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

Dissipation — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

    The price of a gradient

    Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

    part 1 · viscous
  2. Two things moving by a million, and their product standing still. Viscosity, the squared velocity gradient at the dissipation scale, and their product, across six decades of Reynolds number at a fixed large-scale flow. The viscosity falls by a factor of 1e+6; the squared gradient rises by exactly the same factor, because η falls as Re^(−3/4) and u_η as Re^(−1/4); and the dissipation ν(u_η/η)² does not move at all. That is the dissipation anomaly stated as arithmetic: the limit of the dissipation as viscosity vanishes is not the value it takes when viscosity is zero.

    The limit that is not the value

    Dissipation is viscosity times the square of a velocity gradient, so it ought to vanish as the viscosity does. It does not. The gradient rises by exactly the factor the viscosity falls by, the product stands still, and a fluid with no viscosity at all would dissipate nothing — which is why the limit and the value are different numbers.

    part 1 · turbulence
  3. How far downstream the heat is still being made. The dissipation accumulated from a station ahead of a cylinder to a station behind it, as a fraction of the whole of what is made inside the frame, at five Reynolds numbers. At Reynolds number 1 the fluid has finished paying by about a diameter behind the body. At 100 it has not finished at five, and the curve is still climbing at the edge of the picture — the drag is a force on the body, and the heat it stands for is somewhere else.

    Where the heat of a drag is made

    The power it takes to tow a body through a fluid becomes heat, all of it, eventually. None of the interesting words in that sentence are the first four. It is the "eventually" that decides how a wake behaves, how far a disturbance reaches, and why no box drawn round a body contains its own bill.

    part 2 · viscous
  4. The two dissipations, side by side. The strain form on the left and the enstrophy form on the right, for the same field, on the same scale. They have their maxima in different places — 0.46 apart on a box of side 2 pi — and neither is a smoothed version of the other. One says the dissipation is in the strained regions and the other says it is in the rotating ones, which is nearly a complete disagreement about what a turbulent flow is doing.

    Equal on average, and nothing else

    The rate at which a fluid turns motion into heat can be written two ways, and every textbook says the two are equivalent. Their averages are equal to fourteen decimal places. Point by point they are uncorrelated, and their maxima are in different places.

    part 2 · turbulence
  5. The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

    A dissipation that lags its production

    Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

    part 3 · turbulence

All series