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

Minimum-dissipation — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The parabola is the cheapest shape the walls allow. The dissipation of the profile u = A(1 − |y/h|ⁿ) carrying a fixed flux between fixed walls, against the exponent n, divided by the parabola's. Every member of the family satisfies the same boundary conditions and carries the same fluid; they differ only in shape. The minimum is at n = 2 exactly, which is not a coincidence — it is Helmholtz's theorem, and the parabola is a solution of the equations because it is the least dissipative shape rather than the other way round.

    The cheapest shape the walls allow

    The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.

    part 1 · viscous
  2. Every velocity guesses low and every stress guesses high. Two bounds on the flow rate of a duct, in units of G a⁴/μ, against the number of mesh cells per half-side: from above, the best stress field in equilibrium with the pressure gradient (red); from below, the best velocity that vanishes on the walls (green). For the square the series value 0.562308 (grey) lies between them at every mesh. For the L-shaped duct, which has no closed form, they close on 0.21399 to 0.21415.

    A flow pinned between two guesses

    The minimum-dissipation principle says the true flow is the cheapest one the walls allow, so any guessed velocity carries too little. It has a twin that nobody teaches: any guessed stress in balance with the pressure carries too much, and it needs no wall condition at all. Between the two, the flow through a duct with no formula is pinned down to as many figures as anyone wants.

    part 2 · viscous

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