Numerical analysis of the influence of turbulence on exchange processes between porous-medium and free flow

Transcription

1 Universität Stuttgart - Institut für Wasser- und Umweltsystemmodellierung Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Prof. Dr.-Ing. Rainer Helmig Diploma thesis Numerical analysis of the influence of turbulence on exchange processes between porous-medium and free flow Submitted by Thomas Fetzer Matrikelnummer Stuttgart, December 5, 2012 Examiners: Supervisors: Prof. Dr.-Ing. Rainer Helmig Apl.-Prof. Dr.-Ing. Holger Class Dipl.-Ing. Klaus Mosthaf Dipl.-Inf. Dipl.-Math. Christoph Grüninger

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3 I I hereby certify that I have prepared this thesis independently, and that only those sources, aids and advisors that are duly noted herein have been used and/or consulted. Stuttgart, December 5 Thomas Fetzer

4 Acknowledgements We want to thank the German research foundation (DFG) for funding the international research training group Non-Linearities and Upscaling in Porous Media - NUPUS and the research unit Multi-Scale Interfaces in Unsaturated Soil - MUSIS. II

5 Abstract Flow and transport processes in domains composed of a porous medium and an adjacent free-flow region appear in a wide range of industrial, medical and environmental applications. One fundamental example is the evaporation from unsaturated soil under the influence of atmospheric processes. In a preliminary work, Stokes flow and isothermal conditions have been applied to simulate these evaporation processes. However, there is a discrepancy between the simulated evaporation rates and rates which have been measured in laboratory experiments. The main goal of this thesis is to extend the given model to turbulent conditions. For this purpose averaged turbulence models, such as Reynolds-averaged Navier-Stokes are reviewed and discussed for their applicability and limitations. Suitable concepts are implemented in the existing DuMu x framework. In a first step the implemented models will be applied to a pipe flow test case. Then the effects on the transport behavior of water vapor across the soil-atmosphere interface will be analyzed and discussed. Finally simulations will be compared to evaporation data measured in wind tunnel experiments in cooperation with the research unit MUSIS and the group of Dani Or at the Institute of terrestrial ecosystems, ETH Zürich. III

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7 Contents 1 Introduction Motivation Structure of this thesis Fundamentals Scales and definitions Fluids Flow characteristics Equilibrium processes Porous medium concept Model concepts Mathematical-physical model Physical model Numerical model Conceptual model Results and discussion Pipe flow test case Coupled results Interpretation Summary and Outlook 57 A Mathematical and physical basics 59 A.1 Mathematical operators A.2 Dimensionless numbers B Eddy viscosity models 61 C Eddy diffusivity and eddy conductivity models 64 V

8 List of Figures 1.1 Free flow transport processes Turbulent motion in a pool of water Evaporation in a porous medium Derivation of the REV concept Parallel plate concept for Newtonian fluids Boundary layer development Velocity profiles in a pipe flow Laminar and turbulent flow characteristics Semilogarithmic velocity profile Vapor pressure REV concept for porosity and saturation Capillary pressure and relative permeability Reynolds decomposition and averaging Hierarchy of turbulence models Prandtl s mixing length Theoretical mixing lengths Surface roughness considerations Box-method discretization Eddy model implementation Eddy model discretization Pipe flow test case experimental setup Pipe flow boundary conditions Pipe flow velocity and eddy viscosity profiles Pipe flow semilogarithmic plot Pipe flow flow development Pipe flow grid convergence Coupled model experiment and simulation setup Coupled model boundary conditions Drying process of the porous medium Evaporation rate for the eddy viscosity models VI

9 LIST OF FIGURES VII 4.11 Eddy viscosity profiles Velocity profiles Viscous sublayer estimation Evaporation rate and velocity distribution for rough surfaces Evaporation rate for the Reynolds analogy Evaporation rate for the eddy diffusivity models Eddy diffusivity profiles Turbulent Peclet numbers Evaporation rate for the eddy conductivity models Eddy conductivity profiles Coupled model grid convergence Effect of wet pore pattern on the evaporation rate

10 List of Tables 4.1 Pipe flow simulations details Coupled model soil, grid and initial parameters Surface roughness parameters Coupled model computation times VIII

11 Nomenclature Latin letters A area [m 2 ] Ã coefficient matrix A + model constant [ ] a an arbitrary variable B + model constant [ ] C (Chezy) constant C CP model constant [ ] C kleb model constant [ ] C wake model constant [ ] c p specific heat capacity [ J /kg K] c s specific heat capacity of solid phase [ J /kg K] D diffusion coefficient [ m2 /s] D t eddy diffusivity [ m2 /s] d diameter [m] d spatial discretization [m] F, f function F kleb model function F wake model function g gravity constant [ m /s 2 ] h specific enthalpy [ J /kg] h height [m] K model constant [ ] K intrinsic permeability [m 2 ] K α hydraulic conductivity [ m /s] k r relative permeability [ ] k s equivalent sand grain roughness [ [m] k st Strickler value m 1/3 /s] L k Kolmogorov micro scale [m] l length [m] l e entrance length [m] M molar mass [ kg /mol] IX

12 Nomenclature X Ma Mach number [ ] m mass [kg] m model constant [ ] n number [ ] n d Deissler constant [ ] n vg Van Genuchten parameter [ ] Pr Prandtl number [ ] p pressure [ kg /m s 2 ] p 0 reference pressure [ kg /m s 2 ] p c capillary pressure [ kg /m s 2 ] q source/ sink term r radius [m] r hy hydraulic radius [m] R universal gas constant [ ] Re Reynolds number [ ] Ri Richardson number [ ] S saturation [ ] S r residual saturation [ ] Sc Schmidt number [ ] T temperature [K] or [ C] t time [s] U perimeter [m] u main flow velocity component [ m /s] u vector of unknowns u, v, w velocity components [ m /s] u outer velocity [ m /s] u τ friction velocity [ m /s] u e specific inner energy [ J /kg] V volume [m 3 ] v velocity vector [ m /s] X mass fraction [ ] x mole fraction [ ] x coordinate vector [m] x, y, z coordinate components [m] y distance normal to the wall [m] y cross wall distance of the switching point [m] Greek letters α phase index α vg Van Genuchten parameter [ 1 /Pa] β model variable [ ]

13 Nomenclature XI γ model variable [ ] difference δ boundary layer thickness [m] δ 99 boundary layer thickness with u = 0.99 u [m] δ vs viscous sublayer thickness [m] ɛ t eddy coefficient [ m2 /s] κ Karman constant [ ] κ component λ heat conductivity [ W /m K] λ t eddy conductivity [ W /m K] λ model constant [ ] µ dynamic viscosity [ kg /m s] µ t dynamic eddy viscosity [ kg /m s] µ inner, µ outer dynamic eddy viscosities in the Baldwin and [ kg /m s] Lomax model ν kinematic viscosity [ m2 /s] ν t kinematic eddy viscosity [ m2 /s] φ porosity [ ] ϱ density [ kg /m 3 ] τ shear stress [ kg /m s 2 ] τ tortuosity [ ] Θ dimensionless number [ ] ω vorticity [ 1 /s] Subscripts and superscripts char characteristic d related to diameter ff freeflow g gas phase l liquid m constant for Meier and Rotta s model max maximum value min minimum value n non-wetting pm porous medium T temperature t turbulent w wetting w water x, y, z used coordinate α phase index

14 Nomenclature XII κ component index value at wall value at boundary layer edge + dimensionless quantity Others a ã a ā a a T vector tensor temporal average mean value of quantity quantity fluctuation transposed tensor/ vector Laplace operator nabla operator Abbreviations DNS DuMu x DUNE ETH FE IP MUSIS PDE RANS REV RSM SCV direct numerical simulation DUNE for Multi -phase, -physics, -component, -scale Flow in Porous Media Distributed and Unified Numerics Environment Eidgenössische Technische Hochschule finite elements integration point research unit Multi-Scale Interfaces in Unsaturated Soil partial differential equation Reynolds-averaged Navier-Stokes representative elementary volume Reynolds stress models sub control volume

15 Chapter 1 Introduction When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first. Werner Heisenberg 1.1 Motivation The phase transition of water and therefore the transfer of latent heat is a phenomenon, which is observed and used in different situations. A prominent example for the evaporation process is the human perspiration, which is essential for survival. Within this process a water-rich secrete is produced by sweat glands, then on the human skin the water evaporates into the air. This evaporation process absorbs energy and the temperature of the human body decreases. The cooling effect, due to the transfer process between a porous medium and a free flow region was already used in the ancient world to cool meals and drinks. Stored in porous containers made of clay, the water slowly diffuses from the inner to the container surface. There it evaporates and cools the food stored inside. At the surface, the conditions of the free flowing air are relevant, because evaporation only happens, if the air is not fully water saturated. For fully water saturated air, the water on the surface cannot evaporate and no cooling is achieved. Regarding the processes of transport and mixing in free flow, two different characteristic flow states are discerned. First the well-ordered and little interacting laminar flow, which occurs mainly at low velocities or in bounded flows. In contrast to that, turbulent flow is more chaotic and better mixed, it is typical for high velocities or gas flow. Depending on the flow behavior, water or substances in the air are either distributed throughout the flow field or stay in position, see fig Going back to the problems with connected porous-medium and free flow; not all of them are dealing with turbulent free flow. Transport of agents through blood vessels of a human body is normally laminar. The interaction of free flow in the vessels and 1

16 1.1 Motivation 2 laminar velocity profile advection diffusion phase transition turbulent mean velocity profile turbulent mixing latent heat Figure 1.1: Transport of particles in a free flow. (a) under laminar, isothermal conditions. (b) under turbulent, non-isothermal conditions. porous-medium flow through the cell tissue also plays a role for transporting agents or other substances [1]. A more technical example are fuel cells. Fuel cells are composed of two free flow channels separated by a semipermeable membrane. Oxygen and hydrogen have to diffuse from the gas channel through the membrane into the reaction layer, there they react to water. Afterwards the water has to be transported out again [39]. These two examples are small scale and therefore rather laminar problems. Remembering the evaporation and its interaction with the atmosphere, this is a problem influenced by turbulence. Turbulence itself is fascinating, but also difficult to comprehend. Leonardo da Vinci tried to visualize turbulent flow in a pool of water (see fig. 1.2), but also described it in words: Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which is caused by the weight if the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to the random and reverse motion. [15]. One example of turbulent flow and transport combined with a porous media concept is flow and heat transfer around buildings. The wall of a building is regarded to be a porous medium and due to the thermal gradient, heat is transferred to the outside and then transported away by wind [10]. Ultimately the focus is on the evaporation process from soil to atmosphere. This is relevant, as the water content in the ground is important for plants to grow and survive. Nevertheless evaporation has an effect on the terrestrial surface as well as on various climate processes. The air flow is one of the key factors on the evaporation rate and a laminar description of the adjacent free flowing gas phase is not sufficient to reproduce

17 1.2 Structure of this thesis 3 Figure 1.2: Turbulent motion in a pool of water drawn by da Vinci in those mechanisms. From a geoscientific point of view it could be argued, that the vegetation or leaf interface is most important. But this is not true. On bare land, flow and transport processes are dominated by fundamental physics, still driven by different factors, see fig. 1.3.All in all evaporation and transpiration (water is transferred from the leaves to the air) are contributing equally to total rate [19]. In addition to that, radiation, surface roughness, temperature and pressure conditions are of interest. In this thesis the existing model for coupling porous-medium and free flow processes presented in [28] is extended to turbulent conditions. It is motivated by gaining a better prediction of the transfer fluxes between the two domains. For this purpose different algebraic turbulence models for flow and transport will be presented and evaluated. Simulation results will be compared to laboratory experiments done by the research unit named MUSIS, Multi-Scale Interfaces in Unsaturated Soil [29], and the group of Dani Or at the Institute of terrestrial ecosystems, ETH Zürich. Their goal is to create methods for describing water-gas-interface processes and to apply them on hydrological relevant scales. 1.2 Structure of this thesis In chapter 2 some basic explanations and definitions pertaining scales, fluids and flow are be given. In the subsequent chapter 3, the focus is on the mathematical equations, the physical assumptions, the numerical details and the coupling conditions. Further a couple of algebraic turbulence models are introduced. Chapter 4 presents the results of the performed simulations. This comprises results for a pipe flow test case as well as results for the coupled model and experimental results. In the last chapter, chapter 5, the work is briefly summarized and concluded with an outlook. In the appendices, p. 59 ff, details to mathematical operators and dimensionless numbers are given. The used algebraic turbulence models are also summarized there.

18 1.2 Structure of this thesis 4 time abstraction model reality processes direct vaporization patch of wet pores, flow through pores diffusion Figure 1.3: Schematic sketch of the relevant processes for water transfer to the free flowing gas phase. Differences between model and reality at different points in time.

19 Chapter 2 Fundamentals I have yet to see any problem, however complicated, which, when you looked at it the right way, did not become still more complicated. Poul Anderson This chapter gives a fundamental knowledge about different flow characteristics, especially free flow and flow in porous media. Basic concepts pertaining fluids, equilibrium processes and the porous media are explained. 2.1 Scales and definitions In general multiphase flow processes require the presence of fluid phases and are mainly driven by forces like pressure or temperature gradients as well as gravity. When investigating a multicomponent system, interactions like dissolution, degassing, diffusion processes or chemical reaction may be of interest. Almost every solid or any material with a structure containing void space can be regarded as a porous medium. The lowest and most fundamental level on which physical processes can be described, is the molecular scale. This means that properties, behavior and interaction of one molecule are specified. A molecular consideration is only possible for small scale applications. Considering a larger quantity of molecules, it is possible to define fluid properties like density, viscosity, etc. In this case the fluid is a so called continuum, but still molecular processes have to be accounted. With respect to porous media, on the microscale the resolution is high enough to distinguish different pores. Fluid flow on the macroscale can be described by the Navier-Stokes equations. A next averaging step leads to the macroscale. On this scale individual pores or phase interfaces are no longer noticeable. Again, processes on the smaller scales influencing the behavior on the macroscale and have to be accounted. The concept of a representative elementary volume is derived in [4], therefore the macroscale is often denoted as REV scale. New parameters have to be considered and an appropriate length scale has 5

20 2.2 Fluids 6 φ consideration as porous medium not possible possible heterogeneous φ homogeneous micro scale macroscale field scale volume fracture Figure 2.1: Derivation of the REV concept, according to [4]. Here, the averaging of the pore volume φ. to be applied, so that the quantities describing the system will neither fluctuating too much, nor will heterogeneities be smeared out, see also fig Flow on this scale can be described by Darcy s law. 2.2 Fluids Following the definitions mentioned above, basic considerations pertaining a fluid are given Phases In multiphase flow, the different phases α are considered to have sharp interface, resulting in discontinuities of fluid properties. Furthermore two phases are considered to be immiscible. Different phases occur because of different states; like solid, fluid or gas phases. Different polarity, like for oil and water or changes in pressure or temperature can also be a reason Components In a multicomponent system, each phase consists of at least one component κ. Components are considered, if the modeler wants to simulate exchange processes between different phases. In a multiphase systems, components are neglected and therefore not

21 2.2 Fluids 7 balanced. For the evaporation process, the two phases fluid and gas, with their two components water and air are considered. Air dissolves in water and the other way around water evaporates into the gas phase. In this case, air is a pseudo component, which in reality consists of various components, but here a constant composition is assumed for simplicity Mass and mole fraction Being aware of the fact, that one phase consists of several components, it is now possible to express the ratio of mass X in one phase by Xα κ m κ n α = X m 1 α + m 2 α + + m n α κ = 1, (2.1) α and respectively the corresponding mole fraction x x κ α = m κ α/m κ m 1 α/m 1 + m2 α/m 2 + mn α/m n κ=1 n x κ α = 1. (2.2) In these equations m denotes the mass, whereas M is the molar mass of a substance. The relation between those two is given by κ=1 Xα κ = x κ M κ α, (2.3) M α which is a non-linear equation, because M α is depending on the composition of the phase Ideal gas law The ideal gas law is a fundamental thermodynamic law, which relates basic fluid and system properties. This equation is used to calculate the density of the gas phase. R is the universal gas constant Viscosity p g = ϱ g R M g T g (2.4) Some type of fluids can be considered as a system of parallel moving plates, see fig The velocity difference is due to internal friction between these plates. Plate B is moving with a constant velocity, whereas A is a fixed wall. The dynamic viscosity µ defines the ratio of the shear stress and the velocity gradient. τ = µ dv dx (2.5)

22 2.3 Flow characteristics 8 u B y dy A du Figure 2.2: Parallel plate concept for a Newtonian fluid, after [1]. This is called Newton s law of viscosity. Fluids showing this linear behavior between shear stress and velocity gradient are called Newtonian fluids. Air and water are Newtonian fluids. The viscosity is a fluid property and depends on pressure and temperature. The link between the dynamic viscosity µ and the kinematic viscosity ν is given by the density ϱ. ν = µ (2.6) ϱ 2.3 Flow characteristics This section provides basic principles of fluid flow, mainly for free flow processes. The concept of dimensionless numbers is also helpful in porous-medium flow Dimensionless numbers Dimensionless numbers help to identify important processes, dominating forces or relevant scales. They are used to compare different flow configuration based on the same phenomena. Further they help to make or judge assumptions concerning the physical model. Nevertheless, these numbers should be taken as a qualitative information. The Reynolds number for example relates inertia and viscous forces. The magnitude gives a hint about the turbulence of a flow. The larger the Reynolds number is, the more turbulent or the more likely turbulent is the flow. Free gas flows are almost always turbulent. Flow with Re < 1 is called creeping flow and typical for porous-medium applications. Re = u l ν = u l ϱ µ = inertia forces viscous forces (2.7) The Schmidt number balances the diffusion of a component in a phase with the viscosity. Sc = ν = µ (2.8) Dα κ ϱ Dα κ The Prandtl number does the same for heat conduction and viscosity. Pr = ν a = µ c p λ (2.9)

23 2.3 Flow characteristics 9 outer layer laminar laminar boundary layer turbulent boundary layer turbulent log layer viscous sublayer Figure 2.3: Development of the boundary layer for flow along a flat plate. In the appendix A.2 different dimensionless numbers, used in this thesis, can be found Boundary layers The boundary layer theory was developed by Ludwig Prandtl in 1904 [33]. Within the laminar boundary layer viscous and inertia forces are in the same range of magnitude [34]. The laminar boundary layer thickness is proportional to the square root of the flow distance. Additionally, the boundary layer is normally a small region near the wall. Outside this boundary layer, the flow is considered to be frictionless and the velocity to be constant u. After a certain time or distance, the flow might get turbulent. The total turbulent boundary layer thickness is given by the distance y [37]. u(y) = 0.99 u y = δ 99. (2.10) An additional effect is, that the boundary layer splits up into two parts. The turbulent boundary layer and the viscous sublayer. In the viscous sublayer the flow is still laminar. This should kept in mind, as the viscous dominated region influences flow and transport near the wall with the characteristic of a laminar flow, even if the main flow is turbulent Laminar flow Laminar flow can be regarded as a system of well ordered layers with little interaction, see fig It is characterized by a high momentum diffusion, but a low momentum advection and diffusion of the transported components. This implicates, that viscous forces are dominating over inertia forces. For pipe flow, the velocity profile has a parabolic shape (see fig. 2.4) and is considered to be laminar in a range of Re d < Mathematically it is described by the whole Navier-Stokes-equation or the simplified Stokes-equation, in which the inertia term is neglected.

24 2.3 Flow characteristics 10 Figure 2.4: Steady state velocity profiles for laminar (left) and turbulent (right) flow in a pipe. The turbulent profile was calculated according to the power law, eq y a y u y b u/u ū/ū (a) Dye experiment after Reynolds (1883), done by W. Dubs (1939), taken from [37]. [a] is a laminar water flow at a Re d = 1150, [b] at turbulent conditions Re d = (b) Velocity fluctuations of a turbulent flow at Re x profiles were taken at the same position, superposed and finally a mean velocity profile was generated, taken from [8]. Figure 2.5: Laminar and turbulent flow characteristics Turbulent flow In contrast to laminar flow, turbulent flow is characterized by different factors. Turbulent flow is always irregular and chaotic. It is described by the Navier-Stokes equations, but only on a theoretical level. The second point is the non-zero vorticity and vortex stretching, which is why turbulence is always rotational and three-dimensional. These vortices or eddies are spreading over a wide-ranging length scale. The largest eddies are on the order of the domain size, the smallest eddies are determined by viscosity and measure even less than millimeters. In this process, energy is transferred from larger eddies to smaller ones. Ultimately the eddies are transformed into heat, which is called dissipation. The most conspicuous phenomenon of turbulent motion is the enhanced mixing of the transported substances. Experiments to visualize the flow structure were done by Osborne Reynolds already in 1883, see fig Until now the transition from laminar to turbulent flow is not fully understood. Turbulence starts with little disturbances and local turbulence, which gradually become larger and dominate the fluid flow more than viscosity. Smaller eddies arise at the wall and move into the center of the flow field. This leads to the characteristic flow with

25 2.3 Flow characteristics viscous sublayer buffer layer log layer outer layer u + = ū/uτ, [ ] u + = y + u + = 2.5 ln (y + ) u + = 1 κ ln (y+ ) + C Laufer 1954, Re max,d = y + = u τ, y/ν [ ] Figure 2.6: Dimensionless, semilogarithmic velocity profile for a turbulent boundary layer, after [8, 41]. a laminar background flow, overlain by turbulent motion, back flow and vertical flow, which da Vinci tried to visualize in fig The terms fully developed turbulence or fully developed flow denote, that the mean velocity profile is uniform from this point on. For pipe flow the whole domain is under influence of the wall friction, resulting in an intersection of the two boundary layers [40]. As mentioned above, in a turbulent flow field different regions are discerned, fig For turbulent flow especially the viscous sublayer and the log layer. The viscous forces are dominating in the viscous sublayer, whereas the inertia forces dominate the behavior in the log layer. An analytical solution for the flow in these parts is derived in [34]. The following equation is obtained for the viscous sublayer. τ xy, = µ [ ū y ] = u 2 τ, ϱ ū u τ, = u τ, y ν (2.11) The log layer is described by this equation. ū y = 1 u τ, ū = 1 ( ) κ y u τ, κ ln uτ, y + C (2.12) With u τ, = τ xy, /ϱ as the wall friction velocity. The Karman constant κ = 0.4 and the integration constant C = 5.5 are experimental results for smooth pipes from [30]. ν

26 2.4 Equilibrium processes 12 vapor pressure [P a] Henry s law real behavior Raoult s law 0 mole fraction x 1 l [ ] 1 Figure 2.7: Vapor pressure above a liquid phase with two components. The real behavior in contrast to Henry s and Raoult s law are shown. The power law gives an analytical approximation to the turbulent velocity profile [5]. 2.4 Equilibrium processes ( ū = 1 r ) 1/7 (2.13) ū max r max Local thermodynamic equilibrium is assumed in this work, which means all phases in a REV have the same temperature pressure is in equilibrium, but can jump due to capillary forces components have the same chemical potential It is assumed, that the following postulations are valid: Fick s law for diffusion, Dalton s law for the sum of partial pressures, Raoult s law for the vapor pressure above an ideal solution and Henry s law for dissolution of gases in liquids, see also fig They are not explained in detail here and can be found in most thermodynamic textbooks or in [17]. 2.5 Porous medium concept The last section presents a concept for modeling the porous medium itself and the flow through it. These concepts are necessary to account for processes occurring below the considered scale, see section 2.1. This procedure is called upscaling.

27 2.5 Porous medium concept 13 void solid micro scale averaging REV scale gas fluid solid Figure 2.8: The transition and averaging from microscale to REV scale pertaining porosity and saturation, after [17] Porosity The porosity φ is the ratio of the void or non-solid volume to the total volume of a REV. φ = V pores V total (2.14) This is the portion of the soil, which can be occupied by fluids and is contributing to the flow. The porosity is assumed to be constant, although it changes possibly, due to external forces, reactions, etc. In fig. 2.8 the porosity equates the ratio of the white area to the total area Saturation On the macroscale the exact distribution of different phases is no longer known. The saturation S α of a REV with a certain phase is defined in a similar manner as the porosity, see fig S α = V α (2.15) V pores For multiphase conditions, the saturations of all phases sum up to one. This means, that there is no empty space left in the pore volume, the following closure relationship is obtained. n S α = 1 (2.16) α=1

28 2.5 Porous medium concept Darcy s law and Permeability One important experiment concerning the porous media concept, was performed by Henry Darcy in Darcy applied a difference in piezometric height, meaning a pressure difference and measured the flow through soil samples. The result is a proportionality factor between the pressure difference and the seepage velocity, depending on soil and fluid. v α = k rα K ( p α ϱ α g) (2.17) µ α This fundamental macroscale equation is called Darcy s law for multiphase flow. The hydraulic conductivity K α consists of the intrinsic permeability K (property of the porous medium), the relative permeability k rα (property of the fluid system), the viscosity ν α and the gravity constant g. K α = k rα K ϱ αg µ α (2.18) The physical meaning of K as a tensor is the possibility of different permeabilities and therefore different velocities in various directions, e.g. in layered structures p c -S w relationship One main physical behavior, that has to be upscaled is the one of capillary pressure. Capillary effects occur, because a state of minimized free energy is aimed. On the microscale, the phenomena of capillary pressure is known as the effect of water rising in small tubes. In natural systems, water is normally the wetting phase, trying to cover as much of the solid surface as possible. On hydrophobic material water is the non-wetting phase and tries to minimize the covered surface. Due to this behavior a pressure discontinuity occurs at the phase edge. p c = p n p w (2.19) Considering the porous domain as a bundle of tubes with various diameters, it is possible to obtain a connection between the saturation and the capillary pressure. This was done by Van Genuchten and Brooks-Corey, see fig. 2.9a. A low wetting saturation results in an occupation of the smaller pores by the wetting fluid and an increase of the capillary pressure k rα -S w relationship The relative permeability accounts for the presence of a different phase, which reduces the flow of both phases. An immobile residual saturation S rα can also occur. The relative permeability k rα is derived from the p c -S w -relation and shown in fig. 2.9b. This relationship is depending on the composition of the fluid system.

29 2.5 Porous medium concept 15 capillary pressure pc [Pa] wetting phase saturation S w [ ] Van Genuchten Brooks-Corey relative permeability krα [ ] wetting phase saturation S w [ ] wetting non-wetting (a) Capillary pressure - saturation (b) Relative permeability - saturation Figure 2.9: Macroscale approaches for capillary pressure and relative permeability according to Van Genuchten and Brooks-Corey, after [9].

30 Chapter 3 Model concepts The best model of a cat is another cat, and preferably the same cat. Norbert Weiner and Arturo Rosenblueth A basic knowledge for modeling flow processes has now been provided, but still the methods to simulate flow problems are missing. In this chapter necessary mathematical and physical equations for describing and calculating multiphase flow systems will be derived. The assumptions made in the individual steps will also be pointed out. Further a short introduction to the numerics and the applied discretization scheme is given. The last section contains information about the conceptual model, especially the coupling concept. 3.1 Mathematical-physical model By figuring out the relevant physical processes and with the knowledge, how to put them into equations, it is possible to get a mathematical description of the problem. A detailed derivation of the system of equations goes beyond the scope of this thesis, so the interested reader will be referred to textbooks [5, 17, 34, 37, 40]. In this section a general form of the equations is given. From this set of equations the models in the subdomains are derived. Generally multiphase flow problems can be described by so called balance equations. Four different types of balance equations are considered to describe the flow system in this thesis. These are the conservation laws for total mass, momentum, components and energy. The equations for total mass and components are not linear independent, the sum over all components balances gives the total mass balance again. 16

31 3.1 Mathematical-physical model Total mass balance The first balance equation describes the overall change of mass in the system. This balance equation has to be set up for all phases α Momentum balance ϱ α t + (ϱ αv α ) q α = 0 (3.1) A general equation for balancing the momentum ϱv is given by: (ϱ α v α ) } t {{ } storage + (ϱ α v α v α ) τ + p }{{} α }{{} advection surface forces ϱ α g }{{} external forces = 0 (3.2) This equation is also valid for non-newtonian fluids and gravity is assumed to be the only external force Component balance In a multicomponent model, the respective transport has to be modeled for all components κ. (ϱ α Xα) κ + (ϱ α v α X } t α) κ (D καϱ ) M κ {{}}{{} α x κ α qα κ = 0 (3.3) M α }{{} advection }{{} source/sink storage diffusion This equation already includes Fick s law for binary diffusion. Having the supplementary equation given by eq. 2.1, for K components only K 1 equations will be needed Energy balance The last equation balances the energy. Thermodynamic equilibrium is assumed, which means that all phases possess the same temperature, so only one equation is needed. The structure of the energy balance is similar to the component balance. (ϱ α u e α) t + (ϱ α v α h α ) (λ α T ) q T α = 0 (3.4) The heat conductivity λ acts like the diffusion coefficient or the viscosity. In this case dissipation, the viscous transfer from kinetic energy to heat, is neglected. The link between the enthalpy h and the inner energy is given by: h = u e + p ϱ (3.5)

32 3.2 Physical model Physical model The system of equations is simplified for the possibility of simulating turbulent flow. Different assumptions pertaining free and porous-medium flow will be made. The idealizations made for the free flow will still be valid for the porous medium, which is discussed in the subsequent part Turbulent free flow For modeling turbulent free flow, it is only theoretically possible to make direct use of the equation presented in section 3.1. Even the smallest eddies have to be modeled correctly, otherwise physically unrealistic results are obtained. The main question is the estimation of the smallest eddies. Referring to the Kolmogorov energy cascade [20], energy is transferred from larger to smaller eddies and finally dissipates into heat. The kinematic viscosity is a crucial quantity and determines the Kolmogorov microscale L k, the smallest possible eddy size. If L k is considered to be about 20 µm, an equidistant discretization of d i = 10 µm would be necessary. Furthermore, eddies are always three dimensional, so this would need n cells for simulating a cube of 0.5 m length [38], which is not feasible. Turbulent flow simulated by this direct numerical simulation, also called DNS, accurately reproduces turbulent motion like eddies, backflow, etc. The approaches presented and used in this thesis are rather modeling the effects of turbulence, the turbulent flow structure is no longer visible Reynolds-averaged Navier-Stokes - RANS One possibility to solve this problem is the so called Reynolds decomposition. Any transported quantity a is considered to consist of a temporal constant mean value ā and the fluctuations about this mean value a, see fig. 3.1a. This temporal averaging is comparable to the spatial averaging, which is done for the REV concept. The interval t should be chosen such, that the mean value of a quantity is constant, but also a change of the mean value due to changing flow conditions is captured, see fig. 3.1b. In a next step these approximations are inserted in equations 3.1, 3.2, 3.3 and 3.4. Then the whole equation is averaged over time. A long overbar indicates a temporal averaging. By definition it follows, that u = 1 t t 0 u dt = ū ū = ū and u = 0. Above, this is shown for the velocity component u, but can be done in the same matter for the other quantities (v, w, p, X, T ). For two multiplied turbulent quantities this averaging process is more complicated. This is shown for the product uv, which is part

33 3.2 Physical model 19 u u u = ū + u v = v + v ū ū accelerated flow v steady state flow t (a) t t min t max t (b) Figure 3.1: (a) The Reynolds decomposition and averaging of turbulent quantities, here the velocity components u and v. (b) The mean value development of the horizontal velocity over time. of the advection term in the momentum balance. uv = ū v + ūv + u v + u v = ū v + u v The two terms in the middle cancel out, because the time average of any fluctuations a times a constant value ā is zero. The product of two fluctuations in contrast is typically not zero. This procedure and the following assumptions lead to the Reynolds-averaged Navier-Stokes equations. steady state turbulence, meaning the mean value is in a steady state constant viscosity density is pressure independent ϱ p Newtonian fluids and no dilatation incompressibility regarding shear stress = 0, which is valid for Ma < 0.3 The system of equations for turbulent free flow can now be written in a simplified manner. The assumption of a steady state turbulence is not applied on the storage term. ϱ α t + (ϱ α v α ) q α = 0 (3.6) (ϱ α v α ) t + (ϱ α v α v α + ϱ α v αv α) (µα v) + p α ϱ α g = 0 (3.7)

34 3.2 Physical model 20 ( ϱ α Xκ α ) t + (ϱ ) ) α v α Xκ α + ϱ α v αx α κ (Dαϱ κ M κ α x κ α qα κ = 0 (3.8) M α (ϱ α ū e α) + (ϱ ( α v α hα + ϱ α v t αh α) λα T ) qα T = 0 (3.9) The occurrences of only one instantaneous quantity are replaced with the mean value. In the advection term, the equations contain new terms with fluctuating quantities, which are advective accelerating terms. Nevertheless they are responsible for a stronger mixing and called turbulent stresses or Reynolds stresses τ t,α. ϱ α v αv α = τ t,α (3.10) In order to compute all mean values, the Reynolds stresses have to be described. This means, that six new unknown for the momentum balance and one for the component and the energy balance have to be fixed. But no supplementary equation was gained by the Reynolds decomposition procedure, so other methods have to be found to solve this closure problem Principles of the eddy viscosity concept Jospeh Boussinesq was the first, who tackled this closure problem [6]. Boussinesq assumed, that turbulence is acting in strict analogy to viscous stresses. This reduces the number of unknown from six stress components to only one additional viscosity. In general, the additional term ϱv a is formally appended to viscosity or diffusion terms, although it origins in the advective part. This is justified phenomenologically and accounts for the enhanced mixing due to turbulence. Additional to the assumptions of the Reynolds averaging, following assumptions are made. Boussinesq assumption µ t,α is only a flow property µ t,α is a scalar, turbulence is isotropic (strictly only the case for high Reynolds numbers) Now it is possible to get a description for those Reynolds stresses. ϱ α v αx κ α ϱ α v αv α = µ t,α v (3.11) = ϱ α D t,α M κ M α x κ α (3.12) ϱ α v αh α = λ t,α T (3.13) The new quantities are called eddy viscosity µ t or ν t, eddy diffusivity D t and eddy conductivity λ t or generally eddy coefficients. Eddy accounts for the fact, that they are

35 3.2 Physical model 21 DNS RANS RSM Eddy viscosity models algebraic 1 PDE 2 PDEs Prandtl, Van Driest,... Spalart and Almaras,... k ɛ, k ω,... Figure 3.2: Hierarchy of turbulence models, after [35]. flow properties, depending on turbulent flow structure and behavior and not on the fluid itself. Again a description for these three additional parameters has to be found, whereof the eddy diffusivity and the eddy conductivity are coefficients for transport of scalar values. For this reason they are gained by the same equation. The Reynolds-averaged Navier-Stokes equations are derived and now different complex models for calculating the Reynolds stresses or the eddy coefficients can be chosen, see fig The eddy viscosity models are characterized by the number of partial differential equations for additional transported quantities. With these equations the production, dissipation, diffusion and transport of kinetic turbulent energy k and/or dissipation ɛ is modeled. The Reynolds stress models, called RSM, model each entry of the stress tensor individually Zero equation or algebraic models The simplest category of turbulence models are the algebraic equations. They do not model transport effects of turbulent quantities, the application is limited and the parameters of these models are problem dependent. Further they are mainly based on geometrical information, but still it is necessary to compute them numerically. In this thesis, gas is the only free flowing phase, so the subscript α is dropped to make the equations more readable. In some cases α is still used to point out, that the fluid quantity is meant. For the eddy viscosity models, the following assumptions are made. equilibrium between turbulence production and dissipation, which means fully developed turbulence u = v one main flow direction, ν t depends on one coordinate

36 3.2 Physical model 22 y, v turbulent velocity profile l l v u u v u ( v ) < 0 u v < 0 u Figure 3.3: Prandtl s mixing length. dimensionless analysis yields ν t = l char u char Eddy viscosity models In a next step several eddy viscosity models will be presented. The presented eddy viscosity models are only a portion of numberless concepts, which can be found in literature. This list does not claim completeness, but wants to present different approaches and considerations. Prandtl [34] was the one, who proposed a mixing length concept for the eddy viscosity. Most of the models introduced in this thesis use the mixing length approach as basis. The underlying concept is, that one fluid element traveling the way of the mixing length loses its individuality compared to the surrounding, see fig This is in analogy to kinetic gas theory, which means, that eddies behave like molecules in gas at low density conditions. The mixing length l char is depending on the wall distance. The characteristic velocity is also depending on mixing length and supposed to be u char = l char ū/ y [34, 41]. Those two characteristic values give the Prandtl mixing length expression. µ t = ϱl 2 ū y (3.14) The mixed length is defined as l = κy, with the Karman constant κ = 0.40 or 0.41 and the distance from the wall y. The Karman constant is well known through measurements [8, 34], but still depends on the problem [41]. This equation is only valid for wall bounded flows, additionally it is restricted to smooth surfaces. Van Driest [12] made a first key modification to Prandtl s approach. Based on the theoretical background from G. Stokes, who postulated, that the amplitude of motion are weakened by an exponential function for increasing distances from the oscillating

37 3.2 Physical model 23 wall. Van Driest derived the following equation, based on eq. 3.14, by multiplying the wall distance with a dampening function. In case of a fixed wall and oscillations parallel to the wall, the dampening is observed in vicinity near the wall. µ t = ϱκ 2 y [ 2 1 exp ( y + /A +)] 2 ū y (3.15) The constant A + = 26 was fitted to data of Laufer [22]. Van Driest admits, that also A + = 27 is possible to fit better to data of Nikuradse [30]. Nevertheless both are experimental data for smooth walls and Newtonian fluids. It is argued, that τ xy y 3 for y going to 0, [41]. But for Van Driest s modification it is τ xy y 4 for y 0. Hanna et al. s modification [16] is fixing that τ xy y 3 for y 0, based on Van Driest s model and with help of an eddy diffusivity description by Notter and Schleicher [31]. µ t = ϱκ 2 y 2 [1 exp ( y+ /A + )] 2 ū 1 exp ( B + y + ) y (3.16) A + is same as above, but B + is accounting for drag reduction and becomes B + = 0.26 for zero drag reduction. For the case of mass and heat transfer, this formula is in good agreement with data about 2 < Θ <. Θ is the Schmidt or Prandtl number of the fluid. Michel et al. s hypothesis [27] in [37] is similar to Escudier s [13] consideration to limit the mixing length in respect to the boundary layer thickness. In this approach the mixing length is independent of y for y > 0.6 δ. ( κ y )] 2 µ t = ϱλ 2 δ [tanh 2 ū λ δ y (3.17) λ = is a constant, δ is numerically calculated by eq Perrels and Karelse s [32] approach is motivated by modeling the effects of stratification in rivers, in contrast to the others, which are motivated by more technical applications. The mixing length approach now can account for surface roughness, but also a two layer approach with a fixed discontinuity is applied. { } ϱκ 2 (y + y) 2 ū y 0 y < 0.25 y µ t = ϱκ 2 (0.25 y max + y) 2 ū for max (3.18) 0.25 y y max y y max The surface roughness coefficient y is expressed with help of the Chézy formula given in appendix B. The discontinuity instead is arbitrarily chosen, but motivated by fitting to measurements.

38 3.2 Physical model y [m] Prandtl Van Driest Hanna et al. Michel et al. Perrels and Karelse l mix [m] Figure 3.4: Theoretical mixing lengths of different models. Deissler s near wall law [11] was motivated by the lack of a good expression for high Pr and Sc and a bad description in the near wall region y + < 5, because the fluid viscosity is dominating this region. Deissler assumes, that for laminar flow near wall, the kinematic viscosity cannot be neglected. Further with a dimensionless analysis he proposes µ t = βf ( ) β ν. After an integration this yields the following equation. [ ( µ t = β 1 exp β )]. (3.19) µ With β = ϱn 2 dūy and n d = by fitting to his own data and the data of Laufer [22]. He also limits the valid range to y + 26 and applies an analytic log law for y + > 26. An Indirect formulation of the eddy viscosity is given by [37]. The magnitude of the eddy viscosity is strongly depending on its position and the wall friction velocity, whereas the shape of the function is fixed. Like Deissler s model, the indirect model is not depending on the local velocity gradient. The coefficient γ = 1 r r max µ t = κ 6 (1 γ2 )(1 + 2γ 2 )u τ, r max ϱ (3.20) is 1 at the wall and 0 at the centerline of the tube. Baldwin and Lomax [2] developed a complex eddy viscosity model, which is used to model separation and wakes in the flow field and the effects of the boundary layer. It is also a two layer model, using the Van Driest equation in the near wall region and modifications done by Clauser and Klebanoff in the outer, accounting for the fact of intermittency, meaning that the outer flow is sometimes turbulent and sometimes laminar. Furthermore it can be used to model a three dimensional flow field. The switching point between the two eddy viscosity models is given by the wall-nearest point, fulfilling µ inner = µ outer. The layer approach is depending on a bundle of constants and equations, so a detailed description is given in the appendix.

39 3.2 Physical model Eddy diffusivity and conductivity models One difference between laminar and turbulent flow is the velocity profile. The eddy viscosity models are accounting for this. A further main characteristic of turbulent flow is the enhanced mixing, due to turbulent motion. The eddy diffusivity and the eddy conductivity are used for describing this effect. Because both descriptions transport scalar quantities with comparable processes, the same equations are used for modeling the eddy coefficient ɛ t. Θ denotes the Schmidt or Prandtl number. D t = ɛ t, Θ t = Sc t and λ t = ɛ t c p ϱ, Θ t = Pr t (3.21) The temperature conductivity is defined as a = λ /c pϱ. The kinematic viscosity ν, the diffusion coefficient D and the temperature conductivity a have the same dimension [ m2 /s]. Although the same assumption as for the eddy viscosity are made for the eddy transport models, it is not clear whether Θ t is completely independent of Θ and therefore of the fluid. Further there is no definite conclusion, if and how the eddy coefficients are related to each other [8]. For component transport, diffusion strictly denotes only transport due to concentration gradients. With the eddy diffusivity description, the overall phenomenological process of mixing is called dispersion, which in a mathematical sense means D = D α + D t. Reynolds analogy [5] is based on the assumption of a similarity between turbulent momentum and turbulent heat transfer. Reynolds stated his assumption already in The analogy means that eddy coefficients are comparable; they are flow quantities and mainly depending on flow behavior and position. One approach is to define a proportional constant between two eddy coefficients. Then this dimensionless number has to be fixed according to the problem. ɛ t = ν t Θ t = µ t Θ t ϱ (3.22) For Θ t = 1 this is called the Reynolds analogy. The values for Pr t can also be assumed 0.7 to 0.9 for gas flow in conduits and 0.85 for pipe flow. Normally the range of values for gas is from 0.5 to 1, [5, 8]. Fluid Prandtl numbers are about 0.72, Schmidt numbers about 0.63 for water vapor in air. Meier and Rotta [26] adapted the Van Driest formula, so that it fits to supersonic flow in near wall region. The two constants κ and A + from Van Driest s formula are modified. ɛ t = κ 2 my [ 2 1 exp ( )] y + /A + 2 ū m y (3.23) The modification is made in a way, that new variables are obtained by defining Θ at the wall and in the free stream. Θ t, = [κ/κ m ] 2

40 3.2 Physical model 26 Θ t, = [ (κ/κ m ) ( A + m/a )] Meier and Rotta fitted those numbers to temperature experiments, with Pr t, = 0.86 and Pr t, = 1.34, resulting in κ m = 0.44 for κ = 0.41 and A + m = for A + = 26. Rohsenow and Cohen [36] modified an equation by Martinelli (1941), who was interested in flow of liquid metals and therefore neglected ν but not λ in the turbulent core. He integrated equations for T T in the different sections y + < 5, 5 < y + < 30 and 30 < y + and afterwards inserted them in each other. Rohsenow and Cohen modified this approach by data fitting for < Pr < 0.05 and assumed eddies with linearly fluctuating temperature passing through the fluid and an infinite surface heat conductivity. Finally they obtained the following eddy diffusivity relation: [ 1 ɛ t = 416Θ 15 6 { ( )} ] π 2 π 4 n exp n 2 ν 4 t (3.24) Θ n=1 In this approach the eddy coefficients D t and λ t are flow and fluid properties, depending on molecular Prandtl or Schmidt numbers. The Exponential dampening of the eddy coefficient due to stratification and buoyancy is proposed by [32]. The eddy diffusivity is calculated based on the eddy viscosity. ɛ t = e ( m Ri) ν t (3.25) The Richardson number accounts for the stratification. This is an empirical correlation with m = 18. Various approaches can be found for m or the exponential dampening [23] Roughness effects In literature different approaches for considering a rough wall can be found [21]. In this thesis a shifting parameter is used, as proposed by Rotta (1962) and also discussed in [8]. Effects of this additional length are, that the velocity increases slower in vertical direction near the wall and the viscous sublayer thickness is enlarged. With this approach a step towards reality is done, but still turbulent free flow in and through the pores are not reproduced. The roughness elements are considered to lay on an impermeable plate, see fig y rough = y smooth + y (3.26) y = 0.9 ν u τ, [ k + s k + s exp ( k + s /6 )] (3.27) k + s = k su τ, ν (3.28)

41 3.2 Physical model 27 ks (a) Smooth surface. (b) Rough surface. (c) Real behavior. Figure 3.5: Different methods for describing a rough surface and the effects on free and porous-medium flow. k s is the equivalent sand grain roughness height. For round sand grains this is the diameter. This approach is only valid for < k s + < 2000 and not used for calculating eddy diffusivity or eddy conductivity. It is also not used for Baldwin and Lomax, Perrels and Karrelse and the indirect formulation Higher order models Towards a more precise description of flow and transport, the use of higher order eddy viscosity models is proposed. Additional partial differential equations have to be accounted in this case, see fig On the one hand, domain shapes can be chosen more arbitrarily and flow properties are described more exactly, but on the other hand more boundary conditions have to be fixed and also the computational effort increases. This work focuses on the evaporation processes with a geometrically simple domain and subsurface depending processes, so only algebraic turbulence models will be investigated Stokes free flow Some simulations were done for Stokes flow, which neglects the advective momentum transport. The changes due to turbulence modeling can be figured out more precisely in contrast to this flow. no inertia term no dilatation incompressible, Newtonian fluids no gravity

42 3.2 Physical model 28 With these assumptions the Navier-Stokes equations can be simplified to the unsteady Stokes equations. ϱ α t + (ϱ αv α ) q α = 0 (3.29) (ϱ α X κ α) t (ϱ α v α ) µ α v α + p α = 0 (3.30) t + (ϱ α v α Xα) κ (D καϱ ) M κ α x κ α qα κ = 0 (3.31) M α (ϱ α u e α) + (ϱ α v α h α ) (λ α T ) qα T = 0 (3.32) t For a detailed description and derivation be referred to [28] Porous-medium flow Until now, only a free flowing phase was considered. In chapter 2 it was already indicated, that for porous-medium flow different aspects have to be considered. In the porous medium a two phase flow is simulated, in contrast to the one phase free flow. Namely the phases water and gas with the components water and air are used. With regard to the porous medium, following assumptions are made. constant pore volume local thermodynamic equilibrium ideal gas phase The main simplification for porous-medium flow is, that the momentum balance is replaced by the Darcy law for multiphase flow. Additional assumptions have to be made for applying the Darcy law. creeping flow, Re < 1 no momentum transfer due to mass transfer no momentum transfer by viscous shear gravity as only external force incompressible, Newtonian fluids rigid solid phase

43 3.3 Numerical model 29 Further now has to be accounted for saturation and porosity. This yields the new set of equations. n phases α n phases α n phases α k rα { φ ( ϱ } αs α ) + (ϱ α v α ) q α = 0 (3.33) t K ( p α ϱ α g) = v α, α {g,w} (3.34) µ α (D κα,pmϱ ) } M κ α x κ α qα κ = 0 (3.35) M α { φ (ϱ αs α Xα) κ + (ϱ α v α X κ t α) { φ (ϱ } αs α u e α) + (ϱ α h α v α ) + (1 φ) ( ϱ sc s T ) (λ pm T ) q T = 0 t t (3.36) According to the assumption of thermodynamic equilibrium, λ pm denotes a bulk heat conductivity, depending on saturation. The diffusion coefficient D κ α,pm is a parameter of the component, the phase and the soil. D κ α,pm = τφs α D α (3.37) 3.3 Numerical model Now all necessary equations for multiphase, multicomponent flow systems are derived. The system of equations consists of coupled and highly non-linear partial differential equations. Those cannot be solved in an analytical way and therefore discretization schemes in space and time have to be applied. In this thesis the numerical simulator DuMu x is used. The system of equations is solved for the primary variables contained in the vector of unknowns u. The secondary variables are calculated by supplementary equations Discretization in space In order to solve the balance equations given in section 3.2.1, and 3.2.3, the whole domain is divided into a finite number of cells or elements. In this thesis both discretization methods are used in the so called box-method. For a detailed description and derivation is referred to [14]. Other methods are given in [17]. The box-method is characterized by having the flexibility of a finite element method, but also mass conservation is ensured by a finite volume approach. First the finite element mesh is constructed in the model domain. Then the barycenters of neighboring finite elements are connected via the midpoint of the edge between them. With this procedure a new box B is created around each node of the finite element and the original element is divided in sub control volumes. All sub control volumes

44 3.3 Numerical model 30 finite volume mesh finite element mesh node i box B i node l i IP d ij box B i IP d il sub control volume scv d i node j element E d Figure 3.6: Discretization scheme of the box-method. belonging to a certain node i are forming the box B i. Fluxes are evaluated at the integration points, called IP. These points lay on the edge of the boxes, between the barycenter and the edge of an element, see fig Afterwards the primary variables are evaluated with an interpolation method like it is used in finite element approaches. All in all the box-method mesh can be regarded as a node-centered finite volume mesh on finite elements. Many multiphase flow problems and also the problems posed in this thesis have an advective character. For stability reasons a fully upwind scheme is used, taken the secondary variables for the advective transport term from the upstream node. However through this proceeding sharp fronts are smeared out [17] Discretization in time For the time discretization a fully implicit method is applied. With this method the equations are solved for the primary variables at a new time level t i+1, whereas for explicit methods the equations are solved based on the current solution t i. The storage term of each balance equations is given by u t ui+1 u i t i+1 t i = à i+1 u i+1, (3.38) with u as the vector containing the unknowns and à as the coefficient matrix. In contrast to explicit schemes or mixed methods, the fully implicit technique has no time step constraints. It is unconditionally stable and no oscillations occur [9] Numerical simulator DuMu x The numerical simulator DuMu x is chosen as simulation tool. DuMu x stands for DUNE for Multi -phase, -physics, -component, -scale Flow in Porous Media. It simulates mul-

45 3.4 Conceptual model 31 tiphase, multicomponent flow processes in porous media as well as free flows. DuMu x can be used for a variety of different applications. It is an open source software written in C++ [14]. DuMu x is a module of DUNE, the Distributed and Unified Numerics Environment [3]. DUNE includes modules from grid management tools over finite element shape functions to different solver methods. It provides a highly sophisticated platform for solving partial differential equations. In this thesis the direct solver SuperLU is used for solving the system of linear equations [24]. The Newton solver is used for the non-linear system of equations. SuperLU offers the possibility of solving large, sparse or non-symmetric matrices. Converging iterative solvers have not be found yet. 3.4 Conceptual model The two last pending problems are, how the different eddy models are implemented and how the two flow regimes are coupled. The first task is to define the phrase wall distance and implement a routine for obtaining properties at the wall and extreme values in the flow. Different assumptions are made for the two model domains, so in a second step appropriate boundary or coupling conditions have to be found and applied, to connect the subdomains Implementation of the eddy coefficient model For the calculation of the eddy coefficients, information not only depending on the current sub control volume or element is needed. The proposed solution is to set up an additional one dimensional mesh. This mesh covers all values in a discrete interval in direction of the main flow, see fig Depending on the current position, the information is stored in the corresponding wall interval, so the neighbor relationship at one point has not to be known. For the proposed eddy approaches, variables of the flow field, the flow values u max, y cross, F max, are needed as well as parameters at the wall, the wall values ϱ, τ xy,, ν. It is suggested to use at least the same number of wall intervals, as number of cells in flow direction. This avoids the wall intervals form influencing the solution. Nevertheless an interpolation is necessary to have flux and wall values given in every interval. This is only a solution for rectangular problems, for more difficult geometries other implementation possibilities are available [25]. Further it is possible to update the wall intervals more often, if these values are suspected to change too fast. For steady state

46 3.4 Conceptual model 32 pre time step starts reset flow values u max = y cross = F max = 0 reset wall values ϱ = τ xy, = ν = 0 do interpolation loop over all sub control volume faces loop over all boundary faces at the wall interpolate for wall intervals with no flow values overwrite flow values in relevant interval if necessary (new max. or min.) calculate average wall values in relevant interval interpolate for wall intervals with no wall values pre time step ends Figure 3.7: Implemented routine for calculating the values of a wall interval. This is done only once per time step. problems, this is not the case. The literature of the eddy models mainly provides the vague term wall distance. For flow along a flat plate the definition is trivial. In pipe flows, like it is modeled here, only the smallest distance to a point on a wall is considered, the influence of the other wall is neglected. Pertaining the simulation these assumptions are made. rectangular shaped domain each point is only influenced by its nearest wall point, for pipe flow the centerline cuts the domain in two halves heat capacity and conductivity of a mixture are weighted in respect to the mass ratio of its component heat capacity and conductivity are not depending on temperature, values for T = 20 C are taken Coupling concept In general, two different approaches are available. The first applies a transition zone between the porous medium and the free flow region within continuously varying parameters. Only one set of equation is used for the whole domain [7]. The second approach comprises a sharp interface between the two domains. The coupling conditions used in thesis consider sharp interfaces. They are based on the concept proposed

47 3.4 Conceptual model 33 FE node, box center: volume variables (φ, ϱ,...) SCV IP flux variables ( v, v x,...) wall interval volume variables (ϱ, ν ) flux variables (u max, [ ū y ], y cross, F max ) Figure 3.8: Implementation of the wall intervals onto to box-method [14]. by [28]. This interface is not able to store mass, momentum or energy, further it has no thickness. The coupling conditions account for microscale processes and can also be transferred to macroscale. Motivated by the assumption of thermodynamic equilibrium, three different coupling conditions preserving mechanical, chemical and thermal equilibrium are derived as far as possible. One main difference to the proposed proceeding is the model concept for the free flow region. In [28], only Stokes flow is considered, in the thesis the conditions have to be extended to fit also for turbulent flows Mechanical equilibrium At the interface a balance of forces has to be formulated. The mechanical equilibrium consists of two components, the normal and tangential component. For the tangential velocity in the free flow, no-slip conditions are applied. u ff g = 0 (3.39) For the porous medium the same assumption as in section are made. The viscous forces are included somehow in the permeability and an equilibrium of momentum fluxes for the gas phases is aimed. Balancing momentum comprising capillary effects, this ends up with the following expression. ) ] ff [(ϱ g v g v g + p g Ĩ µ g v g µ g,t v g n n = p pm g (3.40) This relation can either be used as a Neumann condition for free flow or a Dirichlet condition for pressure in the porous medium part. Further the fluxes have to be continuous across the interface. Two phases are considered in the porous medium, whereas the gas phase in the free flow has to integrate both fluxes, which means a direct evaporation of liquid water. [(ϱ g v g ) n] ff = [(ϱ g v g + ϱ l v l ) n] ff (3.41)

48 3.4 Conceptual model Chemical equilibrium For coupling the component mass balances it would be easiest, to assume the same chemical potential for the component water in both domains at the interface. But this is already violated by the postulated mechanical equilibrium, because of the possible pressure discontinuity in the gaseous phase. The next option and the approach chosen for this thesis, is to assume continuity of the mole fraction (or the mass fraction) for both components air and water x κ,ff g and respectively the component fluxes for each component [( ϱg v g X κ g ϱ g D g X κ g ϱ g D t,g X κ g ) n ] ff = [( ϱ g v g X κ g ϱ g D g,pm X κ g + ϱ l v l X κ l ϱ l D l,pm X κ l = x κ,pm g, (3.42) ) n ] pm. (3.43) The diffusion coefficient are depending on different parameters, D g is the diffusion coefficient of the gas phase and is a function of T and p, whereas the eddy diffusivity D g,t is mainly a flow property, depending on the corresponding approach. D pm is a function of the soil and the fluid Thermal equilibrium The last coupling condition is the one for the energy balance. temperature itself can be achieved by A continuity of the T ff = T pm, (3.44) whereas the following equation ensures continuity of energy fluxes [(ϱ g h g v g λ g T λ t,g T ) n] ff = [(ϱ g h g v g + ϱ l h l v l λ pm T ) n] pm. (3.45)

49 Chapter 4 Results and discussion Get your facts first, and then you can distort them as much as you please. Mark Twain All necessary tools and methods for describing and simulating coupled porous-medium and free flow systems are now explained. Before presenting results of the coupled model, the code will be tested and applied on an one phase pipe flow problem. 4.1 Pipe flow test case As test case for the algebraic turbulence models and evaluation of their impact on the flow field, an experiment done by John Laufer is simulated [22]. In this experiment, the velocity of a fully developed pipe flow was measured with hot wire anemometers and with attention on turbulence near the wall. A part of the setting is shown in fig Measurements were done for Reynolds numbers of , based on diameter and maximum velocity. Figure 4.1: Experimental setup for velocity measurements in a fully developed pipe, taken from [22]. 35

50 4.1 Pipe flow test case 36 ū = 2.5 m s x = 10 m y = m p0 = Pa Figure 4.2: Boundary conditions and domain size of the pipe flow test case Model setup The pipe is modeled in a two dimensional way, with domain sizes of 10 m length and m height. In x-direction the discretization is constant with 0.1 m. In y-direction the cells are coarser near the center line and finer at the wall, see fig The smallest cell has a height of m, the grading is applied with factor 1.1, meaning the next cell is 1.1 times higher than the precedent. All in all the grid possesses nodes. The simulation starts with an initial time step of 0.5 s and ended after 1000 s, when steady state turbulence is reached. The inflow and initial velocity is constant over height, but set to zero at the upper and lower boundaries, which is called no-slip condition. On the right an outflow condition is chosen for the momentum balance. The pressure is set as Dirichlet condition at the right boundary and outflow for the mass balance everywhere else Evaluation All plots are taken after 1000 s, at m in x-direction, if not denoted otherwise. The position, at which the flow field is evaluated differs from the one Laufer took his data. This is due to the additional rough pipe segment in the experiment, which boosts the turbulence. Comparing the velocity profiles it is mainly important, that the flow is fully developed and in steady state. An error source are the pressure oscillations in x-direction. This may lead to slightly oscillating velocity profiles, even if the pressure difference within one oscillation is only 0.1 percent of the absolute pressure.

51 4.1 Pipe flow test case y/h [ ] 0.1 y/h [ ] ū/ū max [ ] ū/ū max [ ] (a) Velocity profiles at x = 0.45 m. 0.5 (b) Velocity profiles at x = m µ g y/h [ ] y [m] ν t /(u τ, R) [ ] µ t [ kg /m s] (c) Dimensionless eddy viscosity profiles. Laufer 1954, Re max,d = power law, n = 7 Prandtl Van Driest Hanna et al. (d) Near wall eddy viscosity values, with the fluid viscosity µ g. Michel et al. Perrels and Karelse Deissler indirect formulation Baldwin and Lomax Figure 4.3: Velocity and eddy viscosity profiles of the pipe flow test case. The velocity distribution in the lower half of the pipe can be seen in fig If no eddy viscosity model is applied, this would lead to unphysical results. The profiles with a higher velocity gradient near the wall underestimate the maximum velocity, whereas those with a smaller gradient overestimate it. For a short distance of 0.45 m, which is almost the total length of the coupled problem, it is hard to judge the correctness of

52 4.1 Pipe flow test case 38 u + = ū/uτ, [ ] Laufer 1954, Re max,d = Prandtl Van Driest Hanna et al. Michel et al. Perrels and Karelse Deissler indirect formulation Baldwin and Lomax y + = u τ, y/ν [ ] Figure 4.4: Semilogarithmic velocity profile, scaled by the wall friction velocity, taken for a fully developed flow. each model. Fig. 4.4 is a semilogarithmic plot, scaled by the wall friction velocity u τ,. There it can be seen, that the Baldwin and Lomax model reproduces the experimental data best. The lower curves do not reproduce the near wall region, but show the same increasing velocity at the beginning of the outer region. The dimensionless values for the eddy viscosity differ strongly between the models, see fig Only the indirect formulation reproduces the measurements, by its definition. The Deissler model does not show any dampening and only for the Baldwin and Lomax model the dampening is not too strong. Models with a dampening near wall have lower eddy viscosities there. In case of the Reynolds analogy and an evaporation process, diffusion near the free-flow-soil-interface is important, so the influence has to be investigated. For models depending on the velocity gradient, the eddy viscosity is zero on the centerline. The length, a flow needs to get fully turbulent is given by DNS-simulations [40]. Experimental relations are also available [37]. l e = 1.6Re 1/4 max,dd 5.91 m (4.1) l e = 4.4Re 1/6 max,dd 6.59 m (4.2)

53 4.1 Pipe flow test case 39 y [m] ū/(10 ū max ) [ ] at x + = x/x max [ ] velocity profile boundary layer thickness Figure 4.5: Development of the dimensionless velocity profile and the boundary layer thickness δ for the Baldwin and Lomax model. Model Time [s] l e [m] Baldwin and Lomax Deissler Indirect Michel et al Hanna et al Perrels and Karelse Prandtl Van Driest Table 4.1: Simulation details of the pipe flow test case. Determination of l e by achieving a constant boundary layer thickness, the experimental result is 6.59 m, whereas DNS simulations predict 5.91 m. After this entrance length l e flow is uniform in x-direction. The simulated entrance lengths are given in table 4.1. Despite the fact, that all models are unable to predict the value, only the Baldwin and Lomax model overestimates it. The model by Michel is depending on the boundary layer itself, the Baldwin and Lomax model comprises a varying two layer approach. For this reason, they are able to describe this entrance behavior. The exactness of estimating the entrance length is taken as a factor to judge the models in their ability to rebuild flow behavior for a short, not fully developed distance, which is the case in the coupled problem. The artifacts in fig. 4.5 are due to the fact, that more wall intervals than cells in x direction are used. Evaluating the effects of the grid resolution is necessary, because on the one hand

54 4.1 Pipe flow test case Laufer 1954, Re max,d = refinement steps 1 refinement step 2 refinement steps u + = ū/uτ, [ ] y + = u τ, y/ν [ ] Figure 4.6: Influence of the grid on the solution of the Baldwin and Lomax model. In this graph, the curves for 1 and 2 refinement steps coincide. computational effort decreases with less cells, but on the other hand the results are less precise. For turbulent flows, the viscous sublayer is important. This means, that a couple of cells have to lay inside the viscous sublayer, but it is not feasible to keep this discretization throughout the whole domain. The term refinement step means, that all cells of the domain are cut into four smaller ones. Only the Baldwin and Lomax model is evaluated, because it depends strongly on the fluid properties at the wall as well as the exactness of the switching point in the flow field. The results in fig. 4.6 show, that the solution is only improved a little by further refinement steps and also only far away from the wall. Although no additional partial differential equation is solved, some computational relevant tendencies can be recognized. The models calculating the eddy viscosity independently from the local velocity gradient are a bit faster, but once a steady state is achieved, all models show the same speed. Nevertheless the simulation time depends on the initial guess of the velocity profile and with a bad one, the Deissler model even fails to converge.

55 4.2 Coupled results 41 (Navier-)Stokes wind channel Interface y = 0.75 m p0 = Pa X = ū = 3.5 m /s x = 0.5 m Darcy soil Figure 4.7: Experimental setup, the two domains are not coupled yet (left). dimensional model domain for the simulation (right). Two 4.2 Coupled results The main purpose of this thesis is to couple the two preliminary mentioned flow regimes and evaluate the influence of an adjacent turbulent free flow on computed transfer fluxes between porous-medium and free flow. An additional goal is to present the results in respect to measured data, which was done in cooperation with the ETH Zürich beforehand. The experimental setup is shown in fig Sand was filled in a box at the bottom of the round wind channel. Flow straighteners have been installed in a distance of 25 cm to enforce the flow to a rectangular velocity profile. Velocity was measured to be 3.5 m /s and hence the characteristic pipe diameter based Reynolds number is In the experiment there was a minimal height difference between the bottom of the wind channel and the top of the sand box, which is not the case in the model. The amount of evaporated water was measured by the weight difference of the sand box Model setup For the simulation a graded grid is used. The parameters for the grid, soil and initial conditions are given in table 4.2. The boundary types are shown in fig Only for the model comprising the eddy conductivity, non-isothermal conditions are applied. The level of complexity will be increased by enabling the eddy coefficient models one after an other. For a better general understanding, the drying process is shown over time in fig. 4.9.

56 4.2 Coupled results 42 Neumann component energy Neumann component energy Neumann component energy Dirichlet mass outflow momentum component energy coupling all coupling all Dirichlet energy (a) If noted denoted otherwise, the standard boundary type is Neumann. Figure 4.8: Boundary conditions of the coupled model (left). If no boundary condition is given and also at the corner points, the boundary conditions are: outflow for mass and Dirichlet for the other balance equations. The standard boundary type is Neumann for the porous-medium domain (right). soil parameters grid parameters initial conditions K = m 2 grading= 1.16 T = K Φ = 0.41 cells x = 50 p 0 = Pa S rw = cells y = 120 S w,initial = 0.90 S rn = d x = m d grain,min = m d y,min = m d grain,max = m α vg = n vg = 6.9 Table 4.2: Soil, grid and initial parameters for experiment and simulation Influence of eddy viscosity models The eddy viscosity model is the most fundamental one, having a strong influence on the flow field and on the other eddy models later on. No comparison is done between laminar and turbulent flow, but between turbulent flow modeled with the Stokes equations and the Reynolds-averaged Navier-Stokes equations. If no additional diffusion is applied, only the two characteristic velocity profiles are compared. In fig the evaporation rate of water from the soil to the free flow is shown over time. The rate contains both, advective and diffusive fluxes through the interface. Additionally to the curves obtained by different eddy viscosity models, the evaporation rate for Stokes flow and the measured rate are shown.

57 4.2 Coupled results 43 ū X H 2O g S w (a) t = 0 d (b) t = 1 d (c) t = 1 d (d) t = 0 d, x = 0.4 m Figure 4.9: Drying process of the porous medium. For the porous medium the water saturation and for the free flow the mass fraction of water is given. On the right the characteristic profiles for saturation, mass fraction and velocity are shown qualitatively evaporation rate of water [ mm d ] time [d] experimental data Stokes Prandtl Van Driest Hanna et al. Michel et al. Perrels and Karelse Deissler indirect formulation Baldwin and Lomax Figure 4.10: Evaporation rate over time for different eddy viscosity models (left) with detailed view of the gray rectangle (right). In the first period until approximately 2 days, liquid water is still present at the interface. This means with Raoult s law, that the partial pressure of water in the gas phase equals the saturation vapor pressure and therefore the maximum possible amount of water molecules in the gas phase. For a present liquid water phase, only the humidity deficit between the arriving gas mixture and the saturation vapor pressure is important and therefore the evaporation rate is constant.