MATHEMATICAL DERIVATIONS
5.1 Introduction
As discussed in the earlier chapter of this ship resistance study, which incorporate with the effect of lateral drift due to wind and/ or current (for severe case) as the main mission, an initial investigation will be concentrated first. As the early phase, the study about this ship resistance by taking the lateral drift effect is developed by approximate resistance prediction method. At this stage, method approached by Holtrop and Mennen is selected due to wider range of types, sizes and limitation of ships/ vessels can be applied.
5.2 Holtop’s and Mennen’s Derivation
As far as ship resistance prediction is concerned, the original ship resistance prediction formula which was developed by Holtrop and Mennen (1982) is used as the main reference and guidance.
RTotal = RF(1+k) + RAPP + RW + RB + RTR + RA (5.1)
Where;
RF : frictional resistance according to the ITTC 1957 friction formula
(1+k) : form factor describing the viscous resistance of the hull form in relation to RF
RAPP : resistance of appendages
RW : wave- making and wave- breaking resistance
RB : additional pressure resistance of bulbous bow near the water surface RTR : additional pressure resistance of immersed transom stern
RA : model- ship correction resistance
Base on this mathematical formulation, together with the literatures, the effect of lateral drift in the ship resistance prediction using the method is investigated and developed. In deriving the ship resistance prediction with lateral drift effects formula, it is determined that the element/ parameter of ship’ velocity, VS principally is the main point of concerned. It is viewed that due to the severe lateral drift which is caused by the wind and/ or current, the vector of the ship’s velocity is modified, depending on the drift angle produced, β. As a result, there exists components of velocity, which represented by the longitudinal velocity, VS (L) and lateral velocity, VS (T). The drift angle, β presents due to the effect of drift, and it will be the main variables in influencing the velocity’s components. With the presence of lateral drift effect, it will modify the parameter of ship velocity (into longitudinal and lateral component) in the Holtrop’s and Mennen’s, and consequently, the related equations with the velocity parameter will be modified as well.
Modifying the existing formulae of the selected ship resistance prediction method, the ship’s velocity, VS parameter (due to the action of drift angle, β) started with the Frictional Resistance, RF. The Frictional Resistance, RF is broke down into
longitudinal component, RF (L) and lateral component, RF (T). With the same concept of frictional resistance determination, the ITTC 1957 is applied in determining the Frictional Resistance longitudinally and laterally.
Table 5.1: Frictional Resistance Component due to Drift Angle, β
F
Longitudinal Component Lateral Component
F
Besides the effect of lateral drift (drift angle, β) due to wind, when concerning the case of severe effect, there has another element that need to be considered. The significant lateral drift effect (severe case) which is caused by current is to be highlighted as well. This current element is said gave a severe case in lateral drift due to its velocity which acts onto the moving ship. As a result, a part of Frictional Resistance, RF due to ship velocity, VS, it is identified that there has additional frictional resistance interacts with the ship’s hull which is due to the velocity of current, VC. It is known as Frictional Resistance due to current velocity, RF(C). The value will be taken into account and combined with the existing Frictional Resistance due to ship’s velocity, RF(S). The additional resistance which due to the current is said only affects at this frictional resistance component since at the other components of resistance, namely appendages resistance (RAPP), wave- making resistance (RW), bulbous bow resistance (RB), immersed transom resistance (RTR) and model- ship correlation resistance (RA) the effect is not so significant. It is found that
the values produced by these components of resistance are very small and considerably neglected.
Frictional Resistance due to current, RF(C), mainly is determined depending on the current velocity, VC, as well as the current direction angle. In this study, current direction angle is the direction of the current (with its velocity) acting on the moving ship and it is represented by α. The various values of current direction angle, α will break down the current velocity components into longitudinal current velocity, VC (L)
and lateral current velocity, VC (T). Thus, it also will give the various effect of lateral drift (severe) with different current direction angle. Similarly applying the ITTC 1957 of frictional coefficient, the additional frictional resistance RF(C), which is due to the current velocity (longitudinally and laterally) is considered and determined as followed.
Table 5.2: Frictional Resistance Component due to Current Direction angle,α (In severe case)
Longitudinal Component Lateral Component
F
By referring to the main reference of ship resistance prediction method, the other component of resistance which will be influenced by the modified ship’s velocity is the wave making and wave breaking resistance, RW. In RW calculation, the component of ship’s velocity mainly influences the Froude’s number, Fn parameter and also coefficient of m2 as shown below.
Table 5.3: Wave Making Resistance Component due to Drift Angle, β
( )
Longitudinal Component Lateral Component
ν
Longitudinal Component Lateral Component
ν
The other component of resistance prediction which is modified due to the ship’s velocity parameter is called additional pressure resistance of bulbous bow near the water surface, or indicated as RB. In this component of resistance, it is affecting the parameter of Fni and the modified equation related as followed;
Table 5.4: Bulbous Bow Resistance Component due to Drift Angle, β )
Longitudinal Component Lateral Component
2
Same goes to the resistance component due to the additional pressure resistance of immersed transom stern, RTR. The additional resistance due to the immersed transom part is modified due to the modified c6 coefficient, which is influenced by the FnT as described below.
Table 5.5: Immersed Transom Resistance Component due to Drift Angle, β
6
Longitudinal Component Lateral Component
2
The ship’s velocity also one of the parameters in predicting the model-ship correlation resistance, known as RA. Due to that, it also modifies the model-ship correction resistance, RA as stated below;
Table 5.6: Model Correlation Resistance Component due to Drift Angle, β
A
A V SC
R =0.5ρ 2
Longitudinal Component Lateral Component
A S
L
A V SC
R ( ) =0.5ρ( cosβ)2 RA(T) =0.5ρ(VSsinβ)2SCA
By taking into account and modifying all the related resistance components, coefficients and functions which influenced by the ship’s velocity due to drift angle, β, and the additional frictional resistance due to acting current velocity, RF(C) at various angle, α, the problem of ship resistance prediction with lateral drift effect can be solved. There are slightly differences with the original procedure, which due to the presence of ship’s and current velocity component (longitudinal and lateral). The
prediction of ship resistance using the proposed prediction formula can be performed by solving it separately; longitudinally and laterally.
Applying the original Holtrop’s and Mennen’s prediction approach as the guideline, it is explored by considering the lateral drift effect due to wind and severe current. Both of these effects (small drift angle due to wind and severe drift angle due to current) are calculated separately at each component (longitudinal and lateral).
The modified procedure in determining the total ship resistance, RTotal are written as follows:
RTotal (longitudinally) = RF (L)’(1+k) + RAPP + RW + RB + RTR + RA (5.2)
Where RF (L)’= RFS (longitudinal) + RFC (longitudinal) (5.3)
RTotal (lateral) = RF (T)’(1+k) + RAPP + RW + RB + RTR + RA (5.4)
Where RF (T) ‘= RFS (lateral) + RFC (lateral) (5.5)
RFS = Frictional Resistance due to ship’s velocity, VS
RFC = Frictional Resistance due to current’s velocity, VC
The results of each component are combined in view of trigonometric relationship to obtain the Total Ship Resistance, RTOTAL with severe drift effects (at various angles).
The proposed trigonometric relation for this Total Ship Resistance, RTOTAL is written as follows:
RTOTAL= (RTotal(longitudinally)2+(RTotal(laterally))2 (5.6)
CHAPTER VI