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NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM By Krishna Chaitanya Ravi Bachelor of Technology in Mechanical Engineering Vellore Institute of Technology Vellore, Tamil Nadu, India 2006 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2010 COPYRIGHT c By Krishna Chaitanya Ravi December, 2010 NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Thesis Approved: Dr. Frank W. Chambers Thesis Advisor Dr. Afshin J. Ghajar Dr. Khaled A. Sallam Dr. Mark E. Payton Dean of the Graduate College iii ACKNOWLEDGMENTS I would like to take this opportunity to thank my advisor Dr. F. W. Chambers for his support, patience, guidance throughout my research and coursework. His encouragement to learn and implement new concepts has always helped me in having an empty mind to learn things. He was always present to solve problems and guide me in the right path. I am also indebted to Dr. Sallam and Dr. Ghajar for their time and patience for reviewing the thesis document and attending my examination. Dr. Ghajar has been very supportive and helpful in many ways throughout the course of study in Oklahoma State University. Dr. Sallam has been an excellent teacher and his simplistic visualization approach towards learning complex systems is admirable. I also like to thank my fellow researchers Deshpande, Karthik, Alok, and Netaji for their assistance and help in research and outside the lab. Unconditional love and support from my parents Madhusudhan Rao, Vijaya Lakshmi and my brother Goutham in all fronts made my achievements possible. I am grateful to my grandparents Rama Krishnaiah and Kota Ratnama for their constant love and respect for me as an individual. I like to thank all my friends not limited to Ashvin, Ashwin, Kapil, Sai, Kamal, Bhargav, Vikas, Harsha, Sudheer, Durga, Muthappa, Rao, Chinnmay, Raagini, Akshata, Varun, Sunil, Sarvana and so on who were always there to share fun, enjoyment, and also during times of trouble. I thank God for all the necessary. iv TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Literature Survey 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Study of Particulate Flow in Backward Facing Step . . . . . . . . . . 8 2.4 Study of Backward Step with Porous Medium Insert . . . . . . . . . 13 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Numerical Method 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.1 ReynoldsAveraged Navier Stokes . . . . . . . . . . . . . . . . 21 3.3 k − Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 Turbulent Kinetic Energy(k) . . . . . . . . . . . . . . . . . . . 22 3.3.2 Modeling Dissipation . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.3 Numerical Treatment of k − Model . . . . . . . . . . . . . . 22 3.3.4 Variations of k Turbulence Model . . . . . . . . . . . . . . . 23 3.4 k! Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 v 3.5 SST Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6.1 Wall Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.7 Discrete Phase Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.8 Porous Medium Modeling . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Modeling Process 30 4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Grid Generation and Adaptation . . . . . . . . . . . . . . . . . . . . 30 4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.1 Discretization Schemes . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Grid Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4.1 Turbulence Model Comparison . . . . . . . . . . . . . . . . . . 37 5 Results and Discussions 41 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Flow Field Estimation for Re = 6550 and Re = 10000 . . . . . . . . 41 5.3 Flow Field Estimation for Filter at 4.25s after the Step . . . . . . . . 50 5.4 Flow Field Estimation for Filter at 6.75s after the Step . . . . . . . . 54 5.5 Discrete Phase Model for No Filter Case . . . . . . . . . . . . . . . . 58 5.5.1 Particle Tracks Re = 6550 No Filter Case . . . . . . . . . . . 58 5.5.2 Particle Tracks Re = 10000 No Filter Case . . . . . . . . . . . 61 5.6 Discrete Phase Model for Filter Case . . . . . . . . . . . . . . . . . . 63 5.6.1 Filter located at 4.25s after the step . . . . . . . . . . . . . . . 63 5.6.2 Filter Located at 6.75s after the step . . . . . . . . . . . . . . 68 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Conclusions and Recommendations 75 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vi 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A Appendix: Stokes Number 81 vii LIST OF TABLES Table Page 3.1 Modeling Constants Used for k . . . . . . . . . . . . . . . . . . . . 22 3.2 Modeling Constants Used for k RNG from Yakhot et al. (1992) . . 24 3.3 Modeling Constants Used for k! from Wilcox (2006) . . . . . . . . . 25 3.4 Modeling Constants Used for SST Turbulence Model . . . . . . . . . 26 4.1 Mesh Size Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Boundary Conditions Implemented . . . . . . . . . . . . . . . . . . . 32 4.3 Velocities at Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Comparison of Recirculation Zone Length and Grid Sizes . . . . . . . 37 4.5 Comparison of Recirculation Zone Length with Various Turbulence Models and Wall Functions . . . . . . . . . . . . . . . . . . . . . . . 40 A.1 Stokes Number for Re 6550 . . . . . . . . . . . . . . . . . . . . . . . 81 A.2 Stokes Number for Re 10000 . . . . . . . . . . . . . . . . . . . . . . . 82 viii LIST OF FIGURES Figure Page 1.1 Computational Geometry With Different Boundary Conditions . . . . . . 2 2.1 Aspect Ratio Effects on Recirculation Zone Length from Papadopoulos and Otugen (1995). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Recirculation Length remaining Constant for Fully Developed Turbulent Flow Past the Step from Tihon et al (2001). . . . . . . . . . . . . . . . . 7 2.3 Particle Concentration Contours behind the step with varying particle diameters. Case (a) dp = 7μm, Case (b) dp = 15μm, Case (c) dp = 25μm, and Case (d) dp =45μm from Tu (1997) . . . . . . . . . . . . . . . . . . 10 2.4 Particle Distribution of Various Diameters from Yu et. al (2004) . . . . . 11 2.5 Effect of Slip Velocity on Particle of Diameter 100 μm from Yu et. al (2004) 11 2.6 Effect of Slip Velocity on Particle of Diameter 200 μm from Yu et. al (2004) 12 2.7 Effect of Gravity in ydirection on Particles from Yu et. al (2004). . . . . 12 2.8 Stream Traces for Variation of Darcy Constant from Chan and Lien (2005). 14 2.9 Stream Traces for Variation of Forchheimer Constant from Chan and Lien (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.10 Stream Traces for Variation of Porous Media Thickness Constant from Chan and Lien (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Near Wall Region Grid Point . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Implemented Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Wall Adaptations at Step . . . . . . . . . . . . . . . . . . . . . . . . . 32 viii 4.3 Experimental Data of Pressure Drop vs. Flow Velocity . . . . . . . . . . 33 4.4 Velocity Profile at Step for Re 6550 No Filter Case . . . . . . . . . . . . 35 4.5 Velocity Profile at X=3.75s after Step for Various Mesh Sizes, Re 6550 No Filter Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 Velocity Profile at X=6.25s after step for various Mesh Sizes, Re 6550 No Filter Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.7 Comparison of Velocity Profiles at Step for Re 6550 No Filter case for Various Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.8 Comparison of Velocity Profiles at X=3.75s after the Step for Re 6550 No Filter Case for Various Turbulence Models . . . . . . . . . . . . . . . . . 38 4.9 Comparison of Velocity Profiles at X=6.25s after the Step for Re 6550 No Filter case for Various Turbulence Models . . . . . . . . . . . . . . . . . 39 5.1 Velocity Magnitude Contours after the Step for Re = 6550 . . . . . . . . 43 5.2 Velocity Magnitude Contours after the Step for Re = 10000 . . . . . . . . 43 5.3 Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 6550 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 6550 . . . . . . . . . . . . . . . . . . . 47 5.6 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 10000 . . . . . . . . . . . . . . . . . . 47 5.7 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re = 6550 . . . . . . . . . . . . . . . . . . . 49 5.8 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 . . . . . . . . . . . . . . . . . . . 49 ix 5.9 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =6550 for Filter at X= 4.25s . . . . . . . . 51 5.10 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =10000 for Filter at X= 4.25s . . . . . . . 51 5.11 Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 4.25s 53 5.12 Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 4.25s 53 5.13 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =6550 for Filter at X= 6.75s . . . . . . . . 55 5.14 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 for Filter at X= 6.75s . . . . . . . 55 5.15 Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 6.75s 57 5.16 Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 6.75s 57 5.17 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . . 59 5.18 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . 59 5.19 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . . 60 5.20 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 61 5.21 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 62 5.22 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 62 5.23 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . . 64 x 5.24 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . 64 5.25 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . 65 5.26 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 66 5.27 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 67 5.28 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 67 5.29 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . . 69 5.30 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . 69 5.31 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . 70 5.32 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . . 71 5.33 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . 72 5.34 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . 72 xi NOMENCLATURE C 1 k Modeling Constant C 2 k Modeling Constant C2 Inertial Resistance Factor Cμ k Modeling Constant CD Coefficient of Drag D Distance of Porous Medium from the Step dp Particle size, (μm) g Gravitational Acceleration,(m s2 ) H Upstream Channel Length, (mm) k Turbulent kinetic energy,(m2 s2 ) Res Reynolds number based on Step Height s Step Length, (mm) St Stokes number T Thickness of Porous Medium, (mm) Umax Maximum Velocity at Step, (m/s) Xr Recirculation Zone Length, (mm) Xc Corner Recirculation Zone Length, (mm) Permeability of Medium, (m2) Turbulent dissipation,(m2 s3 ) RNG Expansion Parameter μ Dynamic viscosity, (Pas) μt Turbulent viscosity, (Pas) xii Density of fluid, ( kg m3 ) k k Modeling Constant k Modeling Constant F Fluid Response Time v Particle Momentum Response Time w Wall Shear Stress,( N m2 ) ! Turbulence Frequency ABBREVIATIONS AR Aspect Ratio ER Expansion Ratio CFD Computational Fluid Dynamics DES Detached Eddy Simulation DPM Discrete Phase Model DNS Direct Numerical Simulation LES Large Eddy Simulation RANS Reynolds Averaging of NavierStokes RNG Renormalization Group SST Shear Stress Transport xiii CHAPTER 1 Introduction The motivation for the current study is the air filtration device commonly found on automobiles for providing clean air to the engine. Due to the constraints of space and design of other components, air filters are given a reduced priority on the design point of view. Many air filter housings have sudden expansion of area, and from the fluid mechanics point of view whenever there is area expansion generally there is a recirculation zone. The best examples are backward step flow, flow over a bluff body etc. Predicting the flow behavior is much more complex when a recirculation zone is present due to the fact the mean flow might be laminar, transitional, or turbulent and the recirculation zone characteristics might be different. The filter modeling can be achieved in two ways: microscopically considering all the intricate details of the filter or utilizing a computationally suitable macroscopic model. The current research focuses on the study of particle behavior in a recirculation zone and a filter modeled as porous media. Generally the particles, depending on the Stokes number, get trapped in the recirculation zone or just escape the recirculation zone because of high inertia and lift forces. 1.1 Problem Statement The backward facing step is chosen as the geometry for the current study due to the presence of a distinct recirculation zone and the existence of a huge experimental and computational database. The backward facing step geometry is selected based on experiments performed by Yao (2000). The geometry of the backward step is shown 1 in Figure 1.1 Figure 1.1: Computational Geometry With Different Boundary Conditions • s  Step Length (25mm) • H  Upstream Channel Length (50mm) • D  Distance of porous medium from the Step. • T  Thickness of the Filter or Porous Medium. • Xr  Recirculation Zone Length • Xc  Corner Recirculation Zone Length The Characteristics of the backward facing step studied here are Expansion Ratio (ER) = s H = 0.5 The Reynolds number used is based on the definition of Yao (2000), and is based on maximum velocity at the step and with Res = Umaxs (1.1) 2 1.2 Objective The goal behind the current study was to investigate whether the recirculation zone can be beneficial for filtration design. The current study focuses on study of mono disperse particulate flow in a backward step flow with a filter medium. The backward facing step is modeled as 2D with the dimensions provided in Figure 1.1. The current work focuses on particulate sizes of 1μm, 20μm, 50μm and the particle properties have been chosen close to ultrafine test case ISO 12103, A1 Ultrafine from Powder Technology Incorporated (PTI). The Reynolds numbers studied were of 6550 and 10000 for the carrier phase or the main flow carrying the dispersed phase. The filter location is also varied, with two locations, at X=4.25s and 6.75s. The particle trajectories were studied for no filter and filter cases. 3 CHAPTER 2 Literature Survey 2.1 Introduction The backward facing step flow is a widely studied flow geometry both experimentally and numerically due to the boundary layer detachment, the presence of a large recirculation zone and the shear layer. The backward facing step flow has also proven to be a good test for the turbulence models due to the complexity in flow structures and the various regimes. This literature review focuses on experiments and numerical investigations pertinent to the current study. The goal of the literature review is to provide foresight on the numerical results we should expect after CFD analysis and also provide some parameters for grid independence test. The literature review can be broadly classified into three sub sections: • Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step Geometry • Study of Particulate Flow in Backward Facing Step Geometry • Study of Backward Step with Porous Medium Insert 2.2 Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step Kim et al. (1980) performed experiments on Backward Facing Step with two expansion ratios, 1.33 and 1.5. The Reynolds number based on s, see Fig 1.1 (Res) is 40,000 4 and 61333 approximately. The recirculation length, xr/s, zone found experimentally is 7 ± 1. The authors also observed shear stress and turbulence intensities reaching maxima at the reattachment point and subsiding afterwards. Armaly et al. (1983) performed experiments on backward facing step flow using Laser Doppler Anemometry. The backward facing step expansion ratio (ER) was 1:1.94. The experiments were performed for Reynolds numbers varying from 70 to 8000 encompassing all three regimes: laminar, transitional and turbulent flow. For Armaly et al. (1983), the Reynolds number is defined as Re = V (D)/ where D the hydraulic diameter of the inlet section = 2s and V is 2 3Umax, with Umax maximum velocity at the inlet section. The experiment results showed additional recirculation zones at the opposite wall at certain Reynolds numbers along with the main recirculation zone. The recirculation zone length increased with Reynolds number up to the transition regime and then decreased as the flow became turbulent. The flow also behaved as twodimensional for laminar and turbulent cases due to the high aspect ratio in the zdirection but in the transitional regimes the flow behaved threedimensionally. Driver and Seegmiller (1985) performed experiments on backward facing step flow using Laser Doppler Anemometry for flow velocities and laser interferometer skinfriction apparatus to calculate skin friction. The expansion ratio was 1.13 and the flow Reynolds number based on h and velocity is 37500. Driver and Seegmiller (1985) also performed numerical modeling using various turbulence models. The k , kmodified and ASM failed to predict recirculation zone length. The ASM with modified provided good estimates of the recirculation zone length and also the velocity profiles along the length of the channel. Adams and Johnston (1988a) performed experiments on backward step flow with an expansion ratio of 1.25 and Reynolds number of 36000 based on s (see Fig 1.1). The authors’ aim was to address the problem of sudden pressure rise at reattachment as a function of ER. The authors’ results, along with compilation of data from other authors work, showed that as the flow becomes fully turbulent, the re 5 circulation pressure gradient is independent of upstream Reynolds number. This also can be interpreted in terms of recirculation length being almost constant with fully developed turbulent regimes and consistent with findings of Armaly et al. (1983). Papadopoulos and Otugen (1995) performed experiments on backward step flow with expansion ratio of 2 and flow Reynolds number of 26500 based on s (see Fig 1.1). The authors’ aim was to find the effect of aspect ratio(AR) on the recirculation zone length, velocity, and wall pressure. The Figure 2.1 shows the variation of recirculation zone length with AR. They varied AR from approximately 5 to 30 and found as the AR increases there is a gradual increase of recirculation zone length. The recirculation zone length attains constant length after AR of 6. The recirculation zone length xr/s attained is of the order 8. The effect of small AR is that the flow is three dimensional along the total channel and for even higher aspect ratios the flow behaved at certain regions as three dimensional. Figure 2.1: Aspect Ratio Effects on Recirculation Zone Length from Papadopoulos and Otugen (1995). 6 Tihon et al. (2001) performed experiments in backward facing step of expansion ratio of 1.4 for varied Reynolds numbers based on s (see Fig 1.1) from 1000 to 12000. The authors used electro diffusion technique for determining the wall shear stress and estimating the recirculation zone lengths. The authors provided good experimental values for normal recirculation zone length as well as the corner recirculation zone length. The authors have experimentally shown that after flow reaches fully turbulent regime the recirculation zone length remained constant about xr/s 5.1 and corner recirculation zone about xc/s 1.75 consistent with Armaly et al. (1983) and Adams and Johnston (1988a) findings. Figure 2.2 shows the recirculation zone attaining constant length for fully developed turbulent flow behind the step. The authors have also presented a good survey of backward step experiments with varying expansion ratios and aspect ratios. Figure 2.2: Recirculation Length remaining Constant for Fully Developed Turbulent Flow Past the Step from Tihon et al (2001). 7 2.3 Study of Particulate Flow in Backward Facing Step Ruck and Makiola (1988) studied the dispersion of particles in a backward facing step of expansion ratio of 1:2. The Reynolds numbers studied based on s and velocity were 15000 and 64000 and particle diameters studied were from 1 μm to 70 μm. The particles used by the author were oil and starch. They observed the velocity field increase as particle size increased. This also can be understood as the particle size increases the Stokes number increases, due to which there is modulation of turbulence. The recirculation zone length also decreased as the particle size increased. The authors make a note that for different particle diameters it can be seen that particle diffusivity of energy is different from eddy diffusivity even for micronsized particles. Fessler and Eaton (1997) studied particle response in a backward facing step of expansion ratio of 5:3 with main flow direction vertically downward. The Reynolds number at the inlet of the step is 18400 based on the channel centerline velocity and step size s. The particle diameters varied from 25μm to 150μm and were made of copper and glass. The authors observations were particles accelerated downstream due to negligible effect of pressure gradient experienced by the particles compared to the fluid. The velocity decreased because the fluid was more responsive and adapted to adverse pressure gradient. The particle response is also consistent in the shear layer just after the sudden expansion where the fluid velocity fluctuations were large. The authors noted the high inertia of particle remembers stream wise distribution of velocities and can be easily seen in the vicinity of the recirculation zone. They also noted that the particles’ response is negligible as the particle Reynolds number increased. Fessler and Eaton (1999) studied the modification of turbulence by particles. The experimental setup of Fessler and Eaton (1997) was used. The Reynolds number based on the step height and channel centerline velocity was 18400. The particle diameters varied from 70μm to 150μm. The mass loading of the particle was 3% to 8 40%. The particle Stokes numbers were 1. The authors observed modulation of carrier phase increased (main flow which carries the dispersed flow) with the increase of particle Reynolds number, Stokes number and the mass loading. The authors also suggest that there exists a limiting particle Reynolds number and Stokes number for modulation. The authors also suggest as particle size increases the attenuation decreases and turbulence modulation increases. Hetsroni (1989) reviewed particle turbulence interaction in detail and showed as the particle diameter increased the turbulence intensity increases due to the help of vortex shedding from the particle. Tu (1997) performed a two fluid model simulation of particle flow in backward step flow and Tjunction. The primary phase turbulence was resolved using k RNG. An interphase momentum transfer equation was derived based on the Favreaveraged momentum equation of particulate phase. The author compared the results to the experiments of Ruck and Makiola (1988) with particle size of 70 μm and Reynolds numbers based on step height s which is 64000. The results matched experiments reasonably. The author also performed various simulations varying particle sizes from 7μm to 45 μm. The particle concentration in the recirculation zone was minimal compared to that in the main flow as the particle diameter increased which can be seen in Figure 2.3. Yu et al. (2004) performed numerical simulation of gas particle flow using LES for resolving primary phase and Lagrangian tracking for particles. The particles were glass spheres with diameters varying from 2200 μm. The particle tracks and dispersion were consistent with the literature of Fessler and Eaton (1997) and Tu (1997). The smaller particles (Figure 2.4) with smaller Stokes number tend to enter the recirculation zone but as the particle size and Stokes number increase the particles entering the recirculation zone decreases. The authors also considered the effect of slip velocity between the particle and the primary phase. The slip velocity was defined as the ratio of particulate phase velocity to the carrier phase velocity at the inlet 9 Figure 2.3: Particle Concentration Contours behind the step with varying particle diameters. Case (a) dp = 7μm, Case (b) dp = 15μm, Case (c) dp = 25μm, and Case (d) dp =45μm from Tu (1997) . of the channel. The authors observed the effect of slip velocity on small particles is negligible which can be seen in Figure 2.5. The effect of slip velocity on large particles is moderate as the slip velocity reduces from 0 to 1 which can be seen in Figure 2.6. The authors also displayed the effect of gravity on particles in Figure 2.7. The authors considered the gravity in x and y directions, x direction is the flow direction. The xdirection gravity produced similar results to no gravity case. The y direction gravity produced settlement of particles of 100 and 200 μm. 10 Figure 2.4: Particle Distribution of Various Diameters from Yu et. al (2004) . Figure 2.5: Effect of Slip Velocity on Particle of Diameter 100 μm from Yu et. al (2004) . 11 Figure 2.6: Effect of Slip Velocity on Particle of Diameter 200 μm from Yu et. al (2004) . Figure 2.7: Effect of Gravity in ydirection on Particles from Yu et. al (2004). . 12 2.4 Study of Backward Step with Porous Medium Insert Consider the step flow with porous medium, and how porous medium properties like thickness, porosity, etc. affect the recirculation zone length. Chan and Lien (2005) performed numerical modeling of step flow with a porous medium immediately behind the step. The authors used the k model for the flow and the porous medium was modeled using the DupuitForchheimer equation by incorporating an additional sink term in the Navier Stokes equations. The authors chose the experimental setup of Driver and Seegmiller (1985) as the geometry for numerical modeling and modeled it as a 2D backward step. The authors studied various cases of the porous medium, varying the thickness of the porous medium, the Darcy constant, and Forchheimer constant. The authors first lowered the Darcy constant increasing streamwise resistance to the flow and keeping the other factors constant, shown in Figure 2.8. They observed that with increasing resistance along the flow direction the recirculation zone vanished and the porous medium laminarized the flow. Then the authors kept the Darcy constant and the thickness of the porous medium constant and varied the Forchheimer constant, shown in Figure 2.9. They observed that by increasing the Forchheimer constant the effect was seen primarily on the flow leaving the porous medium. The flow past the porous medium was turbulent and laminarized with increase in the Forchheimer constant. Finally the authors varied the thickness of the porous medium, shown in Figure 2.10. They observed that, increasing the thickness of the porous zone length increased the resistance to flow and decreased the recirculation length. 13 Figure 2.8: Stream Traces for Variation of Darcy Constant from Chan and Lien (2005). . 14 Figure 2.9: Stream Traces for Variation of Forchheimer Constant from Chan and Lien (2005) . 15 Figure 2.10: Stream Traces for Variation of Porous Media Thickness Constant from Chan and Lien (2005) . 16 2.5 Summary The single phase numerical and experimental studies of backward facing step flow in the literature suggest the following: • The recirculation length strongly dependent on the expansion ratio • For flow fully developed turbulent flow past the step, recirculation length is independent of Reynolds number • Flow is three dimensional in the transitional regime from laminar to turbulent and low aspect ratio • The pressure gradient of the recirculation zone is independent of upstream Reynolds number • Recirculation length increases from laminar to transitional regime and then decreases The previous studies of particulate flow in backward facing step flow leads us to the following conclusions. The smaller particles with lower Stokes number follow the flow like tracer particles. The heavier particles with higher Stokes number escape the recirculation zone due to their inertia. The increase of concentration of secondary phase or particles has influence on primary phase velocity . The increase of concentration of smaller particles leads to decreases in turbulence intensity due to small particles promoting more dissipation of energy. On the other hand when the larger particles concentration increases the turbulence intensity increases due to vortex shedding of larger particles. The effect of gravity is predominantly on the heavier particles. 17 The only previously studied case of porous media and backward facing step was done by Chan and Lien (2005) numerically. The authors placed the porous media adjacent to the step. The findings were increasing viscous resistance or Darcy’s constant the flow laminarized and recirculation zone disappeared. Increasing the inertial resistance or Forchheimer constant the recirculation area decreased but not the length. The increment of thickness of porous media had similar results to increasing the Darcy’s constant and decreasing recirculation zone length. 18 CHAPTER 3 Numerical Method 3.1 Introduction The numerical modeling of the current study can be broadly classified into the following major categories • Turbulence modeling of carrier phase • Discrete phase modeling of the particles • Modeling of the porous region Along with the above fundamental modeling topics, numerical difficulties also exist, the major aspects like grid generation, pressurevelocity coupling, transfer of momentum from carrier phase to discrete phase, and wall treatment of turbulence by the turbulence model. The following content provides the basic equations, models and strategies for the aforementioned topics. 3.2 Turbulence Modeling turbulence flow phenomena is one of the areas which brought computational fluid dynamics predominance as a design tool. Modeling or simulating turbulent flows is a very challenging area due to the existence of various length scales and time scales across the flow. One of the most important areas in turbulence modeling is the resolution of stresses near the wall and carrying this information to the core of the flow. There exists no universal common tool for modeling or simulating varied 19 flow types. Modeling always incorporates finding some of the closure coefficients from experimental data. The experimental data set is limited in respect to Reynolds number as well as types of flow. Hence the accuracy of the model varies from one situation to another in respect to flow types. The current strategies to model or simulate turbulence are • Reynolds Averaging of Navier Stokes(RANS) • Filtering Approach (Large Eddy Simulation(LES)) • Hybrid Models • Direct Numerical Simulation(DNS) The RANS models use the concept of Reynolds averaging. The new terms which arise are modeled using correlations or coefficients obtained from experiments. The widely used RANS models are k − , k − !, Reynolds Stress Modeling etc. In the filtering or LES approach a filter is used to cut off eddies of various lengths. The large scale eddies are simulated and small scale eddies are modeled. Hybrid methods use a different strategy in which the core of the fluid is simulated using LES and near the walls RANS models are used to resolve stresses. Discrete Eddy Simulation (DES) is the best example of a hybrid model. Direct Numerical Simulation does not involve any modeling; it is just forward time stepping Navier Stokes Equations and limited to very small Reynolds number due to the computational power required to resolve the very smallest eddies. The selection criteria for turbulence models is well presented in Pope (2000). They are • Level of Description • Completeness • Cost and ease of use 20 • Range of Applicability • Accuracy 3.2.1 ReynoldsAveraged Navier Stokes The Reynoldsaveraged NavierStokes equations for incompressible flow from Pope (2000) DhUji Dt = @ @xi " μ DhUii @xj + DhUji @xi ! − hpi ij − huiuji # (3.1) The term huiuji is defined as the Reynolds stress.The Reynolds stress term is generally conceptualized as the momentum transfer by the fluctuating components of the velocity field at the molecular level. From 3.1 all the terms are know and pressure can be obtained from the continuity equation except the Reynolds stress, hence the problem of closure arises. The closure to this problem is achieved by modeling the Reynolds stress using different strategies. All the k − , k − ! and other variation of these models assume a simple relation between the Reynolds stresses, kinetic energy and mean strain rate. This is called the Boussinesq Constitutive Relation: uiuj = 2 3 k ij − t @Ui @xj + @Uj @xi ! (3.2) 3.3 k − Model The k − model is the first of the two equation models and a widely used model for simulating industrial flows. In k− we solve both the kinetic energy equation and the dissipation equation which is obtained by taking the moment of the Navier Stokes. The k− has been well validated for many cases and has proven efficient on the basis of the cost of computational power and ease of programming. 21 3.3.1 Turbulent Kinetic Energy(k) The turbulent kinetic energy is equal to half the trace of the Reynolds stress matrix i.e k = 0.5 hui.uii . k transport equation is represented as follows: Dk Dt = r. T krk + P − (3.3) P = −huiuji @ hUii @xj , = 1 2 t @ui @xj + @uj @xi ! (3.4) where P represents turbulent kinetic energy production, represents the dissipation, T = Cμ k2 is the turbulent viscosity and D/DT = @/@t + U.r 3.3.2 Modeling Dissipation The eddy dissipation is obtained by taking the moment of the Navier Stokes equations as follows 2μ @ui @xj @ @xj (3.5) The corresponding result after some manipulations results in the following epsilon equation D Dt = r. T r + C 1 P k − C 2 2 k (3.6) The standard values of these constants are shown in the following table Cμ C 1 C 2 k 0.09 1.44 1.92 1.0 1.3 Table 3.1: Modeling Constants Used for k Turbulence Model from Wilcox (2006) 3.3.3 Numerical Treatment of k − Model The k − model can easily be implemented to an existing Navier Stokes Solver. The following shows the implementation of k − model for a two dimensional case from 22 Shyy et al. (1997). The kand equations can be treated as convective, diffusive reaction type equations and modeled. The equations are as follows @ @t ( ) + @ @x (u ) + @ @y (v ) = @ @x @ @x ! + @ @y @ @y ! + R1 + R2 (3.7) where = 8>>< >>: + t k for the k equation + t for the equation (3.8) R1 = 8>>< >>: tR + t k for the k equation C 1 t R K for the equation (3.9) where R = 2 2 4 @u @x !2 + @u @y !2 3 5 + " @u @y ! + @v @x !#2 (3.10) and R2 = 8>>< >>: − − C K K for the k equation −C 2 2 K − −C 2 K for the equation (3.11) where K and are evaluated at the previous step. 3.3.4 Variations of k Turbulence Model Like any model, the k does fail to model flows accurately in some cases. The variations of k turbulence models which work better in some cases include the Renormalization group (RNG) and Realizable k model. RNG K The RNG model accounts for rapidly changing strain rates in the flows. The RNG model incorporates an additional term with expansion parameter = Sk/ where S is the average strain rate. The k equation remains the same and a new added term is incorporated to the dissipation. The model equations for RNG from Yakhot et al. (1992) are as shown below 23 Dk Dt = 2μTS2 ij − + @ @xi Kμ @K @xi ! (3.12) where Sij is the mean strain rate given by Sij = 1 2 @Ui @xj + @Uj @xi ! (3.13) the equation for epsilon is as follows D Dt = C 1 μTSij 2 k − C 2 2 k + @ @xi μ @ @xi ! − R (3.14) where R = 2μSij @ui @xi @ui @xi . R is computed as follows R = Cμ 3(1 − / 0) 1 + 3 ! 2 k ! (3.15) where = Sk/ and the values of constants proposed in Yakhot et al. (1992) are C C 1 C 2 k 0 0.085 1.42 − R/Cnu 3 1.68 0.7179 0.7179 4.38 .0125 Table 3.2: Modeling Constants Used for k RNG from Yakhot et al. (1992) 3.4 k! Model The k! model is also a two equation model, but based on k and !. The ! is enstrophy or RMS of fluctuating vorticity ! and regarded as ratio of k where k is turbulent kinetic energy, is the dissipation. Solving the k! model is similar to k ; instead of we solve for !. The governing equations for incompressible turbulent flows are as follows from Wilcox (2006). The Turbulence Kinetic Energy Equation @k @t + Uj @k @xj = ij @Ui @xj − k! + @ @xj " (μ + μt) @k @xj # (3.16) The Enstrophy transport equation @! @t + Uj @! @xj = ! k ij @Ui @xj − k!2 + @ @xj " (μ + μt) @! @xj # (3.17) The closure constants are as follows 24 5 9 3 40 9 100 1 2 1 2 Table 3.3: Modeling Constants Used for k! from Wilcox (2006) 3.5 SST Model The SST model is well defined in Menter et al. (2003). The SST model is a combination of k and k! model using a blending function. The blending function makes the SST model behave as k! as we approach the wall and k as we move to core of the flow. The modified k equations is as follows @( k) @t + @( Uik) @xi = P˜k − k! + @ @xi " (μ + kμt) @k @xi # (3.18) The modified ! equation is as follows @( !) @t + @( Ui!) @xi = S2 − !2 + @ @xi " (μ + wμt) @! @xi # + 2 (1 − F1) w2 1 ! @k @xi @! @xi (3.19) The blending equation F1 is given as follows F1 = tanh 8< : ( min " max pk !y , 500 y2! ! , 4 !2k CDk!y2 #)49= ; (3.20) The cross diffusion term is defined as follows CDkw = max 2 !2 1 ! @k @xi @! @xi , 10−10 ! (3.21) The turbulent viscosity is defined as follows Vt = a1k max (a1!, SF2) (3.22) where S is the invariant measure of strain and F2 is the second blending function. F2 = tanh 2 4 " max 2pk !y , 500 y2! !#23 5 (3.23) 25 The production of turbulence is limited as follows Pk = μt @Ui xj @Ui @xj + @Uj @xi ! ! P˜k = min (Pk, 10 · k!) (3.24) The closure constants are as follows 1 !1 kl 2 2 5 9 3 40 9 100 1 2 0.85 0.44 0.0828 Table 3.4: Modeling Constants Used for SST Turbulence Model 3.6 Wall Treatment The wall treatment is one of the important features that controls the numerical prediction of the turbulence models for flows with walls. The amount of computational power required to resolve the stresses near the wall by having a very fine mesh near the wall is very high. To reduce computational costs and the failure of turbulence models to model low local Re flows near the different wall, some of the strategies proposed for modeling are as follows: 3.6.1 Wall Function Standard Wall Function The wall function method was first proposed by Launder and Spalding (1974). The theory was based on the universal law of the wall. The wall function model deploys empirical formula to model the flow near the wall. In the above figure the point P is the computational cell located at yp from the wall.The wall shear stress is evaluated as follows w = @U @y UP − UW yp (3.25) 26 Figure 3.1: Near Wall Region Grid Point Up ( / )w C 1/4kp 1/2 = 1 ln 2 64 Eyp C 1/2kp 1/2 3 75 (3.26) where kp is the turbulent kinetic energy at point p, Cμ modeling constant from the k equations, E is a function of the wall roughness equal to 9.0 for smooth wall, is the wall shear stress on the wall in the direction of Up and is Von Karman’s constant. There is a debating issue regarding the universality of wall functions, one reason being that is Von Karmans constant is really a constant. The issue is neglected and is beyond the scope in the current research. 3.7 Discrete Phase Modeling Discrete Phase Modeling belongs to the class of EulerianLagrangian strategy to model multiphase flows. Discrete phase model can be achieved in two ways depending on the coupling of the flows and particles. Oneway coupled flows are those in which particles have no influence on the carrier phase velocity. The Eulerian phase is resolved and particle tracks are integrated over the known velocity field. In twoway coupled multiphase flows there is interaction between particles and carrier fluid. The carrier phase is unresolved and particle trajectories are integrated over the instantaneous resolved carried fluid velocity per time step. The governing particle equation from Crowe et al. (1998) is as follows dv dt = f v (u − v) + g 1 − c d ! + Fi (3.27) 27 Here m refers to mass of the particle, f is the drag force factor f = CDRer 24 (3.28) v is the particle response time v = dD2 18μc (3.29) Saffman’s Lift Force The small particles experience lift force in turbulent flows. The Saffman’s lift force term for small particles equation was developed by Saffman (1965). This equation was generalized by ?) as follows ~F = 2K 1/2dij Sdp(dlkdkl)1/4 (~u − ~up) (3.30) where K is Saffman’s Constant Lift force Coefficient, S is the ratio of particle density to fluid. 3.8 Porous Medium Modeling The porous medium is treated as homogeneous and the pressure drop contributed by the porous medium is handled by an additional sink in the Navier Stokes Equation. The sink term is modeled with the DupuitForchheimer equation. The Dupuit Forchheimer equation from Nield and Bejan (2006) is as follows rP = − v + CF − 1 2 vv (3.31) where is the kinematic viscosity, is the permeability. The first term incorporates the Darcy Law where is the Darcy’s Constant and the second term is called the Forchheimers term to incorporate inertial effects due to the porous medium. The porous medium can be modeled in FluentTMusing two approaches 1. Porous Jump 2. Porous Zone Formulation 28 Porous Jump The porous jump formulation was chosen because the region of interest is not in the porous region and the non availability of DPM boundary condition for porous zone like trap, reflect, and escape. Preliminary computations performed with a porous zone showed small effects on velocities, hence the porous jump was chosen as the choice of model. The porous jump incorporates the additional sink term through the following equation on a line in 2D geometry and plane in 3D geometry. The porous jump equation is as follows P = μ + C2 1 2 V 2 T (3.32) where is the permeability of the medium, C2 is the inertial factor and T is the thickness of the porous medium. 29 CHAPTER 4 Modeling Process 4.1 Geometry The geometry was created in ICEM CFDTMwith the dimensions displayed in Figure 1.1. The 2D section of the experimental results was modeled due to flow being predominantly 2D due to high aspect ratio used in the experiments by Yao (2000) and also from the view of computational power. 4.2 Grid Generation and Adaptation The grid was generated in ICEM CFDTMconsisting of fully structured quadrilateral cells. The boundary layer mesh was difficult to generate in this geometry due to the sharp edge at the corner and also to maintain fully structure grid. Hence the methodology described in Schafer (2006) was implemented to achieve the required y+ ut μ for the wall functions. The methodology is as follows: • Perform calculation on a mesh of optimum number of grid points. • Calculate y+ from the previous results for the wall nodes. • Adapt the grid at the wall to resolve the mesh at the wall. Perform calculations based on adapted mesh. • Iteratively check the solution until the required y+ is achieved for the wall function in use. 30 The test mesh sizes were of fully structured grids. The mesh size was selected based on the capture of recirculation zone along with implementation of wall adaptation for each case. The boundary adaptation was done until the wall y+ condition for respective wall function y+ is reached. The following table gives the mesh sizes used. The following Figure 4.1 shows the mesh after the mesh adaptations near the boundary in the Figure 4.2 Mesh Quality Number of Nodes Coarse 9500 Intermediate 17613 Fine 77684 Table 4.1: Mesh Size Terminology Figure 4.1: Implemented Mesh 31 Figure 4.2: Wall Adaptations at Step 4.3 Boundary Conditions The different boundary conditions used for modeling the backward step regions shown in Figure 1.1 are presented in following table: Bakward Facing Step Geometry Regions Boundary Condition Inlet Velocity Inlet Outlet Outflow Filter Porus Jump Boundaries No Slip Table 4.2: Boundary Conditions Implemented Velocity Inlet The following velocities were given as input based on Reynolds number calculations and turbulent intensity was specified as 5%. 32 Re at the Step U(m/s) at the Inlet 6550 3.83 10000 5.84 Table 4.3: Velocities at Inlet Porous Jump The pressure drop across the filter media was experimentally found by ?) for various flow rates. The data was curve fitted and constants were used to determine the modeling constants of the porous jump. The following figure shows the pressure drop profile for the mean velocity variation. The curve fit equation for the pressure drop Figure 4.3: Experimental Data of Pressure Drop vs. Flow Velocity obtained was P = 88.744V 2 + 115.57V (4.1) 33 The above constants were used to find and C2 in porous jump equations. The values obtained were C2 = 10.28 × 103m−1 and = 2.389 × 10−9m2. Particle Properties The particle were modeled as inter particle and the density of the particles used was 550kg/m3 close to the density of ultrafine test case ISO 12103, A1 Ultrafine of 500kg/m3 from Powder Technology Incorporated (PTI). The input velocity given to particles was 2m/s for Re=6550 and 4m/s for Re=10000, the flow rate of the particles was 1.8e − 5kg/s. The turbulent dispersion modeling inputs used for Random Walk model are given below. The Time scale constant is the Lagrangian Time scale and needs to found iteratively as the particle trajectories are dispersive in nature. The maximum number of steps provides the number of integrations need to be performed for each particle trajectory calculation. The length scale was specified as an estimate of Lagrangian length scale. Number of Tries 2 Time Scale Constant 0.15 The tracking parameters used are presented below Max Number of Steps 10000 Specified Length Scale 10mm 4.3.1 Discretization Schemes The various discretizations schemes used were 34 Momentum Second Order Upwind Turbulent Kinetic Energy Second Order Upwind Turbulent Dissipation Rate Second Order Upwind Pressure Standard 4.4 Grid Independence The grid independence was performed for no filter case of Reynolds Number 6550. The following Figures show the velocity profiles at the step, X=3.75s and X=6.25s after step comparison to experimental velocity profiles of Yao (2000). Figure 4.4: Velocity Profile at Step for Re 6550 No Filter Case 35 Figure 4.5: Velocity Profile at X=3.75s after Step for Various Mesh Sizes, Re 6550 No Filter Case Figure 4.6: Velocity Profile at X=6.25s after step for various Mesh Sizes, Re 6550 No Filter Case 36 The recirculation zone lengths achieved are as follows. Mesh Quality Recirculation Length Coarse Mesh 6.15s Intermediate Mesh 6.25s Fine Mesh 4.72s Table 4.4: Comparison of Recirculation Zone Length and Grid Sizes Comparing the velocity profiles and recirculation zone length the intermediate mesh was chosen as the appropriate mesh size. The fine mesh was numerical diffusive in nature and under predicted the recirculation zone length. The reason for this diffusion is unknown.The convergence criteria specified was 1E06. 4.4.1 Turbulence Model Comparison The various turbulence models and wall functions were compared for the Re 6550 case. The intermediate mesh was chosen as the mesh size for the cases. The following Figures 4.7, 4.8, and 4.9 show comparison of various turbulence models with estimation of velocity profiles at step and X=3.75s , X=6.25s after the step. The turbulence models predicted the velocity at the step almost the same which can be seen in Fig 4.7. The turbulence models also predicted the velocity profile at X=3.75s after the step similarly except for k RNG with enhanced wall treatment. At X=6.25s after the step all the models overpredict the velocities near the top wall. 37 Figure 4.7: Comparison of Velocity Profiles at Step for Re 6550 No Filter case for Various Turbulence Models Figure 4.8: Comparison of Velocity Profiles at X=3.75s after the Step for Re 6550 No Filter Case for Various Turbulence Models 38 Figure 4.9: Comparison of Velocity Profiles at X=6.25s after the Step for Re 6550 No Filter case for Various Turbulence Models 39 The Table 4.5 shows the estimation of recirculation zone length by various turbulence models. The experimental findings of recirculation zone length was found to be 6.5s (Yao (2000)). Comparing the plots and recirculation zone length k RNG with NonEquilibrium Wall Functions was chosen as the model for the rest of the study. Kim et al. (2005) have done comparison studies for various turbulence models and wall function in detail for the geometry of Driver and Seegmiller (1985) and have come to same conclusion k RNG with NonEquilibrium Wall Functions is better of the all models for backward facing step. Grid Independence for Re=10000 was not performed due very small change in Reynolds number and also from literature flow fully developed turbulent past step flow downstream look almost similar irrespective of upstream Reynolds number when expansion ratio is kept constant. Turbulence Model and Wall Function Recirculation Zone Length(s) k RNG Standard Wall Function 6.25 k RNG Non Equilibrium Wall Function 6.24 k RNG Enhanced Wall Treatment 6.35 k! 6.44 SST 6.73 RSM Standard Wall Function 6.53 Table 4.5: Comparison of Recirculation Zone Length with Various Turbulence Models and Wall Functions 40 CHAPTER 5 Results and Discussions 5.1 Introduction The current chapter presents the results of various cases. The primary phase Reynolds numbers chosen are Re= 6550 and Re = 10000 and the filter at two locations, 4.25s, 6.75s, based on the experiments performed by Yao (2000). The monodisperse particle sizes chosen for study are diameters 1μm, 20μm and 50μm based on the range of dust particle sizes and for comparison of results to Yu et al. (2004). The results are presented in the order estimation of primary phase flow field with and without the filter, DPM tracks of particles with and without filter, and mixture model modeling of the airdust mixture without the filter. For the evaluation of models, gradually increasing the complexity will give better understanding of the results. The Stokes number of particles for the Reynolds numbers is 1 and are presented in Appendix A. 5.2 Flow Field Estimation for Re = 6550 and Re = 10000 Velocity Magnitude Contours of No Filter Case Figure 5.1 shows the velocity contours of Re 6550 case and Figure 5.2 shows the velocity contours of Re 10000. The velocity contours aid in better understanding of the flow and also help in estimating the functioning of the turbulence model. The contours clearly show the flow separation at the step, the shear layer and the recirculation zone. The flow contour also shows gradual flow adjustment to fully developed 41 flow downstream. The placement of the outflow boundary at 30s downstream is justified by the previous finding of fully developed flow. The length of the recirculation zone (Xr) is 6.24, in good standing with the literature for fully developed flow at the step, for example Yao (2000) and Armaly et al. (1983). The comparison of contours also shows that when the flow is fully developed after the step the flow characteristics almost remain same and independent of the Reynolds number and depend only on the expansion ratio consistent with findings of Armaly et al. (1983), Adams and Johnston (1988a), and Tihon et al. (2001). 42 Figure 5.1: Velocity Magnitude Contours after the Step for Re = 6550 Figure 5.2: Velocity Magnitude Contours after the Step for Re = 10000 43 Velocity Profile Comparison at the Step The following plots show the velocity profiles at the step; Figure 5.3 for Re 6550 and Figure 5.4 for Re 10000. The velocity profiles are fully developed turbulent. The velocity profiles near the wall differ from the experimental profiles. The variation can be attributed to the wall functions used which are based on standard data sets and upon the experimental conditions. 44 Figure 5.3: Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 6550 Figure 5.4: Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 10000 45 Velocity Profile Comparison at the X=3.75s Figures 5.5 and 5.6 show the velocity profiles at X=3.75s for Re 6550 and Re 10000 respectively. The location behind the step shows a decrease in the maximum velocity from that at the step. The decrease in the velocity is due to expansion of the flow at the step. The models under predict velocities slightly near the top wall and in the recirculation zone. This can be again attributed to wall functions and numerical error. The velocity profiles match very closely in the mid section of the channel at the current location. 46 Figure 5.5: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 6550 Figure 5.6: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 10000 47 Velocity Profile Comparison at the X=6.25s Figures 5.7 and 5.8 show the velocity profiles at X=6.25s for Re 6550 and Re 10000 respectively. The location of current velocity profiles is approximately near the end of the recirculation zone. The velocity profiles show the flow reattaching to the bottom solid wall and adjusting to be a fully developed profile. The model captures the velocities near the bottom wall very well and near the top wall, as for the location of X= 3.75s there is a deviation from the experimental values. 48 Figure 5.7: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re = 6550 Figure 5.8: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 49 5.3 Flow Field Estimation for Filter at 4.25s after the Step The current section discusses for the filter placed at 4.25s. The numerical results are compared to experimental results of Yao (2000). Velocity Profiles at X=3.75s Figures 5.9 and 5.10 show the velocity profile comparison between the CFD results and experimental results. The experimental data shows the velocity profile adjusting to the center of the flow for the current filter location when compared to the no filter case. The velocity profile for Figure 5.9 is normalized by the maximum velocity at the step and for the Figure 5.10 is normalized by the the maximum velocity at the current location. The normalization was due to absence of raw experimental data. The turbulence model and the porous zone model together give good estimation of the velocity flow field near the bottom wall. The velocity magnitude calculation near the top wall shows large deviation from the experimental data and also the velocity profile. The error can be attributed to previous error for no filter case at current location, and also filter being approximated as porous jump. The filter surface is actually a pleated air filter. Due to the pleated design the surface makes the flow experience more surface area and may dampen the flow structures. The modeling of the whole filter is very difficult as greater computational effort is needed and there are limitations in meshing. The sharp edges give very high aspect ratio cells which result in divergence of the solution even if modern algorithms are employed. Considering the limitations, the current model predicted the velocities acceptably. 50 Figure 5.9: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =6550 for Filter at X= 4.25s Figure 5.10: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =10000 for Filter at X= 4.25s 51 Velocity Contours The velocity contours shown in Figures 5.11 and 5.12 show the truncation of recirculation zone produced by the filter. The area of recirculation zone also increased in the Y direction reducing the area between the step and the filter to almost half the original. The current filter location shows the strong influence on the flow field due to placement of filter in the recirculation zone. The placement also shows secondary recirculation on the top wall near the filter. The flow past the filter the model predicts a turbulent reattachment. 52 Figure 5.11: Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 4.25s Figure 5.12: Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 4.25s 53 5.4 Flow Field Estimation for Filter at 6.75s after the Step The following section discusses the results for filter placed at 6.75s after the step. The placement of the filter at this location has negligible effect on the recirculation zone. Velocity Profiles Figures 5.13 and 5.14 show the velocity profiles at the current location show the same tendency of the velocity profiles adjusting to the center of the channel. The recirculation zone is least affected at this location in comparison to the area increases in the Y direction for the filter located at X=4.25s. The experimental and CFD results match better than the previous location of filter at 3.75s. 54 Figure 5.13: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =6550 for Filter at X= 6.75s Figure 5.14: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 for Filter at X= 6.75s 55 Velocity Contours Comparing the velocity contours in Figures 5.15, 5.16 with filter at 6.75s to those for the no filter case, Figures 5.1 and 5.2 the effect of the filter on the recirculation zone is negligible. 56 Figure 5.15: Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 6.75s Figure 5.16: Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 6.75s 57 5.5 Discrete Phase Model for No Filter Case The following sections shows the particle tracks for the two Reynolds numbers 6550 and 10000 for the no filter case. The steady state DPM model was employed owing to the fact that the concentration of secondary phase was negligible and the particles do not participate in chemical reaction nor do they transfer in any mass. The solved flow field was used and particle trajectories were integrated on the known primary phase velocities. The particle residence time which is the time for which a particle resides in the domain of calculation, was used to track particle trajectories. The particle tracks help us visualize the particle response to different flow structures. 5.5.1 Particle Tracks Re = 6550 No Filter Case The DPM tracks (Figures 5.17, 5.18, and 5.19) show that as the particle diameter and Stokes numbers increase the number of particles entering the recirculation zone is minimized. The phenomena is consistent with the literature for example, Ruck and Makiola (1988), Fessler and Eaton (1997), Fessler and Eaton (1999), Tu (1997), and Yu et al. (2004). The larger particles which have higher momentum do not follow the flow and escape from fluid vortices of low momentum. The maximum particle residence times varied from 1.4 sec to 1.9 sec. When the particle diameter and Stokes number increase we can observe that the particles take longer time to escape from the recirculation zone once they enter. 58 Figure 5.17: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, No Filter case Figure 5.18: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, No Filter case 59 Figure 5.19: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, No Filter case 60 5.5.2 Particle Tracks Re = 10000 No Filter Case Figures 5.20, 5.21, and 5.22 show similar tendencies as the case of Re=6550. The maximum particle residence time is varied from 0.86 sec to 1.8 sec. The larger particles which enter the recirculation zone find it harder to escape the recirculation zone, as in the previous case. For the case of 50μm particle, the particles recirculate longer in the recirculation zone compared to the Re 6550 case. This is due to the particles entering the recirculation zone taking longer time to achieve momentum for cross stream escape to the main flow. Figure 5.20: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, No Filter case 61 Figure 5.21: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, No Filter case Figure 5.22: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, No Filter case 62 5.6 Discrete Phase Model for Filter Case The following section discusses the particle trajectories for filter placed at 4.25s and 6.75s after the step for Re of 6550 and 10000. 5.6.1 Filter located at 4.25s after the step The following section presents the particles tracks for filter located at 4.25s. The particle boundary condition at the filter is treated as trap. Re = 6550 Figures 5.23, 5.24, and 5.25 show the particle tracks for primary phase Re 6550. The cases where the particle diameter is 1μm, and 20μm belong to the low Stokes number category. In all the cases the effective area for filtration is reduced to around 60 % of porous zone due to development of recirculation zone. The effect of the recirculation zone reduces the filter efficiency and life time drastically. In the case of 50μm particle, the bottom region of the filter is hardly used. 63 Figure 5.23: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 4.25s Figure 5.24: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 4.25s 64 Figure 5.25: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 4.25s 65 Re = 10000 Figures 5.26, 5.27, and 5.28 show the particle tracks for Re 10000 of primary phase. When the Reynolds number of the primary phase is increased to 10000 there is slight increase in Stokes number for 1μm and 20μm particles. The effective area of filtration, where the particles impact the media is increased compared to the previous case of Re 6550. Comparing the 1μm, 20μm with the 50μm, hardly any particles enter the recirculation zone. From the filtration design point of view if we could control the carrier phase velocity and average Stokes number we could use the recirculation zone to transport particles to the filter. The position or location of the filter also plays an important role; this criterion will be explained in the next section for the filter located at 6.75s after the step. Figure 5.26: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, Filter located at 4.25s 66 Figure 5.27: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, Filter located at 4.25s Figure 5.28: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, Filter located at 4.25s 67 5.6.2 Filter Located at 6.75s after the step The section contains particle tracks for the filter located at 6.75 for the two Reynolds numbers. The section helps us in understanding the placement of filter and effects of filtration with the recirculation zone present. Re 6550 Figures 5.29, 5.30, and 5.31 present the particle tracks for varying diameters. The placement of the filter at the end of the recirculation zone promoted more particles to enter the recirculation zone. The particles had to travel a greater distance compared to the filter at 4.25s case, effectively losing momentum and getting trapped in the recirculation zone. There are two distinct patterns observed when comparing particle tracks of the filter at 4.25s from Figures 5.23, 5.24, and 5.25 and the current case, in which we can observe strong particle settlement at the corner beneath the step. The second observation is that the top recirculation zone observed in velocity contours Figure 5.15 at the filter also holds particles for longer time. 68 Figure 5.29: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 6.75s Figure 5.30: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 6.75s 69 Figure 5.31: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 6.75s 70 Re 10000 The particles also behaved similarly for Re 10000 as for the case of Re 6550 as shown in Figures 5.32, 5.32, 5.32. The particles tend to settle near the corner of the step and particles were found to get trapped in the recirculation zone developed on the top wall due to the filter. Figure 5.32: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re =10000, Filter located at 6.75s 71 Figure 5.33: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re =10000, Filter located at 6.75s Figure 5.34: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re =10000, Filter located at 6.75s 72 5.7 Summary CFD modeling was performed for particulate flow in a backward facing step with an expansion ratio of 1:2 with a porous medium located at X=4.25s and X=6.75s location. The particle Stokes number in this study used was 1 and the particle concentrations were low. The Reynolds numbers used in the current study are fully developed turbulence conditions after the step of 6550 and 10000. The following inferences may be drawn from the CFD modeling done for No Filter, Filter Cases and DPM for aforementioned cases. The CFD results of no filter case matched well with experimental results considering the flow conditions at the step being fully developed turbulent for CFD model and not fully developed for the experimental case. There has been a consistent deviation in the CFD model and experimental case at the top wall and bottom wall. This can be attributed to wall functions used and the flow conditions at the step. The recirculation zones are in good agreement. The CFD results of the filter cases did match the experimental results acceptably considering the model approximation of the pleated filter as a one dimensional porous jump. For the filter placed near the step at 4.25s, the recirculation zone got truncated and divided the region between the filter and step in almost half. For the filter placed at 6.75s after the step, the CFD results and experimental data match well when compared to the 4.25s case. With the filter place at a location beyond the recirculation zone of the no filter case, the filter did not appreciably affect the recirculation zone length. From the DPM of particles for the no filter case we can clearly see that as the particle size and Stokes number increases, fewer particles enter the recirculation zone. For the particle tracks realized by particle residence time for the filter placed at 4.25s 73 the effective area of filtration was reduced to around 60%. With the filter moved to a location of 6.75s after the step, the effective area of filtration changed slightly compared to the 4.25s case and also promoted more time for the particles to enter the recirculation zone. For the filter cases the maximum particle residence time can be observed for the filter placed at the more downstream location. Analyzing the motion of particles with respect to size, the recirculation zone can effectively be used to transport most of the heavier particles bypassing the recirculation zone. 74 CHAPTER 6 Conclusions and Recommendations 6.1 Conclusions The DPM was utilized to study trajectories of particle diameters of 1μm, 20μm and 50μm for a backward facing step with porous medium for carrier phase Reynolds numbers of 6550 and 10000. The particle track trajectories helped in visualizing the particle flight in the carrier phase medium and aided in better understanding of filtration. The current study gave rise to the following conclusions: 1. The effect of porous medium on the recirculation zone is predominant when the porous medium is placed in the recirculation zone of the no filter case. 2. The effect of the porous medium on the recirculation zone is small when the medium is placed downstream of the recirculation zone zone of the no filter case. 3. The recirculation zone tends to trap some of the smaller particles and thereby increase the filtration time of the particulate flow 4. The recirculation zone decreases the effective area of filtration to around 60% with filter surface towards the bottom wall receiving few particles. 5. The recirculation zone can be predominantly used to transport most of the heavier particles to the porous medium. 6. The current geometry, if modified with some pre filtration techniques like electrostatic precipitation,the modified configuration can effectively increase filtra 75 tion by removing small particles in the recirculation zone, thereby enhancing filter life. 6.2 Recommendations 1. The flow field estimation accuracy can increased by using LES and DES to model the flow field. 2. The differences between the filter case experimental and modeling results can be minimized by developing an empirical correlation for pressure drop based on the number of filter pleats. This empirical correlation can incorporated into Fluent using a User Defined Function. 3. The effect of the higher concentration of particles was not studied in the current research. The feasibility of applying the current geometry for higher concentration particle flows can be studied using EulerianEulerian models. 76 Bibliography Adams, E. and J. Johnston (1988a). Effects of separation shear layer on the reattachment flow structure. part 1: Pressure and turbulence quantities. Experiments in Fluids, 6, 400–408. Adams, E. and J. Johnston (1988b). Effects of separation shear layer on the reattachment flow structure. part 2: Reattachment length and wall shear stress. Experiments in Fluids, 6, 493–499. Armaly, B. F., F. Durst, J. C. F. Pereira, and B. Schonung (1983). Experimental and theoretical investigation of backwardfacing step flow. Journal of Fluid Mechanics, 127, 473–496. Benavides, A. and B. van Wachem (2009, June). EulerianEulerian prediction of dilute turbulent gasparticle flow in a backwardfacing step. International Journal of Heat and Fluid Flow, 30 (3), 452–461. Chan, E. C. and F.S. Lien (2005). Permeability effects of turbulent flow through a porous insert in a backwardfacingstep channel. Transport in Porous Media, 59, 47–71. Crowe, C., M. Sommerfeld, and Y. Tsuji (1998). Multiphase Flows with Droplets and Particles. CRC Press. Deen, N. G., M. V. Annaland, M. A. Van der Hoef, and J. A. M. Kuipers (2007). Review of discrete particle modeling of fluidized beds. Chemical Engineering Science, 62 (12), 28–44. Driver, D. M. and H. L. Seegmiller (1985). Features of a reattaching turbulent shear layer in divergent channel flow. AIAA Journal, 23, 163–171. 77 Fessler, J. R. and J. K. Eaton (1997). Particle response in a planar sudden expansion flow. Experimental Thermal and Fluid Science, 15 (4), 413 – 423. Fessler, J. R. and J. K. Eaton (1999). Turbulence modification by particles in a backwardfacing step flow. Journal of Fluid Mechanics, 394, 97–117. Hetsroni, G. (1989). Particlesturbulence interaction. International Journal of Multiphase Flow, 15 (5), 735 – 746. Kim, J., S. J. Kline, and J. P. Johnston (1980). Investigation of a reattaching turbulent shear layer: Flow over a backwardfacing step. Journal of Fluids Engineering, 102, 302–308. Kim, J. Y., A. J. Ghajar, C. Tang, and G. L. Foutch (2005). Comparison of nearwall treatment methods for high Reynolds number backwardfacing step flow. International Journal of Computational Fluid Dynamics 19, 493–500. Launder, B. and D. Spalding (1974). The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 3, 269–289. Menter, F. R., M. Kuntz, and R. Langtry (2003). Ten years of industrial experience with the SST turbulence model. Heat and Mass Transfer, 4, 625. Ming, Xu. Wei, G. J. L. (2007). A discrete particle model for particlefluid flow with consideration of subgrid structures. Journal of Chemical Engineering Science. 62, 2302–2308. Mohanarangam, K., J. Tu, and L. Chen (2008, December). Numerical study of particle dispersion behind a sudden expansion geometry and its effect on step heights. Computers & Chemical Engineering, 32 (12), 3187–3197. Mohanarangam, K. and J. Y. Tu (2007). Twofluid model for particleturbulence interaction in a backwardfacing step. AIChE Journal, 53 (9), 2254–2264. Nield, D. and A. Bejan (2006). Convection in Porous Media. Springer New York. 78 Papadopoulos, G. and M. V. Otugen (1995). Separating and reattaching flow structure in a suddenly expanding rectangular duct. Journal of Fluids Engineering, 117 (1), 17–23. Pope, S. B. (2000). Turbulent Flows. Cambridge University. PTI. http://www.powdertechnologyinc.com/products/testdust/testdust.php, accessed August 2010. Ruck, B. and B. Makiola (1988). Particle dispersion in a singlesided backwardfacing step flow. International Journal of Multiphase Flow, 14 (6), 787 – 800. Saffman, P. G. (1965). The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics, 22 (02), 385–400. Schafer, M. (2006). Computational Engineering Introduction to Numerical Methods. Springer Berlin Heidelberg. Shih, T.H., W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu (1995). A new k eddy viscosity model for high Reynolds number turbulent flows. Computers & Fluids, 24 (3), 227 – 238. Shyy, W., S. Thakur, H. Ouyang, and J. Liu (1997). Computational Techniques for Complex Transport Phenomena. Cambridge University,. Tian, Z. F., J. Y. Tu, and G. H. Yeoh (2005). Numerical simulation and validation of dilute gasparticle flow over a backwardfacing step. Aerosol Science and Technology, 39 (4), 319 – 332. Tihon, J., J. Legrand, and P. Legentillhomme (2001). Nearwall investigation of backwardfacing step flows. Experiments in Fluids, 31, 484–493. Tu, J. Y. (1997). Computation of turbulent twophase flow on overlapped grids. Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 32, 175–193. 79 Wang, B., H. Zhang, and X. Wang (2006). Large eddy simulation of particle response to turbulence along its trajectory in a backwardfacing step turbulent flow. International Journal of Heat and Mass Transfer, 49 (12), 415–420. Wilcox, D. C. (2006). Turbulence Modeling for CFD (3rd ed.). DCW Industries Inc. Yakhot, V., S. A. Orszag, S. Thangam, T. B. Gatski, and C. G. Speziale (1992). Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids AFluid Dynamics, 4 (7), 1510–1520. Yao, S. (2000). Two Dimensional Backward Facing Step Flow Preceding an Automotive Filter. Ph. D. thesis, Oklahoma State University. Yu, K. F., K. S. Lau, and C. K. Chan (2004). Numerical simulation of gasparticle flow in a singlesided backwardfacing step flow. J. Comput. Appl. Math, 163 (1), 319–331. 80 APPENDIX A Appendix: Stokes Number The Stokes numbers have been calculated as follows to be consistent with the literature, following Fessler and Eaton (1999). The Stokes number is the ratio of particle momentum response time, v to the fluid response time F as follows St = v F (A.1) where v is given as follows v = pd2 p 18μc (A.2) and F is given as follows, the characteristic length is based on the largest vortex present. F = 6.25s Umax (A.3) The Stokes number for Re = 6550 case is given in following table Particle Diameter Stokes Number 1μm 4.13 ×10−5 20μm 0.017 50μm 0.103 Table A.1: Stokes Number for Re 6550 81 The Stokes number for Re = 10000 case is given in following table Particle Diameter Stokes Number 1μm 6.4 ×10−5 20μm 0.026 50μm 0.160 Table A.2: Stokes Number for Re 10000 82 VITA Krishna Chaitanya Ravi Candidate for the Degree of Master of Science Thesis: NUMERICAL PREDICTION OF PARTICULATE FLOWOVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Major Field: Mechanical and Aerospace Engineering Biographical: Personal Data: Born in Kothagudem, Andhra Pradesh, India on January 05, 1985. Education: Received the B.S. degree from Vellore Institute of Technology,Vellore,Tamil Nadu, India ,2006, Mechanical Engineering Completed the requirements for the degree of Master of Arts with a major in Mechanical and Aerospace Engineering Oklahoma State University in December, 2010. Experience: Internship : August 2006  August 2007 Cusp Technologies, Hyderabad, Andhra Pradesh, India Teaching Assistant: Jan 2008May 2010 Department of Mechanical Engineering Oklahoma State University Research Assistant: Jan 2008 Aug 2010 Department ofMechanical and Aerospace Engineering Oklahoma State University Professional Memberships: Member of American Society of Mechanical Engineers (ASME) Member of American Society of Heating, Refrigerating and Air Conditioning Engineers (ASHRAE) Name: Krishna Chaitanya Ravi Date of Degree: December, 2010 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Pages in Study: 82 Candidate for the Degree of Master of Science Major Field: Mechanical and Aerospace Engineering Scope and Method of Study:The current research focuses on CFD modeling of turbulent particulate flows involving separation and filters or porous media. The backward step geometry was selected due to the large experimental data sets present in the literature. The backward step dimensional parameters used are expansion ratio 1:2, entrance length of 2 step heights and channel length of 30 step heights from the step. The mesh was developed in ICEM CFDTMand modeling was done using the FluentTMcommercial CFD package. The carrier phase turbulence was modeled using the k RNG model. The particles were modeled using the discrete phase model and particle dispersion was modeled using stochastic tracking. The porous jump condition was used at the porous medium. The boundary conditions employed are uniform velocity at the inlet with 5% turbulent intensity, outflow for outlet boundary condition, and noslip boundary conditions at the walls. The particle boundary condition implemented at the walls is “reflect” and at the filter is “trap”. Findings and Conclusions : The numerical results of no filter case matched experimental results with good accuracy. The current model agreed well with these results for Reynolds Numbers (Re) of 6550 and 10000 when compared in terms of the recirculation zone length and velocity profiles at 3.75 and 6.25 step heights downstream of the step. The presence of porous media showed a profound effect on the recirculation zone length. The velocity profile comparison for the filter cases do not validate very well with experimental results due to the initial conditions and the porous jump boundary condition. The current study focuses on prediction of dust particle trajectories of sizes 1μm to 50μm for different carrier phase Reynolds Number with the porous medium located at 4.25 and 6.75 step heights downstream of the step. The particle trajectories were observed for the aforementioned cases and filter locations. As particle size increased, the number of particles entering the recirculation zone decreased. The placement of the filter far from the step promoted more particles getting trapped in the recirculation zone. ADVISOR’S APPROVAL:
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Title  Numerical Prediction of Particulate Flow over a Backward Facing Step Preceding a Filter Medium 
Date  20101201 
Author  Ravi, Krishna Chaitanya 
Keywords  Cfd, Discrete Phase Model, Numerical, Porous Medium, Turbulence 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  The current research focuses on CFD modeling of turbulent particulate flows involving separation and filters or porous media. The backward step geometry was selected due to the large experimental data sets present in the literature. The backward step dimensional parameters used are expansion ratio 1:2, entrance length of 2 step heights and channel length of 30 step heights from the step. The mesh was developed in ICEM CFD and modeling was done using the Fluent commercial CFD package. The carrier phase turbulence was modeled using the kepsilon RNG model. The particles were modeled using the discrete phase model and particle dispersion was modeled using stochastic tracking. The porous jump condition was used at the porous medium. The boundary conditions employed are uniform velocity at the inlet with 5% turbulent intensity, outflow for outlet boundary condition, and noslip boundary conditions at the walls. The particle boundary condition implemented at the walls is "reflect" and at the filter is "trap". The numerical results of no filter case matched experimental results with good accuracy. The current model agreed well with these results for Reynolds Numbers (Re) of 6550 and 10000 when compared in terms of the recirculation zone length and velocity profiles at 3.75 and 6.25 step heights downstream of the step. The presence of porous media showed a profound effect on the recirculation zone length. The velocity profile comparison for the filter cases do not validate very well with experimental results due to the initial conditions and the porous jump boundary condition. The current study focuses on prediction of dust particle trajectories of sizes 1μm to 50μm for different carrier phase Reynolds Number with the porous medium located at 4.25 and 6.75 step heights downstream of the step. The particle trajectories were observed for the aforementioned cases and filter locations. As particle size increased, the number of particles entering the recirculation zone decreased. The placement of the filter far from the step promoted more particles getting trapped in the recirculation zone. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM By Krishna Chaitanya Ravi Bachelor of Technology in Mechanical Engineering Vellore Institute of Technology Vellore, Tamil Nadu, India 2006 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2010 COPYRIGHT c By Krishna Chaitanya Ravi December, 2010 NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Thesis Approved: Dr. Frank W. Chambers Thesis Advisor Dr. Afshin J. Ghajar Dr. Khaled A. Sallam Dr. Mark E. Payton Dean of the Graduate College iii ACKNOWLEDGMENTS I would like to take this opportunity to thank my advisor Dr. F. W. Chambers for his support, patience, guidance throughout my research and coursework. His encouragement to learn and implement new concepts has always helped me in having an empty mind to learn things. He was always present to solve problems and guide me in the right path. I am also indebted to Dr. Sallam and Dr. Ghajar for their time and patience for reviewing the thesis document and attending my examination. Dr. Ghajar has been very supportive and helpful in many ways throughout the course of study in Oklahoma State University. Dr. Sallam has been an excellent teacher and his simplistic visualization approach towards learning complex systems is admirable. I also like to thank my fellow researchers Deshpande, Karthik, Alok, and Netaji for their assistance and help in research and outside the lab. Unconditional love and support from my parents Madhusudhan Rao, Vijaya Lakshmi and my brother Goutham in all fronts made my achievements possible. I am grateful to my grandparents Rama Krishnaiah and Kota Ratnama for their constant love and respect for me as an individual. I like to thank all my friends not limited to Ashvin, Ashwin, Kapil, Sai, Kamal, Bhargav, Vikas, Harsha, Sudheer, Durga, Muthappa, Rao, Chinnmay, Raagini, Akshata, Varun, Sunil, Sarvana and so on who were always there to share fun, enjoyment, and also during times of trouble. I thank God for all the necessary. iv TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Literature Survey 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Study of Particulate Flow in Backward Facing Step . . . . . . . . . . 8 2.4 Study of Backward Step with Porous Medium Insert . . . . . . . . . 13 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Numerical Method 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.1 ReynoldsAveraged Navier Stokes . . . . . . . . . . . . . . . . 21 3.3 k − Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 Turbulent Kinetic Energy(k) . . . . . . . . . . . . . . . . . . . 22 3.3.2 Modeling Dissipation . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.3 Numerical Treatment of k − Model . . . . . . . . . . . . . . 22 3.3.4 Variations of k Turbulence Model . . . . . . . . . . . . . . . 23 3.4 k! Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 v 3.5 SST Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6.1 Wall Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.7 Discrete Phase Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.8 Porous Medium Modeling . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Modeling Process 30 4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Grid Generation and Adaptation . . . . . . . . . . . . . . . . . . . . 30 4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.1 Discretization Schemes . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Grid Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4.1 Turbulence Model Comparison . . . . . . . . . . . . . . . . . . 37 5 Results and Discussions 41 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Flow Field Estimation for Re = 6550 and Re = 10000 . . . . . . . . 41 5.3 Flow Field Estimation for Filter at 4.25s after the Step . . . . . . . . 50 5.4 Flow Field Estimation for Filter at 6.75s after the Step . . . . . . . . 54 5.5 Discrete Phase Model for No Filter Case . . . . . . . . . . . . . . . . 58 5.5.1 Particle Tracks Re = 6550 No Filter Case . . . . . . . . . . . 58 5.5.2 Particle Tracks Re = 10000 No Filter Case . . . . . . . . . . . 61 5.6 Discrete Phase Model for Filter Case . . . . . . . . . . . . . . . . . . 63 5.6.1 Filter located at 4.25s after the step . . . . . . . . . . . . . . . 63 5.6.2 Filter Located at 6.75s after the step . . . . . . . . . . . . . . 68 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Conclusions and Recommendations 75 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vi 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A Appendix: Stokes Number 81 vii LIST OF TABLES Table Page 3.1 Modeling Constants Used for k . . . . . . . . . . . . . . . . . . . . 22 3.2 Modeling Constants Used for k RNG from Yakhot et al. (1992) . . 24 3.3 Modeling Constants Used for k! from Wilcox (2006) . . . . . . . . . 25 3.4 Modeling Constants Used for SST Turbulence Model . . . . . . . . . 26 4.1 Mesh Size Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Boundary Conditions Implemented . . . . . . . . . . . . . . . . . . . 32 4.3 Velocities at Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Comparison of Recirculation Zone Length and Grid Sizes . . . . . . . 37 4.5 Comparison of Recirculation Zone Length with Various Turbulence Models and Wall Functions . . . . . . . . . . . . . . . . . . . . . . . 40 A.1 Stokes Number for Re 6550 . . . . . . . . . . . . . . . . . . . . . . . 81 A.2 Stokes Number for Re 10000 . . . . . . . . . . . . . . . . . . . . . . . 82 viii LIST OF FIGURES Figure Page 1.1 Computational Geometry With Different Boundary Conditions . . . . . . 2 2.1 Aspect Ratio Effects on Recirculation Zone Length from Papadopoulos and Otugen (1995). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Recirculation Length remaining Constant for Fully Developed Turbulent Flow Past the Step from Tihon et al (2001). . . . . . . . . . . . . . . . . 7 2.3 Particle Concentration Contours behind the step with varying particle diameters. Case (a) dp = 7μm, Case (b) dp = 15μm, Case (c) dp = 25μm, and Case (d) dp =45μm from Tu (1997) . . . . . . . . . . . . . . . . . . 10 2.4 Particle Distribution of Various Diameters from Yu et. al (2004) . . . . . 11 2.5 Effect of Slip Velocity on Particle of Diameter 100 μm from Yu et. al (2004) 11 2.6 Effect of Slip Velocity on Particle of Diameter 200 μm from Yu et. al (2004) 12 2.7 Effect of Gravity in ydirection on Particles from Yu et. al (2004). . . . . 12 2.8 Stream Traces for Variation of Darcy Constant from Chan and Lien (2005). 14 2.9 Stream Traces for Variation of Forchheimer Constant from Chan and Lien (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.10 Stream Traces for Variation of Porous Media Thickness Constant from Chan and Lien (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Near Wall Region Grid Point . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Implemented Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Wall Adaptations at Step . . . . . . . . . . . . . . . . . . . . . . . . . 32 viii 4.3 Experimental Data of Pressure Drop vs. Flow Velocity . . . . . . . . . . 33 4.4 Velocity Profile at Step for Re 6550 No Filter Case . . . . . . . . . . . . 35 4.5 Velocity Profile at X=3.75s after Step for Various Mesh Sizes, Re 6550 No Filter Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 Velocity Profile at X=6.25s after step for various Mesh Sizes, Re 6550 No Filter Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.7 Comparison of Velocity Profiles at Step for Re 6550 No Filter case for Various Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.8 Comparison of Velocity Profiles at X=3.75s after the Step for Re 6550 No Filter Case for Various Turbulence Models . . . . . . . . . . . . . . . . . 38 4.9 Comparison of Velocity Profiles at X=6.25s after the Step for Re 6550 No Filter case for Various Turbulence Models . . . . . . . . . . . . . . . . . 39 5.1 Velocity Magnitude Contours after the Step for Re = 6550 . . . . . . . . 43 5.2 Velocity Magnitude Contours after the Step for Re = 10000 . . . . . . . . 43 5.3 Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 6550 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 6550 . . . . . . . . . . . . . . . . . . . 47 5.6 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 10000 . . . . . . . . . . . . . . . . . . 47 5.7 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re = 6550 . . . . . . . . . . . . . . . . . . . 49 5.8 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 . . . . . . . . . . . . . . . . . . . 49 ix 5.9 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =6550 for Filter at X= 4.25s . . . . . . . . 51 5.10 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =10000 for Filter at X= 4.25s . . . . . . . 51 5.11 Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 4.25s 53 5.12 Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 4.25s 53 5.13 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =6550 for Filter at X= 6.75s . . . . . . . . 55 5.14 Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 for Filter at X= 6.75s . . . . . . . 55 5.15 Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 6.75s 57 5.16 Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 6.75s 57 5.17 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . . 59 5.18 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . 59 5.19 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, No Filter case . . . . . . . . . . . . . . . . . . . 60 5.20 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 61 5.21 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 62 5.22 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, No Filter case . . . . . . . . . . . . . . . . . . 62 5.23 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . . 64 x 5.24 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . 64 5.25 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 4.25s . . . . . . . . . . . . . . 65 5.26 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 66 5.27 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 67 5.28 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, Filter located at 4.25s . . . . . . . . . . . . . . 67 5.29 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . . 69 5.30 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . 69 5.31 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 6.75s . . . . . . . . . . . . . . 70 5.32 Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . . 71 5.33 Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . 72 5.34 Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re =10000, Filter located at 6.75s . . . . . . . . . . . . . . 72 xi NOMENCLATURE C 1 k Modeling Constant C 2 k Modeling Constant C2 Inertial Resistance Factor Cμ k Modeling Constant CD Coefficient of Drag D Distance of Porous Medium from the Step dp Particle size, (μm) g Gravitational Acceleration,(m s2 ) H Upstream Channel Length, (mm) k Turbulent kinetic energy,(m2 s2 ) Res Reynolds number based on Step Height s Step Length, (mm) St Stokes number T Thickness of Porous Medium, (mm) Umax Maximum Velocity at Step, (m/s) Xr Recirculation Zone Length, (mm) Xc Corner Recirculation Zone Length, (mm) Permeability of Medium, (m2) Turbulent dissipation,(m2 s3 ) RNG Expansion Parameter μ Dynamic viscosity, (Pas) μt Turbulent viscosity, (Pas) xii Density of fluid, ( kg m3 ) k k Modeling Constant k Modeling Constant F Fluid Response Time v Particle Momentum Response Time w Wall Shear Stress,( N m2 ) ! Turbulence Frequency ABBREVIATIONS AR Aspect Ratio ER Expansion Ratio CFD Computational Fluid Dynamics DES Detached Eddy Simulation DPM Discrete Phase Model DNS Direct Numerical Simulation LES Large Eddy Simulation RANS Reynolds Averaging of NavierStokes RNG Renormalization Group SST Shear Stress Transport xiii CHAPTER 1 Introduction The motivation for the current study is the air filtration device commonly found on automobiles for providing clean air to the engine. Due to the constraints of space and design of other components, air filters are given a reduced priority on the design point of view. Many air filter housings have sudden expansion of area, and from the fluid mechanics point of view whenever there is area expansion generally there is a recirculation zone. The best examples are backward step flow, flow over a bluff body etc. Predicting the flow behavior is much more complex when a recirculation zone is present due to the fact the mean flow might be laminar, transitional, or turbulent and the recirculation zone characteristics might be different. The filter modeling can be achieved in two ways: microscopically considering all the intricate details of the filter or utilizing a computationally suitable macroscopic model. The current research focuses on the study of particle behavior in a recirculation zone and a filter modeled as porous media. Generally the particles, depending on the Stokes number, get trapped in the recirculation zone or just escape the recirculation zone because of high inertia and lift forces. 1.1 Problem Statement The backward facing step is chosen as the geometry for the current study due to the presence of a distinct recirculation zone and the existence of a huge experimental and computational database. The backward facing step geometry is selected based on experiments performed by Yao (2000). The geometry of the backward step is shown 1 in Figure 1.1 Figure 1.1: Computational Geometry With Different Boundary Conditions • s  Step Length (25mm) • H  Upstream Channel Length (50mm) • D  Distance of porous medium from the Step. • T  Thickness of the Filter or Porous Medium. • Xr  Recirculation Zone Length • Xc  Corner Recirculation Zone Length The Characteristics of the backward facing step studied here are Expansion Ratio (ER) = s H = 0.5 The Reynolds number used is based on the definition of Yao (2000), and is based on maximum velocity at the step and with Res = Umaxs (1.1) 2 1.2 Objective The goal behind the current study was to investigate whether the recirculation zone can be beneficial for filtration design. The current study focuses on study of mono disperse particulate flow in a backward step flow with a filter medium. The backward facing step is modeled as 2D with the dimensions provided in Figure 1.1. The current work focuses on particulate sizes of 1μm, 20μm, 50μm and the particle properties have been chosen close to ultrafine test case ISO 12103, A1 Ultrafine from Powder Technology Incorporated (PTI). The Reynolds numbers studied were of 6550 and 10000 for the carrier phase or the main flow carrying the dispersed phase. The filter location is also varied, with two locations, at X=4.25s and 6.75s. The particle trajectories were studied for no filter and filter cases. 3 CHAPTER 2 Literature Survey 2.1 Introduction The backward facing step flow is a widely studied flow geometry both experimentally and numerically due to the boundary layer detachment, the presence of a large recirculation zone and the shear layer. The backward facing step flow has also proven to be a good test for the turbulence models due to the complexity in flow structures and the various regimes. This literature review focuses on experiments and numerical investigations pertinent to the current study. The goal of the literature review is to provide foresight on the numerical results we should expect after CFD analysis and also provide some parameters for grid independence test. The literature review can be broadly classified into three sub sections: • Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step Geometry • Study of Particulate Flow in Backward Facing Step Geometry • Study of Backward Step with Porous Medium Insert 2.2 Experimentation and Numerical Study on Single Phase Flow in Backward Facing Step Kim et al. (1980) performed experiments on Backward Facing Step with two expansion ratios, 1.33 and 1.5. The Reynolds number based on s, see Fig 1.1 (Res) is 40,000 4 and 61333 approximately. The recirculation length, xr/s, zone found experimentally is 7 ± 1. The authors also observed shear stress and turbulence intensities reaching maxima at the reattachment point and subsiding afterwards. Armaly et al. (1983) performed experiments on backward facing step flow using Laser Doppler Anemometry. The backward facing step expansion ratio (ER) was 1:1.94. The experiments were performed for Reynolds numbers varying from 70 to 8000 encompassing all three regimes: laminar, transitional and turbulent flow. For Armaly et al. (1983), the Reynolds number is defined as Re = V (D)/ where D the hydraulic diameter of the inlet section = 2s and V is 2 3Umax, with Umax maximum velocity at the inlet section. The experiment results showed additional recirculation zones at the opposite wall at certain Reynolds numbers along with the main recirculation zone. The recirculation zone length increased with Reynolds number up to the transition regime and then decreased as the flow became turbulent. The flow also behaved as twodimensional for laminar and turbulent cases due to the high aspect ratio in the zdirection but in the transitional regimes the flow behaved threedimensionally. Driver and Seegmiller (1985) performed experiments on backward facing step flow using Laser Doppler Anemometry for flow velocities and laser interferometer skinfriction apparatus to calculate skin friction. The expansion ratio was 1.13 and the flow Reynolds number based on h and velocity is 37500. Driver and Seegmiller (1985) also performed numerical modeling using various turbulence models. The k , kmodified and ASM failed to predict recirculation zone length. The ASM with modified provided good estimates of the recirculation zone length and also the velocity profiles along the length of the channel. Adams and Johnston (1988a) performed experiments on backward step flow with an expansion ratio of 1.25 and Reynolds number of 36000 based on s (see Fig 1.1). The authors’ aim was to address the problem of sudden pressure rise at reattachment as a function of ER. The authors’ results, along with compilation of data from other authors work, showed that as the flow becomes fully turbulent, the re 5 circulation pressure gradient is independent of upstream Reynolds number. This also can be interpreted in terms of recirculation length being almost constant with fully developed turbulent regimes and consistent with findings of Armaly et al. (1983). Papadopoulos and Otugen (1995) performed experiments on backward step flow with expansion ratio of 2 and flow Reynolds number of 26500 based on s (see Fig 1.1). The authors’ aim was to find the effect of aspect ratio(AR) on the recirculation zone length, velocity, and wall pressure. The Figure 2.1 shows the variation of recirculation zone length with AR. They varied AR from approximately 5 to 30 and found as the AR increases there is a gradual increase of recirculation zone length. The recirculation zone length attains constant length after AR of 6. The recirculation zone length xr/s attained is of the order 8. The effect of small AR is that the flow is three dimensional along the total channel and for even higher aspect ratios the flow behaved at certain regions as three dimensional. Figure 2.1: Aspect Ratio Effects on Recirculation Zone Length from Papadopoulos and Otugen (1995). 6 Tihon et al. (2001) performed experiments in backward facing step of expansion ratio of 1.4 for varied Reynolds numbers based on s (see Fig 1.1) from 1000 to 12000. The authors used electro diffusion technique for determining the wall shear stress and estimating the recirculation zone lengths. The authors provided good experimental values for normal recirculation zone length as well as the corner recirculation zone length. The authors have experimentally shown that after flow reaches fully turbulent regime the recirculation zone length remained constant about xr/s 5.1 and corner recirculation zone about xc/s 1.75 consistent with Armaly et al. (1983) and Adams and Johnston (1988a) findings. Figure 2.2 shows the recirculation zone attaining constant length for fully developed turbulent flow behind the step. The authors have also presented a good survey of backward step experiments with varying expansion ratios and aspect ratios. Figure 2.2: Recirculation Length remaining Constant for Fully Developed Turbulent Flow Past the Step from Tihon et al (2001). 7 2.3 Study of Particulate Flow in Backward Facing Step Ruck and Makiola (1988) studied the dispersion of particles in a backward facing step of expansion ratio of 1:2. The Reynolds numbers studied based on s and velocity were 15000 and 64000 and particle diameters studied were from 1 μm to 70 μm. The particles used by the author were oil and starch. They observed the velocity field increase as particle size increased. This also can be understood as the particle size increases the Stokes number increases, due to which there is modulation of turbulence. The recirculation zone length also decreased as the particle size increased. The authors make a note that for different particle diameters it can be seen that particle diffusivity of energy is different from eddy diffusivity even for micronsized particles. Fessler and Eaton (1997) studied particle response in a backward facing step of expansion ratio of 5:3 with main flow direction vertically downward. The Reynolds number at the inlet of the step is 18400 based on the channel centerline velocity and step size s. The particle diameters varied from 25μm to 150μm and were made of copper and glass. The authors observations were particles accelerated downstream due to negligible effect of pressure gradient experienced by the particles compared to the fluid. The velocity decreased because the fluid was more responsive and adapted to adverse pressure gradient. The particle response is also consistent in the shear layer just after the sudden expansion where the fluid velocity fluctuations were large. The authors noted the high inertia of particle remembers stream wise distribution of velocities and can be easily seen in the vicinity of the recirculation zone. They also noted that the particles’ response is negligible as the particle Reynolds number increased. Fessler and Eaton (1999) studied the modification of turbulence by particles. The experimental setup of Fessler and Eaton (1997) was used. The Reynolds number based on the step height and channel centerline velocity was 18400. The particle diameters varied from 70μm to 150μm. The mass loading of the particle was 3% to 8 40%. The particle Stokes numbers were 1. The authors observed modulation of carrier phase increased (main flow which carries the dispersed flow) with the increase of particle Reynolds number, Stokes number and the mass loading. The authors also suggest that there exists a limiting particle Reynolds number and Stokes number for modulation. The authors also suggest as particle size increases the attenuation decreases and turbulence modulation increases. Hetsroni (1989) reviewed particle turbulence interaction in detail and showed as the particle diameter increased the turbulence intensity increases due to the help of vortex shedding from the particle. Tu (1997) performed a two fluid model simulation of particle flow in backward step flow and Tjunction. The primary phase turbulence was resolved using k RNG. An interphase momentum transfer equation was derived based on the Favreaveraged momentum equation of particulate phase. The author compared the results to the experiments of Ruck and Makiola (1988) with particle size of 70 μm and Reynolds numbers based on step height s which is 64000. The results matched experiments reasonably. The author also performed various simulations varying particle sizes from 7μm to 45 μm. The particle concentration in the recirculation zone was minimal compared to that in the main flow as the particle diameter increased which can be seen in Figure 2.3. Yu et al. (2004) performed numerical simulation of gas particle flow using LES for resolving primary phase and Lagrangian tracking for particles. The particles were glass spheres with diameters varying from 2200 μm. The particle tracks and dispersion were consistent with the literature of Fessler and Eaton (1997) and Tu (1997). The smaller particles (Figure 2.4) with smaller Stokes number tend to enter the recirculation zone but as the particle size and Stokes number increase the particles entering the recirculation zone decreases. The authors also considered the effect of slip velocity between the particle and the primary phase. The slip velocity was defined as the ratio of particulate phase velocity to the carrier phase velocity at the inlet 9 Figure 2.3: Particle Concentration Contours behind the step with varying particle diameters. Case (a) dp = 7μm, Case (b) dp = 15μm, Case (c) dp = 25μm, and Case (d) dp =45μm from Tu (1997) . of the channel. The authors observed the effect of slip velocity on small particles is negligible which can be seen in Figure 2.5. The effect of slip velocity on large particles is moderate as the slip velocity reduces from 0 to 1 which can be seen in Figure 2.6. The authors also displayed the effect of gravity on particles in Figure 2.7. The authors considered the gravity in x and y directions, x direction is the flow direction. The xdirection gravity produced similar results to no gravity case. The y direction gravity produced settlement of particles of 100 and 200 μm. 10 Figure 2.4: Particle Distribution of Various Diameters from Yu et. al (2004) . Figure 2.5: Effect of Slip Velocity on Particle of Diameter 100 μm from Yu et. al (2004) . 11 Figure 2.6: Effect of Slip Velocity on Particle of Diameter 200 μm from Yu et. al (2004) . Figure 2.7: Effect of Gravity in ydirection on Particles from Yu et. al (2004). . 12 2.4 Study of Backward Step with Porous Medium Insert Consider the step flow with porous medium, and how porous medium properties like thickness, porosity, etc. affect the recirculation zone length. Chan and Lien (2005) performed numerical modeling of step flow with a porous medium immediately behind the step. The authors used the k model for the flow and the porous medium was modeled using the DupuitForchheimer equation by incorporating an additional sink term in the Navier Stokes equations. The authors chose the experimental setup of Driver and Seegmiller (1985) as the geometry for numerical modeling and modeled it as a 2D backward step. The authors studied various cases of the porous medium, varying the thickness of the porous medium, the Darcy constant, and Forchheimer constant. The authors first lowered the Darcy constant increasing streamwise resistance to the flow and keeping the other factors constant, shown in Figure 2.8. They observed that with increasing resistance along the flow direction the recirculation zone vanished and the porous medium laminarized the flow. Then the authors kept the Darcy constant and the thickness of the porous medium constant and varied the Forchheimer constant, shown in Figure 2.9. They observed that by increasing the Forchheimer constant the effect was seen primarily on the flow leaving the porous medium. The flow past the porous medium was turbulent and laminarized with increase in the Forchheimer constant. Finally the authors varied the thickness of the porous medium, shown in Figure 2.10. They observed that, increasing the thickness of the porous zone length increased the resistance to flow and decreased the recirculation length. 13 Figure 2.8: Stream Traces for Variation of Darcy Constant from Chan and Lien (2005). . 14 Figure 2.9: Stream Traces for Variation of Forchheimer Constant from Chan and Lien (2005) . 15 Figure 2.10: Stream Traces for Variation of Porous Media Thickness Constant from Chan and Lien (2005) . 16 2.5 Summary The single phase numerical and experimental studies of backward facing step flow in the literature suggest the following: • The recirculation length strongly dependent on the expansion ratio • For flow fully developed turbulent flow past the step, recirculation length is independent of Reynolds number • Flow is three dimensional in the transitional regime from laminar to turbulent and low aspect ratio • The pressure gradient of the recirculation zone is independent of upstream Reynolds number • Recirculation length increases from laminar to transitional regime and then decreases The previous studies of particulate flow in backward facing step flow leads us to the following conclusions. The smaller particles with lower Stokes number follow the flow like tracer particles. The heavier particles with higher Stokes number escape the recirculation zone due to their inertia. The increase of concentration of secondary phase or particles has influence on primary phase velocity . The increase of concentration of smaller particles leads to decreases in turbulence intensity due to small particles promoting more dissipation of energy. On the other hand when the larger particles concentration increases the turbulence intensity increases due to vortex shedding of larger particles. The effect of gravity is predominantly on the heavier particles. 17 The only previously studied case of porous media and backward facing step was done by Chan and Lien (2005) numerically. The authors placed the porous media adjacent to the step. The findings were increasing viscous resistance or Darcy’s constant the flow laminarized and recirculation zone disappeared. Increasing the inertial resistance or Forchheimer constant the recirculation area decreased but not the length. The increment of thickness of porous media had similar results to increasing the Darcy’s constant and decreasing recirculation zone length. 18 CHAPTER 3 Numerical Method 3.1 Introduction The numerical modeling of the current study can be broadly classified into the following major categories • Turbulence modeling of carrier phase • Discrete phase modeling of the particles • Modeling of the porous region Along with the above fundamental modeling topics, numerical difficulties also exist, the major aspects like grid generation, pressurevelocity coupling, transfer of momentum from carrier phase to discrete phase, and wall treatment of turbulence by the turbulence model. The following content provides the basic equations, models and strategies for the aforementioned topics. 3.2 Turbulence Modeling turbulence flow phenomena is one of the areas which brought computational fluid dynamics predominance as a design tool. Modeling or simulating turbulent flows is a very challenging area due to the existence of various length scales and time scales across the flow. One of the most important areas in turbulence modeling is the resolution of stresses near the wall and carrying this information to the core of the flow. There exists no universal common tool for modeling or simulating varied 19 flow types. Modeling always incorporates finding some of the closure coefficients from experimental data. The experimental data set is limited in respect to Reynolds number as well as types of flow. Hence the accuracy of the model varies from one situation to another in respect to flow types. The current strategies to model or simulate turbulence are • Reynolds Averaging of Navier Stokes(RANS) • Filtering Approach (Large Eddy Simulation(LES)) • Hybrid Models • Direct Numerical Simulation(DNS) The RANS models use the concept of Reynolds averaging. The new terms which arise are modeled using correlations or coefficients obtained from experiments. The widely used RANS models are k − , k − !, Reynolds Stress Modeling etc. In the filtering or LES approach a filter is used to cut off eddies of various lengths. The large scale eddies are simulated and small scale eddies are modeled. Hybrid methods use a different strategy in which the core of the fluid is simulated using LES and near the walls RANS models are used to resolve stresses. Discrete Eddy Simulation (DES) is the best example of a hybrid model. Direct Numerical Simulation does not involve any modeling; it is just forward time stepping Navier Stokes Equations and limited to very small Reynolds number due to the computational power required to resolve the very smallest eddies. The selection criteria for turbulence models is well presented in Pope (2000). They are • Level of Description • Completeness • Cost and ease of use 20 • Range of Applicability • Accuracy 3.2.1 ReynoldsAveraged Navier Stokes The Reynoldsaveraged NavierStokes equations for incompressible flow from Pope (2000) DhUji Dt = @ @xi " μ DhUii @xj + DhUji @xi ! − hpi ij − huiuji # (3.1) The term huiuji is defined as the Reynolds stress.The Reynolds stress term is generally conceptualized as the momentum transfer by the fluctuating components of the velocity field at the molecular level. From 3.1 all the terms are know and pressure can be obtained from the continuity equation except the Reynolds stress, hence the problem of closure arises. The closure to this problem is achieved by modeling the Reynolds stress using different strategies. All the k − , k − ! and other variation of these models assume a simple relation between the Reynolds stresses, kinetic energy and mean strain rate. This is called the Boussinesq Constitutive Relation: uiuj = 2 3 k ij − t @Ui @xj + @Uj @xi ! (3.2) 3.3 k − Model The k − model is the first of the two equation models and a widely used model for simulating industrial flows. In k− we solve both the kinetic energy equation and the dissipation equation which is obtained by taking the moment of the Navier Stokes. The k− has been well validated for many cases and has proven efficient on the basis of the cost of computational power and ease of programming. 21 3.3.1 Turbulent Kinetic Energy(k) The turbulent kinetic energy is equal to half the trace of the Reynolds stress matrix i.e k = 0.5 hui.uii . k transport equation is represented as follows: Dk Dt = r. T krk + P − (3.3) P = −huiuji @ hUii @xj , = 1 2 t @ui @xj + @uj @xi ! (3.4) where P represents turbulent kinetic energy production, represents the dissipation, T = Cμ k2 is the turbulent viscosity and D/DT = @/@t + U.r 3.3.2 Modeling Dissipation The eddy dissipation is obtained by taking the moment of the Navier Stokes equations as follows 2μ @ui @xj @ @xj (3.5) The corresponding result after some manipulations results in the following epsilon equation D Dt = r. T r + C 1 P k − C 2 2 k (3.6) The standard values of these constants are shown in the following table Cμ C 1 C 2 k 0.09 1.44 1.92 1.0 1.3 Table 3.1: Modeling Constants Used for k Turbulence Model from Wilcox (2006) 3.3.3 Numerical Treatment of k − Model The k − model can easily be implemented to an existing Navier Stokes Solver. The following shows the implementation of k − model for a two dimensional case from 22 Shyy et al. (1997). The kand equations can be treated as convective, diffusive reaction type equations and modeled. The equations are as follows @ @t ( ) + @ @x (u ) + @ @y (v ) = @ @x @ @x ! + @ @y @ @y ! + R1 + R2 (3.7) where = 8>>< >>: + t k for the k equation + t for the equation (3.8) R1 = 8>>< >>: tR + t k for the k equation C 1 t R K for the equation (3.9) where R = 2 2 4 @u @x !2 + @u @y !2 3 5 + " @u @y ! + @v @x !#2 (3.10) and R2 = 8>>< >>: − − C K K for the k equation −C 2 2 K − −C 2 K for the equation (3.11) where K and are evaluated at the previous step. 3.3.4 Variations of k Turbulence Model Like any model, the k does fail to model flows accurately in some cases. The variations of k turbulence models which work better in some cases include the Renormalization group (RNG) and Realizable k model. RNG K The RNG model accounts for rapidly changing strain rates in the flows. The RNG model incorporates an additional term with expansion parameter = Sk/ where S is the average strain rate. The k equation remains the same and a new added term is incorporated to the dissipation. The model equations for RNG from Yakhot et al. (1992) are as shown below 23 Dk Dt = 2μTS2 ij − + @ @xi Kμ @K @xi ! (3.12) where Sij is the mean strain rate given by Sij = 1 2 @Ui @xj + @Uj @xi ! (3.13) the equation for epsilon is as follows D Dt = C 1 μTSij 2 k − C 2 2 k + @ @xi μ @ @xi ! − R (3.14) where R = 2μSij @ui @xi @ui @xi . R is computed as follows R = Cμ 3(1 − / 0) 1 + 3 ! 2 k ! (3.15) where = Sk/ and the values of constants proposed in Yakhot et al. (1992) are C C 1 C 2 k 0 0.085 1.42 − R/Cnu 3 1.68 0.7179 0.7179 4.38 .0125 Table 3.2: Modeling Constants Used for k RNG from Yakhot et al. (1992) 3.4 k! Model The k! model is also a two equation model, but based on k and !. The ! is enstrophy or RMS of fluctuating vorticity ! and regarded as ratio of k where k is turbulent kinetic energy, is the dissipation. Solving the k! model is similar to k ; instead of we solve for !. The governing equations for incompressible turbulent flows are as follows from Wilcox (2006). The Turbulence Kinetic Energy Equation @k @t + Uj @k @xj = ij @Ui @xj − k! + @ @xj " (μ + μt) @k @xj # (3.16) The Enstrophy transport equation @! @t + Uj @! @xj = ! k ij @Ui @xj − k!2 + @ @xj " (μ + μt) @! @xj # (3.17) The closure constants are as follows 24 5 9 3 40 9 100 1 2 1 2 Table 3.3: Modeling Constants Used for k! from Wilcox (2006) 3.5 SST Model The SST model is well defined in Menter et al. (2003). The SST model is a combination of k and k! model using a blending function. The blending function makes the SST model behave as k! as we approach the wall and k as we move to core of the flow. The modified k equations is as follows @( k) @t + @( Uik) @xi = P˜k − k! + @ @xi " (μ + kμt) @k @xi # (3.18) The modified ! equation is as follows @( !) @t + @( Ui!) @xi = S2 − !2 + @ @xi " (μ + wμt) @! @xi # + 2 (1 − F1) w2 1 ! @k @xi @! @xi (3.19) The blending equation F1 is given as follows F1 = tanh 8< : ( min " max pk !y , 500 y2! ! , 4 !2k CDk!y2 #)49= ; (3.20) The cross diffusion term is defined as follows CDkw = max 2 !2 1 ! @k @xi @! @xi , 10−10 ! (3.21) The turbulent viscosity is defined as follows Vt = a1k max (a1!, SF2) (3.22) where S is the invariant measure of strain and F2 is the second blending function. F2 = tanh 2 4 " max 2pk !y , 500 y2! !#23 5 (3.23) 25 The production of turbulence is limited as follows Pk = μt @Ui xj @Ui @xj + @Uj @xi ! ! P˜k = min (Pk, 10 · k!) (3.24) The closure constants are as follows 1 !1 kl 2 2 5 9 3 40 9 100 1 2 0.85 0.44 0.0828 Table 3.4: Modeling Constants Used for SST Turbulence Model 3.6 Wall Treatment The wall treatment is one of the important features that controls the numerical prediction of the turbulence models for flows with walls. The amount of computational power required to resolve the stresses near the wall by having a very fine mesh near the wall is very high. To reduce computational costs and the failure of turbulence models to model low local Re flows near the different wall, some of the strategies proposed for modeling are as follows: 3.6.1 Wall Function Standard Wall Function The wall function method was first proposed by Launder and Spalding (1974). The theory was based on the universal law of the wall. The wall function model deploys empirical formula to model the flow near the wall. In the above figure the point P is the computational cell located at yp from the wall.The wall shear stress is evaluated as follows w = @U @y UP − UW yp (3.25) 26 Figure 3.1: Near Wall Region Grid Point Up ( / )w C 1/4kp 1/2 = 1 ln 2 64 Eyp C 1/2kp 1/2 3 75 (3.26) where kp is the turbulent kinetic energy at point p, Cμ modeling constant from the k equations, E is a function of the wall roughness equal to 9.0 for smooth wall, is the wall shear stress on the wall in the direction of Up and is Von Karman’s constant. There is a debating issue regarding the universality of wall functions, one reason being that is Von Karmans constant is really a constant. The issue is neglected and is beyond the scope in the current research. 3.7 Discrete Phase Modeling Discrete Phase Modeling belongs to the class of EulerianLagrangian strategy to model multiphase flows. Discrete phase model can be achieved in two ways depending on the coupling of the flows and particles. Oneway coupled flows are those in which particles have no influence on the carrier phase velocity. The Eulerian phase is resolved and particle tracks are integrated over the known velocity field. In twoway coupled multiphase flows there is interaction between particles and carrier fluid. The carrier phase is unresolved and particle trajectories are integrated over the instantaneous resolved carried fluid velocity per time step. The governing particle equation from Crowe et al. (1998) is as follows dv dt = f v (u − v) + g 1 − c d ! + Fi (3.27) 27 Here m refers to mass of the particle, f is the drag force factor f = CDRer 24 (3.28) v is the particle response time v = dD2 18μc (3.29) Saffman’s Lift Force The small particles experience lift force in turbulent flows. The Saffman’s lift force term for small particles equation was developed by Saffman (1965). This equation was generalized by ?) as follows ~F = 2K 1/2dij Sdp(dlkdkl)1/4 (~u − ~up) (3.30) where K is Saffman’s Constant Lift force Coefficient, S is the ratio of particle density to fluid. 3.8 Porous Medium Modeling The porous medium is treated as homogeneous and the pressure drop contributed by the porous medium is handled by an additional sink in the Navier Stokes Equation. The sink term is modeled with the DupuitForchheimer equation. The Dupuit Forchheimer equation from Nield and Bejan (2006) is as follows rP = − v + CF − 1 2 vv (3.31) where is the kinematic viscosity, is the permeability. The first term incorporates the Darcy Law where is the Darcy’s Constant and the second term is called the Forchheimers term to incorporate inertial effects due to the porous medium. The porous medium can be modeled in FluentTMusing two approaches 1. Porous Jump 2. Porous Zone Formulation 28 Porous Jump The porous jump formulation was chosen because the region of interest is not in the porous region and the non availability of DPM boundary condition for porous zone like trap, reflect, and escape. Preliminary computations performed with a porous zone showed small effects on velocities, hence the porous jump was chosen as the choice of model. The porous jump incorporates the additional sink term through the following equation on a line in 2D geometry and plane in 3D geometry. The porous jump equation is as follows P = μ + C2 1 2 V 2 T (3.32) where is the permeability of the medium, C2 is the inertial factor and T is the thickness of the porous medium. 29 CHAPTER 4 Modeling Process 4.1 Geometry The geometry was created in ICEM CFDTMwith the dimensions displayed in Figure 1.1. The 2D section of the experimental results was modeled due to flow being predominantly 2D due to high aspect ratio used in the experiments by Yao (2000) and also from the view of computational power. 4.2 Grid Generation and Adaptation The grid was generated in ICEM CFDTMconsisting of fully structured quadrilateral cells. The boundary layer mesh was difficult to generate in this geometry due to the sharp edge at the corner and also to maintain fully structure grid. Hence the methodology described in Schafer (2006) was implemented to achieve the required y+ ut μ for the wall functions. The methodology is as follows: • Perform calculation on a mesh of optimum number of grid points. • Calculate y+ from the previous results for the wall nodes. • Adapt the grid at the wall to resolve the mesh at the wall. Perform calculations based on adapted mesh. • Iteratively check the solution until the required y+ is achieved for the wall function in use. 30 The test mesh sizes were of fully structured grids. The mesh size was selected based on the capture of recirculation zone along with implementation of wall adaptation for each case. The boundary adaptation was done until the wall y+ condition for respective wall function y+ is reached. The following table gives the mesh sizes used. The following Figure 4.1 shows the mesh after the mesh adaptations near the boundary in the Figure 4.2 Mesh Quality Number of Nodes Coarse 9500 Intermediate 17613 Fine 77684 Table 4.1: Mesh Size Terminology Figure 4.1: Implemented Mesh 31 Figure 4.2: Wall Adaptations at Step 4.3 Boundary Conditions The different boundary conditions used for modeling the backward step regions shown in Figure 1.1 are presented in following table: Bakward Facing Step Geometry Regions Boundary Condition Inlet Velocity Inlet Outlet Outflow Filter Porus Jump Boundaries No Slip Table 4.2: Boundary Conditions Implemented Velocity Inlet The following velocities were given as input based on Reynolds number calculations and turbulent intensity was specified as 5%. 32 Re at the Step U(m/s) at the Inlet 6550 3.83 10000 5.84 Table 4.3: Velocities at Inlet Porous Jump The pressure drop across the filter media was experimentally found by ?) for various flow rates. The data was curve fitted and constants were used to determine the modeling constants of the porous jump. The following figure shows the pressure drop profile for the mean velocity variation. The curve fit equation for the pressure drop Figure 4.3: Experimental Data of Pressure Drop vs. Flow Velocity obtained was P = 88.744V 2 + 115.57V (4.1) 33 The above constants were used to find and C2 in porous jump equations. The values obtained were C2 = 10.28 × 103m−1 and = 2.389 × 10−9m2. Particle Properties The particle were modeled as inter particle and the density of the particles used was 550kg/m3 close to the density of ultrafine test case ISO 12103, A1 Ultrafine of 500kg/m3 from Powder Technology Incorporated (PTI). The input velocity given to particles was 2m/s for Re=6550 and 4m/s for Re=10000, the flow rate of the particles was 1.8e − 5kg/s. The turbulent dispersion modeling inputs used for Random Walk model are given below. The Time scale constant is the Lagrangian Time scale and needs to found iteratively as the particle trajectories are dispersive in nature. The maximum number of steps provides the number of integrations need to be performed for each particle trajectory calculation. The length scale was specified as an estimate of Lagrangian length scale. Number of Tries 2 Time Scale Constant 0.15 The tracking parameters used are presented below Max Number of Steps 10000 Specified Length Scale 10mm 4.3.1 Discretization Schemes The various discretizations schemes used were 34 Momentum Second Order Upwind Turbulent Kinetic Energy Second Order Upwind Turbulent Dissipation Rate Second Order Upwind Pressure Standard 4.4 Grid Independence The grid independence was performed for no filter case of Reynolds Number 6550. The following Figures show the velocity profiles at the step, X=3.75s and X=6.25s after step comparison to experimental velocity profiles of Yao (2000). Figure 4.4: Velocity Profile at Step for Re 6550 No Filter Case 35 Figure 4.5: Velocity Profile at X=3.75s after Step for Various Mesh Sizes, Re 6550 No Filter Case Figure 4.6: Velocity Profile at X=6.25s after step for various Mesh Sizes, Re 6550 No Filter Case 36 The recirculation zone lengths achieved are as follows. Mesh Quality Recirculation Length Coarse Mesh 6.15s Intermediate Mesh 6.25s Fine Mesh 4.72s Table 4.4: Comparison of Recirculation Zone Length and Grid Sizes Comparing the velocity profiles and recirculation zone length the intermediate mesh was chosen as the appropriate mesh size. The fine mesh was numerical diffusive in nature and under predicted the recirculation zone length. The reason for this diffusion is unknown.The convergence criteria specified was 1E06. 4.4.1 Turbulence Model Comparison The various turbulence models and wall functions were compared for the Re 6550 case. The intermediate mesh was chosen as the mesh size for the cases. The following Figures 4.7, 4.8, and 4.9 show comparison of various turbulence models with estimation of velocity profiles at step and X=3.75s , X=6.25s after the step. The turbulence models predicted the velocity at the step almost the same which can be seen in Fig 4.7. The turbulence models also predicted the velocity profile at X=3.75s after the step similarly except for k RNG with enhanced wall treatment. At X=6.25s after the step all the models overpredict the velocities near the top wall. 37 Figure 4.7: Comparison of Velocity Profiles at Step for Re 6550 No Filter case for Various Turbulence Models Figure 4.8: Comparison of Velocity Profiles at X=3.75s after the Step for Re 6550 No Filter Case for Various Turbulence Models 38 Figure 4.9: Comparison of Velocity Profiles at X=6.25s after the Step for Re 6550 No Filter case for Various Turbulence Models 39 The Table 4.5 shows the estimation of recirculation zone length by various turbulence models. The experimental findings of recirculation zone length was found to be 6.5s (Yao (2000)). Comparing the plots and recirculation zone length k RNG with NonEquilibrium Wall Functions was chosen as the model for the rest of the study. Kim et al. (2005) have done comparison studies for various turbulence models and wall function in detail for the geometry of Driver and Seegmiller (1985) and have come to same conclusion k RNG with NonEquilibrium Wall Functions is better of the all models for backward facing step. Grid Independence for Re=10000 was not performed due very small change in Reynolds number and also from literature flow fully developed turbulent past step flow downstream look almost similar irrespective of upstream Reynolds number when expansion ratio is kept constant. Turbulence Model and Wall Function Recirculation Zone Length(s) k RNG Standard Wall Function 6.25 k RNG Non Equilibrium Wall Function 6.24 k RNG Enhanced Wall Treatment 6.35 k! 6.44 SST 6.73 RSM Standard Wall Function 6.53 Table 4.5: Comparison of Recirculation Zone Length with Various Turbulence Models and Wall Functions 40 CHAPTER 5 Results and Discussions 5.1 Introduction The current chapter presents the results of various cases. The primary phase Reynolds numbers chosen are Re= 6550 and Re = 10000 and the filter at two locations, 4.25s, 6.75s, based on the experiments performed by Yao (2000). The monodisperse particle sizes chosen for study are diameters 1μm, 20μm and 50μm based on the range of dust particle sizes and for comparison of results to Yu et al. (2004). The results are presented in the order estimation of primary phase flow field with and without the filter, DPM tracks of particles with and without filter, and mixture model modeling of the airdust mixture without the filter. For the evaluation of models, gradually increasing the complexity will give better understanding of the results. The Stokes number of particles for the Reynolds numbers is 1 and are presented in Appendix A. 5.2 Flow Field Estimation for Re = 6550 and Re = 10000 Velocity Magnitude Contours of No Filter Case Figure 5.1 shows the velocity contours of Re 6550 case and Figure 5.2 shows the velocity contours of Re 10000. The velocity contours aid in better understanding of the flow and also help in estimating the functioning of the turbulence model. The contours clearly show the flow separation at the step, the shear layer and the recirculation zone. The flow contour also shows gradual flow adjustment to fully developed 41 flow downstream. The placement of the outflow boundary at 30s downstream is justified by the previous finding of fully developed flow. The length of the recirculation zone (Xr) is 6.24, in good standing with the literature for fully developed flow at the step, for example Yao (2000) and Armaly et al. (1983). The comparison of contours also shows that when the flow is fully developed after the step the flow characteristics almost remain same and independent of the Reynolds number and depend only on the expansion ratio consistent with findings of Armaly et al. (1983), Adams and Johnston (1988a), and Tihon et al. (2001). 42 Figure 5.1: Velocity Magnitude Contours after the Step for Re = 6550 Figure 5.2: Velocity Magnitude Contours after the Step for Re = 10000 43 Velocity Profile Comparison at the Step The following plots show the velocity profiles at the step; Figure 5.3 for Re 6550 and Figure 5.4 for Re 10000. The velocity profiles are fully developed turbulent. The velocity profiles near the wall differ from the experimental profiles. The variation can be attributed to the wall functions used which are based on standard data sets and upon the experimental conditions. 44 Figure 5.3: Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 6550 Figure 5.4: Velocity Profile Comparison between Experimental and Modeled Flow Field at Step for Re = 10000 45 Velocity Profile Comparison at the X=3.75s Figures 5.5 and 5.6 show the velocity profiles at X=3.75s for Re 6550 and Re 10000 respectively. The location behind the step shows a decrease in the maximum velocity from that at the step. The decrease in the velocity is due to expansion of the flow at the step. The models under predict velocities slightly near the top wall and in the recirculation zone. This can be again attributed to wall functions and numerical error. The velocity profiles match very closely in the mid section of the channel at the current location. 46 Figure 5.5: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 6550 Figure 5.6: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re = 10000 47 Velocity Profile Comparison at the X=6.25s Figures 5.7 and 5.8 show the velocity profiles at X=6.25s for Re 6550 and Re 10000 respectively. The location of current velocity profiles is approximately near the end of the recirculation zone. The velocity profiles show the flow reattaching to the bottom solid wall and adjusting to be a fully developed profile. The model captures the velocities near the bottom wall very well and near the top wall, as for the location of X= 3.75s there is a deviation from the experimental values. 48 Figure 5.7: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re = 6550 Figure 5.8: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 49 5.3 Flow Field Estimation for Filter at 4.25s after the Step The current section discusses for the filter placed at 4.25s. The numerical results are compared to experimental results of Yao (2000). Velocity Profiles at X=3.75s Figures 5.9 and 5.10 show the velocity profile comparison between the CFD results and experimental results. The experimental data shows the velocity profile adjusting to the center of the flow for the current filter location when compared to the no filter case. The velocity profile for Figure 5.9 is normalized by the maximum velocity at the step and for the Figure 5.10 is normalized by the the maximum velocity at the current location. The normalization was due to absence of raw experimental data. The turbulence model and the porous zone model together give good estimation of the velocity flow field near the bottom wall. The velocity magnitude calculation near the top wall shows large deviation from the experimental data and also the velocity profile. The error can be attributed to previous error for no filter case at current location, and also filter being approximated as porous jump. The filter surface is actually a pleated air filter. Due to the pleated design the surface makes the flow experience more surface area and may dampen the flow structures. The modeling of the whole filter is very difficult as greater computational effort is needed and there are limitations in meshing. The sharp edges give very high aspect ratio cells which result in divergence of the solution even if modern algorithms are employed. Considering the limitations, the current model predicted the velocities acceptably. 50 Figure 5.9: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =6550 for Filter at X= 4.25s Figure 5.10: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 3.75s after Step for Re =10000 for Filter at X= 4.25s 51 Velocity Contours The velocity contours shown in Figures 5.11 and 5.12 show the truncation of recirculation zone produced by the filter. The area of recirculation zone also increased in the Y direction reducing the area between the step and the filter to almost half the original. The current filter location shows the strong influence on the flow field due to placement of filter in the recirculation zone. The placement also shows secondary recirculation on the top wall near the filter. The flow past the filter the model predicts a turbulent reattachment. 52 Figure 5.11: Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 4.25s Figure 5.12: Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 4.25s 53 5.4 Flow Field Estimation for Filter at 6.75s after the Step The following section discusses the results for filter placed at 6.75s after the step. The placement of the filter at this location has negligible effect on the recirculation zone. Velocity Profiles Figures 5.13 and 5.14 show the velocity profiles at the current location show the same tendency of the velocity profiles adjusting to the center of the channel. The recirculation zone is least affected at this location in comparison to the area increases in the Y direction for the filter located at X=4.25s. The experimental and CFD results match better than the previous location of filter at 3.75s. 54 Figure 5.13: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =6550 for Filter at X= 6.75s Figure 5.14: Velocity Profile Comparison between Experimental and Modeled Flow Field at X = 6.25s after Step for Re =10000 for Filter at X= 6.75s 55 Velocity Contours Comparing the velocity contours in Figures 5.15, 5.16 with filter at 6.75s to those for the no filter case, Figures 5.1 and 5.2 the effect of the filter on the recirculation zone is negligible. 56 Figure 5.15: Velocity Magnitude Contours after the Step for Re = 6550 for Filter at 6.75s Figure 5.16: Velocity Magnitude Contours after the Step for Re = 10000 for Filter at 6.75s 57 5.5 Discrete Phase Model for No Filter Case The following sections shows the particle tracks for the two Reynolds numbers 6550 and 10000 for the no filter case. The steady state DPM model was employed owing to the fact that the concentration of secondary phase was negligible and the particles do not participate in chemical reaction nor do they transfer in any mass. The solved flow field was used and particle trajectories were integrated on the known primary phase velocities. The particle residence time which is the time for which a particle resides in the domain of calculation, was used to track particle trajectories. The particle tracks help us visualize the particle response to different flow structures. 5.5.1 Particle Tracks Re = 6550 No Filter Case The DPM tracks (Figures 5.17, 5.18, and 5.19) show that as the particle diameter and Stokes numbers increase the number of particles entering the recirculation zone is minimized. The phenomena is consistent with the literature for example, Ruck and Makiola (1988), Fessler and Eaton (1997), Fessler and Eaton (1999), Tu (1997), and Yu et al. (2004). The larger particles which have higher momentum do not follow the flow and escape from fluid vortices of low momentum. The maximum particle residence times varied from 1.4 sec to 1.9 sec. When the particle diameter and Stokes number increase we can observe that the particles take longer time to escape from the recirculation zone once they enter. 58 Figure 5.17: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, No Filter case Figure 5.18: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, No Filter case 59 Figure 5.19: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, No Filter case 60 5.5.2 Particle Tracks Re = 10000 No Filter Case Figures 5.20, 5.21, and 5.22 show similar tendencies as the case of Re=6550. The maximum particle residence time is varied from 0.86 sec to 1.8 sec. The larger particles which enter the recirculation zone find it harder to escape the recirculation zone, as in the previous case. For the case of 50μm particle, the particles recirculate longer in the recirculation zone compared to the Re 6550 case. This is due to the particles entering the recirculation zone taking longer time to achieve momentum for cross stream escape to the main flow. Figure 5.20: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, No Filter case 61 Figure 5.21: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, No Filter case Figure 5.22: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, No Filter case 62 5.6 Discrete Phase Model for Filter Case The following section discusses the particle trajectories for filter placed at 4.25s and 6.75s after the step for Re of 6550 and 10000. 5.6.1 Filter located at 4.25s after the step The following section presents the particles tracks for filter located at 4.25s. The particle boundary condition at the filter is treated as trap. Re = 6550 Figures 5.23, 5.24, and 5.25 show the particle tracks for primary phase Re 6550. The cases where the particle diameter is 1μm, and 20μm belong to the low Stokes number category. In all the cases the effective area for filtration is reduced to around 60 % of porous zone due to development of recirculation zone. The effect of the recirculation zone reduces the filter efficiency and life time drastically. In the case of 50μm particle, the bottom region of the filter is hardly used. 63 Figure 5.23: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 4.25s Figure 5.24: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 4.25s 64 Figure 5.25: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 4.25s 65 Re = 10000 Figures 5.26, 5.27, and 5.28 show the particle tracks for Re 10000 of primary phase. When the Reynolds number of the primary phase is increased to 10000 there is slight increase in Stokes number for 1μm and 20μm particles. The effective area of filtration, where the particles impact the media is increased compared to the previous case of Re 6550. Comparing the 1μm, 20μm with the 50μm, hardly any particles enter the recirculation zone. From the filtration design point of view if we could control the carrier phase velocity and average Stokes number we could use the recirculation zone to transport particles to the filter. The position or location of the filter also plays an important role; this criterion will be explained in the next section for the filter located at 6.75s after the step. Figure 5.26: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 10000, Filter located at 4.25s 66 Figure 5.27: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 10000, Filter located at 4.25s Figure 5.28: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 10000, Filter located at 4.25s 67 5.6.2 Filter Located at 6.75s after the step The section contains particle tracks for the filter located at 6.75 for the two Reynolds numbers. The section helps us in understanding the placement of filter and effects of filtration with the recirculation zone present. Re 6550 Figures 5.29, 5.30, and 5.31 present the particle tracks for varying diameters. The placement of the filter at the end of the recirculation zone promoted more particles to enter the recirculation zone. The particles had to travel a greater distance compared to the filter at 4.25s case, effectively losing momentum and getting trapped in the recirculation zone. There are two distinct patterns observed when comparing particle tracks of the filter at 4.25s from Figures 5.23, 5.24, and 5.25 and the current case, in which we can observe strong particle settlement at the corner beneath the step. The second observation is that the top recirculation zone observed in velocity contours Figure 5.15 at the filter also holds particles for longer time. 68 Figure 5.29: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re = 6550, Filter located at 6.75s Figure 5.30: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re = 6550, Filter located at 6.75s 69 Figure 5.31: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re = 6550, Filter located at 6.75s 70 Re 10000 The particles also behaved similarly for Re 10000 as for the case of Re 6550 as shown in Figures 5.32, 5.32, 5.32. The particles tend to settle near the corner of the step and particles were found to get trapped in the recirculation zone developed on the top wall due to the filter. Figure 5.32: Particle Tracks described by Particle Residence Time for Particle Diameter of 1μm for Re =10000, Filter located at 6.75s 71 Figure 5.33: Particle Tracks described by Particle Residence Time for Particle Diameter of 20μm for Re =10000, Filter located at 6.75s Figure 5.34: Particle Tracks described by Particle Residence Time for Particle Diameter of 50μm for Re =10000, Filter located at 6.75s 72 5.7 Summary CFD modeling was performed for particulate flow in a backward facing step with an expansion ratio of 1:2 with a porous medium located at X=4.25s and X=6.75s location. The particle Stokes number in this study used was 1 and the particle concentrations were low. The Reynolds numbers used in the current study are fully developed turbulence conditions after the step of 6550 and 10000. The following inferences may be drawn from the CFD modeling done for No Filter, Filter Cases and DPM for aforementioned cases. The CFD results of no filter case matched well with experimental results considering the flow conditions at the step being fully developed turbulent for CFD model and not fully developed for the experimental case. There has been a consistent deviation in the CFD model and experimental case at the top wall and bottom wall. This can be attributed to wall functions used and the flow conditions at the step. The recirculation zones are in good agreement. The CFD results of the filter cases did match the experimental results acceptably considering the model approximation of the pleated filter as a one dimensional porous jump. For the filter placed near the step at 4.25s, the recirculation zone got truncated and divided the region between the filter and step in almost half. For the filter placed at 6.75s after the step, the CFD results and experimental data match well when compared to the 4.25s case. With the filter place at a location beyond the recirculation zone of the no filter case, the filter did not appreciably affect the recirculation zone length. From the DPM of particles for the no filter case we can clearly see that as the particle size and Stokes number increases, fewer particles enter the recirculation zone. For the particle tracks realized by particle residence time for the filter placed at 4.25s 73 the effective area of filtration was reduced to around 60%. With the filter moved to a location of 6.75s after the step, the effective area of filtration changed slightly compared to the 4.25s case and also promoted more time for the particles to enter the recirculation zone. For the filter cases the maximum particle residence time can be observed for the filter placed at the more downstream location. Analyzing the motion of particles with respect to size, the recirculation zone can effectively be used to transport most of the heavier particles bypassing the recirculation zone. 74 CHAPTER 6 Conclusions and Recommendations 6.1 Conclusions The DPM was utilized to study trajectories of particle diameters of 1μm, 20μm and 50μm for a backward facing step with porous medium for carrier phase Reynolds numbers of 6550 and 10000. The particle track trajectories helped in visualizing the particle flight in the carrier phase medium and aided in better understanding of filtration. The current study gave rise to the following conclusions: 1. The effect of porous medium on the recirculation zone is predominant when the porous medium is placed in the recirculation zone of the no filter case. 2. The effect of the porous medium on the recirculation zone is small when the medium is placed downstream of the recirculation zone zone of the no filter case. 3. The recirculation zone tends to trap some of the smaller particles and thereby increase the filtration time of the particulate flow 4. The recirculation zone decreases the effective area of filtration to around 60% with filter surface towards the bottom wall receiving few particles. 5. The recirculation zone can be predominantly used to transport most of the heavier particles to the porous medium. 6. The current geometry, if modified with some pre filtration techniques like electrostatic precipitation,the modified configuration can effectively increase filtra 75 tion by removing small particles in the recirculation zone, thereby enhancing filter life. 6.2 Recommendations 1. The flow field estimation accuracy can increased by using LES and DES to model the flow field. 2. The differences between the filter case experimental and modeling results can be minimized by developing an empirical correlation for pressure drop based on the number of filter pleats. This empirical correlation can incorporated into Fluent using a User Defined Function. 3. The effect of the higher concentration of particles was not studied in the current research. The feasibility of applying the current geometry for higher concentration particle flows can be studied using EulerianEulerian models. 76 Bibliography Adams, E. and J. Johnston (1988a). Effects of separation shear layer on the reattachment flow structure. part 1: Pressure and turbulence quantities. Experiments in Fluids, 6, 400–408. Adams, E. and J. Johnston (1988b). Effects of separation shear layer on the reattachment flow structure. part 2: Reattachment length and wall shear stress. Experiments in Fluids, 6, 493–499. Armaly, B. F., F. Durst, J. C. F. Pereira, and B. Schonung (1983). Experimental and theoretical investigation of backwardfacing step flow. Journal of Fluid Mechanics, 127, 473–496. Benavides, A. and B. van Wachem (2009, June). EulerianEulerian prediction of dilute turbulent gasparticle flow in a backwardfacing step. International Journal of Heat and Fluid Flow, 30 (3), 452–461. Chan, E. C. and F.S. Lien (2005). Permeability effects of turbulent flow through a porous insert in a backwardfacingstep channel. Transport in Porous Media, 59, 47–71. Crowe, C., M. Sommerfeld, and Y. Tsuji (1998). Multiphase Flows with Droplets and Particles. CRC Press. Deen, N. G., M. V. Annaland, M. A. Van der Hoef, and J. A. M. Kuipers (2007). Review of discrete particle modeling of fluidized beds. Chemical Engineering Science, 62 (12), 28–44. Driver, D. M. and H. L. Seegmiller (1985). Features of a reattaching turbulent shear layer in divergent channel flow. AIAA Journal, 23, 163–171. 77 Fessler, J. R. and J. K. Eaton (1997). Particle response in a planar sudden expansion flow. Experimental Thermal and Fluid Science, 15 (4), 413 – 423. Fessler, J. R. and J. K. Eaton (1999). Turbulence modification by particles in a backwardfacing step flow. Journal of Fluid Mechanics, 394, 97–117. Hetsroni, G. (1989). Particlesturbulence interaction. International Journal of Multiphase Flow, 15 (5), 735 – 746. Kim, J., S. J. Kline, and J. P. Johnston (1980). Investigation of a reattaching turbulent shear layer: Flow over a backwardfacing step. Journal of Fluids Engineering, 102, 302–308. Kim, J. Y., A. J. Ghajar, C. Tang, and G. L. Foutch (2005). Comparison of nearwall treatment methods for high Reynolds number backwardfacing step flow. International Journal of Computational Fluid Dynamics 19, 493–500. Launder, B. and D. Spalding (1974). The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 3, 269–289. Menter, F. R., M. Kuntz, and R. Langtry (2003). Ten years of industrial experience with the SST turbulence model. Heat and Mass Transfer, 4, 625. Ming, Xu. Wei, G. J. L. (2007). A discrete particle model for particlefluid flow with consideration of subgrid structures. Journal of Chemical Engineering Science. 62, 2302–2308. Mohanarangam, K., J. Tu, and L. Chen (2008, December). Numerical study of particle dispersion behind a sudden expansion geometry and its effect on step heights. Computers & Chemical Engineering, 32 (12), 3187–3197. Mohanarangam, K. and J. Y. Tu (2007). Twofluid model for particleturbulence interaction in a backwardfacing step. AIChE Journal, 53 (9), 2254–2264. Nield, D. and A. Bejan (2006). Convection in Porous Media. Springer New York. 78 Papadopoulos, G. and M. V. Otugen (1995). Separating and reattaching flow structure in a suddenly expanding rectangular duct. Journal of Fluids Engineering, 117 (1), 17–23. Pope, S. B. (2000). Turbulent Flows. Cambridge University. PTI. http://www.powdertechnologyinc.com/products/testdust/testdust.php, accessed August 2010. Ruck, B. and B. Makiola (1988). 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(1997). Computation of turbulent twophase flow on overlapped grids. Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 32, 175–193. 79 Wang, B., H. Zhang, and X. Wang (2006). Large eddy simulation of particle response to turbulence along its trajectory in a backwardfacing step turbulent flow. International Journal of Heat and Mass Transfer, 49 (12), 415–420. Wilcox, D. C. (2006). Turbulence Modeling for CFD (3rd ed.). DCW Industries Inc. Yakhot, V., S. A. Orszag, S. Thangam, T. B. Gatski, and C. G. Speziale (1992). Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids AFluid Dynamics, 4 (7), 1510–1520. Yao, S. (2000). Two Dimensional Backward Facing Step Flow Preceding an Automotive Filter. Ph. D. thesis, Oklahoma State University. Yu, K. F., K. S. Lau, and C. K. Chan (2004). Numerical simulation of gasparticle flow in a singlesided backwardfacing step flow. J. Comput. Appl. Math, 163 (1), 319–331. 80 APPENDIX A Appendix: Stokes Number The Stokes numbers have been calculated as follows to be consistent with the literature, following Fessler and Eaton (1999). The Stokes number is the ratio of particle momentum response time, v to the fluid response time F as follows St = v F (A.1) where v is given as follows v = pd2 p 18μc (A.2) and F is given as follows, the characteristic length is based on the largest vortex present. F = 6.25s Umax (A.3) The Stokes number for Re = 6550 case is given in following table Particle Diameter Stokes Number 1μm 4.13 ×10−5 20μm 0.017 50μm 0.103 Table A.1: Stokes Number for Re 6550 81 The Stokes number for Re = 10000 case is given in following table Particle Diameter Stokes Number 1μm 6.4 ×10−5 20μm 0.026 50μm 0.160 Table A.2: Stokes Number for Re 10000 82 VITA Krishna Chaitanya Ravi Candidate for the Degree of Master of Science Thesis: NUMERICAL PREDICTION OF PARTICULATE FLOWOVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Major Field: Mechanical and Aerospace Engineering Biographical: Personal Data: Born in Kothagudem, Andhra Pradesh, India on January 05, 1985. Education: Received the B.S. degree from Vellore Institute of Technology,Vellore,Tamil Nadu, India ,2006, Mechanical Engineering Completed the requirements for the degree of Master of Arts with a major in Mechanical and Aerospace Engineering Oklahoma State University in December, 2010. Experience: Internship : August 2006  August 2007 Cusp Technologies, Hyderabad, Andhra Pradesh, India Teaching Assistant: Jan 2008May 2010 Department of Mechanical Engineering Oklahoma State University Research Assistant: Jan 2008 Aug 2010 Department ofMechanical and Aerospace Engineering Oklahoma State University Professional Memberships: Member of American Society of Mechanical Engineers (ASME) Member of American Society of Heating, Refrigerating and Air Conditioning Engineers (ASHRAE) Name: Krishna Chaitanya Ravi Date of Degree: December, 2010 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: NUMERICAL PREDICTION OF PARTICULATE FLOW OVER A BACKWARD FACING STEP PRECEDING A FILTER MEDIUM Pages in Study: 82 Candidate for the Degree of Master of Science Major Field: Mechanical and Aerospace Engineering Scope and Method of Study:The current research focuses on CFD modeling of turbulent particulate flows involving separation and filters or porous media. The backward step geometry was selected due to the large experimental data sets present in the literature. The backward step dimensional parameters used are expansion ratio 1:2, entrance length of 2 step heights and channel length of 30 step heights from the step. The mesh was developed in ICEM CFDTMand modeling was done using the FluentTMcommercial CFD package. The carrier phase turbulence was modeled using the k RNG model. The particles were modeled using the discrete phase model and particle dispersion was modeled using stochastic tracking. The porous jump condition was used at the porous medium. The boundary conditions employed are uniform velocity at the inlet with 5% turbulent intensity, outflow for outlet boundary condition, and noslip boundary conditions at the walls. The particle boundary condition implemented at the walls is “reflect” and at the filter is “trap”. Findings and Conclusions : The numerical results of no filter case matched experimental results with good accuracy. The current model agreed well with these results for Reynolds Numbers (Re) of 6550 and 10000 when compared in terms of the recirculation zone length and velocity profiles at 3.75 and 6.25 step heights downstream of the step. The presence of porous media showed a profound effect on the recirculation zone length. The velocity profile comparison for the filter cases do not validate very well with experimental results due to the initial conditions and the porous jump boundary condition. The current study focuses on prediction of dust particle trajectories of sizes 1μm to 50μm for different carrier phase Reynolds Number with the porous medium located at 4.25 and 6.75 step heights downstream of the step. The particle trajectories were observed for the aforementioned cases and filter locations. As particle size increased, the number of particles entering the recirculation zone decreased. The placement of the filter far from the step promoted more particles getting trapped in the recirculation zone. ADVISOR’S APPROVAL: 



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