Application of the Immersed Boundary Method to particle ...dutw1479.wbmt.tudelft.nl/~wim/euromech549/presentations/... · Application of the Immersed Boundary Method to particle-laden

Application of the Immersed Boundary Method to particle-laden andp

bubbly flows

T. Kempe, B. Vowinckel, S. Schwarz, C. Santarelli, J. Fröhlich

Institute of Fluid MechanicsInstitute of Fluid MechanicsTU Dresden, Germany

EUROMECH Colloquium 549, Leiden, 17-19 June 2013

Motivation - Particles

[J. Gaffney, University of Minnesota, www.youtube.com]

http://ryanhanrahan.wordpress.com

Bed load transport

Important for multiple applications

Fundamental research

2T. Kempe

Motivation - Particles

Experiments at University of Aberdeen in progress

Our goal:Our goal:

Analysis by direct numerical simulations

Later: comparison with experimentsLater: comparison with experiments

3T. Kempe

Motivation - Bubbles

Our goals:

Bubble - turbulence interactionBubble turbulence interaction

Bubbles in MHD flows

Formation of metal foamsFormation of metal foams

X R • X-Ray measurements [Boden EPM 2009]

• Ar in GaInSn • Ar in GaInSn • No magnetic field

4T. Kempe

1. Basic fluid solver and IBM

2. Collision modeling

3. Bed load transportp

4. Bubble laden flows

PRIME(Phase-Resolving sIMulation Environment)

Solver for continuous phaseIncompressible Navier-Stokes equations

( ) fuuuu+∇=∇+⋅∇+

∂ 21 νp

S

2nd order Finite Volumes

( ) fuuu +∇∇+∇+∂

νρ

pt

8

Staggered Cartesian grid

Explicit 3-step Runge-Kutta for convective terms

Speedup with 2.65·108 grid pointssolid: ideal , dash-dot: PRIME

p p gImplicit Crank-Nicolson for diffusive terms

Highly sophisticated libraries

E

g y pPETSc for parallelization (MPI)HYPRE as solver (Poisson, Helmholtz Eq.)

6

Scaleup with 1.3·105 grid points per processor

T. Kempe

Immersed boundary method [Uhlmann JCP 2005]

Fluid solved on fixed Cartesian gridFluid-solid interface represented by markersPhase coupling by volume forcesPhase coupling by volume forces

Interpolation of velocities to marker pointsDirect forcing d −

=ΓUUF

Spreading to grid points Regularized δ – functions used F

f

tΔF

Grid points and surface markersRegularized δ – function[ Roma, 1999]

P tiPropertiesgood stability order of convergence

L2 - error of velocity,single sphere in

channel flow

1st Order

2nd O d

7

≈1.7 2nd Order

T. Kempe

Mobile particles[Uhlmann JCP 2005]

[Kempe & Fröhlich JCP 2012]

Linear momentum balance:

ud

[Kempe & Fröhlich JCP 2012]

cpfpfp

p VSt

m Fgnτu

+−+⋅= ∫Γ

)( dd

dρρρ Collisions

Angular momentum balance:

Pressure & viscous forces buoyancySphere with 874 marker points

g

cfp

c St

I Mnτrω

+⋅×= ∫ d)(d

dρ

Runge–Kutta (3rd order) for time integration

t Γd

Parallel implementation in PRIME

Master & slave strategy Particle with interface resolution in

8

Master & slave strategy Particle with interface resolution in decomposed domain

T. Kempe

Improved IBM

gfu

+−

−= ∫p

VVt fpp

fp

Ωρρρ

d)(d

d

8

( )

∑

∑

=

=

−−≈ 8

1

8

1,,

mm

mmm

kji

H

φ

φφα

R

Velocity u at Ly /2, Re=10

Stable time integration for previously g p yinaccessible density ratiosAccurate imposition of BC’s

9

Velocity of buoyant & sedimenting particles

[Kempe & Fröhlich, JCP , 2012]

T. Kempe

1. Basic fluid solver and IBM

2. Collision modeling

3. Bed load transportp

4. Bubble laden flows

Normal particle-wall collision with various collision modelsvarious collision models

up

soft-sphere model

repulsive potential, kn=1e6

Repulsive potential

Hard-sphere model

experiment

repulsive potential, kn=1e4

Soft-sphere model

Choice of coefficients ?

Time scale separation for

Surface distance vs. time

hard-sphere model

p

fluid and collision

Δt100≈

Δ

Δ

c

f

tt

M d l bl f l l i l i

11

Models not usable for large-scale simulations

T. Kempe

Adaptive collision time model (ACTM)

Purpose Avoid reduction of Δ tf

Give correct restitution ratio in

outdry u

ue =

Ideas Stretch collision in time to Tc = 10 Δ tf

Optimization to find coefficients

in

ODE of collision process

dd2 ζζ ζ0

dd

dd 2/3

2

2

=++ nnn

nn

p kt

dt

m ζζζ Physically exact

ACTM

ζ

Problem: given : uin , edry , Tc

find : dn , kn

rp0

Fixed-point problem solved by Quasi-Newton scheme

tTc

Quasi Newton scheme

12T. Kempe

linear approaches possible [Breugem 2010]

Oblique collisions

Oblique = normal + tangential

Normal ACTM

q

[Joseph 2004]

Normal ACTM

Tangential force model (ATFM)

Consider only sliding and rolling Consider only sliding and rolling

Critical local impact angle

Sliding: Coulomb friction

RSΨ cpinn

cpint

in gg

,

,=Ψ

Sliding: Coulomb friction

Rolling: Compute force such that relative

nfcol

t FF μ=

Rolling: Compute force such that relative

surface velocity is zero 0=cptg

( )( )( ) tF 2

,,,,

2 ppp

intq

intpp

intq

intpppcol

t RmItRuumI

+Δ

+−−=

ωω

13T. Kempe

Lubrication modeling

Local grid refinement not desiredUnder-resolved fluid in gap

[Gondret, 2002]with lubrication model

without model

explicit expression for lubrication force [Cox & Brenner , 1967]

without model

1

2

6 −

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−= n

qp

qpnf

lubn RR

RRgF ζμπ

Particle-wall collision, St = 27⎠⎝ qp

14

Resolved and modeled contributions during a particle-wall collision

T. Kempe

Adaptive collision model (ACM)Adaptive collision model (ACM)

Phase 1 Phase 2 Phase 3Phase 1 Phase 2 Phase 3

Up,out

Up,in

Approach Surface contact Rebound

Lubrication model Normal ACTMTangential ATFM

Lubrication model

15T. Kempe

Normal collisions – Results [Kempe & Fröhlich JFM 2012]

edry = 0

present Experiment

edry = 0.97

present

[Gondret, 2002]

Steel spheres impacting on glass wall

Flow around sphere impacting on wall,Left: simulation, right: experiment [Eames, 2000]

Fluid time step can be used pppp DuSt

ρτFluid time step can be usedLocal grid refinement is avoidedGood agreement with experiments

f

ppp

f

pStμτ 9

==

16T. Kempe

Effective angle of impact

Oblique collisions – Results [Kempe & Fröhlich JFM 2012]

Effective angle of impact

Effective angle of rebound

cpinn

cpint

in gg

,

,=Ψ

Effective angle of rebound

cpinn

cpoutt

out gg

,

,=Ψcp

outtgcp

inng ,cp

intg ,

Compared with experiment [Joseph, 2004]

Glass spheres, rough, μ f = 0.15Steel spheres, smooth, μ f = 0.01

outtg ,

Glass spheres, rough, μ f 0.15ΨRS = 0.95

present

p μ f ΨRS = 0.25

sliding

ΨRS

rolling

Ψ

17T. Kempe

Stretched collisions with multiple particles

Results with static stiffness

EFluid

Ekin

Significant influence of Δt

Results with ACM

Fi l ti l iti Etotal /20

Final particle positions

EFluid

Ekin

1818T. Kempe

CFL = 1 (stretched) CFL = 0.02 (Hertz)

Results independent of Δt

1. Basic fluid solver and IBM

2. Collision modeling

3. Bed load transportp

4. Bubble laden flows

3D interface resolved

Short history „DNS of sediment transport“

Uhlmann & Fröhlich 2006

3D, interface resolved18 million fluid points560 particles, short runs, no statistics

Osanloo 20082D, mass pointsFixed velocity profile for fluid

Papista 2011

y p

2D, interface resolved200 i l200 particles

3D, interface resolved

Vowinckel 2012

,645 million fluid points8696 particles

This project

3D, interface resolved1.4 billion fluid points40500 pa ticles40500 particles

20T. Kempe

Computational Setup

H/D Lx/H Lz/H Reb Reτ D+

9 24 6 2941 193 21

Two layers of particles:

• Fixed bed: single layer of hexagonal packing

• Sediment bed on top

Free slipp

No slip

Periodic in x- and z- direction 21T. Kempe

Numerics

Nx Ny Nz Ntot D/Δx Δx+ Nproc

4800 240 1200 1.4·109 22.2 0.95 4048 - 16192

true DNS all scales resolved

22T. Kempe

Computational Setup

Case Np,fixed Np,mobil Θ /Θcrit Configuration

Fix 27000 0 0 Two fixed layers

Ref 13500 13500 1.18 One layer of mobile particles

FewPart 13500 6750 1.18 Lower mass loading

LowSh 13500 13500 0.75 Lower mobility

( )gDu

p ρρρ

θ τ

−=

2

Mobility Shields parameter( )gp ρρ

Probability

TransitionalSmooth

θ

of particlemotion

D+f

Rough

DRef,FewPart

LowSh

23T. Kempe

Case Ref Θ /Θcrit = 1.18 [Vowinckel et al. ICMF 2013]

Spanwise oriented dunes

Contour Iso-surfaces of fluctuations Particleu / Uu / Ub

u' / Ub = -0.3' / U 0 3

up / uτ ≥ 4up / uτ< 4fixedu' / Ub = 0.30

24T. Kempe

Case Ref Θ /Θcrit = 1.18

mean fluid velocity <u> fluctuations <u’ u’>

0.8

1

Fix

0.8

1

mean fluid velocity <u>

0.8

1y

/ H

-0.2

0

0.2

0.4

0.6

0.8 FixRefFewPartLowSh

y / H

0.2

0.4

0.6

y / H

0.2

0.4

0.6

<u> / Ub

0 0.5 1-0.2

Mobile or restingFixed bed

<u> / U0 0.5 1

-0.2

0

0.2

<u’u’> / U20 0.02 0.04 0.06-0.2

0

0.2

<u> / Ub<u’u’> / Ub

higher intrusion, larger resistance, more turbulence25T. Kempe

Less particles Θ /Θcrit = 1.18

Streamwise oriented, inactive ridges

Contour Iso-surfaces of fluctuations Particleu / U

u' / Ub = -0.3' / U 0 3

up / uτ ≥ 4up / uτ< 4

u / Ub

fixedu' / Ub = 0.30

26T. Kempe

Less particles Θ /Θcrit = 1.18

Streamwise oriented, inactive ridges

Mobile

Fixed

27T. Kempe

Heavier particles Θ /Θcrit = 0.75

Inactive plane bed with single eroded particles

Contour Iso-surfaces of fluctuations Particleu / U

u' / Ub = -0.3' / U 0 3

up / uτ ≥ 4up / uτ< 4

u / Ub

fixedu' / Ub = 0.30

28T. Kempe

Conclusions so far [Vowinckel et al. ICMF 2013]

many & light particles:

Two parameters: mass loading & mobility (=density)

y g p

dunes with distance 12H

few & light :

inactive ridges

many & heavy :

closed plane bed

Experiments [Dietrich et al. Nature 1989]

Sediment patterns in agreement with experimental evidence29T. Kempe

1. Basic fluid solver and IBM

2. Collision modeling

3. Bed load transportp

4. Bubble laden flows

Bubble shape depends on regime

Reynolds number

[Clift et al. 1978]

number

νeqpdu

=Reν

ρρ 2dg

Eötvös numberσρρ eqfp dg

Eo−

=

31T. Kempe

Representation of bubble shapes

• Spherical Particles as model for bubbles

• Ellipsoidal

time dependent shape X(t)baX =b

a

• Spherical harmonics

∑+= mnmn Ytaatr 0 ),()(),,( θφθφ

Exact evaluation of curvature, normal vector, volume…

∑mn

nmn,

0 ),()(),,( φφ

[Bagha 1981]Coupling of shape to fluid loads by local force blance (vanishing displacement energy)

32T. Kempe

[Schwarz & Fröhlich ICNAAM 2012]

Light particles in vertical turbulent channel flow

Upward flow

• lenght x height x width = 2 2H x H x 1 1H• lenght x height x width 2.2H x H x 1.1H

• periodic in x- und z- direction

• Wall: no-slip

• Fluid Reynolds number

• Mesh

for unladen flow

33T. Kempe

Particle parameters

Fixed shape:

oblate ellipsoid

0 001ρp

Density ratio: 0.001ρf

Density ratio:

Np Deq/H Deq/∆ a/b Void fr. Rep

740 0.05172 ~24 2 ~ 2 % ~214

34T. Kempe

Looking atLooking at

transport of particles by the flow

influence of particles on turbulence

particle-particle interaction

[Santarelli ICMF 2013]

35T. Kempe

Results: fluid phase

mean streamwise velocity turbulent kinetic energy

Symmetry loss and turbulence enhancement

36T. Kempe

Ascent of single bubble in liquid metal [Schwarz 2013]

Argon bubble deq = 4.6 mm in quiescent GaInSn: Eo = 2.5Experiments by [Zhang et al. 2005]

− Cartesian grid: 84 Mio. Pointsg− 9093 Lagrangian surface

markers− “Wobbling” bubble shape

Re Ret

− 60 x 61.5 CPU hours

PRI• Comparison

Unsteady rise velocity, ellipsoidal

0.270.28f / 245369σRe

2871

2879

MEAveraged rise velocity matches exp. dataUnderprediction of standard deviation in ReGood agreement in oscillation frequency0.27

60.28

0f / fref

245369σRe[Schwarz, 2011, 2013]

37T. Kempe

Single bubble in liquid metal with vertical magnetic field

dB 2

ref

eqel

udB

Nρ

σ 20=)( Bjf ×= NL )( Buj ×+Φ−∇= eσ 0=⋅∇ j

)(2 Bu×⋅∇=Φ∇

electric current density jj

)( Bu×⋅∇=Φ∇

By

By

electric potentialjelectric conductivity jeσ

Φ

More rectilinear bubbletrajectory

R d d ill tiReduced oscillationfrequency & amplitude in Re

Large bubbles rise faster, small ones slower

Ny = 0 Ny = 1

small ones slower

Reduced wake vorticity with larger vortices being alignedwith B

Instant. helicity iso-contours

38

with B

Ny = 0 Ny = 1T. Kempe [Schwarz 2013]

Thank you for your attention.

Feedback and questions ? q

Bubble Coalescence

13:10 – 13:30

S Sch a S Tschisgale and J F öhlich S. Schwarz, S. Tschisgale and J. Fröhlich

“Bubble coalescence model for phase-resolving simulations using an Immersed Boundary Method”

39T. Kempe