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Diffuse interface approach to CFD-based simulation of gas-liquid
systems
Diffuse Interface Approach to
CFD-Based Simulation of Gas-Liquid Systems
A. Donaldson, A. Macchi , D.M. Kirpalani$* CSChE 2008, Ottawa, ON
October 21, 2008
Miniaturization and
Multi-phase CFD
• Process intensification and miniaturizationhas led to alternative technologies for multi-phase contacting (i.e. monoliths) • The modular nature of this technology is
better suited to CFD-based optimization, where the process dynamics remain consistent during industrial scale-up. Simulation of interfacial phenomena in • Simulation of interfacial phenomena in
small-scale geometries is complicated by a change in the predominant forces acting on the fluids
on the fluids.
Multi-Phase CFD:
Sharp Interface
Sharp interface tracking methods describe the transition between Sharp interface tracking methods describe the transition between phases as a discontinuity, from which topology is reconstructed to evaluate property transitions and surface forces.Eulerian - VOF Lagrangian
Diffuse Interface
Approach
• The interface between phases is describedas a region of finite width, across which physical properties transition smoothly • Phase interactions are described by
resolving thermodynamic relationships within the artificially thickened interface. • Surface forces are evaluated as a function
of free energy variation, eliminating the need to reconstruct the interface geometry. • Topological changes are handled implicitly
DI Approach:
Current Limitations
5
0
Spontaneous Drop Shrinkage and Continuity Loss:
When the DI approach is used as an interface tracking technique, the contour
0
.
5
is frequently used to delineate between phases
contour is frequently used to delineate between phases. While is conserved, the area within the contour is not. Application To Non-Uniform Meshes and Surface Force Approximationspp pp
Center Cell Nearest Neighbour Second-Nearest Neighbour
Continuity Loss:
Sources and Effects
Initial Conditioning: Loss During Advection:
Surface
Energy
Bulk
Free
Energy
H
Spontaneous Drop Shrinkage:
Vgy
gy
f
Assume: Negligable Volume,
Near Zero
Infinite Volume, Dominant energy
Reality: Finite Volume,
Near Zero Concentration of Free
Energy on Interface
Conventional Approach:
Double-Well Function
Simplified Energy Density Function:
2
21
25
0
W ll D blf
Spontaneous Drop Shrinkage: Critical Radius 2-D
For a set interface width and computational domain volume a critical
0
.
25
1
Well Doublef
0 0.5 1 0 0.5 1For a set interface width and computational domain volume, a critical droplet radius exists, below which a droplet will eventually disappear.
3 1
6
8
6
V
r
C H H Radius r0 Radius r0Yue et al., J. Comp. Phys., 223 (2007)
Proposed Approach:
TVSED Function
TR=0
TR TVSEDf
1
1Temperature Variant SEDFunction: TR=0.5
0 0.5 1 0 0.5 1
TVSEDf
Critical Radius 2D
1 2 14
2
1
1
2
2
1
TR TR R RV
T
T
T
T
Critical Radius – 2D
1 2;
1
2
3
;
2
1
,
2
1
4
1
2
R R R cT
F
T
T
r
2
2
2
Continuity Loss
During Advection
Continuity loss during advection occurs due to the formation of meta-stable mixtures in the bulk phases, caused by poor separation dynamics.Thermodynamic Regions
Diagonal Advection
Simulations
Two-Phase Flow:
Momentum Transfer
Navier-Stokes for Laminar Flow:
UU
U
U
g
U
T
P
Continuity:
g
t
Surface forces accounted for in the pressure tensor
0
U
t
in the pressure tensor.Note: As per Ding et al. (2007), the volume averaged velocity is used to obtain a solenoidal velocity field, thereby simplifying the PISO algorithm used for determining the pressure field
used for determining the pressure field.
Ding, H., Spelt, P.D.M., Shu, C., Diffuse interface model for incompressible two-phase flows with large density ratios, J. of Comp. Phys., 226 (2007), 2078-2095.
Pressure Tensor:
Surface Force Est.
The pressure tensor for diffuse interface simulations is commonly based on the Korteweg tensor, with isotropic terms incorporated into the isotropic pressure, p p ,
p
p
.
2p
P
Surface Force and
Gamma Filtering
By applying a filter to , variations in the pressure tensor can be localized to the center of the interface where is more linear, minimizing the error introduced by low-order estimates of
g y
*
0
.
5
0
.
5
sin
min
max
1
05
2 1 5 . 0 1 , , max min
*2 * * *
*2 cos cos 2 sin
2 * *
* *2 * * p P , 1 1 0 1 2 2 1 2
* * * * cos2 2 2 sin2 ,Surface Force Results
Gamma Filtering VOF Conventional
Gamma Filtering VOF Conventional
Summary
• While the diffuse interface approach to interface tracking has advantages over the sharp interface approach for modeling small-scale flows, it’s application has been limited due to continuity losses and the need for highly structured grids
and the need for highly structured grids.
• The current practice of adopting the double-well function is partially responsible for the observed loss in continuity. By using an energy density for immiscible fluids overall phase continuity is improved density for immiscible fluids, overall phase continuity is improved. • Estimates for pressure can be improved through the use of a filtering
parameter, limiting pressure variations to the central region of the interface where the equilibrium profile is linear
interface where the equilibrium profile is linear.
• These improvements enable the use of the diffuse interface technique on unstructured grids, a requirement for most practical applications
here it ma be of se where it may be of use.
Scalar Transport
Equations:
Convective Cahn-Hilliard Equation:
0
J
U
Diffusive Flux, , maintains the interface profile through chemical potential gradients :
t
J
potential gradients, :
f
J
2 2 2 - Capillary Width - Mixing Energy Density
- Free energy density governing interfacial dynamics and the equilibrium profile of
f
Multi-Phase CFD:
Approach Limitations
The ability of each method to resolve complex multi-phase flow is often limited by shortcomings inherent to the numerical approach.Lagrangian Eulerian
Lagrangian
Front Tracking Sharp Interface (VOF) Diffuse Interface (PF)
Eulerian
• Book keeping • CSF model • Larger grid needed to • Complex Topology
• Coalescence & Breakup
• Coal. & Breakup • Interface smearing
and reconstruction
resolve interface • 4thorder derivative
High order schemes
S d
While a number of these limitations have been addressed the associated
• CSF model • Spontaneous drop shrinkage
While a number of these limitations have been addressed, the associated increase in complexity often results in a method which is too cumbersome
for general application to multi-phase flow problems.
Pressure Profiles:
Method Comparison
Phase Field Grad Gamma
Phase Field Grad Gamma
