Welcome back. In today’s blog post, I am excited to discuss Koopman operator theory, which provides a mathematical framework for representing nonlinear dynamics as a linear system. This theory was originally developed by Columbia University professor Bernard Koopman in 1931 [1], but remained in relative obscurity until the 2000s, when modern computers allowed it to be applied … [Read more...] about Linear Embeddings of Nonlinear Dynamics with the Koopman Operator
An Introduction to Flight
Hydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows
Introduction: Welcome back. Today, I am excited to return to our series on hydrodynamic stability. In previous blogs, we considered the stability of two-dimensional parallel shear flows. Now, we will derive the governing equations for stability analysis of three-dimensional parallel shear flows. Our mathematical work here will give us the tools to consider oblique modes and … [Read more...] about Hydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows
Water Channel Dye Visualization Experiments – Fall 2023
Welcome back. This week, I hope to continue our break from the regular hydrodynamic stability program and share a project I worked on last fall. This work involved water channel dye visualization experiments to reveal the locations of flow separation and reattachment on the cambered NACA 65(1)-412 airfoil at transitional Reynolds numbers Re ∈ [104, 105]. These experiments fit … [Read more...] about Water Channel Dye Visualization Experiments – Fall 2023
Computing Finite-Time Lyapunov Exponent Fields around Airfoils
Welcome back. In this blog, I was hoping to take a short break from hydrodynamic stability, and share an old essay I have written about computing finite-time Lyapunov exponent (FTLE) fields to analyze Lagrangian coherent structures (LCSs): My project was to develop numerical integration code that could compute particle trajectories in unsteady flows. While previous … [Read more...] about Computing Finite-Time Lyapunov Exponent Fields around Airfoils
Hydrodynamic Stability: Resolvent Analysis
Welcome back. In today’s blog post, we will continue our discussion of hydrodynamic stability by discussing resolvent analysis, which is complementary to classical linear stability analysis. While linear stability analysis looks at the growth of small perturbations to an unforced base flow, resolvent analysis finds maximum gain input-output structures in a flow field that … [Read more...] about Hydrodynamic Stability: Resolvent Analysis