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
The Structures of Fluid Flows and Our Efforts to Understand Them
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
The Calculus of Variations, the Euler-Lagrange Equation, and Classical Mechanics
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
Higher Dimensional Integration By Parts and Some Results on Harmonic Functions
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
Separation of Variables and the Method of Characteristics: Two of the Most Useful Ways to Solve Partial Differential Equations
Welcome back. I hope all of my readers are having an excellent start to the new semester. Over the last couple of months, we have discussed partial differential equations (PDEs) in some depth, which I hope has been interesting and at least somewhat enjoyable. Today, we will explore two of the most powerful and commonly used methods of solving PDEs: separation of variables and … [Read more...] about Separation of Variables and the Method of Characteristics: Two of the Most Useful Ways to Solve Partial Differential Equations