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Large Eddy Simulation of Turbulent Flows Using Finite Volume Methods with Structured, Unstructured, and Zonal Embedded Grids Todd Allen Simons Iowa State University

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Large Eddy Simulation of Turbulent Flows Using Finite Volume Methods with Structured, Unstructured, and Zonal Embedded Grids Todd Allen Simons Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1998 Large eddy simulation of turbulent flows using finite volume methods with structured, unstructured, and zonal embedded grids Todd Allen Simons Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mechanical Engineering Commons Recommended Citation Simons, Todd Allen, "Large eddy simulation of turbulent flows using finite volume methods with structured, unstructured, and zonal embedded grids " (1998). Retrospective Theses and Dissertations. 11892. https://lib.dr.iastate.edu/rtd/11892 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. 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UMI A Bell & Howell Infonnadon Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Large eddy simulation of turbulent flows using finite volume methods with structured, unstructured, and zonal embedded grids by Todd Allen Simons A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mechanical Engineering Major Professor: Richard H. Pletcher Iowa State University Ames, Iowa 1998 Copyright © Todd Allen Simons, 1998. All rights reserved. DMI Number: 9841087 mvn IVUcrofonn 9841087 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii Graduate College Iowa State University This is to certify that the Doctoral dissertation of Todd Allen Simons has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Major Professor Signature was redacted for privacy. For the Major Program Signature was redacted for privacy. For the Graduate College Ill TABLE OF CONTENTS NOMENCLATURE vii CHAPTER 1 INTRODUCTION I 1.1 Motivation 1 1.2 Large Eddy Simulation 2 1.2.1 Subgrid Scale Models 2 1.2.2 Isotropic Decaying Turbulence 3 1.2.3 Channel Flow 4 1.3 Discretization Methods 6 1.4 Unstructured Grids and LES " 1.0 Zonal Embedded Grids and LES 8 1.6 Objectives 9 1.6.1 Structured Finite Volume Scheme 10 1.6.2 Unstructured Finite Volume Scheme 10 1.6.3 Zonal Embedded Grids 11 1.7 Dissertation Organization 12 CHAPTER 2 MATHEMATICAL FORMULATION 13 2.1 Compressible Navier-Stokes Equations 13 2.2 Filtering 15 2.3 Favre Filtered Governing Equations 16 2.4 Subgrid Scale Model 18 2.4.1 Gradient-Diffusion Models 18 2.4.2 Smagorinsky 19 iv 2.4.3 Isotropic Part of the SGS Model 20 2.4.4 Dynamic Model 20 2.5 Integral-Vector Form of the Equations 21 CHAPTER 3 FINITE VOLUME FORMULATION FOR HEXAHEDRAL GRIDS 24 3.1 Finite Volume Approach 24 3.2 Dependent Variables 25 3.3 Integral Approximations 25 3.4 Gradient Calculations 26 3.5 Low Mach Number Preconditioning 27 3.6 Artificial Dissipation 29 3.7 Integration Schemes 32 3.7.1 Runge Kutta 32 3.7.2 Local Time Stepping 32 3.7.3 Implicit Residual Averaging 33 3.7.4 Lower-Upper Symmetric Gauss-Seidel 33 3.8 Convergence Criteria 37 3.9 Boundary Conditions 37 3.9.1 Solid Wall Boundary Conditions 37 3.9.2 Periodic Boundary Condition 38 3.10 Unstructured Hexahedral Formulation 39 3.11 Zonal Embedded Grids 41 3.11.1 Blocking Arrangement 43 3.11.2 Interpolation 43 3.11.3 Load Balancing 47 CHAPTER 4 FINITE VOLUME FORMULATION FOR TETRAHEDRAL GRIDS 51 4.1 Finite Volume Approach 51 V 4.1.1 Grid Data Structure 52 4.2 Grid Generation 53 4.3 Geometry Calculations 54 4.4 Integral Approximations 54 4.5 Gradients 55 4.5.1 Gauss Divergence Theorem 55 4.5.2 Gradient Calculation Attributed to Frink 56 4.5.3 .iVodal Calculation 56 4.5.4 Least Squares 58 4.5.5 Gramm-Schmidt Method 59 4.6 Reconstruction 60 4.7 Flux Limiting 61 4.8 Upwinding 63 4.9 Explicit Solution Technique 63 4.10 Implicit Solution Technique 63 4.11 Boundary Conditions 64 CHAPTERS LAMINAR VALIDATION 66 5.1 Lid Driven Cavity 66 5.1.1 Structured Formulation 67 5.1.2 Unstructured Formulation 71 5.2 Laminar Channel 72 5.3 Summary 74 CHAPTER 6 ISOTROPIC DECAYING TURBULENCE 76 6.1 Introduction 76 6.2 Problem Description 76 6.3 Test Filtering 77 6.4 Initial Conditions 78 6.4.1 Structured Mesh 78 vi 6.4.2 Unstructured Mesh 80 6.5 Results 80 6.5.1 Structured Results 81 6.5.2 Unstructured Results 85 6.6 Summary 86 CHAPTER 7 PLANAR TURBULENT CHANNEL 88 7.1 Introduction 88 7.2 Problem Description 88 7.3 Test Filtering 89 7.4 Simulation Details 90 7.5 Results for Velocity Statistics for Single Zone Grid 93 7.6 Results for the Unstructured Hexahedral Formulation 96 7.7 Details of Zonal Embedded Grids 100 7.8 Results for Zonal Embedded Grid at Low Reynolds Number 101 7.9 Analysis of Zonal Embedded Formulation 103 7.10 Grid Study for High Reynolds Number Channel with Smagorinsky Model ... 105 7.11 Results for the High Reynolds Number Grid Study Ill 7.12 Comparison of the Dynamic and Smagorinsky Models 116 7.13 Summary 116 CHAPTER 8 CONCLUSIONS 122 8.1 Summary 122 8.2 Contributions 123 8.3 Recommendations for Future Work 124 APPENDIX A JACOBIAN MATRICES 125 APPENDIX B DERIVATION OF BI-CUBIC INTERPOLATION 127 BIBLIOGRAPHY 132 ACKNOWLEDGMENTS 140 Vll NOMENCLATURE Roman Symbols a. b, c length of triangle edges A area [.4], [J3], [C] inviscid flux Jacobians .4+ constant in Van Driest damping formula .4c channel cross-sectional area (= LyL;) B body force or source term C cost function C discretized convective and viscous flux vector C(f, C,- dynamic subgrid-scale model coefficients Cj Smagorinsky subgrid-scale model coefficient c speed of sound (= y/fRT) Cp constant pressure specific heat Ct, constant volume specific heat D Van Driest damping function D artificial dissipation operator vector d artificial dissipation flux vector B specific total energy (= e 4- u,ti,/2) E, F, G flux vectors E(k) turbulent energy spectrum e specific internal energy (= c^T) VUl F flatness factor Fi Fourier coefficients G filter function [/] identity matrix /xr. /yv. Lagrange multipliers i,],k unit vectors K subgrid-scale term in energy equation k wave number, thermal conductivity, or turbulent kinetic energy L Laplacian operator [L], [D], [U] approximate factorization matrices for LU-SGS scheme Lij resolved or Leonard turbulent stress tensor Lr reference length Lx, Ly, L; dimensions of channel M Mach number or mesh spacing for grid generated turbulence Mij anisotropic part of subgrid-scale stress tensor m mass flow rate number of time steps for turbulent statistics n unit normal vector "xt components of unit normal vector n. i normal and tangential components Pr Prandtl number (= fJ.Cp/k) p thermodynamic pressure Q orthogonal matrix Qj subgrid-scale turbulent heat flux vector qj heat flux vector subgrid-scale turbulent kinetic energy r location of a point in the domain R gas constant or cross-correlation coefficient ix Rx, Ry Lagrangian multipliers R residual vector or upper triangular matrix Tt preconditioned residual vector Re generic Reynolds number Rer Re based on reference quantities (= prVrLr/Hr) Rej bulk Re based on channel half-height (= pbUbS/fXb) Rbt Re based on channel half-height and friction velocity (= pbUrS/m,) fBi^c relaxation factors in LU-SGS scheme s semiperimeter of a triangle 5 magnitude of cell face area vector or skewness factor 5 cell face area vector 5,J strain rate tensor T thermodynamic temperature [T] time derivative Jacobian (=c/U/c?W) t physical time C' contravariant velocity U vector of conserved variables (/j, pu, pv, pw, pE)^ Uo mean velocity for grid generated decaying turbulence Ur friction velocity (= \/t^/Pw) u,V, w Cartesian velocity components in x, y, ^ directions velocity in wall coordinates (= u/ur) V velocity or volume Vr reference velocity w weight function W vector of primitive variables (p, u, v, w, T)"^ x,y,z Cartesian coordinates I/"*" distance to wall in wall coordinates (= SyUr/fw) Greek Symbols a, IT, £ subgrid-scale terms in energy
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