Download Coupling of Free Flow and Flow in Porous Media

Universität Stuttgart - Institut für Wasserbau
Lehrstuhl für Hydromechanik und Hydrosystemmodellierung
Prof. Dr.-Ing. Rainer Helmig
Independent Study
Coupling of Free Flow and Flow in Porous
Media - A Dimensional Analysis
Submitted by
Vinay Kumar
Matrikelnummer 2550493
Stuttgart, June 7, 2011
Examiner: Prof. Dr.-Ing Rainer Helmig
Supervisors: Dipl.-Ing Klaus Mosthaf, Dipl.-Ing Katherina Baber
Contents
1 Introduction
2 Model Concept
2.1 Mathematical model . . . . . . . . . . .
2.2 Equations in the Porous-Medium Region
2.2.1 Mass Balance . . . . . . . . . . .
2.2.2 Energy Balance . . . . . . . . . .
2.2.3 Closure Conditions . . . . . . . .
2.3 Equations of the Free-Flow Region . . .
2.3.1 Mass Balance . . . . . . . . . . .
2.3.2 Momentum Balance . . . . . . . .
2.3.3 Energy Balance . . . . . . . . . .
2.4 Interface Description . . . . . . . . . . .
2.4.1 Mechanical Equilibrium . . . . .
2.4.2 Thermal Equilibrium . . . . . . .
2.4.3 Chemical Equilibrium . . . . . .
1
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3
5
5
5
7
8
8
8
9
9
10
10
11
12
3 Dimensional Analysis
13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Types of Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Model Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Example: Flow of Incompressible Newtonian Fluids . . . . . . . . . . . 15
3.5.1 Velocity and Acceleration of a Fluid Particle . . . . . . . . . . . 16
3.5.2 Equations of Motion of a Fluid . . . . . . . . . . . . . . . . . . 16
3.5.3 Stresses and Deformation . . . . . . . . . . . . . . . . . . . . . 16
3.5.4 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 17
3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6.1 Choice of Variables for Forming Governing Equations . . . . . . 20
3.6.2 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6.3 Importance of Dimensionless Variables Based on Dimensionless Numbers 20
3.6.4 Scales: The Choice of Characteristic Quantities . . . . . . . . . 21
CONTENTS
3.6.5
I
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Dimensionless Equations
4.1 Dimensionless Quantites - Definitions . . . . . . . . . . . .
4.2 Equations of the Porous Medium . . . . . . . . . . . . . .
4.2.1 Dimensionless Darcy Law . . . . . . . . . . . . . .
4.2.2 Dimensionless Transport Equation . . . . . . . . .
4.2.3 Dimensionless Mass Balance . . . . . . . . . . . . .
4.2.4 Dimensionless Energy Balance . . . . . . . . . . . .
4.3 Equations of the Free-Flow Domain . . . . . . . . . . . . .
4.3.1 Dimensionless Transport Equation . . . . . . . . .
4.3.2 Dimensionless Mass Balance Equation . . . . . . .
4.3.3 Dimensionelss Stokes Equation for the Momentumn
4.3.4 Dimensionless Energy Balance . . . . . . . . . . . .
4.4 Interface Conditions . . . . . . . . . . . . . . . . . . . . .
4.4.1 Mechanical Equilibrium - Normal Component . . .
4.4.2 Mechanical Equilibrium - Tangential Component .
4.4.3 Thermal Equilibrium . . . . . . . . . . . . . . . . .
4.4.4 Chemical Equilibrium . . . . . . . . . . . . . . . .
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Balance
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21
22
22
23
23
24
24
25
26
26
26
27
27
28
28
28
29
29
5 Discussion of Characteristic Quantities
31
6 Summary
35
List of Figures
2.1
2.2
Two-domain coupling concept for a single-phase flow system, after [12].
Interface descriptions, after [12] . . . . . . . . . . . . . . . . . . . . . .
II
4
4
Chapter 1
Introduction
Interaction between free-flowing fluids and fluids flowing in porous media systems can
be witnessed in a variety of applications ranging from environmental and technical systems to biological systems. Therefore, the knowledge of flow and transport processes
in such domains is of great importance in understanding the behaviour of such systems.
In the unsaturated zone, the infiltration of rainwater into the soil after a heavy
downpour, evaporation of water from the unsaturated zone influenced by wind and
temperature, and spread of a contaminant spill into the saturated zone can be notable
examples. Such applications, especially evaporation processes and contaminant
spreads, require governing equations that account for phases, components and effects
caused by temperature since there is two-phase compositional flow in the soil and
a single-phase compositional flow in the atmosphere. Such a model concept is
a pre-requisite to understand the movement of phases and components from the
free-flow domain to the porous-medium domain and vice-versa where the influence of
temperature and dissolution processes are prominent [12]
In biological systems, an example would be the trans-vascular exchange between
the blood vessels and the surrounding tissue. The understanding of this exchange
behaviour and the factors influencing it is crucial in understanding the transport
of therapeutic agents and nutrients across the micro-vascular wall and therefore an
important step in understanding the distribution of substances in the human body [12].
The application of the model is at multiple scales which brings about dominance of different sets of forces at each respective scale. Knowing the dominant forces
and their combined influence in all applications gives a clearer picture of the system
behaviour than having solutions only to the governing equations. A knowledge of
the dominant forces and their effects can be well studied with a dimensional analysis
which forms the motivation for this Independent Study.
1
2
In the Independent Study the existing model describing the coupling between
two-phase compositional porous-media flow and single-phase compositional free flow
has been converted into a non-dimensional model. Thereby it will be possible to
know the forces governing the system. It is also possible, from the non-dimensional
implementation, to discuss the choice of the characteristic quantities of the system
and their physical significance.
The aims of the independent study are:
− To perform a dimensional analysis of the considered system of equations to identifiy the dimensionless numbers of interest.
− To discuss the choice of appropriate characteristic quantities with respect to
specific model application.
Chapter 2
Model Concept
To describe the system, there are two possible modelling approaches (see figure 2.2)
present [9, 16].
− The single domain approach,
− The two domain approach.
In the single-domain approach, it is assumed that a single equation holds in both the
free-flow and the porous-medium domain. The equation to be solved for both domains
is the Brinkman equation [2]. The model arises from a superposition of the Darcy’s
law and the Stokes equation. This is approach does not involve coupling conditions.
Because of the single equation used, it makes stresses and velocities continuous in the
entire domain and the transition is denoted by spatial variation of properties such as
permeability and porosity across an equi-dimensional transition zone [12].
In the two-domain approach (figure 2.1), there are different governing equations in the
free-flow and in the porous-medium domains and suitable transfer conditions of mass,
momentum and energy have to be stated at the interface and the continuity of fluxes
normal to the interface have to be satisfied [10]. To give a connection between the horizontal free-flow velocity and the velocity in the porous medium, the Beavers-Joseph
condition [1] is used. This condition was further simplified by the proposal by Saffman
[15] to neglect the velocity in the porous medium. The two-domain approach is chosen
in this study and all explanations of the model refer to the two-domain approach. A
more detailed explanation of the two approaches mentioned and a discussion of the
coupled model is done in [12].
In the current chapter, the model concept and the mathematical model will be explained with the concepts and assumptions made to formulate the equations in each of
the sub domains. In the next chapter, the topic of dimensional analysis and its importance will be explained with the help of an example. The dimensionless equations, the
discussion of dimensionless numbers influencing them and the possible choices of characteristic quantities which determine these dimensionless numbers will be qualitatively
covered in the subsequent separate chapters respectively.
3
4
Figure 2.1: Two-domain coupling concept for a single-phase flow system, after [12].
,1
-$#(.')'/(&0/(*
#1
21
6%$$36*)7
&'%
,-#$%."/$
&
!"#$%&'()*$+#,*
!"#$% &'% ()*'+
0)%)1234$+'15
Figure 2.2: Interface descriptions, after [12]
2.1 Mathematical model
2.1
5
Mathematical model
The mathematical model is derived based on the conceptual model explained in brief
in the previous section 2. Since the model is constructed from a two-domain approach,
the equations describing the conservation of mass, momentum and energy in the two
domains are different and are described separately in the proceeding sections. The
mathematical model can be classified into sets of equations which hold in different
domains [12] as follows:
− Equations in the porous-medium region,
− Equations in the free-flow region,
− Equations at the interface.
2.2
Equations in the Porous-Medium Region
The set of equations describing the processes in the porous medium are formulated
under the following assumptions as in [12]:
− The solid phase (subscript s) is rigid.
− Slow velocities or creeping flow (Re " 1) and hence, the application of the
multiphase Darcy law.
− Dispersion, caused due to differences in flow velocities arising from varying pore
diameters in the porous medium, is neglected due to relatively higher diffusion
in the gas phase and under the assumption of slow flow velocities.
− A two-phase, two-component, compositional flow model is used to describe flow,
transport in and exchange between two phases. Liquid and gas phases are denoted
by subscript l and g respectively and liquid and gas components are denoted by
superscripts w and a respectively.
− Local thermodynamic equilibrium (mechanical, chemical and thermal).
− A non-isothermal model consisting of one energy balance equation and two mass
balance equations (one for total mass and one for the water component in the
gas phase).
− An ideal gas phase according to [8] and [?].
2.2.1
Mass Balance
Two mass-balance equations can be formulated, one for each component in the porous
medium domain. Therefore, for κ ∈ {w, a}
2.2 Equations in the Porous-Medium Region
!
α∈{l,g}
φ
6
!
∂ ($α Xακ Sα )
+ ∇ · Fκ −
qακ = 0,
∂t
(2.1)
α∈{l,g}
where
− $α is the density of phase α,
− Xακ is the mass fraction of component κ in phase α,
− qακ is the source or sink term,
− Sα is the saturation of the phase α,
− φ is the porosity of the porous medium.
The flux term Fκ representing the mass flux of a component is given by
! "
#
κ
Fκ =
$α v α Xακ − Dα,pm
$α ∇Xακ .
(2.2)
α∈{l,g}
The diffusion coefficients in the porous medium for each component κ ∈ {w, a} within
a phase α, Dακ , are equal under the consideration of a binary system where binary
diffusion coefficients of components within a phase are equal [12]. Dακ are determined
by the properties of the soil such as
− Porosity φ,
− Tortuosity τ which can be determined by the method after Millington and Quirk
[11] as:
(φSα )7/3
,
τ=
φ2
and by the properties of the fluid such as
− Binary diffusion coefficient Dα ,
− Saturation Sα .
Then, considering the above, the diffusion coefficient is given from [11] as
Dα,pm = τ φSα Dα .
By addition of the mass balance of the individual components given by (2.1) and
with the assumption of binary diffusion, the total mass balance in the porous medium
domain is given by
!
α∈{l,g}
φ
!
!
∂ ($α Sα )
+∇·
($α v α ) −
qα = 0,
∂t
α∈{l,g}
α∈{l,g}
(2.3)
2.2 Equations in the Porous-Medium Region
7
where v α is the velocity of phase α given by the multiphase Darcy Law:
vα = −
krα
K (∇ pα − $α g) ,
µα
α ∈ {l, g}.
(2.4)
where
− µα is the dynamic viscosity of phase α,
− krα is the relative permeability of the porous medium to phase α, taken here as
a function of the phase saturation Sα given by the Brooks-Corey law [3],
− K is the intrinsic permeability tensor,
− pα is the unknown phase pressure of phase α.
2.2.2
Energy Balance
Under the assumption of local thermal equilibrium justified by the slow flow velocities
in the porous medium, the energy balance equation can be formulated for the storage
of heat in the fluid phase and in the solid phase, heat fluxes due to conduction and
convection, and sources and/or sinks in the porous medium. After [4] the energy
balance is given by
!
α∈{l,g}
φ
∂ ($α uα Sα )
∂ ($s cs T )
+ (1 − φ)
+ ∇ · FT − qT = 0,
∂t
∂t
(2.5)
where
− uα is the specific internal energy,
− cs is the specific heat of the solid phase,
− T is the temperature at local thermal equilibrium,
− FT is the heat flux.
The flux term FT is given by
FT =
!
α∈{l,g}
$α hα v α − λpm ∇ T,
(2.6)
where
− hα is the specific enthalpy of phase α given as a function of phase pressure pα
and temperature T ,
− λpm (Sl ) is the effective heat conductivity which accounts for the combined heat
conduction of the fluid phases and the solid phase.
2.3 Equations of the Free-Flow Region
2.2.3
8
Closure Conditions
The following constitutive relationships help to close the system [12]:
− Saturation : Sl + Sg = 1,
− Capillary Pressure - Saturation relationship pc (Sl ) = pg −pl given by the BrooksCorey law [3],
− Introducing the partial pressure of components in the gas phase pκg , κ ∈ {w, a}
a
and using Dalton’s Law for pg = pg = pw
g + pg ,
a
− Mass and Mole fractions: Xαw + Xαa = xw
α + xα = 1, α ∈ {l, g} and can be
converted to molar masses from the relation:
w
a
a
Xακ = xκα Mκ /(xw
α M + xα M ), α ∈ {l, g}, κ ∈ {w, a}.
2.3
(2.7)
Equations of the Free-Flow Region
Under the assumption of laminar flow, the free-flow region is described mathematically
with the instationary Stokes equation under the assumption of single-phase, compressible gas flow comprising two components, air (a) and water (w).
2.3.1
Mass Balance
Similar to the description of the porous-medium domain, two mass balance equations
can be formulated, one for each component. Therefore, for κ ∈ {w, a}
"
#
∂ $g Xgκ
+ ∇ · Fκ − qgκ = 0,
(2.8)
∂t
where the mass flux is given by
Fκ = $g v g Xgκ − Dgκ $g ∇Xgκ .
(2.9)
The mathematical nomenclature is made in the same way as for the equations in the
porous-medium domain.
The diffusion coefficients of both components are considered to be equal and Fick’s
law is used for diffusion. The equation of state is given by the ideal-gas law as the
density of the gas phase is dependent on temperature, pressure and fluid composition.
Considering the above along with the closure of the sum of mass fractions to unity, the
mass balance of the individual components given in (2.8) can be summed up to get the
mass balance as:
∂$g
+ ∇ · ($g v g ) − qg = 0,
(2.10)
∂t
where qg is the total source or sink term, defined here as the sum of the sources or
sinks of individual components in the domain.
2.3 Equations of the Free-Flow Region
2.3.2
9
Momentum Balance
The momentum balance is formulated by neglecting the inertial term in the NavierStokes equation and considering gravity as the only external force. This can be written
as:
∂ ($g v g )
+ ∇ · Fu − $g g = 0,
(2.11)
∂t
where v g is the velocity of the gas phase. The flux term Fu is given by
Fu = pg I − τ ,
where
− I is the identity tensor,
− τ is the stress tensor defined by Newton’s law of viscosity.
The stress tensor can be defined after [18] as
%
$
2
µg ∇ · v g I
τ g = 2µg Dg −
3
(2.12)
where
− the deformation tensor D =
1
2
"
∇v + ∇v T
#
Substitution of the above into (2.11) the instationary stokes equation is formulated as:
$
%
&
"
#'
∂ ($g v g )
2
T
+ ∇ · pg I − µg ∇v g + ∇v g − ∇
µg ∇ · v g − $g g = 0.
(2.13)
∂t
3
2.3.3
Energy Balance
The energy balance for the free-flow domain is formulated as:
∂($g ug )
+ ∇ · FT − qT = 0,
∂t
(2.14)
where the flux term is given by
FT = $g hg v g − λg ∇T,
(2.15)
The specific enthalpy and internal energy is analogous to the porous medium, given
as a function of pressure and temperature. To close the system, the condition for the
a
mass and mole fractions: Xgw + Xga = xw
g + xg = 1 and the relation for interconversion
between them, mentioned in the porous medium section, is used.
2.4 Interface Description
2.4
10
Interface Description
The interface is physically only a few grains thick [9]. But, for the mathematical
description, the interface is assumed to be a simple interface [6] and unable to store
energy, mass or momentum. The coupling conditions based on the exchange of the
transported properties across the interface in both directions is applied to the model
on the REV scale. Owing to the different equations in the respective domains, strong
coupling conditions at the interface in terms of mechanical, chemical and thermal
equilibrium cannot be perfectly fulfilled. So, a solvable model is constructed based on
assumptions which are physically sensible and yet account for the overall process with
agreeable accuracy [12]
The interface conditions are given by
− Mechanical equilibrium consisting of:
– Continuity of normal stresses, thus resulting in a discontinuity of pressure
of the gas phase,
– Continuity of normal fluxes,
– Beavers-Joseph-Saffman condition for the tangential component of the freeflow velocity,
− Thermal equilibrium consisting of:
– Continuity of temperature and,
– Continuity of normal heat fluxes.
2.4.1
Mechanical Equilibrium
The mechanical forces at the interface are resolved into normal and tangential components. The normal component is given by
(
$
% )
"
#
2
T
σn = (−pg I + τ ) n = −pg I + µg ∇ v g + v g −
µg ∇ · v I n.
3
The continuity of the momentum fluxes represents the mechanical equilibrium of the
gas phases in both domains. The pressure of the gas phase in the porous medium will
balance out the both the pressure of the gas in the free-flow region and its shear stresses.
This balance occurs on the area of contact of the gas phases in the two domains. Under
the assumption of no-slip conditions existing at the solid-fluid interface and a rigid solid
phase, no mechanical equilibrium condition between the gas phase and the solid phase
is needed. Now, an equilibrium condition needs to be derived between the liquid phase
of the porous medium and the gas phase in the free-flow region. This can be formulated
by taking into account the pressure exerted by the liquid in the porous medium along
with the pore scale processes. The pressure of the gas, and its shear stress in the
2.4 Interface Description
11
free-flow region not only has to balance the pressure exerted by the liquid, but also has
to account for balancing the capillary forces existing at the interface of the two fluids.
This balance occurs on the area of contact of the gas phase with the liquid phase [12].
Therefore it is clear that the pressure is discontinuous across this interface and the
resulting discontinuity is defined as capillary pressure [7].
Thus the coupling conditions can be written as separate equations based on concepts
explained in the previous section and then summed up to get the normal component
of the mechanical equilibrium.
n · [Ag ((pg I − τ )n)]ff = [pg Ag ]pm
ff
(2.16a) + (2.16b)
n · [Al ((pg I − τ )n)] = [(pl + pc ) Al ]
* +, ff
n · [((pg I − τ ) n)] = [pg ]
(2.16a)
pm
pg
pm
(2.16b)
(2.16c)
The tangential component of the mechanical equilibrium is given by the BeaversJoseph-Saffman condition [10] after neglecting the small tangential velocity at the
interface in the porous-medium side. This simplification was provided by [15]. This
condition can be formulated by considering that the shear-stress at the interface is
proportional to the slip velocity at the interface after [1]:
√
($
% )ff
ki
vg +
τ n · ti = 0,
αBJ µg
i ∈ {1, . . . , d−1},
(2.17)
where ti , i ∈ {1, . . . , d−1} is the basis of a tangential plane of the interface Γ,
αBJ is the Beavers-Joseph coefficient which has to be valid for a two-phase system and
must be determined experimentally of numerically,
k is the permeability corresponding to the porous-medium component and is given by,
ki = (Kti ) · ti .
The continuity of mass fluxes across the interface completes the mechanical equilibrium
of the system. The fluxes at the interface have to be balanced, but the free-flow region
has a single-phase flow and the porous-medium region has two-phase flow. Therefore
for the liquid phase, it is assumed that there is direct evaporation at the interface and
so the flux in the free-flow domain accounts for the combined fluxes of the two phases
in the porous-medium domain:
[$g v g · n]ff = − [($g v g + $l v l ) · n]pm .
2.4.2
(2.18)
Thermal Equilibrium
Under the assumption of local thermal equilibrium at the interface, two equations can
be formulated describing the thermal equilibirum between the two domains.
Continuity of temperature between the two domains across the interface
[T ]ff = [T ]pm ,
(2.19)
2.4 Interface Description
12
and continuity of heat flux across the interface
[($g hg v g − λg ∇T ) · n]ff = − [($g hg v g + $l (hl + ∆hv )v l − λpm ∇T ) · n]pm .
(2.20)
The continuity of heat flux holds under the same assumption mentioned in the mechanical equlibirum that the water evaporates completely and instantaneously at the
interface. The term ∆hv accounts for the latent heat of vaporization which is required
for phase change of the liquid.
2.4.3
Chemical Equilibrium
The chemical equilibrium is formulated by first considering the chemical potentials
of the water component ψαw , α ∈ {l, g}. On the micro scale, the chemical potentials
can be considered to be in equilibrium based on a pair-wise equilibrium consideration.
However, on the REV scale, the continuity of the normal forces (2.16c) results in a
jump in the pressure of the gas phase and therefore the continuity of chemical potential
cannot be completely fulfilled. The resulting difference in chemical potential is given
by the following equation
ψ ff (pffg ) − ψ pm (ppm
g ) =
.&
'ff /RT
(
$
%)ff (
%)pm
$
w
x
p
p
p
g
g
g
g
RT ln xw
− RT ln xw
= ln & w 'pm
, (2.21)
g
g
p0
p0
xg p g
where p0 is the reference pressure and R is the universal gas constant. This difference
in chemical potential is not known, but, it is assumed that the discontinuity of pressure
has a small minor influence on the chemical equilibrium. Thus the chemical equilibrium
given by the continuity of mole fractions in assumed to be valid.
[xκg ]ff = [xκg ]pm ,
κ ∈ {a, w}.
(2.22)
Furthermore, the continuity of the component fluxes has to be fulfilled. This can be
written as
&"
# 'ff
$g v g Xgκ − Dg $g ∇Xgκ · n =
&"
# 'pm
− $g v g Xgκ − Dg,pm $g ∇Xgκ + $l v l Xlκ − Dl,pm $l ∇Xlκ · n
. (2.23)
Chapter 3
Dimensional Analysis
3.1
Introduction
Dimensional analysis is a technique developed to derive or study governing equations
from the point of view of dimensions of individual parameters responsible for the physical phenomenon under consideration. The physical process can then be described in
terms of equations containing these parameters in systematic arrangement. Such a system is then considered dimensionally homogeneous and the equation holds regardless
of the the system of units used.
There are, in general, two methods followed to derive relations between parameters
influencing a system or to study their proportional influence on the system behaviour.
They are
− Dimensional Analysis,
− Scaling of Equations.
Dimensional analysis is a method of forming governing equations by listing relevant
variables and then finding a relationship using those variables such that the resulting
equation is still dimensionally homogeneous. This means that the quantity quantifying
the force or effect we are trying to describe should have the same dimensions as the
dimensions of the combination of parameters that we have chosen to be influential
to the system. From this, it is clear that the list of the relevant variables should be
comprehensive and independent. This method is applied when there is not enough
knowledge available about the basic laws holding for the system under study. [14]
Describing a real system in terms of a mathematical model developed using the above
mentioned technique is simple but prone to serious errors occurring due to possible
chances of omission of important variables. Therefore, a second method based on governing equations, when available, exists to minimize the error.
For systems having well established concepts and mathematical equations, this procedure can be used to understand the nature of driving forces on different scales of the
13
3.2 Homology
14
model application. With this procedure, even though the equations cannot be solved
analytically, similarity laws can be developed [13].
3.2
Homology
In general, homologous states are states at certain homologous times where at certain
corresponding points on two different bodies, named as homologous points, attributes
such as stresses, deformations and speed are the same. From this definition is can
be gathered that the homologous times for two different bodies, especially of different
dimensions, are usually not the same [17]. Based on the concepts of homology and
similarity, the concept of similitude can be derived considering that if two systems are
similar, then one system can be scaled up or scaled down to behave exactly like the
other system.
3.3
Types of Similarity
For a model to be termed ”similar” to the real system in the complete sense, three
conditions must be satisfied. They are
− Geometric Similarity,
− Kinematic Similarity,
− Dynamic Similarity.
Geometric Similarity is said to have been achieved if the ratios of length dimensions
are the same between the model and the real system [13]. In other words, if the
model can be made to fit exactly to the larger system with sufficient enlargement or
diminishing [17].
Kinematic Similarity is said to have been achieved if the following two conditions are satisfied
− the paths of homologous moving particles have geometric similarity and,
− the ratios of acceleration and velocities at homologous points are constant in
magnitudes and are parallel.
Geometric similarity hold for all types of motion, linear or angular, but it is to
be noted that the motions need not be simultaneous in time.
Dynamic Similarity is said to have been achieved if the model has geometric and
kinematic similarity and in addition, the ratios of forces acting at homologous points
are equal. These forces can be, but are not limited to, the following:
3.4 Model Similitude
15
− Inertia,
− Friction or viscosity,
− Gravity,
− Pressure,
− Elasticity,
− Surface tension.
Apart from these three types of similarities, there exists another type namely Thermal
Similarity. Bodies that have equal or homologous temperatures at homologous times
can be defined as being thermally similar. In general, if two bodies have similar heat
flow pattern, then they can be said to be thermally similar to each other [17].
3.4
Model Similitude
When the relevant variables completely describing the system are listed, they are scaled
and made non-dimensional to get dimensionless equations. Such an equation is independent of the system of units and is relevant to the problem based on the chosen
definitions of the scaling factors. The equation contains only dimensionless variables
and parameter groups as coefficients to these dimensionless equations. The parameter
groups can then be divided by each other to yield dimensionless numbers. These dimensionless numbers show the ratios between individual terms in the governing equation.
For the dimensionless equation to show the effect of these ratios of parameters groups
accurately, the dimensionless variables should be scaled to the order of unity. Then
the magnitude of the term in the equation is represented purely by the ratio of parameter groups which are later defined as standard dimensionless numbers which show the
relative effects of one force to another. This method is called Scaling of Equations.[14]
This method can be used to analyse known equations to get the forces involved in the
system. This will be explained in the following section with an example.
3.5
Example: Flow of Incompressible Newtonian
Fluids
The following example is chosen to demonstrate how dimensional analysis can be of
help to know the various forces which are acting on the system and to establish model
similitude in two systems which are governed by the same differential equations. The
equations of the coupled model are explained in the following chapter.
3.5 Example: Flow of Incompressible Newtonian Fluids
3.5.1
16
Velocity and Acceleration of a Fluid Particle
With the velocity field v of fluid flow known, the acceleration a of particles in the fluid
can be defined by
a=
∂v
∂v
∂v
∂v
+u
+v
+w
∂t
∂x
∂y
∂z
(3.1)
Where u, v and w are velocity components in the x, y and z directions respectively.
3.5.2
Equations of Motion of a Fluid
From Newton’s law of momentum and the definition of acceleration of a fluid particle
(3.1), the equation of motion for a fluid can be written in vector notation as
Dv
,
Dt
where p is the pressure, g is the gravity, τ is the shear stress tensor and
$g − ∇p + ∇ · τ = $
(3.2)
∂ ()
D ()
=
+ (v · ∇) () ,
Dt
∂t
is the material derivative or substantial derivative [13].
3.5.3
Stresses and Deformation
From Newton’s law of viscosity, the rates of deformation are linearly related to the
stress, with the dynamic viscosity µ being the proportionality factor. Therefore, the
normal and shearing stresses can be defined respectively as following after [5]
∂u
∂x
∂v
= 2µ
∂y
τxx = 2µ
(3.3)
τyy
(3.4)
τzz = 2µ
∂w
∂z
(3.5)
∂u ∂v
+
∂y ∂x
%
(3.6)
τxy = τyx = µ
$
τxz = τzx = µ
$
∂w ∂u
+
∂x
∂z
%
(3.7)
τyz = τzy = µ
$
∂v ∂w
+
∂z
∂y
%
(3.8)
3.5 Example: Flow of Incompressible Newtonian Fluids
3.5.4
17
Navier-Stokes Equations
Inserting equations (3.3) through (3.8) in equation (3.2) and with further rearrangement and the use of the continuity equation
∇·v =0
the classical Navier-Stokes equations for an incompressible Newtonian fluid can be
written as
Dv
$g − ∇p + µ∇2 v = $
(3.9)
Dt
This form of the Navier-Stokes equation is chosen only as an example for dimensional
analysis and is not the same equation chosen to describe the coupled model.
This problem is well-posed, as there are four unknowns (u, v, w, p) and four equations.
But, the equations are second order, non-linear partial differential equations and therefore are very complex for analytical methods except for a few specific cases.
Due to the complexity of the Navier-Stokes equations, they are chosen as suitable examples to show how dimensional analysis is applied to a governing equation for analysis
of forces and to establish similarity requirements without solving the equations analytically or numerically. In this case, a two dimensional system is considered and only
one dimension, the y dimension along which gravity g is acting, is shown to illustrate
this method. Similar results can be obtained in all three dimensions. The velocities at
all boundaries and the initial conditions (at time t = 0) are assumed to be known.
For the variables of the equations, namely the velocities u, v, w, the directions (lengths)
x, y, z, the pressure p and the time t, there has to be a reference quantity chosen for
each to make the variables dimensionless. Here, the reference velocity is denoted by vc ,
the reference length along all directions by lc , the reference time by tc and the reference
pressure by pc . These are also called characteristic quantities and have the subscript
3.5 Example: Flow of Incompressible Newtonian Fluids
18
c. The dimensionless variables, denoted by a hat, can now be defined as the following
u
vc
v
v̂ =
vc
w
ŵ =
vc
x
x̂ =
lc
y
ŷ =
lc
z
ẑ =
lc
p
p̂ =
pc
t
t̂ =
tc
ˆ = lc ∇
∇
û =
The characteristic length and time can be chosen independently based on the system
under consideration. The velocity can then be defined as the ratio of the characteristic
length and time scales. Although this is not the only way the characteristic velocity
can be described, it is one of the simplest definitions of the characteristic velocity. The
pressure has to be chosen based on the system under consideration.
Now substituting the above dimensionless variables into the equation for the y direction,
written below in its full component form
$
%
$ 2
%
∂v
∂v
∂v
∂p
∂ v ∂ 2v
$
+u
+v
=−
− $g + µ
+
,
∂t
∂x
∂y
∂y
∂x2 ∂y 2
yields
$
$
vc ∂v̂
vc ∂v̂
vc ∂v̂
+ vc û
+ vc v̂
tc ∂ t̂
lc ∂ x̂
lc ∂ ŷ
)
$vc
t
* +,c -
(
inertia(local)
∂v̂
+
∂ t̂
)
$vc2
l
* +,c -
(
$
%
pc ∂ p̂
vc
=−
− $g + µ 2
lc ∂ ŷ
lc
∂v̂
∂v̂
+ v̂
û
∂ x̂
∂ ŷ
%
$
∂ 2 v̂ ∂ 2 v̂
+
∂ x̂2 ∂ ŷ 2
%
,
(3.10)
=
inertia(convective)
(
)
)( 2
)
(
pc ∂ p̂
∂ v̂ ∂ 2 v̂
µvc
−
. (3.11)
− [$g] +
+
*+,lc ∂ ŷ
lc2
∂ x̂2 ∂ ŷ 2
* +, * +, gravity
pressure
viscosity
The terms enclosed in square parentheses can be taken to represent various forces
acting in the considered system. When these terms are divided by one of the other
3.5 Example: Flow of Incompressible Newtonian Fluids
19
terms in brackets, in this example the convective inertia, a ratio of that force with the
rest is be obtained
)
)( 2
%
(
)
( ) (
)
(
$
∂v̂
pc ∂ p̂
glc
µ
∂ v̂ ∂ 2 v̂
lc ∂v̂
∂v̂
+ v̂
=−
− 2 +
+
(3.12)
+ û
tc vc ∂ t̂
∂ x̂
∂ ŷ
$vc2 ∂ ŷ
vc
$vc lc ∂ x̂2 ∂ ŷ 2
From the above equation standard dimensionless groups can be identified and they
represent ratios of specific forces.
Strouhal N umber (St) =
lc
local inertial f orce
=
t c vc
convective inertialf orce
(3.13)
pc
pressure f orce
=
2
$vc
inertia f orce
(3.14)
vc
inertia f orce
F roude N umber (F r) = √
=
gravity f orce
glc
(3.15)
Euler N umber (Eu) =
Reynolds N umber (Re) =
inertia f orce
$c v c l c
=
µg
viscous f orce
(3.16)
The dimensionless equation (3.12) is not any more helpful in solving the system than
the original Navier-Stokes equation, but the dimensionless equation along with the
dimensionless numbers can establish similarity between two systems and gives the
ratios of different forces which are present in the system. When specific values are
assigned to the parameters comprising these ratios, then dominant forces in the system
can be identified and their effect can be better understood at multiple scales. The
choice of these characteristic quantities varies with the scale of the system and the
physical processes which are occurring in it and so it is necessary to have a clean
understanding of the system while setting the values for these characteristic quantities.
Due to the number of choices available and the complexity involved in choosing the
characteristic quantities, it is explained in more detail in subsequent chapters.
In general if two systems are governed by the Navier-Stokes equations, then their
solutions in terms of the newly introduced dimensionless variables will be the same if
the dimensionless numbers for the two systems are the same. The two systems are
then said to have dynamic similarity.
The system can further be simplified for specific cases, thereby reducing the number
of conditions to be met for similarity. The simplifications can be the following:
− For steady state problems, the Strouhal Number will not play a role.
− The Froude Number is important only for problems involving a free surface or
where gravity is dominant.
− The Euler Number can be reduced to 1 by an appropriate scaling for the reference
pressure. This number is important in problems where cavitation or pressure
differences along the direction is important.
3.6 Remarks
20
− The Reynolds Number gives the ratio of the viscous to the inertial force. At
very low values of the Reynold Number, the flow can be considered as viscous
and therefore inertial effects can be neglected. Conversely at very high values of
Reynolds Number the fluid can be considered inviscid and only inertial effects
can be considered dominant [13].
These dimensionless numbers can also be developed by the Buckingham’s Pi Theorem
but that is not in the scope of this work. Please refer to [13] and [14] for a more
comprehensive description of dimensional analysis.
3.6
3.6.1
Remarks
Choice of Variables for Forming Governing Equations
The choice of variables and number of variables chosen for forming governing equations
by dimensional analysis is very crucial for the success or failure of this method. If there
are lesser variables than required, then the required dimensions cannot be made up by
the variables chosen and the procedure fails. If there are more variables, either the extra
variables are eliminated, or the system becomes insolvable due to lack for equations to
make up for the unknown variables [14].
3.6.2
Dimensions
For dimensional analysis, an important rule to be followed is that the chosen list of
variables should contain, as far as possible, variables with independent dimensions.
In theory, there can be any number of dependent variables which can be expressed as
functions of other independent variables, but choosing only one of them for dimensional
analysis avoids complications [17].
3.6.3
Importance of Dimensionless Variables Based on Dimensionless Numbers
The variables can be included or neglected because of two reasons. One, based on
judgement and two, based on relevance of the variable in the problem. All dimensionless
numbers give the ratio of two forces and based on the magnitude of the dimensionless
number, one of the forces can be said to dominate the system and the other force
can be neglected. The best example is the Reynolds Number which gives the ratio of
the inertia force to the viscous force. Based on the magnitude of the of the Reynolds
Number, one can judge if pure viscous flow or pure potential flow or a combination of
both has to be considered [14].
3.6 Remarks
3.6.4
21
Scales: The Choice of Characteristic Quantities
The choice of scales is very important and is specific to a system. This is influenced to a
great extent by the characteristic scales happening in the system under consideration.
In many systems, there can be different processes happening in different directions,
each requiring a particular scale for the chosen variables. Although a common scale at
the system level will still yield results, there has to be specific scales chosen for each
system when the system behaviour is to be analysed more precisely. [14].
3.6.5
Limits
The choice of scales is also the limitation to dimensional analysis. Due to the nature of
the procedure, involving the choice of suitable scales for variables, there is an inherent
risk that if the scales are not optimal, the newly formulated dimensionless equation
will lack information which is smaller than the chosen scale. Therefore, more emphasis
should be given to choosing variables, their number, dependency and scales for the
particular problem. Another limitation of this procedure is when strong transitions
in flow regimes occur in the system making certain chosen variables irrelevant and
bringing new variables into focus. [14].
Chapter 4
Dimensionless Equations
The procedure of dimensional analysis as explained in the previous chapter is applied
to the equations of the coupled model explained in chapter (2) to identify the dominant
forces and their relative importance in each of the equations.
4.1
Dimensionless Quantites - Definitions
To make the equations non-dimensional, a set of variables has to be chosen and then
scaled to a reference quantity of that variable. The model has two fluid phases, and
hence, two phase pressures and phase velocities. But, there is only one characteristic
pressure and one characteristic phase velocity chosen to obtain the respective dimensionless variables for both the phases. Here the reference quantities are denoted with
the subscript c, indicating they are characteristic quantities of the system. The dimensionless variables are denoted with a hat
l
lc
t
time t̂ =
tc
ˆ = ∇lc
gradient, divergence operator ∇
T
temperature T̂ =
Tc
p
pressure p̂ =
pc
u
internal energy û =
uc
$
density $̂ =
$c
length ˆl =
It is not possible to fix all the characteristic quantities independently of each other,
there are some characteristic quantities which are given as a function of others. The
22
4.2 Equations of the Porous Medium
23
characteristic velocity vc (chosen as a scalar value), time tc and length lc are related to
each other. The characteristic density for the gas phase $c is determined by the ideal
gas law and hence depends on the characteristic phase pressure pc , the characteristic
temperature Tc and the volume from the characteristic length scale lc . The densities of
the solid and liquid phases are assumed to be constant. The characteristic internal energy uc is determined by the characteristic enthalpy which is a function of temperature
[8].
hc (T ) = 1005 (T − 273.15K) ,
and the thermodynamic relation
uc = hc − plc3 .
These variables are then used in the equations of the free-flow and the porous-medium
domains to obtain dimensionless equations and dimensionless numbers. The choice
of the characteristic quantities introduced above can vary the output of the model
significantly since they scale the dimensionless variables. Therefore, the discussion
of possible choices of characteristic quantities is done in the discussion chapter. The
results of the dimensionless analysis are explained in the following sections.
4.2
4.2.1
Equations of the Porous Medium
Dimensionless Darcy Law
The dimensionless form of the Darcy law can be derived from (2.4) as
0
1
ˆ α − Gr
v̂ α = −krα Ca∇p̂
(4.1)
where
− Ca is the capillary number and
− Gr is the gravity number.
Ca =
Kpc
capillary f orce
=
lc v c µα
viscous f orce
(4.2)
K$α g
gravity f orce
=
vc µα
viscous f orce
(4.3)
Gr =
The dimensionless Darcy law is formulated to determine the dimensionless phase velocity as a function of the dimensionless pressure gradient and the capillary and gravity
numbers. Here, it is to be noted that the capillary and gravity numbers are formulated
to contain scalar values of intrinsic permeability and characteristic velocity. Additionally, the characteristic velocity is not only assumed to be independent of directions,
but also assumed to be same for both phases. In a specific flow scenario, under the
4.2 Equations of the Porous Medium
24
assumption of a creeping flow regime (Re " 1), the velocity of phase α is determined
by the capillary and gravity numbers. For fluids with low viscosity and/or for very
high characteristic pressures, the capillary number will be very low in which case the
pressure gradient dominates since the inverse of the capillary number scales the dimensionless pressure gradient. So, at very high characteristic pressures, the effect of
gravity is not so prominent in determining the velocity.
4.2.2
Dimensionless Transport Equation
The dimensionless transport equation is obtained by substituting the expression for
the mass fluxes of component κ from equation (2.2) in equation (2.1)
%%
$
! $ ∂ ($̂α X κ Sα )
! q κ tc
1 ˆ
α
α
ˆ
φ
∇Xα
−
=0
(4.4)
+ ∇ · $̂α v̂ α Xα −
Pe
$cα
∂ t̂
α∈{l,g}
α∈{l,g}
where Pe is the Peclet Number
Pe =
lc v c
advection
=
κ
Dα,pm
dif f usion
(4.5)
and v̂ α is the dimensionless phase velocity (4.1). The Peclet number determines the
ratio of advection to diffusion. A low diffusion coefficient would give high Peclet numbers which makes the effect of the gradient of the mole fraction insignificant compared
to the advective transport of the mole fraction in the system. Larger diffusion would
make the gradient of the mole fraction a significant term in the equation and it will
have to be considered.
4.2.3
Dimensionless Mass Balance
The dimensionless mass balance equation is
!
α∈{l,g}
φ
!
! qα t c
∂ ($̂α Sα )
ˆ ·
=0
+∇
$̂α v̂ α −
$cα
∂ t̂
α∈{l,g}
α∈{l,g}
(4.6)
where v α is the dimensionless phase velocity from Darcy’s law (4.1). The mass balance
equation involves the dimensionless Darcy velocity and therefore, the mass balance
is determined indirectly by the capillary (4.2) and the gravity number (4.3) which
determine the dimensionless phase velocity v α .
4.2 Equations of the Porous Medium
4.2.4
25
Dimensionless Energy Balance
The dimensionless energy balance equation is
0
1
! ∂ ($̂α ûα Sα ) (1 − φ) Tc ∂ $s cs T̂
φ
+
$cα ucα
∂ t̂
∂ t̂
α∈{l,g}


! hα
Tc λpm tc ˆ 
tc
ˆ ·
−
q
$̂α v̂ α −
∇
T̂
= 0 (4.7)
+∇
T
ucα
$cα ucα lc2
$cα ucα
α∈{l,g}
where v α is the dimensionless phase velocity from eq (4.1).
The term
Tc λpm tc
$cα ucα lc2
can be rearranged as
λpm tc
$cα uTcαc lc2
(4.8)
where ucα is the characteristic internal energy of the phase α and Tc is the characteristic
temperature. If uαc is taken as the change in internal energy ∆uαc corresponding to
the change in temperature ∆Tc , then in the limit it is the definition of the specific heat
c, as by definition
)
(
du
dV
∂u
c=
= cv +
.
(4.9)
dT
∂V T dT
For the water phase, the change in volume with the change in temperature can be
considered negligible. The solid phase is assumed rigid while formulating the equations
of the porous medium. For these two cases the fraction of the change in internal energy
with change in temperature can be considered as the specific heat at constant volume cv .
αc
But considering ∆u
as the specific heat at constant volume is not possible for all three
∆T
phases of the system since, for the gas phase, it is evident that this assumption does
not hold, and therefore a further term should be defined which accounts for the volume
expansion of the fluid with change in temperature. The specific heat is also a function
of the absolute temperature and the temperature is a function of time, therefore the
specific heat in the system varies with time. But, as a convenient simplification, the
specific heat is considered to be independent of the absolute temperature by averaging
it over a suitable temperature range applicable to the particular scenario.
If the thermal diffusivity, αT , can be defined as the ratio of the thermal conductivity
and the volumetric heat capacity, then
αT =
λpm
$c c v
and hence, with the above, the Fourier number is defined as
Fo =
conduction
αT t c
=
2
lc
storage
(4.10)
4.3 Equations of the Free-Flow Domain
26
which represents the ratio of thermal conductivity to the heat storage. The equation
(4.8) is of the same form of the Fourier number and gives the same ratio of forces,
namely, heat conduction to heat storage. At high Fourier numbers the thermal conduction dominates over the heat storage and therefore the gradient of temperature
determines the behaviour of the system. At low Fourier numbers, the heat storage is
dominant over conduction, and so the gradient of temperature does not significantly
affect the system as the major part of the heat flow is due to advective processes and
is therefore transported as stored heat with the fluid as given from the advective part
of equation (4.7).
4.3
4.3.1
Equations of the Free-Flow Domain
Dimensionless Transport Equation
The dimensionless transport equation for the free-flow domain is given by
"
#
%
$
∂ $̂Xgκ
qgκ tc
1
κ
κ
ˆ
ˆ
$̂g ∇Xg −
=0
+ ∇ · $̂g Xg v̂ g −
Pe
$c
∂ t̂
(4.11)
The role played by the gradient of the molar fraction of the components in the gas
ˆ gκ is determined by the Peclet number (4.5) which indicates that
phase, the term ∇X
the effect of the gradient is determined by the flow regime. For a very high Peclet
number, the effect of diffusion is much smaller compared to the effect of advection
ˆ gκ does not affect the transport equation as significantly as the
and so the term ∇X
advection term $̂Xgκ v̂ g
4.3.2
Dimensionless Mass Balance Equation
The dimensionless mass balance equation for the free-flow domain is given by
∂ ($̂g )
ˆ · ($̂g v̂ g ) − qg tc = 0
+∇
$c
∂ t̂
(4.12)
The mass balance of the system is not influenced by any forces and therefore there are
no dimensionless numbers which are appearing in the equation owing to the nature of
the equation. The source and sink term is made non-dimensional by dividing by the
ratio of characteristic density and the characteristic time, #tcc , which, by definition, is a
characteristic source or sink in the domain.
4.3 Equations of the Free-Flow Domain
4.3.3
27
Dimensionelss Stokes Equation for the Momentumn
Balance
The dimensionless Stokes equation is given by
(
1)
∂ ($̂g v̂ g )
1 0ˆ
T
ˆ
ˆ
ˆ
∇v̂ g + ∇v̂ g
+ Eu∇ · (p̂g I) − ∇ ·
Re
∂ t̂
%
$
2ˆ
1
1 ˆ
− ∇
∇ · v̂ g − $̂g 2 = 0 (4.13)
3
Re
Fr
The dimensionless form of the instationary Stokes equation has more than one dimensionless number determining the behaviour of momentum transfer in the free-flow
domain. The Euler Number Eu defined by (3.14) gives the ratio of the pressure and
intertial forces. The Reynolds number Re (3.16) gives the ratio of inertial forces to
viscous forces and the Froude number Fr (3.15) gives the ratio of inertia forces to the
gravity forces. Low velocities in the free-flow domain lead to high Euler numbers and
low Froude and Reynolds numbers. This makes the pressure, viscous and gravity terms
in the equation dominant since at slow velocities the force of gravity contributes to the
momentum of the fluid. The gradient of velocity becomes important at low Reynolds
numbers since momentum is lost in overcoming the viscosity to maintain the fluid in
motion. At high Reynolds numbers however, the viscosity of the fluid is not as important as the velocity and therefore at sufficiently high Reynolds number, the flow can
be considered inviscid and the viscosity term may be neglected. This would necessitate
the inclusion of the inertial part of the Navier-Stokes equation but since the free-flow
velocities are assumed to be slow enough to not consider the inertia term, this specific
scenario will not be discussed in depth.
4.3.4
Dimensionless Energy Balance
The dimensionless energy balance equation for the free-flow domain is given by
$
%
∂ ($̂g ûg )
h
T
λ
t
g
c
g
c
ˆ ·
ˆ T̂ − qT tc = 0
+∇
$̂v̂ g −
∇
(4.14)
2
uc
$c ucg lc
$ c uc
∂ t̂
The term
T c λg t c
$c ucg lc2
can again be rearranged to the Fourier number, Fo (4.10), which gives the ratio between
heat conduction and heat storage in the gas phase. The explanation about the effect
of the Fourier number in the free-flow domain is similar to that of the porous medium
domain. At high Fourier numbers the conduction dominates the storage and at low
Fourier numbers, the storage dominates the conduction. Here the specific heat term
which appears in the Fourier number is considered in a way as explained in the energy
balance of the porous medium section (4.2.4). This brings in the volume expansion of
4.4 Interface Conditions
28
the gas with change in temperature. Consequently the specific heat which is associated
with volume expansion similar to equation (4.8).
4.4
4.4.1
Interface Conditions
Mechanical Equilibrium - Normal Component
The normal component of the mechanical equilibrium is given by
(
)
pm
# 2 µg vc
µg vc ˆ "
T
ˆ · v̂ = [p̂]pm pc ,
−p̂g I +
∇ v̂ g + v̂ g −
∇
lc p c
3 lc p c
pffc
where
µg vc
µ g $c v c 2
,
=
lc p c
$c v c l c p c
* +, - * +, 1
Re
(4.15)
(4.16)
1
Eu
can be taken as ratios of two dimensionless numbers namely Euler Number (3.14) and
Reynolds Number (3.16). The Reynolds number and the Euler number together give
the ratio of the viscous forces to the pressure forces. Here, the justification of the
choice of the characteristic pressure is crucial in determining the effect of the term in
the equation. Similarly, the choice of the characteristic pressures in the two domains
also plays a major role in the above equation, because the ratio of the pressures is a
coefficient for the dimensionless pressure term in the porous medium. Additionally,
the pressure and velocity are not completely independent and so the characteristic
pressure cannot take a wide range of values without affecting the choice of the value of
the characteristic velocity significantly. The choice should be made after investigating
specific systems and processes in them which are to be analysed.
4.4.2
Mechanical Equilibrium - Tangential Component
The tangential component is given by the Beavers and Joseph condition (2.17) and its
non-dimensional form can be formulated as given below
√
% )ff
ki ˆ
v̂ g +
∇v̂ g · ti = 0 i ∈ {1, . . . , d−1}
(4.17)
αBJ lc
the above mentioned equation gives only the relation between the tangential velocity
in the free-flow region in terms of the free-flow gas-phase velocity and the permeability
of the porous medium region and the Beavers and Joseph coefficient. Therefore there
are no dimensionless numbers appearing in the equation.
Additionally the flux in the free-flow region should balance the individual phase fluxes
in the porous medium region. This is formulated below
($
($c vc )ff
[($̂g v̂ g ) · n]ff = − [($̂g v̂ g + $̂l v̂ l ) · n]pm
($c vc )pm
(4.18)
4.4 Interface Conditions
29
This again being a balance equation, there are no ratios of forces which arise from the
dimensional analysis. But the ratio of the characteristic fluxes are of much importance
and this particular case will be emphasised in the discussion section.
4.4.3
Thermal Equilibrium
The thermal equilibrium in the entire domain is given by equating the temperatures
of the free-flow and the porous-medium regions and by the continuity of heat fluxes
across the interface.
[Tc ]ff
[T̂ ]ff = [T̂ ]pm ,
(4.19)
pm
[Tc ]
The continuity of the heat fluxes is made non-dimensional and given by
($c hc vc )ff
($c hc vc )pm
($
% )ff
λg T c t c ˆ
$̂g ĥv̂ g −
∇T̂ · n =
$c hc lc2
($
% )pm
0
1
λpm Tc ˆ
− $̂g ĥg v̂ g + $̂l ĥl + ∆ĥv v̂ l −
∇T̂ · n
(4.20)
$c hc lc2
Here, as in the case of the equations of the energy fluxes in the individual domains,
the particular term
λg T c t c
(4.21)
$c hc lc2
can be reformulated as
and by definition
and so at constant pressure
λg t c
$c Thcc lc2
$
∂h
∂T
%
dT +
p
$
∂h
∂T
%
$
=
p
∂h
∂p
%
dp = dh
T
dh
= cp
dT
is the specific heat at constant pressure. With these relations, equation (4.21) takes the
form of the Fourier number, Fo and expresses the relation between the heat conduction
and the heat storage, in the system but now with cp instead of cv .
4.4.4
Chemical Equilibrium
The continuity of mole fractions and the component fluxes represent the coupling condition of for the chemical equilibrium. The continuity of mole fractions is given by
[xκg ]ff = [xκg ]pm ,
κ ∈ {a, w}.
(4.22)
4.4 Interface Conditions
30
The continuity of the component fluxes is obtained for κ ∈ {w, a}
($c vc )ff
($c vc )pm
($
% )ff
1
κ
ˆ
−
$̂g ∇Xg · n =
P eg
($
% )pm
1
1 ˆ κ
κ
κ
κ
ˆ
− $̂g v̂ g Xg −
$̂g ∇Xg + $̂l v̂ l Xl −
$̂∇Xl · n
(4.23)
P eg
P el
$̂g v̂ g Xgκ
This equation contains the Peclet number, Pe and gives the ratio of advection to diffusion. Based on the Peclet number, it is possible to determine if the component fluxes
at the interface are influenced mainly by advection (low Peclet numbers) or mainly
by diffusion (for low Peclet numbers). The Peclet numbers in the porous medium are
determined for the above equation by the characteristic phase velocity in the domain.
From the earlier assumption of binary diffusion and the diffusion coefficients of components within one phase being equal, the Peclet numbers will be the same for the
transport of components.
Chapter 5
Discussion of Characteristic
Quantities
The non-dimensional formulation of the coupling of two-phase compositional porous
media flow and single-phase compositional free flow involves many challenges. Some of
them were briefly introduced in the previous chapter. The form of the dimensionless
equations themselves are not significantly different from the equations with dimensions,
the difference being that the dimensionless variables now have the ratios of forces as
coefficients. These ratios of forces are identified as standard dimensionless numbers.
These dimensionless numbers help to identify the dominating forces and process in
each of the domains. The forces in the dimensionless numbers are defined as functions
of characteristic quantities and parameters which are representative of the processes in
the respective systems. Therefore, choosing a physically meaningful set of characteristic quantities is of great importance in determining the coherence of the dimensionless
model and it’s similarity to the dimensional model and therefore ultimately to the real
system. On the whole, there are a maximum of three possible ways to set characteristic
quantities, once for the free-flow domain, once for the porous-medium domain and once
for the interface.
The characteristic quantities in the free-flow region, the porous-medium region and
the interface individually based on the dominant processes in each of the domains can
be determined. But to ensure that the coupling conditions between the two domains
to be still valid at this interface is something which has to be investigated in greater
detail. One case would be if it was possible to choose one set of quantities for the entire
model. But this option is immediately eliminated since the model contains both free
flow and flow in a porous medium and there are distinctly different flow regimes and
processes which dominate in both regions. The porous medium region is assumed to
have creeping flow (Re " 1) and diffusion in the porous-medium domain might have
a more significant impact than advection. On the other hand, the free-flow region is
assumed to have slow flow velocities, meaning that the advection may contribute more
or less to the transport than diffusion or both phenomena may even be on equal terms
31
32
depending on the Reynolds and the Peclet numbers and their ratios. Therefore it is
highly unlikely that there exists a unique set of characteristic quantities for the entire
model.
Selection of these characteristic quantities requires a clear understanding of the processes. In the entire system, the characteristic quantities which have to be fixed are
− independent variables:
– length,
– time,
– pressure,
– temperature.
− dependent variables:
– density as a function of pressure and temperature,
– specific enthalpy as a function of temperature,
– specific internal energy as a function of specific enthalpy, pressure and volume,
– velocity as a ratio of length and time.
There can be two possible choices for the length scale, one in the longitudinal direction
and one in the transverse direction. Longitudinally, the system length or the length over
which a drop in pressure occurs can be chosen for the length scale. In the transverse
direction, for problems involving no free surfaces, the diameter of the pipe where the
free flow occurs (for example) might be a meaningful choice of the length scale, where
as for problems with free surfaces, the length along a chosen pressure drop or the length
over which a desired velocity gradient exists can be a possible choice. The time scale
can be the characteristic time of a dominant process or any other scale over which a
change in a crucial parameter is observed.
The length and time scales and therefore the velocity, can be chosen based on the scales
of the processes which are occurring in individual domains. Here, it is to be observed
that the velocities in all three directions are not the same in a real system, but choosing
three characteristic lengths and three characteristic times for three directions makes
the system very complex and difficult to analyse. Therefore the topic of scaling in
particular directions is not touched upon in this work and only one single scale is
chosen for all three dimensions.
The characteristic pressure is the characteristic value of the capillary pressure in a twophase system, then the capillary number for the porous medium can be formulated from
this characteristic quantity. In the free-flow region the pressure drop along the system
length, or the pressure drop occurring along any specified length of interest can be
taken. The characteristic pressure is taken as a difference in pressure and not as an
33
absolute value since pressure differences are usually the driving forces for the system
and therefore are of much interest.
The characteristic value for the temperature is specified as a temperature difference,
since a characteristic temperature drop provides more information than an absolute
temperature and, like pressure, the temperature difference is the driving force for heat
transport. The characteristic temperature drop can possibly be
− the temperature drop in the boundary layer, under the assumption that the
system is completely mixed outside the boundary layer,
− a naturally existing temperature difference between the free-flow and the porousmedium domains,
− the temperature drop caused by evaporation effects, linked to the latent heat of
vaporisation.
The characteristic temperature drop cannot be defined in a general manner for all problems, but has to be defined after there is sufficient knowledge of the physical system.
The choice of the temperature can be made considering the physical processes involved
in specific cases.
For the calculation of the characteristic densities, the densities of the solid and liquid
phases do not change and therefore the characteristic values for solid and liquid density
are the same as their actual densities. For the gas phase, the density is derived from the
ideal gas law. Since the characteristic density is determined by the pressure and temperature, and the characteristic values of the pressure and temperature are defined as
differences and not absolute values, a different, absolute value for pressure and temperature need to be chosen to define the characteristic density of the gas phase. This would
mean that a characteristic quantity would be defined as a function of non-characteristic
but representative values of the system. This is done as an engineering-based solution
considering the individual problem which is to be solved. In such a case the ambient
temperature can be taken if the temperature does not show temporal variations. If
the temperature is time-dependent, then an averaged value over a specified time period can be taken. When the example of the unsaturated groundwater zone is taken,
the pressure can be taken to be atmospheric, from the consideration that for shallow
depths the contribution from the hydrostatic pressure to the total gas pressure is not
as significant as the contribution of the atmospheric pressure. For systems with a low
liquid-phase saturation, the gas phase in the porous medium is completely linked to
the gas phase in the free-flow region, in which case it will be under the same pressure
of the free-flow region.
In the free-flow region length and time scales and therefore the velocity can be chosen considering the processes which are occurring. The possible choices for the length
scales can be the system length or the length over which a pressure drop occurs. The
possible time scale can be the time scale of a specific process, based on its dominance
34
in the considered system. The density of the gas phase can be calculated from the pressure and temperature. Considering the same example of the unsaturated groundwater
zone, gas pressure in the free-flow region can be taken to be equal to the atmospheric
pressure and the temperature can be taken, again, as ambient temperature or one of
the temporal averages described for the porous-medium domain. The characteristic
pressure drop can be the the pressure drop over the system length or pressure drop
over the characteristic length scale.
The characteristic internal energy in both domains is given to be a function of the
characteristic enthalpy, the pressure and the volume given by the characteristic length
scale. The characteristic enthalpy is given as a function of the same temperature value
chosen to calculate the density from the ideal gas law.
The interface conditions do not need a separate set of characteristic quantities as they
are basically formulated using the variables of the free-flow and the porous medium
region. In the coupling conditions namely mechanical, thermal and chemical equilibrium conditions, ratios of characteristic quantities of both sub-domains appear in an
equation. Due to the characteristic scales of both sub-domains appearing in the coupling conditions, variables of both the sub-domains need not be scaled with the same
characteristic quantities. This makes the task of choosing characteristic quantities in
individual domains a more meaningful exercise as the processes in separate domains
can be considered exclusive of each other and adequate scales can be derived from
them.
The choice of the characteristic scales and their numerical values depend on various
processes and scales of model application, the choice therefore should be made once a
fairly comprehensive picture of the processes and the model scale is known. The effect
of the characteristic quantities on the output of the model is immense. These quantities have to be chosen and the the output of the model for specific set of values has to
be investigated in more detail so that conclusions can be drawn about the model, its
constituent forces and their dominance. The output of the model for these different
set of chosen characteristic values helps to compare the dimensionless model to the
conventional model in order to justify the most appropriate set of values.
Chapter 6
Summary
In this work the two-domain model for coupling of non-isothermal two-phase twocomponent porous media flow with non-isothermal single-phase two-component free
flow was subjected to a dimensional analysis to identify the role played by constituent
forces. The principal dimensionless numbers appearing in the equations describing flow
and transport of mass and energy in both domains were discussed. The constituent
forces of the dimensionless numbers are formulated from the parameters of the conventional equations which are lumped together during the process of dimensional analysis.
Therefore these forces depend on the values of the characteristic quantities which are
very application specific. Choosing physically-meaningful values to these characteristic
quantities is a challenging task due to the huge number of available choices for each
model application. These choices arise from different processes and the dominance of
different forces in each specific case. The influence of these forces can be better visualised through a numerical analysis which will enable a comparison of driving forces in
different application scenarios, different sets of characteristic quantities, and will also
help to compare the dimensionless numerical model to the conventional model.
35
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