lattice-boltzmann method for non-newtonian and non-equilibrium flows

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


Lattice-Boltzmann method

ˆ ˆ ˆˆ, ,i i i if t t x tc f t x t f

1c

2c

3c


, ; :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


NS equations

0,

,

.p

ut

uu u p

t

TC u T

t

σ

j


Plan of the presentation

uu u p

t

σ

fc f f

t



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


Knudsen number: KnL

Boltzmann equation


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


Kinetic effects:

1. Knudsen slip (Kn),2. Thermal slip (Kn).

Knudsen layer (Kn2)


Kinetic effects:

wT

3. Thermal creep (Kn).


Kinetic effects:

4. Thermal stress flow (Kn2).

1T

2T


Discrete ordinates equation



Collision operator

1

relKBGK model:


Boundary conditions

Ox

Wx

tc


Boundary conditions: bounce-back rule

u


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.


Method of moments

1. Euler set:0Kn ;

2. Grad set:

3. Grad-26:

1Kn ;

4. Grad-45, Grad-71:

2Kn ;

3,4Kn ...

The error:


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 .


M. Young, E.P. Muntz, G. Shiflet and A. Green

Knudsen compressor

4m,2.5EL

4m,5.0EL

r

x


Knudsen compressor

WT


Effect of the boundary conditions


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


Newtonian liquid: 2 1 2 3

s

σ ε

Bingham liquid:, ,

1 2 32

0,

yy

y

y

s


General case: 3

2

2 ε 1 2 3

ss

σ ε

ss

2

3

2

1



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:



Bingham liquid Power-law liquid


Flow of a Bingham liquid in a constant cross-section channel


0, 0.1, 0.15, 0.25.y

Creep flow through mesh of cylinders


Re 24

Flow through mesh of cylinders

Re 40


• 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

lattice-boltzmann method for non-newtonian and non-equilibrium flows

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