A regression-based approach to total ship resistance — components, formulas, applicability ranges, and practical guidance.
Before a ship enters a model test basin, and long before sea trials, designers need a number: how much power will this vessel need to achieve its service speed? The answer comes from resistance prediction, and for the majority of conventional displacement ships, the Holtrop–Mennen method is where that prediction begins.
The method was developed by J. Holtrop and G.G.J. Mennen at the Maritime Research Institute Netherlands (MARIN) and published in two landmark papers — first in 1978, revised and extended in 1982, and further refined in 1984. It is a statistical regression method, calibrated against an extensive database of model test results for a broad range of ship types. The result is a set of formulas that express each component of resistance as a function of hull geometry, speed, and waterline dimensions.
This article walks through the method in full: its theoretical basis, each resistance component, the empirical coefficients, the limits of applicability, and the practical judgment needed to use results responsibly.
Understanding where a method comes from matters as much as knowing how to apply it. The Holtrop–Mennen formulas are not derived from first principles — they are curve-fits to measured data, and their accuracy depends on how well a new vessel resembles the ships in the original dataset.
Holtrop and Mennen first presented their regression approach in the paper "An Approximate Power Prediction Method" (International Shipbuilding Progress, 1982). The database behind it included model tests for tankers, bulk carriers, general cargo ships, container vessels, frigates, and ferries — a wide but not exhaustive set of conventional hull forms. The 1984 update (also in ISP) introduced corrections for the bulbous bow and transom resistance terms, and refined several regression coefficients.
The method expresses total resistance as a sum of physically distinct components, each handled by separate sub-formulas. This structure makes it both transparent and flexible: components that are irrelevant (no bulbous bow, no submerged transom) can be set to zero without corrupting the rest of the calculation.
The Holtrop–Mennen method expresses total bare-hull resistance as the sum of several physically distinct components:
Each term is computed independently using its own sub-formula, all as functions of hull geometry and speed. The following sections treat each component in turn.
Skin friction accounts for the largest share of total resistance in slow, full-form vessels and remains significant even in faster ships. The Holtrop–Mennen method uses the ITTC-1957 model-ship correlation line as its friction baseline.
The wetted surface area S is central to the frictional resistance calculation. Where an exact value is not available from lines plans, Holtrop and Mennen offer the following regression formula:
Skin friction as computed by the ITTC-1957 line assumes a flat plate. Real ship hulls have curved surfaces and three-dimensional flow, which increase the viscous resistance above the flat-plate value. The form factor (1 + k₁) accounts for this.
Holtrop and Mennen derive k₁ from a regression on hull geometry:
| Stern Type | Cstern |
|---|---|
| Pram with gondola | −25 |
| V-shaped sections | −10 |
| Normal sections (most ships) | 0 |
| U-shaped sections with Hogner stern | 10 |
For most conventional merchant ships, Cstern = 0, and (1 + k₁) typically falls between 1.10 and 1.25. Fuller hull forms (higher CB) tend toward larger form factors; fine, slender hulls toward smaller ones.
Rudders, bilge keels, shaft brackets, bossings, stabiliser fins, and thrusters all generate additional resistance. These are grouped as appendage resistance.
When multiple appendages are present, an equivalent factor is used:
| Appendage Type | (1 + k₂) |
|---|---|
| Rudder behind skeg (twin-screw) | 1.5 – 2.0 |
| Rudder behind stern (single-screw) | 1.3 – 1.5 |
| Twin-screw balance rudders | 2.8 |
| Shaft brackets | 3.0 |
| Skeg | 1.5 – 2.0 |
| Strut bossings | 3.0 |
| Hull bossings | 2.0 |
| Shafts | 2.0 – 4.0 |
| Stabiliser fins | 2.8 |
| Dome | 2.7 |
| Bilge keels | 1.4 |
Appendage resistance is often underestimated at the preliminary design stage. On a twin-screw vessel with exposed shafts and large rudders, the appendage contribution can reach 15–20% of total bare-hull resistance.
Wave-making resistance is the dominant concern at higher Froude numbers. It rises steeply with speed and depends strongly on hull form — particularly prismatic coefficient and the position of the longitudinal centre of buoyancy.
The Holtrop–Mennen wave resistance expression is the most complex part of the method. It takes the general form:
The exponential structure of the wave resistance formula means that errors in CP or lcb propagate non-linearly into RW. Small changes in prismatic coefficient or the position of the centre of buoyancy can meaningfully shift the prediction — particularly in the Fn = 0.20–0.30 range where wave-making begins to dominate.
Even when a bulbous bow reduces wave-making resistance at design speed, it can generate additional resistance when it emerges from the water or operates far from its design Froude number. This additional pressure resistance is computed separately.
The emergence parameter PB increases as the bulb rises closer to the waterline. When PB falls below approximately 0.25, RB becomes negligible. The formula captures the fact that a partly-emerged bulb generates spray and pressure forces that add resistance rather than reducing it — a common condition in ballast passages.
At low speeds, an immersed transom creates a hollow behind the vessel and generates a drag-inducing low-pressure zone. As speed increases, the transom eventually "clears" (the water detaches cleanly) and this resistance diminishes.
For vessels with a dry transom at service speed (FnT ≥ 5), this term is zero and the immersed transom area actually reduces wave resistance through the c₅ coefficient in RW. For slower vessels with a wetted transom — common in tugs, ferries, and research vessels — RTR can be a non-trivial component.
The model–ship correlation allowance accounts for the roughness of a full-scale hull surface compared to the smooth model tested in the towing tank. It also incorporates any residual systematic difference between model prediction and ship performance.
CA is positive and adds resistance. It is typically in the range 0.0003–0.0008 for large ships in new-paint condition. The Holtrop formula for CA includes a length dependency that reflects how surface roughness effects scale with ship size.
Because the Holtrop–Mennen method is a regression fit, it is only reliable within the parameter ranges covered by the underlying model test database. Applying it outside these bounds does not necessarily cause an error message — but the result will be increasingly unreliable.
| Parameter | Valid Range | Notes |
|---|---|---|
| Froude number (Fn) | 0.10 – 0.45 | Transition range 0.40–0.45 can be imprecise; planing craft not covered |
| Block coefficient (CB) | 0.55 – 0.85 | Below 0.55: fine warship forms, accuracy degrades |
| Prismatic coefficient (CP) | 0.55 – 0.85 | Consistent with CB range |
| Length–beam ratio (L/B) | 3.9 – 9.5 | Extreme slenderness not well represented |
| Beam–draft ratio (B/T) | 2.1 – 4.0 | Very shallow hulls need care |
| Length of run (LR/L) | 0.50 – 0.75 | Related to stern form quality |
| Half angle of entrance (iE) | 1° – 90° | Extreme values (very fine or blunt bows) reduce accuracy |
The method is well suited to tankers, bulk carriers, general cargo ships, container vessels, and similar conventional forms. It is less reliable for:
| LWL (waterline length) | 182.0 m |
| B (moulded beam) | 30.4 m |
| T (design draft) | 10.8 m |
| ∇ (displaced volume) | 47,500 m³ |
| CB (block coefficient) | 0.800 |
| CP (prismatic coefficient) | 0.818 |
| CM (midship coefficient) | 0.977 |
| CWP (waterplane coefficient) | 0.872 |
| lcb (% L from midships, fwd +) | −1.25% |
| Service speed V | 14.5 knots (7.46 m/s) |
| Froude number Fn | 0.176 |
| ABT (bulb transverse area) | 28.0 m² |
| hB (bulb centroid height) | 4.2 m |
| AT (transom area, immersed) | 0.0 m² (dry transom) |
| Stern type | Normal (Cstern = 0) |
| Appendages | Single rudder, one shaft (estimated SAPP ≈ 45 m²) |
The total resistance RT gives effective (towing) power PE. To reach delivered power at the propeller shaft — the quantity that governs engine selection — a series of efficiency corrections are applied:
Holtrop and Mennen also provide regression formulas for the wake fraction w, thrust deduction t, and relative rotative efficiency ηR, making the method a complete preliminary power prediction package — from hull geometry to installed engine power.
The Holtrop–Mennen method routinely achieves resistance predictions within ±5–10% of model test results for ships that fall squarely within the calibration range. For vessels near the edges of applicability, errors of ±15–20% are possible. Several systematic tendencies are documented in the literature:
| Method | Type | Applicability | Typical Accuracy |
|---|---|---|---|
| Holtrop–Mennen (1982/1984) | Statistical regression | Conventional displacement ships, Fn 0.10–0.45 | ±5–10% (within range) |
| Guldhammer–Harvald | Systematic series | Full displacement ships, older method | ±10–15% |
| Savitsky (1964) | Regression | Planing craft, Fn > 1.0 | ±10–15% |
| van Oortmerssen (1971) | Regression | Small ships, tugs, trawlers, L < 80 m | ±5–10% within type |
| DSYHS (Delft yacht series) | Systematic series | Sailing yachts | ±5–8% |
| CFD (RANS solvers) | Numerical simulation | Any hull form | ±2–5% (validated grids) |
| Towing tank model test | Physical experiment | Any hull form | ±1–3% |
CFD has increasingly displaced regression methods in detailed design, particularly for novel hull forms or optimisation studies. However, Holtrop–Mennen remains the standard for quick estimates, the first iteration of a design spiral, and validation of CFD results in conventional regimes.
The Holtrop–Mennen method has remained in continuous professional use for over four decades — a longevity that reflects both the quality of the original regression work and the practical elegance of its formulation. No single method covers all ship types or all speed regimes, but for the displacement vessels that make up the commercial fleet, it occupies a central place in preliminary design.
Used with an understanding of its origins, its regression basis, and its applicability limits, it is a reliable first estimate. Used without that understanding — applied outside its range, with unchecked inputs, or without sea margins — it will produce numbers that look precise but may be significantly wrong. The method rewards the engineer who reads the original papers, not just the formula sheets.
The primary sources and standard texts that underpin or complement the Holtrop–Mennen method: