Review of Numerical Methods for Multiphase · PDF fileMartin-Luther-Universität Halle-Wittenberg Title Review of Numerical Methods . for Multiphase Flow . M. Sommerfeld . Mechanische

Martin-Luther-Universität Halle-Wittenberg

Title

Review of Numerical Methods for Multiphase Flow

M. Sommerfeld

Mechanische Verfahrenstechnik Zentrum für Ingenieurwissenschaften Martin-Luther-Universität Halle-Wittenberg 06099 Halle (Saale), Germany www-mvt.iw.uni-halle.de


Content of the Lecture

Classification of multiphase flows Classification of numerical methods for multiphase flows. Particle-resolved direct numerical simulations Interface tracking method for bubbly flows Lattice-Boltzmann method (particle resolved) Direct numerical simulations with point-particles Approaches with turbulence modelling (point-particles) Euler/Euler (two fluid) approach

Euler/Lagrange approach

Summary/Conclusions


Classification of Multiphase Flows 1 Multiphase flows may be encountered in various forms:

Transient two-phase flows

Separated two-phase flows

Dispersed two-phase flows

The two-fluid concept is suitable, but requires methods for handling the interface (e.g.Tracking, VOF,.............)


Classification of Multiphase Flows 2

Examples of separated multiphase flows:

slug flow

churn flow

Slug flow in oil-water-gas pipe flow


Classification of Multiphase Flows 3 Dispersed two-phase flow systems and important technical and industrial

applications.

Moreover, numerous examples of multiphase flows may be found in industry, such as: liquid-gas-solids reactors or spray scrubbers (droplets and particles dispersed in a gas flow)

CCoonnttiinnuuoouuss //DDiissppeerrsseedd PPhhaassee

IInndduussttrriiaall//TTeecchhnniiccaall AApppplliiccaattiioonn

Gas-solid flows pneumatic conveying, particle separation in cyclones and filters, fluidised beds, coal combustion

Liquid-solid flows hydraulic conveying, liquid-solid separation, particle dispersion in stirred vessels

Gas-droplet flows spray drying, spray cooling, spray painting, spray scrubbers, spray combustion

Liquid-droplet flows mixing of immiscible liquids, liquid-liquid extraction

Liquid-gas flows bubble columns, aeration of swage water, flotation


Examples of dispersed gas-solid flows:

Dilute two-phase flow

aerodynamic transport

Dense two-phase flow

particle-particle interaction

Classification of Multiphase Flows 2 Classification of Multiphase Flows 4

Gas-Solid Flow; Euler/Lagrange Calculation

(Helland et al. 2000)


Estimation of the interaction between particles based on the inter-particle spacing for a regular cubic arrangement:

Regimes of dispersed two-phase flows:

3/1

PP 6DL

απ

=

1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1

volume fraction [-]

100 10 1inter-particle spacing L / DP

Dilute DispersedTwo-Phase Flow

Dense DispersedTwo-Phase Flow

Two-WayCoupling

One-WayCoupling

Four-WayCoupling

Example: gas-solids flow : η = 1.0 αp = 5 ⋅ 10-4 ρp = 2500 kg/m3

ρF = 1.18 kg/m3

L/DP = 10.15

Classification of Multiphase Flows 5 Classification of Multiphase Flows 5

L

Inter-particle distance for a cubic arrangement

524.0max,P =α


Classification of Multiphase Flows 6 Dispersed multiphase flows are characterised by the following

properties of the particle phase.

Characteristics of the particles: size distribution particle shape, porosity surface and surface structure

Particle motion in fluids: velocity distribution of particles fluctuating velocity of particles

Integral values for characterising two-phase flows: volume fraction of the dispersed phase mass fraction, porosity


Classification of Multiphase Flows 7 Volume fraction of the particle phase and porosity:

Effective density of both phases (or bulk density):

Mixture density:

Number concentration (particles per unit volume):

Mass loading of particles (generally only for gas-solid flows):

V

VNi

Pii

P

∑=α

PPPbP c ρα==ρ ( ) FP

bF 1 ρα−=ρ

( ) PPFPbP

bFm 1 ρα+ρα−=ρ+ρ=ρ

VNn P

P =

( ) FFP

PPP

F

P

U1U

mm

ρα−ρα

==η

P1 α−=ε


Numerical Methods Multiphase Flow 1

Full direct numerical simulation (DNS): Resolution of particle contour and flow around the particle: contour-adapted grid Interface tracking Volume of Fluid (VOF) Lattice Boltzmann (LBM)

Discharge of bulk solids from a silo

Prof. D.D. Joseph

Particle methods: Numerical calculation of the behaviour of bulk solids with and without flow:

Molecular dynamics (MD) Granular dynamics (GD) Discrete element methods (DEM)


Numerical Methods Multiphase Flow 2

Direct numerical simulations (DNS) and large eddy simulations (LES):

Point-particle assumption for turbulence studies

Numerical methods for dispersed multiphase flows (RANS-type methods): Reynolds-averaged conservation equations with turbulence model, point-particle assumption:

Mixture models Euler/Euler (two-fluid) approach Population balance method Euler/Lagrange approach

Homogeneous isotropic turbulence


Numerical Methods Multiphase Flow 3 Application of different numerical approaches:

St = 0.25

Taylor-bubble with VOF Point-particle-DNS for turbulent flow

RANS-calculation for technical systems

Increasing modelling requirements


Particle-Resolved Simulations 3

The volume-of-fluid method (VOF):

Real interface location

VOF (Hirt & Nichols)

0futf

=∇⋅+∂∂ The interface is being reconstructed

based on the solution of a transport equation for the volume fraction.

Young`s VOF


Particle-Resolved Simulations 4 Influence of shear flow on bubble migration calculated by a VOF approach

(Tomiyama et al. 1993)

Eo = 1, Mo = 10-3 Eo = 10, Mo = 10-3

( )σ

ρ−ρ=

2bgf Dg

Eo

( )32

f

gf4fg

Moσρ

ρ−ρµ=

Determination of lift forces


Particle-Resolved Simulations 5 Flow around nearly spherical and

large non-spherical bubbles simulated by VOF (Bothe et al. 2007)

( )σ

ρ−ρ=

2hgf

h

DgEo

( )( ) ( )( )

≤<⋅

=hh

hhBA Eo4:fürEof

4Eo:fürEof,Re121.0htan288.0minC

( ) 474.0Eo0204.0Eo0159.0Eo00105.0Eof h2h

3hh +−−=

Tomiyama correlation


Particle-Resolved Simulations 6 Interface resolved direct numerical simulations allow a detailed analysis of the atomisation process

Numerical grid: 128 × 128 × 896 Mesh size: 2.36 µm

Numerical grid: 256 × 256 × 2048 Mesh size: 1.17 µm

BERLEMONT 2008

Level Set/Ghost Fluid Method


Interface Tracking Approach 1

Direct numerical simulations of the bubble motion using a interface tracking approach (Prof. Tryggvason).

Time-dependent solution of the three-dimensional, Navier-Stokes equations.

The surface tension force is calculated from the curvature of the interface.

Incompressible gas- und liquid-phase:

Equation of state for the density and the viscosity:

( ) ( ) ( )∫ −δ+∇+∇µ⋅∇+ρ−ρ+−∇=ρ⋅∇+∂ρ∂

Ffs

T0 daxxFuugpuu

tu

0ut

=ρ∇⋅+∂

ρ∂0u

t=µ∇⋅+

∂µ∂

0u =∇

Surface tension



Rectangular grid for the flow domain.

Interface grid to mark the position of the interface.

The interface is resolved within 2-3 meshes of the grid resolving the flow

domain.

Density ratio:

Spatial Discretisation: 2nd-order central differences

Time Discretisation: 2nd-order explicit method

20b

f =ρρ

Periodic boundary conditions at the cube faces



Binary mixture of bubbles, instantaneous distribution, case 1, Simulation for 29 small and 29 large bubbles (α = 6 %).

Ga = 600, 1200 Eo = 0.63, 1.0

( )2f

3bfgf Dg

Gaµ

ρρ−ρ=

( )σ

ρ−ρ=

2bgf Dg

Eo

volume ratio: 2 diameter ratio: 1.26


Interface Tracking Approach 4 Binary mixture case 2 (Göz and Sommerfeld 2004):

Volume Ratio: 8 Diameter Ratio: 2

Ga = 900, 7200 Eo = 1, 4

α = 6 %: 40 small, 5 large α = 12 %: 64 small, 8 large


Interface Tracking Approach 5 Direct numerical simulations of bubble motion in a cube by

accounting for interface deformation.

Behaviour of bi-disperse bubble systems (case 2)

α = 2 %: 28 small, 4 large α = 6 %: 40 small, 5 large



Turbulent kinetic energy for case 1 and 2:

temporal development turbulent kinetic energy in dependence of gas hold-up



Bubble Reynolds number in dependence of gas hold-up:

( )bf

bbfd 1UDRe α−

µρ

=

Case 1


Interface Tracking Approach 9 Trajectories of non-spherical bubbles obtained by interface tracking

method (Hua et al. 2007)

zigzag trajectory

helical trajectory


Lattice Boltzmann Method 1

Lattice-Boltzmann method is based on the simulation of discrete fluid elements in order to predict the macroscopic flow system.

The LBM is very robust and suitable for complex geometries. The basic variable of the Boltzmann statistics is the distribution function

f(x, v, t) which declares the number of fluid elements having the velocity v at the location x and time t:

Macroscopic properties are related to the moments of the probability function:

( ) ∫=ρ3v

3vdft,x

( ) ( ) ∫=ρ3v

3vdfvt,xut,x

( ) vdxdt,v,xfm3 3v x

33∫ ∫=


Lattice Boltzmann Method 2 Boltzmann equation (rate of change due to transport and collision) with single

relaxation approach (Bhatnagar Gross Krook (BGK) equation):

The Lattice Boltzmann equation (post collision distribution) arises from the

discretisation of the BGK equation in time, space and velocity:

Macroscopic properties, discretised:

( )),(),(),(),( )0( txftxfttxftttvxf iiiii σσσσσ τ−

∆−=−∆+∆+

∑∑=σ

σρi

i txftx ),(),(

∑∑=σ

σσρi

ii txfvtxutx ),(),(),(

Collision term

txv i

i ∆∆

= σσ

( ) ( ) ( )( )( )t,v,xft,v,xf1t,v,xfvt

0−τ

−=

∇⋅+

∂∂

Propagation term

τ = 0.515



Discrete velocity vectors of the D3Q19 model:

Velocity vectors in the different directions:

0

2

22

2

2

2

2

22

2

1

1

1

1

1

Three-dimensional

Spatial discretisation by regular grid (voxels)

19 discrete velocity directions

Sequential solution: Propagation step Collision step

===σ±±±±±±==σ±±±

==σ

σ

121i,2,c)1,1,2(,c)1,0,1(,c)0,1,1(61i,1c)1,0,0(,c)0,1,0(,c)0,0,1(

1i,0),0,0,0(

iv

txc

∆∆

=

Lattice constant txv i

i ∆∆

= σσ


Lattice Boltzmann Method 4 Discrete equilibrium distribution function (Maxwellian distribution for Kn << 1):

Pressure (equation of state):

From a series expansion around the equilibrium distribution (Chapman-Enskog-Expansion) the dependence of the viscosity on the relaxation parameter (e.g. τ = 0.515 follows:

( )t2c61 2 ∆−τ=ν

−

⋅+

⋅+ω= σσ

σσ 2

2

4

2i

2i)0(

i c2)t,x(u3

c2))t,x(uv(9

c)t,x(uv31)t,x(f

==

==

==

=

121,2,361

61,1,181

1,0,31

i

i

i

σ

σ

σ

ωσ

( ) 2sc)t,x(t,xp ⋅ρ=

3ccs =



Standard wall boundary condition: Curved wall boundary condition:

Local grid refinement:

6 Cells per Agent Particle

Forces over a particle are obtained from a momentum balance (reflection of the fluid elements)


Validation: Particle Sitting on a Wall under Shear Flow

O U T L E T

Moving Wall

Non-Slip

I N L E T

2 4 6 8 10 12 14 16 18 204980

5010

5040

5070

5100

5130

5160

5190

x/D = 18, y/D = 12

C D [ -

]

z / D [ - ]

Res = 0.02

Domain Size x/D = 18, y/D = 18, z/D = 18 Three refinement regions

µGDP

Sρ

=Re

0.01 0.1 1 10 1000.1

1

10

C LS [

- ]

ReS [ - ]

Saffman Leighton & Acrivos sim. Derksen & Larsen sim. Zeng et al. correlation Zeng et al. present LBM

Resolution: 40 grid cells of the finest mesh


Microscale Simulations by LBM Lattice-Boltzmann simulations are performed for a carrier particle (100

µm) randomly covered with the drug particles (3, 5 or 10 µm); Cui et al. 2014:

The particle cluster is centrally fixed in a cubic domain

Determination of fluid forces on the drug particles for flow conditions obtained by RANS calculations

O U T L E T

Symmetric

Symmetric

I N L E T

Re < 100: X = 7.8⋅Dcarrier; Y = Z = 6.5⋅Dcarrier Re > 100: X = 10.4⋅Dcarrier; Y = Z = 9.1⋅Dcarrier


Flow Structure at Different Re

Re = 16

Re = 32

Re = 100

Re = 200

Re = 17.9

Re = 37.7

Re = 104

Re = 202

Measurement of Taneda Present simulations


Force on Drug Particles Depending on Position Angle Total force on all the fine particles in dependence of position angle for four

different random distributions (colored dots) and resulting polynomial fitting curve (Re = 100, coverage degree 50 %, Dfine/Dcarrier = 5/100).

Data points Polynomial fit Re = 100

0 20 40 60 80 100 120 140 160 1800.0

1.0x10-9

2.0x10-9

3.0x10-9

4.0x10-9

F tota

l [N]

Position Angle [degree] Different drug particle distribution


Different Reynolds Number Fitting curves for the normal force on fine particles as a function of position

angle for different Reynolds numbers (coverage degree 50 %, Dfine/Dcarrier = 5/100)

Fitting curve: Re = 70 Re = 140 Re = 200

0 20 40 60 80 100 120 140 160 180

-2.0x10-9

-1.0x10-9

0.0

1.0x10-9

2.0x10-9

Fn [N]

Position Angle [degree]

FvdW ≈ 35 nN

Direct lift-off is not likely to occur, hence detachment occurs by sliding or rolling


Flow Resistance of Agglomerates Resistance coefficients for different agglomerates (Dietzel & So

2013): O1 O2

O3 O4

0

20

40

60

80

100

120

140

160

L_30 C_30 VES

Agglomerate type

Aer

odyn

amic

coe

ffici

ent

[-]

draglift

torque

0

20

40

60

80

100

120

140

160

180

200

c_d c_l

Dra

g an

d lif

t coe

ffic

ient

c_d

, c_l

[-] Sphere

Agg_O1Agg_O2Agg_O3Agg_O4

agglomerates with identical volume equivalent diameter

30 primary particles 50 grids per volume equivalent size porosity of convex hull

(dendrite: 0.78, compact: 0.57) Reynolds number of agglomerate: 0.3


Direct Numerical Simulations 1

Direct numerical simulations (DNS) for dispersed turbulent two-phase flows by considering the particles as point-particles and using a Lagrangian approach to simulate the dispersed phase (all real particles).

The grid needs to resolve all turbulence structures (i.e. Kolmogorov scale). The calculations are limited to smaller flow Reynolds numbers. The particles need to be smaller than the grid size and smaller than the Kolmogorov scale. point-particles !!! The equation of motion needs to be solved by accounting for all relevant particle forces (generally Stokes flow). DNS has been applied mainly to basic turbulence research, in order to analyse the particle behaviour in turbulent flows and to derive closure relations or modelling approaches.


Direct Numerical Simulations 2 Direct Numerical Simulation (DNS): ⇒ time-dependent solution of the three-

dimensional conservation equations by resolving the smallest turbulent scales.

⇒ Continuity equation:

⇒ Momentum equations:

⇒ Stress tensor:

⇒ Kronecker symbol:

( ) 0xu

t i

i =∂ρ∂

+∂

ρ∂

( ) ( )iu,P

j

ij

ij

iji gSxx

px

uutu

iρ++

∂

τ∂−

∂∂

−=∂

ρ∂+

∂ρ∂

ijj

j

i

j

j

iij x

u32

xu

xu

δ∂

∂µ+

∂

∂+

∂∂

−µ=τ

=δ

100010001

ij

Particle source terms are obtained from all fluid dynamic forces acting on all particles in a control volume


Direct Numerical Simulations 3 Direct numerical simulations (963) on

turbulence modification by particles in isotropic turbulence (Bovin et al. 1998):

( ) 38.11,49.4,26.1/

10/T

m02.0L62Re

0KF12

KE

f

=ττ

=τ

==

=φ

λ

Turbulent kinetic energy

Dissipation rate Particle dissipation

St St


Direct Numerical Simulations 4 Analysis of particle preferential accumulation in homogeneous isotropic

turbulence (Scott and Shrimpton 2007):

Domain size: 643

• Enstropy • Velocity field

St = 0.25

St = 4.0


RANS-Approaches 1

The numerical calculation of industrial flow processes is generally based on the Reynolds-averaged Navier-Stokes (RANS) equations.

Turbulence modelling (e.g. k-ε turbulence model, Reynolds-stress model)

Approaches for dispersed multiphase flows:

Euler/Euler Approach

Euler/Lagrange Approach


Modelling Strategy With reduced complexity of the numerical method used, the modelling

(closure) requirements are increasing:

Modelling approach:

The resulting analytical and/or empirical models have to be implemented and validated based on detailed experiments.

Particle resolved simulations (full DNS)

RANS methods with point particles

Modelling of micro-physical phenomena: non-spherical particles particle-wall collisions droplet coalescence bubble break-up

Experimental studies

Direct numerical simulations

Increased modelling


Two-Fluid Approach 1

• Both (multiple) phases are treated as interpenetrating continua. • The properties of the dispersed phase have to be averaged for the control volumes (DP << ∆x). • Similar sets of conservation equations are obtained for both phases, allowing for identical solution algorithms. • Requires considerable modelling work to describe the relevant micro-physics (closure of the conservation equations):

Interaction between the phases Turbulent dispersion of particles (fluid-particle correlation) Wall collisions of particles (wall roughness effect)

• The consideration of size distributions requires the solution of several sets of conservation equations (for each phase). • Numerical diffusion at particle phase boundaries may result in errors. • This approach is especially suitable for high volume fractions of the dispersed phase. • The two-fluid approach might be also coupled to a population balance.

Euler/Euler approach (two-fluid approach)



Classification of multi-fluid models:

Mixture Models

Drift flux model: The slip between the phases is calculated by analytical correlations

Homogeneous model: All phases share the same velocity field (no slip)

Complete Multi-Fluid Model

Reduced turbulence model: Turbulence model only for the continuous phase The fluctuation energy of the dispersed phase is related to the fluid turbulence by appropriate correlations

Multiphase turbulence model: For each phase conservation equations are solved for the turbulence properties including coupling


Two-Fluid Approach 3 The averaging of the conservation equations for a multiphase flow results in

the following set of equations (Simonin 2000):

Continuity equations:

Momentum equation continuous phase:

( ) ( )

( ) ( ) ( )ni,nnni

nn

iffi

ff

mCUxt

0Uxt

=ρα∂∂

+ρα∂∂

=ρα∂∂

+ρα∂∂

( ) ∑ ρα−′′∂

∂ρα−

∂∂

µ+∂∂

−=∂

∂ρα+

∂∂

ραN n

Pinnj,fi,f

jff

jj

i,f2

ij

i,fj,fff

i,fff m

Fuuxxx

UxP

xU

Ut

U

Mass transfer due to collision/coalescence



Momentum equation dispersed phase:

Momentum transfer due to drag force:

Turbulent dispersion described by drift velocity:

( )

( )n

i,nnninni,nn

i,nj,nnnji

nj

i,nj,nnn

i,nnn

mF

gumC

uuxx

Px

UU

tU

ρα+ρα+′+

′′ρα∂

∂−

∂∂

ρ−=∂

∂ρα+

∂∂

ρα

( ) i,ni,ni,fP2PP

Dfi,n VUUm

Df18F +−

ρµ

=

∂α∂

α−

∂α∂

α−=

j

f

fj

n

nij,fni,n x

1x

1DV


Two-Fluid Approach 5 Diffusion coefficient:

Integral time scale of turbulence seen by the particles (Csanady):

j,ni,ffnij,fn uuD ′′τ=

2r

Lllfn

C1T

ξ+=τ

β

2r

Lfn

C41T

ξ+=τ

β

⊥

k32u

r

∆

=ξ

45.0C =β

Requires information on turbulent integral time scale


Two-Fluid Approach 6 Simple correlations for stationary homogeneous isotropic turbulence:

Fluid-Particle velocity correlation:

Kinetic energy of particle fluctuating motion:

In the case of more complex two-phase flows additional transport equations for both properties have to be solved.

η+η′′=′′

r

rj,fi,fj,ni,f 1

uu2uu

η+η′′=′′

r

rj,fi,fj,ni,n 1

uuuu

n

fnr τ

τ=η

Simonin, 1991, 2000


Fluidised Bed

Numerical computation of a bubbling fluidised bed with the two-fluid approach (see Sommerfeld 2013).

dp ~ 140µm

ρp/ρg ~ 11000

Ug ~ 2.5m/s

Δx = 30cm Δx = 14cm Δx = 10cm Δx = 3cm


Euler/Lagrange Approach 1

• The fluid flow is calculated by solving the Reynolds-averaged Navier-Stokes equations (or LES) with an appropriate turbulence model. • The dispersed phase is simulated by tracking a large number of particles through the flow field (representative particles). • A statistically reliable determination of the particle phase properties and source terms requires a large number of particles to be tracked. • This method is a hybrid approach and requires coupling iterations between Eulerian and Lagrangian part. • The particles are point-particles which have to be considerably smaller then the size of the grids (DP << ∆x). • The particle size distribution may be considered with good resolution. • The relevant micro-processes may be : Wall collisions of particles Inter-particle collisions and agglomeration Droplet/bubble coalescence and break-up

• For high particle concentration the standard approaches may cause considerable convergence problems.

Euler/Lagrange Approach


Euler/Lagrange Approach 2 Fluid flow; continuous phase (Sommerfeld 1996; Lain &

Sommerfeld 2012):

constants

Cµ = 0.09

C1 = 1.44

C2 = 1.92

σk = 1.0

σε = 1.0

φ Sφ Sφ,P Γφ

1 - 0 -

Ui iii

jU

j

gxp

xU

x iρ+

∂∂

−

∂

∂Γ

∂∂

P,UiS tµ+µ

k G k − ρεP,kS µ

µσ

+ t

k

ε ( )ερε

kC G Ck1 2−

P,Sεµ

µσ ε

+ t

j

i

i

j

j

itk x

UxU

xU

G∂∂

∂

∂+

∂∂

µ= , ε

ρ=µ µ

2

tkC

General form of the conservation equations in connection with the k-ε turbulence model:

( ) ( ) Pjj

jfj

f SSxx

Uxt φφ ++

∂φ∂

Γ∂

∂=φσ

∂∂

+φσ∂∂

( )ρα−=σ Pf 1 effective density



Dispersed phase:

( ) ( )( )

( )i

PiP

PFPFLR

2p

F

FpFPLS2p

FPi,PiDP

PP

i,PP

F1gmuuuuCD42

uuDCD42

uuuuCmD4

3td

udm

+

ρρ

−+Ω

−×Ω−

πρ+

ω×−πρ

+−−ρ

ρ=

The Lagrangian approach is based on tracking a large number of representative particles through the flow field considering all relevant forces.

drag force gravity/ buoyancy slip-shear lift other forces, e.g. electrostatic slip-rotation lift

Ω⋅Ω

ρ=

ω

R

5pFp

p C2

D2dt

dI

Rotation:

pp u

dtxd

=

Models elementary processes: turbulent dispersion particle-rough wall collision inter-particle collisions



Two-way coupling approach for stationary flows

Eulerian Part

Calculation of the fluid flow without particle phase source terms

Eulerian Part Calculation of the fluid flow with

particle phase source terms: • Converged Solution • Solution with a fixed number

of iterations

yes

Output: Flow field,

Particle-phase statistics

no

Grid generation, Boundary conditions,

Inlet conditions

Lagrangian Part Tracking of parcels without inter-

particle collisions, Sampling of particle phase properties

and source terms

Lagrangian Part Tracking of parcels with inter-particle

collisions, Sampling of particle phase properties

and source terms

Convergence two-way coupling

Coupling Iterations

Under-relaxation of the source terms improves convergence behaviour !!! (Kohnen et al. 1994)

Under-relaxation of source terms:

( ) ( )1i

calculatedPi

P1i

P SS1S +φφ

+φ γ+γ−=


Conclusion Numerical methods for multiphase flows were subdivided into three classes:

Particle resolved numerical simulations (full DNS) become increasingly

important for analysing micro-physical phenomena and providing results for modelling.

Point-particle DNS is mainly used for turbulence studies. For analysing or optimising processes of technical relevance, RANS

approaches are still very important (i.e. Euler/Euler and Euler/Lagrange). However, the importance of LES is continuously increasing. The Lagrangian approach has considerable advantages in modelling

elementary processes and is preferred for systems with particle size distributions.

Full DNS methods resolving the particles; or for interfacial systems. Methods for dispersed multiphase flows with point particle approximation (DNS, LES and RANS). Discrete particle methods (e.g. DPM and DEM) for dense particle systems (with and without flow).

Review of Numerical Methods for Multiphase · PDF fileMartin-Luther-Universität Halle-Wittenberg Title Review of Numerical Methods . for Multiphase Flow . M. Sommerfeld . Mechanische

Documents