Crenshaw Cap. 2 — Forces Affecting the Ship (Naval Shiphandling)

1. Overview — The Six Forces

CRENSHAW, R. S. Naval Shiphandling. 4th ed. Annapolis: Naval Institute Press, 1975. Chapter 2 — Forces Affecting the Ship (the six sources of force; pressure differences and Bernoulli's Theorem; hydrofoils; the propeller and its spiral discharge; side force from single and twin screws; resistance — frictional, wave, eddy, appendage, air and wind; squat; shallow water; total resistance; rough water; the cube rule; wind and current).

Anexo 2-B, Seção I (Manobrabilidade do Navio), item I.1 — CRENSHAW, Naval Shiphandling, 4ª ed., Cap. 2 (único capítulo prescrito). Anexo 2-A, Seção I — forças que atuam sobre o navio: empuxo e resistência, força lateral de hélice simples e geminado, efeito de esteira, águas rasas, squat, resistências (friccional, de onda, de redemoinho, de apêndices, de ar/vento), regra do cubo, vento e corrente.

To predict a ship's movement we must understand the nature and magnitude of the forces acting on her. Crenshaw counts six general sources of force that can bear on a ship independently of any other vessel: the propellers, the rudders, the mooring lines, the ground tackle, the wind, and the current (tidal currents included). The first four are controllable from the ship; the wind and current are not controllable, but can be turned to our aims if handled properly.

Fig. 2-1 The six forces that can bear on a ship, with sample magnitudes for a 2,200-ton destroyer

A vital qualifier opens the chapter: these are forces only, and motion follows only after inertia has played its part. A modern ship distributes many thousand tons over several hundred feet, so it has enormous inertia against linear acceleration and an enormous moment of inertia against rotational acceleration. Resting in water and covered by air — two fluids that resist relative motion — the ship accelerates under an applied force until the fluid resistance produced by the motion balances that force, at which point the velocity becomes constant. This holds for off-centre forces (which turn the ship) just as for forces through the centre of gravity.

2. Basic Principles

All forces in water manifest themselves as pressure differences. Water is incompressible, but applying force builds higher pressure in one region than its surroundings, and water then flows from the high-pressure area to the low-pressure area. Pulling an oar illustrates the whole system: high pressure forms on the driving face, low pressure on the trailing face, water flows from one to the other, and the inertia of the water — resisting through the oar — pushes the boat the opposite way. The lesson generalises: if the ship moves at all, a force is acting somewhere on her structure, and that force can be located by finding the pressure difference that causes it.

2.1 Bernoulli and Dynamic Pressure

At any point in a large body of water there are two pressure components: the static pressure due to depth (the sheer weight of water above), and the dynamic pressure caused by motion in the surrounding water. Static pressure is the same everywhere at a given depth, so it causes no motion and balances out. Bernoulli's Theorem states that at a given depth the sum of static and dynamic pressure is constant; when a hull or a propeller blade sets the water in motion, the static pressure falls by the amount of the dynamic pressure, and that decrease produces our hydrodynamic effects. The magnitude of the dynamic pressure is:

$$P = \frac{\rho V^2}{2g}$$

where $P$ is the dynamic pressure (lbs/ft²), $\rho$ the density of the moving fluid (lbs/ft³), $V$ the velocity of flow (ft/sec) and $g$ the acceleration of gravity (32.2 ft/sec²). The pressure difference is therefore proportional to the density of the fluid and to the square of the velocity.

The same theorem applies to air, but salt water at 64.4 lbs/ft³ is 855 times denser than standard moist air at 0.0752 lbs/ft³, so the dynamic forces of water on the hull vastly exceed those of air. Because dynamic pressure grows with the square of velocity, however, and the wind speed relative to the ship may be far higher than water speeds, strong winds still matter. A useful rule of thumb: air must move about 30 times the velocity of water for equal dynamic pressure — that is, 30 knots of wind equals 1 knot of current.

Water is continuous: it exists as an unbroken body except under extraordinary forces. If a propeller blade moves water away faster than the pressure difference can accelerate new water in behind it, a gap forms — separation. If, in a high-velocity stream, pressure drops to the vaporisation point, water vaporises like boiling — cavitation. Both upset the streamlines and change the resulting forces, and both tend to occur around abrupt underwater changes at high speed or about a rapidly turning propeller blade.

3. Hydrofoils

Several key forces come from hydrofoils, so the chapter defines the terms first. A hydrofoil is any thin, plate-like member — a propeller blade or a rudder — designed to obtain a lift force when inclined to the water flow. The angle of attack is the angle at which the hydrofoil is inclined to the relative free-stream flow. Lift is the reaction component perpendicular to that flow; drag is the component parallel to it.

A flat plate set at an angle in a stream acts as a hydrofoil: water moves aside on the leading side (high pressure) and accelerates in behind the trailing side (low pressure), and that pressure difference is the lift. With smooth flow the force is proportional to the angle of inclination, the dynamic pressure, and the plate area. The flat plate forces an abrupt change of flow, so separation can occur and alter the pressure distribution; a properly shaped foil makes the acceleration gradual and avoids separation.

Fig. 2-2a Flow around a flat plate inclined to the stream — abrupt change and possible separation

Fig. 2-2b Flow around a shaped hydrofoil — gradual acceleration, separation avoided

Several ship members are hydrofoils. The rudder is the obvious one, producing lateral force through the rudder stock to push the stern to port or starboard, proportional to its area, the dynamic pressure, and the angle of attack. The hull itself is a hydrofoil: inclined to the flow it feels a side force and a turning moment that tend to realign it with the flow, and in a steady turn the rate becomes constant when the hull's moment equals the rudder's. The rudder's true angle of attack equals the rudder angle relative to the ship minus the ship's angle of inclination to its real direction of motion. Keels and skegs are hydrofoils too — aligned with the centreline, they produce a turning moment whenever the ship is inclined to the flow; bilge keels (rolling chocks) follow the flowlines in straight running but become inclined and produce a correcting moment when the ship rolls.

A most important property of any hydrofoil: the force is proportional to the square of the velocity of the water actually flowing past it, which may differ greatly from the ship's speed. A rudder directly astern of a propeller turning ahead develops large force from the screw current even before the ship moves; conversely, a back bell sends little current past the rudder, so no appreciable rudder force is felt until the ship is moving smartly through the water.

4. The Propeller

One of the most important sources of force is the ship's own propeller. One might expect turning it ahead to drive the ship straight ahead and turning it astern to drive her straight astern — but this is not necessarily so, and the shiphandler must study propeller action to predict its effect. The design objective is maximum thrust along the shaft line from a given torque. A fixed-blade propeller is optimised for one speed (usually the maximum) but works efficiently at all normal speeds; the ship's actual speed is less than the ideal pitch × rpm because the blade must be inclined to the flow for the water to exert force. Speed varies nearly linearly with shaft rpm until separation and cavitation become pronounced.

Water exerts force perpendicular to the mean face of the inclined blade, so the force is inclined to the shaft rather than along it. Because the blades are arranged symmetrically, the radial components cancel and the net thrust lies along the shaft axis — which is why exact blade balance matters. A small nick or dent unbalances the radial forces (though it barely affects useful thrust), and the long external shafting on weak struts withstands radial forces poorly, so a damaged propeller causes heavy vibration. A propeller works well astern too: the pitch is the same, but the reversed blade section is less efficient, so a given astern rpm needs more power than ahead while delivering roughly the same thrust.

The real flow differs from the ideal parallel discharge. As the stream accelerates through the propeller and the blades impart rotation, the discharge leaves in a spiral. The disturbance grows with the difference between the mean flow through the propeller and the surrounding current: it is least when the propeller is doing the least work, large when the ship is stopped and the screws turn rapidly, and maximum when the ship moves one way and the propellers are driven at full power the other.

Fig. 2-3 Flow pattern about a driving propeller — spiral (helical) discharge

This helical discharge — the tangential motion the rotating blades give the water — is one of the secondary effects of propeller rotation, and the adroit use of those secondary effects is often what makes intricate manoeuvres possible.

5. Side Force from a Single Screw

As a propeller turns it produces, besides axial thrust, an appreciable side force at the stern, which is often the deciding factor in whether a manoeuvre can be done. An isolated propeller deeply submerged in open water feels no appreciable side force — all radial lift components cancel. The real case differs: the propeller is shallowly submerged, close to the hull, and surrounded by shafts, struts and rudders, so the flow across the disc is neither parallel to the axis nor uniform, and side force is always experienced.

5.1 The Wake and the Velocity Diagram

A moving ship drags water along by skin friction, forming a following wake. Near the hull the relative water velocity is very small (the water is carried along); farther out it approaches the ship's speed. Taking the boundary as the point where the wake is 2 percent of ship's speed, the frictional wake grows from zero at the bow to several feet near the stern. A ship making 15 knots with a 3-knot following wake presents only 12 knots to its propeller.

Fig. 2-4 Thickness of the frictional wake — forward velocity falling with distance from the hull

Fig. 2-5 Wake behind a single-screw ship — fore-and-aft velocity contours over the propeller disc

The wake is far from uniform: behind blunt structure it may move bodily forward with the ship, and over the disc the fore-and-aft velocity drops in places to 20 percent of ship's speed. The water near the propeller also moves upward and inward as it closes in behind the stern — an upward component that strongly affects propeller behaviour. Studying a typical blade section, its velocity relative to the water is the resultant of a forward component $V_A$ (ship's speed minus wake) and a tangential component $2\pi r N$ (rotation); the inclination of the resultant $V_o$ to the blade face is the angle of attack, which develops lift and drag and resolves into thrust $T$ and torque $Q$.

Fig. 2-6 Velocity diagram for a propeller blade section — V_A, tangential component and resultant V_o

Because the section meets different wake areas as it rotates, $V_A$, the angle of attack, thrust and torque all vary, so the propeller delivers an uneven thrust and produces side forces. For a single screw these break into four parts, summarised here for a right-hand screw going ahead.

Effect	Mechanism	Net tendency (RH screw, ahead)
Following wake	Strong upper-arc wake → more thrust on the top blade	Stern to port — veers right
Inclination	Upward flow under the stern raises starboard-side angle of attack	Twists ship to the left
Helical discharge	Stronger upper-half spiral strikes the rudder	Turns ship to the left
Shallow submergence	Air-drawing/surface break weakens the upper arc	Stern to starboard — veers left

5.2 The Four Effects (Single Screw)

Following wake effect. In the vertical position behind the hull (blade A) the blade passes through strong following wake, giving a greater angle of attack and more thrust there; the reaction, with a right-hand screw, tends to move the stern to port when going ahead. The lower blade meets much weaker wake over a smaller area, so the upper blade predominates — net stern to port, ship veers right.

Fig. 2-7 Side force from the following wake on a single-screw ship — blades A/B/C/D

Inclination effect. The shaft axis is inclined to the flow because the water moves upward under the stern. As blade B moves down to horizontal it meets water moving up as well as aft — effectively more relative velocity and angle of attack, hence more thrust; on the opposite (port) side, less thrust. The net reaction is a torque twisting the ship to the left.

Helical discharge effect. The spiral discharge strikes the rudder. The part above the propeller hub pushes the stern to starboard, the lower half to port; but the upper discharge is stronger (greater upper-arc angles of attack from the following wake), so the net effect turns the ship to the left. An unsymmetrical rudder, or one not spanning the whole disc, can alter or increase this.

Shallow submergence effect. At light displacement the propeller may break surface, or draw air down with little way on, working as if in a less dense medium in the upper arc. The result is to move the stern to starboard and veer the ship to the left. Going ahead, then, a single-screw ship is subject to several opposing actions; experiment is needed for a given ship, but most single-screw ships tend to turn left when going ahead.

5.3 Getting Under Way and Backing

Getting under way. With the ship at rest just starting to move, the stern usually goes to starboard. Forward motion — and hence wake — is negligible, so only the shallow submergence effect (independent of wake) applies: the churning screw draws air down predominantly into the upper half of the disc, moving the stern to starboard.

Backing. Turning astern with the ship dead in the water, the same cause acts but reversed, so from the propeller alone the stern goes to port. The backing discharge hits the stern: its upper half banks against the starboard counter, its lower half spills under the keel, again driving the stern to port.

Both backing effects carry the stern to port, so a single-screw ship with a right-hand screw has a strong tendency to back to port. The suction current is too diffuse to give the rudder useful bite, so such a ship relies almost entirely on sternway for rudder effect — it must gather appreciable sternway before the rudder can overcome the back-to-port tendency. In short, the side force acts as though the blades bore against the bottom, and its direction depends only on the direction of rotation; only when proceeding ahead can the conflicting forces reduce or even reverse it.

6. Side Force with Twin Screws

In a normal twin-screw ship the propellers turn in opposite directions, so their side forces cancel. By convention the rotation is chosen so the blade tips move outboard during the upper half of travel when driving ahead — calling for a right-hand screw on the starboard shaft and a left-hand screw on the port shaft — so the side force augments the turning moment from the offset shafts. Analysing the four effects against a destroyer's measured wake shows how twin-screw behaviour differs from single-screw.

Fig. 2-8a Wake near a destroyer's propeller — transverse flow components and following-wake percentages

Effect	Behaviour with twin screws
Following wake	Much reduced — blade tips nearest the hull see only ~15% following wake, and most of the disc sees none.
Inclination	Present in all ships — the upward wake plus the downward shaft inclination (the shaft must pass through the bottom to reach the propeller).
Helical discharge	Absent with a single rudder (outside the discharge); with twin spade rudders it is large, since the rudder feels mainly the stronger upper half.
Shallow submergence	Uncommon (tips rarely break surface), but churning and air-drawing still occur, so it counts.

Fig. 2-8b Side view of the wake — the downward inclination of the shaft confirms the inclination effect

Because the only opposing effect — the following wake — is diminished, the side force on a right-hand screw turning ahead is definitely to starboard, and on a left-hand screw definitely to port (reversed when rotation reverses). The forces are large and uniformly in the direction of rotation. Dead in the water the side forces from each propeller equal those of a single-screw ship; when backing they are usually somewhat smaller, because the structure into which the discharge is directed is less extensive.

6.1 Use of Side Force

In all conventional ships a side force appears whenever the propellers turn, almost always in the direction indicated by the rotation — as though the blades bore against a more solid layer in the lower part of their travel. A handler who understands the origin of these forces knows what reaction to expect, and a little experimentation reveals the magnitude and character for a given ship, which the conning officer can turn to good use.

7. Resistance and Power

Driving a ship raises a puzzle: feeble plants move large ships at moderate speed, yet vast power is needed for even small ships above 30 knots, as if a "wall" of resistance appeared near high speed. A destroyer with 60,000 shaft horsepower makes about 35 knots, yet a cruiser of six times the displacement makes the same speed with twice the power. Greater horsepower per ton buys acceleration at low speed, not a higher top speed — so size itself helps, and the larger ship does better.

Ship	Horsepower per ton	Approx. maximum speed
Battleship	3.7	All about the same (~35 kt)
Cruiser	6.5
Destroyer	19

All fluid resistance results from motion and grows as a power of velocity — there is no static friction in the sea. A perfectly streamlined body needs no power merely to displace water (forward and after pressures balance), so the real sources of resistance lie elsewhere and are not obvious. The book treats them in turn.

7.1 Frictional Resistance

Water against the skin is carried along, the next layer dragged less, and so on, forming a boundary layer that thickens from a few molecules at the bow to several feet at the stern. Energy spent imparting this motion is the frictional resistance. Beginning with Froude in England in 1874, experiments expressed it as:

$$R_f = f S V^n$$

where $R_f$ is frictional resistance, $f$ the (dynamic) coefficient of friction, $S$ the total wetted surface, $V$ the ship's speed, and $n$ the power by which friction varies. Froude found $n = 2.00$ for rough surfaces but as low as 1.83 for hard ones, and in 1888 fixed his constant at 1.825 for all sizes; later work nudged it slightly higher.

For the shiphandler it is enough to remember that frictional resistance is proportional to the total wetted surface and approximately to the square of the speed.

7.2 Wave Resistance

A ship on the surface makes waves, and waves carry energy the ship must supply; if we could measure that wave energy we could measure the propulsive power spent creating it. The energy of a single wave is proportional to its breadth and to the square of its height. The principal waves form at the bow and the stern — as if two wave generators travelled one shiplength apart — and the two systems interact, increasing or decreasing the result.

Fig. 2-9a Bow and stern wave systems reinforcing — crest on crest, R_w increases

Fig. 2-9b Bow and stern wave systems cancelling — crest on trough, R_w decreases

When a bow-system crest meets the first stern-system crest, reinforcement raises the waves and the wave resistance $R_w$; when a bow crest meets the first stern trough, cancellation lowers them and $R_w$ falls. The affected waves are the transverse waves, whose crests run perpendicular to the track and travel at ship's speed. Their length is:

$$l = 0.557\,V^2$$

where $l$ is the wave length in feet and $V$ the wave velocity in knots — equal to ship's speed for transverse waves. Doubling the speed therefore quadruples the distance between crests.

Whether reinforcement or cancellation occurs depends on the ratio of wave length $l$ to ship length $L$. Since the two systems start about one shiplength apart, $l/L = 0.557V^2/L$, which varies as $V^2/L$; for convenience its square root is used:

$$\frac{V}{\sqrt{L}}$$

the speed-length ratio ($V$ in knots, $L$ in feet) — a very important index in considering a ship's resistance and power.

No simple formula $R_w = aV^n$ fits, because $n$ would range from 1.5 to 11 across the speed range. Representing the ship by two disturbances, Professor Havelock showed the wave resistance forms a curve of distinct "humps and hollows" as reinforcements and cancellations occur, with a general fall at very high speed (speed-length ratio above 2) — partly because fast disturbances give the water less time to respond, and partly because the ship rides up on its own bow wave.

Fig. 2-10 Wave resistance versus speed-length ratio — humps and hollows (Havelock)

7.3 Eddy, Appendage and Wind Resistance

Eddy resistance arises where flow breaks away from abrupt hull changes — a square sternpost edge — leaving a low-pressure confused area astern that drags the ship back; it varies as the frontal area of the disturbing surface and the square of the velocity, so even small causes matter at high speed. Appendage resistance (shafts, struts, rudders, bilge keels) is mainly frictional, since well-designed appendages run deep and streamlined, so it is roughly proportional to their wetted surface and the square of the velocity.

Air and wind resistance. A complex superstructure defies formula, but experiment shows plain air resistance is only 1½ to 3 percent of the water resistance at maximum speed and can usually be neglected.

Wind resistance cannot be overlooked: a ship steaming at 10 knots into a 20-knot wind (a 30-knot relative wind) may spend as much as 20 percent of her power overcoming it. Maximum resistance ahead may occur with the relative wind on one bow rather than dead ahead, and a beam component forces rudder use to hold course — even a small rudder angle increases drag appreciably.

8. Squat, Shallow Water and Total Resistance

8.1 Squat

As speed rises the ship first sinks bodily; then, at the critical speed (speed-length ratio about 1.2), the bow begins to rise and the stern to sink — she squats. The first bow-wave crest moves aft and buoys the bow up, while the stern settles into the hollow where the bow-system trough augments the stern hollow near the screws. As she squats, resistance increases abruptly.

8.2 Shallow Water Effect

In shallow water a wave made at a given speed is longer than the same-speed wave in deep water, so reinforcements and severe squatting occur at lower speeds, and resistance rises faster as speed increases. Paradoxically, because the wave-resistance curve shifts left, certain very high-speed ships — already operating beyond the peak of that curve — can reach a higher maximum speed in shallow water than in deep.

8.3 Total Resistance and Rough Water

Combining all sources gives the total resistance. The general sinkage at lower speeds and the squatting with abrupt rise at higher speeds are clearly visible, and the hump-and-hollow character is marked in the shallow-water curves. For equal model resistance, a ship can reach a higher speed-length ratio in shallow water than in deep at the same power.

Fig. 2-11 Resistance and changes in level of a 20-foot model — three water depths, showing sinkage and squat

Rough water adds resistance: head seas constantly change the trim and the ship crashes into them, and rolling continually changes the submerged shape. Pitching is the most severe deterrent — it raises every source of resistance and can lose propulsive efficiency as the screws race near the surface. Rolling is less harmful than expected: inclinations up to 20° add only a few percent, though rough seas overall affect resistance markedly.

8.4 The Cube Rule

The mariner's thumb rule that power and fuel for a given speed rise as the cube of the speed holds in certain ranges but not all — especially once wave resistance dominates. Since power equals resistance × speed, the cube rule would require resistance to vary as the square of the speed, which is far from true generally.

Resistance, power and fuel rise drastically with speed, and even more abruptly once squatting begins. Length matters greatly: two similar ships at the same speed-length ratio need power in the ratio of the squares of their lengths, and the higher the ratio — especially beyond the critical 1.2 — the more power is wasted making waves. This is why a deeply submerged submarine, free of surface effects, reaches a higher speed for given power than an equivalent surface ship, and why inventors keep seeking ways to make a hull plane and escape wave drag.

9. Wind

Wind is an important force precisely because it is outside the handler's control and quite changeable — a hazard, but also a very useful aid that can do what engines and rudders alone cannot. Wind normally forces the ship bodily downwind, with a force proportional to the square of the wind velocity, to the cross-sectional area presented normal to the flow, and to the form of the superstructure. Double the relative wind and you quadruple the force; turn a larger cross-section to it, or present many flat irregular surfaces, and the force grows.

Reaction is fairly predictable: a ship with high freeboard and shallow draft responds readily to wind (large sail area, small water resistance), while a deep-draft, streamlined ship feels little. Trim matters by section — a high bow, low stern and trim by the stern let the wind carry the bow downwind, and a lightly loaded ship is more wind-sensitive than a heavily laden one.

Propellers and rudders apply force to the stern only. To turn into a beam wind the handler must move the stern downwind faster than the wind blows the bow downwind; if the bow presents a much larger sail area, the side force needed at the stern can be large. Since the engines' side forces are relatively small, it is often necessary to gain considerable headway before the rudder force suffices to overcome the wind.

10. Current

The last general force is the current. The hull's resistance to current resembles the superstructure's resistance to wind, but is far larger for a given velocity because water is so much denser; streamlining is critical, and a ship's top speed is the speed at which total hull resistance exactly balances the maximum force the propellers can deliver. The resistance is proportional to the square of the current velocity and to the cross-section presented, and inversely dependent on streamlining — so a current from ahead meets far less resistance than the same current on the beam.

Since current is the movement of the water, the ship is normally carried along with it. A handy way to predict behaviour is to treat the water as still and imagine the stationary objects moving at the current's speed in the opposite direction — satisfactory for a steady current, though near docks and buoys the current is not steady (covered later). A current relative to the ship cannot be other than from ahead or astern unless an external force — mooring lines or ground tackle — restrains her.

The handler must always add the current vector to the vector of the ship's motion through the water to find true movement over the ground. As speed through the water changes, the current velocity stays generally constant and must be compensated; and broadside to a current the ship will be swept along unless restrained by external means — in which case the forces required are relatively large.

11. Summary

We have examined the forces that bear on a ship through her environment and her means of propulsion and control, and the reasons they exist, so as to estimate them for a given ship in a given situation. The handler must stay alert for their evidence — watching bunting and rigging for the relative wind, and the water's surface for the true wind and current — so that he can compensate for the harmful effects and exploit the helpful ones.

A thorough understanding of all the forces that can act on the ship is the cornerstone of shiphandling ability: unless the handler understands how they act and how they can be controlled, he cannot hope to handle his ship efficiently.