RSVD-?t: A Unified Algorithm for Streamlined Analysis of Large-scale Flows via Resolvent and Harmonic Resolvent Analyses

Abstract

This thesis introduces a novel approach to resolvent analysis, a powerful framework for understanding coherent structures in turbulent flows. Resolvent analysis characterizes input-output relationships within fluid systems, making it possible to identify the most energetic structures and devise effective control strategies. Despite its potential, the application of resolvent analysis to complex flows has been limited by the high computational cost associated with computing resolvent modes, particularly as the problem dimension increases. To overcome this limitation, we have developed an innovative algorithm that combines randomized singular value decomposition with an optimized time-stepping method to efficiently compute the action of the resolvent operator.

The RSVD-Dt algorithm dramatically reduces the computational demands of resolvent analysis, achieving linear scaling in both CPU and memory requirements relative to the problem size. This advancement opens the door for applying resolvent analysis to large-scale systems such as inherently three-dimensional jets, airfoils, and other complex flows. We further refine strategies to minimize computational expenses and control errors, thereby ensuring that our results remain both accurate and reliable. Beyond the standard resolvent analysis, we extend the capabilities of the RSVD-Dt algorithm to handle harmonic resolvent analysis for time-periodic flows. This extension enables a comprehensive characterization of linearized dynamics in the frequency domain, addressing challenges like the coupling of retained frequencies and the singular nature of the harmonic resolvent operator. We demonstrate the validity of this extension through the analysis of a periodically varying Ginzburg-Landau equation, illustrating its potential for studying time-periodic flows.

The RSVD-Dt algorithm's robustness and efficiency are validated through several case studies. Initially, we apply the algorithm to resolvent analysis of the Ginzburg-Landau system, which validates RSVD-Dt against RSVD-LU. We demonstrate its linear scaling and performance by using a three-dimensional round turbulent jet. For harmonic resolvent analysis, RSVD-Dt is validated against RSVD-LU on a periodic Ginzburg-Landau system. We then evaluate its memory and CPU performance on flow over an airfoil. With the tool fully validated, we apply it to a three-dimensional jet with streaks, observing significant gain increases at lower frequencies due to the interaction of KH modes with the streaks. The second application involves analyzing a three-dimensional twin jet, focusing on the sensitivity of gains to nozzle-to-nozzle spacing. Lastly, we conduct harmonic resolvent analysis using RSVD-Dt to investigate the modes of an axisymmetric screeching jet, which reveals the mechanisms underlying the screeching phenomenon.

Our RSVD-Dt algorithm has been meticulously implemented using the powerful PETSc and SLEPc libraries, ensuring seamless scalability and exceptional efficiency. The code is fully open-source, providing the scientific community with a versatile and freely accessible tool for both resolvent and harmonic resolvent analysis. By making our code available, we aim to foster innovation in fluid dynamics research, inviting researchers to modify, extend, and adapt our algorithm to meet their specific needs, and thereby creating a collaborative environment that accelerates scientific discovery.

While this thesis focuses on applications to turbulent jets, airfoil flows, and the Ginzburg-Landau problem, the algorithm's scalability and accuracy make it an ideal tool for exploring other turbulent phenomena, such as twin jets, elliptical jets, swept wings, and even full vehicle geometries. This work not only advances the field of resolvent analysis by introducing a more efficient and accurate method for computing resolvent and harmonic resolvent modes but also lays the groundwork for future research in fluid dynamics and related disciplines.