Concept

Image vortex — where it appears

A fictitious vortex placed across a boundary so that the combined flow meets the boundary's condition, such as no flow through a wall. It turns a problem with a boundary into one in open fluid that can be solved by superposition.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A hydrofoil loses lift at speed and gains it slowly, with a dip between. The lift of a flat hydrofoil beneath the surface over its value in deep water, against the chord Froude number U/√(gc) on a logarithmic axis, at depths of half a chord, one chord and two. Slow, the surface is a lid and the foil gains lift, as a wing over the ground does. Fast, the surface is a pressure-release boundary and the foil loses it. Between the two it does not simply pass from one to the other: the lift dips well below its fast value where the foil's waves are longest compared with its depth, and then rises through the lid value before settling back on it.

A hydrofoil loses most lift on the way up

A wing near the ground gains lift, and a hydrofoil near the surface is often described as the same thing upside down. At low speed it is: the surface acts as a lid and the foil gains lift. At high speed the surface is a boundary that cannot hold a pressure, and the foil loses lift instead. But the lift does not pass smoothly from one to the other. Between them, where the foil's waves are a few depths long, it falls well below both — to 43 per cent of its deep-water value half a chord down — and a foil boat meets that speed in the middle of its take-off run.

misconceptions · Ground cushion

A phantom biplane partner doubles the induced drag. Induced drag at a given lift, as a multiple of its deep-water value, against the foil's depth in spans. The surface's image of the foil is an identical wing, equally loaded, twice the depth above it, so the real foil pays its own induced drag plus the mutual drag of a biplane with a gap of twice its depth: 1 + σ, with σ Prandtl's biplane factor. The elliptic biplane's 1 + σ and the aspect-ratio-9 foil's lattice both run to two as the depth closes.

A foil under the surface flies with a phantom

At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.

applied · Sailing $The waves a wing makes on the sea cost it almost nothing. The wave drag of a wing skimming deep water, as a fraction of its lift, against speed, at heights of half a metre, one and two. Each curve peaks at the speed √(2gh) — 4.43 m/s at one metre, whatever the chord — and there the wave drag is 4.4·10⁻⁴ of the lift, the density ratio of air to water times Cₗ c/4eh. At a cruise of 50 m/s it is 9.31·10⁻⁶ of the lift.$

A wing over the sea barely touches it

A wing skimming the water presses its whole weight onto the surface beneath it, spread over a width about its own height. The sea is a deformable ground, and one might expect it to give way: a trough under the wing, waves behind, a drag to pay. It barely notices. Air is eight hundred times lighter than water, and that one ratio sets the dent at millimetres, the waves at a few millimetres high, and the wave drag at less than a two-thousandth of the lift even at the worst speed.

misconceptions · Ground cushion

Named alongside it

The objects these essays reach for when they reach for this one.

Free surface Froude number Ground effect Model limit Hydrofoil Biplane Circulation Density Induced drag Kutta condition Misconception Munk stagger

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