Numerical simulation of a square bubble column usingDetached Eddy Simulation and Euler-Lagrange approachCitation for published version (APA):Masterov, M., Baltussen, M. W., & Kuipers, J. A. M. (2018). Numerical simulation of a square bubble columnusing Detached Eddy Simulation and Euler-Lagrange approach. International Journal of Multiphase Flow, 107,275-288. https://doi.org/10.1016/j.ijmultiphaseflow.2018.06.006

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International Journal of Multiphase Flow 107 (2018) 275–288

Contents lists available at ScienceDirect

International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

Numerical simulation of a square bubble column using Detached Eddy

Simulation and Euler–Lagrange approach

M.V. Masterov, M.W. Baltussen

∗, J.A.M. Kuipers

Multiphase Reactors Group, Department of Chemical Engineering & Chemistry, Eindhoven University of Technology, P.O. Box 513, MB Eindhoven 5600,

The Netherlands

a r t i c l e i n f o

Article history:

Received 26 March 2018

Revised 6 June 2018

Accepted 15 June 2018

Available online 18 June 2018

Keywords:

Bubble column

Turbulence

Computational Fluid Dynamics

Detached Eddy Simulation

Large Eddy Simulation

Euler–Lagrange

a b s t r a c t

To accurately simulate industrial sized bubble columns, the accurate prediction of the turbulent structures

is important. Currently used RANS and LES models cannot capture the dynamics of the bubble columns

or require a high grid resolution, respectively. In this work the Detached Eddy Simulation (DES) was used

to combine the advantages of RANS and LES approaches. The DES method was based on Spalart–Allmaras,

k − ε and k − ω SST turbulence models and was used to simulate gas–liquid flow for a bubble column

with a square cross-section. The results are compared with experimental data and Large Eddy Simulation

(LES) results, based on the Vreman and Smagorinsky subgrid-scale models. Profiles of turbulence kinetic

energy and average axial liquid and gas velocities are compared at three heights of the column: near the

sparger, in the middle and at the top of the column. The obtained results are in a very good agreement

with the experimental data and LES simulations, proving ability of DES to capture highly dynamic flow

motion and accurately predict main liquid characteristics in gas–liquid systems.

© 2018 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

1

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. Introduction

Bubble column reactors (BCR) are considered to be a very ver-

atile and efficient type of reactor in many chemical and bio-

hemical processes, such as: Fisher–Tropsch process, waste water

reatment and absorption. Despite its simplicity, a successful de-

ign and scale-up of BCRs requires a detailed understanding of

ultiphase fluid and gas dynamics, which can be obtained using

combined) experimental and numerical approaches, see Table 1 .

Because of the complex flow patterns and the high void frac-

ion it is often difficult to determine the main liquid and gas char-

cteristics experimentally. In large-scale systems, like pilot or in-

ustrial plants, non-intrusive techniques are not capable to capture

he dynamics of the flow in the bulk, due to the opacity of the liq-

id. While intrusive techniques might increase perturbations of the

ow and as a result, influence the data ( Roghair, 2013 ).

To overcome limitations of physical experiments, numerical ap-

roaches are frequently used for a detailed investigation of BCRs.

espite theoretical boundless abilities of numerical techniques, the

imulation of industrial (large scale) BCRs still passes a huge chal-

enge, due to limits in available computational power, quality of

∗ Corresponding author.

E-mail address: m.w.baltussen@tue.nl (M.W. Baltussen).

l

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ttps://doi.org/10.1016/j.ijmultiphaseflow.2018.06.006

301-9322/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article u

losures and models capabilities ( Spalart, 20 0 0 ). Moreover, large

cale BCRs are determined by large amount of gas (and/or solid)

nclusions, which induces turbulence motion on multiple scales

nd therefore increases complexity of the physical description of

he system even more.

The most frequently used technique for industrial scale simu-

ations of two-phase gas–liquid flows is the Euler–Euler (E–E) ap-

roach, which represents the continuous liquid phase and bubbles

s interpenetrating fluids. This approach has been proved to be

uitable for industrial applications, due to its relatively low com-

utational power requirements ( Xiaoping et al., 2017 ). However,

he method demands for careful formulation and treatment of clo-

ures for phases interactions and loses information on the scale of

he individual bubble. As a result it is difficult to take into account

reak-up or coalescence.

As an alternative, the Euler–Lagrange (E–L) approach with di-

ect tracking of the dispersed gas phase may be used. The most

opular method is the Discrete Bubble Modeling (DBM), which ex-

licitly treats the bubble-bubble interaction. However, this deter-

inistic E–L approach is computationally intensive and thus, can

ot be scaled up toward industrial scales ( Darmana, 2006 ).

It is of big interest to develop and investigate methods for

arge-scale systems which will allow to keep bubbles as individual

bjects. Based on the E–L method, Kamath et al. (2017) proposed

nder the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

276 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

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to use Direct Simulation Monte Carlo (DSMC) as an alternative to

DBM. The stochastic nature of DSMC handles bubble-bubble colli-

sions more efficiently in comparison to deterministic DBM. Thus,

the method can in principle be applied to a much larger amount

of bubbles.

Since bubble dynamics is directly coupled with dynamics of the

surrounding liquid it is very important to have a suitable method

for resolution of complex turbulent flows at large scales. Gen-

erally, simulations of large-scale systems are based on Reynolds-

veraged Navier–Stokes (RANS) equations in combination with the

E–E approach ( Laborde-Boutet et al., 2009 ), primarily because of

the low grid density requirements of RANS models in most areas

of the computational domain. However, the time-averaged nature

of RANS equations does not allow for an accurate representation

of highly dynamic flows in bubble column reactors. Therefore, they

may give only rough estimates of the key hydrodynamic parame-

ters. In addition, the semi-empirical nature of most RANS models

limits the area of their applicability, since internal constants were

calibrated only for a particular type of single-phase flows.

The more accurate method for turbulence modeling of BCRs

is Large Eddy Simulation (LES), which resolves large-scale turbu-

lent eddies and models unresolved turbulence using subgrid-scale

(SGS) models. The main disadvantage of LES is the requirement of

a very dense grid, especially in the near wall regions. Although the

method cannot be applied for industrial systems ( Spalart, 20 0 0 ), it

became very popular for simulations of lab-scale columns.

Deen et al. (2001) compared Smagorinsky SGS model with k − εRANS model for E–E simulations of a square bubble column, us-

ing commercial software Ansys CFX-4.3. They investigated the key

features the highly dynamic flow with Bubble Induced Turbulence

(BIT). The RANS model demonstrated an overestimation of the gas

and liquid velocities, while LES simulations provided a good match

with experimental data.

Milelli et al. (2001) also reported poor accuracy of the k − εmodel, which fails in prediction of turbulence kinetic energy and

dissipation rate in E–E simulations and leads to significant error in

the calculated time-averaged liquid velocity.

Zhang et al. (2006) compared Smagorinsky and Vreman SGS

models for the LES method using the E–E approach and experi-

mental data of Deen (2001) . Although the liquid and gas velocity

matched for both models, the Smagorinsky model showed overes-

timation of the eddy viscosity in the near wall regions, where the

Vreman model gave accurate results. In addition, the effect of the

Smagorinsky constant in Vreman model was studied. A good agree-

ment with experiments was found for the range C s = [0 . 08 . . . 0 . 1] ,

which is almost a factor two lower than the default value of the

Smagorinsky constant ( C s = 0 . 25 ).

Hu and Celik (2008) also reported that for gas–liquid systems,

modeled with the E–L approach, the Smagorinsky constant in the

t

Table 1

Other works on bubble columns.

Ref. Type

Deen (2001) Experiment and numerical modeling with LES/Sma

Chien k − ε model, C s = 0 . 1

Zhang et al. (2006) LES based on Smagorinsky and Vreman SGS model

Bai et al. (2011) Experiment and numerical modeling with LES/Sma

C s = 0 . 1

Darmana (2006) Numerical modeling with LES/Smagorinsky SGS mo

Ekambara and Dhotre (2010) Numerical modeling with k − ε, k − ω, RNG and RS

LES/Smagorinsky, C s = 0 . 12

Niceno et al. (2010) Numerical modeling with one-equation model for

McClure et al. (2016) Experiment, effect of sparger design and different s

Rollbusch et al. (2015) Experiment, investigation of effects of different ope

holdup.

GS model should be taken lower than for single-phase flows.

Bai et al., 2011 ) investigated Smagorinsky and Vreman SGS mod-

ls using the DBM. Results were compared with experimental data

rom Deen (2001) and showed that the Vreman model is in better

greement with experimental data. The constant in both models

as set to C s = 0 . 1 .

More fundamental results were reported by Labourasse et al.

2007) and Toutant et al. (2008) , who proposed new subgrid clo-

ures for two-phase LES and reported good agreement with DNS

imulation.

Ma et al. (2015, 2016) used Machine Learning to derive sim-

le models for average bubble-liquid flow quantities. Authors used

odel Averaging Neural Network and linear regression models.

rained systems showed good match with DNS results and demon-

trate good performance. However, obtained models are not yet ap-

licable to complex large-scale simulations.

To reduce high computational expenses of the LES in large-scale

ystems Spalart et al. (1997) proposed the Detached Eddy Simula-

ion (DES) approach, which combines RANS and LES in one model.

asood and Delgado (2014) were the first to study a bubble col-

mn using DES in combination with the E–E approach. The authors

mployed Shear Stress Transport (SST) k − ω model in Ansys CFX-

4.0 and reported quite good agreement with experimental mea-

urements from Deen (2001) .

The interest in the DES method is underpinned by its better ac-

uracy and lower grid requirements in comparison with turbulence

odels based on RANS equations and the LES method, respectively.

hus, DES allows to obtain accurate resolution of turbulent struc-

ures in large-scale systems, where RANS and LES fail or are not

pplicable. In the present work the quality of different DES models

Spalart–Allmaras, k − ε and k − ω SST), coupled with DSMC, is in-

estigated for a gas–liquid flow with respect to experimental data

eported by Deen (2001) .

The paper is structured as follows. In Section 2 treatment

f bubble dynamics is explained. Sections 3 –5 provide explana-

ion of the governing equations and used turbulence models. In

ection 6 the main aspects of DES are described. Section 7 is ded-

cated to implementation details and numerical methods. Results

re presented in Section 8 and finally conclusions are given in

ection 9 .

. Bubble dynamics

For the present work, the developed and validated DSMC

ethod of Kamath et al. (2017) , is used to resolve bubble

ynamics. In this method, the identification of the collision part-

er is treated in a stochastic manner, while collisions themselves

re based on hard sphere collision rules. As in the original paper a

wo-way coupling of the gas–liquid system is employed.

Coupling BCR type

gorinsky SGS model and E–E Square, 0.15 × 0.15 × 0.45 m

s, C s = [0 . 08 . . . 0 . 2] E–E Square, 0.15 × 0.15 × 0.45 m

gorinsky SGS model, E–L Square, 0.15 × 0.15 × 0.45 m

del, C s = 0 . 1 E–L Square, 0.15 × 0.15 × 0.45 m

M models and E–E Cylindrical, 0.30 × 0.90 m

(d × H)

sub-grid scale kinetic energy E–E Square, 0.15 × 0.15 × 0.45 m

uperficial velocities — Cylindrical, 0.39 × 2.00 m

(d × H)

rating conditions on gas — Cylindrical, 0.16 × 1.80 m,

0.30 × 2.63 m, 0.33 × 3.88 m

(d × H)

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 277

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S

The presence of the gas phase is accounted for by the voidage

nd momentum source terms in the governing equations for the

iquid phase. Six forces are applied to each individual bubble: grav-

tational force, far-field pressure force, drag force, lift force, virtual

ass and lubrication force. The wall, drag and lift forces coeffi-

ients are calculated according to Tomiyama (1998) . To account for

warm effects, the drag closure proposed by Roghair et al. (2009) is

mployed.

The DSMC method imposes some restrictions on the size of the

omputational cells used for discretization of the governing equa-

ions: the bubble size should be less than the cell size.

. Governing equations for the liquid phase

In the present work the system of full Navier–Stokes equations

s used to describe the dynamics of the liquid phase. Momentum

nd continuity equations are given by

∂ ρl αl u i

∂t +

∂ ρl αl u j u i

∂x j = −αl

∂ p

∂x i +

∂

∂x j αl τi j + αl ρl g i + F i (1)

∂αl ρl

∂t +

∂αl ρl u i

∂x i = 0 (2)

Here αl is the local liquid volume fraction. The quantity F i rep-

esents an external source term due to the presence of bubbles.

he viscous tensor τ ij is given by

i j = μe f f

(S i j −

2

3

S kk δi j

)(3)

here the effective dynamic viscosity μeff is calculated as a com-

ination of molecular and eddy viscosities

e f f = μ + μt (4)

hereas the rate-of-strain tensor is given by

i j =

1

2

(∂u i

∂x j +

∂u j

∂x i

)(5)

Since we consider isothermal flow the molecular viscosity μ, as

ell as liquid density ρ l and gas density ρg are constants in our

imulations.

The system of Eqs. (1) and (2) remains unclosed until the eddy

iscosity is determined. In the present work five different models

re used to calculate this field. Among them we use two subgrid-

cale models for LES method and three RANS models embedded

nto DES. These models will be described in the following sections.

. SGS models

LES simulations are conducted with the Smagorinsky (SMG) and

reman (VR) subgrid-scale (SGS) models. The SMG model was first

roposed in 1963 ( Smagorinsky, 1963 ) and is commonly accepted

n different applications. However, the SMG may give non-zero val-

es of the eddy viscosity at walls, since effects of solid surfaces

n the subgrid viscosity are not taken into account. The problem

an be overcome by modifying the initial model formulation or by

sing damping functions, which activate in the near wall regions

Piomelli et al., 1984 ). However, the primary interest of this work

s the flow characteristics in the bulk of the domain. Although the

rid refinement would eliminate the problem of the unresolved

oundary layer, it cannot be used in the current study due to re-

trictions imposed by the DSMC method. Therefore, the model was

tilized in its original formulation

t = ρ(C s ) 2 S (6)

here C s is a model constant taken equal to 0.1, is a cubic root

f the grid cell volume and S is a modulus of the rate-of-strain

ensor given by

=

√

2 S i j S i j , S =

1

2

(∂u i

∂x j +

∂u j

∂x i

)(7)

An alternative to the SMG model is the VR ( Vreman, 2004 ) SGS

odel, which has an improved treatment of the wall regions. The

uthor also claimed other advantages, e.g. lack of explicit filtering

nd averaging and rotational invariability. The VR model has been

roven to be accurate in simulations of highly dynamic gas–liquid

ows ( Zhang, 2007; Bai, 2010; Lau, 2013 ). The eddy viscosity is cal-

ulated with

t = 2 . 5 ρC 2 s

√

B β

αi j αi j

(8)

here

i j =

∂u j

∂x i , βi j = 2

m

αmi αm j (9)

β = β11 β22 − β2 12 + β11 β33 − β2

13 + β22 β33 − β2 23 (10)

Here m

is the cell size in m th direction ( Vreman, 2004 ).

. RANS models

.1. Spalart–Allmaras model

The Spalart–Allmaras (SA) model is one of the most simplest

ANS models available today. The model became very popular

ithin the last decades because of its accuracy in many CFD ap-

lications. The SA model consists of only one transport equation

ritten for the field of modified eddy viscosity ˆ ν, resulting in low

omputational expenses and relatively simple implementation.

The authors claimed that the model can be applied to both

ompressible and incompressible flows without any modifications.

n the conservative form Allmaras et al. (2012) the SA model is

iven by

∂ αl ρ ˆ ν

∂t +

∂ αl ρu j ̂ ν

∂x j = αl P

SA − αl D

SA +

1

σ

∂

∂x j

[αl ρ

(ν + ˆ ν

) ∂ ̂ ν

∂x j

]

+ c b2 αl ρ1

σ

∂ ̂ ν

∂x j

∂ ̂ ν

∂x j − (ν + ˆ ν)

1

σ

∂αl ρ

∂x i

∂ ̂ ν

∂x i

(11)

here production and destruction terms are defined by

SA = ρc b1 ̂ S ̂ ν (12)

SA = ρ

(c w 1 f w

− c b1

κ2 f t2

)(ˆ ν

d

)2

(13)

Here d represents the distance to the nearest wall. The modified

ddy viscosity is proportional to the dynamic eddy viscosity as

t = ρ ˆ ν f v 1 (14)

here

f v 1 =

χ3

χ3 + c 3 v 1 , χ =

ˆ ν

ν(15)

Additional terms in the Eq. (11) are given by

ˆ = � +

ˆ ν

κd 2 f v 2 , f v 2 = 1 − χ

1 + χ f v 1 (16)

278 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

Table 2

Spalart–Allmaras model constants.

Const Value Const Value Const Value

c b 1 0.1355 σ 2/3 c b 2 0.622

κ 0.41 c w 2 0.3 c w 3 2

c v 1 7.1 c t 3 1.2 c t 4 0.5

c 2 0.7 c 3 0.9 c w 1 c b1

κ2 +

1 + c b2

σ

Table 3

KE model constants.

Const Value Const Value

C μ 0.09 C 1 1.44

C 2 1.92 σ k 1.0

σ ε 1.3

e

P

τ

S

o

μ

a

w

R

T

t

t

m

i

t

E

D

5

S

T

a

b

a

f w

= g

[1 + c 6 w 3

g 6 + c w

3

6

]1 / 6

, g = r + c w 2 (r 6 − r) (17)

r = min

[ˆ ν

ˆ S κ2 d 2 , 10

](18)

� =

√

2 W i j W i j , W i j =

1

2

(∂u i

∂x j − ∂u j

∂x i

)(19)

The term

ˆ S is limited according to the recommendations given

in Allmaras et al. (2012)

S̄ =

ˆ ν

κ2 d 2 f v 2 (20)

ˆ S =

⎧ ⎨

⎩

� + S̄ , if S̄ ≥ −c 2 �

� +

�(c 2 2 � + c 3 ̄S )

(c 3 − 2 c 2 )� − S̄ , otherwise

(21)

Note, that ˆ S may become equal to zero, and therefore, to pre-

vent problems in the evaluation of Eq. (18) , the term r is explicitly

set to 10. Model constants are taken unchanged from the original

paper ( Spalart and Allmaras, 1992 ), see Table 2 .

At the walls the modified eddy viscosity is set to zero. At the

outflow and pressure prescribed boundaries, the Neumann condi-

tion is imposed.

To improve the stability of the model, the production term P is

treated explicitly, while the destruction term D is used implicitly

in the following form

D

SA =

(c w 1 f w

− c b1

κ2 f t2

)old

(ρ ˆ ν

d 2

)old

ˆ νnew

(22)

where subscripts “new” and “old” refer to data from new and old

time steps, respectively.

5.2. k − ε model

The k − ε (KE) model is one of the most popular turbulence

models in engineering practice because of its robustness and ac-

curacy. This model consists of two transport equations written

for the turbulence kinetic energy k and its dissipation rate ε. In

the present study the version of the model proposed by Lam and

Bremhorst (1981) is used

∂ ραl k

∂t +

∂ ραl u j k

∂x j = αl P

KE − αl D

KE k +

∂

∂x j

[αl

(μ +

μt

σk

)∂k

∂x j

](23)

2

∂ ραl ε

∂t +

∂ ραl u j ε

∂x j = αl C ε1 f 1

ε

k P KE − αl D

KE ε

+

∂

∂x j

[αl

(μ +

μt

σε

)∂ε

∂x j

](24)

The production and destruction terms can be calculated with

quations (25) .

KE = τi j

∂u i

∂x j , D

KE k = ρε, D

KE ε = C ε2 f 2

ρε2

k (25)

Reynolds stresses and the rate-of-strain tensor are given by

i j = μt

(2 S i j −

2

3

S kk δi j

)− 2

3

ρkδi j (26)

i j =

1

2

(∂u i

∂x j +

∂u j

∂x i

)(27)

The turbulent eddy viscosity is calculated from a combination

f model constants and the k and ε fields

t = C μ f μρk 2

ε(28)

According to Lam and Bremhorst (1981) , functions f μ, f 1 and f 2 re defined by

f μ = [ 1 − exp (−0 . 0165 Re y ) ] 2 [ 1 + 20 . 5 / Re t ] (29)

f 1 = 1 + (0 . 05 / f μ) 3 , f 2 = 1 − exp (−Re 2 t ) (30)

here dimensionless parameters Re t and Re y are given by

e t =

ρk 2

με, Re y =

ρ√

k d

μ(31)

Constants specified for the KE model are presented in the

able 3 .

At the walls kinetic energy of turbulence is set to zero, while

he Neumann condition ( (∂ ε/∂ x n ) w

= 0 ) is applied to the dissipa-

ion rate, as recommended by Patel et al. (1985) .

It is well-known that the robustness of the numerical imple-

entation can be increased if negative terms are treated implic-

tly and positive terms are treated explicitly ( Hirsch, 2007 ). Using

he same procedure as for the SA model the destruction terms in

qs. (23) and (24) can be re-written according to Eq. (32) .

KE k =

(ραl ε

k

)old

k new

, D

KE ε = C ε2

(ραl f 2 ε

k

)old

εnew

(32)

.3. k − ω SST model

The last RANS model considered in the present study is a Shear

tress Transport (SST) k − ω model proposed by Menter (1994) .

he model represents a combination of more fundamental k − εnd k − ω models and gives an accurate resolution of the turbulent

oundary layer, while the characteristics of the mean external flow

re captured well. In the present study the SST-2003 ( Menter et al.,

003 ) is implemented. The model is given by two equations for the

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 279

Fig. 1. Geometry of the numerical domain. Black regions show pressure outlet

faces. Black dots at the bottom represent nozzles. Lines of interest are indicated

with gray color.

t

P

E

τ

S

μ

w

h

F

Fig. 2. Time-average RANS/LES regions for SA (a), SST (b) and KE (c) turbulence models

modes, respectively. (For interpretation of the references to color in this figure legend, th

urbulence kinetic energy k and its specific dissipation rate ω

∂ ραl k

∂t +

∂ ραl u j k

∂x j = αl P

SST − αl D

SST k +

∂

∂x j

[αl ( μ + σk μt )

∂k

∂x j

](33)

∂ ραl ω

∂t +

∂ ραl u j ω

∂x j = αl

γ

νt P SST − αl D

SST ω +

∂

∂x j

[αl ( μ + σω μt )

∂ω

∂x j

]

+2(1 − F 1 ) ραl σω2

ω

∂k

∂x j

∂ω

∂x j (34)

The production and destruction terms are given by

SST = τi j

∂u i

∂x j , D

SST k = β∗ρωk, D

SST ω = βρω

2 (35)

The turbulent stress and rate-of-strain tensor are determined by

qs. (36) and (37) .

i j = μt

(2 S i j −

2

3

S kk δi j

)− 2

3

ρkδi j (36)

i j =

1

2

(∂u i

∂x j +

∂u j

∂x i

)(37)

The expression for eddy viscosity is given by

t =

ρa 1 k

max (a 1 ω, SF 2 ) , S =

√

2 S i j S i j (38)

here S is a modulus of the rate-of-strain tensor. The function F 2 ere is calculated by

2 = tanh

( [max

(2

√

k

β∗ωd ,

500 ν

d 2 ω

)]2 )

(39)

at XZ-plane, D = 0 . 075 m. Blue and dark red colors represent pure RANS and LES

e reader is referred to the web version of this article.)

280 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

Fig. 3. Comparison of experimental and simulated profiles of average axial liquid velocity at H = 0 . 1575 m (e–f), H = 0 . 25 m (c–d) and H = 0 . 3375 m (a–b). The results of

LES with different SGS models are presented on the left, while the results of DES models are presented on the right.

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 281

Fig. 4. Comparison of experimental and simulated profiles of average axial liquid

velocity at H = 0 . 25 m.

Fig. 5. Comparison of experimental and simulated profiles of average axial liquid

velocity at H = 0 . 25 m.

C

t

φ

w

F

ζ

a

C

(

P

Table 4

SST model constants.

Const Value Const Value Const Value

σ k 1 0.85 σω1 0.5 β1 0.075

σ k 2 1.0 σω2 0.856 β2 0.0828

β∗ 0.09 κ 0.41 a 1 0.31

γ 1 5/9 γ 2 0.44

T

i

w

l

ω

w

fi

m

(

p

t

t

g

6

s

w

t

m

w

m

t

s

L

w

g

s

R

b

s

s

6

i

S

w

d

S

g

f

c

i

d

onstants σ k and σω in Eqs. (33) and (34) are blended according

o Eq. (40) .

= F 1 φ1 + (1 − F 1 ) φ2 (40)

here the blending function F 1 is defined by

1 = tanh

( (min

[ζ ,

4 ρσω2 k

CD kω d 2

])4 )

(41)

= max

( √

k

β∗ωd ,

500 ν

d 2 ω

)(42)

nd

D kω = max

(2 ρσω2

1

ω

∂k

∂x j

∂ω

∂x j , 10

−10

)(43)

The production term in both Eqs. (33) and (34) is limited

Menter et al., 2003 ) as follows

SST = min (P SST , 10 β∗ρωK) (44)

he standard model constants used in the present study are listed

n the Table 4 .

At the walls, the turbulence kinetic energy k is set to zero

hereas the specific dissipation rate ω is calculated using the fol-

owing expression

wall = 10

6 ν

β1 (d 1 ) 2 (45)

here d 1 is the distance between the wall and the center of the

rst cell. At the outlet or pressure prescribed boundaries, the Neu-

ann boundary condition is imposed for both k and ω fields.

For the sake of stability, the destruction terms in Eqs. (33) and

34) are treated implicitly, while production terms are taken ex-

licitly. We found that the SST model may lead to divergence of

he solution, if the ω field is not properly initialized. Therefore, in

he present study, value ω wall (see Eq. (45) ) is used as an initial

uess.

. Detached Eddy Simulation

The main idea of the DES ( Spalart et al., 1997 ) is employing a

ingle turbulence model, which is acting as SGS model in regions

here the grid resolution is fine enough to resolve turbulent struc-

ures, while in other regions the model is used as a pure RANS

odel. This combination allows for a less refined grid near the

alls compared to LES, decreasing thereby the memory require-

ents and the computational costs. The DES procedure replaces

he RANS length scale ( Spalart et al., 1997 ), which is explicitly pre-

ented in most models, by a modified expression given by

DES = min ( L RANS , C DES ) (46)

here L RANS is an original turbulence length scale of a RANS model,

is the LES filter limit, which is generally equal to the maximum

rid cell size, and C DES is a constant analogous to the Smagorin-

ky one. The main advantage of the DES, compared to other hybrid

ANS/LES approaches ( Sagaut et al., 2006 ), is automatic transition

etween two branches of the model, as given by the Eq. (46) .

The following subsections explain the main aspects with re-

pect to the application of the DES procedure to RANS models, de-

cribed in Section 5 .

.1. DES-SA

In a case of the SA model, the RANS length scale L RANS

s determined by a distance to the nearest wall. Therefore,

palart et al. (1997) suggested to replace d in the original model

ith the modified distance to the nearest wall given by

ˆ = min

(d, C SA

DES )

(47)

hur et al. (1999) performed simulations of the decay of homo-

eneous isotropic turbulence on the basis of the experimental data

rom Comte-Bellot and Corrsin (1971) and determined C SA DES

as 0.65.

Replacing d with modified

ˆ d in every equation of the SA model

an easily lead to an unrealistic distribution of the eddy viscosity

n the numerical domain ( Breuer et al., 2003 ). A fast non-linear

rop of the subgrid viscosity can be generated by the activation

282 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

Fig. 6. Comparison of experimental and simulated profiles of turbulence kinetic energy at H = 0 . 1575 m (e–f), H = 0 . 25 m (c–d) and H = 0 . 3375 m (a–b). The results of LES

with different SGS models are presented on the left, while the results of DES models are presented on the right.

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 283

Fig. 7. Comparison of simulated profiles of average eddy viscosity at H = 0 . 25 m.

o

S

d

w

�

d

6

d

L

a

L

e

m

D

v

6

R

t

D

w

L

L

s

C

i

7

u

a

a

u

m

A

s

c

t

m

f

t

p

i

B

d

f

c

Y

o

m

p

o

d

w

s

a

a

o

r

m

E

w

t

a

t

D

b

b

v

w

v

f the low-Re terms in the LES regions. To overcome this problem,

hur et al. (2003) proposed to use the new LES length scale

ˆ = min

(d w

, �

(νt

ν

)C SA

DES

)(48)

here the limiting function � is given by

2 = min

(

100 , 1 − c b1

c w 1 κ2 f DES w

f v 2

f v 1

)

(49)

Here, term f DES w

denotes the model function f w

calculated usingˆ .

.2. DES-KE

The KE model consists of two equations and L RANS appears in

ifferent terms. The model length scale is given by

KE RANS =

k 3 / 2

ε(50)

nd Eq. (46) is re-written to

KE DES = min

(L KE

RANS , C KE DES

)(51)

According to Strelets (2001) only the dissipative term in the

quation of the turbulence kinetic energy (see Eq. (23) ) should be

odified as follows

KE k,DES = ρ

k 3 / 2

L KE DES

(52)

Strelets (2001) calibrated the model constant C KE DES

and obtained

ery good agreement with experimental data for the value 0.61.

.3. DES-SST

Similar to the KE model, the DES based on the SST replaces the

ANS length scale only in the dissipative term of the equation of

he turbulence kinetic energy (see Eq. (33) ) as follows

SST k,DES = ρ

k 3 / 2

L SST DES

(53)

here L SST DES

is determined by

SST DES = min

(L SST

RANS , C SST DES

)(54)

Here, the L RANS is given by

SST RANS =

k 1 / 2

β∗ω

(55)

Since SST uses the blending function F 1 the constant C SST DES

hould be calculated with Eq. (56)

SST DES = (1 − F 1 ) C

k −εDES + F 1 C

k −ω DES (56)

Strelets (2001) determined C k −εDES

= 0 . 61 and C k −ω DES

= 0 . 78 by cal-

brating constants for both k − ε and k − ω DES models, separately.

. Implementation details

All numerical simulations in the present study are performed

sing the new in-house code “FoxBerry”. Solutions are obtained in

full 3D formulation with implicit treatment of the convection

nd diffusion terms. Discretization is done using the Finite Vol-

me method on an uniform Cartesian grid with staggered arrange-

ent of the primary variables ( Versteeg and Malalasekera, 2007 ).

ll convective fluxes are discretized using the second-order Barton

cheme ( Centrells and Wilson, 1984 ), while diffusive fluxes are dis-

retized using the second-order central difference scheme. To keep

he computational stencil compact, a Deferred Correction (DC)

ethod is used ( Ferziger and Peric, 1999 ). The low-order scheme

or the DC method is the first order upwind scheme. Time integra-

ion is done using a first-order backward Euler’s method. For the

ressure–velocity coupling the SIMPLE algorithm ( Patankar, 1980 )

s used.

All linear systems built after discretization are solved using

iGStab(2) method ( Sleijpen and van der Vorst, 1995 ). No precon-

itioners are used for momentum and scalar equations since only a

ew iterations are required for convergence. However, for pressure

orrection equation an Algebraic Multigrid method ( Henson and

ang, 2002 ) is used as a preconditioner. All linear solvers are part

f the in-house library and have been tested on several single- and

ultiphase problems.

Fig. 1 shows the geometry of the numerical domain used in the

resent study. No-slip boundary condition is imposed on all faces

f the domain, except the top one, where free-slip boundary con-

ition is used. Few cells at the top of the domain’s vertical sides,

hich are indicated by the black regions in Fig. 1 , are set to pres-

ure outlet to prevent mass conservation problems.

The sparger is modeled using 7 × 7 nozzles arranged in a square

nd is located exactly in the middle of the bottom wall. Bubbles

re introduced into the domain simultaneously. Since no break-up

r coalescence model is used, bubble sizes are preserved.

The gas and liquid physical parameters as well as numerical pa-

ameters used for the current study are presented in Table 5 . Di-

ensionless parameters are calculated as follows

o =

| � g | d 2 b ρl

σ, Mo =

| � g | μ4 l ρ

ρ2 l σ 3

, Re b =

v d b ρl

μl

(57)

here d b is the bubble diameter, v is the maximum magnitude of

he average relative bubble velocity, σ is the surface tension, μl

nd ρ l are liquid dynamic viscosity and liquid density, respectively.

Three heights in the column (see gray lines, Fig. 1 ) are used

o compare results of the present work with experimental data of

een (2001) and numerical data from Masood and Delgado (2014) .

To obtain profiles of the average gas velocity, 30 averaging cu-

ic volumes are created along lines of interest. At each time step,

ubble velocities are volume-averaged for every cell as follows

=

∑ N i =0 v i �i ∑ N i =0 �i

(58)

here � is the volume of a bubble presented in the averaging cell,

is bubble velocity and N is the total amount of bubbles presented

284 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

Fig. 8. Comparison of experimental and simulated profiles of average axial gas velocity at H = 0 . 25 m (e–f), H = 0 . 2835 m (c–d) and H = 0 . 324 m (a–b). The results of LES

with different SGS models are presented on the left, while the results of DES models are presented on the right.

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 285

Fig. 9. Comparison of simulated profiles of average gas holdup at H = 0 . 25 m.

Table 5

Simulation parameters used for the bubbly flow simu-

lations in a square bubble column (see Fig. 1 ).

Parameter Value

Column size ( D × L × H ) 0.15 × 0.15 × 0.45 m

Liquid density, ρ l 10 0 0 m

3

Liquid dynamic viscosity, μl 1 . 002 × 10 −3 m · m

Gas density, ρg 1 m

3

Gas dynamic viscosity, μg 1 . 85 × 10 −5 m · m

Bubble diameter, d b 4 m

Superficial gas velocity, v s 4.9 m

Surface tension, σ 72 . 86 × 10 −3 m

Simulation time 200 m

Time step 10 −3 m

Averaging time 10 . . . 200 m

Grid size 30 × 30 × 90

Tolerance 10 −8

Eo 2.15

−log(Mo) 10.6

Re b 800

i

a

w

6

8

8

o

t

t

(

m

b

b

a

a

w

A

c

L

b

d

t

t

a

l

C

s

i

t

d

e

v

8

b

d

t

c

e

t

0

g

m

e

t

c

o

c

t

n

m

l

H

fi

h

s

s

n

p

i

t

M

H

c

d

b

t

t

i

C

o

w

t

p

0

F

n the cell at the time step under consideration. All-volume aver-

ged velocities are averaged in time with number averaging. The

idth of the averaging volume, which plays a crucial role, is set to

cm in order to obtain correct results.

. Results and discussion

.1. RANS/LES regions

For convenience, in the next sections abbreviation DES will be

mitted leaving only pure RANS model’s name as an indicator of

he type of simulation. First, dynamics of the DES branches are de-

ermined. Fig. 2 shows time-average fields of RANS (blue) and LES

dark red) modes at the vertical plane in the middle of the do-

ain. It was assumed that during simulations RANS regions will

e concentrated along walls of the domain and partially inside the

ubble plume, where large scale vortical structures are not present

nd small-scale turbulence cannot be accurately resolved.

The SA model ( Fig. 2 (a)) shows the expected pattern, which is

lso in line with Eq. (48) . Fig. 2 (b) demonstrates that the SST model

as intensively used in RANS mode in the region near the sparger.

lso, a very thin layer of RANS branch can be seen along walls.In

ontrast, the KE model shows very dynamic behavior of RANS and

ES regions. The difference between all models can be explained

y different expressions of RANS length scales, i.e. the SA model

etermines L RANS as a simple function to the nearest wall, while

he KE and SST models calculate L RANS based on the local fields of

he turbulent quantities (see Eq. (50) and (55) ). In addition, C DES is

fixed constant in the SA and KE models, leading to a fixed LES

ength scale during the entire simulation, while in the SST model

DES is a function changing in the time.

The SA model tends to produce a laminar solution, when the

ystem is initialized at rest and without presence of any bubbles

n the domain. Therefore, the model is initialized from fields ob-

ained with the Vreman model after simulating 2.5 s. Since flow

isturbances were necessary only as an initial guess the modified

ddy viscosity field ˆ ν was approximated as 10% of the LES eddy

iscosity field.

.2. Liquid velocity

Fig. 3 shows profiles of the average axial liquid velocity. As can

e seen from Fig. 3 (a), (c) and (e) the SMG model leads to un-

erestimation of the liquid velocity along the middle line at all

hree heights. This might be resolved by a calibration of the model

onstant C s . The VR model shows very good agreement with the

xperimental data at the middle height, H = 0 . 25 m, and overes-

imates axial liquid velocity in the region near the sparger, H = . 1575 m. At the top, H = 0 . 3375 m, both the SMG and VR models

ive similar results. Concluding, the VR model is the most accurate

odel and therefore, will be used as a reference LES simulation.

Fig. 3 (b), (d) and (f) show results obtained with DES mod-

ls. The SA model significantly overestimates liquid velocity for all

hree heights at the center of the domain. Therefore, it is con-

luded that the bubble plume remains narrow and almost does not

scillate. The current implementation of the SA model is not appli-

able for simulations of highly dynamic gas–liquid systems, since

he model demands a high grid density near the wall, which could

ot be satisfied in the current study due to DSMC restrictions as

entioned in Section 2 .

Both the KE and SST models slightly overpredict axial liquid ve-

ocity along the middle line of the domain, at H = 0 . 1575 m and

= 0 . 25 m, but produce better match with the experimental pro-

le at H = 0 . 3375 m, in comparison with LES simulations.

The difference between numerical and experimental results at

eight H = 0 . 3375 m is mainly related to the absence of the free

urface in the present simulations, which leads to changes in pres-

ure distribution at the top of the domain and fully ignores the dy-

amics of the free surface caused by the bubbles leaving the liquid

hase.

The results of Masood and Delgado (2014) demonstrate signif-

cant deviation with the experimental points and current simula-

ions in the near wall regions. However, the velocity profile from

asood and Delgado (2014) is closer to the experimental data at

= 0 . 3375 m, which is again related to different type of boundary

ondition imposed at the top of the domain in both studies. In ad-

ition, at all heights, profiles from Masood and Delgado (2014) are

iased to the left. The reason of this offset is most probably related

o the short period of time-averaging.

The deviation between results obtained with SST models from

he present and reference simulations is caused by differences in

mplementation. In the present study the original blended constant

SST DES

, proposed in Strelets (2001) , was employed, while in the work

f Masood et al. C SST DES

was fixed to the value of 0.99. Besides, the

ork of Masood and Delgado (2014) employs different discretiza-

ion schemes and different approaches for description of phase-

hase interaction.

Comparison of the average axial liquid velocity at height H = . 25 m, obtained with the SST, KE and VR models, is shown in

ig. 4 . Although the VR model is in better agreement with the

286 M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288

Fig. 10. Instant snapshots of the bubble plume for the tested turbulence models at times t = 25 s, 40 s and 55 s. From left to right the results for the SA, SST, KE and VR

models are shown. Bubbles are colored with the magnitude of their velocities, where blue and red colors represent low and hight values, respectively. (For interpretation of

the references to color in this figure legend, the reader is referred to the web version of this article.)

c

e

8

i

e

w

T

l

T

experiment in the middle, it underestimates velocity magnitude on

the right, where KE and SST provide a very good match with the

experimental points. Moreover, the KE model leads to a more cen-

tered profile compared to other models.

Comparison of DES-based results with other LES-based sim-

ulations is shown in Fig. 5 . As a reference data, results from

Deen (2001) ; Darmana (2006) and Bai (2010) are used (see

Table 1 for details). The results obtained with the VR model in the

present study are in better agreement with the experimental data

compared to reference simulations. Among all models, the KE and

SST perform very well, showing good agreement with the exper-

imental data. Slight difference between both models and experi-

mental points may be caused by C constant, which should be

DES

t

onsidered as a numerical parameter, that has to be calibrated for

ach particular type of flow.

.3. Turbulence kinetic energy

The profiles of the turbulence kinetic energy (TKE) are shown

n Fig. 6 . Both SGS models show slightly higher values than the

xperimental data. The turbulence is not fully damped in the near

all regions, which can be caused by the low grid resolution.

The SA model demonstrates significant overestimation of the

KE at the top level, showing intense turbulence generation, re-

ated to the proximity to the top wall. At the other two heights,

KE profiles are too narrow, indicating that the turbulence genera-

ion aside of the bubble plume is very low.

M.V. Masterov et al. / International Journal of Multiphase Flow 107 (2018) 275–288 287

p

h

t

e

S

s

8

c

a

w

c

S

i

o

8

F

m

m

s

t

o

t

w

p

r

K

b

n

8

i

s

V

t

t

D

8

w

n

b

d

p

r

t

g

B

t

w

p

w

9

e

t

i

D

m

p

s

l

c

n

c

a

o

i

a

i

w

m

w

t

c

c

e

i

2

i

s

k

e

t

D

A

t

p

e

l

s

K

b

S

f

0

R

A

B

B

B

In contrast, the KE model shows good agreement with the ex-

erimental data in the region near the sparger. At the middle

eight the results for the KE model are slightly better compared to

he SST model. At the top of the column both KE and SST mod-

ls show similar results. At all three heights, the tested KE and

ST models predict the magnitude of the TKE better in compari-

on with the results of Masood and Delgado (2014) .

.4. Dynamic eddy viscosity

Fig. 7 shows the distribution of the average dynamic eddy vis-

osity for five models tested in the present study. The SMG, SST

nd SA models show growth of the eddy viscosity in the near

all regions. In contrast, the KE model demonstrates adequate de-

rease of μt field. The resulting eddy viscosity obtained with the

A model is high in the middle of the domain, which causes an

ncrease of the internal fluid friction and as a result the damping

f fluctuations in the liquid velocity.

.5. Gas velocity

Calculated profiles of the axial gas velocity are presented in

ig. 8 . At all three heights the SMG model shows better agree-

ent with the experiment than the VR model. However, both

odels overestimate the magnitude of the axial gas velocity. The

ame tendency was shown by different authors for E–L simula-

ions, ( Deen, 2001; Lau, 2013; Zhang, 2007; Bai, 2010 ). The results

f Masood and Delgado (2014) demonstrate significant overestima-

ion of the gas velocity, especially in the near wall regions.

The SA model overpredicts the gas velocity at all three heights,

hile the SST and KE models are in better agreement with the ex-

eriment. The SST model shows a lower axial gas velocity in the

egion near the sparger and at the middle height compared to the

E model.

Presented results demonstrate, that with the lack of proper

oundary layer resolution, the KE and SST models capture the dy-

amics of the gas–liquid flow very well.

.6. Gas holdup

The distribution of the time average gas holdup at H = 0 . 25 m

s presented in Fig. 9 . As can be seen the SST and KE models

how similar results, demonstrating very symmetric profiles. The

R model shows very close to the KE and SST distribution, while

he SMG model demonstrates lower gas fraction in the middle of

he domain. Comparison is done with the results of Masood and

elgado (2014) .

.7. Plume dynamics

The dynamics of the bubble plume directly depends on how

ell turbulent structures are resolved. The SA model leads to sig-

ificant reduction of the liquid velocity perturbations and the bub-

le plume oscillations compared to other models and experimental

ata, as seen in Fig. 10 . Within this period of time, 25–55 s, the

lume only slightly moves from one wall to another, see Fig. 10 (a),

emaining “compact” with very pronounceable core along the en-

ire height of the domain.

The SST and KE models show absolutely different behavior of

as inclusions compared to the SA model, see Fig. 10 (b) and (c).

ubbles are more spread over the domain and are not only concen-

rated near the top of the column. The plumes are more dynamic,

ith significant changes in shape and size. In addition, the core of

lumes is almost fully destroyed at the second half of the domain,

hich is not shown by VR model, see Fig. 10 (d).

. Conclusion

In the present work, the implementation of three RANS mod-

ls (SA, KE and SST) for DES was investigated with respect to

he capability to predict dynamics of a gas–liquid flow. Numer-

cal results were compared with experimental measurements of

een (2001) , with LES simulations based on Smagorinsky and Vre-

an SGS models and with DES-SST results obtained with E–E ap-

roach by Masood and Delgado (2014) .For the first time, DES was

uccessfully applied in combination with an Euler-Lagrange formu-

ation for the gas–liquid dynamics to the investigation of bubble

olumn.

The simple and computationally inexpensive DES-SA model is

ot able to correctly predict mean liquid and gas velocities in the

onsidered type of flow. This model significantly overestimates the

verage axial liquid and gas velocities. In addition, there is a lack

f highly dynamic oscillations of the bubble plume. This behavior

s caused by a wrong estimation of the turbulence kinetic energy

side of the bubble plume, which leads to damping of liquid veloc-

ty fluctuations.

Both DES-SST and DES-KE models show very good agreement

ith experimental data and LES simulations performed with Vre-

an SGS model. Profiles of average axial liquid velocity obtained

ith DES-SST model demonstrate better match with experimen-

al points in comparison with results from Masood et al. The os-

illations of the bubble plume and spreading of bubbles are well

aptured by both models. In addition, DES-KE and DES-SST mod-

ls, produce a better agreement with experimental measurements

n comparison with data from the reference LES simulations ( Deen,

001; Bai et al., 2011; Darmana, 2006 ).

However, it is important to mention that all DES models used

n the present study utilized C DES constants initially obtained for

ingle-phase flows ( Strelets, 2001; Spalart et al., 1997 ). It is well

nown, that these constants should be carefully calibrated for

ach particular code and discretization scheme. Therefore, in fu-

ure work, it is of big interest to investigate the sensitivity of both

ES-SST and DES-KE models to different C DES constants.

cknowledgments

This work was supported by the Netherlands Center for Mul-

iscale Catalytic Energy Conversion (MCEC), an NWO Gravitation

rogramme funded by the Ministry of Education, Culture and Sci-

nce of the government of the Netherlands. Authors also would

ike to thank SURF SARA ( http://www.surfsara.nl ) and NWO for the

upport in using the Cartesius supercomputer. Special thanks to S.

amath for sharing his DSMC-based code used for simulation of

ubble dynamics.

upplementary material

Supplementary material associated with this article can be

ound, in the online version, at doi: 10.1016/j.ijmultiphaseflow.2018.

6.00 6 .

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