Simulation of the Maneuvering Behavior of Ships in ...

31st Symposium on Naval Hydrodynamics Monterey, California, 11-16 September 2016

Simulation of the Maneuvering Behavior of Ships in Adverse Weather Conditions Apostolos Papanikolaou, Nikos Fournarakis, Dionysia Chroni, Shukui Liu, Timoleon Plessas (Ship Design Laboratory-National Technical University of Athens, Greece) Florian Sprenger (MARINTEK, Norway) ABSTRACT The research presented deals with the simulation of ship’s maneuvering performance in adverse weather conditions. A 4-DoF (surge, sway, yaw and roll) nonlinear maneuvering model is formulated and solved in the time domain by use of MATLAB’s software environment, in which several subsystems referring to the calculation of the hull, propeller and rudder forces/moments are integrated. The maneuvering derivatives and forces/moments are being estimated by use of model experimental data, of semi-empirical relationships and the application of potential theory 3D panel codes and CFD-RANS solvers. The wave-induced forces/moments are herein considered through the mean second order wave forces/moments, which are pre-calculated and provided to the maneuvering simulation solver as response surfaces. The validation of the theoretical/numerical approaches is herein demonstrated by in depth studies on the maneuvering coefficients and on typical maneuvers of the DTC standard containership in waves and the KVLCC2 standard tanker. Associated model experiments were conducted by MARINTEK for the DTC containership and CEHIPAR for the KVLCC2 tanker under the EU funded project SHOPERA. INTRODUCTION Following the introduction of guidelines for the calculation of the Energy Efficiency Design Index (EEDI) by the International Maritime Organization (IMO), concerns were expressed by the maritime industry regarding the safety of ship operations, when adopting the requirements on the allowable maximum powering for some ship types and sizes. Namely, while taking the EEDI reference lines into account, the sufficiency of the specified minimum

propulsion power, as well as the controllability of ships for maintaining safe navigation in adverse weather conditions were disputed; this is even more of a concern when considering that during Phase 3 of the EEDI implementation (IMO,2011) , as of 1st Jan. 2025 and onwards, the required EEDI should be further reduced by 30% compared to the present level. The EU funded project SHOPERA (www.shopera.org) addresses these concerns by in depth research on the maneuvering performance and safety of ships in complex environmental and adverse weather conditions. Present standard criteria of IMO consider the maneuvering performance of a ship only in calm water conditions (IMO,1993). The assessment of the maneuvering performance in adverse weather conditions is, however, of high importance, both with respect to ship’s safety and her overall hydrodynamic performance, including the efficiency of operations. It is evident that when a ship is operating in adverse weather conditions, wind and wave-induced forces and moments will act on her and change her hydrodynamic and motion behavior. Therefore, when simulating the maneuvering performance in realistic sea conditions, the consideration of the above effects is essential. Eventually, the classical maneuvering and seakeeping problems of ship hydrodynamics need to be solved in parallel by an integrated timedomain simulation procedure. Some additional complexity of the problem at hand arises when the ship is operating in extremely severe seas and/or in limited waters; this is mainly a maneuvering/steering problem of the ship at low forward speed, rather than a minimum powering problem, as inferred by the IMO MEC212 (2012) EEDI guidelines. This poses new challenges to the current maneuverability criteria, which hold only for calm water conditions, and existing maneuvering assessment methods; and related simulation codes,

which may consider complicated maneuvers at appreciable speed in calm water, but not at low speed in waves. In short, the physics of the problem at hand is different and needs special care. Back to the roots of the maneuvering theory, it was the milestone work of Abkowitz (1964) that initiated the development of various mathematical formulations to estimate maneuvering forces/moments and to solve analytically the equations of governing ship motions in the horizontal plane. Several researchers further developed and refined the general approach of Abkowitz, among them Hirano (1980) with 3-DOF and Son and Nomoto (1981) with 4-DOF equations of motion models. Traditionally, next to simplified semiempirical methods of limited validity/accuracy, standard Planar Motion Mechanism Tests (PMM) and Circular Motion Tests (CMT) were used to reliably estimate the maneuvering derivatives, yet in a costly and time consuming manner. Recent advances in computer hardware and software, namely Computational Fluid Dynamics (CFD), allowed a drastically different approach to the problem, namely the direct numerical simulation of ship's maneuvering. Research works by CuraHochbaum et al (2002, 2006, 2008), Stern et al., (2011), Simonsen et al. (2012), El Moctar et al (2016) and several others suggest that estimation of the hydrodynamic forces and moments acting on ship's hull, as well as the simulation of ship maneuvers in calm water and in waves are nowadays possible with satisfactory accuracy using CFD methods. However, the computational effort invested herein is considerable, thus faster approaches are often being adopted, when assessing ship's performance in early design stages or for assessing ship’s operation. Bailey et al (1997) were the first to introduce a unified seakeeping and maneuvering model, while accounting for the unsteady memory effects by use of Cummins’ convolution integrals. However, this approach was basically linear in nature and did not properly account for the mean second order wave effects which are of paramount importance in the problem. Later on, Fang et al. (2005) proposed a nonlinear model for the simulation of the maneuvering performance of a ship in waves. Their approach to the calculation is similar to that of Bailey, but without the inclusion of the memory effects; again the importance of the mean second order wave loads was not accounted for. Hirano et al (1980) were the first to propose the concept of separation of seakeeping and maneuvering problem, in which the linear wave-induced ship motions are

assumed to occur at much higher frequency, compared to the slowly varying maneuvering motions in the horizontal plane; the interaction of the seakeeping and maneuvering problems is through the mean wave-induced second order forces and moments. Along the same lines, Yasukawa (2006, 2009) adopted the two time-scales model accounting for the low frequency maneuvering and high frequency seakeeping motion; he used a 6-DOF motion formulation and coupled the seakeeping with the maneuvering motions. The second order mean drift forces were computed by the far field approach of Maruo (1963). Likewise, Skejic and Faltinsen (2008) employed a unified seakeeping and maneuvering, slender-body theory approach, using the two time scale modeling for the maneuvering and seakeeping motions and applied it to the simulation of the maneuvering of a Mariner hull in waves. More recently, Seo and Kim (2011) used a similar approach to the ship maneuvering in waves, where a 3D panel method was used in the estimation of the mean 2nd order wave forces in the horizontal plane. Finally, Yasukawa & Yoshimura (2015) presented the MMG maneuvering model proposed by the Japanese Towing Tank Conference (JTTC), which aims at a standardization of the ensuing mathematical model, which is used for the ship maneuvering simulations; this is an important issue when applying different methods to the estimation of the various components of the maneuvering equations while not caring for their proper interpretation in the set-up of the equations of motion. The present paper outlines a major part of our research conducted within the SHOPERA (20132016) project, whose objective is to develop suitable methods/tools and to conduct systematic case studies which will enable the development of improved IMO guidelines on determining the minimum propulsion power of ships to maintain satisfactory maneuverability in adverse weather conditions. The interest is particularly on the maneuverability of large ships (like the herein studied DTC containership and KVLCC2 tanker) at low speed (4 to 6 knots), namely to examine whether such ships have sufficient installed power and maneuverability to escape an upcoming storm when operating in coastal waters. We adopt the two-time scales concept with respect to the interaction of the seakeeping and the maneuvering problem and formulate a 4-DoF (surge, sway, yaw and roll) nonlinear maneuvering model for the ship motions in calm water and in waves. The solution of the resulting set of Ordinary

Differential Equations (ODE) in the time domain is being accomplished by use of MATLAB’s software environment (Chroni et al., 2015). We elaborate on the calculation procedures for the hull, propeller and rudder forces/moments and the maneuvering derivatives in the equations of motion. The maneuvering coefficients are herein alternatively estimated by semi-empirical relationships, potential flow theory 3D panel methods, and advanced CFDRANS solvers. The results are validated against model experimental data for two standard type vessels, namely the DTC containership (ISMT, 2016) and the KVLCC2 tanker (SIMMAN, 2008).

FORMULATION OF THE MANEUVERING PROBLEM

Finally, as shown in Figure 1, δ is the rudder angle (negative for rudder to starboard), ψ and β are the ship’s heading and drift angle respectively, while α is the wave encounter angle with reference to the earthbound system. MANEUVERING PROBLEM Ship’s motions are herein restricted to the horizontal plane motions with the addition of heel, thus to 4 DOF (surge, sway, yaw and heel). We assume the ship advancing with forward speed V in regular waves of certain heading and undergoing a maneuver by action of her rudder and propeller induced forces. The effect of the maneuver on the high frequency seakeeping motions is herein neglected, whereas the effect of the incident waves on ship’s horizontal plane motions is limited to the action of the mean second order wave forces and moments. In this case the equations of motion with respect to the shipbound system are as follows (Lewis, 1989):

m((u-rv)-xG r 2 )+m(zG (2φrcosφ+φsinφ))= X H +X R +X PROP +R(u)+X w +RX m((v-ru)+xG r)+m(zG ((r 2 +φ2 )sinφ-φcosφ)= YH +YR +Yw +RY

Ι xx φ-I xz rcosφ-mxG cosφ  v+ru = K H +K R +K w +zG RY

I

yy

COORDINATE SYSTEMS Two coordinate systems will be used, namely an earthbound system and a second one moving with the vessel. As shown in Figure 1, the earthbound, right handed coordinate system O(i, j, k) with the k–axis pointing downwards, is used for the identification of the position and orientation of the vessel, during a maneuver. The body-fixed system o(x, y, z), advances with the ship’s forward speed V and rotates with rotational speed r; it is mainly used for the calculation of the forces and moments acting on the ship during the maneuver. A second earthbound coordinate system is introduced in order to express the incident wave and the resulting mean second order wave forces acting on the ship; this system is also right-handed with its i-axis positive towards the advance of the wave and its k–axis pointing upwards.

(2) (3)

sinφ2 +Ι zz cosφ  r+I xz  φcosφ-φ2 sinφ  +

mxG  v+ru  +mzG sinφ  u-φv = Figure 1: The coordinate systems

(1)

(4)

N H +N R +N w +M Z where, m is the ship mass, Ixx and Izz are the moments of inertia about the x- and z-axes respectively, xG and zG are the coordinates of the center of ship’s mass (gravity) with respect to the body-fixed coordinate system (i.e. CG = [xG, 0, zG]). φ is the angle of heel (roll). u , v , r and  are the time varying accelerations, which are also defined with respect to the body-fixed coordinate system. Their integration in time, leads to the u, v and r velocity components, whereas the integration of the velocities leads to determination of ship’s position in the earthbound coordinate system. X, Y, K and N represent the surge, sway, heel and yaw direction of force and moment components, respectively, and subscripts H, R, prop and w indicate forces and moments referring to hull, rudder, propeller and wind, respectively. R(u) is the hydrodynamic resistance due to the ship

advance with speed u. Finally, Rx , RY and M Z are the mean second order wave forces and moments in the horizontal plane. The hydrodynamic forces and moments acting on the hull are modeled as nonlinear functions of the accelerations, the velocities and the Eulerian position angles and are herein expressed by a series expansion in terms of the maneuvering coefficients and the hydrodynamic derivatives. The bare hull forces and moments are estimated based on the formulation of Son & Nomoto (1981). It should be noted, however, that 3rd order polynomials were herein used for the derivatives, whereas the geometrical symmetry about the xz-plane was kept by using the appropriate absolute term values: 3

X H =X uu+X v v +X vv v 2 +X vvv v + X r r +X rr r 2 +X rrr r  X vr vr  X  2 3

(5)

YH =Yv v+Yv v+Yvv v v +Yvvv v 3 + Yr r+Yrr r r +Yrrr r 3 +Yφ φ  Yφ φ+Yvvr v 2 r+ Yvrr vr 2 +Yvvφ v 2 φ+Yvφφ vφ2 +Yrrφ r 2 φ+Yrφφ rφ2

(6)

N H =N r r+N v v+N vv v v +N vvv v 3 + N r r+N rr r r +N rrr r 3 +N φ φ+N φ φ+Ν vvr v 2 r+ Ν vrr vr 2 +Ν vvφ v 2 φ+Ν vφφ vφ2 +Ν rrφ r 2 φ+Ν rφφ rφ2

(7)

where ρ, S, u are the sea water density, the wetted surface area of the hull and the ship’s forward speed, respectively. CT is the total resistance coefficient, which depends on ship’s hull form and the ship’s Froude and Reynolds numbers. For the reliable estimation of the resistance coefficient CT the use of model tests is recommended. In case of absence of experimental data for the vessel, the resistance coefficient may be estimated by several semi-empirical methods, while noting that the frictional part of ship’s total resistance is well estimated by ITTC’s 1957 frictional line (Lewis, 1989). Semi-empirical methods predicting the residuary part of ship’s total resistance depend on the vessel’s characteristics and hull form, which must fit to those of the sample of vessels used for the development of the semi-empirical method. Inherently, innovative hull forms cannot be satisfactorily captured by semi-empirical methods. Holtrop’s method (Lewis, 1989) proved quite useful in predicting the resistance of a large variety of traditional hull forms with satisfactory accuracy. Alternatively, the calm water resistance of modern hull forms can be nowadays well predicted by use of CFD-RANS solvers, as will be elaborated in the following. The required propeller thrust T is defined by the following equation

T=(X s +X a ) (1-t) K H =K φ φ+K v +K vv v v +K vvv v +

(10)

3

K r r+K rr r r +K rrr r 3 +Κ φ φ+Κ vvr v 2 r+

where (8) Xs (N)

Κ vrr vr 2 +Κ vvφ v 2 φ+Κ vφφ vφ2 +Κ rrφ r 2 φ+Κ rφφ rφ2 In the following, we neglect the effect of the heel motion and of associated terms, considering that the conducted maneuvering simulations in waves are for large size ships at low advance speed (6 knots full scale, Froude number ≈ 0.05). For the same reasons, 2nd order sway-yaw and all 3rd order coupling terms were herein omitted. PROPULSION AND RESISTANCE There are several methods for estimating the ship’s calm water resistance with varying accuracy. Generally, ship’s calm water total resistance is expressed by:

RT =-0.5ρCT Su 2

(9)

is the resistance in still (calm) water RT at the specified forward speed including resistance due to appendages; Xa (N) is the added resistance in seaway Xa=Xw+Xaw+Xr+Xh, including the added wind resistance Xw, added resistance due to waves Xaw, added rudder resistance proportional to propeller thrust due to manoeuvring in seaway Xr and added resistance due to drift angle in seaway Xh; the above force components are elaborated in the following sections t is the thrust deduction factor, which needs to consider both the effect of hull and the rudder. The generated propeller thrust is obtained by use of the following equation:

T=Zρn2 D4 KT (J)

(11)

where Z, n, D, KT(J) are the number of propellers, the propeller revolutions per second, the propeller diameter and thrust coefficient, respectively. The thrust coefficient is a function of the advance coefficient J and it is dependent on the propeller characteristics and Reynolds number Rn. For the approximation of KT (J), when the propeller open water characteristics are not available, the Wageningen B-series data can be used (v. Lammeren et al., 1969, Oosterveld, 1975). The thrust deduction factor, as well as the wake fraction was herein obtained from available experimental data (see, Moctar et al., 2012). In case of lack of experimental data, the thrust deduction factor and the wake fraction may be estimated by empirical formulas, while the wake fraction can be also reliably estimated by CFDRANS calculations. A comment is due at this point due on the alteration of the thrust deduction and wake fraction when the ship is operating in a seaway. In the presently studied cases, considering large size ships compared to the incident waves (λ/L < 0.5…0.7), we may assume that their motions are negligibly small and the propeller is well submerged, so that the effect of the incoming waves on the propulsive characteristics may be neglected. However, the smaller the ship size compared to the incoming waves is or when the ship is operating in ballast condition, propeller racing due to heavy heave-pitch motions may occur with drastic degradation of the propeller thrust and the propeller onset flow. It is important to account for these effects when studying the dynamics of the propulsion plant and its control in view of propeller racing. RUDDER FORCES The rudder forces XR, YR and moment KR, NR acting on ship’s hull by rudder action are calculated as follows (Lewis, 1989):

X R =-(1-tR )FN sinδ

(12)

YR =-(1+aH )FN cosδ

(13)

K R =-zRYR

(14)

N R =-(xR +aH xH )FN cosδ

(15)

where δ denotes the rudder angle and xR and zR are the coordinates of the center of the area of the rudder; The coefficient αH can be estimated based on model experimental results, which suggest that it can be

expressed as a function of ship’s block coefficient Hirano (1980).

αΗ =0.62(cb -0.6)+0.227

(16)

where cb is ship’s block coefficient. FN is the normal rudder force, which is estimated as:

FN =0.5ρARU R2 ( 6.13Λ (Λ+2.25) )sinαR

(17)

where AR and Λ are the rudder area and the aspect ratio of the rudder respectively; UR and αR are the effective rudder inflow speed and its onset angle to the rudder. UR can be calculated as follows:

U R =U(1-wR )(1+k2 g(s))0.5

(18)

where k2=1.065 if δ < 0 and 0.935 if δ > 0. Finally, αR is calculated as follows: (19)

aR =δ+arcsin( -v-rxR U R ) WIND FORCE The wind force Xw, Yw and moment Kw, Nw acting on the ship hull is estimated as follows:

X w =0.5ρCx AT Vres2

(20)

Yw =0.5ρCY ALVres2

(21)

K w =0.5ρCN (AL 2 /L)Vres2

(22)

N w =0.5ρCN AL LVre2s

(23)

Where Vres = Vwx +u  + Vwy +v  2

2

(24)

is the resultant wind velocity acting on the ship, where Vwx, Vwy are the wind velocity components. The dimensionless CX, CY, CN and CK coefficients are functions of ship’s above water profile and of the relative wind angle. These coefficients may be obtained from published, model experimental data for

the specific vessel type; e.g., they are given by Blendermann (2001) in tabulated form as a function of the relative wind angle for various ship types.

MEAN SECOND ORDER WAVE FORCES AND MOMENTS We assume the ship is advancing with certain speed in regular, monochromatic waves of certain heading to ship’s advance direction (Fig. 1). Her course trajectory will be affected by the wave induced forces and moments and especially the mean second-order drift forces and moments in the horizontal plane, which depend on ship’s hull form, the wave heading and the corresponding frequency of encounter. The longitudinal component of the wave drift forces opposite the direction of ship’s speed of advance is also known as the added resistance of the ship in waves, noting that in following seas conditions, the added resistance may become negative, thus acts as an additional thrust in the direction of ship’s speed of advance. In the present study, the incident wave drift forces and moments are pre-calculated and stored as response surfaces covering the range of ship speeds and wave headings as necessary for the corresponding maneuvering simulation. A 6-DOF, 3D frequency domain panel code is employed for the seakeeping analysis and the calculation of the wave induced forces/moments acting on ships or generally, arbitrarily shaped floating structures (code NEWDRIFT; Papanikolaou 1985; Papanikolaou & Schellin, 1992). This code also calculates the second order mean forces/moments using an improved near-field, pressure integration method (Papanikolaou & Zaraphonitis, 1987). Independently, a far field method was also developed by Liu et al. (2011) to calculate the added resistance and the mean transverse wave drift forces using first order velocity potential values calculated by the 3D panel code NEWDRIFT; namely, following Maruo’s far field approach (Maruo, 1963)：

ρ  -α0 RAW =   π + 8π  - 2 H  k1 ,θ  ρ 2π-α0 +  8π α0

2

  -   k1  k1cosθ-Κcosχ  π 2 α0

3π 2 π 2

dθ 1-4Ωcosθ 2 k  k cosθ-Κcosχ  H  k2 ,θ  2 2 dθ 1-4Ωcosθ

(26)     k j  θ  z+ik j  θ  xcosθ+ysinθ  H k j ,θ =   ds e  n  n  S 





where φ is the velocity potential of the boundary value problem at hand. It is noted that in the short wave region (incident wave length smaller than about half of the ship length, i.e., λ / L < 0.5) the added resistance predicted by potential theory codes is often greatly underestimated, both for full type and fine ship hull forms. This is due to a variety of reasons, such as, insufficient account of the forward speed effect and the swell-up bow effect, wave breaking and viscous effects in general (see Ley et al., 2014), as well as possible limitations of numerical procedures (such as irregular frequencies for some panel codes) in the short wave length range. As a practical approach to this problem, a semi-empirical formula (Liu et al., 2015; Liu and Papanikolaou, 2016b) has been introduced to correct the prediction of added resistance in short waves, namely λ/L < 0.5 (0.7). It takes the following form: (27)

RAW crtd =RAW -RAW D +F1

where RAWcrtd is the corrected prediction of the added resistance in waves of heading α, RAWD is the added resistance due to the diffraction effect and F1 is the short waves semi-empirical correction, which takes the following form:

F1= Fe sinθdl

(28)

C

Where



f  α  1+4 Fn



 0.87  1 Fe = ρgζ 2 secαWL αT   2  CB  2ω0U  2  cosα-cosθcos  θ+α     sin  θ+α  + g  

(29)

where f(α)=cosα for 0 ≤ α < π / 2 and f(α)=0 for π / 2 ≤ α < π, and

aT =1-e-2kT(x)

(30)

(25)

where H(kj, θ) is the Kochin function describing the elementary waves radiated from the ship:

This correction is applied only if F1 >RAWD . The above formula has been further developed in the frame of the SHOPERA project (2013-2016) as a possible new level-1/level-2 empirical method to cover the need of a fast estimation method for the added resistance of ships in

waves, as necessary for both the minimum power assessment (IMO 2013 Interim Guidelines) and the calculation of fw coefficient in EEDI guidelines (IMO MEPC.1/Circ.796). Preliminary validation results were presented in relevant recent publications (Liu and Papanikolaou, 2016a). The mean sway force is herein calculated also by applying the far field method. The mean yaw moment in bow waves has been calculated using the pressure integration method of NEWDRIFT, while for stern quartering seas, the empirical formula of Faltinsen et al. (1980) was applied. This is justified by the relatively short wave conditions tested in SHOPERA. Finally, in this study, we restrict ourselves to regular wave cases enabling the direct validation of results by use of model experimental data. However, when dealing with irregular seas conditions, which are described by a wave spectrum, then we can follow the standard spectral techniques for the superposition of mean second-order wave force effects by use of the corresponding quadratic transfer functions. Validations of the methods employed herein for the estimation of the mean second order forces and moment acting on the DTC standard containership are shown in Figures 2 to 5, while corresponding validations for the KVLCC2 standard tanker are presented in Figures 6 to 8. Comparative experimental results were obtained by MARINTEK for the DTC and CEHIPAR for the KVLCC2 hull in the frame of the EU funded research project SHOPERA (2013-2016).

Figure 3: Longitudinal mean drift force of DTC ship at 0 speed, β=30deg

Figure 4: Transverse mean drift force of DTC ship at 0 speed, β=30deg

Figure 2: Added resistance of DTC ship at Fn=0.14 in head seas

Figure 5: Mean yaw drift moment of DTC ship at 0 speed, β=30deg

HYDRODYNAMIC FORCES AND MOMENTS BY USE OF CFD Assuming proper implementation of codes and usage, CFD computations of hydrodynamic forces and moments may be used in lieu of experimental data, and in all cases, where other calculation methods fail or prove unsatisfactory. DESCRIPTION OF THE CFD METHOD

Figure 6: Longitudinal mean drift force of KVLCC2 ship at 0 speed, β=30deg

Figure 7: Transverse mean drift force of KVLCC2 ship at 0 speed, β=30deg

The CFD computations performed in the context of the present research used the Reynolds Averaged Navier–Stokes (RANS) solver STAR-CCM+ of CD– ADAPCO. The code solves the RANS and continuity integral equations on an unstructured mesh using the Finite Volume technique. For the cases studied, even though it could theoretically be dealt with using the steady RANS equations and modeling, herein the Unsteady RANS Volume Of Fluid (VOF) technique of STAR CCM+ was preferred; it proved far more robust when solving the unsteady equations of such a problem. For the time-stepping, an iterative convergence was selected to achieve converged steady solutions. A second order difference temporal discretization scheme was used. Spatial discretization was achieved using second order schemes for both the convective and the viscous terms. The SIMPLE method was used by the solver, coupling pressures and fluid velocities, while the Reynolds stress tensor closure was achieved using a realizable (Shih et al., 1997) k–ε turbulence model (Jones et al., 1972), using an all Y+ wall treatment that is capable to treat the flow near the wall depending on the Y+ value achieved by the respective mesh and flow conditions.

Figure 9: The computational domain

Figure 8: Mean yaw drift moment of KVLCC2 ship at 0 speed, β=30deg

In order to analyze the resulting forces and moments, two coordinate systems were used: an earth fixed inertial reference coordinate system and a COG ship fixed coordinate system. All ship motions and hull forces and moments are calculated with reference to the ship fixed frame. For the case

studied, the ship is free to move in heave and pitch, while all other ship motion DOFs are constrained.

well as the corresponding fluid phase (sole water for the bottom side and sole air for the top side) are used at these boundaries. The resulting hull mesh is presented in Figure 10, while the near field meshing is presented in Figure 11. The standard DTC hull was modeled using a model scale of 1:63.65 full scale. UNCERTAINTY ANALYSIS: VERIFICATION AND VALIDATION

Figure 10: The DTC hull mesh (2.5M cells grid)

COMPUTATIONAL DOMAIN

Figure 11: The near field mesh (2.5M cells grid)

The computational domain presented in Figure 9 was developed using the STAR-CCM+ meshing tools. Typically the mesh consists of 2.5M cells. The mesh is dominated by hexahedral cells, while near the hull the cells are polyhedral as a result of the trimming to follow the hull lines. In order to resolve the boundary layer flow (Simonsen et al., 2012) and the near field viscous phenomena, a number of prism layers are attached on the hull wall, where a no–slip condition is applied. Typical boundary conditions are used. The inlet boundary is located two ship lengths in front of the ship. A Dirichlet condition is used at the inlet of the fluid flow, prescribing the inlet velocity as well as the volume fractions of the corresponding fluid phases. The sides are located two ship lengths aft of the hull. On the sides, Neumann symmetry conditions are used, ensuring the field gradient continuity on the sides. The outlet is located three ship lengths from the hull. A Dirichlet pressure condition is used at the outlet, ensuring zero gradients of velocity and volume fraction of fluids, as well as the continuity of hydrostatic pressure. The bottom and top sides are located one and half a ship lengths from the hull respectively, and they are considered as flow inlets. A Dirichlet condition prescribing the inlet velocity as

CFD analysis is a laborious task, yet often the outcome is poor in terms of stability or of accuracy or both! When it comes to the verification of the obtained results, grid sensitivity, i.e. the perturbation of the results when assuming a slight change on grid sizing, is of major importance. Table 1: PMM static drift non-dim surge and sway forces (X΄, Y΄) and yaw moment (N΄) acting on the DTC hull X' β 20 kts 16 kts 6 kts 0 0.000 -6.14E-04 -6.39E-04 -6.62E-04 5 0.087 -6.51E-04 -6.94E-04 -6.95E-04 10 0.174 -6.56E-04 -6.71E-04 -7.20E-04 15 0.259 -8.49E-04 -7.78E-04 -7.58E-04 20 0.342 -9.99E-04 -9.30E-04 -8.14E-04 Y' β v' 20 kts 16 kts 6 kts 0 0.000 0.00E+00 0.00E+00 0.00E+00 5 0.087 4.75E-04 5.36E-04 5.38E-04 10 0.174 1.46E-03 1.54E-03 1.50E-03 15 0.259 2.87E-03 2.87E-03 2.15E-03 20 0.342 4.73E-03 4.60E-03 4.08E-03 N' β v' 20 kts 16 kts 6 kts 0 0.000 0.00E+00 0.00E+00 0.00E+00 5 0.087 3.67E-04 3.70E-04 3.54E-04 10 0.174 7.25E-04 7.03E-04 6.79E-04 15 0.259 1.20E-03 1.14E-03 1.07E-03 20 0.342 1.75E-03 1.65E-03 1.33E-03

Following ITTC recommendations (ITTC, 2002), an uncertainty analysis is required to verify and validate CFD results. We present herein only the verification and validation process for the static drift force cases, even though this was done for all cases studied. Three distinct values for each parameter studied are required for this analysis and therefore, three different grid sizes were used, namely a 695K cell coarse (No. 3), a 1.2M medium (No. 2) and a 2.5M fine grid (No. 1) leading to three solution sets. However, a systematic grid refinement is not possible when using an unstructured grid, since the

refinement areas are not explicitly controlled. In order to overcome this challenge, a grid base size was used and all grid areas were defined according to this size. Refinement is performed by limiting the grid base size using a refinement factor of rG2=2 (ITTC, 2002). In order to subsequently estimate the hydrodynamic derivatives for the different grid sizes, the longitudinal X–force, the transverse Y–force and the yaw moment N around the vertical axis Z with reference at mid–ship, were computed. The results from the CFD PMM static drift analysis are presented in Table 1 while from the Circular Motion Tests (CMT) are presented in Table 2. Forces and moments are non–dimensionalized with the appropriate factors for the forces ( 1 U 2 L2 ) and 2

i) Monotonic convergence 0 < RG < 1. ii) Oscillatory convergence RG < 0. iii) Divergence RG > 1. Table 3:CFD vs EFD results for the static drift cases(6 kts)

β (deg) 0 5 10 15 20 β (deg)

bp

moments ( 1 U 2 L3 ) respectively. 2

bp

PMM static drift analysis was performed for three different ship speeds, namely 20 kts, 16 kts and 6 kts, in order to evaluate how the viscous phenomena affect the calculated hydrodynamic derivative values. Drift angles ranged from 0 to 20 degrees. The results from the CFD PMM drift analysis at 6kts, as well as the respective EFD results from MARINTEK tank tests, within the frame of SHOPERA project, are presented in Table 3. Furthermore, CMT analysis was performed for two different ship speeds, namely 15.5 kts and 6 kts with a rotating radius varying from 20 to 50 meters at model scale. Table 2: CMT non-dim sway force (Y΄) and yaw moment (Ν΄) on DTC hull Y' N' R (m) 15.5 kts 6 kts 15.5 kts 6 kts 50 40 30 20

4.52E-04 7.24E-04 1.09E-03 1.49E-03

6.81E-05 7.81E-05 9.93E-05 1.59E-04

7.28E-04 1.15E-03 1.63E-03 2.76E-03

1.27E-04 1.62E-04 2.27E-04 3.78E-04

Following the ITTC recommended procedure, a parameter and iterative convergence study is required, with systematic refinement ceteris paribus. Such an analysis was performed and the respective results validated the accuracy of the CFD findings, for the grids used and the cases studied. The process and the results obtained for the PMM drift cases is herein described. The variances of solution parameters between coarse and medium grid, eG32=S3-S2, and between medium and fine grid, eG21=S2-S1, are used to estimate the convergence ratio RG=eG21/eG32. As suggested, three convergence conditions may arise depending on the RG value:

0 5 10 15 20

X' CFD -6.62E-04 -6.95E-04 -7.20E-04 -7.58E-04 -8.14E-04 EFD -7.00E-04 -7.50E-04 -8.00E-04 -8.49E-04 -9.42E-04

Υ' CFD 0.00E+00 5.38E-04 1.50E-03 2.15E-03 4.08E-03 EFD 0.00E+00 6.20E-04 1.24E-03 1.86E-03 4.45E-03

Ν' CFD 0.00E+00 3.54E-04 6.79E-04 1.07E-03 1.33E-03 EFD 0.00E+00 3.50E-04 6.99E-04 1.05E-03 1.34E-03

As suggested by ITTC (2002), meeting condition iii) means no uncertainty in the results. For condition ii) uncertainties are estimated while attempting to bound the error based on the oscillation maximum SU and minimum SL, i.e. UG=1/2(SU-SL). For condition i) a generalized Richarson extrapolation (RE) is used to estimate the grid error δREG1 and the order of accuracy pG, which are given as:

δREG1=

eG21 rGpG -1

(31)

And

eG32 ) eG21 pG = ln(rG ) ln(

(32)

where rG is the refinement ratio. As suggested by ITTC (2002), due to the limited number of grid refinements studied, only the leading term of the RE method was estimated, which provides only one term estimate for the error and order of accuracy, according to the above equations 31 and 32. Using the calculated values of δREG1 and pG, one can estimate the grid uncertainty, using either a factor of safety or a correction factor, whereas the later depends on the asymptotic nature of the solution. In short, the correction factor C G is calculated as:

(33)

r pG -1 CG = GpGest rG -1

and pGest is an estimate of the limiting order of accuracy of the first term as spacing size goes to zero. Moreover: i) If CG is close to unity, the numerical error δSN as well as the uncertainty UGC may be calculated as:

δSN =CG δREG1

(34)

2 (35) (2.4(1-CG ) +0.1) δREG1 , 1-CG