Conference on SPH and Particular Methods
for Fluids and Fluid Structure Interaction,
Lille, January 21-22, 2015
Volume penalization approach in LBM for
fluid structure interaction problems
M. Benamour*, E. Liberge*, C. Béghein*
* LaSIE, UMR-7356 CNRS, University of La Rochelle, France
1
Outline
• Lattice Boltzmann method
• Volume penalization method and LBM applied to the 1D Burgers equation
• Volume penalization method and LBM applied to the Navier Stokes equations
• Conclusions
2
Lattice Boltzmann method
Fluid flow computation
 Continuous approach : macroscopic scale
Conservation equations for mass, momentum, energy
 Molecular dynamics simulations : microscopic scale
Calculate atomic positions and velocities using Newton’s second law
 Lattice Boltzmann method : mesoscopic scale
Collection of particles : calculate the distribution function f(x,v,t) : probability of finding at time t : f(x,v,t)
dx dv particles in a small volume dx at position x, in a small velocity volume dv with velocity v
3
Lattice Boltzmann method
Boltzmann Equation
Discretization in phase – space & time
Lattice Boltzmann Equation
𝜕𝑓
𝜕𝑓 𝐹 𝜕𝑓
+𝜉
+
= Ω(𝑓)
𝜕𝑡
𝜕𝑥 𝑚 𝜕𝜉
𝜕𝑓𝛼
𝜕𝑓𝛼 𝐹𝛼 𝜕𝑓𝛼
+ 𝜉𝛼
+
= Ω(𝑓𝛼 )
𝜕𝑡
𝜕𝑥 𝑚 𝜕𝜉
LBE with Bhatnagar-Gross-Krook (BGK) collision operator :
𝑓𝛼 𝑥𝑖 + 𝑒𝛼 ∆𝑡, 𝑡 + ∆𝑡 = 𝑓𝛼 𝑥𝑖 , 𝑡 −
Collision step :
1
𝑒𝑞
𝑓𝛼 𝑥𝑖 , 𝑡 − 𝑓𝛼 𝑥𝑖 , 𝑡
𝜏
𝑓𝛼 𝑥𝑖 , 𝑡 + ∆𝑡 = 𝑓𝛼 𝑥𝑖 , 𝑡 −
Chapman-Enskog-Expansion
Navier Stokes Equations
+ ∆𝑡 𝐹𝛼
1
𝑒𝑞
𝑓𝛼 𝑥𝑖 , 𝑡 − 𝑓𝛼 𝑥𝑖 , 𝑡
𝜏
Macroscopic variable :
Flow density
Streaming step :
𝑓𝛼 𝑥𝑖 + 𝑒𝛼 ∆𝑡, 𝑡 + ∆𝑡 = 𝑓𝛼 𝑥𝑖 , 𝑡 + ∆𝑡
𝜌=
𝑒𝑞
𝑓𝛼 =
𝛼
Momentum
𝑓𝛼
𝛼
𝜌𝑢 =
𝑒𝑞
𝜉𝛼 𝑓𝛼 =
𝛼
𝜉𝛼 𝑓𝛼
𝛼
4
Moving obstacles and LBM
• Practical importance of flows around moving obstacles
• Search of a method easy to implement
Moving obstacles with LBM :
 On a moving mesh : Arbitrary Lagrangian Eulerian (Meldi et al. (2013))
 On a fixed cartesian  Bounce back
(Ladd (1994))
mesh :
 Lagrange multiplier based fictitious methods (Shi and Phan-Thien (2005))
 Immersed boundary method (Feng and Michaelides (2004), Wu and Shu (2010))
In our work:
Volume penalization method Angot et al. (1999)
5
Volume penalization method (1D Burgers equation)
One equation for fluid and solid domains :
𝜕 𝑢
𝜕 𝑢2
𝜕
𝜕𝑢
1
+
=
𝜈
− 𝜒 𝑢 − 𝑢𝑆
𝜕𝑡
𝜕𝑥 2
𝜕𝑥
𝜕𝑥
𝜂
𝝌 : mask function
0 in the fluid domain
1 in the solid domain
𝜼 : permeability coefficient
Very high in the fluid domain
Very small in the solid domain
𝒖𝑺 : velocity of the solid obstacle
𝜒
 In the fluid domain : → 0
𝜂
 In the solid domain :
𝜒
→∞
𝜂
𝜕 𝑢
𝜕 𝑢2
𝜕
𝜕𝑢
+
=
𝜈
𝜕𝑡
𝜕𝑥 2
𝜕𝑥 𝜕𝑥
𝑢 = 𝑢𝑆
6
Lattice Boltzmann method (Burgers equation)
Lattice Boltzmann model for a non linear convection diffusion equation (Shi and Guo (2009))
𝜕𝜙
+ 𝛻 ∙ 𝐵 𝜙 = 𝛻 ∙ 𝛼𝛻𝐷 𝜙 + 𝐹 𝑥, 𝑡
𝜕𝑡
Single relaxation time Bhatnagar-Gross-Krook (BGK) collision operator :
1
𝑒𝑞
𝑓𝛼 𝑥 + 𝑒𝛼 ∆𝑡, 𝑡 + ∆𝑡 = 𝑓𝛼 𝑥, 𝑡 − 𝑓𝛼 𝑥, 𝑡 − 𝑓𝛼 𝑥, 𝑡
𝜏
𝑒𝑞
𝑓𝛼
𝑥, 𝑡 = 𝑤𝛼
𝑒𝛼 ∙ 𝐵
𝐶 − 𝑐𝑆2 𝜙𝐼 : 𝑒𝛼 𝑒𝛼 − 𝑐𝑠2 𝐼
𝜙+ 2 +
𝑐𝑆
2𝑐𝑆4
Macroscopic variable : 𝜙 =
𝑓𝛼 =
𝛼
𝛼=
𝑒𝑞
𝑓𝛼
𝑐𝑆2
𝛼
1
𝜏−
Δ𝑡
2
with
+ ∆𝑡 𝐹𝛼
𝐶 𝜙 = 𝐶0 𝜙 + 𝑐𝑆2 𝐷 𝜙 𝐼
𝐶0 𝜙
𝛼𝛽
=
𝐵𝛼′ 𝜙 𝐵𝛽′ 𝜙 𝑑𝜙
𝐵′ 𝜙 =
𝑒𝛼 ∙ 𝐵′ 𝜙
Forcing term : 𝐹𝛼 = 𝑤𝛼 𝐹 1 + 𝜆
𝑐𝑆2
with
𝜏−1 2
𝜆=
𝜏
7
𝑑𝐵
𝑑𝜙
Lattice Boltzmann method (Burgers equation)
1D Burgers equation :
𝜕 𝑢
𝜕 𝑢2
𝜕
𝜕𝑢
1
+
=
𝜈
− 𝜒 𝑢 − 𝑢𝑆
𝜕𝑡
𝜕𝑥 2
𝜕𝑥
𝜕𝑥
𝜂
𝑢2
𝜙 = 𝑢 ,𝐵 𝜙 =
,𝛼 = 𝜈 ,𝐷 𝜙 = 𝑢
2
D1Q3 model :
𝑒0 , 𝑒1 , 𝑒2 = 0, 𝑐, −𝑐
and
and
1
𝐹 𝑥, 𝑡 = − 𝜒 𝑢 − 𝑢𝑆
𝜂
𝑤0 =
2
1
, 𝑤1 = 𝑤2 =
3
6
3 cases :
 The solid domain does not move
 The motion of the solid domain is imposed
 The fluid force obtained by solving Burgers equation is coupled with the motion of the
solid domain
8
Burgers equation (case 1)
The solid domain does not move :
fluid
solid
𝑥=0
𝑥 = 1.2
𝑥𝑆 = 1
Fluid domain : 0 ≤ 𝑥 ≤ 1
∀𝑡 ∶ 𝑥𝑆 = 1 𝑎𝑛𝑑 𝑢𝑆 = 0
𝑎𝑡 𝑡 = 0𝑠: 𝑢 𝑥, 0 = 𝑠𝑖𝑛 𝜋𝑥
𝑢 0, 𝑡 = 𝑢 1, 𝑡 = 0
Exact solution in the fluid domain :
∞
−𝑛2 𝜋2 𝜈𝑡
𝑏
𝑒
𝑛 sin 𝑛𝜋𝑥
𝑛=1 𝑛
2𝜋𝜈
−𝑛2 𝜋2 𝜈𝑡 cos 𝑛𝜋𝑥
𝑏0 + ∞
𝑛=1 𝑏𝑛 𝑒
𝑢 𝑥, 𝑡 =
1
𝑏0 =
𝑒 − 2𝜋𝜈
−1
1−cos 𝜋𝑥
𝑑𝑥
0
1
𝑏𝑛 = 2
𝑒−
0
2𝜋𝜈 −1 1−cos 𝜋𝑥
cos 𝑛𝜋𝑥 𝑑𝑥
9
Burgers equation (case 1)
The solid domain does not move : solution u at different times
120 lattice cells,
12000 lattice time steps
+ : penalization method and LBM
- : exact solution
10
Burgers equation (case 2)
The motion of the solid domain is imposed
fluid
𝑥=0
𝑎𝑡 𝑡 = 0𝑠:
solid
𝑥𝑆
𝑥 = 1.06
𝑥𝑆 𝑡 = 𝐴 𝑡 + 𝐵 𝑎𝑛𝑑 𝑢𝑆 = 𝐴
0 ≤ 𝑥 ≤ 1 ∶ 𝑢 𝑥, 0 = sin 𝜋𝑥
1 ≤ 𝑥 ≤ 1.06 ∶ 𝑢 𝑥, 0 = 0
𝑢 0, 𝑡 = 0
Solution u at different times :
+ : penalization method and LBM
- : numerical solution (finite difference method
and penalization method)
11
Burgers equation (case 3)
The fluid force obtained by solving Burgers equation is coupled with the motion of the solid domain
fluid
𝑥=0
𝑎𝑡 𝑡 = 0𝑠:
solid
𝑥𝑆
𝑥 = 1.2
0 ≤ 𝑥 ≤ 1 ∶ 𝑢 𝑥, 0 = sin 𝜋𝑥
1 ≤ 𝑥 ≤ 1.2 ∶ 𝑢 𝑥, 0 = 0
𝑢 0, 𝑡 = 0
𝑚 𝑥𝑆′′ 𝑡 + 𝑘 (𝑥𝑆 𝑡 − 𝑋0 ) = 𝐹 𝑥𝑆
m: solid mass, k: spring constant, X0 spring natural length
𝐹 𝑥𝑆 is the fluid force calculated with the solution of Burgers equation :
𝐹 𝑥𝑆 = −𝜈
𝜕𝑢
𝜕𝑥
𝑥𝑆
12
Burgers equation (case 3)
Position of the interface according to time
Solution u at t=1s
+ : penalization method and LBM
- : numerical solution (finite difference method and penalization method)
13
Volume penalization method (Navier Stokes equations)
One equation for fluid and solid domains :
𝜕 𝜌𝑢
𝜌
𝑇
+ 𝛻 ∙ 𝜌𝑢𝑢 = −𝛻𝑝 + 𝜈𝛻 ∙ 𝜌 𝛻𝑢 + 𝛻𝑢
− 𝜒 𝑢 − 𝑢𝑆
𝜕𝑡
𝜂
𝝌 : mask function
0 in the fluid domain
1 in the solid domain
𝜼 : permeability coefficient
Very high in the fluid domain
Very small in the solid domain
𝒖𝑺 : velocity of the solid obstacle
 In the fluid domain :
𝜒
→0
𝜂
 In the solid domain :
𝜒
→∞
𝜂
𝜕 𝜌𝑢
+ 𝛻 ∙ 𝜌𝑢𝑢 = −𝛻𝑝 + 𝜈𝛻 ∙ 𝜌 𝛻𝑢 + 𝛻𝑢
𝜕𝑡
𝑇
𝑢 = 𝑢𝑆
14
Lattice Boltzmann method (Navier Stokes equations)
Lattice Boltzmann method, with the single relaxation time Bhatnagar-Gross-Krook (BGK) collision operator :
1
𝑒𝑞
𝑓𝛼 𝑥 + 𝑒𝛼 ∆𝑡, 𝑡 + ∆𝑡 = 𝑓𝛼 𝑥, 𝑡 − 𝑓𝛼 𝑥, 𝑡 − 𝑓𝛼 𝑥, 𝑡
𝜏
Equilibrium distribution function :
Penalization term
force :
Guo, Zheng, Shi (2002) :
𝑒𝑞
𝑓𝛼
𝑒𝛼 ∙ 𝑢 𝑢𝑢: 𝑒𝛼 𝑒𝛼 − 𝑐𝑠2 𝐼
𝑥, 𝑡 = 𝑤𝛼 𝜌 1 + 2 +
𝑐𝑠
2𝑐𝑠4
𝜌
𝐹 = − 𝜒 𝑢 − 𝑢𝑆
𝜂
1
𝑒𝛼 − 𝑢
𝑒𝛼 ∙ 𝑢
𝐹𝛼 = 1 −
𝑤
+
𝑒𝛼 ∙ 𝐹
2𝜏 𝛼
𝑐𝑠2
𝑐𝑠4
𝜌𝑢 =
𝑒𝛼 𝑓𝛼 +
𝛼
2D simulations
+ ∆𝑡 𝐹𝛼
∆𝑡
𝐹
2
D2Q9 model
15
Flow past a motionless square obstacle
Blockage ratio : 𝐵 =
𝐻
=8
𝐷
𝐿
= 50
𝐷
𝑢𝑚𝑎𝑥 𝐷
𝑅𝑒 =
= 20 , 40 , 100
𝜈
Boundary conditions :
Inlet : parabolic velocity profile
Outlet : zero velocity gradient
Walls : no-slip velocity
Zou and He (1997) boundary condition
(bounce back of the non equilibrium populations)
16
Flow past a motionless square obstacle
Boundary conditions on the square obstacle :
Zou and He (1997)
Penalization method
Comparison with Breuer, Bernsdorf, Zeiser, Durst (2000) :
Lattice Boltzmann simulations,
simple bounce back boundary condition (walls and square obstacle)
 Drag coefficient : 𝐶𝑑 =
 Lift coefficient : 𝐶𝑙 =
𝐹𝑑
1 2
2 𝜌𝑢𝑚𝑎𝑥 𝐷
𝐹𝑙
(Force calculated with the momentum exchange method)
1 2
2 𝜌𝑢𝑚𝑎𝑥 𝐷
 Strouhal number : 𝑆𝑡 =
𝑓𝐷
𝑢𝑚𝑎𝑥
17
Flow past a motionless square obstacle
Results obtained with the finest grid (3000 x 480 cells)
Re=40
Re=100
Streamlines (penalization method)
Re
Penalization
Bounce back (Zou
He)
Breuer et al.
(2000)
20
Lr/D
Cd
1.072
2.503
1.044
2.479
1.044
2.328
40
Lr/D
Cd
2.221
1.801
2.156
1.780
2.156
1.717
100
Cd ave
Cl ave
St
1.396
0.401
0.138
1.362
0.388
0.139
1.378
0.363
0.139
Recirculation length, drag and lift
coefficients, Strouhal number
18
Flow past a motionless cylinder
L1=15D , L2=30D , H=60D
𝑅𝑒 =
𝑢𝐷
= 20 , 40 , 100
𝜈
Boundary conditions :
Inlet : constant velocity profile
Outlet : zero velocity gradient
Horizontal planes : symmetry
boundary conditions
Zou and He (1997) boundary condition
Symmetry boundary conditions
19
Flow past a motionless cylinder
Results obtained with the finest grid (1845 x 2460 cells)
Re=40
Re=100
Streamlines (penalization method)
20
Flow past a motionless cylinder
Comparison with the litterature
Re
Penalization
Wu and Shu (2009)
Tritton (1959)
20
Lr/D
Cd
1.02
2.10
0.93
2.09
2.22
40
Lr/D
Cd
2.36
1.59
2.31
1.57
1.48
Penalization
Wu and Shu (2009)
Williamson (1996)
1.44
0.38
0.165
1.36
0.34
0.16
0.17
Re
100
Cd ave
Cl ave
St
Recirculation length, drag and lift coefficients, Strouhal number
(Wu and Shu : immersed boundary method , and Lattice Boltzmann method
Tritton : experiments
Williamson : experiments)
21
Conclusions
 Summary:
- Volume penalization method combined with LBM method, applied to the computation
of fluid flow around an obstacle
- 1D Burgers equation with motionless obstacles and moving obstacles : satisfactory results
- Navier Stokes equations with motionless obstacles : good agreement with results found in
the literature
 Future work:
- Implementation of grid refinement in the LBM Navier Stokes code
- Apply this code to the computation of flows around moving obstacles
22
THANKS FOR YOUR ATTENTION
23