fluid flow in sea iceseaice.alaska.edu/gi/publications/pringle/06p_fluidflowseaice.pdf · fluid...
Post on 12-Aug-2018
215 Views
Preview:
TRANSCRIPT
Fluid flow in Sea Ice
Daniel PringleARSC / Geophysical Institute
Outline of talk
Sea ice background and motivation
Sea ice porosity and permeability
Introduction to Lattice Boltzmann Methods
Sea ice work so far..
Sea Ice
Thermal and mechanical barrier between Ocean and Atmosphere: complex ocean-ice-atmosphere interactions.
Indicator and agent of local and global climate change.
Studied via numerical modelling:
needs to be faithful to sea ice physics and processes
Arctic Antarctic Habitat/9 - 16 million km2 4 - 19 million km2 Ecosystem
Figure: Thomas & Dieckmann, ‘Sea, Blackwell Science, 2003
Sea Ice length scales
Antarctic cover Pack ice Pancake Biological Pores microbiologylayering (diatom chain)
‘microstructure’
Albedo and Permeability
Arctic Ocean, SHEBA study
α= SW ↑ / SW ↓αOcean ~ 0.07
αSnow ~ 0.85
αPonds < 0.4
αbare ice~ 0.65
Cold ice: impermeable
-ponding
Warm ice: permeable
-drainage
Other effects..
Heat flow
porosity changes allow discharge of head of cold dense brine
contribution to total heat flow ~ several percent.
Nutrient Delivery to resident biology
Sediment and pollutant transport
Brine expulsion during growth
April 26
Sea ice = ‘fresh ice’ + inclusions of brine, air, (salts)
Salinity and temperature as ‘state variables’
(S,T,ρ) → vair, vbrine,vice
Lake Ice
vs.
Sea Ice
Inclusions and Porosity
Constitutive super-cooling at growth front: ‘lamellar interface’
Brine incorporated in inter-lamellar spaces
1 cm
Porosity Evolution
1 mm
- 15 C
- 5 C
- 2 C
Light et al, 2003
Porosity Evolution10 cm
Jeremy Miner1 mm
- 15 C
- 5 C
- 2 C
Light et al, 2003
Dual porosity: 0.1 mm / 1 cm scales; geological analogs.
Permeability
Pk∇−=
μv
Darcy’s Law
-empirical -low flow
v - discharge velocity μ - dynamic viscosity
∇P -Pressure gradient k - permeability
1 Darcy ≈ 1 x 10-12 m2
Gravel ~ 105 – 102 D; Oil reservoir rocks 10 – 10-1 D; Granite 10-6 - 10-7 D
Permeability of sea ice: percolation?
Few experiments - field or lab
Developing percolation theory (Golden)
-critical brine volume fraction, vb = 5 %
Arctic field data (Eicken)
Now: numerical modeling of flow in imaged sea ice samples
Obtain 3D internal structure from X-ray Computed Tomography.
Characterize pore space
Apply Lattice Boltzmann method to model flow: calculate k
Numerical Modeling Overview
e.g. Fontainebleau sandstone, Martys and Hagedorn, 2002
LBM
Sea ice ‘dynamic’: porosity and permeability depend on S,T
XCT
Sea Ice Summary, Specific Objectives
Sea ice porosity from brine, air inclusions.
Dual porosity: primary (0.1 mm) & secondary (1 cm)
Porosity and permeability depend on S,T and history
1.Does sea ice really undergo a ‘percolation’ transition?If so, at what porosity and with what critical exponent?
2. What role do primary and secondary porosity play?
3. Sea ice as an analog to inaccessible geophysical materials near melt transitions: volcanic conduit, lower mantle materials.
Imaged Sea Ice Structures
Lab-grown sea ice: reconstructions of X-ray CT of 1 cm cores
Heaton, Miner, Eicken, Zhu,Golden, in prep (2006)
Brineinclusions
LBM: good at handling complex porous geometries
Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
Kinetic theory
Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
0u
)u(1u)u(u 2
=⋅∇
+∇+−∇=∇⋅+∂∂ Fp
tμ
ρTraditional Comp. Fluid Dynamics
Discretize macroscopic equations(solve for u, ρ, T)
Kinetic theory
Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
0u
)u(1u)u(u 2
=⋅∇
+∇+−∇=∇⋅+∂∂ Fp
tμ
ρTraditional Comp. Fluid Dynamics
Discretize macroscopic equations(solve for u, ρ, T)
Kinetic theory
Lattice Boltzmann MethodsDiscretize kinetics and recover behavior of macroscopic equations
( f(x,t), particle distribution function )
13 9
2
4
56
7 8
Lattice Boltzmann Modeling
),v,x( tf rrLudwig Boltzmann
Maxwell-Boltzmann distribution
Kinetic theory
statistical approach for micro → macro
Particle distribution function
Probability of finding a particle
with position x, velocity v, at time t
Equilibrium Distribution ),v,x( tf eq rr
Lattice Boltzmann Modeling
),x( tfi1
2
3
4
56
7 8
9
D2Q9 lattice i =1:9
particle distribution function in
each lattice direction
Lattice Boltzmann Modeling
),x( tfi
ρ
ρ
∑
∑=
=
ii
ii
f
f
ivu
1
2
3
4
56
7 8
9
D2Q9 lattice i =1:9
macroscopic output
particle distribution function in
each lattice direction
Lattice Boltzmann Modeling
),x( tfi
ρ
ρ
∑
∑=
=
ii
ii
f
f
ivu
1
2
3
4
56
7 8
9
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx(
D2Q9 lattice i =1:9
macroscopic output
stream + collide
particle distribution function in
each lattice direction
LBM ↔ Fluid dynamics
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx(
LBM ↔ Fluid dynamics
Single relaxation time (BGK)
Equilibrium distributions fieq from specific lattice
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
LBM ↔ Fluid dynamics
Single relaxation time (BGK)
( ) ( )t
xΔ
Δ−=
2
21
31υ τ
Equilibrium distributions fieq from specific lattice
These kinetics give incompressible Navier-Stokes dynamics with:
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
τ ↔ kinematic viscosity, ν
Lattice Boltzmann Modeling
Matlab Central file exchange Iain Haslam (U.Durham -1 page 2D)Google: matlab lattice Boltzmann Google: Iain Haslam
LBM RecipeRead obstacle fileIntialize density distributionLoop for time steps / iterations{Stream fluid ‘particles’Check for obstacles (bounce back)Calc. ρ and vα components, and fα
eq
Collide: ie. calculate relaxationcheck convergence
}Output velocity and density distributions.
e.g. single relaxation time
e.g. simple bounce back
Starting simple!
Set parameters
Generate porous medium
Stream - calculate fi(x,t)
Calculate ρ and u(x,t)
Calculate fieq(x,t)
Collide/relax towards eqm
Convergence test
Haslam 1-page 2D matlab code
Take care with obstacle handling and imaging!
Validate against pipe flow and sphere pack geometries.
~ 110 x 130 pixels here; need features ~ 6+ pixels
Balance resolution vs. sample size: 2-scale problem in sea ice!
LBM Summary‘Lattice-land’ kinetics lead (quite amazingly!) to Navier-Stokes behavior
(explicit, O(Δt), O(Δx2), finite difference approx to incompressible N/S)
Ingredients:1. lattice2. Collision operator (feq) 3. Obstacle handling conditions
Pros: Complex geometries, highly parallelizable, multi-phase flow, suspension/tracer transport
Care: needed with implementation: lattice choice, boundary handling, collision operator,validation
1. Obtain 3D internal structure from X-ray Computed Tomographyprimary and secondary pores.
- Jeremy Miner, Hajo Eicken
2. Characterize pore space and connectivity, network modeling- JM, DP, R. Glantz (John Hopkins)
Sea Ice Modeling
3. Lattice Boltzmann Modeling to come: - permeability
- nutrient transport
X-CT Network Modeling
-connectivity
-critical path analysis
SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
Lattice Boltzmann Methods> lattice-land kinetics which recover Navier-Stokes behaviour
(~ explicit, O(Δt), O(Δx2), finite difference approx to incomp. N/S)> computationally intensive, but amenable to parallelization
(some caution with boundary handling etc.)
SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
Lattice Boltzmann Methods> lattice-land kinetics which recover Navier-Stokes behaviour
(~ explicit, O(Δt), O(Δx2), finite difference approx to incomp. N/S)> computationally intensive, but amenable to parallelization
(some caution with boundary handling etc.)
Numerical approach> model flow in imaged real structures - permeability
- nutrient transport> apply method to volcanic samples, insights to other materials
Acknowledgements
Hajo Eicken, Jeremy Miner
Roland Glantz, John Hopkins University
Greg Newby
ARSC
Barrow Sea Ice Observatorywww.gi.alaska.edu/BRWICE
Lattice Boltzmann Modeling
⎥⎦
⎤⎢⎣
⎡−⋅+⋅+=23u)v(
29u)v(31
22 uwf ii
eqi αρ
Single relaxation time (BGK)
( ) ( )t
xΔ
Δ−=
2
21
31υ τ
Equilibrium distribution (D2Q9 lattice)
1
2
3
4
56
7 8
9
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
τ ↔ kinematic viscosity
1/9 a = 1,2,3,4
wα, = 1/36 a = 5,6,7,8
4/9 a = 9
Recover Navier-Stokes with:
s =
g (s
alt)
/ g (w
ater
)
Temperature [°C]
Effective Medium Approach
)(1
1
bia
i
sii
w
sib
vvvT
v
Tv
+−=
⎟⎠⎞
⎜⎝⎛ −−=
=
ασσ
ρρ
ασ
ρρ
State variables (to be specified)
Salinity
Temperature T
Density ρ (air content)
seaice
salt
mm
=σ
Effective medium property
Component
properties+ +
Geometric model
Weeks & Ackley, 1986
top related