Simulation of multiphase flows

1
Simulation of multiphase flows

• Multiphase/multicomponent fluid systems
• Fluid domain W split in two o more fluid regions
  W1, W2
• Fluids separated by interface G
• Fluid with different fluid properties r1, m1, r2,
  m2
• Interface provided by surface tension s

• Numerical simulation of Multiphase/multicomponent
  flows
• Interface tracking
• flow field solution
• Flow field interface coupling

• Numerical approaches
• Sharp interface approach
• Diffuse interface approach

• Numerical issues
• Large change in physical properties across
  interface (i.e. air water r1/r2?1000 )
• Interface dissolution and generation (i.e.
  droplet breakup/coalescence )
• Jump conditions at the interface

2
Sharp interface approaches

• Basic ideas
• Interface is treated as sharp layer
• Each fluid described by a set of Navier Stokes
  equations
• Fluid properties change sharply across the
  interface
• Boundary conditions at the interface (free
  boundary problem)
• Independent interface tracking

• Interface tracking
• Lagrangian tracking (sharp interface)
• Level set (transport equation of diffuse level
  function)
• Front tracking (sharp interface)
• Volume of fluid (transport equation of diffuse
  fraction function)

3
Sharp interface approaches

• Drawbacks of flow field solution
• Application of a set of boundary conditions at
  the interface
• Sharp variations of fluid properties at the
  interface, infinite gradients
• Particular solution techniques should be
  developed (i.e. ghost fluid methods, )
• Smearing of fluid properties should be introduced
  (i.e. Immersed boundary method)

• Drawbacks of Interface tracking
• Level set, Volume of fluid interface degradation
  and mass leakage (non conservative methods)
• Level set, Volume of fluid Interface
  re-initialization techniques required (remove
  interface degradation)
• Sharp approaches cannot deal interface creation
  and dissolution

4
Diffuse Interface Approach

• Interface is a finite thickness transition layer
• Localized and controlled fluid mixing (even for
  immiscible fluids)
• fluid properties change smoothly from between the
  fluids

r2
5
Phase Field Modelling

• Definition of a scalar order parameter
• Two fluid system represented as a mixture
• The order parameter represents the local mixture
  concentration
• f fM identifies the actual sharp interface

Fluid properties proportional the Order parameter

• State of the system represented by a scalar field
  
• Continuous over the domain
• Smooth variations across the interfaces
• Order parameter function of the position

6
The Cahn Hilliard Equation

• Time evolution of the order parameter gives the
  evolution of the system

• From the PFM, the system is modeled as a mixture
  of two fluids
• The order parameter represents the fluid
  concentration
• Evolution of the concentration given by
  convective diffusion equation

• Mass diffusion flux to be determined
• derivation from evolution of binary mixtures free
  energy
• Thermodynamically consistent
• First derivation Cahn Hilliard (1958, 1959)

• Cahn Hilliard equation o generalized mass
  diffusion equation
• Evolution of an immiscible partially-miscible
  multiphase fluid system
• Interface evolution controlled by a chemical
  potential

7
The free energy functional

• Thermodynamic chemical potential, by definition

• Partial derivative of free energy functional with
  respect to the mixture concentration

• Free energy functional defines the behavior of
  the system under analysis
• Fluid repulsion in bulk fluid regions (bulk free
  energy)
• Controlled fluid mixing in the interfacial
  regions (non-local free energy)

• Bulk Free energy or ideal free energy
• Accounts for the fluid repulsion
• Shows two stable (minima) solutions
• Its simplest form is a double-well potential
• Different formulations can represent more complex
  systems (tri-phase,)

8
The free energy functional

• Non-Local Free energy
• Responsible for the interfacial fluid mixing
• Depends on the order parameter gradients
  (non-local behavior)
• Keeps in account the mixing energy stored into
  the interface

• The chemical potential, using the double-well
  free energy

• The cahn hilliard equation, using the double-well
  free energy

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Interface Properties

• The equilibrium profile of the order parameter
  across the interface
• Free energy is at its minima
• Chemical potential is null
• Two uniform solutions (bulk fluid regions)
• Non-uniform solution normal to the interface

analytical non-uniform solution first derived by
van der waals (1879)
Capillary length
99 of the surface tension is stored in an
interface thickness of 4.164 capillary lengths
10
Interface Properties

• The free energy functional keeps in account the
  mixing energy
• Mixing energy is stored into the interface
• Capillary effects are catch by the model
• Thermodynamic definition of surface tension holds
  at equilibrium

• Coefficients a, b, k, of the free energy
  functional
• Define the surface tension
• Define Capillary width
• Define equilibrium concentration

Cannot be independently defined

• Mobility parameter M of the Cahn-Hillard equation
• Controls the diffusivity in the interface
• Gives the interface relaxation time

• Surface tension definition holds at equilibrium
• Interface should always be at equilibrium
• Relaxation time lower than convective time
• Mobility and interface thickens are not
  independent

scaling law between Interface thickness and
mobility Magaletti (2013)
11
Flow field Coupling

The Cahn-Hilliard equation accounts also for the
convective effects
Convective effects

• Flow field solution
• Navier Stokes / continuity equations system
• Coupling term dependent on the phase field

• The Chan-Hilliard/Navier-Stokes equations system
  has first been derived by Hohenberg and Halperin
  (1977) (model H)
• Phase field surface force yields to the surface
  tension stress tensor
• Phase field dependent viscosity (viscosity ratio
  between fluids)
• Density matched fluids
• Density mismatches require the solution of
  compressible Navier Stokes

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Dimensionless Equations

Dimensionless Cahn-Hilliard equation and Chemical
potential
Dimensionless Navier-Stokes/Continuity
Non-Dimensional groups
Reynolds Number
Cahn number Dimensionless interface thickness
Peclet number Dimensionless interface
relaxation time
Weber number Inertia vs. Surface tension
Dimensionless mobility
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Advantages

• Overcoming of sharp interface models problems
• Absence of boundary conditions on the interface
• Interface creation and dissolution cached
• Interfacial layer do not degrade (conservative)

Level-Set (interface tracking for sharp interface
approaches) interface Degradation
Diffuse Interface Model Conservative interface
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Advantages

• Reliability of the model
• Thermodynamically consistent
• Conservative interfacial layer
• convergence to Sharp interface limit
• Consistent interface tracking and flow field
  coupling

• Flexibility, different phenomena can be analyzed
• Near critical phenomena
• Morphology evolution
• Droplet breakup /coalescence
• .

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Drawbacks

• Diffuse interface approximation
• non physical interface thickness for immiscible
  fluids (Real thickness O(10-6)m)
• Interfacial layer resolution require at least
  three mesh points
• High resolution simulations required

• Cahn Hilliard Numerical solution
• Involves high order operators (up to 4th order)
• thin interfacial layers involve high gradients
• robust numerical algorithms required

16
Droplet under shear flow
Typical two phase flows benchmark, analytical
solution is known

• Newtonian fluids
• matched densities
• matched viscosities
• constant mobility.

• Pseudo-spectral DNS Fourier modes (1D FFT) in
  the homogeneous directions (x and y)?,
• Chebychev coefficients in the
  wall-normal direction (z)?
• Time integration Adams-Bashforth (convective
  terms), Crank-Nicolson (viscous terms)

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Droplet under shear flow
Deformation analysis, comparison with taylor
(1921)
Droplet deforms as a prolate ellipsoid of major
axis L and minor axis b
Taylor law, valid for small Deformations D lt 0.3
Major axis orientation converge to 45
The actual Capillary number depends on droplet
initial radius and shear rate (Taylor 1921)
Deformation Parameter
18
Droplet under shear flow
Deformation analysis, comparison with taylor
(1921)
Re 0.2 Ch 0.05 Pe 20 Grid 128x128x129 ?t
10-5
R/H We Cae
0.5 0.0006 0.032
0.5 0.0012 0.064
0.5 0.0024 0.127
0.5 0.0050 0.255

• Matching with Taylor law
• Correct orientation of the deformed droplet
• Minor discrepancies due to finite Reynolds number
  and interface identification

19
Droplet deformation an breakup In turbulent
flows

• Newtonian fluids
• matched densities
• matched viscosities
• constant mobility.

Dimensionless groups
Governing Equations

• Time-dependent 3D turbulent flow at Ret100
• Wide range of surface tension We 0.1 ? 10
• Pseudo-spectral DNS Fourier modes (1D FFT) in
  the homogeneous directions (x and y)?,
• Chebychev coefficients in the
  wall-normal direction (z)?
• Time integration Adams-Bashforth (convective
  terms), Crank-Nicolson (viscous terms)

20
Droplet deformation an breakup In turbulent
flows
Interface described by three mesh-points
Simulation parameters
Re? 100
We 0.1 ? 10
Pe 1.3?105
Ch 0.035
Nx x Ny x Nz 256 x 128 x 129
d 80 w.u.
Lx x Ly x Lz 1257 x 628 x 200 w.u.
? 3.6 w.u.
Physical parameters Water flow
? 1000 kg/m3
? 1?10-6 m2/s
U? 0.04 m/s
H 25 mm
d 2 mm
? 875 ?m
?p 310 kPa
s 0.038 ? 0.00038 N/m
? 1000 kg/m3
? 1?10-6 m2/s
U? 0.04 m/s
H 25 mm
d 2 mm
? 875 ?m
?p 310 kPa
s 0.038 ? 0.00038 N/m
21
Droplet deformation an breakup In turbulent
flows
Qualitative analysis of deformation and breakup
process
Qian et al. (2006)
Risso and Fabre (1998)
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Droplet deformation an breakup In turbulent
flows
Deformation and breakup
Diameter based Weber number
Deformation parameter Normalized external
surface

• Linear behavior of deformation with Weber number
  (Risso 1998)
• Qualitative agreement with experiments of Risso
  and Fabre (1998)
• Qualitative agreement with numerical Lattice
  Boltzmann results of Qian et al. (2006)

23
Droplet deformation an breakup In turbulent
flows
Deformation behaviour, local curvatures
probability density functions
Undeformed droplet curvature
Increasing Surface tension

• Increasing surface tension reduce local
  deformability
• Increasing principal curvature reduce the
  secondary curvature, incompressible interface

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Droplet deformation an breakup In turbulent
flows
Oil in Water ? 0.038N/m Wed 0.085
? 0.002N/m Wed 1.7
?
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Droplet deformation an breakup In turbulent
flows
Oil in Water ? 0.038N/m Wed 0.085
? 0.002N/m Wed 1.7
? 0.004N/m Wed 0.85
?
26
Droplet deformation an breakup In turbulent
flows
Velocity field interface interactions, Analysis
framework

• Probability density functions of the velocity
  fluctuations

• Statistics across the interface

Analysis along the interface normal direction
n
ZG
27
Droplet deformation an breakup In turbulent
flows
Deformation and breakup

• Fluctuations reduced inside the droplet
• Similar behavior between different We
• Outside the droplet fluctuations pdf similar to
  single-phase channel flow Dinavahi et al.
  Phys. Fluids 7 (1995)

28
Droplet deformation an breakup In turbulent flows
Volume averaged turbulent kinetic energy
Turbulent Kinetic Energy modulation observed for
all surfece tensions.
Different responses from external turbulent
forcing
Turbulent kinetic energy conserved in the wole
channel
?
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Droplet deformation an breakup In turbulent flows
Volume Averaged Mean Total Kinetic Energy
?
30

• Droplet deformation an breakup
• In turbulent flows

Oil in Water ? 0.038N/m Wed 0.085
? 0.002N/m Wed 1.7
?