Added resistance in waves: “Pinkster” solver versus “Keuning 2006” comparison

Sailing yacht race. Yachting sport

1. Introduction

We had estimated the added resistance on waves of a day-boat project, the Dolfi 37, with a solver
using the so-called “Pinkster’s approximation” for the second-order drift forces, and it was of
interest to compare this approach commonly used for ocean engineering but unusual for a
sailboat, with the one proposed by Keuning and al. in 2006.

Dolfi 37 day-boat project wants to be in line with the historical Skerry cruisers, fine elegant
keelboats with a small cabin, which are still highly appreciated by amateurs. Dolfi 37 offers a
renewed version of this program, with a more modern hull and sail plan, with the comfort of a
larger cockpit and of a cabin with sanitary, for day or week-end sailing. With a length of 11.3 m for
a width of 2.4 m, Dolfi 37 shows classic lines and slenderness (L/B = 4.7) proper to this tradition. The
light weight is 2,560 kg, but the weight with loading taken into account in this study is 2,740 Kg.

Loa37.07′11.30 m
Lwl27.56′8.40 m
B max7.87′2.40 m
Bwl6.73′2.05 m
Draught5.41′1.65 m
Light weight5,644 lbs2,560 kg
Ballast2,601 lbs1,180 kg
187.3 ft²17.4 m²
Mainsail238 ft²22.1 m²
Jib214 ft²19.9 m²
Light Genoa323 ft²30 m²

Wave loads on a boat

A practical approach to seakeeping assessment is based on the potential flow formulation for a
perfect fluid. The resulting method is the so-called “diffraction and radiation.” The “diffraction”
formulation allows calculating the first-order (zero-mean) wave excitation loads which are
proportional to the incoming wave amplitude and oscillates at the same frequency as the incoming
waves with phases. The “radiation” formulation covers the calculation of the added masses and
linear wave damping of the floating body experiencing unit motions at the incoming wave
frequency. The first-order motions are the results of the dynamics of the boat in waves, including
the effects of these wave excitation, the effects of the added masses and linear wave damping, and
the effects of the hydrostatic stiffness. Other effects can be included as far as they can be
linearized (for example, damping related to the keel, rudder, and other appendices, some effect of the
sails, etc.).

In irregular multichromatic waves, the floating body response is the superposition of its
response to each individual monochromatic wave. These first-order results in the frequency
domain are the first-order transfer functions.

Second-order wave excitation loads are proportional to the square of the incoming wave
amplitudes, and contains a nonzero mean component as well as components oscillating with
frequencies higher and lower than the incoming waves (in the case of multichromatic waves). The nonzero mean components are called mean-drift forces, including the added resistance in waves (Raw)
when considering the force along the boat’s longitudinal X-axis.

Pinkster in 1980 [1] proposed an approximation to estimate these second-order steady drift loads
based on the same first-order potential flow approach.

These steady drift loads can be expressed as the contribution of second-order pressure calculated
with first-order quantities only. As a sum of products of sinusoidal quantities, the resulting load may
have non-zero mean values. Extra second-order pressure can be computed by solving the second-order problem but does not raise any mean component.

The solver, mostly dedicated to the mooring ships issue (which was the main motivation for investigating such drift forces), can also be used with a forward speed through the encounter wave frequencies for various headings. Here we consider an upwind sailing boat with waves coming at 45° to the boat X-axis. The wetted hull is modeled upright with its appendages. This method does not need any slender hull assumption as in a 2-D strip theory.

Keuning et al. in 2006 [2], in line with previous investigations to estimate the added resistance in waves of sailboats which included towing tank tests of models, proposed a formulation of the Raw/a² transfer function. The transfer function they came up with is a polynomial expression where the coefficients can be derived from a shortlist of boat characteristics: Lw, Bw, Tc, Dc, Cp, and Kyy (the longitudinal radius of gyration).

The approach was first performed with a 2-D linear ordinary strip theory, then confronted and adjusted with experiments of various models of the Delft Systematic Yacht Hull Series (DSYHS) series. Then the formulation was compared with 2D strip theory results for other boats not belonging to the DSYHS and the approximation method was revealed as accurate as the original calculations using the entire ship motions and added resistance in waves calculation routine for a wide range of ship lengths.

Based on these Raw/a²(ω) transfer functions computed by either “Pinkster” or “Keuning 2006” methods (ω being the wave pulsation) and on irregular waves of spectrum S(ω), the mean Raw value is computed using: Raw = ∫ (Raw/a²) * 2 * S(ω) * dω.

List of symbols

  • a: wave amplitude;
  • α: wave heading, 45° from X-axis when the boat is sailing upwind corresponds to 225° heading in “Pinkster” and to 135° in “Keuning 2006”;
  • Bw: waterline beam;
  • Cp: hull prismatic coefficient;
  • D: displacement;
  • Dc: bare hull displacement;
  • Fn: Froude number;
  • g: gravity acceleration;
  • Hs: significant wave height;
  • Lw: waterline length;
  • Raw: added resistance drag;
  • Ryy (in “Pinkster”) or Kyy (in “Keuning 2006”): the longitudinal radius of gyration;
  • S(ω): wave spectrum;
  • ω: wave pulsation;
  • λ: wave length;
  • Tc: hull body draft;
  • Vb: boat speed
  • X, Y, Z-axis: boat axis, respectively longitudinal, transversal, vertical.

2. Transfer functions (Raw/a²) computed with “Pinkster” solver

The (Raw/a²) transfer functions are computed with the “Pinkster” solver for the Dolfi 37 with waves at 45° (upwind conditions), with using λ/Lw in abscissa and for boat speed 0, 2, 4, 6 and 7 Knots, i.e. Fn 0, 0.112, 0.224, 0.336 and 0.392 respectively: 

Yellow : Vb = 0 Knots; Green: Vb = 2 Knots (Fn 0.112); Blue Vb = 4 Knots (Fn 0.224); Red: Vb = 6 knots (Fn 0.336); Black Vb = 7 Knots (Fn 0.392)

One can see that for Fn 0.392, the (black) curve is lower than the one for 0.336 (in red), which is neither logical nor relevant: “Pinkster” solver losses its relevance from such Froude number. Fortunately, most of the sailboats are limited to Fn ~ 0,35 sailing upwind on calm water and to Fn from 0.30 to 0.35 when sailing upwind on waves, so the Fn 0.392 case is of no use in the VPP process.

One can see also that on the left side, for Fn > 0.2 and λ/Lw < 0.7, the curves are going to zero and even to negative (negative drag = thrust), which also is not logical: we have turned these values to zero before the integration with S(ω). For such Fn values, this issue may be due to an insufficiently dense meshing of the hull.

3. Transfer functions (Raw/a²) computed with “Keuning 2006”

The original coefficients file provided by “Keuning 2006” includes, among other, coefficients for wave direction of α =120° and α =140°. But it does not include coefficients for the case under study, i.e., α = 135° (equivalent to 45° heading waves). Therefore, we have computed the coefficients for α = 135° interpolating linearly the coefficients provided for α =120° and α =140°. We have also built a dedicated calculation spreadsheet that computes, for α = 135°, the coefficients for any Fn between 0.2 and 0.45 and Ryy/Lw between 0.2 and 0.3.

4. Comparison of the transfer functions obtained with “Pinkster” and “Keuning 2006” methods

The value of the transfer functions (Raw/a²) computed with the “Pinkster” and “Keuning 2006” methods have been drawn together with JONSWAP sea spectra (ϒ = 3.3) computed for:

  • Hs = 0.2 m (Wind 6 Knots & Fetch : 10 km);
  • Hs = 0.4 m (Wind 8 Knots & Fetch : 21 km);
  • Hs = 0.6 m (Wind 10 Knots & Fetch : 30 km);
  • Hs = 0.8 m (Wind 12 Knots & Fetch : 37 km);
  • Hs = 1.0 m (Wind 14 Knots & Fetch : 43 km).

The calculations consider the following Dolfi 37 loading state:

Displacement (kg)2,750Hull Displacement (m³)2.52446
Lw (m)8.62Bw (m)2.075
Tc (m)0.3777Ryy (m)2.03
Cp0.530Lw / Bw4.15
Bw / Tc5.49Ryy / Lw0.235
Lw / Dc1⁄36.33

4.1. Comparison for α = 135°, Kyy/Lw = 0.235 and Fn = 0.224 (Vb = 4 knots)

Red continuous: “Pinkster” solver; Red points: “Keuning 2006”; Blue: S(w) for Hs = 0.2, 0.4, 0.6, 0.8, and 1.0 m

One can see a good agreement for the Raw/a² peak (position and magnitude) and two differences:

  • for λ/Lw < 0.7, the “Keuning 2006” value tends to 8,000 while the “Pinkster” drops to zero. Yet, for these values of λ/Lw, the JONSWAP spectra are also close to zero, meaning that the resistance in waves (Raw) on this range of λ/Lw should be low (except perhaps for small waves of Hs = 0.2 m).
  • for 1.1 < λ/Lw < 2.1, the “Keuning 2006” values are lower than the “Pinkster” one in a zone where the JONSWAP spectra are not negligible, especially for Hs = 0.4 m.

4.2. Comparison for α = 135°, Kyy/Lw = 0.235 and Fn = 0.336 (Vb = 6 knots)

Red continuous: “Pinkster”; Red points: “Keuning 2006”; Blue: S(w) for Hs = 0.2, 0.4, 0.6, 0.8, and 1.0 m

Besides the same remarks as above, we also note the peak with “Keuning 2006” is a bit higher than with “Pinkster”.

4.3. Comparison for α = 135°, Kyy/Lw = 0.235 and Fn = 0.392 (Vb = 7 knots)

Red continuous: “Pinkster”; Red points: “Keuning 2006”; Blue: S(w) for Hs = 0.2, 0.4, 0.6, 0.8, and 1.0 m

Here, the “Pinkster” curve for Fn 0.392 is not relevant because it is lower than the “Pinkster” one for Fn 0.336 as already mentioned. This fact makes this curve also radically lower than the “Keuning 2006” one for Fn 0.392.

5. Added resistance in waves (Raw) computation with JONSWAP spectra

We have calculated the added resistance in waves for Dolfi 37 for Vb = 4 Knots (Fn = 0.224) and 6 Knots (Fn = 0.336) considering the JONSWAP spectra mentioned above:

Wind (Knots)68101214
Fetch (km)1021303743
Significative wave height, Hs (m)

The results are given in % of the Displacement 2,750 kg:

Raw (% Displacement) at Fn 0.224 (Vb = 4 Knots)

λ/Lw <
0.7 < λ/Lw <
λ/Lw >
Keuning 20060.120.310.350.400.44
λ/Lw <
0.7 < λ/Lw <
λ/Lw >

Raw (% Displacement) at Fn 0.336 (Vb = 6 Knots)

λ/Lw <
0.7 < λ/Lw <
λ/Lw >
Keuning 20060.130.440.480.550.60
λ/Lw <
0.7 < λ/Lw <
λ/Lw >

From the tables above we can say:

  • For Fn = 0.224, the concordance is quite good. We note that, as expected, bigger values are obtained with “Keuning 2006” for Hs = 0.2 m. As we have seen before, this is due to the contribution of the left side of the λ/Lw range (when < 0.7).
  • For Fn = 0.336, in addition to the left side effect, “Keuning 2006” contributes with higher values in the central zone where 0.7 < λ/Lw < 2.0.
  • Both “Pinkster” and “Keuning 2006” barely contribute to the total Raw (maximum value obtained is 3%) for λ/Lw > 2.0.

6. Conclusions

About the use of such “Pinkster” solver, the comparison shows two weak points :

  • the Raw/a² values for very small waves such as λ/Lw < 0.7 and Fn > 0.2;
  • the Raw/a² peak value when the Froude becomes too high (Fn > 0.34).

One can wonder if these points could be improved with a more dense meshing of the hull. Fortunately, the first point gives a small difference for Raw except on very small waves (Hs = 0.2 m). The second point is not a drawback for our purpose because Fn < 0.34 is a usual range for a sailboat sailing upwind on waves, and we have shown the agreement between the two approaches considered is good to quite good within this range.

About the use of formulation “Keuning 2006” for Raw/a², with just Kyy, Lw, Bw, Tc, Dc, and Cp as input data: the comparison shows a very good agreement in the useful range Fn 0.20 to 0.34, with an independent direct computation not using the 2D Strip theory method and carried out with a sailboat (Dolfi 37) of shape and characteristics different from the DSYHS models having been used for its establishment.


[1] Low frequency second-order wave exciting forces on floating structures | TU Delft Repositories
Pinkster J. A. Thesis – 1980.

[2] «An Approximation Method for the Added Resistance in Waves of a Sailing Yacht» by
J.A. Keuning, K.J. Vermeulen, H.P. ten Have – MDY06 Madrid march 2006.

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