When trying to solve a problem, it can be useful to examine a very simplified case of that problem. Then, solutions to that simple case can be leveraged to solve increasingly complex cases until the final, most complicated case is solved. We can employ this strategy to investigate fluid flow.

When first trying to understand how fluids work, one simplification that can be made is assuming that a fluid is inviscid, or has no viscosity. Since viscosity is a measure of how much resistance a flow gives to moving freely, such ideal, inviscid flows would move about quickly without any resistance. Moreover, we might assume that a fluid flow is laminar. This means that the flow is well-behaved, moving in straight lines without any complicated swirls or zig-zags. It is also important to mention here that laminar flow is the opposite of turbulent flow, which is characterized by crazy, seemingly-random motions. In inviscid, laminar cases, we can write nice, compact equations. However, when trying to solve real-world problems, we eventually have to consider both viscosity and turbulence.

When accounting for viscosity, we can imagine that our airplane (or any other object we are investigating) is sitting in a bubble, or boundary layer, of viscous flow. In turn, this boundary layer is surrounded by inviscid flow. Since we are often only concerned about flow patterns directly near the surface of an object, it can save computational energy to limit our consideration of viscosity to just a small boundary layer.

We apply this theory of boundary layers to estimate the friction on wings due to the viscous effects of the air around them. This information is important to know because the friction on a plane’s wings contributes to the total drag that it experiences, which in turn hinders its ability to fly. To tackle this problem, we can make one final assumption that the wings are geometrically close to a flat plate. The reason for this assumption is that it is substantially easier to write equations for the friction on a flat surface than it is to write equations for the friction on more complex topologies. When considering the friction on a flat plate surrounded by a viscous boundary layer, we can write one equation for the case that we have laminar flow (friction = 1.328 / (Re_{L})^{ ½} ) and one equation for the case that we have turbulent flow (friction = 0.074 / (Re_{L})^{⅕} ). In these equations, the variable Re_{L}^{ }is a Reynold’s number that uses the streamwise length, L, of the flat plate as the length scale.

This problem of modeling friction on flat surfaces using boundary layers thus demonstrates that engineering is an art of approximation. At every step of the process, we make assumptions that allow us to carefully reformulate situations into renditions that can be understood. In future posts, I hope to explore more cases in how careful approximations can yield helpful results.

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