The Wake Field: What the Propeller Actually Swims In

Our reference ship, the Panamax bulker of about 75,000 tonnes deadweight that runs through this whole specialisation, makes 14.0 knots at her design draught of 12.80 m. That is 7.202 metres per second. Her propeller is 7.00 m across and turns at 84 rpm. Ask what speed the water arrives at that propeller and most answers will be 7.202 m/s, or, from someone better read, a little less. There is no single answer. At the top of the disc, 2.10 m below the waterline, the water is dragged along with the hull hard enough that it may pass the blade at 0.40 of ship speed, under 3 m/s. Seven metres lower it is doing 0.85. (That 2.10 m is our teaching ship's figure, and it is shallow: a real Panamax carries her shaft about 3.7 m above the keel, so her top tip runs nearer 5.6 m down. Nothing in this lesson's argument turns on it; lesson 8 comes back to the heights.) The propeller does not swim in a stream. It swims in a map, and every blade reads the whole map once per revolution.

Three flows meet at the disc

The wake splits conventionally into three contributions with different physics and, more importantly, different scaling behaviour.

The potential wake is what an ideal fluid would produce on its own: the streamlines pushed aside by the midbody have to close in again over the after body, and near the centreline that closing flow moves forward with the ship. It is set by hull geometry, largely recovered as the flow closes, and over parts of the disc it changes sign and speeds the local flow up instead.

The viscous wake is the retarded, vorticity-filled layer grown along the whole hull, together with the bilge vortices that roll up over a full after body. At 14 knots over 217.5 m of length, her Reynolds number is

Re = V·L/ν = 7.202 × 217.5 / (1.19 × 10⁻⁶) = 1.32 × 10⁹

where V is ship speed in m/s, L the length between perpendiculars in m and ν the kinematic viscosity of seawater, about 1.19 × 10⁻⁶ m²/s at 15 °C. A flat plate at that Reynolds number carries a boundary layer between 1 and 2 m thick at its trailing edge. A stern with a block coefficient of 0.83, a skeg and a pair of bilge vortices carries a retarded region two or three times thicker, comparable with the 3.50 m propeller radius. This is the dominant term on a full hull, and it is the term that refuses to scale.

The wave wake is the orbital velocity of the ship's own stern wave system where it passes through the disc. It is a function of Froude number and of nothing else:

Fr = V/√(g·L) = 7.202/√(9.81 × 217.5) = 0.156

with g the acceleration of gravity, 9.81 m/s². At Fr = 0.156 that contribution is small; on a fine 25-knot hull near Fr = 0.28 it is not. Because a model is towed at the ship's Froude number, it is the one contribution a tank reproduces faithfully.

The mix therefore depends on the hull: a full bulker's wake is large and viscous-dominated, while a fine, fast hull's is much smaller, with the potential and wave terms taking a far larger share of it.

The map, and the trough at twelve o'clock

The wake field is a map of velocity over the propeller disc, drawn in cylindrical coordinates with r the radius and θ the angle, and with θ = 0 taken vertically upward, at top dead centre. The axial component is plotted non-dimensionally as va(r,θ)/V, the local axial velocity divided by ship speed.

On a single-screw ship with a full after body the deficit is concentrated in a narrow vertical sector at the top of the disc, which lies closest behind the thickest part of the hull, deepest inside the boundary layer, and directly astern of the skeg and the shaft. Representative numbers for a full hull: va/V falls to between 0.3 and 0.5 in the core of that sector, while at 0.7R away from it the same ship gives 0.8 or better.

Note the vocabulary trap: this region is universally called the wake peak, and a peak of wake fraction is a deficit, so the velocity there is at its minimum.

What one blade feels in one revolution

Take the 0.7R station, where blade loading is conventionally read: on a 7.00 m propeller, r = 2.45 m. The rotational velocity the section sees is

2π·n·r = 2π × 1.400 × 2.45 = 21.55 m/s

with n the rotation rate in revolutions per second, 84.0 rpm being 1.400 rev/s. That number is fixed; the axial inflow is not. The angle at which the water arrives is the arctangent of one over the other:

β = arctan(va / (2π·n·r))

Away from the wake peak, va = 0.80 × 7.202 = 5.76 m/s and β = 15.0 degrees. In the peak, va = 0.40 × 7.202 = 2.88 m/s and β = 7.6 degrees. The geometric pitch of the blade was fixed in the foundry, so the whole of that 7.4 degree swing appears as a change of angle of attack, once every revolution, on a section whose mean incidence is only a few degrees.

Almost everything unpleasant about a propeller follows from that sentence. Blade thrust and torque fluctuate at shaft rate; summed over four blades, the shaft and the plating above the propeller see excitation at blade rate, 4 × 1.400 = 5.6 Hz, and at its harmonics. The suction-side pressure minimum peaks at the same instant, so sheet cavitation grows and collapses near 12 o'clock once per revolution, which drives the pressure pulses on the after body and the erosion near the leading edge and tip. Skew exists to soften exactly this, by staggering the angular position at which successive radii enter the deficit.

The tangential and radial components are smaller, and traditional design from a systematic series chart uses the axial component alone. They still matter, because they act on the denominator of that arctangent. A swirl component of 0.10V, or 0.72 m/s, turning with the blade cuts the effective rotational velocity at 0.7R from 21.55 to 20.83 m/s and moves β by about half a degree. At 0.3R, where 2π·n·r is only 9.24 m/s, the same 0.72 m/s moves β by about two degrees. Swirl bites hardest near the root, which is where devices that reorganise the inflow are aimed.

Twin screws, and an awkward fact about measurement

A twin-screw ship carries her shafts out on V-brackets and I-brackets, so the discs sit outside the thickest part of the boundary layer. The wake is lower and, more usefully, far more uniform: what deficit remains sits in the wakes of the brackets and the shafts rather than in one deep vertical sector, so the once-per-revolution excursion is milder and blade-rate excitation smaller. Lesson 4 will show that a mild wake is not an unmixed blessing.

How is any of this known? In a towing tank, with the propeller removed and the model under tow, a rake of pitot tubes on the shaft line sweeps from hub to tip, and the model is run again and again to build the circumference up in about 5 degree steps. A five-hole head returns all three velocity components; a single-hole head returns the axial one. Behind a working propeller the total velocity has to be measured optically, by laser-Doppler velocimetry in a cavitation tunnel or by particle image velocimetry in research work.

At full scale, on a trading ship, essentially nothing is measured: there is no routine full-scale wake survey. So only two things here are measurements, the field over a towed model and the total field behind a working propeller. The propeller's own induced velocity, and the background field it would have seen had it not disturbed the flow, are constructions; splitting one field into the other two is exact only in potential flow, and a ship's stern is not.

Monday morning

Open the model-basin report for one of your ships at the wake survey and read two things: how deep the trough is in the top sector, and whether the plot is a model-scale field or one already scaled. If that ship carries a vibration complaint, an erosion pattern near the blade tips, or a hull pressure-pulse figure a yard has queried, that trough is where the explanation starts. Next lesson we collapse the whole map into a single number, and confront the difference between the wake the tank measures and the wake the propeller actually feels.