Multicomponent Multiphase LB Models

1
Multicomponent Multiphase LB Models
Single Component Multiphase
Single Phase (No Interaction)
Attractive
Interaction Strength
Number of Components
Nature of Interaction
Multi- Component Multiphase
Repulsive
Miscible Fluids/Diffusion (No Interaction)
Immiscible Fluids
Inherent Parallelism
Low
High
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Adding a component/substance

• Often just need another loop
• for( subs0 subsltNUM_FLUID_COMPONENTS subs)
• for( j0 jltLY j)
• for( i0 iltLX i)

3
One composite u for feq calculation(Eqn. 95 in
Sukop and Thorne note error in 2006 printing)

• // Compute density, Eq. (97), and the sums used
  (below)
• // in the velocities.
• for( subs0 subsltNUM_FLUID_COMPONENTS subs)
• for( j0 jltLY j)
• for( i0 iltLX i)
• rhoijsubs 0.
• u_xijsubs 0.
• u_yijsubs 0.
• if( !is_solid_nodeji)
• for( a0 alt9 a)
• rhoijsubs ftemp_ija
• u_xijsubs exaftemp_ija
• u_yijsubs eyaftemp_ija

4
One composite u for feq calculation

• // Compute the composite velocity and
  individual velocities.
• for( j0 jltLY j)
• for( i0 iltLX i)
• if( !is_solid_nodeji)
• ux_sum u_xij0/tau0 u_xij1/tau1
• uy_sum u_yij0/tau0 u_yij1/tau1
• if( rhoij0 rhoij1 ! 0)
• // Composite velocity, Eq. (95).
• uprime_x ( ux_sum) / ( rhoij0/tau0
  rhoij1/tau1)
• uprime_y ( uy_sum) / ( rhoij0/tau0
  rhoij1/tau1)
• else uprime_x 0. uprime_y 0.
• // Individual velocities, Eq. (96),
  x-direction.
• if( rhoij0 ! 0) u_xij0 u_xij0
  / rhoij0

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Interparticle Forces

• // Compute fluid-fluid interaction force,
  equation (98),
• // (assuming periodic domain).
• //
• // We begin by computing psi even though in
  this implementation
• // it is the same as rho. A different function
  of rho could
• // be substituted here.
• for( subs0 subsltNUM_FLUID_COMPONENTS subs)
• for( j0 jltLY j)
• for( i0 iltLX i)
• if( !is_solid_nodeji)
• psisubsji rhosubsji

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Interparticle Forces

• // Compute the summations in Eq. (98).
• for( subs0 subsltNUM_FLUID_COMPONENTS subs)
• for( j0 jltLY j)
• jp ( jltLY-1)?( j1)( 0 )
• jn ( jgt0 )?( j-1)( LY-1)
• for( i0 iltLX i)
• ip ( iltLX-1)?( i1)( 0 )
• in ( igt0 )?( i-1)( LX-1)
• Fxtemp 0.
• Fytemp 0.

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Interparticle Forces

• if( !is_solid_nodeji)
• if( !is_solid_nodej ip) // neighbor
  1
• Fxtemp Fxtemp WMex1psisubsj
  ip
• Fytemp Fytemp WMey1psisubsj
  ip
• if( !is_solid_nodejpi ) // neighbor
  2
• Fxtemp Fxtemp WMex2psisubsj
  pi
• Fytemp Fytemp WMey2psisubsj
  pi
• if( !is_solid_nodej in) // neighbor
  3
• Fxtemp Fxtemp WMex3psisubsj
  in
• Fytemp Fytemp WMey3psisubsj
  in
• if( !is_solid_nodejni ) // neighbor
  4
• Fxtemp Fxtemp WMex4psisubsj
  ni
• Fytemp Fytemp WMey4psisubsj
  ni
• if( !is_solid_nodejpip) // neighbor
  5
• Fxtemp Fxtemp WDex5psisubsj
  pip
• Fytemp Fytemp WDey5psisubsj
  pip
• if( !is_solid_nodejpin) // neighbor
  6
• Fxtemp Fxtemp WDex6psisubsj
  pin

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Interparticle Forces

• Fxsubsji Fxtemp
• Fysubsji Fytemp
• / for( i0 iltLX i) /
• / for( j0 jltLY j) /
• / for( subs0 subsltNUM_FLUID_COMPONENTS
  subs) /
• // Compute the final interaction forces of Eq.
  (98) using
• // the summations computed above.
• for( j0 jltLY j)
• for( i0 iltLX i)
• if( !is_solid_nodeji)
• Fxtemp Fx1ji
• Fx1ji -Gpsi1jiFx0ji
  
• Fx0ji -Gpsi0jiFxtemp
• Fytemp Fy1ji
• Fy1ji -Gpsi1jiFy0ji
  
• Fy0ji -Gpsi0jiFytemp

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Complementary Densities

• 6,000 ts

Domain 5X100 Periodic boundary
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Complementary Densities

• 2,500 ts

Domain 100X100 Periodic boundary
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Computing big U (aka ueq)

• define BIG_U_X( u_, rho_) \
• (u_) \
• lattice-gtparam.tausubs \
• lattice-gtforcesubsn.force0/(rho_
  ) \
• lattice-gtparam.tausubs \
• lattice-gtforcesubsn.sforce0/(rho
  _) \
• lattice-gtparam.tausubs \
• lattice-gtparam.gforcesubs0
• define BIG_U_Y( u_, rho_) \
• (u_) \
• lattice-gtparam.tausubs \
• lattice-gtforcesubsn.force1/(rho_
  ) \
• lattice-gtparam.tausubs \
• lattice-gtforcesubsn.sforce1/(rho
  _) \
• lattice-gtparam.tausubs \
• lattice-gtparam.gforcesubs1

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Multicomponent Multiphase LBM

• Separate distributions
• Repulsive interaction

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Phase (fluid-fluid) separation
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Laplace Law

• Interfacial tension (as opposed to surface
  tension between a liquid and its own vapor)

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Metastability






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MCMP LBM with Surfaces

• Like SCMP except each fluid phase can interact
  with surface
• Two surface interaction parameters, one
  fluid/fluid
• Youngs Equation

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MCMP SForce

• for( j0 jltLY j)
• jp ( jltLY-1)?( j1)( 0 )
• jn ( jgt0 )?( j-1)( LY-1)
• for( i0 iltLX i)
• ip ( iltLX-1)?( i1)( 0 )
• in ( igt0 )?( i-1)( LX-1)
• if( !is_solid_nodeji)
• sum_x0.
• sum_y0.
• if( is_solid_nodej ip) // neighbor 1
• sum_x sum_x WMex1
• sum_y sum_y WMey1
• if( is_solid_nodejpi ) // neighbor 2
• sum_x sum_x WMex2
• sum_y sum_y WMey2
• if( is_solid_nodej in) // neighbor 3

• if( is_solid_nodejpip) // neighbor 5
• sum_x sum_x WDex5
• sum_y sum_y WDey5
• if( is_solid_nodejpin) // neighbor 6
• sum_x sum_x WDex6
• sum_y sum_y WDey6
• if( is_solid_nodejnin) // neighbor 7
• sum_x sum_x WDex7
• sum_y sum_y WDey7
• if( is_solid_nodejnip) // neighbor 8
• sum_x sum_x WDex8
• sum_y sum_y WDey8
• for( subs0 subsltNUM_FLUID_COMPONENTS
  subs)
• sforce_xsubsji
  -Gadssubssum_x
• sforce_ysubsji
  -Gadssubssum_y

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MCMP surface forces

• A surrounded by itself
• FA G rArB
• A surrounded by solid
• FadsA GadsArA
• FadsA FA leads to
• Since complimentary density is low, Gads should
  be small relative to G

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90-degree contact angle
Multicomponent fluids interacting with a surface
when G 0.1 and Gads1 Gads2 -0.01.
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(No Transcript)
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45 Contact Angle
Wetting fluid must have lowest Gads
Multicomponent fluids interacting with a surface
when G 0.1, Gads1 -0.02, and Gads2 0.0507.
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2 Phase Flow Analytical Solution
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Co- and Counter-current flows
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Countercurrent air and water
Pressure gradient in air phase
Pressure gradient in water phase
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Density and Viscosity Contrasts

• Large density and viscosity contrasts are a major
  challenge of LBM research.
• McCracken and Abraham (2005) pressure in
  standard multicomponent LB models is p (r1
  r2)cs2, where cs is the speed of sound
• Significance is that for total pressure to be
  constant, the sum of the densities of the 2
  species must be constant
• Not the case in real gasses, where differing
  molecular weights lead to constant pressures
  despite different densities