PROPULSION PERFORMANCE PREDICTION METHOD FOR MULTI SHAFT VESSELS

A method is suggested for estimating the propulsion performance of a multi-shaft vessel. Possibility of applying one total thrust deduction coefficient to all ship shafts is justified and associated issues are discussed. Multi-shaft vessels are generically defined as vessels equipped with different types of propulsive units with non-similar geometry particulars or non-similar operating conditions, if geometry is the same. The paper suggests an iteration algorithm to estimate propulsion ship performance from standard data inputs obtained in model tests. An example of such calculations is given.


Introduction
Propulsion performance prediction based on model test data is a part of the ship theory first addressed by William Froude [1] in his publications back in the middle of the 19th century, who introduced the concept of equivalent flat plate and assumption of equal residual resistance coefficient for model and ship. It laid foundation for further in-depth research regarding water resistance to ship motion. William Froude and his son carried out the first experiments on model propellers to study their interaction with ship hull models.
A range of various ship theory aspects related to ship propulsion predictions from model tests were developed further in the 20th century [2]. A notable milestone on this way was the ITTC'78 performance prediction method for single-shaft vessels [3,4]. Ever since, persistent efforts have been continued for further elaboration of this method, for example [5,6], with its fourth revision published in 2017 [7].
One of the recent shipbuilding trends is construction of multi-shaft vessels enabling designers to achieve larger displacements and power/weight ratios within conventional draught constraints. This tendency is reflected in the ITTC Propulsion Committee recommendations regarding triple shaft vessels [8,9]. Unfortunately, these guidelines [9] have been published without prior across-the-board discussions in specialist journals. The purpose of this publication is to develop a method for estimation of multi-shaft vessel propulsion performance. The method suggested here is functional and free from deficiencies of the work [9]. In the authors' opinion,  [7] to the multi-shaft vessels.

Multi-shaft vessel definition.
In shipbuilding a multi-shaft vessel is traditionally understood as a ship propelled with three or more propulsive units. However, analysis of many modern ship designs indicates that this definition is not aligned with practical demands. For the purposes of this study the multishaft ship is understood as a vessel having at least two different types of propulsors. Propulsive units of a multi-shaft ship differ either by geometry particulars or by operating conditions at the same geometry. The ship can be equipped with N different types of propulsors. The number of units in each propulsor type is specified as ZP1, …, ZPi ,…, ZPN. Then, the total number of propulsor units is Examples for illustration of the introduced definitions are given below. A common twin-shaft vessel with two propellers (port & starboard) is not considered as a multi-shaft ship in this study. However, a similar twin-screw ship having some special hull design features, like moonpool, asymmetrical with respect to the ship centerline, falls into the multi-shaft vessel category. The point is that a moonpool on one of the ship sides would alter the propeller/hull interaction coefficients so that operating conditions for the propulsors would be different. In this example 12 2; 1 Another case: a propulsive system with one screw propeller on shaft in the CL plane and one azimuthing thruster installed behind the propeller should be treated as a multi-shaft system [10]. Here also we have 12 2; 1 For a common triple shaft vessel it is 12 2; 3; 1; 2 One of the vivid examples is described in Ref. [11] containing model test data for a port icebreaker Aker ARC 130А equipped with three Azipod units [12], with two thrusters at the stern and one thruster in the bow. Model experiments have provided conclusive evidences that the bow thruster slipstream has a strong effect on one of the stern units and significantly modify flow conditions around the starboard propulsor. Thus, each thruster of this triple shaft vessel has its own individual operating conditions. For the port icebreaker Aker ARC 130А we have A multi-shaft vessel may be outfitted with different types of propulsors of any number. According to usual practice the number of propulsive unit types is . All propulsors under consideration here should be amenable to open-water tests to enable evaluation of their hydrodynamic characteristics in isolation from hull. Each propulsive unit is supposed to include a screw propeller.
The approach described below is applicable to any multi-shaft vessel.

Determination of propeller/hull interaction coefficients for a multi-shaft vessel.
a. Interaction coefficients.
The propeller/hull interaction coefficients are determined by self-propelled model tests with an additional towing force FD. Self-propelled model tests are to be performed at the same shaft-wise power ratios as in full scale. This is easy to achieve by measuring propulsor torque _ Mi Q and using a well-known formula,

N 
Once revolution rates are adjusted for all the propulsors, the next stage is to perform selfpropulsion model tests with subsequent determination of propeller revolution rate for type 1 propulsors at which traction force of the towing carriage is equal to additional towing force FD. After this, revolution rates for other propulsor types _ MD i n can be calculated as per the obtained ratios for revolution rates of different propulsors.
In the overall analysis of all self-propelled test results the data obtained in the behind conditions and open-water propeller curves are processed using standard procedures recommended by ITTC 1978 [7]. In accordance with these recommendations the total thrust deduction coefficient is found: It can be noted that it is preferable to use open-water test data at relatively low revolutions for analysis of self-propelled model test results. These low revolutions should be chosen as close as possible to self-propulsion test conditions. In this case the relative rotative efficiency ηR_i is close to unity.
b.Total thrust deduction coefficient and partial thrust deduction coefficients. Towing resistance RT of three-shafted ships can be calculated as follows: where -total thrust deduction coefficient determined from model test data; -thrust deduction coefficients of the central and side propellers, respectively; -thrusts of central
(3) specifies a certain straight line in the coordinates of partial thrust deduction coefficients. This line can be obtained by giving various values to t2 at constant t, T1 & T2 and by calculating corresponding values of t1. Any pair of the partial coefficients belonging to this straight line provides the same towing resistance (total effective thrust) of the vessel. It should be noted that among possible pairs of partial coefficients we always have the following pair . It can be proven by direct substitution into Eq. (3). For this reason the total thrust deduction coefficient should enable correct estimation of ship's towing resistance (total effective thrust) in propulsion performance calculations. In this connection the authors believe it no advisable to introduce a concept of "partial thrust deduction coefficients".
c.Scale effect of wake fraction. In [9] for taking account of the scale effect the formula for wake fraction is used, which is proposed in [7] and derived for single shaft vessels and then adapted to the twin shaft vessels. This formula is as follows: Formula (4) is far from perfect and has long been criticized by researchers. At the time when this formula was derived the single-shaft merchant vessels used to have V-or U-shaped stern frames. It was presumed that propulsion performance could be improved by increasing the wake fraction. At present the merchant vessels are mainly designed with the buttock-flow stern resulting in lower towing resistance and drastic reduction in wake fraction. The wake fraction has become lower than the thrust deduction t. In these cases Eq. (4) gives paradox results when the full-scale wake fraction is higher than in model conditions. It follows directly from Eq. (4): at , . For avoiding such unrealistic results Ref. [13] suggests to apply a "provisional measure" assuming that To overcome these difficulties it is required to estimate the scale effect from formulae that do not include the thrust deduction coefficient. An example of such formulae is given below. It is derived from test data obtained for models with increased hull roughness [14].
where CVcoefficient of viscous resistance due to friction of hull plating. 12 , tt 12 ,  [15], as well as practical experience confirm that Eq.(5) is workable and provides satisfactory accuracy of results for practical applications.
For multi-shaft case Knowing the propeller-hull interaction coefficient t, wTS_i, ηRi, open-water characteristics of propulsors KTS_i , KQS_i and towing resistance of vessel RTS one can pass to propulsion performance estimations for the multi-shaft vessel.

Propulsion prediction method.
The main idea of the proposed method is that the thrust deduction coefficient is not divided between shafts, but a total thrust deduction coefficient is applied, which is estimated from self-propelled model test data. Therefore, the following relation can be written, which is based on the obvious equality between the ship's total effective thrust TE and resistance of water to ship motion RTS: where RTS-ship resistance at given speed VS. Ti thrust of i -th propulsor. It should be noted that the term of effective thrust TE_i means effective part of thrust.
For further reasoning we need to know the fractions of thrust of each type of propulsors i  expressed as: The initial data inputs for this method are ships' hydrodynamic resistance, hull/propeller interaction coefficients and open-water propeller curves found from model tests and extrapolated to full scale for a given ship speed .
Calculations are performed based on the iterative procedure using the thrust load coefficient , making it possible to meet the specified law of delivered power distribution by shafts of a multi-shaft vessel. Theoretically, the power distribution by shafts can be arbitrary. However, the same power is delivered to propulsors of the same type. The symbol denotes fractions of the total power delivered to each propulsor of the i-th type. The values of these fractions are defined by: where -power delivered to each propulsor of the i-th type. In this case it is supposed that the thrust of propulsors is distributed in accordance with the delivered power ratio.
Eq. (11) can be used to find the revolution rate , consumed power and thrust for each i-th propulsor type at the initial iteration step. (16) Then, the next iteration is calculated starting from Eq. (11).
The iteration process is completed when for all i with a given accuracy.

Case study.
The proposed method is applied to calculate propulsion performance of a shallow-water icebreaker. The principal dimensions of this vessel are given in Table 1. The ship is equipped with two side screw propellers in bossings and one pulling podded thruster in the middle. Diameter of all three propellers is 5.3 m. A ship model was made to λ = 26.5 scale to perform the entire cycle of traditional model experiments. Fig.1 shows a photo of the icebreaker model stern.  The sea water density was assumed S = 1025 kg/m 3 .
Hydrodynamic characteristics of the podded thruster are shown in Fig.4, and hydrodynamic characteristics of side propeller are given in Fig. 3.