Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Flow curves — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Streamlines and pathlines are not the same curve. In an unsteady flow the line tangent to the velocity everywhere at one instant, and the track a single particle actually follows, are different curves. They coincide only when the flow is steady, which is the hypothesis most figures forget to state.

    Streamlines are not the paths particles take

    Three different curves get drawn through a flow and they are routinely treated as one. In steady flow they coincide, which is why the confusion survives; in unsteady flow they are as different as a photograph and a long exposure.

    part 1 · kinematics
  2. The line the dye actually draws. A streakline in an oscillating uniform stream: everything released from the origin over the last 3.4 units of time, drawn where it has got to at one instant. The parameter along it is release time, not distance and not time of flight, which is why it is a record rather than a curve of the field. The mean speed of 1 exceeds the amplitude of 0.6, so the fluid never reverses and the filament is single-valued in x.

    The line the dye actually draws

    The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

    part 2 · kinematics
  3. The one curve of the four that measures a gradient. A row of particles released along a line at one instant, seen at 4 later times, in a layer profile. Each line has been displaced by the local velocity times the elapsed time and by nothing else, so dividing the displacement back out returns the profile exactly — checked here to machine precision at every one of two hundred heights. A streamline, a pathline and a streakline each report where fluid went; this one reports how fast neighbouring fluid was going relative to its neighbour, which is the quantity a boundary layer is made of and the one the other three never show.

    The curve that measures a gradient

    Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

    part 3 · kinematics

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