Tag: Euler equations

  • Sonic Booms and Urban Canyons

    Sonic Booms and Urban Canyons

    In the days of the Concorde — thus far the world’s only supersonic passenger jet — noise complaints from residents kept the aircraft from faster-than-sound travel except over the open ocean. With many pursuing a new generation of civil supersonic aircraft, researchers are looking at how those sonic booms could interact with those of us on the ground.

    In this study, researchers simulated the shock waves from aircraft interacting with single and multiple buildings on the ground. They found that the presence of a building increases the perceived sound level of the boom by about 7 dB at the most. But the most interesting results are what happens between multiple buildings.

    If the street between buildings is wide enough, they each act independently, as if they were single buildings. But for narrower streets, the acoustics waves reflect and diffract between the buildings, creating a resonance that makes the acoustic echoes last longer. The effect is especially pronounced for a sonic boom traveling across a series of buildings, which mimics the layout of a dense city full of urban canyons. (Image credit: Concorde – M. Rochette, simulation – D. Dragna et al.; research credit: D. Dragna et al.)

    Acoustic waves reflect and propagate through 2D urban canyons with widths of 10 meters (top), 20 meters (middle), and 30 meters (bottom).
    Acoustic waves reflect and propagate through 2D urban canyons with widths of 10 meters (top), 20 meters (middle), and 30 meters (bottom).
  • Blowing Up Euler

    Blowing Up Euler

    The mathematics of fluid dynamics still have many unknowns, which makes them an attractive playground for mathematicians of all stripes. One perennial area of interest is the Euler equations, which describe an ideal (i.e., zero viscosity), incompressible fluid. Mathematicians suspect that these equations may produce impossible answers — vortices with infinite velocities, for example — under just the right circumstances, but so far no one has been able to prove the existence of such singularities.

    A recent Quanta article delves into this issue and the race between researchers using traditional methods and those using new deep learning techniques. Will the singularities be found and who will get there first? It’s well worth a read, whether theoretical mathematics is your thing or not. (Image credit: S. Wilkinson; see also Quanta; submitted by Jo V.)

  • Breaking the Euler Equations

    Breaking the Euler Equations

    Mathematicians like to break things. Or, more exactly, they like to know when the equations we use to describe physics break down. One popular target in fluid mechanics are the Euler equations, which describe the motion of frictionless, incompressible flows. Mathematicians have been on the hunt for centuries for situations where these equations predict singularities, points where the velocity or vorticity of a fluid change infinitely quickly. Since that can’t happen in reality (at least as far as we understand it), these singularities indicate weaknesses in our mathematical description and may help uncover fundamental flaws in our understanding.

    Despite centuries of effort, the Euler equations withstood mathematical assault… until recently. Since 2013, a series of mathematicians have been successfully chipping away at the Euler equations’ seeming perfection with a series of scenarios that seem to lead to singularities. One is similar to stirring a cup of tea, except that you stir the upper part of the cup in one direction and the bottom half in the opposite. As the flow develops, a singularity occurs where the secondary flows of these two stirring motions collide. For more, check out these two articles over at Quanta. (Image credit: L. Fotios; see also Quanta Magazine 1, 2)