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1. Overview — ship propulsion
Every ship in motion experiences resisting forces from the water and the air that must be overcome by a thrust (thrust) supplied by some propulsive mechanism. This chapter deals with how that thrust is generated, with what efficiency, and how the propeller interacts with the hull. The source is the Principles of Naval Architecture (Lewis/SNAME), and the technical terms that appear on the examination are kept exactly as the book uses them.
The standard propulsor is the screw propeller (screw propeller). Since the mid-nineteenth century it has dominated marine propulsion owing to decisive advantages over the paddle wheel: it is not materially affected by normal changes in service draft, it is well protected against damage, it does not increase the overall beam of the ship, and it can run much faster while still keeping good efficiency — allowing smaller, lighter, faster-running engines.
1.1 The propulsion chain
The energy follows a chain from the engine to the ship: the propelling machinery generates power → the shaft transmits it → the propeller converts it into thrust → the thrust drives the hull against the resistance. There are losses at every link, and a distinct definition of power corresponds to each stage. Understanding this chain is what the syllabus calls the definition of propulsion (item 2.1).
1.2 Types of propelling machinery
The choice of machinery weighs weight, space, first cost, reliability, length of life, flexibility, cost of upkeep and of fuel, and its suitability for the type of propeller to be used. The book summarizes the main options and their typical fuel consumptions:
Machinery
Salient characteristic
Typical consumption (kg oil/kWh)
Steam reciprocating engine
Exceptional controllability, easily reversed, RPM matches that of the screw propeller; but heavy and with high fuel consumption
≈ 0.70 (triple-expansion)
Steam turbine
Uniform turning effort, excellent for large-unit power; non-reversible and RPM far too high → requires reversing turbine and reduction gears
≈ 0.30 (large units)
Diesel engine (internal-combustion)
Directly reversible, compact, low fuel consumption; but heavier and more expensive; torque limited by the maximum cylinder pressure
≈ 0.20
Turbo-electric drive
Eliminates direct shafting and the reversing turbine; great maneuvering flexibility; freedom of arrangement
—
Nuclear reactor
Operates at full load indefinitely without refueling; weight of reactor + shielding ≈ that of boilers + the fuel oil it replaces
—
Gas turbine
Light, dispenses with boilers, comes up to full load in ~15 min; burns much fuel; common in naval ships
—
Matching the diesel to the propeller: the diesel's torque is limited by the maximum cylinder pressure, so it delivers maximum power only at maximum RPM. With time, resistance grows (fouling) and the propeller thrust falls; the designer must choose the propeller pitch so that, over the life of the ship, the engine is neither overloaded nor unable to reach its full capability.
2. The power chain and propulsive efficiency
The various engine types are not rated on the same basis, because measuring the output of each requires different instruments. Hence the set of power definitions below — the backbone of syllabus topic 2.1.
2.1 The powers in the chain
Symbol
Name
What it is / where it is measured
$P_I$
Indicated power
Indicated power, measured inside the cylinders by a pressure indicator (steam reciprocating engines)
$P_B$
Brake power
Brake power, measured at the crankshaft coupling by a mechanical/hydraulic/electrical brake (diesel)
$P_S$
Shaft power
Shaft power, measured aboard by a torsionmeter as close to the propeller as possible (turbines)
$P_D$
Delivered power
Power delivered to the propeller; less than $P_S$ by the friction loss in the shaft bearings and the stern tube
$P_T$
Thrust power
Thrust power: the propeller advances through the water at the speed of advance $V_a$ delivering thrust $T$
$P_E$
Effective power
Effective (useful) power, which overcomes the resistance $R_T$ at speed $V$
Effective (useful) power and thrust power:
$$P_E = R_T \cdot V \qquad\qquad P_T = T \cdot V_a$$
Brake power from torque $Q$ and rotation $n$ (rev/s):
$$P_B = 2\pi \, n \, Q$$
Unit conversion used in the book: $1\ \text{hp} = 0{,}7457\ \text{kW}$ (in English units, $1\ \text{hp} = 550\ \text{ft·lb/s}$). The shear modulus of the steel shafting is taken as $G = 8{,}35 \times 10^{7}\ \text{kN/m}^2$.
The shaft power read by the torsionmeter depends on the position of the instrument: it should be as close as possible to the stern tube to approach the power actually delivered. The shaft transmission loss is taken as ~2% for ships with machinery aft and ~3% for those with machinery amidships.
2.2 The efficiencies and the quasi-propulsive coefficient
Efficiency, in engineering, is the ratio of the useful power obtained to the power expended. In the ship, the useful power is the effective power $P_E$ (overcoming the resistance). The most meaningful measure compares $P_E$ with the power actually delivered to the propeller, $P_D$ — a ratio named the quasi-propulsive coefficient (QPC):
The QPC breaks down into three factors that organize the whole chapter — each studied further on:
$$\eta_D = \eta_0 \cdot \eta_H \cdot \eta_R$$
$\eta_0$ — open-water efficiency (open-water efficiency): of the propeller working in isolation, away from the hull (Section 2).
$\eta_H$ — hull efficiency (hull efficiency): effect of wake and thrust deduction (Section 4).
$\eta_R$ — relative rotative efficiency (relative rotative efficiency): the difference between the propeller behind the hull and in open water (Section 4).
Why not use an "overall propulsive efficiency" $P_E/P_I$ or $P_E/P_S$ directly? Because mechanical efficiencies, gear losses and transmission losses vary from ship to ship and even with the instantaneous load. Isolating the hydrodynamics of the hull-propeller combination requires the QPC, which separates the mechanical loss from the hydrodynamic loss.
3. Momentum theory — the propeller as an actuator disk
The propeller generates thrust by accelerating the fluid in which it works. By Newton's third law, the water pushed astern reacts by pushing the ship ahead: the change in the momentum of the fluid per unit time equals the force that produced it. Historically, two independent lines explained this — the momentum theories (changes of momentum in the fluid) and the blade-element theories (forces on the blade sections). The circulation theory (Betz, Prandtl) united the two.
Each theory, alone, failed: the momentum theory gave the limiting efficiency but did not give the shape of the propeller; the blade-element theory predicted the effect of the shape but led to the incorrect result that the ideal efficiency would be 100%.
3.1 The actuator disk
In the ideal conception, the propeller is a disk able to impart a sudden increase of pressure to the fluid passing through it (Rankine 1865, Froude 1889). Assumptions: uniform acceleration over the whole disk (uniformly distributed thrust) and a frictionless fluid. Consider the disk of area $A_0$ advancing at speed $V_a$ in still fluid; it is equivalent to a fixed disk in a uniform stream $V_a$ (Fig. 1 pressure and velocity scheme at the propeller disk, momentum theory).
Far ahead (section 1), the velocity is $V_a$. Far astern (section 3), the accelerated column of water (the race) has velocity $V_a(1+b)$. At the disk itself (section 2), the velocity is already $V_a(1+a)$, where $a$ is the axial inflow factor. By Bernoulli's law, the pressure drops as the flow approaches the disk, jumps at the disk and drops again in the race.
Thrust = change of momentum per second (mass flow × velocity gain):
$$T = \rho \, A_0 \, V_a(1+a)\cdot b\,V_a$$
From the equality between useful work and the kinetic-energy gain it follows that half of the velocity increase is acquired BEFORE the disk:
$$a = \tfrac{1}{2}\,b$$
Ideal efficiency (axial losses only):
$$\eta_i = \frac{V_a}{V_a(1+a)} = \frac{1}{1+a}$$
Defining the thrust-loading coefficient $C_T = T / (\tfrac{1}{2}\rho A_0 V_a^2)$, the ideal efficiency becomes:
$$\eta_i = \frac{2}{1+\sqrt{1+C_T}}$$
Conclusion of great practical importance: the lower the loading coefficient $C_T$, the higher the efficiency. Since $C_T$ falls as the disk area grows, the propeller of larger diameter tends to be the most efficient, other conditions being equal. This is the principle behind large, slow-turning propellers.
Zero speed of advance ($V_a = 0$) gives zero efficiency, but the propeller still delivers thrust and absorbs power — the bollard pull condition, measured in a dock trial, which serves as a measure of the thrust capacity at zero speed.
3.2 Including the rotation of the race
The real disk also imparts rotation to the water (angular velocity in the same direction as the disk). A rotational inflow factor $a'$ appears, analogous to the axial one. The kinetic energy of rotation in the race is lost energy, so it reduces the ideal efficiency by the factor $(1-a')$:
$$\eta_{\text{ideal, with rotation}} = \frac{1-a'}{1+a}$$
4. Blade-element and circulation theory
4.1 The blade element
Here the blade is divided into strips from the leading edge to the trailing edge (Fig. 2 geometric definitions of the propeller blade). On each strip the forces are evaluated from the relative velocity and the section shape, resolved into thrust $dT$ (forward) and torque $dQ$ (plane of rotation). Integrating from the hub to the tip (Fig. 3 blade loading curves $dT/dr$ and $dQ/dr$) gives the total $T$ and $Q$. Most of the thrust and torque is generated in the outer part of the blade, with a maximum around $r \approx 0{,}7R$.
The force on the section resolves into lift ($L$, normal to the flow) and drag ($D$, along the flow) (Fig. 4 section forces: lift, drag and $C_L$, $C_D$ curves vs angle of incidence). Dimensionless coefficients are used:
$$L = C_L \cdot \tfrac{1}{2}\rho A V^2 \qquad D = C_D \cdot \tfrac{1}{2}\rho A V^2 \qquad \tan\gamma = \frac{D}{L} = \frac{C_D}{C_L}$$
$C_L$ grows linearly with the angle of incidence $\alpha$ for small angles.
Zero lift does not occur at $\alpha = 0$ but at a small negative angle, the angle of zero lift $\alpha_0$.
The drag $C_D$ is small and nearly constant for small $\alpha$, and shoots up when $C_L$ begins to fall (stall).
The $L/D$ ratio is greatest at a small angle of incidence (~3 to 6°) — which is why efficient blades work at small incidence.
The common profile shape is the NACA 66 (modified) with mean line $a=0{,}8$ (Fig. 5 symbols defining the airfoil shape). Lift results from the pressure difference between the faces: the pressure reduction on the back contributes more to lift than the pressure increase on the face (Fig. 6 pressure distribution on the blade section).
Naming convention for the blade: the surface facing aft, which undergoes the pressure increase when propelling the ship ahead, is the face; the forward side is the back. The face is the high-pressure (pushing) side; the back is the low-pressure (suction) side.
4.2 Pitch, geometric helix and slip
The blade face is, in the simplest case, a portion of a helicoidal surface — generated by a straight line that advances along the axis while it rotates (Fig. 7 definition of a helix). The advance in one complete turn is the pitch (pitch, $P$). The angle between the helix and the plane normal to the axis is the pitch angle $\Phi$ (Fig. 9 definition of the pitch angle):
$$\tan\Phi = \frac{P}{2\pi r}$$
The pitch is not always the same at every radius; the pitch at $0{,}7R$ is taken as the representative mean pitch (point of maximum lift). In a rigid medium the propeller would advance $P$ per turn; in a real fluid there is "slip": it advances less, and the difference is the slip (Fig. 10 definition of slip):
The inflow factors $a$ and $a'$ accelerate the water axially (from $V_a$ to $V_a(1+a)$) and slow the relative rotation (from $2\pi n r$ to $2\pi n r(1-a')$), reducing the effective angle of incidence (Fig. 11 blade velocity diagram). For that reason the simple blade-element theory, ignoring the induced velocities, was considerably in error.
4.3 Circulation theory
Modern design methods rest on the vortex theory (Lanchester, 1907). Around a cylinder with imposed circulation in a uniform stream, the velocity distribution becomes asymmetric, generating a transverse force — the Magnus effect (Fig. 12c flow over a cylinder with circulation — Magnus effect), used to propel the Flettner rotor ship. Without circulation, the flow is symmetric and there is no force ().
Transverse force (Kutta-Joukowski equation), valid for any shape — the shape enters only through the circulation $\Gamma$:
$$L = \rho \, V_0 \, \Gamma$$
Thus the blade can be treated as a lifting line endowed with circulation, without having to define the shape until the end of the calculation. The lines have continuations at the tips — the tip vortices (Fig. 13 wing vortex with constant circulation) — because the fluid leaks from the face to the back at the extremities. When the circulation varies along the span, free vortices form along the trailing edge, a vortex sheet (Fig. 14 line integral, Fig. 15 vortex system with varying circulation).
Betz's theorem: the losses from induced velocities are a minimum when the helicoidal vortex sheet is pushed astern as though it were a rigid sheet. Hence the practical design rule: for maximum efficiency, design the blades so that the inflow velocity is the same at every element. The induced velocity at the disk is half that existing far behind the propeller.
5. Interaction between hull and propeller
Everything seen so far applied to the propeller in open water, advancing in undisturbed water. Behind the hull, conditions change greatly: the propeller works in water already disturbed by the passage of the hull. This is syllabus topic 2.3.
5.1 Wake
The water around the stern has acquired a motion forward, in the same direction as the ship — the wake. For that reason the propeller does not advance relative to the water at the ship's speed $V$, but at a lower speed, the speed of advance $V_a$. The wake has three causes:
Frictional: the drag of the hull drags along a current that grows toward the stern (positive wake, forward).
Potential (streamline): near the stern the streamlines close in, the pressure rises and the relative velocity falls (positive wake).
Wave (orbital): at the crests the orbital motion is forward (positive); at the troughs, aft (negative). It can be positive or negative depending on whether a crest or a trough is at the propeller.
The total is almost always positive; the exception is very fast ships (destroyers, ~34 knots), where it may be nil or slightly negative. There are two definitions of the wake fraction:
Taylor (fraction of the SHIP's speed) — now universal:
Examination caution: the two definitions do NOT give the same number. A Taylor wake of 50% means the wake speed = 50% of the ship's speed; in Froude notation, 50% means the wake = 33% of the ship's speed. Old data (especially British) use Froude; the current literature uses Taylor. Relation: $w = w_F/(1+w_F)$.
The wake measured without the propeller (by Pitot tubes, giving iso-wake curves) is the nominal wake; with the propeller present, the induced inflow reduces it, giving the effective wake — generally 3 to 4 points lower than the nominal. In a single-screw ship the wake is most intense in the upper part of the disk (Fig. 17 wake diagram of a single-screw ship, $C_b = 0{,}65$); in a twin-screw ship the mean is lower, but it concentrates behind the bossings or struts (Fig. 18 wake diagrams of a twin-screw ship, with struts and with bossings). The shape of the afterbody is what most affects the wake pattern (Fig. 19 comparison of wake fields according to afterbody form).
The non-uniformity of the wake is undesirable: it produces periodic forces and moments on the blades → hull vibration, and a periodic variation of the angle of attack → it favors cavitation (noise and erosion). Hence the importance of the stern lines, the appendages and the propeller clearances. The choice of the number of blades $Z$ helps to avoid resonances.
5.2 Real and apparent slip
Real slip (uses the speed of advance $V_a$):
$$s_R = 1 - \frac{V_a}{P\,n}$$
Apparent slip (uses the ship's speed $V$ — logged aboard):
$$s_A = 1 - \frac{V}{P\,n}$$
The real slip is the only true guide to performance, but it requires knowing the effective wake; the apparent slip, which needs only $V$, $n$ and $P$, is the one usually entered in the log.
5.3 Augment of resistance and thrust deduction
When towing the hull, there is a high-pressure region at the stern with a forward component that reduces the resistance. With the ship self-propelled, the propeller accelerates the water there, reduces that pressure and the forward component, increasing the resistance and the thrust required. This is usually seen as a thrust deduction: the propeller gives a thrust $T$, but only $R_T$ is available to overcome the resistance.
The value of $t$ depends greatly on the appendages: for a 122 m cargo ship with no rudder/sternpost $t = 0{,}20$ (augment effect only); with a plate rudder and a square sternpost it rose to $0{,}29$. When using published values, it is essential to know under what model conditions they were obtained.
5.4 Hull efficiency and the quasi-propulsive coefficient
The hull efficiency compares the work done on the ship ($R_T \cdot V$) with that of the propeller ($T \cdot V_a$):
The relative rotative efficiency $\eta_R$ compares the propeller's performance behind the hull with that in open water (at the same $V_a$, $T$, $n$): $\eta_R = Q_0/Q$. It differs from 1 because the non-uniform wake alters the conditions at each blade section and the turbulence behind the hull is greater. It is around 0.95–1.0 in twin-screw ships and 1.0–1.1 in single-screw ships.
Putting it all together, the quasi-propulsive coefficient breaks down — the key to understanding and estimating propulsive performance (already anticipated in Section 2):
$$\eta_D = \eta_H \cdot \eta_R \cdot \eta_0$$
hull efficiency × relative rotative efficiency × open-water efficiency of the propeller.
✎ Editorial note (bridge of understanding): the open-water efficiency $\eta_0$ is obtained from a test of the isolated propeller, from the dimensionless thrust and torque curves $K_T = T/(\rho n^2 D^4)$ and $K_Q = Q/(\rho n^2 D^5)$ as functions of the advance coefficient $J = V_a/(nD)$, with $\eta_0 = \dfrac{J}{2\pi}\cdot\dfrac{K_T}{K_Q}$. These coefficients are detailed by the book in Section 3 (outside the syllabus scope), but $\eta_0$ is indispensable to close $\eta_D$.
6. Geometry of the screw propeller
To design a propeller one must go beyond the basic definitions (syllabus topic 2.4). The starting point is nearly always a helicoidal surface.
6.1 The pitch surface
The helicoidal surface is swept by a straight line whose point $A$ advances along the axis while the line rotates. The curves traced are helices, all with the same advance per turn — the same pitch $P$. When the pitches vary with radius or the radial line is curved, one obtains the pitch surface, described by the shape of the radial reference line and by the pitches of the helices at various radii — that is enough to describe any practical propeller. Unrolling the cylinder of radius $r$, the helix becomes a straight line and the pitch angle satisfies $\tan\Phi = P/(2\pi r)$. The radius of curvature of the helix at that point is:
$$\rho_{\text{helix}} = \frac{r}{\cos^2\Phi}$$
6.2 The propeller drawing and the blade outlines
The design drawing has four parts (Fig. 25 propeller drawing in four views): (a) side elevation — shows the rake (longitudinal inclination of the generating line) and the variation of maximum thickness from tip to root; (b) expanded outline with the section shapes; (c) pitch distribution (if not uniform); (d) transverse view with the developed outline and the skew.
There are three distinct blade outlines — a classic examination distinction:
Outline
What it is
Projected
Projection of the blade onto a transverse plane (the "shadow" seen along the axis)
Developed
Section widths set out on helical arcs (radius of curvature $r/\cos^2\Phi$)
Expanded
Sections laid flat in the plane, with the pitch lines parallel to the axis
The section shape changes the outlines for the same total chord width: airfoil sections (nose and tail raised from the pitch face) give projected and developed outlines different from those of a flat-faced section (Fig. 26 effect of section shape on the blade outlines).
6.3 Dimensionless ratios
The propeller's characteristics are expressed by dimensionless ratios ($A_0 = \pi D^2/4$ is the disk area):
Ratio
Symbol
Definition
Pitch ratio
$P/D$
pitch divided by diameter
Developed area ratio
DAR
$A_D/A_0$ (developed blade area / disk area)
Projected area ratio
PAR
$A_P/A_0$
Expanded area ratio
EAR
$A_E/A_0$
Mean width ratio
MWR
mean blade width / $D$
6.4 Constructional details
For many years propellers had 3 or 4 blades (4 almost universal in single-screw ships), held to be more efficient. With increasing power, designers went to 5, 6 or more blades to increase the area (delay cavitation) without blades too wide, reduce the thrust per blade and damp vibration (more blades = smaller force per blade and higher frequency, helping to escape resonance).
Hub: cylindrical or conical, diameter from $0{,}15D$ to $0{,}25D$.
Solid vs built-up: blades cast integral with the hub (solid) or separate and bolted on (built-up — easy replacement and small pitch adjustment, but more expensive, heavier and a larger hub).
Pitch ratio $P/D$: ~0.6 (tugs, heavily loaded) up to 2.0+ (fast launches). In single-screw ships the pitch is usually reduced near the hub (wake concentrated in the inner radii) and, in heavily loaded propellers, also at the tip (reduces tip-vortex cavitation).
Rake aft: increases the clearance to the hull/bossings, benefits efficiency and reduces periodic forces (vibration).
Skew: makes the leading edge enter the wake concentrations more gradually, reducing the periodic forces.
Section thickness is a conflict of requirements: small for efficiency, but larger for structural strength and to delay certain kinds of cavitation. The developed area ratio ranges from $0{,}35$ to more than $1{,}0$ in very fast ships.
Material
Characteristics
Manganese bronze / nickel-aluminium alloys
Tough blades, high polish, erosion-resistant → keep high efficiency. Nickel-aluminium is lighter, with higher allowable stress → thin, light blades.
Cast iron
Cheap, but low tensile strength (thick blades), corrodes in salt water, low resistance to cavitation erosion. Used in tugs and icebreakers, because it breaks "clean" on striking an obstacle, without damaging the hull/machinery.
6.5 Blade strength
The minimum thickness must satisfy the classification societies. In the simplest method the blade is treated as a cantilever beam, computing the stress at a typical section near the root; the ABS adopts the simplified Schoenherr formula with the strength section at 0.25 of the radius. In highly skewed propellers (skew above ~40°) the beam method no longer holds, and the risk of static divergence appears — the blade flexes increasing the effective pitch and, with it, the loading (Fig. 28 dependence of the maximum stress on skew and iso-stress (Von Mises) curves in the astern condition).
7. Cavitation
Cavitation appears in heavily loaded propellers: beyond a certain critical rotation, there is a progressive breakdown of the flow and a loss of thrust; at the extreme, it prevents the ship from reaching its speed. Before that, it shows up as noise, vibration and erosion of the blades, struts and rudders (syllabus topic 2.5). The earliest recorded case was the British destroyer Daring (1894), which reached only 24 knots instead of 27 until it received propellers with 45% more blade area.
7.1 Mechanism and the cavitation number
On a section at a small angle of attack (Fig. 29 flow and pressure around an airfoil), the fluid divides at a stagnation point $S$ (zero velocity, maximum pressure). The dynamic (stagnation) pressure is $q = \tfrac{1}{2}\rho V^2$. On the back the velocity increases and the pressure falls. When the local pressure falls to the vapor pressure $p_v$, the water "boils" and cavities form — the water cannot sustain tension. Hence the cavitation number:
where $p_0$ is the total static pressure (hydrostatic + atmospheric). Cavitation begins when the pressure reduction on the back reaches $(p_0 - p_v)$.
Margin of safety: the vapor pressure of fresh water at 14 °C is small ($1{,}70$ kN/m²), but seawater has dissolved air and nuclei that anticipate cavitation — it can occur at local pressures up to ~$17$ kN/m². The most resistant sections are those with the most uniform pressure distribution and the gentlest peak. Since the pressure is lowest when the blade is at the top, transient cavitation appears first in the upper part of the disk.
7.2 Types of cavitation
Knapp classifies cavitation in general into travelling (free bubbles in the stream), fixed (attached to the body), vortex (in the low-pressure core of a vortex) and vibratory (from pressure pulses). On propellers the fixed and vortex types predominate. By appearance:
Type
Where / how
Fig.
Sheet
Leading edge, on the back under a positive angle (or on the face under a negative angle); may cover the whole back; stable in uniform flow, unstable in the wake
Fig. 30 stable sheet cavitation
Bubble
Mid-chord / maximum thickness, at shock-free entry; large bubbles that grow and contract
Fig. 31 bubble cavitation
Cloud
Behind stable sheets; a mist of small bubbles
Fig. 32 cloud cavitation
Tip vortex
Leakage face→back at the tip; starts unattached and then attaches
Fig. 33 unattached tip vortexFig. 34 attached tip vortex
Hub vortex
Combined root vortices; a "thick cord" with strands = the number of blades
Fig. 35 hub cavitation
7.3 Cavitation-tunnel testing
Similarity requires the same cavitation number $\sigma$ between model and ship (besides Froude, Reynolds and slip $J$ — Froude and Reynolds are incompatible, as for the hull). In an ordinary tank the atmospheric pressure is not scaled down, so a variable-pressure facility is needed: the cavitation tunnel (the first was Parsons's, 1897) or MARIN's variable-pressure towing tank (Fig. 36 MARIN variable-pressure towing tank). In practice the test is run with $\sigma$ 15 to 25% lower than the ship's, to compensate for the unsimulated wake.
The results are presented like the open-water curves, but with a set per $\sigma$ (Fig. 37 characteristic curves in a cavitation tunnel). Newton's diagram plots on $J$–$\sigma$ axes the boundaries of each cavitation type (Fig. 38 development of cavitation patterns (Newton)).
7.4 Harmful effects
Performance (thrust breakdown): cavitation reduces the suction peak at the leading edge and spreads it over the chord (Fig. 39 pressure distribution at various $\sigma$); the $L/D$ ratio falls after an initial rise (Fig. 40 $C_L$, $C_D$, $C_L/C_D$ vs $\alpha$), bringing down thrust, torque and efficiency (Fig. 41 $K_T$, $K_Q$, $\eta_0$ vs $J$ — propeller B5-75). To hold the speed, more power and rotation are needed.
Erosion: the collapse of the bubbles at the surface generates shock waves (and re-entrant micro-jets) of very high energy over tiny areas → a pitted surface (Fig. 42 cavitation erosion on the blade), which can reach severe damage with loss of material (Fig. 43 severe erosion damage). In high-power single-screw ships bending of the trailing edge also occurs (Fig. 44 trailing edge bent by cavitation). Erosion and corrosion feed on each other.
Vibration and noise: the pressure fluctuations induced on the afterbody are amplified by cavitation by a factor of 1 to 10 (or more). The high-frequency noise reduces detection by sonar — which is why warships need a high speed of cavitation inception.
Prevention: choose suitable material and coatings; but the best is to avoid harmful cavitation — making the wake uniform (stern lines, clearances, rake, alignment of bossings/struts). When unavoidable, a supercavitating propeller is designed (fully developed cavity), so that the bubbles collapse in the water, away from the blade surface.
7.5 Criteria to avoid cavitation
The oldest criterion (Barnaby, from the Daring) limited the pressure to $76{,}7$ kN/m² of projected area. Burrill's diagram (Fig. 45 simple Burrill cavitation diagram) plots the thrust-loading coefficient $\tau_c$ against the local cavitation number at $0{,}7R$, with limit lines for warship, merchant-ship and tug/trawler propellers; the line of 5% back cavitation became the practical design target. Keller's formula gives a first estimate of the expanded area ratio (EAR).
Limitation: criteria such as Burrill's and Keller's formula do not reflect the wake nor the blade geometry (pitch, camber, thickness). They should be used with caution, as first guidance only.
8. Other propulsion devices
The screw propeller drives the vast majority of ships, but there are other devices with advantages in special situations (syllabus topics 2.6 and 2.7).
8.1 Jet propulsion and pump-jet
It is the oldest mechanical type: a pump inside the hull draws in water and expels it astern as a jet of higher velocity — a reaction device, like the propeller, but with the moving parts internal (Fig. 86 jet-propulsion scheme). The efficiency tends to 1 as the jet velocity $V_j$ approaches that of the ship, but then the thrust tends to zero (unless the area $A$ is enormous) — the same dilemma as the propeller. Since obtaining a large $A$ is easier outside the hull, the external propeller is preferable.
In practice the jet's efficiency is low (pump, inlet and duct losses), below that of the ordinary propeller. Other drawbacks: loss of internal volume, clogging (debris/weed) and non-uniform inlet flow. Main advantage: maneuvering — the steerable jet acts as a rudder and, if it gives astern thrust, dispenses with engine reversal; it also brings quietness and no appendage drag.
In the pump-jet, the impeller is external inside a duct with guide vanes; the duct widens up to the impeller, the velocity falls and the pressure rises, allowing an impeller of larger diameter with lower $C_T$ → higher efficiency and delayed cavitation/noise. The price is the resistance of the duct itself.
8.2 Paddle wheels and vertical-axis propellers
Paddle wheels (side or stern), especially with feathering blades, can match the efficiency of the propeller where the draft restricts its diameter; they are best placed over the crest of the wave.
Vertical-axis propellers (cycloidal) have vertical blades on a disk that revolves about a vertical axis, each blade also turning about its own axis (Fig. 89 model with vertical-axis propellers). Two types:
Type
Blade motion
Fig.
Kirsten-Boeing
Each blade makes half a revolution about its own axis per revolution of the disk
Fig. 87 Kirsten-Boeing propeller
Voith-Schneider
Each blade makes one full revolution; the point C (eccentricity < 1) is shifted to set the thrust direction
Fig. 88 Voith-Schneider propeller
Great advantage: the thrust serves to steer and stop the ship without stopping or reversing the engine — ideal for restricted and crowded waters, which demand much steering effort at low speed. Disadvantage: much lower efficiency (about 30 to 40% below the propeller) (Fig. 90 comparison of open-water efficiencies, Fig. 91 open-water results, 6-blade cycloidal motion).
8.3 Controllable-pitch propeller (CPP)
In the controllable-pitch propeller (CPP) each blade is mounted on a spindle in the hub and the pitch can be altered and even reversed with the propeller turning, by an internal hydraulic mechanism.
Ideal for ships with widely varying conditions (tugs, trawlers) and non-reversible machinery: by reducing the pitch when towing, the engine keeps full RPM and power without overloading the cylinder (important in the diesel).
In ferries that stop/start/reverse frequently, full astern comes from reversing the pitch with the engine always in the same direction.
Dispenses with the reversing mechanism and the astern turbine → saves weight and cost, and makes emergency reversal much faster.
It is almost as efficient as the fixed-blade propeller in the chosen condition, the only difference being the larger hub (housing the mechanism). When the pitch is changed, all sections turn through the same angle, so the pitch face ceases to be a true helicoid.
8.4 Tandem, contra-rotating and overlapping
When the diameter is restricted, the loading factor rises, the efficiency falls and cavitation increases; dividing the load among several propellers relieves this. Tandem: propellers on the same shaft, same direction. Contra-rotating: coaxial shafts turning in opposite directions, recovering the rotational energy of the race (the after one is smaller, to follow the contraction of the race). A gain of ~7% less power than the twin-screw, but with the weight and complication of gearing, coaxial shafts and sealing; used in torpedoes to balance the torque.
Overlapping propellers (twin-screw with fields that cross) reduce the power by 5–8% relative to the single-screw and 20–25% relative to the conventional twin-screw; optimum spacing between centers ~$0{,}7D$. There can be interference of the tip vortices (Fig. 99 interference of the tip vortices of overlapping propellers).
8.5 Supercavitating propellers
When the cavity covers the whole back (which is no longer wetted), the propeller operates in the supercavitating regime: the back no longer generates additional lift, but the face keeps increasing the thrust with rotation (Fig. 92 cavitation development and $T$, $Q$, $\eta$ curves vs RPM). Decisive advantage: no erosion on the back (the bubbles collapse in the water, downstream) and less vibration. Wedge sections of Tulin are used, with a very thin leading edge to ensure clean separation (Fig. 93 wedge sections for a supercavitating propeller); the criterion is $\sigma \le 0{,}045$ at $0{,}7R$ (Fig. 94 supercavitating-propeller design chart, Fig. 95 zones of practical use).
They are ~10% less efficient than the conventional propeller on a liner, but for the 40 to 80 knot range they may be the only viable option (the conventional one would not even be designable). There are serious strength problems (high thrust + thin edge). Inclined shafts can increase the efficiency (Fig. 98 characteristics with the effect of shaft angle). Alternative: the ventilated propeller (air injected on the back).
The partially submerged propeller (surface-piercing) is attractive above ~40 knots (topic 2.7):
The air-filled cavities do not collapse violently → less erosion.
It sits directly behind the ship → little resistance from shafts/supports, and allows a larger diameter than under the bottom.
Since the erosion risk falls, smaller area ratios are acceptable → less friction → higher efficiency (above 60% in open water is possible).
Drawbacks: the blade strength undergoes large stress variations — load ~zero at the top and maximum at the bottom (more than in the conventional propeller) → attention to fatigue; and the torque is very sensitive to cavitation/ventilation with submergence, requiring the pitch to be chosen so as to keep the torque within limits at low speed (Fig. 97 measured strain vs angle of rotation).
Among the energy-saving devices, Grim's vane wheel (free-running paddle wheel) stands out: a larger free-running wheel behind the propeller absorbs energy from the race at the inner radii and returns it as thrust at the outer radii, outside the race — producing the total thrust with less power.
