Front tracking simulations on liquid-liquid systems; an investigation of the drag force on droplets

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Ivo Roghair, Wouter Dijkhuizen, Martin van Sint Annaland and Hans Kuipers

Front Tracking simulations on liquid-liquid

systems; an investigation of the drag

force on droplets

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12/06/08 I. Roghair, CFD2008 2

• Introduction

• Objectives

• Numerical simulations

– Grid dependency study

– Drag force study

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Introduction

Multi-level modelling strategy for multiphase flow

Direct numerical simulations Discrete element model Multi-fluid continuum model

Closures for:

- Drag, lift, virtual mass - Swarm effects

- Mass transfer coefficients Medium scale structures

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Introduction

Direct Numerical Simulations (DNS)

• Fully resolved

– Based only on fundamental equations for

fluid flow

• Navier-Stokes + continuity equation for incompressible flow

– Can be used to derive closures for forces on

• Bubbles • Droplets • Particles

• Only valid when grid independence can be

shown!

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Front tracking

• Incompressible fluids

• Fixed Eulerian grid

• Interface consists of Lagrangian marker

points that build up a triangular mesh

– Points are moved with the interpolated fluid flow

– Straightforward surface tension force calculation

• Advantages

– Calculation of surface tension force with

sub-grid accuracy.

– No numerical coalescence of dispersed

phase elements

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12/06/08 I. Roghair, CFD2008 6

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Drag force

F_D F_G F_P F_L F_VM Droplet velocity Σ F F_D

Forces acting on a

droplet

Stationary force balance

in the rise direction

m_b dvb dt = FG F P F D FL FVM=

∑

F _c−_dg  6 deq 3 −1 2CDc  4 deq 2 u_{d , z}− u_{c , z}2=0 CD= 4__c−_d g d_eq 3 _c_u_{d , z}− u_{c , z}

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12/06/08 I. Roghair, CFD2008 8

Drag force

• Determine drag force coefficient by

different averaging procedures

– Average rise velocity, then determine C

_D

– Determine C

_D

as a function of time, average

this value

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Drag force

C_D= 24 Re C_D= 16 Re



1 2 116 Re 3.315 Re0.5



C_D=max

[

min

[

16 Re10.15 Re 0.687_ , 48 Re

]

, 8 3 Eo Eo4

]

Re=cuddeq _c Eo= _c−_d g d_eq2  C_D=max

[

24 Re 10.15 Re 0.687_ , 8 3 Eo Eo4

]

Correlations from literature (bubbly flow)

Rigid sphere:

Mei et al. (1994):

Tomiyama (1998):

– Pure

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12/06/08 I. Roghair, CFD2008 10

Drag force

• Experiments and simulations on drag

force for bubbly flow

From:

Wouter Dijkhuizen, PhD thesis, University of Twente, 2008

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Objectives

• Investigate the behavior of the Front

Tracking model for liquid-liquid systems

• Simulate droplets in an infinite quiescent

liquid to derive drag force closures

• Investigate the relation between

gas-liquid and gas-liquid-gas-liquid drag force and

their dependencies

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Grid dependency

•Vary resolution in droplet, domain 5 times droplet size •Vary resolution in droplet, keep domain at 1003 _cells

•Keep resolution in droplet at 20 cells, vary domain size Simulation parameters: ρ_c = 1000 kg/m3_, _μ c = 10 -3_Pa·s ρ_d = 800 kg/m3_, _μ d = 10-1 Pa·s σ = 52.9 mN/m, d_eq = 1 mm t_end = 1 s dt = 10-5_s

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Grid dependency

•Vary resolution in droplet, domain 5 times droplet size •Vary resolution in droplet, keep domain at 1003 _cells

•Keep resolution in droplet at 20 cells, vary domain size

30

100

20 6

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Grid dependency

•Vary resolution in droplet, domain 5 times droplet size

•Vary resolution in droplet, keep domain at 1003 _cells

•Keep resolution in droplet at 20 cells, vary domain size

100

20 8

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Grid dependency

•Vary resolution in droplet, domain 5 times droplet size •Vary resolution in droplet, keep domain at 1003 _cells

•Keep resolution in droplet at 20 cells, vary domain size

50

100

20 20

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Drag force simulations

• Used settings:

– 20 grid cells in droplet diameter

– 100

3

_{grid cells in domain}

• Variation of continuous phase viscosity

between 0.001 - 0.2 Pa·s

• Variation of equivalent droplet diameter

between 0.2 – 5 mm

• “Dodecane droplet in water” system:

– ρ_c = 1000 kg/m3;

– ρ_d = 746 kg/m3; μ_d = 1.34·10-3 Pa·s – σ = 0.0529 N/m;

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Drag force simulations

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Drag force simulations

• Variation of dispersed phase viscosity

between 10

-3

– 10

-1

Pa·s

• Variation of equivalent droplet diameter

between 0.2 – 7 mm

• Physical properties

– ρ_c = 1000 kg/m3; μ_c = 10-1 Pa·s – ρ_d = 800 kg/m3;

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Drag force simulations

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Drag force simulations

• Due to volume losses more detailed

simulations:

– Computational grid 150

3

_cells

– 30 cells within droplet diameter

– Higher surface tension

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Drag force simulations

Simulation parameters: ρ_c = 1000 kg/m3_{; μ} c = 10 -3_Pa·s ρ_d = 800 kg/m3_{; μ} d = 10 -1_Pa·s σ = 0.1 N/m; d_eq = 0.5 - 7 mm

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Drag force simulations

Simulation parameters: ρ_c = 1000 kg/m3_{; μ} c = 10 -3_Pa·s ρ_d = 800 kg/m3_{; μ} d = 10 -3 _{- 0.5 Pa·s} σ = 0.1 N/m; d_eq = 1 mm

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Conclusions and outlook

• Front tracking model can simulate dispersed liquid phases but a high resolution is required

• Volume loss strongly depending on droplet resolution • Correlations of Mei et al. and Tomiyama for bubbly

flow are well predicted

– Some overshoot due to wall effects

• Transition of free-slip to no-slip condition as a function of μ_d shown

• Outlook:

– Eo dependence of drag force coefficient – Droplet and bubble swarms

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Thank you

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Front tracking

 Surface tension is

mapped from the

interface mesh to the Eulerian grid. a b c m Fc F_b F_a n_a n_b nc t_m,a t_m,c t_m,b  F_a =σ

_

t _m,a×n_a



 F _b=σ

_

t_m,b×n_b



 F _c=σ

_

t _m,c×n_c

Front tracking simulations on liquid-liquid systems; an investigation of the drag force on droplets