A NUMERICAL STUDY ON HYDRODYNAMIC INTERACTION AMONG OBLIQUE SHIP HULL, PROPELLER AND RUDDER
Susumu Tanaka (Akishima Laboratories, Japan)
Koyu Kimura (Akishima Laboratories, Japan)
Abstract; For accurate computer simulations of ship manoeuvrability, it will be required to estimate high accurate hydrodynamic interaction among ship hull, propeller and rudder as well as hydrodynamic forces acting on a bare hull, taking account of detailed ship form such as stern profile and frame line. Use of CFD(Computational Fluid Dynamics) , particularly viscous flow simulation method can deal with complicated hull form. CFD programs are already in use at design stage for the prediction of propulsive performance.
Recently CFD technique has been applied to the ship manoueuvrability problem. It will be important for ship designers to evaluate a ship's propulsion performance and manoeuvrability by using a common tool. Today calculations for interaction among hull, propeller and rudder by CFD have been made. CFD, however, is still under development for a manoeuvring ship.
In this paper, we discuss on hullpropellerrudder interaction for a ship in several oblique motions by using CFD technique. Interaction is essential for the mathematical model of ship manoeuvring motion. An emphasis is placed here on rudder forces in oblique motion. The flow around an oblique ship with constant rudder angle is computed using a grid system with rudder. A propeller is expressed by infinitely bladed propeller. Calculated rudder forces without propeller are compared with captive model test data.
1. INTRODUCTION
The introduction of the IMO provisional steering standards has made it necessary to perform quantitative assessments of vessel maneuverability at the design stage. Except for standards governing stopping performance, we must be able to predict, with high precision, the maneuvering motion of a vessel in turning trials at the maximum rudder angle at a given level of main engine power output, and in 10° and 20° Z tests.
It is already possible to make fairly accurate predictions of a vessel's maneuverability, which are sufficient for practical use, using simulations involving numerical models based on maneuvering motion descriptions from the Mathematical Modeling Group (MMG). However, the accuracy of these predictions is affected by the influence of hydrodynamic forces acting on the hull, propeller, and rudder, as well as the accuracy with which we can determine the hydrodynamic interactions between the hull, propeller, and rudder.
Particularly with regard to the evaluation of course stability, the sensitivity of the input hydrodynamic data tends to increase as the course stability of the ship deteriorates [1]. The accuracy of estimating maneuverability is most affected by the linear microcoefficient of the ship hull hydrodynamics, and secondly, by the hydrodynamic coefficient (the hydrodynamic data that indicates the effects of the ship's hull on the rudder). Thus, it is important to be able to predict hydrodynamic interactions with high accuracy from the initial design stage.
Theoretically, interactions between the ship hull, propeller, and rudder are closely linked to the viscosity of the flow field. In the past, this has often discouraged the use of theoretical analysis in place of conventional flow modeling[2]. A new approach, known as Computational Fluid Dynamics (CFD), solves directly for the flow field and can be used to model the complex flows around the hull involving threedimensional separation. CFD has been employed in maneuverability studies of interactions between the hull and propeller and between the hull and rudder.
Takada, for instance, has used a CFDbased propeller model to calculate hydrodynamics during oblique motion and turning maneuvers while the propeller turns [3]. Similarly, Miyazaki et al.[4] employed CFD to investigate the interactions between the hull and rudder during oblique motion (a phenomenon of maneuvering motion that requires further investigation). They demonstrated that CFD can be used to calculate, with reasonable accuracy, the hydrodynamics acting on the hull and rudder. Other researchers have investigated the application of CFD techniques to maneuverability studies by applying CFD to assess hydrodynamics over time, without recourse to hydrodynamics expressions based on conventional numerical models, and at the same time, deploying techniques based on motion calculation[5][6]. It is believed that such CFD techniques can also be applied to maneuvering motion calculations to permit direct incorporation of factors such as the memory effect of hydrodynamics and the impact of interaction into calculations. Further applied research into the use of CFD in maneuverability analysis is expected to lead to further breakthroughs.
The idea of developing hydrodynamic expressions based on MMG numerical modeling (as used in maneuverability performance analyses) is physically significant. Because this approach uses simple expressions wherever possible[7], it can be used to generate a macrolevel description of a phenomenon, as exemplified by the average rudder inflow angle and the average rudder inflow velocity. The advantages of this approach can be applied, for example, to validation processes for CFD techniques (which will become increasingly sophisticated). It will be possible to perform an initial check on CFD results by identifying hydrodynamics in the form of integration values from the vast volumes of detailed information generated by CFD and analyzing this data in accordance with the expressions used in numerical models. Detailed information on the flow field and hydrodynamics at any given time or position provided by CFD simulations can be used to evaluate the numerical models currently in use and improve the accuracy of same, as well as improving the accuracy of maneuvering motion analysis.
This paper assesses the application of CFD techniques (the subject of applied research in several fields) to the problem of interactions among the ship hull, propeller and rudder, in order to determine the usefulness of CFD techniques in this area. Flow calculations were performed for a tanker ship (Esso Osaka) to determine the flow around rudder behind oblique hull without propeller or with propeller. We calculated the interactive forces on the hull generated by steering, as well as the rudder normal force characteristics behind the ship hull, and compared calculation results with observations from scale model tests[2].
2. MATHEMATICAL MODEL OF MANNOEUVRING MOTION AND INTERACTION FORCES AMONG THE SHIP HULL, PROPELLER AND RUDDER
2.1 Coordinate System and Hydrodynamics Acting on the Rudder
Figure 1 shows the coordinate system for maneuvering motion and for the hydrodynamics acting on the rudder. The coordinate system adheres to the conventions of the MMG model, taking the center of gravity of the hull as the origin and employing a fixed hull coordinate system for maneuvering motion. Here, U is the ship velocity, u, v, and r are the velocity components, β is the drift angle, δis the rudder angle, FN is the rudder normal force, UR is the effective rudder infrow velocity, UR and VR are the components of the effective rudder inflow velocity, and aR is the effective rudder inflow angle.
Fig.1 Coordinate Systems of Manoeuvring Motion
2.2 Expression of Rudder Force
Rudder force represents an important controlling force during maneuvering. The rudder, which is normally located at the stern of the ship, is significantly affected by the motion of the hull and hull form.
The following expressions can be used to describe the rudder force, the longitudinal hydrodynamic force XR lateral hydrodynamic force YR and moment NR.
XR =  (1tR)FNsinδ
YR =  (1 + aH)FNcosδ (1)
NR =  (xR + aHxH)FNcosδ
where, tR, aH, and xH are the coefficients of interaction generated by the rudder that act upon the hull.
The rudder normal force FN is defined as follows[8].
αR = δ + δ0  vR/uR (2)
uR = U(1  wR)[1 + C・g(s)]^{1/2}
vR = Uγ(β  2xR・r)
g(s) = ηK｛ 2  (2  K)s｝s/(1  s)^{2}
η = Dp /hR
κ = 0.6(1  wp)/(1  WR) (3)
s = 1  u(1  wp )/(nP)
WR = WRO exp( 4.0β^{2}R)
βR = β  2χR・r
where, p is the water density, AR is the surface area of the rudder, ∧is the aspect ratio of the rudder, C is a constant that represents the difference in rudder force when turning to either side, xR is the distance in the forward direction from the center of the ship to the rudder position, WRO is the effective wake coefficient at the rudder position when the ship is proceeding straight ahead, DP is the diameter of the propeller, HR is the height of the rudder and P is the propeller pitch.
3. CFD CALCULATION METHOD
Flow field calculations were performed using NICE[9], a form of CFD code. A modified BL (BaldwinLomax) model was used as the turbulence model. The effects of free surface were ignored. Table 1 lists the principal particulars of the Esso Osaka, the 278,000 DWT ship used for the calculations. Figure 2 shows the calculation grid system for the hull (with rudder but without propeller). The grid partitioning was I × J × K = 101 × 69 × 41,for a total of approximately 290,000 units.
Table.1 Principal particulars of ship.
Items 

Length between perpendiculars 
325.00 
(m) 
53.00 
Breadth moulded(m) 
21.73 
Draught moulded at midship(m) 
0.8293 
Block coefficient 
27,680 
Wetted surface area(m^{2}) 
319,040 
Displacement(t) 

Rudder 

Breadth(m) 
9.0 
Height(m) 
13.85 
Aspect ratio 
1.539 
Effective rudder area ratio 
1/56.66 

Fig.2 Grid system around stern with rudder
4. RESULTS OF CFD CALCULATIONS OF INTERACTION BETWEEN THE HULL AND RUDDER
In order to validate the CFD approach in a hydrodynamic field, we performed flow field calculations during rudder angle changes for both straightahead and oblique motion, using a Reynolds number of 1.0 × l06 (equivalent to a scale model trial). The results are shown in Figure 3.
Fig.3 
An example of computed flow around a ship with a rudder in a manoeuvring motion. 
The CFD calculation results for the interactive force from the rudder acting on the ship hull and the rudder normal force characteristics behind the ship were then compared to results from the scale model trials.
