multi phase turbulent flow

8/2/2019 Multi Phase Turbulent Flow

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Multiphase Turbulent Flow

Ken Kiger - UMCP


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Overview Multiphase Flow Basics

General Features and Challenges

Characteristics and definitions

Conservation Equations and Modeling Approaches

Fully Resolved

Eulerian-Lagrangian

Eulerian-Eulerian Averaging & closure

When to use what approach?

Preferential concentration

Examples

Modified instability of a Shear Layer Sediment suspension in a turbulent channel flow

Numerical simulation example: Mesh-free methods in multiphase flow


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What is a multiphase flow? In the broadest sense, it is a flow in which two or more

phases of matter are dynamically interacting Distinguish multiphase and/or multicomponent

Liquid/Gas or Gas/Liquid

Gas/Solid

Liquid/Liquid

Technically, two immiscible liquids are multi-fluid, but are often referred to asa multiphase flow due to their similarity in behavior

Single component Multi-component

Single phase Water

Pure nitrogen

Air

H20+oil emulsions

Multi-phase Steam bubble in H20

Ice slurry

Coal particles in airSand particle in H20


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Dispersed/Interfacial Flows are also generally categorized by distribution of the

components

separated or interfacial

both fluids are more or less contiguous

throughout the domain

dispersed One of the fluids is dispersed as non-

contiguous isolated regions within the

other (continuous) phase.

The former is the dispersed phase,

while the latter is the carrier phase.

One can now describe/classify thegeometry of the dispersion:

Size & geometry

Volume fraction


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Bubbly Pipe Flow heat exchangers in power plants, A/C units

Gas-Liquid Flow


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Aeration:-produced by wave action

- used as reactor in chemical processing- enhanced gas-liquid mass transfer

Gas-Liquid Flow (cont)


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Ship wakes detectability

Cavitation noise, erosion of structures

Gas-Liquid Flow (cont)


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Weather cloud formation

Biomedical inhalant drug delivery

Liquid-Gas Flow

http://www.mywindpowersystem.com/2009/07/wind-power-when-nature-gets-angry-the-worst-wind-disasters-of-the-world/

Vukasinovic, Glezer, Smith (2000)


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Energy production liquid fuel combustion

Biomedical inhalant drug delivery

Gas-Liquid Flowhttp://converge

cfd.com/applications/engine

/sparkignited/

Album of fluid motion, Van Dyke

Image courtesy A. Aliseda


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Gas-Solid FlowEnvironmental avalanche, pyroclastic flow, ash plume, turbidity currents


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Gas-Solid (dense)Granular Flow collision dominated dynamics; chemical processing

http://jfi.uchicago.edu/~jaeger/group/

http://www.its.caltech.edu/~granflow/homepage.html


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Chemical production mixing and reaction of immiscible liquids

http://www.physics.emory.edu/students/kdesmond/2DEmulsion.html

Liquid-Liquid


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Sediment Transport pollution, erosion of beaches,

drainage and flood control

Solid-Liquid


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Settling/sedimentation,

turbidity currents

Solid-Liquid

http://www.physics.utoronto.ca/~nonlin/turbidity/turbidity.html


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Material processing generation of particles & composite materials

Energy production coal combustion

Solid-Gas


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Aerosol formation generation of particles & environmental safety

Solid-Gas


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Classification by regime Features/challenges

Dissimilar materials (density, viscosity, etc)

Mobile and possibly stochastic interface boundary Typically turbulent conditions for bulk motion

Coupling One-way coupling: Sufficiently dilute such that

fluid feels no effect from presence of particles.

Particles move in dynamic response to fluidmotion.

Two-way coupling: Enough particles are presentsuch that momentum exchange betweendispersed and carrier phase interfaces altersdynamics of the carrier phase.

Four-way coupling: Flow is dense enough thatdispersed phase collisions are significantmomentum exchange mechanism

Depends on particle size, relative velocity, volume

fraction


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Viscous response time To first order, viscous drag is usually the dominant force on the

dispersed phase

This defines the typical particle viscous response time

Can be altered for finite Re drag effects, added mass, etc. as appropriate

Stokes number:

ratio of particle response time to fluid time scale:

mpd vp

dt= 3pmD

ru -

rvp( )

u =0 for t< 0

Ufor t 0

dvp

dt=18m

rpD2U- vp( ) =

1

tpU- vp( )

vp =U1- exp - t tp( )[ ]

u, vp

t

U

p =rpD2

18m

St=tp

tf


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Modeling approach? How to treat such a wide range of behavior?

A single approach has not proved viable

Fully Resolved : complete physics

Eulerian-Lagrangian : idealized isolated particle

Eulerian-Eulerian : two co-existing fluids Computation

alEffort

Modelin

gEffort


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Fully Resolved Approach Solve conservation laws in coupled domains

1. separate fluids Each contiguous domain uses appropriate transport coefficients Apply boundary jump conditions at interface

Boundary is moving and may be deformable

2. single fluid with discontinuous properties

Boundary becomes a source term

Examples

Stokes flow of single liquid drop

Simple analytical solution

Small numbers of bubbles/drop

Quiescent or weakly turbulent flow

G Tryggvason, S Thomas, J Lu, B Aboulhasanzadeh (2010)


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Eulerian-Lagrangian Dispersed phase tracked via individual particles

Averaging must be performed to give field properties

(concentration, average and r.m.s. velocity, etc.)

Carrier phase is represented as an Eulerian single fluid

Two-way coupling must be implemented as distributed source term

M. Garcia,http://www.cerfacs.fr/cfd/FIGURES/IMAGES/vort_stokes2-

MG.jpg

Collins & Keswani (2004)


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Two-Fluid Equations Apply averaging operator to mass and momentum equations

Drew (1983), Simonin (1991)

Phase indicator function

Averaging operator

Assume no inter-phase mass flux, incompressible carrier phase

Mass

Momentum

( )

=kx

kxtx

i

i

ikphaseoutsideif0

phaseinsideif1, 0, =

+

j

kjI

k

xu

t

t

a krk( ) +

xja krkUk,j( ) = 0

krkUk,it

+Uk,jUk,ixj

= - a k

P1x i

+a krkgi +

xja ktk,ij( ) -

xja krk ui uj k( ) +Ik,i

kijk phaseintensorstressviscousaveraged, =

on)contributipressuremean(less

transportmomentuminterphaseMean, =ikI

ijkijkk gG =,fractionvolume,kk = ijijij Ggg ,2=


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Two-Fluid Equations (cont) Interphase momentum transport

For large particle/fluid density ratios, quasi-steady viscous drag is by far the

dominant term

For small density ratios, additional force terms can be relevant

Added mass

Pressure term

Bassett history term

For sediment, 2/1 ~ 2.5 > 1 (k=1 for fluid, k=2 for dispersed phase) Drag still first order effect, but other terms will likely also contribute

2

,1,1,24

3irr

Dkii

d

CII vv==

iiir uu ,1,2, =v CD =24 1+ 0.15Rep

0.687[ ]

Rep 1

1Re

drp

v

=


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Closure requirements Closure

Closure is needed for:

Particle fluctuations Particle/fluid cross-correlations

Fluid fluctuations

Historically, the earliest models used a gradient transport model Shown to be inconsistent for many applications

Alternative: Provide separate evolution equation for each set of terms Particle kinetic stress equation

Particle/fluid covariance equation

Fluid kinetic stress equation Also required for single-phase RANS models

Also will require third-moment correlations models to complete the closure


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Simpler Two-Fluid Models For St < 1, the particles tend to follow the fluid motion with

greater fidelity

Asymptotic expansions on the equation of motion lead to a closedexpression for the particle field velocity, in terms of the local fluid velocityand spatial derivatives (Ferry & Balachandran, 2001)

Where

This is referred to as the Eulerian Equilibrium regime (Balachandar 2009).

Also, similar to dusty gas formulation by Marble (1970)

For larger St, the dispersed phase velocity at a point can be

multivalued!

St=tp

tf

w =tpg

=3

2r+1


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When is a given approach best? What approach is best, depends on:

D/: Particle size and fluid length scales (typically Kolmogorov)

p/f: Particle response time and fluid time scales

Total number of particles: Scale of system

, = p: Loading of the dispersed phase (volume or mass fraction)

Balachandar (2009)

p

tf=

2r+1

36

1

f Re( )

d

h

2

p/f= 1000p/f= 25

p/f= 2.5

p/f= 0


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From early studies, it was observed that inertial particles can be

segregated in turbulent flows

Heavy particles are ejected from regions of strong vorticity

Light particles are attracted to vortex cores

Small Stapprox. shows trend

Taking divergence of velocity

Preferential Concentration

Wood, Hwang & Eaton (2005)

St = 1.33 St = 8.1

up= - St1- b( ) Sij2

- W2

[ ]


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PDF formulation Consequences of inertia

Implies history of particle matters

Particles can have non-unique velocity

How can models account for this?

Probability distribution function

f(x,u,t) = phase space pdf

Instantaneous point quantities come as moments ofthe pdf over velocity phase space

n x, t( ) = f xp ,up , t( )-

+

dup

ft= - x upf( ) - up apf( )

x ( ) = xi

up ( ) = up,i

up =dxp

dt

ap =dup

dt

=1

mp

3pmDru -

rvp( ) +

1

2

mfDvu

Dt

-drvp

dt

+ mfDvu

Dt

- mp 1-rg

rp

rg

Fevier, Simonin & Squires (2005)

up x, t( ) = upf xp ,up , t( )-

+

dup

up

f


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Examples Effect of particles on shear layer instability

Particle-Fluid Coupling in sediment transport

Case studies in interface tracking methods


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Effect of particles on KH Instabilty How does the presence of a dynamics dispersed phase influence

the instability growth of a mixing layer?

x

y

U

U


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Results Effect of particles

At small St, particles follow flow

exactly, and there is no dynamicresponse. Flow is simply a heavierfluid.

As St is increased, dynamic slipbecomes prevalent, and helps damp

the instability

At large St, particles have noresponse to perturbation and arestatic

Effect is stronger, for higher loadings,but shear layer remains weaklyunstable

1/St


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Mechanism Particles damp instability

Particles act as a mechanism to

redistribute vorticity from the coreback to the braid, in opposition tothe K-H instability

Limitations

Results at large St do not captureeffects of multi-value velocity

Fluid streamfunction

Dispersed concentration

up

Meiburg et al. (2000)


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Sediment Transport in ChannelFlow

Planar Horizontal Water Channel

4 36 488 cm, recirculating flow

Pressure gradient measurements show fully-developed by x =250 cm

Particles introduce to settling chamber outlet across span

Concentration Mean Velocity

g

x

y2h


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Both single-phase and two-phase experiments conducted

Carrier Fluid Conditions Water, Q =7.6 l/s

Uc= 59 cm/s, u= 2.8 cm/s, Re =570

Flowrate kept the same for two-phase experiments

Tracer particles: 10 m silver-coated, hollow glass spheres, SG= 1.4

Dispersed Phase Conditions

Glass beads: (specific gravity, SG= 2.5)

Standard sieve size range: 180 < D< 212 m

Settling velocity, vs =2.2 to 2.6 cm/s

Corrected Particle Response Time, = 4.5 ms

St+ = p/+~ 4

Bulk Mass Loading: dM/dt= 4 gm/s, Mp/Mf ~5 10-4

Bulk Volume Fraction, = 2 10-4

Experimental Conditions

p


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Concentration follows a power law

Equivalent to Rouse distribution for

infinite depth

Based on mixing length theory, but still

gives good agreement

Mean Concentration Profile

a

o

a

o

o

o y

y

yh

y

y

yh

yC

yC

=

)(

)(

00.0)/00.0)(00.0(

/00.0

=== scmscmuvas


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Mean Velocity

Particles alter mean fluid profile Skin friction increased by 7%; qualitatively similar to effect of fixed roughness

Particles lag fluid over most of flow Observed in gas/solid flow (much large Stokes number likely not same reasons)

Particles on average reside in slower moving fluid regions?

Reported by Kaftori et al, 1995 forp/f= 1.05 (current is heavier ~ 2.5)

Organization of particles to low speed side of structures a la Wang & Maxey (1993)?

Particles begin to lead fluid near inner region transport lag across strong gradient


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Particle Slip Velocity,

Streamwise direction Particle-conditioned slip (+) is generally small in outer flow Mean slip () and particle conditioned slip are similar in near wall region

Wall-normal direction Mean slip () is negligible Particle-conditioned slip (+) approximately 40% of steady-state settling velocity (2.4 cm/s)

ppfuu

ppfuu

ppfuu

ppfuu

ppfvv ppf vv

vs


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Particle Conditioned Fluid Velocity

Average fluid motion at particle locations:

Upward moving particles are in fluid regions moving slower than mean fluid

Downward moving particles are in fluid regions which on average are the same as the fluid

Indicates preferential structure interaction of particle suspension

downpfu

,

uppfu

,

fu

fv

downpfv

,

uppfv

,


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Suspension Quadrant Analysis Conditionally sampled fluid velocity fluctuations

Upward moving particles primarily in quadrant II

Downward moving particles are almost equally split in quadrant III and IV

Persistent behavior Similar quadrant behavior in far outer region

Distribution tends towards axisymmetric case in outer region


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Expected Structure: Hairpin packets Visualization of PIV data in single-phase boundary layer

Adrian, Meinhart & Tomkins (2000), JFM

Use swirl strength to find head of hairpin structures Eigenvalues of 2-D deformation rate tensor, swirl strength is indicated by magnitude of complexcomponent

Spacing ~ 200 wall units

Packet growth angle can increase or decrease, +10 on average

Packets were observed in 80% of images (Re = 7705)

Swirl

strength

contours

Hairpin

packets


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Event structures: Quadrant II hairpin

Similar structures found Appropriate spacing

Not as frequent Re effects? (Re = 1183)

Smaller field of view?

Evidence suggests packets contribute to particle suspension

Q2 & Q4

contribution

s

Swirl

Strength


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Turbulence budget for particle stresses (Wang, Squires, Simonin,1998)

Production by mean shear

Transport by fluctuations

Momentum coupling to fluid (destruction)

Momentum coupling to fluid (production)

Particle Kinetic Stress

t+U2,m

xm

u2,i u2,j 2 = P2,ij +D2,ij + P 2,ij

d + P 2,ijp

P2,ij = - u2,i u2,mU2,j

xm- u2,j u2,m

U2,i

xm

D2,ij = -1

a 2

xma 2 u2,i u2,j u2,m 2[ ]

2,ijp =

r1

r2

3

4

Cd

dvr u1,i u2,j + u1,j u2,i[ ]

2

2,ijd

= -r1

r2

3

2

Cd

dvr u2,i u2,j

2

0


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Streamwise Particle/Fluid Coupling: d2,11, p2,11 Compare results to Wang, Squires, & Simonin (1998)

Gas/solid flow (2

/1

=2118), Re

= 180, No gravity, St+~700

Computations, all 4 terms are computed; Experiments, all but D2,ijcomputed

Interphase terms are qualitatively similarSimilar general shapes, d11 > p11

Quantitative difference Magnitudes different: d11 /

p

11~1.3 vs 3, overall magnitudes are 10 to 20 times greater

Interphase terms are expected to increase with decreased St+

Dominant interphase transfer () greatly diminishes importance of mean shear (P) Turbulent transport (D) has opposite sign because of small shear production (P)

Particle Kinetic Stress Budget

0.2

0.1

0

-0.1

-0.2

Experiment, solid/liquid Wang, et al, solid/gas


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Particle Transfer Function Useful to examine steady-state particle response to 1-D oscillating flow

of arbitrary sum of frequencies

Represent uas an infinite sum of harmonic functions Neglect gravity (DC response, not transient), Basset history term (higher order effect)

u t( ) = L f w( )0

exp iwt( )dw

vp t( ) = Lp w( )0

exp iwt( )dw

2mp

3pmD

dvp

dt= 2 u - vp( ) +

mf

mp

mp

3pmD

du

dt-dvp

dt

+ 2

mf

mp

mp

3pmD

du

dt

du t( )

dt= iwL f w( )

0

exp iwt( )dw

2 rpD

2

18mLpiw exp iwt( )

0

dw = 2 L f - Lp( )exp iwt( )0

dw + rfrp

rpD2

18miw 3L f - Lp[ ]exp iwt( )dw

0

2iStLp = 2 L f - Lp( ) + igSt 3L f - Lp[ ]

dvp t( )

dt = iwLp w( )0

exp iwt( )dw

St=tp

tf=rpD

2w

18m g =

rf

rp


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Particle Transfer Function This can be rearranged:

Examine

Liquid in air, gas in water, plastic in water

vpru=LpLp

*

L fL f*

1 2

=A

2 +B2

A2 +1

1 2

( )

+=

+

=

2

32

2

B

St

A

= tan- 1Im L

p/L

f( )

Re Lp/L

f( )

= tan- 1A(1- B)

A2 +B


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Liquid particles in Air

Liquid drops in air require St~0.3 for 95% fidelity (1 m ~ 15 kHz)

Size relatively unimportant for near-neutrally buoyant particles

Bubbles are a poor choice: always overrespond unless quite small

multi phase turbulent flow

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