a. vikhansky, lattice-boltzmann method for non-newtonian and non-equilibrium flows lattice-boltzmann...
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A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Alexander VikhanskyDepartment of Engineering,
Queen Mary, University of London
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Lattice-Boltzmann method
ˆ ˆ ˆˆ, ,i i i if t t x tc f t x t f
1c
2c
3c
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
, ; :f t x c
fc f f
t
Boltzmann equation
2, , 3 . f c dc u cf c dc RT c u f c dc
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
NS equations
0,
,
.p
ut
uu u p
t
TC u T
t
σ
j
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Plan of the presentation
uu u p
t
σ
fc f f
t
ˆ ˆ ˆˆ, ,i i i if t t x tc f t x t f
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Plan of the presentation
uu u p
t
σ
fc f f
t
ˆ ˆ ˆ, ,i i i if t t x tc f t x t f
ii i i
fc f f
t
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Knudsen number: KnL
Boltzmann equation
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
2, , 3 . eq eq eqf c dc u cf c dc RT c u f c dc
Kneq neqf c f c f c
2, . neq neqc u c u f c dc c u c u f c dc σ j
Chapman-Enskog expansion
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Kinetic effects:
1. Knudsen slip (Kn),2. Thermal slip (Kn).
Knudsen layer (Kn2)
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Kinetic effects:
wT
3. Thermal creep (Kn).
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Kinetic effects:
4. Thermal stress flow (Kn2).
1T
2T
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Discrete ordinates equation
ˆ ˆ ˆˆ, ,i i i if t t x tc f t x t f
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Collision operator
1
relKBGK model:
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Boundary conditions
Ox
Wx
tc
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Boundary conditions: bounce-back rule
u
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Method of moments
1. Euler set: , ,u T
2. Grad set: , , , ,u T σ j
– 5 equations;
– 13 equations;
3. Grad-26, Grad-45, Grad-71.
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Method of moments
1. Euler set:0Kn ;
2. Grad set:
3. Grad-26:
1Kn ;
4. Grad-45, Grad-71:
2Kn ;
3,4Kn ...
The error:
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Simulation of thermophoretic flows
Velocity set:
1,0,0 , 0, 1,0 , 0,0, 1 ,
2, 2,0 , 0, 2, 2 , 2,0, 2 ,
1, 1, 1 , 0,0,0 .
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
M. Young, E.P. Muntz, G. Shiflet and A. Green
Knudsen compressor
4m,2.5EL
4m,5.0EL
r
x
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Knudsen compressor
WT
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Effect of the boundary conditions
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Semi-implicit lattice-Boltzmann method for non-Newtonian flows
1 2
3 3t
i i ic c u u s ε
From the kinetic theory of gases:
1 1ˆ2 2
neq neqi i i i i i if c c f c c
σ τ s
Constitutive equation: σ = σ ε
1 3
2 2
s τ = σ s
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Newtonian liquid: 2 1 2 3
s
σ ε
Bingham liquid:, ,
1 2 32
0,
yy
y
y
s
Semi-implicit lattice-Boltzmann method for non-Newtonian flows
General case: 3
2
2 ε 1 2 3
ss
σ ε
ss
2
3
2
1
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Semi-implicit lattice-Boltzmann method for non-Newtonian flows
Velocity set (3D):
1,0,0 , 0, 1,0 , 0,0, 1 ,
1, 1,0 , 0, 1, 1 , 1,0, 1 .
Velocity set (2D): 0,0 , 1,0 , 1,0 , 1, 1 .
Equilibrium distribution:
Post-collision distribution:
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Semi-implicit lattice-Boltzmann method for non-Newtonian flows
Bingham liquid Power-law liquid
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Flow of a Bingham liquid in a constant cross-section channel
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
0, 0.1, 0.15, 0.25.y
Creep flow through mesh of cylinders
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
Re 24
Flow through mesh of cylinders
Re 40
A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows
• Continuous in time and space discrete ordinate equation is used as a link from the LB to Navier-Stokes and Boltzmann equations. This approach allows us to increase the accuracy of the method and leads to new boundary conditions.
• The method was applied to simulation of a very subtle kinetic effect, namely, thermophoretic flows with small Knudsen numbers.
• The new implicit collision rule for non-Newtonian rheology improves the stability of the calculations, but requires the solution of a (one-dimensional) non-linear algebraic equation at each point and at each time step. In the practically important case of Bingham liquid this equation can be solved analytically.
CONCLUSIONS