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

Transition — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The laminar line does not end; the flow leaves it. Friction factor against Reynolds number in a pipe. The laminar law f = 64/Re is exact and is drawn continuing past the transitional Reynolds number, faintly, because it remains a solution there — the flow simply stops taking it. The turbulent branch is Blasius' correlation and begins where experiments find transition, not where any calculation puts it.

    The solutions stop being chosen

    Hagen and Poiseuille's pipe profile is an exact solution of the Navier–Stokes equations at every Reynolds number, and it is linearly stable at every Reynolds number. Something else happens at 2300 anyway, and it is not that the solution stopped being one.

    part 1 · turbulence
  2. Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put.

    The number that is not a number

    Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.

    part 2 · turbulence
  3. Stable in every mode, and 100 times larger first. The energy of the worst-case disturbance against time, on a logarithmic scale, at four Reynolds numbers and at the one the slider selects. Time is in units of Re, which is what makes the four curves the same shape; what changes with Reynolds number is the height, and it changes as the square. Every eigenvalue of this operator is negative throughout, so nothing that grows here is an instability in the sense a stability analysis reports.

    Every mode decays and it grows anyway

    A stability analysis asks whether any mode of a flow grows, and for pipe flow the answer is no, at every Reynolds number, which the pipe disagrees with. The missing ingredient is that the modes are not perpendicular — a disturbance made of two nearly parallel decaying pieces can grow by a factor of Re²/16 before it dies.

    part 3 · turbulence
  4. Two lifetimes, crossing. The mean time for a turbulent puff to decay and the mean time for it to split into two, against Reynolds number, on a logarithmic axis spanning thirty orders of magnitude. Below the crossing puffs die faster than they multiply; above it they multiply faster than they die, and the flow stays turbulent.

    A puff that does not know how old it is

    A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.

    part 4 · turbulence

All series