Characterisation of Laminar Flow in Periodic Porous Structures

Characterisation of laminar ﬂow in periodic porous structures

Pedro Dinis Caseiro Jorge

Thesis to obtain the Master of Science Degree in Aerospace Engineering

Supervisors: Prof. José Carlos Fernandes Pereira Prof. José Manuel da Silva Chaves Ribeiro Pereira

Examination Committee Chairperson: Prof. Fernando José Parracho Lau Supervisor: Prof. José Manuel da Silva Chaves Ribeiro Pereira Member of the Committee: Prof. João Manuel Gonçalves de Sousa Oliveira

November 2017 ii To my parents

iii iv Acknowledgments

I would like to thank supervisors Prof. Jose´ Chaves Ribeiro Pereira and Prof. Jose´ Carlos Fernandes Pereira for the invaluable guidance and experience they have provided me. A mention should also be made for KIT for all the support given during my stay, in particular to Prof. Miguel Mendes who not only guided me during two months but offered any help necessary living abroad. To my LASEF colleagues, a big thank you for the camaraderie. To my family, friends and girlfriend a big thank you for all the help during these years.

v vi Resumo

Uma soluçao˜ detalhada para as equaçoes˜ de Navier-Stokes 3-D e´ obtida para uma celulas´ 3-D periodicas´ de variados parametrosˆ geometricos´ e e´ desenvolvida uma correlaçao˜ de base teorica´ e comparada com a equaçao˜ de Forcheimmer para obter os parametrosˆ das permeabilidades Darcianas e nao-˜ Darcianas. Estes sao˜ apenas baseados em parametrosˆ geometricos´ assim como a tortuosidade.

Foi descoberto um coeficiente constante c1 semelhante para geometrias periodicas´ e nao˜ periodicas,´ contudo a dependenciaˆ da tortuosidade torna necessarias´ simulaçoes˜ ao escoamento ou o uso de correlaçoes˜ da literatura para a tortuosidade, cuja capacidade de previsao˜ foi considerada insuficiente. E´ achado um coeficiente tendo em conta a porosidade, tortuosidade e o racio´ entre o tamanho do poro e o tamanho caracter´ıstico sendo este bem ajustado como funçao˜ da porosidade. A correlaçao˜ preveˆ com sucesso a permeabilidade de Darcy quando comparada com correlaçoes˜ da literatura. Quanto a` porosidade nao-Darciana,˜ foram achados dois coeficientes constantes, para espumas periodicas´ e nao˜ periodicas.´ E´ proposta uma relaçao˜ entre tortuosidade e o racio´ entre area´ projectada espec´ıfica e porosidade para escoamento em anguloˆ com bons resultados comparando com a literatura.

Palavras-chave: Espuma periodica,´ meios porosos, CFD, laminar

vii viii Abstract

A detailed solution of the 3-D Navier-Stokes equation was obtained for periodic 3-D cells of varied geometric parameters and a theoretical based correlation is developed and compared with the Forcheimmer equation to obtain Darcy and non-Darcy permeabilities. These are solely based on the foams geometric parameters along with tortuosity. A constant coefficient c1 is found to be similar for both periodic and non-periodic foams although the dependency of tortuosity makes necessary simulating fluid flow or the use of tortuosity correlations which were found to be defective for periodic porous media. A coefficient is found taking into account porosity, tortuosity and the ratio of pore to characteristic length and is found to be a well fitted function of porosity. The correlation successfully predicted the Darcy permeability when compared to the literature correlations. Regarding non Darcy porosity, two constant coefficients were found respectively for periodic and non periodic foams. A relation between tortuosity and the ratio of specific projected area to porosity is found for the angled flow geometries with good results compared to literature correlations.

Keywords: periodic foam, porous media, CFD, laminar

ix x Contents

Acknowledgments...... v Resumo...... vii Abstract...... ix List of Tables...... xiii List of Figures...... xv Nomenclature...... xviii Glossary...... xix

1 Introduction 1 1.1 Motivation...... 1 1.2 Historical Background...... 2 1.3 Literature Review...... 2 1.4 Objectives...... 3 1.5 Outline of the Thesis...... 3

2 Fundamentals and modeling of transport in porous media5 2.1 Scale...... 5 2.2 Volume Averaging...... 6 2.2.1 Volume Averaged Continuity Equation...... 7 2.3 Momentum Balance...... 7 2.3.1 Volume Averaged Momentum Equation...... 7 2.3.2 Darcy Law...... 8 2.3.3 Forcheimmer Law...... 8 2.3.4 Ergun Equation...... 9 2.3.5 Foam Correlations...... 10 2.4 Numerical Method...... 11

3 Implementation 13 3.1 Numerical model...... 13 3.1.1 Star-CCM+ R ...... 13 3.1.2 Derivation of the correlation...... 15 3.1.3 Veriﬁcation and Validation...... 16

xi 4 Results 25 4.1 REV Validity...... 25 4.2 Different Geometries and Foam Anisotropy...... 26 4.3 Coefficients Computation...... 27 4.4 Darcy and Forcheimmer regimes...... 33 4.5 Force type and velocity flow field...... 37 4.6 Porosity and Tortuosity Effects...... 40 4.7 Correlation Analysis...... 47 4.7.1 Darcy Permeability...... 47 4.7.2 Non-Darcian Permeability...... 52 4.8 Unsteady flow on a periodic REV...... 56

5 Conclusions 59 5.1 Achievements...... 60 5.2 Future Work...... 60

Bibliography 61

xii List of Tables

3.1 Mesh base sizes and cell count...... 17 3.2 Correlation and Simulation Results...... 22 3.3 Drag Coefﬁcient for Re = 20 ...... 23 3.4 Drag Coefﬁcient for Re = 200 , (*) in Belov...... 24 3.5 Results for periodic 2D array of cylinders...... 24

4.1 Mesh Name and Characteristics...... 26 4.2 Specific projected and surface area for different arrangements at = 0.6 ...... 29 −4 4.3 Coefficients for 0.6 porosity and uD = 3 × 10 ...... 30 4.4 Coefficients for 0.6 Porosity and Hooman-Dukhan, Lacroix and Calmidi correlations... 34 4.5 Coefficients for 0.6 Porosity and Hooman-Dukhan, Lacroix and Calmidi correlations... 37 4.6 Force type and coefficient for 100 at 0.6 porosity...... 38 4.7 Transition Reynolds for ABC geometry and different porosities...... 41 4.8 Pressure and friction forces for Re = 1 for 0.9 Porosity...... 42 4.9 Pressure and friction forces for Re = 29 for 0.9 porosity...... 43 4.10 Geometric parameters of the correlation foams...... 47 4.11 Geometric Parameters of the tested foams...... 50

4.12 Correlations maximum absolute Error[%] of K1 for the tested foams...... 50

4.13 k1 Correlation variations error...... 52 4.14 Tortuosity correlations maximum absolute error [%] for the tested foams...... 52 4.15 Correlations average of absolute error for both regimes...... 53

4.16 Sensitivity Study for the K2 correlation...... 53 4.17 Convergence study for different array arrangements...... 57

xiii xiv List of Figures

2.1 REV (Left), REV deﬁnition (Right)...... 6

2.2 Cubic Cell (Left), aca vs (Right) REV deﬁnition (right)...... 11

3.1 Geometry used in Star CCM+: (Left): Fluid medium (Right): Solid medium...... 17

3.2 Grid convergence for Res = 10: Forces(Left) and Velocities(Right)...... 18

3.3 Grid convergence for Res = 80: Forces(Left) and Velocities(Right)...... 18

3.4 Normalized axial velocity PDF for different meshes with Res = 10 ...... 19

3.5 Normalized axial velocity PDF comparison for four different meshes with Res = 10 .... 19

3.6 Normalized velocity magnitude PDF for different meshes with Res = 20 ...... 20

3.7 Normalized velocity magnitude PDF comparison for four different meshes with Res = 20 . 20 3.8 Porosity convergence study...... 21 3.9 Domain schematic (Left) and Mesh sample (Right) for cross flow cylinder problem.... 22 3.10 Velocity Profile for Re = 20(left) and Re = 200 (Right)...... 23 3.11 Domain schematic and mesh for 2D cylinder array problem...... 24 3.12 Velocity field for Re = 10 (Left) and Re = 200 (Right) for 2D cylinder array problem.... 24

4.1 Full periodic Structure...... 25 4.2 Possible REVs...... 26 4.3 Projected areas in the direction: 100 (Left) and 111(Right)...... 26 4.4 Projected areas in the direction: AAA (Left) and ABC (Right)...... 27 4.5 Dimensionless pressure gradient vs Reynolds for 4 different arrangements...... 28 4.6 Pressure gradient vs Reynolds for 4 different arrangements...... 28 4.7 Axial velocity PDF for different Reynolds numbers for 100 and 111...... 30 4.8 Streamlines for 100 geometry at Reynolds: 1.8 (Left) and 60 (Right)...... 31 4.9 Streamlines for ABC geometry at Reynolds: 1.8 (Left) and 12.1 (Right)...... 31 4.10 Axial velocity PDF for different Reynolds numbers for AAA and ABC...... 31 4.11 Full steady range axial velocity PDF for different Reynolds numbers for: 100 and 111 (Left) AAA and ABC (Right)...... 32 4.12 Flow diagram on cylinder array...... 32 4.13 Normalized pressure gradient vs Re ...... 33 4.14 Both terms magnitude (Left) Second term percentage of total (Right)...... 33

xv 4.15 Variation of inertial to total force ratio with respect to Fo...... 34

4.16 CD Assumption (Left) Measurements and prediction of ∆P/(LU) (Right)...... 35 4.17 Pressure drop of foams vs packed bed...... 35 4.18 Pressure drop vs Predicted pressure drop for the different geometries...... 36 4.19 RUC used by Du Plessis to model metallic foams...... 37 4.20 Pressure to total force ratio for 0.6 porosity geometries...... 38 4.21 Wall shear field for 100 (Left) and 111 (Right) for 0.6 Porosity...... 39 4.22 Pressure vs Reynolds for 111 and 0.6 porosity...... 39 4.23 Velocity field for 0.6 porosity for 100 geometry at Reynolds: 1.8 (Left) and 60 (Right)... 39 4.24 Velocity field for 0.6 porosity for ABC geometry at Reynolds: 1.8 (Left) and 60 (Right)... 40 4.25 DPG vs Reynolds for 4 different porosities for 100 (Left) and ABC (Right)...... 40 4.26 Permeability vs porosity...... 41 4.27 Pressure to total force ratio 100 (Top) and ABC (Bottom) for 100 and ABC...... 42 4.28 Inertial coefficient vs porosity for all arrangements...... 43 4.29 Axial velocity PDF for ABC and Re = 0.16 (Left) and Re = 17 (Right) for different porosities 44 4.30 Tortuosity vs Reynolds for 100 geometry (Left) and ABC (Right) for different porosities.. 44 4.31 Average Tortuosity vs porosity...... 45 4.32 Particle path and residence time for Re = 30, 0.6 porosity AAA, flow from left, xy plane, from top to bottom, left to right, 100s, 300s, 700s and 10000s...... 45 4.33 Particle path and residence time, Re = 17, 0.9 porosity, flow from right, for ABC (Left) and AAA (Right)...... 46

4.34 α1 as a function of porosity...... 48 4.35 Different shapes of strut studied by Eric Werzner...... 49 4.36 Foam studied by Calado...... 49 4.37 Sensitivity Analysis...... 51

4.38 c1 for periodic and non periodic foams...... 51

4.39 fβ for periodic and non periodic foams...... 52

4.40 Error value of k1 and comparison with literature for all tested foams...... 53

4.41 c2 and c3 for periodic and non periodic foams...... 54

4.42 Tortuosity correlation fτ for angled ﬂow...... 54

4.43 Error value of k2 and comparison with literature for all tested foams...... 55

4.44 Error value of k2 and comparison with literature for 111 and ABC...... 55 4.45 Velocity ﬁeld for low Re 10x10 cylinder array...... 56 4.46 Velocity ﬁeld for high Re 10x10 cylinder array...... 56

xvi Nomenclature

Roman Symbols

2 Ap projected solid area (m ) 2 Asf solid-ﬂuid surface area (m ) c1 Darcian constant c2 non-Darcian constant for periodic media c3 non-Darcian constant for non-periodic media dp,i characteristic pore diameter in ﬂow direction (m) ds characteristic strut diameter (m) F o forcheimmer number

F1 Darcian drag force (N)

F2 non-Darcian drag force (N) fp active fraction of Ap fsf active fraction of Asf fτ tortuosity function fβ constant parameter for Darcian coefficient g body force 2 k1 Darcian permeability (m ) k2 non-Darcian permeability (m) L characteristic unit cell length p pressure (Pa) −1 Sp specific projected area (m ) −1 Sv specific surface area (m ) uD superficial velocity (m/s) up intrincic pore velocity (m/s) v velocity vector 3 Vf fluid volume (m ) 3 VT total volume (m )

Greek Symbols

α1 Darcian coefﬁcient

α2 non-Darcian coefﬁcient β characteristic boundary layer thickness adimensionalised by strut diameter δ characteristic boundary layer thickness (m) ε porosity

xvii µ dynamic viscosity (Pa.s) ν kinematic viscosity ρ density (kg/m3) τ tortuosity

Subscripts

D with respect to cylinder diameter, Darcy regime i axial direction p pore

xviii Glossary

AIAA is the world’s largest technical society dedicated to the global aerospace profession.

CFD Computational Fluid Dynamics is a branch of fluid mechanics that uses numerical methods and algorithms to solve problems that involve fluid flows.

DNS Direct Numerical Simulation is a simulation in which the Navier- Stokes are solved without any turbulence model

DPG Dimensionless Pressure Gradient is the pressure gradient divided by density and squared Darcian velocity.

FVM Finite Volume Method is a method of solving partial differential equation in algebraic form.

LBM Lattice Boltzman Method is a method for ﬂuid simulation with col- lision models.

LES Large Eddy Simulation is a mathematical model for turbulence used in CFD.

NPG Dimensionless Pressure Gradient is the pressure gradient divided by density and Darcian velocity.

PDE Differential equation that incorporates multivariable functions and their partial derivatives.

PDF Probability Density Function is the sum of counts divided by the total count and interval width. The integral of the function must be equal to one.

PPI Pores Per Inch is a measure of the number of cells per inch of foam usually used by manufacturers.

REV Representative Elementary Volume is the right sized volume with which the averaging process produces representative values.

RUC Representative Unit Cell is a geometric model used to model metallic foams

xix xx Chapter 1

Introduction

1.1 Motivation

Porous materials are familiar to everyone as they can be found in everyday life. Apart from metals, some plastics and dense rocks, all solids and quasi-solids are porous to some extent. As such, the study of the theory and characteristics inherent to the material itself as well as fluid flow through or around them assumes great importance. This is demonstrated by the vast number of engineering applications where porous media are used and studied (e.g. porous burners, filters, water and mineral migration, insulation, oil and gas flow in reservoirs [1].

Motivated by real world engineering applications, scientists have put some effort into finding correlations between the pressure drop and the geometrical properties of the medium (e.g. porosity, specific area, pore diameter) by experimental analysis and posterior fitting [2,3,4,5]. Furthermore, with the increasingly accessible computational power and image techniques (MRI, CT scans) it is now possible to simulate flow in the scale of the structure’s pore and quickly obtain data to analyse. Successful correlations would make obtaining data for pressure drop possible without the need for simulations or experiments with progressively decreasing error.

Furthermore, periodic ideal media have been studied as approximations for real porous media [6,7,8] with the implicit problem of introducing a modeling error. However, ideal foams have the advantage of being easier to study as well as easy to design and optimise for speciﬁc problems. With the advancements in 3D printing, projecting and building foams with a deﬁned structure (e.g. periodic) is becoming a possibility not only to the academia but also a viable option to the industry.

With this in mind the decision of studying the inertial and Darcy regime was made due not only to the increase in computational power that would be required to study the unsteady regime but also because experimentally it is easier to study the latter and extremely costly to study the ﬁrst two. This is due to the extremely low pressure gradients that happen at low Reynolds number, that would require extremely precise and expensive instruments that are easily replaced with low cost simulations.

1 1.2 Historical Background

The study of porous media started in the 19th century with Darcy’s experimental approach to studying beds of sand [9]. His former classmate, Dupuit, worked almost simultaneously on the same topic applying Darcy’s law to quantitative studies of fluid flow in the subsurface [10]. However, the approach of Darcy was restricted by an upper limit of validity regarding strut Reynolds number. Trying to tackle this issue and finding a more general expression, Forcheimmer proposed adding a quadratic term as well as a third order term in the average velocity at high velocities as a means to correct Darcy’s equation [11]. Forcheimmer’s hypothesis was later demonstrated experimentally by Ergun [12]. Ergun expanded on the hydraulic radius model by Carman-Kozeny [13], which defines a characteristic length of the porous media based on the void volume and surface area, to construct a correlation over a wider range of Reynolds numbers that in fact depends on the square of the average fluid velocity outside the Darcian validity region. Macdonald [14] posteriorly reexamined the Ergun correlation and tested it against a large number of experimental data, for low porosity media, and concluded his work with nothing more than slight modifications attesting to the validity of the parameters used. Ten years earlier, Beavers and Sparrow [15] research on nickel foams was one of the first cited research regarding high porosity media. Whitaker [16] introduced the Local Volume-Averaging Method which involves the integration over representative elementary volume, REV, of the conservation equations with semi-empirical treatments.Ene and Sanchez-Palencia [17] introduced a similar approach, the Homogenization Method, founded on studying periodic solutions of differential equations which is more restrictive by the use of the implicit periodic structure. Dybbs and Edwards [18] did a major hydrodynamics experimental study in porous media defining a Reynolds number with a characteristic pore length and superficial velocity and were able to distinguish between four different regimes in the Reynolds spectrum (Darcy, Inertial, Unsteady and Turbulent).

1.3 Literature Review

The focus of the current study is ceramic porous foams. Either ideal and real commercial foams have diverse applications. To illustrate this, applications as volumetric solar receivers [19] have been studied where the porous media is heated by concentrated solar radiation and then transfers the energy to flowing ambient air passing through a heat exchanger for a conventional steam turbine process. Other applications include porous burners [20] that allow for higher flammability and higher flame speeds which also compacts flame size and reduces emissions. This is achieved because the mixture is promoted and energy is transfered from the combustion products to the fresh mixture. Regarding ideal porous media, Kuwahara [6] used a periodic array of squares (one of the standard numerical models for porous media) and Large Eddy Simulation, (LES), to analyze macroscopic parameters such as turbulent kinetic energy and pressure drop. The range of Reynolds encompassed the inertial, unsteady and turbulent regime and the porosity was varied for the same geometric shapes. A correlation was obtained for the dimensionless pressure drop at high Reynolds (Re≥3000) which only

2 differed slightly from Ergun’s equations for packed beds. The authors emphasized the need of more numerical experiments to better understand the transition from unsteady to turbulent. Hutter [7] also used LES and periodic media (array of cylinders) to study flows at high Reynolds numbers (Re≥1200) varying not only the porosity but also the ligament shape which was found to have an impact on turbulent kinetic energy and pressure drop of 30 %. Kundu [8] also used cylinder arrays to study the influence of turbulence models and the effects of different porosity on fluid flow. Making reference to previous work [21] on arrays of square cylinders a comparison is made between both geometries. Regarding correlations there are numerous studies.In one of them, Edouard [4] presents the state of the art at 2008, comparing the most used correlations with experimental data and their validity.It was found that Du Plessis [22] and Lacroix [23] correlations were the most adapted to predicting pressure drop on porous foams with the majority of the data fitting within a standard error of 30%. Dietrich [2], on the other hand, used over 2500 experimental values reported by over 20 authors to arrive at a correlation that is valid for a range of Reynolds from 10-1 to 105. The geometric parameters needed for both Lacroix and Dietrich correlations are the porosity and the mean particle diameter, however for Du Plessis the tortuosity is involved. Further literature review regarding the numerical method can be seen on section 2.4.

1.4 Objectives

The current work’s focus can be branched in the following steps:

• Set an appropriate computational domain for the ﬂuid ﬂow analysis of the real porous media

• Develop different geometries for periodic porous media

• Cover the Darcy and inertial regime for both domains

• Determination of the relevant physical parameters for the range of Reynolds number studied

• Develop a correlation that covers the inertial and Darcian region and comparison with what is available in the literature

1.5 Outline of the Thesis

Chapter 2 will provide the theoretical background regarding local volume averaging techniques, different scales, equilibrium criteria and balance equations as well as relevant parameters paramount to the study of porous media. It will further present a description of the analytical ﬂow models and an overview of the numerical approach will also be provided. Chapter 3 covers the numerical method as well the veriﬁcation and validation process. Furthermore, a derivation of the correlation is presented.

3 In chapter 4 the REV validity will be discussed and the results regarding physical processes and correlation predictions will be analysed. Furthermore, other correlations will be used to attest to the reliability of the formula. In the last chapter a brief description of the objectives achieved will be exhibited as well as future work suggestions

4 Chapter 2

Fundamentals and modeling of transport in porous media

2.1 Scale

The procedure to analyse the fluid flow is heavily reliant on the scale of the mathematical model considered. At a microscopic scale the microscopic balance equations of continuity and transport of momentum and energy can be used as only a few small pores are considered (pore-scale). However, it has been usual to consider volume average Navier-Stokes equations which stem from the aforementioned microscopic ones when averaged over a small REV. This approach, which looses some of the microscopic information as result of filtering, is used in order to calculate macroscopic averaged quantities.

The Knudsen number (Kn), deﬁned as λ Kn = , (2.1) L is a dimensionless number where λ is the mean free path and L is the representative physical length, and is relevant to access if modeling ﬂuid as a continuum is correct. At small pore size and low gas pressure the mean free path of the molecules might be of the same order of the system dimension and velocity slip phenomenon might be relevant resulting in higher permeabilities[24]. In the work that follows the continuum hypothesis will be assumed.

Kaviany [24] presented a relation from which the representative different lengths present in a porous medium are approximately governed by,

1 K 2 << d < l << L. (2.2)

1 K 2 is referred as the screening length, which predicts the boundary layer thickness, i.e. gives the order of magnitude over which the velocity disturbance decays, d is the pore diameter, l is the REV characteristic dimension and L is the system dimension.

Taking this into account, the REV size must be sufﬁciently large so a change in volume in any given

5 Figure 2.1: REV (Left) [24], REV definition (Right) [25] position of the media would not affect the quantity being averaged (I in Fig 2.1). However, there is also an upper limit to the size of the REV as macroscopic heterogeneities cause fluctuation on the averaged quantity (III in Fig 2.1). When the averaging is done, the fluid and solid phase become indistinguishable as the spatial resolution is destroyed. The description of the system is done with quantities as porosity and intrinsic permeability i.e. effective parameters.

2.2 Volume Averaging

Let Ψ be a quantity associated with the ﬂuid within the REV; there are two averages to deﬁne. The volumetric phase average, 1 Z hψi = ψdV, (2.3) V Vf and the volumetric intrinsic phase average,

1 Z hψif = ψdV. (2.4) Vf Vf

In the above equation, ψf is the value of ψ in the fluid phase and is, by definition, zero in the solid phase. The difference between both lies in the domain of averaging. While hψi is averaged over the entire domain and hence might be non-zero on the solid phase,hψif is intrinsic to the volume phase it is averaging. Both operations are integrated only over the fluid phase domain and may be related with porosity using

hψi = hψif . (2.5)

Introducing the concept of ﬂuctuation of a volume averaged property, φˆ, meaning the deviation of a value from the average, the value a property holds in a point can be described as

ψ = hψif + ψˆ (2.6) where, by deﬁnition, hψˆif = 0.

6 For further developments, it will be important to introduce the theorem for the volume average of a gradient, 1 Z h∇ψi = ∇hψi + ψ.ndA (2.7) V Asf

where n is the unit normal to the area Asf of the solid-ﬂuid interface.

2.2.1 Volume Averaged Continuity Equation

The continuity equation is ∂ρ + ∇.ρv = 0. (2.8) ∂t

Volume averaging, the equation becomes

∂hρi + ∇.hρvi = 0, (2.9) ∂t assuming the velocity on the pore-wall is zero (no slip condition) and a rigid static matrix.

2.3 Momentum Balance

This section describes the macro-scale momentum balance equations for ﬂuid ﬂow through porous media.

2.3.1 Volume Averaged Momentum Equation

Using the same procedure used before on the continuity equation a macroscopic Navier-Stokes equation can be derived starting from a microscopic one. Assuming constant properties and negligible viscous dissipation, the equation is

∂(ρv) + ∇.(ρvv) = −∇p + ∇.(µ∇v) + ρg, (2.10) ∂t where p is the absolute pressure and µ is the dynamic viscosity of the ﬂuid. Volume averaging and assuming the porosity does not change over the volume leads to [26]

∂(ρhvi) 1 Z 1 Z +∇.(ρhvvi)+ ρ(u-w).ndA = −∇hpi+∇.hµ∇vi+ (−p+µ∇v)ndA+ρg. (2.11) ∂t V Asf V Asf

Assuming a no slip condition, the interphase mass transfer is zero (third term on the LHS) as u = w. Following the decomposition stated in equation (2.6), the term containing the average of the product can be changed to the product of averages. It is then possible to arrive at,

∂(ρhvif ) 1 Z + ∇.(ρhvif hvif ) = −∇(hpif ) + ∇.hµ∇vf i + (−p + µ∇v)ndA + ρg. (2.12) ∂t V Asf

7 The hydrodynamic dispersion ∇.(ρhvîhvî) was neglected for simplicity as in high porosities the intrinsic average velocity is close to the microscopic velocity and then the deviation becomes very small but might be non-neglectable for lower porosities. The third term on the RHS might be understood as the form and viscous drag per unit volume by virtue of the solid matrix. The model above and its assumptions hold to only certain regimes (Stokes and laminar) based on the Reynolds number. The Reynolds number is defined as the ratio of inertial to viscous forces and can be calculated by,

ρu d Re = d p (2.13) µ where ud is the Darcian velocity, dp the average pore diameter and ρ is the ﬂuid density. According to Dybbs and Edwards [18], the four ﬂow regimes in a typical porous matrix based on Reynolds number are:

• Re < 1, Creeping ﬂow

• 1 − 10 < Re < 150, Forchheimer

• 150 < Re < 300, Laminar unsteady

• Re > 300 Turbulent

2.3.2 Darcy Law

Starting with creeping flow, Henry Darcy’s experimental work with respect to groundwater flow [9] has great importance. The steady-state unidirectional flow experiments through a uniform bed indicated that the flow rate was linearly dependent on the hydraulic head, or, neglecting gravitational effects and generalizing for three dimensions, µ ∆P = − v (2.14) K where ∆P is the pressure drop, v is the superficial velocity, flow rate divided by area and µ is the kinematic viscosity. K is the permeability tensor which accounts for the capacity of a fluid to flow past the porous media. K is second order if the porous matrix is anisotropic and constant if isotropic. It is possible to derive Darcy’s law [16] from the Navier-Stokes equations provided that the inertial and time dependent effects are neglected. As follows, Darcy’s law is valid in the creeping flow assumption (Re << 1) and Re ≈ 1 is an upper limit of validity for the Darcy law. Deviation from the creeping flow regime implies a need for a different relation for pressure drop and superficial velocity.

2.3.3 Forcheimmer Law

With the departure from the low velocities regime, the inertial forces can not be neglected and the form drag has to be considered as it is of the same order as the friction drag. Hazen [27] ﬁrst proposed modiﬁcations to include the effects of temperature on viscous effects. However, it was Forcheimmer

8 in 1901 [11] that predicted that a non-linear term would have to be added to the Darcy law in order to account for the aforementioned effects and represent the kinetic energy of the ﬂuid,

∆P µu ρu2 − = + (2.15) L k1 k2 where k1 and k2 are referred to as Darcian and non-Darcian permeability parameters, respectively.These parameters are assumed to be a function of the foam geometry and the Reynolds number. A slightly different form of the equation above,

∆P µu Ceρ − = + √ |ud|ud, (2.16) L K K was proposed by Joseph [28] based on the work of Beavers et al. [29]. Ce is the Ergun coefficient and is strongly dependent on the flow regime. For low velocities Ce is small and can be neglected reducing the equation to Darcy law. When the inertial effects become important, as velocity increases, the flow enters the Forcheimmer regime. There are a variety of criteria that can be used to predict the deviation from the Darcian regime. Du Plessis [30] provided an expression which only depends on the porosity of the porous medium,

1 50.8φ(1 − φ) 3 Rec = 1 . (2.17) 1.9[1 − (1 − φ) 3 ]

Other authors like Zeng and Grigg [31] proposed a new dimensionless number to deﬁne the departure from the regime, the Forcheimmer number, F o, deﬁned as the ratio of the pressure gradient to the viscous resistance, √ C ρu K F o = e . (2.18) µ

A non-Darcy effect of 10% gives a Forcheimmer number of 0.11, considered the critical Forcheimmer number.

2.3.4 Ergun Equation

Another common equation that is seen in the literature of porous media is the equation proposed by Ergun in 1952 [12], 2 ∆P (1 − ) µ (1 − )ρ 2 − = A 3 2 u + B 3 u , (2.19) L dp dp where is the porosity. For spherical particles A and B are 150 and 1.75,respectively, depending both on particle geometry as stated by Macdonald [14], who also recommended replacing 3 by 3.6. The equation is based on the assumption that the space between spheres can be modeled as cap- illaries with the first term being the viscous effect and the second the inertial. The constants would be correction factors to account for the geometrical difference between the assumption and the real geometry. Bird [32] stated that superimposing the Blake-Kozeny and the Bruke-Plummer equations (first and second terms on the RHS, respectively, which account for laminar and turbulent flow, respectively) one

9 would arrive at the Ergun equation. As such the equation describes the behavior of two extremes the creeping and turbulent ﬂow, the two asymptotic solutions, but not the Forcheimmer regime which is thought of as a transitional regime. To tackle this, Du Plessis [33] derived, analytically, a momentum transport equation for fully developed laminar ﬂow where for the high Reynolds numbers the inertial term was modeled as a form drag.

2.3.5 Foam Correlations

The large porosity values (0.6-0.95) and high surface area along with complex struts with heterogeneous shapes and dimensions make the correlations derived for packed beds erroneous for porous foams. The foams permeability is in the order of 1 × 10−8 m2 set by side with 1 × 10−2 m2 for packed beds. Analogous to Ergun equations for packed beds other relations might be derived for open cell foams either analytically or experimentally. To some degree of uncertainty these relations detail foam characteristics such as permeability, pressure drop, form/inertia coefficient and tortuosity as functions of other parameters of the foam, easier to obtain, such as porosity and specific surface area. Several researchers have tried to tackle this issue, yet, by virtue of the problem’s geometric com- plexity, there is no accord concerning the definition of structural characteristics. The main morphological characteristics are the pore diameter, dp, the strut thickness ds and the porosity, defined as

V = f , (2.20) V where Vf is the fluid’s volume and V the total volume, V = Vf + Vs. Various researchers reported theoretical geometrical models that simplify the real foams geometry, and facilitate relations for the morphological characteristics. Lu [34] proposed a cubic cell model which is also the model considered in this work and it can be seen on figure 2.2. Other cell models are also possible; Bhatacharya [35] modelled the unit cell as a dodecahedron while Fourie and DuPlessis idealized a tetrakaidecahedron. This will naturally influence the correlations for the specific surface area, which is dependent on the cell model of figure 2.2 and since this will be a parameter for the correlation of pressure drop, further error is being introduced in what could be a sound correlation. A tortuosity model was developed by DuPlessis [22] and later confirmed experimentally by Bhatacharya [35] (who also developed a model for tortuosity) for high porosities. Tortuosity is the quotient of the real length that a particle travels and the length in a straight line from the initial position to final position and is a measure of recirculation and diffusion in the medium. In regards to pressure drop, some authors [36] define a friction factor characterized by leaving the inertial term as a constant, the Ergun coefficient, which is obtained by dividing the pressure drop equation by the coefficient of the inertial term. For the Ergun equation the friction factor is

∆PD 3 (1 − ) f = p = 150 + 1.75, (2.21) k Lρu2(1 − ) Re where Re is strongly dependent on pore diameter that can be deﬁned very differently in the literature

10 Figure 2.2: Cubic Cell [34] (Left) aca vs [4](Right)

depending on the geometrical relationships used. This means that for a given pore size, ds and porosity different dp can be obtained and consequently different values for Reynolds number.This can explain the notorious differences encountered in the literature when different authors plot the friction factor as function of the Reynolds number. Other options include fitting of the Forcheimmer extended Darcy equation and obtaining the permeability and inertia coefficient as done by Vafai and Tien [37], obtaining the permeability and inertia coefficient by modeling them as function of geometrical parameters as done by Calmidi[38] and other theoretical models that depart from the internal flow assumption. Fourie and DuPlessis [39] followed this approach and developed a model for the inertial coefficient in an array of tetrakaidecahedra as function of porosity, tortuosity and form drag coefficient CD. Bhat- acharya [35], on the other hand, with an experimental and theoretical work developed a correlation which used the same parameters and also a shape factor. The derivation was based on the theory of flow over bluff bodies. Both works used DuPlessis model [22] to predict the permeability.

2.4 Numerical Method

Numerical simulations in porous foams might be divided in two branches, the ideal periodic simulations where one admits fully developed ﬂow and simulates a REV with periodic conditions [40], and random foams. In the latter, the domain might be obtained by a computer generated random structure or cap- tured by imaging techniques of real commercial foams (CT or MRI scans). Regarding numerical methods both Lattice-Boltzman and Finite Volume Methods (FVM) have been used with great success. Krishnan [41] studied the laminar ﬂow and heat transfer in a periodic open-cell structure solving all the spatial and temporal scales with Direct Numerical Simulation (DNS). The results compare reasonably

11 well with the existing experimental and semiempirical models for porosities higher than 94%. Karimian and Straatman [42] tried to characterize a graphitic foam with an ideal structure and simulated flow in the laminar regime concluding the work with developed semi-heuristic models for pressure drop and heat transfer. Habisreuther [43] developed a random structure and simulated laminar flow in it as well as on a scanned (MRI) ceramic foam in a Reynolds ranging from 3 to 373. The pressure drop data from the random structure did not compare well against the data for commercial ceramic porous foams however, the scanned foam results were in good agreement. On the other hand, the Lattice-Boltzmann technique was used by von der Schulenburg [44] who used MRI to provide both the domain data and also the 3-D pore scale velocity. The LB flow simulation was then used to investigate the effects of foam compression on the flow with great success as the results compared well with experiments. Tabor [45] used yet another imaging technique, this time to obtain the structural domain of a plastic foam. With a micro-CT scan the authors were able to preform a steady state laminar flow simulation

(Res (based on strut)< 35) and obtain results where the pressure drop suitably correlates with the ﬂow velocity. Regulski [46] studied, both numerical and experimentally, the pressure drop in 10, 20 and 30 PPI alumina foams of porosity between 75-79%. The simulation was conducted using LB method on a geometry with 360 million nodes. A sensitivity analysis was conducted and a difference of over 20% was found between the highest and lowest porosity (obtained with the different image processing thresholds) which demonstrates the inﬂuence that the process of image processing might have on obtaining accurate results. An excellent agreement between simulation and experiments was achieved and the results were compared to external data and correlation to different results.

12 Chapter 3

Implementation

3.1 Numerical model

3.1.1 Star-CCM+ R

In order to construct the domain of the ideal porous media and solve the flow equations Star-CCM+ R software was used. With a wide variety of packages, this tool can be used to solve problem regarding fluid flow, heat transfer as well as solid mechanics. The problem that is going to be solved is described by partial differential equations (PDE), namely the Navier-Stokes equations. In order to solve them the software solves discretised equations, using the FVM, meaning that algebraic relations representing the PDE are derived and that these, as approximations, will inherently conduce to error, the residual. The field of some quantity φ is calculated in the domain with the general equation,

d Z I I Z (ρφV )0 + ρφ(v).dA = Γ∇φ.dA + SφdV, (3.1) dt V A A V with the ﬁrst term on the LHS being the transient term, the second, the convective term, and on the RHS the diffusive and source terms, respectively the ﬁrst and second. is the void fraction. In order to solve the problem a discretisation is in order leaving the equation as [47],

d X X (ρφV ) + ρφ(v.a) = (Γ∇φ.dA) + (S V ) , (3.2) dt 0 f f φ 0 f f

In order to write the discrete equations as functions of the cell variables several schemes will be employed. The transient term, which is zero during a steady state solution, is approximated by a second order temporal scheme that uses both the current time level and the previous two, except on the ﬁrst time level where only two levels are available and then the ﬁrst order scheme, Euler Implicit, is used. The source term is simply approximated by the product of the integrand, Sφ, on the cell centroid, and the corresponding cell volume. This approximation is of second order. The convection and diffusion term

13 are approximated by second order upwind and second order central differences respectively. After building or importing the geometry of the porous media, the domain is discretised by the mesh construction. The user may then opt between available mesh models such as : Polyhedral, Tetrahedral, Trimmer, Extruder and Prism Layer. A polyhedral cell typology is used in this work. It is characterised for its stability and accuracy compared to tetrahedral and is also less diffusive. It also contains approximately 5 times fewer cells for the same surface. The tetrahedral model is a simple mesh useful to make comparisons with legacy models while trimmer is hexahedrally dominant and one that has minimal cell skewness and particularly useful for external aerodynamic flows given its ability to be refined in the wake region. Extruder is typically used for inlet and boundaries as it extends the mesh volume beyond the original domain so a more representative domain is obtained. The prism layer mesher, adds prismatic cell layers in the vicinity of wall boundaries, improving the accuracy of the flow solution by predicting the velocity or temperature gradients normal to the wall. Regarding physical models the following flow assumptions were chosen:

• Steady or Unsteady depending on the predicted (based on Res) regime

• Three Dimensional

• Laminar

• Constant density

• Segregated

The segregated ﬂow assumption along with SIMPLE (Semi-Implicit Method Pressure Linked Equa- tions) introduced by Patankar [48] is the iterative solver used to solve the NS equations approximations presented above. This is a guess and correct procedure for the calculation of the pressure on a staggered grid arrangement and can be described with the following steps:

• Data input (Boundary conditions, guessed ﬁelds)

• Solving the discretised momentum equations to compute intermediate velocity ﬁeld, u∗, v∗, w∗

• Solving of pressure correction equation to obtain, p0

• Correct pressure using p = p∗ + αp0, where α is the under relaxation factor

• Calculate u, v, w from the star values with the velocity correction formulas,

• Calculate the normalized residual and test for convergence

• If convergence was not attained assume the new p∗ = p and return to step 2.

The aforementioned normalised residual is a value monitored to evaluate the convergence of the solution. It is calculated considering the ratio of the average residual after k iterations and its value at the ﬁrst iteration. The local residual is given by

X rφ = | aiφi + Si − apφp|. (3.3)

14 In order to obtain an indication of the convergence across the whole domain a global residual is deﬁned which is the sum of the local residuals over all M control volumes in the domain.

3.1.2 Derivation of the correlation

As stated before, several attempts to find correlations have been made either by experimentally fitting the data obtained to Forchheimer like equations or using the theory of bluff bodies to predict the flow; this work used the latter. A physically meaningful correlation was derived by starting with the momentum equation for a Newtonian fluid and starting by neglecting the time dependent term as well as body forces. For 1-D, the equation is

1 ∂p ∂u ν ∂2u = u + , (3.4) ρ ∂x ∂x ρ ∂x2 meaning that the pressure drop would equal the sum of viscous and pressure forces acting on the matrice, ∂p F F + F = = p f , (3.5) ∂x L3 L3 where L3 is the total volume of the REV. Knowing that the friction force would be dependent on a characteristic area, the area chosen was the surface area, the dynamic viscosity and a characteristic velocity divided by a characteristic length. Regarding the pressure term, the characteristic area was 1 2 assumed as the projected area, and by form drag theory the rest of the term is given by 2 ρu . Dividing all terms by ρ, the equation one ends up with is

1 ∂p Af ν uf Ap 1 2 = + up. (3.6) ρ ∂x VT Lf VT 2 | {z } | {z } Sv Sp

Recognizing the uncertainty in the variables, factors ff and fp are added to each term. Assuming the characteristic velocities are the pore velocity and the characteristic length is the thickness of the boundary layer, δ, one arrives at the equation,

1 ∂p S νu S u2 = f V P + f p P . (3.7) ρ ∂x f δ p 2 2

uP uD Since uD = uPi and τ = the following can be written, uP = τ. Substituting in the previous uPi equation and incorporating the 1 factor into 1 . Making further simplifications, the boundary layer thick- 2 α2 q ness is of the order of δ ≈ ν as can be seen on White, [49]. Defining β as the ratio δ and defining uP d d 1 = ff the resulting equation is α1 β

2 1 ∂p 1 Svτ 1 Spτ 2 = 2 νuD + 3 uD. (3.8) ρ ∂x α1 ε ds α2 ε

Comparing to Darcy-Forcheimmer equation the expressions for Darcian and non-Darcian permeabilities:

15 ∂p 1 1 2 = µuD + ρuD (3.9) ∂x k1 k2 2 ε ds k1 = α1 (3.10) Svτ 2ε3 k2 = α2 2 (3.11) Spτ

3.1.3 Veriﬁcation and Validation

Errors and uncertainty are inescapable aspects of CFD. These become increasingly important when used in a wide range of industries where attention is focused on value for money and potential wrong decisions based on CFD results can compromise projects. The consequences include wasting money, time and effort and at worst failure of components. It then becomes necessary to establish rigorous methods to evaluate the level of confidence in the results. The AIAA guide for verification and validation [50] has had an important effect in the definition of these terms. Verification has been described by Roache [51] as ”solving the equations right”, meaning that the errors will be quantified and verifying that the model implementation represents the developer’s concep- tual description of the model and its solution. The errors are not due to lack of knowledge but instead by discretisation errors, roundoff or iterative convergence. In this work, the iterative convergence criteria was fixed at 10−6 for all residuals. Validation, on the other hand, accesses whether the model used is an accurate representation of the real world. This was defined by Roache as ”Solving the right equations” and the process quantifies the uncertainty. Its sources stem from inaccuracies due to an approximated geometry, material properties or boundary conditions as well as physical model uncertainty, as a result of inadequate representation of physical processes such as turbulence or simplifying hypothesis(e.g. steady flow, incompressible flow). It then requires highly accurate experimental measurements to validate a model. To perform the verification, a grid independence study was executed on a representative elementary volume(REV) of a unit, 1m x 1m x 1m, cubic cell, as shown on figure 2.2. The cell was designed to have 90% porosity and was idealized to be under fully developed flow, the effect of wall boundaries constrain- ing the medium was not considered as the flow was assumed to be periodic.The boundary conditions are periodic pairs of internal interfaces in the faces of the cube the motion of the fluid being caused by a momentum source per unit volume. This arrangement was made in order to simplify changing the flow direction without the need of geometry change. In order to evaluate grid convergence a group of seven meshed was evaluated. The mesh base sizes, which can be consulted on table 3.1, start at 0.1m and was further refined to an end base size of 0.015m. All of the meshes were considered with polyhedral mesh and prism layer being the discretization of the circle, points per radian, adjusted in order to avoid slender prism cells. Regarding the solid surface walls the no-slip condition was assumed. Air was considered for all simulations with specific properties of: ρ = 1.18 kgm3 and µ = 1.86 × 10−5Nsm−2. A computational domain schematic is shown in figure 3.1.

16 Figure 3.1: Geometry used in Star CCM+: (Left): Fluid medium (Right): Solid medium

Mesh Number Base Size(m) Number of Cells 1 0.1 12960 2 0.07 21457 3 0.05 61962 4 0.04 82808 5 0.026 172623 6 0.02 225772 7 0.015 390219

Table 3.1: Mesh base sizes and cell count

In order to evaluate whether convergence is attained the following quantities were calculated for each mesh: the maximum, Darcy, and pore velocities, the pore velocity in the momentum source direction and the pressure and friction forces. The probability density function was calculated for both the axial velocity as well as velocity magnitude. The results are displayed on ﬁgure 3.2 for Res = 10 and on ﬁgure 3.3 for

Res = 80. It can be inferred that, for both regimes, there is no significant advantage in refining the mesh 5 further. The difference between mesh 5 and 7, for Res = 10, regarding pressure force is of 0.14% and regarding maximum velocity of 0.24%. For Res = 80 the difference regarding pressure force rises to 0.80% while the friction force is 0.30%. Further evidence of this can be seen on figures from 3.4 to 3.7 where the probability density function (PDF) of the normalized axial velocity as well as the normalized velocity magnitude are plotted. The simulation data was exported and then processed with MATLAB, plotting the normalized velocities within 100 intervals for the different mesh values. It can be seen that the difference between mesh 5, 6 and 7 is negligible and the results are converged.

17 Figure 3.2: Grid convergence for Res = 10: Forces(Left) and Velocities(Right)

Figure 3.3: Grid convergence for Res = 80: Forces(Left) and Velocities(Right)

18 Figure 3.4: Normalized axial velocity PDF for different meshes with Res = 10

Figure 3.5: Normalized axial velocity PDF comparison for four different meshes with Res = 10

19 Figure 3.6: Normalized velocity magnitude PDF for different meshes with Res = 20

Figure 3.7: Normalized velocity magnitude PDF comparison for four different meshes with Res = 20

20 Figure 3.8: Porosity convergence study

In order to finish the verification study, the progress of the value of porosity, , can be calculated numerically and then compared with the analytical value given by the volume of the connected three cylinders divided by the REV total volume. The results can be seen on figure 3.8 to evaluate whether the domain is well represented. Even though the errors are negligible regarding domain representation, even for mesh 1, mesh 5 is the one used taking into account the previous convergence studies. In order to validate the results, the permeability of the medium will be calculated using some correlations derived from the physical domain. Calmidi and Mahajan [38] considered an empiric correlation, where the ratio of strut diameter to pore diameter, df is empirically fitted taking into account a shape Dp factor, G, for various ligament shapes. Its validity is confined to = 0.9 − 0.97 and pore densities of 5-40 PPI. The correlation equation is

2 −0.224 df −1.11 K = 0.00073Dp(1 − ) ( ) , (3.12) Dp where, d r1 − 1 f = 2 (3.13) Dp 3π G and 1− G = 1 − e 0.04 (3.14)

On the other hand, Hooman and Dukhan [52] developed a semianalytical correlation that bases itself on a scale analysis at pore level and hydraulic resistance networks. Its porosity range of validity is larger than Calmidi’s, = 0.75 − 0.97 but regarding pore densities is more restricted, 10-40 PPI. The equation is √ 2 K = 0.054Dp 1 − . (3.15)

The results are collected and the result of calculating the permeability using a regression with com- plementary data for different Reynolds is added to table 3.2. It was also added to the table the ﬁnal result of this work correlation as well as the Dietrich [2] experimental correlation given by Hg = 110Re+1.45Re2 .The Hagen number(Hg) is a dimensionless number that can be understood as a dimensionless pres- 1 dp L3 sure drop and is deﬁned as Hg = ρ dx ν2 As can be seen on table 3.2 the results are, in majority, of the same order of magnitude even though

21 Re = 10

Author K1 Error K2 Error Simulation Regression 2.61 × 10−2 — 4.22 — Calmidi-Mahajan 6.47 × 10−3 75.2% — — Hooman-Dukhan 1.53 × 10−2 41.4% 1.23 70% Dietrich 3.86 × 10−2 47.98% 1.22 71.15%

Table 3.2: Correlation and Simulation Results.

Figure 3.9: Domain schematic (Left) and Mesh sample (Right) for cross flow cylinder problem the errors can not be neglected. The correlations of Hooman-Dukhan and Dietrich give very similar results, compared to each other. The discrepancy in results will be analysed later in this work on chapter 4. The correlation comparison results do not guarantee confidence in the results so another approach will have to be used. Following the AIAA guidelines and approaching the problem using ”building blocks” to validate this problem, the flow across a cylinder was studied. This is mainly due to the extensive research done on the subject and also the many results available for comparison. The cross cylinder flow was studied in the domain and mesh represented in figure 3.9 both for low Reynolds, steady laminar regime (Re = 20) and for laminar unsteady regime (Re = 200). The velocity field can be seen on figure 3.10 with the expected vortex shedding on the high Reynolds simulation. In order to avoid wall effects the domain has approximately 27 cylinder diameters in height, each side, 14 in front of the cylinder and 28 in its wake as can be seen on figure 3.9. The mesh was converted to 2D not only to avoid expensive computation time but also due to the extensive literature existing with this mesh arrangement. It is also important to acknowledge that results depend not only on the mesh refinement and convergence but also on the domain size. It appears that different blockage values have an influence on the quantities evaluated even though mesh convergence was attained. This was is consistent with what was referred by Domingos [53]. As stated on chapter2, the geometric parameters to be used on correlations do not gather con- sensus, specifically the length scale for the Reynolds number, which can make it hard to perform a comparison between different works.Possibilities range from the root of permeability, both strut and pore

22 Figure 3.10: Velocity Proﬁle for Re = 20(left) and Re = 200 (Right)

Author CD Present Work 2.06 Fornberg [54] 2.00 Dennis and Chang [55] 2.05 Tritton(Experimental) [56] 2.09

Table 3.3: Drag Coefﬁcient for Re = 20 diameter, and the hydraulic diameter which takes into account the inverse of the speciﬁc surface area,

SV , as in Dietrich [2]. Since a well defined periodic REV is being used and useful comparisons between porous media and cross cylinder flow regarding flow regimes can be made, the cylinder diameter was the chosen length scale.

uDs The velocity was set to obtain Res = ν = 20 with Ds being the cylinder diameter and equal to 0.22m as the REV studied before. The boundary conditions used were symmetry plane walls, velocity inlet and pressure outlet. The mesh parameters were chosen to be the same as mesh 5. The results can be seen on table 3.3. Setting the Reynolds number now to Re = 200 another regime is tackled. With the same mesh parameters the results are exposed on table 3.4. It is possible, analysing both tables, to conclude that the results are in conformity with the literature. To further validate the solution the problem of a 2D array of circles is studied. Taking a 2D square domain and circle with a ratio of P/D= 1.5 as seen on figure 3.11, with periodic conditions in every square face one can validate the mesh used. The results are exposed on table 3.5 alongside with the numerical results of Kevlahan [61] where it can be seen that for both regimes the errors are not greater than 2.5%, which, taking into account the scarcity of other literature, is considered acceptable. Taking all the verification and validation tests into account and evaluating the agreement as sufficient, the model assumptions hold and the errors in the discretisation process are reasonable for the proposed analysis.

23 Author CD St Present Work 1.28 0.18 Berger and Wille(Experimental) [57] – 0.18-0.19 Belov [58] 1.19 0.193 Rogers, Kwak* 1.23 0.185 Miyake et al.* 1.34 0.196 Liu [59] 1.31 0.192 Pan and Damodaran [60] 1.37 0.192

Table 3.4: Drag Coefﬁcient for Re = 200 , (*) in Belov

Figure 3.11: Domain schematic and mesh for 2D cylinder array problem

Figure 3.12: Velocity ﬁeld for Re = 10 (Left) and Re = 200 (Right) for 2D cylinder array problem

Author Reynolds CD Difference(%) St Difference(%) Present Work 10 29.890 0.3 — — Kevlahan [61] 10 29.8±0.2 — — — Present Work 200 2.623 2.45 1.071 1.27 Kevlahan [61] 200 2.56±0.2 — 1.085±0.045 —

Table 3.5: Results for periodic 2D array of cylinders

24 Chapter 4

Results

4.1 REV Validity

As discussed on chapter2, the REV size must fulfill certain characteristics in order to represent the pore scale flow. However, when dealing with periodic idealized structures the case is slightly different given the periodicity of the structure. When dealing with this structures one only has to guarantee that the simulation domain repeats itself and in doing so forms the complete structure. As such, both REVs presented in figure 4.2 are valid options to represent the idealized structure as depicted in figure 4.1

It is also important to note that this simplification can only be ge assumed when the flow is steady. Since the flow is steady the flow in any one of this REVs of the porous media will not change. The same can not be said about the unsteady regime. If vortex shedding occurs it will most likely not be periodic and the shedding direction will vary on space, differing for most REVs. An argument could be made that statistically the flow would ”even out” and the average would tend to be the same in every REV over time. This will be discussed further on this thesis.

Figure 4.1: Full periodic Structure

25 Figure 4.2: Possible REVs

Name Characteristics 100 The cubic cell represented in figure 4.2 111 Same geometry as above but the flow direction is the space diagonal of the REV cube AAA Cubic cell with modified strut shapes to ellipses with different orientations ABC Same geometry as above but the flow direction is the space diagonal of the REV cube

Table 4.1: Mesh Name and Characteristics

4.2 Different Geometries and Foam Anisotropy

In order to study the influence of different geometric parameters three different geometries were used and the anisotropy of two of them was analysed. For all geometries the porosity was varied between 0.6 and 0.9 with 0.1 increments. This arrangement of geometries will hopefully allow the analysis of the influence of: porosity, projected area, specific surface and tortuosity. These are the parameters that were set as variables to influence the permeabilities, and so the pressure drop, on the correlation. Table 4.1 contains a brief description of every geometry and the respective nomenclature. Figure 4.3 shows the area projection in the direction 100 and 111. The projected area in the direction of the flow will increase and consequently an effect on tortuosity is expected to occur and will be quantified later in this thesis. Figure 4.4 shows the last geometry, AAA and ABC obtained scaling the axis and maintaining the volume of the circle in order to have the major axis of the ellipse to equal 125% of the circle diameter

Figure 4.3: Projected areas in the direction: 100 (Left) and 111(Right)

26 Figure 4.4: Projected areas in the direction: AAA (Left) and ABC (Right) and the minor 75%. The ellipses major and minor axis are different for each strut and the arrangement is clear on the figure. The first arrangement allows for the study of the influence of strut shape on transition Reynolds as well as tortuosity while the second will make it possible to study the anisotropy and consequences it brings to transition and pressure drop.

4.3 Coefﬁcients Computation

As discussed in chapter2 when high velocity flow is present in the porous media, pressure drop follows the Forcheimmer equation. For this equation to be meaningful and representative, accurate results of the intrinsic permeability, k1, and the inertial coefficient, k2, must be obtained either by estimates given by correlations of geometric parameters or by data obtained from pore scale simulations. To do the latter the simulation data will have to be fitted to the Forcheimmer equation given by eq(2.15). The imposed pressure source terms were spaced equally in a logarithmic scale and plotted against the logarithm of the obtained velocities. This procedure is required to guarantee the resolution on the Darcy regime. The fit was then made to minimize the sum of the square of the difference between the estimated and simulated value. This approach guaranteed results with a difference of up to 10% from second order polynomial fittings forced through the origin. The solution procedure for the simulations was completed as follows:

i) Imposing a momentum source in every cell of the domain

ii) Obtaining the pressure and shear force on the structure

iii) Obtaining characteristic velocities in the domain: maximum, Darcy, pore, and axial pore velocity

The pore velocity, up was calculated as a volume average quantity and can be written in discrete form, depending on velocity magnitude, umag, by:

PN u .V ol u =< u >f = i mag,i i . (4.1) p mag PN i V oli

Regarding the axial pore velocity, upi the formula is identical, however, the considered velocity is the velocity component in the direction of the ﬂow. The Darcy velocity was calculated involving the REV faces, via the volumetric ﬂux for each direction (pair of faces).

27 Figure 4.5: Dimensionless pressure gradient vs Reynolds for 4 different arrangements

Figure 4.6: Pressure gradient vs Reynolds for 4 different arrangements

N X uDi = uaxial,i.Ai.ni (4.2) i with the volumetric ﬂux for each face the Darcy velocity is computed calculating the magnitude of the vector as

q u = u2 + u2 + u2 . (4.3) D Dx Dy Dz

In order to study the anisotropy two flow angles relative to the structure were considered, 0o and 45o. Here the coefficients, obtained for a porosity of 0.6 are presented for the pairs of arrangements studied and the dimensionless pressure gradient (DPG) and pressure gradient are plotted in figure 4.5 and 4.6, respectively.

2 The DPG equation can be obtained simply dividing the Forcheimmer equation by ρ and uD as

1 dp 1 ν 1 2 = + , (4.4) ρ dx uD k1UD k2

28 Name SP SV 100 0.55 2.58 111 1 2.58 AAA 0.55 2.72 ABC 1 2.72

Table 4.2: Speciﬁc projected and surface area for different arrangements at = 0.6

and dividing and multiplying the ﬁrst term on the RHS by the strut diameter ds, one arrives at

d 1 1 dp∗ = + . (4.5) k1 ReD k2

Several aspects can be drawn from the plots on figures 4.5 and 4.6. Firstly, the anisotropy in the 1-series foam influences the Forcheimmer term by noticeably deviating from the linear regime. This can be attributed to the increase in projected area making the term not negligible for the same velocity. This geometric parameters are listed on table 4.2 One can also conclude that modifying the strut shapes from cylindrical to ellipsoidal shapes translates the curve up. This means that for the same velocity the pressure gradient is higher and hence the permeability is lower. Additionally, the curves 100 and 111 coincide for the lower Reynolds number because their specific surface is the same and the velocity of the flow is still not considerable. Con- sequently, the flow is constrained by the same obstacles and it is not until latter (when the projected 2 area starts to become meaningful, due to higher uD) that the flow will feel this change. This effect is evident since for the same pressure gradient the velocities obtained are the same and so the intrinsic permeability, k1, is the same for both arrangements. The same can not be said regarding ABC and AAA. In this geometry, the specific surface is greater and therefore for the same imposed pressure gradient the velocity will be lower. This is translated into a decrease in k1 that can explain the translation of the curve upwards, since for low velocities the second term on the RHS is negligible. The equality of k1 for both ABC and AAA and the negligibility (due to higher 1st term) of the constant term for low velocities is also the reason why the curves overlap in this regime.

The effect of the inertial coefficient, k2, only becomes notorious when the velocity turns the first term on the RHS low enough to become of the same order of magnitude of 1 . This happens when k2 −4 o uD ≈ 3 × 10 , Re ≈ 9 for the arrangements at 45 . The coefficients are presented in table 4.3. Conclusions about the possible onset of unsteadiness can also be drawn. For both geometries when the flow is angled, hints of unsteadiness happen at lower imposed pressure gradients. Comparing different struts for the same type of flow one can see that, as expected, the ellipse shaped struts anticipate transition for both angled and aligned flow. Figure 4.5 sheds some light on the hypothesis of a transition criterion based on the root of permeability or Reynolds number alone. It is often used as a rule of thumb that the transition to the inertial regime happens at ReD ≈ 10, however, from figure 4.5 this is only the case for the angled flow arrangements.

29 1 d 1 Name K1 K2 K2 K1 ReD 100 8.85 × 10−3 1.36 0.74 5.94 111 8.85 × 10−3 0.66 1.52 5.94 AAA 7.82 × 10−3 1.12 0.90 6.72 ABC 7.82 × 10−3 0.57 1.76 6.72

−4 Table 4.3: Coefﬁcients for 0.6 porosity and uD = 3 × 10

Figure 4.7: Axial velocity PDF for different Reynolds numbers for 100 and 111

For aligned flow this seems not to be the case as no influence of quadratic effects can be detected from the plots. The PDF of the axial velocity for the 1-series arrangements is plotted on figure 4.7. It is possible to observe the difference in flow regime as the pressure gradient increases. The first two increments in pressure jump did not have a noticeable effect on the velocity field as the flow is still in the Darcy regime.

This similarity in the velocity field justifies the coincidence of k1. The last Reynolds number evaluated has very different velocity fields depending on the flow direction with two peaks on the opposite side of the axis. The recirculation zones are also consistent with what was expected. For the aligned direction only in the inertial regime does the velocity start to assume negative values characteristic of inertial recircu- lative flow. Figures 4.8 and 4.9 show the streamlines. From the axial velocity PDF one can also agree that for the same Reynolds the velocities are more uniform for the 100 arrangement while on the 111 arrangement the negative velocities cover double the negative range. It is also worthy of note that for all the pressure gradients investigated the angled flow originates some notorious presence of negative velocities. For the A-series, PDF plotted on figure 4.10, the comparisons between angled and aligned flow still hold, however comparing between geometries there are some noteworthy differences. Firstly by modifying the strut shape the velocity field becomes less uniform as the range of normalised velocities is increased, specially in the negative velocity areas. The range of negative velocities roughly doubles

30 Figure 4.8: Streamlines for 100 geometry at Reynolds: 1.8 (Left) and 60 (Right)

Figure 4.9: Streamlines for ABC geometry at Reynolds: 1.8 (Left) and 12.1 (Right)

Figure 4.10: Axial velocity PDF for different Reynolds numbers for AAA and ABC

31 Figure 4.11: Full steady range axial velocity PDF for different Reynolds numbers for: 100 and 111 (Left) AAA and ABC (Right)

Figure 4.12: Flow diagram on cylinder array [64] for the respective geometries with the shape modification and the peaks are also slightly smoothed for the ellipsoidal struts. The PDFs for the full range of Reynolds are plotted on figure 4.11. It is apparent from the PDF for 100 geometry that the curves converge as the Reynolds increase, as can be seen by the negligible difference between Re = 30 and Re = 60. After Re ≈ 60, the residuals and velocity field start oscillating so a change in regime from steady to unsteady seems to occur for 100 geometry while for a single cylinder it occurs typically at Re ≈ 50 [62]. This can be explained by several factors. The results of a delayed transition to unsteadiness are concordant with both Kevlahan’s [61] and Price’s [63] results from simulation and experiments with arrays of cylinders where unsteadiness occurs around Re ≈ 100 for both of them. Kevlahan also tackled the angled flow problem and the results are concordant with this work’s quali- tative analysis, the transition to unsteadiness occurs at Re ≈ 60, earlier than for aligned flow. This might be explained by the obstruction of the free flow lane as depicted on figure 4.12. It is expected that the values are different since a steady solver is used, however, qualitatively the conclusions are sound. Regarding the AAA PDF, the flow is only steady until Re ≈ 30. The difference can be attributed not to the overall structure projected area but to the geometry the ellipsoidal strut. If the major ellipse axis is perpendicular to the x-axis transition occurs earlier as studied by Jackson [65], who predicted a critical Reynolds equal to Re = 39 for a 2-D ellipse with the major to minor axis ratio used in this work. If the minor axis is perpendicular to the flow the inverse happens. In the AAA arrangement there will always be one strut with major axis perpendicular to the flow and so the transition occurs earlier.

32 Figure 4.13: Normalized pressure gradient vs Re

Figure 4.14: Both terms magnitude (Left) Second term percentage of total (Right)

4.4 Darcy and Forcheimmer regimes

In the previous section it was shown that the permeability, k1, is only influenced by specific surface area without influence of flow direction or projected area. However, for the inertial coefficient less definitive conclusion were drawn. In an attempt to study the transition between both regimes, the pressure drop equation was normalized with Darcy’s velocity in order to have the Darcy term as a constant.

1 dp 1 ν u = + D . (4.6) ρ dx uD k1 k2 The first term on the LHS of equation (4.6) will be called normalized pressure gradient (NPG) from now on. The quantity can be seen on figure 4.13 plotted against the Reynolds number. Figure 4.14 shows the influence of both terms together with the percentage of the second term in respect to the sum of both. Figure 4.14 shows that the inertial coefficient deviates earlier in the angled rather than aligned directions and the hypothesis that transition occurs at the same Reynolds number is disproved. Another typical transition indicator is the Forcheimmer number as defined by E.Werzner [66]. This number can be viewed as a Reynolds number whose characteristic length is the ratio of permeability and inertial coefficient,

33 Figure 4.15: Variation of inertial to total force ratio with respect to Fo

Name K1Sim. K2Sim. K1H-D K2H-D K1Lacrx K2Lacrx K1Cal. K2Cal. 100 8.85 × 10−3 1.36 2.06 × 10−2 2.65 1.83 × 10−3 0.44 2.47 × 10−3 0.217 111 8.85 × 10−3 0.66 2.06 × 10−2 2.65 1.83 × 10−3 0.44 2.47 × 10−3 0.217 AAA 7.82 × 10−3 1.12 2.06 × 10−2 2.65 1.83 × 10−3 0.44 2.47 × 10−3 0.217 ABC 7.82 × 10−3 0.57 2.06 × 10−2 2.65 1.83 × 10−3 0.44 2.47 × 10−3 0.217

Table 4.4: Coefﬁcients for 0.6 Porosity and Hooman-Dukhan, Lacroix and Calmidi correlations

u k F o = D 1 . (4.7) νk2

F o Having this coefficient in account the inertial to total forces ratio can be expressed by 1+F o [66]. Fig- ure 4.15 shows its dependence from the Forcheimmer number. This dimensionless group is important because the curve will collapse and a definitive transition criteria can be defined. The criteria to the departure of Darcy flow is ambiguous but will be considered to be F o = 0.11 since neglecting the inertial term until that Fo would imply an error of under 10 % but neglecting it further would lead to a rapid increase. The criterion for inertia dominated flow is obvious and equal to F o = 1. F o This is when 1+F o equals 0.5, meaning that inertial and viscous forces are equally important and above that inertia will dominate. The coefficients obtained in the previous section, listed in table 4.3 are compared with the coefficients obtained from literature correlations that use porous foam model with the unit cubic cell. Table 4.4 lists the values obtained by Hooman-Dukhan’s correlation [3] and shows a reasonable agreement with the simulation. However, it has no account of tortuosity or projected area and then has no way of taking into account the fact that the flow is angled and gives the same output for every geometry. The overestimation of permeabilities might be related to two main reasons. Firstly, in the derivation of the correlation, a simplifying assumption regarding the drag coefficient of cylinders was made and it was 6 assumed to be CD = 1+ Re . It can be seen on figure 4.16 that this underestimates the drag and therefore

34 Figure 4.16: CD Assumption (Left) Prediction and experimental values of of ∆P/(LU) (Right) [3]

Figure 4.17: Pressure drop of foams vs packed bed [23] would have an effect of over predicting the permeability. Secondly, in the article the experimental values of pressure drop were constantly higher meaning that the permeability was overestimated giving strength to the argument made. It is to be expected that at higher Reynolds number, as the approximation of drag coefﬁcient is better adjusted, the results show a better agreement. Comparing the coefﬁcients with the Lacroix correlation [23] we obtain the results presented on table

4.4. The correlation is based on the Ergun equation conserving the same coefficients E1 and E2 but adjusting the particle diameter of a sphere to a cylinder (conserving the specific surface area) making an analogy between a pack of sphere and a cubic unit cell. This approach has several drawbacks. To begin with we can see that there is also no way for the correlation to account the directionality of the flow and the coefficients are constant for every geometry. Apart from this problem, making an analogy based on the conservation of specific surface area induces errors since, as the authors state, the pressure drop on foams for the same specific surface area is lower than for a pack of beds as can be seen on figure 4.17. Taking this into account the correlation would underpredict the permeability for foams and that is actually what happens in this case. Calmidi’s correlation [38] again used the cubic cell to approximate a porous metal foam structure. After realizing experiments on the foam a fitting was done and a correlation derived. The coefficients

35 Figure 4.18: Pressure drop vs Predicted pressure drop for the different geometries can be seen on table 4.4. The same problems rise regarding the directionality of the flow. Other than that, the underprediction of the coefficients can be justified due to the different regime the experiment was done and also the high porosity of the foam.

Lu [34] also used the cubic cell correlation to derive a pressure drop expression. The plot can be found on ﬁgure 4.18 for the different geometries, comparing with the simulation curve. The correlation makes use of a friction factor to account for pressure drop in cylinders given by the experimental Holman- 0.008(a/d) −0.15 Jakob model f = (0.044 + (a/d−1)0.43+1.13(d/a) )Re . This experimental model was not derived for the regime of this work and hence the disparity in results.

It is obvious from observation of the figure that the error in the prediction of pressure drop lowers as Reynolds increases. This is in conformity with the argument given above. To finish this brief comparison the coefficients from correlations of Du Plessis [67] and Dietrich [2] are presented on table 4.5. Dietrich just takes into account the hydraulic diameter, related with specific surface area, and hence the coefficients will disregard the flow direction and be equal for the pairs of geometry with the same SV . The correlation comes from fitting experimental data for vastly different ceramic and metal foams at 10 < Re < 3900. For that reason, it can be justified that the disparity occurs, however, it is also important to take notice that the results are better predictions than those detailed before on table 4.4.

Du Plessis on the other hand, modeled metallic foams as a Representative Unit Cell (RUC) as it can be seen on ﬁgure 4.19. The equation takes into account the tortuosity which will be useful to distinguish between the ﬂow directions and the effects it has on the permeability. The authors also present a relation

36 Name K1Sim. K2Sim. K1Du Pl. K2Du Pl. K1 Dietrich K2 Dietrich 100 8.85 × 10−3 1.36 8.16 × 10−2 0.10 5.08 × 10−3 0.25 111 8.85 × 10−3 0.66 8.47 × 10−3 0.35 5.08 × 10−3 0.25 AAA 7.82 × 10−3 1.12 1.25 × 10−1 0.082 4.55 × 10−3 0.24 ABC 7.82 × 10−3 0.57 8.07 × 10−3 0.36 4.55 × 10−3 0.24

Table 4.5: Coefﬁcients for 0.6 Porosity and Hooman-Dukhan, Lacroix and Calmidi correlations

Figure 4.19: RUC used by Du Plessis [22] to model metallic foams between tortuosity and porosity. The actual values of tortuosity were used instead of the relation. The results are accurate, only for the angled ﬂow cases. It is also worth to note the sensitivity to the tortuosity, in particular if the values are low. For the 100 and AAA geometry the differences in tortuosity are roughly 1% (1.02 and 1.03 respectively) however the coefﬁcient changes by almost 34%. This results reinstate the need of a correlation suited for the Darcy and laminar regime for ideal porous media.

4.5 Force type and velocity ﬂow ﬁeld

Taking into account the equations derived in Chapter 2, in particular equation (2.12) the terms regarding viscous and pressure forces were calculated based on the equations,

Z Fv = − µ∇v.ndS, (4.8) Asf

Z Fp = P.ndS. (4.9) Asf Table 4.6 is presented for the particular case of 100 at a porosity of 60%. It is obvious from the table that at low Reynolds the forces acting on the structure are not only due to shear but mostly due to pressure. This ratio is dependent on the structure itself and not on the flow velocity (for low Reynolds). This can be explained by the theory of Stokes flow where by virtue of low Reynolds number the inertial effects can be neglected but the pressure gradient can still be significant. Schampheleire et al [68] studied this process for a unit cell and compared the case of a sphere subjected to Stokes flow and staggered cylinders. For the former, pressure accounts for one third of the

37 Fp Re Fp Fv Fp+Fv 0.05 1.77 × 10−9 1.25 × 10−9 0.59 0.50 1.78 × 10−8 1.25 × 10−8 0.59 1.24 4.50 × 10−8 3.09 × 10−8 0.59 1.94 7.23 × 10−8 4.81 × 10−8 0.60 2.99 1.17 × 10−7 7.36 × 10−8 0.61 4.56 1.92 × 10−7 1.11 × 10−7 0.63 6.90 3.15 × 10−7 1.64 × 10−7 0.66 10.40 5.18 × 10−7 2.41 × 10−7 0.68 15.66 8.51 × 10−7 3.52 × 10−7 0.71 23.55 1.39 × 10−6 5.13 × 10−7 0.73 35.32 2.28 × 10−6 7.49 × 10−7 0.75 69.47 4.81 × 10−6 1.36 × 10−6 0.78

Table 4.6: Force type and coefﬁcient for 100 at 0.6 porosity

Figure 4.20: Pressure to total force ratio for 0.6 porosity geometries forces while for the latter it accounts for 63%. The present work’s case is consistent with the latter with an influence of 59%. Figure 4.20 shows the influence of geometry on the ratio of pressure and viscosity forces. It is clear that the ellipsoidal geometries have a greater pressure ratios and comparing this with figure 4.14 the behaviour seems counter intuitive since the inertial term on the aligned directions starts to grow later than the angled directions in opposition with pressure ratios. For high Reynolds, differences between the pressure force of 111 and 100 or ABC and AAA are negligible although the viscous forces are greater at angled directions than at aligned directions. The relative percentage of pressure force to viscous forces is then lower on the angled direction than at the aligned direction. This can be seen on figure 4.21, where the surface integral of shear stress is higher on 111. Figure 4.22 shows the evolution of pressure force that may be approximated by a second order polynomial for the whole range of Reynolds numbers, however, for low Reynolds numbers (until Re ≈ 3) the curve behavior is not second order but yet linear. Figures 4.23 and 4.24 show how linear or non-linear behaviour is dependent on flow field.

38 Figure 4.21: Wall shear ﬁeld for 100 (Left) and 111 (Right) for 0.6 Porosity

Figure 4.22: Pressure vs Reynolds for 111 and 0.6 porosity

Figure 4.23: Velocity ﬁeld for 0.6 porosity for 100 geometry at Reynolds: 1.8 (Left) and 60 (Right)

39 Figure 4.24: Velocity ﬁeld for 0.6 porosity for ABC geometry at Reynolds: 1.8 (Left) and 60 (Right)

Figure 4.25: DPG vs Reynolds for 4 different porosities for 100 (Left) and ABC (Right)

4.6 Porosity and Tortuosity Effects

Up to here, all the geometries studied had a porosity equal to 0.6. In this chapter the influence of changing the porosity to a greater value is going to be evaluated. To evaluate the effect of porosity on pressure drop the DPG is plotted on figure 4.25 against Reynolds number for 100 and ABC. As expected with an increase in porosity the pressure drop decreases. This means that increasing porosity one increases permeability. The curve topology is conserved throughout the increase in porosity, being dependent only on the geometry (see figure 4.26) and the assumption that k1 does not depend on SP still holds for every porosity. To account for the effects of porosity on the transition to the inertial regime the Forcheimmer number

νk2F o is used. From the deﬁnition of Forcheimmer number, equation (4.7), it can be said that uD = and k1 hence the transition to Forcheimmer regime based on this deﬁnition is given by the equation

k2 Retr = dsF o . (4.10) k1

Table 4.7 lists the results for the ABC geometry. It can be seen that the transition Reynolds number decreases with the increased porosity. To gain a better understanding of this phenomena the evolution

40 Figure 4.26: Permeability vs porosity

Porosity Retr D 0.6 3.58 0.49 0.7 2.79 0.41 0.8 1.88 0.32 0.9 1.31 0.22

Table 4.7: Transition Reynolds for ABC geometry and different porosities of the fraction of pressure to total force is plotted on figure 4.27 for both 100 and ABC. It is possible to see that the fraction of pressure force is constant, under Stokes flow, for each porosity and that the departure from this state is first made for the higher porosities, confirming the results obtained using equation (4.10). The pressure force ratio increases as the porosity decreases and this happens not only in ratio but also in magnitude, for both forces, as would be expected from the increases in projected and specific surface area.

Regarding the angled direction the same conclusions can be drawn. It is also apparent that the values are less dispersed for the angled directions and are greater for each porosity when compared with the aligned ones. This is explained by the ellipsoidal strut shape and not by the ﬂow direction. In fact, for the same Reynolds number, the aligned geometries have higher pressure fractions than their angled counterparts as it can be seen on table 4.8, the angled ﬂow promotes more viscous and lower pressure forces.

To access the effect of porosity on the non-Darcian permeability k2, the value is plotted on ﬁgure

4.28. The effect is analogous to the Darcian permeability and the k2 increases as the porosity increases. The angled directions are homogeneous with no major difference between modiﬁed and non modiﬁed cylinder struts. The same can not be said about the aligned directions where the ellipsoidal shaped struts have considerably lower permeabilities, particularly for higher porosities.

Taking into account what was discussed for ﬁgure 4.28 it would be expected that the ellipsoidal shape

41 Geometry FP FV ABC 2.35 2.21 AAA 2.37 2.18 111 2.27 2.26 100 2.29 2.24

Table 4.8: Pressure and friction forces for Re = 1 for 0.9 Porosity

Figure 4.27: Pressure to total force ratio 100 (Top) and ABC (Bottom) for 100 and ABC

42 Figure 4.28: Inertial coefﬁcient vs porosity for all arrangements would promote pressure forces and hence it might be reasonable to argue that these would take more importance as the Reynolds numbers increase and the inertial regime is in order. Comparing the forces acting on the structure at Re = 29 table 4.9 is presented. In this case, for the same velocity the pressure gradient is higher for the ellipsoidal structure and hence the permeability will have to be lower.

Geometry FP FV 100 1.20 0.60 AAA 1.22 0.59

Table 4.9: Pressure and friction forces for Re = 29 for 0.9 porosity

To understand how ﬂow differs by increasing the porosity of ABC, the PDF were plotted for two Reynolds, 0.16 and 17 and the results are presented on ﬁgure 4.29. It is apparent from observation that increasing porosity narrows the curve, however it can not be neglected that maintaining a constant

Reynolds number for different porosities results in the average velocity, up, being lower for lower porosities since the characteristic length is increasing. It can also be said that with decreasing porosities the density of negative axial velocity increases which is a indicator of recirculation. With increasing Reynolds number the negative values stretch to more negative ones and the curve topology for different porosities appears to be more uniform for lower velocities, but less uniform in general. As discussed in chapter2 many correlations, including this work’s, use tortuosity in order to account for differences that can not be easily attributed to geometry. Between many possible deﬁnitions tortuosity might be described by the ratio between the actual streamline and the straight line distance through the medium in the direction of the ﬂow.

L τ = (4.11) Ls

43 Figure 4.29: Axial velocity PDF for ABC and Re = 0.16 (Left) and Re = 17 (Right) for different porosities

Figure 4.30: Tortuosity vs Reynolds for 100 geometry (Left) and ABC (Right) for different porosities

To use this deﬁnition one would have to trace the particle trajectories and compare them with the straight line distance. An alternative was proposed by Duda et al. [69] which had into account the variables already calculated. The equation is

< U > τ ≤ , (4.12) < Ux > where <>denotes a spatial average over the pore space. Using the equation proposed by Duda et al. the effects of Reynolds on tortuosity are studied. In figure 4.30 it can be seen that tortuosity is constant when the structure is under Stokes flow. However, when velocity is increased the tortuosity rises, only after decreasing, through a graph valley. In the ABC structure this regime is not achieved since the flow is under transition to unsteadiness which occurs before the tortuosity can recover the initial value of Stokes flow, even though the tendency is clearly to increase. Figure 4.31 plots the tortuosity against the porosity for all arrangements studied. It is obvious that aligning the flow with the structure tends to lower tortuosity and that as porosity decreases tortuosity increases. The former was explained in the previous section and the latter can be explained due to the higher fluid constraints which makes it impossible for streamlines not to curve. The weakness of the statement proposed by Duda, equation (4.12), is due to the flow possibility

44 Figure 4.31: Average Tortuosity vs porosity

Figure 4.32: Particle path and residence time for Re = 30, 0.6 porosity AAA, ﬂow from left, xy plane, from top to bottom, left to right, 100s, 300s, 700s and 10000s

45 Figure 4.33: Particle path and residence time, Re = 17, 0.9 porosity, flow from right, for ABC (Left) and AAA (Right) of being compressible or reentrant. For the former, our assumptions would not pose a problem since, as explained in chapter3, a constant density model was assumed. Against the latter it can be argued that the reentrant and recirculating flow occupies a smaller volume fraction and contributes with lower velocities, so the effect on the averaged value might not be relevant. However, in order to evaluate the error present in this method a particle tracking approach is followed. To achieve this, close to 20000 massless particles were launched from the inlet of the 100 geometry and their trajectories were tracked. Only particles without excessive residence time are accounted for, having this particles also to enter and leave the domain excluding, this way, the recirculation zones from the calculation. The streamlines can be seen on figure 4.32 for Re = 30. It is visible that the particles enter a first recirculation zone, which is the tail of the recirculation zone downwind of the cylinder, forced by the inlet feed. However, the vortex behind the cylinder is completely empty. Applying the quotient of the actual length divided by the system length the tortuosity was computed to be 1.01, a significant deviation from the previously computed value of 1.06 with the average velocity formula. This is because of the vast portion of the flow that is under recirculation that influences the flow nature and stabilizes it as studied by Liu [70]. With 6 recirculation bubbles in the REV, the fluid travels through 4 clear channels, feeding the recirculation bubble through shear force with no contribution to mixing. It then means that for the range of Reynolds numbers studied as Reynolds numbers increase and the channels are formed tortuosity will decrease until the flow becomes unsteady. This is also the reason why for lower porosities the transition is delayed. For 0.9 porosity the recirculation bubble wake does not connect with the next cylinder before transition occurs around reynolds 50. For lower porosities this is not the case and the flow is stabilized being able to be steady until a Reynolds number based on pore velocity of more than 100. This particular behaviour means that this geometry, 100, and the results regarding tortuosity and transition that are based on it might not be representative of conventional porous media. Regarding the angled directions, the problem of high volume recirculation areas does not appear as

46 Geometry LLi dp ds Sv Sp 111-09 1.000 1.732 1.512 0.220 1.661 0.686 0,901 111-08 1.000 1.732 1.408 0.324 2.162 0.876 0,801 111-07 1.000 1.732 1.322 0.410 2.439 0.968 0,701 111-06 1.000 1.732 1.242 0.490 2.582 1.000 0,602 ABC-09 1.000 1.732 1.512 0.220 1.706 0.686 0,907 ABC-08 1.000 1.732 1.408 0.324 2.242 0.876 0,810 ABC-07 1.000 1.732 1.322 0.410 2.551 0.968 0,713 ABC-06 1.000 1.732 1.242 0.490 2.725 1.000 0,615 100-09 1.000 1.000 0.780 0.220 1.661 0.392 0,901 100-08 1.000 1.000 0.676 0.324 2.162 0.543 0,801 100-07 1.000 1.000 0.590 0.410 2.439 0.652 0,701 100-06 1.000 1.000 0.510 0.490 2.582 0.740 0,602 AAA-09 1.000 1.000 0.780 0.220 1.706 0.392 0,907 AAA-08 1.000 1.000 0.676 0.324 2.242 0.543 0,810 AAA-07 1.000 1.000 0.590 0.410 2.551 0.652 0,713 AAA-06 1.000 1.000 0.510 0.490 2.725 0.740 0,615

Table 4.10: Geometric parameters of the correlation foams the flow, in its vast majority, works around the structure with very small recirculation areas, as it can be seen in figure 4.9 and by studying it with particle tracking, resulting in figure 4.33. As for the AAA geometry the conclusions drawn for 100 will follow, as depicted on figure 4.33

4.7 Correlation Analysis

The validity of the correlation will now be tested with the results obtained above and results offered by Eric Werzner et al., PhD [66] and Andre´ Calado [71]. This analysis will be done for the permeability and non-Darcian permeability separately, as k1 is independent of k2 and is treated as a constant for the whole range of Reynolds.

4.7.1 Darcy Permeability

As it was illustrated in ﬁgure 4.22, for low Reynolds number(Re ≤ 3) the pressure force varies linearly with the the Reynolds number, and the same can be said for the shear force. It is then clear that only one simulation is needed to calculate k1 which will be used for the whole spectrum of Reynolds. For the Darcy regime the equation that is dealt with is

1 ∂p 1 SV τ = 2 νuD, (4.13) ρ ∂x α1 d and all the uncertainty about the ﬂow is put in the variables 1 and τ. Since τ is provided by the α1 simulation results one can investigate a possible relation between the fraction 1 in the studied cases. α1

µuD Firstly, k1 is calculated based on a simply rearranged Darcy equation, k1sim = dp . The geometric dx parameters of all the arrangements used can be seen on table 4.10. The parameter α1 is then calculated,

47 Figure 4.34: α1 as a function of porosity

as a constant, by rearranging equation (4.13).

k1sim α1 = 2 . (4.14) ds Sv τ

This coefﬁcient is then calculated for all the studied geometries and the average is calculated. The computed average α1 is equal to 0.205 and the standard deviation, σ, equal to 0.06, approximately 30% of the average value.

In order to test the validity of this coefficient several foams were studied. Calado [71] used a ceramic commercial foam of 10 PPI as seen on figure 4.36. Eric Werzner, on the other hand, analysed several ceramic foam filters with a range of 10-30 PPI and two different materials, Al2O3 − C and Al2O3. This foams also vary in strut shape factor that can be either 0.9 or 1.1. The geometric parameters can be seen on table 4.11 for the various foams and the foams can be seen on figures 4.35.

Werzner performed all simulations under a pressure gradient in the x-direction while Calado tested the anisotropy of the foam by testing the 3 main directions. Their results of permeability were tested against the results of the correlation and the results are presented on figure 4.34. It is visible that the parameters are not constant and visibly dependent on porosity. For the same porosity there is still some dispersion so the effect of tortuosity might have to be enhanced. This might be explained by assuming that β ∝ C τ . This is based on the inability of the boundary layer to grow if the cylinders are too close (low porosity) and the use of high tortuous flow surfaces to limit boundary layer growth (e.g. golf balls) and hence the inverse proportionality. It was then found that in order to find a constant coefficient c1 the data is better fitted if a dependency of the constant on the ratio of pore diameter to characteristic length dp/Li is considered.

This hypothesis was tested with the promising outcome of 21% maximum error for the k1 of the anisotropic foam of Calado, on the y-direction. This would mean that α1 is not in fact a constant but yet deﬁned by equation (4.15). Regarding the results of the coefﬁcient c1 the maximum deviation from the mean was for the same simulation of Calado with a difference of 27%.

48 Figure 4.35: Different shapes of strut studied by Eric Werzner [66] (Top) and different materials and PPI (Bottom)

Figure 4.36: Foam studied by Calado

dp,i dp,i α1 = c1 = c1 (4.15) τ dp,i + ds τ Li

The errors presented were for real foams. If periodic foams are accounted for, DuPlessis correlation maximum error increases greatly to 801% errors. A sensitivity analysis was done to the exponent of the parameters dp , and τ having a 10% variation of the total coefficient value within each one meaning Li that, the exponent of dp can vary between 0.9-1.1, between 2.7-3.3 and τ 1.8-2.2. This results in 27 Li (3x3x3) combinations for which the maximum error was computed. The results are presented on figure 4.37 and the combinations order is present on table 4.12 It is clear from inspection that the tortuosity exponent is not particularly relevant for most cases. On the other hand, regarding porosity, the correlation is extremely sensible and there is a general tendency of lower errors for higher porosity exponent. Regarding the exponent of the fraction Li , it follow the same dp trend of porosity with lower errors for higher exponent. With this considered the final Darcy correlation is taken to be 2 1.1 1 ∂p 1 SV τ (dp/Li) = 3.3 νuD. (4.16) ρ ∂x c1 ds and by plotting the value of c1 it is visible that the coefficients fit nicely to a constant curve valued c1 = 0.461 as seen on figure 4.38. Table 4.13 shows the error of the variations of the correlation tested

49 Geometry LLi dp ds Sv Sp PPI xx-dir 5,33E-03 5,00E-03 4,30E-03 7,00E-04 653,000 309,0 0,830 10 yy-dir 5,33E-03 6.10E-03 5.40E-03 7,00E-04 653,000 211.0 0,830 10 zz-dir 5,33E-03 4,90E-03 4,20E-04 7,00E-04 653,000 326,0 0,830 10 k0.9-05 1 1.467 1.124 0.344 2.822 2.012 0.500 20 k0.9-06 1 1.467 1.160 0.307 2.910 1.818 0.600 20 k0.9-07 1 1.467 1.201 0.266 2.837 1.380 0.700 20 k0.9-08 1 1.467 1.250 0.217 2.578 1.026 0.800 20 k0.9-09 1 1.467 1.314 0.154 2.042 0.670 0.900 20 k1.1-05 1 1.467 1.124 0.344 2.740 1.968 0.500 20 k1.1-06 1 1.467 1.160 0.307 2.833 1.588 0.600 20 k1.1-07 1 1.467 1.201 0.266 2.770 1.294 0.700 20 k1.1-08 1 1.467 1.250 0.217 2.517 0.980 0.800 20 k1.1-09 1 1.467 1.314 0.154 1.982 0.649 0.900 20 E1 1 1.512 1.315 0.198 2.727 0.675 0.801 10 E2 1 1.489 1.244 0.245 2.740 0.755 0.778 20 E3 1 1.529 1.342 0.186 2.916 0.683 0.784 10 E4 1 1.495 1.307 0.188 2.786 0.722 0.787 30 E4 1 1.472 1.288 0.185 2.813 0.707 0.805 30

Table 4.11: Geometric Parameters of the tested foams

Number Li Exponent Exponent τ Exponent 1 1 3 2 2 1 3 1.8 3 1 3 2.2 4 1 2.7 2 5 1 2.7 1.8 6 1 2.7 2.2 7 1 3.3 2 8 1 3.3 1.8 9 1 3.3 2.2 10 0.9 3 2 11 0.9 3 1.8 12 0.9 3 2.2 ......

....

Table 4.12: Correlations maximum absolute Error[%] of K1 for the tested foams

50 Figure 4.37: Sensitivity Analysis

Figure 4.38: c1 for periodic and non periodic foams so far and the next variation taken. The previous approach lets us evaluate the parameters that inﬂuence the permeability. However, to be able to use the correlation without any simulation or information about tortuosity, a correlation relating tortuosity with physical parameters will need to be used. Several correlations were tried and the results are explicit on table 4.14. Another approach was used by considering the boundary layer parameter β to include τ 2 and trying 1.3 1.1 dp to relate this parameter, fβ = 2 , against porosity so that the pressure drop could be calculated τ Li from

1 ∂p 1 SV ν 1 = 2 uD. (4.17) ρ ∂x c1 d fβ

3.401212 By fitting the simplified geometries structure’s data a good correlation was found with fβ = 0.0298e as it can be seen on figure 4.39 with a strong determination coefficient R = 0.9984. With this approach and applying this to all data, the maximum error was 30%, almost half of the best literature correlation

51 Correlation variation Max. error [%] Avg. abs. error [%] Theoretical 80 31 Sens. Analysis 17 7 fβ 31 9.

Table 4.13: k1 Correlation variations error

Correlation Maximum error [%] Du Plessis [72] 45 Yang [73] 77 Vallabh [74] 15 Yu and Li [75] 43 Chen [76] 51

Table 4.14: Tortuosity correlations maximum absolute error [%] for the tested foams tested, by Dietrich. The correlation average absolute error can be seen on table 4.15 compared to literature correlations. A detailed visualisation of the error for all geometries and correlations is present on ﬁgure 4.40 with a limited y-axis (Lacroix predictions for Calado foams k1 are severely off). It can be seen that there is no general tendency to over or under predict by the correlations with exception of Lacroix’s correlation which systematically under predicts the coefﬁcients unless for the Calado ceramic foam where it vastly over predicts the permeability. It is also clear that the ceramic foam studied by Calado is the most prone to errors with all correlations having peak values of error when analysing it. The correlation (equation 4.17) outperforms the literature correlations maintaining a low error through periodic and non periodic foams.

4.7.2 Non-Darcian Permeability

The Forcheimmer equation will require both Darcy and non-Darcy permeability in order to compute pressure drop. In order to invert the equation and arrive at k2 the results of k1 from simulation will

Figure 4.39: fβ for periodic and non periodic foams

52 Regime Correlation Dietrich[2] Lacroix[23]

k1 9 24 757 k2 31 49 33

Table 4.15: Correlations average of absolute error for both regimes

Figure 4.40: Error value of k1 and comparison with literature for all tested foams be used to avoid error propagation. The non-Darcian permeability will then be computed from two simulations, one in the Darcy regime and one in the inertial and not from a ﬁtting of a more extensive collection of points. This should mean that k2 is treated as dependent of Reynolds number.

The procedure to ﬁne tune k2 is identical to the Darcian counterpart and a sensitivity analysis was performed in order to better the ﬁt to the data. Table 4.16 presents this analysis where the error tends to reduce when the tortuosity exponent rises and the porosity exponent is lower predicting with minimum error for the case of 2.7 and τ 2.2.

The c2 coefﬁcient was calculated for the range of porous foams with equation (4.18).

k c = 2sim 2 2.7 (4.18) 2 2.2 τ Sp

Figure 4.41 shows the plotted coefﬁcients and it is apparent that the coefﬁcients are roughly grouped in

Number Exponent τ Exponent Maximum absolute error [%] 1 3 2 84.33 2 3 1.8 89.07 3 3 2.2 79.75 4 2.7 2 75.11 5 2.7 1.8 79.75 6 2.7 2.2 70.63 7 3.3 2 94.67 8 3.3 1.8 99.53 9 3.3 2.2 89.98

Table 4.16: Sensitivity Study for the K2 correlation

53 Figure 4.41: c2 and c3 for periodic and non periodic foams

Figure 4.42: Tortuosity correlation fτ for angled ﬂow

two groups, periodic and non periodic foams. The coefficient for periodic foams is c2 = 1.49 and for the non periodic c3 = 0.96. If this coefficients are used to calculate k2 with equation (4.19) one obtains the error bar graphs presented in figure 4.43 and comparing with the literature correlations and the average error as can be seen on table 4.15. The y-axis is limited as (Du Plessis predictions for Calado foams k2 are severely off)

2.7 c2 k2 = 2.2 (4.19) Spτ

As for k1, to achieve a closed formulation a tortuosity equation would need to be found. To achieve a usable correlation for periodic porous media the tortuosity of 111 and ABC was related with the fraction of specific projected area to porosity with a second order polynomial function with great success. Figure 4.42 shows the fitting of the form of equation (4.20) with a correlation coefficient R = 0.992 and coefficients a2 = 0.02095, a1 = 0.12244 and a0 = 0.97039.

S S τ = a ( p )2 + a p + a (4.20) 2 1 0

54 Figure 4.43: Error value of k2 and comparison with literature for all tested foams

Figure 4.44: Error value of k2 and comparison with literature for 111 and ABC

Figure 4.44 shows a bar graph comparing the correlation with the tortosity relation and the literature correlations for the angled foams. It can be argued that the simplicity and acceptable errors of Dietrich or Lacroix correlations make it desirable. However, the errors predicting periodic foams are signiﬁcant and the correlation is solely based on experimental data ﬁtting while this work’s correlation has a theoretical background.

The averaged values are listed on table 4.15 where it is obvious that Lacroix’s correlation as a slight advantage predicting permeability on periodic foams. A possible explanation for the increase in error from k1 to k2 is the fact that the simulations are not well into the inertial regime which handicaps the evaluation of the k2 weight. This would justify the better results from correlations derived from higher, experimental Reynolds.. An attempt to study this regime will be studied on the next section.

Summing up, the proposed correlations with no need of physical parameters are given for both permeabilities as : dp 1 1 2 = µuD + ρuD (4.21) dx k1 k2

55 Figure 4.45: Velocity ﬁeld for low Re 10x10 cylinder Figure 4.46: Velocity ﬁeld for high Re 10x10 cylinder array array

 2 c1 dsfβ (3.4012) k1 = , where fβ = 0.0298e and c1 = 0.461  Sv   2.7  c Sp 2 Sp k2 = 2.2 , where τ = 0.02095( ) + 0.12244 + 0.97039  Spτ   1.49, if Periodic media (4.22)     c =       0.96, if Random foams

4.8 Unsteady ﬂow on a periodic REV

Given the expected relevance of the study of higher Reynolds inertial flow to the correlation, the study of unsteady flow was hypothesised. However, due to the nature of the flow field the problem of periodicity appears. Under steady regime the flow is periodic and by virtue of the periodicity of the structure one can impose periodic boundary conditions without loosing accuracy or oversimplifying the model. This can be seen on figure 4.45, where an array of 2D cylinders under low Reynolds number flow is studied and it is clear that by imposing periodic boundaries the calculated flow field is still similar in the vicinity of every cylinder. If the same was assumed under unsteady laminar flow one would be forcing the flow field to be equal over all domain, meaning that, in the case of a cylinder, one would force shedding in the same direction for all the domain. This is clearly not the case as it can be seen on figure 4.46, where periodic boundary conditions are assumed in the faces of the square. This problem does not occur under high enough Reynolds turbulence since the flow will be statistically steady and the model would be sound. It can be argued, however, that even though the field is not the same it will be statistically similar, meaning that by averaging the fluctuations over enough time the two approaches would give similar

56 Cylinder arrangement up Absolute error [%] 10x10 0.0810 [-] 5x5 0.0811 0.1 3x3 0.0820 1.2 2x2 0.0780 3.7 1x1 0.0110 35.8

Table 4.17: Convergence study for different array arrangements results. To test this hypothesis a 10 by 10 cylinder array was simulated and assumed to be the correct solution. The forces (viscous and pressure) on all cylinders were monitored and averaged by summing the force total of the 100 cylinders and dividing by the total number of cylinder as well as averaging in time. The average velocity on the ﬂuid phase was also monitored. The case was then compared with the 1 cylinder domain with periodic boundary conditions and posteriorly by increasing the number of cylinders in the domain the evolution of average quantities was monitored. The results are presented on table 4.17. It is clear that by assuming periodic boundaries and imposing the same pressure gradient the pore velocity increases meaning that permeability is over predicted. The results suggest that the case of 3x3 domain gives a good enough approximation of the solution. The computational effort of validating this results with a 3-D model (e.g. 100 geometry) surpasses the scope of this thesis and will be considered for future work.

57 58 Chapter 5

Conclusions

Modelling of pore scale laminar flow was detailed within a representative elementary volume with periodic boundaries. Different variations of the unit cubic cell were modelled in order to access the influence of different geometric parameters on fluid flow. The finite volume method of Star-CCM+ code is adequate to solve pore scale flow details using polyhedral cells. The concept of volume averaging ensured meaningful computational values and the computed parameters, k1 and k2 were compared against literature values. The effects of anisotropy on transition to inertial regime and unsteadiness were studied. The transition to inertial regime happens at lower Reynolds under angled flow when compared to aligned which disproves the hypothesis of a transition criteria based on Reynolds or Darcy permeability and the relevance of a Forcheimmer number criterion was highlighted. It is proven that by modifying the structure orientation unsteadiness predicted start changes from a Reynolds of Re ≈ 20 (111) to Re ≈ 60 (100). Anisotropy proved to have no effect in Darcy permeability. However, lower non-Darcy permeabilities were registered for angled flow. The effects on force nature and tortuosity were also studied with increased viscous force ratio (at high Re) and increased tortuosity values for angled flow. The relative influence of the equations terms is compared to the force nature and a direct relation between pressure force and inertial term is established. It is also established that pressure evolution with Reynolds number is linear under Stokes flow. The influence of geometric parameters,such as projected area and specific surface, on pressure drop are analysed. The projected area has no influence on Darcian permeability but is relevant in the inertial regime since an increase in projected area means a decrease in non-Darcian permeability. It is found that an increase in specific surface translates the curve upwards and lowers permeability to both Darcy and non-Darcy. The velocity probability density functions were analysed and relations about the PDF topology and permeability were established. The incidence of possible recirculation zones is analysed for different Reynolds numbers and foam typology and there is clear evidence of channeling under aligned flow for both geometries.The validity of traditional average velocity formulas for tortuosity is evaluated using particle tracking. Using this method, the computed tortuosity values is significantly lower due to massive

59 recirculation areas and filtering of reentrant particles and excessive residence times. The effects of porosity are studied and it is proven that lowering porosity has a stabilizing effect on unsteadiness. By use of a modified Reynolds number based on Forcheimmer characteristic velocity a delay on transition to inertial regime is found. Higher values of porosity lead to higher permeabilities, Darcian and non-Darcian, for every geometry and flow arrangement. A theoretical correlation for the Darcian and non-Darcian permeability is developed based on the Navier-Stokes with simplifying assumptions. The correlation is tested against periodic and non-periodic random foams and fine tuned for posterior use. For the Darcy permeability a correlation is developed based solely on geometric parameters with a maximum error of 28% roughly half the error compared with the best literature correlation and an average absolute error of 9.21%, compared with 23.5 for the best correlation. The non-Darcy permeability correlation presented maximum errors of 70% compared to 124% for the best correlation found and an average absolute error of 31% similar to other correlations tested. The present correlation surpassed several shortcomings of the existing correlations, being more accurate and for a wider range of applications.

5.1 Achievements

The full process of geometry creation to numerical simulation was performed and subsequent grid dependence and validation studies were completed in order to ensure proper discretisation and model assumptions. An extensive study of the laminar Reynolds range is performed for different ﬂow directions and geometries. A correlation is developed and compared to literature correlations with valuable results for Darcy permeability.

5.2 Future Work

Related to the present topic a few concepts are presented:

• Quantify other directions anisotropy and how it related with the studied two.

• Study of ﬂow instability in periodic foams: Critical Reynolds number and determining ﬂow structure (strut shape, wake)

• Study of the domain validity for a 3-D REV: Continue the study started in this work for three dimensions

• Correlation extension to the laminar unsteady and turbulent regime using parameters like Strouhal number and kinetic turbulent energy, respectively

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