Hydrodynamic study of heeled double-stepped planing hulls using CFD and 2D+T method

Ocean Engineering 196 (2020) 106813

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Hydrodynamic study of heeled double-stepped planing hulls using CFD and 2DþT method Rasul Niazmand Bilandi a, Abbas Dashtimanesh b, *, Sasan Tavakoli c a

Department of Engineering, Persian Gulf University, Bushehr, Iran Estonian Maritime Academy, Tallinn University of Technology, 11712, Tallinn, Estonia c Department of Infrastructure Engineering, The University of Melbourne, Parkville, 3031, VIC, Australia b

A R T I C L E I N F O

A B S T R A C T

Keywords: Doubled-stepped planing hull Asymmetric planing 2DþT method CFD simulation Running attitudes

In the current paper, we have developed a method, based on 2DþT theory, to model the performance of doubledstepped planing hulls in asymmetric conditions. We have performed Computational Fluid Dynamics (CFD) simulations to evaluate the difference between the results of the 2DþT method and CFD. We have validated 2DþT and CFD simulations. The quantitative comparison between the results of both methods shows they predict almost similar heeling moment, resistance and trim angle for a doubled-stepped planing hull. Results of nonstepped and doubled-stepped planing hulls are compared against each other, demonstrating that an increase in heel angle has less influence on the performance of the doubled-stepped planing hull. The heeling moment of a double-stepped planing hull is found to be smaller than a heeling moment of a non-stepped planing hull at early planing speeds, but, by the increase in speed, heeling moment of doubled-stepped planing hulls becomes slightly larger.

1. Introduction Planing hulls with a transverse notch from one chine to another are known as stepped boats, which can reach high-speeds in the sea (Garland 2010, Morabito and Pavkov, 2014). These vessels are used for a wide range of purposes (e. g. sports and pleasure). The step on the bottom of these hulls causes an air cavity just behind it (Dashtimanesh et al., 2019a), resulting in a smaller wetted surface (and resistance), while the boat (not always) may become less stable, i.e., the air venti­ lation underneath the bottom can trigger transverse instabilities. Modelling of the hydrodynamic behavior of stepped planing hulls in non-zero heel conditions (asymmetric planing) improves our under­ standing about the transverse instability of these vessels (while low in­ formation is available as discussed by Morabito et al., 2014) and provides us with a pattern by which we can avoid instabilities in the early stage design (see the published paper by Xu et a. 1999). To model the non-zero heel condition for a planing boat, it is required to compute the hydrodynamic pressure (or forces) acting on the boat, and then to find the equilibrium condition of the vessle in calm water (Ghadimi et al., 2017a). Empirical methods which are developed using a large variety of towing tank tests (see Savitsky, 1964) are recognized as the first option that can be used for modeling of the

asymmetric planing motion (Judge, 2014). These methods have shown good ability in modeling of (non-stepped) planing hulls in calm water conditions. The method developed by Savitsky (1964), which predicts the calm water performance of a hard-chine planing hull is a good example of empirical methods used for simulations of planing motion. The high potential of empirically developed methods in modeling of the heeled motion of a non-stepped planing hull has been previously observed (Lewandowski, 1997;Ghadimi et al., 2017; Judge, 2014). But for the case of stepped planing hulls, there are still some challenges. To establish any model for hydrodynamic simulation of a stepped planing boat, the wake behind the step is needed to be found (Savitsky and Morabito, 2010). It has been seen that empirical models can be further developed for performance prediction of stepped planing hulls (a zero-heel angle condition) in calm water (Dashtimanesh et al., 2016, 2017) when empirical equations of the free surface or linear wake theory are used to model water rise behind each step (Savitsky and Morabito, 2010; Dashtimanesh et al., 2016, 2017). But the flow pattern becomes asymmetric when a planing boat rests at a non-zero heel angle, which can result in a different pattern of the hydrodynamic pressure (Iafrati and Broglia, 2008). Under the action of such a condition, empirical equations, like those of Savitsky (1964), are not applicable for the lifting surface locating behind of the step.

* Corresponding author. E-mail address: [email protected] (A. Dashtimanesh). https://doi.org/10.1016/j.oceaneng.2019.106813 Received 29 June 2019; Received in revised form 1 December 2019; Accepted 2 December 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Ocean Engineering 196 (2020) 106813

The 2DþT method, or so-called 2.5D method, which has been used to model the dynamic motions (Zarnick 1978, 1979; Martin, 1976; Payne, 1995a) and the performance of planing vessels (Ghadimi et al., 2017a), is another method that can be applied to model the problem (asym­ metric motion of a planing hull). This method is developed using a two-dimensional water entry problem, which is then extended to a three-dimensional (3D) solution. Zarnick (1979) developed an early nonlinear mathematical method for the dynamic motion of a planing boat in waves by using 2DþT theory. At a very early stage, simulations of this method were based on the added mass theory (Von Karman, 1929). Concerns about the accuracy of this method in prediction of the sectional force managed some researchers to present more accurate solutions for sectional forces (see e. g. in Akers, 1999, Troesch and Hicks, 1994, Hicks et al., 1995; Garme, 2005, Payne, 1994, Payne, 1995b, Van Deyzen 2008; Ghadimi et al., 2013b). Later, the pressure-based simu­ lations, like the methods presented by Wagner (1932) and Algarin and Tascon (2011), were used to compute the sectional hydrodynamic forces (e.g. Zhao et al., 1997; Sun and Faltinsen, 2007, Ghadimi et al., 2016a). 2DþT method can also use available simulations of water entry with asymmetric flow pattern (Toyoma, 1993, Xu, 1998, Xu et al., 1999, Xu and Troesch, 1999 a and b, Korobkin and Melenica, 2005, Algarin and Tascon, 2011, Niazmand Bilandi et al., 2018), and obliquespeed (Judge et al., 2004, Gu et al., 2014, Izadi et al. (2018). These solutions has been found to have the potential to be used for simulation of other motions of a planing hull like yawed/heeled steady performance (Ghadimi et al., 2013a; Ghadimi et al., 2016b, 2016c, Tavakoli and Dashtimanesh, 2019; Tavakoli et al., 2017a, 2017b, 2017c, 2017d, 2018a; Tavakoli and Dashtimanesh, 2018; Dashtimanesh et al., 2019b), and maneuvering (Tavakoli et al., 2018b, 2018c; Tavakoli and Dashtimanesh, 2019) mo­ tions. Most recently, 2DþT method has been used to study the perfor­ mance of double-stepped planing hulls in calm water (Niazmand Bilandi et al., 2019a). Niazmand Bilandi et al. (2019a) have found that this method works with proper accuracy for stepped planing hulls advancing in symmetric conditions. If this very last effort is adopted for the asymmetric conditions (like the work of Ghadimi et al., 2017a which is particular to the asymmetric motion of non-stepped hulls), it can pro­ vide simulation for the asymmetric motion of a doubled-stepped planing boat. Last but not least, numerical simulations of the viscous flow around a planing hull has been used to model the planing motion by some authors using Computational Fluid Dynamics (CFD) approach (see e. g. in Dashtimanesh et al. (2019b)), De Marco et al. (2017), Di Caterino et al. (2018), Esfandiari et al., 2019). Application of CFD in the field of naval hydrodynamics has seen to be accelerated in the last two-decades (2017, Khojasteh and Kamali). For the case of double-stepped planing hulls, fair accuracy in prediction of the wetted surface of the vessel and the dy­ namic trim angle has been reported by Dashtimanesh et al. (2019b). While the 2DþT theory can be used to simulate the problem, CFD can also be used to solve the fluid field around a planing vessel (Iafrati and Broglia (2010)). Whereas CFD models have high potential to model the problem (planing motion), they are time-consuming and need High Performance Computers (HPC) in some cases (where the number of grids becomes large and motions are involved, Stern et al., 2015). Therefore, if it is not necessary, and mathematical models provide us with accurate results, we can neglect CFD simulations in the early stage design of the vessel. For the case of a heeled doubled-stepped planing boat, we are not sure about this issue. A comparison between CFD simulation and 2DþT simulation can provide an answer for this question. In the current paper, 2DþT and CFD methods are used to model the performance of heeled stepped planing hulls. The 2DþT method is developed by using the solution of the water entry of an asymmetric wedge and the linear wake profile assumption. CFD is used to solve the governing equations on the viscous fluid flow around the heeled vessel. The paper is organized as follows. In the next section, the problem is defined, and the governing equations on the problem are presented. In Section 3, the 2DþT method is developed. The set-up for the CFD model

is described in Section 4. Validation of both 2DþT and CFD method are presented in Section 5. Section 6 provides the main results of the paper. First, it is investigated how different the results of the CFD and 2DþT models are. Then, the influences of the step on the performance of a heeled planing hull are evaluated. A summary of the results is presented in Section 7. 2. Problem definition 2.1. Hydrodynamic aspect In the current research, it is assumed that a hard-chine prismatic planing hull is moving forward with the velocity of V in the calm water condition at a fixed heel angle of ϕ as shown in the upper panel of Fig. 1. The vessel has two steps, the front (or first) step, located at Ls1 (longi­ tudinal position of the first step with respect to the transom), and the rear (or second) step, located at Ls2 (longitudinal position of the second step with respect to the transom), with heights of H1 (step height with respect to the bottom of the front body), and H2 (step height with respect to the bottom of the middle body). Boat has a constant beam of B and a constant deadrise angle of β in its entire length. When the boat locates at a heel angle of ϕ, deadrise angles of port (shown by subscript p) and starboard (shown by subscript s) are found as

Fig. 1. A sketch of the problem. A heeled hard-chine planing hull with two steps locates at the trim angle of θ is moving forward with the velocity v as shown in upper panel. Two coordinate systems are considered. Oxyz (the hy­ drodynamic frame which moves forward with vessel speed) is placed under/ above the CG on calm water line. The x-axis is parallel to the calm water and positive forward, while the z-axis is normal to the calm water and positive downward. Governing equations on the motion of the vessle are formulated in this frame. Gξηζ is the body-frame, attached on the CG of the vessel. The middle panel shows a cross section of the vessel. Each side of the vessel (starboard and port) has a different deadrise angle. Dashed black line shows the calm water line in both upper and middle panels. Forces acting on the bottom of the vessel are displayed in lower panel. F refers to the force caused by the fluid pressure (which is normal to the planing surfaces) and R refers to the resistance. The weight of the vessel is acting on the CG. Dashed red line shows the free surface around/behind each planing surface. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 2

R.N. Bilandi et al.

�

Ocean Engineering 196 (2020) 106813

βp ¼ β ϕ : βs ¼ β þ ϕ

Southampton (Taunton et al. (2010)). The performance of the double-stepped planing hull is computed (sub-section 5.1) by both methods to check the validity of simulations. Also, the heeling motion of the model c and the model c2 is computed to explore the effects of two steps on the performance and heeling moment of a planing vessel (sub-section 6.2). The principal characteristics of these two models are shown in Table 1. Note that LCG and VCG respectively refer to longitu­ dinal and vertical positions of the CG. Their body plan is shown in panels a and b of Fig. 2. The third planing hull (Table 2), introduced in the United States Naval Academy, is a non-stepped planing hull previously studied by Judge and Judge (2013). The asymmetric motion of this planing hull is modeled by using both 2DþT method and CFD to assess the validity of them in the prediction of the heeling moment. The principal charac­ teristics of this model is reported in Table 2. Its body plan is shown in panel c of Fig. 2.

(1)

The boat reaches an equilibrium in the calm water, i.e. it eventually locates at a constant (dynamic) trim angle (shown by θ) and its CG lo­ cates at a fixed vertical position with respect to the calm water. In such a condition, hydrodynamic and hydrostatic forces, caused by the fluid pressure, support the weight of the vessel (which is denoted with Δ). The resistance force including fictional, hydrodynamic and spray compo­ nents acts on the vessel. A heeling moment (MHi) keeps the vessel at the fixed heel angle of ϕ. In response to this moment, a restoring force due to pressure is produced. The vessel dynamic obeys the Newton’s Second Law. By assuming two coordinate systems, a body-fixed frame, shown by Gξηζ, and a hy­ drodynamic frame (which is located on the calm water and advances forward but doesn’t have any heave and pitch motions), denoted with Oxyz, the governing equations on the motion are found as ! 3 X 0¼ ðRxi þ Fxi þ T i¼1 3 X 0¼ ðRzi þ Fzi þ Δ i¼1 3 X 0¼ ðMHi þ Fϕi

3. 2DþT method The current mathematical approach incorporates the linear wake assumption, the wave profile behind the step, and the 2DþT theory to solve the problem. The mathematical approach is developed by assuming flow is ideal. A final iterative procedure is developed to predict the targeted parameters, including running attitudes, resistance force and heeling moment.

!

!

(2)

i¼1

! 3 X 0¼ ðRθi þ Fθi :

3.1. Linear wake assumption

i¼1

In the above equation, subscripts x, z, ϕ, θ, denote the forces (or moments) components in the surge, heave, roll and pitch directions, respectively. Tis the thrust force and is assumed to pass through CG and be parallel to the calm water line, i.e., this force doesn’t affect the per­ formance of the vessel. Note that effects of thrust force on performance of the vessel have been reported to be negligible (Savitsky, 1964; Gha­ dimi et al. 2015, 2016b). Finally, we are computing performance of some planing models (tested in the towing tank) in the current research (details are given in the sub-Section 2.2). All these tests are performed without any thrust force. It is reasonable to neglect thrust force contri­ bution when we want to replicate these experiments. But the effects of thrust force can be applied in the future to make the method more general. It is assumed that the double-stepped planing hull is divided into three lifting surfaces, including the fore (or front) surface, the middle surface and, the aft (or rear) surface. Each of these surfaces behaves, not absolutely but almost, like a planing hull (see Dashtimanesh et al., 2018). The wetted keel lengths of the surfaces are denoted with LKF, LKM and LKA (note that K refers to keel), and are given by LKF ¼ LK

LS2

LK A ¼ Ls2

Ldry2

Ldryi ¼

Hi ; tanðθ þ τiþ1 Þ

(6)

ði ¼ 1; ​ 2Þ

where, Hi is the step height, θ is the dynamic trim angle, and τi is the local trim angle. It is assumed that the longitudinal sections have different wake profiles, which result in a local deadrise angle, denoted with βi . The deadrise angle behind each step in the non-zero heel angle is found by � βpi ¼ βi þ βLi ϕ ; ​ ði ¼ 1; ​ 2; ​ 3Þ (7) βsi ¼ βi þ βLi þ ϕ where, βp is deadrise angle of port, βs is deadrise angle of starboard, βL is local deadrise angle, and ϕ is heel angle.

(3)

Ls1

LKM ¼ LS1

For the case of a stepped planing hull, a linear wake assumption is used (Fig. 3a). It is assumed that the wake behind each step is parallel to the calm water line (Niazmand Bilandi et al., 2019a, b). Using this assumption, the ventilated length behind each step is found as

Ldry1

(4) (5)

Table 1 Principal characteristics of model c2 and model c (Taunton et al. (2010)).

where, LK is the overall wetted length. Ldry is the ventilation length behind each step, computation of which is explained in the next section. Forces acting on these surfaces and the entire vessel are needed to be found to compute the final equilibrium condition of the vessel. The method for computing forces and moments is different from traditional techniques used for ships. 2.2. Investigated vessels In this study, three different planing hulls (models) have been studied (they are modeled by CFD and 2DþT methods). The first and second planing hulls are non-stepped (model c) doubled-stepped (model c2) planing models, previously introduced in the University of 3

Model Characteristics

Value

L (m) B (m) Drafts (m) β (� ) Δ (N) VCG (m) LCG (from transom) (m) Ls1 /L (only for C2)

2 0.46 0.09 22.5 243.40 0.1 0.66 0.31

H1 (only for C2)

0.015

Ls2 /L (only for C2)

0.158

H2 (only for C2)

0.008

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Ocean Engineering 196 (2020) 106813

Fig. 2. Body planes of model c (panel a) and c2 (panel b) which were introduced by Taunton et al. (2010) as well as the body plan of the non-stepped model (panel c) studied by Judge and Judge (2013). Table 2 Principal characteristics of the planing hull used by Judge (2014). Model Characteristics

Value

L (m) B (m) Drafts (m) β ð∘Þ

1.54 0.448 0.5, 0.75, 1, 1.25, 1.5 20

Δ (N) VCG (m) LCG (from transom) (m) Trim angle (� ) Keel wetted length (m)

135.10 0.134 0.3048 3.3 0.59

2.9 0.57

3.2. Asymmetric planing and 2DþT theory 3.2.1. Conversion from 3D problem to 2D problem in time domain The heeled doubled-stepped planing boat is assumed to pass through a fixed transverse plane (Tavakoli et al., 2018d) as shown in Fig. 3b. The three-dimensional (3D) hydrodynamic problem is changed into three water entry problems for heeled (asymmetric) wedge sections (Find the technical commented on the 2DþT method in the published papers by Sun and Faltinsen (2010, 2011)). For each planing surface, a water entry problem with a vertical speed of wi ¼ v sinðθ þ τi Þ;

ði ¼ 1; ​ 2; ​ 3Þ

(8)

is established. The 2D water entry problem is solved from time 0 (which corresponds to the intersection of the keel and calm water in the considered planing surface) to tp (which corresponds to the transom/ step section), given by tpi ¼

LKi ; v

ði ¼ F; ​ M; ​ AÞ

(9)

Longitudinal position (with respect to the intersection of the calm water and keel) corresponding to each time is found by ξs i ¼

vt ; ​ ði ¼ 1; ​ 2; ​ 3Þ: cosðθ þ τi Þ

Fig. 3. The schematic of the 2DþT method. (a) shows the basics of the linear wake theory. The wake is assumed to behave linearly as a function of distance. Surfaces behind the step have local trim angles of τi (i ¼ 1, 2) due to threedimensional effects. A local deadrise angle is considered to apply threedimensional effects of the wake as well. Note that the dashed red line refers to the free surface of the water. (b) shows the schematic of the 2DþT theory. A 2D fixed earth fixed plane (black line) is in the path of the vessel. As the vessel passes through it, the problem is changed into a water entry of heeled wedge with a constant speed of wi ¼ v sinðθ þ τi Þ. This process is the same as water entry problem and depends on the chine wetting condition. Both chines can be dry (left), one chine can be wet (middle) and both chines can be wet (right). The water entry problem provides sectional forces and moments acting on each planing surface (panel c). Technical information about the hydrodynamic of an asymmetric can be found in Javanmardi et al. (2018). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

(10)

3.2.2. 2D forces The pressure acting on the wall of the wedge is found using Wagner Water entry solution as ! ! wi ðci c_i þ ð μi þ yi Þμ_ i Þ w2i ð μi þ yi Þ2 pi ¼ ρw ; ​ i¼1; ​ 2; ​ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 2 c2i ð μi þ yi Þ2 c2i ð μi þ yi (11) where ρw is fluid density, yi is the lateral distance from wedge apex. c and μ are the half-wetted beam and asymmetric parameter, varying in time. A dot over these parameters refers to the time rate of them (Ghadimi et al., 2019a). These two parameters are calculated as

μi ¼ 0:5ðcPi

cSi Þ; ​ ði ¼ 1; ​ 2; ​ 3Þ

(12)

4

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R.N. Bilandi et al.

Integrating the hydrodynamic pressure over the wedge wall (Fig. 3c), the 2D hydrodynamic forces are found as 0 1 1 Z Z V @ A f HDi ¼ pi cosðβPi þ βLi dl þ pi cosðβSi þ βLi Þdl; ​ i ¼ 1; ​ 2; ​ 3A

(13)

ci ¼ 0:5ðcPi þ cSi Þ; ​ ði ¼ 1; ​ 2; ​ 3Þ

where cPi and cSi are the half-wetted beam of the port and starboard respectively. During the water entry of an asymmetric wedge, three different phases, depending on the chine wetting condition, occur (see Fig. 3b):

SP

(21)

Phase 1: two dry chines

� � π 1 1 c_i ¼ wi þ ; ​ ði ¼ 1; ​ 2; ​ 3Þ tanðβPi þ βLi Þ tanðβSi þ βLi Þ 4 �

1 tanðβPi þ βLi Þ1

4

π μ_ i ¼ wi

�

4

1 tanðβPi þ βLi Þ

� 1 ; ​ ði ¼ 1; ​ 2; ​ 3Þ tanðβSi þ βLi Þ � 1 ; ​ ði ¼ 1; ​ 2; ​ 3Þ tanðβSi þ βLi Þ

(15) (16)

μi þ bji

SS

Hydrostatic force acting each section is found using fHSi ¼ ρw gðAPi þ ASi Þ ; ​ ​ ðj ¼ P; ​ SÞ ​ ði ¼ 1; ​ 2; ​ 3Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

(17)

(24)

Aj i

where Aj (j ¼ P, S) is the wetted area of each section given by 8 0 10 1 c2ji > > �; cji < bji B > Aji ¼ > C B C > tan βji þ βLi > B CB C < B CB C Bj ¼ P; ​ SC Bi ¼ 1; 2; 3C B CB C > > B C B C 2 > > @ A@ A bji > > �; ​ cji � bji : Aji ¼ tan βji þ βLi Also, the center of hydrostatic pressure (CpHSi) has is given � � b*P b*Si CpHSi ¼ i ; i ¼ 1; 2; ​ 3 3 3

are needed to be solved, where,

�2

@i ¼ 1; ​ 2; ​ 3A

where, superscripts V and H refer to the vertical and horizontal forces, and l is the distance of any point from wedge apex. dl is the differential of l. The center of hydrodynamic force (CpHDi) is found by R p ldl s þs i CpHDi ¼ R P S ; ​ ði ¼ 1; ​ 2; ​ 3Þ (23) p dl sP þsS i

(18)

�2 �32

pi sinðβSi þ βLi Þdl;

1

(22)

Phase 2 and 3 respectively refer to the condition one and two chines are touched by water. In this condition, the set of equations 0 1 � �qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffi2ffi 2 ci c_i þ μi þ bji μ_ i c2i μi þ bji ¼ wi ; @j ¼ P; SA ði ¼ 1; 2; 3Þ �2 μi þ bji

μi þ bji

0

Z

pi sinðβPi þ βLi Adl SP

Phase 2 and 3: Wetted chines

� 2 2 ci 3

1

Z f HHDi ¼

None of the chines has been touched by the water in this phase. c, μ and their time rates are given by � � π 1 1 ci ¼ wi ti þ ; ​ ði ¼ 1; ​ 2; ​ 3Þ (14) tanðβPi þ βLi Þ tanðβSi þ βLi Þ 4

π μi ¼ wi ti

SS

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi� � 2� � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi 2 � � c þ c μ þ b B i j 2 i i i � ¼ wi Bti 2 c2i μi þ bji þ 2ci ln�� �2 � @ � � μi þ bji

(25)

(26)

1 C tcwji C A;

(19)

ðj ¼ P; ​ SÞ ði ¼ 1; 2; 3Þ

bj is the half-beam of the boat at side j. tcwi is time at which the water reaches chine, and is found by � � � 2 bji tan βji þ βLi tcwji ¼ ; j ¼ P; ​ S ði ¼ 1; 2; 3Þ (20) wi π

(

Also ti refers to the time. Details of the equations can be found in Algarin and Tascon (2011).

Summation of hydrodynamic and hydrostatic moments gives 2D rolling moment as

3 X

mi ¼

� f VHDi CpHHDi

� � VCG sin ϕ þ f HHDi CpVHDi

� VCG cos ϕ þ fHSi CpHSi

where b*ji (j ¼ P, S) is b*ji ¼ cji ; cji < bji

b*ji ¼ bjj ; ​ cji � bji

� VCG sin ϕ ; ​ ði ¼ 1; ​ 2; ​ 3Þ

i¼1

5

! j ¼ P; ​ S

! i ¼ 1; ​ 2; 3

(27)

(28)

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Ocean Engineering 196 (2020) 106813

where, superscripts V and H denote component refer to the vertical and horizontal directions, respectively.

3.2.4. Frictional forces The frictional drag forces acting on the boat is assumed to be composed of two terms: frictional force acting on the pressure area and frictional force acting on the whisker spray area (see Savitsky et al., 2007; Begovic and Bertorello, 2012; Ghadimi et al., 2014, 2015). The frictional force acting on the pressure and spray areas can be calculated using the equation

3.2.3. 3D forces Integrating the 2D forces along the entire length of the boat, threedimensional forces are computed. But to consider the effects of transom/step, a transom correction function, introduced by Garme (2005) is implemented. This function is given by � � � 2:5 ξji ξi ; ði ¼ 1; ​ 2; ​ 3Þ (29) Ctri ¼ tanh 0:34BFnB

1 1 Ri ¼ ρSwi v2 Cfi þ ρSsi v2 Cfi ; ​ ði ¼ 1; ​ 2; ​ 3Þ 2 |fflfflfflfflfflffl {zfflfflfflfflfflffl } 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl} Pressure ​ area

j ξi

where is the longitudinal position of the step/transom in the bodyfixed frame. Note that it is assumed that each step reduces 2D forces in the way a transom does (Niazmand Bilandi et al., 2019a, b). By implementing the transom reduction function, the hydrodynamic force acting on each planing surface in the longitudinal direction is found by Z Fx i ¼ f vHDi Ctri ðξÞsinðθ þ τi Þdξ; ði ¼ 1; ​ 2; ​ 3Þ (30)

(34)

Spray ​ area

where Cfi is the frictional drag coefficient of each planing surface and has been determined using ITTC 78.Swi and Ssi respectively refer to the pressure and spray areas and have been computed using equations Swi ¼ SwPi þ SwSi ; ​ ði ¼ 1; ​ 2; ​ 3Þ;

(35)

Ssi ¼ SsPi þ SsSi ; ​ ði ¼ 1; ​ 2; 3Þ:

(36)

The wetted pressure and spray areas for each side of planing surfaces are found by equations (37) and (38), respectively.

L wi

The vertical force, summation of hydrodynamic pressure and hy­

Z Swji ¼

ξcji

ξki

cji � dξ þ cos βji þ βLi

Z

ξT

​ or ​ S

ξcji

drostatic pressure, is found as Z Fz i ¼ f vHDi Ctri ðξÞcosðθ þ τi Þdξ L wi

bji � dξ; ðj ¼ P; ​ SÞ ði ¼ 1; 2; 3Þ cos βji þ βLi

Z

q

fHSi Ctri ðξÞdξ;

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl }|fflfflfflfflfflffl�fflfflfflfflfflffl��� fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl�{ � ξi tan 1:5 tan 1 cji ξi cji dξ � Ssji ¼ cos β þ β ξki ji Li � Z ξT ​ or ​ S bji cji � dξ; þ cos βji þ βLi ξcji

ði ¼ 1; ​ 2; ​ 3Þ

Z

L wi

(31) The rolling and pitching moments are computed as Z Fϕi ¼ mi Ctri ðξÞdξ; ði ¼ 1; 2; 3Þ

(32)

Z

Z f vHDi Ctri ðξÞdξ

L wi

2

ξ

fHSi Ctri ðξÞdξ;

ði ¼ 1; 2; 3Þ:

ξcji

(38)

where ξT ​ or ​ S refers to the longitudinal location of transom and step in the body-fixed frame. The pitching moments caused by frictional forces are found by

Lwi

Fθ i ¼ ξ i

(37)

(33)

L wi

� 0

cP B tanðβPi þ βLi Þ @

� � VCG SwPi þ

cS tanðβSi þ βLi Þ

� 1 VCG SwSi C Aþ

6 VCG 6 Swpi þ SwSi 6 6 6 � � Rθi ¼ Ri 6 0� ðbPi cPi Þ ðbSi cSi Þ qP qS 6 þ VCG SsPi þ þ 6B cosðβSi þ βLi Þ cosðβSi þ βLi Þ 4 B cosðβPi þ βLi Þ cosðβPi þ βLi Þ @ Sspi þ SsSi

6

3 7 7 7 7 � 17 7​; VCG SsSi 7 C7 C5 A

0

1

B C B C B C B C B ​ Bi ¼ 1; ​ 2; ​ 3C C B C B C @ A

(39)

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Fig. 4. Guideline for prediction of the running attitudes of heeled double-stepped planing hulls by using 2DþT theory.

Note that for phases 2 and 3, cP ¼ bP , cS ¼ bS and qP ¼ bP , qS ¼ bS . Total resistance of the vessel is found by 3 X

RT ¼

Rxi þ Fxi ; ​ ði ¼ 1; 2; 3Þ:

Table 3 Summary of boundary conditions.

(40)

Boundary

Velocity

Pressure

Volume fraction

Hull surface

u¼0

∂n p ¼ 0

∂n α ¼ 0

3.3. Guideline

Inlet water

u ¼ vi

∂n p ¼ 0

α¼1

Inlet air

u ¼ vi

∂n p ¼ 0

∂n α ¼ 0

The guideline for determination of the running attitudes of a heeled stepped hull by 2DþT method has been shown in Fig. 4. The following steps are needed to be taken.

Outlet

∂n u ¼ 0

p¼0

∂n α ¼ 0

Top

u ¼ vi

∂n p ¼ 0

∂n α ¼ 0

Bottom

u ¼0

∂n p ¼ 0

α¼1

Water inlet

u ¼ vi

∂n p ¼ 0

α¼1

Air inlet

u ¼ vi

∂n p ¼ 0

α ¼0

i¼1

Side wall

� STEP1: A trim angle is guessed. � STEP2: A keel wetted length is guessed.

Fig. 5. The sketch of computational domain and boundaries used to numerically solve the asymmetric planing problem: left and right panels respectively show the front and side views of the domain. Gray and white colors denote on the water and air, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 7

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� STEP3: Water entry problem for each planing surface is solved (section 3.2.2). � STEP4: Resistance force is computed (Equation (40)). � STEP5: The error of the heave equation (Fz þ Δ < ε) is checked. The keel wetted length is re-guessed (STEP 2) until the left side of heave equation converges to zero, i.e., the lift and weight agree. � STEP6: The error of pitch equation (Fθ þ Rθ < ε) is checked. The trim angle is re-guessed (STEP 1) until the left side of pitch equation converges to zero, i.e., the pitching moment vanishes.

�

(42)

�T where, u ¼ ux ðx; y; z; tÞ; uy ðx; y; z; tÞ; uz ðx; y; z; tÞ is fluid velocity, ρeff ðx; y; z; tÞ is the effective density, pðx; y; z; tÞ is the pressure, Tðx; y; z; tÞ is the viscous stress tensor, f b ¼ ð0; 0; gÞ is the force due to gravity. The stress tensor should be calculated with the Realizable k-ε model. The air-water flow is solved by using the Volume of Fluid method (Hirt and Nichols, 1981). The governing equation on the volume fraction parameter is given by

4. CFD model

∂t α þ r:ðαuÞ ¼ 0:

4.1. Computational approach

(43)

where volume fraction parameter, α, varies between 0 (pure air) and 1 (pure water) and is used to find effective parameters (Dashtimanesh et al., 2018). Once the equations are solved, the forces and moments acting on the vessels are found as {\bf F\vphantom{F}}¼{∬_{S}}\left(\vphantom{{{{p}\ {\ }}}}\rightpþ{\bf σ\vphantom{σ}}\left)\vphantom{{{{p}\ {\ }}}}\right{\bf n\vphantom{n}}dS \fleqno \tf="TTe692faf0"\hbox{(44)}

The numerical modelling is performed by assuming that fluid is Newtonian, viscous, and two-phases (air and water). Continuity and Navier Stokes (NS) equations govern on the fluid field as r:u ¼ 0

�

∂t ρeff u þ r: ρeff u uT ¼ rp þ r:T þ ρeff fb

(41)

ZZ ðp þ σÞn � r dS

M¼ S

(45)

where σ is the normal stress tensor, and S is the total area of the boat. n is the normal unit vector. The longitudinal components of the force, Fx, is set to be the total resistance. The vertical component of the force is the lift. The moment also includes heeling and pitching moments. During the simulation, heave and pitch directions are set to be free and the equilibrium condition is found. All equations are solved by Finite Volume Method using a commer­ cial CFD code, Siemens PLM Star-CCMþ. Details of the numerical approach can be found in Siemens PLM Star-CCMþ (User’s Guide Version 14.02.010, CD-adapco, 2019). 4.2. Computational domain To provide the set up for the numerical simulation, a computational domain is defined as shown in Fig. 5. An inlet boundary and an outlet boundary are defined at both ends of the domain. Water and air enter from the inlet boundary condition, located at the right end of the domain. A no-slip boundary condition is set on the bottom and upper surfaces. Two side boundaries are defined, which behave like an inlet patch. The walls of the vessel are defined to behave like a wall, on which no-slip condition is satisfied. A summary of boundaries is shown in Table 3. Moreover, the dimensions of the domain and the location of the boat are set using the recommendations of ITTC (see ITTC 7.5-02-02-01). The boat transom is located at 3 LPP far from the inlet. The depth of the computational domain is 3.5 LPP. The outlet boundary should be located 3 LPP from the hull. The domain is 4 Lpp wide. Water (light gray) is initially set to be at the rest (note that technical infor­ mation about dimension of the domain can be found in Zou et al., 2019). 4.3. Gridding methodology and mesh convergence To discretize the numerical domain, the morphing mesh approach is combined with rigid dynamic motion. The numerical grid is generated by using different types of grids including 1) surface remesher 2) prism layer mesher, and 3) trimmer (Dashtimanesh et al., 2019b). The smallest grids are defined near the free surface and the vessel, which are of in­ terest. An overview of the mesh is shown in the upper panel of Fig. 6. The wall function approach is used to model treatments of the fluid near the walls of the vessel. In particular, the All Wall yþ model is used. yþ is set to vary between 30 and 300, as shown in the middle panel of the Fig. 6. Four different mesh including coarse (corresponds to 432151

Fig. 6. Grids used for modelling the problem. (a) shows the computational grid. This grid shows a morphing mesh with 822221 elements and a close-up view of the grids near the vessel is displayed. Mesh sizes near the free surface of water is 0.825 cm. (b) shows the mesh convergence study. Mesh study is performed for the model c2 studied by Taunton et al. (2010) at FrB ¼ 4.16. (c) yþ for the 10� heeled double stepped hull at FrB ¼ 4.16. 8

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Fig. 7. Comparison of the results obtained by 2DþT method (circles) and CFD model (triangles) against experimental results (squares) of Taunton et al. (2010): (a) dynamic trim, (b) resistance and (c) wetted surface.

against experimental data. The errors in prediction of any parameter (like running attitudes or forces) is computed using � � �Aexp A2DþT ​ or ​ CFD � � � 100 (46) E2DþT ​ or ​ CFD % ¼ �� � Aexp

Table 4 Errors of the 2DþT model and CFD model in prediction of the running attitudes and resistance of the model C2. Speed

Errors in prediction of different parameters (%) Trim angle

V (m/s) 4.05 5.1 6.25 7.11 8.13 9.18 10.13 11.13 12.05

FrB 1.9 2.4 2.94 3.35 3.83 4.32 4.77 5.24 5.67

E2DþT% 41 2 12 18 23 29 29 33 1

Resistance ECFD% 10 9 10 12 10 11 10 15 9

E2DþT% 20 21 18 16 7 3 1 5 8

Wetted Surface ECFD% 4 2 1 4 13 7 11 6 8

E2DþT% 23 14 40 7 45 24 53 36 59

where exp stands for experimental measurements.

ECFD% 1 42 33 1 42 16 50 28 51

5.1. Double stepped planing hull with zero heel angle Running attitudes and resistance of a doubled stepped planing hull (model C2) are found using the 2DþT method and CFD model. Results, including the dynamic trim angle, resistance, and the wetted surface, are presented in Fig. 7. The experimental data are also shown in the Figure. The comparison of the predicted dynamic trim angle against experi­ mental measurements (Fig. 7a) shows that both CFD and 2DþT models have reasonable accuracy in prediction of trim. Trim angles computed by the CFD model and 2DþT method are seen to follow the experimental data. But, CFD (circles) is more accurate than 2DþT method (triangles). The CFD model and the 2DþT method predict the resistance of the double-stepped planing boat with proper accuracy (Fig. 7b). At highspeeds, the 2DþT model provides better accuracy. The graphs corre­ sponding to the predicted and measured wetted surface diverge from the experimental method. Note that, the experimental results (squares) fluctuate. To sum, the CFD model used in the current research predicts the trim angle and resistance with average errors of 10.9% and 6.22% respec­ tively (Table 4). One of the main reasons for the errors of the current CFD model is the technique (morphing mesh) used to model the dynamic motion of the boat. The previous CFD researches have shown that other motion techniques (like overset technique) can decrease the errors in the prediction of the trim angle (under 8 percent in most of the cases) when the equilibrium condition of a planing hull is numerically simulated (details can be found in De Luca et al., 2016; De Marco et al., 2017, and Sukas et al., 2017). Besides, the errors in prediction of the resistance can

grids), medium (625451 grids), fine (802221 grids) and finest (1011102 grids) are considered. Mesh study is performed, and it is found that the results converge for the medium mesh size. A sample of the mesh study for resistance prediction of a double-stepped planing hull (at beam Froude number of 4.36) is illustrated in the lower panel of Fig. 6. 5. Comparison analysis There is no available experimental laboratory data presenting the performance of double-stepped planing hulls moving forward at the non-zero heel angle. To assess the accuracy of both 2DþT and CFD models in the prediction of the performance of a heeled double-steeped vessel, two different validation studies are performed. The first valida­ tion study focuses on the performance of a doubled-stepped planing hull moving forward in symmetric condition. Running attitudes and resis­ tance of the vessel are found and compared against experimental data. The second validation study focuses on the modeling of a heeled nonstepped ponding hull advancing in calm water. The heeling moment and normal force acting on the vessel are computed and compared 9

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Fig. 8. Lift force of a heeled stepless boat advancing at beam Froude Number of 3.6 at different draft. Triangle, circle and square symbols respectively refer to results of CFD model, 2DþT method, and Experiments of Judge (2014). Table 5 Errors of the 2DþT method and the CFD model in predicting lift of a planing hull at different heel angles. η3

0.5

ϕ 0 5 10 15 20

E2DþT% 8.5 11 7.7 6.7 12.5

0.75 ECFD% 3.7 4.3 12 5.3 5.7

E2DþT% 5.5 7 6 9 9

1 2.2 3 4.4 4.5 3.3

ECFD%

1.25

E2DþT% 10 7 7.5 6.5 6.7

also be attributed to the dynamic technique used in the current research. The over-predictions in the trim angle (which are resulted from the morphing mesh technique) lead in over-prediction of the resistance (induced-drag caused by hydrodynamic pressure). In sum, the morphing mesh technique has caused an extra error in the prediction of the trim angel and resistance (which has also been reported by Dashtimanesh et al., 2018), which is needed to be further investigated in the future. Note that, performing simulations with fixed trim angle and pitch can highly increase the accuracy of CFD model in computation of the resistance (see e. g. in Ferrando et al., 2015, and Ghadimi et al., 2019b).

ECFD% 4.7 2.5 2 9 7.7

E2DþT% 5.5 4 2.4 7.6 9.8

1.5 ECFD% 7.5 4 3 2 3.4

E2DþT% 6.9 6.2 7 8.5 4.8

ECFD% 11.5 11 6.5 3 4

But such an assumption cannot help us to evaluate the effects of asym­ metric condition on running attitudes of the vessel (which is the main aim of the current paper). The 2DþT method predicts the trim angle with an average error of 20.8%. Note that, this average error corresponds to the beam Froude Numbers ranging from 1.9 to 5.6. The smallest considered speed, 4.05 m/s, corresponds to the beam Froude Number of 1.80, which is cate­ gorized as a semi-planing speed (not planing). The 2DþT method is not expected to work for this speed. But we have performed simulations at all speeds to see how the method works for all experiments. The largest 10

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Fig. 9. Heeling moment of a heeled stepless boat advancing at beam Froude Number of 3.6 at different draft. Triangle, circle and square symbols respectively refer to results of CFD model, 2DþT method, and Experiments of Judge (2014). Table 6 Errors of the 2DþT method and the CFD model in predicting the heeling moment of a planing hull at different heel angles. η3

0.5

ϕ 5 10 15 20

E2DþT% 25 12 6.9 15

0.75 ECFD% 10 13 1.5 8.5

E2DþT% 7.5 5.5 7.6 11.8

1 ECFD% 13 5.3 5.6 15

1.25

E2DþT% 2.5 7.8 3.9 3.6

error is seen to occur at the mentioned speed (2.07 m/s). If we only consider the range of applicability of the 2DþT method (Fr > 2.0), the average error drops to 18.3%. One of the main reasons for the error is the method used to apply the effects of step on the sectional forces (Eq. (29)). This function is originally developed to apply the effects of the transom on the sectional force of a stepless boat. But for the case of a stepped boat, while the Kutta condition (zero pressure) governs on the step, step effects on the sectional force in its vicinity is less significant (compared with a transom of a step boat) because the wetted length of the lifting surface (locating in front of the step) is much smaller than the

ECFD% 4.4 12 7.1 3.2

E2DþT% 3.5 5 3.4 6.7

1.5 ECFD% 7 4.8 3.6 1

E2DþT% 4 7 4.3 9.5

ECFD% 6 8.2 7.6 4.2

wetted length of the stepless boat. When we use this function (Eq. (29)), the center of pressure might be predicted to be closer to the transom than what it is, which may lead to under-prediction of the trim angle to some degrees. As a result, the induced drag may also be under-predicted. Besides, for the case of a stepped boat, side wetted area near the chine of the middle lifting surface may appear due to vortices. This area is found not to result in any remarkable hydrodynamic pressure, while it can marginally contribute to the resistance. Subsequently, the resistance may be under-predicted to some degrees. It is necessary to provide new methods to predict the transom/step reduction function with better 11

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Fig. 10. Comparison of the computed trim angle (trim vs. FrB) by 2DþT model (circles) and CFD model (triangles). Results are related to the double-stepped planing hull (model C2): (a) shows the trim angle of zero-heel angle. (b), (c) and (d) respectively show the results for the heel angles of 5, 10 and 15� .

Fig. 11. Comparison of the computed resistance (resistance vs. FrB) by 2DþT method (circles) and CFD model (triangles). Results are related to the double-stepped planing hull (model C2): (a) shows the resistance of zero-heel angle. (b), (c) and (d) respectively show the results for the heel angles of 5, 10 and 15� .

12

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accuracy in the future. Moreover, to increase the accuracy of the current method, especially in the prediction of the resistance, some simplified methods for prediction of the wetted area located near the chine are needed to be proposed.

The heeling moment of a heeled planing hull at FrB ¼ 3.6 is computed by both 2DþT and CFD methods. Comparisons of the predictions against experimental measurements are shown in Fig. 9. As evident, the 2DþT and CFD method predict the heeling moment with almost good accuracy in all cases. Some errors appeared at large heel angel of 20� , which equal to the deadrise angle of the vessel. In this condition, the starboard of the vessel is touched by the vessel at all the wetted sections of the vessel. Both 2DþT and CFD methods lead to errors in this condition. Increasing the accuracy of simulations at large heel angles should be considered in future studies. In the next section, where the main results are presented, this heel angle is not modeled by both methods. The errors corresponded to the 2DþT and CFD methods in prediction of the heeling moment are shown in Table 6. The good accuracy of both CFD and 2DþT methods in prediction of running attitude of a doubled-stepped planing boat and forces acting on a heeled non-stepped planing hull show that both methods can be used to model performance of a heeled double-stepped planing hull.

5.2. Heeled non-stepped planing hull The asymmetric motion of a heeled planing hull (experiments of Judge and Judge, 2013) is replicated by using 2DþT and CFD models. The vertical force and the heeling moment are computed and compared against experimental data. The performance of the vessel at different drafts (η3 ¼0.5, 0.75, 1, 1.25 and 1.5), computed by

η3 ¼

Tcurrent ; Tnominal

(51)

is found. Simulations are performed for FrB ¼ 3.6 (corresponding to the speed of 7.54 m/s). Fig. 8 displays the predicted and measured values of the lift force vs. heel angle at different drafts. The lift force increases by the increase in the heel angle at three smaller drafts. At the two large drafts, the vertical force decreases by the increase in the heel angle. The panels of Fig. 8 illustrate that the results of both CFD and 2DþT data fit with experi­ mental data, especially at three smaller drafts. At the two larger drafts, errors appear which can be (not defiantly) attributed to the larger wetted surface of the vessel, i.e. larger pressure area can be a source of the error in the predictions. This point needs to be investigated in detail in the near future. The errors in the prediction of the lift force of the heeled non-stepped planing boat are presented in Table 5, showing that for most of the cases, the vertical force is predicted by both 2DþT method and CFD with an error under 10 percent.

6. Results 6.1. Comparison between 2DþT method CFD A heeled doubled-stepped planing hull is modeled by both 2DþT and CFD methods. Results are compared against each other to see how different the results of these methods are. Three different beam Froude Numbers, including 2.94 (corresponding to the speed of 6.25 m/s), 3.83 (corresponding to the speed of 8.13), and 4.32 (corresponding to the speed of 9.18 m/s) along with three different heel angles of 5, 10 and 15� are considered. The symmetric conditions are also modeled. Note that the considered speeds correspond to the cases for which the errors of

Fig. 12. Comparison of the computed for wetted surface (for wetted surface vs. FrB) by 2DþT method (circles) and CFD model (triangles). Results are related to the double-stepped planing hull (model C2): (a) shows the for wetted surface of zero-heel angle. (b), (c) and (d) respectively show the results for the heel angles of 5, 10 and 15� .

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Fig. 13. Comparison of the computed for heeling moment (for heeling moment vs FrB) by 2DþT method (circles) and CFD model (triangles). Results are related to the double-stepped planing hull (model C2): (a) shows the for heeling moment of zero-heel angle. (b), (c) and (d) respectively show the results for the heel angles of 5, 10 and 15� .

CFD and 2DþT methods have seen to be smaller (see Table 4). The computed trim angles of the double-stepped planing hull in symmetric and asymmetric planing conditions are shown in Fig. 10. Results of both 2DþT method (circles) and CFD (triangles) agree with each other in most of the cases. Results of 2DþT method are seen to follow the experimental data at heel angles of 5 and 10� . But, at the heel angle of 15� , trim angle of 2DþT method and CFD model diverge from each other, which shows that for the case of a doubled stepped planing hull at large heel angles, 2DþT method predicts larger normal forces. This fact will be discussed in detail later. The predicted values of the resistance of the model c2 in asymmetric conditions are found using the 2DþT and CFD methods. The results are displayed in Fig. 11. The values of the predicted resistance by the 2DþT method are found to be larger than the predicted values of the resistance by CFD in zero-heel condition, as was seen earlier (Fig. 7). At the nonzero heel angles, the results of 2DþT and CFD methods are observed to be closer to each other and fit for most of the cases. This point shows that, in asymmetric condition, 2DþT method and CFD model provide similar results for the resistance of a double-stepped planing hull, while they might not fit at the zero-heel angel condition. It was earlier dis­ cussed that the emergence of the wetted area near the chine of the vessel (which are not computed by the 2DþT method) can lead to this discrepancy between the computed resistance by the 2DþT method and CFD model. But when the vessel locates at a heel angle of ϕ, this area is found to become smaller (which will be shown later) or (even) vanish in some cases. Subsequently, the resistance computed by CFD and 2DþT models match. Fig. 12 displays the computed wetted surface of the doubled-stepped planing hull (model C2) by both 2DþT and CFD methods. Results of this Figure provide evidence that predicted wetted surface by CFD and 2DþT methods agree with each other in most of the cases. Results of both methods diverge from each other at the largest beam Froude Number (FrB ¼ 4.32) and heel angle (ϕ ¼ 15� ). It should be noted that the

difference between predicted wetted surface by 2DþT and CFD methods is larger at the largest heel angle, imposing that when heel angle in­ creases, both methods show different behavior in the prediction of wetted surface. It has been attempted to study the reason for this dif­ ference. The related study is shown in the upcoming discussions in this sub-section. Fig. 13 shows the computed heeling moment of the double-stepped planing hull at different heel angles. The heeling moment of the doubled-steeped planing hull is seen to decrease by the increase in Froude number. Numerical predictions of CFD (triangles) agree with predictions of 2DþT theory (circles) at most of the cases. The largest differences between the results (heeling moment) of 2DþT and CFD methods are observed at the largest heel angle, where the 2DþT method predict smaller heeling moment. Note that, these differences are not significant and are about 24 percent in the most case (panel c). By the increase in the speed, the predictions made by the CFD model and 2DþT model fit. Overall, by the increase in the speed, the discrepancy between the heeling moment predicted by the CFD model and 2DþT becomes smaller. Note that at the higher-speeds, the contribution of hydrody­ namic pressure becomes dominant. In such a situation, the CFD model and 2DþT method predict similar values for the heeling moment. The top views of the wetted surface of the doubled-stepped planing hull, obtained by 2DþT and CFD methods, are shown in Fig. 14. Note that the wetted surface of the zero-heel condition is not shown in this figure. The results show that the wetted surface of the middle planing surface is negligible. This surface is almost dry in all cases, and a small proportion of the area of this surface is wetted by water. This implies that this surface doesn’t contribute to generation of forces significantly. The CFD model predicts extra wetted surfaces near the chine of the second and third surfaces, which are not captured by the 2DþT model as explained earlier. These wetted areas are seen to emerge when the chine of the fore planing surface is not fully wetted (see some similar discus­ sion in the thesis of Lee et al. (2014)). De Marco et al. (2017) have the 14

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Fig. 14. Comparison of predicted wetted surface of heeled doubled-stepped planing hull (model C2) by 2DþT method (left) and CFD model (right).

attributed appearance of these wetted areas to the vortices path. These wetted surfaces are seen not to have any significant contribution in the hydrodynamic lift. But frictional forces caused by the shear stresses can occur there. 2DþT model cannot predict such forces, and thus it can compute smaller resistance in the cases these wetted surfaces are large. As seen, by the increase in speed and heel angle, these wetted surface (the wetted surface around the chine) become smaller. This fact is consistent with what was observed in Fig. 11, i.e. the difference between the predicted resistance by the CFD model and 2DþT model becomes smaller by the increase in the heel angel. Moreover, at larger heel angles (e. g. heel angle of 15� ), the predicted wetted surface of the fore planing hull by the 2DþT method is larger, which is consistent with the pre­ sented results in Fig. 12 (where the 2DþT computes larger wetted

surface). Overall, the 2DþT method was found to have an average error of 18.4% in the prediction of the trim angle of the planing hull in zero-heel. The predicted trim angle by the 2DþT method in the zero-heel condition was highly different from the CFD model, i.e., the 2DþT model underpredicts the trim angle while CFD model over-predicts the trim angle at symmetric planing condition. But it was seen that the results of both methods agree in the asymmetric conditions at most of cases. The only significant differences of the predicted trim angle were observed at the heel angle of 15� (Fig. 10d), which were attributed to predicted wetted surface by the 2DþT model (Fig. 12d). While it was seen that CFD and 2DþT models predict different values for resistance in the symmetric condition, both methods were found to 15

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Fig. 14. (continued).

Fig. 15. Dynamic trim angle (trim angles vs. FrB) of heeled non-stepped (filled circles) and double-stepped (unfilled circles) planing hulls. (a), (b) and (c) respectively show the results for the heel angle of 5, 10 and 15� . Note that all computations are performed using 2DþT method.

compute similar resistance (see Fig. 11) in asymmetric condition. The analysis of the wetted surface pattern on the bottom of the stepped planing hulls showed that when wetted surfaces near the chine of the middle and rear bodies are notable, predicted resistance by the CFD and 2DþT models diverge. Such a condition is more probable in the

symmetric condition. But by the increase in the speed and heel angle, these wetted surfaces near the chine are smaller, and thus the predicted resistance by the CFD and 2DþT models fit. Overall, the predicted trim angle, resistance, wetted surface and heeling moment predicted by CFD and 2DþT agree with each other at most of the cases. The only difference 16

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Fig. 16. Resistance (resistance vs. FrB) of heeled non-stepped (filled circles) and double-stepped (unfilled circles) planing hulls. (a), (b) and (c) respectively show the results for the heel angle of 5, 10 and 15� . Note that all computations are performed using 2DþT method.

Fig. 17. Wetted surface (wetted surface vs. FrB) of heeled non-stepped (filled circles) and double-stepped (unfilled circles) planing hulls. (a), (b) and (c) respectively show the results for the heel angle of 5, 10 and 15� . Note that all computations are performed using 2DþT method.

17

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Fig. 18. Heeling moment (heeling moment vs. FrB) of heeled non-stepped (filled circles) and double-stepped (unfilled circles) planing hulls. (a), (b) and (c) respectively show the results for the heel angle of 5, 10 and 15� . Note that all computations are performed using 2DþT method.

is observed at large heel angle (ϕ ¼ 15� ), where wetted surface and trim angle predicted by CFD and 2DþT diverge from each other.

provide almost similar results for the asymmetric planing (see discussed in sub-Section 6.2), only the 2DþT method is used to simulate the problem. Note that, the 2DþT method has been observed to work properly for the non-stepped planing hull previously (Ghadimi et al., 2017a). The dynamic trim angle of the heeled non-stepped (filled circles) and the double-stepped planing hulls (unfilled circles) are shown in Fig. 15. The trim angle is seen to decrease by the increase in heel angle for both hulls. Note that, Ghadimi et al., (2017) have similarly observed that the trim angle of a heeled planing hull is affected and reduces by the in­ crease in heel angle. The trim angle of the heeled double-stepped planing hull is less affected by the increase in heel angle especially at smaller beam Froude Numbers, e. g. at the beam Froude Number of 2.0,

6.2. Effect of adding two steps on performance of heeled planing hull Effects of steps on the performance of a heeled planing hull are studied by simulating the asymmetric motion of model C (non-stepped planing hull) and C2 (doubled-stepped planing hull), and then comparing results (including dynamic trim angle, resistance, wetted surface and heeling moment) against each other. Simulations are per­ formed for the beam Froude Number ranging from 1.9 (corresponding to the speed of 4.05 m/s) to 5.24 (corresponding to the speed of 11.3 m/s) and three heel angles of 5, 10 and 15� . Since 2DþT and CFD methods

Fig. 19. Hydrostatic (left) and hydrodynamic (right) moments of the non-stepped (filled circles) and double-stepped (unfilled circles) planing hulls at beam Froude Number of 2.94 (circles with lines) and 5.24 (circles with dashed lines). Results are computed using 2DþT method. 18

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5-degrees change in the heel angle of non-stepped planing hull causes a 1.0-degree reduction in trim angle (compare filled circles of Fig. 15a, b, and c), while for the case of a double-stepped planing hull this reduction is seen to be under 0.2� . The hydrodynamic force acting on the boat causes this behavior. For the case of a doubled-stepped planing boat the dominant force supporting the weight of the vessel is the hydrodynamic force at small beam Froude Numbers and hydrostatic force doesn’t contribute remarkably, whereas for the case of a non-stepped planing hull (see the pressure distribution discussion in Dashtimanesh et al., 2018), hydrostatic force has larger contribution. An asymmetric condi­ tion (i.e. non-zero heel angle) increases the hydrodynamic pressure significantly for the case of a non-stepped planing hull. Further discus­ sion is presented in the rest of this sub-section. Fig. 16 shows the values of computed resistance for the non-stepped (filled circles) and the stepped (unfilled) planing hulls in the asymmetric condition. The values of the resistance of both vessels are seen to in­ crease by an increase in the speed as expected for any planing vessel (Fridsma, 1969). The presence of non-zero heel angles affect both vessels in different ways. For the case of the non-stepped planing hull, non-zero heel angles reduce the resistance at two smaller beam Froude Numbers (FrB ¼ 2.0 and 3.19), e. g. at beam Froude Number of 2.0, the resistance of the non-stepped planing hulls decreases from 2.0 to 3.19 as the heel angle increases from 5 to 10. At larger beam Froude Numbers (FrB � 3.83), the resistance of the non-stepped planing hull increases by the increase in heel angle. But, for the case of a double-stepped planing hull, an increase in the heel angle causes slight reductions (up to 10 percent) in resistance at all beam Froude Numbers. These results provide the evidence that the presence of two steps can manage a vessel to reach high-speeds even in heeled planing motion (because the resistance is not affected and is still far smaller than non-stepped vessel). A study of the wetted surface of both vessels at non-zero heel angles, which will be presented later, improve our understanding of the resistance results. The wetted surface of the doubled-stepped and the non-stepped planing hulls are presented in Fig. 17. The values of the wetted sur­ face of both hulls reduce by the increase in beam Froude Number in the non-zero heel condition. An increase in heel angel reduces the wetted surface of the non-stepped planing hull at two smaller beam Froude numbers (FrB ¼ 1.9 and 2.94). At the larger beam Froude numbers, heel angle increases the wetted surface of the vessel. This shows that when the speed is increased and the vessel rests at a non-zero heel angle, larger wetted surfaces are needed to support the weight of the vessel. For the case of a doubled-stepped planing hull, however, an increase in the value of the heel angle results in the reduction of the wetted surface, which is not significant. Note that, these results are consistent with the resistance results (Fig. 16). Comparisons between the heeling moment of both hulls at different heel angels and beam Froude numbers are shown in Fig. 18. Heeling moments of both vessels decrease by the increase in speed. Note that, Balsamo et al. similarly observed that the heeling moment of a planing hull decreases by the increase in the speed of the vessel. At heel angle of 5� , the heeling moment of the non-stepped planing hull (filled circles) is larger than heeling moment of double-stepped planing hulls (unfilled circles). At the heel angle of the 10� , the heeling moment of the non-stepped planing hull is larger than the heeling moment of the double-stepped planing hulls at beam Froude Numbers ranging from 1.9 to 3.83. At larger beam Froude Numbers, the heeling moment of the double-stepped planing hull becomes larger. At heel angle of 15� , the heeling moment of the double-stepped planing hull is larger than the heeling moment of a non-stepped planing hull at all speeds. Overall, the results reveal that by the increase in speed and heel angle, the heeling moment of the double-stepped planing hull becomes larger than that of a non-stepped planing hull. The reason underlying this behavior is the hydrodynamic pressure which causes the dominant force in supporting the weight of the vessel. When speed is increased and heel angle grows, a larger difference between forces in starboard and port of the vessel ap­ pears (the double-stepped planing hull has three maximum pressure

area while the non-stepped has one, which take larger forces in larger heel angle and speeds). More evidence regarding this fact is presented in the Fig. 19. To increase the understanding of effects of heel angle on the heeling moment of both hulls, hydrodynamic heeling moment and hydrostatic heeling moment of both hull at two heel angles of 5 and 15� and beam Froude Numbers of 2.94 and 5.24 are computed and shown in Fig. 19. The hydrostatic moment (left panel of Fig. 19) of the doubled-stepped (filled circles) planing hull is smaller. At the larger beam Froude Num­ ber (circles with dashed lines), hydrostatic heeling moment significantly decreases. The difference between the hydrostatic heeling moment of the doubled-stepped hull and the non-stepped planing hull become very small at this beam Froude Number (in comparison with what it was at smaller beam Froude Number, solid lines with circles). Hydrodynamic heeling moments (right panel) of both vessels decrease by the increase in speed (solid lines are located above the dashed lines). The double-stepped planing hull has larger heeling mo­ ments at all conditions. The interesting point is that, at larger heel angles and higher beam Froude Numbers, the hydrodynamic heeling moment of the doubled-stepped planing hull is significantly larger than the hy­ drodynamic heeling moment of the non-stepped planing hull, which results in higher heeling moment (which is the summation of hydrostatic and hydrodynamic heeling moments) for the doubled steeped planing hull. This fact supports what was observed in Fig. 18. 7. Conclusion In the current paper, hydrodynamic of heeled doubled-stepped planing hulls were studied. It was aimed to provide a numerical method based on 2DþT theory for performance prediction of heeled double-stepped planing hull and investigate whether results of this method and CFD fit or not. A secondary aim of the current research was to improve the understating of the effects of adding two steps on the bottom of a planing hull on its performance in non-zero heel conditions. The 2DþT method was developed by using the theoretical solutions of water entry of a solid wedge. A CFD set-up was developed using FVM and VOF. The validity of both methods was assessed by comparing their results against previous experimental data in two steps. In the first step, the performance of a doubled-stepped planing hull was predicted. 2DþT and CFD were found to provide predictions for the dynamic trim angle, resistance and, wetted surface with reasonable accuracy. In the second step, heel motion of a non-stepped planing hull was modeled, and it was shown that 2DþT and CFD methods accurately predict the forces acting on the vessel. Hydrodynamic of a heeled doubled-stepped planing hull was modeled using both the 2DþT method and CFD. CFD and the 2DþT methods provided similar heeling moment and resistance at all speed and heel angles. There was an agreement between mathematically and numerically computed trim angle and wetted surface at heel angels of the 5 and 10� . But, at a heel angle of 15� , the trim angle and wetted surface computed by CFD and 2DþT method diverged (not significantly) from each other by the increase in the speed. But these differences didn’t affect the heeling moment and resistance. Also, while the predicted resistance by the 2DþT method and CFD model were seen to be different in symmetric conditions, they fit in the asymmetric condition. The wetted surfaces near the chine of the middle and rear lifting surfaces, which were not computed by the 2DþT method, were seen to become very small in the asymmetric condition, which decreases the difference between predicted resistance by the 2DþT method and CFD model in asymmetric condition. 2DþT method was used to model both doubled-stepped and nonstepped planing hulls. The following conclusions were made. � When a non-stepped planing hull locates at a non-zero heel angle, its dynamic trim angle is decreased at smaller speeds, while it is increased at larger speeds. But when a double-stepped planing hull 19

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locates at a non-zero heel angle, its trim angle is (not significantly) reduced. � At high speeds, an increase in the heel angle of a non-stepped planing hull increases the resistance of the vessel since the wetted surface increases. But, for the case of a doubled stepped planing hull, resis­ tance and the wetted surface of the vessel are found to decrease at all speeds by the increase in heel angle. The presence of three maximum pressure areas helps the double-stepped to keep its speed at the higher range in non-zero heel conditions. � Heeling moment of a non-stepped planing hull is larger than the heeling moment of the doubled-stepped planing hull at early planing speeds. But, by the increase in the speed, the hydrostatic heeling moment vanishes, while hydrodynamic heeling moment becomes dominant. In this condition, the doubled-stepped planing hull, which is supported by larger hydrodynamic pressure, has a slightly larger heeling moment.

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Authors contributions Rasul Niazman Bilandi: He implemented the previously developed mathematical models and has had contribution in development of them for heeled condition. He also developed the numerical model and per­ formed validation and verification studies. He has also contribution in analyzing the results and writing the initial draft of the paper. Abbas Dashtimanesh: He developed the idea and concepts related to the mathematical modelling of stepped hulls and presenting linear wake theory. He also developed the basic formulation and computer program. He has had contribution in writing and revising the paper and analyzing the results. He is the supervisor and leader of the performed research. Sasan Tavakoli: He has developed the basis of mathematical model for heeled condition. The paper mainly has been written by him. He has analyzed the results in details and had a critical contribution on enhancing the quality of the paper. Declaration of competing interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments ST is supported by a Melbourne Research Scholarship awarded by University of Melbourne (264516). References Akers, R.H., 1999. Dynamic analysis of planing hulls in vertical plane. In: Proceedings of the Society of Naval Architects and Marine Engineers, New England Section. Algarin, R., Tascon, O., 2011. Hydrodynamic modeling of planing boats with asymmetry and steady condition. In: Proceedings of the 9th International Conference on High Performance Marine Vehicles (HIPER 11), Naples, Italy. Balsamo, F., Milanesi, S., Pensa, C., 2001. Rolling dynamic in planing and semi-planing range. In: Proceedings of the 6th International Conference on Fast Sea Transportation (FAST 2001). Begovic, E., Bertorello, C., 2012. Resistance assessment of warped hullform. Ocean. Eng. 56, 28–42. CD-adapco, 2019. STAR CCMþ User’s Guide Version 14.02.010. Dashtimanesh, A., Enshaei, H., Tavakoli, S., 2019. Oblique-asymmetric 2Dþ T model to compute hydrodynamic forces and moments in coupled sway, roll, and yaw motions of planing hulls. J. Ship Res. 63 (1), 1–15. Dashtimanesh, A., Esfandiari, A., Mancini, S., 2018. Performance prediction of twostepped planing hulls using morphing mesh approach. J. Ship Prod. Des. 34, 236–248, 10.5957/JSPD.160046. Dashtimanesh, A., Roshan, F., Tavakoli, S., Kohansal, A., Barmala, B., 2019. Effects of step configuration on hydrodynamic performance of one-and doubled-stepped planing flat plates: a numerical simulation. In: Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment. https://doi.org/10.1177/1475090219851917. Published Online.

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