Nonlinear Disturbance Evolution in Boundary Layers Using the One-Way Navier-Stokes Equations Citation Sleeman, Michael Kenneth Elliott (2026) Nonlinear Disturbance Evolution in Boundary Layers Using the One-Way Navier-Stokes Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/wwv8-v242. https://resolver.caltech.edu/CaltechTHESIS:08132025-212958945 Abstract Laminar-turbulent transition prediction for boundary-layer flows is a pacing item in engineering design. This work extends the one-way Navier Stokes (OWNS) equations to support nonlinear interactions between waves of different frequencies which enables nonlinear disturbance evolution in spatially-developing shear flows, with the goal of predicting transition for a reduced computational cost relative to direct numerical simulation (DNS). The OWNS approach linearizes the Navier-Stokes equations about a user-specified equilibrium solution, and then evolves disturbances to the equilibrium solution by solving a spatial initial-value problem in the frequency domain. OWNS yields a reduced computational cost compared to global linear stability analysis, while also conferring numerous advantages over the parabolized stability equations (PSE) that we seek to extend to nonlinear OWNS (NOWNS). We validate NOWNS for two- and three-dimensional disturbances to a low-speed Blasius boundary layer by comparing to DNS results from the literature. We further demonstrate that NOWNS can be used to for transition prediction since it accurately predicts the onset of laminar-turbulent transition in low-speed boundary-layer flows, relative to DNS. Subsequently, we extend the approach to high-speed boundary-layer flows, where we apply it to study oblique-wave breakdown of Mack's first and second modes. Finally, we formulate a greedy algorithm for choosing optimal OWNS recursion parameters, which achieves rapid error convergence and a net decrease in computational cost compared to previous approaches to recursion parameter selection. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: boundary-layer transition; subsonic; supersonic; hypersonic; reduced-order modeling; one-way marching; spatial-initial-value problem; spatial-integration; first-order hyperbolic equation; Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Mechanical Engineering Minor Option: Applied And Computational Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Colonius, Tim Thesis Committee: Blanquart, Guillaume (chair) Hunt, Melany L. Bae, H. Jane Colonius, Tim Defense Date: 1 August 2025 Funders: Funding Agency Grant Number The Boeing Company CT-BA-GTA-1 Record Number: CaltechTHESIS:08132025-212958945 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08132025-212958945 DOI: 10.7907/wwv8-v242 Related URLs: URL URL Type Description https://doi.org/10.2514/1.J064909 DOI Article adapted for Chapter 3 https://doi.org/10.2514/6.2024-3530 DOI Article adapted for Chapter 3 https://doi.org/10.2514/6.2023-3273 DOI Article adapted for Chapter 3 https://arxiv.org/abs/2506.02320 arXiv Article adapted for Chapters 2 and 4 ORCID: Author ORCID Sleeman, Michael Kenneth Elliott 0000-0001-5949-9289 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17622 Collection: CaltechTHESIS Deposited By: Michael Sleeman Deposited On: 21 Aug 2025 21:43 Last Modified: 28 Aug 2025 09:11 Thesis Files PDF - Final Version See Usage Policy. 9MB Repository Staff Only: item control page

Nonlinear disturbance evolution in boundary layers using the one-way Navier-Stokes equations Thesis by Michael Kenneth Elliott Sleeman In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mechanical Engineering CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2026 Defended August 1, 2025 ii © 2026 Michael Kenneth Elliott Sleeman ORCID: 0000-0001-5949-9289 All rights reserved iii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor Professor Tim Colonius for his thoughtful guidance, feedback, and encouragement throughout my time at Caltech. Professor Colonius is deeply knowledgeable about fluid mechanics and scientific computing, and I appreciate the time and effort he spent deepening my knowledge and understanding of these subject areas. I would also like to thank my committee members, Professors Guillaume Blanquart, Jane Bae, and Melany Hunt, for their valuable feedback on my thesis. My academic journey began at the University of British Columbia, where I had the privilege of working in Professor Gwynn Elfring’s research group. His courses in fluid mechanics and scientific computing inspired my interest in academic research and his encouragement was critical to the start of my academic journey. I began my graduate studies in Professor Masayuki Yano’s research group at the University of Toronto. I would like to thank Professor Yano for his patient mentorship, and for introducing me to finite element methods, model reduction, and other foundational methods in scientific computing. The skills I developed during my Master’s studies were instrumental to my success at Caltech. During my time at Caltech, I have been fortunate to form lasting friendships with many fellow students who have enriched both my academic and personal life. I am especially grateful to Alex Mori Carroll, with whom I studied fluid mechanics in first year; Ethan Epperly, who deepened my understanding of mathematics and shared many thoughtful research discussions and lunches; Liam Heidt, whose feedback on my research and technical presentations was always insightful; Ethan Eichberger, who has built social connections within the Colonius group and the Mechanical and Civil Engineering department; and Kasper Petersen, who reminded me to enjoy life outside of the academic setting. I also wish to acknowledge previous researchers who developed and extended the One-Way Navier-Stokes (OWNS) approach and generously shared their expertise and time. This includes Omar Kamal, Aaron Towne, Georgios Rigas, Matthew Lakebrink, and Vishal Anantharaman. I would like to thank all members, past and present, of the Colonius group, whose camaraderie and collaboration made my time at Caltech all the more rewarding. iv I am also thankful for my friendships with Eugene, Sohini, and Nick at the University of Toronto, and with Charles, Sun Jun, and Ben at the University of British Columbia. Beyond academia, I am deeply grateful to my lifelong friends, including Marcus, Michael, Aaron, Jonathan, Thomas, Chris, Lan, and Kyra, who have known me since high school or earlier. Your continued friendship, whether through shared meals, walks in the park, hikes in the mountains, or bike rides through the city, has brought me joy whenever I have returned home. I owe immense thanks to my family for their unwavering love and support. I would like to thank my mother for encouraging me to pursue mechanical engineering, both during my undergraduate and graduate studies. Her empathy and emotional support throughout my research journey have been invaluable. I am grateful to my father for cultivating my interest in computing and helping to develop my programming skills. I thank both my parents for instilling in me the values of resilience and a growth mindset, qualities that have shaped my academic and personal life. I am also thankful to my brother, Matthew, whose perspective and strength have taught me many valuable lessons, and whose presence I deeply appreciate. I owe a special thanks to my grandmother, Margaret, and my Uncle Barry, whose curiosity and thoughtful questions about my research and education have always made me feel supported and understood. Finally, I would like to thank Luna for her steadfast love and support. Your patience, understanding, and unwavering belief in me have been critical to both my success at Caltech and my growth beyond it. I am so grateful to have you in my life, and I am proud that we have supported each other in achieving our goals. v ABSTRACT Laminar-turbulent transition prediction for boundary-layer flows is a pacing item in engineering design. This work extends the one-way Navier Stokes (OWNS) equations to support nonlinear interactions between waves of different frequencies which enables nonlinear disturbance evolution in spatially-developing shear flows, with the goal of predicting transition for a reduced computational cost relative to direct numerical simulation (DNS). The OWNS approach linearizes the Navier-Stokes equations about a user-specified equilibrium solution, and then evolves disturbances to the equilibrium solution by solving a spatial initial-value problem in the frequency domain. OWNS yields a reduced computational cost compared to global linear stability analysis, while also conferring numerous advantages over the parabolized stability equations (PSE) that we seek to extend to nonlinear OWNS (NOWNS). We validate NOWNS for two- and three-dimensional disturbances to a low-speed Blasius boundary layer by comparing to DNS results from the literature. We further demonstrate that NOWNS can be used to for transition prediction since it accurately predicts the onset of laminar-turbulent transition in low-speed boundary-layer flows, relative to DNS. Subsequently, we extend the approach to high-speed boundarylayer flows, where we apply it to study oblique-wave breakdown of Mack’s first and second modes. Finally, we formulate a greedy algorithm for choosing optimal OWNS recursion parameters, which achieves rapid error convergence and a net decrease in computational cost compared to previous approaches to recursion parameter selection. vi PUBLISHED CONTENT AND CONTRIBUTIONS M. Sleeman and T. Colonius. (2025). “Greedy recursion parameter selection for the One-Way Navier–Stokes (OWNS) equations.” Submitted in 2025. M. Sleeman, M. Lakebrink, and T. Colonius. (2025). “Simulation of K-type and H-type Transition Using the Nonlinear One-Way Navier-Stokes Approach.” In: Tenth IUTAM Symposium on Laminar-Turbulent Transition. M. Sleeman, M. Lakebrink, and T. Colonius. (2025). “Boundary layer stability analysis using the nonlinear One-Way Navier-Stokes approach”. In: AIAA Journal 2025 63:8, 3145-3159. doi: 10.48550/arXiv.2409.17274. M. Sleeman, M. Lakebrink and T. Colonius. (2024) “The Nonlinear One-Way Navier-Stokes (NOWNS) Approach for Boundary-Layer Transition”. In: AIAA AVIATION 2024 Forum. doi: 10.2514/6.2024-3530. M. Sleeman, M. Lakebrink and T. Colonius. (2023) “Nonlinear stability of wallbounded flows using the One-Way Navier-Stokes (OWNS) Equations”. In: AIAA AVIATION 2023 Forum. doi: 10.2514/6.2023-3273. The second, fourth, and fifth contributions are conference proceedings: the fifth contribution presents the development and validation of the methodology, while the fourth contribution showcases the advantages of the methodology relative to existing methods. The first contribution is a journal article that develops a greedy algorithm for recursion parameter selection, which must be carefully chosen to ensure the OneWay Navier-Stokes (OWNS) approximation converges. We show that the greedy algorithm achieves faster convergence and yields a reduced computational cost relative to existing recursion parameter selection methods. The second contribution demonstrates that the method can be used to predict laminar-turbulent transition in low-speed boundary-layer flows, while the third contribution is a journal article that combines the fourth and fifth contributions into a single paper. The first contribution appears in Chapter 2 where it develops criteria for the numerical method to converge, and Chapter 4 where it develops and demonstrates the greedy algorithm. The third, fourth, and fifth contributions appear in Chapter 3 where the numerical method is developed and demonstrate, while the second contribution appears in Chapter 5 where we demonstrate that the method can be used to predict transition in low-speed boundary-layer flows. vii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Boundary-layer flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Laminar-turbulent transition in boundary-layer flows . . . . . . . . . 1.3.1 A brief history of laminar-turbulent transition . . . . . . . . 1.3.2 Notions of stability . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Types of disturbances . . . . . . . . . . . . . . . . . . . . . 1.3.4 Paths to transition for compressible boundary-layer flows . . 1.4 Methods for hydrodynamic stability analysis . . . . . . . . . . . . . 1.4.1 Locally-parallel linear stability theory . . . . . . . . . . . . 1.4.2 Linear global stability analysis . . . . . . . . . . . . . . . . 1.4.3 The Parabolized Stability Equations . . . . . . . . . . . . . 1.4.4 The One-Way Navier-Stokes equations . . . . . . . . . . . . 1.4.5 Empirical methods for transition prediction . . . . . . . . . 1.5 Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . 1.6 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: One-way Marching of Linear Hyperbolic Equations . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 OWNS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Exact projection equations . . . . . . . . . . . . . . . . . . 2.2.2 Approximate projection using OWNS-P . . . . . . . . . . . 2.2.3 Approximate projection using OWNS-R . . . . . . . . . . . 2.3 Recursion parameter selection . . . . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Nonlinear OWNS Methodology . . . . . . . . . . . . . . . . . . 3.1 Linear OWNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Semi-discrete equations . . . . . . . . . . . . . . . . . . . . 3.1.2 Parabolization using the OWNS projection procedure . . . . 3.1.3 Approximate projection operator . . . . . . . . . . . . . . . 3.1.4 Fully-discrete equations . . . . . . . . . . . . . . . . . . . . 3.1.5 Streamwise diffusion terms . . . . . . . . . . . . . . . . . . 3.2 Nonlinear OWNS . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fully-discrete equations . . . . . . . . . . . . . . . . . . . . iii v vi vi xi xv 1 1 1 2 2 3 4 5 6 7 8 8 9 9 9 10 11 11 12 15 17 20 24 24 25 25 26 26 28 28 29 29 31 viii 3.2.2 Pseudo-spectral method . . . . . . . . . . . . . . . . . . . . 3.2.3 Nonlinear solution procedure and computational cost . . . . 3.2.4 Special treatment of the zero-frequency modes . . . . . . . 3.2.5 Spanwise symmetry . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 3.2.7 Effects of the mean-flow distortion . . . . . . . . . . . . . . 3.2.8 Recursion parameters . . . . . . . . . . . . . . . . . . . . . 3.2.9 Modeling errors . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 2D evolution of a Tollmien-Schlichting wave . . . . . . . . 3.3.2 3D oblique-wave breakdown . . . . . . . . . . . . . . . . . 3.4 Advantages of the NOWNS procedure . . . . . . . . . . . . . . . . . 3.4.1 High amplitude evolution of TS wave . . . . . . . . . . . . 3.4.2 High amplitude oblique-wave breakdown . . . . . . . . . . 3.4.3 Low amplitude oblique-wave breakdown with random noise 3.4.4 Blowing/suction strip . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: A Greedy Approach to Recursion parameter selection . . . . . . 4.1 Greedy algorithm for recursion parameter selection . . . . . . . . . . 4.1.1 Objective function . . . . . . . . . . . . . . . . . . . . . . 4.1.2 A summary of the greedy algorithm . . . . . . . . . . . . . 4.1.3 Numerical stability of the recursive filter . . . . . . . . . . . 4.1.4 Greedy algorithm and spatial marching . . . . . . . . . . . 4.1.5 Cost trade-off . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Demonstration of the greedy algorithm for a single station . . . . . . 4.2.1 Convergence of the objective function . . . . . . . . . . . . 4.2.2 Stability of the march . . . . . . . . . . . . . . . . . . . . . 4.2.3 Accuracy of the projection operator . . . . . . . . . . . . . 4.2.4 3D low-speed boundary-layer flow . . . . . . . . . . . . . . 4.3 Demonstration of the greedy algorithm for spatial marching . . . . . 4.3.1 Linear calculation . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Nonlinear calculation . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter V: The NOWNS approach for low-speed boundary-layer transition prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 K-type transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 H-type transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Skin friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Investigations into the failure of the NOWNS march . . . . . . . . . 5.4.1 Discretization errors . . . . . . . . . . . . . . . . . . . . . 5.4.2 Modeling errors in NOWNS . . . . . . . . . . . . . . . . . 5.4.2.1 Negative upstream streamwise velocity . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VI: NOWNS for Stability and Transition Analysis of High-speed Boundary-Layer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 34 35 35 35 36 36 36 36 37 39 39 40 41 44 46 48 48 48 49 50 50 51 52 52 54 55 57 58 58 58 59 61 61 62 64 65 66 66 67 67 69 ix 6.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Demonstration of greedy algorithm for high-speed flows . . . . . . . 69 6.2.1 Error convergence at the domain inlet . . . . . . . . . . . . 70 6.2.2 Linear calculation for Mack’s second mode . . . . . . . . . 70 6.3 Extension of NOWNS to high-speed flows . . . . . . . . . . . . . . 71 6.3.1 Temperature-dependent fluid properties . . . . . . . . . . . 72 6.3.2 Linearizing about the corrected mean flow . . . . . . . . . . 73 6.3.3 Disturbance amplitude . . . . . . . . . . . . . . . . . . . . 73 6.4 2D nonlinear evolution of Mack’s second mode . . . . . . . . . . . . 74 6.5 Oblique breakdown of Mack’s first mode . . . . . . . . . . . . . . . 74 6.5.1 Small amplitude . . . . . . . . . . . . . . . . . . . . . . . . 77 6.5.2 Large amplitude . . . . . . . . . . . . . . . . . . . . . . . . 78 6.6 Oblique breakdown of Mack’s second mode . . . . . . . . . . . . . 78 6.6.1 Small amplitude . . . . . . . . . . . . . . . . . . . . . . . . 79 6.6.2 Large amplitude . . . . . . . . . . . . . . . . . . . . . . . . 79 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Chapter VII: Conclusions and Future Work . . . . . . . . . . . . . . . . . . 84 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2.1 Transition prediction . . . . . . . . . . . . . . . . . . . . . 85 7.2.2 High-speed boundary-layer flows . . . . . . . . . . . . . . . 85 7.2.3 Recursion parameter selection . . . . . . . . . . . . . . . . 86 7.2.4 Numerical filtering . . . . . . . . . . . . . . . . . . . . . . 86 7.2.5 Free-shear flows . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2.6 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . 86 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Appendix A: Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . 95 Appendix B: Similarity solutions . . . . . . . . . . . . . . . . . . . . . . . . 98 B.1 Blasius equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Appendix C: Special cases of singular 𝐴 . . . . . . . . . . . . . . . . . . . . 100 Appendix D: Proofs for OWNS-P and OWNS-R . . . . . . . . . . . . . . . . 101 D.1 Proofs for OWNS-P . . . . . . . . . . . . . . . . . . . . . . . . . . 101 D.2 Proofs for OWNS-R . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Appendix E: A variational approach to OWNS . . . . . . . . . . . . . . . . . 110 E.1 Petrov-Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . 111 E.1.1 Constant-coefficient system . . . . . . . . . . . . . . . . . . 111 E.1.2 Variable-coefficient system . . . . . . . . . . . . . . . . . . 112 Appendix F: Jacobian of the NOWNS equations . . . . . . . . . . . . . . . . 114 Appendix G: Design choices for NOWNS . . . . . . . . . . . . . . . . . . . 116 G.1 Streamwise diffusion terms . . . . . . . . . . . . . . . . . . . . . . 116 G.2 Inclusion of the streamwise pressure gradient . . . . . . . . . . . . . 116 G.3 Comparison of nonlinear solution procedures . . . . . . . . . . . . . 117 G.4 Linearization about the baseflow vs. the corrected mean flow . . . . . 119 G.4.1 Equations linearized about the corrected mean flow . . . . . 119 G.4.1.1 Results for K-type transition . . . . . . . . . . . . 120 x Appendix H: Heuristic recursion parameter sets . . . . . . . . . . . . . . . . 122 H.1 Recursion parameters for non-zero frequencies . . . . . . . . . . . . 122 H.2 Zero-frequency modes . . . . . . . . . . . . . . . . . . . . . . . . . 123 Appendix I: Labeling eigenvalues according to Briggs’ criterion . . . . . . . 125 Appendix J: Transient growth in low-speed boundary-layer flows . . . . . . . 128 J.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 xi LIST OF ILLUSTRATIONS Number Page 1.1 Sketch of convective and absolute stability from Figure 7.6 in Schmid and Henningson (2001): (a) 𝑥 − 𝑡 plane diagram of an absolutely unstable disturbance; (b) 𝑥−𝑡 plane diagram of a convectively unstable disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Contours of the 𝑢- and 𝑣-velocity disturbance contours for the 2D TS wave, demonstrating that it evolves with a well-defined wavelength that is constant in the streamwise direction. . . . . . . . . . . . . . . 5 1.3 Figure 4.1 from Schmid and Henningson (2001) illustrating transient growth due to non-orthogonal superposition of two decaying vectors. 5 1.4 Figure 1 from Fedorov (2011), adapted from Morkovin et al. (1994), showing transition paths in boundary-layer flows. . . . . . . . . . . . 6 3.1 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for 2D evolution of TS wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 𝑢-and 𝑣-velocity profiles at streamwise coordinate 𝑅𝑒 𝑥 = 7.80 × 105 for 2D evolution of TS wave. . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for the smallamplitude oblique-wave breakdown case at frequency 𝐹 = 86 × 10−6 , spanwise wavenumber 𝑏 = 2/9 × 10−3 , with initial amplitude of √ −3 𝑢′(1,1) max (𝑥 0 ) = 2 × 10 . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 𝑢-velocity profiles at streamwise coordinate 𝑅𝑒 𝑥 = 4.69 × 105 for small-amplitude oblique-wave breakdown. . . . . . . . . . . . . . . 40 3.5 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for 2D evolution of TS wave with 𝑢′(1) max (𝑥 0 ) = 0.4%. . . . . . . . . . . . . . . . . . . . . 41 3.6 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for high amplitude oblique-wave breakdown. . . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Contours of the instantaneous 𝑢-velocity for high amplitude obliquewave breakdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Oblique-wave inlet boundary condition with and without random noise. 44 3.9 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for low amplitude oblique-wave breakdown with and without random noise. . . . . . . 45 xii 3.10 3.11 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 6.1 Contour plots of the 𝑢- and 𝑣 velocities of the oblique wave, with and without random noise. . . . . . . . . . . . . . . . . . . . . . . . . . 45 Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for K-type transition 47 Convergence of the objective function for greedy and heuristic parameter selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Convergence of the objective function, split into upstream- and downstreamgoing components, for greedy and heuristic parameter selection. . . . 53 Recursion parameters plotted against spectrum for heuristic and greedy recursion parameter selection with 𝑁 𝛽 = 20. . . . . . . . . . . . . . . 54 Comparison of OWNS spectrum with heuristic parameter selection for 𝜂 = 0 and 𝜂 = 1000. According to Briggs’ criterion, I (𝛼) < 0 for large values of 𝜂 indicates a downstream-going mode. For 𝜂 = 0, we wish to have I (𝛼) ≥ 0 for the march to be stable. We achieve a stable march using OWNS-P, but not for OWNS-R. . . . . . . . . . . 55 Error convergence as a function of 𝑁 𝛽 for OWNS-P and OWNS-R for the 2D low-speed boundary-layer flow. . . . . . . . . . . . . . . . 56 𝑗 For OWNS-R, the error in computing 𝛽∗ increases with increasing 𝑁 𝛽 . 57 Error convergence as a function of 𝑁 𝛽 for OWNS-P and OWNS-R for the low-speed oblique wave case. . . . . . . . . . . . . . . . . . . 57 NOWNS for oblique-wave breakdown, comparing greedy and heuristic recursion parameter selection with 𝑁 𝛽 = 6. Although NOWNS with greedy selection is sufficiently converged, heuristic selection is not, and the solution blows-up. . . . . . . . . . . . . . . . . . . . . . 59 𝑢′-velocity amplitudes for K-type transition, compared to the DNS of Sayadi et al. (2013). . . . . . . . . . . . . . . . . . . . . . . . . . 62 𝑢-velocity contours at 𝑦/𝛿inlet = 0.6 for K-type transition, demonstrating the aligned Λ-vortex structure. . . . . . . . . . . . . . . . . 63 𝑢′-velocity amplitudes for H-type transition, compared to the DNS of Sayadi et al. (2013). . . . . . . . . . . . . . . . . . . . . . . . . . 63 𝑢-velocity contours at 𝑦/𝛿inlet = 0.6 for H-type transition, demonstrating the staggered Λ-vortex structure. . . . . . . . . . . . . . . . 64 Skin friction coefficient for K- and H-type transition, compared to the DNS of Sayadi et al. (2013). . . . . . . . . . . . . . . . . . . . . . . 65 Convergence of the error for greedy and heuristic parameter selection for 2D high-speed boundary-layer flow. . . . . . . . . . . . . . . . . 71 xiii 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 Temperature disturbance amplitudes for 2D nonlinear evolution of Mack’s second mode for a Mach 4.5 flat-plate boundary-layer flow, computed using NOWNS and compared to linear OWNS. . . . . . . 75 Comparison of temperature disturbance amplitudes for NOWNS calculation linearized about base flow and corrected mean flow for 2D nonlinear evolution of Mack’s second mode for a Mach 4.5 flat-plate boundary-layer flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Temperature contours for first and second harmonic of Mack’s second mode with heuristic and greedy recursion parameter selection. . . . . 76 Temperature profiles at domain outlet for first and second harmonics of Mack’s second mode with heuristic and greedy recursion parameter selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Temperature disturbance amplitudes for small amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Temperature disturbance amplitudes for large amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. 78 Skin friction coefficient for large amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. . . . . . 79 Temperature disturbance amplitudes for small amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundarylayer flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Temperature disturbance amplitudes for large amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundary-layer flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Skin friction coefficient for large amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundary-layer flow. . . . 81 Contours of 𝑢-velocity at 𝑦 = 0.00189m for oblique breakdown of Mack’s second mode, exhibiting streaks towards the end of the domain 81 Temperature contours in the 𝑦 − 𝑧 plane at the domain outlet for oblique breakdown of Mack’s second mode. . . . . . . . . . . . . . . 82 Pressure contours at 𝑧 = 0 for oblique breakdown of Mack’s second mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Pressure contours at 𝑦 = 0 for oblique breakdown of Mack’s second mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 xiv G.1 𝑢-velocity amplitudes with and without streamwise diffusion terms for the 2D evolution of the TS wave. . . . . . . . . . . . . . . . . . . 117 G.2 2D validation case with and without streamwise pressure gradient for zero-frequency modes. . . . . . . . . . . . . . . . . . . . . . . . . . 117 G.3 K-type transition case with and without streamwise pressure gradient for zero-frequency modes. . . . . . . . . . . . . . . . . . . . . . . . 118 G.4 Comparison of solution procedures for the nonlinear system of equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 G.5 Amplitude of 𝑢′ v.s. streamwise coordinate, Re𝑥 , for fundamental breakdown. Compare linearizing NOWNS about 𝒒¯ vs. 𝒒¯ + 𝒒′00 . . . . . 121 I.1 Spectrum of TS wave at Re𝑥 = 1.6 × 105 for different values of 𝜂. . . 126 J.1 Norm of optimal inlet boundary condition, compared to random and TS wave inlet boundary conditions for 𝐹 = 50 × 10−6 , between branches I and II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 J.2 Optimal inlet boundary condition for 𝐹 = 50 × 10−6 , compared to TS wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 J.3 Optimal vs. TS gain for a range of frequencies. . . . . . . . . . . . . 132 J.4 Difference in optimal and TS gains for a range of frequencies. . . . . 132 xv LIST OF TABLES Number Page 3.1 Three approaches to parabolizing the zero-frequency modes . . . . . 35 4.1 Speed-up achieved by using greedy parameter selection with OWNSP (P-G) and OWNS-R (R-G) relative to heuristic parameter selection with OWNS-P (P-H) for different relative errors in the 𝑁-factor for the 3D low-speed boundary-layer flow. . . . . . . . . . . . . . . . . 59 6.1 Speed-up achieved by using greedy parameter selection with OWNSP (P-G) and OWNS-R (R-G) relative to heuristic parameter selection with OWNS-P (P-H) for different relative errors in the 𝑁-factor for the 2D high-speed boundary-layer flow. . . . . . . . . . . . . . . . . 71 H.1 Recursion parameter sets for subsonic boundary layer flows. . . . . . 124 H.2 Recursion parameter sets for subsonic boundary layer flows for zerofrequency modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 xvi NOMENCLATURE (𝑥, 𝑦, 𝑧) 𝜈 (𝑢, 𝑣, 𝑤) 𝑝 𝜌 𝑇 𝜇 𝑎 Ma 𝑐𝑝 𝛾 𝑘 = = = = = = = = = = = = 𝛿0∗ streamwise, wall-normal, and spanwise coordinates specific volume velocity (streamwise, wall-normal, and spanwise) pressure density, where 𝜌 = 1/𝜈 temperature dynamic viscosity sound speed Mach number specific heat capacity at constant pressure heat capacity ratio thermal conductivity √︃ ∗ ∗ 𝑥 𝜇 ∗ = inlet Blasius length scale, defined as 𝛿0 = 𝜌∗0 𝑈∞∗ 𝑅𝑒 = Reynolds number in terms of 𝛿0∗ , defined as 𝑅𝑒 = 𝑃𝑟 𝜔 𝐹 = = = Prandtl number, defined as 𝑃𝑟 = 𝑘 ∗ 𝑝 temporal frequency 𝜔∗ 𝜇∗ dimensionless temporal frequency, defined as 𝐹 = 𝜌∗ 𝑈∞∗2 𝛽 𝑏 = = spanwise wavenumber 𝛽∗ 𝜇∗ dimensionless spanwise wavenumber, defined as 𝑏 = 𝜌∗ 𝑈∞∗ 𝛼 𝑅𝑒 𝑥 = = streamwise wavenumber (complex-valued) 𝜌∗ 𝑈 ∗ 𝑥 ∗ Reynolds number in terms of 𝑥, defined as 𝑅𝑒 𝑥 = ∞𝜇∗∞ ∞ ∞ 𝜇∗ 𝑐∗ ∗ 𝑎 ∗ 𝛿∗ 𝜌∞ ∞ 0 ∗ 𝜇∞ ∞ ∞ 𝑀 = 𝑁 = (𝑚, 𝑛) = Subscripts ∞ = 𝑚𝑛 = +−0 = Superscripts ∗ = ′ = ‡ = ∞ ∞ ∞ denotes the number of temporal Fourier modes denotes the number of spanwise Fourier modes denotes Fourier mode with frequency 𝑚𝜔 and wavenumber 𝑛𝛽 free-stream value a quantity associated with frequency 𝑚𝜔 and wavenumber 𝑛𝛽 plus, minus, and zero characteristics dimensional quantity a disturbance to the equilibrium solution a matrix or a vector containing the OWNS auxiliary values xvii Other symbols ¯ (·) = a quantity associated with the equilibrium solution (·) = a complex conjugate value ˆ (·) = a quantity in Fourier space Abbreviations MFD = mean flow distortion BC = boundary condition PSE = Parabolized Stability Equations NPSE = Nonlinear Parabolized Stability Equations NRBC = non-reflecting boundary condition OWNS = One-Way Navier-Stokes NOWNS = Nonlinear One-Way Navier-Stokes LST = Locally-parallel linear stability theory GSA = linear global stability analysis TS = Tollmien-Schlichting, referring to the Tollmien-Schlichting wave MM = referrings to Mack’s second mode 1 Chapter 1 INTRODUCTION 1.1 Boundary-layer flows In fluid flows, the boundary layer is a thin region near a solid surface, such as an airplane wing, where viscous effects dominate.A flow is laminar if it flows in parallel layers with no disruption between those layers, while it is turbulent if it is characterized by abrupt changes to the flow. In aerospace design, delaying or preventing laminar-turbulent transition is desirable, as turbulence increases skin friction and heating. Despite extensive theoretical, experimental, and numerical studies, controlling transition remains a challenge. 1.2 Governing equations For low-speed boundary-layer flows, we formulate the governing equations in terms of the specific volume, 𝜈, the velocity, 𝒖, and the pressure, 𝑝. Then, the nondimensional, compressible Navier-Stokes equations for an ideal gas can be written as 𝐷𝜈 − 𝜈(∇ · 𝒖) = 0, 𝐷𝑡 𝐷𝒖 1 + 𝜈∇𝑝 = 𝜈∇ · 𝜏, 𝐷𝑡 𝑅𝑒 𝐷𝑝 1 + 𝛾 𝑝(∇ · 𝒖) = (𝛾 − 1) [(∇𝒖) : 𝜏] 𝐷𝑡 𝑅𝑒 𝛾 (𝜈∇2 𝑝 + 2∇𝜈 · ∇𝑝 + 𝑝∇2 𝜈), + 𝑃𝑟 𝑅𝑒 (1.1a) (1.1b) (1.1c) (1.1d) for the stress tensor   2  𝜏 = 𝜇 ∇𝒖 + (∇𝒖)𝑇 − 𝜇 − 𝜅 (∇ · 𝒖)𝐼, 3 (1.1e) where 𝜇 is the dynamic viscosity and 𝜅 is the bulk viscosity. We take the bulk viscosity to be zero (𝜅 = 0) and we introduce the Reynolds number, the Prandtl number, and the Blasius length scale √︄ ∗ 𝜇∗ ∗ 𝛿∗ 𝑎 ∗ ∗ 𝑥∗ 𝑐 𝜇∞ 𝜌∞ 𝑝,∞ ∞ ∞ 0 0 ∗ 𝑅𝑒 = , 𝑃𝑟 = , 𝛿 = 0 ∗ ∗ ∗ 𝑈∗ . 𝜇∞ 𝑘∞ 𝜌∞ ∞ ∗ denotes the free-stream specific volume, 𝜇 ∗ the free-stream dynamic Here 𝜈∞ ∞ ∗ ∗ viscosity, 𝑘 ∞ the free-stream thermal diffusivity, 𝑐 𝑝,∞ the free-stream specific heat 2 capacity at constant pressure, and 𝑎 ∗∞ the free-stream sound speed, while 𝑥, 𝑦, 𝑧, correspond to the streamwise, transverse and spanwise directions, respectively. To ∗ , in avoid confusion with the kinematic viscosity, we use the free-stream density, 𝜌∞ the definition of our dimensionless variables. At low Mach numbers, temperature fluctuations are minimal, so we take the fluid properties (𝜇, 𝑘, and 𝑐 𝑝 ) to be constant. Throughout this work, we also use the following dimensionless quantities: 𝑅𝑒 𝑥 = ∗ 𝑈∗ 𝑥∗ 𝜌∞ ∞ , ∗ 𝜇∞ 𝑦= 𝑦∗ , 𝛿0∗ 𝐹= ∗ 𝜔∗ 𝜇∞ , ∗ 𝑈 ∗2 𝜌∞ ∞ 𝑏= ∗ 𝛽∗ 𝜇∞ ∗ 𝑈∗ , 𝜌∞ ∞ where 𝑅𝑒 𝑥 is the streamwise coordinate, 𝐹 is the temporal frequency, and 𝑏 is the spanwise wavenumber. 1.3 Laminar-turbulent transition in boundary-layer flows We briefly review the developments in boundary-layer transition prediction and introduce key definitions and concepts relevant to the transition process. 1.3.1 A brief history of laminar-turbulent transition Orr (1907) and Sommerfeld (1908) independently formulated a linear eigenvalue problem, now known as the Orr–Sommerfeld (OS) equation, to study viscous instabilities in two-dimensional (2D) parallel flows. Meanwhile, Prandtl (1904) hypothesized the existence of the boundary layer and employed scaling arguments to simplify the Navier–Stokes equations within this region. Building on Prandtl’s theory, Blasius (1908) derived a similarity solution for the steady boundary layer that develops on a semi-infinite flat plate aligned with a constant unidirectional flow. Later, Tollmien (1929) and Schlichting (1933) independently solved the OS equation for the Blasius boundary-layer profile. Their analyses revealed the presence of at most one unstable mode, now known as the Tollmien–Schlichting (TS) wave, and enabled the prediction of the critical Reynolds number at which the TS wave becomes unstable. Early experiments attributed transition to high levels of freestream turbulence rather than exponential growth of the TS wave, but Schubauer and Skramstad (1947) confirmed the existence of TS waves by using damping screens to ensure low levels of freestream turbulence, showing excellent agreement with theoretical predictions. Schubauer and Skramstad initially investigated naturally occurring disturbances from small freestream fluctuations, later introducing a vibrating ribbon to control their frequency and amplitude. In a similar setup, Klebanoff et al. (1962) placed evenly spaced tape strips along the plate’s leading edge to generate oblique waves 3 at the same frequency as the fundamental TS wave. Although initially weaker, these oblique waves interact with the TS wave, amplifying both themselves and their higher harmonics. This interaction leads to the formation of aligned Λ-vortex structures, characteristic of the transition path now known as the fundamental (or Ktype, for Klebanoff) path. Similar experiments by Kachanov and Levchenko (1984) identified the subharmonic path in which the oblique waves have half the temporal frequency of the fundamental TS wave, while the Λ-vortex structure is staggered and transition is delayed relative to the fundamental path. Herbert (1988) developed a secondary stability theory, using Floquet analysis, to study these transition paths, so the subharmonic path is also called the H-type path after Herbert. For high-speed boundary-layer flows, Mack (1984) identified unstable eigenmodes now known as Mack modes. Mack’s first mode is the high-speed analog of the TS wave, as both are viscous in nature. However, the TS wave is most unstable for two-dimensional disturbances (𝛽 = 0), while Mack’s first mode is most unstable for oblique disturbances (𝛽 ≠ 0). Mack’s first mode is the primary disturbance leading to transition for Mach numbers up to 4.5, while the higher Mack modes, which are non-viscous, are the primary instabilities leading to transition for Mach numbers higher than 4.5. 1.3.2 Notions of stability Disturbances to the flow lead to flow instabilities, which can be classified either as absolute or convective instabilities according to the definitions of Huerre and Monkewitz (1985): Definition 1.3.1 (absolute instability) Consider a disturbance introduced at 𝒙 = 0 and 𝑡 = 0, then it is an absolute instability if its amplitude grows as 𝑡 → ∞ for all 𝒙. Definition 1.3.2 (convective instability) Consider a disturbance introduced at 𝒙 = 0 and 𝑡 = 0, then it is a convective instability if it is swept away from 𝒙 = 0 such that the system returns to its unperturbed state as 𝑡 → ∞. Figure 1.1 compares absolute and convective instabilities in the 𝑥 − 𝑡-plane using Figure 7.6 from Schmid and Henningson (2001). Note that the TS wave, and other disturbances to boundary-layer flows, are convective in nature, so this work will focus on these types of instabilities. 4 Figure 1.1: Sketch of convective and absolute stability from Figure 7.6 in Schmid and Henningson (2001): (a) 𝑥 − 𝑡 plane diagram of an absolutely unstable disturbance; (b) 𝑥 − 𝑡 plane diagram of a convectively unstable disturbance. 1.3.3 Types of disturbances We divide disturbances into the following three categories: (i) modal, (ii) multimodal, and (iii) non-modal. Definition 1.3.3 (modal disturbance) For a specified frequency, a disturbance that propagates in the streamwise direction with a well-defined streamwise wavelength. Definition 1.3.4 (multi-modal instabilities) For a specified frequency, there are multiple modal disturbances, each evolving with different streamwise wavelengths. Definition 1.3.5 (non-modal instabilities) The linearized Navier-Stokes operator is non-normal so that its eigenvectors are not mutually orthogonal. Therefore, the superposition of linearly stable modes can temporarily increase in magnitude, even as each mode decays, leading to transient growth. The TS wave is a modal disturbance, as illustrated in Figure 1.2, which shows contours of the streamwise (𝑢) and wall-normal (𝑣) velocity disturbances. These contours demonstrate that the TS wave evolves with a well-defined streamwise wavelength, characteristic of a modal instability. Low-speed boundary-layer flows typically experience only modal disturbances, while multi-modal disturbances are characteristic of high-speed flows (Fedorov, 2011). Figure 1.3 uses Figure 4.1 from Schmid and Henningson (2001) to illustrate transient growth. Both vectors Φ1 and Φ2 are decaying, but their superposition temporarily grows in time due to the mutual non-orthogonality. Appendix J investigates transient growth for low-speed boundary-layer flows. 5 (a) 𝑢-velocity contour (b) 𝑣-velocity contour Figure 1.2: Contours of the 𝑢- and 𝑣-velocity disturbance contours for the 2D TS wave, demonstrating that it evolves with a well-defined wavelength that is constant in the streamwise direction. Figure 1.3: Figure 4.1 from Schmid and Henningson (2001) illustrating transient growth due to non-orthogonal superposition of two decaying vectors. 1.3.4 Paths to transition for compressible boundary-layer flows Initially disturbances enter the boundary-layer flow through a process known as receptivity, in which energy from the free stream enters and excites instability waves inside the boundary layer (Morkovin, 1969). Following receptivity, there are multiple paths to transition, as discussed by Morkovin et al. (1994). Figure 1.4 reproduces Figure 1 from Fedorov (2011), adapted from Morkovin et al. (1994), which presents the different transition paths. We summarize them as follows: • Modal path: if the initial disturbances are sufficiently small, then modal growth is the dominant growth mechanism, and the disturbances grow exponentially before reaching triggering secondary instabilities and reaching the nonlinear 6 Figure 1.4: Figure 1 from Fedorov (2011), adapted from Morkovin et al. (1994), showing transition paths in boundary-layer flows. regime. • Transient path: if the modal disturbances are initially linearly stable, then non-orthogonality can transient growth to increase disturbance amplitude prior to reaching the region of the flow where the modal disturbances are linearly unstable. If the transient growth is sufficiently large, then secondary instabilities can be triggered without exponential growth. • Bypass path: if the free-stream disturbances are sufficiently large, or if the transient growth is sufficiently large, then bypass transition can be triggered, where the nonlinear regime is reached directly (without exponential growth and secondary instabilities). This work will focus on the modal path to transition, while future work could investigate transient growth. 1.4 Methods for hydrodynamic stability analysis In hydrodynamic stabiltiy analysis we study the evolution of a time-varying distur¯ Therefore, we decompose the bance, 𝒒′, to a time-invariant equilibrium solution, 𝒒. 7 flow into terms associated with 𝒒¯ and 𝒒′, as described in appendix A, yielding the stability equations (A.6), for the vector of primitive variables 𝒒 = (𝜈, 𝑢, 𝑣, 𝑤, 𝑝). 1.4.1 Locally-parallel linear stability theory The theory developed by Tollmien and Schlichting is now known as locally-parallel linear stability theory (LST). In LST, the flow is assumed locally parallel, meaning the base flow does not vary in the streamwise direction (𝜕𝑥 𝒒¯ = 𝜕𝑥𝑥 𝒒¯ = 0), and the wall-normal velocity vanishes (𝑣¯ = 0), so the boundary-layer thickness remains constant. We consider the spatial stability problem with the following wave-like ansatz for the disturbance:  ∫ 𝑥  ′ ˆ 𝒒 (𝑥, 𝑦, 𝑧, 𝑡) = 𝒒(𝑦) exp 𝑖 𝛼( 𝑥) ¯ 𝑑 𝑥¯ + 𝑖(𝛽𝑧 − 𝜔𝑡) , (1.2) 𝑥0 where 𝒒ˆ is the eigenfunction, 𝜔 the temporal frequency, 𝛼 the streamwise wavenumber, 𝛽 the spanwise wavenumber, and 𝑥 the streamwise coordinate. While stability studies of low-speed boundary layers often express the streamwise location in terms √ of 𝑅 = 𝑅𝑒 𝑥 , we use 𝑅𝑒 𝑥 throughout this work. Tollmien and Schlichting conducted temporal stability analysis, which assumes that disturbances grow in time. However, disturbances to boundary-layer flows grow in the streamwise direction, so later studies instead solved the spatial stability problem. In this approach, the streamwise coordinate 𝑥, spanwise wavenumber 𝛽, and temporal frequency 𝜔 are specified, with 𝛽 and 𝜔 real-valued. The complex ˆ streamwise wavenumber 𝛼 and eigenfunction 𝒒(𝑥) must then satisfy ˆ ˆ 𝑖𝛼 𝒒(𝑥) = 𝐿(𝜔, 𝛽, 𝑥) 𝒒(𝑥), (1.3) where 𝐿 (𝜔, 𝛽, 𝑥) is the linear operator governing spatial stability. Disturbances are unstable if 𝛼𝑖 < 0, stable if 𝛼𝑖 > 0, and neutrally stable if 𝛼𝑖 = 0. The TS wave exhibits two neutral stability points, termed branches I and II, and is stable except within 𝑥 ∈ (𝑥 I , 𝑥II ). The growth of the TS wave amplitude is given by   ∫ 𝑥 𝐴(𝑥) =− 𝛼𝑖 ( 𝑥) ¯ 𝑑 𝑥, ¯ (1.4) ln 𝐴0 𝑥0 where 𝐴(𝑥) is the disturbance amplitude and 𝐴0 is the disturbance amplitude at branch I. Building on Tollmien and Schlichting’s initial work for incompressible flows, Mack (1984) extended the theory to hypersonic boundary layers. We further note that for H- and K-type transition, although LST can be used to predict the growth of the TS and oblique waves separately, the growth of the oblique wave is affected substantially by the TS wave. Therefore, Herbert (1988) introduced a secondary stability theory, using Floquet analysis, to account for this effect. 8 1.4.2 Linear global stability analysis LST is computationally efficient, but neglects history and non-parallel effects, which can be included using linear global stability analysis (GSA). In GSA, we specify an inlet disturbance at 𝑥 In and/or a forcing function for 𝑥 ∈ [𝑥 In , 𝑥Out ], and solve for the resulting disturbance. However, this approach entails high computational and memory costs because a large linear system of equation, O (𝑁𝑥 × 𝑁 𝑦 × 𝑁 𝑧 ) must be solved. Here 𝑁𝑥 , 𝑁 𝑦 , and 𝑁 𝑧 denote the resolution in 𝑥, 𝑦, and 𝑧, respectively. Moreover, the disturbances associated with high-speed boundary-layer flows evolve with short wavelength over long streamwise domains, so that global methods are particularly expensive for these flows (Fedorov, 2011). 1.4.3 The Parabolized Stability Equations The linear parabolized stability equations (PSE) also include history and non-parallel effects, and entail much lower computational and memory costs than GSA (Bertolotti et al., 1992). PSE uses a wavelike ansatz ′ ˆ 𝑦)𝑒 𝒒 (𝑥, 𝑦, 𝑧, 𝑡) = 𝒒(𝑥, ∫𝑥 𝑖 𝑥 𝛼( 𝑥)𝑑 ¯ 𝑥¯ 𝑖(𝛽𝑧−𝜔𝑡) 0 𝑒 , so that  ∫𝑥 𝜕𝒒′  𝜕 𝒒ˆ 𝑖 𝛼( 𝑥)𝑑 ¯ 𝑥¯ 𝑖(𝛽𝑧−𝜔𝑡) = + 𝑖𝛼 𝒒ˆ 𝑒 𝑥0 𝑒 , 𝜕𝑥 𝜕𝑥  ∫𝑥 𝜕 𝒒ˆ 𝜕 2 𝒒′  𝜕𝛼 𝑖 𝛼( 𝑥)𝑑 ¯ 𝑥¯ 𝑖(𝛽𝑧−𝜔𝑡) 2 = 𝑖 𝒒ˆ + 2𝑖𝛼 − 𝛼 𝒒ˆ 𝑒 𝑥0 𝑒 , 2 𝜕𝑥 𝜕𝑥 𝜕𝑥 2 = 0. In addition, nonlinear analyses can be ˆ where we have assumed that 𝜕 2 𝒒/𝜕𝑥 performed using the nonlinear PSE (NPSE). PSE has proven effective for boundary-layer flows (Bertolotti et al., 1992; Herbert, 1997; Paredes et al., 2015; Chang et al., 1993; Chang and Malik, 1994; Paredes et al., 2016a; Joslin et al., 1993) and mixing layers (Day, 1999; Day et al., 2001). However, to achieve a stable march, PSE relies on ad hoc regularization techniques (Li and Malik, 1996; Li and Malik, 1997; Broadhurst and Sherwin, 2008) to suppress (rather than eliminate) upstream-propagating waves. This introduces a minimum step size requirement for numerical stability, which limits the streamwise resolution. Furthermore, Towne et al. (2019) demonstrated that PSE is generally not accurate for multi-modal disturbances because it can track only the most dominant disturbance, and that the other disturbances are neither necessarily damped away, nor properly evolved. While some studies have reported success when applying PSE to non-modal disturbances (transient growth) (De Tullio et al., 2013; Paredes et al., 2016a; Hack 9 and Moin, 2017), others have documented limitations and challenges (Cheung and Lele, 2007; Rodríguez et al., 2018). Although PSE has been applied to hypersonic boundary-layer flows (Pruett et al., 1995; Paredes et al., 2016b; Lakebrink et al., 2017), such flows often feature multiple discrete modes and disturbance branches that can synchronize, which proves challenging for PSE (Towne et al., 2019; Fedorov, 2011). 1.4.4 The One-Way Navier-Stokes equations Towne and Colonius (2015) developed the linear One-Way Navier–Stokes (OWNS) equations, which use recursive filtering to construct efficient and well-posed one-way approximations to linear hyperbolic systems. This original formulation, now known as OWNS outflow (OWNS-O), is based on non-reflective boundary conditions for outflow boundaries and is limited to homogeneous (unforced) equations. To address this, Towne et al. (2022) introduced OWNS projection (OWNS-P), which extends the method to inhomogeneous (forced) equations. Subsequently, Zhu and Towne (2023) developed OWNS recursive (OWNS-R), offering a more computationally efficient approach for inhomogeneous equations compared to OWNS-P. OWNS overcomes several limitations of the Parabolized Stability Equations (PSE), particularly by correctly removing all upstream-traveling waves. As a result, it avoids the minimum step-size requirement and accurately captures non-modal and multi-modal instabilities (Towne et al., 2019). The method has been applied to low-speed boundary-layer flows by Rigas et al. (2017a) and extended to high-speed boundary-layer flows by Kamal et al. (2020). 1.4.5 Empirical methods for transition prediction The linear methods discussed above can be combined with empirical correlations to predict transition. For example, the 𝑒 𝑁 method (Smith et al., 1956; Van Ingen et al., 1956) computes the 𝑁-factor   𝐴(𝑥) 𝑁 = max ln , (1.5) 𝑥 𝐴0 and predicts transition for 𝑁 > 9. While effective for low-speed flows, the method faces challenges for high-speed flows (Fedorov, 2011). 1.5 Frequency domain analysis Time-marching, with finite difference discretization in the spanwise direction, allows all frequencies and wavenumbers to be considered simultaneously, as long as they are 10 resolved properly using a sufficiently refined grid and time step. If the disturbances are periodic, then they can be represented using a Fourier series 𝒒′ (𝑥, 𝑦, 𝑧, 𝑡) = 𝑀 𝑁 ∑︁ ∑︁ 𝒒ˆ 𝑚𝑛 (𝑥, 𝑦, 𝑧)𝑒𝑖(𝛽𝑧−𝑚𝜔𝑡) , (1.6) 𝑚=−𝑀 𝑛=−𝑁 where 𝜔 and 𝛽 are the fundamental frequency and fundamental wavenumber, respectively. We do not discuss in detail conditions under which Fourier series converge, but note that increasing 𝑀 and 𝑁 will generally lead to a better approximation. In flight and wind tunnel tests, multiple modes and frequencies are excited simultaneously, which broadband forcing can replicate. However, this complicates analysis due to overlapping transition scenarios (e.g., oblique-wave breakdown, fundamental, or subharmonic transition). Although not representative of experiments, selectively forcing dominant instabilities at specific frequencies and spanwise wavenumbers enables focused study of individual transitions. While NOWNS could be applied to study periodic broadband disturbances, this would require many Fourier components to converge. Instead, NOWNS is better suited for selective forcing of dominant frequencies and wavenumbers. We demonstrate throughout this work that NOWNS can rapidly and accurately simulate such transition scenarios. We further note that NOWNS is well-suited to do so, compared to time domain analysis, since time domain analysis must time march through a transient to reach to a periodic state, which require small time-steps and many periods of the fundamental frequency to achieve. 1.6 Outline of thesis The goal of this work is to develop the nonlinear OWNS (NOWNS) procedure and use it to predict laminar-turbulent transition for boundary-layer flows. Chapter 2 reviews OWNS-P and OWNS-R, while Chapter 3 extends OWNS-P to include nonlinear effects and validates it for low-speed boundary-layer flows. Chapter 5 applies NOWNS to transition prediction for low-speed boundary-layer flows. Chapter 4 develops a greedy algorithm for recursion parameter selection, which is demonstrated for high-speed boundary-layer flows in Chapter 6. We make recommendations for future work and concluding remarks in chapter 7. 11 Chapter 2 ONE-WAY MARCHING OF LINEAR HYPERBOLIC EQUATIONS This section reviews the OWNS-P and OWNS-R formulations in preparation for the exposition of NOWNS in Chapter 3, and for the greedy recursion parameter selection algorithm in Chapter 4. 2.1 Introduction Many physical phenomena, such as wave propagation, can be modeled by (linear) systems of hyperbolic partial differential equations (PDEs), solvable either as timedomain initial boundary value problems or frequency-domain elliptic boundary value problems. When wave propagation is predominantly unidirectional, one-way (parabolic) wave equations can be used to evolve the downstream-going solution. For accuracy and well-posedness, the one-way equation must retain the downstreamgoing waves, while not supporting any of the upstream-going waves. One-way wave equations can be obtained by factoring the dispersion relation in Fourier–Laplace space into upstream- and downstream-going factors, and then retaining only the downstream-going branch. Since the resulting equation contains a square root of the Fourier-Laplace variables, the transformation back to physical space results in a nonlocal integro-differential equation, which can be localized using rational approximations of the square root (Lee et al., 2000). Although these methods are accurate, well-posed, and efficient for simple wave equations (Trefethen and Halpern, 1986; Halpern and Trefethen, 1988) such as the equations for geophysical migration of seismic waves (Claerbout, 1976, 1985) and underwater acoustics (Collins, 1989; Jensen et al., 1995), they can only be applied in cases where the eigenvalues can be determined analytically, since the dispersion relation must be factored analytically. Methods for one-way marching that do not depend on this factorization have been developed by Guddati (2006) for acoustic and elastic wave equations, and by Towne and Colonius (2015) for general hyperbolic systems. To avoid factoring the dispersion relation, the method of Towne and Colonius first discretizes the transverse directions, yielding a semi-discrete ordinary differential equation (ODE) in the marching direction (2.7). An eigendecomposition of the 12 discretized operator, 𝑀 in (2.7), then enables classification of its eigenvectors as upstream- or downstream-going via Briggs’ criterion (Briggs, 1964), allowing construction of a one-way equation. However, to avoid taking the eigendecomposition, a recursive filter based on work by Givoli and Neta (2003) and Hagstrom and Warburton (2004) for non-reflective boundary conditions (NRBCs) at “outflow” boundaries is used to construct an approximation to the one-way equation. The framework has been extensively refined and generalized for the linearized Navier-Stokes equations (Towne et al., 2022; Kamal et al., 2020, 2021, 2022; Rigas et al., 2017a). For simplicity, we will refer to the general framework as the One-Way Navier-Stokes (OWNS) equations, but note that many of the techniques are broadly applicable to other hyperbolic systems. The OWNS projection method (OWNS-P) generalizes the approach to handle inhomogeneous (forced) equations (Towne et al., 2022). Both OWNS-O and OWNS-P involve high computational costs compared to PSE, which motivated the development of the more efficient OWNS recursive method (OWNS-R) (Zhu and Towne, 2023). We also note that a similar method, which uses recursive filters to approximate the eigenvectors of 𝑀 for wave propagation in complex media, has been developed by Rudel et al. (2022). 2.2 OWNS equations Here we will develop the OWNS approach for a general linear system of hyperbolic equations. First we note that a hyperbolic system is defined as in Definition 2.2.1. Definition 2.2.1 (Hyperbolic system of first-order equations) Consider the following system of first-order partial differential equations for 𝑁 unknown functions 𝒒′ = (𝑞′1 , . . . , 𝑞′𝑁 ), 𝒒 = 𝒒(𝑥, 𝒚, 𝑡) where 𝒚 ∈ R𝑑−1 and 𝝉 𝑗 ∈ C 1 (R𝑁 , R𝑁 ) are once continuously differentiable (nonlinear) functions: 𝑑 𝜕𝒒′ 𝜕 𝑗 ′ ∑︁ 𝜕 𝑗 ′ + 𝝉 (𝒒 ) + 𝝉 (𝒒 ) = 0. 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝑗 𝑗=2 (2.1) We introduce the Jacobians  𝜕𝜏1𝑗  𝜕𝑞 ′  1  . 𝐵 𝑗 =  ..  𝑗  𝜕𝜏𝑁  𝜕𝑞 ′  1 𝑗 ··· .. . 𝜕𝜏1  𝜕𝑞 ′𝑁  ..  .  , 𝑗 𝜕𝜏  · · · 𝜕𝑞𝑁′  𝑁 𝑗 = 1, . . . , 𝑁, 𝐴 = 𝐵1 , (2.2) 13 and we say that (2.1) is hyperbolic if for all 𝛾1 , . . . , 𝛾𝑑 ∈ R the matrix 𝛾1 𝐴 + 𝛾2 𝐵 2 + · · · + 𝛾 𝑑 𝐵 𝑑 has only real eigenvalues and is diagonalizable. Consider the constant-coefficient hyperbolic system of equations 𝑑 𝜕𝒒′ 𝜕𝒒′ ∑︁ 𝜕𝒒′ 𝐵𝑗 +𝐴 + + 𝐶𝒒′ = 𝒇 , 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝑗 𝑗=2 (2.3) in 𝑑 spatial-dimensions for 𝐴, 𝐵 𝑗 , 𝐶 ∈ R𝑁×𝑁 with 𝐴 and 𝐵 𝑗 defined as in (2.2), with forcing function 𝒇 ∈ R𝑁 where 𝒇 = 𝒇 (𝑥, 𝒚, 𝑡). Remark 2.2.2 System (2.3) is a general first-order linear hyperbolic system of equations. We delay writing explicit expressions for 𝐴, 𝐵 𝑗 and 𝐶 until Chapter 3 where we specialize the approach to the Navier-Stokes equations, for the operators presented in Appendix A. Applying finite differences (or other discretizations) in (𝑦 2 , . . . , 𝑦 𝑑 ) yields 𝑑 𝜕𝒒′ 𝜕𝒒′ ∑︁ +𝐴 + 𝐵 𝑗 𝐷 𝑗 𝒒′ + 𝐶𝒒′ = 𝒇 , 𝜕𝑡 𝜕𝑥 𝑗=2 (2.4) for the difference operator 𝐷 𝑗 . Since (2.3) is hyperbolic, we know that 𝐴 is diagonalizable with real eigenvalues 𝐴˜ = 𝑇 𝐴𝑇 −1 and eigenvectors 𝑇 −1 , which we use to define the characteristic variables 𝝓 = 𝑇 𝒒′. Next we write 𝑑 𝜕𝝓 ˜ 𝜕𝝓 ∑︁ ˜ 𝜕𝝓 ˜ = 𝒇𝜙 , +𝐴 + 𝐵𝑗 + 𝐶𝝓 𝜕𝑡 𝜕𝑥 𝑗=2 𝜕𝑦 𝑗 (2.5) where 𝐵˜ 𝑗 = 𝑇 𝐵 𝑗 𝑇 −1 for 𝑗 = 2, . . . , 𝑑, 𝐶˜ = 𝑇𝐶𝑇 −1 , and 𝒇𝜙 = 𝑇 𝒇 . We assume that 𝐴˜ is non-singular (we treat the singular case in Appendix C) so that " # 𝐴 0 ++ 𝐴˜ = (2.6) 0 𝐴−− for 𝑁+ × 𝑁+ diagonal matrix 𝐴++ with 𝐴++ > 0 and 𝑁− × 𝑁− diagonal matrix 𝐴−− < 0, where 𝑁+ + 𝑁− = 𝑁. Taking a Laplace transform in time, and Fourier transforms in space for 𝑗 = 2, . . . , 𝑑, we obtain 𝑑 𝝓ˆ = −𝑀 (𝑠) 𝝓ˆ + 𝒇ˆ𝜙 , 𝑑𝑥 (2.7) 14 where 𝝓ˆ and 𝒇 represent 𝝓 and 𝒇𝜙 , respectively, in the frequency domain, while 𝑑   ∑︁ 𝑖𝜔 𝑗 𝐵˜ 𝑗 + 𝐶˜ . 𝑀 (𝑠) = 𝐴˜ −1 𝑠𝐼 + (2.8) 𝑗=2 For the problems considered in this work, the matrix 𝑀 (𝑠) is diagonalizable, which simplifies the exposition of the OWNS methodology. Therefore, we assume 𝑀 (𝑠) to be diagonalizable and note that the defective case is handled by Towne and Colonius (2015). We use Briggs’ criterion (Definition 2.2.3) to classify eigen-vectors as upstream- or downstream-going, while Proposition 2.2.4 tells us that 𝑀 (𝑠) has precisely 𝑁+ and 𝑁− downstream- and upstream-going eigenvalues. Definition 2.2.3 (Briggs’ criterion) Consider a wave with complex wave-number 𝑖𝛼 = 𝑖𝛼𝑟 − 𝛼𝑖 for some 𝑠 = 𝑖𝜔 + 𝜂 with real 𝜔 and 𝜂. Take 𝜂 → ∞, then according to Briggs’ criterion, this wave is downstream-going if 𝛼𝑖 → ∞, while it is upstreamgoing if 𝛼𝑖 → −∞ (Briggs, 1964). Proposition 2.2.4 For R (𝑠) > 0, the matrix 𝑀 (𝑠) has precisely 𝑁+ and 𝑁− downstream- and upstream-going eigenvalues, respectively, according on Briggs’ criterion (Towne and Colonius, 2015). Proof: The eigenvalues are the solutions to the characteristic equation det( 𝐴˜ −1 ) −1 det(𝑀 − 𝑖𝛼𝐼) = det(−𝑠𝐼 − 𝑑 ∑︁ ˜ 𝑖𝜔 𝑗 𝐵˜ 𝑗 − 𝑖𝛼 𝐴˜ − 𝐶). 𝑗=2 The eigenvalues are continuous functions of 𝑠 so taking the limit R (𝑠) → ∞ yields 𝑁+ eigenvalues with I (𝛼) > 0 and 𝑁− eigenvalues with I (𝛼) < 0. □ We then partition 𝑀 into upstream- and downstream-going blocks as " # h i 𝐷 i −1 0 h ++ 𝑀 = 𝑉+ 𝑉− , 𝑉+ 𝑉− 0 𝐷 −− (2.9) where the columns of 𝑉+ ∈ C𝑁×𝑁+ and 𝑉− ∈ C𝑁×𝑁 − the eigenvectors associated with the downstream- and upstream-going eigenvalues of 𝑀, and we have dropped the argument 𝑠 for brevity. Next we introduce the coefficients 𝝍ˆ + ∈ C𝑁+ and 𝝍ˆ − ∈ C𝑁 − such that 𝝓ˆ = 𝑉+ 𝝍ˆ + + 𝑉− 𝝍ˆ − , (2.10) 15 while for notational convenience, we define 𝑖 (+) = 1, . . . , 𝑁+ , 𝑖 (−) = 𝑁+ + 1, . . . , 𝑁, (2.11) to denote indices associated with downstream- and upstream-going modes, respectively. 2.2.1 Exact projection equations OWNS-O (Towne and Colonius, 2015) has two notable drawbacks: (i) it does not allow a forcing function, and (ii) it couples upstream- and downstream going modes through 𝑑𝑉/𝑑𝑥 for flows that vary in the marching direction. These limitations can be overcome by OWNS-P (Towne et al., 2022) and OWNS-R (Zhu and Towne, 2023). Here we introduce the exact one-way projection approach, while we subsequently introduce the approximations to this approach using OWNS-P and OWNS-R. Definitions 2.2.5 and 2.2.8 introduce the exact projection matrix and its associated one-way equations, while Proposition 2.2.10 demonstrates that these one-way equations are consistent with the global equations, even for systems varying in the marching direction. This is in contrast to the OWNS-O approach, which is consistent if and only if it is applied to constant coefficient systems (Towne and Colonius, 2015). Definition 2.2.5 We define the one-way projection matrix " # h i 𝐼 h i −1 ++ 0 , 𝑃 = 𝑉+ 𝑉− 𝑉+ 𝑉− 0 0 (2.12) which partitions the solution into downstream- and upstream-going components as " #" # h i 𝐼 0 𝝍ˆ + ++ 𝝓ˆ′ = 𝑃 𝜙ˆ = 𝑃𝑉 𝝍ˆ = 𝑉+ 𝑉− = 𝑉+ 𝜓+ , (2.13a) 0 0 𝝍ˆ − " #" # h i 0 0 𝝍ˆ + 𝝓ˆ′′ = [𝐼 − 𝑃] 𝜙ˆ = [𝐼 − 𝑃]𝑉 𝝍ˆ = 𝑉+ 𝑉− = 𝑉− 𝝍ˆ + . (2.13b) 0 𝐼−− 𝝍ˆ − For brevity define " # 𝐼++ 0 𝐸= , 0 0 so that 𝑃 = 𝑉 𝐸𝑉 −1 . Proposition 2.2.6 𝑃 is a projection matrix (Towne et al., 2022). (2.14) 16 Proof: We see that 𝑃2 = 𝑉 𝐸𝑉 −1𝑉 𝐸𝑉 −1 = 𝑉 𝐸𝑉 −1 = 𝑃, where 𝐸 2 = 𝐸, so that 𝑃2 = 𝑃, and (2.12) is a projection matrix by definition since 𝑃 = 𝑃2 . □ Proposition 2.2.7 𝑃 commutes with 𝑀 (Towne et al., 2022). Proof: Diagonal matrices commute with each other so that 𝐸 𝐷 = 𝐷𝐸 and 𝑃𝑀 = 𝑉 𝐸 𝐷𝑉 −1 = 𝑉 𝐷𝐸𝑉 −1 = 𝑀 𝑃. □ Definition 2.2.8 (exact one-way projection equations) The projection matrix, 𝑃, can be used to obtain one-way equations 𝜕 𝝓ˆ′ ˆ = 𝑃[𝑀 𝝓ˆ′ + 𝒈], 𝜕𝑥 𝜕 𝝓ˆ′′ ˆ = [𝐼 − 𝑃] [𝑀 𝝓ˆ′′ + 𝒈], 𝜕𝑥 (2.15a) (2.15b) for the downstream- and upstream-going modes, respectively (Towne et al., 2022). Proposition 2.2.9 The exact one-way projection equations are well-posed as spatial initial value problems according to the criterion of Kreiss (1970). Proof: See the discussion by Towne and Colonius (2015) and Towne et al. (2022), the criterion developed by Kreiss (1970), and the review by Higdon (1986). □ Proposition 2.2.10 The exact one-way projection equations are consistent with the global equations. Proof: If 𝝓ˆ′ and 𝝓ˆ′′ are solutions to (2.15a) and (2.15b), respectively, then summing these equations yields 𝜕 ( 𝝓ˆ′ + 𝝓ˆ′′) ˆ + [𝐼 − 𝑃] [𝑀 𝝓ˆ′′ + 𝒈] ˆ = 𝑃[𝑀 𝝓ˆ′ + 𝒈] 𝜕𝑥 = 𝑀 𝑃[ 𝝓ˆ′ + 𝝓ˆ′′] + 𝑀 [𝐼 − 𝑃] [ 𝝓ˆ′ + 𝝓ˆ′′] + 𝒈ˆ ˆ = 𝑀 [ 𝝓ˆ′ + 𝝓ˆ′′] + 𝒈, 17 so that 𝝓ˆ′ + 𝝓ˆ′′ is a solution to (2.7). If 𝝓ˆ is a solution to (2.7), then by linearity 𝜕 𝝓ˆ = 𝑀 𝝓ˆ + 𝒈ˆ 𝜕𝑥 ˆ ˆ 𝜕 [𝑃 𝝓] 𝜕 [(𝐼 − 𝑃) 𝝓] + = 𝑀 𝑃 𝝓ˆ + 𝑃 𝒈ˆ + 𝑀 (𝐼 − 𝑃) 𝝓ˆ + (𝐼 − 𝑃) 𝒈ˆ 𝜕𝑥 𝜕𝑥 𝜕 𝝓ˆ′ 𝜕 𝝓ˆ′′ ˆ + (𝐼 − 𝑃) [𝑀 𝝓ˆ′′ + 𝒈], ˆ + = 𝑃[𝑀 𝝓ˆ′ + 𝒈] 𝜕𝑥 𝜕𝑥 so that 𝑃 𝝓ˆ and (𝐼 − 𝑃) 𝝓ˆ are solutions to (2.15a) and (2.15b), respectively. □ Remark 2.2.11 The exact OWNS projection framework can be re-interpreted using a variational formulation, as shown in Appendix E. 2.2.2 Approximate projection using OWNS-P To avoid taking the eigendecomposition of 𝑀, OWNS-P uses a recursive filter to approximate the projection matrix (2.12). Definition 2.2.12 introduces this recursive filter, while Proposition 2.2.13 recasts it in a matrix form, where the matrix is a projection matrix by Proposition 2.2.14. Proposition 2.2.15 provides a criterion for OWNS-P convergence, while Proposition 2.2.16 provides a necessary condition for convergent recursion parameters to exist. The error introduced by this approximation is bounded in Proposition 2.2.17, while Proposition 2.2.20 shows that the approximate projection matrix does not generally commute with 𝑀. Definition 2.2.12 The action of 𝑃 can be applied approximately using the OWNS-P recursive filter −𝑁 𝝓ˆ+ 𝛽 = 0 𝑗 (𝑀 − 𝑖𝛽−𝑗 𝐼) 𝝓ˆ− 𝑗 − (𝑀 − 𝑖𝛽+ 𝐼) 𝝓ˆ− 𝑗−1 = 0, (2.16a) 𝑗 = 1, . . . , 𝑁 𝛽 − 1, (2.16b) ˆ (𝑀 − 𝑖𝛽−0 𝐼) 𝝓ˆ0 − (𝑀 − 𝑖𝛽+0 𝐼) 𝝓ˆ−1 = (𝑀 − 𝑖𝛽−0 𝐼) 𝝓, (2.16c) (𝑀 − 𝑖𝛽+ 𝐼) 𝝓ˆ 𝑗 − (𝑀 − 𝑖𝛽−𝑗 𝐼) 𝝓ˆ 𝑗+1 = 0, (2.16d) 𝑗 𝑗 = 0, . . . , 𝑁 𝛽 − 1, 𝑁 𝝓ˆ− 𝛽 = 0, (2.16e) 𝑗 where (𝛽+ , 𝛽−𝑗 ) are the recursion parameters and 𝝓ˆ 𝑗 the auxiliary variables for 𝑗 = 0, . . . , 𝑁 𝛽 − 1 (Towne et al., 2022). Proposition 2.2.13 The recursive filter (2.16) can be recast in matrix form −1 𝑃 𝑁 𝛽 = 𝑉 𝑅 𝑁 𝛽 𝐸 𝑅 −1 𝑁𝛽 𝑉 , (2.17a) 18 for " −1𝑉 𝐹 −1 𝐼++ 𝐹++𝑉++ +− −− 𝑅 𝑁 𝛽 = −1 −1 𝐹−−𝑉−−𝑉−+ 𝐹++ 𝐼−− # −1 , (2.17b) where 𝐹, which is a diagonal matrix " # 𝐹++ 0 𝐹= , 0 𝐹−− 𝐹𝑘 = 𝑁Ö 𝛽 −1 𝛼 𝑘 − 𝛽+ 𝑗=0 𝛼 𝑘 − 𝛽− 𝑗 𝑗 , 𝑘 = 1, . . . , 𝑁, (2.17c) as shown by Towne et al. (2022). Proof: See Appendix D.1. □ Proposition 2.2.14 𝑃 𝑁 𝛽 is a projection matrix (Towne, 2016). Proof: Note that −1 −1 −1 −1 = 𝑃𝑁𝛽 , = 𝑉 𝑅 𝑁 𝛽 𝐸 𝑅 −1 𝑃2𝑁 𝛽 = 𝑉 𝑅 𝑁 𝛽 𝐸 𝑅 −1 𝑁𝛽 𝑉 𝑁𝛽 𝑉 𝑉 𝑅𝑁𝛽 𝐸 𝑅𝑁𝛽 𝑉 so that it is a projection matrix by definition. □ Proposition 2.2.15 𝑃 𝑁 𝛽 converges to 𝑃 if and only if lim 𝑁Ö 𝛽 −1 𝑁 𝛽 →∞ 𝑗 |𝛼𝑚 − 𝛽+ | |𝛼𝑛 − 𝛽−𝑗 | 𝑗 𝑗 𝑗=0 |𝛼𝑚 − 𝛽− | |𝛼𝑛 − 𝛽+ | = 0, (2.18) for all pairs (𝛼𝑚 , 𝛼𝑛 ) such that 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) (Towne et al., 2022). Proof: See Appendix D.1. □ Proposition 2.2.16 Recursion parameters such that 𝑃 𝑁 𝛽 converges to 𝑃 exist if and only if 𝛼𝑚 ≠ 𝛼𝑛 for all 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) . In particular, if 𝑁+ ≤ 𝑁− , then choose 𝑁 𝛽 = 𝑁+ with 𝛽+𝑛−1 = 𝛼𝑛 , 𝛽−𝑗−1 ≠ 𝛼𝑛 , 𝑛 = 1, . . . , 𝑁+ , 𝑗 = 1, . . . , 𝑁+ , while if 𝑁− < 𝑁+ , then choose 𝑁 𝛽 = 𝑁− with 𝛽−𝑛−1 = 𝛼𝑛 , 𝑗−1 𝛽+ ≠ 𝛼𝑛 , 𝑛 = 𝑁+ + 1, . . . , 𝑁, 𝑗 = 1, . . . , 𝑁− . 19 Proof: See Appendix D.1. □ Proposition 2.2.17 (OWNS-P error bound) The error introduced by the OWNS-P approximation is bounded by
−1 −1 −1 ∥𝑃 𝑁 𝛽 − 𝑃∥ ≤ ∥𝑉 ∥∥𝐹++ ∥∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥ + ∥𝑉−− 𝑉−+ ∥ ∥𝑉 −1 ∥ + O (𝜖 2 ), (2.19a) −1 ∥ < 𝜖 where for ∥𝐹++ ∥∥𝐹−− −1 −1 𝜖 = min{𝜖, ˆ ∥𝑉++ 𝑉+− ∥ −1/2 ∥𝑉−− 𝑉−+ ∥ −1/2 }, (2.19b) for some small 𝜖ˆ > 0 such that 𝜖ˆ ≪ 1. Here we have used the Euclidean vector norm √︁ ∥𝒗∥ 2 = |𝑣 1 | 2 + · · · + |𝑣 𝑁 | 2 , 𝒗 ∈ C𝑁 , (2.19c) and its induced matrix norm. ∥ 𝐴𝒗∥ 2 , 𝒗≠0 ∥𝒗∥ 2 ∥ 𝐴∥ 2 = sup 𝐴 ∈ C𝑁×𝑁 , (2.19d) where we have dropped the subscript for brevity, so that ∥ · ∥ = ∥ · ∥ 2 . Proof: See Appendix D.1. □ 𝑗 Proposition 2.2.18 If 𝛽+ = 𝛼𝑚 for any downstream-going mode, it will be retained accurately by the OWNS-P filter; if 𝛽−𝑗 = 𝛼𝑛 for any upstream-going mode, it will be removed. Proof: See Appendix D.1. □ Remark 2.2.19 Proposition 2.2.17 shows that the error ∥𝑃 𝑁 𝛽 − 𝑃∥ scales with ∥𝐹++ ∥∥𝐹−− ∥ for sufficiently small ∥𝐹++ ∥∥𝐹−− ∥, while Proposition 2.2.18 shows it is always possible to achieve zero error for any upstream- or downstream-going mode. Proposition 2.2.20 𝑃 𝑁 𝛽 commutes with 𝑀 if and only if 𝑅 −1 𝑁 𝛽 = 𝐼. 20 Proof: If 𝑅 −1 𝑁 𝛽 = 𝐼, then −1 𝑃 𝑁 𝛽 𝑀 = 𝑉 𝑅 𝑁 𝛽 𝐸 𝑅 −1 𝑁 𝛽 𝑉 𝑀 = 𝑃𝑀 = 𝑀 𝑃 = 𝑀 𝑃 𝑁 𝛽 . −1 If 𝑅 −1 𝑁 𝛽 ≠ 𝐼, then 𝑅 𝑁 𝛽 is not diagonal and does not commute with 𝐷, so that 𝑅 −1 □ 𝑁 𝛽 𝐷 ≠ 𝐷 𝑅 𝑁 𝛽 , and 𝑃 𝑁 𝛽 does not commute with 𝑀. Remark 2.2.21 Proposition 2.2.14 establishes that 𝑃 𝑁 𝛽 is a projection operator for any choice of recursion parameters, while Proposition 2.2.20, in conjunction with Proposition 2.2.15, establishes that 𝑃 𝑁 𝛽 commutes with 𝑀 if and only if the recursion parameters are chosen such that 𝑃 𝑁 𝛽 = 𝑃. Proposition 2.2.22 The error in treating 𝑃 𝑁 𝛽 and 𝑀 as if they commute is bounded by −1 −1 ∥𝑃 𝑁 𝛽 𝑀 − 𝑀 𝑃 𝑁 𝛽 ∥ ≤ 2∥𝑉 ∥∥𝐹++ ∥∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥
−1 + ∥𝑉−− 𝑉−− ∥ ∥𝑉 −1 ∥∥𝑀 ∥ + O (𝜖 2 ), (2.20) −1 ∥ < 𝜖 and 𝜖 as in Proposition 2.2.17. for ∥𝐹++ ∥∥𝐹−− Proof: See Appendix D.1. □ Remark 2.2.23 Although 𝑃 𝑁 𝛽 and 𝑀 do not generally commute, the introduced by treating them as if they commute is small if ∥𝑃 − 𝑃 𝑁 𝛽 ∥ is small. 2.2.3 Approximate projection using OWNS-R OWNS-R constructs approximations to (2.12), but with a reduced computational cost relative to OWNS-P. Definition 2.2.24 presents the OWNS-R recursive filter. Unlike OWNS-P, Proposition 2.2.27 shows that the resulting matrix is not generally a projection, while Proposition 2.2.28 shows that it always commutes with 𝑀. Proposition 2.2.30 provides a criterion for convergence, while Proposition 2.2.31 provides a necessary condition for convergent parameters to exist. While Zhu and Towne (2023) suggested that the OWNS-P recursion parameters would also work for OWNS-R, we show in Proposition 2.2.32 that this is not always true. In addition, Proposition 2.2.38 shows that repeated applications of the OWNSR matrix leads to unbounded growth or decay unless the approximation is fully converged. We then conclude by discussing the practical implementation of OWNSR in Definition 2.2.40 and Proposition 2.2.41. 21 Definition 2.2.24 The action of 𝑃 can be applied approximately using the matrix 𝑃 𝑁(𝑅) = (𝐼 + 𝑐𝑍) −1 , 𝛽 (2.21) for 𝑍= 𝑁𝛽 Ö (𝑀 − 𝑖𝛽+𝑗 𝐼)(𝑀 − 𝑖𝛽−𝑗 𝐼) −1 , (2.22) 𝑗=1 where 𝑐 is a freely chosen parameter (Zhu and Towne, 2023). Proposition 2.2.25 If we define the approximate eigenvalues 𝐸 𝑁 𝛽 = (𝐼 + 𝑐𝐹) −1 , (2.23) then 𝑃 𝑁(𝑅) = 𝑉 𝐸 𝑁 𝛽 𝑉 −1 (Zhu and Towne, 2023). 𝛽 Proof: See Appendix D.2 □ Remark 2.2.26 Zhu and Towne (2023) showed that setting 𝑐 = 0 gives 𝐸 𝑁(𝑘)𝛽 = 1, so that the march is unstable since all upstream-going modes are retained. Similarly, taking 𝑐 → ∞ gives 𝐸 𝑁(𝑘)𝛽 = 0, so that the march is stable but inaccurate. Therefore, they recommend taking 𝑐 ≈ 1, and so we take 𝑐 = 1 for the rest of this paper. Proposition 2.2.27 The matrix 𝑃 𝑁(𝑅) is a projection matrix if and only if 𝐸 𝑁(𝑘)𝛽 = 1 𝛽 or 𝐸 𝑁(𝑘)𝛽 = 0 for 𝑘 = 1, . . . , 𝑁. Proof: See Appendix D.2. □ Proposition 2.2.28 The matrix 𝑃 𝑁(𝑅) commutes with 𝑀 for any choice of recursion 𝛽 parameters. Proof: Note that 𝑃 𝑁(𝑅) 𝑀 = 𝑉 𝐸 𝑁 𝛽 𝐷𝑉 −1 = 𝑉 𝐷𝐸 𝑁 𝛽 𝑉 −1 = 𝑀 𝑃 𝑁(𝑅) , 𝛽 𝛽 since 𝐸 𝑁 𝛽 and 𝐷 are both diagonal matrices. □ Remark 2.2.29 OWNS-P always produces a projection matrix (Proposition 2.2.14) but it does not always commute with 𝑀 (Proposition 2.2.20). Conversely, OWNS-R may not yield a projection matrix (Proposition 2.2.27) but it always commutes with 𝑀 (Proposition 2.2.28). 22 Proposition 2.2.30 𝑃 𝑁(𝑅) converges to 𝑃 if and only if 𝐸 𝑁(𝑘)𝛽 → 1 for 𝑘 = 𝑖 (+) and 𝛽 𝐸 𝑁(𝑘)𝛽 → 0 for 𝑘 = 𝑖 (−) (Zhu and Towne, 2023). Proof: Clearly 𝑃 𝑁(𝑅) → 𝑃 if and only if 𝐸 𝑁 𝛽 → 𝐸, while by definition, 𝐸 (𝑘) = 1 𝛽 for 𝑘 = 𝑖 (+) and 𝐸 (𝑘) = 0 for 𝑘 = 𝑖 (−) , so that 𝐸 𝑁 𝛽 → 𝐸 if and only if 𝐸 𝑁(𝑘)𝛽 → 1 for 𝑘 = 𝑖 (+) and 𝐸 𝑁(𝑘)𝛽 → 0 for 𝑘 = 𝑖 (−) . □ Proposition 2.2.31 Recursion parameters such that 𝑃 𝑁(𝑅) converges to 𝑃 exist if and 𝛽 (+) (−) only if 𝛼𝑚 ≠ 𝛼𝑛 for all pairs of 𝑚 = 𝑖 and 𝑛 = 𝑖 . In particular, choose 𝛽+𝑚−1 = 𝛼𝑚 , 𝛽−𝑚−1 ≠ 𝛼𝑛 , 𝑚, 𝑛 = 𝑖 (+) , (2.24a) 𝛽−𝑛−1 = 𝛼𝑛 , 𝛽+𝑛−1 ≠ 𝛼𝑚 , 𝑚, 𝑛 = 𝑖 (−) . (2.24b) Proof: See Appendix D.2. □ Proposition 2.2.32 The minimal parameter set such that 𝑃 𝑁 𝛽 = 𝑃 for OWNS-P (Proposition 2.2.16) yields 𝑃 𝑁(𝑅) ≠ 𝑃 for OWNS-R. 𝛽 Proof: If 𝑁+ < 𝑁− , then 𝑁 𝛽 = 𝑁+ , with 𝛽+𝑛−1 = 𝛼𝑛 and 𝛽−𝑛−1 ≠ 𝛼𝑛 for 𝑛 = 𝑖 (+) , and for the upstream-going modes Î 𝑁+ 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) (𝑘) ≠ 0, 𝐸 𝑁 𝛽 = Î𝑁 Î 𝑁+ 𝑗 𝑗 + (𝛼 − 𝛽 ) + 𝑐 (𝛼 − 𝛽 ) − 𝑘 𝑘 + 𝑗=1 𝑗=1 𝑘 = 𝑖 (−) , since 𝛼 𝑘 ≠ 𝛽−𝑗 for 𝑘 = 𝑖 (−) and 𝑗 = 𝑖 (+) . Then 𝑃 𝑁(𝑅) ≠ 𝑃 since 𝐸 𝑁(𝑘)𝛽 ≠ 0 for 𝑘 = 𝑖 (−) . 𝛽 □ Remark 2.2.33 Zhu and Towne (2023) claimed that the minimal parameter set ensuring OWNS-P convergence also apply to OWNS-R, but Proposition 2.2.32 disproves this, while Proposition 2.2.31 identifies a different minimal set for OWNSR. Proposition 2.2.34 The error introduced by OWNS-R is bounded by  −1 ∥𝑃 − 𝑃 𝑁(𝑅) ∥ ≤ max |𝑐|∥𝐹++ ∥, ∥𝐹−− ∥ ∥𝑉 ∥∥𝑉 −1 ∥ + O (𝜖 2 ), 𝛽 (2.25) where ∥𝐹++ ∥, ∥𝐹−− ∥ < 𝜖 for small 𝜖 > 0 such that 𝜖 ≪ 1 (Zhu and Towne, 2023). 23 Proof: See Appendix D.2. □ 𝑗 Proposition 2.2.35 If 𝛽+ = 𝛼𝑚 for any downstream-going mode, it will be retained accurately by the OWNS-R filter; if 𝛽−𝑗 = 𝛼𝑛 for any upstream-going mode, it will be removed. Remark 2.2.36 Proposition 2.2.34 establishes that the error in introduced by −1 ∥} for sufficiently small ∥𝐹 ∥ and ∥𝐹 ∥, OWNS-R scales with max{|∥𝐹++ ∥, ∥𝐹−− ++ −− while Proposition 2.2.18 shows it is always possible to achieve zero error for any upstream- or downstream-going mode, while Proposition 2.2.18 shows it is always possible to achieve zero error for any upstream- or downstream-going mode. −1 ∥, so that we have Remark 2.2.37 The OWNS-P error converges with ∥𝐹++ ∥∥𝐹−− −1 ∥ = 0 or ∥𝐹 ∥ = 0, while the OWNS-R error scales convergence with either ∥𝐹−− ++ −1 −1 ∥ = ∥𝐹 ∥ = 0. Thus, OWNS-R with max{|∥𝐹++ ∥, ∥𝐹−− ∥} so that we must have ∥𝐹−− ++ generally requires a larger and more carefully chosen parameter set. Proposition 2.2.38 Repeated application of 𝑃 𝑁(𝑅) causes unbounded growth if |𝐸 𝑁(𝑘)𝛽 | > 𝛽 1 and decay to zero if |𝐸 𝑁(𝑘)𝛽 | < 1 for any 𝑘 = 1, . . . , 𝑁. Therefore, repeated applications introduce additional error unless |𝐸 𝑁(𝑛)𝛽 | < 1 for all 𝑛 = 𝑖 (−) , and 𝐸 𝑁(𝑚) =1 𝛽 (+) for all 𝑚 = 𝑖 . Proof: See Appendix D.2. □ Remark 2.2.39 By Proposition 2.2.38, 𝑃 𝑁(𝑅) should be applied only once for OWNS𝛽 R to avoid solution blow-up or decay. Definition 2.2.40 The action of 𝑃 𝑁(𝑅) is applied to 𝝓ˆ by solving for 𝝓ˆ′𝑁 𝛽 as 𝛽 1ˆ 𝝓ˆ0 = 𝝓, 2 𝑗 ˆ𝑗 (𝑀 − 𝑖𝛽∗ ) 𝝓 = (𝑀 − 𝑖𝛽−𝑗 ) 𝝓ˆ 𝑗−1 , (2.26a) 𝑗 = 1, . . . , 𝑁 𝛽 , 𝝓ˆ′𝑁 𝛽 = 𝝓ˆ 𝑁 𝛽 , (2.26b) (2.26c) 𝑗 where 𝛽∗ are defined such that 2 𝑁𝛽 Ö 𝑗=1 𝑗 (𝛼 − 𝛽∗ ) = 𝑁𝛽 Ö 𝑗=1 (𝛼 − 𝛽−𝑗 ) + 𝑁𝛽 Ö 𝑗=1 𝑗 (𝛼 − 𝛽+ ). (2.26d) 24 Proposition 2.2.41 The action of 𝑃 𝑁(𝑅) can be applied using (2.26). 𝛽 Proof: See Appendix D.2. □ Remark 2.2.42 In theory, the OWNS-R error decreases with increasing 𝑁 𝛽 if both −1 ∥ decrease. In practice, computing 𝛽∗ numerically (e.g., via roots ∥𝐹++ ∥ and ∥𝐹−− 𝑗 in MATLAB) introduces rounding errors that can prevent convergence. 2.3 Recursion parameter selection All three OWNS formulations, as well as the method by Rudel et al., require selecting a set of recursion parameters that govern the convergence, stability, and accuracy of the approximation. These parameters are, loosely speaking, estimates of key eigenvalues (or their branches) representing upstream and downstream modes. Previous work has selected them heuristically, based on the Euler equations linearized about a uniform flow Towne and Colonius (2015). We refer to this as the heuristic approach, and note that it avoids computing the eigenvalues of 𝑀, but requires flow-specific tuning (e.g., different parameters for low- and high-speed boundary layers) and often results in slow error convergence. We delay a more detailed discussion of recursion parmaeter selection until Chapter 4, where we develop a greedy algorithm that automates parameter selection, while also ensuring rapid convergence and stability. 2.4 Summary We have presented the OWNS-P and OWNS-R formulations for hyperbolic equations. Whereas OWNS-P yields a projection matrix that generally does not commute with 𝑀, OWNS-R yields a matrix that commutes with 𝑀 but is not generally a projection. Although they can use similar recursion parameters, Proposition 2.2.32 shows that there exists parameter sets for which OWNS-P is converged while OWNSR is not. In general, OWNS-R requires more careful parameter selection for stability and accuracy. Finally, Proposition 2.2.18 and 2.2.35 show that it is always possible to choose recursion parameters such that zero error is achieved for any upstreamor downstream-going mode. We will show in Chapter 3 how the procedure can be applied to the Navier-Stokes equations, and extended to include nonlinear effects. 25 Chapter 3 NONLINEAR OWNS METHODOLOGY This chapter has been adapted from Sleeman et al. (2023), Sleeman et al. (2024b), and Sleeman et al. (2025). Here we develop the NOWNS methodology and validate it by comparing to direct numerical simulation (DNS) data for low-speed Blasius boundary-layer flows. The OWNS-P approach presented in Chapter 2 was applied to hyperbolic equations, but the Navier-Stokes equations are not hyperbolic, so we will first demonstrate how to apply one-way marching to the Navier-Stokes equations, as done by Towne et al. (2022). 3.1 Linear OWNS In Appendix A, we rewrite the Navier-Stokes equations (1.1) in operator form (A.6) for the vector of primitive variables 𝒒 = (𝜈, 𝑢, 𝑣, 𝑤, 𝑝), where the nonlinear terms, ¯ is defined as in (A.5). 𝐹 (𝒒′), are defined as in (A.4), while the linear operator, L ( 𝒒), To simplify the exposition, we temporarily neglect the streamwise diffusion terms by introducing the exogeneous forcing function 𝒇 = 𝒇forcing + 𝒇viscous where ¯ 𝒇viscous = −𝐵𝑥 ( 𝒒) 𝜕 2 𝒒′ 𝜕 2 𝒒′ 𝜕 2 𝒒′ 𝜕𝒒′ ¯ ¯ ¯ − 𝐵𝑥𝑥 ( 𝒒) − 𝐵 ( 𝒒) − 𝐵 ( 𝒒) , 𝑥𝑦 𝑥𝑧 𝜕𝑥 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑥 2 represents streamwise diffusion terms associated with the disturbance variable, while 𝒇forcing is an arbitrary forcing function. We then neglect the nonlinear term and rewrite equation (A.6) to obtain the linear stability equation 𝜕𝒒′ ¯ ¯ ′ + 𝒇. 𝐴𝑥 ( 𝒒) = L ( 𝒒)𝒒 𝜕𝑥 (3.1) Previous work OWNS has often neglected 𝒇viscous because this simplifies the procedure without significantly impacting its accuracy (Towne and Colonius, 2015; Rigas et al., 2017a; Kamal et al., 2020), but we have found that this term impacts more significantly the nonlinear calculation, as discussed in Appendix G.1, so we re-introduce it (approximately) in Section 3.1.5. In NOWNS, we consider a system of linear OWNS equations coupled together through the nonlinear term, which acts as an inhomogeneous forcing function taking the place of 𝒇 in (3.1). The OWNS-O approach (Towne and Colonius, 2015) supports only homogeneous equations, so we must instead consider either the OWNS-P 26 (Towne et al., 2022) or the OWNS-R (Zhu and Towne, 2023) approaches. OWNSR entails a reduced computational cost compared to OWNS-P, but we have found OWNS-P to be more robust, so we use it to develop the NOWNS approach. 3.1.1 Semi-discrete equations Throughout this work, we assume that our disturbances are periodic in the spanwise direction with wavenumber 𝛽 and in time with frequency 𝜔 so that we can expand our disturbances as Fourier series ∞ ∑︁ ′ 𝒒 (𝑥, 𝑦, 𝑧, 𝑡) = 𝒒ˆ 𝑚𝑛 (𝑥, 𝑦)𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) . (3.2) 𝑚,𝑛=−∞ In the linear case, all Fourier modes evolve independently, and we consider a single ˆ 𝑦)𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) , while we discretize in the disturbance of the form 𝒒′ (𝑥, 𝑦, 𝑧, 𝑡) = 𝒒(𝑥, wall-normal direction using a 4th-order central finite differences which we represent using 𝐷 ≈ 𝜕/𝜕𝑦. Our semi-discrete linear operator is then ¯ = 𝑖𝜔𝐼 − [ 𝐴 𝑦 ( 𝒒) ¯ + 𝐵 𝑦 ( 𝒒)]𝐷 ¯ ¯ + 𝐵 𝑧 ( 𝒒)] ¯ 𝐿( 𝒒) − 𝑖𝛽[ 𝐴𝑧 ( 𝒒) (3.3) ¯ − 𝐵 𝑦𝑦 ( 𝒒)𝐷 ¯ 2 + 𝛽2 𝐵 𝑧𝑧 ( 𝒒) ¯ − 𝑖𝛽𝐵 𝑦𝑧 ( 𝒒) ¯ = 0, − 𝐶 ( 𝒒) (3.4) and we obtain 𝜕𝒒′ ¯ ′ + 𝒇, = 𝐿( 𝒒)𝒒 (3.5) 𝜕𝑥 a system of ODEs in 𝑥 comprising 𝑁𝑣 variables, where 𝑁𝑣 = 5𝑁 𝑦 (𝑁𝑣 = 4𝑁 𝑦 ) in 3D (2D), for the 𝑁 𝑦 grid points in 𝑦. ¯ 𝐴𝑥 ( 𝒒) 3.1.2 Parabolization using the OWNS projection procedure The above ODEs in 𝑥 contain components that propagate both upstream and downstream, and cannot be stably integrated without further intervention. Therefore, we remove upstream effects using a projection operator that we apply in the characteristic variables, 𝝓 = 𝑇 𝒒′, where 𝑇 are the eigenvectors of 𝐴𝑥 , while 𝐴˜ 𝑥 = 𝑇 𝐴𝑥 𝑇 −1 are the eigenvalues. Here we have dropped the argument 𝒒¯ for brevity. We transform our equations to characteristic variables as 𝜕 𝝓ˆ ˜ ˆ ˆ 𝐴˜ 𝑥 = 𝐿 𝝓 + 𝒇𝜙 . 𝜕𝑥 (3.6a) with −1 𝜕𝑇 𝐿˜ = 𝑇 𝐿𝑇 −1 − 𝑇 𝐴𝑥 𝜕𝑥 ˆ𝒇𝜙 = 𝑇 𝒇ˆ (3.6b) (3.6c) 27 where 𝐿˜ and 𝑓ˆ𝜙 denote the linear operator and forcing function in characteristic variables, respectively. We then re-organize the diagonal matrix 𝐴˜ 𝑥  𝐴˜ ++ 0 0     𝐴˜ 𝑥 =  0 𝐴˜ −− 0  ,    0 ˜ 00  0 𝐴   (3.7) for the 𝑁+ positive eigenvalues 𝐴˜ ++ > 0, the 𝑁− negative eigenvalues 𝐴˜ −− < 0, and the 𝑁0 zero eigenvalues 𝐴˜ 00 = 0, where 𝑁0 + 𝑁− + 𝑁+ = 𝑁𝑣 . For a boundary-layer flow, 𝐴˜ = 0 at the wall (𝑢¯ = 0) or at a sonic point (𝑢¯ − 𝑐¯ = 0). We further define " # " # " # ˜ ++ 0 ˆ 𝐴 ˆ = 𝝓± , 𝐿˜ +0 𝐴˜ ±± = , 𝝓 𝐿˜ ±0 = , 0 𝐴˜ −− 𝝓ˆ0 𝐿˜ −0 " # " # h i ˆ𝜙,± 𝐿˜ ++ 𝐿˜ +− 𝒇 𝐿˜ ±± = , 𝐿˜ 0± = 𝐿˜ 0+ 𝐿˜ 0− , 𝒇ˆ𝜙 = , ˜𝐿 −+ 𝐿˜ −− 𝒇ˆ𝜙,0 and so that our equations become 𝜕 𝝓ˆ± ˜ ˆ = 𝐿 ±± 𝝓± + 𝐿˜ ±0 𝝓ˆ0 + 𝒇ˆ𝜙,± , 𝐴˜ ±± 𝜕𝑥 0 = 𝐿˜ 0± 𝝓ˆ± + 𝐿˜ 00 𝝓ˆ0 + 𝒇ˆ𝜙,0 , (3.8a) (3.8b) which is a differential algebraic equation (DAE) of index 1. We can use the algebraic −1 [ 𝐿 ˜ 0± 𝝓ˆ± + 𝒇ˆ𝜙,0 ] so that we can re-write our constraint (3.8b) to obtain 𝝓ˆ0 = − 𝐿˜ 00 DAE as an ODE 𝜕 𝝓ˆ± ˆ = 𝑀 𝝓ˆ± + 𝒈, (3.9) 𝜕𝑥 −1 [ 𝐿 −1 [ 𝒇ˆ ˜ ±± − 𝐿˜ ±0 𝐿˜ −1 𝐿˜ 0± ], and 𝒈ˆ = 𝐴˜ ±± ˜ ±0 𝐿 −1 𝒇ˆ𝜙,0 ]. for 𝑀 = 𝐴˜ ±± 𝜙,± − 𝐿 00 00 The upstream- and downstream-going modes of (3.9) can be determined based on the eigenvalues of 𝑀, according to Brigg’s criterion from definition 2.2.3 (Briggs, 1964) to introduce the projection operator from definition 2.2.5, as described in (Towne and Colonius, 2015; Towne et al., 2022). Then we obtain the linear OWNS equation from definition 2.2.8, yielding the equation for the downstream-going solution 𝜕 𝝓ˆ′± ˆ = 𝑃[𝑀 𝝓ˆ′± + 𝒈], (3.10) 𝜕𝑥 where we used that 𝑃 and 𝑀 commute, as shown by Towne et al. (2022). 28 3.1.3 Approximate projection operator Exact OWNS is prone to numerical error and entails a high computational cost to compute the eigenvalues and eigenvectors of the linear operator. Therefore, instead apply the projection operator approximately using a recursive filter (Towne and Colonius, 2015; Towne et al., 2022). Whereas in definition 2.2.12 we apply the ˆ here we instead apply the filter to the residual. We define the filter to the variable 𝜙, −1 [ 𝐿 ˜ ±± 𝝓ˆ± + 𝐿˜ ±0 𝝓ˆ0 + 𝒇ˆ𝜙,± ], and 𝒓ˆ0 (𝝓) = 𝐿˜ 0± 𝝓ˆ± + 𝐿˜ 00 𝝓ˆ0 + 𝒇ˆ𝜙,0 residuals 𝒓ˆ± (𝝓) = 𝐴˜ 𝑥,±± based on (3.8), and following the approach of Towne et al. (Towne et al., 2022), we can apply 𝑃 approximately to the residual using the recursive filter 𝒓ˆ+(−𝑁𝑏 ) = 0 ( 𝑗) ˜ 𝒓ˆ (− 𝑗) − ( 𝐿˜ − 𝑖𝛽+ 𝐴) ˜ 𝒓ˆ (− 𝑗−1) = 0, ( 𝐿˜ − 𝑖𝛽−( 𝑗) 𝐴) (3.11a) 𝑗 = 1, . . . , 𝑁 𝑏 − 1 ˜ 𝒓ˆ (0) − ( 𝐿˜ − 𝑖𝛽+(0) 𝐴) ˜ 𝒓ˆ (−1) = ( 𝐿˜ − 𝑖𝛽−(0) 𝐴) ˜ 𝒓ˆ ( 𝐿˜ − 𝑖𝛽−(0) 𝐴) ( 𝑗) ˜ ( 𝑗) ˜ 𝒓ˆ ( 𝐿˜ − 𝑖𝛽+ 𝐴) 𝒓ˆ − ( 𝐿˜ − 𝑖𝛽−( 𝑗) 𝐴) 𝒓ˆ0(0) = 𝒓ˆ0 , ( 𝑗+1) = 0, (3.11b) (3.11c) (3.11d) 𝑗 = 0, . . . , 𝑁 𝑏 − 1 𝒓ˆ−(𝑁𝑏 ) = 0, (3.11e) (3.11f) where 𝒓ˆ0 = 0 when the algebraic constraint (3.8b) is satisfied. We introduce the vector auxiliary variables 𝒓ˆaux ∈ C 𝑁aux , and the approximate projection operators 𝑃1 ∈ C𝑁aux ×𝑁𝑣 , 𝑃2 ∈ C𝑁aux ×𝑁aux , 𝑃3 ∈ R𝑁± ×𝑁aux , where 𝑃1 𝒓ˆ and 𝑃2 𝒓ˆaux give the right- and left-hand sides of (3.11), respectively, while 𝑃3 extracts 𝒓ˆ±(0) from 𝒓ˆaux as 𝒓ˆ±(0) = 𝑃3 𝒓ˆaux . The action of the approximate projection operator on the DAE (3.8) can expressed compactly as 𝜕 𝝓ˆ′± = 𝑃3 𝒓ˆaux , 𝜕𝑥 " # −1 [ 𝐿 ˜ ±± 𝝓ˆ′± + 𝐿˜ ±0 𝝓ˆ′ + 𝒇ˆ𝜙,± ] 𝐴˜ ±± 0 𝑃2 𝒓ˆaux = 𝑃1 , 𝐿˜ 0± 𝝓ˆ′± + 𝐿˜ 00 𝝓ˆ′0 + 𝒇ˆ𝜙,0 0 = 𝐿˜ 0± 𝝓ˆ′± + 𝐿˜ 00 𝝓ˆ′0 + 𝒇ˆ𝜙,0 . 3.1.4 Fully-discrete equations We define 𝐴‡ , 𝐿 ‡ ∈ R (𝑁𝑣 +𝑁aux )×(𝑁𝑣 +𝑁aux ) and 𝝓ˆ‡ , 𝒇ˆ𝜙‡ ∈ C𝑁𝑣 +𝑁aux such that  𝐼±± 0 0     𝐴 ‡ =  0 0 0 ,    0 0 0    0 Δ𝑥𝑃3   " 0 # " #   −1 𝐿 −1 𝐿   𝐴˜ ±± 𝐴˜ ±± ±± ±0 ‡  𝐿 =  𝑃1 𝑃1 −𝑃2  , 𝐿 0± 𝐿 00      𝐿 0± 𝐿 00 0   (3.12a) (3.12b) (3.12c) 29 and    " 0 #   ˜ −1 ˆ   ˆ𝒇 ‡ = 𝑃1 𝐴±± 𝒇𝜙,±  . 𝜙  𝒇ˆ𝜙,0     ˆ𝜙,0   𝒇   Then, following the previous work on linear OWNS (Rigas et al., 2017a; Kamal et al., 2020), we apply an 𝑠-order BDF scheme to obtain the linear OWNS equation  𝝓′   ±   ‡ ˆ 𝝓 =  𝝓0  ,    𝒓 aux    [𝑐 (0) ‡ 𝐴 −𝐿 ‡(𝑘 𝑥 +1) ] 𝝓ˆ‡(𝑘 𝑥 +1) = − 𝑠−1 ∑︁ 𝑐 (𝑙 𝑥 ) 𝐴‡ 𝝓ˆ‡(𝑘 𝑥 +1−𝑙 𝑥 ) + 𝒇ˆ𝜙‡(𝑘 𝑥 +1) , (3.13) 𝑙 𝑥 =1 for 𝑘 𝑥 = 1, . . . , 𝑁𝑥 , where 𝑁𝑥 denotes the number of streamwise stations, while 𝑠 denotes the order of the BDF scheme and 𝑐 (𝑙 𝑥 ) for 𝑙 𝑥 = 0, . . . , 𝑠 − 1 denotes the marching coefficients. 3.1.5 Streamwise diffusion terms Following discretization in the wall-normal direction and trasnformation to characteristic variables, our streamwise diffusion terms become 𝜕 2 𝝓ˆ 𝜕 𝝓ˆ ˜ ˆ 𝒇ˆ𝜙,viscous = − 𝐵˜ 𝑥𝑥 2 − 𝐵˜ 𝑥 − 𝐵 𝝓, 𝜕𝑥 𝜕𝑥 −1 for 𝐵˜ 𝑥𝑥 = 𝑇 𝐵𝑥𝑥 𝑇 −1 , 𝐵˜ 𝑥 = 2𝑇 𝐵𝑥𝑥 𝜕𝑇𝜕𝑥 + 𝑇 [𝐵𝑥 + 𝐵𝑥𝑦 𝐷 + 𝑖𝛽𝐵𝑥𝑧 ]𝑇 −1 , and 𝐵˜ = 2 −1 −1 𝑇 𝐵𝑥𝑥 𝜕𝜕𝑥𝑇 2 + 𝑇 [𝐵𝑥 + 𝐵𝑥𝑦 𝐷 + 𝑖𝛽𝐵𝑥𝑧 ] 𝜕𝑇𝜕𝑥 . We discretize the second-derivative using a second-order backward difference, while we discretize the first-derivative using the BDF scheme to obtain 𝑠−1  ˆ (𝑘+1) − 2𝝓ˆ (𝑘) + 𝝓ˆ (𝑘−1)  ∑︁ (𝑙) ˜ ˆ (𝑘+1−𝑙) ˆ𝒇 (𝑘+1) = − 𝐵˜ 𝑥𝑥 𝝓 − 𝑐 𝐵𝑥 𝝓 − 𝐵˜ 𝝓ˆ (𝑘+1) , 𝜙,viscous 2 (Δ𝑥) 𝑙=0 and we add this term back into the fully-discrete OWNS equations (3.13). 3.2 Nonlinear OWNS Whereas infinitesimal disturbances (in the linear case) evolve independently from each other so that each Fourier mode can be considered separately, finite amplitude disturbances (in the nonlinear case) are coupled through the nonlinear term. Since it is not feasible to consider an infinite number of Fourier modes, we truncate the Fourier (3.2) series 𝒒′ (𝑥, 𝑦, 𝑧, 𝑡) = 𝑀 𝑁 ∑︁ ∑︁ 𝑚=−𝑀 𝑛=−𝑁 𝒒ˆ 𝑚𝑛 𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) , (3.14) 30 resulting in (2𝑀 + 1) × (2𝑁 + 1) Fourier modes. However, we require that 𝒒′ be real-valued, which yields the constraint 𝑞ˆ 𝑚𝑛 = 𝑞ˆ −𝑚𝑛 , so that we need only track (𝑀 + 1) × (2𝑁 + 1) Fourier modes. We can additionally introduce a spanwise symmetry condition, as described in Section 3.2.5, which further reduces the number of modes to (𝑀 + 1) × (𝑁 + 1). The nonlinear terms in primitive variables, 𝐹 (𝒒′), are defined in (A.4), and we transform them to primitive variables in a way that mimics the transformation of the linear terms, to obtain the nonlinear terms in characteristic variables, 𝐹˜ (𝝓). We use our assumption of periodicity to obtain the Fourier series ˜ 𝒒)𝝓 ¯ = 𝐿( ˜ 𝑭(𝝓) = 𝒇˜𝜙 = 𝑀 𝑁 ∑︁ ∑︁ 𝑚=−𝑀 𝑛=−𝑁 𝑀 𝑁 ∑︁ ∑︁ 𝑚=−𝑀 𝑛=−𝑁 𝑀 𝑁 ∑︁ ∑︁ 𝐿ˆ 𝑚𝑛 𝝓ˆ𝑚𝑛 𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) , (3.15) 𝑭ˆ𝑚𝑛 (𝝓)𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) (3.16) 𝒇ˆ𝜙,𝑚𝑛 𝑒𝑖(𝑛𝛽𝑧−𝑚𝜔𝑡) . (3.17) 𝑚=−𝑀 𝑛=−𝑁 Note that 𝑓˜𝜙 is the forcing function in characteristic variables defined in (3.6c). The Fourier modes are mutually orthogonal, yielding the following system of equations: 𝜕 𝝓ˆ𝑚𝑛 ˆ ˆ = 𝐿 𝑚𝑛 𝝓𝑚𝑛 + 𝑭ˆ𝑚𝑛 (𝝓) + 𝒇ˆ𝜙,𝑚𝑛 , 𝐴˜ 𝜕𝑥 ∀𝑚 ∈ Z 𝑀 , ∀𝑛 ∈ Z𝑁 , (3.18) for Z 𝑀 ≡ {𝑥 ∈ Z| − 𝑀 ≤ 𝑥 ≤ 𝑀 } and Z𝑁 ≡ {𝑥 ∈ Z| − 𝑁 ≤ 𝑥 ≤ 𝑁 }, where Z is the set of all integers. We follow a procedure that mimics the linear OWNS approach to obtain 𝜕𝝓′±,𝑚𝑛 = 𝑃ˆ𝑚𝑛 [ 𝑀ˆ 𝑚𝑛 𝝓ˆ′±,𝑚𝑛 + 𝒈ˆ 𝑚𝑛 (𝑃𝝓′± )], 𝜕𝑥 ∀𝑚 ∈ Z 𝑀 , ∀𝑛 ∈ Z𝑁 . (3.19) In the linear case, 𝑃 and 𝑀 commute so that two one-way parabolic equations can be solved to recover the full elliptic solution (Towne et al., 2022). However, this property does not apply in the nonlinear case because the 𝑃 does not commute with the nonlinear term (𝑃𝒈(𝝓± ) ≠ 𝑃𝒈(𝑃𝝓± ) in general), so that (3.19) removes the upstream effect, 𝝓′′, from the nonlinear term, and when we sum the upstreamand downstream-going equations together, we do not recover the elliptic equation. This is a reasonable choice for convective instabilities, where the disturbances travel primarily in one direction, and we verify a posteriori that we match closely DNS results in the literature. 31 3.2.1 Fully-discrete equations ‡ We define 𝐿 ‡𝑚𝑛 ∈ R (𝑁𝑣 +𝑁aux )×(𝑁𝑣 +𝑁aux ) and 𝝓ˆ‡𝑚𝑛 , 𝒇ˆ𝑚𝑛,𝜙 ∈ C𝑁𝑣 +𝑁aux , as in the linear case, ‡ for all 𝑚 ∈ Z 𝑀 and 𝑛 ∈ Z𝑁 , and further introduce the nonlinear term 𝑭ˆ𝑚𝑛 ∈ C𝑁𝑣 +𝑁aux ‡ such that it mimics the definition of the forcing function 𝒇ˆ𝑚𝑛,𝜙 . Then we discretize the first-derivatives in both the linear and nonlinear terms using the BDF scheme yielding the fully-discrete nonlinear system of equations 𝑠−1 ∑︁ ‡(𝑘 𝑥 +1) ‡(𝑘 𝑥 +1) 𝑥 +1−𝑙) 𝑥 +1) ˆ‡(𝑘 𝑥 +1) 𝑐 (𝑙 𝑥 ) 𝐴‡ 𝝓ˆ‡(𝑘 = 𝐿ˆ ‡(𝑘 𝝓𝑚𝑛 + 𝑭ˆ𝑚𝑛 + 𝒇ˆ𝜙,𝑚𝑛 , 𝑚𝑛 𝑚𝑛 (3.20) 𝑙 𝑥 =0 for 𝑚 ∈ Z 𝑀 , 𝑛 ∈ Z𝑁 , and 𝑘 𝑥 = 1, . . . , 𝑁𝑥 . 3.2.2 Pseudo-spectral method We employ a pseudo-spectral method whereby we solve our equations in Fourier space, while we evaluate the nonlinear terms in physical space. Given the Fourier coefficients 𝝓ˆ𝑚𝑛 for 𝑚 = 0, . . . , 2𝑀 and 𝑛 = 0, . . . , 𝑁, we can use the inverse discrete Fourier transform (iDFT) to compute the solution in physical space as 𝝓 𝑘𝑙 = 2𝑀 2𝑁 1 1 ∑︁ ∑︁ ˆ 𝑖2𝜋𝑘𝑚/(2𝑀+1) 𝑖2𝜋𝑙𝑛/(2𝑁+1) 𝝓𝑚𝑛 𝑒 𝑒 . 2𝑀 + 1 2𝑁 + 1 𝑚=0 𝑛=0 (3.21) for discrete times and spanwise locations associated with indices 𝑘 = 0, . . . , 2𝑀 and 𝑙 = 0, . . . , 2𝑁, respectively. Then, we can evaluate the nonlinear terms in physical ˜ 𝑚𝑛 ), and employ the discrete Fourier transform (DFT) to compute space, 𝑭˜𝑚𝑛 = 𝑭(𝝓 the Fourier components of the nonlinear terms as 𝑭ˆ 𝑘𝑙 = 2𝑀 ∑︁ 2𝑁 ∑︁ 𝑭˜𝑚𝑛 𝑒 −𝑖2𝜋𝑚𝑘/(2𝑀+1) 𝑒 −𝑖2𝜋𝑛𝑙/(2𝑁+1) , (3.22) 𝑚=0 𝑛=0 for 𝑘 = 0, . . . , 2𝑀 and 𝑙 = 0, . . . , 2𝑁. In practice, the DFT and iDFT are performed using fast Fourier transform (FFT) libraries. 3.2.3 Nonlinear solution procedure and computational cost We must ensure that the residual ‡(𝑘 𝑥 +1) 𝒓ˆ𝑚𝑛 = 𝑠−1 ∑︁ ‡(𝑘 𝑥 +1) ‡(𝑘 𝑥 +1) 𝑥 +1−𝑙) 𝑥 +1) ˆ‡(𝑘 𝑥 +1) 𝑐 (𝑙 𝑥 ) 𝐴‡ 𝝓ˆ‡(𝑘 − 𝐿ˆ ‡(𝑘 𝝓𝑚𝑛 − 𝑭ˆ𝑚𝑛 − 𝒇ˆ𝜙,𝑚𝑛 𝑚𝑛 𝑚𝑛 (3.23) 𝑙 𝑥 =0 is converged for every Fourier mode (𝑚 ∈ Z 𝑀 and 𝑛 ∈ Z𝑁 ) at every step of the march (𝑘 𝑥 = 1, . . . , 𝑁𝑥 ). We will use Newton’s method to iteratively force the norm 32 of the residual to zero in both an absolute and relative sense to a tolerance of 10−10 : ‡(𝑘 𝑥 +1) ∥ 𝒓ˆ𝑚𝑛 ∥ < 10−10 , ‡(𝑘 𝑥 +1) ∥ 𝒓ˆ𝑚𝑛 ∥ < 10−10 . Í𝑁 ˆ (𝑘 𝑥 +1) ∥ 2 𝑚=−𝑀 𝑛=−𝑁 ∥ 𝝓 𝑚𝑛 √︃ Í𝑀 We concatenate the residual and solution vectors as  𝒓ˆ‡(𝑘 𝑥 +1)   𝑚,−(𝑁−1)    (𝑘 𝑥 +1) .. , 𝑅𝑚 =  .   ‡(𝑘 𝑥 +1)    𝒓ˆ  𝑚,𝑁−1  𝝓ˆ‡(𝑘 𝑥 +1)   𝑚,−(𝑁−1)    (𝑘 𝑥 +1) .. , Φ𝑚 =  .   ‡(𝑘 𝑥 +1)  ˆ 𝝓   𝑚,𝑁−1  and  𝑅 (𝑘 𝑥 +1)   0   .  (𝑘 𝑥 +1) 𝑅 =  ..  ,  (𝑘 𝑥 +1)   𝑅2𝑀    In 2D with 𝑀 = 1 we obtain  𝒓ˆ‡(𝑘+1)   𝒓ˆ‡(𝑘+1)  0   0  ‡(𝑘+1)   𝒓ˆ‡(𝑘+1)  (𝑘+1) 𝑅 =  𝒓ˆ1 = ,  ‡(𝑘+1)   1   𝒓ˆ‡(𝑘+1)   𝒓ˆ   1  2  𝑚 = 0, . . . , 2𝑀, Φ (𝑘 𝑥 +1)   0   .  (𝑘 𝑥 +1) Φ =  ..  .  (𝑘 𝑥 +1)  Φ2𝑀    𝝓ˆ‡(𝑘+1)  𝝓ˆ‡(𝑘+1)  0  0   ˆ‡(𝑘+1)  𝝓ˆ‡(𝑘+1)  (𝑘+1) Φ = 𝝓1 = ,  ‡(𝑘+1)   1 𝝓ˆ  𝝓ˆ‡(𝑘+1)   2   1  (3.24) (3.25) (3.26) where we have used that ‡(𝑘+1) ‡(𝑘+1) 𝒓ˆ𝑚 = 𝒓ˆ2𝑀+1−𝑚 , 𝝓ˆ‡(𝑘+1) = 𝝓ˆ‡(𝑘+1) 𝑚 2𝑀+1−𝑚 , 𝑚 > 𝑀. Using Newton’s method, we obtain the Newton iterations 𝜕𝑅 (𝑘+1),𝜎 ΔΦ (𝑘+1),𝜎+1 = −𝑅 (𝑘+1),𝜎 , 𝜕Φ (𝑘+1),𝜎 𝜎 = 0, 1, . . . , 𝜎(𝑘+1) , (3.27) where we take Φ (𝑘+1),0 = Φ (𝑘),𝜎(𝑘 ) for 𝜎(𝑘) denoting the number of Newton iterations at streamwise station 𝑘, while the Jacobian 𝜕𝑅 (𝑘+1),𝜎 /𝜕Φ (𝑘+1),𝜎 is as defined in (F.6). Neglecting matrix blocks that are not on the diagonal, and writing the expression for 3D disturbances, we obtain the Newton iterations ‡(𝑘+1),𝜎 ‡(𝑘+1),𝜎 [𝑐 (0) 𝐴‡ − 𝐿ˆ ‡(𝑘+1),𝜎 + 𝐽ˆ00 ]Δ𝝓ˆ‡(𝑘+1),𝜎+1 = − 𝒓ˆ𝑚𝑛 , 𝑚𝑛 𝑚𝑛 (3.28) for 𝑚 = −𝑀, . . . , 𝑀, 𝑛 = −𝑁, . . . , 𝑁, and 𝜎 = 0, 1, . . . , 𝜎(𝑘+1) . If we neglect ‡(𝑘+1),𝜎 𝐽00 , then we obtain ‡(𝑘+1),𝜎 [𝑐 (0) 𝐴‡ − 𝐿ˆ ‡(𝑘+1),𝜎 ]Δ𝝓ˆ‡(𝑘+1),𝜎+1 = − 𝒓ˆ𝑚𝑛 , 𝑚𝑛 𝑚𝑛 (3.29) for 𝑚 = −𝑀, . . . , 𝑀, 𝑛 = −𝑁, . . . , 𝑁, and 𝜎 = 0, 1, . . . , 𝜎(𝑘+1) . To solve (3.28) or (3.29), we mimic the NPSE solution procedure (Bertolotti, 1991; Chang et al., 33 1993; Day, 1999): we take the lower-upper (LU) decomposition of [𝑐 (0) 𝐴‡ − 𝐿ˆ ‡(𝑘+1) ] 𝑚𝑛 ‡(𝑘+1),𝜎+1 ˆ to solve for Δ𝝓𝑚𝑛 , then we update the nonlinear term, and repeat until the residual is converged. We typically prefer solving (3.28) over (3.29) because it reduces the number of iterations to convergence, without increasing the computational cost. Alternatively, if we accept a larger computational cost, then we can further reduce the number of iterations by solving (3.27). The linear OWNS system (3.13) comprises 𝑁𝑣 + 𝑁aux = 𝑁𝑣 + (2𝑁𝑣 + 1)𝑁 𝑏 equations, where 𝑁𝑣 scales with 𝑁 𝑦 , so that the computational cost to solve this system using a direct multifrontal solver (LU decomposition) scales as O (𝑁 𝑦𝑎 𝑁 𝑏𝑎 ), where 𝑎 is a problem dependent coefficient that depends on the sparsity pattern. Theoretically, 1 < 𝑎 ≤ 3 and we typically observe 𝑎 ≈ 1.5 for 2D problems and 𝑎 ≈ 2 for 3D ones, see (Towne et al., 2022) for further details. The cost to integrate over 𝑁𝑥 stations is then O (𝑁𝑥 𝑁 𝑦𝑎 𝑁 𝑏𝑎 ), while a global method entails a cost of O (𝑁𝑥𝑎 𝑁 𝑦𝑎 ), so that OWNS is more efficient for 𝑁 𝑏 ≪ 𝑁𝑥 . The nonlinear OWNS system (3.20) comprises (𝑀 + 1) × (2𝑁 + 1) × (𝑁𝑣 + 𝑁aux ) equations, and can be solved using Newton’s method (3.27), which entails a cost of O (𝑀 𝑎 𝑁 𝑎 𝑁 𝑦𝑎 𝑁 𝑏𝑎 ). However, this cost can be reduced by instead solving (3.28) or (3.29), since we can perform the LU decomposition of [𝑐 (0) 𝐴‡ − 𝐿ˆ ‡(𝑘+1) ] separately for each Fourier mode, yielding 𝑚𝑛 𝑎 𝑎 a cost of O (𝑀 𝑁 𝑁 𝑦 𝑁 𝑏 ). Therefore, the cost to integrate the NOWNS equations is O (𝑁𝑥 𝑀 𝑁 𝑁 𝑦𝑎 𝑁 𝑏3 ) using (3.28) or (3.29), while it increases to O (𝑁𝑥 𝑀 𝑎 𝑁 𝑎 𝑁 𝑦𝑎 𝑁 𝑏3 ) for Newton’s method (3.27), as compared to the cost O (𝑁𝑥𝑎 𝑀 𝑎 𝑁 𝑎 𝑁 𝑦𝑎 ) for nonlinear global methods. Although (3.28) and (3.29) entail a lower computational cost, these methods fail for strong nonlinearities and we must instead employ Newton’s method (3.27). In practice we implement a hybrid approach whereby we first solve (3.28) to harness its reduced computational cost, and then if more than O (100) iterations have elapsed, we switch to Newton’s method to harness its better convergence properties. We compare the performance of these three methods in Appendix G.3. To reduce the computational cost of Newton’s method, we re-use the LU factors from the first iteration as a pre-conditioner for the Generalized Minimal Residual Method (GMRES), and we note that a similar procedure was performed for linear OWNS in (Araya et al., 2022). We also tested the block-Jacobi relaxation method, which entails a cost of O (𝑀 𝑁 𝑁 𝑦𝑎 𝑁 𝑏𝑎 ), and although we found this approach converged quickly in the early stages of the march, it failed as the nonlinearity grew stronger and we did not pursue relaxation methods further. 34 3.2.4 Special treatment of the zero-frequency modes We have three options for obtaining a stable march for the zero-frequency modes, as depicted in table 3.1. In the first approach, which mimics how the zero-frequency modes are handled by NPSE (Chang et al., 1993; Day, 1999), we neglect the streamwise pressure gradient, 𝜕 𝑝 0𝑛 /𝜕𝑥, and the streamwise diffusion terms, 𝜕 2 𝑞 0𝑛 /𝜕𝑥 2 , associated with the zero-frequency modes. However, in general we would prefer to avoid neglecting terms, and to instead parabolize these equations using the OWNS approach. In the second approach, we include the streamwise diffusion terms (but exclude the streamwise pressure gradient) associated with the zero-frequency modes, and parabolize the equations for all modes using OWNS. This approach yields a stable spatial march that agrees well with DNS, but offers no advantages over the first approach: for all of the cases examined in this paper, neglecting the streamwise diffusion terms associated with the zero-frequency modes does not change substantially the results of the NOWNS calculation. In the third approach, we include both the streamwise diffusion terms and the streamwise pressure gradient associated with the zero-frequency modes. Although the OWNS approach yields a stable spatial march, we have found that this approach produces inferior comparisons to DNS solutions from the literature, as discussed in Appendix G.2. We are applying NOWNS to boundary-layer flows where we wish to maintain zero mean streamwise pressure gradient. The base flow has zero streamwise pressure gradient, while the corrected mean flow will have zero streamwise pressure gradient if 𝜕 𝑝 0𝑛 /𝜕𝑥 = 0. In general, we will have 𝜕 𝑝 0𝑛 /𝜕𝑥 ≠ 0, but as long as it is small, the streamwise pressure gradient of the corrected mean flow will be nearly zero. It is possible that neglecting 𝜕 𝑝 0𝑛 /𝜕𝑥 leads to better agreement with the zero mean streamwise pressure gradient condition, which is why we observe better agreement with DNS. We further note that for flows with non-zero streamwise pressure gradients, we will likely also wish to neglect 𝜕 𝑝 0𝑛 /𝜕𝑥 = 0 since the base flow will have the desired streamwise pressure gradient. In summary, including the streamwise pressure gradient for the zero-frequency modes worsens agreement with DNS, while including the streamwise diffusion terms for the zero-frequency modes has nearly no impact on the solution. Since neglecting these terms is more computationally efficient while still providing excellent agreement with DNS, we recommend that these terms be neglected. 35 1 2 3 Include 𝜕 𝑝0𝑛 /𝜕𝑥? No No Yes Include 𝜕2 𝑞0𝑛 /𝜕𝑥 2 ? No Yes Yes Parabolized using OWNS? No Yes Yes Agreement with DNS? Yes Yes No Table 3.1: Three approaches to parabolizing the zero-frequency modes 3.2.5 Spanwise symmetry If the disturbances are symmetric, then we can enforce a symmetry condition to reduce the number of equations from (𝑀 + 1) × (2𝑁 + 1) to (𝑀 + 1) × (𝑁 + 1). ′ ′ , 𝑢ˆ ′ ′ ′ ′ All variables have even-symmetry (𝜈ˆ𝑚,−𝑛 = 𝜈ˆ𝑚,𝑛 𝑚,−𝑛 = 𝑢ˆ 𝑚,𝑛 , 𝑣ˆ 𝑚,−𝑛 = 𝑣ˆ 𝑚,𝑛 , 𝑝ˆ′𝑚,−𝑛 = 𝑝ˆ′𝑚,𝑛 ), with the exception of the 𝑤-velocity which has odd-symmetry (𝑤ˆ ′𝑚,−𝑛 = −𝑤ˆ ′𝑚,𝑛 ). 3.2.6 Boundary conditions At the wall, we impose no-slip isothermal boundary conditions (𝑢′ = 𝑣 ′ = 𝑤 ′ = 𝑇 ′ = 0) and solve for the specific volume, 𝜈′, using the (nonlinear) continuity equation. At the far-field boundary, we impose 1D (in 𝑦) inviscid Thompson characteristic boundary conditions to minimize spurious numerical reflections (Thompson, 1987), which we implement using the linearized boundary-layer flow equations. Some previous work on NPSE has used similar characteristic far-field boundary conditions (Day, 1999). Chang et al. (1993) used the far-field boundary condition 𝒒ˆ ′𝑚𝑛 (𝑦 max ) = 0 for 𝑚, 𝑛 ≠ 0. As the boundary layer must be allowed to grow in the wall-normal direction (due to nonlinear interactions), they used 𝜕 𝑣ˆ ′00 /𝜕𝑦 = 0 at 𝑦 max for the mean-flow distortion (MFD). The characteristic far-field boundary conditions are advantageous because they allow us to use the same boundary conditions for all Fouriers modes, instead of handling the MFD as a separate case. 3.2.7 Effects of the mean-flow distortion In NOWNS, the disturbances interact to excite the MFD, so that the corrected mean ¯ We have experimented with linearizing flow, 𝒒¯ + 𝒒ˆ 00 , differs from the baseflow, 𝒒. about both the baseflow and the corrected mean flow and found that it does not have a large impact on the NOWNS calculation, as discussed in Appendix G.4. Linearizing about the corrected mean flow increases the computational cost of NOWNS because the projection operators change between iterations, since the MFD changes, so the LU factorization must be updated. On the other hand, linearizing about the baseflow allows us to use the same LU factorization at each iteration because the baseflow is not affected by changes in the MFD. Since it is more computationally efficient to 36 linearize only about the baseflow, we choose this approach moving forward. 3.2.8 Recursion parameters The choice of recursion parameters is described in Sleeman et al. (2024b), and matches the recursion parameters used by Rigas et al. (2017a), which are based on the recursion parameters originally developed by Towne and Colonius (2015). We briefly discuss our strategy for selecting the recursion parameters when 𝜔 = 0 in Appendix H. 3.2.9 Modeling errors NOWNS removes upstream effects, which introduces a modeling error, since the real flow does allow waves to propagate in the upstream direction. We discuss this modeling error and its effect on the NOWNS methodology in more detail in Section 5.4.2. 3.3 Validation We validate NOWNS by applying it to 2D and 3D Blasius boundary layer flows, where the base flows are generated following the procedure outlined in Appendix B, for which there are existing DNS and NPSE results in the literature. We choose a Mach number of Ma∞ = 0.1 to study flows near the incompressible limit. We refer to modes according to their temporal frequency and their spanwise wave number as (𝑚, 𝑛), where 𝑚 refers to the frequency 𝜔𝑚 = 𝑚𝜔 and 𝑛 refers to the spanwise wave number 𝛽𝑛 = 𝑛𝛽. To be consistent with previous literature, we measure the amplitude of disturbances as ′ 𝑢′(𝑚,𝑛) max (𝑥) = 𝑐 𝑚,𝑛 max |𝑢 𝑚,𝑛 (𝑥, 𝑦)|, (3.30a) 𝑦 for    1   √  𝑐 𝑚,𝑛 = 2     2  3.3.1 𝑚 = 𝑛 = 0, 𝑚 = 0, 𝑛 ≠ 0; 𝑛 = 0, 𝑚 ≠ 0, (3.30b) otherwise. 2D evolution of a Tollmien-Schlichting wave We consider the test case developed by Bertolotti et al. (1992) which has been widely used in the literature as a validation case for NPSE (Joslin et al., 1991; Joslin et al., 1993; Paredes et al., 2015). This case examines the evolution of a TollmienSchlichting (TS) wave excited at the inlet at a frequency 𝐹 = 86×10−6 and amplitude 37 Figure 3.1: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for 2D evolution of TS wave. 𝑢′(1) max (𝑥 0 ) = 0.25%. All other Fourier components initially have zero-amplitude and are generated through nonlinear interactions with the TS wave. The grid extends over the domain 𝑅𝑒 𝑥 ∈ [1.6 × 105 , 106 ] and 𝑦 ∈ [0, 75] with 4000 stations evenly spaced in 𝑥 and 150 grid points in 𝑦, with the majority of the grid points clustered towards the wall, while the Fourier series is truncated at 𝑀 = 5 temporal modes. Figure 3.1 compares NOWNS to DNS and NPSE for 𝑢′(𝑚,𝑛) max (𝑥), while the 𝑢- and 𝑣-velocity profiles for the MFD and TS waves are shown in Figure 3.2. Excellent agreement is obtained; the discrepancy for the MFD of 𝑣 can be attributed to the Dirichlet boundary conditions used in the DNS. 3.3.2 3D oblique-wave breakdown Next we consider the oblique-wave breakdown case studied by Joslin et al. (1993), where transition is initiated by two oblique waves with opposite wave angle. They defined two cases–with small and large initial amplitude. In this section, for validation purposes, we consider the small amplitude case for which NPSE was previously successful. Oblique-wave breakdown has also been studied using both experiment and spatial DNS by Berlin et al. (1999), while it was studied using NPSE for compressible flows by Chang and Malik (1994). We further note that whereas fundamental and 38 (a) 𝑢-velocity for MFD (b) 𝑢-velocity for TS wave (c) 𝑣-velocity for MFD (d) 𝑣-velocity for TS wave Figure 3.2: 𝑢-and 𝑣-velocity profiles at streamwise coordinate 𝑅𝑒 𝑥 = 7.80 × 105 for 2D evolution of TS wave. subharmonic transition can be studied using Herbert’s secondary stability theory (Herbert, 1988), no such theory exists for oblique-wave breakdown, so that either experiment or numerical simulation is necessary to study this transition scenario (Joslin et al., 1993; Berlin et al., 1999). √ −3 The oblique waves have amplitude 𝑢′(1,1) max (𝑥 0 ) = 2 × 10 at the inlet at a frequency 𝐹 = 86 × 10−6 and spanwise wavenumber 𝑏 = 2/9 × 10−3 , while the grid extends over the domain 𝑅𝑒 𝑥 ∈ [2.74 × 105 , 6.08 × 105 ] and 𝑦 ∈ [0, 75] with 2, 000 stations evenly spaced in 𝑥 and 150 grid points in 𝑦, while the Fourier series is truncated at 𝑀 = 3 and 𝑁 = 4. Figure 3.3 compares NOWNS to DNS and NPSE for 𝑢′(𝑚,𝑛) max (𝑥), while Figure 3.4 compares the 𝑢-velocity profiles at 𝑅𝑒 𝑥 = 4.69 × 105 ; we see that we have excellent agreement between the DNS and NOWNS results for all modes. 39 Figure 3.3: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for the small-amplitude −6 , spanwise wavenumber oblique-wave breakdown case at frequency 𝐹 = 86 × 10 √ ′(1,1) 𝑏 = 2/9 × 10−3 , with initial amplitude of 𝑢 max (𝑥0 ) = 2 × 10−3 . 3.4 Advantages of the NOWNS procedure In this section, we apply NOWNS in three scenarios where NPSE is known to break down, to demonstrate that NOWNS can succeed where NPSE fails. 3.4.1 High amplitude evolution of TS wave It is well-known that NPSE can fail for sufficiently strong nonlinearities (Day, 1999; Towne et al., 2019). Here we consider a 2D test case from Scholten et al. (2024), where we modify the validation case discussed in Section 3.3.1, such that we march over the domain 𝑅𝑒 𝑥 ∈ [1.6 × 105 , 8.59 × 105 ] with 2663 stations evenly spaced in 𝑥, while the wall-normal domain remains the same as before, and the Fourier series truncate at 𝑀 = 10 temporal modes. Scholten et al. (2024) compute a high-fidelity solution using a harmonic Navier-Stokes equation (HNSE) solver, and demonstrate that NPSE agrees with HNSE with an inlet amplitude of 𝑢′(1) max (𝑥 0 ) = 0.4%, but that the NPSE march fails. In Figure Scholten et al. (2024), we plot the NOWNS amplitudes against those from HNSE and demonstrate that NOWNS marches farther than NPSE, while still being accurate, but that it too fails (by fail, we mean that the Newton solver fails to converge). This highlights that NOWNS can march farther than NPSE, while still being accurate, and that the failure of NPSE may not be related to its minimum step-size requirement or the regularization techniques it uses to suppress the upstream-going waves. As discussed in 3.2.3, the quasi-Newton method is more computationally efficient than the full Newton’s method, but takes 40 (a) MFD (b) Oblique mode (c) Vortex mode (d) First harmonic in 𝑧 Figure 3.4: 𝑢-velocity profiles at streamwise coordinate 𝑅𝑒 𝑥 = 4.69 × 105 for smallamplitude oblique-wave breakdown. more iterations to converge. In addition, the full Newton’s method is more likely to converge for stronger nonlinearities. In Figure 3.5, we indicate where each of the the quasi-Newton and full Newton methods fail. Even without employing the full Newton’s method, NOWNS is able to march farther downstream than NPSE. 3.4.2 High amplitude oblique-wave breakdown Here we apply NOWNS the high amplitude oblique-wave breakdown case studied by Joslin et al. (1993). We modify the setup of Section 3.3.2 by increasing the amplitude √︁ −2 of the oblique wave at the inlet by a factor of 10 to 𝑢′(1,1) max (𝑥 0 ) = (2) × 10 , while we change the streamwise domain so that it comprises 2,000 grid points over the domain 𝑅𝑒 𝑥 ∈ [2.73529 × 105 , 4.9 × 105 ], and the Fourier series are truncated at 𝑀 = 𝑁 = 7. We plot the amplitudes of the 𝑢-velocity as a function of streamwise station in Figure 3.6. First we note that we have excellent agreement between the 41 Figure 3.5: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for 2D evolution of TS wave with 𝑢′(1) max (𝑥 0 ) = 0.4%. amplitudes predicted by NOWNS, NPSE, and DNS for the early stages of transition. However, NPSE begins to fail towards the end of the domain, while NOWNS is able to march all the way to the end of the DNS calculation. The DNS calculation is under-resolved near the end of the domain (Joslin et al., 1993), which may explain the discrepancy between the DNS and NOWNS calculations. We further note that although the oblique-wave, (1, 1), is initially the dominant instability, it is rapidly overtaken by the vortex mode, (0, 2), leading to streaks, which we observe in the contour plots of the 𝑢-velocity (𝑢 = 𝑢¯ + 𝑢′), shown Figure 3.7. The maximum amplitude of the oblique-wave occurs at 𝑦 = 2.55 (Figure 3.7a), while the maximum amplitude of the vortex mode occurs at 𝑦 = 1.31 (Figure 3.7b). We also note that the QN solver fails before the NPSE solver. However, the NPSE calculation becomes inaccurate earlier than the QN solver fails. 3.4.3 Low amplitude oblique-wave breakdown with random noise Here we demonstrate for the low amplitude oblique wave breakdown case of Section 3.3.2 that NOWNS is robust to random noise applied to the inlet boundary condition. Given the eigenfunction from the locally parallel linear stability theory, 𝒒 LST , we add random noise, 𝒒 noise , to obtain the inlet condition 𝒒 LST + 𝜀 𝒒˜ noise . We choose complex random noise such that 𝒒˜ noise = 𝒒˜ noise,𝑟 + 𝑖 𝒒˜ noise,𝑖 , 𝒒˜ noise,𝑟 , 𝒒˜ noise,𝑖 ∼ U[0,1] , 42 Figure 3.6: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for high amplitude oblique-wave breakdown. where U[𝑎,𝑏] represents the uniform distribution over the interval [𝑎, 𝑏]. We then normalize the noise to obtain 𝑞 noise , such that the maximum amplitude of the 𝑢-velocity noise is equal to the free-stream 𝑢-velocity, 𝑈∞ . We recall that the √ √ amplitude of 𝒒 LST is 2 × 10−3 , and we choose 𝜀 = 2 × 10−5 . In Figure 3.8, we plot the profile of the 𝑢- and 𝑣-velocities, as well as the thermodynamic variables. We see that the random noise has a relatively small effect on the 𝑢-velocity, and a slightly more pronounced effect on the 𝑣-velocity, while it has a larger impact on the thermodynamic variables. Like the 𝑢-velocity profile, the random noise has a relatively small effect on the 𝑤-velocity profile, so we omit this plot. The NOWNS march succeeds even if we introduce large disturbances to the velocity fields, but the 𝑢′ amplitudes differ substantially from the noise-less case due to the large perturbations. Therefore, we instead introduce relatively large random disturbances to the thermodynamic variables, and relatively small ones to the velocity field. In Figure 3.9, we see that the amplitudes predicted by NOWNS for the noisy inlet condition agree closely with those without noise. We also see that although NPSE initially is able to accurately predict the evolution of the 𝑢-velocity amplitudes, it eventually becomes inaccurate (especially for 𝑢′(0,0) and 𝑢′(1,1) ) before failing. We also plot the contours of the real part of the 𝑢- and 𝑣-velocities of the oblique wave, with and without noise in Figure 3.10. We see that despite the noisy inlet condition, NOWNS evolves the 𝑢-velocity of the oblique wave such that it matches closely the case without noise. On the other hand, the 𝑣-velocity is more 43 (a) 𝑦 = 2.55 (b) 𝑦 = 1.31 Figure 3.7: Contours of the instantaneous 𝑢-velocity for high amplitude obliquewave breakdown. 44 Figure 3.8: Oblique-wave inlet boundary condition with and without random noise. affected by the numerical noise, yet we still obtain good qualitative agreement. The other modes (e.g., the vortex mode) are evolved accurately by the NOWNS calculation with noisy inlet condition, and the contour plots with and without the numerical noise are indistinguishable from each other, and so are not plotted here. 3.4.4 Blowing/suction strip Blowing/suction strips are frequently used to study laminar-turbulent boundary layer transition in low-speed boundary layer flows (Fasel et al., 1990; Rist and Fasel, 1995; Sayadi et al., 2013; Huai et al., 1997; Rigas et al., 2021). Here, we introduce disturbances by specifying a non-zero wall-normal velocity such that 𝑣(𝑦 = 0) = 𝑓 (𝑥, 𝑧, 𝑡), for some function 𝑓 (𝑥, 𝑧, 𝑡) that is periodic in 𝑡 and 𝑧. NPSE does not support blowing/suction strips because they introduce non-modal disturbances, demonstrating an advantage of NOWNS. However, we note that Herbert’s second stability theory yields an inlet boundary condition so that NPSE (and NOWNS) 45 Figure 3.9: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for low amplitude oblique-wave breakdown with and without random noise. Figure 3.10: Contour plots of the 𝑢- and 𝑣 velocities of the oblique wave, with and without random noise. 46 can be used to study fundamental (K-type) and subharmonic (H-type) transition (Herbert, 1988, 1997). Rist and Fasel (1995) used DNS with a blowing/suction strip to study K-type transition (Rist and Fasel, 1995), while similar studies were performed by Sayadi et al. (2013) using DNS (Sayadi et al., 2013), and by Rigas et al. (2021) using a harmonic balance method (HBM) (Rigas et al., 2021). The blowing/suction strip is given by 𝑓 (𝑥, 𝑧, 𝑡) = 5 × 10−3 sin(𝜔𝑡)𝑣 𝑎 (𝑥) + 1.3 × 10−4 cos(𝛽𝑧)𝑣 𝑠 (𝑥), (3.31a) where   0, 𝑅𝑒 𝑥 ≤ 𝑅𝑒 𝑥 (𝑥 1 )      15.1875𝜉 5 − 35.4375𝜉 4 + 20.25𝜉 3 , 𝑅𝑒 𝑥 (𝑥 1 ) < 𝑅𝑒 𝑥 ≤ 𝑅𝑒 𝑥 (𝑥 𝑚 )  𝑣 𝑎 (𝑅𝑒 𝑥 ) =
 −𝑣 2𝑅𝑒 (𝑥 ) − 𝑅𝑒 , 𝑅𝑒 𝑥 (𝑥 𝑚 ) < 𝑅𝑒 𝑥 ≤ 𝑅𝑒 𝑥 (𝑥 2 )  𝑎 𝑥 𝑚 𝑥     0, 𝑅𝑒 𝑥 (𝑥 2 ) < 𝑅𝑒 𝑥  (3.31b)   0, 𝑅𝑒 𝑥 ≤ 𝑅𝑒 𝑥 (𝑥 1 )     4 3   −3𝜉 + 4𝜉 , 𝑅𝑒 𝑥 (𝑥 1 ) < 𝑅𝑒 𝑥 ≤ 𝑅𝑒 𝑥 (𝑥 𝑚 ) 𝑣 𝑠 (𝑅𝑒 𝑥 ) = (3.31c)
 𝑣 2𝑅𝑒 (𝑥 ) − 𝑅𝑒 , 𝑅𝑒 (𝑥 ) < 𝑅𝑒 ≤ 𝑅𝑒 (𝑥 )  𝑠 𝑥 𝑚 𝑥 𝑥 𝑚 𝑥 𝑥 2     0, 𝑅𝑒 𝑥 (𝑥 2 ) < 𝑅𝑒 𝑥  for 𝑅𝑒 𝑥 (𝑥 1 ) = 1.3438 × 105 , 𝑅𝑒 𝑥 (𝑥 2 ) = 1.5532 × 105 , 𝑥 𝑚 = (𝑥 1 + 𝑥 2 )/2, and 𝜉 = (𝑅𝑒 𝑥 − 𝑅𝑒 𝑥 (𝑥1 ))/(𝑅𝑒 𝑥 (𝑥 𝑚 ) − 𝑅𝑒 𝑥 (𝑥 1 )). We choose 𝐹 = 110 × 10−6 and 𝑏 = 0.423 × 10−3 with 𝑀 = 𝑁 = 4, while the grid extends over 𝑅𝑒 𝑥 ∈ [1.33956 × 105 , 2.72 × 105 ] and 𝑦 ∈ [0, 60], with 1,300 stations in 𝑥 and 100 grid points in 𝑦. Figure 3.11 shows excellent agreement between the 𝑢′ amplitudes of NOWNS and the DNS of Rist and Fasel (1995) (Rist and Fasel, 1995). We note that in the early stages of the march there is disagreement between the DNS and NOWNS calculations because the blowing/suction strip causes upstream effects that NOWNS neglects by construction. However, these disturbances are convective in nature, and the amplitudes predicted by NOWNS rapidly converge to those predicted by DNS as the march progresses downstream. 3.5 Summary In this chapter we developed the NOWNS approach and validated for low-speed flat-plate boundary-layer flows by comparing to DNS data in the literature for the 2D evolution of a TS, 3D oblique-wave breakdown, and a fundamental (K-type) transition case. In addition, we demonstrated that NOWNS supports stronger nonlinearities than NPSE. For the 2D high amplitude evolution of a TS wave, we find 47 Figure 3.11: Amplitude of 𝑢′ v.s. streamwise coordinate, 𝑅𝑒 𝑥 , for K-type transition that the NOWNS march remains accurate until the point where the Newton solver fails to converge, while for the 3D high amplitude oblique-wave breakdown case, NOWNS marches all the way to the end of the DNS calculation. We also show that NOWNS is robust to numerical noise in the inlet boundary-condition, and that a blowing/suction strip can be used to introduce disturbances, which is advantageous because blowing/suction strips are frequently used in DNS studies but cannot be used in NPSE calculations. 48 Chapter 4 A GREEDY APPROACH TO RECURSION PARAMETER SELECTION As discussed in Chapters 2 and 3, previous work on OWNS has chosen recursion parameters heuristically, based on the Euler equations, linearized about a uniform flow. This section proposes a greedy algorithm for recursion parameter selection, and we demonstrate that it achieves rapid error convergence relative to the heuristic approach for low-speed boundary-layer flows. This section is adapted from Sleeman and Colonius (2025). 4.1 Greedy algorithm for recursion parameter selection Assuming 𝛼𝑚 ≠ 𝛼𝑛 for all 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) , propositions 2.2.16 and 2.2.31 define recursion parameter sets that ensure convergence. However, these sets are large and require the eigenvalues of 𝑀, so previous work has used heuristic parameter selection to avoid this eigendecomposition while maintaining stability and accuracy with 𝑁 𝛽 ≪ 𝑁. Although it has been used successfully, heuristic selection converges slowly, so we propose a greedy algorithm to accelerate convergence. 4.1.1 Objective function 𝑗 𝑗 𝑁 𝑗 𝛽 We define 𝛽± = {𝛽+ , 𝛽−𝑗 } and the recursion parameter set Ξ𝑁 𝛽 ≡ {𝛽± } 𝑗=0 , as well as the functions Ĵ+ (𝛼; Ξ𝑁 𝛽 ) ≡ 𝑁Ö 𝛽 −1 𝑗 |𝛼 − 𝛽+ | , 𝑗 |𝛼 − 𝛽 | − 𝑗=0 and Ĵ (𝛼𝑚 , 𝛼𝑛 ; Ξ𝑁 𝛽 ) ≡ Ĵ− (𝛼; Ξ𝑁 𝛽 ) ≡ 𝑁Ö 𝛽 −1 |𝛼 − 𝛽−𝑗 | , 𝑗 𝑗=0 |𝛼 − 𝛽+ | 𝑁Ö 𝛽 −1 |𝛼𝑚 − 𝛽+ | |𝛼𝑛 − 𝛽−𝑗 | 𝑗=0 |𝛼𝑚 − 𝛽− | |𝛼𝑛 − 𝛽+ | (4.1a) 𝑗 𝑗 . 𝑗 (4.1b) Then, by proposition 2.2.17 the OWNS-P error scales as J (Ξ𝑁 𝛽 ) ≡ max 𝑚=𝑖 (+) ,𝑛=𝑖 (−) −1 Ĵ (𝛼𝑚 , 𝛼𝑛 ; Ξ𝑁 𝛽 ) = ∥𝐹++ ∥∥𝐹−− ∥, while by proposition 2.2.34 the OWNS-R error scales as  J (𝑅) (Ξ𝑁 𝛽 ) ≡ max max Ĵ+ (𝛼𝑚 ; Ξ𝑁 𝛽 ), max Ĵ− (𝛼𝑚 ; Ξ𝑁 𝛽 ) 𝑚=𝑖 (+) 𝑛=𝑖 (−)  −1 = max ∥𝐹++ ∥, ∥𝐹−− ∥ . (4.1c) (4.1d) 49 4.1.2 A summary of the greedy algorithm By Propositions 2.2.17 and 2.2.34, the OWNS-P and OWNS-R errors scale linearly with J (Ξ𝑁 𝛽 ) and J (𝑅) (Ξ𝑁 𝛽 ) for small values, respectively. Thus, for two recursion parameter sets Ξ𝑁(1)𝛽 and Ξ𝑁(2)𝛽 , we expect Ξ𝑁(1)𝛽 to introduce less error if J (Ξ𝑁(1)𝛽 ) < J (Ξ𝑁(2)𝛽 ). We seek Ξ𝑁 𝛽 such that J is minimized for 𝑁 𝛽 ≪ 𝑁, which is well-posed as an optimization problem due to the continuity of the objective function (a quotient of polynomials with a non-zero denominator). However, identifying the optimal set is computationally intractable because we must consider 𝑁− ! 𝑁+ ! 𝑁 𝛽 !(𝑁+ − 𝑁 𝛽 )! 𝑁 𝛽 !(𝑁− − 𝑁 𝛽 )! combinations of recursion parameters, since we must choose 𝑁 𝛽 recursion parameters from the 𝑁+ downstream-going and the 𝑁− upstream-going eigenvalues. Therefore, we use the greedy algorithm outlined in algorithm 1, which we briefly summarize below, to choose quasi-optimal parameter sets. Given a recursion parameter set Ξ𝑁greedy with 𝑁greedy < 𝑁 𝛽 , we can add two new 𝑁 𝑁 recursion parameters, (𝛽+ greedy , 𝛽− greedy ), to reduce the error. We compute 𝑚˜ = arg max Ĵ+ (𝛼𝑚 ; Ξ𝑁greedy ), 𝑖=𝑖 (+) 𝑁 𝑛˜ = arg max Ĵ− (𝛼𝑛 ; Ξ𝑁greedy ), 𝑖=𝑖 (−) 𝑁 and choose 𝛽+ greedy = 𝛼𝑚˜ and 𝛽− greedy = 𝛼𝑛˜ . This choice does not necessarily lead to the optimal error decrease, but it locally eliminates the largest source. Remark 4.1.1 Previous work on OWNS combined heuristic parameter selection with mode-tracking, where important discrete modes (e.g., Kelvin-Helmholtz in freeshear flows or Mack’s second mode in high-speed boundary layers) were added to the recursion parameter set to improve approximation accuracy (Towne and Colonius, 2015; Towne et al., 2022; Zhu and Towne, 2023). While previous methods manually identified eigenvalues to track, the present work proposes an automatic approach with minimal user intervention. Remark 4.1.2 The greedy algorithm can be modified so that it is initialized with user-specified recursion parameters instead of randomly chosen recursion parameters. For example, choose 𝛽+0 corresponding to Mack’s second mode, and 𝛽−0 as the upstream-going mode closest to the real axis. 50 Algorithm 1 Greedy recursion parameter selection 𝛽+0 ← 𝛼𝑚 for randomly chosen 𝑚 = 𝑖 (+) 𝛽−0 ← 𝛼𝑛 for randomly chosen 𝑛 = 𝑖 (−) Ξ𝑁greedy ← {𝛽±0 } where 𝛽±0 = {𝛽+0 , 𝛽−0 } 𝑁greedy ← 1 while 𝑁greedy < 𝑁 𝛽 do 𝑚˜ ← arg max𝑚=𝑖 (+) Ĵ+ (𝛼𝑚 ; Ξ𝑁greedy ) 𝑛˜ ← arg max𝑛=𝑖 (−) Ĵ− (𝛼𝑛 ; Ξ𝑁greedy ) 𝑁 𝛽± greedy ← {𝛼𝑚˜ , 𝛼𝑛˜ } 𝑗 𝑁greedy Ξ𝑁greedy +1 ← {𝛽± } 𝑗=0 𝑁greedy ← 𝑁greedy + 1 end while 4.1.3 Numerical stability of the recursive filter OWNS solves equations of the form (𝑀 − 𝑖𝛽−𝑗 ) 𝝓ˆ 𝑗+1 = (𝑀 − 𝑖𝛽+ ) 𝝓ˆ 𝑗 , 𝑗 𝑗 = 0, . . . , 𝑁 𝛽 − 1. If the exact solution is given by 𝝋ˆ 𝑗+1 , then the rounding error introduced by solving the linear system is bounded by ∥(𝑀 − 𝑖𝛽−𝑗 𝐼) 𝝓ˆ 𝑗+1 − (𝑀 − 𝑖𝛽+ 𝐼) 𝝓ˆ 𝑗 ∥ ∥ 𝝓ˆ 𝑗+1 − 𝝋ˆ 𝑗+1 ∥ 𝑗 ≤ 𝜅(𝑀 − 𝑖𝛽 𝐼) − ∥(𝑀 − 𝑖𝛽+ 𝐼) 𝝋ˆ 𝑗 ∥ ∥ 𝝋ˆ 𝑗+1 ∥ 𝑗 ∥𝑀 − 𝑖𝛽−𝑗 𝐼 ∥∥ 𝝓ˆ 𝑗+1 − 𝝓ˆ 𝑗 ∥ + |𝛽+ − 𝛽−𝑗 |∥ 𝝓ˆ 𝑗 ∥ ≤ 𝜅(𝑀 − 𝑖𝛽−𝑗 𝐼) . ∥(𝑀 − 𝑖𝛽+ 𝐼) 𝝋ˆ 𝑗 ∥ 𝑗 If 𝛽+ = 𝛽−𝑗 , then ∥ 𝝓ˆ 𝑗+1 − 𝝓ˆ 𝑗 ∥ = O (𝜖mach ), where 𝜖mach is machine epsilon, and 𝑗 the error is O (𝜖mach 𝜅(𝑀 − 𝑖𝛽−𝑗 𝐼)). However, this error grows as |𝛽+ − 𝛽−𝑗 | grows, 𝑗 and accumulates for 𝑗 = 0, . . . , 𝑁 𝛽 − 1. Therefore, it is preferable for |𝛽−𝑗 − 𝛽+ | 𝑗−1 𝑗 to be small, so we re-order the recursion parameters such that |𝛽+ | < |𝛽+ | and |𝛽−𝑗−1 | < |𝛽−𝑗 | for 𝑗 = 1, . . . , 𝑁 𝛽 − 1. We further note that the error bounds and objective functions do not depend on the order of the recursion parameters, so we are free to re-order them, which is necessary when using greedy selection. Remark 4.1.3 Heuristic parameter selection does not explicitly sort the recursion parameters, but in practice they are chosen such that each pair has the same 𝑗 magnitude (|𝛽+ | ≈ |𝛽−𝑗 |). 4.1.4 Greedy algorithm and spatial marching For systems where 𝑀 varies slowly in 𝑥, the recursion parameters computed at the inlet can be reused downstream. However, they generally need to be updated to 51 keep the error low, which can be done by tracking them as they evolve downstream, as shown in algorithm 2. When the number of upstream- and downstream-going characteristics changes, such as supersonic boundary-layer flows when the number of subsonic and supersonic grid points changes, it can be advantageous (although not necessary) to perform the greedy algorithm again to keep the error low. Consider the recursion parameters, Ξ𝑁(𝑖−1) , at station 𝑖 − 1, which are eigenvalues 𝛽 (𝑖−1) of 𝑀 . If the number of upstream- and downstream-going characteristics is the same at stations 𝑖 −1 and 𝑖, then given 𝑀 (𝑖) , the recursion parameters from Ξ𝑁(𝑖−1) can 𝛽 be used with, for example, MATLAB’s eigs function to update to Ξ𝑁(𝑖)𝛽 . For large, sparse 𝑀, this is much less costly than solving for the complete eigenspectrum. If the number of characteristics changes, the greedy algorithm is performed again, as shown in algorithm 2. Algorithm 2 Recursion parameter tracking Compute 𝑁+(1) and 𝑁−(1) Compute Ξ𝑁(1)𝛽 using algorithm 1 with the eigenvalues of 𝑀 (1) for 𝑖 = 2, . . . , 𝑁𝑥 do Compute 𝑁+(𝑖) and 𝑁−(𝑖) if 𝑁+(𝑖) = 𝑁+(𝑖−1) then for 𝑗 = 0, . . . , 𝑁 𝛽 do 𝑗 𝛽˜± ← 𝛽± from Ξ𝑁(𝑖−1) 𝛽 Compute 𝛽+ by finding eigenvalue of 𝑀 (𝑖) closest to 𝛽˜+ Compute 𝛽−𝑗 by finding eigenvalue of 𝑀 (𝑖) closest to 𝛽˜− 𝑗 𝑗 𝛽± ← {𝛽+ , 𝛽−𝑗 } end for 𝑗 𝑁 𝛽 −1 Ξ𝑁(𝑖)𝛽 ← {𝛽± } 𝑗=0 else Compute Ξ𝑁(𝑖)𝛽 using algorithm 1 with the eigenvalues of 𝑀 (𝑖) end if end for 𝑗 4.1.5 Cost trade-off OWNS avoids computing the eigendecomposition of 𝑀 by using a recursive filter, whereas the greedy algorithm selects recursion parameters based on these eigenvalues. While this may seem contradictory, the greedy approach can reduce the overall computational cost by requiring fewer recursion parameters, thereby lowering the cost of solving the OWNS system. Moreover, the full eigenspectrum is needed only at a small number of stations, potentially just at the inlet, and the selected eigenval- 52 ues can be efficiently updated by tracking them downstream. Moreover, as noted in remark 4.1.1, this approach has previously been employed for eigenvalues chosen manually by the user. For linear calculations, OWNS-R can reduce computational cost relative to OWNSP, though selecting appropriate recursion parameters is more challenging. We show that OWNS-P with greedy selection outperforms OWNS-P with heuristic selection, and that OWNS-R with greedy selection is more efficient than both. In optimization tasks, such as the resolvent analysis by Towne et al. (2022), the initial cost of computing recursion parameters can be amortized over many optimization iterations. For nonlinear calculations, we demonstrate that NOWNS with greedy selection is more efficient than with heuristic selection. Remark 4.1.4 This work focuses on flows with a single inhomogeneous direction (normal). Computing the eigenvalues of 𝑀 for flows with two inhomogeneous directions (normal and spanwise) may not be feasible, making heuristic selection the preferred method for these flows. 4.2 Demonstration of the greedy algorithm for a single station Here we demonstrate the greedy algorithm for a two-dimensional (2D) subsonic boundary-layer flow, where the disturbance frequency is 𝐹 = 86 × 10−6 , while the wall-normal extent is 𝑦 ∈ [0, 75] with 𝑁 𝑦 = 150 grid points, which matches the parameters used by Bertolotti et al. (1992). For now, we focus only on comparing the greedy and heuristic parameters at 𝑅𝑒 𝑥 = 1.6 × 106 . We show the following: 1. Greedy parameter selection yields a much more rapid convergence of the objective functions for both OWNS-P and OWNS-R. 2. Heuristic parameter selection yields a stable march for OWNS-P, while the same recursion parameters yield an unstable march for OWNS-R. 𝑗 3. Rounding errors when computing 𝛽∗ for OWNS-R lead to increasing error with increasing 𝑁 𝛽 , while the OWNS-P error is zero to machine zero-precision. 4.2.1 Convergence of the objective function Figure 4.1a shows that although the J decreases with increasing 𝑁 𝛽 for both greedy and heuristic parameter selection, it decreases far more rapidly for greedy selection (although greedy selection leads to larger values of J for small 𝑁 𝛽 ). In contrast, for 53 (a) OWNS-P (b) OWNS-R Figure 4.1: Convergence of the objective function for greedy and heuristic parameter selection. Figure 4.2: Convergence of the objective function, split into upstream- and downstream-going components, for greedy and heuristic parameter selection. OWNS-R, heuristic selection leads to an increase in J (𝑅) , while greedy selection leads to a rapid decrease. Figure 4.2 plots both Ĵ+ and Ĵ− , showing that although Ĵ− is decreasing for heuristic selection, Ĵ+ is increasing, which causes J (𝑅) to increase. This highlights that OWNS-P will always have a lower error than OWNS-R small −1 ∥, since ∥𝐹++ ∥ and ∥𝐹−− −1 −1 ∥𝐹++ ∥∥𝐹−− ∥ < max{∥𝐹++ ∥, ∥𝐹−− ∥}, −1 ∥𝐹++ ∥, ∥𝐹−− ∥ < 1. Figure 4.3 shows the greedy and heuristic recursion parameter sets for 𝑁 𝛽 = 20. The greedy algorithm places parameters along the branches of acoustic and vortical waves, focusing near the origin where the eigenvalues are closest, and uses the discrete Tollmien-Schlichting (TS) wave as a recursion parameter. The heuristic approach targets the same eigenvalues but is less efficient, with many parameters far from their targets, which explains its slower convergence. 54 (a) Heuristic (b) Greedy Figure 4.3: Recursion parameters plotted against spectrum for heuristic and greedy recursion parameter selection with 𝑁 𝛽 = 20. 4.2.2 Stability of the march All upstream-going modes must be removed for the march to be stable. Figure 4.4a demonstrates for heuristic parameter selection that 𝑃 𝑁 𝛽 removes all upstream-going waves from 𝑀, while Figure 4.4b demonstrates that 𝑃 𝑁(𝑅) does not. For 𝜂 = 0, 𝛽 we observe that the OWNS-R spectrum has many eigenvalues with I (𝛼) < 0, and taking 𝜂 = 1000 allows us to verify that these are upstream-going modes according to Briggs’ criterion. Therefore, the exist parameter sets where the march is stable for OWNS-P, but unstable for OWNS-R. Repeating this analysis with greedy selection yields stable marches for both OWNS-P and OWNS-R. Remark 4.2.1 Zhu and Towne (2023) noted that OWNS-P and OWNS-R approximate the 𝑉 and 𝐸, respectively. Therefore, we expect OWNS-P to better remove upstream-going modes since it makes no approximations to 𝐸, which is consistent with the above observations. Remark 4.2.2 Figures 3 and 6 from Zhu and Towne (2023) show that the OWNSR operator removes all upstream-going modes for the dipole and jet test cases, while Figure 10 suggests it still supports upstream-going modes for their supersonic boundary-layer case. Thus, our observations in Figure 4.4b for the low-speed boundary-layer flow align with those of Zhu and Towne (2023) for their supersonic case. 55 (a) OWNS-P, 𝑁 𝛽 = 20 (b) OWNS-R, 𝑁 𝛽 = 55 Figure 4.4: Comparison of OWNS spectrum with heuristic parameter selection for 𝜂 = 0 and 𝜂 = 1000. According to Briggs’ criterion, I (𝛼) < 0 for large values of 𝜂 indicates a downstream-going mode. For 𝜂 = 0, we wish to have I (𝛼) ≥ 0 for the march to be stable. We achieve a stable march using OWNS-P, but not for OWNS-R. 4.2.3 Accuracy of the projection operator Consider a solution 𝝓ˆ = 𝑉+ 𝝍ˆ + + 𝑉− 𝝍ˆ − , where 𝑃 𝝓ˆ = 𝑉+ 𝝍ˆ + by definition. We choose random coefficients 𝝍ˆ and compute the relative errors ∥𝑃 𝑁(𝑅) [𝑉+ 𝝍+ + 𝑉− 𝝍− ] − 𝑉+ 𝝍+ ∥ 𝛽 ∥𝑃 𝑁 𝛽 [𝑉+ 𝝍+ + 𝑉− 𝝍− ] − 𝑉+ 𝝍+ ∥ , ∥𝑉+ 𝝍+ ∥ , ∥𝑉+ 𝝍+ ∥ (4.2) for OWNS-P and OWNS-R, respectively. Similarly, we consider specifically the ability to retain only the TS wave and compute the relative errors ∥𝑃 𝑁(𝑅) 𝝓TS − 𝝓TS ∥ 𝛽 ∥𝑃 𝑁 𝛽 𝝓TS − 𝝓TS ∥ ∥𝝓TS ∥ , ∥𝝓TS ∥ . (4.3) Figure 4.5a shows for OWNS-P that greedy selection yields rapid error convergence, and that machine zero precision is achieved at 𝑁 𝛽 = 29, while heuristic selection yields slow convergence. For OWNS-R, we observe that the error is large compared to OWNS-P, and that although the error initially decreases with increasing 𝑁 𝛽 , it begins to grow again around 𝑁 𝛽 = 45 for greedy selection and 𝑁 𝛽 = 55 for heuristic. Figures 4.5a and 4.5b show that whereas the OWNS-P projection error is (nearly) a monotonically decreasing function of 𝑁 𝛽 , the OWNS-R projection error is an increasing function of 𝑁 𝛽 for sufficiently large values of 𝑁 𝛽 (𝑁 𝛽 ≈ 45 and 𝑁 𝛽 ≈ 55 for greedy and heuristic parameter selection, respectively). Figure 4.5b shows that although J (𝑅) is decreasing rapidly for greedy selection, the projection error is 56 (a) OWNS-P (b) OWNS-R (c) OWNS-P, TS wave only (d) OWNS-R, TS wave only Figure 4.5: Error convergence as a function of 𝑁 𝛽 for OWNS-P and OWNS-R for the 2D low-speed boundary-layer flow. 𝑗 increasing due to rounding error when computing 𝛽∗ . We demonstrate this in Figure 4.6 by plotting an estimate for the relative error in the polynomial factorization, which we compute by choosing random complex numbers 𝑐 𝑘 = 𝑎 𝑘 + 𝑖𝑏 𝑘 with 𝑎 𝑘 , 𝑏 𝑘 ∈ [−1, 1] for 𝑘 = 1, . . . , 100 and computing the error in the polynomial approximation as Î𝑁 𝛽 Î𝑁 𝛽 Î𝑁 𝛽 𝑗 𝑗 |2 𝑗=1 (𝑐 𝑘 − 𝛽∗ ) − 𝑗=1 (𝑐 𝑘 − 𝛽−𝑗 ) − 𝑗=1 (𝑐 𝑘 − 𝛽+ )| max . (4.4) Î𝑁 𝛽 Î𝑁 𝛽 𝑗 𝑗 𝑘=1,...,100 | 𝑗=1 (𝑐 𝑘 − 𝛽− ) − 𝑗=1 (𝑐 𝑘 − 𝛽+ )| Figure 4.6 shows that this error is an increasing function of 𝑁 𝛽 , which explains why adding more recursion parameters for sufficiently large values of 𝑁 𝛽 increases, rather than decreases the OWNS-R projection error. OWNS-R can lead to a reduced computational cost relative to OWNS-P, but this is helpful only if the OWNS-R march is stable and accurate. Therefore, future work should investigate how to perform the polynomial factorization such that the effects of rounding errors on the accuracy of the OWNS-R approximation are minimized. 57 𝑗 Figure 4.6: For OWNS-R, the error in computing 𝛽∗ increases with increasing 𝑁 𝛽 . (a) OWNS-P (b) OWNS-R Figure 4.7: Error convergence as a function of 𝑁 𝛽 for OWNS-P and OWNS-R for the low-speed oblique wave case. 4.2.4 3D low-speed boundary-layer flow We investigate the oblique-wave case studied by Joslin et al. (1993), with disturbance frequency and wavenumber are 𝐹 = 86×10−6 and 𝑏 = 0.222×10−3 , respectively, for the wall-normal domain 𝑦 ∈ [0, 60] with 𝑁 𝑦 = 100 at 𝑅𝑒 𝑥 = 2.74 × 105 . Figure 4.7 shows that greedy selection yields rapid convergence of the objective function and projection error. Moreover, OWNS-P achieves machine-zero error with 𝑁 𝛽 = 24, 𝑗 while OWNS-R does not, due to rounding errors in computing 𝛽∗ . We further note that the 3D case converges with fewer recursion parameters because the upstreamand downstream-going branches are more widely separated than in the 2D case, which makes it easier to choose recursion parameters that keep both ∥𝐹++ ∥ and −1 ∥ small, as noted by Towne and Colonius (2015). ∥𝐹−− 58 4.3 Demonstration of the greedy algorithm for spatial marching The greedy algorithm is advantageous because it yields a rapidly convergent OWNS approximation without the need to manually choose flow-dependent recursion parameters. However, this requires that the eigenvalues of 𝑀 be computed, which adds a cost that is not present in the heuristic approach. Here we demonstrate that the overall computational cost of spatial-marching using OWNS can be reduced using the greedy selection (algorithm 1) and tracking (algorithm 2) procedures. 4.3.1 Linear calculation We first demonstrate, for the oblique-wave case studied by Joslin et al. (1993) and discussed in Section 4.2.4, that greedy parameter selection reduces computational cost in both linear and nonlinear calculations. The problem parameters remain the same as before, and we consider the streamwise domain 𝑅𝑒 𝑥 ∈ [2.74, 6.08] × 105 with 𝑁𝑥 = 2000 stations. We compare the number of recursion parameters required for convergence and measure speed-up based on total wall-clock time, including all steps of the greedy selection process. Due to instability in OWNS-R with heuristic parameter selection for the linear evolution of the oblique wave, we compare OWNS-P and OWNS-R with greedy selection to OWNS-P with heuristic selection. To ensure meaningful comparisons, we select 𝑁 𝛽 as the smallest value that yields 10%, 1%, and 0.1% relative error in the 𝑁-factor, measured against the OWNS-P greedy calculation with 𝑁 𝛽 = 30. We choose to measure error using the 𝑁-factor. Table 4.1 summarizes the data, and shows that the speed-up of OWNS-R with greedy parameter selection is most pronounced when the desired error is 1%, while for OWNS-P with greedy parameter selection it is 0.1%. If a tight error tolerance is desired, then OWNS-P is preferable, while OWNS-R is preferable for looser error tolerances. We further that the greedy approach yields a net decrease in computational cost, despite the added cost of computing the eigenvalues, as it allows convergence with fewer recursion parameters. 4.3.2 Nonlinear calculation For the nonlinear calculations, we truncate the Fourier series at 𝑀 = 3 temporal and 𝑁 = 4 spanwise modes, while we specify at the inlet that the oblique wave √ 2 × 10−3 . We again select 𝑁 𝛽 as the smallest value has amplitude 𝑢′(1,1) max = for which the relative error in the 𝑁-factor is 0.1% with respect to the reference solution. For NOWNS, heuristic and greedy parameter selection require 𝑁 𝛽 = 18 and 𝑁 𝛽 = 6, respectively, resulting in a speed-up of 3.83 for the greedy approach 59 Error 10% 1% 0.1% P-H 11 17 18 𝑁𝛽 P-G 5 5 5 R-G 29 29 36 P-H 1.0 1.0 1.0 Speed-up P-G R-G 2.3 3.6 3.9 6.0 4.2 4.9 Table 4.1: Speed-up achieved by using greedy parameter selection with OWNS-P (P-G) and OWNS-R (R-G) relative to heuristic parameter selection with OWNS-P (P-H) for different relative errors in the 𝑁-factor for the 3D low-speed boundarylayer flow. Figure 4.8: NOWNS for oblique-wave breakdown, comparing greedy and heuristic recursion parameter selection with 𝑁 𝛽 = 6. Although NOWNS with greedy selection is sufficiently converged, heuristic selection is not, and the solution blows-up. relative to the heuristic. Figure 4.8 compares solutions obtained using NOWNS with both parameter selection methods using 𝑁 𝛽 = 6. While greedy selection yields a converged solution, the heuristic does not and the solution diverges. 4.4 Summary We demonstrated that our proposed greedy algorithm can yield faster convergence and a reduced computational cost, relative to heuristic parameter selection, for linear and nonlinear stability analyses of boundary-layer flows. We have focused on flows with a single inhomogeneous direction (the wall-normal direction), and we note that the approach may not extend directly to flows with two inhomogeneous directions (wall-normal and spanwise) due to the increased grid size. Future work should explore how to adapt the greedy algorithm for such cases. Additionally, future work should investigate the development of a posteriori error estimates so that we can estimate the error introduced by a recursion parameter set without needing to 60 calculate the eigen-spectrum of 𝑀, which may also make it possible to indirectly implement the greedy algorithm without solving for the eigenvalues of 𝑀. 61 Chapter 5 THE NOWNS APPROACH FOR LOW-SPEED BOUNDARY-LAYER TRANSITION PREDICTION In Chapter 3 we demonstrated that NOWNS can replicate the K-type transition scenario simulated using DNS by Rist and Fasel (1995). More recently, Sayadi et al. (2013) computed the complete transition to turbulence for the H- and K-type paths. We show that NOWNS matches the 𝑢-velocity amplitudes predicted by Sayadi et al. (2013) in the early stages of transition, and that it is able to qualitatively reproduce their 𝑢-velocity contour plots demonstrating the aligned (staggered) Λvortex structure that is characteristic of K-type (H-type) transition. The NOWNS march fails before the flow becomes fully-turbulent, but we plot the mean skin friction coefficient to demonstrate that NOWNS is able to accurately predict the initial rise in skin-friction coefficient, relative to DNS, during the onset of laminarturbulent transition. This section is adapted from Sleeman et al. (2024a). We further note that Fasel et al. (1990) and Rist and Fasel (1995) have previously compared their DNS results to experimental results in the literature. Since we are applying NOWNS to these DNS cases, we forego discussions of experimental results and refer to Fasel et al. (1990) and Rist and Fasel (1995). 5.1 K-type transition For K-type transition, we use the blowing/suction function specified in (3.31), which corresponds to the function used in Rigas et al. (2021) and Rist and Fasel (1995). However, to match the calculations of Sayadi et al. (2013), we modify the parameters such that 𝑅𝑒 𝑥 (𝑥 1 ) = 1.36 × 105 , 𝑅𝑒 𝑥 (𝑥 2 ) = 1.56 × 105 , 𝐹 = 110 × 10−6 and 𝑏 = 0.419 × 10−3 to match Sayadi et al. (2013). We further choose 𝑀 = 4 temporal modes and 𝑁 = 8 spanwise modes, with 1,300 stations over the streamwise domain 𝑅𝑒 𝑥 ∈ [1.36 × 105 , 2.73 × 105 ], and 100 grid points in the wall-normal direction for 𝑦 ∈ [0, 60]. Figure 5.1 shows excellent agreement between the 𝑢′ amplitudes of NOWNS and the DNS Sayadi et al. (2013), where the gray strip denotes the blowing/suction strip. Recall that we refer to modes according to their temporal frequency and their spanwise wave number as (𝑚, 𝑛), where 𝑚 refers to the frequency 𝜔𝑚 = 𝑚𝜔 and 𝑛 refers to the spanwise wave number 𝛽𝑛 = 𝑛𝛽, while the amplitudes are 62 Figure 5.1: 𝑢′-velocity amplitudes for K-type transition, compared to the DNS of Sayadi et al. (2013). defined as (3.30). We note that in the early stages of the march there is disagreement between the DNS and NOWNS calculations because the blowing/suction strip causes upstream effects that NOWNS neglects by construction. However, these disturbances are convective in nature, and the amplitudes predicted by NOWNS rapidly converge to those predicted by DNS as the march progresses downstream. We also note that Sayadi et al. (2013) do not plot the 𝑢′ amplitudes for the entire streamwise domain, which is why the NOWNS amplitudes extend farther downstream. Figure 5.2 shows the 𝑢-velocity contours at 𝑦/𝛿inlet = 0.6, as in Sayadi et al. (2013). We observe the aligned Λ-vortex structure that is characteristic of K-type transition, and good qualitative agreement with Sayadi et al. (2013). 5.2 H-type transition For H-type transition we use the blowing/suction strip function 𝑓 (𝑥, 𝑧, 𝑡) = 1.5 × 10−3 sin(2𝜔𝑡)𝑣 𝑎 (𝑥) + 1.5 × 10−5 sin(𝜔𝑡) cos(𝛽𝑧)𝑣 𝑎 (𝑥), (5.1) where for 𝑣 𝑎 (𝑥) defined in (3.31b). We choose 𝑅𝑒 𝑥 (𝑥 1 ) = 1.65 × 105 and 𝑅𝑒 𝑥 (𝑥2 ) = 1.81 × 105 , with 2𝐹 = 124 × 10−6 and 𝑏 = 0.419 × 10−3 . To match the parameters of Sayadi et al. (2013), we choose 𝑀 = 8 temporal modes and 𝑁 = 14 spanwise modes, with 2,370 stations over the streamwise domain 𝑅𝑒 𝑥 ∈ [1.65 × 105 , 5.16 × 105 ], and 100 grid points in the wall-normal direction for 𝑦 ∈ [0, 60]. As in the 63 Figure 5.2: 𝑢-velocity contours at 𝑦/𝛿inlet = 0.6 for K-type transition, demonstrating the aligned Λ-vortex structure. Figure 5.3: 𝑢′-velocity amplitudes for H-type transition, compared to the DNS of Sayadi et al. (2013). K-type transition case, Figure 5.3 demonstrates that the amplitudes predicted by NOWNS agree well with those from DNS, for the streamwise domain where DNS results are available, while Figure 5.4 demonstrates the staggered Λ-vortex structure characteristic of H-type transition. 64 Figure 5.4: 𝑢-velocity contours at 𝑦/𝛿inlet = 0.6 for H-type transition, demonstrating the staggered Λ-vortex structure. 5.3 Skin friction Boundary-layer transition is characterized by a rapid rise of the skin friction drag. We will measure the skin friction drag using the dimensionless skin-friction coefficient 𝐶𝑓 = 1 𝜏𝑤∗ , ∗ ∗ 2 2 (𝑈∞ ) /𝜈∞ 𝜏𝑤∗ = 𝜇∗ 𝜕𝑢 ∗ , 𝜕𝑦 ∗ where 𝜏𝑤∗ is the wall shear stress. We rewrite this in terms of the dimensionless variables as √︄ ∗ 𝜇∗ ∗ ∗ 𝜇∗ ∗ ∗ 𝜈 2𝜈∞ 𝜈 𝜕𝑢 𝜕𝑢 𝑈∞ 2𝜇 𝜕𝑢 𝜕𝑢 ∞ ∞ ∞ ∗ = 2𝜇 = √︁ , 𝐶𝑓 = 𝜇 = 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 𝜕𝑦 𝛿0𝑈∞ 𝜕𝑦 𝑈∞ 𝑥0 𝜇∞ 𝜈∞ 𝜕𝑦 (𝑈∞ ) 𝑅𝑒 𝑥 (𝑥 0 ) 𝜕𝑦 so that from NOWNS we obtain the time- and spanwise-averaged skin friction coefficient as ∫ 𝑡 ∫ 𝑧+1/(2𝜋𝛽) 1 1 2𝜇 𝜕𝑢 2𝜇(𝑇¯ + 𝑇ˆ00 ) 𝜕 [𝑢¯ + 𝑢ˆ 00 ] 𝐶 𝑓 ,ave. = d𝑧¯d𝑡¯ = √︁ . √︁ 𝑇 2𝜋𝛽 𝑡+𝑇 𝑧 𝜕𝑦 𝑅𝑒 𝑥 (𝑥0 ) 𝜕𝑦 𝑅𝑒 𝑥 (𝑥 0 ) Next, the laminar value can be obtained from the Blasius solution, while we use the turbulent correlation from White (2006). In summary, we plot the following skin friction coefficients: 2𝜇(𝑇¯ + 𝑇ˆ00 ) 𝜕 [𝑢¯ + 𝑢ˆ 00 ] 𝐶 𝑓 ,ave. = √︁ , 𝜕𝑦 𝑅𝑒 𝑥 (𝑥 0 ) 0.664 𝐶 𝑓 ,lam. = √ , 𝑅𝑒 𝑥 𝐶 𝑓 ,turb. = 0.455 2 . ln (0.06𝑅𝑒 𝑥 ) Figure 5.5 compares these values against the DNS values from Sayadi et al. (2013), and we see that for both H- and K-type transition NOWNS correctly predicts the initial rise in the skin friction coefficient, but following the initial rise, the NOWNS 65 Figure 5.5: Skin friction coefficient for K- and H-type transition, compared to the DNS of Sayadi et al. (2013). predicts a much more rapid growth than DNS, before eventually failing. When we say the NOWNS march fails, we mean that the Newton solver fails to converge to the desired error tolerance. Transition is characterized by a rise in skin friction coefficient, in which it initially overshoots the turbulent correlation, before converging to this value downstream (see Figure 5.5). If NOWNS overshot the turbulent value before failing, we could likely say that it predicted transition. However, NOWNS fails before reaching this value, so it is not clear when we can say that it predicts transition. 5.4 Investigations into the failure of the NOWNS march We discuss the root cause of NOWNS’ failure, separating errors into modeling and discretization errors. We also note that we say that NOWNS fails when the Newton’s method fails to converge. However, it becomes inaccurate before it fails, since the skin friction coefficient no longer agrees with the value predicted by DNS. Even if we modified the nonlinear solution method so it continued to converge, it would be inaccurate. 66 5.4.1 Discretization errors Section 1.4.3 discusses PSE’s minimum step-size requirement, which prevents it from properly resolving all streamwise length scales, while also making it more challenging for the nonlinear solver to converge. NOWNS does not have this minimum step-size requirement, yet it still becomes inaccurate following the onset of transition, before eventually failing (the Newton solver does not converge to the desired error tolerance). This suggests that the minimum step-size requirement is not necessarily the root cause of NPSE’s failure, since taking arbitrarily small steps using NOWNS still leads to failure. To ensure that the failure is not due to a lack of resolution, we doubled the resolution in 𝑥, 𝑦, 𝑀, and 𝑁, and we found that the results did not change substantially. We also performed the calculation without enforcing symmetry in 𝑧, and again found that the result did not change substantially. Prior to the development of the greedy algorithm (see Chapter 4), we were concerned that the recursion parameters were not properly tuned following the onset of transition, since they were originally developed for a Blasius profile. However, the greedy algorithm allows us to automatically choose recursion parameters that achieve any desired error tolerance, even following the onset of transition. Increasing 𝑁 𝛽 does not change the results substantially, so that recursion parameter selection is also not likely the root cause of failure. 5.4.2 Modeling errors in NOWNS Proposition 2.2.10 shows that the linear OWNS approach can evolve the upstreamand downstream-going modes independently of each other, and recover the global solution as the sum of the upstream- and downstream-going solutions. However, this property is lost for the nonlinear equations, since we lose the effect of the upstream-going solution on the nonlinear term: 𝑃𝐹 ( 𝝓ˆ′ + 𝝓ˆ′′) ≠ 𝑃𝐹 ( 𝝓ˆ′). If ∥𝑃[𝐹 ( 𝝓ˆ′ + 𝝓ˆ′′) − 𝐹 ( 𝝓ˆ′)] ∥ is small, then we expect NOWNS to be accurate, while it becomes less accurate as this term becomes larger. The 3D Euler equations have characteristics 𝑢, ¯ 𝑢, ¯ 𝑢, ¯ 𝑢¯ − 𝑐, ¯ and 𝑢¯ + 𝑐. ¯ For a subsonic boundary-layer flow, we have 𝑢¯ > 0 and 𝑢¯ < 𝑐, ¯ so that 𝑢, ¯ 𝑢, ¯ 𝑢, ¯ 𝑢¯ + 𝑐¯ > 0, 𝑢¯ − 𝑐¯ < 0. The OWNS approach removes upstream-going modes associated with 𝑢¯ − 𝑐, ¯ which is a modeling error, since the NOWNS model is not consistent with flow it is 67 trying to model. It is possible that modes associated with this characteristic are important following the onset of transition, and that NOWNS cannot be accurate since it neglects these upstream-going modes, which is a limitation of one-way spatial-marching. ¯ In In linear OWNS, we linearize the projection operator about the base flow, 𝒒. contrast, for nonlinear OWNS we have the choice to linearize about either the base flow or the corrected mean flow, 𝒒¯ + 𝒒ˆ ′00 . Appendix G.4 compares these two approaches for K-type transition and finds that it does not substantially affect the calculation. However, the mean flow distortion calculated by NOWNS neglects upstream effects by construction, which represents a modeling error. Therefore, it is possible that the results would change more substantially if we linearized about the mean flow computed using a global solver. 5.4.2.1 Negative upstream streamwise velocity For Blasius boundary layers, the 𝑢-velocity is downstream-going throughout the whole domain, so that 𝑢¯ > 0 everywhere. In nonlinear calculations, we must also consider the corrected mean flow, 𝑢¯ + 𝑢ˆ′00 , which arises due to nonlinear interactions of the mean flow distortion, 𝑢ˆ′00 , with the base flow. For the cases considered in this work, we find that the corrected mean flow is also downstream-going throughout the whole domain, 𝑢¯ + 𝑢′00 > 0, but that the flow can instantaneously have a negative 𝑢-velocity, 𝑢¯ + 𝑢′ < 0. For the 2D high-amplitude TS wave evolution from Section 3.4.1, as well as the H- and K-type transition scenarios considered here, we find that the negative 𝑢-velocity precedes the failure of the NOWNS march. For the 2D case from Section 3.4.1, we find that NOWNS is able to (accurately) march farther downstream than NSPE before failing. In Section 3.4.2, we also applied NOWNS to a large-amplitude oblique-wave breakdown case where NPSE fails. We found that although NOWNS is not accurate compared to DNS, it is able to march farther downstream than NPSE, and does not fail. We further note that no negative 𝑢-velocity is observed for this case. It appears that the failure of the NOWNS march is correlated with instantaneous 𝑢-velocities. 5.5 Summary We have demonstrated that NOWNS accurately simulates the early stages of transition by matching the 𝑢-velocity amplitudes predicted by DNS, while correctly predicting the initial rise in the skin friction coefficient, showing that NOWNS can 68 be used to predict transition onset analytically. Our investigation into the cause of NOWNS’ failure suggests that it is due to a modeling error, since NOWNS neglects upstream effects by construction. Future work should develop a criterion with which to predict transition using NOWNS, and should consider potential modifications to the routine such that it can march farther downstream while matching accurately DNS results. 69 Chapter 6 NOWNS FOR STABILITY AND TRANSITION ANALYSIS OF HIGH-SPEED BOUNDARY-LAYER FLOWS This chapter presents the extension of NOWNS to high-speed flows, along with a demonstration of the greedy algorithm from Chapter 4, and some of the results are adapted from Sleeman and Colonius (2025). High-speed boundary-layer transition is challenging to study experimentally using wind and shock tunnel measurements because of the high noise levels (Schneider, 2001), while they are not necessarily representative of flight tests because freestream fluctuations introduce higher amplitude disturbances than those typically experienced in flight. Quiet tunnels can relieve some of these problems, but are limited by the Reynolds number they can achieve. Therefore, NOWNS could be a valuable tool for studying transition mechanisms in high-speed flows. 6.1 Governing equations Whereas for low-speed flows we performed our analysis in terms of the primitive variables (𝜈, 𝒖, 𝑝), for high-speed flows we will instead use (𝜌, 𝒖, 𝑇), as was done in (Kamal, 2023). The non-dimensional compressible Navier-Stokes equations for an ideal gas can be written as 𝜕𝜌 + ∇ · (𝜌𝒖) = 0, (6.1a) 𝜕𝑡 𝐷𝒖 1 𝜌 + ∇𝑝 = ∇ · 𝜏, (6.1b) 𝐷𝑡 𝑅𝑒
𝐷𝑇 𝐷 𝑝 1 1 𝜌𝑐 𝑝 − = ∇ · 𝑘 (𝑇)∇𝑇 + 𝜏 : ∇𝒖, (6.1c) 𝐷𝑡 𝐷𝑡 𝑃𝑟 𝑅𝑒 𝑅𝑒 for the stress tensor (1.1e). We assume that 𝑃𝑟 and 𝑐 𝑝 are constant, while 𝜇 is a function of temperature according to Sutherland’s law  ∗  3/2 𝑇 ∗ + 𝑆 ∗ ∞ ∗ 𝑇 𝜇∗ (𝑇 ∗ ) = 𝜇∞ , (6.2) 𝑇∞∗ 𝑇 ∗ + 𝑆∗ ∗ (𝑇) = 𝜇 ∗ (𝑇)𝑐 ∗ /𝑃𝑟. where 𝑆 ∗ = 110.4K, so that 𝑘 ∞ 𝑝 6.2 Demonstration of greedy algorithm for high-speed flows We consider the Mach 4.5 boundary-layer flow over an adiabatic flat plate studied by Ma and Zhong (2003), which was used by Zhu and Towne (2023) as a validation 70 case for the OWNS-R formulation. The flow conditions are 𝑀∞ = 4.5, 𝑇∞∗ = 65.15 ∗ 𝑈 ∗ /𝜇 ∗ = K, 𝑝 ∗∞ = 728.44 Pa, 𝑃𝑟 = 0.72, and unit Reynolds number 𝑅𝑒 ∗∞ = 𝜌∞ ∞ ∞ 7.2 × 106 / m. 6.2.1 Error convergence at the domain inlet First we compare the greedy and heuristic approaches at 𝑅𝑒 𝑥 = 5.625 × 103 , where Figure 6.1 shows the convergence of the error as a function of 𝑁 𝛽 for OWNSP and OWNS-R. We again observe that OWNS-P achieves machine zero error, while OWNS-R does not. We also highlight that for OWNS-R only, we excluded downstream-going eigenvalues with 𝛼𝑖 > 100 from the selection procedure, which improves the accuracy of the approximation. Excluding these modes does not significantly affect the accuracy of the approximation because they decay rapidly. We also note that heuristic parameter selection does a better job of capturing Mack’s second mode (MM), which is the dominant instability for high-speed boundary-layer flows. For low-speed flows, the eigenvalue associated with the TS wave is far (in the complex plane) from the rest of the eigenvalues of 𝑀, while for high-speed flows, MM is close to the other eigenvalues. Therefore, the greedy algorithm does not choose MM as a recursion parameter, explaining the slower convergence. In contrast, Kamal (2023) places the heuristic recursion parameters such that many are clustered near MM. As described in Remark 4.1.1, the user could specify MM as an eigenvalue to track, improving the performance of the greedy approach. 6.2.2 Linear calculation for Mack’s second mode We use OWNS with greedy and heuristic recursion parameter selection to perform a linear stability calculation of Mack’s second mode over the streamwise domain 𝑅𝑒 𝑥 ∈ [5.625, 12.1] × 105 with 𝑁𝑥 = 4, 500 streamwise stations. We compute the 𝑁-factor based on the temperature disturbance and find that the greedy approach leads to a cost increase relative to heuristic recursion parameter selection. The reference solution is the greedy OWNS-P calculation with 𝑁 𝛽 = 20, and table 6.1 shows that heuristic recursion parameter selection generally converges with fewer recursion parameters, which is consistent with our observations in section 6.2.1 that the error in Mack’s second mode converges more quickly using heuristic, rather than greedy, parameter selection. Although greedy selection will generally achieve faster 71 (a) OWNS-P (b) OWNS-R (c) OWNS-P, MM only (d) OWNS-R, MM only Figure 6.1: Convergence of the error for greedy and heuristic parameter selection for 2D high-speed boundary-layer flow. Error 10% 1% 0.1% P-H 1 1 2 𝑁𝛽 P-G 6 6 7 R-G 17 21 22 P-H 1.0 1.0 1.0 Speed-up P-G R-G 0.34 0.83 0.34 0.83 0.32 0.82 Table 6.1: Speed-up achieved by using greedy parameter selection with OWNS-P (P-G) and OWNS-R (R-G) relative to heuristic parameter selection with OWNS-P (P-H) for different relative errors in the 𝑁-factor for the 2D high-speed boundarylayer flow. convergence, heuristic selection can be better when parameters are carefully chosen. We note that if the greedy algorithm is initialized with Mack’s second mode, then it also converges with 𝑁 𝛽 = 1 for OWNS-P. 6.3 Extension of NOWNS to high-speed flows Kamal et al. (2020) extended linear OWNS from low- to high-speed boundary-layer 72 flows, and we follow a similar procedure for NOWNS. 6.3.1 Temperature-dependent fluid properties For the low-speed flows discussed in Chapters 3, 4, and 5, we assumed that the fluid properties were constant, while they are function of temperature for high-speed flows. Let Φ(𝑇) denote a temperature-dependent fluid property, where 𝑇 = 𝑇¯ + 𝑇 ′. In linear OWNS we take a Taylor series expansion about 𝑇¯ and discard all terms higher than linear order to obtain ¯ + Φ(𝑇) = Φ(𝑇) 𝜕Φ 𝑇 ′ + O (𝑇 ′2 ). 𝜕𝑇 𝑇=𝑇¯ (6.3a) However, we wish to retain these terms in NOWNS, so we instead write ¯ + 𝜕Φ ¯ 𝑇 ′), Φ(𝑇) = Φ(𝑇) 𝑇 ′ + Φ′ (𝑇, 𝜕𝑇 𝑇=𝑇¯ (6.3b) for ¯ 𝑇 ′) = Φ(𝑇¯ + 𝑇 ′) − Φ(𝑇) ¯ − 𝜕Φ Φ′ (𝑇, 𝑇 ′. 𝜕𝑇 𝑇=𝑇¯ As an example, we consider the quadratic product (6.3c) ¯ + 𝒒′ × Φ(𝑇) ¯ + 𝒒¯ × 𝜕Φ 𝑇′ ( 𝒒¯ + 𝒒′) × Φ(𝑇¯ + 𝑇 ′) = 𝒒¯ × Φ(𝑇) ¯ 𝜕𝑇 𝑇=𝑇 | {z } | {z } Discarded terms Linear terms i ¯ 𝑇 ′) , ¯ 𝑇 ′) + 𝒒′ × 𝑇 ′ + Φ′ (𝑇, + 𝒒¯ × Φ′ (𝑇, 𝜕𝑇 𝑇=𝑇¯ | {z } h 𝜕Φ Nonlinear terms where terms without a disturbance variable are discarded (the baseflow satisfies the governing equations), while terms that are higher than first-order in the disturbance variables are used only by nonlinear OWNS. Therefore, in the nonlinear calculation (after simplification), we write ¯ + 𝒒¯ × 𝜕Φ 𝑇′ ( 𝒒¯ + 𝒒′) × Φ(𝑇¯ + 𝑇 ′) = 𝒒′ × Φ(𝑇) 𝜕𝑇 𝑇=𝑇¯ | {z } ′ Linear terms ′ ¯ ′ (6.4) ¯ . + 𝒒¯ × Φ (𝑇, 𝑇 ) + 𝒒 × [Φ(𝑇) − Φ(𝑇)] | {z } Nonlinear terms We can simplify further to obtain ¯ + [ 𝒒¯ + 𝒒′] × [Φ(𝑇) − Φ(𝑇)], ¯ ( 𝒒¯ + 𝒒′) × Φ(𝑇¯ + 𝑇 ′) = 𝒒′ × Φ(𝑇) (6.5) which matches the expression commonly employed in high-speed NPSE (Chang et al., 1993; Chang and Malik, 1994). 73 6.3.2 Linearizing about the corrected mean flow In Chapter 3, we presented NOWNS for projection operators linearized about the ¯ In Appendix G.4, we compared results for low-speed K-type transition base flow, 𝒒. ¯ and the corrected mean flow, 𝒒¯ + 𝒒ˆ 00 , and we found linearized about the base flow, 𝒒, that the difference was not substantial. However, we might expect the difference to be more pronounced for high-speed flows because the number of upstream- and downstream-going modes can be different for 𝒒¯ and 𝒒ˆ 00 , as explained below. The number of upstream- and downstream-going modes are determined by the ¯ characteristics of 𝐴( 𝒒) √︃ √︃ ¯ 𝑢¯ + 𝛾𝑅𝑔𝑇, ¯ 𝑢, ¯ 𝑢, ¯ 𝑢, ¯ 𝑢¯ − 𝛾𝑅𝑔𝑇, or 𝐴( 𝒒¯ + 𝒒ˆ 00 ) 𝑢¯ + 𝑢ˆ 00 , 𝑢¯ + 𝑢ˆ 00 , 𝑢¯ + 𝑢ˆ 00 , 𝑢¯ + 𝑢ˆ 00 − √︃ 𝛾𝑅𝑔 (𝑇¯ + 𝑇ˆ00 ), 𝑢¯ + 𝑢ˆ 00 + √︃ 𝛾𝑅𝑔 (𝑇¯ + 𝑇ˆ00 ). ¯ and For the low-speed flows considered in this work, the characteristics of 𝐴( 𝒒) 𝐴( 𝒒¯ + 𝒒ˆ 00 ) have the same signs, since 𝑢¯ and 𝑢ˆ 00 have the same sign and 𝑇ˆ00 is small. In contrast, for high-speed flows, 𝑇ˆ00 can grow large, so that sign( 𝑢¯ − √︃ ¯ ≠ sign( 𝑢¯ + 𝑢ˆ 00 − 𝛾𝑅𝑔𝑇) √︃ 𝛾𝑅𝑔 (𝑇¯ + 𝑇ˆ00 )). Therefore, the base flow and corrected mean flow will have different numbers of upstream- and downstream-going modes, and we might expect that linearizing about the corrected mean flow instead of the base flow will have a larger impact than in the low-speed cases. However, we find again that the difference between linearizing about 𝒒¯ and 𝒒ˆ 00 is negligible for high-speed flows. 6.3.3 Disturbance amplitude Whereas for low-speed flows we have measured disturbance amplitude in terms of the 𝑢-velocity, we will instead measure it in terms of the temperature as ′(𝑚,𝑛) 𝑇max (𝑥) = 𝑐 𝑚,𝑛 ′ max |𝑇𝑚,𝑛 (𝑥, 𝑦)|, 𝑦 𝑇∞ (6.6a) for    1   √  𝑐 𝑚,𝑛 = 2     2  𝑚 = 𝑛 = 0, 𝑚 = 0, 𝑛 ≠ 0; 𝑛 = 0, 𝑚 ≠ 0, otherwise. (6.6b) 74 6.4 2D nonlinear evolution of Mack’s second mode We make the calculation from 6.2.2 nonlinear by specifying that Mack’s second mode has an amplitude of 10% based on the temperature disturbance, relative to the free-stream value, while we truncate the Fourier series at 𝑀 = 10, with 𝑁 𝛽 = 12 recursion parameters chosen using the greedy algorithm. We plot the temperature disturbance amplitudes for both the linear and nonlinear OWNS calculations in Figure 6.2. We see that the two calculations initially match, but as the disturbances grow, the linear calculation exhibits larger growth of the fundamental disturbance, while the MFD grows substantially for the nonlinear calculation. Figures 6.4a and 6.4b show that the heuristic parameter selection yields reasonable temperature profiles for the MFD and fundamental disturbance, respectively. However, Figures 6.4c and 6.4e show that there are non-physical grid effects present near the sonic line for heuristic parameter selection, while these are not present when greedy selection is used, as shown in Figures 6.4d and 6.4f. We additionally compare the temperature profiles at the domain outlet for the first and second harmonics of Mack’s second mode in Figure 6.5. Although heuristic parameter selection outperforms greedy (in terms of computational cost) for the linear problem, the same set of heuristic parameters do not work in the nonlinear case. As discussed above, the high-speed heuristic recursion parameters have been tailored to track Mack’s second mode, but they do so at the expense of the other modes, and this effect is noticeable for the higher harmonics in nonlinear calculations. Figure 6.3 compares the amplitudes computed when linearized about the base flow to the amplitudes computed when linearized about the corrected mean flow. The results are nearly indistinguishable, but this may be due to the small disturbance amplitude. 6.5 Oblique breakdown of Mack’s first mode While TS waves are most unstable for 𝛽 = 0, Mack’s first and second modes are most unstable when 𝛽 ≠ 0, so we extend our analysis to 3D disturbances. 3D disturbances to Mach 6 flat-plate boundary-layer flows have been studied using DNS by Leinemann et al. (2021) and Franko and Lele (2013). Both studies introduce disturbances using a blowing/suction strip. Both groups investigated oblique breakdown of Mack’s first mode and fundamental transition for Mack’s second mode, while Franko and Lele (2013) additionally studied oblique breakdown of Mack’s second 75 Figure 6.2: Temperature disturbance amplitudes for 2D nonlinear evolution of Mack’s second mode for a Mach 4.5 flat-plate boundary-layer flow, computed using NOWNS and compared to linear OWNS. Figure 6.3: Comparison of temperature disturbance amplitudes for NOWNS calculation linearized about base flow and corrected mean flow for 2D nonlinear evolution of Mack’s second mode for a Mach 4.5 flat-plate boundary-layer flow. mode. We will use the conditions of Leinemann et al. (2021), where the free-stream Mach number is 𝑀∞ = 6, the stagnation pressure and temperature are 956.3 kPa and 420 K, respectively, while the unit Reynolds number is 𝑅𝑒 ∞ = 10.8224 × 106 / m. We assume the fluid to be a perfect gas with 𝛾 = 1.4, while we also assume a constant 𝑃𝑟 = 0.71. The dynamic viscosity is computed according to Sutherland’s law. Although both Leinemann et al. (2021) and Franko and Lele (2013) introduced disturbances using a blowing/suction strip, which NOWNS can do, we will introduce our disturbance using LST. The disturbance frequency is 𝑓 ∗ = 35 kHz while the spanwise wavenumber is 𝛽∗ = 330/m. We use a fourth-order Padé-filter (Gaitonde and Visbal, 2000) with 𝛼 𝑓 = 0.3 to remove numerical oscillations. Increasing the wall-normal resolution does not affect the solution substantially, indicating that we 76 (a) Heuristic 𝑇0′ (b) Heuristic 𝑇1′ (c) Heuristic 𝑇2′ (d) Greedy 𝑇2′ (e) Heuristic 𝑇3′ (f) Greedy 𝑇3′ Figure 6.4: Temperature contours for first and second harmonic of Mack’s second mode with heuristic and greedy recursion parameter selection. 77 Figure 6.5: Temperature profiles at domain outlet for first and second harmonics of Mack’s second mode with heuristic and greedy recursion parameter selection. Figure 6.6: Temperature disturbance amplitudes for small amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. have enough grid-points to resolve all physically-relevant waves. 6.5.1 Small amplitude We consider first a low-amplitude case, with an amplitude of 1.4% based on temperature. The domain is 𝑥 × 𝑦 ∈ [0.15 m, 0.55 m] × [0 m, 0.025 m] with 𝑁𝑥 = 2, 500 and 𝑁 𝑦 = 200, while we 𝑀 = 3 and 𝑁 = 4 temporal and spanwise Fourier modes, respectively, with 𝑁 𝛽 = 12 recursion parameters chosen using the greedy algorithm. We plot the disturbance amplitudes in Figure 6.6, where we observe that the amplitude of the oblique wave grows more slowly in the nonlinear calculation, as compared to the linear calculation. Additionally, the vortex mode (0, 2) grows to be the largest disturbance near the end of the domain. We find that the difference in linearizing about the corrected mean flow instead of the base flow is negligible, and so we do not compare the results. 78 Figure 6.7: Temperature disturbance amplitudes for large amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. 6.5.2 Large amplitude We increase the amplitude to 14%, while we truncate the domain to 𝑥 × 𝑦 ∈ [0.15 m, 0.45 m] × [0 m, 0.025 m] with 𝑁𝑥 = 2000 and 𝑁 𝑦 = 200. We increase the number of Fourier modes to 𝑀 = 𝑁 = 7, while we maintain the same number of recursion parameters. We plot the temperature amplitudes in Figure 6.7, where we see again that the amplitude of the oblique wave grows more slowly in the nonlinear calculation, while the vortex mode (0, 2) grows to be nearly the largest disturbance near the end of the march. We further note that the march fails at 𝑥 = 0.288m, corresponding to station 919. We see from Figure 6.8 that the rise in the skin friction coefficient is small, indicating the transition has only just begun. Future work should explore why NOWNS fails so early into the transition process compared to the low-speed flows studied in Chapter 5. We also note that linearizing about the corrected mean flow instead of the base flow has a negligible impact on the solution. 6.6 Oblique breakdown of Mack’s second mode Mack’s first mode is the dominant instability at lower frequencies, while Mack’s second mode is dominant for higher frequencies. Leinemann et al. (2021) investigated fundamental transition of Mack’s second mode for the same Mach 6 boundary-layer flow for which they investigated oblique breakdown of Mack’s first mode, but with a frequency of 𝑓 ∗ = 95 kHz and a spanwise wavenumber 𝛽∗ = 500/m. Rather than study fundamental transition, we consider oblique breakdown of Mack’s second mode. We use a fourth-order Padé-filter (Gaitonde and Visbal, 2000) with 𝛼 𝑓 = 0.3 to remove numerical oscillations. 79 Figure 6.8: Skin friction coefficient for large amplitude oblique breakdown of Mack’s first mode for Mach 6 flat-plate boundary-layer flow. 6.6.1 Small amplitude We consider first a low-amplitude case, with an amplitude of 1.4% based on temperature. The domain is 𝑥 × 𝑦 ∈ [0.15 m, 0.55 m] × [0 m, 0.025 m] with 𝑁𝑥 = 2, 000 and 𝑁 𝑦 = 200, while we 𝑀 = 3 and 𝑁 = 4 temporal and spanwise Fourier modes, respectively, with 𝑁 𝛽 = 12 recursion parameters chosen using the greedy algorithm. We plot the disturbance amplitudes in Figure 6.6, where we observe that the amplitude of the oblique wave grows more slowly in the nonlinear calculation, as compared to the linear calculation. Additionally, the vortex mode (0, 2) grows to be the largest disturbance near the end of the domain. Note that NOWNS calculation ¯ Figure 6.9 compares the linear calculation is linearized about 𝒒¯ + 𝒒 𝑐 instead of 𝒒. to the nonlinear calculation linearized about the corrected mean flow. We see that the vortex mode (0, 2) becomes the largest disturbance, while the oblique wave has a smaller amplitude in the nonlinear calculation than in the linear calculation. 6.6.2 Large amplitude We increase the amplitude to 14%, while we truncate the domain to 𝑥 × 𝑦 ∈ [0.15 m, 0.45 m] × [0 m, 0.025 m] with 𝑁𝑥 = 2000 and 𝑁 𝑦 = 200. We increase the number of Fourier modes to 𝑀 = 𝑁 = 7, while we maintain the same number of recursion parameters. We plot the temperature amplitudes in Figure 6.10, where we see again that the amplitude of the oblique wave grows more slowly in the nonlinear calculation, while the vortex mode (0, 2) quickly overtakes the oblique wave to be the largest disturbance. We further note that the march fails at 𝑥 = 0.0.350m, corresponding to station 1336. We see from Figure 6.11 that the rise in the skin 80 Figure 6.9: Temperature disturbance amplitudes for small amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundary-layer flow. Figure 6.10: Temperature disturbance amplitudes for large amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundary-layer flow. friction coefficient is larger than for the oblique breakdown of Mack’s first mode in figure 6.8, but still smaller than the low-speed cases. We again find that linearizing about the corrected mean flow instead of the base flow has a negligible impact on the solution. We plot the 𝑢-velocity contours in figure 6.12, where we observe that streaks associated with the (0, 2) mode develop towards the end of the domain. Figure 6.13 plots the temperature contour in the 𝑦 − 𝑧 plane at the domain outlet, while figure 6.14 plots the pressure contour in the symmetry plane and figure 6.15 plots the pressure contour at the wall. 6.7 Summary We have demonstrated that greedy parameter selection works well for both linear and nonlinear calculations in high-speed boundary-layer flows. Although heuristic 81 Figure 6.11: Skin friction coefficient for large amplitude oblique breakdown of Mack’s second mode for Mach 6 flat-plate boundary-layer flow. Figure 6.12: Contours of 𝑢-velocity at 𝑦 = 0.00189m for oblique breakdown of Mack’s second mode, exhibiting streaks towards the end of the domain parameter selection outperforms greedy for tracking Mack’s second mode in the linear calculation, greedy outperforms heuristic in the nonlinear calculation because other modes are introduced by the nonlinear term for the higher harmonics, and these are not properly evolved by the heuristic recursion parameters. In principle, it is possible to modify the heuristic parameters to enable accurate nonlinear calculations, but the greedy algorithm circumvents this problem by automatically choosing parameters that yield a rapidly convergent error in the OWNS approximation. We have demonstrated that NOWNS can be used to study the nonlinear evolution of Mack’s first and second modes in high-speed boundary-layer flows, and we have compared results obtained when linearized about the base flow to results obtained 82 Figure 6.13: Temperature contours in the 𝑦 − 𝑧 plane at the domain outlet for oblique breakdown of Mack’s second mode. Figure 6.14: Pressure contours at 𝑧 = 0 for oblique breakdown of Mack’s second mode. when linearized about the corrected mean flow. For both the oblique breakdown of Mack’s first and second mode, the march fails before a significant rise in the skin friction coefficient occurs. Future work should attempt to modify the routine such that NOWNS can march farther into the transitional regime. 83 Figure 6.15: Pressure contours at 𝑦 = 0 for oblique breakdown of Mack’s second mode. 84 Chapter 7 CONCLUSIONS AND FUTURE WORK 7.1 Conclusions This thesis developed the nonlinear One-Way Navier-Stokes (NOWNS) equations based on the OWNS projection (OWNS-P) approach. Chapter 1 provided background information on boundary-layer flows and methods for hydrodynamic stability analysis. Although linear methods can be used to predict transition in low-speed boundary-layer flows by combining them with an empirical correlation, this approach is less accurate for high-speed flows, particularly with complex geometries. In principle, laminar-turbulent transition can be predicted using DNS and LES, but these methods are limited by their high computational costs. NPSE is a low-cost alternative, but it fails for strong nonlinearities, is inaccurate for non-modal and multi-modal instabilities, and has a minimum step-size for numerical stability. In contrast NOWNS does not have a minimum step-size requirement and is accurate for non-modal and multi-modal instabilities, and succeeds for stronger nonlinearities than NPSE. Chapter 2 introduced the OWNS-P (Towne et al., 2022) and OWNS-R (Zhu and Towne, 2023) approaches, while Chapter 3 developed NOWNS based on OWNS-P. We validated NOWNS against DNS for low-speed boundary-layer flows in cases where NPSE is already successful, and then we highlighted three strengths of NOWNS relative to NPSE: 1. It can handles larger disturbances than NPSE, although it also fails for sufficiently large disturbances. 2. It can handle large amounts of numerical noise, while NPSE fails. 3. It can handle arbitrary disturbances, such as those introduced using a blowing/suction strip, while NPSE requires the disturbances to specified as a (modal) inlet boundary condition. The OWNS approximation relies on a recursive filter to remove upstream-going waves, which requires that recursion parameters be specified. Previous work has chosen these parameter heuristically, based on the Euler equations linearized about 85 a uniform flow. Chapter 4 develops a greedy algorithm for recursion parameter selection, which we demonstrate to have better convergence properties than the heuristic approach, leading to a new decrease in computational cost. In Chapter 5 we demonstrate that for the H- and K-type transition scenarios commonly observed in low-speed boundary-layer flows, NOWNS is able to accurately predict the onset of transition, relative to DNS, before the march eventually fails. We do not believe this failure is due to discretization errors, and is instead due to modeling errors related to upstream effects which are removed (neglected) by construction. Chapter 6 extends the NOWNS procedure to high-speed boundary layer flows and demonstrates the importance of the greedy algorithm for high-speed NOWNS. In particular, we show that heuristic parameter selection leads to a non-physical grideffect along the sonic line. We further demonstrate that NOWNS has the potential to predict transition in high-speed boundary-layer flows by applying it oblique breakdown of Mack’s first and second modes for a Mach 6 isothermal flat-plate boundary-layer flow. 7.2 Future Work The results from this thesis suggest several avenues for future work on nonlinear disturbance evolution using NOWNS. 7.2.1 Transition prediction We have developed and validated NOWNS for low-speed flows, showing that it can accurately predict transition onset. While DNS predicts an overshoot of the turbulent correlation, followed by agreement with the turbulent correlation, NOWNS fails before this can occur. Therefore, future work should develop a criterion to enable NOWNS to predict transition without skin-friction overshoot. Alternatively, NOWNS should be modified so that it can march far enough downstream to observe skin-friction overshoot. 7.2.2 High-speed boundary-layer flows High-speed flows are characterized by multi-modal disturbances with short wavelengths and long streamwise domains over which they are unstable, so that NOWNS is well-suited to study these disturbances. Therefore, NOWNS should be validated for high-speed boundary-layer flows and applied to study realistic flight geometries, such as the HiFIRE-5 flight vehicle. In particular, it should be used to study 86 stationary and traveling crossflow vortices. Moreover, it should be used to identify optimal (worst-case) disturbances for high-speed boundary-layer flows in a multifidelity optimization framework, where NOWNS is used as a low-fidelity model to be queried rapidly, while a high-fidelity model (such as DNS/LES) is used to ensure accuracy. 7.2.3 Recursion parameter selection In our initial work to extend NOWNS to high-speed flows, we observed non-physical grid-like effects along the sonic line, which we eliminated using the greedy algorithm. In principle, it is possible to choose heuristic recursion parameters such that the grid-like effects disappear, which should be another focus of future work. In addition, the recursion parameters chosen by the greedy algorithm could be used as a guide for choosing better recursion parameters using the heuristic approach. 7.2.4 Numerical filtering For the high-speed flows, we used a fourth-order Padé-filter (Gaitonde and Visbal, 2000) with 𝛼 𝑓 = 0.3 to remove numerical oscillations. 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URL https://www. sciencedirect.com/science/article/pii/S0021999122008075. 95 Appendix A NAVIER-STOKES EQUATIONS Given the non-dimensional Navier-Stokes equations (1.1), we define the vector 𝒒 = (𝜈, 𝑢, 𝑣, 𝑤, 𝑝) and write 𝜕𝒒 𝜕𝒒 𝜕𝒒 + [ 𝐴𝑥 (𝒒) + 𝐵𝑥 (𝒒)] + [ 𝐴 𝑦 (𝒒) + 𝐵 𝑦 (𝒒)] 𝜕𝑡 𝜕𝑥 𝜕𝑦 2 𝜕𝒒 𝜕 𝒒 𝜕2 𝒒 𝜕2 𝒒 + [ 𝐴𝑧 (𝒒) + 𝐵 𝑧 (𝒒)] + 𝐵𝑥𝑥 (𝒒) 2 + 𝐵 𝑦𝑦 (𝒒) 2 + 𝐵 𝑧𝑧 (𝒒) 2 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 2 2 2 𝜕 𝒒 𝜕 𝒒 𝜕 𝒒 + 𝐵𝑥𝑦 (𝒒) + 𝐵𝑥𝑧 (𝒒) + 𝐵 𝑦𝑧 (𝒒) = 0, 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 (A.1) where 𝐵 denotes viscous terms while 𝐴 denotes inviscid terms. Next we decompose ¯ and a time-varying disturbance the flow into a time-invariant equilibrium solution, 𝒒, ′ ′ variable, 𝒒 , such that 𝒒 = 𝒒¯ + 𝒒 , which we use to obtain 𝜕𝒒′ 𝜕𝒒′ 𝜕𝒒′ 𝜕𝒒′ ¯ + 𝐵𝑥 ( 𝒒)] ¯ ¯ + 𝐵 𝑦 ( 𝒒)] ¯ ¯ + 𝐵 𝑧 ( 𝒒)] ¯ + [ 𝐴𝑥 ( 𝒒) + [ 𝐴 𝑦 ( 𝒒) + [ 𝐴𝑧 ( 𝒒) 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 2 ′ 2 ′ 2 ′ 𝜕 𝒒 𝜕 𝒒 𝜕 𝒒 ¯ ′ + 𝐵𝑥𝑥 ( 𝒒) ¯ ¯ ¯ + 𝐶 ( 𝒒)𝒒 + 𝐵 𝑦𝑦 ( 𝒒) + 𝐵 𝑧𝑧 ( 𝒒) (A.2) 2 2 𝜕𝑥 𝜕𝑦 𝜕𝑧 2 𝜕 2 𝒒′ 𝜕 2 𝒒′ 𝜕 2 𝒒′ ¯ ¯ ¯ + 𝐵𝑥𝑧 ( 𝒒) + 𝐵 𝑦𝑧 ( 𝒒) = 𝑭(𝒒′), + 𝐵𝑥𝑦 ( 𝒒) 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 ¯ such that where we have defined 𝐶 ( 𝒒) 𝜕 𝒒¯ 𝜕 𝒒¯ + [ 𝐴 𝑦 (𝒒′) + 𝐵 𝑦 (𝒒′)] 𝜕𝑥 𝜕𝑦 2 𝜕 2 𝒒¯ 𝜕 𝒒¯ 𝜕 𝒒¯ + [ 𝐴𝑧 (𝒒′) + 𝐵 𝑧 (𝒒′)] + 𝐵𝑥𝑥 (𝒒′) 2 + 𝐵 𝑦𝑦 (𝒒′) 2 𝜕𝑧 𝜕𝑥 𝜕𝑦 2 2 2 𝜕 𝒒¯ 𝜕 𝒒¯ 𝜕 𝒒¯ 𝜕 2 𝒒¯ + 𝐵 𝑧𝑧 (𝒒′) 2 + 𝐵𝑥𝑦 (𝒒′) + 𝐵𝑥𝑧 (𝒒′) + 𝐵 𝑦𝑧 (𝒒′) , 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 𝜕𝑧 ¯ ′ = [ 𝐴𝑥 (𝒒′) + 𝐵𝑥 (𝒒′)] 𝐶 ( 𝒒)𝒒 (A.3) and the nonlinear term 𝜕𝒒′ 𝜕𝒒′ − [ 𝐴 𝑦 (𝒒′) + 𝐵 𝑦 (𝒒′)] 𝜕𝑥 𝜕𝑦 ′ 2 ′ 𝜕𝒒 𝜕 𝒒 𝜕 2 𝒒′ − [ 𝐴𝑧 (𝒒′) + 𝐵 𝑧 (𝒒′)] − 𝐵𝑥𝑥 (𝒒′) 2 − 𝐵 𝑦𝑦 (𝒒′) 2 𝜕𝑧 𝜕𝑥 𝜕𝑦 2 ′ 2 ′ 2 ′ 𝜕 𝒒 𝜕 𝒒 𝜕 𝒒 𝜕 2 𝒒′ − 𝐵 𝑦𝑦 (𝒒′) 2 − 𝐵𝑥𝑦 (𝒒′) − 𝐵𝑥𝑧 (𝒒′) − 𝐵 𝑦𝑧 (𝒒′) . 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 𝜕𝑧 𝑭(𝒒′) = −[ 𝐴𝑥 (𝒒′) + 𝐵𝑥 (𝒒′)] (A.4) 96 We further define the linear operator 𝜕 𝜕𝒒′ 𝜕𝒒′ ¯ + 𝐵 𝑦 ( 𝒒)] ¯ ¯ + 𝐵 𝑧 ( 𝒒)] ¯ ¯ ′ − [ 𝐴 𝑦 ( 𝒒) − [ 𝐴𝑧 ( 𝒒) − 𝐶 ( 𝒒)𝒒 𝜕𝑡 𝜕𝑦 𝜕𝑧 2 ′ 2 ′ 2 ′ 𝜕 𝒒 𝜕 𝒒 𝜕 𝒒 ¯ ¯ ¯ − 𝐵 𝑦𝑦 ( 𝒒) − 𝐵 𝑧𝑧 ( 𝒒) − 𝐵 𝑦𝑧 ( 𝒒) = 0, 2 2 𝜕𝑦𝜕𝑧 𝜕𝑦 𝜕𝑧 ¯ =− L ( 𝒒) (A.5) yielding ¯ 𝐴𝑥 ( 𝒒) 𝜕𝒒′ 𝜕𝒒′ ¯ ′ + 𝑭(𝒒′) − 𝐵𝑥 ( 𝒒) ¯ = L ( 𝒒)𝒒 𝜕𝑥 𝜕𝑥 𝜕 2 𝒒′ 𝜕 2 𝒒′ 𝜕 2 𝒒′ ¯ ¯ ¯ − 𝐵𝑥𝑥 ( 𝒒) − 𝐵 ( 𝒒) − 𝐵 ( 𝒒) . 𝑥𝑦 𝑥𝑧 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑥 2 (A.6) The operators for the first derivatives (without viscous terms), 𝐴𝑥 , 𝐴 𝑦 , and 𝐴𝑧 , as well as the operators for the first derivatives (with viscous terms) 𝐵𝑥 , 𝐵 𝑦 , and 𝐵 𝑧 , are given by  𝑢 −𝜈 0 0 0       0 𝑢 0 0 𝜈    𝐴𝑥 (𝒒) =  0 0 𝑢 0 0  ,    0 0 0 𝑢 0     0 𝛾𝑝 0 0 𝑢     𝑣 0 −𝜈 0 0       0 𝑣 0 0 0    𝐴 𝑦 (𝒒) =  0 0 𝑣 0 𝜈  ,    0 0 0 𝑣 0     0 0 𝛾𝑝 0 𝑣     𝑤    0  𝐴𝑧 (𝒒) =  0   0   0  0 𝑤 0 0 0 0 −𝜈 0   0 0 0   𝑤 0 0  ,  0 𝑤 𝜈   0 𝛾 𝑝 𝑤   0    0 2𝛾  𝐵𝑥 (𝒒) = − 0 𝑃𝑟 𝑅𝑒   0  𝜕𝑝   𝜕𝑥  0    0 2𝛾  𝐵 𝑦 (𝒒) = − 0 𝑃𝑟 𝑅𝑒   0  𝜕𝑝   𝜕𝑦  0    0 2𝛾  𝐵 𝑧 (𝒒) = − 0 𝑃𝑟 𝑅𝑒   0  𝜕𝑝   𝜕𝑧 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   0   0  ,  0   𝜕𝜈  𝜕𝑥  0   0   0  ,  0   𝜕𝜈  𝜕𝑦  0   0   0  .  0   𝜕𝜈  𝜕𝑧  97 For the second derivatives 𝐵𝑥𝑥 , 𝐵 𝑦𝑦 , 𝐵 𝑧𝑧 , 𝐵𝑥𝑦 , 𝐵𝑥𝑧 , and 𝐵 𝑦𝑧 we have 0 0 0 0   0 0   0 − 34 𝑅𝑒𝜈 0 0  𝜈 0 0  , 0 − 𝑅𝑒 𝐵𝑥𝑥 (𝒒) =  0 𝜈  0 0 0 − 𝑅𝑒 0   − 𝛾𝑝 0 𝛾𝜈  0 0 −  𝑅𝑒𝑃𝑟 𝑅𝑒𝑃𝑟  0 0   00 − 0𝜈 00 0 0   𝑅𝑒  𝜈 0 − 43 𝑅𝑒 0 0  , 𝐵 𝑦𝑦 (𝒒) =  0 𝜈  0 0 0 − 𝑅𝑒 0   − 𝛾𝑝 0 𝛾𝜈  0 0 − 𝑅𝑒𝑃𝑟  𝑅𝑒𝑃𝑟  0 0   00 − 0𝜈 00 0 0   𝑅𝑒 𝜈  0 − 𝑅𝑒 0 0  𝐵 𝑧𝑧 (𝒒) =  0 , 𝜈  0 0  0 0 − 34 𝑅𝑒  − 𝛾𝑝 0 0 𝛾𝜈  0 − 𝑅𝑒𝑃𝑟   𝑅𝑒𝑃𝑟 0 0 0 0 0 𝜈 0 0  0 0 − 3𝑅𝑒 𝜈 𝐵𝑥𝑦 (𝒒) =  0 − 3𝑅𝑒 0 0 0  , 0 0 0 0 0 0 0 0 0 0  0 0 0 0𝜈 0   0 0 0 − 3𝑅𝑒 0  𝐵𝑥𝑧 (𝒒) =  0 0𝜈 0 0 0  ,  0 − 3𝑅𝑒 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝜈 𝐵 𝑦𝑧 (𝒒) =  0 0 0𝜈 − 3𝑅𝑒 0  .  0 0 − 3𝑅𝑒 0 0  0 0 0 0 0 Finally, the operator 𝐶 is given by 𝜕𝜈 𝜕𝜈 𝜕𝜈   −∇·𝒖 0 𝜕𝑥 𝜕𝑦 𝜕𝑧  𝜕𝑝 1 2 1  𝜕𝑢 𝜕𝑢 𝜕𝑢  𝜕𝑥 − 𝑅𝑒 ∇ 𝑢− 3𝑅𝑒 [𝜕𝑥 𝑥 𝑢+𝜕𝑥 𝑦 𝑣+𝜕𝑥𝑧 𝑤] 𝜕𝑥 𝜕𝑦 𝜕𝑧  0  𝜕𝑝 1 2 1  𝜕𝑣 𝜕𝑣 𝜕𝑣  , 0 𝐶 (𝒒) =  𝜕𝑦 − 𝑅𝑒 ∇ 𝑣− 3𝑅𝑒 [𝜕𝑥 𝑦 𝑢+𝜕𝑦 𝑦 𝑣+𝜕𝑦𝑧 𝑤] 𝜕𝑥 𝜕𝑦 𝜕𝑧  𝜕 𝑝 1 1 𝜕𝑤 𝜕𝑤 𝜕𝑤  𝜕𝑧 − 𝑅𝑒 ∇2 𝑤− 3𝑅𝑒 [𝜕𝑥𝑧 𝑢+𝜕𝑦𝑧 𝑣+𝜕𝑧𝑧 𝑤] 𝜕𝑥 𝜕𝑦 𝜕𝑧  0   𝛾 𝛾 𝜕𝑝 𝜕𝑝 𝜕𝑝 2 2   − 𝑅𝑒𝑃𝑟 ∇ 𝑝 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝛾∇·𝒖− 𝑅𝑒𝑃𝑟 ∇ 𝜈   while the nonlinear term is a vector comprising the following components: 𝜕𝜈 𝜕𝜈 𝜕𝑢 𝜕𝑣 𝜕𝑤 𝜕𝜈 +𝑣 +𝑤 −𝜈 −𝜈 −𝜈 , 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑝 𝜕𝑢 𝜕𝑢 𝜕𝑢 1 1  𝜕 2𝑢 𝜕2𝑣 𝜕2𝑤  2 𝐹2 (𝒒) = 𝜈 −𝑢 −𝑣 −𝑤 + 𝜈∇ 𝑢 + + + , 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑅𝑒 3𝑅𝑒 𝜕𝑥 2 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑣 𝜕𝑣 𝜕𝑣 1 1  𝜕 2𝑢 𝜕2𝑣 𝜕2𝑤  𝜕𝑝 −𝑢 −𝑣 −𝑤 + 𝜈∇2 𝑢 + + 2+ 𝐹3 (𝒒) = 𝜈 , 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑅𝑒 3𝑅𝑒 𝜕𝑥𝜕𝑦 𝜕𝑦 𝜕𝑦𝜕𝑧 𝜕𝑝 𝜕𝑤 𝜕𝑤 𝜕𝑤 1 1  𝜕 2𝑢 𝜕2𝑣 𝜕2𝑤  𝐹4 (𝒒) = 𝜈 −𝑢 −𝑣 −𝑤 + 𝜈∇2 𝑢 + + + 2 , 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑅𝑒 3𝑅𝑒 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 𝜕𝑧 𝛾 [𝜈∇2 𝑝 + 2∇𝑝 · ∇𝜈 + 𝑝∇2 𝜈] − 𝒖 · ∇𝑝 − 𝛾 𝑝∇ · 𝒖. 𝐹5 (𝒒) = 𝑅𝑒𝑃𝑟 𝐹1 (𝒒) = 𝑢 98 Appendix B SIMILARITY SOLUTIONS B.1 Blasius equation A time-invariant equilibrium solution (base flow) can be obtained for flat-plate boundary-layer flows by solving the Blasius equation (Blasius, 1908). We assume a 2D steady flow where 𝜕 𝑝/𝜕𝑥 = 0, and the streamwise length scale is much larger than the wall-normal length scale, yielding the equations 𝜕 (𝜌 ∗ 𝑢 ∗ ) 𝜕 (𝜌 ∗ 𝑣 ∗ ) + = 0, 𝜕𝑥 ∗ 𝜕𝑥 ∗  𝜕𝑢 ∗ 𝜕  𝜕𝑢 ∗  𝜕𝑢 ∗  𝜌 ∗ 𝑢 ∗ ∗ + 𝑣 ∗ ∗ = ∗ 𝜇∗ ∗ , 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑦  𝜕ℎ∗   ∗ 2 ∗ ∗ 𝜕ℎ 𝜕 𝜇 𝜕ℎ∗  ∗ 𝜕𝜇 𝜌∗ 𝑢∗ ∗ + 𝑣 ∗ ∗ = ∗ , + 𝜇 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝑃𝑟 𝜕𝑦 ∗ 𝜕𝑦 ∗ (B.1a) (B.1b) (B.1c) where 𝜌 ∗ = 1/𝜈 ∗ is the density and ℎ∗ is the enthalpy. In addition, the 𝑦momentum equation yields 𝜕 𝑝/𝜕𝑦 so that the pressure is uniform throughout the flow. We introduce self-similar variables using the Howarth–Dorodnitsyn transformation (Schlichting and Gersten, 2017) √︄ ∗ 𝑈 ∗ ∫ 𝑦∗ 𝜌∗ √︁ 𝜌∞ ∞ ∗ 𝜌 ∗ 𝑈 ∗ 𝑥 ∗ 𝑓 (𝜂), 𝑑 𝑦 ˜ , 𝜓 = 𝜂= 2𝜇∞ (B.2) ∞ ∞ ∗ 𝑥∗ ∗ 2𝜇∞ 𝜌 0 ∞ where 𝜓 is a stream function such that 𝜌∗𝑢∗ = 𝜕𝜓 , 𝜕𝑦 ∗ 𝜌∗ 𝑣∗ = − 𝜕𝜓 , 𝜕𝑥 ∗ so that conservation of mass is satisfied by construction. Next √︄ ∗ 𝑈∗ 𝜂 𝜕𝜂 𝜌 ∗ 𝜌∞ 𝜕𝜂 ∞ = − , = ∗ ∗ 𝑥∗ , 𝜕𝑥 ∗ 2𝑥 ∗ 𝜕𝑦 ∗ 𝜌∞ 2𝜇∞ while 𝑢∗ = so that 1 𝜕𝜓 1 √︁ ∗ ∗ ∗ ∗ ′ 𝜕𝜂 ∗ ′ = 2𝜇∞ 𝜌∞𝑈∞ 𝑥 𝑓 . = 𝑈∞ 𝑓 , ∗ ∗ ∗ ∗ 𝜌 𝜕𝑦 𝜌 𝜕𝑦 𝜕𝑢 ∗ ∗ ′′ 𝜕𝜂 = 𝑈∞ 𝑓 , ∗ 𝜕𝑥 𝜕𝑥 ∗ 𝜕𝑢 ∗ ∗ ′′ 𝜕𝜂 = 𝑈∞ 𝑓 , ∗ 𝜕𝑦 𝜕𝑦 ∗ 99 and conservation of 𝑥-momentum becomes 𝜕  𝜌 ∗ 𝜇∗ ′′  + 𝑓 𝑓 ′′ = 0. ∗ 𝜇∗ 𝑓 𝜕𝜂 𝜌∞ ∞ We assume that the temperature is a function of 𝜂 only so that 𝑇 ∗ = 𝑇 ∗ (𝜂), while for an ideal gas 𝑑ℎ∗ = 𝑐∗𝑝 𝑑𝑇 ∗ and 𝑎 ∗∞ = 𝛾𝑅 ∗𝑇∞∗ so that the conservation of energy equation becomes ∗ ∗ ∗ 𝜕  1 𝜇∗ 𝜌 ∗ ∗ ′  2 𝜇 𝜌 𝑇∞ ′′ 2 ∗ ′ (𝑇 ) + (𝛾 − 1)Ma ∞ ∗ ∗ ( 𝑓 ) + 𝑓 (𝑇 ) = 0. ∗ 𝜌∗ 𝜕𝜂 𝑃𝑟 𝜇∞ 𝜇 𝜌 ∞ ∞ ∞ Assuming that 𝜇∗ ∝ 𝑇 ∗ we obtain in non-dimensional form 𝑓 ′′′ + 𝑓 𝑓 ′′ = 0, (B.3a) 1 ′′ 𝑇 + 𝑓 𝑇 ′ + (𝛾 − 1)Ma2∞ ( 𝑓 ′′) 2 = 0. 𝑃𝑟 (B.3b) We must enforce the no-slip and no-penetration boundary conditions so that 𝑓 (0) = 𝑓 ′ (0) = 0, while in the far-field we must have lim𝜂→∞ 𝑢(𝜂) = 𝑈∞ so that lim𝜂→∞ 𝑓 ′ (𝜂) = 1. At the wall we satisfy either isothermal, 𝑇 ′ (0) = 0, or adiabatic, 𝑇 (0) = 0, boundary conditions, while in the free-stream we must have lim𝜂→∞ 𝑇 (𝜂) = 1. We solve this ODE for ( 𝑓 , 𝑓 ′, 𝑓 ′′, 𝑇, 𝑇 ′) using a shooting method, which we then use to compute the base flow as a function of 𝑥 and 𝑦. Then the base flow (𝜈, 𝑢, 𝑣, 𝑝) can be obtained using √︂ 1 1 𝜕𝑓 , 𝑣(𝜂) = 𝑇 [𝜂 𝑓 ′ − 𝑓 ], 𝑝 = , (B.4) 𝜈(𝜂) = 𝑇 (𝜂), 𝑢(𝜂) = 𝜕𝜂 2𝑥𝑅𝑒 𝛾 where the expression for 𝑝 comes from the ideal gas law and by using that 𝑝 is constant, while for a given 𝑥, we solve for 𝑦 in terms of 𝜂 as ∫ 𝜂 √ 𝑦(𝜂) = 2𝑥𝑅𝑒 𝑇 𝑑 𝜂. ˜ (B.5) 0 100 Appendix C SPECIAL CASES OF SINGULAR 𝐴 If 𝐴 is singular, then we obtain  𝐴++ 0 0     𝐴˜ =  0 𝐴−− 0     0 0 𝐴00   (C.1) for diagonal 𝐴++ ∈ R𝑁+ ×𝑁+ with 𝐴++ > 0, 𝐴−− ∈ R𝑁 − ×𝑁 − with 𝐴−− < 0, and 𝐴00 ∈ R𝑁0 ×𝑁0 with 𝐴00 = 0, where 𝑁+ + 𝑁− + 𝑁0 = 𝑁. Then we define " # 𝐴 0 ++ 𝐴˜ ±± = (C.2a) 0 𝐴−− and " # 𝐿 ±± 𝐿 ±0 𝐿˜ (𝑠) = 𝐿 0± 𝐿 00 " # Í Í𝑑 ˜ ˜ 𝑠𝐼±± + 𝑑𝑗=2 𝑖𝜔 𝑗 𝐵˜ 𝑗,±± + 𝐶˜±± 𝑖𝜔 𝐵 + 𝐶 𝑗 𝑗,±0 ±0 𝑗=2 = , Í𝑑 Í𝑑 ˜ ˜00 ˜ ˜ 𝑖𝜔 𝐵 + 𝐶 𝑠𝐼 + 𝑖𝜔 𝐵 + 𝐶 𝑗 𝑗,0± 0± 00 𝑗 𝑗,00 𝑗=2 𝑗=2 (C.2b) so that 𝑑 𝝓ˆ± = 𝐿 ±± 𝝓ˆ± + 𝐿 ±0 𝝓ˆ0 + 𝒇ˆ𝜙,± 𝐴˜ ±± 𝑑𝑥 0 = 𝐿 0± 𝝓ˆ± + 𝐿 00 𝝓ˆ0 + 𝒇ˆ𝜙,0 . (C.3a) (C.3b) The zero eigenvalues, 𝐴˜ 00 , correspond to points in the base flow where the streamwise velocity is exactly zero or sonic. In practice, unless a grid point is placed exactly on the sonic line, we will have 𝐴˜ 00 associated with wall boundary-conditions only, so that 𝐿 00 will be invertible and we can eliminate 𝝓ˆ0 as −1 𝝓ˆ0 = −𝐿 00 (𝐿 0± 𝝓ˆ± + 𝒇ˆ𝜙,0 ). (C.4) We then define −1 −1 𝑀 = 𝐴˜ ±± (𝐿 ±± − 𝐿 ±0 𝐿 00 𝐿 0± ), −1 ˆ −1 ˆ 𝒈ˆ = 𝐴˜ ±± ( 𝒇𝜙,± − 𝐿 ±0 𝐿 00 𝒇𝜙,0 ), (C.5) to obtain the ODE 𝑑 𝝓ˆ ˆ = 𝑀 𝝓ˆ± + 𝒈, 𝑑𝑥 which can be treated as in the non-singular case. (C.6) 101 Appendix D PROOFS FOR OWNS-P AND OWNS-R D.1 Proofs for OWNS-P Proof: [Proof of Proposition 2.2.13] First note that (2.16b) through (2.16d) can be diagonalized using (𝑀 − 𝑖𝛽𝐼) 𝝓ˆ = (𝑉 𝐷𝑉 −1 − 𝑖𝛽𝑉𝑉 −1 )𝑉𝝍 = 𝑉 (𝐷 − 𝑖𝛽𝐼)𝝍, where 𝑉 has full rank while (𝐷 − 𝑖𝛽𝐼) is diagonal so that each scalar component of 𝑁𝛽 {𝝍 𝑗 } 𝑗=−𝑁 and 𝝍 can be treated separately as 𝛽 −𝑗 𝑗 − 𝑗−1 (𝛼 𝑘 − 𝛽−𝑗 )𝜓 𝑘 − (𝛼 𝑘 − 𝛽+ )𝜓 𝑘 𝑗 = 1, . . . , 𝑁 𝛽 − 1, = 0, 0 (𝛼 𝑘 − 𝛽−0 )𝜓 0𝑘 − (𝛼𝑘 − 𝛽+0 )𝜓 −1 𝑘 = (𝛼 𝑘 − 𝛽− )𝜓 𝑘 , 𝑗 𝑗 𝑗+1 (𝛼𝑘 − 𝛽+ )𝜓 𝑘 − (𝛼 𝑘 − 𝛽−𝑗 )𝜓 𝑘 𝑗 = 0, . . . , 𝑁 𝛽 − 1, = 0, for 𝑘 = 1, . . . , 𝑁. For the downstream-going modes we obtain 𝜓 0𝑘 − 𝜓 𝑘 = 𝑁 𝜓𝑘 𝛽 = 𝑁Ö 𝛽 −1 𝑗 𝛼 𝑘 − 𝛽+ 𝑗 𝑗=0 𝛼 𝑘 − 𝛽− 𝑁Ö 𝛽 −1 −𝑁 −𝑁 𝜓 𝑘 𝛽 = 𝐹𝑘 𝜓 𝑘 𝛽 , (D.1a) 𝜓 0𝑘 = 𝐹𝑘 𝜓 0𝑘 , (D.1b) 𝑗 𝛼 𝑘 − 𝛽+ 𝑗 𝑗=0 𝛼 𝑘 − 𝛽− so that −𝑁 𝐹++ 𝝍+ 𝛽 = 𝝍+0 − 𝝍+ , 𝐹++ 𝝍+0 = 𝝍+ 𝛽 , 𝑁 (D.1c) for 𝑘 = 1, . . . , 𝑁+ . Similarly, for the upstream-going modes we obtain 𝜓 0𝑘 = −𝑁 𝜓𝑘 𝛽 = 𝑁Ö 𝛽 −1 𝛼𝑘 − 𝛽−𝑗 𝑗 𝑗=0 𝛼 𝑘 − 𝛽+ 𝑁Ö 𝛽 −1 𝛼𝑘 − 𝛽−𝑗 𝑗 𝑗=0 𝛼 𝑘 − 𝛽+ 𝜓 𝑘 𝛽 = 𝐹𝑘−1 𝜓 𝑘 𝛽 , 𝑁 (D.2a) (𝜓 0𝑘 − 𝜓 𝑘 ) = 𝐹𝑘−1 (𝜓 0𝑘 − 𝜓 𝑘 ), (D.2b) 𝑁 so that −1 𝐹−− 𝝍− 𝛽 = 𝝍−0 , 𝑁 −𝑁 −1 𝐹−− (𝝍−0 − 𝝍− ) = 𝝍− 𝛽 , (D.2c) 102 for 𝑘 = 𝑁+ + 1, . . . , 𝑁. Next recall that " # " #" # " # 𝝓ˆ+ 𝑉++ 𝑉+− 𝝍+ 𝑉++ 𝝍+ + 𝑉+− 𝝍− = = 𝝓ˆ− 𝑉−+ 𝑉−− 𝝍− 𝑉−+ 𝝍+ + 𝑉−− 𝝍− so that (2.16a) and (2.16e) are equivalent to −𝑁 −𝑁 −𝑁 𝝓ˆ+ 𝛽 = 𝑉++ 𝝍+ 𝛽 + 𝑉+− 𝝍− 𝛽 = 0, 𝑁 𝑁 𝑁 𝝓ˆ− 𝛽 = 𝑉−+ 𝝍+ 𝛽 + 𝑉−− 𝝍− 𝛽 = 0. or −𝑁 −𝑁 −1 𝝍+ 𝛽 = −𝑉++ 𝑉+− 𝝍− 𝛽 , −1 𝝍− 𝛽 = −𝑉−− 𝑉−+ 𝝍+ 𝛽 . 𝑁 𝑁 −1 to obtain We pre-multiply by 𝐹++ and 𝐹−− −𝑁 −𝑁 −1 −1 −1 𝝍+0 − 𝝍+ = 𝐹++ 𝝍+ 𝛽 = −𝐹++𝑉++ 𝑉+− 𝝍− 𝛽 = −𝐹++𝑉++ 𝑉+− 𝐹−− (𝝍−0 − 𝝍− ), −1 −1 −1 −1 −1 𝝍−0 = 𝐹−− 𝝍− 𝛽 = −𝐹−− 𝑉−−𝑉−+ 𝝍+ 𝛽 = −𝐹−− 𝑉−−𝑉−+ 𝐹++ 𝝍+0 , 𝑁 𝑁 and rearranging yields −1 −1 0 −1 −1 𝝍+0 + 𝐹++𝑉++ 𝑉+− 𝐹−− 𝝍− = 𝝍+ + 𝐹++𝑉++ 𝑉+− 𝐹−− 𝝍− , −1 −1 𝐹−− 𝑉−−𝑉−+ 𝐹++ 𝝍+0 + 𝝍−0 = 0, which in matrix form becomes " #" # " #" # −1𝑉 𝐹 −1 0 −1𝑉 𝐹 −1 𝐼++ 𝐹++𝑉++ 𝝍 𝐼 𝐹 𝑉 𝝍+ +− −− ++ ++ ++ +− −− + = . −1𝑉 −1𝑉 𝐹 𝐹−− 𝐼−− 𝝍−0 0 0 𝝍− −− −+ ++ 0 −1 Using 𝑅 𝑁 𝛽 from (2.17b), the above is equivalent to 𝑅 −1 𝑁 𝛽 𝝍 = 𝐸 𝑅 𝑁 𝛽 𝝍, so that −1 ˆ ˆ 𝝓ˆ0 = 𝑉 𝑅 𝑁 𝛽 𝐸 𝑅 −1 𝑁 𝛽 𝑉 𝝓 = 𝑃 𝑁 𝛽 𝝓0 . Thus, we have shown that the action of the filter (2.16) can be applied using the projection matrix (2.17a). □ Proof: [Proof of Proposition 2.2.15] Clearly 𝑃 𝑁 𝛽 → 𝑃 if and only if 𝑅 −1 𝑁 𝛽 → 𝐼, while 𝐹 is diagonal so that −1 −1 (𝐹++𝑉++ 𝑉+− 𝐹−− )𝑚𝑛 = −1 −1 (𝐹−− 𝑉−−𝑉−+ 𝐹++ )𝑛𝑚 = 𝑁Ö 𝛽 −1 𝑗 𝛼𝑚 − 𝛽+ 𝛼𝑛 − 𝛽−𝑗 𝑗 𝑗 𝑗=0 𝛼𝑚 − 𝛽− 𝛼𝑛 − 𝛽+ 𝑁Ö 𝛽 −1 𝑗 𝛼𝑚 − 𝛽+ 𝛼𝑛 − 𝛽−𝑗 𝑗 𝑗 𝑗=0 𝛼𝑚 − 𝛽− 𝛼𝑛 − 𝛽+ and 𝑅 −1 𝑁 𝛽 → 𝐼 if and only if (2.18) holds. −1 (𝑉++ 𝑉+− )𝑚𝑛 , −1 (𝑉++ 𝑉+− )𝑛𝑚 , □ 103 Proof: [Proof of Proposition 2.2.16] First we show that if 𝛼𝑚 ≠ 𝛼𝑛 for all 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) then we can always choose recursion parameters such that the OWNS-O approximation converges. Without loss of generality, assume that 𝑁+ ≤ 𝑁− . Then 𝑁Ö + −1 𝑗 |𝛼𝑚 − 𝛽+ | |𝛼𝑛 − 𝛽−𝑗 | = 0, 𝑗 𝑗 𝑗=0 |𝛼𝑚 − 𝛽− | |𝛼𝑛 − 𝛽+ | ∀𝑚 = 𝑖 (+) , 𝑛 = 𝑖 (−) , were we have used 𝛼𝑚 − 𝛽+𝑚−1 = 0, ∀𝑚 = 1, . . . , 𝑁+ , 𝛼𝑚 − 𝛽−𝑗−1 ≠ 0, ∀𝑚, 𝑗 = 1, . . . , 𝑁+ , 𝑗−1 𝛼𝑛 − 𝛽 + ≠ 0, ∀ 𝑗 = 1, . . . , 𝑁+ , ∀𝑛 = 1, . . . , 𝑁− . An identical procedure for 𝑁− < 𝑁+ proves that 𝛼𝑚 ≠ 𝛼𝑛 for all 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) is a sufficient condition for convergence. If 𝛼𝑚˜ = 𝛼𝑛˜ for some some 𝑚˜ = 𝑖 (+) and 𝑛˜ = 𝑖 (−) , then −1 −1 (𝐹−− 𝑉−−𝑉−+ 𝐹++ )𝑛˜ 𝑚˜ = 𝑁Ö 𝛽 −1 𝑗 𝛼𝑚˜ − 𝛽+ 𝛼𝑛˜ − 𝛽−𝑗 𝑗 𝑗 𝑗=0 𝛼𝑚˜ − 𝛽− 𝛼𝑛˜ − 𝛽+ −1 −1 (𝑉−− 𝑉−+ )𝑛˜ 𝑚˜ = (𝑉−− 𝑉−+ )𝑛˜ 𝑚˜ so that 𝑅 −1 𝑁 𝛽 ≠ 𝐼, no matter how the recursion parameters are chosen, proving the necessity of the condition. □ Theorem D.1.1 (Block matrix inverse) If a matrix is partitioned into four blocks such that " # 𝑅++ 𝑅+− 𝑅= , (D.3a) 𝑅−+ 𝑅−− then it can be inverted blockwise as " # −1 " # −1 ] −1 ] 𝑅 𝑅 [𝑅 [𝑅 ++ +− ++ +− 𝑅 −1 = = , 𝑅−+ 𝑅−− [𝑅 −1 ] −+ [𝑅 −1 ] −− (D.3b) where −1 𝑅−+ ) −1 , [𝑅 −1 ] ++ = (𝑅++ − 𝑅+− 𝑅−− (D.3c) −1 −1 [𝑅 −1 ] +− = −(𝑅++ − 𝑅+− 𝑅−− 𝑅−+ ) −1 𝑅+− 𝑅−− , (D.3d) −1 −1 [𝑅 −1 ] −+ = −𝑅−− 𝑅−+ (𝑅++ − 𝑅+− 𝑅−− 𝑅−+ ) −1 , (D.3e) −1 −1 −1 −1 [𝑅 −1 ] −− = 𝑅−− + 𝑅−− 𝑅−+ (𝑅++ − 𝑅+− 𝑅−− 𝑅−+ ) −1 𝑅+− 𝑅−− . (D.3f) 104 Theorem D.1.2 (Neumann series) For a matrix 𝑅, if the sum to a finite value, then ∞ ∑︁ −1 𝑅𝑘 . (𝐼 − 𝑅) = Í∞ 𝑘=0 𝑅 𝑘 converges (D.4) 𝑘=0 The proof of Proposition 2.2.17 relies on two theorems. Theorem D.1.1 provides an expression for the inverse of a block matrix, while theorem D.1.2 provides conditions −1𝑉 𝐹 −2𝑉 −1𝑉 𝐹 ) −1 ∥ is finite. under which ∥(𝐼++ − 𝐹++𝑉++ +− −− −− −+ ++ Proof: [Proof of Proposition 2.2.17] First note that −1 ∥𝑃 − 𝑃 𝑁 𝛽 ∥ = ∥𝑉 𝐸𝑉 −1 − 𝑉 𝑅 −1 𝑁𝛽 𝐸 𝑅𝑁𝛽 𝑉 ∥ −1 = ∥𝑉 (𝐸 − 𝑅 −1 𝑁 𝛽 𝐸 𝑅 𝑁 𝛽 )𝑉 ∥ −1 ≤ ∥𝑉 ∥∥𝐸 − 𝑅 −1 𝑁 𝛽 𝐸 𝑅 𝑁 𝛽 ∥∥𝑉 ∥, while by theorem D.1.1 for the inverse of a block matrix " #" #" # −1 ] −1 ] [𝑅 [𝑅 𝐼 0 [𝑅 ] [𝑅 ] ++ +− ++ 𝑁 ++ 𝑁 +− 𝑁𝛽 𝑁𝛽 𝛽 𝛽 𝑅 −1 𝑁𝛽 𝐸 𝑅𝑁𝛽 = −1 ] [𝑅 −1 ] [𝑅 0 0 [𝑅 ] [𝑅 𝑁 𝛽 −+ 𝑁 𝛽 ] −− 𝑁 𝛽 −+ 𝑁 𝛽 −− " #" # [𝑅 −1 [𝑅 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] +− 𝑁 𝛽 ] ++ 0 = [𝑅 −1 0 0 𝑁 𝛽 ] −+ 0 " # −1 ] [𝑅 ] [𝑅 −1 ] [𝑅 ] [𝑅 ++ 𝑁 ++ ++ 𝑁 +− 𝛽 𝛽 𝑁𝛽 𝑁𝛽 = −1 −1 [𝑅 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] +− so that " 𝐸 − 𝑅 −1 𝑁𝛽 𝐸 𝑅𝑁𝛽 = −1 [𝑅 −1 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] ++ − 𝐼++ [𝑅 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] +− [𝑅 −1 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] ++ [𝑅 −1 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] +− # . By the triangle inequality −1 −1 ∥𝐸 − 𝑅 −1 𝑁 𝛽 𝐸 𝑅 𝑁 𝛽 ∥ ≤ ∥ [𝑅 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] ++ − 𝐼++ ∥ + ∥ [𝑅 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] +− ∥ −1 + ∥ [𝑅 −1 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] ++ ∥ + ∥ [𝑅 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] +− ∥, while [𝑅 𝑁 𝛽 ] ++ = 𝐼++ −1 −1 [𝑅 𝑁 𝛽 ] +− = 𝐹++𝑉++ 𝑉+− 𝐹−− −1 −2 −1 −1 [𝑅 −1 𝑁 𝛽 ] ++ = (𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−−𝑉−− 𝑉−+ 𝐹++ ) , −1 −1 −1 −2 −1 −1 [𝑅 −1 𝑁 𝛽 ] −+ = −𝐹−−𝑉−− 𝑉−+ 𝐹++ (𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−−𝑉−− 𝑉−+ 𝐹++ ) , 105 so that −1 −1 −1 −1 ∥ [𝑅 −1 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] ++ − 𝐼++ ∥ ≤ ∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥ × ∥𝐹−−𝑉−− 𝑉−+ 𝐹++ ∥ −1 −2 −1 × ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) −1 ∥, −1 −1 ∥ [𝑅 −1 𝑁 𝛽 ] ++ [𝑅 𝑁 𝛽 ] +− ∥ ≤ ∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥ −1 −2 −1 × ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) −1 ∥, −1 −1 ∥ [𝑅 −1 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] ++ ∥ ≤ ∥𝐹−−𝑉−− 𝑉−+ 𝐹++ ∥ −1 −2 −1 × ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) −1 ∥, −1 −1 −1 −1 ∥ [𝑅 −1 𝑁 𝛽 ] −+ [𝑅 𝑁 𝛽 ] +− ∥ ≤ ∥𝐹−−𝑉−− 𝑉−+ 𝐹++ ∥ × ∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥ −1 −2 −1 × ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) −1 ∥. Combining these results yields −1 −2 −1 −1 −1 −1 ∥𝐸 − 𝑅 −1 𝑁 𝛽 𝐸 𝑅 𝑁 𝛽 ∥ ≤ ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−−𝑉−− 𝑉−+ 𝐹++ ) ∥ ∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥
−1 −1 −1 −1 −1 −1 + 2∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥∥𝐹−− 𝑉−−𝑉−+ 𝐹++ ∥ + ∥𝐹−− 𝑉−−𝑉−+ 𝐹++ ∥ . Next we note that ∞ ∑︁ −1 −2 −1 (𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) 𝑘 𝑘=0 is finite if −1 −1 −1 ∥𝐹++ ∥∥𝐹−− ∥ < (∥𝑉++ 𝑉+− ∥∥𝑉−− 𝑉−+ ∥) −1/2 . Take some 𝜖ˆ > 0 such that 𝜖ˆ ≪ 1 and define −1 −1 𝜖 = min{𝜖, ˆ ∥𝑉++ 𝑉+− ∥ −1/2 ∥𝑉−− 𝑉−+ ∥ −1/2 }. −1 ∥ < 𝜖, theorem D.1.2 for Neumann series applies and Then assuming ∥𝐹++ ∥∥𝐹−− −1 −2 −1 ∥(𝐼++ − 𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ) −1 ∥ ∞ ∑︁ −1 −2 −1 ∥𝐹++𝑉++ 𝑉+− 𝐹−− 𝑉−−𝑉−+ 𝐹++ ∥ 𝑘 ≤ 𝑘=0 ≤ ∞ ∑︁ −1 −1 2 −1 𝑉−+ ∥) 𝑘 (∥𝐹++ ∥ 2 ∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥∥𝑉−− 𝑘=0 −1 2 −1 −1 −1 4 = 1 + ∥𝐹++ ∥ 2 ∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥∥𝑉−− 𝑉−+ ∥ + O (∥𝐹++ ∥ 4 ∥𝐹−− ∥ ), while −1 −1 −1 −1 ∥𝐹++𝑉++ 𝑉+− 𝐹−− ∥ ≤ ∥𝐹++ ∥∥𝐹−− ∥∥𝑉++ 𝑉+− ∥, −1 −1 −1 −1 ∥𝐹−− 𝑉−−𝑉−+ 𝐹++ ∥ ≤ ∥𝐹++ ∥∥𝐹−− ∥∥𝑉−− 𝑉−+ ∥, 106 so that 2 −1 2 −1 −1 4 ∥𝐸 − 𝑅 −1 𝑁 𝛽 𝐸 𝑅 𝑁 𝛽 ∥ ≤ [1 + ∥𝐹++ ∥ ∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥∥𝑉−− 𝑉−+ ∥ + O (𝜖 )] −1 −1 −1 × ∥𝐹++ ∥∥𝐹−− ∥ [∥𝑉++ 𝑉+− ∥ + ∥𝑉−− 𝑉−+ ∥ −1 −1 −1 + 2∥𝐹++ ∥∥𝐹−− ∥∥𝑉−− 𝑉−+ ∥∥𝑉++ 𝑉+− ∥]
−1 −1 −1 = ∥𝐹++ ∥∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥ + ∥𝑉−− 𝑉−+ ∥ + O (𝜖 2 ), and
−1 −1 −1 ∥𝑃 𝑁 𝛽 − 𝑃∥ ≤ ∥𝑉 ∥∥𝐹++ ∥∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥ + ∥𝑉−− 𝑉−+ ∥ ∥𝑉 −1 ∥ + O (𝜖 2 ), −1 ∥ < 𝜖. where we have used ∥𝐹++ ∥∥𝐹−− □ Proof: [Proof of Proposition 2.2.22] Note that ∥𝑃𝑀 − 𝑀 𝑃∥ = 0 while ∥𝑃 𝑁 𝛽 𝑀 − 𝑀 𝑃 𝑁 𝛽 ∥ = ∥(𝑃 𝑁 𝛽 − 𝑃) 𝑀 + 𝑀 (𝑃 − 𝑃 𝑁 𝛽 )∥ ≤ 2∥𝑃 𝑁 𝛽 − 𝑃∥∥𝑀 ∥ −1 −1 ≤ 2∥𝑉 ∥∥𝐹++ ∥∥𝐹−− ∥ ∥𝑉++ 𝑉+− ∥
−1 + ∥𝑉−− 𝑉−+ ∥ ∥𝑉 −1 ∥∥𝑀 ∥ + O (𝜖 2 ), by Proposition 2.2.17. □ Proof: [Proof of Proposition 2.2.18] A downstream-going mode associated with 0 − 𝜓 = 0, while by equation (D.1a) we have 𝛼𝑚 is retained accurately if 𝜓𝑚 𝑚 0 𝜓𝑚 − 𝜓𝑚 = 𝑗 𝑁Ö 𝛽 −1 𝑗 𝛼𝑚 − 𝛽 + 𝑗 𝑗=0 𝛼𝑚 − 𝛽− −𝑁 𝜓𝑚 𝛽 = 0 𝑗 since 𝛼𝑚 = 𝛽+ for some 𝛽+ , while 𝛼𝑚 ≠ 𝛽−𝑗 for all 𝛽−𝑗 . An upstream-going mode associated with 𝛼𝑛 is removed if 𝜓𝑛0 = 0, while by equation (D.2a) we have 𝜓𝑛0 = 𝑁Ö 𝛽 −1 𝛼𝑛 − 𝛽−𝑗 𝑗 𝑗=0 𝛼𝑛 − 𝛽+ 𝑗 𝑁 𝜓𝑛 𝛽 = 0 𝑗 since 𝛼𝑛 = 𝛽−𝑗 for some 𝛽−𝑗 , while 𝛼𝑛 ≠ 𝛽+ for all 𝛽+ . □ 107 D.2 Proofs for OWNS-R Proof: [Proof of Proposition 2.2.25] First note that 𝐸 𝑁(𝑘)𝛽 =  1+𝑐 𝑁𝛽 𝑗 Ö 𝛼 𝑘 − 𝛽  −1 + 𝑗 𝑗=1 𝛼 𝑘 − 𝛽− Î𝑁 𝛽 𝑗 𝑗  𝑗=1 (𝛼 𝑘 − 𝛽− ) + 𝑐 𝑗=1 (𝛼 𝑘 − 𝛽+ ) −1 = Î𝑁 𝛽 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) Î𝑁 𝛽 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) = Î𝑁 , Î𝑁 𝛽 𝑗 𝑗 𝛽 (𝛼 − 𝛽 ) + 𝑐 (𝛼 − 𝛽 ) − 𝑘 𝑘 + 𝑗=1 𝑗=1  Î𝑁 𝛽 for 𝑘 = 1, . . . , 𝑁, then observe that 𝑃 𝑁(𝑅) = (𝐼 + 𝑐𝑍) −1 = [𝑉 (𝐼 + 𝑐𝐹)𝑉 −1 ] −1 = 𝑉 (𝐼 + 𝑐𝐹) −1𝑉 −1 = 𝑉 𝐸 𝑁 𝛽 𝑉 −1 . 𝛽 □ Proof: [Proof of Proposition 2.2.27] First note that 𝑃 𝑁(𝑅) 𝑃 𝑁(𝑅) = 𝑉 𝐸 𝑁 𝛽 𝑉 −1𝑉 𝐸 𝑁 𝛽 𝑉 −1 = 𝑉 𝐸 𝑁2 𝛽 𝑉 −1 , 𝛽 𝛽 so that 𝑃 𝑁(𝑅) is a projection matrix if and only if 𝐸 𝑁2 𝛽 = 𝐸 𝑁 𝛽 . Since 𝐸 𝑁 𝛽 is diagonal, 𝛽 we can consider each entry separately so that we must have 𝐸 𝑁2 𝛽 ,𝑘 = 𝐸 𝑁 𝛽 ,𝑘 for 𝑘 = 1, . . . , 𝑁. We know that 𝐸 𝑁2 𝛽 ,𝑘 = 𝐸 𝑁 𝛽 ,𝑘 if and only if 𝐸 𝑁 𝛽 ,𝑘 = 0 or 𝐸 𝑁 𝛽 ,𝑘 = 1, concluding the proof. □ Proof: [Proof of Proposition 2.2.31] First we show that if 𝛼𝑚 ≠ 𝛼𝑛 for all pairs of 𝑚 = 𝑖 (+) and 𝑛 = 𝑖 (−) , then we can always choose recursion parameters such that the OWNS-R approximation converges. Note that Î𝑁 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) (𝑘) = 1, 𝑘 = 𝑖 (+) , 𝐸 𝑁 𝛽 = Î𝑁 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) + 0 0 𝐸 𝑁(𝑘)𝛽 = = 0, 𝑘 = 𝑖 (−) , Î𝑁 𝑗 0 + 𝑐 𝑗=1 (𝛼 𝑘 − 𝛽+ ) Î Î 𝑗 where 𝑁𝑗=1 (𝛼 𝑘 − 𝛽−𝑗 ) ≠ 0 for 𝑘 = 𝑖 (+) and 𝑁𝑗=1 (𝛼 𝑘 − 𝛽+ ) ≠ 0 for 𝑘 = 𝑖 (−) by construction. Next we show that if 𝛼𝑚˜ = 𝛼𝑛˜ for some 𝑚˜ = 𝑖 (+) and 𝑛˜ = 𝑖 (−) , then there does not exist any choice of recursion parameters such that 𝑃 𝑁(𝑅) → 𝑃. By contradiction, suppose 𝛽 108 there exists recursion parameters such that 𝑃 𝑁(𝑅) converges to 𝑃 when 𝛼𝑚˜ = 𝛼𝑛˜ for 𝛽 ˜ ˜ some 𝑚˜ = 𝑖 (+) and 𝑛˜ = 𝑖 (+) . Then we must have 𝐸 𝑁( 𝑚) → 1 and 𝐸 𝑁( 𝑛) → 0, but 𝛽 𝛽 ˜ ˜ 𝐸 𝑁( 𝑚) = 𝐸 𝑁( 𝑛) and 1 ≠ 0, leading to a contradiction. 𝛽 𝛽 □ Proof: [Proof of Proposition 2.2.34] Note that ∥𝑃 − 𝑃 𝑁(𝑅) ∥ = ∥𝑉 𝐸𝑉 −1 − 𝑉 𝐸 𝑁 𝛽 𝑉 −1 ∥ ≤ ∥𝑉 ∥∥𝐸 − 𝐸 𝑁 𝛽 ∥∥𝑉 −1 ∥, 𝛽 where for the downstream-going modes 𝐸 (𝑘) Î𝑁 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) (𝑘) − 𝐸 𝑁 𝛽 = 1 − Î𝑁 Î𝑁 𝑗 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) + 𝑐 𝑗=1 (𝛼 𝑘 − 𝛽+ ) Î 𝑗 𝑐 𝑁𝑗=1 (𝛼 𝑘 − 𝛽+ ) = Î𝑁 Î𝑁 𝑗 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) + 𝑐 𝑗=1 (𝛼 𝑘 − 𝛽+ ) = 𝑐𝐹𝑘 , 1 + 𝑐𝐹𝑘 for 𝑘 = 𝑖 (+) , while for the upstream-going modes Î𝑁 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽− ) (𝑘) (𝑘) 𝐸 − 𝐸 𝑁 𝛽 = 0 − Î𝑁 Î𝑁 𝑗 𝑗 𝑗=1 (𝛼 𝑘 − 𝛽+ ) 𝑗=1 (𝛼 𝑘 − 𝛽− ) + 𝑐 =− 𝐹𝑘−1 𝐹𝑘−1 + 𝑐 , for 𝑘 = 𝑖 (−) . Assuming that |𝐹𝑘 | < 𝜖 ≪ 1 for 𝑘 = 𝑖 (+) and |𝐹𝑘−1 | < 𝜖 ≪ 1 for 𝑘 = 𝑖 (−) we obtain |𝐸 (𝑘) − 𝐸 𝑁(𝑘)𝛽 | = |𝑐||𝐹𝑘 | + O (|𝐹𝑘 | 2 ), 𝑘 = 𝑖 (+) |𝐸 (𝑘) − 𝐸 𝑁(𝑘)𝛽 | = |𝐹𝑘−1 | + O (|𝐹𝑘−1 | 2 ), 𝑘 = 𝑖 (−) . Since 𝐸 − 𝐸 𝑁 𝛽 is a diagonal matrix we obtain ∥𝐸 − 𝐸 𝑁 𝛽 ∥ = max  max |𝑐||𝐹𝑘 | + O (𝜖 2 ), 𝑘=1,...,𝑁+ max 𝑘=𝑁+ +1,...,𝑁 |𝐹𝑘−1 | + O (𝜖 2 )  −1 = max |𝑐|∥𝐹++ ∥, ∥𝐹−− ∥ + O (𝜖 2 ), so that  −1 ∥𝑃 − 𝑃 𝑁(𝑅) ∥ ≤ ∥𝑉 ∥ max |𝑐|∥𝐹++ ∥, ∥𝐹−− ∥ ∥𝑉 −1 ∥ + O (𝜖 2 ). 𝛽 □ 109 𝑗 Proof: [Proof of Proposition 2.2.35] If 𝛽+ = 𝛼𝑚 for downstream-going mode 𝛼𝑚 , then 𝐸 𝑁(𝑚) = 1. Taking 𝝍 such that 𝜓 𝑘 = 0 for 𝑘 ≠ 𝑚 and 𝜓𝑚 = 1, we have 𝛽 𝑃 𝑁(𝑅) 𝑉𝝍 = 𝑉 𝐸 𝑁 𝛽 𝝍 = 𝑉𝝍. Similarly, if 𝛽−𝑗 = 𝛼𝑛 for upstream-going mode 𝛼𝑛 , 𝛽 then 𝐸 𝑁( 𝛽 𝑛) = 0. Taking 𝝍 such that 𝜓 𝑘 = 0 for 𝑘 ≠ 𝑛 and 𝜓𝑛 = 1, we have 𝑃 𝑁(𝑅) 𝑉𝝍 = 𝑉 𝐸 𝑁 𝛽 𝝍 = 0. 𝛽 □ Proof: [Proof of Proposition 2.2.38] Note that (𝑃 𝑁(𝑅) ) 𝑛 = 𝑉 𝐸 𝑁𝑛 𝛽 𝑉 −1 , 𝛽 and that    0     (𝑘) 𝑛 lim |𝐸 𝑁 𝛽 | = 1 𝑛→∞     ∞  |𝐸 𝑁(𝑘)𝛽 | < 0, |𝐸 𝑁(𝑘)𝛽 | = 1, |𝐸 𝑁(𝑘)𝛽 | > 1. If |𝐸 𝑁(𝑘)𝛽 | < 1, then mode 𝑘 will be removed by repeated applications of 𝑃 𝑁(𝑅) , 𝛽 which is desirable for upstream-going modes, but not for downstream-going modes. In contrast, if |𝐸 𝑁(𝑘)𝛽 | > 1, then mode 𝑘 will grow without bound through repeated applications of 𝑃 𝑁(𝑅) , which is undesirable for both upstream- and downstream-going 𝛽 modes. If 𝐸 𝑁(𝑘)𝛽 = 1, then mode 𝑘 will be accurately retained. Therefore, repeated applications of 𝑃 𝑁(𝑅) introduces error unless unless |𝐸 𝑁(𝑛)𝛽 | < 1 for all 𝑛 = 𝑖 (−) , and 𝛽 𝐸 𝑁(𝑚) = 1 for all 𝑚 = 𝑖 (+) . 𝛽 □ Proof: [Proof or Proposition 2.2.41] Note that Î𝑁 𝛽 Î𝑁 𝛽 𝑗 𝑗 1 𝑗=1 (𝛼 𝑘 − 𝛽− ) 𝑗=1 (𝛼 𝑘 − 𝛽− ) (𝑘) 𝐸 𝑁 𝛽 = Î𝑁 = Î𝑁 , Î𝑁 𝛽 𝑗 𝑗 𝑗 𝛽 𝛽 2 (𝛼 − 𝛽 ) + (𝛼 − 𝛽 ) (𝛼 − 𝛽 ) − 𝑘 𝑘 𝑘 ∗ + 𝑗=1 𝑗=1 𝑗=1 so that 𝑃 𝑁(𝑅) = 𝑉 𝐸 𝑁 𝛽 𝑉 −1 𝛽 𝑁𝛽 1 Ö 𝑗 = 𝑉 (𝐷 − 𝛽∗ 𝐼) −1 (𝐷 − 𝛽−𝑗 𝐼)𝑉 −1 2 𝑗=1 𝑁𝛽 1Ö 𝑗 = (𝑀 − 𝛽∗ 𝐼) −1 (𝑀 − 𝛽−𝑗 𝐼), 2 𝑗=1 which can be-rearranged into (2.26). □ 110 Appendix E A VARIATIONAL APPROACH TO OWNS We will now consider a variational approach to OWNS. Consider the equation for the downstream-going modes in characteristic variables 𝜕 𝝓ˆ′± ˆ = 𝑃𝑀 𝝓ˆ′± + 𝑃 𝒈. 𝜕𝑥 (3.10) We note that " 𝑀 = 𝑉 𝐷𝑉 −1 = 𝑉 𝐷𝑈 = 𝑉++ 𝑉+− 𝑉−+ 𝑉−− #" 𝐷 ++ 0 0 𝐷 −− #" 𝑈++ 𝑈+− 𝑈−+ 𝑈−− # , where 𝑈 = 𝑉 −1 , while the projection operator is defined such that " #" #" # " # 𝑉++ 𝑉+− 𝐼++ 0 𝑈++ 𝑈+− 𝑉++𝑈++ 𝑉++𝑈+− 𝑃= = . 𝑉−+ 𝑉−− 0 0 𝑈−+ 𝑈−− 𝑉−+𝑈++ 𝑉−+𝑈+− 𝑀 has full rank (𝑁± ), while 𝑃 is rank-deficient unless there are no upstream-going characteristics (𝑁− = 0), since rank(𝑃) = 𝑁+ ≤ 𝑁± , while " # 𝐻 𝑉 𝐻 𝑉 𝐻 𝑈𝐻 𝑈 ++ ++ −+ ++ 𝑃𝐻 = . 𝐻 𝐻 𝐻 𝑈𝐻 𝑈+−𝑉++ 𝑉−+ +− We further define the left and right bases " # h i 𝑉++ 𝑃𝐿 = , 𝑃 𝑅𝐻 = 𝑈++ 𝑈+− , 𝑉−+ so that " 𝑃 𝐿 𝑃 𝑅𝐻 = 𝑉++ 𝑉−+ 𝑃 𝑅𝐻 𝑃 𝐿 = h # h i " 𝑈++ 𝑈+− = " # while i 𝑈++ 𝑈+− 𝑉++ 𝑉−+ 𝑉++𝑈++ 𝑉++𝑈+− 𝑉−+𝑈++ 𝑉−+𝑈+− # = 𝑃, = 𝑈++𝑉++ + 𝑈+−𝑉−+ = 𝐼++ , where we have used " # " # 𝑈++𝑉++ + 𝑈+−𝑉−+ 𝑈++𝑉+− + 𝑈+−𝑉−− 𝐼++ 0 𝑈𝑉 = = 𝐼±± . = 𝑈−+𝑉++ + 𝑈−−𝑉−+ 𝑈−+𝑉+− + 𝑈−−𝑉−− 0 𝐼−− It is shown in Towne et al. (2022) that 𝑃2 = 𝑃, so that 𝑃 is a projection. 111 E.1 Petrov-Galerkin method The (Bubnov-)Galerkin method requires an orthogonal projection, while the PetrovGalerkin method does not. If 𝑀 is a normal operator, then 𝑀 = 𝑉 𝐷𝑉 𝐻 by the spectral theorem so that 𝑃 is self-adjoint and thus orthogonal. In general, 𝑀 is non-normal, so we consider only the Petrov-Galerkin approach. We consider the space V of elliptic solutions V ≡ {𝝓± | 𝝓± is bounded}, which we use to define the trial space V+ of downstream-going solutions V+ ≡ {𝝓± ∈ V | 𝝓± = 𝑃𝝓± }, and the test space W+ ≡ {𝒘 ∈ V | 𝒘 = 𝑃 𝐻 𝒘}. Any solution in the trial space and can be expressed as 𝝓± = 𝑃𝝓± = 𝑃 𝐿 𝑃 𝑅𝐻 𝝓± = 𝑃 𝐿 𝜶, 𝜶 = 𝑃 𝑅𝐻 𝝓 ∈ C𝑁+ , while any test function can be expressed as 𝒘 = 𝑃 𝐻 𝒘 = 𝑃 𝑅 𝑃 𝐿𝐻 𝒘 = 𝑃 𝑅 𝜷, 𝜷 = 𝑃 𝐿𝐻 𝒘 ∈ C𝑁+ . We now consider the weak form of problem (3.9): find 𝝓ˆ± ∈ V+ such that 𝒘𝐻 𝜕 𝝓ˆ± ˆ = 𝒘 𝐻 𝑀 𝝓ˆ± + 𝒘 𝐻 𝒈, 𝜕𝑥 ∀𝒘 ∈ W+ . Choosing 𝜷𝑖 = 𝒆ˆ𝑖 , where 𝒆ˆ𝑖 is the unit vector associated with the 𝑖th axis, we state our problem: find 𝜶 ∈ C𝑁+ such that 𝑃 𝑅𝐻 E.1.1 𝜕 (𝑃 𝐿 𝜶) ˆ = 𝑃 𝑅𝐻 𝑀 𝑃 𝐿 𝜶 + 𝑃 𝑅𝐻 𝒈. 𝜕𝑥 (E.1) Constant-coefficient system If the problem has constant coefficients (in the streamwise direction), then it is straightforward to show that if 𝜶 is a solution to (E.1), then 𝑃 𝐿 𝜶 is a solution to (3.10). We left multiply (E.1) by 𝑃 𝐿 to obtain 𝜕 (𝑃 𝐿 𝜶) ˆ = 𝑃[𝑀 𝑃 𝐿 𝜶 + 𝒈], 𝜕𝑥 (E.2) 112 where we have used that 𝑃𝑃 𝐿 𝜶 = 𝑃 𝐿 𝜶 and that 𝑃 is not a function of 𝑥. We take the difference of (E.2) and (3.10) to obtain
𝜕 (𝑃 𝐿 𝜶 − 𝝓′± ) 𝜕 − 𝑃𝑀 [𝑃 𝐿 𝜶 − 𝝓′± ] = − 𝑀 [𝑃 𝐿 𝜶 − 𝝓′± ] = 0 𝜕𝑥 𝜕𝑥 𝜕 and we use that 𝜕𝑥 − 𝑀 has full rank to assert that the above equation is satisfied if and only if 𝑃 𝐿 𝜶 = 𝝓′± . We can similarly show that the projection of 𝝓′± into the space V+ , 𝜶± = 𝑃 𝑅𝐻 𝝓′± , satisfies (E.1). First we note that 𝝓′± = 𝑃𝝓′± = 𝑃 𝐿 𝑃 𝑅𝐻 𝝓′± = 𝑃 𝐿 𝜶± , then we left multiply (3.10) by 𝑃 𝑅𝐻 to obtain 𝑃 𝑅𝐻 𝜕 (𝑃 𝐿 𝜶± ) ˆ = 𝑃 𝑅𝐻 [𝑀 𝑃 𝐿 𝜶± + 𝒈]. ˆ = 𝑃 𝑅𝐻 𝑃[𝑀 𝑃 𝐿 𝜶± + 𝒈] 𝜕𝑥 (E.3) The difference of (E.1) and (E.3) yields 𝜕 (𝑃 𝐿 [𝜶 − 𝜶± ]) − 𝑃 𝑅𝐻 𝑀 𝑃 𝐿 [𝜶 − 𝜶± ] = 𝑃 𝑅𝐻 𝑃 𝑅𝐻 𝜕𝑥 𝜕 𝜕𝑥  − 𝑀 𝑃 𝐿 [𝜶 − 𝜶± ] = 0,
𝜕 and we use that 𝑃 𝑅𝐻 𝜕𝑥 − 𝑀 𝑃 𝐿 has full rank to assert that 𝜶 = 𝑃 𝑅𝐻 𝝓′± = 𝜶± . Therefore, we conclude for a constant-coefficient system that (E.1) and (3.10) are equivalent problems. E.1.2 Variable-coefficient system If the coefficients vary in 𝑥, then we can show that the projection of 𝝓′± onto V+ satisfies (E.1) using the procedure outline for the constant-coefficient system. However, we cannot show that 𝑃 𝐿 𝜶 is a solution to (3.10). Let us assume that 𝑃 𝐿 𝜶 is a solution to (3.10), then we left multiply (E.1) by 𝑃 𝐿 to obtain 𝑃 𝜕 (𝑃 𝐿 𝜶) ˆ = 𝑃𝑀 𝑃 𝐿 𝜶 + 𝑃 𝑅𝐻 𝒈. 𝜕𝑥 (E.4) If we take the difference of the above equation with (3.10), then we obtain 𝑃 𝜕 (𝑃 𝐿 𝜶) 𝜕𝝓′± − − 𝑃𝑀 [𝑃 𝐿 𝜶 − 𝝓′± ] = 0, 𝜕𝑥 𝜕𝑥 (E.5) and we can no longer assert that 𝑃 𝐿 𝜶 = 𝝓′± . Let us instead consider the problems that result from discretizing in the streamwise direction. Following discretization with an 𝑠-order BDF scheme we obtain the fully-discrete form of (3.9) 𝑠−1 ∑︁ 𝑗=0 (𝑛+1− 𝑗) 𝑐 ( 𝑗) 𝝓± = Δ𝑥 [𝑀 (𝑛+1) 𝝓±(𝑛+1) + 𝒈 (𝑛+1) ]. 113 At station 𝑛 + 1, the Petrov-Galerkin approach yields the problem 𝑐 (0) 𝜶 (𝑛+1) = Δ𝑥𝑃 𝑅(𝑛+1)𝐻 [𝑀 (𝑛+1) 𝑃 𝐿(𝑛+1) 𝜶 (𝑛+1) + 𝒈 (𝑛+1) ] − 𝑠−1 ∑︁ (𝑛+1− 𝑗) 𝑐 ( 𝑗) 𝑃 𝑅(𝑛+1)𝐻 𝝓± . 𝑗=1 (E.6) Similarly, if we apply the OWNS projection operator, 𝑃 (𝑛+1) , to the discrete form of (3.9) we obtain 𝑐 (0) 𝝓′(𝑛+1) = Δ𝑥𝑃 (𝑛+1) [𝑀 (𝑛+1) 𝝓′(𝑛+1) + 𝒈 (𝑛+1) ] − ± ± 𝑠−1 ∑︁ ′(𝑛+1− 𝑗) 𝑐 ( 𝑗) 𝑃 (𝑛+1) 𝝓± . (E.7) 𝑗=1 We left multiply (E.6) by 𝑃 𝐿(𝑛+1) and take the difference with (E.7) to obtain
(E.8) 𝑐 (0) 𝐼±± − Δ𝑥𝑀 (𝑛+1) [𝑃 𝐿(𝑛+1) 𝜶 (𝑛+1) − 𝝓±(𝑛+1)′] = 0, and we use that 𝑐 (0) 𝐼±± −Δ𝑥𝑀 (𝑛+1) has full rank to assert that 𝑃 𝐿(𝑛+1) 𝜶 (𝑛+1) = 𝝓±(𝑛+1)′. Next we left multiply (E.7) by 𝑃 𝑅𝐻 (𝑛+1) and take the difference with (E.6) to obtain
𝑐 (0) 𝐼++ − Δ𝑥𝑃 𝑅𝐻 (𝑛+1) 𝑀 (𝑛+1) 𝑃 𝐿(𝑛+1) [𝜶±(𝑛+1) − 𝜶 (𝑛+1) ] = 0, (E.9) and we use that 𝑐 (0) 𝐼 − Δ𝑥𝑃 𝑅𝐻 (𝑛+1) 𝑀 (𝑛+1) 𝑃 𝐿(𝑛+1) has full rank to assert that 𝜶±(𝑛+1) = 𝜶 (𝑛+1) . Therefore, we conclude for a constant-coefficient system, as well as a constant-coefficient system, that (E.6)and (E.7) are equivalent problems. Let us now consider the OWNS system that results from discretizing (3.10) 𝑐 (0) 𝝓′(𝑛+1) = Δ𝑥𝑃 (𝑛+1) [𝑀 (𝑛+1) 𝝓′(𝑛+1) + 𝒈 (𝑛+1) ] − ± ± 𝑠−1 ∑︁ ′(𝑛+1− 𝑗) 𝑐 ( 𝑗) 𝑃 (𝑛+1− 𝑗) 𝝓± . (E.10) 𝑗=1 The only difference between system (E.7) and (E.10) are how the projection operator is applied to the derivative term. We left multiply (E.6) by 𝑃 𝐿(𝑛+1) and take the difference with (E.10) to obtain [𝑐 (0) 𝐼±± − Δ𝑥𝑀 (𝑛+1) ] [𝑃 𝐿(𝑛+1) 𝜶 (𝑛+1) − 𝝓±(𝑛+1) ] + 𝑠−1 ∑︁ (𝑛+1− 𝑗) 𝑐 ( 𝑗) [𝑃 (𝑛+1) − 𝑃 (𝑛+1− 𝑗) ]𝝓± = 0. (E.11) 𝑗=1 We can no longer assert that 𝜶 (𝑛+1) and 𝝓±(𝑛+1)′ are equal because of the left-over spatial-marching term. We see that while (E.6)and (E.7) are equivalent systems, the system (E.10) is not. We return now our attention to (E.5). If we discretize using an 𝑠-order BDF scheme, then we obtain (E.11), showing that Petrov-Galerkin approach is not generally equivalent to the OWNS approach when the coefficients vary. However, the OWNS approach assumes slowly-varying coefficients so that the Petrov-Galerkin and OWNS approaches should yield similar results. 114 Appendix F JACOBIAN OF THE NOWNS EQUATIONS Here we derive the Jacobian of the fully-discrete system of NOWNS equations (3.20) for 2D disturbances. Note that this analysis readily extends to 3D disturbances. We introduce the 2D residual ‡(𝑘 𝑥 +1) 𝒓ˆ𝑚 = 𝑠−1 ∑︁ 𝑥 +1−𝑙) 𝑥 +1) 𝑐 (𝑙) 𝐴‡ 𝝓ˆ‡(𝑘 − 𝐿ˆ ‡𝑚 𝝓ˆ‡(𝑘 − 𝑭ˆ𝑚‡(𝑘 𝑥 +1) − 𝒇ˆ𝑚‡(𝑘 𝑥 +1) , 𝑚 𝑚 (F.1) 𝑙=0 for 𝑚 ∈ Z 𝑀 , which mimics the 3D residual defined in (3.23). We take its derivative 𝑥 +1) to obtain with respect to 𝝓ˆ‡(𝑘 𝑝 ‡(𝑘 𝑥 +1) 𝜕 𝒓ˆ𝑚 𝜕 𝑭ˆ𝑚‡(𝑘 𝑥 +1) (0) ‡ 𝑥 +1) ˆ ‡(𝑘 = (𝑐 𝐴 − 𝐿 )𝛿 − , 𝑚𝑝 𝑚 ‡(𝑘 𝑥 +1) 𝑥 +1) ˆ 𝜕 𝝓ˆ‡(𝑘 𝜕 𝝓 𝑝 𝑝 𝑚, 𝑝 = 0, . . . , 2𝑀, (F.2) Using the definition DFT, it can be shown (for 2D disturbances) that 2𝑀 ∑︁ 𝑝−𝑚 𝜕 𝑭ˆ𝑚‡(𝑘 𝑥 +1) 1 𝜕 𝑭˜ ‡(𝑘 𝑥 +1) = 𝑒𝑖2𝜋𝑙 2𝑀+1 , ‡(𝑘 +1) ‡(𝑘 +1) 𝝓=𝝓𝑙 2𝑀 + 1 𝑙=0 𝜕𝝓 𝑥 𝜕 𝝓ˆ 𝑝 𝑥 (F.3) which allows us to write  0 0 0  " # # "   (𝑘 +1) (𝑘 𝑥 +1) ˜ (𝑘 𝑥 +1) ) −1 𝐽ˆ(𝑘 𝑥 +1)  𝜕 𝑭ˆ𝑚‡(𝑘 𝑥 +1)  (𝑘 𝑥 +1) ( 𝐴˜ ±±𝑥 ) −1 𝐽ˆ±±,𝑝−𝑚 (𝑘 𝑥 +1) ( 𝐴±± ±0,𝑝−𝑚 , 𝑃 0 𝑃 = (𝑘 +1) (𝑘 +1) 1,𝑚 1,𝑚 𝑥   𝑥 ˆ ˆ 𝑥 +1) 𝐽 𝐽   𝜕 𝝓ˆ‡(𝑘 0±,𝑝−𝑚 00,𝑝−𝑚 𝑝   (𝑘 +1) (𝑘 +1) 𝑥 𝑥 ˆ ˆ   𝐽 𝐽 0 0±,𝑝−𝑚 00,𝑝−𝑚   (F.4) for 𝑚, 𝑝 = 0, . . . , 2𝑀. We define 𝜕 𝑭ˆ𝑚‡(𝑘 𝑥 +1) ‡(𝑘 𝑥 +1) ˆ 𝐽 𝑝−𝑚 = − ‡(𝑘 +1) 𝜕 𝝓ˆ 𝑝 𝑥 (F.5) for 𝑚, 𝑝 = 0, . . . , 2𝑀 and taking 𝑀 = 1, we obtain the Jacobian 𝜕𝑅 (𝑘+1),𝜎 /𝜕Φ (𝑘+1),𝜎 shown in (F.6), where 𝜎 denotes the index for the Newton iteration, while 𝑅 (𝑘+1),𝜎 and Φ (𝑘+1),𝜎 are the concatenated residual and solution vectors defined in (3.26). ‡(𝑘 𝑥 +1) ‡(𝑘 𝑥 +1) Note that we have used 𝐽ˆ𝑚‡(𝑘 𝑥 +1) = 𝐽ˆ𝑚+2𝑀+1 and 𝐽ˆ𝑚‡(𝑘 𝑥 +1) = 𝐽ˆ−𝑚 for 𝑚 = 0, . . . , 𝑀.  (0) ‡   (𝑐 𝐴 − 𝐿ˆ ‡(𝑘 𝑥 +1) ) + 𝐽ˆ‡(𝑘 𝑥 +1),𝜎  𝐽ˆ1‡(𝑘 𝑥 +1),𝜎 𝐽ˆ1‡(𝑘 𝑥 +1),𝜎 0 0     ‡(𝑘 +1),𝜎 ‡(𝑘 +1),𝜎 ‡(𝑘 +1) ‡(𝑘 +1),𝜎 𝑥 𝑥 𝑥 𝑥 (0) ‡ ˆ ˆ ˆ ˆ =  𝐽1 𝐽1 (𝑐 𝐴 − 𝐿 1 ) + 𝐽0 (𝑘+1),𝜎   𝜕Φ   ‡(𝑘 +1),𝜎 ‡(𝑘 +1),𝜎 ‡(𝑘 +1) ‡(𝑘 +1),𝜎 𝑥 𝑥 𝑥 𝑥 (0) ‡   𝐽ˆ1 𝐽ˆ1 (𝑐 𝐴 − 𝐿ˆ −1 ) + 𝐽ˆ0   𝜕𝑅 (𝑘+1),𝜎 (F.6) 115 116 Appendix G DESIGN CHOICES FOR NOWNS G.1 Streamwise diffusion terms Here we demonstrate for the 2D validation case discussed in Section 3.3.1 that although streamwise diffusion effects have a minimal impact on the linear calculation (Figure G.1a), the impact is more pronounced in the nonlinear case (Figure G.1b). We see that prior to the second neutral stability point, the calculations with and without the streamwise diffusion terms yield similar amplitudes, but that downstream of this point, the amplitudes of the calculation including the streamwise viscous terms has much lower amplitudes (particularly for the higher harmonics in the nonlinear case). G.2 Inclusion of the streamwise pressure gradient Unlike the NPSE approach, the NOWNS approach has no minimum step size requirement for the march to be numerically stable, but it is still necessary to neglect the streamwise pressure gradient for the zero-frequency modes. If the streamwise pressure gradient for these modes is retained, the NOWNS march remains stable but becomes inaccurate, as demonstrated in Figure G.2. We plot the 𝑢-velocity amplitudes with and without 𝜕𝑥 𝑝 0 in Figure G.2a, where we see that we have reasonable agreement for 𝑚 ≠ 0, but disagreement for 𝑚 = 0. In Figure G.2b we plot the profile of the 𝑣 0 , and we notice that the profile predicted by NOWNS when 𝜕𝑥 𝑝 0 is included is substantially different from the profiles predicted by NPSE and DNS. If we neglect 𝜕𝑥 𝑝 0 , but include the streamwise diffusion terms 𝜕𝑥𝑥 𝑞 0 , then we must project the MFD and we find that we have good agreement between the DNS and NOWNS calculations. Therefore, we can conclude that the recursion parameters we are using for the zero-frequency modes are valid. However, as discussed above, the calculation remains stable but becomes inaccurate when we include 𝜕𝑥 𝑝 0 . Although our march remains stable, there including 𝜕𝑥 𝑝 0 leads to inaccuracies in the march for unknown reasons. In Figure G.3a, we plot the 𝑢-velocity amplitudes computed by NOWNS with and without the pressure gradient for the zero-frequency modes. Mode (1,0) is tracked reasonably accurately, but the other modes are not. In particularly, the (1,2), 117 10 -2 10 -5 10 -3 10 -10 2 4 6 8 10 2 (a) Linear calculation 4 6 8 10 (b) Nonlinear calculation Figure G.1: 𝑢-velocity amplitudes with and without streamwise diffusion terms for the 2D evolution of the TS wave. 8 6 10 -5 4 2 NOWNS DNS NPSE 10 -10 2 4 6 8 10 0 -2 0 2 4 6 8 10 -5 (a) 𝑢-velocity amplitudes (b) 𝑣-velocity profile Figure G.2: 2D validation case with and without streamwise pressure gradient for zero-frequency modes. (1,1), and (0,1) modes and the MFD have higher amplitudes than they should. In Figure G.3b, we see that we have better qualitative agreement in the early stages of the march between the NOWNNS and DNS calculations when the streamwise pressure gradient terms are include for the zero-frequency modes. However, we have worse quantitative agreement in the later stages of the march, which in turn causes the larger amplitudes of the (1,2), (1,1) and (0,1) modes observed in Figure G.3a. G.3 Comparison of nonlinear solution procedures We have discussed three procedures for solving the nonlinear system of equations: (i) Newton’s method, (ii) a quasi-Newton method that includes part of the nonlinear Jacobian, and (iii) a quasi-Newton method that excludes the Jacobian of the nonlinear 118 10 -2 10 -4 10 -5 10 -6 10 -10 1.4 1.6 1.8 2 10 -8 2.2 (a) 𝑢-velocity amplitudes 1.4 1.6 1.8 2 2.2 (b) 𝑢-velocity amplitudes for MFD Figure G.3: K-type transition case with and without streamwise pressure gradient for zero-frequency modes. 20 140 15 Quasi-Newton (linear) Quasi-Newton 120 100 10 80 60 5 40 Newton Quasi-Newton 20 0 2 4 6 8 10 0 1.5 (a) Comparison of Newton’s method and the quasi-Newton method for the 2D validation case 2 2.5 (b) Comparison of the linear and nonlinear quasi-Newton methods for K-type transition Figure G.4: Comparison of solution procedures for the nonlinear system of equations. term. We plot the iterations to converge as a function of streamwise station in Figure G.4. For the 2D validation case, we compare the iterations to convergence for the quasi-Newton method and Newton’s method in Figure G.4a, which shows that Newton’s method converges in fewer iterations. In Figure G.4b, we make the same comparison for linear and nonlinear quasi-Newton methods for K-type transition. We see that farther downstream, where the nonlinearity is stronger, the nonlinear quasi-Newton method converges in fewer iterations than the linear version. However, for sufficiently strong nonlinearities, it is necessary to use Newton’s method. 119 G.4 Linearization about the baseflow vs. the corrected mean flow Whereas in linear stability analysis, the mean flow is determined only by the baseflow, ¯ in the nonlinear case the disturbances interact to excite the MFD, 𝒒′00 , which 𝒒, yields the corrected mean flow, 𝒒¯ + 𝒒′00 . In linear OWNS, we linearize the projection operators about the baseflow, while in nonlinear OWNS, we can choose to instead linearize about the corrected mean flow. G.4.1 Equations linearized about the corrected mean flow Consider nonlinear equations of the form N (𝒒) = 0. (G.1) To linearize about the base flow, we take 𝒒 = 𝒒¯ + 𝒒′ and assume that the base flow ¯ = 0 and satisfies the governing equations, so that N ( 𝒒) ¯ + 𝐿 ( 𝒒)𝒒 ¯ ′ + 𝐹 ( 𝒒, ¯ 𝒒′) = 𝐿( 𝒒)𝒒 ¯ ′ + 𝑭( 𝒒, ¯ 𝒒′), N ( 𝒒¯ + 𝒒′) = N ( 𝒒) ¯ ′ and 𝑭( 𝒒, ¯ 𝒒′) are the linear and nonlinear terms, respectively. This where 𝐿 ( 𝒒)𝒒 yields ¯ ′ + 𝑭( 𝒒, ¯ 𝒒′) = 0. 𝐿( 𝒒)𝒒 (G.2) If we linearize about the corrected mean flow, then we define 𝒒˜ ′ = 𝒒′ − 𝒒ˆ 00 so that N ( 𝒒¯ + 𝒒ˆ 00 + 𝒒˜ ′) = N ( 𝒒¯ + 𝒒ˆ 00 ) + 𝐿( 𝒒¯ + 𝒒ˆ 00 ) 𝒒˜ ′ + 𝑭( 𝒒¯ + 𝒒ˆ 00 , 𝒒˜ ′) ¯ + 𝐿 ( 𝒒) ¯ 𝒒ˆ 00 + 𝑭( 𝒒, ¯ 𝒒ˆ 00 ) + 𝐿 ( 𝒒¯ + 𝒒ˆ 00 ) 𝒒˜ ′ + 𝑭( 𝒒¯ + 𝒒ˆ 00 , 𝒒˜ ′), = N ( 𝒒) yielding ¯ 𝒒ˆ 00 + 𝑭( 𝒒, ¯ 𝒒ˆ 00 ) + 𝐿 ( 𝒒¯ + 𝒒ˆ 00 ) 𝒒˜ ′ + 𝑭( 𝒒¯ + 𝒒ˆ 00 , 𝒒˜ ′) = 0. 𝐿 ( 𝒒) (G.3) In the current form, we must update the LU decomposition after each Newton iteration, so we decrease the computational cost by linearizing about a fixed corrected mean flow. We take the MFD from the previous streamwise station and label it 𝒒 𝑐 . Then we take 𝒒˜ ′ = 𝒒′ − 𝒒 𝑐 to obtain ¯ 𝑐 + 𝑭( 𝒒, ¯ 𝒒 𝑐 ) + 𝐿 ( 𝒒¯ + 𝒒 𝑐 ) 𝒒˜ ′ + 𝑭( 𝒒¯ + 𝒒 𝑐 , 𝒒˜ ′) = 0. 𝐿 ( 𝒒)𝒒 If 𝒒 𝑐 = 𝒒ˆ 00 , then we recover (G.3). We consider three approaches: 1. Solve (G.3) by updating 𝒒ˆ 00 after each Newton iteration, or (G.4) 120 2. Take 𝒒 𝑐 equal to the MFD from the previous station and then solve (G.4). Update 𝒒 𝑐 by taking 𝒒 𝑐 = 𝑞˜ 00 + 𝒒 𝑐 each time the Newton solver converges. We say that the solver has converged when ∥ 𝒒˜ 00 ∥ is sufficiently small. 3. Take 𝒒 𝑐 equal to the MFD from the previous station and then solve (G.4). Do not update 𝒒 𝑐 unless 𝐴( 𝒒¯ + 𝒒 𝑐 ) and 𝐴( 𝒒¯ + 𝒒 𝑐 + 𝒒˜ 00 ) have characteristics with different signs. For the sake of computational efficiency, we take the third approach. This ensures that the signs of our characteristics are consistent with those of the corrected mean flow, without an increase in computational cost relative to linearizing about the base flow. G.4.1.1 Results for K-type transition For the K-type transition case of Section 3.4.4, we perform the NOWNS calculation again, but we instead linearize about the corrected mean flow. In Figure G.5, we compare the amplitudes computed by the two approaches to NOWNS and we see that in the early stages of the march, when the amplitudes are small, the two calculations are nearly identical, but that they begin to differ slightly as the disturbance amplitudes increase. The differences are small, but are most notable for the MFD, 𝒒′00 , and the vortex mode, 𝑞′01 , after Re𝑥 = 2.5 × 105 , which corresponds to an MFD amplitude of roughly 3% of 𝑈∞ . The two NOWNS marches continue to yield similar results until the end of the calculation at Re𝑥 = 2.74 × 105 , where the MFD amplitude is 11% of 𝑈∞ . Since linearizing about the corrected mean flow has a negligible impact on the solution, we recommend linearizing about the base flow for the sake of simplicity. 121 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 1.4 1.6 1.8 2 2.2 2.4 2.6 Figure G.5: Amplitude of 𝑢′ v.s. streamwise coordinate, Re𝑥 , for fundamental breakdown. Compare linearizing NOWNS about 𝒒¯ vs. 𝒒¯ + 𝒒′00 . 122 Appendix H HEURISTIC RECURSION PARAMETER SETS This thesis requires recursion parameter sets for the following low- and high-speed boundary-layer flows. We use the heuristic recursion parameters for high-speed boundary-layer flows from Kamal (2023), and the heuristic recursion parameters for low-speed boundary-layer flows from Rigas et al. (2017a), which we present below. H.1 Recursion parameters for non-zero frequencies The recursion parameters are chosen based on the eigenvalues of the Euler equations linearized about a uniform flow, as described in Towne and Colonius (2015). The eigenvalues are 𝑖𝛼𝑐 = 𝑖𝑘 , 𝑀¯ 𝑥 𝑖𝛼𝑎1 (𝑧) = 𝑖𝑘 − 𝑀¯ 𝑥 + 𝜇(𝑧) , 1 − 𝑀¯ 𝑥2 𝑖𝛼𝑎2 (𝑧) = 𝑖𝑘 − 𝑀¯ 𝑥 − 𝜇(𝑧) , 1 − 𝑀¯ 𝑥2 where 𝑀¯ 𝑥 = 𝑢/ ¯ 𝑐¯ is the local streamwise Mach number, 𝑘 = 𝜔/𝑐¯ is the streamwise wave number, 𝑧 is a composite wave-number (for the transverse directions), and the function 𝜇(𝑧) is given by √︃ 𝜇(𝑧) = 1 − (1 − 𝑀¯ 𝑥2 )𝑧 2 . For subsonic boundary layer flows, we use the same recursion parameters as in (Rigas et al., 2017a), which differ slightly from those developed originally in (Towne and Colonius, 2015), so we will briefly explain the new choice of recursion parameters. The present choice of recursion parameters are separated into the following groups: (i) vortical modes, which replace convective modes, (ii) fast and slow stream evanescent acoustic modes, and (iii) fast and slow stream propagating acoustic modes, where fast stream modes are associated with the free-stream streamwise Mach number, while slow stream modes are associated with a small Mach number (inside the boundary layer). The recursion parameters for the vortical modes of the present study match closely the parameters for convective modes in (Towne and Colonius, 2015), while the recursion parameters for the propagating acoustic modes are distributed the same way in both the present study and in (Towne and Colonius, 2015). The parameters for the evanescent acoustic modes are distributed in a slightly different way, so we focus our discussion on these parameters. 123 To distribute the parameters for the evanescent acoustic modes, we define 𝜂𝑐 = 𝑘 , 1 − 𝑀¯ 𝑥 𝜂𝑚 = 1.5 𝐿𝑦 , Δ𝑦 where 𝐿 𝑦 is the (dimensionless) extent of the domain in the wall-normal direction, while Δ𝑦 is the (uniform) wall-normal grid-spacing. These define the spacing parameter 𝜂 (ℎ) = 𝜂𝑚 +  ℎ  𝜂𝑚 − 𝜂𝑐 − 𝜂𝑚 , 𝜂𝑐 + 0.1 𝑁𝑒 − 1 𝑁𝑒 − 1 ℎ = 0, . . . , 𝑁 𝑒 − 1, which in turn defines 𝜇 (ℎ) = 𝜇(𝜂 (ℎ) ), ℎ = 0, . . . , 𝑁 𝑒 − 1. This contrasts with the choice 𝜇 (ℎ) = 𝜇max ℎ , 2𝑁 𝑒 ℎ = 0, . . . , 2𝑁 𝑒 − 1, used in Towne and Colonius (2015), where 𝜇max = 𝜇(𝑧max ). Here, 𝑧max represents the maximum transverse wave number supported by the semi-discrete Euler equations (wave numbers larger than 𝑧 max need not be considered because they are not supported by the semi-discrete equations). The present approach considers a range of 𝑧 near 𝑧max (distributed according to 𝜂 (ℎ) ) which is then used to define 𝜇 (ℎ) , while the approach presented in Towne and Colonius (2015) considers only one 𝑧 to get 𝜇max , which is then used to define 𝜇 (ℎ) spaced linearly over [0, 𝜇max ]. The present approach is advantageous because 𝜇(𝑧) is nonlinear in 𝑧. The recursion parameter sets for a subsonic boundary layer flow are summarized in Table H.1. The fast-stream values are denoted 𝑘 1 and 𝑀¯ 𝑥,1 , while the slow-stream values are denoted 𝑘 2 and 𝑀¯ 𝑥,2 . The acoustic modes must be computed for both the fast- and slow-stream values. H.2 Zero-frequency modes The recursion parameter sets for non-zero frequency modes match those used in Sleeman et al. (2024b). They depend on 𝜔 through the streamwise wave number 𝑘 = 𝜔/𝑐, ¯ and when 𝑘 = 0, the recursion parameters associated with the vortical and propagating acoustic waves go to zero, leaving only the evanscent acoustic waves, as shown in Table H.2. 124 Type Vortical Spacing 𝑏 (ℎ) = 𝑀¯𝑘1 + 𝑁ℎ𝜔 𝑥,1 Evanescent acoustic 𝑘1
𝑘2 ¯ 𝑥,2 − 𝑀 ¯ 𝑥,1 𝑀 ℎ = 0, . . . , 𝑁 𝜔 − 1 √︁ 𝑘 2 − (1 − 𝑀¯ 𝑥2 ) (𝜂 (ℎ) ) 2 𝜇 (ℎ) = 𝛽+ (1 + 𝑖)𝑏 ( 𝑗 ) ( 𝑗) 𝛽 −( 𝑗 ) −2𝑘2 𝑀2 − 𝑏( 𝑗) 1−𝑀22 ¯ 𝑥 𝑘+𝜇 ( 𝑗) −𝑀 ¯ 𝑥2 1− 𝑀 ¯ 𝑥 𝑘− 𝜇 ( 𝑗) −𝑀 ¯ 𝑥2 1− 𝑀 ℎ = 0, . . . , 𝑁𝑒 − 1 Propagating acoustic 𝜃 (ℎ) = 𝑁ℎ𝑝 𝜋/2 ℎ = 0, . . . , 2𝑁 𝑝 − 1 ¯ +cos 𝜃 𝑘 − 𝑀𝑥1− ¯2 𝑀 (2 𝑗) 𝑥 ¯ 𝜃 𝑘 − 𝑀𝑥 −cos ¯2 1− 𝑀 (2 𝑗+1) 𝑥 Table H.1: Recursion parameter sets for subsonic boundary layer flows. Type Evanescent acoustic Spacing √︁ 𝜇 (ℎ) = −(1 − 𝑀¯ 𝑥2 )(𝜂 (ℎ) ) 2 ℎ = 0, . . . , 𝑁 𝑒 − 1 𝛽+ ( 𝑗) 𝛽−( 𝑗) 𝜇( 𝑗) 1− 𝑀¯ 𝑥2 −𝜇 ( 𝑗 ) 1− 𝑀¯ 𝑥2 Table H.2: Recursion parameter sets for subsonic boundary layer flows for zerofrequency modes. 125 Appendix I LABELING EIGENVALUES ACCORDING TO BRIGGS’ CRITERION To evaluate J (·) and J (𝑅) (·), we must know whether an eigenvalue is upstream- or downstream-going, which we can do using Briggs’ criterion (see definition 2.2.3). By proposition 2.2.4, 𝑀 (𝑠) has 𝑁+ downstream- and 𝑁− upstream-going eigenvalues so that taking 𝜂 → ∞ yields 𝑁+ eigenvalues with I (𝛼) > 0 and 𝑁− eigenvalues with I (𝛼) < 0. In principle, we can choose a large value of 𝜂 such that there 𝑁+ eigenvalues with I (𝛼) > 0 and 𝑁− eigenvalues with I (𝛼) < 0, and then slowly decrease 𝜂 to zero while tracking the eigenvalues (since they are a continuous function of 𝜂) to assign a label of upstream- or downstream-going. However, for problems of interest, the sign of I (𝛼) remains the same for most eigenvalues as 𝜂 increases, as shown in Figure I.1a. For this configuration, the sign of I (𝛼) changes for only one eigenvalue, as shown in Figure I.1b. Since the sign of I (𝛼) does not change as 𝜂 increases for most eigenvalues, we choose the smallest 𝜂 > 0 such that there 𝑁+ eigenvalues with I (𝛼) > 0 and call it 𝜂opt (see algorithm 3). Then we match each eigenvalue associated with 𝜂opt to the closest eigenvalue associated with 𝜂 = 0, where we measure distance as |𝛼(𝜂opt − 𝑖𝜔) − 𝛼(−𝑖𝜔)| (see algorithm 4). There is no guarantee that this algorithm will always correctly label the eigenvalues, but we have found this to work well in-practice. Moreover, if an eigenvalue is labeled incorrectly, then the OWNS march will not be well-posed and will fail. Therefore, any failure in labeling the eigenvalues will be easy to identify. 126 (a) (b) Figure I.1: Spectrum of TS wave at Re𝑥 = 1.6 × 105 for different values of 𝜂. Algorithm 3 Bisection search for smallest 𝜂 such that there are 𝑁+ eigenvalues with I (𝛼) > 0 and 𝑁− eigenvalues with I (𝛼) < 0 Input: (real) frequency 𝜔 𝜂max ← 1, 𝜂min ← 0 while 𝜂max − 𝜂min > 𝜂max × 10−3 do 𝜂 ← (𝜂min + 𝜂max )/2 𝜶 ← eigenvalues of 𝑀 (𝜂 − 𝑖𝜔) 𝑁˜ + ← number of 𝛼 such that I (𝛼𝑖 ) > 0 for 𝑖 = 1, . . . , 𝑁 𝑁˜ − ← number of 𝛼 such that I (𝛼𝑖 ) < 0 for 𝑖 = 1, . . . , 𝑁 if 𝑁˜ + = 𝑁+ and 𝑁˜ − = 𝑁− then 𝜂max ← 𝜂 else 𝜂min ← 𝜂 end if end while etaopt ← 𝜂 return 𝜂opt 127 Algorithm 4 Apply Briggs’ criterion to label eigenvalues Given a (real) frequency 𝜔, compute 𝜂opt using algorithm 3 𝜶opt ← eigenvalues of 𝑀 (𝜂opt − 𝑖𝜔) 𝜶zero ← eigenvalues of 𝑀 (−𝑖𝜔) 𝑁˜ +zero ← number of 𝛼zero such that I (𝛼zero,𝑖 ) > 0 for 𝑖 = 1, . . . , 𝑁 𝑁˜ −zero ← number of 𝛼zero such that I (𝛼zero,𝑖 ) < 0 for 𝑖 = 1, . . . , 𝑁 opt 𝑁˜ + ← number of 𝛼opt such that I (𝛼opt,𝑖 ) > 0 for 𝑖 = 1, . . . , 𝑁 𝑁˜ −opt ← number of 𝛼opt such that I (𝛼opt,𝑖 ) < 0 for 𝑖 = 1, . . . , 𝑁 opt 𝑁flip ← | 𝑁˜ +zero − 𝑁˜ + | 𝑁count ← 0, 𝑖 ← 0, 𝜶+ ← {}, 𝜶− ← {} while 𝑁count < 𝑁flip do 𝑖 ←𝑖+1 𝑗 ← arg min 𝑗=1,...,𝑁 |𝛼zero,𝑖 − 𝛼opt, 𝑗 |. if sign(I (𝛼zero,𝑖 )) ≠ sign(I (𝛼opt, 𝑗 )) then 𝑁count ← 𝑁count + 1 if sign(I (𝛼zero,𝑖 )) > 0 then 𝜶− ← append 𝛼zero,𝑖 else 𝜶+ ← append 𝛼zero,𝑖 end if else if sign(I (𝛼zero,𝑖 )) > 0 then 𝜶+ ← append 𝛼zero,𝑖 else 𝜶− ← append 𝛼zero,𝑖 end if end if end while 128 Appendix J TRANSIENT GROWTH IN LOW-SPEED BOUNDARY-LAYER FLOWS OWNS can be used to find optimal (worst-case) disturbances, as done by Rigas et al. (2017b) and Towne et al. (2022). We also note that Monokrousos et al. (2010) have investigated optimal disturbances for 3D low-speed boundary-layer flows using time-stepping methods, while Tempelmann et al. (2010) have used PSE to investigate optimal disturbances for incompressible boundary-layer flows, and Tempelmann et al. (2012) did the same for compressible boundary-layer flows. We follow the procedure introduced by Rigas et al. (2017b), adapted to a low-speed boundary-layer flow, where we use the kinetic energy ∫ 𝑦 2 𝐻 ∥𝒒∥ 𝐾 = 𝒒 𝑊 𝐾 𝒒 ≈ [ 𝜌𝑢 ¯ ′2 + 𝜌𝑣 ¯ ′2 + 𝜌𝑤 ¯ ′2 ]𝑑𝑦, (J.1) 0 to measure the size of our disturbances. Then we can seek, for example, the inlet boundary condition that leads to the largest kinetic energy at the domain outlet: J (𝒒 in , qout ) ≡ ∥𝒒 out ∥ 2𝐾 ∥𝒒 in ∥ 2𝐾 (J.2) . We introduce the inner product ∫ ∫ ∫ 𝐻 ⟨𝝍± , 𝝓± ⟩ = 𝝍± 𝑊 𝝓± 𝑑𝑥 ≈ 𝝍±𝐻 𝝓± 𝑑𝑦𝑑𝑥, 𝑥 𝑥 (J.3) 𝑦 and note that our solution must satisfy the forward problem 𝜕𝝓± = 𝑃𝑀 𝝓± , 𝜕𝑥 𝑇 −1 𝝓± (𝑥 in ) = 𝒒 in , (J.4) which allows us to introduce the Lagrangian L (𝝓±,in , 𝝓±,out , 𝝓± , 𝝍± ) = J (𝑇 −1 𝝓±,in , 𝑇 −1 𝝓±,out ) − ⟨𝝍± , 𝜕𝝓± − 𝑃𝑀 𝝓± ⟩. (J.5) 𝜕𝑥 129 Integration-by-parts yields ∫ ∫ 𝜕𝝓± 𝐻 𝜕𝝓± ⟨𝝍± , − 𝑃𝑀 𝝓± ⟩ = 𝝍± 𝑊 𝑑𝑥 − 𝝍±𝐻 𝑊 𝑃𝑀 𝝓± 𝑑𝑥 𝜕𝑥 𝜕𝑥 𝑥 𝑥 ∫ 𝐻 ∫ 𝝍± =− 𝑊 𝝓± 𝑑𝑥 − (𝑀 𝐻 𝑃 𝐻 𝝍± ) 𝐻 𝑊 𝝓± 𝑑𝑥 𝑥 𝜕𝑥 𝑥 𝐻 𝐻 + 𝝍±,out 𝑊 𝝓±,out − 𝝍±,in 𝑊 𝝓±,in 𝜕𝝍± 𝐻 𝐻 = −⟨ + 𝑀 𝐻 𝑃 𝐻 𝝍± , 𝝓± ⟩ + 𝝍±,out 𝑊 𝝓±,out − 𝝍±,in 𝑊 𝝓±,in , 𝜕𝑥 so that 𝜕𝝍± + 𝑀 𝐻 𝑃 𝐻 𝝍± , 𝝓± ⟩ 𝜕𝑥 𝐻 𝐻 − 𝝍±,out 𝑊 𝝓±,in . 𝑊 𝝓±,out + 𝝍±,in L (𝝓±,in , 𝝓±,out , 𝝓± , 𝝍± ) = J (𝑇 −1 𝝓±,in , 𝑇 −1 𝝓±,out ) + ⟨ Variations with respect to each variable must be zero, yielding the following constraints: 𝜕𝝓± − 𝑃𝑀 𝝓± = 0, 𝜕𝑥 𝜕𝝓± ⟨∇𝝓 , 𝛿𝝍⟩ = 0 ⇒ + 𝑀 𝐻 𝑃 𝐻 𝝓± = 0, 𝜕𝑥 ⟨∇𝝓out , 𝛿𝝓out ⟩ = 0 ⇒ 𝝍±,out = 𝑇 −𝐻 𝑊 𝐾𝑇 −1 𝝓±,out (J.6b) ⟨∇𝝓in , 𝛿𝝓in ⟩ = 0 ⇒ 𝝓±,in = (𝑇 −𝐻 𝑊 𝐾𝑇 −1 ) −1 𝝍±,in . (J.6d) ⟨∇𝝍 , 𝛿𝝓⟩ = 0 ⇒ (J.6a) (J.6c) The solution procedure is summarized as follows: 1. Solve the direct equation (J.6a). 2. Solve for the adjoint outlet boundary condition (J.6c), rescaled to unit norm. 3. Solve the adjoint equation (J.6b). 4. Solve for the inlet boundary condition (J.6d), rescaled to unit norm. J.1 Preliminary results The Tollmien-Schlichting (TS) wave is the only unstable mode in low-speed boundarylayer flows. Therefore, it is the optimal modal inlet boundary condition. Moreover, given a sufficiently long streamwise distance over which the TS wave is unstable, we expect any other disturbance to eventually turn into a TS wave, since any components that are colinear with the TS wave will be amplified. However, since the linearized Navier-Stokes operator is non-normal, there should exist an inlet boundary condition 130 that causes non-modal (transient) growth, which leads to more growth than the TS wave on its own. We verify that our results agree with expectation for 2D disturbances to the Blasius profile at 𝑀𝑎 = 0.1 with disturbance frequency 𝐹 = 50 × 10−6 . The TS wave is unstable over 𝑅𝑒 𝑥 ∈ [3.75 × 105 , 1.43 × 105 ], and so we seek the optimal inlet boundary condition over this streamwise domain with 𝑦 ∈ [0, 150], 𝑁𝑥 = 6000 and 𝑁 𝑦 = 200. Figure J.1 plots the norms associated with the optimal, random, and TS wave boundary conditions, showing that the optimal solution experiences transient growth, before turning into a TS wave and evolving like a TS wave. Similarly, the random inlet boundary condition also turns into a TS wave, but more slowly than the optimal. To assess how similar a disturbance is to a TS wave, we measure it using the inner product TS Score of 𝒒 ≡ 𝒒 𝑯 𝑊 𝒒 TS , ∥𝒒∥ 𝐾 ∥∥𝒒 TS ∥ 𝐾 (J.7) which is zero if 𝒒 is orthogonal to the TS wave, and 1 if it is the TS wave. We see that the optimal inlet boundary condition becomes a TS wave much more quickly than the random inlet boundary condition. Figure J.2 compares the optimal inlet boundary condition to the TS wave. We see that the optimal inlet condition has a large amplitude 𝑢-velocity disturbance near the surface of the plate, while both the 𝑢- 𝑣-velocity disturbances are smaller farther from the wall. We repeat this analysis for a range of frequencies, and plot the gain as a function of 𝐹 in Figure J.3, where we see that both the optimal and TS gains decrease with increasing 𝐹 at roughly the same rate. Figure J.4 plots 𝐺 optim − 𝐺 TS and shows that the transient growth decreases with increasing 𝐹. 131 Figure J.1: Norm of optimal inlet boundary condition, compared to random and TS wave inlet boundary conditions for 𝐹 = 50 × 10−6 , between branches I and II. (a) (b) Figure J.2: Optimal inlet boundary condition for 𝐹 = 50 × 10−6 , compared to TS wave. 132 Figure J.3: Optimal vs. TS gain for a range of frequencies. Figure J.4: Difference in optimal and TS gains for a range of frequencies.