To the Graduate Council: I am submitting herewith a thesis written by Nicholas G. Currier entitled “A Hybrid Method for Flows in Local Chemical Equilibrium and Nonequilibrium.” I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Computational Engineering. We have read this thesis and recommend its acceptance: Daniel G. Hyams, Ph.D. Associate Professor, Computational Engineering Chairperson/Principal Adviser W. Roger Briley, Ph.D. Professor, Computational Engineering Robert S. Webster, Ph.D. Associate Research Professor, Computational Engineering Accepted for the Graduate Council: Stephanie Bellar, Ph.D. Interim Dean of The Graduate School (Original signatures are on file with official student records) A Hybrid Method for Flows in Local Chemical Equilibrium and Nonequilibrium A Thesis Presented for The Master of Science Degree The University of Tennessee at Chattanooga Nicholas G. Currier August 2010 °c by Nicholas G. Currier, 2010 All rights reserved. ii iii Dedication I would like to dedicate this thesis to my parents, Frederick and Janice Currier, who instilled in me a love of learning and have always encouraged me to pursue my dreams. iv Acknowledgements Foremost, I would like to thank my soon-to-be wife, Maria, for her unending support. Her encouragement has helped me immensely in the pursuit of this degree. A very sincere thanks goes to my adviser, Dr. Daniel Hyams, for his guidance and for the many long conversations we have had regarding this work. Without his help, this thesis would not have been possible. I would like to thank my other committee members, Dr. Roger Briley and Dr. Robert Webster, for their guidance both regarding this work and in general. Dr. Kidambi Sreenivas also deserves my appreciation for his help and advice on several issues that came to light during this process. v Abstract The primary objective of this work is to develop a more efficient chemically active compressible Euler equation solver. Currently, a choice between the physical accuracy of a finite-rate solver or the computational efficiency of an equilibrium flow solver must be made. The number of species modeled continues to increase with available computational resources. A method of further leveraging the increase in computational power is desired. The hybrid chemistry scheme proposed here attempts to maintain the accuracy of finite-rate schemes while retaining some of the cost savings associated with equilibrium chemistry solvers. The method given uses a full finite-rate flux in regions where chemistry is slow compared to the advection rate and an equilibrium chemistry scheme in regions where the chemistry outpaces the fluid transport. Control volume switching is based on a locally defined Damk¨ohlernumber. This method could be extremely useful for full reaction path modeling or the tracking of a very large number of species. The cost of symmetric Gauss-Seidel iterations grows like the number of species plus four, quantity squared. Thus, eliminating the increased cost of solving for a large number of unknowns in regions where it is unjustified can be very useful. Tenasi, a University of Tennessee SimCenter research code, is used as a base for the new solver. The hybrid method is implemented and tested with an explicit solution technique in one dimension. In combination with a five species air chemistry model, a high-temperature shock tube is used as a verification test case. Results are compared with those from pure equilibrium, full finite-rate, perfect gas Euler, and exact perfect vi gas Riemann solvers. Timings are also given, suggesting the cost savings that would be possible should the hybrid method be extended using implicit algorithms. vii Contents List of Tables x List of Figures xi Nomenclature xii 1 Introduction 1 2 Computational Formulation 4 2.1 Governing Equations . 4 2.1.1 Finite-rate Chemistry Regime . 5 2.1.2 Equilibrium Chemistry Regime . 8 2.1.3 Hybrid Chemistry Regime . 15 2.2 HLLC Flux . 20 2.3 Spatial Discretization . 22 2.4 Temporal Discretization . 24 2.5 Explicit Solution Technique . 25 2.6 Chemistry . 26 2.6.1 Thermodynamic Data . 26 2.6.2 Reaction Rates . 29 3 Implementation Issues 31 viii 4 Numerical Verification 34 4.1 Problem Setup . 34 4.2 Refinement Study . 35 4.3 Verification . 39 4.4 Results of Hybrid Regime . 45 4.5 Performance . 47 5 Conclusions 49 Future Work 50 Bibliography 51 A Definition of Mapping Jacobians 57 A.1 For Finite-rate Regime . 58 A.2 For Equilibrium Regime . 60 Vita 62 ix List of Tables 4.1 Performance comparison - Implicit solve (seconds) . 47 4.2 Performance comparison - Explicit solve (seconds) . 47 x List of Figures 2.1 Illustration of fluxes used near hybrid interfaces . 18 2.2 Illustration of residual contributions near hybrid interfaces . 19 2.3 Illustration of equilibrium cell residual shifting and summation operation 20 4.1 Refinement study: perfect gas compressible Euler equations - density plot . 36 4.2 Refinement study: perfect gas compressible Euler equations - velocity plot . 37 4.3 Refinement study: perfect gas compressible Euler equations - pressure plot . 38 4.4 Comparison: density comparison plot of four regimes . 40 4.5 Comparison: velocity comparison plot of four regimes . 41 4.6 Comparison: pressure comparison plot of four regimes . 42 4.7 Species mass fractions N2 and O2 comparison . 43 4.8 Species mass fractions N and O comparison . 43 4.9 Species mass fractions NO comparison . 44 4.10 Percentage of cells switched from equilibrium as solution progresses . 46 xi Nomenclature αr,k collisional efficiency for species k and reaction r β extrapolated state coefficient n ∆qi nonconservative variable update for cell i and time level n ∆t time step δ variable jump vector W˙ chemical source term vector w˙ i mass production rate for species i Γ effective concentration γ specific heat ratio nˆx, nˆy, nˆz face area vector components in x, y, and z directions ~nˆ normalized face area vector R Ω control volume integral R ∂Ω control surface integral κ, φ extrapolation parameters λ eigenvalue xii M mass contraint V volume 00 νi,r product stoichiometric coefficient for reaction r and species i 0 νi,r reactant stoichiometric coefficient for reaction r and species i < residual ρ mixture density ρi density of species i σ equilibrium solver update scaling factor θ covariant velocity,n ˆxu +n ˆyv +n ˆzw + at R˜ mixture gas constant ~n control volume face area vector U~ flow field velocity vector, uˆi + vˆj + wkˆ ~ ˆ V control volume face velocity vector, Vxˆi + Vyˆj + Vzk ξ1, ξ2 temporal accuracy parameters ζ equilibrium solver maximum allowed relative change in species a1, a2, a3, a4, a5, a6 thermochemistry curvefit coefficients ai,j number of particles of element i in species j at control volume face velocity in direction of normal cL, cR speed of sound to the left and right of a face cp specific heat at constant pressure xiii cv specific heat at constant volume d equilibrium solver equation of state Da local Damk¨ohlernumber et specific total energy Ea activation energy F conservative flux vector h specific enthalpy o hi standard state enthalpy for species i ht specific total enthalpy Kb,r backward reaction rate for reaction r Kc,r equilibrium rate constant for reaction r Kf,r forward reaction rate for reaction r Mk molecular weight of species k NE number of elemental species NS number of species P pressure Pstd standard pressure = 1 atm Q conservative variable vector q nonconservative variable vector R specific mixture gas constant xiv Ri specific gas constant Runiv universal gas constant s specific entropy o si standard state entropy for species i SL,SR,S∗ wave speed estimates T temperature u, v, w flow field velocity components in x, y, and z directions Vx,Vy,Vz grid velocities in x, y, and z directions x, y, z Cartesian coordinate directions Yi mass fraction z00 sum of product stoichiometric coefficients z0 sum of reactant stoichiometric coefficients xv Chapter 1 Introduction The importance of high-fidelity Computational Fluid Dynamics (CFD) simulations has grown enormously in recent years. The search for new methods to increase modeling accuracy is of great interest, not only in academia, but also in industrial settings [1]. CFD is capable of modeling physics that are either too costly or too complicated to address with physical experiments. This ability helps drive the rapid dissemination of high technology solutions for public use. One of the many facets included in this search is reacting chemistry. The addition of chemical reactions to fluid models can lead to new observations and more efficient design in areas such as combustion [2], hypersonic flow [3], and atmospheric propagation of chemically active components. The physics modeled by including chemical reactions in CFD are not trivial nor easily dismissed. In hypersonic flows, the addition of chemically reactive terms can significantly affect shock stand-off distances [4]. This change in shock characteristic can affect dynamic loading of a vehicle and therefore the required aerodynamic or structural design. The investigation of chemically active species as they are transported through Earth’s atmosphere is a complex problem. Pollutant oxidation and breakdown is also difficult to observe and computer modeling stands as one of the more useful methods for investigating these environmental issues. The recent increase in concern for environmental emissions 1 makes these simulations very helpful in understanding our effect on the planet’s ecosystems [5, 6]. The utility of modeling chemistry in the context of combustion is obvious. Design of combustors is highly dependent on chemical kinetics, as well as flow field character [2]. Despite the usefulness of these methods, the increased fidelity of the modeling does not come without additional cost.