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

Bernoulli's equation — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The hypotheses Bernoulli's equation needs. The equation is correct and its hypotheses are strict. Most misuse is not a wrong formula but a right formula carried across a streamline, through a machine, or into a region where viscosity dominates.

    Where Bernoulli's equation applies

    The equation is right. Its hypotheses are strict, and almost all misuse is a correct formula carried somewhere it does not hold — across streamlines, through a fan, or into the one layer where friction is the whole story.

    part 1 · misconceptions
  2. The energy is all there the whole way: enthalpy spent to buy speed. At each Mach number, the fraction of the stagnation enthalpy still held as heat and the fraction converted into directed motion. The two always sum to the total, which is the compressible replacement for Bernoulli's equation — an energy statement rather than a pressure one. At Mach 2 more than four fifths of the heat has become speed.

    Energy instead of pressure

    Bernoulli's equation is a statement that pressure and speed trade against each other at fixed density. Take the fixed density away and the trade is between heat and speed instead — and what survives is not a pressure at all.

    part 2 · compressible
  3. One sheared stream, and the total pressure across it. A parallel shear flow is an exact steady solution of the Euler equations, and the momentum equation requires its static pressure to be uniform. So the total pressure is entirely the dynamic pressure, which varies with the speed — by three and a fifth dynamic heads across this layer, on a flow where the static pressure does not vary at all.

    Four Bernoullis and one name

    "Bernoulli's equation" names at least four statements with four different constants, three domains of validity and one shared reputation for being misapplied. A single sheared stream separates the first three: its total pressure is constant along every streamline, varies by three dynamic heads across them, and its static pressure never moves at all.

    part 3 · misconceptions
  4. Two totals across a shock. The ratio of the total temperature and the ratio of the total pressure across a normal shock, against the shock's Mach number. One of them is one at every Mach number, to the last bit of double precision; the other falls to under a hundredth by Mach eight.

    Two totals, one of which a shock cannot touch

    Across a normal shock the total temperature ratio is 1.000000000000000 at every Mach number, and the total pressure ratio falls to 0.0085 by Mach 8. One of the two records the energy that has been added to the gas and nothing else; the other records every irreversibility on the way.

    part 3 · compressible

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