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to my family

The research for this thesis has been performed in the time when I was PhD student at the

Chair of Engineering Thermodynamics of the Technische Universität Bergakademie Freiberg from

autumn 2009 until summer 2013 and during research visits at Oxford Centre for Collaborative

Applied Mathematics of Oxford University in spring 2011 and winter 2012.

I would like to express my gratefulness to Prof. Dr.-Ing. habil. U. Groß who gave me the

opportunity to work at his chair and providing me the freedom to research what I am interested

in. Furthermore, I am thankful to Prof. Dr.-Ing. habil. R. Schwarze for overtaking the second

report and all the comments on the manuscript.

There is one person I would like to thank especially for his steady support and readiness for

discussion, all the useful hints and comments, and of course for inviting me to college dinners:

Dr. T. Reis. This treatise would never have been ﬁnished without you! I am very indebted to Dr.

P. J. Dellar and Dr. R. Ledesma Aguilar for all the discussions and help during my time in Oxford.

In addition, I would like to say ‘Thank you!’ to Prof. Dr. S. Ray and Dr.-Ing. A. Al-Zoubi for

the nice lattice Boltzmann theory sessions with their enlightening discussions, and I would like

to thank Mrs A. Miehe, Dr.-Ing. I. Riehl, and Dr.-Ing. R. Wulf for explaining me all the technical

things which are not related to lattice Boltzmann methods and the time I could share with.

I am very thankful to ‘my’ students who did throughout excellent work. They are in alphabet-

ical order: Mr R. Behrend, Mr C. Berkholz, Mr D. Dietzel, Mr S. Gierth, Mr F. Hartmann, Mr M.

Heinrich, and Mr A. Zschutschke. Thank you all!

Special thanks goes to the computing centre of the TU Bergakademie Freiberg, who provided

access to the high-performance computing facilities, where all the computations have been carried

out. I would like to thank Mr M. Brislin for his thorough proof reading of the manuscript.

Finally, I would like to thank my parents and sister, and especially my wife and daughters for

their steady motivation, support, and never-ending patience when I had somewhat unintelligible

priorities.

Liquid ﬁlm ﬂow is an important ﬂow type in many applications of process engineering. For sup-

porting experiments, theoretical and numerical investigations are required. The present state of

the art is to model the liquid ﬁlm ﬂow with Navier–Stokes-based methods, whereas the lattice

Boltzmann method is employed here. The ﬁnal model has been developed within this treatise by

means of a two-phase ﬂow and a heat transfer model, and boundary and initial conditions. All

these sub-models have been applied to simple test cases.

It could be found that the two-phase model is capable of solving ﬂow phenomena with a large

density ratio which has been shown impressively in conjunction with wall boundary conditions.

The heat transfer model was tested against spectral method results with a transient non-uniform

ﬂow ﬁeld. It was possible to ﬁnd optimal parameters for computation. The ﬁnal model has been

applied to steady-state ﬁlm ﬂow, and showed very good agreement to OpenFOAM simulations.

Tests with transient ﬁlm ﬂow demonstrated that the model is also able to predict these ﬂow

phenomena.

Flüssigkeitsﬁlmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur

Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand

der Technik ist es, Navier–Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-

Boltzmann-Methode verwendet wird. Das ﬁnale Modell wurde unter Verwendung eines Zwei-

phasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbe-

dingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft.

Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichte-

unterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen

gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines

transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten

sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen,

dass das Model auch solche Strömungen abbilden kann.

liquid ﬁlm ﬂow, two-phase ﬂow, lattice Boltzmann method, LBM, boundary conditions, moment

method, advection-diffusion problem, heat transfer

2.1 Modelling ﬂuid ﬂow on different scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Balance equations of mass, momentum, and energy . . . . . . . . . . . . . . . . . . . . 4

2.3 Kinetic gas theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Lattice Boltzmann equation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Interface physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Non-dimensional numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Macroscopic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Macroscopic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Hydrodynamic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Thermal boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1 Final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2 Implementation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 Steady-state laminar-waveless liquid ﬁlm ﬂow . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Outlook: Transient laminar-wavy liquid ﬁlm ﬂow . . . . . . . . . . . . . . . . . . . . . 86

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B.1 Lattice symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.2 Equilibrium density distribution moments . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.3 Equilibrium energy density distribution moments . . . . . . . . . . . . . . . . . . . . . 103

B.4 Forcing moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.5 Calculating the stream function from the velocity ﬁeld . . . . . . . . . . . . . . . . . . 104

B.6 Derivation of the bounce-back scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.7 Theoretical solution of steady-state advection–diffusion problems . . . . . . . . . . . 105

B.8 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Common symbols are summarised in this chapter, whereas symbols being rarely employed are

deﬁned in the corresponding sections. Vectors are typeset with bo l d i t a l ic s t y p e, tensors with

and unit tensors of any rank with bold type. Generally, vector notation is preferred

over index notation, but the latter is utilised for difﬁcult terms. In these cases, Greek subscripts

(α,β,γ,δ, ...) are employed for the Cartesian coordinates and summation is assumed over re-

peating indices. The Latin subscript qis used for the discrete velocity directions in velocity space

V. Variables and functions of the source code, as well as ﬁle and directory names are displayed

with .

Latin symbols

cpspeciﬁc heat at constant pressure J/(kg K)

csspeed of sound in LB units lu/ts

Ccollision term

eqmicroscopic velocity vector in LB units: eq∈Vlu/ts

Eerror vector: E= (Ex,Ey,Ez)T

f,¯

fdensity distribution function, modiﬁed distribution function kg/m3

Ffrequency of disturbance 1/s

Fforce N

F

qforce term

g,¯

genergy distribution function, modiﬁed distribution function J/m3

ggeneral acceleration m/s2

hheat transfer coefﬁcient W/(m2K)

igeneral index or counter

Iunit tensor: Iαβ =1 for α=β,Iαβ =0 for α6=β

kthermal conductivity W/(m K)

kwave number vector: k= (kx,ky,kz)T1/m

Llength m

˙

mmass ﬂux kg/(m2s)

nunit normal vector

ppressure N/m2

q,Qvelocity direction with q∈ {0, 1, . . . , Q−1}

˙

qheat ﬂux vector: ˙

q= (˙

qx,˙

qy,˙

qz)TW/m2

relaxation tensor: =diag(R0,R1, . . . , RQ−1)

Rradius m

Rspeciﬁc gas constant J/(kg K)

swidth of a layer m

stress tensor N/m2

ttime s

Ttemperature K

uvelocity vector: u= (ux,uy,uz)Tm/s

Vvelocity space V⊂RQ

wqweighting factors for velocity directions q

xlocation vector: x= (x,y,z)Tm

Greek symbols

αthermal diffusivity m2/s

β,βAcompressibility, artiﬁcial compressibility Nm10/kg4

δﬁlm thickness m

δ

αβ Kronecker delta: δ

αβ =1 for α=β,δ

αβ =0 for α6=β

"relative amplitude of disturbance

ηdynamic viscosity kg/(m s)

θcontact angle rad

κinterfacial tension parameter N m6/kg2

Λ,Λ

T‘magic parameter’, thermal ‘magic parameter’ in LB units ts2

µchemical potential J/mol

νkinematic viscosity m2/s

Πimoment of a distribution function with i∈ {0, α,β, . . .}

ρdensity kg/m3

σinterfacial tension N/m

σTinterfacial tension coefﬁcient N/(m K)

τ,τTrelaxation time, thermal relaxation time in LB units ts

φgeneral quantity

ϕphase index

ψstream function m2/s

ψHelmholtz free-energy density J/m3

ΨHelmholtz free-energy J

ξinterface width m

ξcontinous microscopic velocity vector: ξ= (ξx,ξy,ξz)Tm/s

Non-dimensional numbers

Bi Biot number

Bo Bond number

Ka Kapitza number

Ma Mach number

Mn Marangoni number

Nu Nusselt number

Oh Ohnesorge number

Pr Prandtl number

Re Reynolds number

We Weber number

Operators (φis a general variable)

kφkEuclidean norm: kφk=qPiφ2

i

l2(φ)relative Euclidean norm: l2=kφ−φthk2/kφthk2

〈φ〉volume average: 〈φ〉=1/VRφdx

∂ipartial derivative: ∂i=∂ /∂ iwith i∈ {t,x,y,α,β,γ, . . .}

∂i j partial derivative: ∂i j =∂2/(∂i∂j)with i,j∈ {t,x,y,α,β,γ, . . .}

Dtsubstantial derivative: Dtφ=∂tφ+ (u·∇)φ

∇spatial gradient operator: ∇= (∂x,∂y,∂z)T

∆difference operator; ∆6=∇2

Subscripts

0 initial

b bulk

l liquid

s solid

sc scaling value

th theory

v vapour

Superscripts

∗non-dimensional quantity

?ratio of liquid and vapour values of a given quantity, e.g., ρ?=ρl/ρv

B biased

C central

eq equilibrium

M mixed

T transposed

Acronyms

BC boundary condition

D2Q5 two-dimensional lattice stencil with ﬁve velocity directions

D2Q9 two-dimensional lattice stencil with nine velocity directions

HC Neumann’s type heat ﬂux condition

LBE lattice Boltzmann equation

LB(M) lattice Boltzmann (method)

MLNUPS million lattice node updates per second

MRT multiple-relaxation time

SRT single-relaxation time

TC Dirichlet’s type temperature condition

TRT two-relaxation time

‘[...]die Wissenschaft hebt allen Glauben auf und verwandelt ihn in Schauen.’

Johann Gottlieb Fichte (1762–1814)

Liquid ﬁlm ﬂow is a special type of multi-phase ﬂow, characterised by a thin liquid layer at a

wall and some gas in the remainder of domain (for a graphical representation see Fig. 1.1). The

layer thickness is typically below a few millimetres. Contrary to pure-gas ﬂows, liquid ﬁlm ﬂows

provide superior heat and mass transfer conditions due to the thin liquid layer, which improves

the performance even better when being wavy or turbulent and phase–change takes place.

Given the advantages, liquid ﬁlm ﬂows are widely in use in energy and process engineering.

One can ﬁnd them in the condenser of power stations, in chemical industries for rectiﬁcation and

distillation, in food industries for densiﬁcation of milk and juice, in cooling devices utilising heat

pipes (also known as thermo-siphons), and geothermal heat exchangers based on phase–change

processes. The typical geometry of heat exchanger utilising liquid ﬁlm ﬂow is a tube, a ﬂat-plate,

or a packed bed.

The experimental and theoretical investigation of liquid ﬁlm ﬂow dates back to Nußelt [1]

who created a model for laminar-waveless liquid ﬁlms, assuming negligible shear stress at the

liquid–vapour interface. Since then, a lot of experimental work has been carried out, utilising

shadow graphs [3–7], interferometry [8–10], ﬂuorescence [11,12], needle [13], and particle

image velocimetry [11,12,14]. For a detailed overview see Groß [15].

x

y

z

y=0

δ(x,z,t)

n

tz

tx

Three-dimensional sketch of a

wavy liquid ﬁlm ﬂowing down a vertical

wall with the ﬁlm thickness δ, and the in-

terfacial unit normal nand unit tangents

txand tz(adopted from [2])

However, with improved computer resources, numerical simulations of liquid ﬁlm ﬂow have

become also quite popular, especially due to fact that they are cheaper than experimental test

series in many cases. Theoretical and numerical studies were carried out on different levels and

complexities. One-dimensional ﬁlm thickness evolution equations were utilised by [2,16–23],

one- and two-dimensional integrated boundary conditions equations by [24–28], and the full two-

or three-dimensional balance equations for mass, momentum, and if applicable energy, by [29–

34]. For a detailed overview see Hantsch and Gross [35]. As can be seen from the dimensionality,

the latter models provide most detailed insight with a maximum of generality, but at a relatively

large computational cost.

In order to reduce the computational cost, it is worth considering substitutes for the full models

which are based on the balance equations. One of those substitutes might be the lattice Boltzmann

method (LBM) utilising the discretised Boltzmann equation on a discrete set of velocity directions,

which is the so-called lattice stencil. Due its kinetic origin, the method is local and has therefore

an intrinsically parallel algorithm. By now this method has proven to be reliable and efﬁcient

and it is widely employed in many applications of single and multi-phase ﬂow. [36–40]However,

literature on applied multi-phase models is rare, as the authors mostly concentrate on drop and

bubble ﬂow. Hence, describing liquid ﬁlm ﬂow phenomena with lattice Boltzmann methods is

still an open ﬁeld of research.

Based on earlier work by the present author [41,42], the objective of this treatise is to develop

a lattice Boltzmann model capable solving thermal two-phase ﬂow as being required for liquid

ﬁlm ﬂow. For each of the main components of the model, ﬂuid ﬂow, heat transfer, and boundary

conditions, a literature review will present the state of the art, followed by a description of the

necessary equations, and being ﬁnalised with some numerical test cases.

In order to facilitate this objective, this treatise is organised as follows: the theoretical back-

ground for ﬂuid ﬂow modelling, LBM, an boundary conditions will be provided in Chap. 2. Chap-

ter 3is dedicated to two-phase ﬂuid ﬂow and Chap. 4to heat transfer modelling. In Chap. 5,

the hydrodynamic and thermal boundary conditions will be provided and tested. The ﬁnal model

and its application for laminar-waveless and laminar-wavy liquid ﬁlm ﬂow will be illustrated in

Chap. 6. In Chap. 7, the treatise will be summarised and in the appendices A–D, there will be

some additional ﬁgures and derivations presented, as well as some details on the implementation,

the parameters for the simulations and some typical physical values.

The theoretical background for this treatise will be given within this chapter.

At ﬁrst, different scales for modelling ﬂuid ﬂow are discussed, and, second, the

balance equations for mass, momentum, and energy will be presented. The third

section is dedicated to the kinetic gas theory, and the fourth one to the lattice

Boltzmann equation method. The last two sections deal with interface physics

and with the scaling of the system by means of non-dimensional numbers.

Fluid ﬂow can be modelled on different scales – micro-, meso-, and macro-scale. Each of these

scales has its own set of differential equations, describing the relevant physical effects.

On the micro-scale it is both possible and necessary to track individual particles. This approach

might be necessary for very dilute gases, but it is presently impossible to simulate liquids and even

gases in standard conditions in realistic geometries. For instance, a cubic centimetre of pure liquid

water with a temperature of 300 Kelvin at a pressure of 105Pascals, has approximately 3.3 ·1022

molecules. The resulting system of equations is much too large for present computer capabilities.

Averaging over all particles on a meso-scopic scale has the main advantage of reducing the

computational effort dramatically with the disadvantage of loosing the information of individual

particle’s momentum and location. Since this information is usually not interesting for engi-

neering applications, the averaging does not sacriﬁce the generality of the resulting Boltzmann

equation related to transient and inhomogeneous ﬂow. The Boltzmann equation, an integro-

differential equation, is difﬁcult to solve due to its complicated collision integral and high dimen-

sionality of phase–space. Although many interesting things can be learned from analysing the

Boltzmann equation, but it is still impractical for real world applications.

In order to simulate ﬂuid ﬂow in geometries which are of practical relevance, the state of

the art is to employ the balancing equations for mass, momentum, and energy on a macroscopic

scale. The computational effort is least compared with micro and meso-scale, but the validity

of the equations is limited to continua, which implies that the characteristic length scale of the

geometry has to be much larger than the mean free path of the particles within the ﬂuid. In most

engineering cases this requirement is fulﬁlled, but in vacua, very small channels, ﬂow situations

with nano-meter sized particles, or shock waves, the balancing equations are no longer valid. In

this work, neither extremely small structures nor strong vacua are of interest. Hence, these cases

are not considered in the following.

As mentioned above, the present standard in simulating ﬂuid ﬂow employs the balance equations

for mass, momentum, and energy, [43–45]

∂tρ+∇·(ρu) = 0, (2.1a)

∂t(ρu) + ∇·(ρuu ) = −∇p+∇·+ρg, (2.1b)

∂t(ρh) + ∇·(ρuh) = −∇·˙

q+Dtp+:∇u, (2.1c)

respectively. Herein, ρ,u,p, and hare the ﬂuid mass density1, velocity, pressure, and speciﬁc

enthalpy, respectively, and , ˙

q, and gare the stress tensor, heat ﬂux vector, and an acceleration

vector, respectively. The operator Dtis the substantial or material derivative and reads, applied

on the pressure, Dtp=∂tp+ (u·∇)p.

The system of Eqs. (2.1) is not closed, until the stress tensor, the heat ﬂux vector, and the speciﬁc

enthalpy are speciﬁed. Following [43,44], the stress tensor of a Newtonian ﬂuid can be written

as:

=Sαβ =η∂αuβ+∂βuα+ζ∂γuγδ

αβ , (2.2)

with Kronecker’s delta δ

αβ (δ

αβ =1 if α=β,δ

αβ =0 if α6=β), the dynamic viscosity η, and, by

employing Stokes’ hypothesis, the volume dilatation ζ=−2/3η[44]. Assuming Fourier’s law of

thermal conduction, the heat ﬂux vector reads [43,46]

˙

q=−k∇T, (2.3)

wherein kis the thermal conductivity and Tis the temperature. The speciﬁc enthalpy is deﬁned

by the Caloric law as:

dh=dh(T,p) = cpdT, (2.4)

with the speciﬁc heat at constant pressure cp, assuming isobaric conditions. To ﬁnally close the

system (2.1), some suitable equation of state has to be employed for linking the thermodynamic

pressure pwith the density ρ. This might be the ideal gas law:

p=ρRT , (2.5)

or the van der Waals equation of state [47]2

p=RT

1

ρ−b−aρ2, (2.6)

with the speciﬁc gas constant R, and the van der Waals coefﬁcients aand b.

1In the remainder of this treatise, the term ‘density’ always refers to the mass density.

2The equation of state shown here differs from that in the reference since it is reformulated and transformed from

molecular to mass-based quantities.

The above balance equations (2.1) can be simpliﬁed by assuming incompressible ﬂow with con-

stant material parameters and negligible viscous heating. By virtue of Eqs. (2.2) and (2.4), the

following equations can be derived:

∇·u=0, (2.7a)

∂tu+ (u·∇)u=−1

ρ∇p+ν∇2u+g, (2.7b)

∂tT+ (u·∇)T=α∇2T, (2.7c)

with the kinematic viscosity ν=η/ρ and the thermal diffusivity α=k/(ρcp). According to

Schwarze [48], Eqs. (2.7) can be solved numerically in a coupled or sequential manner. In the

sequential approach – which is the standard for incompressible ﬂow – the pressure cannot be

calculated directly from the governing equations. Some effort is necessary which can be adduced

by a Poisson equation for the pressure or a predictor–corrector scheme. Both approaches require

time-intensive iterations until the pressure is correctly determined. Further details of these ap-

proaches can be found, e.g., in [48–50].

The kinetic gas theory is a logical predecessor of the lattice Boltzmann method (LBM). However,

a detailed derivation of Boltzmann’s equation and the discussion of related kinetic gas theory

features is left out here, since it is beyond the scope of the present work and since there are some

excellent textbooks illustrating this, e.g., see de Groot and Mazur [46], Chapman and Cowling

[51], Vincenti and Kruger [52]. Instead, a quick journey through kinetic gas theory is provided

here.

The Boltzmann equation can be derived in two different ways. Firstly, by averaging the Liou-

ville equations, as stated above, and secondly, by balancing density distribution functions3at

the boundary of a control volume in phase–space. The phase–space is the product of the vol-

ume ∆V=∆x∆y∆zwith the velocity volume ∆ξ=∆ξx∆ξy∆ξz. Loosely written, this im-

plies that all particles having a location x= ( x±1/2∆x,y±1/2∆y,z±∆z)T, and a velocity

ξ= (ξx±1/2∆ξx,ξy±1/2∆ξy,ξz±1/2∆ξz)T, belong to the same phase–space.

In brief, the Boltzmann equation balances the change of a density distribution function fdue

to advection, forcing, and collision as follows: [51,52]

∂f

∂t+ξ·∂f

∂x+F·∂f

∂ξ=C(f), (2.8)

C(f) = Z

ξ1Z

ACf0f0

1−f f1ξ1−ξdACdξ1. (2.9)

Herein, ξ=u+c,u,c, and Fare the vectors of the absolute velocity, ﬂow velocity, thermal velocity

and a general force, respectively, C(f)is the collision term, ACis the collisional cross–section, and

3Most references in kinetic gas theory use particle distribution functions, e.g., [46,51–53], but in this work, density

distribution functions are employed. The difference between both is that the molecular mass of a particle is already

included in the density distribution function. However, in the majority of the LB models the latter expression is

employed, hence it is also utilised here.

dξ1=dξ1, xdξ1, ydξ1, zis the differential velocity volume. The integration over velocity space is,

unless otherwise stated, within the bounds −∞ and ∞, and the primed density distribution

functions f0and f0

1represent the conditions after a collision of a density distribution function f

with its collision partner f1. Some explanation concerning how to derive the collisional cross–

section can be found in Vincenti and Kruger [52].

Macroscopic quantities can be calculated directly from the density distribution function fby

employing moments, which are generally deﬁned as follows: [53]

Π=Zφfdξwith φ∈1, c,ξ,1/2c2,1/2ξ2,cc,ξξ,1/2c2c, . . . . (2.10)

Herein, cc and ξξ are dyadic products of the respective vectors. Inserting some of the values for

φinto Eq. (2.10), the following special moments can be derived: [53]

ρ=Z1fdξ=Z1feqdξ(2.11a)

ρu=Zξfdξ=Zξfeqdξ(2.11b)

ρuu +pI−=Zξξ fdξ(2.11c)

ρuu +pI=Zξξ feq dξ(2.11d)

3/2ρRT =1/2Zc2fdc=1/2Zc2feqdc(2.11e)

˙

q=1/2Zc2cfdc(2.11f)

Note, that Eqs. (2.11a), (2.11b), and (2.11e) are collision invariant, and hence integrals over

both fand feq lead to the same result. Hänel [53]provides a table with all the physical moments

of ideal gases on page 60 as Tab. 3.1.

The analytical and even numerical solution of the collision integral, Eq. (2.9), is far too compli-

cated for non-trivial applications. A very common simpliﬁcation has been suggested by Bhatnagar

et al. [54].4Assuming small deviations from local equilibrium, the authors proposed that the colli-

sion term is proportional to the difference between the current and the corresponding equilibrium

state,

CSRT =−1

τ(f−feq), (2.12)

with the relaxation time τ. It is possible to demonstrate the relation of this relaxation time to the

kinematic viscosity of the ﬂuid by means of a formal procedure, which is called Chapman–Enskog

4The ﬁrst letters of each author’s surname, Bhatnagar, Gross, and Krook, lead to the acronym BGK which is commonly

used. In this treatise, however, the acronym SRT will be employed due to the single-relaxation time.

(a) (b)

e6

e3

e7e4

e8

e1

e5

e2

e0

Lattices for two dimensions with (a) ﬁve (D2Q5) and (b) nine velocity directions (D2Q9)

expansion.[52,53]It is ν=τRT , where Rand Tare the speciﬁc gas constant and the tempera-

ture, respectively. The equilibrium state – also known as Maxwellian state – is characterised by

Maxwell’s distribution function and reads:

feq =ρ

(2πRT )2/3exp

−ξ−u2

2RT

. (2.13)

Equation (2.13) can be derived from Boltzmann’s equation considering collisions only and by

employing the above-shown moments (2.11a) and (2.11e). [51,53]With a formal procedure,

known as H-theorem, it is possible to prove that this collision term is in accordance with the

second law of thermodynamics (see, e.g., [46,51–53]).

The lattice Boltzmann equation (LBE) can be derived from the Boltzmann equation with an in-

termediate step: the discrete Boltzmann equation. Contrary to the common discretisation, e.g.,

in ﬁnite difference methods, the term ‘discretisation’ implies here that possible velocity directions

are discretised. Whilst a ﬂuid particle in the Boltzmann equation may ﬂy in any direction ξ, the

directions are now limited to eqwith q∈ {0, 1, . . . , Q−1}. A set of velocity directions is called lat-

tice stencil, or just lattice for short.5Common two-dimensional lattices with the usual numbering

of the ﬁve (D2Q5) and nine (D2Q9) velocity directions are demonstrated in Fig. 2.1. The lattices

exhibit some useful symmetry properties, illustrated in Sec. B.1, leading to numerical values for

weighting factors wqand the lattice-depending speed of sound csby employing a Gauss–Hermite

quadrature. [55]Table 2.1 provides the numerical values of c2

s,wq, and eqof D2Q5 and D2Q9

lattices.

With the velocity vectors eqintroduced above, the discrete Boltzmann equation reads: [56]

∂fq

∂t+eq·∂fq

∂x+F·∂fq

∂eq

=Cq. (2.14)

5There exists a typical nomenclature of the lattices: DDQQ, whereby Dis the number of spatial coordinates and Qis

the number of velocity directions.

Properties of D2Q5 and D2Q9 lattices [55]

0 1 2 3 4 5 6 7 8

lattice D2Q5:c2

s=1/3

weighting factor, wq1/31/61/61/61/6

velocity component, eq,x0 1 0 -1 0

velocity component, eq,y0 0 1 0 -1

lattice D2Q9:c2

s=1/3

weighting factor, wq4/91/91/91/91/91/36 1/36 1/36 1/36

velocity component, eq,x0 1 0 -1 0 1 -1 -1 1

velocity component, eq,y0 0 1 0 -1 1 1 -1 -1

Herein, the density distribution functions are fq, corresponding to the directions qwith the ve-

locity vectors eqbelonging to the velocity space Vas eq∈V⊂RQ. The discrete collision term Cq

will be elaborated subsequently.

The SRT collision term for the Boltzmann equation mentioned above exists for the discrete Boltz-

mann equation, too:

Cq, SRT =−1

τfq−feq

q(2.15)

and is also quite common here. The equilibrium distribution function feq

qcan be derived from

the Maxwell distribution by a Taylor expansion [53]:

feq

q=ρwq

1+u·eq

RT +u·eq2

2(RT )2−u2

2RT

(2.16)

with direction-dependent weights wq, neglecting terms higher order terms of uand hence of the

Mach number Ma. The term RT can be related to the speed of sound with c2

s=RT [53].

Beside the SRT collision term, there exists a more generic formulation, the so-called generalised

or multiple-relaxation time collision term (MRT): [56–58]

Cq, MRT =−−1· ·(Π−Πeq). (2.17)

Herein, Πand Πeq are the vectors of the moments to be employed in their general and equilib-

rium form, respectively. A restriction for the moments is that they have to be mutually orthogo-

nal, which can be realised by a Gram–Schmidt orthogonalisation [58]or by employing Hermite

polynomials [56]. Due to the discretisation in velocity space, leading to Qdensity distribution

functions fq,Qdegrees of freedom arise, which have to be ﬁxed by Qmoments. Among those

moments, all physical relevant ones should be utilised, and the remaining ones chosen in such a

way that the system is most stable and accurate. Lallemand and Luo [57], and d’Humières et al.

[58]provide some details on this issue and also the corresponding tensor which is necessary

to map the density distribution functions to the moments by virtue of:

Π=·f=Π0,Π1, . . . , ΠQ−1T, and (2.18)

Πeq =·feq =Πeq

0,Πeq

1, . . . , Πeq

Q−1T. (2.19)

The mapping back to density distribution functions is carried out with the inverse of as provided

by:

f=−1·Π. (2.20)

The tensor is a diagonal tensor and contains the relaxation rates corresponding to the respective

moments:

=diag R0,R1, . . . , RQ−1. (2.21)

Equations for the entries Rqwill be provided subsequently.

Summarising the MRT collisional term, it can be stated that in order to carry out the collision,

the density distribution functions are transformed to moments, which are relaxed to the equilib-

rium moments with their corresponding relaxation rates, and, ﬁnally, the term in parentheses in

Eq. (2.17) is converted back to density distribution functions. With the cost of approximately 20%

more computational time of a SRT implementation, one has the freedom to tune the relaxation of

different moment differently in order to signiﬁcantly enhance the accuracy and stability.

Beside single and multiple-relaxation time schemes, there are other methods, such as two-

relaxation time (TRT)[59], and entropic schemes [60]. The advantages of the TRT scheme over

SRT and MRT are that it can be more stable than SRT, but with less computational cost than

MRT. The idea behind the TRT operator is, to use different relaxation times for symmetric and

anti-symmetric parts of the density distribution functions. The TRT collisional operator reads in

reformulated form6[59]:

Cq, TRT =−ω+fq−feq

q−ω−fq0−feq

q0, (2.22)

where the directions q0are related to qas eq=−eq0. The coefﬁcients ω±are related to the

relaxation time τas:

ω±=1/2R+(τ)±R−(Λ/τ), (2.23)

by employing the so-called ‘magic parameter’ Λ. Ginzburg et al. [61]give a discussion on the

numerical value of Λand discovered that the TRT scheme is most stable for Λ=1/4whilst it

coincides with the SRT scheme in the case of Λ=τ2.7The equations for the relaxation terms R±

will be provided subsequently.

The lattice Boltzmann equation (LBE) can be derived from the discrete Boltzmann equation by in-

tegration along a characteristic line, which is the discrete velocity vector eq.[62,63]Neglecting

the force for this procedure – it will be reinserted afterwards – the discrete Boltzmann equa-

tion (2.14) can be integrated. It should be stressed that the left-hand side can be integrated

analytically and correctly, whereas the right-hand side has to be integrated numerically. The

numerical integration should be carried out with the trapezium rule for obtaining the desired

accuracy. Finally,

fq(x+eq∆t,t+∆t)−fq(x,t) = ∆t

2Cq(x+eq∆t,t+∆t) + Cq(x,t)+O∆t3(2.24)

6Ginzburg [59]presented the operator in a didactically better form (the symmetric and anti-symmetric parts can be

seen directly), which is, if being implemented in such a manner, slower than the one given here.

7It can be easily shown that the second term on the right-hand side of Eq. (2.22) vanishes as soon as Λ=τ2.

can be derived. The result of the integration has the undesired feature to be an implicit algebraic

equation. In order to avoid the implicitness, He et al. [64]proposed to change the variables as

follows:

¯

fq=fq−∆t

2Cq=fq+∆t

2τfq−feq

qwith Eq. (2.15) (2.25)

and reformulated:

fq=¯

fq+∆t

2Cq=

¯

fq+∆t

2τfeq

q

1+∆t

2τ

. (2.26)

Employing the ﬁrst term of Eq. (2.25) and the second term of Eq. (2.26) and applying both on

Eq. (2.24), the explicit LBE can be derived:

¯

fq(x+eq∆t,t+∆t)−¯

fq(x,t) = −∆t

τ+1/2∆t¯

fq−feq

q+O∆t3. (2.27)

Comparing the coefﬁcient of the collision term in Eq. (2.27) with the MRT collision term (2.17),

it can be discovered that the relaxation rates Rqmay be deﬁned as:

Rq=∆t

τq+1/2∆t. (2.28)

The variables τqare the relaxation rates of the corresponding qth moment which have to be

chosen in such a way that the physical moments have the correct values and that the non-physical

moments have a less disturbing inﬂuence. In the case of τq=τ,∀q∈ {0, 1, . . . , Q−1}, the MRT

collisional operator reduces to the SRT one.

The force term which has been neglected above requires some closer investigation: following Luo

[65], the force term can be approximated with a series expansion in eqas:

F·∂fq

∂eq≈ −wqF·eq−u

RT +eq·u

(RT )2eq, (2.29)

satisfying

X

q

F·∂fq

∂eq

=0, (2.30a)

X

q

eqF·∂fq

∂eq

=−F, (2.30b)

X

q

eqeqF·∂fq

∂eq

=−(F u +u F ). (2.30c)

With a different argumentation it can be stated whereby assuming that the gradient of fqis

dominated by the leading order equilibrium distribution function. Hence, the force term can be

approximated with [66,67]:

F·∂fq

∂eq≈F·∂feq

q

∂eq

=−F·eq−u

RT feq

q. (2.31)

It can be easily demonstrated that both expressions for the force term, Eqs. (2.29) and (2.31),

are equivalent when neglecting any terms which are non-linear in uin Eq. (2.31).

Finally, the lattice Boltzmann equation (LBE) reads:

¯

fq(x+eq∆t,t+∆t)−¯

fq(x,t) = Cq+F·eq−u

RT feq

q, (2.32)

with the SRT,TRT and MRT collision terms:

Cq, SRT =−∆t

τ+1/2∆t¯

fq−feq

q, (2.33)

Cq, TRT =−ω+¯

fq−feq

q−ω−¯

fq0−feq

q0, (2.34)

Cq, MRT =−−1· ·¯

Π−Πeq. (2.35)

The hydrodynamic moments of density distribution functions can be obtained analogous to the

moments in section 2.3. However, the change of the variables has to be considered correctly. In

order to distinguish the moments of ¯

ffrom those of f, they are provided now as ¯

Π. Summation,

instead of integration as for the Boltzmann equation, leads to the following moments: [68]

¯

Π0=ρ=X

q

¯

fq(2.36a)

¯

Πα=ρuα=X

q

eq,α¯

fq+∆t/2Fα(2.36b)

Πeq

αβ =pδ

αβ +ρuαuβ=X

q

eq,αeq,βfeq

q(2.36c)

¯

Παβ =pδ

αβ +ρuαuβ−1+∆t

2τη∂αuβ+∂βuα=X

q

eq,αeq,β¯

fq(2.36d)

A comparison of Eqs. (2.36b) and (2.36d) with their corresponding ones Eqs. (2.11b) and (2.11c)

reveals extra terms. The one in Eq. (2.36b) arises because forces are considered here. The one

in Eq. (2.36d) is due to the change of variables from fqto ¯

fq. Summation of Eq. (2.26) leads to

this extra term.

The LBM algorithm is rather simple, as it is a repetitive sequence of streaming, boundary con-

ditions, and collision. Hereby, the left-hand side of Eq. (2.32) is the so-called streaming step,

and right-hand side is the collision step plus the force. The boundary conditions have to be ap-

plied between streaming and collision in order to update all density distribution functions before

collision. Further details on the algorithm can be found in section 6.2.2.

All physical quantities, such as density, velocity, pressure, and diffusivities, in the LBE (2.32) and

in its moments (2.36) are provided in LB units. These units have to be converted into physical ones

in order to simulate correctly. The conversion can be performed, in general, in two ways. Firstly,

by employing conversion factors for time, length, and mass and secondly, by non-dimensional

numbers. The conversion factors are computed with a ratio of the physical to LB value of the

respective quantities. Complex units can be converted by combinations of the three basic factors.

Details on this can be found, e.g., in Lätt [69]and Hantsch [41].

The conversion with non-dimensional numbers, however, is preferred in this treatise, since it

is much more convenient to show results dimensionless. In general, it is not possible to proved a

certain set of non-dimensional numbers to be employed for conversion, since their choice depends

on the physical system. The non-dimensional numbers deﬁning a system can be obtained from the

governing equations and the necessary boundary conditions as will be illustrated in section 2.6.

Multi-phase ﬂow of immiscible ﬂuids is characterised by the presence of an interface separating

the ﬂuids. The investigation of capillary effects dates back to Laplace and Young who created

this ﬁeld of research at the beginning of the 19th century. Laplace provided a mechanical point of

view on generally curved interfaces, known as Laplace’s theorem: [47,70]

∆p=σK=σ1

R1

+1

R2, (2.37)

relating the difference of static pressure on both sides of the interface ∆pto the curvature K

through the interfacial tension σ. Both radii R1and R2have to be measured in mutually orthog-

onal planes. A practical consequence of Eq. (2.37) is that small radii lead to a large pressure

difference which, therefore, drives the ﬂuid to regions with larger radii. This implies that small

drops and bubbles empty themselves into larger ones. [70]

The interfacial tension σwith its standard unit N/m may be understood as a force which is

necessary to increase the interface by unit length. Multiplying a meter to both numerator and

denominator, and employing the relation J =Nm, the interfacial tension unit J/m2indicates

that σcan be also understood to be the energy which is necessary to increase the interface by

unit area. [47,70]From a thermodynamic point of view, the interfacial tension can be deﬁned

through thermodynamic potentials by differentiation with respect to the interface area at other-

wise constant conditions: [47]

σ=∂U

∂A{S,V,n}=const.

=∂F

∂A{T,V,n}=const.

=∂Ω

∂A{T,V,µ}=const.

(2.38)

Herein, the potentials U,F, and Ωare the internal energy, free-energy, and grand potential, re-

spectively, and T,S,n, and µare the temperature, entropy, amount of substance, and chemical

potential, respectively. Depending upon the system under consideration and its constant vari-

ables, a convenient potential has to be selected. Usually the interfacial tension is derived from

the free-energy as proposed by van der Waals. [47]Following his ideas, the interfacial tension σ

is related to the excess free-energy density ψ=F/Awith:

σ=

∞

Z

−∞

Ψ(y)dy, and (2.39)

Ψ=ψ(y)−ψ(ρl,v,T), (2.40)

provided that:

∞

Z

−∞ ρ(y)−ρl,vdy=0 (2.41)

holds. Herein, yis the coordinate perpendicular to the interface. [47]Note that Eq. (2.39) states,

that the interfacial tension – here treated as energy – is the integrated free-energy excess at the

interface. There might be an excess of free-energy at the interface as can be found in Eq. (2.40),

where ψ(y)is the position-dependent free-energy density and ψ(ρl,v,T)are those corresponding

to the temperature Tand the saturation densities ρland ρvin the bulk phases. A mass excess at

the interface, e.g., via adsorption, is neglected here as can be observed in Eq. (2.41).

From a theoretical point of view it is now important to model the excess free-energy density

Ψ. Van der Waals provided (cited after [47]):

Ψ(y) = p−µρ +ψ(ρ,T) + 1/2a∂yρ(y)2, (2.42)

with the pressure p, the chemical potential µ, the density ρ, the temperature T, and the coefﬁcient

awhich is independent of derivatives of ρwith respect to yof any order. Further discussion of

the modelling of Ψcan be found, e.g., in Rowlinson and Widom [47].

At ﬂuid–ﬂuid–solid interfaces exists a three-phase line, where both ﬂuids and the solid surface

meet each other. From a mechanical point of view, three interfacial tensions balance there, leading

to the well-known Young equation: [47,70]

cos θeq =σsv −σsl

σ. (2.43)

Herein, σsv,σsl, and σ=σlv are the interfacial tensions between solid and vapour, solid and

liquid, and liquid and vapour8, respectively. The equilibrium contact angle9θeq indicates whether

a liquid wets a solid surface or not. In cases of θ=0 one says that the liquid totally wets the

surface, whilst 0 < θ < 1/2πand 1/2π < θ < π are called ‘mostly wetting’ and ‘mostly non-

wetting’ regime, respectively. The contact angle θ=1/2πis referred to as neutral wetting.

It can be observed from Eq. (2.43) that beside the interfacial tension σ, the surface energy

against the two ﬂuid phases is important. When σsv > σsl then cos θ > 0, the liquid tends to wet

the surface (i.e., θ < 1/2π) in order to minimise the total free-energy of the system. The opposite

is the case when σsv < σsl, the ﬂuid tends to de-wet the surface (i.e., θ > 1/2π). De Gennes et

al. [70]provide a brief explanation how surface energies σsl and σsv are inﬂuenced by the solid

material and therefore which forces have to be considered. This discussion, however, is beyond

the scope of this work. Here, the equilibrium contact angle θeq is treated as a property of certain

solid–liquid–vapour combination and is intended to be known.

Extensive further information on wetting may be found in the textbooks of Rowlinson and

Widom [47], and de Gennes et al. [70], as well as in the review article of de Gennes [72]. A

special discussion on critical point wetting is provided by Cahn [73], and on diffuse interface

hydrodynamics by Anderson et al. [74]and Jacqmin [75].

8The interfacial tension/energy between vapour and liquid is just symbolised with σfor simplicity.

9Beside the equilibrium or static contact angle, there are also dynamic contact angles deﬁned, which are known as

the so-called advancing and receding contact angles. They occur as soon as the three-phase line moves over an

imperfect substrate. [70]. Jamet et al. [71]provide a brief discussion of the implementation of variable contact

angles in transient systems. In the remainder of this treatise, the term ‘contact angle’ refers always to the equilibrium

one, unless otherwise stated.

There are certain conditions which have to be satisﬁed at a ﬂuid–ﬂuid interface. These are with

respect to mass, momentum, and energy conservation [76,77]:

mass: ˙

m=ρl(ul−ui)·n=ρv(uv−ui)·n(2.44a)

normal momentum: −(pl−pv) + n·(l−v)·n−˙

m(ul−uv)·n=σK(2.44b)

tangential momentum: n×(l−v)·n=∇

iσ(2.44c)

energy: ˙

m∆hlv =kl∇T·n−kv∇T·n,Tl=Tv(2.44d)

Herein, n=∇ϕ/|∇ϕ|,K=∇·n,∇

i= (I−nn)·∇,σ, and uiare the interfacial unit normal

vector, interfacial curvature, interfacial gradient operator, interfacial tension, and the interfacial

velocity, respectively. The variable ˙

mis the mass ﬂux due to phase change in kg/(m2s),∆hlv is

the latent heat of liquid–vapour phase–change, and ϕis the phase index which is deﬁned as:

ϕ=ρ−ρv

ρl−ρv

, (2.45)

with the saturation liquid ρland vapour ρvdensities.

The general boundary conditions at ﬂuid–ﬂuid interfaces, Eqs. (2.44), can be reduced by assuming

negligible phase–change and vapour stress tensor. Therefore, [2]

mass: ui·n=ul·n=uv·n=∂tδ+u·∇δ, (2.46a)

normal momentum: −(pl−pv) + n·l·n=σK, (2.46b)

tangential momentum: n×l·n=∇

iσ, (2.46c)

energy: −kl∇T·n=h(Ti−Tv). (2.46d)

Herein, δis the ﬁlm thickness, his the heat transfer coefﬁcient, and pvand Tvare considered to

be constant in the entire vapour phase.

Fluid–solid boundaries are usually treated as impermeable surfaces which implies that us·ns=0.

The velocity component along the surface is constant – with consequences for the viscous stress

tensor – and might have any value. In this treatise, only cases will be considered where us=

0holds. Assuming perfect hydrodynamic and thermodynamic accommodation it can be stated

that [76–78]:

uv,l =us, (2.47a)

Tv,l =Ts, (2.47b)

−kv,l∇Tv,l ·ns=−ks∇Ts·ns, (2.47c)

n=nscos θ+ntsin θ. (2.47d)

A geometrical representation for Eq. (2.47d), following a description provided by Brackbill et al.

[78], is illustrated in Fig. 2.2. There are three unit normal vectors drawn: nis the normal vector

of the interface, nsthat of the surface, and ntis the projection of ninto the plane of the solid

surface.

Liquid drop located on a solid surface: nsand nare unit normal vectors of the surface and

interface, respectively, and ntis the projection of ninto the plane of the surface.

The representation of ﬂuid ﬂow and heat transfer with non-dimensional numbers features su-

perior comparability to other systems over dimensional representation.