non-drag forces in gas-liquid bubbly flows and validation ...€¦ · 30. june 2004 joined fzr...

30. June 2004 Joined FZR & CFX Workshop on Multiphas e Flows: Simulation, Experiment & Application, Dresden, Germany

Slide 1

Non-drag Forces in Gas-Liquid Bubbly Flows and Validation of

Existing Formulations

(*) ANSYS GermanyD-83624 [email protected]

Th. Frank *, J.-M. Shi +, E. Krepper +

(+) FZ RossendorfInst. of Safety ResearchD-01328 Dresden


Slide 2

Contents

• Introduction• Non-drag forces in bubbly flows

– Lift force– Wall lubrication force– Turbulent dispersion force

• Model formulations• Validation with experimental data

– Steady state FZR-074 test case– Further test cases

• Summary & Discussion


Slide 3

• Finely disperse (121)• Bubbly flow

– Void maximum near the wall (039)

– Transition region (083)– Centred void fraction

maximum (118)– Centred void fraction

maximum bimodal (129)

• Slug flow (140)• Annular flow (215)

Different Types of Bubbly Flows

Test case FZR-074: dilute bubbly flow with near wall maximum of void f raction

Experiments by Prasser et al., FZR


Slide 4

Eulerian Modelling of Multiphase Flow

Averaged conservation equations• Mass, momentum, energy equation for each phase• turbulence model equations (e.g. k-εεεε / k-ωωωω SST model)

( ) ( ) 0k k k k kr rt

ρ ρ∂ + ∇ =∂

U

( ) ( ) ( )kk k k k k k k k k k kr r r P r

tρ ρ∂ + ∇ ⋅ = − ∇ − ∇ ⋅ Π + +

∂U U U F I

• additional interfacial forces important for accurat e predictions of e.g. gas-liquid flows

• non-drag force terms need empirical closure

� � � � �drag lift turbulentwall virtual mass

dispersionlubrication

L WL Tk D D VM= + + + +F F FI F F


Slide 5

Additional lateral forces –Lift force

Physical mechanism:• acts on particles, droplets and

bubbles in shear flows– due to asymmetric wake– due to deformed asymmetric particle

shape

• sign change of bubble lift indicated by measurements

• found in DNS results (Ervin & Trygvasson)

largeellipsoidal bubble

lift force

smallspherical bubble

lift force

fluid vel.


Slide 6

Additional lateral forces –Lift force

Modelling:

Many available correlations for

( )L L G L L G LC r ρ= − ×∇×F U U U

Solid Particles:• McLaughlin • Saffman & Mei• Moraga et al. (asym. wake)

Bubbles:• Mei & Klausner• Legendre & Magnaudet• Tomiyama (shape deform.)

(Re , Re , )L L PC C Eo∇=


Slide 7

Tomiyama’s Lift force correlation

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

Bubble diameter [mm]

Lift

For

ce C

oeff.

C_L

[-]

Tomiyama C_L (u_slip=0.01 m/s)Tomiyama C_L (u_slip=0.05 m/s)C_L (Tomiyama), 0<Eo_d<10C_L (Tomiyama), 10<Eo_dC_L (Tomiyama, orig.), 10<Eo_D

modified Eo d

number:

horizontal bubble length scale:

[ ]3 2

min 0.288 tanh(0.121 Re ), ( ) 4

( ) 0.00105 0.0159 0.0204 0.474 4 10.0

0.27 10.0

P d d

L d d d d d

d

f Eo Eo

C f Eo Eo Eo Eo Eo

Eo

� ⋅ <�= = − − + ≤ ≤��− >�

2( )L G Hd

g dEo

ρ ρσ−=

0.757 1/3(1 0.163 )H Pd d Eo= + ⋅


Slide 8

Antal’s model (1991):1 2

1 2

max 0,

0.01 ; 0.05

W Wwall

P W

W W

C CC

d y

C C

� �= +� �

� �

= − =

( ) 2

WL wall G L rel rel W W WC r ρ= − −F U U n n n�

Additional lateral forces –Wall lubrication force

Surface tension prevents bubbles from approaching solid walls very closeààà à near wall area of low gas void fractionààà à modelled by a wall force, pushing

bubbles away from walls

wall lubr.force

fluid vel.

gas void fraction


Slide 9

Tomiyama‘s wall lubrication force model

Tomiyama’s model (1998):

��

��

−−⋅=

22 )(

11

2)Eo(

WW

PWwall yDy

dCC

0.933 0.179

( )

1 5

0.007 0.04 5 33

0.179 33

W

Eo

C Eo

e Eo

Eo Eo

Eo

− +

=

� ≤ ≤�= + ≤ ≤�� <�

σρρ PPF dg )(

Eo−=

• geometry dependend model due to pipe diameter D !

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40

Eotvos number [-]

W.L

.F. C

oeffi

cien

t [-]

Exponential expression

Original Linear Expression

Changed Linear Expression

Constant Expression


Slide 10

Deficiencies of prior wall lubrication force models

Antal / CFX-5.7:– FW ~ 1 / yW

– geometry independent– FW too small to

balance F L and FTD in some validation cases

– small influence on flow by change in model parameters

Tomiyama:– FW ~ 1 / yW

2

– amplitude depends on Eo

– contains the pipe diameter ààà à not applicable to

complex geometry– no adjustable model

parameter


Slide 11

Proposed modified wall lubrication force formulation

Proposed modified formulation:

• geometry independent formulation• preserved dependency of amplitude on Eo number

(from Tomiyama’s model)• variable potential law FW ~ 1 / yW

p with : p~1.5-2• CWC – cut-off coefficient; C WD – damping coefficient

• recovers the behavior of Tomiyama’s model with:CWC=10.0; CWD=6.8; p=1.7

��

�

��

�

�

��

�

��

�

�

��

��

⋅

−⋅⋅= −1

11

,0max)Eo( p

PWC

WW

PWC

W

WDWwall

dC

yy

dC

y

CCC


Slide 12

Comparison of wall lubrication force models

e.g. FZR-074 test case

-400.00

-350.00

-300.00

-250.00

-200.00

-150.00

-100.00

-50.00

0.00

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]W

all L

ubric

atio

n F

orce

[kg

m-2

s-2

]

Antal W.L.F. (-0.01, 0.05), 2nd grid level

Antal W.L.F. (-0.01,0.05), 4th grid level

Tomiyama W.L.F.

Modified W.L.F. (10, 6.8, 1.7)

-400.00

-350.00

-300.00

-250.00

-200.00

-150.00

-100.00

-50.00

0.00

20.00 21.00 22.00 23.00 24.00 25.00 26.00

Radius [mm]

Wal

l Lub

ricat

ion

For

ce [k

g m

-2 s

-2]


Antal W.L.F. (-0.01, 0.05), 4th grid level

Tomiyama W.L.F.

Modified W.L.F. (10.0, 6.8, 1.7)


Slide 13

CTD≅≅≅≅0.1,…,0.5 (?)

à Equalization of the gas volume fraction distribution

à CTD depends on Stokes numberà different attempts for accurate

derivation of C TD models:• Lahey et al. (1993) • Lopez de Bertodano (1998)• Drew & Passman (1999, 2001)• Moraga et al. (2003), …

TD TD L L GC K rρ= − ∇F

Additional lateral forces: Turbulent dispersion force

Drew & Lahey (RPI model) formulation:

turb. dispersion

force

fluid vel.

gas void fraction


Slide 14

TD TD L L GC k rρ= − ∇F

Turbulent dispersion force –The Favre Averaged Drag (FAD) model

Issa & Gosman, Carrica et al. and Burns (FAD model):

1 1 3 1

4 1LG tL relL

TD D GracerL L L rL P LL G G

C C UkC C

d kr r rµ ν

σ ρ ε σ

= + =� �−�

• derivation of F TD from double averaging of the interphase drag term in momentum equation

• general form of turbulent dispersion force:

• for disperse two-phase flow (r G+rL=1) we can establish equivalence relation to the Drew & Lahey (RPI) model :

tTD

r

r rD A

r rβα α

αβ αβα β α

νσ

∇ ∇= −� ��

F


Slide 15

Additional lateral forces: The virtual mass force

• accounts for the acceleration of the fluid mass surrounding the bubble/particle

• important for transient / strongly accelerated gas- liquid flows, where ρρρραααα/ρρρρββββ<<1

• CVM=0.5 (analytical) or dependent on acceleration number A C (e.g. Odar&Hamilton, Cook&Harlow, Niemann&Laurien)

G LVM VM G L

D DC r

Dt Dtρ = − −� ��

U UF


Slide 16

Force balance analysis for a generic 3-phase test case

• generic 3-phase test case ààà à Virtual mass force neglectable• FAD TD and lift forces of opposite sign for both bu bble size classes• VM force neglectable; W.L.F. very small for 2 nd disperse phase

-200.00

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

0.0 5.0 10.0 15.0 20.0 25.0

Radius [mm]

For

ce C

ompo

nent

[kg

m^-

2 s^

-2]

Grid 2: Air1 TD ForceGrid 2: Air1 W.L.F.Grid 2: Air1 Lift F.Grid 2: Air1 VM F.Grid 2: Air2 TD ForceGrid 2: Air2 W.L.F.Grid 2: Air2 Lift F.Grid 2: Air2 VM F.

jL=1.0167 m/sjG=0.04236 m/sαααα1=2%αααα2=2%dP1=4.0,…,4.8mmdP2=6.0,…,7.0mm

Grace drag

Tomiyama lift

Tomiyama W.L.F.

FAD TD

Sato model

SST turb. model

∆∆∆∆t=0.002s

2500 Iterations


Slide 17

Two-phase bubbly flow measurements at FZR – the MT-Loop test facility

Sparger

Wire-

mesh

sensor

MT-Loop test facility at FZR


Slide 18

Validation test case definition

• MT-Loop test matrix • validation focused on FZR-074; many other test cases investigated• FZR-074: j L=1.0167m/s, j G=0.0368m/s, d P=4.8-5.2mm


Slide 19

Numerical meshes - 3d grid topology (2nd grid level of refinement)


Slide 20

The 3d grid hierarchy

• Scaling factor between grids is 2 (equal to ~2 1/3 in each coordinate direction)

• ICEM/CFD generated 3d grids with edge parameters:

178

140

111

89

70

56

e

512.000784020206

256.000603216165

129.402472513134

64.000392010103

32.0003016882

15.7442613661

No. of CV’s

dcbagrid level

d

a

b

c

e

yx

z

x


Slide 21

Best Practice Guidelines (BPG) conform study: Wall refinement and cell aspect ratios

0

5

10

15

20

25

30

1 2 3 4 5 6

3D Grid Level of Refinement

Cel

l Siz

e R

atio

[-] Max. horizontal cell size ratio

Max. vertical cell size ratio

Max. cell aspect ratio (z/x)


Slide 22

BPG conform study: Wall function characteristics (SST model)

0

5

10

15

20

25

30

35

1 2 3 4 5 6

3D Grid Level of Refinement

y+ [-

]

y+ Values

Solver y+ Values


Slide 23

BPG conform study: Iteration error

• dependence of gas hold-up vs. convergence criteria; grid dependence; ∆∆∆∆t=0.01s, 750 iterations

0.025

0.0275

0.03

0.0325

0.035

1.00E-02 1.00E-03 1.00E-04 1.00E-05 1.00E-06

Convergence Criteria on Max. Residuum [-]

Gas

Hol

d-up

[-]

3d Grid, Level 13d Grid, Level 23d Grid, Level 33d Grid, Level 43d Grid, Level 5


Slide 24

BPG conform study: Convergence in dependence on physical time scale

• dependence of gas hold-up on physical time scale;

convergence & grid dependence

0.01

0.015

0.02

0.025

0.03

4 1 0.5 0.1 0.05 0.025 0.01 0.005 0.003

Physical Time Scale [s]

Gas

Hol

d-up

[-]

2d Grid, Level 1

3d Grid, Level 1

3d Grid, Level 1 (revised)

converged solution


Slide 25

BPG conform study: Convergence in dependence on additional physical models

• dependence of gas hold-up on physical time scale• convergence depends on setup of non-drag forces

0.015

0.02

0.025

0.03

4 1 0.5 0.1 0.05 0.01 0.005 0.0025 0.001

Physical Time Scale [s]

Gas

Hol

d-up

[-]

3d Grid, Level 1 (TD=0.1)

3d Grid, Level 1 (TD=0.5)

3d Grid, Level 1 (TD=0.1,non-drag forces)

strong convergence criterion satisfied (max. air momentum residuals < 1.e-5; ~4000 iterations)

gas hold-up reaches steady state; air mass flow imbalance < 0.008%(~1200 iterations)

convergence


Slide 26

BPG conform study: Computing resource requirements

32.52512.0006

15.75100.52 h3 (NEC)16.26256.0005

8.2752.75 h2 (HPC)8.22129.4024

4.0525.82 h2 (HPC)4.0764.0003

1.7711.30 h1 (HPC)2.0332.0002

1.006.38 h1 (HPC)1.0015.7441

FactorCPU-hProcs.FactorCV’sGrid Level


Slide 27

Comparison of FAD vs. RPI turbulent dispersion models - I

• Simulation of Air-Water 2-phase flow; FZR-074• Turbulent dispersion force : RPI TD (C TD=0.5) vs. FAD TD• k-εεεε vs. SST turbulence model

Grace drag

Tomiyama lift

Tomiyama W.L.F.

Sato model

∆∆∆∆t=0.002s

2500 Iterations

0

1

2

3

0 5 10 15 20 25

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-

] 3d Grid Level 2: k-eps + RPI TD (0.5)

3d Grid Level 2: k-eps + FAD TD

3d Grid Level 2: SST + RPI TD (0.5)

3d Grid Level 2: SST + FAD TD

Air Volume Fraction (Experiment)


Slide 28

Comparison of FAD vs. RPI turbulent dispersion models - II

• Simulation of Air-Water 2-phase flow; FZR-074• Turbulent dispersion force : RPI TD (C TD=0.5) vs. FAD TD• k-εεεε vs. SST turbulence model

Grace drag

Tomiyama lift

Antal W.L.F.

Sato model

∆∆∆∆t=0.002s

2500 Iterations

0

1

2

0 5 10 15 20 25

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-]


3d Grid Level 1: RPI TD (0.5)

3d Grid Level 2: RPI TD (0.5)

3d Grid Level 1: FAD TD

3d Grid Level 2: FAD TD


Slide 29

Validation & Model parameter variation – ITest case FZR-074

• investigation of grid dependence of the solution (3d grid hierarchy)• solution converges with grid refinement• grid independent solution reached on 3 rd grid level

Grace drag

Tomiyama lift

Tomiyama W.L.F.

FAD TD

Sato model

SST turb. model

∆∆∆∆t=0.002s

2500 Iterations

0

1

2

0 5 10 15 20 25

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] Air Volume Fraction (Experiment)

3d Grid Level 1

3d Grid Level 2

3d Grid Level 3

3d Grid Level 4


Slide 30

Validation & Model parameter variation – IITest case FZR-074

• grid independent solution still not reached on 4 th grid level• volume fraction wall peak predicted too close to th e wall• amplitude of wall peak too high; Antal W.L.F. too w eak

Grace drag

Tomiyama lift

Antal W.L.F.

FAD TD

Sato model

SST turb.model

∆∆∆∆t=0.002s

2500 Iterations

0

1

2

0 5 10 15 20 25

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-]


3d Grid Level 1

3d Grid Level 2

3d Grid Level 3

3d Grid Level 4


Slide 31

Validation & Model parameter variation – IIITest case FZR-074

• comparison of wall lubrication force models• 2nd grid level of mesh refinement• Tomiyama and modified W.L.F. give almost identical results

Grace drag

Tomiyama lift

FAD TD

Sato model

SST turb.model

∆∆∆∆t=0.002s

2500 Iterations

0.0

0.5

1.0

1.5

2.0

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

norm

aliz

ed v

olum

e fr

actio

n [-

]Experiment FZR-074


Tomiyama W.L.F.

Modified W.L.F. (10.0, 6.8, 1.7)


Slide 32

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-040: RPI TD (0.35)

FZR-040: FAD TD

FZR-040: Experiment

0.00

1.00

2.00

3.00

4.00

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-042: RPI TD (0.35)

FZR-042: FAD TD

FZR-042: Experiment

0.00

0.50

1.00

1.50

2.00

2.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-039: RPI TD (0.35)

FZR-039: FAD TD

FZR-039: Experiment

0.00

0.50

1.00

1.50

2.00

2.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-038: RPI TD (0.35)

FZR-038: FAD TD

FZR-038: Experiment

Other test cases: FZR-038 – FZR-042, increasing water superficial velocity

0.00

0.50

1.00

1.50

2.00

2.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-038: RPI TD (0.35)

FZR-038: FAD TD

FZR-038: Experiment

0.00

0.50

1.00

1.50

2.00

2.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-039: RPI TD (0.35)

FZR-039: FAD TD

FZR-039: Experiment

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-040: RPI TD (0.35)

FZR-040: FAD TD

FZR-040: Experiment

0.00

1.00

2.00

3.00

4.00

0.00 5.00 10.00 15.00 20.00 25.00

Radius [mm]

Nor

mal

ized

Air

Vol

ume

Fra

ctio

n [-] FZR-042: RPI TD (0.35)

FZR-042: FAD TD

FZR-042: Experiment


Slide 33

Accuracy of measurements and numerical predictions

• measurement accuracy depends on wire mesh sensor resolution and measurement errors

• numerical simulation is subject to round-off, iteration, solution and model errors

ààà à good agreement between experiments & CFD

Figures by courtesy of Dr. E. Krepper, FZR


Slide 34

Summary & Discussion

• Non-drag forces have been implemented in CFX-5.6/5.7

• Model closure correlations for disperse bubbly and particle flows available via User Fortran routines

• Tomiyama lift and wall lubrication force formulations result in good agreement of CFD results with FZR MT-Loop meas urements

• Modified W.L.F. formulation gives geometry independent model with same accuracy

• Validation carried out for:

‡ MT-Loop test matrix with different air/water superf icial velocities

‡ upward & downward flows; transient change of fluid mass flow

‡ bubble diameter range d P=0.5,…,10.0 mm leading to wall and core peaking in the volume fraction profiles

‡ polydispersed air-water flows with up to 5 disperse phases


Slide 35

Summary & Discussion (cont.)

• Turbulence modeling has significant impact on phase volume fraction distributionààà à best results with SST turb. model and FAD TD modelààà à FAD TD model is a significant improvement over RPI modelààà à FAD TD model became default in CFX-5.7

• Further validation: FZR TOPFLOW experiments (D=194.1 mm)

• Further development:

‡ higher volume fractions

‡ breakup & coalescence

‡ inhomogeneous MUSIG model for polydispersed flows

‡ phase change models (boiling, condensation)


Slide 36

Thank you !

non-drag forces in gas-liquid bubbly flows and validation ...€¦ · 30. june 2004 joined fzr...

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