An Introduction to Flight

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

I would like to extend a sincere and warm welcome back to our “Introduction to Flight” blog series. Today, we will return to fluid dynamics, which is a difficult, yet immensely fascinating subject. In our everyday lives, fluid flows are all around us, from the coffee in our cups, to the wind in the trees, to the air we breathe. In the laboratory, we can visualize flows by … [Read more...] about The Structures of Fluid Flows and Our Efforts to Understand Them

Welcome back. This week, we will take a short break from partial differential equations and have a brief foray into the calculus of variations, a field of mathematics that is concerned with optimizing functionals. A functional, essentially, is a real-valued function that takes functions in as inputs. For example, F[f] = ∫[0,1] f dx, is a functional that takes in a function, f, … [Read more...] about The Calculus of Variations, the Euler-Lagrange Equation, and Classical Mechanics

Welcome back. Integration by parts is a very useful technique that usually shows up in introductory calculus courses. It allows us to efficiently integrate the product of two functions by transforming a difficult integral into an easier one. When working with a single variable, the integration by parts formula appears as follows: ∫[a,b] g(x) (df/dx) dx = g(b)f(b) - g(a)f(a) … [Read more...] about Higher Dimensional Integration By Parts and Some Results on Harmonic Functions