L09 LectureSlides 2015-Spring CHY

1AerodynamicsandFlow L9

Chungen Yin

[email protected]; 30622577

Spring 2015

Turbulent Reacting Multiphase Flows (5 lectures)

2

L9:TurbulentNonreactingMultiphaseFlows Turbulent particle dispersion Turbulence modulation Particle-wall interaction

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3Turbulentparticledispersion(1)

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Small particle A particle is referred to as small, if its diameter is smaller than the Kolmogorov

length scale;

as medium, if its diameter is between the Kolmogorov scale and the integral scale;

In dilute, particle-laden flows of interest, the majority of particles are small, based on this definition.

)(21 vuvuAC

dtvdm pfDp

vtdxd

For small, heavy particles/droplets in dilute two-phase flow,

The particle trajectory & velocity can be determined at each time, provided the knowledge of the flow field at that moment

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Turbulentparticledispersion(2)

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Number density

Position

The interaction between turbulent eddies and immersed small particles is referred to as turbulent particle dispersion.

Observation: dispersive effect on particles released from the same location; Key parameters: (1) particle size, with respect to eddy size; (2) fluid and

particle properties, e.g., fluid viscosity & density and particle density; (3) flow properties, e.g., distribution of turbulent kinetic energy.

inertia dominated viscosity dominated

5Stochastictracking(1). Randomvel.fluctuation

5

A Gaussian distributed random velocity fluctuation is used: also known as discrete random walk (DRW) model or eddy lifetime model.

)(21 vuvuAC

dtvdm pfDp

uUu

In DRW model, each eddy is characterized by

A Gaussian distributed random velocity fluctuation, u, v, w; A time scale

Random velocity fluctuations for k- and k-3/2''' kwvu : normally distributed random number

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Stochastictracking(2).Interactiontime

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),(minninteractio ce tt

A time scale, accounting for the Characteristic eddy lifetime and Crossing trajectory effect, is used as the integral time.

The time for particle to cross an eddy: estimated from eddy size/drift velocity

For small particle (moving with fluidzero drift velocity): the integral time becomes fluid Lagrangian integral time

kAfL

eL

fLe 2 )ln(rfLe orr : a uniform random number [0, 1]

18,with1ln

23/2pp

vev

evc

dkBLvu

Lt

7Stochastictracking(3)

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In literature, a range of values have been suggested for A and B:

23ntconstraithewith41.0135.0

BAA

For assumptions of isotropic turbulence and that the characteristic size of the eddy is the Kolmogorov scale, the constants A and B can be approximated by

4/34/3 ; 23

CBCA )09.0( C

Over the interaction time, , the particle is assumed to interact with the fluid phase eddy with a given instantaneous velocity. When

this time is reached, a NEW value of will be applied to generate

new random velocity fluctuations ( new instantaneous velocity).

ninteractiot

DRW model constants (A & B)

(FLUENT: A=0.15 by default, 0.3 recommended for RSM)

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Particlecloudtracking(1)

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Number density

Position

Track the statistical evolution of a cloud of particles about a mean

trajectory: the concentration of particles about a mean trajectory is

represented by a Gaussian PDF whose variance is based on the degree

of particle dispersion due to turbulent fluctuations.

9Particlecloudtracking(2)

9

The PDF used is derived from Taylors analysis:

)(,)(),,,( ttxftzyxP ii

Its possible to define a normalized Lagranian autocorrelation function:

)(

)()(),(

22

2121

tu

tututtR

i

iipL

This can be assumed to exhibit an exponential decay.

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Example

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Instantaneous particle dispersion from simulation of plane wake

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4

0

-4

-88

4

0

-4

-80 4 8 12 16 20 0 4 8 12 16 20

Particle Stokes number: St=0.01

St=1

St=10 St=100

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Example(cont.)

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1. Large-scale vortex structures are important controlling mechanisms for

the particle dispersion process.

2. Particle dispersion levels tend to maximize at intermediate values of

Stokes numbers, 0(1)0(10).

3. Particles of intermediate-St tend to concentrate preferentially near the

outer boundaries of large-scale vortex structures.

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Turbulentmodulation:Whatisit

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Turbulence properties: characterized by, such as, turbulent kinetic energy, Reynolds stresses, spectra, or two-point correlations.

Turbulence modulation: one or more of the statistical properties of the carrier phase turbulence is changed by the presence of particles.

Most frequently, refers to changes in the carrier-phase turbulent kinetic energy (either enhanced or decayed).

However, TKE changes do not fully describe the changes to the turbulence.

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Turbulentmodulation:Whyisitimportant

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Attenuation of TKE as a function of mass loading ratio on the center-plane of fully developed channel flow for glass beads (150 m) and copper beads (70 m).

p 5.510-5; a mean inter-particle distance of 20dp. Dilute flowSuch a reduction (by a factor >7) in TKE may completely change the characters of the turbulence and the behavior of a reactor.

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Turbulencemodulation:Keyparameters

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Then even more parameters are important!

(A lot of turbulence length scales!)f

pld

f

pp

dvu

Re

f

vSt

Mass loading ratio, Particle volume fraction, Density ratio

If 2nd phase: one size, spheres. Additional dimensionless parameters

If 2nd phase: polydisperse, non-spherical

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Turbulencemodulation:Mechanisms

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The exact mechanisms: not very well understood; the available

theories often can NOT predict the level or even sign of the change in

TKE.

Some basic mechanisms: all them require the particles be large

enough that they can not follow the flow and there is a substantial

instantaneous relative velocity between two phases.

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Turbulencemodulation:Mechanism1

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Through the carrier-phase mean velocity

The effects of carrier mean flow distortions on the carrier turbulence can be

predicted by single-phase turbulence models.

ipj

i

jij

ij

i Fxu

xxp

xuu

tu

,)()(

If the mean carrier-phase velocity field is changed by the addition of particles,

the mean strain field and the turbulence production rate will also be changed.

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Unsteady particle wakes behind relatively large particles.Important when p is not small and Rep is in the vortex shedding regime.

(Experimental) Dye streaklines produced by dye introduced at cylinder surface (Rep=140)

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Extra dissipation of turbulence by particles.

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Turbulencemodulation:Mechanism3(cont.)

19

Som

e

Fluctuating kinetic energy of the particles

Producing local flow distortions around each particle

A particle that cannot respond to fluctuations exerts a force on the fluid that opposes the relative motion. When the particles are heavy, the relative motion is produced mostly by the carrier-phase fluctuations.

The cloud of dispersed particles produces a non-uniform force field that instantaneously opposes the carrier-phase velocity fluctuations, extracting energy from the turbulence.

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Preferential concentration & sweeping of particles.

Preferential concentration occurs when particles (with particle time constants comparable to eddy time scales) are swept out of vortex cores and concentrated in convergence zones.

Particles apply an angular impulse opposing the vortex rotation as they are spun out of a simple vortex.

For St1, the ratio of the angular impulse to the initial angular momentum of the vortex is 1.3 MLR / St. This would act to suppress vortices (whose time scales are close to the particle time constant).

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Eddy distortion.

Particles are expected to produce significant local distortion of the small-scale motions.

The fact that the particle diameter is comparable to the Kolmogorov scale (dp/) indicates that the particles experience significant velocity gradients rather than the locally uniform flow assumed in most models. This means that the particles experience forces transverse to the relative velocity.

The fact that the average particle spacing (Lbetween-p) is so large means the force applied is not continuous. This is likely to produce a significant distortion of the energy-containing eddies and could lead to a higher turbulent dissipation rate.

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Howtostudymodulation:Analytical(1)

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A common approach to modeling the effects of particles on turbulence

is to treat the particles as applying a continuous force field onto the

fluid phase where the force is the reaction force (to the particle drag).

(1) Write down the modified N-S equations, by including fluid-particle inter-phase

forces

(2) Apply Reynolds decomposition; subtract the mean flow component; square

& average the fluctuating component to obtain the transport equation for k.

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23

1,1

1

11 1p

jjjj Fx

uxx

pxuu

tu

x1-momentum:

2,2

2

22 1p

jjjj Fx

uxx

pxuu

tu

x2-momentum:

3,3

3

33 1p

jjjj Fx

uxx

pxuu

tu

x3-momentum:

(1)

(2)

(3)

Eqns) RANS(')3(')2(')1(RANS of derivationin thoseas operationssimilar Use

321 uEquEquEq

23

22

21 ''' uuu The remaining is a transport equation for , i.e., k.

kpj

j Sxku

tk

,)('

T

Start with instantaneous NS equations

Operations

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o The 1st main deficiency: The assumption that the force applied onto the fluid is linearly proportional to the fluid/particle velocity difference.

Most cases in which turbulence attenuation is significant involve Rep>1.

The instantaneous force on a particle is affected by the local straining field.

When dp Kolmogorov scale, the flow around particle is highly non-uniform. The force applied by the particle onto the fluid is almost never aligned with the

instantaneous velocity difference.

o The 2nd main deficiency: The analysis assumes that energy transferred to the particle is eliminated from consideration.

When a force is applied to a particle, that particle accelerates and creates localflow distortion. The assumption is that these fluid motions are at very smallscale and are quickly dissipated by viscosity. However, the particles are oftenlarger than the smallest scales of turbulence. In this case, it is moreappropriate to assume the particleturbulence interaction as energy transferbetween different scales of velocity fluctuation.

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Howtostudymodulation:Experimental(1)

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Heavily rely on laser-based instrumentation

LDA (Laser Doppler anemometry)

The challenge is to discriminate between signals from carrier-phase tracer particles and the generally stronger signals from the dispersed particles.

PDA (Phase Doppler anemometry)

Measure particle size and velocity simultaneously;

More robust but more difficult and expensive; Generally work well for phase separation if there is a significant difference between the tracer and the smallest dispersed particles.

PIV (Particle image velocimetry)

As with LDA, the challenge is to measure the fluid phase using tracer particles

Used extensively for multiphase flow research in recent years;

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Howtostudymodulation:Experimental(2)

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Compilation of experimental data on turbulence modulation

Data cover: Jets Pipe flows Gas-particle Gas-liquid

D : particle diameter

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k : Turbulent length scale

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Howtostudymodulation:Numerical

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Difficult: due to an extremely large range of length scales

Not valid unless the particles are significantly smaller than both the turbulence Kolmogorov scale and the grid spacing.

Earlier efforts: Point-force coupling scheme

Adapt single-phase DNS codes for isotropic flows by using simple drag laws to compute the force applied onto each particle by the flow, and applying the reaction force back onto the fluid as a variable body force.

Recent efforts: Fully resolve the flow around a single particle in a small region of turbulent flow

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Turbulencemodulation:Asummary

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In gas flows, very small particles do not affect the flow. Particles with sizes of the order of Kolmogorov scale can cause significant attenuation. The attenuation reaches a maximum for Stk50 & Rep~the order of 10.

In liquid flows laden with solid particles, turbulence attenuation is rarely observed. Small particles very low Stokes numbers little effect on the flow. Large particles increase turbulence (wake effects).

Dispersed particles in the dilute regime can cause very large changes in the turbulence levels in typical shear flows.

As particle size increases, particle wake effect plays an increasing role, first decreasing turbulence attenuation and eventually causing large turbulence augmentation.

Turbulence modulation is much weaker in flows such as free shear flows.

Models: very poor, inconclusive; significant further research needed.

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Particlewallinteraction

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Lift force Lubrication

Mechanical interaction(in the absence of a fluid)

captured by wall van der Waals force (very small particles)

Collision (massive particle)rebounds but loses kinetic energy due to friction & inelasticity effects

y

v

Hydrodynamic interaction

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Sphericalparticlesmoothwallcollision

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ASSUMPTIONS:

Hard sphere; Particle deformation is neglected (i.e., throughout the collision process, the

distance between the particle center of mass and the contact point is constant and equal to the particle radius);

Coulombs friction law applies to particles sliding along a wall; Once a particle stops sliding, there is no further sliding.

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Sphericalparticlesmoothwallcollision(cont.)

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Pre-collision Compression period (1)

(from the contact with wall to the moment when Vy=0)

Recovery period (2)

(from the moment with an upward Vy to the moment no contact with wall)

Post-collision

Four periods

Pre-collision Post-collision

21tt dtFJ

y

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)0(

)2(

)0(

)2(

X

Xx

Y

Yy V

Ve

VV

e

)0(

)2(

Y

Y

V

Ve

)1(

)2(

Y

Y

J

Je

)0(

)2(

V

Ve

Coefficient of restitution (e): Four definitions

3

1

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33

y

sliding no:0, zx vv Three cases: Depending on the period in which the particle slides

along the wall.

Case1:Theparticlestopsslidinginthecompressionperiod.Case2:Theparticlestopsslidingintherecoveryperiod.Case3:Theparticlecontinuestoslidethroughoutthecompressionandrecoveryperiods.

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Sphericalparticlesmoothwallcollision:Case1

34Impulsion equations (6*3=18 eqns)

In t

he t

able

: 9*

3=2

7 u

nkn

own

s

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0)()( )()()()( kaviav sXsZsZsX (2 Equations)

0)()( )1()1()1()1()1( kavjviav XZYZX (3 Equations)

0)()( )2()2()2()2( kaviav XZZX (2 Equations)

Conditions at the end of different periods (2+3+2=7 eqns)

kavjviavrvU XZYZX )()(

Particle surface velocity:

(1) At the end of sliding period: tangential components of surface velocity are 0

(2) At the end of compression period: all components of surface velocity are 0

(3) At the end of recovery period: also no sliding.

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)( )()()1()2( rYs

YYY JJeJeJ (1 Equation)

kJfiJfkJiJ sYZs

YXsZ

sX

)()()()( (2 Equations)

Coefficient of restitution, e (1 eqn)

Coulombs law: (friction F)= (normal F)*(friction coefficient)(2 eqns + 2 unknowns)

),( ZX : direction cosines of the approach velocity in X-Z plan

(1 eqn)122 ZX

Totally 18+7+1+2+1=29 eqns for 27+2=29 unknowns: closed!

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Impulsion equations

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Impulsion equations

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Sphericalparticlesmoothwallcollision:Solution

3 9

Condition

Translationalvelocity

Angularvelocity

)1(72)0(

efvvY 0

)1(72 )0( v

vef

Y

)(

)(

)0(5

2)0(75)2(

)0()2(

)0(5

2)0(75)2(

Xa

ZZ

YY

Za

XX

vv

vev

vv

av

av

XZ

YY

ZX

)2()2(

)0()2(

)2()2(

)0()0()2(

)0()2(

)0()0()2(

)1(

)1(

YZZZ

YY

YXXX

vefvv

vev

vefvv

)0(25)0()2(

)0()2(

)0(25)0()2(

)1(

)1(

YXaZZ

YY

YZaXX

vef

vef

a:sphereradius;e:coefficientofrestitution;f:frictioncoefficient;X,Z:directioncosinesoftheapproachvelocityinXZplane;:translationalvelocitymagnitude.v

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Exercises/Practices

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(Optional) Repeat Tut. 19 Modeling evaporating liquid spray Follow the tutorial (Fluent15.0_Tutorial-19.pdf) step-by-step from

section 19.4.2, & reproduce the results as given in the tutorial.

The mesh (sector.msh) and the guide are available in the folder Files for lecture 9. Practice earlier for this non-reacting flow! More practices to come, for, e.g., reacting multiphase flows.

To install, license and use FLUENT on your PC, please visit\\Et-fs1.aau.dk\Fileshares\software_students\FLUENTand follow the guides there!

Calculation: A 100-micron nylon particle bounces off a flat, steel wall. The particle speed is 10 m/s and the angle of incidence is 30. The particle has no initial rotation and moves in the x-y plane. The coefficient of friction is 0.2 and the coefficient of restitution is 0.8. Find the rebound speed, the angle of rebound and spin rate (with correct sign). Here, X=1.0.

L09 LectureSlides 2015-Spring CHY

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