A wake held at the stern keeps the bulb and loses the lean

The stern’s wake puts the bulb at the bow broke the one symmetry that made Michell’s thin-ship integral look unlike a ship. Without a wake the integral cannot tell bow from stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at either end. That essay let the water slow down as it passed the hull, weighting each of the centreplane sources that make the waves by the local inflow, and both of the asymmetries real ships have followed. The least-resistance hull went fuller aft, and a bulb at the bow cut the waves by much more than the same bulb at the stern.

It did so with the simplest wake there is. The inflow fell in a straight line from the ship’s speed at the bow to a fraction $1 - w$ of it at the stern, slowing the water along the whole length of the hull. That essay called this its weakest link. A real ship’s nominal wake — the velocity its boundary layer leaves in the plane where the propeller turns — is close to the ship’s speed over most of the hull and falls only over the last tenth or so, where the boundary layer has grown thick and the hull is closing in. It is not uniform in depth, either: on a single-screw ship with a full stern it is deepest near the keel, where the flow under the bilges converges. The question left was whether a wake of that shape does what the linear one did.

It does half of it. Held at the stern, with the same wake at the propeller, the wake keeps a good part of the bulb’s preference for the bow, and loses most of the hull’s lean aft. The two asymmetries that looked like one consequence of viscosity turn out to come from different water.

A wake with a shape #

The wake here weights each source by $W = 1 - w_s\,f(\xi)\,g(\zeta)$, where $\xi$ runs from $-1$ at the stern to $+1$ at the bow and $\zeta$ from 0 at the waterline to 1 at the keel. The length part is an exponential, $f = e^{-(1+\xi)/\delta}$, which is one at the stern and falls by a factor of $e$ every $\delta$; with $\delta = 0.1$, a twentieth of the hull, the wake has fallen to a seventh of its stern value a tenth of the way forward. The depth part is one of three: uniform, $g = 1$; deepest at the keel, $g = \zeta^2$; or deepest at the waterline, $g = 1 - \zeta^2$.

To compare it fairly with the linear wake, the strength $w_s$ is fitted so that the wake averaged over a propeller disc at the stern, from a fifth of the draught down to the keel, is 0.3 — a typical nominal wake fraction for a single-screw merchant ship. The linear wake is given the same 0.3 at the stern. So a propeller placed at the stern sees the same average wake from both. The first figure shows how different they are everywhere else. The linear wake slows the water by fifteen per cent at midships and by three-quarters of its stern value at the start of the run. The shaped wake deepest at the keel slows nothing along nineteen-twentieths of the hull and then, in the last few per cent of the length, takes the water at nine-tenths of the draught down to 41 per cent of the ship’s speed while barely touching it near the waterline.

None of this costs the calculation its exactness. The wake is separable, so its part of each wave’s amplitude — the Kochin function — separates too, into a transform along the hull and one down it. The one along the hull is the Fourier transform of the basis’s slopes against an exponential, and it has the same recurrence as the transforms without a wake, with a complex exponent in place of an imaginary one. The one down the hull is a combination of the same depth integrals Michell’s integral already used. The wave resistance is still quadratic in the hull’s offsets, and the least-resistance hull is still one linear solve.

Most of the lean disappears #

The second figure solves for the least-resistance waterline at Froude numbers from 0.25 to 0.5, holding the Wigley hull’s length, draught and displacement and its depth profile, and plots where its volume is centred. With the linear wake the volume sits aft at every speed, 2.8 per cent of the half-length at a Froude number of 0.30 and nearly five at 0.5. The shaped wakes, with the same wake at the propeller, move it much less. Uniform in depth, the shaped wake moves the volume 0.86 per cent aft at 0.30 — under a third of the linear wake’s shift. Deepest at the waterline, 1.5 per cent. Deepest at the keel, the shape a single-screw stern actually has, 0.11 per cent: a twenty-fifth of the linear wake’s, and at a Froude number of 0.25 the volume even moves very slightly forward.

The ordering of the three is the depth decay of a ship’s waves. A source at depth $z$ makes a wave of wavenumber $k$ weighted by $e^{kz}$, so the sources near the waterline make most of the waves and the sources near the keel make few. A wake that slows the waterline’s sources changes the waves a great deal; one that slows the keel’s sources changes them little, and so changes the hull that minimises them little. That is why the wake deepest at the keel, which is what propeller designers see, is the one that matters least to the waves.

The lean is made along the run #

The obvious suspicion is that the shaped wake is simply weaker overall. It is, but the third figure shows that the strength is not the point — the length is. It holds the propeller’s wake at 0.3 and spreads the wake over an increasing length $\delta$ of the hull, from a fortieth to the whole half-length, and plots the centre of volume at a Froude number of 0.30.

Confined to the last fortieth of the hull, the uniform wake moves the volume 0.35 per cent aft. Over the last twentieth, 0.86. Spread over the whole half-length, 2.3 per cent — close to the linear wake’s 2.8, which slows the water over the whole hull. The curves rise steeply at first and then flatten, because what matters is how much of the run — the afterbody where the hull closes in and its sources are sinks — is made by slowed water. A wake confined to the last few per cent slows the sinks at the very end, where the hull is already fine and its slope small; it is the sinks over the whole run, slowed together, that let the optimum move volume aft.

The fourth figure shows the same thing along the hull. It plots how far each least-resistance hull’s sectional area moves from the symmetric hull Michell’s integral prefers without a wake. The linear wake takes volume out of the whole forebody, by up to 2.5 ten-thousandths of the length squared, and puts it into the whole afterbody by as much; the change crosses zero almost exactly at midships. The shaped wake uniform in depth adds about a third as much, almost all of it in the aft third of the hull, and takes it thinly from the forebody. It is a hull with a slightly fuller stern, not a hull whose whole run has been reshaped.

So the linear wake’s lean was not, as it looked, the stern’s wake acting on the stern. It was the run’s boundary layer acting on the whole run. A real afterbody’s boundary layer does slow the water along the run, more gently than at the propeller, and a complete wake would carry both parts: a slow fall along the run and a deep pit at the stern. This calculation separates them and finds that the pit, which is what the propeller designer measures and calls the wake, is the smaller contributor to the hull’s shape.

The bulb keeps more of its end #

The bulb is different, and the reason is local. A bulb is a compact body, and its wave is made by the water where it sits. At the bow the shaped wake is nothing at all — $f(1) = e^{-20}$ — so a bow bulb makes exactly the wave it made with no wake. At the stern, the same bulb sits in the deepest part of the shaped wake, and its wave is weakened by whatever the wake is at its depth.

The fifth figure varies the bulb’s depth at a Froude number of 0.30. With no wake the bow and stern curves coincide, and the best depth is about 0.6 of the draught, where the bulb cuts the Wigley hull’s wave resistance to just over half. With either shaped wake the bow curve barely moves — it moves at all only because the hull’s own stern waves are weakened, which changes what the bulb is cancelling — and its best depth does not move. The stern curve moves a great deal, and differently for the two wakes. With the wake deepest at the keel, the stern bulb gets worse the deeper it goes: at nine-tenths of the draught it recovers 20 per cent of the wave resistance against the bow bulb’s 42. With the wake deepest at the waterline, the stern bulb’s resistance hardly changes with its depth, between 0.63 and 0.66 of the bare hull’s from half the draught down, so its handicap is largest where the bow bulb is best and shrinks below it.

This answers the question the earlier essay put precisely: does a bulb belong low at the bow because that is where the wave is, or because that is where the wake is not? Both, and they decide different things. The wave sets the bulb’s depth: the best depth at the bow is the same whatever the wake does, because the bow sees none. The wake sets the bulb’s end: whether the bow beats the stern depends on how much the stern’s water is slowed at the bulb’s depth. With the wake deepest at the keel, that preference grows the lower the bulb goes.

At the shallowest depth in the figure, 0.3 of the draught, the stern bulb is slightly better than the bow’s under either wake. A shallow bulb makes a strong short wave, and at the bow that wave is too strong for the hull’s own bow wave to cancel; the wake weakens it at the stern into something closer to the right size. It is a small effect at a depth no designer would choose, but it shows that the wake does not simply favour the bow — it weakens whatever sits in it.

A preference against a lean #

The sixth figure sets the two results side by side across speeds. It plots the stern bulb’s resistance over the bow bulb’s, both at 0.6 of the draught, from a Froude number of 0.27 to 0.45. With the linear wake the stern bulb costs 1.56 times the bow bulb’s resistance at 0.30. The shaped wakes make it 1.18 times when deepest at the keel, 1.30 when uniform and 1.39 when deepest at the waterline: they keep a third to two-thirds of the linear wake’s preference for the bow. They kept a twentieth to a half of its lean aft. The bulb’s preference survives the change of wake far better than the hull’s shape does, which is what a local effect against a global one should do.

The keel wake’s preference is not unconditional. Below a Froude number of about 0.27, where the bulb’s wave and the hull’s are timed differently, that wake prefers the stern bulb slightly. All three preferences shrink at higher speeds, where the bulb’s wave is long and deep and the wake at its station matters less to its amplitude, falling to between 1.05 and 1.12 at a Froude number of 0.45.

What the calculation was checked against #

The new transform along the hull was checked against a direct quadrature of the integral it replaces, at three wavenumbers and two wake lengths, to eight parts in a billion. The depth transforms were checked two ways: with a uniform depth weight they reproduce the transforms Michell’s integral already uses, and the keel and waterline weights add up to the uniform one, both to rounding. With a nominal wake of zero the least-resistance hull’s centre of volume is at midships to $10^{-16}$, Michell’s symmetry. The disc average was integrated directly, giving a local wake strength of 0.726 at the keel for the wake deepest there, so that the slowest water anywhere moves at 27 per cent of the ship’s speed. The checks refuse a nominal wake above one, a depth shape that is not one of the three, a wake length of zero, and a fit that would need the inflow at the keel to reverse.

Everything else is the earlier essays’: the basis of hull shapes, the bending penalty that keeps the optimum a smooth hull, the volume constraint, and the bulb’s Kochin function, whose phase the stern’s wake puts the bulb at the bow checked against the integral without a wake to two parts in a hundred trillion.

What the calculation leaves out #

The wake along the run. The calculation’s point is to separate the stern’s pit from the run’s gentle fall; a real wake has both. A wake model with the two together — a slow linear fall plus a stern pit — would put the linear wake’s lean back almost in full and add the pit’s bulb preference on top.

The wake’s girth. Real wake fields vary round the hull’s girth as well as down it, and the single-screw wake’s deepest part is often near the top of the propeller disc, under the stern’s counter, as well as near the keel. The two depth shapes here bracket that; neither is a measured field.

Viscosity in the waves themselves. A weighted source is the cheapest way to put the boundary layer into a potential theory, as a wake that keeps the drag and forgets the body did for a wake far downstream. It treats the slowed water as making weaker waves of the same shape, and ignores the boundary layer’s own displacement, which thickens the stern as the water sees it — the effect the thin layer and how thick is thin found sets where the flow believes a wall is.

Thin-ship theory. Michell’s integral is linear in the hull’s beam, and its predictions are best for slender hulls at moderate speeds. The shaped wake changes none of that.

Two consequences of one boundary layer #

The hull Michell’s integral prefers found a theory that could shape a ship’s ends but could not tell them apart, and hull speed is not the hump found that the integral’s humps and hollows, not a single speed, are what a hull is designed against. The stern’s wake puts the bulb at the bow put the asymmetry in through viscosity and found both of a real ship’s asymmetries at once. This calculation finds that they are separable after all: the afterbody’s fullness responds to the slowed water along the run, and the bulb’s position to the slowed water at the stern. A ship that has both owes them to the same boundary layer at two different stages of its growth.

That is a reminder about the angle that does not care and its cousins. The Kelvin wedge is the same for every ship because it is made far from the hull, where the boundary layer has nothing to say. Close to the hull, where the waves are made, the boundary layer has a great deal to say, and what it says depends on where along the hull and how deep the water has been slowed — not on a single wake fraction. The number that really is one is the Froude number’s own threshold, and it is exact; the wake fraction is not that kind of number, and the shape behind it matters as much as its value.

Who worked it out #

Michell’s integral is of 1898 and its inversion for least resistance the subject of a long literature since the 1930s, of which Weinblum’s and Kotik’s work are the landmarks. Weakening the stern’s sources to put the boundary layer into wave resistance is an old device of thin-ship theory, and how much of a ship’s stern wave the viscous layer suppresses was argued over from the 1960s to the 1980s. Measured nominal wake fields, with their deep pit behind a single-screw stern, are the propeller designer’s starting point, and the wake fraction of 0.3 is a round value from Harvald’s and Taylor’s tables. The bulbous bow itself goes back to Taylor’s battleship Delaware of 1910, and Inui’s experiments of the 1960s fixed its modern form.

Still open: a bulb that moves the hull #

Every bulb here was added to a fixed hull. The least-resistance hull and the best bulb were never solved together, and with a shaped wake they should be: a bow bulb cancels part of the bow’s wave, which frees the forebody to carry volume it was avoiding, and that could undo some of the aft lean the wake produced. The next calculation adds the bulb’s radius and depth to the quadratic problem as two more unknowns at each end, holds the total displacement including the bulb, and asks whether the hull and bulb found together are fuller aft or fuller forward than the hull alone, and whether a bulb ever appears at the stern when the hull is free to change around it.