An Introduction to Flight

Welcome back. Today, we will explore a concept that is central to the study of linear partial differential equations: that of the fundamental solution. As its name suggests, a fundamental solution is a very special solution to a PDE that serves as a building block for more general solutions. In this blog, my goal is to derive the fundamental solution to Laplace’s … [Read more...] about The Fundamental Solution of Laplace’s Equation and its Importance in Physics

Welcome back. This week, I wanted to discuss the Laplace transform, which is an incredibly useful tool for solving linear ordinary and partial differential equations. The primary reason for its utility is that it converts differential equations in terms of one unknown function into algebraic equations in terms of another unknown function that can be solved relatively quickly. … [Read more...] about The Laplace Transform: A More Powerful Extension of the Fourier Transform

Welcome back. I hope my readers are having a great third week of the semester. In this installment of “Introduction to Flight,” I am very excited to discuss finite-time Lyapunov exponents (FTLEs), which are powerful tools in the fluids research community. When we think of turbulence, we might imagine an infinitely jumbled mess of fluid moving erratically throughout space. This … [Read more...] about Finite-Time Lyapunov Exponents: A Tool to Discover Coherent Structures in Turbulent Flows

Welcome back. I hope that all of my readers have had an excellent summer and a great start to the fall semester. Today, I wanted to gently continue our discussion of aerospace engineering by reviewing a core concept that underpins nearly every macroscopic fluid flow analysis. This concept is the continuum hypothesis, which states that a fluid can be treated as one smooth, … [Read more...] about The Continuum Hypothesis

Welcome back. This week, I have prepared a topic that will hopefully be both interesting and exciting. We will cover a computational technique called resolvent analysis that allows us to understand how to best control a fluid flow so that it acts in a desirable way. The specific problem can be outlined as follows: For the last eight decades, most aircraft have flown at high … [Read more...] about Resolvent Analysis: A Revolutionary Technique for Understanding Turbulent Flows