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a)

b)

Fig. 5.1 Particle velocities in different reference frames in the context of an Eulerian

continuous-phase grid: a) Lagrangian vectors based on particle positions (xp), b) Eulerianparticle velocity vectors based on average over a control volume centered at a discrete fluid

grid nodes (xf,i) .

Particle

node

Particle

path line

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a) b)

Fig. 5.2 Schematic of Lagrangian point-force particles in a two-dimensional Eulerian

continuous-phase grid showing: a) interpolation of fluid velocity of the surrounding nodes tothe particle position at x

p, and b) summation of particle volumes in a computational volume

to compute volume fraction associated with a node xi.

Interpolate u@p at particle location Collect for cell volumeassociated with a node

xixp

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a)

b)

Fig. 5.3 Two-dimensional Eulerian grid which contains discrete particles in adjoiningcomputational control volumes: a) Np,1 allowing a continuum approximation, and b)

Np,~1 so that a continuum approximation is notappropriate.

xx

lp-p

np,i np,i+1

xx

lp-p

np,i+1/2

np,i+1/2

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a)

b)

Fig. 5.4 Lagrangian particlewall interaction outcomes shown with solid lines for: a)absorbing (sticking), b) accommodation (rolling and/or sliding), and c) reflection (bouncing).Also shown for b) and c) are Eulerian no-flux boundary condition outcomes with dashed

lines.

vout = 0 (e = 0)

vout = - evin

vout = - evin

vout = - evin

c)

vin

vin

vin

vin

v

out

= 0 (e = 0)

vout = 0 (e= 0)

vin

vin

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Fig. 5.5 Comparison of Eulerian mixed-fluid and separated fluid treatments for a

computational cell in a multi-phase domain.

Separated-fluid treatment

vu and both used throughout Includes relative velocity effects

such as drag, lift, St influence etc.

PDEs needed for each phase field Ideal for computationally small

particles (dx)

um

x

u

v

x

Physical description

Continuum

descriptions

Mixed-fluid treatment

um throughout (wu) Does not employ particle diameter,

shape, relative velocity, etc.

One set of PDEs for mixed-fluid Ideal forvery small particles with

negligible inertia (St1)

= p/x3

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a)

b)

c)

Fig. 5.6 Particle point-force treatments in turbulent shear flow for particles released at the

arrow location: a) actual distribution based on fully-resolved turbulence (St1), b) Eulerian

mean diffusion based on p and steady RANS solution (St=0), and c) Lagrangian stochasticdiffusion based on steady RANS solution (St=0) with random numbers to represent

turbulent fluctuations.

Mean Eulerianparticle

concentration

contours

StochasticLagrangian

particle

trajectories

Physicalinstantaneous

particle

concentration

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TechniqueInitial

Conditions?

Turbo-

phoresis?

Non-Linear

Drag Bias?

Prefer.

Bias?

Cluster

Bias?

Mixed-Fluid

v=um(St1) No No No No NoWeakly-Sepr. Avg.

term= + +...v u w (St

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a) b)

Fig. 5.7 Different representations for particle treatment based on particle size in relation to

continuous-fluid grid resolution for a point-force representation and a distributed-forcerepresentation.

Distributed-force treatment

Allows d ~x Distributes interphase force of particle on

fluid to a distributed region

Interphase force on particle based on either:a) surface/volume averages of fluid char.b) semi-resolved fluid disturbances

Ideal for many moderate-size particles

ZFr

u@p streamlines

Point-force treatment

Requires d

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Fig. 5.8 Resolved-surface approaches showing: a) schematic of particle in a computationaldomain along with b) near-surface close-ups of a GIM mesh and c) of a IIM mesh

superimposed on the marker function distribution.

xx

D

Gridded Interface Method

(mesh along particle surface)

Immersed Interface Method

(mesh independent of particle)

Ap

Ustreamlines with f

V, p

n

U, f

V, p

UmV

UmU

Resolved-surface treatments

Requires dx (high CPU/particle) Particle surface force automatically captured by flow around particle Ideal for com lex and/or deformin article sha es and com lex flows

I

a)

c)b)

F=0

F=1

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a)

b)

Fig. 5.9 Examples of resolved-surface velocity fields relative to particle centroid velocity: a)flow past a solid spherical particle using GIM forU vectors (Kurose and Komori, 1999), and

b) deforming bubble near an eddy center using IIM forUm vectors (Loth et al. 1997).

x

y

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Numerical

approachContinuous-phase momentum PDEs Dispersed-phase momentum Eqs.

Eulerian withmixed-fluid ( ) ( )m m m m m m m,i j,tp K + = + u u u g

throughout domain and where dx

Eulerian

weakly-separated

point-force fordispersed-phase

( ) ( )

( ) ( )

f f,t

2

f f p f

1 1

1 p

+ =

+

u uu

g u g

throughout the domain

term ...= + +v u w throughout the domain

Eulerian

point-force fordispersed-phase

( ) ( )

( )

f f,t

2

f f surf

1 1

1 p

+ =

+

u uu

g u F

throughout the domain

( ) ( )

( )

p p,t

p surf coll

+ =

+ +

v vv

g F F

throughout the domain

Lagrangian

point-force for

dispersed-phase

( ) ( )

( )

f f,t

2

f f p surf

1 1

1 p n

+ =

+

u uu

g u F

throughout the domain

p p p surf collt = v g + F + Fd d

wheresurf D L S H ...= +F F + F + F + F + F

along particle trajectories

Lagrangian

resolved-surface w/

gridded interface forcontinuous-phase

( )f ,t f 2

f fP

+ =

+

U U U

g U

outside of particle volume

p p p surf collt = v g + F + Fd d

where ( )surf ,i ij ij j pF P K n A= + along particle trajectories

Resolved-surface

with one-fluid &immersed interface

( )m m,t m m m m m m,ijP K + = + U U U g with d andUmU outside the particle andUmVinside the particle

Table 5.1 Forms of the continuous-phase and dispersed-phase momentum equations forvarious multi-phase techniques (assuming constant density and viscosity of both phases and

no interphase mass transfer).

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Fig. 5.10 Physics-based diagram for selecting computational approaches for continuous-

phase and dispersed-phase.

Continuous-phase

appr

oaches

Dispersed-phasefieldapproaches

2-eqn. models

x ~ lmin/10

Turbulent flow

Q: Turbulent dispersion critical?

Transitional

flow

Time-averaged approach

Q: Anisotropy critical?

Reynolds stress

x ~ lmin/10Euler eqs.

x ~ lmin/10

Increasing Domain (D and ReD)

Increasing particle size (d and Rep)

DNS

x ~lmin/10DNS

x ~ LES

x ~ G

NoYes

NoYes

Resolved-eddy approach

Q: St critical?

NoYes

Inviscid flow

ReD=0

Laminar

flow

Q: Two-way coupling more

critical than wall reflection?

Lagrangiandistributed-force

treatment

GriddedInterface

Method

ImmersedInterface

Method

Eulerian

point-force

treatment

Lagrangian

point-force

treatment

Distributed-force Resolved-surface

Q: Particle surface stresses more

critical than deformation?

d x d ~x d x

Yes No

Eulerian

mixed-fluid

treatment

Point-force

Q: Relative velocity negligible (wu)?

Yes

No

Yes No

Q: Relative acceleration small (dw/dtDu/Dt)?

weakly-separated-fluid treatment

Yes No

separated-fluid treatment

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1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

10000 100000 1000000 10000000 100000000

DNS estimate

LES estimate

2-D RANS estimateDNS cases

LES cases

2-D RANS cases

104

105

106

107

108

f

1010

ReD

109

108

107

106

105

104

103

Fig. 5.11 Number of continuous-phase nodes for internal flows as a function of

macroscopic Reynolds number (based on streamwise length of domain).

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a)

b)

c)

Fig. 5.12 Comparison of particle treatments for:

a) Eulerian approach defined on Eulerian computational nodesb) Lagrangian approach defined on particle centroidsc) Lagrangian resolved-surface approach with a surface-fitted grid

Each particle pathdescribed by an ODE:

dv/dt =f(u,w)

larger Np

Particle velocity component at each nodedescribed by a dispersed-phase PDE:

v/t =f(u,w)

Each grid node described by a

continuous-fluid PDE:dv/dt =f(U)

More CPU/particle

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Fig. 5.13 Computational particle approaches as a function of the number of particles in the

domain (Np), number of fluid nodes in the domain (Nf), non-dimensional particle response

time (St), particle diameter (d), and continuous-phase grid resolution (x).

larger Np

Resolved-surface treatment for each

particle with gridded interface or

immersed interface method (8)

Np~1-100dx

(any St)

Distributed-force Lagrangian for

dispersed-phase (7.3)Np