Simulation of multiphase flowsSimulation of multiphase flows

Multiphase/multicomponent fluid systems•Fluid domain split in two o more fluid regions 1, 2

•Fluids separated by interface •Fluid with different fluid properties

•Interface provided by surface tension

Numerical approaches•Sharp interface approach•Diffuse interface approach

Numerical simulation of Multiphase/multicomponent flows

•Interface tracking •flow field solution •Flow field interface coupling

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

Sharp interface approachesSharp 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 approachesSharp 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

Errors in curvature computation

Level set interface degradation

•Jump conditions are not correctly computed•Re-initialization introduce errors•mass leakage still persist

Diffuse Interface ApproachDiffuse 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

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Phase Field ModellingPhase Field Modelling

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

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

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m

Bulk fluid 1Bulk fluid 2

Interfacial layer Interface position

The Cahn Hilliard EquationThe Cahn Hilliard EquationTime 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

Cahn Hilliard Equation

The free energy functionalThe free energy functional

Thermodynamic chemical potential, by definition

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,…)

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

BulkFree energy

Non-LocalFree energy

The free energy functionalThe 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 PropertiesInterface 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 PropertiesInterface 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 of the free energy functional•Define the surface tension •Define Capillary width •Define equilibrium concentration

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

Cannot be independently defined

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 CouplingFlow field Coupling

The Cahn-Hilliard equation accounts also for the 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

Convective effects

Phase field surface force

Dimensionless EquationsDimensionless 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

AdvantagesAdvantages

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

Errors in curvature computation

Surface tension effects applied by a smeared surface force. No interfacial boundary conditions

AdvantagesAdvantages

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

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

DrawbacksDrawbacks

Diffuse interface approximation •non physical interface thickness for immiscible fluids (Real thickness O()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 flowDroplet under shear flow

• 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)

1. Newtonian fluids;2. matched densities;3. matched viscosities;4. constant mobility.

Dimensionless groups

Dimensionless Governing Equations

Typical two phase flows benchmark, analytical solution is known

boundary conditions

Droplet under shear flowDroplet under shear flow

Deformation Parameter

Deformation analysis, comparison with taylor (1921)

Taylor law, valid for small Deformations D < 0.3

The actual Capillary number depends on droplet initial radius and shear rate (Taylor 1921)

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Droplet deforms as a prolate ellipsoid of major axis L and minor axis b

Major axis orientation converge to 45°

Droplet under shear flowDroplet 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 = 0.2Ch = 0.05 Pe = 20Grid 128x128x129t = 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 breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

• Time-dependent 3D turbulent flow at Re=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)

1. Newtonian fluids;2. matched densities;3. matched viscosities;4. constant mobility.

Dimensionless groups

Governing Equations

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Simulation parameters:

Physical parameters:Water flow

Interface described by three mesh-points

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Qualitative analysis of deformation and breakup process

Qian et al. (2006)

Risso and Fabre (1998)

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Breakup

No Breakup

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)

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Deformation behaviour, local curvatures probability density functions

IncreasingSurface tension

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

Undeformed droplet curvature

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Oil in Water = 0.038N/m Wed = 0.085

= 0.002N/m Wed = 1.7

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Oil in Water = 0.038N/m Wed = 0.085

= 0.002N/m Wed = 1.7 = 0.004N/m Wed = 0.85

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Velocity field interface interactions, Analysis framework

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Z Pdf of Velocity fluctuations inside the droplet

Pdf of Velocity fluctuations outside the droplet

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ZG

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•Probability density functions of the velocity fluctuations

•Statistics across the interface

Analysis along the interface normal direction

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn 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 breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Turbulent Kinetic Energy modulation observed for all surfece tensions.

Different responses from external turbulent forcing

Turbulent kinetic energy conserved in the wole channel

Volume averaged turbulent kinetic energy

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Volume Averaged Mean Total Kinetic Energy

Droplet deformation an breakupDroplet deformation an breakupIn turbulent flowsIn turbulent flows

Oil in Water = 0.038N/m Wed = 0.085

= 0.002N/m Wed = 1.7

t1t2

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