Coupling of Free Flow and Flow in Porous Media - A Dimensional Analysis

Transcription

1 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 Stuttgart, June 7, 2011 Examiner: Prof. Dr.-Ing Rainer Helmig Supervisors: Dipl.-Ing Klaus Mosthaf, Dipl.-Ing Katherina Baber

2 Contents 1 Introduction 1 2 Model Concept Mathematical model Equations in the Porous-Medium Region Mass Balance Energy Balance Closure Conditions Equations of the Free-Flow Region Mass Balance Momentum Balance Energy Balance Interface Description Mechanical Equilibrium Thermal Equilibrium Chemical Equilibrium Dimensional Analysis Introduction Homology Types of Similarity Model Similitude Example: Flow of Incompressible Newtonian Fluids Velocity and Acceleration of a Fluid Particle Equations of Motion of a Fluid Stresses and Deformation Navier-Stokes Equations Remarks Choice of Variables for Forming Governing Equations Dimensions Importance of Dimensionless Variables Based on Dimensionless Numbers Scales: The Choice of Characteristic Quantities

3 CONTENTS I Limits Dimensionless Equations Dimensionless Quantites - Definitions Equations of the Porous Medium Dimensionless Darcy Law Dimensionless Transport Equation Dimensionless Mass Balance Dimensionless Energy Balance Equations of the Free-Flow Domain Dimensionless Transport Equation Dimensionless Mass Balance Equation Dimensionelss Stokes Equation for the Momentumn Balance Dimensionless Energy Balance Interface Conditions Mechanical Equilibrium - Normal Component Mechanical Equilibrium - Tangential Component Thermal Equilibrium Chemical Equilibrium Discussion of Characteristic Quantities 31 6 Summary 35

4 List of Figures 2.1 Two-domain coupling concept for a single-phase flow system, after [12] Interface descriptions, after [12] II

5 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 adimensionalanalysis which forms the motivation for this Independent Study. 1

6 2 In the Independent Study the existing model describing the coupling between two-phase compositional porous-media flow and single-phase compositionalfreeflow 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 quantitiesofthesystem 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.

7 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 couplingconditions. 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 analysisanditsimportance will be explained with the help of an example. The dimensionless equations, the discussion of dimensionless numbers influencing them and the possiblechoicesofcharacteristic quantities which determine these dimensionless numbers will be qualitatively covered in the subsequent separate chapters respectively. 3

8 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]

9 2.1 Mathematical model 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 atwo-domainapproach, the equations describing the conservation of mass, momentum andenergyinthetwo 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) isrigid. 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. Atwo-phase,two-component,compositionalflowmodelisusedtodescribeflow, 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). Anon-isothermalmodelconsistingofoneenergybalanceequation 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 [?] Mass Balance Two mass-balance equations can be formulated, one for each component in the porous medium domain. Therefore, for κ {w, a}

10 2.2 Equations in the Porous-Medium Region 6 where α {l,g} φ (ϱ αx κ αs α ) t + F κ α {l,g} q κ α =0, (2.1) ϱ α is the density of phase α, X κ α is the mass fraction of component κ in phase α, q κ α is the source or sink term, S α is the saturation of the phase α, φ is the porosity of the porous medium. The flux term F κ representing the mass flux of a component is given by F κ = ( ) ϱα v α Xα κ Dα,pm κ ϱ α Xα κ. (2.2) α {l,g} The diffusion coefficients in the porous medium for each component κ {w, a} within aphaseα, D κ α,areequalundertheconsiderationofabinarysystemwherebinary diffusion coefficients of components within a phase are equal [12]. D κ α 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: τ = (φs α) 7/3 φ 2, and by the properties of the fluid such as Binary diffusion coefficient D α, Saturation S α. Then, considering the above, the diffusion coefficient is given from [11] as D α,pm = τφs α D α. 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 (ϱ αs α ) t + α {l,g} (ϱ α v α ) α {l,g} q α =0, (2.3)

11 2.2 Equations in the Porous-Medium Region 7 where v α is the velocity of phase α given by the multiphase Darcy Law: where v α = k rα µ α K ( p α ϱ α g), α {l, g}. (2.4) µ α is the dynamic viscosity of phase α, k rα is the relative permeability of the porous medium to phase α, takenhereas a function of the phase saturation S α given by the Brooks-Corey law [3], K is the intrinsic permeability tensor, p α is the unknown phase pressure of phase α Energy Balance Under the assumption of local thermal equilibrium justified by theslowflowvelocities 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 where α {l,g} φ (ϱ αu α S α ) t u α is the specific internal energy, +(1 φ) (ϱ sc s T ) t c s is the specific heat of the solid phase, Tisthetemperatureatlocalthermalequilibrium, F T is the heat flux. The flux term F T is given by where F T = α {l,g} + F T q T =0, (2.5) ϱ α h α v α λ pm T, (2.6) h α is the specific enthalpy of phase α given as a function of phase pressure p α and temperature T, λ pm (S l )istheeffective heat conductivity which accounts for the combined heat conduction of the fluid phases and the solid phase.

12 2.3 Equations of the Free-Flow Region Closure Conditions The following constitutive relationships help to close the system [12]: Saturation : S l + S g =1, Capillary Pressure - Saturation relationship p c (S l )=p g p l given by the Brooks- Corey law [3], Introducing the partial pressure of components in the gas phase p κ g, κ {w, a} and using Dalton s Law for p g = p g = p w g + p a g, Mass and Mole fractions: X w α + X a α = x w α + x a α = 1, α {l, g} and can be converted to molar masses from the relation: X κ α = x κ αm κ /(x w αm w + x a αm a ), α {l, g}, κ {w, a}. (2.7) 2.3 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) andwater(w) 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} ( ) ϱ g Xg κ + F κ qg κ =0, (2.8) t where the mass flux is given by F κ = ϱ g v g X κ g D κ g ϱ g X κ g. (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 fractionstounity,the mass balance of the individual components given in (2.8) can be summed up to get the mass balance as: ϱ g t + (ϱ gv g ) q g =0, (2.10) where q g is the total source or sink term, defined here as the sum of the sources or sinks of individual components in the domain.

13 2.3 Equations of the Free-Flow Region Momentum Balance The momentum balance is formulated by neglecting the inertial term in the Navier- Stokes equation and considering gravity as the only external force. Thiscanbewritten as: (ϱ g v g ) + F u ϱ g g =0, (2.11) t where v g is the velocity of the gas phase. The flux term F u is given by where I is the identity tensor, F u = p g I τ, τ is the stress tensor defined by Newton s law of viscosity. The stress tensor can be defined after [18] as ( ) 2 τ g =2µ g D g 3 µ g v g I (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: (ϱ g v g ) + [p ( ) ( )] 2 g I µ g vg + v T g t 3 µ g v g ϱ g g =0. (2.13) Energy Balance The energy balance for the free-flow domain is formulated as: where the flux term is given by (ϱ g u g ) t + F T q T =0, (2.14) F T = ϱ g h g v g λ g 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 mass and mole fractions: Xg w + Xg a = x w g + x a g =1andtherelationforinterconversion between them, mentioned in the porous medium section, is used.

14 2.4 Interface Description 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 Mechanical Equilibrium The mechanical forces at the interface are resolved into normal and tangential components. The normal component is given by [ σn =( p g I + τ ) n = p g I + µ g ( ( ) ] ) 2 v g + v T g 3 µ g v I n. 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 porousmediumwill 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 shearstressinthe

15 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 interfaceofthetwofluids. This balance occurs on the area of contact of the gas phase with theliquidphase[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 normalcomponent of the mechanical equilibrium. n [A g ((p g I τ )n)] ff =[p g A g ] pm (2.16a) n [A l ((p g I τ )n)] ff =[(p l + p c ) A }{{} l ] pm (2.16b) p g (2.16a) + (2.16b) n [((p g I τ ) n)] ff =[p g ] pm (2.16c) The tangential component of the mechanical equilibrium is given by the Beavers- Joseph-Saffman condition [10] after neglecting the small tangential velocity at the interface in the porous-medium side. This simplification was providedby[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 v g + τ n t i =0, i {1,...,d 1}, (2.17) α BJ µ g where t i, i {1,...,d 1} is the basis of a tangential plane of the interface Γ, α BJ 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, k i =(Kt i ) t i. 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: Thermal Equilibrium [ϱ g v g n] ff = [(ϱ g v g + ϱ l v l ) n] pm. (2.18) 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)

16 2.4 Interface Description 12 and continuity of heat flux across the interface [(ϱ g h g v g λ g T ) n] ff = [(ϱ g h g v g + ϱ l (h l + h v )v l λ pm 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 h v accounts for the latent heat of vaporization which is required for phase change of the liquid Chemical Equilibrium The chemical equilibrium is formulated by first considering the chemical potentials of the water component ψ w α, 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 (p ff g) ψ pm (p pm g )= [ RT ln ( x w g )] ff [ ( p g RT ln x w g p 0 )] pm [ ] p g x w ff ) RT g p g =ln( [ ] p 0 x w pm, (2.21) g p g where p 0 is the reference pressure and R is the universal gas constant. Thisdifference in chemical potential is not known, but, it is assumed that the discontinuityofpressure 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. [x κ g] ff =[x κ g] pm, κ {a, w}. (2.22) Furthermore, the continuity of the component fluxes has to be fulfilled. This can be written as [( ) ] ϱg v g Xg κ D g ϱ g Xg κ ff n = [( ϱ g v g X κ g D g,pm ϱ g X κ g + ϱ l v l X κ l ) ] D l,pm ϱ l Xl κ pm n. (2.23)

17 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

18 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:

19 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 thegoverningequation. For the dimensionless equation to show the effect of these ratios of parameters groups accurately, the dimensionless variables should be scaled to theorderofunity. 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 numberswhichshowthe 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.

20 3.5 Example: Flow of Incompressible Newtonian Fluids 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 t + u v x + v v y + w v z Where u, v and w are velocity components in the x, y and z directions respectively Equations of Motion of a Fluid (3.1) 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 ϱg p+ τ = ϱ Dv Dt, (3.2) where p is the pressure, g is the gravity, τ is the shear stress tensor and D () Dt = () +(v )(), t is the material derivative or substantial derivative [13] 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] τ xx =2µ u x τ yy =2µ v y τ zz =2µ w z ( u τ xy = τ yx = µ y + v x ( w τ xz = τ zx = µ x + u z ( v τ yz = τ zy = µ z + w y ) ) ) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8)

21 3.5 Example: Flow of Incompressible Newtonian Fluids 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 ϱg p+µ 2 v = ϱ Dv (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) andfourequations. 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 chosenassuitableex- amples 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)areassumedtobeknown. For the variables of the equations, namely the velocities u, v, w, thedirections(lengths) x, y, z, thepressurep and the time t, therehastobeareferencequantitychosenfor each to make the variables dimensionless. Here, the reference velocityisdenotedbyv c, the reference length along all directions by l c,thereferencetimebyt c and the reference pressure by p c. These are also called characteristic quantities and have the subscript

22 3.5 Example: Flow of Incompressible Newtonian Fluids 18 c. Thedimensionlessvariables,denotedbyahat,cannowbedefined as the following û = u v c ˆv = v v c ŵ = w v c ˆx = x l c ŷ = y l c ẑ = z l c ˆp = p p c ˆt = t t c ˆ = l c 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 yields ϱ ϱ ( v t + u v x + v v y ) = p y ϱg + µ ( ) 2 v x + 2 v, 2 y 2 ( vc ˆv t c ˆt + v cû v c ˆv l c ˆx + v cˆv v ) c ˆv = p ( c ˆp l c ŷ l c ŷ ϱg + µv c 2ˆv lc 2 ˆx + 2ˆv ), (3.10) 2 ŷ 2 [ ] ϱvc t c }{{} inertia(local) ˆv ˆt + [ ] ϱv 2 c l c }{{} inertia(convective) ( û ˆv ) ˆv +ˆv = ˆx ŷ [ pc l c ] }{{} pressure ˆp ŷ }{{} [ϱg] + gravity [ ] µvc l 2 c }{{} viscosity [ 2ˆv ˆx + 2ˆv ]. (3.11) 2 ŷ 2 The terms enclosed in square parentheses can be taken to represent various forces acting in the considered system. When these terms are divided byoneoftheother

23 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 [ ] ( lc ˆv t c v c ˆt + û ˆv ) [ ] [ ] [ ][ ˆv pc ˆp +ˆv = ˆx ŷ ϱvc 2 ŷ glc µ 2ˆv + vc 2 ϱv c l c ˆx + 2ˆv ] (3.12) 2 ŷ 2 From the above equation standard dimensionless groups can be identified and they represent ratios of specific forces. Strouhal Number (St)= l c t c v c = Euler Number (Eu)= p c ϱv 2 c Froude Number(Fr)= local inertial force convective inertialforce = v c glc = Reynolds Number (Re) = ϱ cv c l c µ g = pressure force inertia force inertia force gravity force inertia force viscous force (3.13) (3.14) (3.15) (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 systemsandgivesthe 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.

24 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 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] 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 oneofthemfordimensional analysis avoids complications [17] 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 ofthedimensionless number, one of the forces can be said to dominate the system and theotherforce 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 oftheofthereynolds Number, one can judge if pure viscous flow or pure potential flow or a combination of both has to be considered [14].

25 3.6 Remarks 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] 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].

26 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) toidentifythedominant 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 twofluidphases,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, indicatingtheyarecharacteristicquantitiesofthesystem. The dimensionless variables are denoted with a hat length ˆl = l l c time ˆt = t t c gradient, divergence operator ˆ = l c temperature ˆT = T T c pressure ˆp = p p c internal energy û = u u c density ˆϱ = ϱ ϱ c It is not possible to fix all the characteristic quantities independently of each other, there are some characteristic quantities which are given as afunctionofothers. The 22

27 4.2 Equations of the Porous Medium 23 characteristic velocity v c (chosen as a scalar value), time t c and length l c are related to each other. The characteristic density for the gas phase ϱ c is determined by the ideal gas law and hence depends on the characteristic phase pressure p c,thecharacteristic temperature T c and the volume from the characteristic length scale l c.thedensitiesof the solid and liquid phases are assumed to be constant. The characteristic internal energy u c is determined by the characteristic enthalpy which is a function of temperature [8]. h c (T )=1005(T K), and the thermodynamic relation u c = h c pl 3 c. These variables are then used in the equations of the free-flow andtheporous-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 Equations of the Porous Medium Dimensionless Darcy Law The dimensionless form of the Darcy law can be derived from (2.4) as ) ˆv α = k rα (Ca ˆ ˆp α Gr (4.1) where Ca is the capillary number and Gr is the gravity number. Ca = Kp c l c v c µ α = Gr = Kϱ αg v c µ α = capillary force viscous force gravity force viscous force (4.2) (4.3) 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

28 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 verylowinwhichcasethe 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 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) ( φ (ˆϱ ( αxαs κ α ) + ˆ ˆϱ α ˆv α X α 1 )) ˆt Pe ˆ X α qαt κ c =0 (4.4) ϱ cα α {l,g} α {l,g} where Pe is the Peclet Number Pe = l cv c D κ α,pm = advection diffusion (4.5) and ˆv α is the dimensionless phase velocity (4.1). The Peclet number determinesthe 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 insignificantcompared 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 equationanditwill have to be considered Dimensionless Mass Balance The dimensionless mass balance equation is α {l,g} φ (ˆϱ αs α ) ˆt + ˆ α {l,g} ˆϱ αˆv α α {l,g} q α t c ϱ cα =0 (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.

29 4.2 Equations of the Porous Medium Dimensionless Energy Balance The dimensionless energy balance equation is ( ) φ (ˆϱ αû α S α ) (1 φ) T ϱ s c s ˆT c + ˆt ϱ cα u cα ˆt α {l,g} + ˆ h α ˆϱ αˆv α T cλ pm t c ˆ u cα ϱ cα u cα l ˆT t c q c 2 T =0 (4.7) ϱ cα u cα α {l,g} where v α is the dimensionless phase velocity from eq (4.1). The term T c λ pm t c ϱ cα u cα lc 2 can be rearranged as λ pm t c u ϱ cα (4.8) cα T c lc 2 where u cα is the characteristic internal energy of the phase α and T c is the characteristic temperature. If u αc is taken as the change in internal energy u αc corresponding to the change in temperature T c,theninthelimititisthedefinitionofthespecificheat c, asbydefinition c = du [ ] u dt = c v + V T dv dt. (4.9) 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 c v. But considering uαc T as the specific heat at constant volume is not possible for all three 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 heatisalsoafunction of the absolute temperature and the temperature is a function oftime,thereforethe 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, α T,canbedefinedastheratioofthethermalconductivity and the volumetric heat capacity, then α T = λ pm ϱ c c v and hence, with the above, the Fourier number is defined as Fo = α T t c l 2 c = conduction storage (4.10)

30 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 doesnotsignificantly 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 fromtheadvectivepart of equation (4.7). 4.3 Equations of the Free-Flow Domain Dimensionless Transport Equation The dimensionless transport equation for the free-flow domain is given by (ˆϱX ) g κ ( + ˆ ˆϱ g Xg κ ˆv g 1 ) ˆt Peˆϱ ˆ X g g κ qκ g t c =0 (4.11) ϱ c The role played by the gradient of the molar fraction of the components in the gas phase, the term ˆ X g κ is determined by the Peclet number (4.5) which indicates that the effect of the gradient is determined by the flow regime. For averyhighpeclet number, the effect of diffusion is much smaller compared to the effect of advection and so the term ˆ X g κ does not affect the transport equation as significantly as the advection term ˆϱX g κ ˆv g Dimensionless Mass Balance Equation The dimensionless mass balance equation for the free-flow domain is given by (ˆϱ g ) ˆt + ˆ (ˆϱ gˆv g ) q gt c ϱ c =0 (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, ϱc t c,which,bydefinition,isa characteristic source or sink in the domain.

31 4.3 Equations of the Free-Flow Domain Dimensionelss Stokes Equation for the Momentumn Balance The dimensionless Stokes equation is given by (ˆϱ gˆv g ) + Eu ˆ (ˆp g I) ˆ ˆt [ 1 Re ( ) ] ˆ ˆv g + ˆ ˆv T g 2 ( 3 ˆ 1 Re ˆ ˆv g ) 1 ˆϱ g 2 =0 (4.13) 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 higheulernumbersand low Froude and Reynolds numbers. This makes the pressure, viscous and gravity terms in the equation dominant since at slow velocities the force of gravitycontributestothe 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 Dimensionless Energy Balance The dimensionless energy balance equation for the free-flow domain is given by ( (ˆϱ g û g ) hg + ˆ ˆϱˆv g T ) cλ g t c ˆ ˆt u c ϱ c u cg l ˆT q T t c =0 (4.14) c 2 ϱ c u c The term T c λ g t c ϱ c u cg lc 2 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 oftheporousmedium domain. At high Fourier numbers the conduction dominates the storageandatlow 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

32 4.4 Interface Conditions 28 the gas with change in temperature. Consequently the specific heatwhichisassociated with volume expansion similar to equation (4.8). 4.4 Interface Conditions Mechanical Equilibrium - Normal Component The normal component of the mechanical equilibrium is given by [ ˆp g I + µ gv c ˆ (ˆv ] ) g + ˆv T 2 µ g v c g ˆ ˆv =[ˆp] pm p pm, (4.15) l c p c 3 l c p c where µ g v c = µ 2 g ϱ c v c, (4.16) l c p c ϱ c v c l }{{ c p }}{{ c } 1 1 Re 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 sothecharacteristic 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 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 ˆv g + ˆ ˆv g t i =0 i {1,...,d 1} (4.17) α BJ l c the above mentioned equation gives only the relation between thetangentialvelocity 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 individualphasefluxes in the porous medium region. This is formulated below (ϱ c v c ) ff (ϱ c v c ) pm [(ˆϱ gˆv g ) n] ff = [(ˆϱ gˆv g +ˆϱ lˆv l ) n] pm (4.18) c p ff c