Multiphase and Reactive Flow Modelling

BMEGEÁTMW07

K. G. SzabóDept. of Hydraulic and Water Management

Engineering,

Faculty of Civil Engineering

Spring semester, 2012

Basic notions and terminology

Ordinary phases:– Solid– Liquid– Gaseous

preserves shape

Fluid phases

deform

preserve volume

Condensed phases

expands

There also exist extraordinary phases, like plastics and

similarly complex materials

The property of fluidity serves in the definition of fluids

Properties of solids:• Mass (inertia),

position, translation• Extension (density, volume),

rotation, inertial momentum• Elastic deformations (small, reversible

and linear), deformation and stress fields• Inelastic deformations (large, irreversible

and nonlinear), dislocations, failure etc.Modelled features:1. Mechanics

• Statics: mechanical equilibrium is necessary• Dynamics: governed by deviation from

mechanical equilibrium

2. Thermodynamics of solids

Properties and models of solids

Mass point model

Rigid body model

The simplestcontinuum model

Even more complex models

Key properties of fluids:• Large, irreversible deformations• Density, pressure, viscosity, thermal conductivity, etc. (are

these properties or states?)

Features to be modelled:1. Statics

• Hydrostatics: definition of fluid (inhomogeneous [pressure and density])

• Thermostatics: thermal equilibrium (homogenous state)

2. Dynamics1. Mechanical dynamics: motion governed by deviation from

equilibrium of forces2. Thermodynamics of fluids:

• Deviation from global thermodynamic equilibrium often governs processes multiphase, multi-component systems

• Local thermodynamic equilibrium is (almost always) maintained

Models and properties of fluids

Only continuum models are appropriate!

Modelling Simple Fluids

• Inside the fluid:– Transport equations

Mass, momentum and energy balances

5 PDE’s for

– Constitutive equationsAlgebraic equations for

• Boundary conditionsOn explicitly or implicitly specified surfaces

• Initial conditions

),(),(),,( rrur

tTttp and

),,(),,(),,( TpkTpTp

Primary (direct)field variables

Secondary (indirect) field variables

Note

Thermodynamical representations

• All of these are equivalent:can be transformed to each other by appropriate formulæ

• Use the one which is most practicable:e.g., (s,p) in acoustics: s = const ρ(s,p) ρ(p).

We prefer (T,p)

Representation (independent variables) TD potential

enthropy and volume (s,1/ρ) internal energy

temperature and volume (T,1/ρ) free energy

enthropy and pressure (s,p) enthalpy

temperature and pressure (T,p) free enthalpy

Some models of fluids•

•

In both of these, the heat transport problem can be solved separately (one-way coupling):

• Mutually coupled thermo-hydraulic equations:

• Non-Newtonian behaviour etc.

const,const

),,(),,(),,( TpkTpTp

const),p(

Stoksean fluid

compressible (or barotropic) fluid

models for complex fluids

general simple fluid

fluid dynamical equations

heat transport equation (1 PDE)

fluid dynamical equations

heat transport equation

Phase transitions

• Evaporation, incl.– Boiling– Cavitation

• Condensation• Freezing• Melting• Solidification• SublimationAll phase transitions

involve latent heat deposition or release

Typical phase diagrams of a pure material:

In equilibrium 1, 2 or 3 phases can exist together

Complete mechanical and thermal equilibrium

Sev

eral

sol

id p

hase

s(c

ryst

al s

truc

ture

s) m

ay e

xist

1

Conditions of local phase equilibriumin a contact point

in case of a pure material• 2 phases:

T(1)=T(2)=:T

p(1)=p(2)=:p

μ(1)(T,p)= μ(2)(T,p)

Locus of solution:a line Ts(p) or ps(T), the saturation temperature or pressure (e.g. ‘boiling point´).

• 3 phases:T(1)=T(2) =T(3)=:T

p(1)=p(2)=p(3) =:p

μ(1)(T,p)= μ(2)(T,p) = μ(3)(T,p)

Locus of solution:a point (Tt,pt), the triple point.

Multiple components

Almost all systems have more than 1 chemical components

Phases are typically multi-component mixtures• Concentration(s): measure(s) of composition

There are lot of practical concentrations in use, e.g.– Mass fraction (we prefer this!)

– Volume fraction (good only if volume is conserved upon mixing!)

Concentration fields appear as new primary field variables in the equation:One of them (usually that of the solvent) is redundant, not used.

k k

kkkk mmcmmcmmcmmc 1,,, 2211

k k

kkkk VV,VV,VV,VV 12211

N,,k),t(ck p2for

r

Note

Notations to be used(or at least attempted)

• Phase index (upper): – (p) or– (s), (l), (g), (v), (f) for solid, liquid, gas, fluid, vapour

• Component index (lower): k• Coordinate index (lower): i, j or t

Examples:

• Partial differentiation:),,(, 321 zyxit

)()()( ,, pi

pk ucs

Material properties in multicomponent mixtures

• One needs constitutional equations for each phase

• These algebraic equations depend also on the concentrations

For each phase (p) one needs to know:– the equation of state– the viscosity– the thermal conductivity– the diffusion coefficients

,c,c,T,pD

,c,c,T,pk

,c,c,T,p

,c,c,T,p

ppp,k

ppp

pppk

ppp

21

21

21

21

Conditions of local phase equilibriumin a contact point

in case of multiple components• Suppose N phases and K components:• Thermal and mechanical equilibrium on the interfaces:

T(1)=T(2) =T(3)=:T

p(1)=p(2)=p(3)=:p• Mass balance for each component among all phases:

(N-1)K equations for 2+N(K-1) unknowns

NK

NNN

KKKKK

NK

NNNKK

NK

NNNKK

c,,c,c,p,Tc,,c,c,p,Tc,,c,c,p,T

c,,c,c,p,Tc,,c,c,p,Tc,,c,c,p,T

c,,c,c,p,Tc,,c,c,p,Tc,,c,c,p,T

2122

22

1211

21

11

21222

22

12

211

21

11

2

21122

22

12

111

21

11

1

Phase equilibriumin a multi-component mixture

Gibbs’ Rule of Phases, in equilibrium:

If there is no (global) TD equilibrium:additional phases may also exist

– in transient metastable state or

– spatially separated, in distant points

22components phases K#N#

TD limit on the # of phases

Miscibility

The number of phases in a given system is also influenced by the miscibility of the components:

• Gases always mix →Typically there is at most 1 contiguous gas phase

• Liquids maybe miscible or immiscible →Liquids may separate into more than 1 phases(e.g. polar water + apolar oil)1. Surface tension (gas-liquid interface)2. Interfacial tension (liquid-liquid interface)(In general: Interfacial tension on fluid-liquid interfaces)

• Solids typically remain granular

Topology of phases and interfaces

A phase may be• Contiguous

(more than 1 contiguous phases can coexist)

• Dispersed:– solid particles, droplets

or bubbles– of small size– usually surrounded by

a contiguous phase• Compound

Interfaces are• 2D interface surfaces

separating 2 phases– gas-liquid: surface– liquid-liquid: interface– solid-fluid: wall

• 1D contact lines separating 3 phases and 3 interfaces

• 0D contact points with 4 phases, 6 interfaces and 4 contact lines

Topological limit on the # of phases(always local)

Special Features to Be Modelled

• Multiple components →– chemical reactions– molecular diffusion of constituents

• Multiple phases → inter-phase processes– momentum transport,– mass transport and– energy (heat) transfer

across interfaces.

(Local deviation from total TD equilibrium is typical)

Class 3 outline

• Balance equations

• Mass balance — equation of continuity

• Component balance

• Advection

• Molecular diffusion

• Chemical reactions

• extensive quantity: F

• density: φ=F/V=ρ∙f

• specific value f=F/m

• molar value f=F/n

• molecular value F*=F/N

Differential forms of balance equations

Conservation of F:• equations for the

density– general

– only convective flux

• equation for the specific value

0

conservedisif

0

if

0

v

v

vj

j

ffD

m

tt

t

F

Ft

These forms describe passive advection of F

Class 4

• Diffusion — continued– further diffusion models– the advection—diffusion equations

• Chemical reactions– the advection—diffusion—reaction equations– stochiometric equation– reaction heat– chemical equilibrium– reaction kinetics– frozen and fast reactions

Incomplete without class

notes

!

Further diffusion models

Thermodiffusion and/or barodiffusion

Occur(s) at

• high concentrations

• high T and/or p gradients

For a binary mixture:

coefficient of thermodiffusion

coefficient of barodiffusion

:

:diff

p

T

pT

kD

kD

ppkTTkcD

jAnalogous cross effects

appear in the heat conduction equation

Further diffusion models

Nonlinear diffusion model

Cross effect among species’ diffusion

Valid at• high concentrations• more than 2 components• low T and/or p gradients

(For a binary mixture it falls back to Fick’s law.)

fractionmole:

massmolarmean:

tcoefficiendiffusionbinary:

if

kk

k

kk

k

k

kk

ks ks

s

kk

kk

kkkk

k

kk

cM

Mx

MxM

D

K

kD

x

M

M

D

xK

xKK

M

M

0

adj~

det

~~

KK

Kj

The advection–diffusion equations

kkktkt

kkkt

cccD

m

cc

jv

jv

1

conservedissince

advective flux

diffusive flux

local rate of change

kcD 2e.g.

The concentrations are conserved but not passive quantities

The advection–diffusion–reaction equations

rateproductionspecificlocal

conservedissince

densityrateproductionmass

kkktkt

kkkt

cccD

m

cc

jv

jv

1

advective flux

diffusive flux reactive source terms

local rate of change

The concentrations are not conserved quantities

Class 5

• Mathematical description of interfaces– implicit description– parametric description (homework)– normal, tangent, curvature– interface motion

• Transport through interfaces– continuity and jump conditions– mass balance– heat balance– force balance

Incomplete without class

notes

!

Interfaces and their motion

• Description of interface surfaces:– parametrically– by implicit function– (the explicit description is the common case of

the two)

• Moving phase interface:(only!) the normal velocity component makes sense

New primary(?)

field variables

Incomplete without class

notes

!

Description of an interface by an implicit function

dVzyxtFzyxtFzyxtzyxt

dVzyxtFzyxtFzyxtfdAzyxtf

RR

FF

zyxtF

),,,(),,,(),,,(),,,(

),,,(),,,(),,,(),,,(

11

2

1

0),,,(

21

vdAv

n

n

:integralssurfaceIIType

:integralssurfaceIType

curvature)(mean

normal)(unit

Equation of motion of an interface given by implicit function

• Equation of interface• Path of the point that

remains on the interface (but not necessarily a fluid particle)

• Differentiate• For any such point the

normal velocity component must be the same

• Propagation speed and velocity of the interface

u

tu

FuF

FtFttFdt

d

ttF

t

tF

t

t

nu

rn

n

rr

r

r

r

)(

0

0)())(,(

0))(,(

)(

0),(

Only the normal component makes sense

Parametric description of an interface and its motion

Homework:Try to set it up analogously

Mass balance through an interface

Steps of the derivation:• describe in a reference frame that moves

with the interface (e.g. keep the position of the origin on the interface)

• velocities inside the phases in the moving frame

• mass fluxes in the moving frame• flatten the control volume onto the

interfaceIncomplete

without class notes

!

0nu

nuu

nuunuu

:componentstangentialtheFor

:interfacethethroughfluxmassnetThedef

mass

0

2211

j Incomplete without class

notes

!

0nu

0u

nu

nununu

:componentstangential

:transfermass(net)Without

mass

0

0 21 uj

0 FFt u

0 FFt u

The kinematical boundary conditions

continuity of velocity

conservation of interface

This condition does not follow from mass conservation

Mass flux of component k in the co-moving reference frame:

Case of conservation of component mass:

kkkkkk cc juuuuuuuu

Diffusion through an interface

• on a pure interface(no surface phase, no surfactants)

• without surface reactions(not a reaction front)

The component flux through the interface:

njnj

nuunuu

nj

nuu

2211

2211

0

0

kkkk

kkkkk

kk

kk

jcjc

j

jc

massmass

def

mass

Momentum balance through an interface

Effects due to• surface tension• surface viscosity• surface

compressibility• mass transfer

Surface tension

• The origin and interpretation of surface tension

Incomplete without class

notes

!

ijjiij uu

S

Sp

Sp

nttnτt

nτn 2

2

Incomplete without class

notes

!

Dynamical boundary conditionswith surface/interfacial tension

• Fluids in rest– normal component:

• Moving fluids without interfacial mass transfer– normal component:– tangential components:

The viscous stress tensor:

Modifies the thermodynamic phase equilibrium conditions

The heat conduction equationThe equation• Fourier’s formula

– (thermodiffusion not included!)• Volumetric heat sources:

– viscous dissipation– direct heating– chemical reaction heat

Boundary conditions• Thermal equilibrium• Heat flux:

– continuity (simplest)– latent heat (phase transition of

pure substance)Even more complex cases:– chemical component diffusion– chemical reactions on surface– direct heating of surface

T

qTTc tp

heat

heatheat

j

ju

heat

massheat

heat

jn

jn

jn

jL

T

0

0

Jump conditions

Approaches of fine models

Phase-by-phase• Separate sets of

governing equations for each phase

• Each phase is treated as a simple fluid

• Describing/capturing moving interfaces

• Prescribing jump conditions at the interfaces

One-fluid• A single set of governing

equation for all phases• Complicated

constitutional equations• Describing/capturing

moving interfaces• Jumps on the interfaces

are described as singular source terms in the governing equations

Phase-by-phasemathematical models

1. A separate phase domain for each phase2. A separate set of balance equations for each

phase domain, for the primary field variables introduced for the single phase problems, supplemented by the constitutional relations describing the material properties of the given phase

3. The sub-model for the motion of phase domains and phase boundaries(further primary model variables)

4. Prescribing the moving boundary conditions:coupling among the field variables of the neighbouring phase domains and the interface variables

,,,

,,,

,,,

)(

)(

)(

pTk

pT

pT

p

p

p

,,

,,,,

r

rur

tT

ttpp

pp

0, r

tF 0, r

tF 0, r

tFe.g.

The one-fluid mathematical model

1. A single fluid domain2. Characteristic function for each phase3. Material properties expressed by the

properties of individual phases and the characteristic functions

4. A single set of balance equations for the primary field variables introduced for the single phase problems, supplemented by discrete source terms describing interface processes

5. The sub-model for the motion of phase domains and phase boundaries(further primary model variables)

p

p

p

1)r,t(

0ro1)r,t(

p

pp

p

pp

p

pp

kk )()(

)()(

)()(

,,

,,,,

r

rur

tT

ttp equivalentsomethingro)r,t(p

Specific methods• MAC: (Marker-And-Cell)• VOF: (Volume-of-Fluid)• level-set• phase-field• CIP

Numerical implementationsof interface sub-models

Main categories• Grid manipulation• Front capturing:

implicit interface representation

• Front-tracking:parametric interface representation

• Full Lagrangian E.g. SPH

Front tracking methodson a fixed grid

by connected marker points(Suits the parametric

mathematical description)• In 3D: triangulated unstructured

grid represents the surface

Tasks to solve:• Advecting the front• Interaction with the grid

(efficient data structures are needed!)

• Merging and splitting (hard!)

Incomplete without class

notes

!

MAC(Marker-And-Cell method)

• An interface reconstruction — front capturing — model (the primary variable is the characteristic function of the phase domain, the interface is reconstructed from this information)

• The naive numerical implementation of the mathematical transport equation :– 1st (later 2nd) order upwind differential scheme

• Errors (characteristic to other methods as well!):– numerical diffusion in the 1st order– numerical oscillation in higher orders

0ut

Due to the discontinuities of the function

MAC

nj

nj

nj

nj CC

h

tuCC 1

1

Incomplete

without class notes

!

VOF (Volume-Of-Fluid method)

1D version (1st order explicit in time):• Gives a sharp interface, conserves mass• Requires special algorithmic handling

The scheme of evolution:

VOF in 2D and 3D

PLIC:Piecewise LinearInterface Construction

SLIC:Simple LineInterface Construction

Hirt & Nichols

Numerical steps of VOF

1. Interface reconstruction within the cell

1. determine n• several schemes

2. position straight interface

2. Interface advection• several schemes exist,

goals:• conserve mass exactly

• avoid diffusion

• avoid oscillations

3. Compute the surface tension force in the Navier–Stokes eqs.• several schemes

Implementation of VOF in• Any number of phases can be

present• The transport equation for is

adapted to allow– variable density of phases– mass transport between

phases• Contact angle model at solid

walls is coupled• Special (`open channel´)

boundary conditions for VOF• Surface tension is

implemented as a continuous surface force in the momentum equation

• For the flux calculations ANSYS FLUENT can use one of the following schemes:– Geometric ReconstructionGeometric Reconstruction:

PLIC, adapted to non-structured grids

– Donor-AcceptorDonor-Acceptor:Hirt & Nichols, for quadrilateral or hexahedral grid only

– Compressive Interface Compressive Interface Capturing Scheme for Capturing Scheme for Arbitrary Meshes (CICSAM)Arbitrary Meshes (CICSAM):a general purpose sheme for sharp jumps (e.g. high ratios of viscosities) for arbitrary meshes

– Any of its standard schemes(probably diffuse and oscillate)

The level set method[hu: nívófelület-módszer]

dVFFS

dVzyxtFzyxtFzyxtzyxt

dVzyxtFzyxtFzyxtfdAzyxtf

RR

FuF

FF

zyxtF

t

()2

),,,(),,,(),,,(),,,(

),,,(),,,(),,,(),,,(

11

2

1

0

0),,,(

21

vdAv

n

n

n

• the interface is implicit• F is continuous

– standard advection schemes work fine

• the curvature can be obtained easily

• the effect of surface tension within a cell can be computed

• Ifthen the computational demand can be substantially decreased

The level set method

dVFFS

dVzyxtFzyxtFzyxtzyxt

dVzyxtFzyxtfdAzyxtf

FRR

uF

F

zyxtFzyxtF

t

()2

),,,(),,,(),,,(),,,(

),,,(),,,(),,,(

11

2

1

0

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

2

21

vdAv

n

n

Signed distance functionas an implicit level-set function

222

sgn

01

0

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

FhF

FFS

FFS

FFSF

FF

zyxtFzyxtF

t

u

• What kind of function is it? Signed distance from the interface!

• Alas, is not conserved.

• Generating F: τ is pseudo-time (t is not changed)

• Apply alternatively!• Unfortunately, mass is not

conserved in the numeric implementation.

• A better numeric scheme

1F

FF

ttFH

F

FF

F

F

F

FFF

F

FH

0

10

,,

1

cos2

1

2

10

1

sin2

1

22

10

rr

if

if

if

if

if

if

Numerical implementation of the interfacial source terms in the transport equations

Only first order accurate in h

With ε = 1.5h, the interface forces are smeared out to a three-cell thick band

• For example, the normal jump condition due to surface tension can be expressed as an embedded singular source term in the Navier–Stokes equation:

– contribution to a single cellin a finite volume model:

• Other source terms (latent heat, mass flux) in the transport equations can be treated analogously.

nτgv FSpDt 2

cell

dVFFS 2

C.f. VOF

Level set demo simulations

Evaluation criteria for comparison

• Ability to– conserve mass/volume

exactly– numerical stability– keep interfaces sharp

(avoid numerical diffusion and oscillation)

• Ability and complexity to model– more than 2 phases– phase transitions– compressible fluid phases

• Demands on resources– number of equations– grid spacing– grid structure– time stepping– differentiation schemes

• Limitations of applicability– grid types– differential schemes– accuracy

Not only for VOF and Level Set

Recommended books

• Stanley Osher, Ronald Fedkiw: Level Set Methods and Dynamic Implicit SurfacesApplied Mathematical Sciences, Vol. 153 (Springer, 2003). ISBN 978-0-387-95482-0– Details on the level setlevel set method

• Grétar Tryggvason, Ruben Scardovelli, Stéphane Zaleski: Direct Numerical Simulations of Gas–Liquid Multiphase Flows (Cambridge, 2011). ISBN 9780521782401– Modern solutions in VOFVOF and front trackingfront tracking

SPHSmoothed Particle Hydrodynamics

• The other extreme — a meshless method:The fluid is entirely modelled by moving representative fluid particles — fully Lagrangian

• There are no– mesh cells– interfaces– PDE– field variables

• Everything is described via ODE’s

SPH simulation of hydraulic jump

Fr1 = 1.37

Fr1 = 1.88

Fr1 = 1.15

SPH simulation of dam-break

Liquid vs. liquid-gas simulation

Ent

rapp

ed a

irV

oid

bubb

leV

acuu

mA

ir

Evaluation of SPH

Advantages• Conceptually easy• Best suits problems

– in which inertia dominates (violent motion, transients, impacts)

• FSI modelling

– with free surface or liquid–gas interface

• Interface develops naturally

• Computationally fast– Easy to parallelise– Can be adapted to GPU’s

Disadvantages• High number of particles• Hard to achieve

incompressibility• Some important boundary

conditions are not realised so far