Amateur Aerodynamics

Last time, we saw how engineers construct differential equations to describe physical phenomena such as fluid flows, representing these real things in the language of mathematics . You may have found yourself wondering, “How do we solve these?” Generally, engineers can use one of two methods. First, we can attempt to solve the equations or systems like any other math problem, the way you’re probably used to doing since as far back as you can remember. This involves manipulating the equations and transforming them using mathematical operations to arrive at an answer. We call this the analytical approach. Implementing a number of simplifying assumptions (constant density, constant pressure, no viscosity, etc.) leads to a scalar equation derived from the Navier-Stokes equations that is solvable analytically. In class last spring, we used this  “1-dimensional linear advection” equation to check the numerical solutions we developed later on.   Analytical Solutions   If we ...

I started writing this article before the end of spring semester, back in May 2025, and quickly found that it grew to monstrous proportions because there's so much to say about the topic of simulation and numerical approximation that it simply can't fit in a concise blog post. I found myself honing in on aerodynamics simulations specifically (perhaps because that was the topic of one of my elective courses) and, after some helpful advice from a friend, decided to split it into several posts. I intend this series of articles to take the form of three parts and explain, mainly in general terms but with plenty of examples (many drawn directly from the numerical methods and computational aerodynamics coursework I've recently completed), the mathematics behind modeling and simulation: how does it work? How well does it represent real things like airflows? At the end of it, if all goes to plan, you should understand that all computational fluid dynamics (CFD) simulations are wron...

When two objects in contact with each other slide, a friction force develops resisting the motion. In classical physics, we distinguish between static friction (which resists the onset of motion as some force is applied) and kinetic friction (which resists movement after it has begun). You’ve noticed this before: the last time you tried to push a heavy box across the floor, say, you had to give it a push to get started and overcome the static friction (and if the static friction was great enough, it perhaps tipped over instead of sliding). When you stopped pushing, it didn’t continue sliding indefinitely; it gradually slowed and required some force to keep moving (although not as much as it took to get started). In both scenarios, your pushing on the box was required to overcome its static or kinetic friction. Upright, the fluid-filled can slides when pushed but as energy is dissipated due to friction with the table surface, it quickly comes to a stop. Cars are a bit more complicated b...