On the Calculation of Propulsive Characteristics of a Bulk-Carrier Moving in Head Seas

Journal of Marine Science and Engineering

Article On the Calculation of Propulsive Characteristics of a Bulk-Carrier Moving in Head Seas

S. Polyzos and G. Tzabiras * School of Naval Architecture & Marine Engineering, National Technical University of Athens, 15780 Athens, Greece; [email protected] * Correspondence: tzab@ﬂuid.mech.ntua.gr; Tel.: +30-210-772-1107

Received: 14 September 2020; Accepted: 8 October 2020; Published: 9 October 2020

Abstract: The present work describes a simpliﬁed Computational Fluid Dynamics (CFD) approach in order to calculate the propulsive performance of a ship moving at steady forward speed in head seas. The proposed method combines experimental data concerning the added resistance at model scale with full scale Reynolds Averages Navier–Stokes (RANS) computations, using an in-house solver. In order to simulate the propeller performance, the actuator disk concept is employed. The propeller thrust is calculated in the time domain, assuming that the total resistance of the ship is the sum of the still water resistance and the added component derived by the towing tank data. The unsteady RANS equations are solved until self-propulsion is achieved at a given time step. Then, the computed values of both the ﬂow rate through the propeller and the thrust are stored and, after the end of the examined time period, they are processed for calculating the variation of Shaft Horsepower (SHP) and RPM of the ship’s engine. The method is applied for a bulk carrier which has been tested in model scale at the towing tank of the Laboratory for Ship and Marine Hydrodynamics (LSMH) of the National Technical University of Athens (NTUA).

Keywords: self-propulsion in waves; CFD; towing tank tests

1. Introduction The problem of a self-propelled ship is one of the most challenging in Naval Architecture since the Engineer is facing an unsteady Three-dimensional (3D) flow, characterized by high Reynold’s number, significant viscous effects, the presence of free surface and the complex interaction between the hull and the propeller. Taking into account incident waves makes the problem even harder. Traditionally, the problem of the self-propelled ship in a seaway was tackled experimentally in model scale by separating the problem of the self-propelled ship in calm water from the problem of added resistance in waves. Thus, the solution could only be approximated. Even when a self-propelled scale model is used, the extrapolation of the results to the full-scale ship is problematic due to the scale effect on viscous resistance [1]. Computational Fluid Dynamics (CFD) methods solving the Reynolds Averaged Navier–Stokes (RANS) equations have become of age and are widely used to predict the resistance and propulsion characteristics of ships [2]. Time-dependent RANS solvers have also been developed and can yield promising results with regards to the 6-degrees-of-freedom (6DOF) motions of a ship as well as the added resistance in waves [3]. Solving the time-dependent self-propulsion problem of a ship in waves has proven to be quite time consuming, thus hybrid methods have been proposed. Such methods calculate the effect on the propulsive characteristics of the ship motions, via semi-empirical methods for the prediction of the effect on the ship wake [4]. In recent years, several methods have been proposed in order to solve the problem of the self-propelled ship in incident waves [3,5–8]. In most cases, only regular waves are considered due to

J. Mar. Sci. Eng. 2020, 8, 786; doi:10.3390/jmse8100786 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 2 of 25

complete problem of the self‐propelled ship in irregular waves [9]. However, these methods require J.significant Mar. Sci. Eng. computing2020, 8, 786 power in order to obtain accurate results. Therefore, some simplifications2 of are 24 required in order to achieve reliable results and to reduce the computational cost. In this respect, we present a hybrid method for predicting the propulsion characteristics of a ship in irregular head seas thein this high work, computational which combines cost of experimental such methods. data There with are CFD some calculations. publications that attempt to solve the completeHaving problem as a starting of the self-propelled point experimental ship in model irregular‐scale waves resistance [9]. However, and sea‐keeping these methods measurements, require significantwe numerically computing solve power the full in‐ orderscale topropulsion obtain accurate problem, results. where Therefore, the thrust some requirements simplifications on arethe requiredpropeller in are order based to achieve on the reliableextrapolation results andof the to reducemodel‐scale the computational resistance measurements. cost. In this respect,In this weextrapolation, present a hybrid the scale method effects for are predicting taken into the account. propulsion Also, characteristics the required thrust of a ship is obtained in irregular by solving head seasthe self in this‐propulsion work, which problem, combines i.e., experimentalby calculating data the withwake CFD fraction calculations. and thrust deduction ratio. The actualHaving flow problem as a starting is solved point by experimental employing the model-scale in‐house developed resistance RANS and sea-keeping solver in conjunction measurements, with wean numericallyactuator disk solve model the full-scale for the propulsion propeller. problem, The method where theis thrustused requirementsto calculate onthe the propulsion propeller arecharacteristics based on the of extrapolation a bulk carrier of thetraveling model-scale at a steady resistance forward measurements. speed in irregular In this extrapolation, head waves. the scale effects are taken into account. Also, the required thrust is obtained by solving the self-propulsion problem, i.e.,2. The by calculatingNumerical the Method wake fraction and thrust deduction ratio. The actual flow problem is solved by employing the in-house developed RANS solver in conjunction with an actuator disk model for the The employed method has been developed at the Laboratory for Ship and marine propeller. The method is used to calculate the propulsion characteristics of a bulk carrier traveling at a Hydrodynamics (LSMH) of the National Technical University of Athens (NTUA). It solves the RANS steady forward speed in irregular head waves. equations in a body fitted structured mesh as shown in Figure 1. The three regions (I), (II) and (III) 2.refer The to Numerical three different Method grid topologies. Region (I) consists of a C‐O grid following a sequence of transverse planes which intersect on the Zc‐axis. Region (II) represents an H‐O grid which is formed The employed method has been developed at the Laboratory for Ship and marine Hydrodynamics by transverse planes to the longitudinal symmetry x‐axis, while region (III) forms a typical H‐H grid (LSMH) of the National Technical University of Athens (NTUA). It solves the RANS equations in a body which is generated when a dry or wet transom stern exists, [2]. The C‐O topology is applicable for fitted structured mesh as shown in Figure1. The three regions (I), (II) and (III) refer to three di fferent bulbus bow configurations. The H‐O grid is the standard grid in most CFD applications around ship grid topologies. Region (I) consists of a C-O grid following a sequence of transverse planes which hulls. Finally, the H‐H is suggested for transom sterns exhibiting discontinuity when the transom is intersect on the Zc-axis. Region (II) represents an H-O grid which is formed by transverse planes to the wetted [10]. longitudinal symmetry x-axis, while region (III) forms a typical H-H grid which is generated when a In any case, the 2D grids on the transverse planes are generated following the conformal dry or wet transom stern exists, [2]. The C-O topology is applicable for bulbus bow configurations. mapping technique as described in [10]. In order to apply as far as possible high grid resolutions, the The H-O grid is the standard grid in most CFD applications around ship hulls. Finally, the H-H is three blocks are created in a restricted domain underneath the predetermined free surface boundary suggested for transom sterns exhibiting discontinuity when the transom is wetted [10]. (FS). The free surface is calculated according to the hybrid potential viscous flow method [2].

Figure 1. The body-ﬁtted block mesh around the hull.

In any case, the 2D gridsFigure on the 1. The transverse body‐fitted planes block are mesh generated around followingthe hull. the conformal mapping technique as described in [10]. In order to apply as far as possible high grid resolutions, the three blocks are created in a restricted domain underneath the predetermined free surface boundary (FS). The free surface is calculated according to the hybrid potential viscous ﬂow method [2]. J. Mar. Sci. Eng. 2020, 8, 786 3 of 24

The RANS code employs a sequence of orthogonal curvilinear systems bounded on each transverse plane. In such a system the three velocity components are denoted as (ui, uj, ul) and the ui- momentum equation reads: ( ) = 1 ∂p + 2 + 2 + C ui h ∂x ρuj Kji ρul Kli ρuiujKij ρuiulKil σii σjj Kji − i i − − − (1) 1 ∂σii 1 ∂σij 1 ∂σil +(σii σll)Kli + σij 2Kij + Klj + σil 2Kil + Kjl + + + − hi ∂xi hj ∂xj hl ∂xl where (hi, hj, hl) are the metrics, Kij = (1/hihj)∂hi/∂xj the curvatures of the orthogonal system, ρ the ﬂuid density, p the static plus the hydrostatic pressure, σij the stress tensor and the left-hand side of Equation (1) represents the convective terms:   ∂ h h u Φ ∂ h h u Φ ∂ h h u Φ  ∂Φ ρ  j l i i l j i j l  C(Φ) = ρ +  + +  (2) ∂t hihjhl  ∂xi ∂xj ∂xl 

The stress tensor components are associated to the deformation tensor through h i 1 ∂ui σii = 2µ eii = 2µ + ujKij + u K e e hi ∂xi l il (3) hj ∂ uj hi ∂ ui σij = µ eij = µ + e e hi ∂xi hj hj ∂xj hi

The eﬀective viscosity in Equation (3) is deﬁned as

µe = µ + ρνt (4) where νt stands for the turbulent (or eddy) viscosity which is calculated according to the two-equation k-ω-SST turbulence model. The equation for the kinetic energy of turbulence k reads 1 ∂ hjhl ∂k ∂ hihl ∂k ∂ hihj ∂k C(k) = σkµt + σkµt + σkµt hihjhl ∂xi hi ∂xi ∂xj hj ∂xj ∂xl hl ∂xl (5) +G β∗ρωk − while the equation for the speciﬁc dissipation ω reads 1 ∂ hjhl ∂ω ∂ hihl ∂ω ∂ hihj ∂ω C(ω) = σωµ + σωµ + σωµ hihjhl ∂xi t hi ∂xi ∂xj t hj ∂xj ∂xl t hl ∂xl γ (6) + G β ρω2 + 2ρ(1 F ) σω 1 ∂k ∂ω vt ∗ 1 ω 2 ∂x ∂x − − hj j j

According to the k-ω-SST model, the turbulent kinematic viscosity is calculated according to the formula a k ν = 1 (7) t max[a1ω, rot(→c ) F2]

In the above Equations (5)–(7), the constants σk, σω, β* and the functions F1, F2 are defined as per the original paper of Menter [11], while G represents the generation term of the turbulence kinetic energy. Essentially, this model separates the flow-field above the body surface in two regions, which are determined according to the local flow parameters and the normal distance from the surface. The k-ω-SST model has been developed in order to calculate effectively complex flow fields, like those appearing in stern flows past full ship forms. The momentum (RANS) and the turbulence model equations are solved numerically by applying the finite volume method in a staggered grid arrangement. In the resulting non-linear algebraic equations, the convective terms are approximated according to the second order MUSCL scheme [12] associated with the minmod limiter function [13], while the diffusive and the source terms are J. Mar. Sci. Eng. 2020, 8, 786 4 of 24 approximated using central differences. The solution follows the SIMPLE algorithm, where the RANS equations are solved following a marching scheme, while the pressure (continuity) equation is treated as fully elliptic. Dirichlet boundary conditions are applied on the external boundary N of Figure1 for the flow variables. The velocity components and the pressure on this boundary are calculated by the potential flow solution performed around the hull and the real free surface [14]. On the free surface, FS in Figure1, the kinematic condition is fulfilled (zero normal velocity) while for the pressure and the turbulence quantities Neumann conditions are applied. In order to calculate the flow variables above the solid boundary in a uniform way, the standard wall function method is employed. On the exit plane D, open-type conditions hold [15]. Finally, on the flow symmetry plane, along the ship symmetry axis, the normal velocity component is set equal to zero while the normal derivatives of all other variables vanish (Neumann conditions). The exchange of the flow-variables on the matching planes between different blocks is based on a second order interpolation. A number of domain sweeps is required to obtain convergence of the momentum and continuity equations. Convergence is achieved when the sum of the absolute residuals of the finite volume equations over all nodes becomes lower than a pre-specified criterion. In the present study, this criterion is set equal to 5 10 4ρV 2A for the momentum equations and equal to 5 10 4ρV A for the × − S M × − S M continuity equation, where VS represents the ship’s speed and AM is equal to the transverse area covered by the grid at the mid-ship section. The self-propulsion problem is solved by adopting the actuator disk model. To introduce this model, the effective wake is approximated according to the open water numerical experiment as in [16,17]. The propeller thrust is calculated following an iterative procedure so that its final value equals the total resistance of the ship. This procedure is also followed when the free surface problem is solved, since the stern wave is influenced by the propeller operation. Regarding modeling the propulsion in waves, the fundamental simplification adopted in the described method, is that the ship moves in steady forward speed, under rough head sea conditions without performing any other motion, Under this assumption, the propeller thrust should be equal to the hydrodynamic resistance of the ship at any time, i.e.,

TP(t) = RT,S(t) (8) where TP denotes thrust and RT,S(t) the resistance. In addition, it is assumed that RT,S(t) can be decomposed in two components as:

RT,S(t) = RCW,S(t) + RAD,S(t) (9)

In the above relation RCW,S(t) represents the hydrodynamic resistance of the body moving in still water and RAD,S(t) the added resistance due to the varying wave loads in sea route. As an extension of the Froude hypothesis, RAD,S(t) is considered as a wave resistance component depending only on the Froude number and is calculated as

1 R (t) = C (t) ρ S U 2 (10) AD,S W,AD × 2 × × × S where CW,AD(t) is the corresponding resistance coeﬃcient, S is the wetted surface of the ship in still water and US the ships’ speed. Since CW,AD(t) depends on the Froude number, it can be calculated by performing towing tank experiments at model scale. In these experiments, we can measure independently the total resistance of the model RT,CW,M in calm water as well as the history of RT,M(t) at a given sea state, always at steady forward speed. J. Mar. Sci. Eng. 2020, 8, 786 5 of 24

Then, the coeﬃcient CW,AD(t) can be derived through

RT,M(t) RT,CW,M CW,AD(t) = − (11) 1 ρ S U 2 2 × × × M Therefore, the added resistance in Equation (9) can be calculated at any time step, taking into account the scale diﬀerence in the time domain, i.e.,

ts = √λ tm (12) × where ts symbolizes the time in the ship scale, tm the time in the model scale and λ is the geometrical scale ratio. The reason that the component RCW,S(t) in Equation (9) is a function of time, is that any change in the propeller thrust induces a different thrust deduction ratio and has to be calculated by performing an iterative procedure for each time step. In this procedure, the actuator disk model is based on the adoption of body forces that are exerted on fluid particles passing through the plane of the propeller blades. In the present approximation, only the axial body forces are taken into account, being included as source terms in the axial momentum equation. The radial distribution of the body forces is based on the lifting line method, as described in [18]. The body forces are integrated in the actual control volumes as described in [15,17]. The lifting line code requires two fundamental parameters, i.e., the propeller thrust and the velocities at infinite. Both of them are defined by performing a self-propulsion iterative procedure at each time step. In each internal iterative step, the propeller thrust TP(t) is assumed to be equal to the body resistance of the previous step and changes the stern flow field accordingly. Then, a new total resistance RCW,S(t), Equation (9), is calculated and the procedure stops when for three successive time steps self-propulsion is achieved, i.e., Equation (8) holds. Since the flow rate through the propeller blades changes as the iterations are performed, the required lifting line input data are changed accordingly. In addition, at a certain internal step an open water numerical experiment is performed as described by [16] in order to calculate the effective wake fraction at infinite as a function of the flow rate passing through the propeller disk. The abovementioned procedure is straight forward and provides the history of TP(t) almost independently of the exact distribution of body forces in each time step. Numerical experiments have shown that only an approximate prediction of the circulation distribution along the blades is required, which more or less remains similar under variable lifting line input conditions. However, when we need to calculate the required engine characteristics, we have to employ the lifting line output as concerned the required torque and rpm. To adapt these data, we assume that the propeller operates in a quasi-steady condition and therefore the steady lifting line approximation produces reasonable estimates for the required SHP and RPM. In a further simplification of the problem, we can also consider the time averaged values for the resistance components of (9):

RT = RCW(t) + RAD(t) (13) Then the time-dependent propulsion problem is simplified into a steady-state one and the iterative procedure described in the previous paragraph is employed for a single time step. The overall convergence algorithm is presented in Table1. At first, for time t = 0, the steady state problem is solved and, then, the solution proceeds to the next time steps to solve the unsteady flow at random waves. In each time step, the flow equations are solved under a constant propeller thrust for NS = 30 iterations and then the thrust is adjusted to be equal to the new ship resistance. This procedure is followed until the thrust remains unchanged for 3 successive K steps. Then, the time changes to t = t+ δt, where δt is specified for the particular case, and the solution stops when t becomes higher than a given tmax value. J. Mar. Sci. Eng. 2020, 8, 786 6 of 24 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 6 of 25

iterations and then the thrust is adjusted to be equal to the new ship resistance. This procedure is Table 1. The general algorithm. followed until the thrust remains unchanged for 3 successive K steps. Then, the time changes to t = t+ δt, whereSTEP δt is specified for the particular case, and the solution stops when t becomes higher than a given tmax1 value. time = 0: Solve the steady-state (cam water) problem 2 time = time + δt, K = 0 Table 1. The general algorithm. 3 If time > tmax GO TO 9 STEP 4 K = K + 1 1 time = 0: Solve the steady‐state (cam water) problem 5 NS = 0 2 time = time + δt, K = 0 63 If time NS = > NStmax+ 1 GO TO 9 4 Solve momentumK = K equations + 1 75 Solve pressure correction equationNS = 0 and correct velocity 6 Solve turbulenceNS model= NS + equations 1 8Solve IF NS momentum= 30 GO TO equations 10 7 Solve pressure correction equation and correct velocity 9 GO TO 5 Solve turbulence model equations 10 Calculate the body resistance R (t) and set propeller thrust T = R (t) 8 IF NST,S = 30 GO TO 10 P T,S 11 If T remains constant for 3 successive K-steps GO TO 2 9 P GO TO 5 12 GO TO 4 10 Calculate the body resistance RT,S(t) and set propeller thrust TP = RT,S(t) 1311 If TP remains END constant OF THE for PROCEDURE 3 successive K‐steps GO TO 2 12 GO TO 4 3. Results 13 END OF THE PROCEDURE

3.1. Characteristics3. Results of the Bulk-Carrier

The3.1. bulk-carrier Characteristics under of the investigation Bulk‐Carrier features a bulbous bow and a transom stern as shown in Figure2. The main characteristics of the ship are shown in Table2. In this Table, L represents the The bulk‐carrier under investigation features a bulbous bow and a transom sternBP as shown in length between perpendiculars, L the length overall, L the waterline length at full load condition Figure 2. The main characteristicsOA of the ship are shownWL in Table 2. In this Table, LBP represents the (FL), B thelength ship’s between beam, perpendiculars, T the draft at full LOA loadthe length condition overall, (FL), LWL ∆thethe waterline corresponding length at displacementfull load condition and CB the block(FL), coeB theﬃ ship’scient. beam, The design T the draft speed at full of load the shipcondition is equal (FL), Δ tothe 14.82 corresponding knots resulting displacement in a Froude and 9 number basedCB the onblock L coefficient.equal to 0.180The design and a speed Reynolds of the number ship is equal equal to to 14.82 1.18 knots10 resulting. in a Froude WL × number based on LWL equal to 0.180 and a Reynolds number equal to 1.18 × 109.

FigureFigure 2. The 2. hullThe hull shape. shape.

Table 2. Characteristics of the ship. Table 2. Characteristics of the ship. Length between Perpendiculars L 179.595 m Length between PerpendicularsBP LBP 179.595 m Length overallLength overall LOA L187.608OA 187.608 mm Waterline lengthWaterline length L WL L183.291WL 183.291 mm BeamBeam BB 23.700 23.700 mm DraughtDraught TT 10.150 10.150 mm DisplacementDisplacement Δ ∆ 37,23837,238 tntn Block coefficient CB 0.841 Block coeﬃcient CB 0.841

Ship Speed US 14.82 kn Froude number Fn 0.180 Reynold’s number Re 1.18 109 × J. Mar. Sci. Eng. 2020, 8, 786 7 of 24

3.2. Model-Scale Experiments In order to evaluate the resistance and sea-keeping characteristics of the ship, a λ = 1/35 scale model was constructed at LSMH. The main characteristics of the model are shown in Table3. The model was tested in various displacements, trim and speeds [19,20] but in the present work we are concerned with the full load condition and the design speed. Following the Froude scaling method, where the Froude number Fn = 0.180 is kept equal between the ship and the model, the resulting model velocity is UM = 1.289 m/s as shown in Table3.

Table 3. Characteristics of the model.

Scale λ 1/35

Length between perpendiculars LBP 5.131 m

Length overall LOA 5.360 m

Waterline length LWL 5.237 m Beam B 0.677 m Draught T 0.290 m

Model Speed UM 1.289 m/s Froude number Fn 0.180 Reynold’s number Re 7.68 106 ×

All experiments were conducted at the towing tank of LSMH-NTUA. The main characteristics of the facility are presented in Table4. For resistance, propulsion and sea-keeping experiments, the models are towed along the tank via a carriage that houses all measuring equipment. The model is towed via a dynamometer, measuring the drag, also allowing for two degrees of freedom in pitch and heave. The tank is also equipped with a paddle-type wave generator able to create harmonic and irregular waves with a maximum amplitude of 0.20 m and a frequency range 0.3–2.0 Hz. The experimental uncertainty has been calculated for previous resistance measurements at the towing tank of LSMH-NTUA, that utilized the same experimental setup [21]. The uncertainty interval for the resistance measurements has been calculated to be less than 2% for the highest resistance values.

Table 4. Characteristics of NTUA towing tank.

Length 90.00 m Breadth 4.60 m Depth 3.00 m Maximum carriage speed 5.30 m/s

The experiments were performed previously and presented in [19,20]. Initially the calm water resistance of the model was measured. Then, the resistance and the motions of the model were measured for several incident waves, both harmonic and irregular. The waves were generated using the calibrated paddle-type wave maker. The model velocity was ﬁxed during each measurement. In the present work, we are concerned with the case of irregular waves, following the Bretschneider spectrum [22]: ( ) α β 3.11 2π S(ω) = exp − α = 8.11 g2/1000β = ω = (14) 5 4 2 T ω ω × (HS) P where HS = 5.0 m is the significant wave height and TP = 12.0 s is the spectrum peak period, both at full scale. Additionally, ωis the angular frequency, g is the gravitational acceleration, π= 3.14 and α, β spectrum shape parameters. For stochastic procedures such as irregular waves and the corresponding J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 25     S()  5 exp  4 S()   exp  4  5  2  8.11g2 /1000  8.11g /1000

3.11 (14)   3.11 2 (14)   2 H S H S   2   2 T TP J. Mar. Sci. Eng. 2020, 8, 786 P 8 of 24

where HS = 5.0 m is the significant wave height and TP = 12.0 s is the spectrum peak period, both at where HS = 5.0 m is the significant wave height and TP = 12.0 s is the spectrum peak period, both at full scale. Additionally, ω is the angular frequency, g is the gravitational acceleration, π = 3.14 and motionsfull ofscale. a ship, Additionally, around 20 ω minis the of angular recordings frequency, are required g is the togravitational obtain an accurateacceleration, representation π = 3.14 and of all α,β spectrum shape parameters. For stochastic procedures such as irregular waves and the waveα components.,β spectrum shape In model parameters. scale thus, For the stochastic experiments procedures have tosuch run as for irregular 200 s. Since waves the and total the length corresponding motions of a ship, around 20 min of recordings are required to obtain an accurate of thecorresponding tank is 90 m motions and the of carriage a ship, around needs time20 min to of accelerate recordings and are decelerate, required to theobtain time an ofaccurate the model representation of all wave components. In model scale thus, the experiments have to run for 200 s. at a speedrepresentation of 1.289 of m all/s iswave 30 s.components. In order to In obtain model scale enough thus, data, the experiments the measurements have to run were for conducted 200 s. Since the total length of the tank is 90 m and the carriage needs time to accelerate and decelerate, the Since the total length of the tank is 90 m and the carriage needs time to accelerate and decelerate, the in a totaltime ofof eightthe model tests at each a speed under of 1.289 sequential m/s is 30 segments s. In order of to a specificobtain enough wave data, time the series measurements corresponding time of the model at a speed of 1.289 m/s is 30 s. In order to obtain enough data, the measurements to thewere abovementioned conducted in a spectrum.total of eight In tests Figure each3 under, the λsequential= 1/35 scale segments model of ofa specific the examined wave time bulk-carrier series were conducted in a total of eight tests each under sequential segments of a specific wave time series is presentedcorresponding during to resistance the abovementioned measurements spectrum. in theIn Figure Towing 3, the Tank λ = 1/35 of LSMH-NTUA. scale model of the In examined Figures4 –11, corresponding to the abovementioned spectrum. In Figure 3, the λ = 1/35 scale model of the examined the waterbulk‐carrier elevation is presentedη is presented, during resistance as measured measurements during experimentsin the Towing #1–#8.Tank of Finally,LSMH‐NTUA. in Figure In 12, bulk‐carrier is presented during resistance measurements in the Towing Tank of LSMH‐NTUA. In the powerFigures density 4–11, the spectrum water elevation that results η is frompresented, the combined as measured water during elevation experiments measurements #1–#8. Finally, is compared in Figures 4–11, the water elevation η is presented, as measured during experiments #1–#8. Finally, in Figure 12, the power density spectrum that results from the combined water elevation measurements to theFigure theoretical 12, the shape power of density the Bretschneider spectrum that spectrumresults from [22 the]. combined water elevation measurements is compared to the theoretical shape of the Bretschneider spectrum [22]. is compared to the theoretical shape of the Bretschneider spectrum [22].

Figure 3. Scale model of the bulk‐carrier in the towing tank of LSMH‐NTUA. FigureFigure 3. Scale 3. Scale model model of of the the bulk-carrier bulk‐carrier in the the towing towing tank tank of ofLSMH LSMH-NTUA.‐NTUA.

Figure 4. Water elevation η time series for Experiment #1. J. Mar. Sci. Eng. 2020, 8, x FORFigureFigure PEER 4. REVIEW 4.Water Water elevation elevation η η timetime series series for for Experiment Experiment #1. #1. 9 of 25

FigureFigure 5. 5.Water Water elevation elevationη η timetime series for for Experiment Experiment #2. #2.

Figure 6. Water elevation η time series for Experiment #3.

Figure 7. Water elevation η time series for Experiment #4.

Figure 8. Water elevation η time series for Experiment #5.

J.J. Mar. Sci.Sci. Eng.Eng. 20202020,, 88,, xx FORFOR PEERPEER REVIEW 99 ofof 2525

J. Mar. Sci. Eng. 2020, 8, 786 9 of 24

FigureFigure 5.5. Water elevationelevation η η timetime seriesseries forfor Experiment #2.#2.

FigureFigureFigure 6. 6.6.Water Water elevation elevationelevation η η η timetime seriesseries series forfor for Experiment Experiment #3.#3. #3.

FigureFigureFigure 7. 7.7.Water Water elevation elevationelevation η η η timetime seriesseries series forfor for Experiment Experiment #4.#4. #4.

J. Mar. Sci. Eng. 2020, 8, x FORFigureFigure FigurePEER 8. REVIEW 8.8.Water Water elevation elevationelevation η η η timetime seriesseries series forfor for Experiment Experiment #5.#5. #5. 10 of 25

FigureFigure 9. 9.Water Water elevation elevation η η timetime series for for Experiment Experiment #6. #6.

Figure 10. Water elevation η time series for Experiment #7.

Figure 11. Water elevation η time series for Experiment #8.

Figure 12. Power Density Spectrum, Bretschneider vs. Measured.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 25

J. Mar. Sci. Eng. 2020, 8, 786Figure 9. Water elevation η time series for Experiment #6. 10 of 24 Figure 9. Water elevation η time series for Experiment #6. Figure 9. Water elevation η time series for Experiment #6.

Figure 10. Water elevation η time series for Experiment #7. Figure 10. Water elevation η time series for Experiment #7. FigureFigure 10. 10. WaterWater elevation elevation η η time seriesseries for for Experiment Experiment #7. #7.

Figure 11. Water elevation η time series for Experiment #8. FigureFigure 11. 11. WaterWater elevation elevation η η timetime series series for for Experiment Experiment #8. #8. Figure 11. Water elevation η time series for Experiment #8.

FigureFigure 12. 12. PowerPower Density Density Spectrum, Spectrum, BretschneiderBretschneider vs. vs. Measured. Measured. Figure 12. Power Density Spectrum, Bretschneider vs. Measured. Figure 12. Power Density Spectrum, Bretschneider vs. Measured.

In Table5, the measured data for the calm water resistance R T,CW,M as well as the average value of the total resistance at waves RT,M(t) are presented. Additionally, in Table5, the calculated values of the time-average added resistance in waves and the corresponding coeﬃcient are presented, following Equations (9) and (11). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 25

In Table 5, the measured data for the calm water resistance RT,CW,M as well as the average value In Table 5, the measured data for the calm water resistance RT,CW,M as well as the average value of the total resistance at waves R(t)T,M are presented. Additionally, in Table 5, the calculated of the total resistance at waves R(t)T,M are presented. Additionally, in Table 5, the calculated valuesJ. Mar. Sci.of Eng.the 2020time, 8,‐ 786average added resistance in waves and the corresponding coefficient11 of 24are values of the time‐average added resistance in waves and the corresponding coefficient are presented, following Equations (9) and (11). presented, following Equations (9) and (11). Table 5. Model scale resistance measurements. Table 5. Model scale resistance measurements. Table 5. Model scale resistance measurements. Calm water resistance RT,CW,M 24.127 Nt Calm water resistance RT,CW,M 24.127 Nt Resistance inCalm Waves, water Experiment resistance #1 RT,M,1 RT,CW,M 32.37624.127 NtNt Resistance in Waves, Experiment #1 RT,M,1 32.376 Nt ResistanceResistance in Waves, in Waves, Experiment Experiment #2 #1 RT,M,2 RT,M,1 30.93732.376 NtNt Resistance in Waves, Experiment #2 RT,M,2 30.937 Nt T,M,2 ResistanceResistance in Waves, in Waves, Experiment Experiment #3 #2 RT,M,3 R 32.05230.937 NtNt Resistance in Waves, Experiment #3 RT,M,3 32.052 Nt Resistance in Waves, Experiment #3 RT,M,3 32.052 Nt Resistance in Waves, Experiment #4 RT,M,4 33.726 Nt Resistance in Waves, Experiment #4 RT,M,4 33.726 Nt ResistanceResistance in Waves, in Waves, Experiment Experiment #5 #4 R RT,M,4 33.47233.726 NtNt Resistance in Waves, Experiment #5 T,M,5 RT,M,5 33.472 Nt Resistance in Waves, Experiment #5 RT,M,5 33.472 Nt ResistanceResistance in Waves, in Waves, Experiment Experiment #6 #6 RT,M,6 RT,M,6 33.19933.199 Nt Resistance in Waves, Experiment #6 RT,M,6 33.199 Nt ResistanceResistance in Waves, in Waves, Experiment Experiment #7 #7 RT,M,7 RT,M,7 31.34631.346 Nt Resistance in Waves, Experiment #7 RT,M,7 31.346 Nt ResistanceResistance in Waves, in Waves, Experiment Experiment #8 #8 RT,M,8 RT,M,8 31.12531.125 Nt Resistance in Waves, Experiment #8 RT,M,8 31.125 Nt AverageAverage resistance resistance in waves in waves R RT,M,av 32.27932.279 Nt Average resistance in waves T,M,av RT,M,av 32.279 Nt AverageAverage added added resistance resistance in waves in waves R RAD,M,av 8.1528.152 Nt Average added resistance in waves AD,M,avRAD,M,av 8.152 Nt −3 AverageAverage added added resistance resistance coeﬃcient coefficient in waves in waves C CW,AD,av1.764 1.76410 ×3 10 −3 Average added resistance coefficient in wavesW,AD,av CW,AD,av 1.764× −× 10

In Figures 13–20, the total resistance RT,M(t) time histories are presented as measured during the InIn Figures Figures 13–20, 13–20 the, the total total resistance resistance R RT,MT,M(t)(t) time time histories histories are are presented presented as as measured measured during during the experiments.the experiments. The average The average resistance resistance in waves, in waves, RT,MR, T,Mand, andthe calm the calm water water resistance, resistance, RT,CW,MRT,CW,M, are, experiments. The average resistance in waves, RT,M , and the calm water resistance, RT,CW,M , are alsoare plotted. also plotted. also plotted.

FigureFigure 13. 13. Resistance time series for Experiment #1. Figure 13. ResistanceResistance time seriesseries for for Experiment Experiment #1. #1.

FigureFigure 14. 14. ResistanceResistance time series forfor Experiment Experiment #2. #2. Figure 14. Resistance time series for Experiment #2.

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FigureFigureFigure 15. 15.15.Resistance Resistance time timetime series series forfor for Experiment Experiment #3.#3. #3.

FigureFigureFigure 16. 16.16.Resistance Resistance time timetime series series forfor for Experiment Experiment #4.#4. #4.

FigureFigureFigure 17. 17.17.Resistance Resistance time timetime series series forfor for Experiment Experiment #5.#5. #5.

FigureFigureFigure 18. 18.18.Resistance Resistance time timetime series series forfor for Experiment Experiment #6.#6. #6.

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FigureFigure 19. 19.Resistance Resistance time series for for Experiment Experiment #7. #7.

FigureFigure 20. 20.Resistance Resistance time series for for Experiment Experiment #8. #8.

3.3. Computations3.3. Computations at Model at Model Scale Scale ComputationsComputations at model at model scale scale were, were, at at first, first, performedperformed in in order order to to validate validate the thenumerical numerical method method withwith respect respect to the to the calm calm water water resistance resistance experiment experiment in the the towing towing tank. tank. Since Since the the hybrid hybrid method method has has beenbeen adopted adopted for for the the calculation calculation of of the the free free surface,surface, the the potential potential flow flow solver solver was, was, at first, at first, applied. applied. The calculationThe calculation domain domain on on the the free free surface surface extended extended from from X/L X /=L − −0.65= 0.65to X/L to = X 2.2,/L where= 2.2, L where stands L for stands the ship’s length and X = 0 at the fore perpendicular. Besides, the external− boundary was placed at a for the ship’s length and X = 0 at the fore perpendicular. Besides, the external boundary was placed distance equal to L from the center line. A total number of 46,000 quadrilateral elements were used, at a distance equal to L from the center line. A total number of 46,000 quadrilateral elements were of which 35,000 covered the free surface and 11,000 the wetted surface of the ship. During the used,calculations, of which 35,000 the ship covered was free the to sink. free surfaceOnce the and potential 11,000 solution the wetted has been surface accomplished, of the ship. the viscous During the calculations,flow code the has ship been was applied free toin sink.order Onceto predict the potentialthe free boundary solution around has been the accomplished, transom stern and the the viscous flowwake code region, has been in model applied‐scale, in thus order obtaining to predict a free the surface free boundary corrected for around the viscous the transom effects at the stern stern. and the wakeThe region, same external in model-scale, boundaries thus were obtaining used in the a computations, free surface correctedwhile the common for the boundary viscous ebetweenffects at the stern.the The two same solutions external located boundaries at X/L = 0.85. were The usedgrid around in the the computations, body consisted while of 9.5M the grid common nodes, while boundary between670 transverse the two solutions sections and located 180 grid at points X/L = in0.85. the transverse The grid direction around thewere body introduced. consisted In Figure of 9.5M 21, grid nodes,the while water 670 elevation transverse contour sections is presented and 180 for gridthe ship points at model in the scale, transverse at a Froude direction number were Fn = introduced. 0.180. In Figure 22, the free surface around the bow and the stern of the model is plotted. In Figure 21, the water elevation contour is presented for the ship at model scale, at a Froude number Fn = 0.180. In Figure 22, the free surface around the bow and the stern of the model is plotted. The corrected free surface, considered now as a fixed boundary, is introduced in the potential flow code in order to calculate the velocity and the pressure components on the external boundary. Next, the viscous flow solver was employed to perform accurate resistance calculations underneath the free surface. The calculation domain extended up to X/L = 1.5, while the external boundary was placed at a distance of 0.33 L. Grid dependence tests have been conducted for three grids, in order to study their influence on the numerical solution. The grid nodes for each grid are presented in Table6, where N1,N2 and N3 denote the number of nodes along the longitudinal, the circumferential and the transverse directions, respectively, whereas the second term stands for the transom grid. The coarse grid comprises about 5.344 M, the medium 9.208 M and the fine 15.799 M grid points and the ratio between the grid resolutions in successive refinements was about 1.2 in each direction. J.J. Mar. Mar. Sci. Sci. Eng. Eng.2020 2020,,8 8,, 786 x FOR PEER REVIEW 1414 of of 24 25 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 25

Figure 21. Water elevation contours, model scale, Fn = 0.180. Figure 21. Water elevation contours, model scale, Fn == 0.180.

(a) (b) (a) (b) FigureFigure 22. 22.Water Water elevation elevation around around the the bow bow ( a(a)) and and the the stern stern ( b(b),), model model scale, scale, Fn Fn= =0.180. 0.180. Figure 22. Water elevation around the bow (a) and the stern (b), model scale, Fn = 0.180.

The corrected free surface,Table considered 6. Characteristics now as a of fixed the used boundary, grids. is introduced in the potential flowThe code corrected in order free to calculate surface, theconsidered velocity nowand theas a pressure fixed boundary, components is introduced on the external in the boundary. potential flow code in order toGrid calculate the velocity N1 andN2 theN3 pressure componentsTotal No of on Nodes the external boundary. Next, the viscous flow solver was employed× to× perform accurate resistance calculations underneath Next,the free the surface. viscous Theflowcoarse calculation solver was645 domain employed57 125extended +to122 perform up59 to104 accurate X/L = 1.5, resistance while5.3442 the M calculations external boundary underneath was the free surface. The calculation domain× × extended× up ×to X/L = 1.5, while the external boundary was placed at a distancemedium of 0.33 L. Grid774 dependence68 150 + 148 tests71 have125 been conducted9.2083 M for three grids, in order to placedstudy theirat a distance influence of on0.33 the L. numericalGrid dependence× × solution. tests The× have grid× beennodes conducted for each gridfor three are presented grids, in order in Table to study their influence onfine the numerical928 81 solution.180 + 178 The85 grid150 nodes for 15.7994each grid M are presented in Table 6, where N1, N2 and N3 denote the× number× of nodes× along× the longitudinal, the circumferential and 6,the where transverse N1, N2 directions,and N3 denote respectively, the number whereas of nodes the along second the term longitudinal, stands for the the circumferential transom grid. andThe The calculated total resistance coefficients C (friction), C (pressure) and C (total) with the three thecoarse transverse grid comprises directions, about respectively, 5.344 M, the whereas mediumF the 9.208 second M and Pterm the stands fine 15.799 for the TM transomgrid points grid. and The the grids are compared in Table7 and they are defined as coarseratio between grid comprises the grid about resolutions 5.344 M, in successivethe medium refinements 9.208 M and was the about fine 15.7991.2 in eachM grid direction. points and the ratio between the grid resolutions in successive refinements was about 1.2 in each direction. The calculated total resistance coefficients CF (friction), CP (pressure) and CT (total) with the three RF RP RT The calculated total resistanceC = coefficientsC = CF (friction), CCP (pressure)= and CT (total) with the three(15) grids are compared in TableF 7 and they2 are Pdefined as 2 T 2 grids are compared in Table 7 and1/2 theyρSU Sare defined1/2 asρSU S 1/2ρSUS RRRFPT where S its wettedCCCFPT surfaceRRR underneath222 the fixed calm water free surface and U S the ship’s speed,(15) CCC1/2FPT SU 1/2 SU 1/2 SU while the total resistanceFPT RT is the222SSS sum of the friction component RF and the pressure component(15) RP, 1/2 SUSSS 1/2 SU 1/2 SU

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i.e., RT = RF + RP. The difference in CT between the medium and the coarse grid is about 0.3% and about 0.67% between the fine and the medium, while convergence is not monotonic. Therefore, it was considered that the fine grid could be reliable for the comparison purposes. In this case, the calculated total resistance was equal to 24.426 Nt, whereas the measured at the towing tank was 24.340 Nt, resulting in a difference of 0.35% which can be considered satisfactory as regards the applicability of the employed method.

Table 7. Calculated resistance coeﬃcients at model scale (Fr = 0.18, Re = 7.68 106). × 3 3 3 Grid CF 10 CP 10 CT 10 × × × coarse 3.226 1.918 5.144 medium 3.206 1.903 5.110 ﬁne 3.186 1.940 5.126

3.4. Calm Water Computations at Full Scale Computations at full scale were, at first, conducted in calm water in order to calculate the resistance of the ship which is the reference for the following added wave resistance tests. The non-dimensional free surface was taken from the model scale calculations, assuming that the Reynolds effect on the stern waves does not play a significant role for the purposes of the present work. The Reynolds number at full scale was equal to 1.18 109 and the same grid densities of Table6 were examined, × changing suitably the distances of the adjacent to the wall nodes to lie in the desirable region regarding the wall function approach. Table8 compares the three grid densities with respect to the integrated resistance coefficients. Unlike the model scale results, the differences in CT appear higher mainly owing to the coarser distribution of nodes in the normal direction, since many grid points are now concentrated near the body surface were the boundary layer is thinner due to the Reynolds effect. The convergence is monotonic and the difference in CT between the coarse and the medium grid is 0.86%, while between the medium and the fine grid is 0.55%. By applying the Richardson extrapolation as suggested by ASME [23], the uncertainty of CT at the fine grid is estimated lower than 0.03%.

Table 8. Calculated resistance coeﬃcients at ship scale (Fr = 0.18, Re = 1.18 109). × 3 3 3 Grid CF 10 CP 10 CT 10 × × × coarse 1.487 1.672 3.159 medium 1.510 1.622 3.132 ﬁne 1.512 1.603 3.115

Next, self-propulsion calculations were carried out. The propeller body forces were calculated according to the lifting line approximation, as described previously, and 25 iterative steps were needed to obtain convergence of the thrust, so that it becomes equal to the ship resistance. The main propulsive characteristics are presented in Table9. By comparing the values of the resistance coe fficients to those of Table8 with the fine grid, it is evident that the increase of C T is mainly due to the increase of CP which is about 38%. The thrust deduction factor 1-t is influenced accordingly. The nominal wake fraction 1-wn appears close to the effective wake fraction 1-we which is derived according to the open water numerical experiment, [16]. Finally, the delivered horsepower (DHP) and the propeller revolutions (RPM) are calculated by the lifting line code where the input data are the propeller thrust and the mean effective wake. J. Mar. Sci. Eng. 2020, 8, 786 16 of 24

Table 9. Calculated propulsive characteristics at ship scale.

3 3 3 CT 10 CF 10 CP 10 1-t 1-wn 1-we DHP (kW) RPM × × × 3.733 1.516 2.217 0.834 0.631 0.656 7306 120.68

In the time-dependent computations when the added resistance is taken into account, it is assumed that the computational grid remains the same as the one examined in calm water, while only the propeller thrust changes according to the varying wave resistance. Among the deviations from the real situation that arise with this simplification, a significant problem is related to the operation of the propeller under a continuously varying wave surface. In order to explore the influence of the relevant approximation on the propulsive performance, a numerical experiment was performed by comparing two different situations: the ship moving in calm water and the double body case where the free surface is taken as a flat plane. Evidently, in the second situation the hydrodynamic resistance will be substantially lower since the wave resistance is not taken into account. Therefore, it should be interesting to calculate the propulsion characteristics of the double body, if this difference is taken as an additional load on the propeller thrust. The grid density adopted for the double model was equal to that of the finer grid around the actual ship. Computations at full scale without propeller in operation have shown that the resistance of the double hull was about 39% lower than that of the actual ship, mainly due to the difference in the integrated pressure resistance component. Then, by adding the corresponding load on the propeller thrust, self-propulsion computations were performed and the percentage differences between the ship and the double body case, regarding the thrust, the wake fractions, the advance coefficient J, the propeller efficiency ηP, DHP and RPM are depicted in Table 10. It is noticeable that all these propulsive characteristics vary below 2%, indicating that the propeller thrust is the dominant parameter which affects the flow development well underneath the free surface.

Table 10. Diﬀerences in the propulsive characteristics between the real ship and the double body.

Actual Ship Double Body % Diﬀ. T (Nt) 7.8126 105 7.7377 105 0.96 P × × 1-t 0.834 0.842 0.96 − 1-wn 0.631 0.634 0.47 − 1-we 0.655 0.650 0.76 J 0.444 0.443 0.22

ηP 0.533 0.532 0.19 DHP (kW) 7306 7180 1.72 RPM 120.68 119.91 0.64

In order to support this argument, the isowakes (i.e., the ratio of the axial velocity component to the ship’s speed) on the propeller plane are plotted in Figure 23, corresponding to the computations without (a) and with propeller in operation (b). On each plot, the data on the left-hand side correspond to the double model while on the right-hand side, the data correspond to the case with free surface. Evidently, the differences between the two cases, double hull vs. free surface, are insignificant on the propeller disk, while deviations are observed close to the free surface. Therefore, these trends justify the small differences in Table 10, concerning the nominal and the effective wake fractions. Naturally, this comparison is only indicative since in reality the ship moves in random waves, which may affect more drastically the flow past the propeller. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 17 of 25

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(a) (b)

Figure 23. Axial velocity isowake curves on the propeller plane; on the left is the double body case and on the right the ship in calm water, for (a) the towed ship and (b) the self‐propelled ship.

(a) (b) 3.5. Computations at Head Sea State Figure 23.23. AxialAxial velocityvelocity isowake isowake curves curves on on the the propeller propeller plane; plane; on on the the left left is the is the double double body body case case and In orderonand to the on study right the right the the ship the effect ship in calm in ofcalm water, the water, for added (a for) the (a ) towedresistance the towed ship andship when ( andb) the ( bthe) self-propelled the ship self‐propelled is advancing ship. ship. in steady speed under random head‐sea conditions, computations in the time domain have been performed following 3.5. Computations at Head Sea State the assumptions3.5. Computations presented at Head in Sea Equations State (8)–(11) utilizing the resistance time series as measured in In order to study the effect of the added resistance when the ship is advancing in steady speed the towing tank.In order Thus, to study eight the numerical effect of the experiments added resistance were when conducted, the ship is advancing each lasting in steady 30 s. speedThe time step under random head-sea conditions, computations in the time domain have been performed following in the calculationsunder random was head Δ‐seat = conditions,1 s and in computations each time stepin the self time‐ propulsiondomain have beenwas performedachieved following in lower than 10 thethe assumptions presented in Equations (8)–(11) utilizing the resistance time series as measuredmeasured inin iterativethethe steps, towing while tank. tank. Thus,15 Thus, domain eight eight numerical numericalsweeps experiments were experiments performed were were conducted, conducted,per iteration. each each lasting Naturally, lasting 30 s. 30 The s. at Thetime the time step end of this procedurestepin the per in calculations the time calculations step, was the Δ wast =propeller 1∆ st and= 1 in s and eachthrust in time each equals step time self stepthe‐propulsion self-propulsionhydrodynamic was achieved was resistance achieved in lower in thanof lower the 10 ship. In Figure 24,thaniterative corresponding 10 iterativesteps, while steps, 15to while domainthe numerical 15 sweeps domain were sweepsexperiment performed were performed#1, per the iteration. time per Naturally,histories iteration. at Naturally,of thethe end thrust of at this the coefficient end of this procedure per time step, the propeller thrust equals the hydrodynamic resistance of the CT = (TPprocedure/(0.5ρSU Sper2)) andtime thestep, added the propeller wave resistancethrust equals C Wthe = hydrodynamic(CW,AD(t)) are resistanceplotted. ofEvidently, the ship. InC T follows ship.Figure In 24, Figure corresponding 24, corresponding to the numerical to the numerical experiment experiment #1, the time #1, histories the time of histories the thrust of coefficient the thrust the variations of CW but their difference2 is not constant, since the thrust deduction factor varies in coefficient C = (T2 /(0.5ρSU )) and the added wave resistance C = (C (t)) are plotted. Evidently, CT = (TP/(0.5TρSUS ))P and theS added wave resistance CW = (CW,AD(t))W are W,ADplotted. Evidently, CT follows time. It Cshouldfollows be the noticed variations here of C thatbut a their cut di‐offfference for the is not negative constant, thrust since the components thrust deduction is factorapplied, since theT variations of CW but theirW difference is not constant, since the thrust deduction factor varies in negativevariestime. thrust It in should time. values It be should imply noticed be reverse here noticed that propeller here a cut that‐off a cut-oforoperation theff fornegative the which negative thrust is impossible thrustcomponents components inis realapplied, is life. applied, since Thus far, the results dependsincenegative negative thruston the thrust values approach values imply imply reverse of the reverse propeller actuator propeller operation disk operation and which represent which is impossible is impossible the “real” in real in real life.effects life.Thus Thus under far, the far, the basic the results depend on the approach of the actuator disk and represent the “real” effects under the basic assumptionsresults ofdepend the proposed on the approach method. of the actuator disk and represent the “real” effects under the basic assumptions of thethe proposedproposed method.method.

Figure 24. Time histories of the thrust CT(t) and added wave resistance CW(t) coefficients, numerical experiment #1. Figure 24. Time histories of the thrust CT(t) and added wave resistance CW(t) coefficients, Figure 24. Time histories of the thrust CT(t) and added wave resistance CW(t) coefficients, numerical numericalNext, the experimentquasi‐steady #1. approximation is followed in order to estimate the powering demands. experiment #1. In Figure 25, the flow rate through the propeller disk 1‐wd and the effective wake fraction 1‐we are

Next, the quasi‐steady approximation is followed in order to estimate the powering demands. In Figure 25, the flow rate through the propeller disk 1‐wd and the effective wake fraction 1‐we are

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 25 plotted versus time. The first one is the direct result of the solution of the RANS equations, while the secondJ. Mar. Sci.oneJ. Mar.Eng. is Sci. found2020 Eng., 82020, byx FOR, 8an, 786 PEERopen REVIEW‐water numerical experiment, so that the velocity at infinite18 of 24produces18 of 25 wd through the disk under the known TP [16]. Apparently, the two curves do not present the same trends,plotted while versusNext, 1‐ wtime. thee is quasi-steady Thealways first lower, one approximation is asthe expected, direct is result followed than of in 1the‐w order dsolution. Besides, to estimate of 1the‐ thew dRANS poweringexhibits equations, demands.unusually while high the values,second which Inone Figure is are found 25 due, the by to flow anthe rateopen high through‐water values the numerical of propeller the added disk experiment, 1-w waved and resistance the so ethatffective the and wake velocity their fraction memory at infinite 1-we are effects. produces In d P anyw case,throughplotted 1‐w ethe and versus disk TP time. are under the The crucialthe first known one parameters is the T direct[16]. result beingApparently, of thethe solution input the data of two the in RANScurves the equations,lifting do not line present while code the or the in samethe opentrends, water secondwhile propeller one1‐w ise foundis characteristics always by an open-waterlower, whenever as numericalexpected, they experiment, than are available. 1‐w sod. thatBesides, theIn Figures velocity 1‐wd at 26exhibits infinite and produces27, unusually the isowake high wd through the disk under the known TP [16]. Apparently, the two curves do not present the same contoursvalues, whichthat are are associated due to the with high the values calculation of the ofadded the fraction wave resistance wd are plotted and their correspondingly memory effects. for aIn trends, while 1-we is always lower, as expected, than 1-wd. Besides, 1-wd exhibits unusually high localany maximumcase,values, 1‐we which and aT are Plocal are due theminimum to crucial the high ofparameters values the added of the being added wave the waveresistance input resistance data component. andin the their lifting memory Evidently, line e ffcodeects. the or thrust in the followingopen waterIn the any propeller case,resistance 1-we andcharacteristics history TP are the affects crucial whenever drastically parameters they being the are thewake available. input field data inandIn the Figures enhances lifting line 26 code andstrongly or27, in the the isowake axial velocitycontours componentopen that water are propellerassociated when it characteristics takes with high the calculationvalues. whenever they of arethe available. fraction In w Figuresd are plotted 26 and 27correspondingly, the isowake for a local maximumcontours that and are a associatedlocal minimum with the of calculation the added of the wave fraction resistance wd are plotted component. correspondingly Evidently, for a the thrust followinglocal the maximum resistance and ahistory local minimum affects of drastically the added wave the resistance wake field component. and enhances Evidently, strongly the thrust the axial following the resistance history affects drastically the wake field and enhances strongly the axial velocity component when it takes high values. velocity component when it takes high values.

Figure 25. Time histories of real (1‐wd) and effective (1‐we) wake fractions, numerical experiment #1.

Figure 25. Time histories of real (1-wd) and eﬀective (1-we) wake fractions, numerical experiment #1. Figure 25. Time histories of real (1‐wd) and effective (1‐we) wake fractions, numerical experiment #1.

Figure 26. Axial velocity isowake curves on the propeller plane for a local maximum of the added resistance. Figure 26. Axial velocity isowake curves on the propeller plane for a local maximum of the added resistance.

Figure 26. Axial velocity isowake curves on the propeller plane for a local maximum of the added resistance.

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Figure 27. Axial velocity isowake curves on the propeller plane for a local minimum of the added resistance. FigureFigure 27. 27. Axial Axial velocity velocity isowake isowake curves curves on on the the propeller propeller plane plane for for a a local local minimum minimum of of the the added added resistance. The resultsresistance. of the lifting line code concerning the propeller DHP and RPM are plotted on Figures 28–35 for numerical experiments #1–#8. As observed, a quite unrealistic behavior is observed TheThe results results of of the the lifting lifting line line code code concerning concerning the the propeller propeller DHP DHP and and RPM RPM are are plotted plotted on on Figures Figures for DHP,28–35 exhibiting28–35 for for numerical numerical prohibitive experiments experiments high values#1–#8. #1–#8. As atAs theobserved, observed, positive a a quite quite peaks unrealistic unrealistic of CT. behavior Thebehavior same is is observed conclusion observed for for holds T for RPMDHP, whichDHP, exhibiting exhibiting presents prohibitive prohibitive peaks of high high about values values 300% at at the the as positive positive regards peaks peaks the of steadyof C C.T .The The case. same same conclusion It conclusion should holds beholds taken for for into account thatRPMRPM this which which behavior presents presents represents peaks peaks of of about about the 300% propeller300% as as regards regards demands, the the steady steady whereas case. case. It It it should should is impossible be be taken taken into forinto account the account engine to that this behavior represents the propeller demands, whereas it is impossible for the engine to respond, becausethat this itbehavior has not represents the necessary the propeller margin demands, and, besides, whereas it cannot it is impossible follow the for extremelythe engine suddento respond,respond, because because it it has has not not the the necessary necessary margin margin and, and, besides, besides, it it cannot cannot follow follow the the extremely extremely sudden sudden peaks. Therefore, sustaining a constant speed seems to be an impossible situation, rendering the peaks.peaks. Therefore, Therefore, sustaining sustaining a a constant constant speed speed seems seems to to be be an an impossible impossible situation, situation, rendering rendering the the correspondingcorrespondingcorresponding towing towing towing tank experiments tank tank experiments experiments questionable questionable questionable unless unless unless approximate approximate estimations estimations estimations are are made. aremade. made.

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Figure 28.FigureFigureTime 28. 28. history Time Time history history of the of of requiredthe the required required DHP DHP DHP ( a( a)(a) and )and and RPM RPM ( b ((b),), numerical numerical numerical experiment experiment experiment #1. #1. #1.

(a) (b)

Figure 29.FigureTime 29. history Time history of the of requiredthe required DHP DHP (a ()a) and and RPM ( (bb),), numerical numerical experiment experiment #2. #2.

(a) (b)

Figure 30. Time history of the required DHP (a) and RPM (b), numerical experiment #3.

(a) (b)

Figure 31. Time history of the required DHP (a) and RPM (b), numerical experiment #4.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 20 of 25

(a) (a) (b) Figure 29. Time history of the required DHP (a) and RPM (b), numerical(b )experiment #2.

Figure 29. Time history of the required DHP (a) and RPM (b), numerical experiment #2. J. Mar. Sci. Eng. 2020, 8, 786 20 of 24

(a) (b) Figure 30. Time(a history) of the required DHP (a) and RPM (b), numerical(b )experiment #3.

FigureFigure 30. 30.Time Time history history of of the the required required DHP DHP (a )(a and) and RPM RPM (b (),b numerical), numerical experiment experiment #3. #3.

J. Mar. Sci. Eng. 2020, 8, x FOR(a PEER) REVIEW (b) 21 of 25 FigureFigure 31. 31.Time Time history(a history) of of the the required required DHP DHP (a ()a and) and RPM RPM (b (),b numerical), numerical( experimentb )experiment #4. #4.

Figure 31. Time history of the required DHP (a) and RPM (b), numerical experiment #4.

(a) (b)

FigureFigure 32. 32.Time Time history history of of the the required required DHP DHP (a )(a and) and RPM RPM (b ),(b numerical), numerical experiment experiment #5. #5.

(a) (b)

Figure 33. Time history of the required DHP (a) and RPM (b), numerical experiment #6.

(a) (b)

Figure 34. Time history of the required DHP (a) and RPM (b), numerical experiment #7.

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(a) (b) (a) (b) Figure 32. Time history of the required DHP (a) and RPM (b), numerical experiment #5. Figure 32. Time history of the required DHP (a) and RPM (b), numerical experiment #5. J. Mar. Sci. Eng. 2020, 8, 786 21 of 24

(a) (b) (a) (b) Figure 33. Time history of the required DHP (a) and RPM (b), numerical experiment #6. FigureFigure 33. 33.Time Time history history of of the the required required DHP DHP (a) ( anda) and RPM RPM (b ),(b numerical), numerical experiment experiment #6. #6.

(a) (b) J. Mar. Sci. Eng. 2020, 8, x FOR (PEERa) REVIEW (b) 22 of 25 FigureFigure 34. 34.Time Time history history of of the the required required DHP DHP (a )(a and) and RPM RPM (b (),b numerical), numerical experiment experiment #7. #7. Figure 34. Time history of the required DHP (a) and RPM (b), numerical experiment #7.

(a) (b)

FigureFigure 35. 35.Time Time history history of of the the required required DHP DHP (a )(a and) and RPM RPM (b ),(b numerical), numerical experiment experiment #8. #8.

InIn order order to to investigate investigate the the e ﬀeffectect of of the the selected selected time time step, step, the the numerical numerical experiment experiment #1 #1 was was repeatedrepeated using using a time a time step step of 0.1 of s. 0.1 In Figures. In Figure 36, the 36, time the histories time histories of the thrust of the coe thrustﬃcient coefficient CT are plotted CT are forplotted the ∆t for= 1 the s and Δt = the 1 s∆ andt = 0.1the s Δ tests.t = 0.1 It s is tests. obvious It is thatobvious by using that by the using lower the value lower of value∆t some of Δ peakst some peaks of the CW,AD can be captured that have a direct influence on the total resistance coefficient. This is clearly shown around t = 15 s and t = 25 s. It is noticeable that the existence of such peaks in CT do not affect the rest of the time history. As a direct consequence of this CT behavior, the calculated values for the required DHP and propeller RPM are influenced accordingly, as shown in Figure 37. However, the observed differences due to Δt do not essentially change the basic conclusions concerning the prohibitive values of DHP and RPM.

Figure 36. Comparison of the time history of the thrust (CT) coefficient, numerical experiment #1.

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(a) (b)

Figure 35. Time history of the required DHP (a) and RPM (b), numerical experiment #8.

J. Mar. Sci.In Eng.order2020 to, 8 ,investigate 786 the effect of the selected time step, the numerical experiment 22#1 of was 24 repeated using a time step of 0.1 s. In Figure 36, the time histories of the thrust coefficient CT are plotted for the Δt = 1 s and the Δt = 0.1 s tests. It is obvious that by using the lower value of Δt some of the CW,AD can be captured that have a direct influence on the total resistance coefficient. This is peaks of the CW,AD can be captured that have a direct influence on the total resistance coefficient. This clearly shown around t = 15 s and t = 25 s. It is noticeable that the existence of such peaks in CT do not is clearly shown around t = 15 s and t = 25 s. It is noticeable that the existence of such peaks in CT do affect the rest of the time history. As a direct consequence of this CT behavior, the calculated values for thenot required affect the DHP rest and of propellerthe time history. RPM are As influenced a direct consequence accordingly, of as shownthis CT inbehavior, Figure 37 the. However, calculated thevalues observed for the di ffrequirederences DHP due to and∆t propeller do not essentially RPM are changeinfluenced the accordingly, basic conclusions as shown concerning in Figure the 37. prohibitiveHowever, values the observed of DHP anddifferences RPM. due to Δt do not essentially change the basic conclusions concerning the prohibitive values of DHP and RPM.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 23 of 25

Figure 36. Comparison of the time history of the thrust (CT) coeﬃcient, numerical experiment #1. Figure 36. Comparison of the time history of the thrust (CT) coefficient, numerical experiment #1.

(a) (b) FigureFigure 37. 37.Comparison Comparison of the of time the history time history of the required of the DHP required (a) and DHP RPM ( (ab)), and numerical RPM experiment(b), numerical #1. experiment #1. Finally, it is interesting to study the self-propulsion of the ship at steady conditions, regarding the mean addedFinally, wave it is resistance interesting as anto study additional the self load‐propulsion on the propeller of the thrust. ship at According steady conditions, to the experimental regarding data, the mean added wave resistance component is calculated equal to 1.764 10-3 and the corresponding the mean added wave resistance as an additional load on the propeller× thrust. According to the propulsiveexperimental characteristics data, the mean are presented added wave in Tableresistance 11. Comparing component the is calculated results with equal those to 1.764 of Table × 109 ‐for3 and thethe calm corresponding water predictions, propulsive we can characteristics find an average are presented increase of in about Table 77% 11. inComparing DHP and 17.5%the results in RPM. with Thesethose differences of Table 9 are for stillthe substantiallycalm water predictions, higher than we the can usual find margins an average of the increase ship’s engine, of about implying 77% in thatDHP thereand will17.5% be in an RPM. actual These loss of differences speed at the are examined still substantially sea state. higher than the usual margins of the ship’s engine, implying that there will be an actual loss of speed at the examined sea state. Table 11. Calculated propulsive characteristics at ship scale.

3 Table3 11. Calculated3 propulsive characteristics at ship scale. CT 10 CF 10 CP 10 1-t 1-wn 1-we DHP (kW) RPM × × × 5.736CT × 10 1.5183 CF × 103 2.481 CP × 103 0.542 1‐t 0.6691‐wn 129471‐we DHP 141.72 (kW) RPM 5.736 5.736 1.518 2.481 0.542 0.669 12947 141.72 5.736

4. Conclusions A simplified numerical method has been employed for calculating the propulsion characteristics of a bulk‐carrier in waves. Under the assumption that the ship moves at steady forward speed, the added resistance in time is estimated through the towing tank experiments and the propeller thrust is adjusted accordingly, while propeller action is modeled according to the actuator disk approach. At first, computations were performed at model scale in still water in order to validate the method. Then, similar computations at full scale were conducted in order to calculate the ship’s resistance which was the basis for determining the added resistance in waves and the required propeller thrust. Then, unsteady RANS computations were applied at a random sea state characterized by a prescribed power spectrum. In order to study the variation of propulsive parameters (including DHP and RPM) in time, it was assumed that the ship moved steadily and, besides, the propeller performance could be approximated by a quasi‐steady procedure. The estimation of these parameters in a certain time period indicated that the propeller demands vary in an unrealistic way as regards the engine response. This behavior is rather due to the abrupt changes of the wave resistance which impose extremely high loads on the propeller. Therefore, the basic assumption of the steady forward speed in waves is subject to criticism. Another crucial issue, related to the aforementioned study, is that the towing tank experiments in random waves at constant speed appear questionable as regards the self‐propulsion behavior of a real ship. However, they are very useful to estimate the mead added resistance and, consequently, to

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4. Conclusions A simpliﬁed numerical method has been employed for calculating the propulsion characteristics of a bulk-carrier in waves. Under the assumption that the ship moves at steady forward speed, the added resistance in time is estimated through the towing tank experiments and the propeller thrust is adjusted accordingly, while propeller action is modeled according to the actuator disk approach. At ﬁrst, computations were performed at model scale in still water in order to validate the method. Then, similar computations at full scale were conducted in order to calculate the ship’s resistance which was the basis for determining the added resistance in waves and the required propeller thrust. Then, unsteady RANS computations were applied at a random sea state characterized by a prescribed power spectrum. In order to study the variation of propulsive parameters (including DHP and RPM) in time, it was assumed that the ship moved steadily and, besides, the propeller performance could be approximated by a quasi-steady procedure. The estimation of these parameters in a certain time period indicated that the propeller demands vary in an unrealistic way as regards the engine response. This behavior is rather due to the abrupt changes of the wave resistance which impose extremely high loads on the propeller. Therefore, the basic assumption of the steady forward speed in waves is subject to criticism. Another crucial issue, related to the aforementioned study, is that the towing tank experiments in random waves at constant speed appear questionable as regards the self-propulsion behavior of a real ship. However, they are very useful to estimate the mead added resistance and, consequently, to predict the mean propulsion demands in a seaway. Such a numerical experiment was also performed in order to compute approximately the required engine margins for the examined bulk carrier.

Author Contributions: S.P. and G.T. conceived and designed the numerical method; performed the simulations; analyzed the data and co-wrote this paper. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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