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Resistance Components Deep Dive

The total hull resistance of a displacement ship at a given speed is conventionally decomposed (after William Froude, 1874) into four principal components: frictional resistance ($R_F$), proportional to the wetted surface area and approximately to $V^2$, the dominant component (60 to 80%) for slow-speed merchant ships; residual resistance ($R_R$), comprising wave-making resistance (energy radiated as a Kelvin wave system around the ship), viscous pressure resistance (form-drag from the boundary layer separation pattern at the stern) and wave-breaking resistance (energy lost in spilling and breaking waves), scaling more steeply (approximately $V^4$ to $V^6$) at higher Froude numbers; appendage resistance ($R_{app}$), from rudders, bilge keels, propeller bossings, bow thrusters, stabiliser fins; and air resistance ($R_A$), from windage on the above-water area. Accurate prediction of each component is the foundation of propulsion power calculation, fuel consumption forecasting, design optimisation, and CII performance prediction. The principal tools are empirical methods (Holtrop-Mennen 1982/1984, Hollenbach 1998, Series 60 systematic series, BSRA series, MARIN series), model testing at recognised model basins (MARIN, HSVA, SSPA, NMRI, Krylov) and modern CFD (Reynolds-averaged Navier-Stokes solvers with free-surface volume-of-fluid). ShipCalculators.com hosts the principal computational tools: the frictional resistance calculator, the Holtrop-Mennen total resistance calculator, the wave-making resistance calculator, the appendage resistance calculator, the air resistance calculator, the Hollenbach calculator, the shallow water resistance calculator and the STAWAVE-2 added resistance calculator. A full listing is available in the calculator catalogue.

Contents

Background

Froude decomposition

William Froude in 1874 proposed the foundational decomposition of total resistance into a frictional component (computed from a flat-plate analogy) and a residual component (the difference between total measured resistance and the frictional contribution). The decomposition is hierarchical:

$$ R_T = R_F + R_R + R_{app} + R_A $$

For modern detailed analysis, the residual component is further decomposed:

$$ R_R = R_W + R_{VP} + R_{WB} $$

where $R_W$ is wave-making, $R_{VP}$ is viscous pressure, $R_{WB}$ is wave-breaking. This is captured by the form factor (1+k) method of Hughes (1954) and the ITTC, where the viscous resistance ($R_F + R_{VP}$) is computed as $(1+k) \cdot R_F$ with $k$ a hull-form-specific form factor.

Speed regimes

The relative importance of the components depends on the Froude number $F_n = V / \sqrt{g \cdot L_{WL}}$:

  • $F_n < 0.20$ (slow displacement: most VLCCs, Capesize bulkers): $R_F$ dominates (70 to 85% of $R_T$); $R_W$ small.
  • $F_n = 0.20$ to $0.30$ (medium displacement: container ships, LNG carriers, ferries): $R_F$ approximately 50 to 65%, $R_W$ approximately 15 to 30%, climbing rapidly with speed.
  • $F_n = 0.30$ to $0.50$ (fast displacement: high-speed ferries, naval frigates): $R_W$ becomes dominant (40 to 60%); $R_F$ proportionally smaller.
  • $F_n > 0.50$ (planing regime): different physics; the hull rises onto a planing surface.

This article focuses on the displacement regime ($F_n < 0.40$) typical of merchant ships.

Frictional resistance ($R_F$)

Calculation

The frictional resistance is calculated as:

$$ R_F = \frac{1}{2} \rho V^2 S C_F $$

where $\rho$ is water density, $V$ is ship speed, $S$ is wetted surface area, and $C_F$ is the frictional coefficient.

The friction coefficient is calculated from the Reynolds number $R_n = V L / \nu$ (where $\nu$ is kinematic viscosity, approximately $1.2 \times 10^{-6}$ m²/s for seawater at 15 °C). The standard correlations are:

  • ITTC 57: $C_F = 0.075 / (\log_{10} R_n - 2)^2$.
  • Schoenherr: $0.242 / \sqrt{C_F} = \log_{10}(R_n \cdot C_F)$, implicit equation.

For typical merchant ships at service speed, $R_n \approx 10^8$ to $10^9$ (turbulent boundary layer) and $C_F \approx 0.0015$ to $0.0020$.

Form factor

The form factor (1+k) captures the additional viscous pressure resistance beyond the flat-plate friction. For typical merchant ships:

  • Slender vessels (containers, LNG): $1+k \approx 1.10$ to $1.20$.
  • Full-form vessels (VLCCs, Capesize): $1+k \approx 1.20$ to $1.35$.
  • Fine vessels (naval, racing): $1+k \approx 1.05$ to $1.15$.

The form factor is determined from low-speed model tests where wave-making is negligible, by extrapolation of the resistance curve back to zero Froude number.

Roughness allowance

In service, hull fouling and surface roughness add an additional friction increment. The roughness allowance is typically 0.0003 to 0.0010 added to the smooth-hull $C_F$, depending on the hull condition. See wetted surface area for the full fouling treatment.

Wave-making resistance ($R_W$)

Physics

A ship moving through water generates a Kelvin wave system: a pattern of transverse and divergent waves emanating from the bow and stern. The energy carried by these waves is irrecoverable (radiated to infinity); the work done in generating them is the wave-making resistance.

The wave system is geometrically constrained: the Kelvin half-angle is approximately 19.5 degrees regardless of ship speed. Within this half-angle, the wave pattern depends on the ship’s hull geometry and Froude number.

Speed dependence

Wave-making resistance rises steeply with speed because:

  • The wave system amplitude scales with $V^2$.
  • The wave system extent (longitudinally) increases with $V^2$.
  • The wave energy radiation rate scales as approximately $V^4$ to $V^6$.

The result is the characteristic resistance hump and hollow pattern: at certain Froude numbers, the bow and stern wave systems interfere constructively (producing a hump) or destructively (producing a hollow). The principal hump is at $F_n \approx 0.50$, with a smaller hump at $F_n \approx 0.30$. Most merchant ship operating points avoid the principal hump but operate in the rising-resistance region.

Bulbous bow effect

A bulbous bow reduces wave-making resistance through wave cancellation: the bulb generates a pressure wave that is approximately 180 degrees out of phase with the stem wave, reducing the net wave amplitude radiated outward. Properly designed, a bulb can reduce $R_W$ by 5 to 15% at the design speed.

Transom stern effect

Modern flat or near-flat transom sterns (typical of post-2000 designs) generate additional wave-making resistance from the dead-water region behind the transom. At slow speeds the transom drags a wake region; at higher speeds the flow detaches cleanly. The transition occurs at approximately $F_n \approx 0.30$ to $0.35$ for typical hull forms.

Calculation

Wave-making resistance is calculated by:

  • Empirical methods (Holtrop-Mennen, Hollenbach, Series 60): empirical correlations based on hull form coefficients and Froude number.
  • Slender-body theory (Michell 1898): analytical method valid for very slender hulls; accurate for fine-form vessels.
  • 3D potential flow (Dawson method, NM4 method): numerical solution of the wave-making problem; accurate for general hull forms.
  • CFD with free surface: full Navier-Stokes solution with volume-of-fluid free surface; most accurate but computationally intensive.

Viscous pressure resistance ($R_{VP}$)

The viscous pressure resistance arises from the boundary layer separation pattern at the stern. A vessel with significant stern flare or a sharp run-up creates a separated wake that produces drag through pressure differential between the stern (low pressure) and bow (high pressure). The viscous pressure resistance is captured in the form factor $(1+k)$ but can be separated by careful analysis.

For modern hull forms with smooth, well-designed stern lines, $R_{VP}$ is typically 5 to 15% of $R_F$. For vessels with poor stern lines or excessive transom area, $R_{VP}$ can rise to 25 to 40% of $R_F$.

Wave-breaking resistance ($R_{WB}$)

Some of the wave-making energy is lost in spilling and breaking waves, particularly at the bow at higher Froude numbers. Wave-breaking resistance is conventionally lumped with wave-making resistance; the distinction is used principally in research and detailed CFD analysis.

Appendage resistance ($R_{app}$)

Components

The principal appendage components contributing to resistance:

  • Rudder: typically 1.5 to 2.5% of $L \times T$ as wetted area; resistance contribution approximately 1 to 3% of total.
  • Bilge keels: longitudinal fins for roll damping; resistance contribution approximately 0.5 to 2%.
  • Propeller bossings, struts, bracketry: for twin-screw vessels; 1 to 4% of total.
  • Bow thruster tunnels: 0.2 to 1% of total, depending on configuration.
  • Stabiliser fins: when retracted, 0.1 to 0.3% of total.
  • Sea chests, anodes, transducer fairings: typically 0.1 to 0.3% of total.

Calculation

Each appendage is computed individually using its wetted area and a characteristic friction or form factor. The Holtrop-Mennen and Hollenbach methods include standard appendage allowances; for unusual configurations, dedicated CFD analysis or model tests are warranted.

Air resistance ($R_A$)

Calculation

The air resistance is:

$$ R_A = \frac{1}{2} \rho_{air} V_{ar}^2 A_T C_{X} $$

where $\rho_{air}$ is air density (1.225 kg/m³ at sea level), $V_{ar}$ is the apparent wind speed (vector sum of ship speed and true wind), $A_T$ is the projected above-water transverse area, and $C_X$ is the longitudinal force coefficient.

Coefficients

The air force coefficient $C_X$ depends on the wind heading angle and the vessel’s above-water geometry. The principal references:

  • Isherwood (1972): empirical wind coefficients for typical merchant ship superstructure types.
  • Blendermann (1994): refined wind coefficients for various vessel types.
  • Wind tunnel testing: bespoke measurement for novel designs.

Importance

Air resistance is small for most cargo ships at design speed (typically 2 to 5% of $R_T$). It is significant for:

  • Container ships: high above-water area from container stacks; 5 to 12% of $R_T$.
  • Cruise ships: high passenger superstructure; 10 to 20% of $R_T$.
  • Car carriers / RoRo vessels: very high above-water area; 8 to 15% of $R_T$.
  • High-speed vessels: square-of-speed scaling makes air resistance proportionally larger at higher speeds.

For wind-assisted propulsion installations (Flettner rotors, wing sails), the wind force is intentionally generated to provide thrust; the air resistance framework is the basis for the wind assist calculation.

Added resistance in waves

In waves, the vessel experiences additional added resistance beyond the calm-water value, from:

  • Wave reflection at the bow: incident waves are partially reflected, generating a force opposing the ship motion.
  • Diffraction: the bow disturbs the incident wave field, generating additional radiation.
  • Drift force: oblique waves generate a sideways force that the rudder must counteract.

The added resistance is computed by Salvesen (1978), Boese (1970), Maruo (1957) or the modern STAWAVE-2 (ITTC 2014) empirical formulae, or by CFD. See seakeeping for the operational implications.

Shallow water resistance

In water depths comparable to or less than the vessel’s draught, additional resistance arises from:

  • Frictional resistance increase: the constrained flow has higher local velocity, increasing friction.
  • Wave-making resistance increase: shallow water modifies the wave system geometry, typically increasing wave-making resistance.
  • Squat-induced resistance: the squat sinkage increases the wetted surface area.

The shallow water resistance increase can be 10 to 50% above the deep-water value at typical channel transit conditions. The Schlichting (1934) and Lackenby (1963) corrections are the standard empirical methods.

Calculation methodology

Holtrop-Mennen (1982, 1984)

The Holtrop-Mennen formula is the dominant empirical method for early-design and preliminary resistance prediction:

  • Inputs: principal dimensions, hull form coefficients, design speed.
  • Outputs: total resistance and components (friction, wave-making, appendage, air).
  • Accuracy: typically ±5 to ±10% for well-behaved hull forms in the validity range.
  • Coverage: most merchant ship types and Froude number ranges.

Hollenbach (1998)

The Hollenbach formula is a more recent regression-based method, particularly accurate for modern container ships and bulk carriers in the slow-steaming Froude number range.

Series 60 / BSRA / MARIN

These are systematic series of model tests on standardised hull forms with parametric variations of the principal coefficients. The results allow rapid resistance estimation by interpolation.

Model testing

Direct model testing at a recognised basin (MARIN, HSVA, SSPA, NMRI, Krylov, MOERI) provides the most accurate calm-water resistance for a specific hull. Typical model test campaign costs USD 200,000 to USD 800,000 and takes 6 to 12 weeks.

CFD

Modern CFD with free-surface modelling (volume-of-fluid) provides accurate resistance prediction for any hull form. Cost is approximately USD 30,000 to USD 200,000 per hull form depending on complexity and validation requirements. CFD is increasingly used in newbuild design and for retrofit performance prediction.

Implementation in design

Newbuild design

For newbuild design, the standard resistance prediction process:

  1. Preliminary: Holtrop-Mennen or Hollenbach for early-design optimisation.
  2. Detail design: CFD-based optimisation of bow form, stern form, bulbous bow.
  3. Validation: model testing at one or more basins for the final design.
  4. Sea trials: full-scale validation at delivery.

Operational performance prediction

For in-service performance prediction (e.g. CII compliance projection, trim optimisation, voyage planning):

  • The vessel-specific resistance characteristics from sea trials and ongoing measurement are used.
  • The ISO 19030 framework normalises measured shaft power to standard conditions.
  • Variations from baseline (fouling, weather, draught variations) are accounted for.

See also

Stability and naval architecture

Operational and technical efficiency

Marine fuels

Regulatory frameworks

Ship types

Calculators

References

  • Froude, W. On Experiments with HMS Greyhound. Transactions of the Institution of Naval Architects, 1874.
  • Hughes, G. Friction and form resistance in turbulent flow. Transactions of the Institution of Naval Architects, 1954.
  • Holtrop, J. and Mennen, G. G. J. An approximate power prediction method. International Shipbuilding Progress, 1982.
  • Holtrop, J. A statistical re-analysis of resistance and propulsion data. International Shipbuilding Progress, 1984.
  • Hollenbach, K. U. Estimating resistance and propulsion for single-screw and twin-screw ships. Ship Technology Research, 1998.
  • ITTC. 1957 Friction Coefficient Correlation Line. International Towing Tank Conference, 1957.
  • ITTC. Recommended Procedures and Guidelines: 1978 ITTC Performance Prediction Method. International Towing Tank Conference, 2017.
  • Schoenherr, K. E. Resistance of flat surfaces moving through a fluid. SNAME, 1932.
  • Isherwood, R. M. Wind resistance of merchant ships. Transactions RINA, 1972.
  • Blendermann, W. Wind loading of ships: Collected data from wind tunnel tests in uniform flow. Hamburg Ship Model Basin, 1994.
  • ITTC. Recommended Procedures and Guidelines: STAWAVE-2 Empirical Formula for Added Resistance in Waves. International Towing Tank Conference, 2014.
  • Schneekluth, H. and Bertram, V. Ship Design for Efficiency and Economy, 2nd edition. Butterworth-Heinemann, 1998.
  • Lewis, E. V. (editor). Principles of Naval Architecture, Volume II: Resistance, Propulsion and Vibration. SNAME, 1988.
  • Bertram, V. Practical Ship Hydrodynamics. Butterworth-Heinemann, 2nd edition, 2012.

Further reading

  • Larsson, L. and Eliasson, R. E. Principles of Yacht Design. Adlard Coles Nautical, 4th edition, 2013.
  • Tupper, E. C. Introduction to Naval Architecture. Butterworth-Heinemann, 5th edition, 2013.
  • Schultz, M. P. Effects of coating roughness and biofouling on ship resistance and powering. Biofouling, 2007.