1. Multiphase and Reactive Flow Modelling
BMEGEÁTMW07
K. G. Szabó
Dept. of Hydraulic and Water Management Engineering,
Faculty of Civil Engineering
Spring semester, 2012

2. Introduction Physical phenomena
Major concepts, definitions, notions and terminology
Equilibrium vs. non-equilibrium models
Lagrangean vs. Eulerian description
Dimensionless numbers
Modelling strategies
Update me!

3. Basic notions and terminology
Ordinary phases:
Solid
Liquid
Gaseous
preserve volume
Condensed phases
expands
preserves shape
Fluid phases
deform
There also exist extraordinary phases, like plastics and similarly complex materials
The property of fluidity serves in the definition of fluids

4. Properties and models of solids
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:
Mechanics
Statics: mechanical equilibrium is necessary
Dynamics: governed by deviation from mechanical equilibrium
Thermodynamics of solids
Mass point model
Rigid body model
The simplest continuum model
Even more complex models

5. Models and properties of fluids
Key properties of fluids:
Large, irreversible deformations
Density, pressure, viscosity, thermal conductivity, etc. (are these properties or states?)
Features to be modelled:
Statics
Hydrostatics: definition of fluid (inhomogeneous [pressure and density])
Thermostatics: thermal equilibrium (homogenous state)
Dynamics
Mechanical dynamics: motion governed by deviation from equilibrium of forces
Thermodynamics of fluids:
Deviation from global thermodynamic equilibrium often governs processes multiphase, multi-component systems
Local thermodynamic equilibrium is (almost always) maintained
Only continuum models are appropriate!

6. Modelling Simple Fluids
Inside the fluid:
Transport equations
Mass, momentum and energy balances
5 PDE’s for
Constitutive equations
Algebraic equations for
Boundary conditions
On explicitly or implicitly specified surfaces
Initial conditions
Primary (direct)
field variables
(p,T) represaentattion
Secondary (indirect) field variables

7. Thermodynamical representations
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
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)
Note

8. Some models of fluids Stoksean fluid compressible
In both of these, the heat transport problem can be solved separately (one-way coupling):
Mutually coupled thermo-hydraulic equations:
Non-Newtonian behaviour etc.
compressible
(or barotropic) fluid
fluid dynamical equations
heat transport equation (1 PDE)
general simple fluid
fluid dynamical equations
heat transport equation
models for complex fluids

9. Phase transitions Evaporation, incl. Condensation Freezing Melting
Boiling
Cavitation
Condensation
Freezing
Melting
Solidification
Sublimation
All phase transitions involve latent heat deposition or release

10. Typical phase diagrams of a pure material:
(crystal structures) may exist
Several solid phases
TODO: 1/rho=spec volume, viscosity, chemical potential. Write equations of equilibrium.
In equilibrium 1, 2 or 3 phases can exist together
Complete mechanical and thermal equilibrium

11. Conditions of local phase equilibrium in 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.
This should be recast to the multicomponent case

12. 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.
TODO: concentrations

13. 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:
Note

14. 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

15. Conditions of local phase equilibrium in 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
This should be recast to the multicomponent case

16. Phase equilibrium in 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
TD limit on the # of phases

17. 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)
Surface tension (gas-liquid interface)
Interfacial tension (liquid-liquid interface)
(In general: Interfacial tension on fluid-liquid interfaces)
Solids typically remain granular

18. 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)

19. 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)

20. Class 3 outline Balance equations
Mass balance — equation of continuity
Component balance
Advection
Molecular diffusion
Chemical reactions

21. extensive quantity: F
density: φ=F/V=ρ∙f
specific value f=F/m
molar value f=F/n
molecular value F*=F/N

22. Differential forms of balance equations
Conservation of F:
equations for the density
general
only convective flux
equation for the specific value
These forms describe passive advection of F

24. Incomplete without class notes!
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

25. 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
Analogous cross effects appear in the heat conduction equation

26. 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.)
Make D caligraphic. C.f. flug 8.9

27. The advection–diffusion equations
local rate of change
advective flux
diffusive flux
The concentrations are conserved but not passive quantities

28. The advection–diffusion–reaction equations
local rate of change
advective flux
diffusive flux
reactive source terms
The concentrations are not conserved quantities

29. Incomplete without class notes!
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

30. 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 !

31. Description of an interface by an implicit function

32. 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
Only the normal component makes sense

33. Parametric description of an interface and its motion
Homework: Try to set it up analogously

34. 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 interface
Incomplete without class notes !

35. The kinematical boundary conditions
Incomplete without class notes !
This condition does not follow from mass conservation
The tangential eqs do not follow from the principle of conservation of mass!!
continuity of velocity
conservation of interface

36. Diffusion through an interface
Mass flux of component k in the co-moving reference frame:
Case of conservation of component mass:
on a pure interface (no surface phase, no surfactants)
without surface reactions (not a reaction front)
The component flux through the interface:
The tangential eqs do not follow from the principle of conservation of mass!!

37. Examples
Impermeability condition
Surface reaction

38. Momentum balance through an interface
Effects due to
surface tension
surface viscosity
surface compressibility
mass transfer

39. Incomplete without class notes
Surface tension
The origin and interpretation of surface tension
Incomplete without class notes !

40. Dynamical boundary conditions with surface/interfacial tension
Fluids in rest
normal component:
Moving fluids without interfacial mass transfer
tangential components:
The viscous stress tensor:
Modifies the thermodynamic phase equilibrium conditions
Incomplete without class notes !

41. The heat conduction equation
The 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
Jump conditions

42. Boundary conditions on moving interfaces
Physical balance equations imply conditions on the interface elements:
continuity conditions
jump conditions
Other conditions prescribed to obtain a well set mathematical model
With and without mass transfer
Incomplete without class notes !

43. 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
1-fluid = embedded interface

44. Phase-by-phase mathematical models
A separate phase domain for each phase
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
The sub-model for the motion of phase domains and phase boundaries (further primary model variables)
Prescribing the moving boundary conditions: coupling among the field variables of the neighbouring phase domains and the interface variables
II/13Bp6f1

45. The one-fluid mathematical model
A single fluid domain
Characteristic function for each phase
Material properties expressed by the properties of individual phases and the characteristic functions
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
The sub-model for the motion of phase domains and phase boundaries (further primary model variables)
II/13Bp6f1

46. The sub-models of phase motion (interface sub-models)
The choice of the mathematical level sub-model is influenced by the available effective numerical methods.

47. Numerical implementations of interface sub-models
Main categories
Grid manipulation
Front capturing: implicit interface representation
Front-tracking: parametric interface representation
Full Lagrangian
Specific methods
MAC: (Marker-And-Cell)
VOF: (Volume-of-Fluid)
level-set
phase-field
CIP
E.g. SPH

48. Front tracking methods
on 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!)
II/13Bp11f7

49. 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
Incomplete without class notes !
Due to the discontinuities of the function

50. Incomplete without class notes
MAC
Incomplete without class notes !

51. 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:

52. VOF in 2D and 3D SLIC: Simple Line Interface Construction PLIC:
Piecewise Linear Interface Construction
Hirt & Nichols

53. Numerical steps of VOF Interface reconstruction within the cell
determine n
several schemes
position straight interface
Interface advection
several schemes exist, goals:
conserve mass exactly
avoid diffusion
avoid oscillations
Compute the surface tension force in the Navier–Stokes eqs.
several schemes

54. 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 Reconstruction: PLIC, adapted to non-structured grids
Donor-Acceptor: Hirt & Nichols, for quadrilateral or hexahedral grid only
Compressive Interface Capturing Scheme for 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)
References in FLUENT Users Guide: [Youngs [412]], [145], [Ubbink [377]]

55. The level set method [hu: nívófelület-módszer]
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

56. The level set method
If then the computational demand can be substantially decreased

57. Signed distance function as an implicit level-set function
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

58. Only first order accurate in h
Numerical implementation of the interfacial source terms in the transport equations
Missing function graph!
With ε = 1.5h, the interface forces are smeared out to a three-cell thick band
Only first order accurate in h

59. 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 cell in a finite volume model:
Other source terms (latent heat, mass flux) in the transport equations can be treated analogously.
C.f. VOF

60. Level set demo simulations

61. Evaluation criteria for comparison
Not only for VOF and Level Set
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

62. Recommended books
Stanley Osher, Ronald Fedkiw: Level Set Methods and Dynamic Implicit Surfaces Applied Mathematical Sciences, Vol. 153 (Springer, 2003). ISBN
Details on the level set method
Grétar Tryggvason, Ruben Scardovelli, Stéphane Zaleski: Direct Numerical Simulations of Gas–Liquid Multiphase Flows (Cambridge, 2011). ISBN
Modern solutions in VOF and front tracking
ISBN:

63. SPH Smoothed 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

64. SPH simulation of hydraulic jump
Fr1 = 1.15
Fr1 = 1.37
Fr1 = 1.88

65. SPH simulation of dam-break

66. Liquid vs. liquid-gas simulation
Air
Entrapped air
Vacuum
Void bubble

67. 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