

respectively. We show that the heat flux is slightly enhanced

for the Rayleigh number range 1.3 × 10

8

_{–2.3 × 10}

9

_{and Prandtl}

number of 7, which can be attributed to droplet induced mixing.

Introduction

The motions of bubble columns agitate the surrounding liquid

and can give rise to enhanced heat and momentum transport.

This phenomenon is important for various engineering

applica-tions, for example in stirred tank reactors [4], water treatment

plants, steel industry [6] and mechanical flotation cells [10]. In

this study, we focus on the influence of bubble columns on the

heat transport within a liquid. Our model setup is the VNC flow

(figure 1a), similar to [11, 8, 9]. The bubbles are idealised as

light droplets and are allowed to freely rise within the domain

(figure 1b). The underlying mechanism of interest is droplet

induced mixing, generally assumed to be controlled by (i) the

capture and transport by droplet/bubble wake, and (ii)

disper-sion by droplet/bubble-induced turbulence [1].

Since we are interested in large droplets, i.e. droplet diameter

larger than the Kolmogorov length scales, fully-resolved

simu-lations are necessary in order to account for the inhomogeneous

hydrodynamic forces acting at the droplet interface. To achieve

this, we perform direct numerical simulations (DNS) coupled

with the immersed boundary method (IBM) and interaction

potential approach, which are versatile numerical

methodolo-gies to simulate fully-coupled fluid flows with deformable

in-terfaces, e.g. [13, 7]. IBM offers some computational

advan-tages over existing numerical methods for multiphase flows

(e.g. volume-of-fluid, level-set and front tracking).

For

in-stance, in IBM, the underlying discretised grid is fixed and no

sharp interfaces need to be resolved [12]; this method therefore

allows simulations of multiphase flows to be easily scaled-up.

Flow Setup

The setup for single-phase VNC is a buoyancy driven flow

con-fined in a cell (see figure 1a), with hot and cold vertical walls (at

z/L

z

= 0 and 1) and also adiabatic horizontal walls (at x/L

z

= 0

and 2.4). The flow is governed by the continuity equation,

(b)

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(a)

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z

g

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z/L

z

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y/L

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z

−0.

2

−0.

1

0

0.1

0.2

Hot

Cold

Figure 1: Schematic of vertical natural convection (VNC) in a

cell, visualising the temperature field: (a) single-phase VNC,

and (b) two-phase VNC. The hot and cold walls are at z/L

z

= 0

and 1, respectively. g is the direction of gravity.

∂

i

u

i

= 0, and the momentum and temperature equations are

re-spectively given by,

∂

t

u

i

+ u

j

∂

j

u

i

= −

1

ρ

re f

∂

i

p

+ δ

i1

gβ(θ − θ

re f

) + ν∂

2

j

u

i

+ f

i

, (1)

∂

t

θ

+ u

j

∂

j

θ

= κ∂

2

_j

θ,

(2)

where ∂

t

≡ ∂/∂t, ∂

i

≡ ∂/∂x

i

, (i, j = 1, 2, 3) and repeated indices

imply summation. We define θ

re f

as the reference temperature,

β as the thermal expansion coefficient of the fluid, ν as the

kine-matic viscosity and κ as the thermal diffusivity, all assumed to

be independent of temperature. The aspect ratio of the setup

Γ

= L

x

/L

z

is chosen to be 2.4, which models the dimensions

of the water column experiment performed by [5]. No-slip and

no-penetration boundary conditions are imposed on the

veloc-ity at all walls. Periodic boundary conditions are imposed on u

i

,

p

and θ in the y-direction. The Rayleigh, Nusselt and Prandtl

numbers are respectively defined as

Ra

≡ gβ∆L

3

z

/(νκ),

Nu

≡ f

w

L

z

/(∆κ),

Pr

≡ ν/κ,

(3)

where, ∆ ≡ T

h

− T

c

is the temperature difference between the

vertical walls, f

w

≡ κ|dθ/dz|

w

is the wall heat flux and

(·)|

w

denotes the wall value. For all simulations, we set the Pr value

equal to 7, corresponding to water. For confined thermal

con-vection problems, the aspect ratio Γ is also an additional

param-eter (see for example in [15]), however, in this study we restrict

our analysis to the Rayleigh number defined based on L

z

and for

a fixed Γ value. In equation (1), f

i

is a time-dependent source

term resulting from the immersed boundaries, which imposes

back-reaction forces from the dispersed droplets phase onto the

liquid carrier phase.

In this study, we consider droplets that are one order of

magni-tude larger than the Kolmogorov length scale η ≡

(ν

3

_/hεi)

1/4

_,

where hεi ≡ νh(∂u

i

/∂x

j

)

2

i is the volume-averaged kinetic

en-ergy dissipation rate and h·i denotes volume-averaged

quanti-ties, and so, the hydrodynamic forces acting on the droplet are

inhomogeneous along the droplet interface and need to be fully

resolved. To accomplish this, the droplet interface is

discre-tised as an unstructured mesh that is composed of triangular

el-ements [12, 3], which are dynamically linked as a spring-mesh

system with the aim of minimising the total potential energy of

the mesh. This approach is referred to as the interaction

poten-tial approach. Then, the external (hydrodynamic) forces from

the fluid and internal forces from the mesh are calculated based

on the centroid of the triangular elements, i.e. the Lagrangian

markers. For additional details of the interaction potential

ap-proach, we refer the reader to [12]. The relevant interaction

potential parameters used in this study are reported in the next

section.

Numerical Details

The liquid phase is solved using direct numerical simulations

(DNS) by a second-order accurate finite-difference scheme and

marched in time using a fractional-step approach [14]. The

streamwise, spanwise and wall-normal domain sizes are L

x

=

9.6L

y

= 2.4Lz, whereas Ra = 1.3 × 10

8

-2.3 × 10

9

. The grid

spacing is equal in the x- and y-directions, and is stretched

us-ing a clipped-Chebychev-type clusterus-ing in the z-direction in

order to resolve the steep near-wall gradients at the hot and cold

walls. The resolutions are selected such that max[∆x

i

/δ

ν

]

. 3.3

(see table 1), where ∆x

i

are the grid spacings in each ith

direc-tion and δ

ν

≡ ν/u

τ

is the viscous length scale which is based on

the shear velocity scale u

τ

≡ (ν|du/dz|

w

)

1/2

. Here,

(·) denotes

xy-plane-averaged statistics.

For the dispersed phase, the volume fraction, α

= 5×10

−3

. The

deformability of the droplets is set by the Weber number, We ≡

ρU

_∆

2

d/σ, which measures the inertia forces relative to surface

tension forces, where ρ is the density of the liquid phase, U

∆

≡

(gβ∆L

z

)

1/2

is the free-fall velocity, d is the droplet diameter

and σ is the surface tension. Because σ is an additional input

parameter for our study, it is prescribed by tuning the constants

of the phenomenological spring-mesh system described in the

previous section. According to the selection criteria described

in [12], we select the following dimensionless spring constants:

the in-plane elastic constant k

e

∗

= 0.03, the bending constant,

k

∗

_b

= k

∗

e

/60, volume constant k

∗

v

≈ 3 × 10

3

k

e

∗

and area constant

k

∗

a

= k

∗

e

. For this study, We ≈ 3 × 10

−2

, corresponding to a

relatively stiff droplet. In addition, we set the density ratio of

the droplet relative to the fluid, ˆρ ≡ ρ

d

/ρ = 0.99.

The simulations are split into two stages. First, the single-phase

simulations are run for at least 200 dimensionless turnover

times, where a turnover time is defined by the free-fall period,

L

z

/U

∆

, so that a statistically stationary state is first achieved.

Then, the droplets are placed randomly inside the domain,

where each droplet is initiated with a vertical velocity

compo-nent that corresponds to its respective height thus mimicking the

velocity distribution of rising bubbles in a water column. Upon

reaching 2 grid points from the top boundary, the droplets are

removed and placed randomly at the bottom of the domain–this

approach guarantees a constant volume fraction throughout our

simulations. The statistics are sampled for at least 100

dimen-sionless turnover times (the first 100 dimendimen-sionless turnover

times are discarded).

Presently, we restrict our simulations to droplets with interfaces

that are surfactant-contaminated (no-slip) and thermally passive

(no temperature boundary condition) as these conditions can

Ra

α

∆x

+

∆y

+

∆z

+

_min

∆z

+

_max

T

s

U

∆

/L

z

(×10

8

₎

_(×10

−3

₎

1.3

-

1.1

1.1

0.17

1.6

202

2.3

-

1.4

1.3

0.21

2.0

289

4.1

-

1.7

1.6

0.26

2.5

400

7.2

-

2.1

2.0

0.32

3.1

400

13.0

-

1.8

1.7

0.19

2.6

300

23.0

-

2.2

2.1

0.23

3.3

200

1.3

5.0

0.7

0.7

0.08

1.1

240

2.3

5.0

0.9

0.9

0.10

1.4

240

4.1

5.0

1.1

1.1

0.12

1.7

228

7.2

5.0

1.4

1.4

0.15

2.1

220

13.0

5.0

1.7

1.7

0.19

2.6

127

23.0

5.0

2.1

2.1

0.23

3.2

148

Table 1: Simulation parameters of the present DNS cases. The

number of grid points are n

x

× n

y

× n

z

= 960 × 96 × 384 for

all cases, except for Ra

6 7.2 × 10

8

_{for the single phase cases,}

where n

x

× n

y

× n

z

= 640 × 64 × 256. T

s

is the simulation

sam-pling time.

be handled easily from a numerical point-of-view. But, even

with these simplified boundary conditions, we emphasise that

the simulations are still two-way coupled. The diameter of the

droplet d

= 0.08L

z

and the number of grid points for the

Eu-lerian grid are chosen such that the droplet diameter is at least

20 times larger than the Eulerian grid spacings, which occurs

in the bulk of the cell due to grid stretching. Thus, the same

resolutions (n

x

× n

y

× n

z

= 960 × 96 × 384) are employed for

the two-phase simulations. Since d is kept constant, d/η ≈ 8

for the lowest Ra and and ≈ 19 for the highest Ra in our study.

In addition, the values of the Reynolds number of the droplet,

Re

d

≡ (u

d

− hui)d/ν lie in the ranges ≈15 to 70 for the lowest

Ra

and ≈70 to 300 for the highest Ra, where u

d

is the vertical

velocity of the droplet (hui is zero, statistically). Note that the

current Re

d

values are much lower than in physical experiments

(e.g. Re

d

≈ 600 as in [5]). For all simulations with a dispersed

phase, each droplet is discretised using 5120 Lagrangian

mark-ers. The simulation parameters for this study are summarised in

table 1.

Influence of Buoyant Droplets on Mean Statistics

To establish a reference, the single phase mean temperature

pro-files

(θ − θ

re f

)/∆ and single phase mean vertical velocity

pro-files u/U

∆

are shown in figure 2. Similar to VNC in a

doubly-periodic channel [8], the mean profiles of the single phase case

are statistically antisymmetric about the xy-plane at z

= 0.5L

z

.

However, when VNC is confined in a cell, u|

z/L

z

=0.5

is

statisti-cally zero with no persistent shear. In the cell centre, the vertical

velocity is stabilised by a vertical temperature gradient, as can

be seen in the visualisation in figure 1a.

With α

= 5 × 10

−3

, we observe subtle changes in the mean

ve-locity and temperature profiles (see dashed lines in figure 2).

The maxima and minima of the near-wall mean velocities are

farther from the walls at lower Ra and some distortion of the

mean profiles can also be observed, which is consistent with

the distortions in the two-phase mean temperature profiles

re-ported in [5]. In particular, we observe some attenuation of the

near-wall overshoots (deviations from the mean bulk velocity

and mean bulk temperature, e.g. at z/L

z

≈ 0.04 in figure 2a),

suggesting that mixing mechanisms in VNC are important even

with the introduction of light drops.

0

0.05 0.1 0.15 0.85 0.9 0.95

1

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0

0.05 0.1 0.15 0.85 0.9 0.95

1

-0.50

0.0

0.50

Figure 2: (a) Profiles of mean vertical velocity for pure VNC

(solid lines) and multiphase VNC (dashed lines).

(b)

Pro-files for mean temperature.

Note that because the bulk

re-gion is statistically zero, only near-wall statistics are shown, i.e.

0

6 z/L

z

6 0.15 and 0.85 6 z/L

z

6 1. The influence of the

two-phase VNC is subtle.

When α

= 5 × 10

−3

, the near-wall temperature gradients

be-come steeper at higher Ra values. These trends are not easily

seen in figure 2b, but we can inspect the trends by plotting the

compensated values of Nu, as shown in figure 3. Note that since

the droplets are thermally passive in our simulations, it can be

shown that

(dθ/dz)|

z/L

z

=0

= (dθ/dz)|

z/L

z

=1

= const. by

inte-grating equation (2) over the domain. Therefore, we compute

Nu

for the droplets-laden VNC by averaging the temperature

gradients at the hot and cold walls, in accordance with

equa-tion (3). The values of Nu is comparable at lowest Ra but with

increasing Ra, Nu increases by up to 10% relative to the

single-phase Nu. For comparison, the Nu values for the single single-phase

experiments of [5] are also plotted in figure 3; these

experimen-tal values are much higher than the single phase results of the

present study because the liquid in the experiments has a free

surface.

Although we observe an enhancement in Nu, the relative

in-crease is less prominent compared to experiments (for example,

[5] reported approximately one order of magnitude increase in

Nu). To explain this difference, we define the droplet Froude

number

Fr

2

≡

U

2

∆

gL

z

α/( ˆρ)

=

Ra

Ra

d

(4)

where Ra

d

≡ αgL

3

z

/(νκ ˆρ) is the droplet Rayleigh number,

which is inspired by the analogy with thermal convection (see

for example in [2]). The Froude number characterises the

rel-ative strength of thermal driving to the driving of the flow by

the rising droplets. In the experiments of [5], Fr

2

≈ 1.5 × 10

−6

,

whereas in our simulations Fr

2

_{= 100. Evidently, the present}

simulations are dominated by thermal buoyancy and the flow is

in a different regime than the experiments of [5].

10

8

10

9

10

10

0.24

0.28

0.32

0.36

0.4

0.44

0.48

Figure 3: Plot of compensated Nu versus Ra for the pure VNC

case (open circles) and dispersed VNC case (crosses). With

buoyant droplets, the VNC flow exhibit an increase in Nu at

higher Ra, but still lower than the Nu trends of pure VNC with

free-slip boundary conditions, as shown by the experimental

and DNS data from [5] (solid blue and solid red symbols,

re-spectively).

Influence of Buoyant Droplets on Turbulent Statistics

0

0.05

0.1

0.15 0.85

0.9

0.95

1

0

0.02

0.04

0.06

0

0.05 0.1 0.15 0.85 0.9 0.95

1

0

0.05

0.1

0.15

0.2

0.25

Figure 4: Similar plots as in figure 2, but now for (a)

root-mean-square of vertical velocity fluctuations u

0

rms

≡ (u

0

u

0

)

1/2

, and (b)

root-mean-square of temperature fluctuations θ

0

_rms

≡ (θ

0

_θ

0

₎

1/2

_.

The two-phase flow has a more pronounced effect on the

turbu-lent statistics: in (a) vertical velocity fluctuations are increased

due to larger intensity of vertical liquid fluctuations [1], in (b)

temperature fluctuations are lower due to capture and transport

of colder fluid in the droplets’ wakes.

Moving on to the influence on the fluctuating quantities, in

fig-ure 4a we find that the root-mean-square (r.m.s. ) of the vertical

velocity fluctuations, u

0

rms

≡ (u

0

u

0

)

1/2

, are increased not only

in the near-wall regions but also in the bulk of the flow. The

magnitudes of the maxima close to the colder wall are between

6%–26% higher than the maxima close to the hotter wall. A

straightforward explanation of the higher magnitudes near the

colder wall is that velocity fluctuations are increased due to the

rising drops that oppose the downwards mean flow. In figure 4b,

the profiles of temperature r.m.s. are lower than the single phase

results. This can be explained by the capture and transport of

the colder fluid at the bottom of the cell by the droplet wakes,

as can be observed in figure 1b.

Droplet Distribution

Figure 5: Normalised p.d.f. of droplets distribution as a

func-tion of wall-normal locafunc-tion, averaged over all Ra cases. The

light blue shade shows the standard deviation of the p.d.f. at the

respective z-location.

In figure 5, we quantify the distribution of the droplets as a

function of wall-normal location z using a normalised

proba-bility density function (p.d.f.), which is averaged in x- and

y-directions as well as in time. The distribution is roughly close to

uniform throughout the domain, implying a relatively

homoge-neous distribution of droplets throughout the simulation domain

and presumably a uniform contribution to the vertical velocities

in the bulk flow, as seen in figure 4a.

Conclusions

Using light droplets, we investigated the influence of mixing on

the heat flux in two-phase VNC for a range of Rayleigh

num-bers from 1.3 × 10

8

_{to 2.3 × 10}

9

_{and Prandtl number of 7. Since}

the droplets are larger than the Kolmogorov scales, we perform

fully resolved direct numerical simulations using the IBM

cou-pled with the interaction potential approach.

The motion of the rising droplets distort the near-wall mean

statistics, and the influence becomes more pronounced at higher

values of Ra, as highlighted by the increasing Nu values. The

vertical velocity fluctuations are also increased compared to the

single-phase case. This increase is especially prominent in the

bulk region of the flow, which is linked to the homogeneous

dis-tribution of droplets. Collectively, we show that the presence of

light droplets with a small droplet Rayleigh number (as

com-pared to experiments) can contribute to mixing and enhanced

heat transport in VNC.

Acknowledgements

This work is part of the research programme of the

Founda-tion for Fundamental Research on Matter with project number

16DDS001, which is financially supported by the Netherlands

Organisation for Scientific Research (NWO). The simulations

were carried out on the national e-infrastructure of SURFsara, a

subsidiary of SURF cooperation, the collaborative ICT

organi-zation for Dutch education and research. We also acknowledge

PRACE for awarding us access to MareNostrum based in Italy

at the Barcelona Supercomputing Center (BSC) under PRACE

project number 2017174146.

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