Lattice Boltzmann methods: Applications - TUM · 2019. 7. 22. · Lattice Boltzmann methods: Applications Porous media ow Dissolution process of a sugar ball Heat transfer (!programming

Lehrstuhl fürNumerische Mathematik

Lattice Boltzmann methods: Applications

Porous media flowDissolution process of a sugar ball

Heat transfer (→ programming tutorial 3)

July 23, 2019

TUM, Munich

Lehrstuhl fürNumerische Mathematik

Porous media flow

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Lehrstuhl fürNumerische Mathematik 0 Approach to porous media simulation

Meso- to Macro-scale approach to porous media

• Here: No “Lattice Boltzmann DNS”

• We do not resolve individual porous mediaparticle with the mesh

• Look at the porous medium “from above”

• Only consider the effect of the mediumspresence on particle collisions

• Additional force contribution to thecollision operator

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Lehrstuhl fürNumerische Mathematik 1 A LBM model of porous media flow

Porous media flow

• Generalized Navier-Stokes model [P. Nihiarasu et al., 1997]

∇ · u = 0

∂tu + (u · ∇)(u

ε

)= −1

ρ∇(εp) + νe∆u + F

• Porosity ε ∈ (0,1]

• Effective viscosity νe = Jν

• Total body force F due topresence of porous medium+ external forces

• Force model for porous medium

F = −ενK

u− εFε√K|u|u + εG︸︷︷︸

linear drag︸︷︷︸

nonlinear drag

• External forces G

• Fluid shear viscosity ν

• Permeability K of porousmedium

• Forchheimer term Fε

• Parameter relations

Fε =1.75√150ε3

K =ε3d2p

150(1− ε)2Re =

U · Lν

Da =K

L2

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Lehrstuhl fürNumerische Mathematik 1 A LBM model of porous media flow

Force treatment with LBM

• Modified LBM-Update equation [Zhaoli Guo, T.S. Zhao, 2002]

fi(t+ ∆t, x+ ci∆t) = fi(t, x)−1

τ· (fi(t, x)− f eq

i (t, x)) + ∆tFi

Collision now also incorporates the influence of forces.

• Force contribution in porous medium setting

Fi = ρωi

(1− 1

2τ

)(3(F · ci) +

9

ε(u · ci) · (F · ci)−

3

ε(u · F)

)

• Redefinition of velocity

u =1

ρ

∑i

cifi +∆t

2F

• F depends on u itself → nonlinear relation

• F(u) Quadratic equation → solvable!

• Macroscopic velocity can be computed in every stepJJ J I II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Lehrstuhl fürNumerische Mathematik 2 Numerical examples

Numerical example I - Poiseuille flow

• Geometry and flow setting:

0

d

x2

x1

• Channel of width H

• Completely filled with porousmedium

• Flow is driven by external force Galong channel direction

• No-slip boundary conditions atchannel walls• Analytical solution

u =

(u(y)

0

)satisfies

νeε

∂2u

∂y2+G− ν

Ku− Fε√

Ku2 = 0

For Fε = 0 (no nonlinear drag) thesolution is (r =

√νε/(Kνe)):

u =GK

ν

(1−

cosh [r(y −H/2)]

cosh(rH/2)

)

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Numerical example I - Poiseuille flow

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Numerical solution with F =0

Brinkman-solutionNumerical solution with F 0

ε = 0.7

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.9

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.95

• Good numerical approximation ofBrinkmann solution

• Forchheimer term Fε hasincreasing influence with higherporosity

• For ε → 1 we get back to thePoiseuille-channel profile

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Numerical example II - Couette flow

• Geometry and flow setting:x2

0

H

x1


• Completely filled with porousmedium

• Flow is driven by prescribed velocityU0 in channel direction at topboundary

• No-slip boundary condition atbottom wall

• Analytical solution

u =

(u(y)

0

)with (in case of Fε = 0): u = U0

sinh(ry)

sinh(rH)

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Numerical example II - Couette flow

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.96

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.99

• Again good numerical approximation of Brinkmann solution

• Also again the Forchheimer term Fε (nonlinear drag) slows the flow down

• For ε→ 1 we get back the linear Couette profile

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Numerical example III - Free-flow to Darcy-flow

• Geometry and flow setting:

0

d

x2

x1


• Only a fraction r is filled withporous medium

• Flow is again driven by external forceG along channel direction

• No-slip boundary conditions atchannel walls

0

d

x2

x1

• Channel can also contain an obstacle

• For example a building

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0 2 4 6 8 10 12-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.9

0 0.5 1 1.5 2 2.5 3 3.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.95

• Poiseuille-profile in the free flowregion

• Good accordance with analyticalsolution in porous media region

• Transition zone around theinterface

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0 2 4 6 8 10 12-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.9

0 0.5 1 1.5 2 2.5 3 3.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4



ε = 0.95

• Poiseuille-profile in the free flowregion

• Good accordance with analyticalsolution in porous media region

• Transition zone around theinterface 0 20 40 60 80 100

10

20

30

40

50

60

70

80

90

100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

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Lehrstuhl fürNumerische Mathematik 3 Literature

Main Literature

• General LBM

Timm Kruger et al - The Lattice Boltzmannmethod

• Good introduction

• Covers a variety of applications and scenarios

• Code examples included

• About the presented porous media simulations

• Guo, Zhaoli, and T. S. Zhao. ”Lattice Boltzmann model for incompressibleflows through porous media.” Physical Review E 66.3 (2002): 036304.

• Fattahi Evati, Ehsan ”High performance simulation of fluid flow in porousmedia using Lattice Boltzmann method” Dissertation, Technische UniversitatMunchen, 2017

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Lehrstuhl fürNumerische Mathematik 4 Dissolution of a candy ball

Dissolution process of acandy ball

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Lehrstuhl fürNumerische Mathematik 5 Motivation

Motivation

Do you remember the “jawbreaker” hard candy from the 90s?

Figure 1: Jawbreaker-candy consisting of multiple layers of sugar formedto a ball. Image sources: http://asusvilla.wonecks.net/2013/05/21/history-of-jawbreaker-rock-candy/ and

https://www.tumblr.com/search/jawbreaker%20candy

The question everyone was curious about is:

xxxxxxxxxxxxxxxx“How many licks does it take to get to the center?”

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http://asusvilla.wonecks.net/2013/05/21/history-of-jawbreaker-rock-candy/


Lehrstuhl fürNumerische Mathematik 5 Motivation

Motivation

Do you remember the “jawbreaker” hard candy from the 90s?

Figure 2: Jawbreaker-candy consisting of multiple layers of sugar formedto a ball. Image sources: http://asusvilla.wonecks.net/2013/05/21/history-of-jawbreaker-rock-candy/ and


The question everyone was curious about is:

xxxxxxxxxxxxxxxx“How many licks does it take to get to the center?”

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http://asusvilla.wonecks.net/2013/05/21/history-of-jawbreaker-rock-candy/


Lehrstuhl fürNumerische Mathematik 6 Experimental setup

Simplification of the “experiment”

• We cannot simulate e.g. tongue roughness or chemical properties of saliva

• Simplify the situation to a uniform candy ball being suspended into a flow channel

• Simulate dissolution due to convection and diffusion

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Figure 3: Fluid flow around the sugar ball is visualized by laser refraction of particles.Video link: https://www.youtube.com/watch?v=OvtZ7jpNCUI

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https://www.youtube.com/watch?v=OvtZ7jpNCUI

Lehrstuhl fürNumerische Mathematik 7 Mathematical formulation

Mathematical model

• The flow is again described by the Navier-Stokes-system

∇ · u = 0

∂u

∂t+ (u · ∇)u = −1

ρ∇p+ ν ·∆u

with no-slip BC at the sugar balls surface

• The “spreading” of sugar into the fluid is described by a coupled (!) Convection-Diffusion-equation

∂ρh∂t

+ u · ∇ρh = D ·∆ρh

where ρh is the sugar density between 0 (no sugar) and 1 (pure sugar). At the sugarballs surface a BC of ρh = 1 is imposed.

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Lehrstuhl fürNumerische Mathematik 7 Mathematical formulation

Mathematical model

• The flow is again described by the Navier-Stokes-system

∇ · u = 0

∂u

∂t+ (u · ∇)u = −1

ρ∇p+ ν ·∆u

with no-slip BC at the sugar balls surface

• The “spreading” of sugar into the fluid is described by a coupled (!) Convection-Diffusion-equation

∂ρh∂t

+ u · ∇ρh = D ·∆ρh

where ρh is the sugar density between 0 (no sugar) and 1 (pure sugar). At the sugarballs surface a BC of ρh = 1 is imposed.

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Lehrstuhl fürNumerische Mathematik 8 Simulation of the sugar convection / diffusion

Simulation without shape change of the sugar ball

Figure 4: Simulation of ρh for different values of D = 0.1, 0.02, 0.002.

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Lehrstuhl fürNumerische Mathematik 9 LBM implementation of Convection-Diffusion-equation

How to solve Convection-Diffusion equation with LBM

• For the flow simulation ones uses populations fi, i = 1, ..., 9, equilibrium populationsf eqi , performs collision and streaming and boundary treatment.

• For the simulation of ρh one uses a second set of populations hi and conducts thesame steps, i.e.

ρh =∑i

hi and v =∑i

cihi

and collision and streaming etc. for hi whenever they are applied to fi.

• Some differences:

1. Different boundary conditions −→ different boundary treatment2. Different relaxation frequency ω → ωh and different material parameters ν → D3. The equilibrium heq

i is formed with u and not v in order to couple the sugardensity with the flow, i.e.

heqi = ωiρh

(1 + 3(ci · u) +

9

2(ci · u)2 − 3

2|u|2

)JJ J I II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16





ρh =∑i

hi and v =∑i

cihi





heqi = ωiρh

(1 + 3(ci · u) +

9

2(ci · u)2 − 3

2|u|2

)JJ J I II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16





ρh =∑i

hi and v =∑i

cihi





heqi = ωiρh

(1 + 3(ci · u) +

9

2(ci · u)2 − 3

2|u|2

)JJ J I II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Lehrstuhl fürNumerische Mathematik 10 Shape change of the sugar ball

Shape change of the sugar ball

• The shape of the sugar ball changes due to the sugar dissolving in the fluid.

• The normal velocity by which the surface moves is given by:

vn = −D ∇ρh · n+ k u · n

• Hence we need the quantities n and ∇ρh via a “post-processing” step

• In the discrete setting, we have to think about how to implement e.g. a normalrecession speed of vn = 0.5 geometrically, since we can not cancel a half cell forexample...

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vn = −D ∇ρh · n+ k u · n



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vn = −D ∇ρh · n+ k u · n



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vn = −D ∇ρh · n+ k u · n



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Dealing with the recession rate

• For us ∆t = 1, hence if (always for one certain cell) in timestep t we have vn = 0.7,in timestep t+ 1, we have vn = 1.5, etc. ...

• We sum up all those values (for the given cell) and let the cell “break away” as awhole as soon as a certain “breakaway value” B is reached.

• In this way we can still use our discrete lattice while still incorporating the differentspeed by which certain parts of the sugar ball vanish.

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Final results

Figure 5: Shape of the sugar ball after: 6751, 8851, and 11401 timesteps, compared toexperimental results.

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Lehrstuhl fürNumerische Mathematik 11 Thermal LBM

Thermal LBM simulation:Rayleigh-Benard

convection

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Lehrstuhl fürNumerische Mathematik 12 Theory: Boussinesq Approximation

Thermo-coupled Navier-Stokes equations

Want to couple a temperature field to the still incompressible Navier-Stokes equations.Do not want to take into account the energy equation or compressibility terms.

• Assume a linear temperature (T ) dependency of the density ρ

ρ = ρ0 − βρ0(T − T0), ρ0, T0 are reference values

• Variations in density lead to a flow (hot air streams up) → One way coupling

• Consider the body force due to gravity on the rhs of Navier-Stokes: F = ρg, where

g = (0,−9.81)>

F

ρ0= g︸︷︷︸

Gravity pulling downoften neglected again

−Heat gradient pulling up (due to− in front)︷︸︸︷

β(T − T0)g

• Temperature enters the Navier Stokes equation as a force term on the rhs

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Thermo-coupled Navier-Stokes equations

Need an evolution equation for the temperature field

• Heat is transported by the flow → Backwards coupling

• Use a convection-diffusion equation / convection enhanced heat equation

∂T

∂t+ u · ∇T − κ∆T = 0

where u is the velocity field from the Navies-Stokes equation.

• Those assumptions are known as Boussinesq approximation.

• They take into account effects caused by buoyancy

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Navier-Stokes equations with Boussinesqapproximation

• Choosing the reference temperature T0 = 0 for simplicity

• Neglect the downwards pulling gravity term g, since we are only interested in thebuoyant force

• One arrives at the following system

∂u

∂t+ (u · ∇)u+

1

ρ0∇p− ν∆u = g − βT g

∇ · u = 0

∂T

∂t+ u · ∇T − κ∆T = 0

• With suitable boundary and initial conditions

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Lehrstuhl fürNumerische Mathematik 13 LBM Adaption of buoyancy terms and convection/diffusion

Characteristic number for thermal flow configurations

In order to characterize a thermal flow, the different processes that participate have tobe set into relation.

• Reynolds number:

Re =UL

ν∼ inertial forces

viscous forces

• Prandtl number:

Pr =ν

κ∼ viscous diffusivity

heat diffusivity

• Rayleigh number:

Ra = Re2 Pr ∼ thermal transport via diffusion

thermal transport via convection

where U is a characteristic velocity scale, L a characteristic length scale, ν the kinematicviscosity and κ the heat conductivity of the material.

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Treatment via Lattice Boltzmann method

Have to incorporate the temperature field and the buoyancy force term on the rhs ofNavier-Stokes.

• Introduce second set of populations gi, i = 1, ..., 9

• Define the macroscopic temperature density

T =

9∑i=1

gi compare to ρ =

9∑i=1

fi

• A “temperature momentum” is not needed, since the temperature is transported bythe flow u→ use the same velocity as for u (coupling).

• Define the temperature equilibrium analogously to the flow equilibrium using thesame velocity u

geqi = ωiT

(1 + 3(ci · u) +

9

2(ci · u)2 − 3

2|u|2)

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Treatment via Lattice Boltzmann method

Have to incorporate the temperature field and the buoyancy force term on the rhs ofNavier-Stokes.

• Remember the update equation for the i-th population

fi(x+ ∆t ci, t+ ∆t) = fi(x, t)−∆t

τν(fi(x, t)− f eq

i (x, t)) + Ji

• Choose the buoyancy force term as follows [1]:

Ji = 3ωiρ(x, t) βT (x, t)(−g · ci

)• The populations for the temperature have their own update equation:

gi(x+ ∆t ci, t+ ∆t) = gi(x, t)−∆t

τκ(gi(x, t)− geq

i (x, t))

• Dependency of the relaxation parameters on viscosity and heat conductivity (∆t = 1normalization) by Chapman-Enskog: τν = 3ν + 1

2, τκ = 3κ+ 12

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Simulation of a Rayleigh-Benard convection cell• Initial flow at rest and isothermal

• Heating at the bottom of the domain

• Cooling at the top of the domain

• Bottom and top are solid walls

• Periodic flow and temperature BCleft and right

� = �bottom

� = �top

periodic

u = �

u = �

periodic

• Boundary conditions for u = 0: Bounce back scheme

• Boundary conditions for T = Tgiven: Anti bounce back scheme [2]

gopp(i)(x, t+ ∆t) = −g∗i (x, t) + 2ωiTgiven

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Lehrstuhl fürNumerische Mathematik 14 Numerical results

Formation of a Benard cell

Velocity and temperature fields for Ra = 50000, Nx = 100,Ny = 50t = 3000

t = 20000

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Influence of the Rayleigh number

• Analyze the behaviour for different values of the Rayleigh number

• Small Rayleigh numbers → diffusion dominated heat transfer

• Larger Rayleigh numbers → convection dominated heat transfer

Stable temperature fields after t = 20000 time stepsRa = 1000 Ra = 5000

Ra = 50000 Ra = 100000

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Turbulent behavior for high Rayleigh numbers

Turbulent behavior at Ra = 1 · 106

t = 4000 t = 5000

t = 6000 t = 7000

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Karman vortex street with heated object

An other example is the Karman vortex street:

• Flow around an object within a channel (see last tutorial)

• New: The object is heated/cooled

• Inspect heat transport in the flow around the object

Ny = 100, Nx = 400, Re = 120t = 30000

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Temperature field of a hot object at: t = 0, 1000, 2000

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Temperature field of a cold object at: t = 0, 1000, 2000

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Lehrstuhl fürNumerische Mathematik 15 Conclusion

Conclusion

• Temperature appears in a convection diffusion equation coupled to Navier-Stokes

• The Boussinesq approximation considers buoyancy effects via a force term on the rhsof Navier-Stokes

• In LBM the quantity of a convection diffusion equation is represented by a secondset of populations

• Coupling via u appearing in the equilibrium of the second set of populations

• Incorporation of force terms via modification of the collision operator

• Behavior of heat flows depending on characteristic Rayleigh number

Overall astonishingly easy implementation of coupled PDE system.

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Lehrstuhl fürNumerische Mathematik 16 References

References

References

[1] KAO, P.-H.; YANG, R.-J. Simulating oscillatory flows inRayleigh-Benard convection using the lattice Boltzmann method.International Journal of Heat and Mass Transfer, 2007, 50. Jg.,Nr. 17-18, S. 3315-3328.

[2] KRUGER, Timm, et al. The lattice Boltzmann method. SpringerInternational Publishing, 2017, 10. Jg., S. 978-3.

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Documents

Lattice Boltzmann methods: Applications - TUM · 2019. 7. 22. · Lattice Boltzmann methods: Applications Porous media ow Dissolution process of a sugar ball Heat transfer (!programming