Development of a Nonlinear Parabolized Stability Equation (NPSE) Analysis Tool for Span- wise Invariant Boundary Layers Technische Universiteit Delft S.H.J. Westerbeek Development of a Nonlinear Parabolized Stability Equation (NPSE) Analysis Tool for Spanwise Invariant Boundary Layers by S.H.J. Westerbeek to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Thursday January 16, 2020 at 10:00 AM. Student number: 4350693 Project duration: Feb 4, 2019 – January 16, 2020 Supervisors: Dr. M. Kotsonis Ir. J. Casacuberta Puig Thesis committee: Dr. M. Kotsonis, Aerodynamics Dr. H.M. Schuttelaars, Applied Mathematics Dr. S.J. Hulshoff, Aerodynamics Ir. J. Casacuberta Puig, Aerodynamics Preface The thesis before you describes the development of a nonlinear stability analysis tool for crossflows at the Delft University of Technology. The research group under Dr. Marios Kotsonis that specializes in flow transi- tion and control contains a numerical department that aims to deliver a transition analysis tool. This thesis contributes to this goal by developing a nonlinear solver to be used in further research and development of laminar airfoils. This thesis is also a depiction of my development as a person and a researcher as it can be considered my first deep dive into the physics of a specific phenomenon. During this dive, I was given opportunities to follow my instinct and try new solution methods on many occasions, which I consider a blessing. This the- sis carries my name first and foremost, however, I have to thank many people without whom its realization would have been impossible. I would like to thank my supervisors Marios Kotsonis and Jordi Casacuberta Puig for their openness, ex- citement and their seemingly infinite knowledge of boundary layer transition, stability, and flow control. In addition, I would like to thank Alberto Rius Vidales, Koen Groot and Theo Michelis for the discussions we had concerning boundary layer stability. It is a blessing to be a part of the team and discuss the complicated problems we encounter in stability analyses. It is truly inspiring to see the physics at play in a wind tunnel while working on the code and simultaneously be able to see a DNS simulation of that same flow with in- credible detail. I want to thank Marios and Jordi especially for not just helping me in this journey that is the thesis, but also investing time in my personal development as a scientist and supporting my ideas. The time and effort you put in this project is greatly appreciated. I have learned so much over the past year through working with you. The words in this thesis could not possibly comprise all of it. For the first time, I felt like time was moving faster than I wanted it to. Working in this team and on the topic of boundary layer transition has sparked much more excitement in me than I ever thought possible. On the road to this thesis, I spent many weekends of my first year in the library to study extensively the many books available there. Accompanied by the library boys Sid, Tomàs, Robert and many more. Thank you for making these weekends so much more than just studying. And thank you to the friends I have known for longer than I can remember. Roeland, Jessica, Jeroen, Bart, Dave, Laurens, Bryan, Sicco, Rahul, Jill, Janiek and Inge thank you for all the joy and laughter you bring. I appreciate you so much and I enjoyed clearing our minds through hard work at the gym, a night at the club, a movie night or a relaxing holiday. I must also thank the coffee ambassadors Joey, Emmy, Boaz and Angelica, and my second family Timara, Arthur, Shané, Logan and Robin who took a complete stranger in for four months during my internship. You are wonderful individuals and every one of you is such a breath of fresh air. Thank you for being you. Finally, I want to thank Peter, Jacqueline, and Elise, my family who has always supported, cared for and in- spired me. My father is the most hardworking man I have ever met and I can only admire his loyalty, dedi- cation, and strength in everything he does. The hard work, positive attitude, witty remarks of my mother are equally inspiring and I could not have wished for better parents. My sister has received the best of both, being strong, determined and intelligent. She should soon be a Bachelor of Science in her own right. Over dinner, they would listen to my findings of phenomena they had hardly any clue existed before I decided to spend a good part of the past year trying to model it. Still, they would, each in their own way, be able to help and support me. Coming home and being able to reset is something I do not take for granted and want to thank you for. It was your support and love that made this project a success. Thank you. S.H.J. Westerbeek Delft, January 8, 2020 iii Summary The laminar-to-turbulent transition of boundary layers is associated with a significant increase in friction drag. To reduce fuel costs of airliners, a laminar boundary layer over the wing surface is preferred. Laminar airfoil technology shows promising results in delaying the laminar-turbulent transition in order to reduce the friction drag of wings. With this technology, the instabilities present in the boundary layer are manipulated to delay or prevent transition. For further research into flow control, a computationally efficient method is required that allows for the prediction of nonlinear growth of perturbations in swept wing boundary layers. For this purpose, a nonlinear stability analysis tool was developed and improved in this thesis. The nonlinear evolution of stationary crossflow instabilities was simulated using the Nonlinear Parabolized Stability Equations (NPSE). The NPSE, due to their accumulative nature, are sensitive to minor changes in base flow variations, boundary conditions and initial conditions. This thesis therefore also investigated the importance of primary and forced mode introduction. In this investigation, several different introduction methods were examined in addition to a newly introduced method that accounts for the history of a forced mode through a first-order backward derivative of its amplitude while neglecting a possible change in shape function which is of a significantly smaller order. An NPSE solver was set up and its application was validated through the simulation of Tollmien-Schlichting waves in a flat plate Blasius boundary layer. The literature shows two different results for this case. Both cases could be matched closely by varying the primary mode introduction method with either Orr-Sommerfeld or Weakly Nonparallel Local (WNL) stability techniques. Both cases were validated in their respective literature with a Direct Numerical Simulation (DNS) which are hypothesized to be initiated using the same initial con- dition as the NPSE; hence, both studies were able to match their DNS results. From this study, the sensitivity of the NPSE to the primary mode introduction was found to be caused by the duality of eigenvalues resulting from the direct solution of the WNL stability problem. Although results closely match the ILST simulation, the initially stable state of the primary mode following WNL initial conditions results in a lower amplitude of forced modes. This reduces the strength of nonlinear forcing downstream of the inflow. The initially stable state is hypothesized to be the cause of this mismatch. The new forced mode introduction method that is presented for nonlinear mode introductions using an In- homogeneous Linear Parabolized Stability Equation (ILPSE) solver was applied to both cases using different assumptions for the amplitude at the stage prior to introduction. This method accounts for nonparallel ef- fects of the base flow, nonlinear forcing and amplitude growth for a newly forced mode. A reduction of initial transients can be seen as a result, which is indicative of physical introduction. For the cases examined in this thesis, however, this introduction always resulted in lower amplitudes of all harmonics introduced through this method downstream of their introduction compared to modes initiated with an assumed zero amplitude at the introduction, as is often presented in the literature. Far downstream results with the newly introduced mode show more transient behaviour as nonlinear interactions strengthen which is hypothesized to be a de- layed effect of this introduction. The NPSE tool was able to simulate the linear and nonlinear growth of crossflow and Tollmien-Schlichting instabilities in a spanwise invariant crossflow or Blasius boundary layer respectively. For very small pertur- bation amplitudes, the results match with the LPSE. As the amplitudes of all harmonics in the system grow, a deviation from the linear result, due to increased nonlinear interactions, can be seen. The nonlinear inter- actions lead to nonlinear saturation of all modes in the system. The results were tested to have converged in streamwise and wall-normal direction and the importance of including higher harmonics for the conver- gence of the primary mode was investigated and proven. A DNS comparison was performed that confirmed the accuracy of the NPSE in the prediction of stationary crossflow instabilities until the far nonlinear regime. It cannot be said with certainty whether the mismatch near the end of the domain was a result of harmonic convergence, grid convergence or that it is a delayed v vi Summary effect of initial conditions. The difference in amplitude of the second harmonic shortly after its introduction is hypothesized to be caused by the fact that the DNS is elliptical and can correct for an introduction error in time. The NPSE can only correct for the error while marching. The forced mode introduction presented in this thesis proved accurate as higher harmonics were generated in the NPSE simulation at an amplitude and growth rate that matches the DNS.