Ch.V — Resistance · PNA Vol. II

1. Overview & why it matters (§1)

Chapter V of Lewis Vol. II treats ship resistance — the force required to tow the hull at a given speed in smooth water. It is one of the two pillars (with propulsion, Ch. VI) on which the powering of a ship is built. For the PSCPP candidate, the chapter is the quantitative bridge between Crenshaw\'s qualitative "Forces Affecting the Ship" and the operational handling of resistance changes on board: shallow-water squat, crosswind drift, fouling penalties, appendage losses.

The edital scopes four sections:

• §1 Introduction — definitions and the four-component decomposition.
• §3 Frictional Resistance — Froude\'s planks, Reynolds, Schoenherr, ITTC, form factor.
• §4 Wave-Making Resistance — Kelvin pattern, Michell\'s theory, interference.
• §5 Other Components — eddy, air/wind, appendages, trim, shallow water, leeway.

Sections 2 (Dimensional Analysis) and 6–9 (model tests, hull form, high-speed craft) are out of edital scope and not treated here.

Source: LEWIS, Edward V. Principles of Naval Architecture, 3rd ed., Vol. II. Jersey City: SNAME, 1988–1989. Chapter V — Resistance, Sections 1, 3, 4, 5.

Exam scope: Anexo 2-B, item I.2 (Manobrabilidade do Navio / Ship Manoeuverability). Anexo 2-A: Manobrabilidade.

2. The four components of resistance (§1)

The naval architect\'s task is to ensure that, within the limits of other design requirements, the hull form and propulsion arrangement are the most efficient in the hydrodynamic sense. The contractual benchmark is the ship\'s ability to achieve a specified speed at a specified power in smooth water on trial — i.e. delivering the required effective power. To attack the problem, the total calm-water resistance — for a bare hull — is split into four conventional components.

The "residuary" packaging

Components (b) wave-making and (c) eddy/separation are commonly lumped as "residuary resistance" because Froude\'s extrapolation method computes them together as the difference between total and equivalent-plank friction.

Why model tests dominate practice

The components interact in complicated ways. Most practical knowledge of ship resistance comes from small-scale model tests in towing tanks. Theory (especially of wave-making) is invaluable for interpreting model results and guiding model research — but cannot yet replace experiments for design-quality predictions.

3. Submerged body & d'Alembert\'s paradox (§1)

The simplest possible case: a streamlined body of revolution moving in a straight horizontal line at constant speed, deeply immersed in an unlimited ocean.

3.1 Perfect-fluid analysis

If the fluid is assumed inviscid (a "perfect" fluid), there is no friction and no eddy-making. With no free surface, no waves form. The pressure distribution around the body is purely from potential flow:

• Near the nose: pressure is increased above hydrostatic.
• Along the middle: pressure is decreased below hydrostatic.
• At the stern: pressure is again increased.
• By Bernoulli\'s law, the velocity distribution is the inverse of the pressure distribution.

Pressure forces act normal to the hull everywhere. Over the forebody they have components against motion; over the afterbody they assist motion. The resultant force is exactly zero.

3.2 Real-fluid corrections

In a real fluid, the body experiences three additional sources of drag:

• Frictional resistance — tangential viscous shear on the wetted surface.
• Form drag / viscous pressure drag — the boundary layer alters the virtual shape and length of the stern, reducing the pressure recovery there.
• Eddy / separation drag — when curvature or attitude is too abrupt, flow detaches and eddies form in the wake.

3.3 The boundary layer

Fluid in contact with the hull moves with it; momentum diffuses outward into adjacent layers. The result is a thin region of disturbed flow — the boundary layer — that grows from the bow to the stern. By convention, the boundary-layer thickness is the distance from the hull at which the local velocity reaches 99 % of the potential-flow velocity outside the layer.

The boundary layer carries away momentum continuously, drawing energy from the body. In wind-tunnel work, measuring the velocity profile in the wake behind a streamlined model is a standard way to measure frictional drag — a method also used in ship tanks via pitot surveys.

4. Surface ship — wave-making preview (§1)

A surface ship suffers all the same viscous effects as a submerged body, plus a fundamentally new component: wave-making resistance.

4.1 Pressure pattern modified by the free surface

The hull still creates the underlying high-low-high pressure pattern of the submerged-body case. But the presence of the free surface modifies it:

• Greater pressure at the bow — produces the visible bow-wave build-up.
• Smaller pressure increase at the stern (in/just below the surface) than around a submerged body.
• The resulting added resistance corresponds to the drain of energy into the wave system, which spreads outward astern and has to be continuously recreated.

4.2 Wave systems originate from three places

Wave systems are formed at:

• The bow — always.
• The shoulders — where the waterline changes slope appreciably.
• The stern — always.

Each origin generates both a divergent family (crests at an angle to course) and a transverse family (crests normal to course). Their interference produces the rich C_T(Fn) signature that section 4 will analyse in depth.

5. Frictional resistance — Froude\'s foundation (§3.1–3.2)

Even on smooth, new ships, friction accounts for 80–85 % of total resistance in slow-speed ships and ~50 % in high-speed ships. Any roughness — corrosion, fouling, paint defects — raises friction appreciably and the propulsive performance is also affected. Friction is therefore the largest single component of ship resistance and has been the most-studied throughout the history of naval architecture.

5.1 Froude\'s plank experiments at Torquay (1872, 1874)

Having concluded that the model–ship extrapolation problem could be solved only by splitting resistance into two components, William Froude set out to characterise the friction of smooth planks at his Torquay tank.

• Planks length range: 0.61 m (2 ft) to 15.2 m (50 ft).
• Speed range: 0.5 m/s (1.67 fps) to 4.1 m/s (13.3 fps).
• Surface finishes tested: varnish, paraffin, calico, fine/medium/coarse sand.

Froude\'s empirical formulation:

Key empirical results:

• For smooth varnish: n drops from 2.0 (2 ft plank) to 1.83 (50 ft plank).
• For rough sand: n stays at 2.0 for all lengths.
• For a given surface: f decreases with increasing length.
• For a given length: f increases with surface roughness.

5.2 The Greyhound trials (1874)

To extrapolate plank coefficients to ship length, Froude towed the HMS Greyhound — a 52.58 m (172\'6") wooden sloop with copper sheathing. Towing-tank predictions were compared with full-scale tow results.

Measured ship resistance was everywhere higher than predicted from the model, the difference being almost constant in R/V² across speeds. Froude attributed this to the copper-sheathed hull being "equivalent to smooth varnish over 2/3 of the wetted surface and to calico over the rest" — i.e. real ships have effective surface roughness somewhere between idealised smooth and rough.

6. Friction — two-dimensional formulations (§3.3–3.4)

6.1 Reynolds and the regimes (1880s)

Osborne Reynolds\' classical pipe experiment introduced dye on the centreline of a glass tube. At low velocity, the dye filament stayed parallel to the axis (laminar flow). Above a critical velocity V_c, the dye broke up into eddies and filled the tube (turbulent flow). Reynolds found that the transition was governed by the dimensionless ratio — the Reynolds number:

The critical velocity for a pipe of diameter D is approximately V_c ≈ 2000·ν/D.

6.2 Blasius (1908) — laminar flat plate

Blasius integrated across the laminar boundary layer to find the total resistance of a smooth plank:

6.3 Prandtl and von Karman (1921) — turbulent flat plate

For turbulent flow the resistance varies as V to a power slightly less than 2 (~1.83 in the Froude–Gebers data range).

6.4 Schoenherr (1932) — ATTC 1947 line

Karl Schoenherr collected most of the plank-test data then available — Froude\'s, Gebers\', 6.1 m and 9.1 m EMB planks, Kempf\'s small-plate measurements on a 76.8 m pontoon — and fitted them to the Prandtl–von Karman implicit form M/√C_F + A·log10(Rn·C_F) = constant. Setting M = 0 and A = 0.242, he arrived at the famous Schoenherr line:

Schoenherr line: 0.242/√C_F = log10(Rn·C_F). Adopted by ATTC in 1947 as the standard friction line. Smooth-surface only.

The Schoenherr coefficients apply to a perfectly smooth hull. For real ships with plate seams, welds, rivets, paint roughness, a correlation allowance C_A must be added:

ATTC default: C_A = +0.0004 for clean, new vessels, to be modified for special cases and clearly stated in the report. (SNAME 1948 ATTC resolution.)

7. Friction — ITTC & form-factor approach (§3.5–3.6)

7.1 The towing-tank conferences

The ICSTS (International Conference of Ship Tank Superintendents, founded 1932 in Europe) and the ATTC (American Towing Tank Conference) progressively standardised friction-line practice:

• 1935 — ICSTS adopts Froude\'s f-values; wetted surface = mean girth × LWL.
• 1947 — ATTC adopts the Schoenherr mean line with C_A = +0.0004 default.
• 1948 — ICSTS\'s 5th Conference (London): unanimity to depart from Froude\'s coefficients, but cannot agree on a single substitute. Skin Friction Committee created.
• 1957 — ICSTS becomes ITTC (International Towing Tank Conference). At Madrid, adopts the ITTC 1957 model–ship correlation line.
• 1963 — ITTC at London: new viscosity values v adopted; correlation allowance gets the symbol C_A.
• 1978 — ITTC Performance Committee endorses form-factor method for conventional ship forms.

7.2 ITTC 1957 correlation line

ITTC 1957 line: C_F = 0.075 / (log10 Rn − 2)². Explicitly labelled "only an interim solution to this problem for practical engineering purposes." NOT a true friction line — a correlation line for model→ship extrapolation.

7.3 Hughes\' two-dimensional line (1954)

Hughes carried out plank/pontoon resistance experiments up to 77.7 m length ($Rn \approx 3 \times 10^{8}$) covering a wide range of aspect ratios. He extrapolated to infinite aspect ratio to obtain what he considered the minimum turbulent resistance for plane smooth surfaces in two-dimensional flow:

Hughes 1954: C_F0 = 0.066 / (log10 Rn − 2.03)². Claims to be a true 2-D friction line. ITTC 1957 gives ~12 % higher C_F over the engineering range.

7.4 Granville generalisation (1977)

Granville derived from boundary-layer velocity profiles the general form C_F = a / (b·log10 Rn − c)². With a = 0.0776, b = 1.88, c = 60, he showed the 1957 ITTC line can be considered a turbulent flat-plate friction line. Above $Rn = 10^{7}$, the 1957 ITTC, ATTC 1947 and Granville lines agree closely.

7.5 The form-factor (three-dimensional) approach

Hughes proposed splitting model resistance into viscous and wave-making parts. At very low Froude number, wave-making vanishes; the residual is purely viscous and parallel to the flat-plate line. The form factor (1+k) captures the difference:

Compared with the Froude method (which transfers the whole residuary unchanged), the form-factor approach reduces only the viscous part in the transfer to ship Rn. Result: lower ship predictions, larger required C_A, and avoidance of the negative C_A values sometimes encountered with the Froude method on long welded ships.

7.6 Prohaska\'s determination

Drawback: unwanted Reynolds scale effects can intrude at very low Fn. Empirically-derived (1+k) values from hull-form correlations are sometimes used instead.

8. Wave-making — Kelvin pattern (§4.1–4.2)

Wave-making resistance is the net fore-and-aft force on the ship due to fluid pressures acting normally on the hull — exactly as frictional resistance results from tangential fluid forces. It depends on the shapes adopted for the area curve, waterlines and transverse sections.

8.1 Lord Kelvin\'s pressure-point analysis (1887, 1904)

Kelvin considered the simplest model: a single travelling pressure point on the water surface. He showed that the wave system this point generates has a characteristic geometric structure — the Kelvin wedge:

• A system of transverse waves following behind the point.
• A series of divergent waves radiating outward from the point.
• The whole pattern is contained within two straight lines from the point making angles of 19°28\' with the line of motion (each side).

8.2 Transverse waves — length formula

The transverse waves move with the ship at speed V. They therefore have the deep-water free-wave length appropriate to that speed:

Waves in the immediate vicinity of a model are a little shorter; they reach the asymptotic length L_w about two wavelengths astern.

8.3 Divergent waves — length formula

The divergent waves have a different speed along the line normal to their crests. If the line normal to a divergent crest makes angle θ with the ship\'s course, the relevant speed is V·cos θ, and the corresponding wave length is:

8.4 The real ship wave system

Three points along the hull generate Kelvin-like sub-patterns:

• Bow — most prominent. Visible large divergent waves stepping back in echelon, plus transverse waves filling the centre.
• Shoulders — only if the waterline changes slope appreciably; usually subsumed within the bow disturbance.
• Stern — separate divergent and transverse patterns, harder to distinguish.

The transverse-wave crests bend back as they approach the divergents and finally coalesce with them at the bounding lines, forming sharp cusps — the highest points in the system. Eventually well astern, the divergents become the more prominent.

9. Wave-making — theory & interference (§4.3–4.5)

9.1 Humps and hollows in C_T(Fn)

At low speeds the resistance is almost wholly viscous and the coefficient C_T = R_T / ½ρSV² decreases with Fn because friction varies less than as V². With further increase, C_T rises more and more rapidly; near Fn ≈ 0.45 (V_k/√L = 1.5), the resistance may vary as $V^{6}$.

Riding on this overall increase are humps and hollows caused by interference between the bow-system and stern-system waves. The wave length changes with V, so the relative position of crests and troughs from the two systems shifts; sometimes they reinforce (hump), sometimes cancel (hollow).

9.2 Michell\'s thin-ship theory (1898)

The pioneering analytical attack on wave-making was made by John Henry Michell — the thin-ship theory. For a slender hull moving over an inviscid free surface, he derived the velocity potential and integrated the fore-and-aft pressure components over the hull surface.

Michell's restrictive assumptions:

• (a) The fluid is non-viscous and the flow irrotational; motion specified by a velocity potential Φ satisfying boundary conditions.
• (b) The hull is narrow compared with its length, so the slope of the surface relative to the centreline plane is small.
• (c) The waves generated have heights small compared with their lengths.
• (d) The ship experiences no sinkage or trim.
• (e) Hull boundary conditions are applied to the centreline plane (not the actual hull surface), so the result is for a vanishingly thin ship.
• (f) The free-surface boundary condition is applied to the original flat, undisturbed surface, neglecting wave-distortion.

Michell\'s work was overlooked for ~50 years until rescued by Havelock (1951).

9.3 Havelock\'s energy method (1923+)

An alternative path, developed by Thomas Havelock, calculates the wave pattern far astern generated by the ship and obtains the wave-making resistance from the energy flow needed to maintain the wave system.

Both methods (Michell\'s pressure integration and Havelock\'s energy in the wake) lead to the same final mathematical expression for inviscid, incompressible flow.

9.4 Eggert\'s experiment (1939)

P.M. Eggert measured the normal pressure distribution over the ends of a model and integrated the longitudinal components. The resulting resistance agreed well with measured residual resistance after subtracting friction.

9.5 Wigley\'s wedge analysis (1931, 1942)

C.M. Wigley examined a double-wedge body with a parallel mid-body. He showed the wave profile along the hull is the sum of five components:

• (a) A symmetrical disturbance with peaks at bow and stern and a trough along the middle, dying away ahead and astern.
• (b) The bow free-wave system — beginning with a crest.
• (c) The forward-shoulder free-wave system — beginning with a trough.
• (d) The after-shoulder free-wave system — beginning with a trough.
• (e) The stern free-wave system — beginning with a crest.

For ship-shaped hulls (no sharp corners), the same 5-component structure applies, but the shoulder systems are smeared along the entrance/run rather than tied to single points.

9.6 Wigley\'s hump positions

For his wedge model, Wigley calculated humps/hollows of C_W at the following Fn = V/√(gL) values:

• First hump: Fn ≈ 0.25.
• Second hump: Fn ≈ 0.32.
• Last hump: Fn ≈ 0.45–0.5.

The C_W curve is a steady increase (∝ $V^{4}$, eq. 23) plus four oscillating curves due to interference. At very high speeds (Fn ≫ 0.5), the oscillations cancel and C_W has no further humps — but by then sinkage and trim have become significant.

Up to Fn 0.4, transverse waves drive the hump-hollow positions. Above Fn 0.4, divergent waves contribute more strongly.

9.7 Speed-dependence at extremes

• At very low Fn (~0.1): R_W varies approximately as tan²(half-entrance angle), but its absolute value is small.
• At very high Fn (> 1.0): R_W varies approximately as displacement² — shape is relatively unimportant, what matters is total displacement carried on a given length.

10. Wave-making — viscosity & scale effect (§4.6–4.8)

10.1 Viscosity\'s indirect effect on wave-making

Calculations of wave-making resistance have so far been unable to incorporate viscosity directly. Havelock (1923, 1935) and Wigley (1938) investigated viscous corrections.

One effect: the boundary layer separates the hull from the potential-flow pattern with which the theory deals. The layer grows from stem to stern; outside it, flow behaves potential. Havelock (1926) suggested treating this as a "virtual lengthening of the afterbody":

This explains why for an unsymmetrical hull (different lines forward vs aft), the theoretical R_W in a non-viscous fluid is the same in either direction, while measurements differ. With viscosity correction, the calculated values also differ.

Inui (1980) also adds two empirical coefficients in his wave-making work — one for virtual lengthening, one for the effect of viscosity on wave height.

10.2 Scale effect on C_W

The wave-making coefficient C_W is normally assumed constant between model and ship (it depends only on Fn). But viscosity makes C_W increase with model size — calculated curves are higher than measured curves, with greater oscillations.

Wigley (1962) traced the differences to three causes:

• (a) Simplifications in the math (linearisations) — errors decrease with increasing Fn.
• (b) Neglect of viscosity\'s effect on R_W — errors depend on Reynolds number, decrease with model size.
• (c) Wave-motion effect on R_F — negligible at low Fn, important above Fn 0.35 (sinkage/trim).

10.3 Practical scale-effect estimate

Wigley estimated the difference for a typical case: extrapolating from a 4.88 m model to a 121.9 m ship at Fn = 0.245. The ship resistance was underestimated by about 9 % using the standard constant-C_W assumption. The effect:

• Disappears at very low Fn.
• Disappears above Fn 0.45 (where viscosity-related errors become small).

10.4 Comparison of calculated vs. observed C_W

Direct comparison between calculated R_W and measured R_W is hard, because only total resistance R_T is measured. R_W must be extracted by subtracting estimated friction, viscous-pressure drag, and eddy-making — all subject to considerable uncertainty.

• In all cases, the humps and hollows on measured curves occur at higher Fn than on calculated curves, by 2–8 % — consistent with virtual lengthening.
• Theory overestimates the humps at Fn 0.25 and 0.32 and the intervening hollow.
• At Fn ≈ 0.5 (last hump), calculated C_W is less than measured — likely due to neglected sinkage and trim.

11. Eddy, form drag & wave-breaking (§5.1)

11.1 Form drag — three causes

The intercept C_a in Fig. 25 — the gap between the model total-resistance curve and the equivalent-plank friction line — is called residuary form-resistance. Three physical causes:

• Curvature effect — the streamlines along a shaped body are longer than along an equivalent plank, so the average velocity is higher and skin friction higher. Pressure and velocity changes are greater for fuller, stumpier forms.
• Viscous pressure drag — the boundary layer reduces the forward-thrust component of the pressure at the afterbody (where pressure was supposed to recover above hydrostatic).
• Separation resistance — if curvature near the stern is too abrupt, bilge radius too hard, after sections too U-shaped, flow leaves the hull at a separation point and the intervening space fills with eddies. Reduces total afterbody pressure further.

Young (1939) measured form-drag percentages for bodies of revolution in air; for surface ships:

• Warships: C_a = 5–15 % of C_F0_M.
• Full cargo ships: C_a up to 40 % or more.
• Optimum L/D for bodies of revolution: 5–7.

11.2 Bow wave-breaking

Especially for bluff hull forms, the flow ahead of the bow becomes irregular and complex, leading to a breaking wave. At very low Fn (below ~0.10) this does not happen — the surface at the stem simply rises by V²/2g per Bernoulli. As speed increases, the rise gives way to a breaking wave at the bow.

Wave-breaking research timeline:

• Taneda et al (1969) — flow observations on a fishery training ship and scaled model.
• Baba (1969) — coined "wave breaking" term, presented momentum-loss measurements via wake survey. Proposed a hydraulic-jump model.
• Dagan et al (1969) — theoretical study of 2-D flow past a blunt body.
• Ogilvie (1973) — analytical results for a fine wedge-shaped bow; universal curve for bow-wave shape.
• Taniguchi et al (1966) / Inui et al (1979) / Miyata et al (1980) / Kayo et al (1981) — criteria for avoiding wave-breaking based on the entrance half-angle of the waterline.

11.3 Taylor\'s instability criterion (G.I. Taylor, 1950)

Taylor showed that at a certain speed, the free surface becomes unstable and breaks when the centrifugal acceleration V²/R (R = radius of curvature of curved streamlines) exceeds a critical value:

11.4 Kayo et al (1981)

Systematic experiments on the effect of shear on the free surface led Kayo et al to conclude that bow wave-breaking can be considered flow separation at the free surface.

12. Air & wind resistance (§5.2)

12.1 True wind vs apparent wind

The true wind is the wind due to natural causes; it would exist whether or not the ship were there. Zero true wind = still air. The apparent wind is the vector sum of the true wind and the ship\'s velocity — what the ship actually feels.

Most superstructure resistance is due to eddy-making, varies with the square of speed, and Reynolds-number effects can be neglected. For a ship in still air:

12.2 Taylor\'s flat-plate analogy (1943)

D.W. Taylor proposed treating the above-water profile as a flat plate normal to flow:

• Width B (= beam of the ship).
• Height B/2.
• Drag coefficient C_D = 1.28 (wind-tunnel value).

Then:

12.3 Hughes\' systematic tests (1930, 1932)

L. Hughes towed models of the above-water hull and erections upside-down in water at different speeds and angles to simulate various relative wind strengths and directions. Three models: a tanker, a cargo ship, and an Atlantic liner.

Key findings:

• The dimensionless coefficient F / (V_r²) is independent of speed (for any given relative-wind angle ψ).
• Maximum wind force F occurs for beam wind (most area exposed).
• BUT maximum ahead component of wind resistance (Fcos ψ) occurs at ~30° off the bow, because area exposed grows faster than cos ψ shrinks.
• "Equivalent area" rule: A_T = 0.3 × A_long_hull + A_super. Main hull below weather deck has only ~0.3 the specific resistance of superstructure frontal area.

12.4 Velocity gradient (BSRA/BMT, 1960+)

Near the sea surface the wind is slowed by the boundary layer. Hughes\' tests had no gradient. Shearer et al (1960) at BSRA simulated the gradient and found K reduced by:

• 28 % for a passenger liner.
• 45 % for tanker and cargo ship (lower superstructure → more affected by lower wind speeds).

When applying gradient-measured K to a ship moving in still air, head-wind resistance is underestimated by ~40 % (tanker), ~25 % (loaded cargo), ~21 % (passenger liner).

12.5 Wilson et al (1970)

Analysed available wind data and defined a wind drag coefficient for head wind:

12.6 Effects of crosswinds, leeway, yaw

A strong beam wind makes the ship make leeway — develop a side velocity that leads to a hydrodynamic side force and additional drag. Wagner (1967) proposed a method to compute the resulting "effective wind resistance". Jorgensen et al (1966) showed leeway can also influence wake and propulsive efficiency.

Van Berlekom (1981): in a thorough analysis, found:

• The order of magnitude of direct wind force ≈ added wave resistance (similar orders).
• Leeway effect is less important than direct force.
• Wind coefficients are very dependent on frontal and lateral areas exposed.
• Yawing moment depends mainly on the position of the main superstructure.

12.7 Worked impact (Hughes 1932 — Table 3)

In a 40-knot head wind, the slow tanker loses 3.27 knots (32.7 %) of speed; the fast Atlantic liner loses only 1.73 knots (6.9 %). Streamlining the superstructures reduces these losses to 2.2 knots (22 %) and 1.2 knots (4.8 %) respectively. Streamlining saves about 30 % of ahead-wind resistance for all three ship types.

13. Added resistance in waves (§5.3)

Winds are seldom encountered without wind-generated waves, sometimes from distant storms. Such waves approaching from ahead cause appreciable added resistance, from two sources:

• Diffraction — the moving hull scatters the encountered waves; the modified pressure field on the hull produces a net resistance.
• Indirect effect — pitching and heaving caused by the waves; in beam or quartering seas, also rolling and yawing. Each motion mode adds resistance. Required rudder action against yaw also adds.

The detailed evaluation of added resistance in waves is treated in Chapter VIII, Vol. III of PNA — both model tests and theoretical methods. Out of edital scope for Vol. II Chapter V.

14. Appendage resistance (§5.4)

Principal appendages: in single-screw ships bilge keels and rudder; in multi-screw ships also bossings or open shafts and struts, and possibly two rudders. All add resistance — best determined by model experiments.

14.1 Bilge keels

The lines of flow along the bilge are measured (dye, small flags); bilge keels are aligned with them. With proper alignment:

• Additional resistance ≈ added wetted surface only: 1–3 % of main-hull resistance.

Peck (1976) decomposes bilge-keel resistance into:

• (1) Skin friction due to additional wetted surface: ½ρSV² · C_F.
• (2) Interference drag at the bilge-keel/hull junction: depends on hull-to-keel angle. At z = +y (no keel): 0. At z = 0 (flush plate keel): equal to skin friction drag. Linear interpolation in between.

14.2 Bossings (multi-screw)

MARIN tests (Lap, 1956) on twin-screw 6-m models:

• Wetted-surface increase: 1–5 %.
• Specific friction higher than main hull (bossing curvature larger).
• Total resistance increase: 5–9 % of frictional resistance.

14.3 Open shafts + struts

• Wetted-surface increase: 0.9–4.0 %.
• Well-designed struts: resistance increase 6–9 % of frictional resistance.
• Extra intermediate struts on very long open shafts: up to 16–18 % of bare-model resistance.

14.4 Rudders

Rudder resistance can be measured (model tests) or calculated (airfoil drag coefficients, appropriate Reynolds and length). Critical operational detail:

For ships with twin rudders, model tests determine the optimum zero setting — the closing-in of streamlines at the stern means this setting is not necessarily parallel to the centreline.

14.5 Drag formulas — Peck, Hoerner, Kirkman

Multiple empirical relations exist; some Reynolds-dependent, some not. The Reynolds-independent ones cannot capture scale effects.

• Peck (1976) — control surfaces: R = ½ρV² · S · [1.1·C_F + Δ_terms]. Substitute 1.1V for rudder in slipstream.
• Hoerner (1965) — spray drag (rudder/strut piercing the surface), palm drag (strut palm mounted on hull), interference drag between control surface and hull (∝ (t/c)²).
• Kirkman et al (1980) — foil-type appendages and cylinder cross-flow drag. Foil interference and cylinder pressure-and-friction drag derived from boundary-layer theory.

14.6 The Lucy Ashton geosim experiments

BSRA used the old Clyde paddle steamer Lucy Ashton (58 m, paddles removed, propelled by aircraft jet engines on deck) to measure full-scale appendage resistance directly. Geosim models (lengths 2.74 to 9.14 m) tested at NPL. Appendage resistance never exceeded 7 % of bare-hull on the geosims.

Best angle for bossings (from streamline tests): 40° to horizontal. But Lucy Ashton bossings were fitted at 20° (across the flow) — measured resistance increased by ~5 % up to 12 knots, then declined.

14.7 Mandel\'s 1953 ranges (Table 5)

Values are appendage resistance as percentage of bare-hull resistance, from model tests, no scale correction.

15. Trim effects (§5.5)

Owing to changes in pressure distribution at different speeds, the ship will rise or sink bodily and trim. At low speeds, slight settling; at Fn ≈ 0.30 or thereabouts, the bow begins to rise appreciably, the stern continues to sink, and the ship takes on decided stern trim.

15.1 Symptom vs cause

15.2 Stern trim effects

Adding stern trim in the at-rest condition typically:

• Low speeds: increases resistance — increased after draft makes the stern virtually fuller, raising form and separation drag.
• High speeds: decreases resistance — finer effective entrance, reduced wave-making.

15.3 Ballast condition

It is usually necessary to carry considerable stern trim in ballast to ensure adequate propeller immersion. Effects similar to the above: higher resistance at low speeds, less at high speeds.

At level trim in ballast, wetted-surface-per-unit-displacement is much higher, so friction is higher. But residuary is lower (finer form at reduced draft). For most ships (except very fast ones), total resistance per unit displacement is greater, but total resistance and power are lower; a ballast ship therefore makes a higher speed at the same power.

15.4 Planing craft

The reductions of resistance possible by trim changes in large displacement ships are small. In high-speed planing craft, by contrast, the position of the centre of gravity and the resultant still-water trim have a most important influence on performance.

16. Shallow-water effects (§5.6)

16.1 Two physical mechanisms

Resistance in shallow water is changed by two distinct mechanisms — both contributing to squat and to the resistance peak near the critical speed:

• Potential-flow change. A restricted depth (and possibly width) forces water passing below the hull to speed up more than in deep water — greater pressure reduction, greater sinkage, trim, and resistance.
• Wave-pattern change. The dispersion relation for surface waves in finite depth h changes the wave length and angle of the system as V/√(g·h) approaches unity.

16.2 Wave speed in finite depth

The critical speed is therefore:

Below V_c: subcritical zone (where most displacement ships operate). Above V_c: supercritical zone (destroyers, cross-channel ferries, similar types).

16.3 Kelvin angle in shallow water

Havelock (1908) studied a point pressure impulse over a free surface of finite depth:

• V < 0.4·√(g·h): wave-pattern angle stays at 19°28\' (deep-water Kelvin).
• V → √(g·h): angle α → 90°. System collapses into a single transverse crest perpendicular to motion.
• V > √(g·h): only divergent waves, no transverse. Wave pattern bounded by sin²α = g·h / V².

16.4 Schlichting\'s method (1934)

For laterally unrestricted shallow water at subcritical speeds, Schlichting estimated speed loss in two steps:

1. Wavelength correction — find V_1 in deep water producing the same wave length L_w as the ship would generate in shallow water at the deep-water speed V_∞. Schlichting assumed wave-making R is the same in both cases.
2. Potential-flow correction — further drop V_p due to depth restriction, function of √A_x / h (A_x = maximum cross-sectional area of hull).

Total shallow-water speed V_h = V_1 − V_p. Below V/√(g·h) ≈ 0.4, the wave-pattern part is negligible and all speed loss is potential-flow.

16.5 Landweber\'s extension (1939) — restricted channels

For laterally restricted channels, Landweber proposed using the hydraulic radius:

For a ship in a canal, the ship\'s own cross-section affects the hydraulic radius: R_H = (A_canal − A_x) / (wetted perimeter of channel + wetted girth of hull at that section).

16.6 Onset of detectable shallow-water effect

Speed at which power begins to increase appreciably (from observations on HMS Cossack at Watts 1909):

16.7 Practical DTRC data (Norley 1948)

Norley tested model resistance/propulsion at DTRC on Liberty, Victory, T-2 ocean tanker, T-1 inland tanker. Generalised charts show sinkage at bow > sinkage at stern, increasing with speed and decreasing with depth.

For these models, the safe depth/draft ratios are:

16.8 High-speed ship signature

A destroyer in shallow water (Rota): as speed approaches V_c, trim by stern and resistance both rise very rapidly; pass V_c and both fall quickly, with resistance remaining nearly constant before rising again at a lower rate than in deep water. At V/√(g·L) ≈ 0.48, both trim and resistance curves cross those for deep water; above this, the shallow-water curves are below deep-water (less resistance!).

This characteristic is well borne out in full-scale trials (HMS Cossack 1909): bow wave grew rapidly up to ~22 knots, became unstable and fell to about half its maximum height; at 28 knots the stern wave had practically disappeared.

17. Leeway and heel — sailing yachts (§5.7)

Resistance increases when the ship has a fixed yaw and/or heel angle, particularly marked in sailing ships and yachts sailing to windward. Steady yaw arises when the vessel counteracts an aerodynamic side force (on sails or superstructure) with an equal-and-opposite hydrodynamic side force on hull + keel.

17.1 Typical angles

• Yacht leeway: 3°–6° (with efficient keels). Larger if keels are inefficient.
• Heel: up to 30° for yachts sailing to windward.
• For ordinary ships with asymmetric properties or mis-aligned shaft brackets: leeway ≤ 1–2°, induced by hydrodynamic moment from rudder counterforce.

17.2 Induced drag

Lift is generated by deflecting flow over angle α_i downward (sideways) from its undisturbed direction. The force generated is at right angles to the deflected flow; the component along the original motion direction — F · sin(α_i) — is the induced drag R_i.

For small α_i (sin α_i ≈ α_i) and elliptical span loading:

The induced resistance of a sailing yacht ≈ sum of induced resistances of hull, keel, and rudder.

17.3 Plan-form effects

• Elliptical loading minimises induced drag.
• Taper ratio (lower/upper chord ratio) 0.3–0.4 ≈ elliptical loading; gives only 1–2 % additional induced drag.
• Sweep-back redistributes loading toward the tip; increases induced drag ∝ 1/cos(sweep angle). For 30° sweep, taper ratio ~0.15 needed for near-elliptical loading.
• Rectangular plan forms often have highest lift-to-drag ratios despite higher induced drag (because they produce more lift).

17.4 Heel asymmetry and wave-making

The immersed hull of a heeled ship is asymmetric, the lee side being bluffer. This:

• Increases wave-making resistance (asymmetric flow).
• In long-overhang yachts: partly compensated by the increase in effective wave-making length as the hull heels.
• For yacht J-class Rainbow tested at MARIN: resistance increase with heel is marginal at 6–9 knots due to wave-making-length increase.

17.5 Large-yaw separation

At yaw angles > ~5°, the flow along the aftbody usually separates, and resistance increases markedly. Even on sailing yachts with block coefficients around 0.4, this occurs because the flow on the windward side forward of the rudder separates.