Simulation of multiphase flows
Y
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
G
W1
W2
Numerical approaches
•Sharp interface approach
•Diffuse interface approach
X
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
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
Navier Stokes
fluid 1
Navier Stokes
fluid 2
Stress and velocity
boundary condition s at the
interface
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)
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
Level set interface degradation
Errors in
curvature
computation
•Jump conditions are not correctly computed
•Re-initialization introduce errors
•mass leakage still persist
Diffuse Interface Approach
r2
r
r
Interface is a finite thickness transition layer
•Localized and controlled fluid mixing (even for immiscible fluids)
•fluid properties change smoothly from between the fluids
r1
r2
Y
Y
r1
X
X
Phase Field Modelling
Bulk fluid 1
Bulk fluid 2
Interfacial layer
Interface position
f=f+
f= fm
f= f-
Y
Fluid properties proportional the
Order parameter
f
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
State of the system represented by a
scalar field
•Continuous over the domain
•Smooth variations across the
interfaces
•Order parameter function of the
position
X
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
Cahn Hilliard equation o generalized mass diffusion equation
•Evolution of an immiscible & partially-miscible multiphase fluid system
•Interface evolution controlled by a chemical potential
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
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,…)
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
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
Uniform solution 1
Uniform solution 2
Mono-dimensional
Non-uniform solution
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
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)
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
Phase field surface force
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
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
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)
Surface tension effects applied by a smeared surface
force. No interfacial boundary conditions
Errors in
curvature
computation
Level-Set (interface tracking
for sharp interface
approaches) interface
Degradation
Diffuse Interface Model
Conservative interface
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
•….
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
4th order operator ensures the
Conservation of interfacial layer
Droplet under shear flow
Typical two phase flows benchmark, analytical solution is known
1.
2.
3.
4.
Newtonian fluids;
matched densities;
matched viscosities;
constant mobility.
Dimensionless Governing Equations
boundary conditions
Dimensionless groups
•
•
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)
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 < 0.3
Major axis orientation
converge to 45°
X
The actual Capillary number
depends on droplet initial radius
and shear rate (Taylor 1921)
Deformation Parameter
Droplet under shear flow
Deformation analysis, comparison with taylor (1921)
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
Re = 0.2
Ch = 0.05
Pe = 20
Grid 128x128x129
t = 10-5
•Matching with Taylor law
•Correct orientation of the deformed droplet
•Minor discrepancies due to finite Reynolds number and interface identification
Droplet deformation an breakup
In turbulent flows
1.
2.
3.
4.
Newtonian fluids;
matched densities;
matched viscosities;
constant mobility.
Dimensionless groups
Governing Equations
•
•
•
•
Time-dependent 3D turbulent flow at Ret=100
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)
Droplet deformation an breakup
In turbulent flows
Interface described by
three mesh-points
Simulation parameters:
Ret
100
We
0.1  10
Pe
1.3105
r
1000 kg/m3
Ch
0.035

110-6 m2/s
Nx x Ny x Nz
256 x 128 x 129
Ut
0.04 m/s
d
80 w.u.
H
25 mm
Lx x Ly x Lz
1257 x 628 x 200 w.u.
d
2 mm


875 mm
3.6 w.u.
Δp
310 kPa
σ
0.038  0.00038 N/m
Physical parameters:
Water flow
Droplet deformation an breakup
In turbulent flows
Qualitative analysis of deformation and breakup process
Qian et al. (2006)
Risso and Fabre
(1998)
Droplet deformation an breakup
In turbulent flows
Deformation and breakup
Diameter based Weber number
Breakup
Deformation parameter –
Normalized external surface
No Breakup
•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)
Droplet deformation an breakup
In turbulent flows
Deformation behaviour, local curvatures probability density functions
Increasing
Surface tension
Undeformed
droplet curvature
•Increasing surface tension reduce local deformability
•Increasing principal curvature reduce the secondary curvature, incompressible
interface
Droplet deformation an breakup
In turbulent flows
Oil in Water s = 0.038N/m Wed = 0.085
s = 0.002N/m Wed = 1.7
s
Droplet deformation an breakup
In turbulent flows
Oil in Water s = 0.038N/m Wed = 0.085
s = 0.004N/m Wed = 0.85
s = 0.002N/m Wed = 1.7
s
Droplet deformation an breakup
In turbulent flows
Velocity field interface interactions, Analysis framework
•Probability density functions of the velocity fluctuations
Z
Pdf of Velocity fluctuations
inside the droplet
Y
X
Pdf of Velocity fluctuations
outside the droplet
•Statistics across the interface
Z
ZG
Analysis along the interface
normal direction
Y
X
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)]
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
s
Droplet deformation an breakup
In turbulent flows
Volume Averaged Mean Total Kinetic Energy
s
Droplet deformation an breakup
In turbulent flows
Oil in Water s = 0.038N/m Wed = 0.085
n
t2
Z Y
t1
s = 0.002N/m Wed = 1.7
X
s