ThesisPDF Available

A Lattice Boltzmann Equation Model for Thermal Liquid Film Flow

Authors:
  • CLOUD&HEAT Technologies GmbH
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: eqVlu/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, . . . , Q1}
˙
qheat flux vector: ˙
q= (˙
qx,˙
qy,˙
qz)TW/m2
relaxation tensor: =diag(R0,R1, . . . , RQ1)
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 VRQ
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=kφφthk2/kφthk2
φvolume average: φ=1/VRφdx
ipartial derivative: i=∂ /∂ iwith i∈ {t,x,y,α,β,γ, . . .}
i j partial derivative: i j =2/(ij)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?=ρlv
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 [37], interferometry [810], 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 [2,1623],
one- and two-dimensional integrated boundary conditions equations by [2428], 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. [3640]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 AD, 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, [4345]
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 =ηαuβ+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=kT, (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
ρbaρ2, (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
ρp+ν2u+g, (2.7b)
tT+ (u·)T=α2T, (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 [4850].
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=xyzwith the velocity volume ξ=ξxξyξz. Loosely written, this im-
plies that all particles having a location x= ( x±1/2x,y±1/2y,z±z)T, and a velocity
ξ= (ξx±1/2ξx,ξy±1/2ξy,ξz±1/2ξz)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
1f f1ξ1ξdACdξ1. (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., [46,5153], 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 φ1, 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(ffeq), (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
ξu2
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., [46,5153]).
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, . . . , Q1}. 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 eqVRQ. 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
τfqfeq
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 )2u2
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): [5658]
Cq, MRT =1· ·(ΠΠeq). (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, . . . , ΠQ1T, and (2.18)
Πeq =·feq =Πeq
0,Πeq
1, . . . , Πeq
Q1T. (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, . . . , RQ1. (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+fqfeq
qωfq0feq
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), (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+eqt,t+t)fq(x,t) = t
2Cq(x+eqt,t+t) + Cq(x,t)+Ot3(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=fqt
2Cq=fq+t
2τfqfeq
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+eqt,t+t)¯
fq(x,t) = t
τ+1/2t¯
fqfeq
q+Ot3. (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+1/2t. (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, . . . , Q1}, 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·equ
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
eqF·feq
q
eq
=F·equ
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+eqt,t+t)¯
fq(x,t) = Cq+F·equ
RT feq
q, (2.32)
with the SRT,TRT and MRT collision terms:
Cq, SRT =t
τ+1/2t¯
fqfeq
q, (2.33)
Cq, TRT =ω+¯
fqfeq
qω¯
fq0feq
q0, (2.34)
Cq, MRT =1· ·¯
ΠΠeq. (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 +ρuαuβ1+t
2τηαuβ+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.
=
A{T,V,µ}=const.
(2.38)
Herein, the potentials U,F, and are 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) = pµρ +ψ(ρ,T) + 1/2ayρ(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(ului)·n=ρv(uvui)·n(2.44a)
normal momentum: (plpv) + n·(lv)·n˙
m(uluv)·n=σK(2.44b)
tangential momentum: n×(lv)·n=
iσ(2.44c)
energy: ˙
mhlv =klT·nkvT·n,Tl=Tv(2.44d)
Herein, n=ϕ/|ϕ|,K=·n,
i= (Inn)·,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=ρρv
ρlρv
, (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: (plpv) + n·l·n=σK, (2.46b)
tangential momentum: n×l·n=
iσ, (2.46c)
energy: klT·n=h(TiTv). (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 [7678]:
uv,l =us, (2.47a)
Tv,l =Ts, (2.47b)
kv,lTv,l ·ns=ksTs·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.