Ship resistance and powering

Background and history

Systematic study of ship resistance began in Britain during the mid-nineteenth century. William Froude (1810-1879) conducted the first controlled towing-tank experiments at Torquay from 1868, and later at the purpose-built Admiralty tank at Haslar from 1872. His central insight, published in 1874 and known as Froude’s hypothesis, was that total resistance RT could be decomposed into two independent components: frictional resistance Rf, governed by the wetted surface area and the viscous character of the flow, and residuary resistance Rr, containing wave-making and the remaining pressure effects, which scaled with hull geometry and speed-length ratio. The practical value of this decomposition was that frictional resistance could be estimated analytically from flat-plate friction data, allowing residuary resistance to be isolated from model tests and extrapolated to ship scale using Froude’s law of similitude.

Froude’s similitude law states that residuary resistance scales when the Froude number Fn = V / √(g × L) is held constant between model and ship, where V is speed in m/s, g is gravitational acceleration (9.81 m/s²), and L is waterline length in metres. Simultaneously, viscous frictional resistance scales with the Reynolds number Re = V × L / ν, where ν is the kinematic viscosity of water (approximately 1.139 × 10⁻⁶ m²/s in seawater at 15°C). Because a model and ship at matched Froude number will have very different Reynolds numbers, the two similarity requirements cannot be satisfied simultaneously at model scale - this is the fundamental scaling problem in experimental naval architecture.

Osborne Reynolds (1842-1912) and William Froude’s son Robert Edmund Froude further refined the frictional resistance correlation in the 1880s. The Froude friction line, expressed as the frictional resistance coefficient Cf as a function of Re, was widely used in British practice for decades. In 1932, K.E. Schoenherr of the United States Navy proposed a revised flat-plate friction formula that became standard in North American practice, given in its implicit form; see the Schoenherr friction coefficient formula page for the full expression.

The critical modern watershed came at the eighth International Towing Tank Conference (ITTC) in Madrid in 1957, which adopted a new standard friction line for model-to-ship extrapolation, the ITTC 1957 model-ship correlation line. This line, expressed as Cf0 = 0.075 / (log₁₀ Re − 2)², replaced the earlier Froude and Schoenherr lines in international practice and remains the foundation of the standard ITTC-78 performance prediction method today. The ITTC friction line calculator implements this formula directly.

G. Hughes (1954) proposed the “form factor” concept to account for the three-dimensional viscous pressure drag of a real hull, distinguishing between the friction of an equivalent flat plate (Cf0) and the total viscous resistance of the actual hull form. The form factor k modifies the frictional term so that total viscous resistance equals (1 + k) × Cf0 × ½ρ × S × V², where ρ is water density and S is wetted surface area.

The Holtrop-Mennen regression method, published by Jan Holtrop and G.G.J. Mennen of MARIN in 1982 and updated in 1984, systematised the prediction of ship resistance from hull form coefficients using a statistical database of model tests. This method remains the most widely used analytical tool for preliminary design resistance prediction. The full Holtrop-Mennen calculation suite is available at ShipCalculators.com through calculators for form factor (k1), wave resistance, and appendage resistance, with the combined method in the Holtrop-Mennen resistance calculator.

Components of total resistance

The ITTC 1957 decomposition

The standard ITTC-78 decomposition expresses total calm-water resistance RT as:

RT = (1 + k) × Rf0 + Rw + Ra + Raa

where Rf0 is the frictional resistance of an equivalent flat plate at the same Reynolds number, (1 + k) is the form factor that inflates this to account for hull three-dimensionality, Rw is wave-making resistance, Ra is the roughness allowance (model-ship correlation allowance), and Raa is air resistance. Each component has a distinct physical origin, a different dependence on speed, and a different sensitivity to hull geometry.

Frictional resistance

Frictional resistance arises from the tangential viscous stresses exerted by water on the wetted hull surface. Its magnitude depends on wetted surface area S, ship speed V, Reynolds number Re, and the roughness of the hull surface. The ITTC 1957 friction coefficient Cf0 = 0.075 / (log₁₀ Re − 2)² gives the flat-plate baseline; the form factor (1 + k) applied on top accounts for the extra viscous pressure drag due to hull curvature.

For a typical 200 m Panamax vessel at 14 knots, Re is of order 2 × 10⁹, Cf0 is approximately 1.4 × 10⁻³, and frictional resistance represents roughly 70 to 80% of total calm-water resistance. At lower speeds this fraction rises; at high Froude numbers (Fn > 0.30) wave-making resistance grows steeply and frictional resistance becomes the smaller share.

Wetted surface area is a primary design variable. The Mumford formula S ≈ L × (1.7 × T + B × Cb), where T is draught, B is beam, and Cb is block coefficient, provides a first-order estimate from principal dimensions alone. The wetted surface area calculator applies this formula directly.

Wave-making resistance

Wave-making resistance Rw arises from the energy carried away by the wave system generated at the hull-water interface. A ship moving through calm water creates a Kelvin wave system - a pattern of transverse and divergent waves contained within a half-angle of approximately 19°28’ regardless of ship speed (for deep water). The energy radiated in this wave system is supplied by the ship’s propulsion, appearing as a resistance force.

The dimensionless wave resistance coefficient Cw = Rw / (½ρ × S × V²) is a strong function of Froude number Fn. At Fn ≈ 0.20 to 0.25 a pronounced hump in Cw appears, corresponding to the condition where the transverse bow wave and stern wave are approximately one ship-length apart and reinforce each other. Further humps appear near Fn ≈ 0.32 and Fn ≈ 0.50; hollows in Cw at intermediate Froude numbers represent destructive interference of bow and stern wave systems.

Most displacement ships operate at Fn between 0.15 and 0.30. A bulk carrier at 15 knots on a 200 m hull has Fn ≈ 0.30, placing it near the primary hump - the reason a speed increase beyond the design speed produces a disproportionate resistance penalty. Container ships at 22 knots on a 330 m hull operate near Fn ≈ 0.22, where wave-making is already appreciable. The steep rise of Rw with Fn is the physical basis of slow steaming as a fuel-saving strategy: a 10% speed reduction from 22 to 19.8 knots reduces wave-making resistance far more than proportionally, yielding a disproportionately large fuel saving.

The wave-making resistance component from the Holtrop-Mennen method accounts for the effects of hull fullness (block coefficient Cb), prismatic coefficient Cp, longitudinal centre of buoyancy position, and half-angle of waterplane entry on wave generation. The wave steepness calculator provides a complementary tool for assessing the limiting steepness of generated waves.

Bulbous bow effects

A bulbous bow introduces an additional wave source at the forward end of the hull that, at the design Froude number, generates waves of opposite phase to the main bow wave system, thereby reducing total wave-making resistance. The cancellation is tuned to a specific speed and draught: at off-design conditions the bulb may actually increase resistance. Bulbous bows are effective on vessels operating at Fn between approximately 0.18 and 0.32, encompassing most large merchant ships. The cross-sectional area, protrusion length, and vertical position of the bulb relative to the waterline are the primary design variables, with optimisation typically performed in a towing tank or by CFD.

Roughness allowance and correlation allowance

The roughness allowance Ra accounts for the fact that a real ship hull is rougher than the smooth flat plate used to derive Cf0. The ITTC-78 procedure uses a standard added resistance coefficient ΔCf to bridge model test results (smooth model) to full-scale ship (rough hull) performance. The hull roughness ΔCf formula relates the average hull roughness AHR (in microns) to the incremental friction coefficient: ΔCf = 0.044 × [(AHR / L)^(1/3) − 10 × Re^(−1/3)] + 0.000125.

A new vessel leaves dry dock with AHR typically around 80 to 150 μm. After five years of service without dry-docking, AHR commonly reaches 300 to 350 μm, with heavily fouled hulls exceeding 500 μm. Each 25 μm increase in AHR above the new-hull baseline adds roughly 0.5 to 0.75% to frictional resistance on a large ship - a fuel penalty that accumulates continuously between dry-dockings. The correlation allowance calculator and skin friction area calculator enable these assessments.

Air resistance

Air resistance Raa acts on the above-water hull, superstructure, cranes, and deck cargo. For laden bulk carriers and tankers in calm air, Raa is typically 2 to 4% of calm-water hull resistance. For container ships carrying large numbers of boxes above deck, particularly on the transoceanic East-West trades, the windage area can be substantial and Raa in a head wind may represent 8 to 10% of total resistance. The air resistance calculator computes Raa from the transverse projected area, a drag coefficient, and relative wind speed, using the method from the air resistance formula page.

Form factor and the Prohaska method

The form factor k quantifies how much the three-dimensional viscous resistance of a real hull exceeds the flat-plate friction baseline. Its determination requires either CFD analysis or low-speed model resistance tests over a range of Froude numbers, using the Prohaska regression method.

Prohaska (1966) observed that at very low Fn (below 0.10), wave-making resistance is negligible and total resistance approximates (1 + k) × Cf0. By plotting Ct0 / Cf0 against Fn⁴ / Cf0 and extrapolating to Fn = 0, the intercept gives (1 + k) directly. The Prohaska form factor calculator implements this regression with the associated formula documentation.

Typical values of k for displacement hulls range from 0.10 to 0.20 for slender fine-form vessels (large container ships, destroyers), 0.15 to 0.25 for medium-fullness hulls (general cargo, ro-ro), and 0.20 to 0.35 for full-form vessels (VLCCs, bulk carriers). The Holtrop-Mennen regression provides its own statistical estimate of k based on hull form coefficients; this is implemented in the Holtrop k1 form factor calculator.

Resistance prediction methods

Taylor-Gertler series

The Taylor Standard Series, developed by David W. Taylor of the US Navy beginning around 1910 and later revised by Gertler in 1954, was the first systematic parametric series of model resistance data. The series covers a range of speed-length ratios, prismatic coefficients, and beam-draught ratios, enabling interpolation of residuary resistance for hull forms geometrically similar to the Taylor parent. The Taylor-Gertler resistance calculator provides access to this classical method.

Holtrop-Mennen regression

The Holtrop-Mennen (1982, 1984) method predicts total calm-water resistance from principal hull dimensions and form coefficients: length L, beam B, draught T, block coefficient Cb, prismatic coefficient Cp, waterplane area coefficient Cwp, midship section coefficient Cm, longitudinal centre of buoyancy LCB, and bulb and transom geometry parameters. The method was calibrated against an extensive database of MARIN model tests covering tankers, bulk carriers, container ships, general cargo ships, and ro-ro vessels. See the Holtrop-Mennen formula reference for the full system of equations.

The method decomposes resistance into: bare hull viscous resistance using the ITTC 1957 line with a statistical form factor, wave-making resistance from a four-term Kaper series, appendage resistance estimated from individual appendage types (bilge keels, fin stabilisers, rudder, shaft brackets, bossings, thrusters), model-ship correlation allowance, and transom resistance when Fn > 0. Bulb parameters enter the wave resistance term. The regression is most reliable for hull parameters within the calibration range; extrapolation to unusual hull forms (ultra-wide bulk carriers, some high-speed craft) should be treated with caution.

For design studies the Holtrop-Mennen method is fast enough to explore the parametric design space in minutes, making it the standard tool for preliminary resistance estimation. The Holtrop-Mennen calculator enables this exploration, with individual component calculators for wave resistance and appendage resistance.

Hollenbach 1998

Hollenbach (1998) published an updated regression method for single-screw and twin-screw displacement ships that addresses some of the accuracy limitations of Holtrop-Mennen at higher block coefficients and for vessels outside its original calibration range. The Hollenbach method explicitly accounts for the stern form, propeller-hull clearance, and rudder type. It is particularly useful for modern wide-beam ultra-large container ships and current-generation VLCCs whose principal dimensions fall outside the 1984 Holtrop-Mennen database.

Shallow-water resistance

In confined waters - rivers, canals, shallow coastal fairways - blockage effects and the interaction between the ship’s wave system and the seabed produce substantial resistance increases. The Schlichting-Lackenby method provides a first-order estimate of the speed loss in shallow water by correcting calm-water deep-water resistance for the increased effective blockage. The shallow-water resistance calculator and its formula page implement this correction. Froude number in shallow water uses water depth h as the length scale: Fnh = V / √(g × h). Near Fnh = 1.0 (critical speed), resistance rises sharply and the classical shallow-water squat effect becomes severe.

Froude speed scaling

When scaling model resistance results to full-scale predictions, speed must be scaled to maintain constant Fn: Vs = Vm × √(λ), where λ is the scale ratio. The Froude speed scaling calculator converts model speed to ship speed for any scale ratio, while the Reynolds number calculator and Froude number calculator provide the dimensionless parameters for scaling analysis.

Propulsive efficiency

The efficiency chain

Effective power PE is the product of total calm-water resistance and ship speed: PE = RT × V. This is the power that an ideal propulsion system would need to supply to the water. A real ship requires substantially more power at the engine output shaft because of losses throughout the propulsion train.

The quasi-propulsive coefficient ηD (also called the propulsive efficiency or QPC) links effective power to the power delivered to the propeller: PD = PE / ηD. The QPC decomposes into three factors: ηD = ηO × ηR × ηH, where ηO is the open-water propeller efficiency (the efficiency of the propeller operating in uniform flow behind its own self-induced inflow), ηR is the relative rotative efficiency (accounting for the difference between open-water and behind-hull propeller performance due to the non-uniform wake field), and ηH is the hull efficiency.

The quasi-propulsive coefficient calculator and its formula page decompose ηD into its three components.

Hull efficiency

Hull efficiency ηH = (1 − t) / (1 − w), where t is the thrust deduction fraction and w is the wake fraction. The wake fraction w represents the retardation of the water entering the propeller disc due to the hull boundary layer and potential flow effects: the effective inflow speed to the propeller is VA = V × (1 − w), lower than ship speed. For a single-screw ship, nominal wake fractions typically range from 0.20 to 0.40. The wake fraction calculator and Harvald wake estimation provide methods for estimating w from hull form coefficients.

Thrust deduction fraction t accounts for the increase in hull resistance that occurs when the propeller is working behind the hull, due to the reduced pressure field at the stern. Typical values are 0.10 to 0.25 for single-screw ships. The thrust deduction calculator implements Harvald’s regression for t as a function of hull form. The resulting hull efficiency ηH is commonly between 0.98 and 1.08 for single-screw ships; values above unity indicate that the wake gain exceeds the thrust deduction penalty.

The hull efficiency calculator and its formula reference consolidate these relationships.

Open-water propeller efficiency

Open-water efficiency ηO depends on advance coefficient J = VA / (n × D), where n is rotation rate in rev/s and D is propeller diameter in metres, and on the non-dimensional thrust and torque coefficients KT and KQ: ηO = KT × J / (2π × KQ). The propeller open-water efficiency calculator evaluates this relationship, and the individual KT and KQ formula pages document the Wageningen B-series polynomials commonly used for systematic propeller series analysis.

For design-point operation of a well-matched propeller, ηO is typically 0.60 to 0.70. The optimal propeller diameter calculator maximises ηO subject to a cavitation constraint defined by the Keller criterion.

Relative rotative efficiency

The relative rotative efficiency ηR accounts for the fact that a propeller operating in the non-uniform, partially tangential wake behind a ship hull does not perform identically to the same propeller in uniform open water. For single-screw ships ηR is typically 0.97 to 1.02; the slight gain above unity is possible because the tangential wake components can partially align the inflow with the propeller blade angles, reducing hydrodynamic losses. For twin-screw ships ηR is usually 0.97 to 1.00.

Shaft and transmission efficiency

Delivered power at the propeller shaft PD is related to engine brake power PB through shaft efficiency ηS (bearing and seal losses, typically 0.98 to 0.99 for conventional single shaft arrangements) and, where a gearbox is fitted, transmission efficiency ηT (typically 0.97 to 0.99). The total brake power required is thus:

PB = PE / (ηD × ηS × ηT)

The shaft and delivered power calculator separates shaft horsepower (SHP) and delivered horsepower (DHP) accounting for intermediate shaft losses, with the formula reference documenting the relationship.

Admiralty coefficient

For preliminary design and first-order EEDI assessments, the Admiralty coefficient C provides a compact speed-power relationship without full resistance decomposition:

C = Δ^(2/3) × V³ / PB

where Δ is displacement in tonnes, V is speed in knots, and PB is brake power in kW (or kilowatts, depending on the convention used). The coefficient is approximately constant for geometrically similar ships operating in the same speed regime. Typical values range from about 300 to 700 for large displacement ships. The Admiralty coefficient calculator and its formula page implement this relationship; the tech-level Admiralty power calculator provides the same tool in the technical calculators suite. The underlying method is also documented in the Admiralty power formula reference.

Speed-power curve and design margins

Speed-power relationship

The speed-power curve relates ship speed to required effective or brake power under defined displacement and sea conditions. Because total resistance RT approximately follows a relationship RT ∝ V^n, where n increases from about 2 at low Fn to 4 or more near the primary wave-making hump, brake power scales roughly as PB ∝ V^(n+1). At service speeds for most displacement ships, the exponent n is in the range 2.5 to 3.5, meaning that a 10% speed increase requires 30 to 50% more power. The speed-power curve fitting calculator fits empirical speed-power data to a polynomial model for operational analysis and CII correction factor estimation.

The strong speed-power relationship explains the economic and regulatory significance of slow steaming: when bunker prices are high or the vessel carries a poor CII rating, reducing speed by even a few per cent materially reduces fuel burn and carbon intensity.

Trial condition vs service condition

Ship speed and power are assessed under two reference conditions. Trial condition refers to performance in calm water on the official speed trial, measured by timed runs over a measured distance in deep water with a clean, freshly painted hull and propeller. Service condition refers to average performance over the operational voyage profile, accounting for sea state, wind, current, hull fouling, and propeller roughness.

The gap between trial and service performance is captured in two design margins:

The sea margin (commonly 15% on trial power) is added to calm-water trial power to account for average sea and weather conditions expected on the trade route. A 15% sea margin means the service speed at rated continuous power equals the trial speed at approximately 87% MCR (maximum continuous rating). The specific margin applied depends on the trade: Pacific routes subject to North Pacific winter swells may justify 20%, sheltered coastal trades 10%.

The fouling margin (sometimes included within a combined service margin) accounts for the degradation of hull and propeller surface condition between dry-dockings. The combined trial-to-service power margin for a vessel midway through a five-year class renewal cycle is typically 25 to 35%, confirming that in-service fuel consumption is substantially higher than trial-condition predictions.

Effect of trim, draught, and displacement

Resistance and propulsive efficiency are functions of the loading condition. At reduced draught (ballast condition), wetted surface area decreases but the changed waterplane entry angle and exposed transom geometry may alter wave-making resistance. Trim - the longitudinal difference in draught between bow and stern - affects resistance through its influence on the effective LCB position and the stern wake field.

Trim optimisation is among the lowest-cost efficiency measures available to operators: by redistributing ballast water or fuel to achieve the resistance-minimising trim at the current displacement and speed, power reductions of one to three per cent are achievable on many hull forms. The trim optimisation calculator provides a first-order assessment of the trim effect on resistance. The ESD trim optimisation tool extends this to energy-saving device context.

The effect of displacement change on power at constant speed can be estimated from the Admiralty coefficient. If displacement changes from Δ1 to Δ2 at the same speed, power scales approximately as PB2 / PB1 ≈ (Δ2 / Δ1)^(2/3). This scaling is used in voyage planning when assessing the fuel penalty of carrying excess ballast.

Added resistance in waves

Sources of added resistance

At sea, a ship experiences resistance in excess of its calm-water value due to wave-induced motions and the diffraction and radiation of incident waves. This added resistance RAW has two principal components: motion-induced added resistance (arising from the energy radiated by the ship heaving, pitching, and rolling in response to incident waves) and diffraction-induced added resistance (arising from the reflection and diffraction of incident wave energy by the hull). In head seas, both components are significant; motion-induced resistance dominates for longer ships where pitch and heave are pronounced.

For a practical design estimate, ITTC recommends the STAWAVE-2 method (developed at MARIN) for added resistance in irregular head seas. The method relates RAW to significant wave height Hs, peak period, ship length, beam, and block coefficient. The STAWAVE-2 formula reference documents the method in full.

Added resistance in a representative North Atlantic sea state may increase total resistance by 15 to 30% for a laden Panamax bulk carrier on a trade route from Brazil to Rotterdam. This figure varies substantially with route, season, and loading condition, which is why the 15% sea margin applied in design is sometimes insufficient for high-latitude winter services.

Beaufort scale and resistance increment

Operational data from ISO 19030 monitoring programs confirm a strong correlation between wind force (Beaufort scale) and the resistance increment experienced in service. At Beaufort 5 (17 to 21 knots wind), the combined wind and wave resistance increment on a large tanker is typically 10 to 15% above calm-water. At Beaufort 7 (28 to 33 knots wind), the increment may reach 25 to 40%. These figures are used in voyage planning to assess routing around weather systems and to benchmark in-service performance against corrected baselines.

Hull surface condition

Average hull roughness

Average hull roughness AHR (measured in microns, μm) is the standard metric for the macroscopic roughness of the underwater hull surface. New coating systems applied to grit-blasted steel in dry dock produce an AHR of approximately 80 to 100 μm for conventional antifouling systems and as low as 50 to 70 μm for premium foul-release silicone coatings. As the hull remains in service, corrosion pitting, mechanical damage, and incomplete antifouling depletion cause AHR to rise. After five years without hull cleaning, AHR values of 300 to 500 μm are commonly reported; biofouling-contaminated hulls can present effective roughness equivalent to 1,000 μm or more if hard fouling organisms attach.

The penalty function ΔCf from the ITTC procedure shows that the marginal fuel cost of additional roughness is front-loaded: the first 100 μm increase above a new-hull baseline imposes a larger percentage penalty than a subsequent 100 μm rise. For a VLCC at 15 knots, an AHR of 350 μm versus 100 μm implies an approximately 2.5 to 3.5% increase in required brake power at constant speed - worth several thousand tonnes of HFO per year on a typical 25,000-deadweight-tonne annual fuel consumption.

Antifouling coatings and biofouling

Antifouling coatings suppress the attachment of marine organisms (algae, barnacles, tube worms, mussels) to the submerged hull. Self-polishing copolymer (SPC) coatings biocide-release at a controlled rate as the hull moves through water, providing a polishing action that renews the active surface. Foul-release coatings achieve biofouling resistance through low surface energy rather than biocide release, making them particularly suited to vessels maintaining a minimum speed that prevents settlement.

Biofouling management under the IMO Biofouling Guidelines (MEPC.1/Circ.811, 2011, updated by Res. MEPC.378(80), 2023) requires operators to maintain a Biofouling Management Plan (BMP) documenting coating selection, in-water inspection intervals, and cleaning procedures. Groyne zones - areas of low flow such as sea chests, rudder stocks, and propeller shaft brackets - are particularly prone to fouling accumulation between dry-dockings.

The antifouling hull cleaning ROI calculator quantifies the fuel saving and payback period from hull cleaning, while the coating antifouling depletion calculator estimates remaining antifouling activity as a function of time and average speed.

ISO 19030 hull and propeller performance monitoring

ISO 19030, published in 2016 in three parts, establishes a framework for measuring and monitoring the change in hull and propeller performance of sea-going vessels over time. Part 1 defines general principles; Part 2 specifies the default method using noon report data filtered for acceptable sea state, heading, and loading condition; Part 3 defines alternative methods using shaft power meters and speed-over-ground sensors with higher data quality requirements.

The central metric is the speed-power performance indicator: the change in required shaft power at a reference speed and draught relative to the vessel’s own baseline established at the last dry-docking. A vessel losing 5% performance indicator from its baseline is consuming roughly 5% more fuel per nautical mile at the same speed solely due to hull and propeller deterioration. The ISO 19030 hull performance calculator and its formula reference implement the performance indicator calculation.

ISO 19030 is increasingly referenced in charter party clauses for hull and propeller performance warranties, providing an objective basis for disputes about speed and consumption guarantees.

Experimental and computational methods

Model testing

Towing tank testing has been the primary experimental method for ship resistance measurement since Froude’s Torquay facility. Modern facilities include MARIN (Maritime Research Institute Netherlands, Wageningen), HSVA (Hamburgische Schiffbau-Versuchsanstalt, Hamburg), KRISO (Korea Research Institute of Ships and Ocean Engineering, Daejeon), NMRI (National Maritime Research Institute, Tokyo), MARINTEK (SINTEF Ocean, Trondheim), and the Ship Model Basin (DTMB/NSWCCD) in the United States.

The standard model test procedure follows ITTC recommended procedures. A geometrically similar model at scale 1:λ (typically 1:20 to 1:50 for merchant ships) is towed at a speed corresponding to matched Froude number, and resistance is measured by a dynamometer. The frictional component estimated from the ITTC 1957 line is subtracted to isolate residuary (wave-making) resistance, which is then scaled up to full-scale by multiplying by the density ratio and scale factor cubed (for forces). The full-scale frictional resistance and roughness allowance are then added.

Propulsion tests complement bare-hull resistance tests by measuring thrust, torque, and rpm at self-propulsion, allowing wake fraction, thrust deduction, and relative rotative efficiency to be determined. Wake surveys behind the model quantify the circumferential and radial variation of axial and tangential velocity in the propeller plane, data essential for detailed propeller design and for predicting cavitation inception.

The ITTC EFD-CFD-Sea trial triangle describes the ideal process: experimental fluid dynamics (EFD, i.e. model tests) calibrate the CFD method, CFD extends the parametric range beyond tested configurations, and sea trial measurements validate the combined EFD-CFD prediction at full scale. All three legs of this triangle are required for high-confidence powering predictions.

Computational fluid dynamics

CFD methods for ship hydrodynamics fall into two broad families:

Panel methods (boundary element methods) solve the linearised or non-linear potential flow problem, which neglects viscosity and assumes the flow is irrotational. They are computationally fast and can predict wave patterns, wave resistance, wave elevation, and free-surface effects with reasonable accuracy for smooth hulls at moderate speeds, but cannot capture boundary layer separation, transom stern flows, or the viscous pressure resistance that the form factor represents.

Reynolds-Averaged Navier-Stokes (RANS) solvers solve the time-averaged Navier-Stokes equations with a turbulence closure model (typically the k-ω SST or similar two-equation model). RANS CFD captures the full viscous-inviscid interaction and can predict resistance, wake, and propeller-hull interaction directly. Leading commercial codes used in ship hydrodynamics include ANSYS Fluent, CD-adapco Star-CCM+, and NUMECA Fine/Marine. Open-source tools - notably OpenFOAM - have become widely used in research and increasingly in design offices. ShipFLOW (FLOWTECH International) is a specialist code developed specifically for ship hydrodynamics.

Free-surface modelling in RANS codes commonly uses the Volume of Fluid (VOF) method, which tracks the water-air interface by solving an additional transport equation for phase fraction. The interface sharpening schemes in modern VOF implementations allow accurate prediction of bow wave profiles and transom stern wetting, critical for wave resistance and added resistance assessments.

Dynamic trim (sinkage and trim in response to hydrodynamic forces) is now routinely computed in RANS simulations using dynamic mesh or mesh morphing methods. Neglecting dynamic trim can produce resistance errors of several per cent at high Fn, which is unacceptable for design purposes.

Self-propulsion CFD simulations couple the RANS hull solution with a rotating propeller representation (either body force actuator disc or fully resolved rotating geometry) to predict the propulsion factors directly in the computational domain. These are computationally expensive (multi-million cell, 10 to 100 CPU-hours for a single operating point on modern hardware) but provide detail impossible to obtain from model tests alone, particularly for complex wake-propeller interactions in twin-screw and podded propulsion configurations.

Sea trials

The ship speed trial, conducted by the shipyard on delivery to satisfy the contractual guaranteed speed, is governed by ISO 15016 (2015) for large ships and ITTC Recommended Procedure 7.5-04-01-01.1. The trial requires multiple runs in each direction at constant power in deep water with minimal wind and swell. Corrections are applied for displacement, water temperature, salinity, current, wind, and wave height to referred to the contractual trial condition. Modern speed trials use differential GPS for speed measurement and continuously logged shaft power meters rather than the traditional engine indicator cards.

The gap between trial prediction and measured trial performance is the basis of the builder’s guarantee. Typical contractual allowances are a guaranteed speed at specified power ±0.1 to 0.2 knots, or a guaranteed power at trial speed ±2 to 3%. Excess fuel consumption attributable to higher-than-predicted resistance triggers compensation under the shipbuilding contract.

Resistance, emissions, and regulation

EEDI design assessment

The Energy Efficiency Design Index (EEDI) mandates a minimum level of energy efficiency for new ships expressed as grams of CO₂ per tonne-nautical mile. The denominator of the EEDI formula contains the product of deadweight and reference speed Vref, which is the ship speed at 75% of rated installed power on the summer load line. The numerator contains CO₂ output proportional to engine power and specific fuel oil consumption (SFOC).

Because EEDI decreases with lower power (or lower SFOC or lower Vref), ship designers must balance resistance reduction (enabling lower installed power at target speed) against the risk of installing too little power to maintain a safe minimum speed in adverse conditions. The minimum propulsion power calculator verifies compliance with MEPC.1/Circ.850 minimum propulsion power guidelines.

An accurate resistance prediction is therefore a regulatory deliverable, not just a design performance indicator. Errors in the Holtrop-Mennen or other resistance estimate cascade directly into EEDI non-compliance risk.

EEXI and CII

The Energy Efficiency Existing Ship Index (EEXI) extends the EEDI framework to existing ships above 400 gross tonnes, mandatory from January 2023. EEXI is calculated identically to EEDI but uses installed power rather than contracted power, and compliance is often achieved by fitting an engine power limiter (EPL) or shaft power limiter (ShaPoLi) rather than by hull or propeller modifications.

The Carbon Intensity Indicator (CII) measures annual operational carbon intensity in grams of CO₂ per deadweight tonne per nautical mile. Since CII is an operational metric measured over a calendar year, hull and propeller performance degradation directly reduces the vessel’s CII rating. A ship losing 8% performance indicator under ISO 19030 from its dry-dock baseline is effectively carrying an 8% CII penalty relative to its clean-hull potential. This creates a direct financial incentive for proactive hull cleaning and condition monitoring. The CII attained calculator and CII rating tool quantify the regulatory implications of resistance changes.

SFOC, fuel type, and resistance interaction

The marine diesel engine converts brake power PB into thrust via the shaft-propeller system. Fuel consumption per unit time equals PB × SFOC, where specific fuel oil consumption (SFOC) is expressed in grams per kilowatt-hour. For a given voyage at fixed speed, a resistance increase of x% demands x% more power and therefore x% more fuel. The economic cost of resistance depends on the fuel type: HFO at lower cost-per-energy but higher sulphur requiring scrubber or fuel switch under IMO 2020, MGO, LNG, methanol, or future ammonia each change the absolute fuel cost per unit of excess resistance.

EU ETS implications

From January 2024, ships calling at European Union ports fall within the EU Emissions Trading System. Surrendered EU Allowances (EUAs) are priced by the market (historically EUR 50 to EUR 100 per tonne CO₂ in 2023-2024). Each percentage point of additional resistance-driven fuel consumption above design baseline translates directly into additional EUA surrender obligation, creating a market-based incentive for resistance reduction on EU-trading routes.

Practical performance monitoring

Speed-power performance indicators

In-service performance monitoring compares measured shaft power (from shaft power meters or torquemeter systems) against the expected power from the vessel’s sea-trial-derived speed-power curve, corrected to the current draught, trim, water temperature, and sea state. The correction procedures of ISO 15016 and the ISO 19030 method provide standardised approaches to these corrections.

A performance index above unity (requiring more power than expected) indicates that hull and/or propeller condition has degraded. Trend analysis of the performance index over time reveals the rate of fouling accumulation and enables decision-making on the timing of hull cleaning. A simple threshold criterion - for example, cleaning when the performance index exceeds 1.05 (5% power increase) - balances the cost of the cleaning operation against the accumulated fuel savings.

The ISO 19030 hull performance calculator implements these monitoring calculations, and the antifouling hull cleaning ROI calculator quantifies the economic case for cleaning.

Noon report data and data quality

Noon reports - the daily ship’s position, speed over ground, engine power, and fuel consumption report submitted by the master to the operator - are the most widely available source of in-service performance data. Their accuracy is limited by the precision of GPS speed measurements, the reliability of fuel meter readings, and the completeness of sea state information. ISO 19030 Part 2 specifies minimum data quality filters: wind speed below a threshold (typically Beaufort 5 to 6), heading not more than 45° from the wave direction, and draft within a defined range of the reference condition.

High-quality performance monitoring increasingly uses continuous sensor data rather than noon reports: shaft power meters, propeller-shaft RPM, draught gauges, hull-mounted current sensors, and satellite-derived weather overlays. The resulting data density enables performance tracking at hourly or sub-hourly resolution, identifying transient fouling events and correlating performance changes with route segment, draught, and seasonal fouling rates.

Hull cleaning approaches

In-water hull cleaning is performed by remotely operated vehicles (ROVs) or divers using rotating brush or water-jet tools. Proactive cleaning (groyne cleaning between dry-dockings) targets the high-fouling zone areas while maintaining the main coating film; full hull cleaning removes the fouling layer across the entire underwater hull. Effectiveness depends on the fouling type (soft algae slime versus hard barnacle settlement) and the cleaning method.

The economic optimisation of cleaning intervals requires a model of the fouling accumulation rate (which depends on route, port call frequency, speed, and water temperature) against the cleaning cost and the fuel saving from a performance recovery. Port state environmental regulations in some jurisdictions - notably Australia under the Australian Biofouling Management Requirements and New Zealand’s Craft Risk Management Standard - impose strict controls on what fouling can be legally present on hulls entering their waters, adding a compliance dimension to the cleaning decision.

Related Calculators

• ITTC 1957, Friction Coefficient Calculator
• Holtrop, Form Factor 1+k1 Calculator
• Holtrop, Wave Resistance Estimate Calculator
• Holtrop, Appendage Resistance Calculator
• Holtrop–Mennen Hull Resistance Calculator
• Wetted Surface, Mumford Formula Calculator
• Wave Steepness, Encounter Limit Calculator
• Model-Ship Correlation Allowance ΔC_F Calculator
• Hull Wetted Surface, Taylor Estimate Calculator
• Air Resistance, Above-Water Drag Calculator
• Form Factor (Prohaska) Calculator
• Taylor-Gertler Series, Residuary Resistance Calculator
• Schlichting Shallow-Water Correction Calculator
• Froude Speed Scaling, Model to Ship Calculator
• Reynolds Number Calculator
• Froude Number Calculator
• Quasi-Propulsive Coefficient η_D Calculator
• Wake Fraction, Taylor Approximation Calculator
• Wake Fraction (Harvald 1983) Calculator
• Thrust Deduction, Holtrop Approximation Calculator
• Hull Efficiency η_H Calculator
• Open-Water Efficiency η_O Calculator
• Propeller, Optimal Pitch-Diameter Calculator
• SHP vs DHP, Transmission Efficiency Calculator
• Admiralty Constant, Delivered Power Calculator
• Admiralty Coefficient Power Calculator
• Speed–Power Curve Fit Calculator
• Trim Optimisation Calculator
• Trim Optimization, Fuel Savings Calculator
• Hull Cleaning, ROI Estimate Calculator
• Antifouling Paint, Depletion Rate Calculator
• Hull Performance, Speed Loss ISO 19030 Calculator
• Minimum Propulsion Power Calculator
• CII Attained Calculator
• CII Rating (A–E) Calculator

See also

• Hull form design - discussion of block coefficient, prismatic coefficient, and hull geometry optimisation as they affect resistance
• Block coefficient - definition and typical values of Cb and its influence on resistance and stability
• Marine propeller - propeller design, Wageningen B-series, cavitation, and thrust characteristics
• Marine diesel engine - engine types, MCR, and SFOC relevant to brake power assessment
• Specific fuel oil consumption - SFOC definitions and influence on fuel cost per unit resistance
• Bow thruster and stern thruster - appendage resistance contribution from lateral thrusters
• What is CII - CII framework and the role of in-service resistance in carbon intensity ratings
• What is EEXI - EEXI compliance and the link between installed power and resistance
• What is EEDI - EEDI framework and design-stage resistance requirements
• Slow steaming and CII - the speed-power relationship as the basis for slow-steaming fuel savings
• Heavy fuel oil - HFO properties and the fuel cost of excess resistance
• Marine gas oil - MGO as an alternative fuel and its unit cost implications
• LNG as marine fuel - LNG fuel systems and their brake power delivery characteristics
• IMO 2020 sulphur cap - sulphur regulations driving fuel switching decisions
• EU ETS for shipping - carbon pricing mechanism affecting the economic cost of resistance
• Holtrop-Mennen resistance calculator - full Holtrop-Mennen resistance prediction
• ITTC friction line calculator - ITTC 1957 frictional resistance coefficient
• Admiralty coefficient calculator - speed-power relationship from displacement and speed
• Quasi-propulsive coefficient calculator - total propulsive efficiency from QPC components
• Hull performance ISO 19030 calculator - in-service hull and propeller performance monitoring
• Antifouling hull cleaning ROI calculator - economic assessment of hull cleaning operations
• Speed-power curve fitting calculator - empirical speed-power model fitting from operational data
• ShipCalculators.com calculator catalogue - full index of resistance, propulsion, and powering tools

References

1. Froude, W. (1874). “On experiments with H.M.S. Greyhound.” Transactions of the Institution of Naval Architects, 15, 36-73.
2. ITTC (1957). Proceedings of the 8th International Towing Tank Conference, Madrid. Adoption of the ITTC 1957 model-ship correlation line.
3. Holtrop, J. and Mennen, G.G.J. (1982). “An approximate power prediction method.” International Shipbuilding Progress, 29(335), 166-170.
4. Holtrop, J. (1984). “A statistical re-analysis of resistance and propulsion data.” International Shipbuilding Progress, 31(363), 272-276.
5. Hollenbach, U. (1998). “Estimating resistance and propulsion for single-screw and twin-screw ships.” Ship Technology Research, 45, 72-76.
6. Prohaska, C.W. (1966). “A simple method for the evaluation of the form factor and the low speed wave resistance.” Proceedings of the 11th ITTC, Tokyo.
7. ITTC (2014). Recommended Procedures and Guidelines 7.5-02-03-01.4: 1978 ITTC Performance Prediction Method. International Towing Tank Conference.
8. ISO 15016:2015. Ships and marine technology - Guidelines for the assessment of speed and power performance by analysis of speed trial data. International Organization for Standardization.
9. ISO 19030-1:2016; ISO 19030-2:2016; ISO 19030-3:2016. Ships and marine technology - Measurement of changes in hull and propeller performance. International Organization for Standardization.
10. Schoenherr, K.E. (1932). “Resistance of flat surfaces moving through a fluid.” Transactions of the Society of Naval Architects and Marine Engineers, 40, 279-313.
11. Harvald, S.A. (1983). Resistance and Propulsion of Ships. Wiley-Interscience, New York.
12. IMO MEPC.1/Circ.850 (2015). Guidelines for the assessment of speed and power performance by analysis of speed trial data.
13. ITTC (2021). Recommended Procedures 7.5-04-01-01.1: Full-Scale Measurements, Speed and Power Trials, Analysis of Speed/Power Trial Data. International Towing Tank Conference.
14. IMO MEPC.378(80) (2023). 2023 Guidelines for the control and management of ships’ biofouling. International Maritime Organization.
15. Hughes, G. (1954). “Friction and form resistance in turbulent flow, and a proposed formulation for use in model and ship correlation.” Transactions of the Institution of Naval Architects, 96, 314-376.

Further reading

• Bertram, V. (2000). Practical Ship Hydrodynamics. Butterworth-Heinemann, Oxford.
• Carlton, J.S. (2012). Marine Propellers and Propulsion. 3rd edition. Butterworth-Heinemann, Oxford.
• Lewis, E.V. (ed.) (1988). Principles of Naval Architecture, Volume II: Resistance, Propulsion and Vibration. SNAME, Jersey City.
• Molland, A.F., Turnock, S.R., and Hudson, D.A. (2017). Ship Resistance and Propulsion: Practical Estimation of Ship Propulsive Power. 2nd edition. Cambridge University Press.
• Faltinsen, O.M. (1990). Sea Loads on Ships and Offshore Structures. Cambridge University Press.

External links

• ITTC Recommended Procedures and Guidelines - International Towing Tank Conference standards for model testing and extrapolation
• MARIN - Maritime Research Institute Netherlands, leading towing tank and CFD facility
• HSVA - Hamburg Ship Model Basin, towing tank and ice basin testing
• KRISO - Korea Research Institute of Ships and Ocean Engineering
• IMO Energy Efficiency - IMO energy efficiency regulations including EEDI, EEXI, and CII