Irina Ginzburg and Dominique dHumires Cemagref, Antony, France, - - PowerPoint PPT Presentation
Irina Ginzburg and Dominique dHumires Cemagref, Antony, France, Water and Environmental Engineering cole Normale Suprieure, Paris, France Laboratory of Statistical Physics Some elements of Lattice Boltzmann method for
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and its application to filling process in casting, J.Comp.Phys., 185, 2003
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1
☞1
✄1
1
1 k
☎q
✌1 k
☎0 λk
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1
2
3
4
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2
1
5
6
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1 3
1 3
1 3
1 3
1 12
1 12
1 12
1 12
1 3Λ
αα
✌Slide 13
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h
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q
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13
✑ ✁13
✑ ✁12
13
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1 2
☞2
1
✝1
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100.0 200.0
Re
0.40 0.60 0.80 1.00
k(Re)/k(0) Edwards et. al Bounce−Back Linear Multi−reflection Ghaddar
0.0 100.0 200.0
Re
0.40 0.60 0.80 1.00
k(Re)/k(0) Edwards et. al Bounce−Back Linear Multi−reflection Ghaddar
Error (in percents) of different methods in Stokes regime in 662 box
Edwards, FE Bounce-back Linear Multi-reflection Ghaddar, FE
✁2
✂2
✂2
✁2
3
✁2
✂2
✁3
2
✁1
2
l
✌4πµU k
✆ ✝φ
✞V 4πlk
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0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8
Normalized Knudsen layer y, [l.u] Symmetric part, L=8
0.2 0.4 0.6 0.8 2 4 6 8 10 12 14 16 18
Normalized Knudsen layer y, [l.u] Symmetric part, L=17
1 4: accommodation
3 256, Λ
3 16)
1 4: it decreases
3 2)
1 4: accommodation in boundary node
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1
✡1
✡αβ
✌ ✟αβ
✡physlα,
αβ
✌ ✟αβ
✡physlαlβ
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1, n
✌6
✏ ✌Slide 28
z, [m] x, [m]
Vector field water-table geometry
z, [m] x, [m]
Vector field water-table geometry
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0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 u_z/Ks x, [m]
hs=0, FE hs=-9cm, LB, Node hs=-9cm, LB, Wall
0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 u_z/Ks x, [m]
YLC, hs=-2cm, FE YLC, hs=-2cm, LB SAND, hs=0, FE SAND, hs=0, LB
0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14 16 18 Relative Exfiltration time, [h]
FE, hs=0 LB: hs=0, it=80 LB, hs=0, it=20
0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 Relative Exfiltration time, [h]
FE, hs=-2cm LB, hs=-2cm
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1 4
✍xycqxcqy
✏ ✎1 4
✍αβcqαcqβ
✏ ✎sα
sγ 2
q tq
αβ
✝ ✁αα
✎ ☎αα
αβ
✝ ✁ ✝ ☎αγ
✝ ✁ ☎αα
✌ ☎Slide 31
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8
✑7m, ∆t
✌0.5 1 1.5 2
z, [m]
0.5 1 1.5 2
z, [m]
0.5 1 1.5 2
z, [m]
✑ ✁Slide 35
R∂qPR
✌B∂qPB, Tq
✌q tq
B
q
✌ ✟q
✡B
q
✡R
✌ ☎B zz
R zz
0,5 1 1,5 2 Pressure head h, [m] 0,5 1 1,5 2 2,5 3 Elevation Z, [m]
0,005 0,01 0,015 0,02 0,025 0,03 Interface layer, [m] 0,5 1 1,5 2 2,5 3 Elevation Z, [m]
q
✌q
✌Slide 36
i
✞xx
i
✞xy
i
✞xy
i
✞yy
i
✞zz
✄ ☎ ✆ ✝zz∂zhR
✍ ✂ ✏ ✌zz∂zhB
✍i
✞ ✍ ✒i
✞i
✞ ✌i
✞xy
i
✞xx σ
gB
2
gB
2
1 2sign
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0.25 0.5 0.75 1
1 2 3
Vertical coordinate Z Horizontal coordinate X
0.25 0.5 0.75 1
1 2 3
Vertical coordinate Z Horizontal coordinate X
0.25 0.5 0.75 1
1 2 3
Vertical coordinate Z Horizontal coordinate X
0.25 0.5 0.75 1
1 2 3
Vertical coordinate Z Horizontal coordinate X
xx∂xxφ
zz∂zzφ
xx∂xxφ
zz∂zzφ
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zz
zz
✌ττ
✌ττ, g
✝i
✞ ✌i
✞xy
i
✞xx
✌1 8sign
✍1 2sign
✍q Λ
✂q R
✁ ✌q Λ
✂q B, ΦR qx
✍ ✂ ✏ ✁ ✌qx
✍0,5 1 1,5 2 2,5 Horizontal coordinate X, m
0,1
Exact LM-I TRT-A with eq.+ correction
0,5 1 1,5 2 2,5 Horizontal coordinate X, m 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Exact LM-I TRT-A with eq.+ correction
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q
✍ ☞q
¯ q
✍ ☞¯ q
q
✍ ☞q
¯ q
✍ ☞¯ q
i
✞q
✌ ✟q
✡ ✝i
✞R zz∂zPR
✍ ☞B zz∂zPB
✍ ☞qα
✌qα, i.e. tR q Λ
✂q R∂αPRcqα
✌q Λ
✂q B∂αPBcqα,
✆q
✌q
✍E
✑q
✌R
✁1 2mR q
✁R
✍ ☞R q
✌x
✂y
✂z
✄R q α , δ
✂R q α
✌qα
✍ErR ΛrR α
✑E
✌q
q , rR Λ
✌q B
q R, rR α
✌ ✟Slide 40
i
✞xy
✌ ✟i
✞xx sign
✍q
✍ ☞0,5 1 1,5 2 2,5 Horizontal coordinate X
0,25 0,5
Exact 2D multi-layer 2D LM-I 3D LM-I
zz
zz
✌ττ
✌ττ
d1
d1
✌d2
d2
✌0,5 1 1,5 2 2,5 Horizontal coordinate X
0,25 0,5
Exact 2D multi-layer 2D LM-I 3D LM-I
zz
zz
✌ττ
ττ
✌d1
d1
✌d2
d2
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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 Distance z, [m] Distance x, [m] Uniform grid, 100x100 box
water table
z
✌ ✁z
✌z
z
✌0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 Distance z, [m] Distance x, [m] Anisotropic grid I, 50x130 box, A=8
A=1 MRT, A=8 L_II, A=8 TRT, A=3.666
q
✍ ☞Slide 42
q top
q bottom)
top
bottom)
z
✍z bottom
✍ ☞z top
✍ ☞0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.4 0.6 0.8 1 Distance z, [m]
x
✍x bottom
x top
✡ ✍ ☞0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0005 0.001 0.0015 0.002 Slide 43
q
✍ ☞q
✍ ☞R q
✑q RmR q
✌B q
✑q BmB q
I q
✌1 2
✍R q
✁B q
✏I q
✌2Λ
BΛ
R
Λ
B
R if Λ
✂q R
q B
✌q
q
I
✌2Λ
Λ
R
B
✌I mB q
I mR q
q
✌1 2
✍q
✁q
✏q
✌1 2
✍q
✁q
✏1 2
✍R
✞ ✁B
✞ ✏ ✁1 4
✍B
✞ ✑R
✞ ✏0,2 0,4 0,6 0,8 1
Channel width, m
2 4 6 8 10 12
q R
q B
✌R
B
✌zz
zz
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fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, J. Fluid. Mech. 5, 317, 1959. –
application to numerical analysis,J. Inst. Maths Applics 8, 397, 1971. –
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P.Bar-Yoseph, and M. Shapira, The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders, Phys. Fluids A 2 (1), 45, 1990. –
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computational approach, Phys. Fluids 7 (11), 2563, 1995. –
dissipation, isotropy, Galilean invariance, and stability.,
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Boltzmann models, Phys. Rev. E, 68,066614, 2003. –
layered anisotropic aquifers, Adv Water Resour, 27:1075, 2004. – Beaugendre H., A. Ern, T. Esclaffer, E. Gaume., Ginzburg I., Kao C., A seepage face model for the interaction of shallow water-tables with ground surface: Application of the obstacle-type method. Journal of Hydrology, 329, 1/2:258, 2006. –
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