4 Method of calculation
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Statutory Documents - IMO Publications and Documents - Circulars - Marine Environment Protection Committee - MEPC.1/Circular.796 – Interim Guidelines for the Calculation of the Coefficient fW for Decrease in Ship Speed in a Representative Sea Condition for Trial Use – (12 October 2012) - Annex – Interim Guidelines for the Calculation of the Coefficient fW for Decrease in Ship Speed in a Representative Sea Condition for Trial Use - Part 1: Guidelines for the Simulation for the Coefficient FW for Decrease in Ship Speed in a Representative Sea Condition - 4 Method of calculation

4 Method of calculation

4.1 General

4.1.1 The total resistance in the representative sea condition, RTw , is calculated by adding ΔRw , which is the added resistance due to wind and waves derived at 4.3, to the resistance RT derived following the procedure specified in paragraph 1.1.2.

4.1.2 The ship speed Vw is the value of V where the brake power in the representative sea condition PBw equals to PB , which is the brake power required for achieving the speed of Vref in a calm sea condition.

4.1.3 Where PBw can be derived from the total resistance in the representative sea condition RTw , the properties for propellers and propulsion efficiency ( ηD ) should be derived from the formulas obtained from tank tests or an alternative method equivalent in terms of accuracy, and transmission efficiency ( ηS ) should be the proven value as verifiable as possible.

The brake power can also be obtained from the reliable self-propulsion tests.

4.1.4 The coefficient for decrease of ship speed fw is calculated by dividing Vw by Vref as follows:

fw = Vw /Vref at the point where PB at Vref = PBw at Vw

4.2 Total resistance in a calm sea condition: RT

4.2.1 The total resistance in a calm sea condition is derived following the procedure specified in paragraph 1.1.2 as the function of speed.

4.3 Total resistance in the representative sea condition: RTw

4.3.1 The total resistance in the representative sea condition, RTw , is calculated by adding ΔRwind , which is the added resistance due to wind, and ΔRwave , which is the added resistance due to waves, to the total resistance in a calm sea condition RT .

RTw = RT + ΔRw
= RT + ΔRwind + ΔRwave

4.3.2 Added resistance due to wind: ΔRwind

  • 4.3.2.1 Symbols

    • AL : Projected lateral area above the designated load condition
    • AT : Projected transverse area above the designated load condition
    • B: Ship breadth
    • C: Distance from the midship section to the centre of the projected lateral area (AL ); a positive value of C means that the centre of the projected lateral area is located ahead of the midship section
    • CDwind : Drag coefficient due to wind
    • LOA : Length overall
    • Uwind : Mean wind speed
    • ρa : Air density (1.226(kg/m3))

  • 4.3.2.2 Added resistance due to wind is calculated by the following formula on the basis of the mean wind speed and wind direction given in table 2.1.

  • 4.3.2.3 CDwind should be calculated by a formula with considerable accuracy, which has been confirmed by model tests in a wind tunnel. The following formula is known for the expression of CDwind , for example:

4.3.3 Added resistance due to waves: ΔRwave

  • 4.3.3.1 Symbols

    • H : Significant wave height
    • T : Mean wave period
    • V : Ship speed
    • α : Angle between ship course and regular waves (angle 0(deg.) is defined as the head waves direction)
    • θ : Mean wave direction
    • ζa : Amplitude of incident regular waves
    • ω : Circular frequency of incident regular waves

  • 4.3.3.2 Irregular waves can be represented as linear superposition of the components of regular waves. Therefore added resistance due to waves ΔRwave is also calculated by linear superposition of the directional spectrum ( E ) and added resistance in regular waves (Rwave ).

  • 4.3.3.3 Added resistance in irregular waves ΔRwave should be determined by tank tests or a formula equivalent in terms of accuracy. In cases of applying the theoretical formula, added resistance in regular waves Rwave is calculated from the components of added resistance primary induced by ship motion in regular waves, Rwm and added resistance due to wave reflection in regular waves Rwr as an example.

    Rwave = Rwm + Rwr

    As an example, Rwm and Rwr are calculated by the method in 4.3.3.4 and 4.3.3.5.

  • 4.3.3.4 Added resistance primary induced by ship motion in regular waves

    • (1) Symbols

      • g : Gravitational acceleration
      • Hm : Function to be determined by the distribution of singularities which represent periodical disturbance by the ship
      • V : Ship speed
      • α : Angle between ship course and regular waves (angle 0(deg.) is defined as the head waves direction)
      • ρ : Fluid density
      • ω : Circular frequency of incident regular waves

    • (2) Added resistance primary induced by ship motion in regular waves Rwm is calculated as follows:

  • 4.3.3.5 Added resistance due to wave reflection in regular waves

    • (1) Symbols

      • B : Ship breadth
      • Bf : Bluntness coefficient, which is derived from the shape of water plane and wave direction
      • CU : Coefficient of advance speed, which is determined on the basis of the guidance for tank tests
      • d : Ship draft
      • Froude number (non-dimensional number in relation to ship speed)
      • g : Gravitational acceleration
      • I1 : Modified Bessel function of the first kind of order 1
      • K : Wave number of regular waves
      • K1 : Modified Bessel function of the second kind of order 1
      • LPP : Ship length between perpendiculars
      • V : Ship speed
      • α : Angle between ship course and regular waves (angle 0(deg.) is defined as the head waves direction)
      • αd : Effect of draft and frequency
      • ρ : Fluid density
      • ζa : Amplitude of incident regular waves
      • ω : Circular frequency of incident regular waves

    • (2) Added resistance due to wave reflection in regular waves is calculated as follows:

      where, dl is a line element along the water plane, βw is the slope of line element along the waterline, and domains of integration are shown in the following figure.

      Figure 4.1 Coordinate system for wave reflection

    • (3) Effect of advance speed αU is determined as follows:

      αU = CU (α)Fn

    • (4) The coefficient of advance speed in oblique waves CU (α) is calculated as follows:

      CU (α) = Max[Fs ,Fc ]

      • (i) Bf (α = 0) < Bfc or Bf (α = 0) < Bfs

        Fs = CU (α = 0) – 310 {Bf (α) – Bf (α = 0)}
        FC = Min[CU (α = 0), 10]

      • (ii) Bf (α = 0) ≥ Bfc and Bf (α = 0) ≥ Bfs

        Fs = 68 – 310Bf (α)
        FC = CU (α = 0)

        where

    • (5) The aforementioned coefficient CU (α = 0) is determined by tank tests. The tank tests should be carried out in short waves since Rwr mainly works in short waves. The length of short waves should be 0.5 Lpp or less.

    • (6) Effect of advance speed in regular head waves αU is calculated by the following equation where is added resistance obtained by the tank tests in regular head waves, and Rwm is added resistance due to ship motion in regular waves calculated by 4.3.3.4.

    • (7) Effect of advance speed αU is obtained for each speed of the experiment by the aforementioned equation. Thereafter the coefficient of advance speed CU (α = 0) is determined by the least square method against Fn ; see figure below. The tank tests should be conducted under at least three different points of Fn . The range of Fn should include the Fn corresponding to the speed in a representative sea condition.

      Figure 4.2 Determination of the coefficient of advance speed

Parent topic: Part 1: Guidelines for the Simulation for the Coefficient F W for Decrease in Ship Speed in a Representative Sea Condition
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