Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Boundary Conditions for the LatticeBoltzmann Equation

Bruce M. Boghosian1,2

1Department of Mathematics, Tufts University2American University of Armenia, Yerevan, Armenia (as of September 2010)

DSFD 2010, CNR Rome, 6 July 2010

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Outline

1 Introduction

2 The Prandtl layerZero-velocity boundary condition

3 Particulate models and the LBEBounce-back boundary conditions

4 LBE boundary conditions“Wet” versus “bounce-back” conditionsDiffuse reflectionExtrapolation schemeOff-equilibrium bounce backRegularized methodNon-local version preserving pressure tensorBoundary interpolation scheme

5 Conclusions

6 Bibliography

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Incompressible Navier-Stokes equations

Hydrodynamic velocity u and pressure p

Navier-Stokes equations (Navier, 1823; Stokes, 1845):

Incompressibility in D∇ · u = 0

Kinematic equation in D

∂tu+ u ·∇u = −∇p + ν∇2u

Boundary condition on ∂D (Prandtl, 1904)

u = 0

One vector and one scalar equation for one vector and onescalar unknown (u and p)

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Zero-velocity boundary condition

Ludwig Prandtl (1875-1953)

“A very satisfactory explanation of the physical process in the boundary layer between

a fluid and a solid body could be obtained by the hypothesis of an adhesion of the

fluid to the walls, that is, by the hypothesis of a zero relative velocity between fluid

and wall. If the viscosity was very small and the fluid path along the wall not too long,

the fluid velocity ought to resume its normal value at a very short distance from the

wall. In the thin transition layer however, the sharp changes of velocity, even with

small coefficient of friction, produce marked results.”

– L. Prandtl, in Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg 1904

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Example of irregular singular point

Order of differential equation decreases when ν = 0

Example for ν > 0:

ODE: −νy ′′(x) + y(x) = x with y(0) = y(1) = 0

has solution: yν(x) = x −sinh

(

x√ν

)

sinh(

1√ν

)

Same ODE with ν = 0 has solution y0(x) = x

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

Ν = 0.0001

Width of “boundary layer” ∼ √ν

Note limν→0√ν y ′

ν(x) = −1 but limν→0

√ν y ′0(x) = 0

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Prandtl boundary layer theory

Long-thin approximation near y = 0: ∂x ∼ 0 and uy ∼ 0

∂tux + ux∂xux + uy∂yux = −∂xp + ν∂2xux + ν∂2

yux

∂tuy + ux∂xuy + uy∂yuy = −∂yp + ν∂2xuy + ν∂2

yuy

Pressure is prescribed, rather than self-consistent

Incompressibility condition determines uy0 = ∂xux + ∂yuy

Nonlinear parabolic equation for uxux∂xux = ν ∂2

yux − (∂xp+uy∂yux)︸ ︷︷ ︸

“source′′

Characteristic vertical width goes as√ν ∼ Re−1/2

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Elliptic problem for pressure

Poisson equation in D

∇2p = −∇ · (u ·∇u)

Neumann b.c. on ∂D∂p

∂n= −n ·

(u ·∇u− ν∇2u

)

We used n · u = 0 on ∂DWe forced normal component of velocity at wall to vanish.

We did not use t · u = 0 on ∂DQuestion: How do we make certain that the tangentialcomponent of the velocity also vanishes at the wall?

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Vorticity transport equation

Vorticity ω = ∇× u

Incompressibility 0 = ∇ · uGiven vorticity, the Biot-Savart Law yields velocity

Use vector identity u ·∇u = ∇(12 |u|2) + ω × u to write

∂tu+ ω × u = −∇

(

p +1

2|u|2

)

+ ν∇2u

Use vector identity ∇× (ω × u) = u ·∇ω−ω ·∇u to obtain

∂tω + u ·∇ω − ω ·∇u = ν∇2ω

Pressure is eliminated from the problem

Question: What is boundary condition for vorticity at wall?

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

Vorticity flow at boundary

Vorticity enters at boundary to keep zero tangential velocity

Suppose tangential velocity ∆Ux 6= 0 appears just aboveboundary, y = ǫ > 0

Introduce vorticity “sheet” with circulation ∆Ux at, e.g.,position y = ǫ/2 so tangential velocity vanishes at wall

ω = ∂yux − ∂xuy = ∆Ux δ(y − ǫ

2

)

ux(0) = ∆Ux −∆Ux

∫ǫ

0dy δ

(y − ǫ

2

)= 0

ux=0

ux=DUx

y=Ε�2 C=DUx

BOUNDARY

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Zero-velocity boundary condition

One “conceptual” time step of Navier-Stokes equations

Time loop:1 Suppose that u is known at time t.2 Solve Poisson problem for pressure p thereby eliminating

normal velocity at the boundary.3 Eliminate any residual tangential velocity by introducing a

vortex sheet with appropriate circulation at y = ǫ/2.4 Add contribution of introduced vortex sheet(s) to the velocity

field throughout domain.5 Advance to time t +∆t and return to step 1.

Pass to limit as ǫ,∆t → 0

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bounce-back boundary conditions

Lattice Boltzmann equation

Lattice with symmetries:∑

j Wj = 1,∑

j Wjcjcj = c2s I2,∑

j Wjcjcjcjcj = γc4s I4

Lattice BGK equation

fj (r + cj , t +∆t)−fj (r, t) =1

τ[f eq

j (ρ (r, t) ,u (r, t))− fj (r, t)]

Hydrodynamic variables:

ρ =∑

j fj , ρu =∑

j fjcj

Mach-expanded equilibrium distribution function

f eqj (ρ,u) = ρWj

[

1 +cj · uc2s

+u ·

(cjcj − c2s I2

)· u

2γc4s

]

Viscosity: ν = c2s(τ − 1

2

)

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bounce-back boundary conditions

Bounce-back boundary conditions

Zero-velocity (“no-slip”) boundary condition at wall

Bounce particles back in the direction from which they arrived

Mass is conserved at wall, but not momentum

“Time average” of velocity vanishes

Introduced for lattice gases; also used for LBE, MD, DPD, etc.

Very simple to understand and implement

BOUNDARY

: f0',=, f3>

: f1',=, f4>

: f2',=, f5>

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bounce-back boundary conditions

Accuracy issues

LBE is fully explicit and second-order accurate in bulk.

Bounce-back prescription is only first-order accurate at wall.

Bounce-back does not allow collisional relaxation toward LTE.

Bounce-back does not alter Boltzmann’s H =∑

j fj ln(

fjwj

)

A number of schemes have been developed to restoresecond-order accuracy at wall or, at least, better understandthe nature of the inaccuracy.

First-order accuracy is not good enough!

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bounce-back boundary conditions

Bad ideas abound

Use bounce-back at all walls

Use equilibrium distribution with desired moments at all walls

Ignore gradient (Chapman-Enskog) corrections at your peril!

Gradient corrections appear at order M ∼ Kn

You won’t even get Poiseuille flow right if you do these things.

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

“Wet” versus “bounce-back” conditions

“Wet” versus “bounce-back” conditions

See review article:Latt, Chopard, Malaspinas, Deville, Michler, Phys. Rev. E 77 (2008) 056703

Consider a wall node with wall velocity Uw

The set of lattice vectors is denoted by CSubset pointing into the domain is C−Subset pointing into the wall is C+Subset perpendicular to the normal or speed zero is C0

After propagation but before collision:

ρ+ and ρ0 are knownρ− is unknown

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

“Wet” versus “bounce-back” conditions

Example

D2Q9 model with straight boundary at right

0 1

234

5

6 7 8

C− = {c4, c5, c6} point into domain

C+ = {c1, c2, c8} point into wall

C0 = {c0, c3, c7} are zero or perpendicular to normal

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

“Wet” versus “bounce-back” conditions

“Wet” boundary conditions

Carry out propagation step

Populate ρ− (somehow)

Collide all nodes as though they were interior (hence “wet”)

Total density is determined

ρ = ρ− + ρ0 + ρ+

ρu⊥ = ρ+ − ρ−

Eliminate unphysical density to obtain

ρ =ρ0 + 2ρ+1 + u⊥

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Diffuse reflection

Method I: Diffuse reflection

Inamuro et al. (1995)

Incoming distribution on C+ pointing into wall

Outgoing distribution on C− pointing into domain

Outgoing distribution is assumed to be a local equilibriumwith density ρ′ and velocity Uw + u′t

Determine unknown parameters ρ′, u′ by demanding:

∑j∈C−

f eqj (ρ′,Uw + u′t)−

∑j∈C+

fj = 0 (mass)

∑j∈C−

f eqj (ρ′,Uw + u′t) cj +

∑j∈C+

fjcj = ρ′Uw (velocity)

This method has been demonstrated to achieve second-orderaccuracy on Poisseuille and Taylor-Couette flow.

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Extrapolation scheme

Method II: Extrapolation scheme

Chen & Martınez (1996)

Add one layer of lattice points outside domain, at x = +1

Physical wall is at x = 0

Second-order accuracy requires fj(0) =fj(−1)+fj (+1)

2 +O(c2)

So impose fj(−1) = 2fj(0)− fj(+1).

Propagate normally at every site, including x = +1

Collide normally at every site with x ≤ 0

Makes no assumptions about incoming distribution function

Tested on, inter alia, lid-driven cavity flow

Conserves mass only to O(c2)

Conserves all other hydrodynamic quantities to O(c2)

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Off-equilibrium bounce back

Method III: Off-equilibrium bounce back

Zou, He (1996a,b).

Write fi = f(0)i + f

(1)i

f (0) is equilibrium distribution

f (1) is Chapman-Enskog correction

Use bounce-back on f (1) in some (not all) directions

Use on directions parallel to normal vector

Populate remaining directions using moment constraints

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Regularized method

Method IV: Regularized method

J. Latt, B. Chopard (2007)

Repopulates all directions – not just C−Computes gradient correction to pressure tensor Π(1) based ondistribution components on C+No assumptions made about distribution components on C−“Inverse Chapman-Enskog analysis” then reconstructs alldistribution components consistent with

{ρ,u,Π(1)

}

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Non-local version preserving pressure tensor

Method V: Non-local version preserving pressure tensor

Skordos (1993)

Progenitor of regularized method

Computes gradient correction by symmetric finite difference

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Boundary interpolation scheme

Method VI: Boundary interpolation scheme

I. Ginzburg, D. d’Humieres (1996a,b).

Bounce-back condition is second-order accurate if we supposethat the actual location of the wall is not on a lattice point.

Effective channel width for Poiseuille flow

H =√

h2 + 163 Λ− 1, where Λ may be determined by kinetic

theory.

Actual no-slip boundary is located about halfway betweenboundary node and solid node.

Exact position depends on orientation of wall with respect tolattice, viscosity, etc.

Difficult to apply in practice, but gave rise to interpolation

schemes often used for MRT LBE

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Boundary interpolation scheme

Method VI: Boundary interpolation scheme (continued)

Chun, Ladd (2007).

Boundary node has at least one solid neighbor in direction cj

The actual wall is located a fraction q between the boundarynode r and the solid node r + cj .

If 0 < q < 1/2, interpolate to position that will reflect fromwall to just reach boundary node.

If 1/2 < q < 1, propagate from boundary node, reflectingfrom wall, and then interpolate to lattice sites.

q < 1�2 q > 1�2

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Conclusions

There is complicated physics in the boundary layer.

Much is expected of boundary conditions.

LBE boundary conditions must be at least second-orderaccurate.

Many methods are known which yield this accuracy.

Some use “wet” boundaries, and others use bounce-back.

Some modify unknown populations, and others modify allpopulations.

Some are local, and others are non-local.

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bibilography

1 K. Nickel, “Prandtl’s Boundary Layer from the Viewpoint of aMathematician,” Ann. Rev. Fluid Mech. 5 (1973) 405-428

2 D.P. Ziegler, “Boundary Conditions for Lattice BoltzmannSimulations,” J. Stat. Phys. 71 (1993) 1171-1177

3 P.A. Skordos, “Initial and Boundary Conditions for the LatticeBoltzmann Method,” Phys. Rev. E 48 (1993) 4823-4842

4 T. Inamuro, M. Yoshino, F. Ogino, “A Non-Slip BoundaryCondition for Lattice Boltzmann Simulations,” Phys. Fluids 7(1995) 2928-2930

5 D.R. Noble, S. Chen, J.G. Georgiadis, R.O. Buckius, “AConsistent Hydrodynamic Boundary Condition for the LatticeBoltzmann Method,” Phys. Fluids 7 (1995) 203-209

6 S. Chen, D. Martınez, “On Boundary Conditions in LatticeBoltzmann Methods,” Phys. Fluids 8 (1996) 2527

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bibilography (continued)

1 I. Ginzburg, D. d’Humieres, “Local second-order boundarymethod for lattice Boltzmann models,” J. Stat. Phys. 84(1996) 927-971

2 I. Ginzburg, D. d’Humieres, “Local second-order boundarymethod for lattice Boltzmann models: Part II. Application tocomplex geometries,” unpublished preprint (1996)

3 Q. Zou, X. He, On Pressure and Velocity BoundaryConditions for the Lattice Boltzmann. BGK Model, Phys.Fluids 9 (1997) 1591-1598

4 D. Kandhai, A. Koponen, A. Hoekstra, M. Kataja, J.Timonen, P.M.A. Sloot, “Implementation Aspects of 3DLattice-BGK: Boundaries, Accuracy, and a New FastRelaxation Method,” J. Comp. Phys. 150 (1999) 482-501

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bibilography (continued)

1 S.-D. Feng, R.-C. Ren, Z.-Z. Ji, “Heat Flux BoundaryConditions for a Lattice Boltzmann Equation Model,” Chin.

Phys. Lett. 19 (2002) 79-82

2 A.J. Wagner, I. Pagonabarraga, “Lees-Edwards BoundaryConditions for Lattice Boltzmann,” J. Stat. Phys. 107 (2002)521-537

3 S. Ansumali, I.V. Karlin, “Kinetic Boundary Conditions in theLattice Boltzmann Method,” Phys. Rev. E 66 (2002) 026311

4 I. Ginzburg, D. d’Humieres, “Multireflection BoundaryConditions for Lattice Boltzmann Models,” Phys. Rev. E 68(2003) 066614

5 P. Lallemand, L.-S. Luo, “Lattice Boltzmann Method forMoving Boundaries,” J. Comp. Phys. 184 (2003) 406-421

Introduction The Prandtl layer Particulate models and the LBE LBE boundary conditions Conclusions Bibliography

Bibilography (continued)

1 J.D. Anderson, “Ludwig Prandtl’s Boundary Layer,” Physics

Today (December 2005) 42-48

2 B. Chun, A.J.C. Ladd, “Interpolated boundary condition forlattice Boltzmann simulations of flows in narrow gaps,” Phys.

Rev. E 75 (2007) 066705

3 J. Latt, B. Chopard, O. Malaspinas, M. Deville, A. Michler,“Straight velocity boundaries in the lattice Boltzmannmethod,” Phys. Rev. E 77 (2008) 056703