Why Is a Fluid So Hard to Solve?

Steven Brunton says force equals mass times acceleration is simple to write down, yet a fluid becomes very complicated to solve in any practical sense. A single pendulum is easy to write equations for. A second pendulum on the end gives a more complicated equation and chaos. Chop the air in a room into a billion little boxes, each with a differential equation for momentum, and the system is simple to write down and simple to understand conceptually, while how the fluid actually moves stays very complicated to solve.

He read that airlines could save something like tens of billions of dollars by reducing drag by 1 percent, and drag is maybe easier than turbulence. A relatively small improvement in predicting turbulence, or in simulating it a little faster and more accurately, has a huge economic, ecological, and human health and safety impact across dozens of fields: drag reduction on airplanes, better artificial hearts, and more efficient wind turbines and gas turbines. People live and work in a fluid, and so do the machines.

Inverse design sits close to a map problem: the goal is a wing with high lift and low drag, and the route there is unknown. He does not know how changing the wing's shape will affect the cascade of smaller and smaller vortices in the turbulent fluid over the wing, the cascade that eventually produces the lift and drag. He does not know how to inverse design efficiently, in part because the equations are expensive to simulate or to test and wind tunnels are expensive.