Thesis presented to the Instituto Tecnol´ogico de Aeron´autica, in partial fulfillment of the requirements for the Degree of Master in Science in the Program of Aeronautical and Mechanical Engineering, Field Aerodynamics, Propulsion and Energy. Andr´eFernando de Castro da Silva ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS Thesis approved in its final version by signatories below: Prof. Dr. Marcos Aur´elio Ortega Advisor Prof. Dr. Celso Massaki Hirata Head of the Faculty of the Department of Graduate Studies Campo Montenegro S~aoJos´edos Campos, SP - Brazil 2012 Cataloging-in Publication Data Documentation and Information Division da Silva, Andr´eFernando de Castro ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS / Andr´eFernando de Castro da Silva. S~aoJos´edos Campos, 2012. 146f. Thesis of Master in Science { Program of Aeronautical and Mechanical Engineering. Field of Aerodynamics, Propulsion and Energy. { Aeronautics Institute of Technology, 2012. Advisor: Prof. Dr. Marcos Aur´elio Ortega. 1. Din^amica dos fluidos computacional. 2. M´etodo de Galerkin. 3. Mec^anica dos fluidos. 4. Aerodin^amica. 5. F´ısica. I.Instituto Tecnol´ogico de Aeron´autica. II. ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS BIBLIOGRAPHIC REFERENCE DA SILVA, Andr´eFernando de Castro. ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS. 2012. 146f. Thesis of Master in Science in Aerodynamics, Propulsion and Energy. { Instituto Tecnol´ogico de Aeron´autica, S~aoJos´edos Campos. CESSION OF RIGHTS AUTHOR NAME: Andr´eFernando de Castro da Silva PUBLICATION TITLE: ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS. PUBLICATION KIND/YEAR: Thesis / 2012 It is granted to Aeronautics Institute of Technology permission to reproduce copies of this thesis to only loan or sell copies for academic and scientific purposes. The author reserves other publication rights and no part of this thesis can be reproduced without the authorization of the author. Andr´eFernando de Castro da Silva Rua Aboli¸c~ao, 87, Apto 135, Torre 1 CEP 12.245-500 { S~ao Jos´edos Campos{SP ON THE USE OF ENO-BASED LIMITERS FOR DISCONTINUOUS GALERKIN METHODS FOR AERODYNAMICS Andr´eFernando de Castro da Silva Thesis Committee Composition: Prof. Dr. Amilcar Porto Pimenta Chairperson - ITA Prof. Dr. Marcos Aur´elio Ortega Advisor - ITA Prof. Dr. Jesuino Takachi Tomita Internal Member - ITA Pesq. Dr. M´arcio Teixeira Mendon¸ca External Member - IAE ITA To my lovely wife. Without her help and encouragement, it simply never would have been. Acknowledgements First and foremost, I would like to thank God, the author and finisher of all things, for the wisdom and perseverance that he has been bestowed upon me during this research work. I would like to express my deep gratitude to my advisor Professor Ortega, for their pa- tient guidance, enthusiastic encouragement and useful critiques of this research work. My special thanks to my friend and colleague Professor Moura, for the hours of conversation and discussion that really contributed to the outcome of this work. I cannot find words to express my gratitude to my wife Raquel, whose support, en- couragement, quiet patience and unwavering love were undeniably the bedrock which made this accomplishment possible. I specially thanks my parents for their faith and unconditional love placed in me since my first years. Finally, I wish to thank my relatives and friends for their support and encouragement throughout my study. \It is the glory of God to conceal a matter; to search out a matter is the glory of kings" | Proverbs 25:2 \Computers are incredibly fast, accurate, and stupid; humans are incredibly slow, inaccurate and brilliant; together they are powerful beyond imagination. " | Albert Einstein Resumo Um problema recorrente na simula¸c~aode alta ordem de escoamentos compress´ıveis ´eo surgimento de oscila¸c~oesesp´urias nas proximidades de regi~oes que contemplem mudan¸cas bruscas das propriedades do escoamento tais como as ondas de choque. Se a amplitude dessas oscila¸c~oes for suficientemente grande, tal fen^omeno, conhecido como fen^omeno de Gibbs, pode levar `adiverg^encia da simula¸c~aonum´erica em curso devido ao aparecimento de estados n~ao f´ısicos (densidade ou press~aonegativa, por exemplo). Dentre as t´ecnicas dispon´ıveis para tratar desse problema est´ao uso de limitadores. Tal metodologia foi introduzida ao contexto de alta ordem por heran¸ca dos m´etodos de Volumes Finitos e tendem a ser de mais f´acil implementa¸c~ao. Entretanto, limitadores cl´assicos possuem a desvantagem de reduzir a ordem local `aprimeira ordem, mesmo quando aplicados a regi~oes suaves. A presente tese se prop~oea estudar os limitadores baseados na tecnologia ENO (Essencialmente N~ao Oscilat´orio) propostos pela primeira vez por Shu e Qiu em 2003. Tais esquemas t^em como propriedade manter a alta ordem em regi~oes suaves do escoamento e ainda fornecer transi¸c~oes finas atrav´esde choques. Como parte deste trabalho, tal limitador foi implementado em um c´odigo computacional baseado na metodologia Galerkin Descont´ınuo (GD) para simular as equa¸c~oes de Euler bidimensionais da din^amica dos gases em malhas triangulares n~ao-estruturadas. Testes e simula¸c~oesforam ent~ao realizados de forma a demonstrar que o esquema possui as capacidades propostas. Abstract A recurring problem in high-order simulation of compressible flows is the appearance of spurious oscillations near regions which display abrupt changes of flow properties such as shock waves. If the amplitude of these oscillations is sufficiently high, this phenomenon, known as Gibbs phenomenon, can lead simulation to diverging due to the appearance of non-physical states (negative pressure or density, for example). Among the techniques available to address this problem, it is highlighted the use of limiters. Such methodology was taken to the high-order context by inheriting from Finite Volume methods and tends to be of easier implementation when compared with other possibilities. However, classical limiters have the disadvantage of reducing the local accuracy order to first order even when applied to smooth regions. This thesis proposes itself to study the ENO(Essentially Non-Oscillatory)-based limiting procedure first proposed by Qiu and Shu in 2003. Such schemes have the property of maintaining the high-order accuracy on smooth regions while still providing sharp shock transitions. As part of this work, such limiter was implemented in a computer code based on the Discontinuous Galerkin (DG) methodology in order to simulate the gas dynamics Euler equations in two dimensions on unstructured triangular meshes. Tests and simulations were then performed in order to demonstrate the proposed capacities . List of Figures FIGURE 2.1 { Illustration of the initial data for the Riemann problem. Source: adapted from [10]............................. 34 FIGURE 2.2 { Structure of the solution of the Riemann problem for a linear system of order m. Source: adapted from [10]................. 34 FIGURE 2.3 { Elementary wave solutions for the Riemann problem in the Euler equations: (a) shock wave, (b) contact discontinuity and (c) rarefac- tion wave. Source: adapted from [10].................. 35 FIGURE 2.4 { Illustration of effect of increasing viscosity on shock thickness. Source: adpatad from [47]............................. 52 FIGURE 3.1 { Standard regions for the (a) quadrilateral, and (b) triangular expan- sion in terms of Cartesian coordinates (η1; η2) or (ξ1; ξ2), respectively. Source: adapted from [15]........................ 62 FIGURE 3.2 { Construction of a complete set of two-dimensional modes φp;q(ξ1; ξ2) within the standard triangle for P = 4. Source: adapted from [15].. 64 FIGURE 3.3 { Example of boundary nodes matching between neighbors for P = 1. A total of Q = P + 3 quadrature nodes were distributed along each common interface. By using Gauss-Legendre integration, triangle's corners are not used as nodes. Source: adapted from [54]...... 68 FIGURE 4.1 { The 2D MINMOD limiter diagram. Source: adapted from [84].... 90 FIGURE 4.2 { Reconstruction procedure diagram. Source: from this author..... 93 LIST OF FIGURES x FIGURE 4.3 { Pascal's triangle showing the modes that complete a basis for P = 3. Source: adapted from [15]........................ 95 FIGURE 5.1 { The two-dimensional Gaussian function for the scalar case test in a mesh with 2349 elements. Source: from this author.......... 112 FIGURE 5.2 { Algebraic convergence via h-refinement for the solution prior and after limiting with P=1. The linear fit equations figuring out at the left and right side correspond, respectively, to the original and limited data. Source: from this author................. 113 FIGURE 5.3 { Algebraic convergence via h-refinement for the solution prior and after limiting with P=2. The linear fit equations figuring out at the left and right side correspond, respectively, to the original and limited data. Source: from this author................. 114 FIGURE 5.4 { Algebraic convergence via h-refinement for the solution prior and after limiting with P=3. The linear fit equations figuring out at the left and right side correspond, respectively, to the original and limited data. Source: from this author................. 115 FIGURE 5.5 { Exponential convergence via h-refinement for the solution prior and after limiting with meshes with 595 (right) and 1022 (left) elements. The linear fit equations figuring out at the left and right side corre- spond, respectively, to the original and limited data. Source: from this author................................ 116 FIGURE 5.6 { Exponential convergence via h-refinement for the solution prior and after limiting with meshes with 2349 (right) and 9612 (left) elements. The linear fit equations figuring out at the left and right side corre- spond, respectively, to the original and limited data. Source: from this author................................ 117 FIGURE 5.7 { Mach number field for a (P = 1)-representation for a mesh with 2351 elements.