The stern's wake puts the bulb at the bow

The hull Michell’s integral prefers turned the thin-ship theory of wave resistance into a design tool. Michell’s integral is quadratic in a hull’s offsets, so the least-resistance hull of a given length, draught and displacement is one linear solve, and it came out recognisably a ship: fine-ended for slow speeds, full-ended for fast ones, its prismatic coefficient rising with its design Froude number as ship designers’ tables have always said. Twice it came out not a ship. Freed in depth, it drained its waterline and piled volume at the keel. And it was symmetric fore and aft at every speed, so it could not grow a bulb at the bow — the same bulb at the stern, that essay showed, cuts the waves by exactly as much.

The second failure has a precise cause. Michell’s theory builds a hull’s waves from sources spread over its centreplane, each as strong as the hull’s slope times the speed of the water past it, and in the theory that speed is the ship’s everywhere. Reversing the hull then conjugates the wave amplitudes and changes nothing measurable. The essay’s closing proposal was to break the symmetry with the one thing a real hull has that the theory does not: a stern moving through water its own boundary layer has already slowed.

A stream that slows towards the stern #

A ship’s hull drags a boundary layer, thin at the bow and thickening along the hull to a few per cent of its length at the stern, and the water a propeller or a stern meets is moving past it more slowly than the sea is. The deficit is the wake fraction, the fraction of the ship’s speed the water has lost; for a single-screw merchant ship it is 0.2 to 0.4 at the propeller. A wake keeps the drag and forgets the body — the momentum deficit behind a body is its drag, whatever shape made it — and here that deficit is doing something else: weakening the waves the stern makes.

The model is the smallest change that says so. Each centreplane source keeps its strength, the hull’s slope, times a weight that is the local inflow over the ship’s speed: one at the bow, falling linearly to $1 - w$ at the stern,

$$W(\xi) = 1 - \tfrac{w}{2}(1 - \xi),$$

with ξ running from −1 at the stern to +1 at the bow and w the wake fraction at the stern. A linear fall is the crudest profile that is asymmetric; the essay’s figures are for wake fractions up to 0.4.

The weight keeps the calculation exact. The earlier essay’s hull is a sum of twelve separable shapes, and each shape’s contribution to the wave amplitudes is a Fourier transform along the hull; with the weight, integrating by parts turns each weighted transform into the unweighted one plus a transform of one degree higher, both from the same exact recurrence. The wave resistance is still a quadratic form in the hull’s offsets and the least-resistance hull is still one linear solve. Checked against direct quadrature, the weighted transforms agree to a part in a million.

The optimum leans aft #

The first figure is the answer to the essay’s first question. It draws the sectional area along the least-resistance hull at a Froude number of 0.30 — a fast cargo ship’s speed — holding the Wigley hull’s length, draught, depth profile and displacement, as the earlier essay did. Without a wake the curve is symmetric, as it must be. With a wake fraction of 0.2 it leans aft, and with 0.4 more so: the optimum moves its centre of volume 1.7 and then 4.0 per cent of the half-length towards the stern.

The reason is the weight itself. The stern’s sources are weaker, so a unit of volume near the stern makes smaller waves than the same volume near the bow, and a hull that must carry a fixed displacement puts more of it where it is cheaper. The optimum is not fuller aft because anything about the stern’s shape was asked for; it is fuller aft because the stern’s waves are made by slower water.

Across speeds the shift is systematic. Without a wake the centre of volume sits exactly at midships at every Froude number. With one it is always aft — by one to seven per cent of the half-length, roughly in proportion to the wake fraction, and largest at the slow end, where the waves are short and the bow’s and stern’s sources act most independently.

That is the right direction for fast ships and the wrong one for slow ones. Designers place a fast ship’s centre of buoyancy slightly aft of midships and a slow, full ship’s slightly forward. The slow ship’s forward centre is not a wave-resistance choice at all: at low Froude numbers the waves are a small part of the resistance, and the afterbody is kept fine so that the flow does not separate before it reaches the propeller — the two drags a wing pays, friction and form, deciding the shape. A wave integral, even a wake-weighted one, is the right tool only where the waves dominate, which for a ship is above a Froude number of about 0.25.

The bulb goes to the bow #

The second question was whether a bulb, once the symmetry is broken, goes to the bow. The figure takes the earlier essay’s bulb — a sphere of a fortieth of the hull’s length in radius, at seven-tenths of the draught, just beyond the hull’s end — and puts it at the bow and then at the stern of the Wigley hull, with a wake fraction of 0.25. Without a wake the two curves were one. With it they separate at every speed: at the design Froude number of 0.30 the bow bulb leaves 0.44 of the bare hull’s wave resistance and the stern bulb 0.62.

The mechanism is a matter of what each bulb has to cancel. A bulb is a second source whose waves, at the right speed, arrive in opposite phase to the hull’s end waves nearest it. At the bow the hull’s waves are made by full-speed water and are strong, and a bulb of full strength cancels much of them. At the stern both the hull’s waves and the bulb’s own are made by slowed water, the hull’s stern system is already the weaker, and the bulb’s cancelling wave has less to do and less strength to do it with.

The bulb’s phase relative to the hull’s waves had to be right for this to mean anything, and it was checked. With the wake set to zero, the combined hull and bulb reproduce the earlier essay’s bulb resistance, computed there by a different route, to two parts in $10^{14}$ at both ends of the hull and three speeds. A wrong phase would have given the right answer at one end and the wrong one at the other.

The preference is the wake’s, and nothing else’s #

The figure makes the attribution quantitative. The ratio of the stern bulb’s resistance to the bow bulb’s is exactly one with no wake, at every speed — linear theory by itself has no opinion — and rises with the wake fraction: at a Froude number of 0.30 to 1.31 for a wake fraction of 0.2 and 1.82 for 0.4. Nothing else in the model distinguishes the ends. In this calculation, then, the bulbous bow is a consequence of the stern’s viscosity: the stern is a worse place for a bulb because its water has been slowed by the boundary layer the whole hull has grown.

That is a different explanation from the usual one, and a sharper one. The usual account says the bow is where a ship makes its biggest wave, which is true, but the account never says why the bow’s wave is bigger than the stern’s in a theory where the two ends are interchangeable. The answer here is that it is not bigger in the theory; it is bigger in a ship, because the ship’s stern moves through its own wake.

What the wake does to the waves themselves #

The wake also changes the bare hull’s resistance, and less dramatically than the bulb’s preference suggests. With a wake fraction of 0.2 the Wigley hull’s wave resistance falls to about 0.81 of its no-wake value, and with 0.4 to 0.66, nearly uniformly across speeds. The fall is slightly larger at the humps, where the bow and stern systems reinforce, than in the hollows where they cancel, so the curve flattens a little, but its humps and hollows stay where hull speed is not the hump put them. The wake’s main effect is on where volume should go and where a bulb should sit, not on how large the waves are.

Why the hull leans a little and the bulb a lot #

The two answers differ in size by more than the wake fraction would suggest, and the difference is a property of optima. The least-resistance hull sits at the bottom of a quadratic bowl, and the wake tilts the bowl by an amount proportional to w. The bowl is steep in almost every direction a hull can change — most changes of shape make a great deal more wave — and the tilt acts mainly along one of them, moving volume between the ends. So the minimum slides only a short way: under two per cent of the half-length at a wake fraction of 0.2. What that short slide is worth is another matter, because the resistance at the bottom of the bowl is already small, a twentieth of the Wigley hull’s at this speed. Under the same wake, the symmetric optimum makes 6.5 per cent more wave resistance than the leaned one. A lean too small to see in a lines plan is worth a noticeable fraction of what is left to save.

The bulb comparison is not an optimum; it is a cancellation. A bulb works by making a wave nearly equal and opposite to one the hull makes, and the residue of a near-cancellation is the difference of two large numbers, which is exactly the kind of quantity a modest change to either one moves a great deal. A wake fraction of a quarter weakens the stern bulb’s wave and the stern’s own wave by comparable amounts, but it leaves the bow’s full, and the bow bulb, cancelling a stronger wave with a stronger wave, removes more of the total. So the same weight that barely moves the optimal hull changes the bulb’s value at the stern by forty per cent.

The wake is also the propeller’s #

The wake that weakens the stern’s waves does a second thing a naval architect counts on. A propeller at the stern works in the slowed water, and a propeller that accelerates slow water to the ship’s speed does less work for the same thrust than one working in the free stream: the ratio of the two, the hull efficiency, is the thrust deduction’s complement over the wake’s, $(1 - t)/(1 - w)$, and it is above one for a single-screw ship — the wake returns some of the energy the boundary layer took. So there are two reasons to carry volume aft on a fast single-screw ship, the wave-making one computed here and the propulsive one, and both come from the same slowed water at the stern.

Bulbs do appear at sterns, on some ships, and it is worth being clear that this calculation says nothing against them. A stern bulb is shaped to guide the flow evenly into the propeller disc, improving the propulsive efficiency and reducing vibration, and it is judged by the propeller’s performance rather than by the waves. What the calculation says is that for cancelling waves — the job the bow bulb is fitted for — the stern is the worse place, and the reason is the water the stern is moving through, which is the thin layer a whole hull’s length thick.

What was checked #

Two checks tie the calculation to the one before it. With no wake the combined hull-and-bulb wave amplitude reproduces the earlier essay’s bulb resistance to two parts in $10^{14}$, at the bow and the stern and at three speeds, which fixes the bulb’s phase relative to the hull’s. And the wake-weighted transform of every basis shape agrees with direct quadrature of the weighted integral to a part in a million. With the wake set to zero the least-resistance hull is the earlier essay’s, symmetric, its fore-and-aft terms zero to rounding. The tests refuse a wake fraction above one, a negative one, and a tolerance of zero.

What the picture cannot show #

A wake profile. The weight falls linearly from bow to stern. A real nominal wake is small over most of the hull and grows steeply over the last fifth of it, concentrated behind the stern and near the keel, and it varies across the stern’s breadth. A wake concentrated at the stern would give the stern sources a weaker weight over a shorter stretch; the direction of every effect here would survive, and their sizes would change.

The wake’s own waves. A boundary layer thickens the hull as the water sees it — a displacement thickness — and that thickening is a source distribution of its own. It is left out; it would make the stern’s effective hull fuller, which works the same way as the weight.

Linear waves and a thin ship. Everything is Michell’s theory: small waves, a hull thin compared with its length, no bow wave breaking, no spray. The bulbous bow of a real ship also works partly by reducing a nonlinear breaking bow wave, which no linear theory can see.

Only the Wigley hull and its family. The bulb comparison is on the Wigley hull; the optimum is among the twelve-shape family of the earlier essay, with its depth profile fixed.

A symmetry, and who broke it #

Michell’s integral is from 1898, and the proof that it is unchanged when a hull is reversed is a line of algebra that has been used both to criticise the theory and to explain its successes. The bulbous bow was patented by Taylor in 1911 and explained in wave terms by Wigley in the 1930s and by Inui in the 1960s, whose wave-cancelling bulbs were designed with linear theory; the role of the boundary layer and wake in reducing a stern’s wave-making was measured and modelled, as a “viscous effect on wave resistance”, from the 1960s. The Froude number that organises all of it is the number that, on a ship, really is one only in shallow water; in deep water it is the ratio of the hull’s length to its own wave’s, and the angle that does not care is the Kelvin pattern those waves make.

Still open: a wake with a shape #

The linear weight is the simplest asymmetry, and the calculation’s weakest link. A measured nominal wake — nearly one over most of the hull, falling to a fraction of itself over the last tenth and deepest near the keel — can be written as a weight in both length and depth, and the depth part multiplies the Laplace transforms down the hull as the length part multiplies the Fourier ones. The next calculation uses a stern-concentrated wake of that shape, with its wake fraction fitted to a single-screw hull’s propeller disc, and asks whether the optimum’s centre of volume moves as far aft with it, and whether a bulb’s preference for the bow then depends on the bulb’s depth as well as its end — which would say whether a bulb belongs low at the bow because that is where the wave is, or because that is where the wake is not.