Hydrodynamic Modelling and Response Amplitude Operators

Integrated vessel performance model combining Holtrop-Mennen resistance, STAWAVE-1 added resistance, and simplified RAO for operational seakeeping assessment

Abstract

Weather routing applications require a vessel performance model that translates environmental conditions — wind, waves, swell, and current — into fuel consumption rates and motion safety assessments at each point along a candidate route. This article describes the hydrodynamic model implemented in WindMar's VesselModel and SeakeepingModel classes, which together form the ship performance engine that drives the route optimiser (see A* Pathfinding) and the Monte Carlo uncertainty quantification system (see Monte Carlo). The calm-water resistance prediction follows the Holtrop-Mennen (1984) regression with ITTC-1957 friction coefficients. Added resistance in waves is computed using the STAWAVE-1 method from ISO 15016:2015, while wind resistance follows a projected-area drag model with directional decomposition. Seakeeping assessment employs a single-degree-of-freedom oscillator model for roll, the Ochi (1964) method for slamming probability, and parametric roll detection via encounter frequency analysis. A noon-report calibration module adjusts model coefficients against observed consumption data using L-BFGS-B optimisation.

1. Introduction

The central challenge in maritime weather routing is to predict how a vessel will perform under a given set of environmental conditions. Empirical tables, while historically useful, lack the resolution and generality needed for optimisation across a continuous space of headings, speeds, and sea states. WindMar therefore implements a physics-based performance model grounded in established naval architecture methods, calibrated for the reference vessel class and adaptable to individual hulls through noon-report calibration.

The model is structured as a layered computation pipeline. Calm-water resistance provides the baseline power requirement at a given speed and loading condition. Added resistance from waves, wind, and current effects is computed independently and summed with the calm-water component to yield total resistance. This total resistance is then converted to brake power through the propulsive efficiency chain. Finally, the SFOC curve maps engine load fraction to specific fuel consumption, yielding a fuel rate in metric tonnes per hour. In parallel, the seakeeping model evaluates ship motions and safety margins to flag hazardous conditions that the A* pathfinding algorithm must avoid.

The reference vessel throughout this article is an MR (Medium Range) Product Tanker, a widely operated class in the global petroleum trade. The principal particulars, which serve as defaults in the VesselSpecs dataclass, are summarised below.

MR Product Tanker — Principal Particulars

The significant difference between laden and ballast conditions is characteristic of tanker operations. In ballast, the vessel rides higher, exposing substantially more freeboard and superstructure area to wind (frontal area increases from 450 m² to 850 m²), while wetted surface and displacement are reduced, lowering the calm-water resistance. These competing effects mean that wind resistance can dominate the total resistance budget for a ballast tanker in strong headwinds, a phenomenon that the model captures through condition-dependent parameters.

2. Calm-Water Resistance

The calm-water resistance prediction follows the Holtrop-Mennen (1984) statistical method, one of the most widely used resistance estimation procedures in preliminary ship design and operational performance assessment. The method decomposes total calm-water resistance into frictional, wave-making, and appendage components, each computed from hull form parameters and speed. This decomposition allows independent calibration of each component against sea-trial or noon-report data (see Section 8).

2.1 Physical Constants

2.2 Froude and Reynolds Numbers

The two fundamental dimensionless parameters that characterise the flow regime around the hull are the Froude number and the Reynolds number. The Froude number governs the wave-making resistance, while the Reynolds number governs frictional resistance. For the MR tanker at service speed (14.5 kts laden, approximately 7.46 m/s), the Froude number is approximately 0.18, placing the vessel firmly in the displacement regime where frictional resistance dominates.

2.3 Frictional Resistance

The frictional resistance component is the dominant term for full-form vessels such as tankers operating at moderate Froude numbers. It is computed using the ITTC-1957 model-ship correlation line, which provides the flat-plate friction coefficient as a function of Reynolds number. A hull roughness allowance ΔC_f accounts for paint roughness, biofouling, and other surface imperfections. The form factor (1 + k₁) converts the flat-plate friction to a three-dimensional hull resistance by accounting for the pressure resistance due to hull form.

The form factor regression used in the implementation is a simplified variant of the original Holtrop-Mennen formulation, retaining the primary dependencies on beam-to-length ratio (B/L_pp), beam-to-draft ratio (B/T), and block coefficient (C_b). The lower bound of 0.1 prevents physically unrealistic negative values for unusual hull proportions.

2.4 Wave-Making Resistance

The wave-making component represents the energy radiated away from the hull as surface waves generated by the moving ship. For the Froude number range typical of laden tankers (Fn ≈ 0.15–0.20), wave-making resistance is a secondary contributor, typically 10–15% of frictional resistance. The implementation uses a simplified scaling proportional to the square of the Froude number.

2.5 Appendage Resistance

Appendage resistance accounts for drag from rudder, bilge keels, propeller shaft brackets, and other hull protuberances. It is modelled as a fixed fraction of frictional resistance, a simplification that is acceptable for standard merchant vessel configurations.

2.6 Total Calm-Water Resistance

The total calm-water resistance is the sum of the three components. For the MR tanker at 14.5 knots laden, frictional resistance typically accounts for approximately 80% of the total, with wave-making contributing around 15% and appendages the remaining 5%. This distribution shifts in favour of wave-making at higher speeds, which is why the speed-power relationship exhibits a steeper-than-cubic growth above the design speed.

3. Added Resistance in Waves

When a vessel advances through a seaway, it experiences an additional mean resistance force beyond its calm-water value. This added resistance arises from the interaction between the hull and the incident wave field: the ship radiates and diffracts waves, and the resulting pressure distribution produces a net retarding force. The magnitude depends on significant wave height, wave period, vessel dimensions, and the relative angle between the wave propagation direction and the ship heading.

WindMar computes added wave resistance using the STAWAVE-1 method from ISO 15016:2015 and the ITTC Recommended Procedures (2014). STAWAVE-1 is a simplified empirical formula intended for use in speed-power analysis when detailed hull form data (e.g., waterline offsets for strip-theory calculations) is unavailable. Despite its simplicity, it has been validated against model test data and provides acceptable accuracy for full-form vessels (C_b ≥ 0.75) in moderate sea states.

The bluntness coefficient α_BK characterises the bow waterline shape and its influence on wave reflection. For full-form vessels with block coefficients exceeding 0.75 — which includes the MR tanker in both laden (C_b = 0.82) and ballast (C_b = 0.75) conditions — the coefficient is set to 1.0 as specified in the standard.

3.1 Directional Factor

The directional factor accounts for the heading-dependent nature of added wave resistance. Head seas produce the maximum added resistance, while following seas produce significantly less. The implementation computes the relative angle between the wave propagation direction and the vessel heading, then applies a cosine-based weighting.

This directional model captures the first-order heading dependence observed in both model tests and full-scale measurements. The A* pathfinding algorithm exploits this directional sensitivity by evaluating candidate legs at different headings, preferring routes that minimise head-sea exposure when wave heights are significant. The wave data driving this computation is sourced from the data pipeline, which provides separate wind-wave and swell components with independent directions.

4. Wind Resistance

Aerodynamic resistance arises from the interaction between the relative wind and the above-water structure of the vessel. For tankers, particularly in ballast condition where the freeboard is large, wind resistance can constitute a significant fraction of total resistance. The model decomposes wind drag into a longitudinal (headwind) component that directly opposes forward motion and a transverse (crosswind) component that induces leeway drift, adding an indirect resistance penalty through increased hydrodynamic drag.

4.1 Projected Areas

The near-doubling of frontal area from laden to ballast reflects the exposed hull above the waterline and the higher superstructure silhouette when the vessel rides light. The lateral area increase is less pronounced but still significant, affecting the transverse wind force and hence leeway.

4.2 Apparent Wind and Relative Angle

The wind resistance calculation uses the true wind speed and direction from the weather field data, resolving the relative angle between the wind vector and the ship heading to determine longitudinal and transverse drag components.

4.3 Longitudinal Drag Component

The longitudinal drag coefficient C_x = 0.8 for pure headwind is consistent with wind-tunnel measurements on tanker superstructures. The max(0, ...) clamp ensures that following winds do not produce a negative (propulsive) resistance; the model conservatively assumes zero wind thrust from stern quarters.

4.4 Transverse Drift Component

The transverse component represents the added resistance due to leeway. When a crosswind pushes the vessel sideways, the hull develops a drift angle, and the resulting hydrodynamic side force produces an induced drag component. The 0.1 scaling factor converts the transverse aerodynamic force into an equivalent longitudinal resistance increase, an approximation that avoids the need for explicit drift angle computation while capturing the first-order effect.

4.5 Total Wind Resistance

5. Current Effects

Ocean currents affect vessel performance by altering the speed over ground (SOG) relative to the speed through water (STW). The vessel's propulsion system drives the hull through the water at STW, but the actual ground track velocity — and hence the time to traverse a route leg — depends on the vector sum of STW and the current velocity. Unlike waves and wind, currents do not change the hydrodynamic resistance directly; instead, they modify the effective distance the vessel must travel through the water to cover a given ground distance.

The implementation resolves the current vector into a component along the vessel's track heading. A favourable current (following) increases SOG, reducing transit time and hence total fuel consumption for the leg. An adverse current (opposing) decreases SOG, increasing transit time and fuel. Cross-currents are resolved into their along-track projection, with the perpendicular component assumed to be compensated by the autopilot heading correction at negligible additional fuel cost.

Current data in WindMar is sourced from the CMEMS global physics model or a synthetic fallback, as described in the data pipeline article. For routes in areas with strong tidal or permanent currents (e.g., the Strait of Gibraltar, the Dardanelles), the current contribution to voyage time can be significant — on the order of 10–20% of transit time for narrow passages with 2–3 knot currents.

6. Seakeeping and Response Amplitude Operators

Beyond resistance and fuel consumption, the vessel performance model must assess whether the ship motions induced by the seaway remain within safe operational limits. Excessive roll, pitch, or vertical accelerations can endanger cargo, crew, and structural integrity. The SeakeepingModel class implements a simplified seakeeping assessment based on the concept of Response Amplitude Operators (RAOs), which relate the amplitude of ship motion responses to the amplitude of incident waves as a function of encounter frequency.

6.1 Seakeeping Parameters

The seakeeping model uses the following vessel-specific parameters, defined in the SeakeepingSpecs dataclass with defaults for the MR Product Tanker.

6.2 Encounter Frequency

The encounter frequency is the frequency at which the vessel experiences successive wave crests, which differs from the absolute wave frequency due to the vessel's forward motion. It is the fundamental parameter governing ship motion response in a seaway. When the encounter frequency approaches the natural frequency of a particular motion (e.g., roll), resonance amplification occurs.

The encounter frequency depends on both ship speed and heading relative to the waves. In head seas, the encounter frequency is higher than the wave frequency (the vessel meets crests more frequently), while in following seas, it is lower. This heading dependence is critical for roll resonance avoidance, as the route optimiser can choose headings that keep the encounter frequency away from the natural roll period.

6.3 Roll RAO (Single-DOF Oscillator)

The roll response is modelled as a single-degree-of-freedom damped harmonic oscillator, where the hull is the mass, hydrostatic restoring moment (proportional to GM) provides the stiffness, and viscous effects provide damping. The RAO gives the amplification factor between incident wave slope and roll angle amplitude as a function of the frequency ratio.

At resonance (r = 1), the RAO amplification reaches 1/(2ζ) = 10.0 for the default damping ratio of 0.05. This extreme amplification highlights why roll resonance is one of the most dangerous seakeeping phenomena and why avoidance is a primary routing constraint. The low damping coefficient of 0.05 is characteristic of tankers without active anti-roll devices — bilge keels provide some additional damping, but tankers remain lightly damped compared to vessels with fin stabilisers or anti-roll tanks.

6.4 Decomposed Response

When both wind-waves and swell are present simultaneously, the seakeeping model computes the response to each component independently and combines them using root-sum-square (RSS) combination. This reflects the physical assumption that wind-waves and swell are statistically independent processes with different frequency content and directions.

6.5 Slamming Probability (Ochi Method)

Bottom slamming occurs when the bow emerges from the water and re-enters with high velocity, producing impulsive pressure loads on the hull bottom. The probability of slamming is estimated using the Ochi (1964) method, which relates the probability of bow emergence to the relative freeboard and the vertical motion amplitude at the bow.

The slamming probability is strongly dependent on loading condition. In laden condition with a bow freeboard of 6.0 m, slamming risk is elevated in significant wave heights above 4–5 m. In ballast with 12.0 m freeboard, substantially higher waves are required to cause emergence. The route optimiser uses slamming probability as one of the safety constraints that trigger voluntary speed reduction or route deviation.

6.6 Parametric Roll Detection

Parametric rolling is a resonance phenomenon that can produce extreme roll angles (30–40 degrees) in certain combinations of wave period, ship speed, and heading. It occurs when the encounter period is approximately half the natural roll period, causing periodic variations in the waterplane area and hence the restoring moment. The conditions are most dangerous in head and following seas when the wavelength is comparable to the ship length.

When the parametric roll criterion approaches unity, the seakeeping model flags the condition as dangerous regardless of the conventional roll amplitude calculation. This information is passed to the routing algorithm, which can then select headings and speeds that detune the encounter frequency away from the parametric resonance condition.

6.7 Safety Limits

The seakeeping model classifies motion responses into three severity categories: safe, marginal, and dangerous. These thresholds, defined in the SafetyLimits dataclass, are based on IMO guidance, classification society rules, and operational experience. When a route leg produces motions in the marginal or dangerous range, the A* pathfinding algorithm applies a cost penalty proportional to severity, effectively steering the optimiser away from hazardous conditions.

The roll limit of 15 degrees for the safe category corresponds to the threshold beyond which unsecured cargo may begin to shift and crew operational effectiveness degrades. The dangerous threshold of 30 degrees represents conditions where structural damage, cargo loss, or capsize risk become significant concerns. The acceleration limits at the bridge location are driven by crew habitability and equipment operability requirements. The slamming threshold of 3% follows the Ochi recommendation for structural fatigue avoidance on routine voyages.

7. Propulsion and Fuel Consumption

The propulsion model converts total resistance into brake power demand and then into fuel consumption rate. The chain of efficiencies between the hull tow rope and the engine output shaft accounts for propeller open-water efficiency, hull-propeller interaction, and the relative rotative efficiency of the propeller operating behind the hull. These three factors combine to produce the overall propulsive coefficient.

7.1 Propulsive Efficiency Chain

The hull efficiency exceeding unity (1.05) reflects the beneficial effect of the wake field: the propeller operates in the hull boundary layer where the local inflow velocity is lower than the ship speed, effectively recovering some of the frictional energy loss. This is a well-established phenomenon for single-screw merchant vessels. The overall propulsive coefficient of 0.6825 means that approximately 68% of the engine's shaft power is converted into useful thrust work.

7.2 Brake Power Requirement

The speed-power relationship follows an approximately cubic law, since resistance is roughly proportional to the square of speed and power equals resistance times speed. This cubic relationship has a profound operational implication: a 10% speed reduction yields approximately 27% power savings, making slow steaming an effective fuel-saving strategy that the route optimiser can exploit by selecting speeds below the design point.

7.3 Voluntary Speed Reduction

When the required brake power exceeds the engine's maximum continuous rating (MCR = 8,840 kW), the vessel cannot maintain the commanded speed. The model caps the delivered power at MCR and reports the required (uncapped) power separately, enabling the routing system to identify legs where involuntary speed loss occurs. The calculate_fuel_consumption method returns both power_kw (capped at MCR) and required_power_kw (uncapped) to facilitate this distinction.

7.4 Specific Fuel Oil Consumption (SFOC)

Marine diesel engines do not consume fuel at a constant rate per unit of power output. The specific fuel oil consumption varies with engine load, reaching an optimum in the 75–85% MCR range where thermal efficiency is highest. At lower loads, increased relative heat losses and suboptimal injection timing raise the SFOC, while at loads approaching 100% MCR, thermal stress and detonation avoidance measures also increase consumption.

The asymmetric shape of this curve — steeper below the optimum than above — reflects the disproportionate efficiency penalty of deep part-load operation. At 50% MCR, the SFOC penalty is approximately 3.75% above the MCR value, while at 100% MCR, it is only 1.25% above. This SFOC variation, combined with the cubic speed-power relationship, determines the optimal operating speed for minimum fuel consumption per nautical mile.

7.5 Fuel Consumption Calculation

The calculate_fuel_consumption method returns a comprehensive result including total fuel in metric tonnes, brake power, required (uncapped) power, transit time, and a breakdown of fuel attributable to calm-water resistance, wind resistance, and wave resistance. This breakdown is computed by running the resistance calculation with each environmental component isolated, then expressing each as a fraction of the total. The breakdown enables post-voyage analysis of which environmental factors dominated fuel consumption on each leg — information valuable for both operational debriefing and model calibration.

8. Noon Report Calibration

The physics-based model described in the preceding sections provides a general prediction framework, but individual vessels deviate from the reference hull due to differences in hull form, propeller condition, engine tuning, and hull fouling state. The VesselCalibrator class addresses this gap by adjusting the model's prediction coefficients to match observed fuel consumption data from noon reports — the daily operational records that every merchant vessel compiles.

8.1 Calibration Factors

The calibration operates on four multiplicative correction factors that scale the corresponding resistance or consumption components. Each factor has a nominal value of 1.0 (no correction) and is bounded to prevent physically unreasonable values.

8.2 Optimisation Method

The calibrator uses the L-BFGS-B (Limited-memory Broyden-Fletcher-Goldfarb-Shanno with Bound constraints) algorithm from scipy.optimize to minimise the sum of squared differences between predicted and observed fuel consumption across all available noon reports. A minimum of 5 reports (MIN_REPORTS = 5) is required to ensure statistical significance, and the maximum allowable deviation from the default factors is capped at 0.5 (MAX_FACTOR_DEVIATION = 0.5) to prevent overfitting.

Each noon report provides the vessel's speed over ground, fuel consumption for the reporting period, loading condition, and observed weather (wind speed and direction, wave height and direction, swell height and direction, and current speed and direction). The calibrator computes predicted fuel consumption for each report using the vessel model with the candidate correction factors, and the optimiser iterates until the residual is minimised.

8.3 Hull Fouling Model

Between calibration events, the calm-water resistance is expected to increase due to biological growth on the hull bottom. The calibrator applies a hull fouling penalty that accumulates over time since the last dry-docking or hull cleaning event. The base fouling rate of 1% per month is modulated by regional water temperature multipliers, reflecting the higher biological productivity of tropical waters.

The 20% cap represents an extreme fouling state that would typically trigger a hull cleaning operation. In practice, most operators clean the hull when the fouling penalty reaches 10–15%, as the fuel cost of the additional resistance exceeds the cost of cleaning. The fouling model allows the vessel performance prediction to remain accurate between calibration events by extrapolating the expected degradation based on the vessel's trading pattern.

8.4 Validation

After calibration, the calibrator reports the mean absolute error (MAE) and root mean square error (RMSE) of the predicted fuel consumption against the training set. A well-calibrated model typically achieves a fuel prediction MAE of 2–5% of the observed consumption. When the RMSE exceeds 10%, the system raises a warning suggesting that the noon report data may contain errors (e.g., incorrect fuel figures, missing weather observations) or that the vessel's performance has changed significantly from the modelled baseline (e.g., propeller damage, engine degradation).

References

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