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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