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Characterisation of Laminar Flow in Periodic Porous Structures

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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 soluc¸ao˜ detalhada para as equac¸oes˜ de Navier-Stokes 3-D e´ obtida para uma celulas´ 3-D periodicas´ de variados parametrosˆ geometricos´ e e´ desenvolvida uma correlac¸ao˜ de base teorica´ e comparada com a equac¸ao˜ 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´ simulac¸oes˜ ao escoamento ou o uso de correlac¸oes˜ 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 func¸ao˜ da porosidade. A correlac¸ao˜ preveˆ com sucesso a permeabilidade de Darcy quando comparada com correlac¸oes˜ da literatura. Quanto a` porosidade nao-Darciana,˜ foram achados dois coeficientes constantes, para espumas periodicas´ e nao˜ periodicas.´ E´ proposta uma relac¸ao˜ 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 geo- metric 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
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