Hydrodynamics

A hull moving through water fights two physically different things, and the exam wants you to know why they are kept apart rather than lumped into one number. The first is skin friction: water is viscous, it sticks to the hull surface, a boundary layer forms, and shearing that layer costs energy in rough proportion to wetted area and to roughly the square of speed. The second is wave-making — the hull has to push water aside and lift it into the bow and stern wave systems it drags along with it, a process governed by gravity and by how the ship's speed compares with the speed those self-generated waves travel at. A smaller share, eddy resistance from flow separation around bilges, transoms and appendages, is conventionally folded into the second group as well.

The reason this split matters is that the two components do not scale the same way between a small model and a full-size ship. Friction depends on the Reynolds number — a viscosity effect. Wave-making depends on the Froude number — a gravity-wave effect. A model in the same water as the ship cannot be run at both the ship's Reynolds number and its Froude number at once; the geometry fixes one ratio and the fluid fixes the other, and they pull in opposite directions as the model gets smaller. That single fact is the reason a model test cannot simply be read off and scaled up directly, and it is the entire justification for the extrapolation procedure in the next section.

Since a model cannot match both governing numbers at once, the standard procedure deliberately gives up on Reynolds similarity and keeps Froude similarity: the model is towed at the speed that gives it the same Froude number as the ship — the "corresponding speed". Total resistance is measured directly at that speed. Friction resistance is not measured at all; it is calculated, for the model, from a standard friction line — the ITTC 1957 formula — evaluated at the model's own, fairly low, Reynolds number. Subtracting that calculated friction from the measured total leaves the residuary coefficient C_R, and Froude's hypothesis is the claim that this residuary coefficient, at equal Froude number, is the same for model and ship regardless of scale. That coefficient is transferred unchanged.

Ship friction is then calculated afresh from the same line but at the ship's own Reynolds number, which is typically two to three orders of magnitude higher than the model's. Because the friction line falls with Reynolds number, this means the ship's friction coefficient is noticeably lower than the model's — the ship, despite being far larger, is relatively "slippier" in the friction sense once nondimensionalised. A correlation allowance C_A is then added on the ship side only, to cover full-scale effects a smooth, clean model can never represent — hull roughness, paint, and minor appendage drag not otherwise accounted for.

Corresponding speed follows directly from requiring equal Froude number at a fixed geometric scale ratio λ = L_S/L_M. Since F_n = V/√(gL) and g is the same for both, equal F_n forces V_S/V_M = √(L_S/L_M) = √λ — a single number once the scale is fixed, independent of the ship's actual speed. This is worth internalising as a ratio, not a formula to look up each time: a 1:25 model always runs at one-fifth of the ship's corresponding speed, whatever that speed happens to be.

The area and volume scalings follow from plain geometric similarity — every linear dimension of the ship is λ times the corresponding model dimension, so every area is λ² times larger and every volume λ³ times larger. This is where a surprising number of marks are lost: it is tempting, when converting a wetted surface area from model to ship, to scale it by λ instead of λ² — treating an area as if it were a length. The error understates ship wetted surface by a full factor of λ, which quietly ripples through every friction and coefficient calculation that follows it.

To compare propellers fairly regardless of their size or the rpm they happen to be tested at, thrust, torque and inflow speed are all made nondimensional. The advance coefficient J relates the propeller's forward travel per revolution to its diameter; the thrust and torque coefficients K_T and K_Q do the same for the forces it produces. All three use n in revolutions per second, not rpm, and all three use the speed of advance V_A — the inflow speed the propeller actually sees — not the ship's speed through the water.

An open-water test runs the propeller alone, with no hull ahead of it, across a range of J by varying inflow speed against a fixed rpm (or vice versa), plotting K_T, K_Q and the derived open-water efficiency η_O against J. The resulting curve is a signature of that propeller's geometry — pitch ratio, blade area ratio, number of blades — and both K_T and K_Q fall roughly linearly as J rises from zero (heavy loading, as at bollard pull) toward the lightly loaded end. One useful identity worth carrying into calculations: K_T/J² = T/(ρD²V_A²) has n cancelled out of it entirely, so the operating point on the curve can be located from thrust, diameter and advance speed alone, before the rpm is even known.

A propeller working behind a hull does not see the ship's speed through the water — the hull's own boundary layer and pressure field drag a layer of water along with it, so the propeller advances into water already moving partly in the ship's direction. The wake fraction w captures this: V_A = V_S(1 − w), and for most single-screw merchant hulls w is a meaningful fraction, not a rounding correction. Running the other way, a working propeller accelerates water forward of itself and lowers pressure over the stern, which shows up as extra resistance on the hull — so the thrust the propeller must produce exceeds the bare-hull resistance by the thrust deduction fraction t: T = R_T/(1 − t).

These two interaction effects combine into hull efficiency η_H, which multiplies with open-water efficiency η_O and relative rotative efficiency η_R — a small correction, usually within a few percent of unity, for the propeller working in the hull's non-uniform wake rather than the smooth inflow of an open-water test — to give overall propulsive efficiency η_D. From there the power chain runs outward: effective power P_E = R_T·V_S is the towrope power, the power needed to drag the bare hull with no propulsion device fitted at all; dividing by η_D gives delivered power P_D, what the propeller itself must be supplied; and a further division by shaft and bearing efficiency gives brake power, what the engine must produce. Knowing exactly which of these a question is asking for, and which direction the chain runs, decides whether the final number is right.

Cavitation is fundamentally a pressure problem, not a shape problem. As a blade section moves through the water it accelerates flow over its back (suction side), and by Bernoulli's principle the local pressure there drops. If it drops far enough to reach the vapour pressure of the water at that temperature, the water itself flashes into vapour-filled cavities, which are then swept into a region of higher pressure downstream and collapse violently — the mechanism behind blade erosion, noise, vibration and, at worst, thrust breakdown. The risk is highest where the local flow speed is greatest (further out on the blade, where rotational speed dominates) and where the static pressure available is least (nearer the surface).

Because static pressure falls with reduced depth, the blade passing through the top of its circular sweep — shallowest, closest to the free surface — is the governing case for a cavitation check, not the shaft centreline that a first pass calculation naturally reaches for. The remedy for a marginal cavitation number is, counter-intuitively to some, not primarily a change of blade shape or pitch but of blade area: spreading the same thrust over a larger expanded blade area lowers the local pressure drop needed to generate it, and this loading-versus-area trade-off is exactly what Burrill-type diagrams and similar cavitation-margin checks are built to assess once a propeller has been provisionally selected on efficiency grounds alone.

The three examples below carry the extrapolation procedure, the propeller open-water diagram and a cavitation-margin check through in full, the way each would actually be worked on the exam paper.