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Effect of Grid Topology and Resolution on Computation of Steady and Unsteady Internal Flows

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Effect of Grid Topology and Resolution on Computation of Steady and Unsteady Internal Flows The Pennsylvania State University The Graduate School College of Engineering EFFECT OF GRID TOPOLOGY AND RESOLUTION ON COMPUTATION OF STEADY AND UNSTEADY INTERNAL FLOWS A Thesis in Mechanical Engineering by Steven P. McHale c 2009 Steven P. McHale Submitted in Partial Fulllment of the Requirements for the Degree of Master of Science May 2009 The Thesis of Steven P. McHale has been reviewed and approved* by the follow- ing: Eric G. Paterson Associate Professor of Mechanical Engineering Thesis Advisor Laura L. Pauley, P.E. Arthur L. Glenn Professor of Engineering Education Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Department Head of Mechanical and Nuclear Engineering *Signatures are on le in the Graduate School Abstract Internal ow has been studied under steady and unsteady conditions. The work was motivated by previous research in canine olfaction which showed that the structure of the dog's nose consists of a highly complex pipe network in which the unsteady air ow transports odorant molecules to the sensory region. Due to geometric complexity of the canine turbinates, a body-tted hex- dominant mesher was used, thus generating the question: What is the impact of grid topology and resolution on solution accuracy? The governing equations are dened and reduced to an analytical solution for a circular pipe with both steady and unsteady components. The numerical methods and solution algorithms are described. Simple test cases are performed using an open source computational uid dynamics (CFD) code, OpenFOAM, to understand the underlying physics and computational challenges inherent in internal pipe ows. The characteristic steady, Poiseuille ow was found and a grid study was performed over a range of Womersley numbers (Wo) for the oscillatory, unsteady calculations of the straight pipe. It was shown that the Womersley number directly aects the amount of near wall resolution needed to accurately resolve the Stokes layer. The same calculations were also performed for two other grid topologies to compare the eectiveness of alternate grid types. The nal simulations performed were steady inhalation/exhalation and unsteady respiration of a straight pipe within an external environment. iii Contents List of Figures vii List of Tables ix Nomenclature x Acknowledgments xi 1 Introduction 1 2 Literature Review 4 2.1 Motivation of the Current Work: Canine Olfaction CFD . 4 2.2 Studies of Non-Bifurcating Pipe Flow . 8 2.3 Studies of Bifurcating Pipe Flow . 11 2.4 Studies of Complex Pipe Systems . 13 2.5 Computational Aspects of Internal Flow . 14 3 Governing Equations and Numerical Methods 18 3.1 Governing Equations . 18 3.2 Poiseuille Flow Solution . 20 3.3 Pulsatile Flow Solution . 22 3.3.1 Pulsatile Shear Stress . 25 3.4 Fourier Series . 25 3.5 Numerical Methods . 26 3.5.1 Spatial Discretization . 27 3.5.1.1 Convection term . 28 iv 3.5.1.2 Laplacian Term . 29 3.5.1.3 Source Terms . 29 3.5.2 Temporal Discretization . 30 3.5.2.1 Temporal Derivatives . 30 3.5.2.2 Treatment of Spatial Derivatives in Transient Problems . 30 3.6 Solution Algorithm for the Navier-Stokes Equations . 32 3.6.1 Linearization . 32 3.6.2 Derivation of the Pressure Equation . 32 3.6.3 Pressure-Velocity Coupling . 34 3.6.3.1 PISO . 34 3.6.3.2 SIMPLE . 35 4 OpenFOAM 37 4.1 Using OpenFOAM . 37 4.1.1 Pre-processing grids . 37 4.1.2 Building cases . 38 4.1.3 Post-processing data . 39 4.2 OpenFOAM Solvers and Linear Algebraic Methods . 39 4.2.1 simpleFoam . 39 4.2.2 icoFoam . 40 4.2.3 channelIcoFoam . 40 4.2.4 Fourier adaptations . 40 4.2.5 Linear algebraic methods . 41 4.3 OpenFOAM Boundary Conditions . 41 4.3.1 xedValue . 41 4.3.2 zeroGradient . 41 4.3.3 cyclic . 41 4.3.4 pressureInletOutletVelocity . 42 4.3.5 timeVaryingUniformFixedValue . 42 5 Internal Flow Computations 43 5.1 Grid and Topology Studies . 51 5.1.1 Steady Solution . 51 v 5.1.2 Unsteady Results: Global Error Evaluation . 51 5.1.3 Unsteady Results Radial Error Evaluation . 57 5.2 The Plenum Study . 64 5.2.1 Steady Results . 67 5.2.2 Unsteady Results . 68 6 Conclusions and Recommendations for Future Work 73 Bibliography 76 vi List of Figures 1.1 The internal geometry of the canine nasal passages [4]. 2 2.1 Reynolds number distribution in the canine nasal airway at peak inspiratory ow rate during sning [5]. 6 2.2 Axial distribution of the Womersley number in the canine nasal cavity during sning (f = 5 Hz). For reference, the background shows an appropriately-scaled sagittal section of the canine nasal airway [4]. 7 3.1 Parabolic velocity prole that is characteristic of Poiseuille ow . 21 3.2 Velocity proles at Wo = 9 across one period. Stokes layer is evident near the wall as the overshoot phase lag from the centerline ow. 24 3.3 Fourier decomposition of pulsatile analytical solution into velocity magnitude and phase lag for Wo = {1, 3, 9, 27}. 27 5.1 Comparison of the internal spatial resolution of the (1) coarse, (2) medium, (3) ne, and (4) nest CFD grids in the maxilloturbinate region. Comparable grid resolution is found in the nasal vestibule and ethmoidal region [5]. 44 5.2 Coarse, medium, and ne grids for the grid study. 45 5.3 Unstructured, hybrid, and structured grids used for the topology study. 46 5.4 The grid used for the plenum case shown as the view on the inlet face, along the axial length, and on the plenum face. 47 5.5 Example of statistically steady for Wo = 9, computation was ran for 15 cycles and mass ux was observed before the nal Fourier cycle was computed. 50 5.6 Steady state results for the grid and topology study: velocity proles compared to the analytical solution and error from the wall to centerline. 52 vii 5.7 The maximum value of the Fourier velocity magnitude across a range of Wo for the grid and topology studies, with the error from the analytical solution. 58 5.8 The thickness of the Stokes Layer across a range of Wo for the grid and topology studies, with the error from the analytical solution. 59 5.9 The phase lag of the Fourier velocity at the wall across a range of Wo for the grid and topology studies, with the error from the analytical solution . 60 5.10 The value of the maximum mass ow rate across a range of Wo for the grid and topology studies, with the error from the analytical solution. 61 5.11 The phase lag of the mass ux to the pressure oscillation across a range of Wo for the grid and topology study, with the error from the analytical solution. 62 5.12 Error across the radius to the analytical solution of the Fourier velocity magnitude and phase lag for Wo = 1, 3 . 65 5.13 Error across the radius to the analytical solution of the Fourier velocity magnitude and phase lag for Wo = 9, 27 . 66 5.14 The velocity distribution for the steady inspiration and expiration for the plenum case. The values for the inspiration velocity have been changed from negative to positive for comparison purposes, but in reality have negative value. 68 5.15 Fourier velocity magnitude and phase lag data sampled at four locations along the pipe, Wo = 1. Error is dened as the dierence to the analytical solution. 70 5.16 Wall shear stress compared to the analytical shear stress at twenty instances across one period. 71 5.17 Velocity proles at every T seconds at four locations along the pipe. The t = 20 dash-dot-dot line is the earliest prole in the phase, followed by the long-dash line second, the dash-dot line third, and the solid line is the last prole of the phase in each gure. The inhalation errors increase as the velocity magnitude does. The prole at L=100*D is most aected by the proximity to the plenum. 72 viii List of Tables 5.2 Parameters of each pipe grid. 48 5.3 Physical parameters of grid/topology studies and the plenum case. 48 5.4 Unsteady parameters of the grid/topology studies and the plenum case. 49 ix Nomenclature CFD Computational Fluid Dynamics Re Reynolds number Rec Critical Reynolds number Wo Womersley number NMR Nuclear Magnetic Resonance Imaging GCI Grid Convergence Index LDV Laser-Doppler Velocimetry TCI Time Convergence Index pIOV pressureInletOutletVelocity tVUFV timeVaryingUniformFixedValue GAMG Geometric-Algebraic MultiGrid method PCG Preconditioned Conjugate Gradient method DIC Diagonal Incomplete-Cholesky preconditioner PBiCG Preconditioned Bi-Conjugate Gradient method DILU Diagonal Incomplete Lower Upper preconditioner x Acknowledgments This thesis was supported by the Oce of Naval Research and the Exploratory and Founda- tional program of Penn State University's Applied Research Laboratory. I would rst and foremost like to sincerely thank Dr. Eric Paterson for providing me with an opportunity to explore an area of science that I found so fascinating. I thank you for the knowledge you imparted and the incred- ible patience with which you directed my eorts. I would like to thank Dr. Pauley for reviewing my thesis. I would also like to thank Dr. Gary Settles, Dr. Brent Craven, and Mike Lawson for your assistance to my research. The Computational Mechanics Division at the Water Tun- nel has my thanks, in particular Dr. Joel Peltier, Dr. John Mahay, Dr. Rob Kunz, and Jack Poremba. Thanks to Dr. Hrvoje Jasak for the creation of OpenFOAM. I would like to thank John Pitt for his advice on numerical methods and Cory Smith for his overall support.
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