Modeling heat and fluid flow in porous media David J. Lopez Penha * , Lilya Ghazaryan * , Bernard J. Geurts * , Steffen Stolz †,* & Markus Nordlund † * Dept. of Applied Mathematics † Philip Morris International R&D University of Twente, Enschede Philip Morris Products S.A., Neuchˆ atel The Netherlands Switzerland ECCOMAS CFD 2010, Lisbon, Portugal June 14–17, 2010 D.J. Lopez Penha et al.
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Modeling heat and fluid flow in porous media
David J. Lopez Penha∗, Lilya Ghazaryan∗,
Bernard J. Geurts∗, Steffen Stolz†,∗ & Markus Nordlund†
∗Dept. of Applied Mathematics †Philip Morris International R&D
University of Twente, Enschede Philip Morris Products S.A., Neuchatel
The Netherlands Switzerland
ECCOMAS CFD 2010, Lisbon, Portugal
June 14–17, 2010
D.J. Lopez Penha et al.
Porous media
Amorphous nano-porous material
Source: http://gubbins.ncsu.edu/research.html
• Porous media may have complicated pore geometries
• Difficult to build practicable body-fitted grids
D.J. Lopez Penha et al.
Goals
1. Develop “gridding-free” method for computing heat & fluidflow in porous media
2. Method allows arbitrary pore geometries
D.J. Lopez Penha et al.
Goals
1. Develop “gridding-free” method for computing heat & fluidflow in porous media
2. Method allows arbitrary pore geometries
D.J. Lopez Penha et al.
Outline
1 Modeling fluid flow in porous media
2 Modeling heat flow in porous media
3 Validation tests
4 Application to realistic porous medium
5 Conclusions
D.J. Lopez Penha et al.
Outline
1 Modeling fluid flow in porous media
2 Modeling heat flow in porous media
3 Validation tests
4 Application to realistic porous medium
5 Conclusions
D.J. Lopez Penha et al.
Representing complex geometries
• Cartesian grid representation of fluid & solid domains
• Trade-off: large spatial resolutions — efficient numericalalgorithms
D.J. Lopez Penha et al.
Representing complex geometries
• Cartesian grid representation of fluid & solid domains
• Trade-off: large spatial resolutions — efficient numericalalgorithms
D.J. Lopez Penha et al.
Fluid dynamics
• Incompressible Navier-Stokes equations for fluid & soliddomains:
∇ · u = 0,∂u
∂t+ u · ∇u = −∇p +
1
Re∇2u+ f
• Methodology: immersed boundary method
• Force f: approximates no-slip condition (volume-penalization)
f = −1
ǫΓ(x) · u, ǫ ≪ 1
• Γ(x): phase-indicator function (Γ = 1 in solid; Γ = 0 in fluid)
D.J. Lopez Penha et al.
Fluid dynamics
• Incompressible Navier-Stokes equations for fluid & soliddomains:
∇ · u = 0,∂u
∂t+ u · ∇u = −∇p +
1
Re∇2u+ f
• Methodology: immersed boundary method
• Force f: approximates no-slip condition (volume-penalization)
f = −1
ǫΓ(x) · u, ǫ ≪ 1
• Γ(x): phase-indicator function (Γ = 1 in solid; Γ = 0 in fluid)
D.J. Lopez Penha et al.
Discretization
• Symmetry-preserving finite-volume method
• Staggered grid (uniform Cartesian)
• Implicit time-integration of force f
D.J. Lopez Penha et al.
Outline
1 Modeling fluid flow in porous media
2 Modeling heat flow in porous media
3 Validation tests
4 Application to realistic porous medium
5 Conclusions
D.J. Lopez Penha et al.
Conjugate heat transfer
• Single temperature equation for fluid & solid domains:
∂T
∂t+ u · ∇T =
1
RePr∇ · (α∇T )
• Thermal diffusivity α: discontinuous if αf 6= αs
α(x) = (1− Γ)αf + Γαs
• Solid domains: convective term vanishes =⇒ diffusion only
D.J. Lopez Penha et al.
Conjugate heat transfer
• Single temperature equation for fluid & solid domains:
∂T
∂t+ u · ∇T =
1
RePr∇ · (α∇T )
• Thermal diffusivity α: discontinuous if αf 6= αs
α(x) = (1− Γ)αf + Γαs
• Solid domains: convective term vanishes =⇒ diffusion only
D.J. Lopez Penha et al.
Discretization
• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces
• Auxiliary temperatures {T xi ,j ,T
yi ,j} on cell surfaces:
T xi ,j =
αi ,jTi ,j + αi+1,jTi+1,j
αi ,j + αi+1,j
Tyi ,j =
αi ,jTi ,j + αi ,j+1Ti ,j+1
αi ,j + αi ,j+1
D.J. Lopez Penha et al.
Discretization
• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces
• Auxiliary temperatures {T xi ,j ,T
yi ,j} on cell surfaces:
T xi ,j =
αi ,jTi ,j + αi+1,jTi+1,j
αi ,j + αi+1,j
Tyi ,j =
αi ,jTi ,j + αi ,j+1Ti ,j+1
αi ,j + αi ,j+1
D.J. Lopez Penha et al.
Discretization
• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces
• Auxiliary temperatures {T xi ,j ,T
yi ,j} on cell surfaces:
T xi ,j =
αi ,jTi ,j + αi+1,jTi+1,j
αi ,j + αi+1,j
Tyi ,j =
αi ,jTi ,j + αi ,j+1Ti ,j+1
αi ,j + αi ,j+1
D.J. Lopez Penha et al.
Discretization
• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces
• Auxiliary temperatures {T xi ,j ,T
yi ,j} on cell surfaces:
T xi ,j =
αi ,jTi ,j + αi+1,jTi+1,j
αi ,j + αi+1,j
Tyi ,j =
αi ,jTi ,j + αi ,j+1Ti ,j+1
αi ,j + αi ,j+1
αi ,j ,Ti ,j αi+1,j ,Ti+1,j
αi ,j+1,Ti ,j+1
T xi ,j
Tyi ,j
D.J. Lopez Penha et al.
Outline
1 Modeling fluid flow in porous media
2 Modeling heat flow in porous media
3 Validation tests
4 Application to realistic porous medium
5 Conclusions
D.J. Lopez Penha et al.
Plane-Poisuielle flow with isothermal walls
101
102
10−1
ny
‖uh−
u‖ ℓ
p
101
102
10−4
10−3
10−2
10−1
ny
‖Th−
T‖ ℓ
p• ℓp-norm of error in velocity & temperature (p = {2,∞})
• Velocity: first-order in ℓ∞
• Temperature: second-order in ℓ∞
D.J. Lopez Penha et al.
Porous medium: inline arrangement of squares
x
y
D/2
H
H
Q
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y• Porosity: φ = 0.75 =⇒ 25% solid grid cells
• Literature: Nakayama et al., J. Heat Transfer, 124, 746–753(2002)
D.J. Lopez Penha et al.
Porous medium: inline arrangement of squares
• Gradient average pressure ∂〈p〉f /∂x vs. grid resolution: