Naval Architecture and Ocean Engineering I: Hydrostatics, Stability, Resistance and Propulsion

A ship is described by its main particulars: the length between perpendiculars L_BP (from the forward perpendicular at the stem to the after perpendicular, usually at the rudder stock), the length on the waterline, the moulded breadth B, the moulded depth D and the draught T; freeboard is D − T. The hull is drawn as a lines plan — body plan (transverse sections), half-breadth plan (waterlines) and profile (buttocks) — and tabulated as a table of offsets, the half-breadths at stations and waterlines. A hull that is streamlined, with a fine entrance, a gradual run aft and no abrupt changes of section, keeps the flow attached to the stern, lowers the viscous pressure resistance and delivers an even wake to the propeller.

Archimedes’ principle and the laws of flotation: a floating ship displaces a mass of water equal to its own mass, Δ = ρ∇, and it floats in equilibrium only when the buoyancy through the centre of buoyancy B and the weight through the centre of gravity G act in the same vertical line. The fullness of the underwater body is expressed by the form coefficients: the block coefficient C_B = ∇/(LBT), the midship-section coefficient C_M = A_M/(BT), the prismatic coefficient C_P = ∇/(A_M L) = C_B/C_M and the waterplane coefficient C_WP = A_W/(LB). A ship 100 m × 16 m × 6 m with C_B = 0.70 displaces ∇ = 0.70 × 100 × 16 × 6 = 6720 m³, that is Δ = 1.025 × 6720 = 6888 t in sea water. A tanker has C_B above 0.8, a fast container ship about 0.6 and a warship nearer 0.5.

Hydrostatic calculations integrate the offsets numerically, most often by Simpson’s first rule, ∫y dx ≈ (h/3)(y₀ + 4y₁ + 2y₂ + 4y₃ + … + y_n) for an even number of intervals. A waterplane with half-breadths 0, 4, 6, 4, 0 m at stations 10 m apart has a half-area of (10/3)(0 + 16 + 12 + 16 + 0) = 146.7 m² and an area A_W = 293.3 m²; the same rule on sectional areas gives the volume, and on moments gives the centroids. The results are drawn as hydrostatic curves against draught: displacement, KB, KM, LCB, LCF, TPC and MCT1cm. The tonnes per centimetre immersion TPC = ρA_W/100 (ρ in t/m³) is the mass that sinks the ship 1 cm in parallel; the moment to change trim by one centimetre MCT1cm = Δ·GM_L/(100L), in t·m per cm, turns a trimming moment into trim.

When a ship heels through a small angle φ, the centre of buoyancy moves towards the immersed side and the new buoyancy line meets the centre line at the transverse metacentre M. The righting lever is GZ = GM sin φ, and the equilibrium is stable if M is above G (GM > 0), neutral if they coincide and unstable if M is below G. The metacentric height is built up from the keel: GM = KB + BM − KG, with BM = I_T/∇, I_T being the second moment of the waterplane about the centre line. For a box barge of breadth B and draught T, KB = T/2 and BM = B²/(12T); with B = 12 m, T = 4 m and KG = 4.5 m, GM = 2 + 3 − 4.5 = 0.5 m. A large GM gives a stiff ship with a short, uncomfortable roll; a small GM a tender ship with a long, easy roll and little reserve.

Shift of G. Adding a mass w at height Kg moves G to KG₁ = (Δ·KG + w·Kg)/(Δ + w), towards the added mass; removing it moves G away. A mass shifted a distance d moves G parallel to the shift by GG₁ = w d/Δ; shifted transversely, it heels the ship to tan φ = w d/(Δ·GM). A suspended mass acts as if it were at its point of suspension, because as soon as the ship heels it swings out to hang below that point: lifting a cargo off the deck with the ship’s own derrick raises its effective centre instantly to the derrick head, which is why heavy lifts are a stability case in their own right.

The free-surface effect: a slack tank whose liquid surface is free to move shifts its liquid to the low side as the ship heels, which reduces the righting lever as though G had risen by the free-surface correction FSC = ρ_l i/(ρ_sw ∇) = ρ_l i/Δ, where i = l b³/12 is the second moment of the tank’s free surface about its own centre line and ρ_l the density of the tank liquid. The correction depends on the tank’s breadth cubed and not on how much liquid it holds, and a longitudinal division into two halves reduces it to a quarter. An oil tank 10 m long and 8 m wide (i = 426.7 m⁴, ρ_l = 0.9 t/m³) in a ship of 6888 t costs 0.9 × 426.7/6888 = 0.056 m of GM; the fluid GM is GM − FSC.

The inclining experiment finds the KG of a completed ship, which no calculation can give reliably. With the ship upright, free of free surfaces, moorings slack and all weights accounted for, known masses w are moved across the deck a distance d and the heel is read from long pendulums (deflection a over length l, tan φ = a/l) or a U-tube; then GM = w d/(Δ tan φ), and KG = KM − GM with KM from the hydrostatics. Moving 10 t through 8 m in a 5000 t ship, with a 0.12 m deflection on a 6 m pendulum (tan φ = 0.02), gives GM = 80/(5000 × 0.02) = 0.8 m. The lightship KG found this way is the starting point of every loading condition.

Beyond about 10° the metacentre no longer stays fixed and GZ must be computed from the actual inclined waterplanes (by cross curves, the KN curves, with GZ = KN − KG sin φ). For a wall-sided ship, whose sides are vertical over the range of immersion, the exact result is GZ = sin φ (GM + ½BM tan²φ); with GM = 1.0 m, BM = 4 m and φ = 20°, GZ = 0.342 × (1 + 2 × 0.1325) = 0.433 m. The curve of statical stability (GZ against φ) is read for its initial slope (equal to GM per radian), its maximum GZ and the angle at which it occurs, the angle of vanishing stability where GZ returns to zero, and the range of stability. The area under it up to an angle is the dynamical stability, the work Δ∫GZ dφ needed to heel the ship to that angle, and it is what resists a sudden gust or a wave impact.

A ship with a small negative GM is not necessarily capsized: as it heels, the wall-sided term raises GZ until, at the angle of loll tan φ = √(2|GM|/BM), the lever is zero again and the ship lies there, flopping from one side to the other. With GM = −0.1 m and BM = 5 m, tan φ = √(0.04) = 0.2 and φ = 11.3°. Loll is distinguished from a list caused by an off-centre weight (which has G off the centre line and a positive GM) by the response: the cure for loll is to lower G — by filling low tanks one at a time, the smaller and lower first, never by moving weight across, which would throw a lolling ship over to the other side.

Damage stability asks whether the ship survives flooding. The deterministic approach floods a prescribed number of adjacent compartments and requires the damaged waterline to stay below the margin line (76 mm below the bulkhead deck) with a minimum residual GM and GZ; flooding is computed by lost buoyancy (the damaged volume is removed and the ship sinks until the intact part supports it; Δ and KG unchanged) or by added weight (the flood water is a mass added; both give the same final waterline). A compartment’s permeability μ is the fraction of its volume that water can fill — about 0.95 for accommodation, 0.85 for machinery spaces and lower for cargo. The floodable length curve gives, at each point along the ship, the greatest length centred there that can flood without submerging the margin line; multiplied by the factor of subdivision it gives the permissible length of a compartment. The probabilistic approach sums, over every damage case, the probability of that damage times the probability of surviving it, into an attained subdivision index A, which must be at least the required index R.

Grounding and docking. When a ship touches the blocks in dry dock or takes the ground, an upward force P acts at the keel. It can be treated as a mass P removed at the keel, which raises G by GG₁ = P·KG/(Δ − P), or as a fall of the metacentre by MM₁ = P·KM/Δ; the second, simpler form gives the virtual loss of GM. With P = 100 t, KM = 8 m and Δ = 5000 t the loss is 0.16 m. P grows as the water level falls, so the critical instant is just before the ship takes the blocks along its whole length; a ship with too small a GM, or trimmed heavily by the stern so that P is large before she settles, can fall over on the blocks. The same reasoning applies to a ship aground on a falling tide.

The calm-water resistance of a ship is divided into frictional resistance, the tangential shear on the wetted surface, which dominates slow full ships; viscous pressure (form) resistance, from the thickened boundary layer and any separation at the stern; wave-making resistance, the energy carried away by the ship’s wave system, which dominates at high Froude number and oscillates with humps and hollows as the bow and stern waves interfere; plus air resistance of the above-water body and appendage resistance of rudders, bilge keels, shaft brackets and stabilisers. In a seaway the ship also suffers added resistance in waves, and hull roughness and fouling add to the friction as the ship ages. The form factor k expresses viscous resistance as (1 + k) times the flat-plate friction, and is found from low-speed model tests where wave resistance is negligible (the Prohaska method).

A model cannot match both the Froude number Fn = V/√(gL) and the Reynolds number Re = VL/ν, so Froude’s method runs the model at the corresponding speed V_m = V_s/√λ (equal Fn) and splits the total resistance coefficient C_T = R_T/(½ρSV²) into a frictional part that depends on Re and a residuary part C_R that depends on Fn and is the same for model and ship. The friction line now used is the ITTC-1957 model–ship correlation line, C_F = 0.075/(log₁₀Re − 2)². The steps: C_R = C_Tm − C_Fm at the model’s Re; C_Ts = C_R + C_Fs at the ship’s Re (plus a correlation allowance, where one is used); R_Ts = ½ρ_s S_s V_s² C_Ts, with S_s = λ²S_m; and the effective power P_E = R_T·V, the power needed to tow the bare hull. The 1978 ITTC method refines this with the form factor, scaling (1 + k)C_F rather than C_F alone.

Tank-wall (blockage) effects: in a towing tank of finite breadth and depth the model’s flow is constrained, the water past it speeds up and the measured resistance is too high unless the model is small against the tank section or a blockage correction is applied; shallow water raises resistance for the same reason. Where no model is tested, resistance is estimated from methodical series — families of systematically varied hull forms such as the Taylor standard series and Series 60, presented as C_R against Fn for each C_P and B/T — or from regression methods fitted to many test results, such as Holtrop–Mennen. Advanced marine vehicles escape the displacement-hull wave barrier by lifting the hull out: planing craft ride on dynamic pressure at high Fn, hydrofoil craft lift the hull clear on foils, air-cushion vehicles and surface-effect ships on a cushion of air, and SWATH and catamaran forms trade wave resistance for more wetted surface.

Geometry. A screw propeller is defined by its diameter D, number of blades Z, pitch P (the advance per revolution of the blade’s helical face, often quoted as P/D at 0.7R), the blade area ratio (expanded blade area over disc area), and the rake, skew, section shape and boss diameter. Theories: the momentum (actuator-disc) theory treats the propeller as a disc that accelerates the flow and gives the ideal efficiency η_i = 2/(1 + √(1 + C_T)), C_T = T/(½ρA V_A²) — so a lightly loaded, large-diameter propeller is the more efficient; blade-element theory integrates the lift and drag of each blade section; and lifting-line and lifting-surface theories model the blades as bound vortices shedding helical trailing vortices, which is how modern propellers are designed.

Open-water characteristics. Tested alone in uniform flow (the open-water test), a propeller of diameter D turning at n rev/s and advancing at V_A is described by the advance coefficient J = V_A/(nD), the thrust coefficient K_T = T/(ρn²D⁴), the torque coefficient K_Q = Q/(ρn²D⁵) and the open-water efficiency η₀ = TV_A/(2πnQ) = (J/2π)(K_T/K_Q). A propeller of 5 m diameter at 2 rev/s advancing at 6 m/s has J = 0.6; with K_T = 0.2 and K_Q = 0.03, η₀ = 0.6 × 0.2/(2π × 0.03) = 0.64, and the thrust is 0.2 × 1025 × 2² × 5⁴ = 512.5 kN. K_T and K_Q fall as J rises, and η₀ peaks just before K_T reaches zero.

Hull–propeller interaction. Behind the hull the propeller works in the ship’s wake, so the water reaches it at V_A = V(1 − w), w being the Taylor wake fraction; and its suction on the stern raises the hull’s resistance, so it must deliver a thrust T = R_T/(1 − t), t being the thrust-deduction fraction. The hull efficiency η_H = (1 − t)/(1 − w) is often above 1 — with t = 0.16 and w = 0.20 it is 0.84/0.80 = 1.05 — because the propeller recovers energy from the wake. The relative rotative efficiency η_R corrects the open-water torque for the non-uniform flow behind the hull. The propulsive efficiency (quasi-propulsive coefficient) is η_D = P_E/P_D = η_H η₀ η_R, and the shaft and brake powers follow by dividing by the shaft transmission and gearing efficiencies. The self-propulsion test tows a model with its own propeller running at the ship self-propulsion point to measure w, t and η_R; a resistance of 300 kN at 15 knots (7.716 m/s) needs P_E = 2315 kW and, with η_D = 0.70, a delivered power of 3307 kW.

Propeller cavitation occurs where blade pressure falls to the vapour pressure: sheet cavitation on the suction back near the leading edge, bubble and cloud cavitation further aft, tip-vortex and hub-vortex cavitation in the trailing vortices, and face cavitation on the pressure side at negative angle. Its effects are erosion of the blades, noise, vibration and hull pressure pulses, and, when extensive, a breakdown of thrust. It is avoided by giving enough blade area (the Burrill chart relates the thrust loading of the projected area to the local cavitation number), by skew, by unloading the tip, and by tests in a cavitation tunnel at the ship’s σ. Propellers are selected from series such as the Wageningen B-series, whose K_T–K_Q charts and B_p–δ diagrams give the optimum diameter and pitch for a given power and speed.

Materials, strength and manufacture. Most large propellers are cast in nickel–aluminium bronze or manganese bronze, which resist corrosion and cavitation erosion and cast well; stainless steel serves ice-class and some small propellers, and composites are used for quietness. The blade is checked as a cantilever from the root under the thrust and torque loads, with centrifugal and fatigue loads in the wake’s fluctuating inflow, and classification rules set a minimum root thickness. It is cast in a mould, machined or ground to the design pitch and section, finished to a specified surface class, and statically balanced before fitting.