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

Unsteady bernoulli — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

    The pressure that depends on the past

    Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

    part 1 · inviscid
  2. The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration.

    Water climbs a falling wedge, and the load comes from the climbing

    A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

    part 2 · inviscid
  3. A wedge's slam peaks when the water it has set moving weighs a fifth of the hull. Slamming force per metre of a 10° wedge of half-beam 1 m entering at 5 m/s, against penetration as a fraction of the depth at which the chine wets, for hulls of 0.3, 1 and 3.73 times the added mass of their full width, and at constant speed. The constant-speed force grows to the chine, 700 kN/m; each finite hull's peaks where its recruited water is a fifth of its mass, at 99.2, 181 and 350 kN/m.

    A light hull slams softly, and a round one does not

    A hull striking water slows as it sets the water moving, and the two share one momentum. For a V-shaped bottom that caps the slam: the force peaks when the water recruited weighs a fifth of the hull, and a hull as heavy as the water under its whole width feels a quarter of the force a constant-speed calculation predicts. For a round bottom it caps nothing, because the peak comes at the instant of contact, before the hull has slowed at all.

    part 3 · inviscid
  4. A bottom panel overshoots its static stress, and a lighter hull's panel is loaded less. The largest bending stress anywhere on the panel against time from first contact, for a 10° wedge entering at 5 m/s at constant speed and as free hulls of the full width's added mass and three tenths of it; solid, the panel's own response, dashed, the static stress the same pressure would give a massless panel. Constant speed: 361 MPa against 287 static; Hull = full width's added mass: 245 MPa against 195 static; Hull = 0.3 of it: 132 MPa against 106 static. The water reaches the panel at 4.5 ms and covers it at 11.2 ms at constant speed; the wetted panel's period is 7.84 ms.

    A hull bottom feels the slam it has time to answer

    The slamming pressure on a V-bottom is a narrow ridge at the spray root, a megapascal high, that sweeps across each panel of plating in a few milliseconds — about the panel's own period. So the panel neither feels the ridge's height nor ignores its speed: it overshoots the static stress by up to half, a rule that designs for the peak pressure asks for half as much plating again as it needs, and a light hull's slower, softer slam is gentler on its plating twice over.

    part 4 · inviscid
  5. On a flat bottom the plating's own give slows the water climbing it and takes a seventh off the slam; on a deep V, almost nothing. At the impact speed that loads the one-way panel to 200 MPa at each deadrise: left, the coupled panel's peak stress over the one-way panel's, against deadrise; right, the same against the panel's largest deflection over its rise across its width, w/(ℓ tan β). 4°: 2.27 m/s, ratio 0.847, deflection 3.99 mm; 6°: 2.9 m/s, ratio 0.894, deflection 4.24 mm; 8°: 3.44 m/s, ratio 0.934, deflection 4.44 mm; 10°: 3.94 m/s, ratio 0.972, deflection 4.61 mm; 15°: 4.99 m/s, ratio 1, deflection 4.81 mm; 20°: 5.8 m/s, ratio 0.983, deflection 4.69 mm.

    Plating that gives way slows the water that slams it, but only on a flat bottom

    The slamming pressure on a hull's bottom is a ridge that rides at the edge of the wetted width, and how fast that edge moves depends on how far the water has to climb. A panel of plating that bends inward under the load gives the water further to climb, and while it is moving away it slows the climb itself. Fed back into Wagner's condition, that takes a seventh off the peak stress of a panel on a four-degree bottom loaded to the edge of what aluminium can bear, and almost nothing on a deep V — because what matters is the plating's deflection against the bottom's own rise across the panel.

    part 5 · inviscid

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