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Numerical Examination of Flux Correction for Solving the Navier- Stokes Equations on Unstructured Meshes

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Numerical Examination of Flux Correction for Solving the Navier- Stokes Equations on Unstructured Meshes Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2014 Numerical Examination of Flux Correction for Solving the Navier- Stokes Equations on Unstructured Meshes Dalon G. Work Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Mechanical Engineering Commons Recommended Citation Work, Dalon G., "Numerical Examination of Flux Correction for Solving the Navier-Stokes Equations on Unstructured Meshes" (2014). All Graduate Theses and Dissertations. 2180. https://digitalcommons.usu.edu/etd/2180 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. NUMERICAL EXAMINATION OF FLUX CORRECTION FOR SOLVING THE NAVIER-STOKES EQUATIONS ON UNSTRUCTURED MESHES by Dalon G. Work A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering Approved: Dr. Aaron J. Katz Dr. Robert Spall Major Professor Committee Member Dr. Barton Smith Dr. Mark R. McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2014 ii Copyright © Dalon G. Work 2014 All Rights Reserved iii Abstract Numerical Examination of Flux Correction for Solving the Navier-Stokes Equations on Unstructured Meshes by Dalon G. Work, Master of Science Utah State University, 2014 Major Professor: Dr. Aaron J. Katz Department: Mechanical and Aerospace Engineering This work examines the feasibility of a novel high-order numerical method, which has been termed Flux Correction. It has been given this name because it \corrects" the flux terms of an established numerical method, cancelling various error terms in the fluxes and making the method higher-order. In this work, this change is made to a traditionally second-order finite volume Galerkin method. To accomplish this, higher-order gradients of solution variables, as well as gradients of the fluxes are introduced to the method. Gradi- ents are computed using lagrange interpolations in a fashion reminiscent of Finite Element techniques. For the Euler Equations, Flux Correction is compared against Flux Reconstruc- tion, a derivative of the high-order Discontinuous Galerkin and Spectral Difference methods, both of which are currently popular areas of research in high-order numerical methods. Flux Correction is found to compare favorably in terms of accuracy, and exceeds expectations for convergence rates. For the full Navier-Stokes Equations, the effect of curved elements on Flux Correction are examined. Flux Correction is found to react negatively to curved elements due to the gradient procedure's poor handling of high-aspect ratio elements. (80 pages) iv Public Abstract Numerical Examination of Flux Correction for Solving the Navier-Stokes Equations on Unstructured Meshes by Dalon G. Work, Master of Science Utah State University, 2014 Major Professor: Dr. Aaron J. Katz Department: Mechanical and Aerospace Engineering This work examines the feasibility of a novel high-order numerical method, which has been termed Flux Correction. This is accomplished by comparing it against another high- order method called Flux Reconstruction. These numerical methods are used to solve the Navier-Stokes equations, which govern the motion of fluid flow. High-order numerical methods, or those that demonstrate a third-order and higher solution error convergence rate, are rarely used on unstructured meshes when solving fluid problems. Flux Correction intends to make high-order accuracy available to the larger world of Computational Fluid Dynamics in a simple and effective manner. The advantages and disadvantages of the method can only be discovered when compared against other high-order numerical methods. This work accomplishes this by comparing Flux Correction and Flux Reconstruction in terms of accuracy, numerical dissipation, and solution times. Flux Correction is found to compare favorably in terms of accuracy, and exceed expectations for convergence rates. Flux Correction is also tested on high-order meshes, or meshes that use high-order polynomials in the construction of the unstructured triangle mesh. High-order meshes generate long, thin elements, which are found to negatively impact the convergence and accuracy of Flux Correction. v For my wife. Here's to many more hectic and fun-filled years together. For my God, whose grace has given me a good life. vi Acknowledgments This work is supported by the Computational Research and Engineering for Acquisition Tools and Environments (CREATE) Program, which is sponsored by the U.S. Department of Defense HPC Modernization Program Office, and by the Army Research Office Fluid Dynamics Program. I would like to thank my major professor, Dr. Aaron Katz, for taking a chance on me, and for patiently helping me through the learning process. I can only hope to be as smart as him one day. A special thank you to my friend and mentor Kyle Horne. His enthusiasm for com- putational methods infected me during my undergraduate career, and placed me on the academic path I currently walk. Also to my many friends and family who supported me, even if it was by giving me incredulous stares as I tried to explain what I was working on, then telling me that it sounds like great fun. They have spent many years teaching me how to laugh at myself, how to love, how to cry, and how to care. From such a large group of people I want to single out my parents, who have loved me, cared for me, prayed for me, taught me, and been there for me for over 25 years now. My life is what it is because of them. Last of all I wish to thank my God and my wife. My Lord and Savior Jesus Christ has been my Exemplar, my Rock, and my Confidence since my teenage years. My wife has been the best as I spent many long hours on this project. I don't know why she would ever consent to marry and put up with a nerd like me. I will be eternally grateful for her faith in me, for her ability to remember the things that I forget, and for her amazing ability to pull my head out of the clouds and out of the ground. Here's to many more happy years together! Dalon G. Work vii Contents Page Abstract ::::::::::::::::::::::::::::::::::::::::::::::::::::::: iii Public Abstract ::::::::::::::::::::::::::::::::::::::::::::::::: iv Acknowledgments ::::::::::::::::::::::::::::::::::::::::::::::: vi List of Tables ::::::::::::::::::::::::::::::::::::::::::::::::::: ix List of Figures :::::::::::::::::::::::::::::::::::::::::::::::::: x Acronyms :::::::::::::::::::::::::::::::::::::::::::::::::::::: xi 1 Introduction ::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.1 A Brief History of High-Order Methods....................3 1.1.1 Finite Volume Methods.........................4 1.1.2 Finite Element Methods.........................4 1.1.3 Other Methods..............................5 1.2 Purpose of Thesis.................................7 1.3 Thesis Outline..................................8 2 Background :::::::::::::::::::::::::::::::::::::::::::::::::: 9 2.1 Governing Equations...............................9 2.2 High-Order Meshing............................... 11 2.3 Verification with the Method of Manufactured Solutions........... 16 3 Flux Correction ::::::::::::::::::::::::::::::::::::::::::::::: 17 3.1 The Effect of Truncation Error on Solution Error............... 17 3.2 1D Flux-Correction................................ 19 3.2.1 Traditional Galerkin........................... 20 3.2.2 Flux-Corrected Galerkin......................... 22 3.3 2D Flux Correction................................ 23 3.4 2D Gradient Approximation........................... 25 3.5 Viscous Terms Consideration.......................... 27 3.6 Source Terms................................... 28 3.7 Unsteady Time Terms.............................. 29 3.8 Solution Method................................. 29 3.9 Multigrid..................................... 31 3.10 Boundary Conditions............................... 31 viii 4 Flux Reconstruction ::::::::::::::::::::::::::::::::::::::::::: 33 4.1 Procedure..................................... 34 4.2 Solution and Flux Point Locations....................... 38 4.3 Interface Values.................................. 38 4.4 Correction Functions............................... 38 4.5 Time Integration and values of c ........................ 40 5 Numerical Results ::::::::::::::::::::::::::::::::::::::::::::: 41 5.1 Method of Manufactured Solutions....................... 41 5.2 Cost Studies.................................... 42 5.3 Isentropic Vortex................................. 50 5.4 Unsteady, Viscous Flow over a Circular Cylinder............... 53 6 Conclusions and Future Work ::::::::::::::::::::::::::::::::::: 57 References :::::::::::::::::::::::::::::::::::::::::::::::::::::: 59 Appendix :::::::::::::::::::::::::::::::::::::::::::::::::::::: 64 ix List of Tables Table Page 4.1 The four special values of c for C-RK with p = 2............... 40 5.1 Cylinder Mesh Statistics............................. 53 5.2 Strouhal Numbers................................ 56 x List of Figures Figure Page 1.1 Relationship graph for higher-order methods..................8 3.1 Discretization used in the derivation of the Linear Galerkin Method.... 19 3.2 Node-centered stencil on a two-dimensional unstructured triangular
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