Practical evaluation of sinkage and trim effects on the drag of a common generic freely floating monohull ship

Applied Ocean Research 65 (2017) 1–11

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Practical evaluation of sinkage and trim effects on the drag of a common generic freely floating monohull ship Chao Ma a,b,c , Chenliang Zhang a,b,c , Fuxin Huang d , Chi Yang d , Xiechong Gu a,b,c , Wei Li a,b,c , Francis Noblesse a,b,c,∗ a

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, China c Shanghai Jiao Tong University, Shanghai, China d School of Physics, Astronomy & Computational Sciences, George Mason University, Fairfax, VA, USA b

a r t i c l e

i n f o

Article history: Received 11 July 2016 Received in revised form 20 February 2017 Accepted 10 March 2017 Keywords: Sinkage Trim Drag Monohull ships Practical methods Design

a b s t r a c t A practical method to account for the influence of sinkage and trim on the drag of a freely floating (free to sink and trim) common monohull ship at a Froude number F ≤ 0.45 is considered. The sinkage and the trim are estimated via two alternative simple methods, considered previously. The drag is also estimated in a simple way, based on the classical Froude decomposition into viscous and wave components. Specifically, well-known semiempirical expressions for the friction drag, the viscous pressure drag and the drag due to hull roughness are used, and the wave drag is evaluated via a practical linear potential flow method. This simple approach can be used for ship models as well as full-scale ships with smooth or rough hull surfaces, and is well suited for early ship design and optimization. The method considered here to determine the sinkage and the trim, and their influence on the drag, yields theoretical predictions of the drag of the Wigley, S60 and DTMB5415 hulls that are much closer to experimental measurements than the corresponding predictions for the hull surfaces of the ships in equilibrium position at rest. These numerical results suggest that sinkage and trim effects, significant at Froude numbers 0.25 < F, on the drag of a typical freely floating monohull ship can be realistically accounted for in a practical manner that only requires simple potential flow computations without iterative computations for a sequence of hull positions. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The drag is a critical element of ship design. Accordingly, the prediction of the flow around a ship hull that advances at a constant speed along a straight path, in calm water of large depth and lateral extent, is a classical basic ship hydrodynamics problem that has been widely considered in a huge body of literature. Indeed, a number of alternative methods – including viscous flow computational methods, nonlinear or linear potential flow methods, and semianalytical methods – have been developed to compute the flow around a ship hull. A brief review of these alternative methods can be found in e.g. [1]. The drag of a ship is influenced by several complicated flow features, including flow separation at a ship stern, notably a transom

∗ Corresponding author at: State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, China. E-mail address: [email protected] (F. Noblesse). http://dx.doi.org/10.1016/j.apor.2017.03.008 0141-1187/© 2017 Elsevier Ltd. All rights reserved.

stern, wavebreaking at a ship bow, hull roughness for full-scale ships, and sinkage and trim for freely floating ships (free to sink and trim). The influence of sinkage and trim on the drag is analyzed here for typical generic freely floating ships at Froude numbers F≡

V



gL

≤ 0.45

(1)

where V and L denote the speed and the length of the ship, and g is the acceleration of gravity. 1.1. Influence of sinkage and trim on the drag The pressure distribution around a ship hull surface H that advances at a constant speed V in calm water evidently differs from the hydrostatic pressure distribution around the wetted hull surface H of the ship at rest, i.e. at zero speed V = 0 . Consequently, 0 the ship experiences a hydrodynamic lift and pitch moment, and a that are commonly related vertical displacement and rotation of H 0

2

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

1.2. Practical determination of sinkage and trim

Fig. 1. Profiles of the wetted hull surfaces of the Wigley hull, the S60 model and the DTMB5415 model at rest (blue dashed lines) and in freely floating positions at Froude numbers F = 0.4 (red solid lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

called sinkage and trim. The sinkage and the trim of a freely floating ship have been widely considered in the literature; e.g. [2–12]. The differences between the wetted hull surface H of a ship at 0 rest and the corresponding actual mean wetted ship hull surface H are illustrated in Fig. 1 for three freely floating ship models, specifically the Wigley, S60 and DTMB5415 hulls, at a Froude number of these three models, conF = 0.4. The wetted hull surfaces H 0 sidered hereafter for purposes of illustration and validation, are depicted in Fig. 2. The ‘dynamic’ hull surface H does not differ very much from the ‘static’ hull surface H in Fig. 1. Indeed, [12] shows that the 0 sinkage and the trim can be realistically determined from flow computations for the static hull surface H for Froude numbers 0 F ≤ 0.45. However, the drag of the dynamic hull surface H can be significantly larger than the drag of the static hull surface H at 0 Froude numbers 0.25 < F, as is well documented in the literature; e.g. [2–4,7]. For instance, the theoretical predictions reported further on show that at a Froude number F = 0.45, the Wigley and S60 models experience an increase in (total) drag of about 15%, while the drag of the DTMB5415 model is about 7% higher, due to sinkage and trim effects. A large part of the increase in the drag stems from the wave drag component, predicted further on to increase by about 20% or 16% for the Wigley or S60 models at F = 0.45. These theoretical predictions are consistent with experimental measurements of the residuary drags of the Wigley and S60 models, which increase significantly for 0.25 < F. These examples show that sinkage and trim effects on the drag, notably the wave drag, of a ship can be significant. Moreover, sinkage and trim effects on the drag depend on the hull form. E.g., at F = 0.45, the wave drag and the viscous drag components are found further on to increase by about 20% and 7% for the Wigley model, by about 16% and 11% for the S60 model, and by 5% and 7% for the DTMB5415 model. The influence of sinkage and trim must then be taken into account to realistically determine the drag of a freely floating ship at 0.25 ≤ F, and indeed needs to be considered within the design process, arguably even at early design stages and for hull form optimization. The analysis of sinkage and trim effects on the drag involves two obvious basic tasks: the determination of sinkage and trim, considered in [12], and the determination of the drag, examined here.

As was already noted, alternative methods for evaluating the sinkage and the trim, as well as the drag, experienced by a freely floating ship have been considered in the literature. In particular, the approach considered in [3,4,6–9,11] involves iterative flow computations for a sequence of hull positions. Such iterative flow computations are shown in [12] to be unnecessary for typical monohull ships at Froude numbers F ≤ 0.45, and are not well suited for routine practical applications to early ship design and hull form optimization. In fact, practical methods to determine the sinkage, the trim and the drag of a ship, notably methods that do not require iterative flow computations for a sequence of hull positions, are useful if not necessary at early design stages and for optimization. Ma et al. [12] consider two simple methods, an experimental method and a numerical method, to determine the sinkage and the trim of a typical freely floating monohull ship that advances in deep water at a Froude number F ≤ 0.45. The numerical method only involves linear potential flow computations for the ship at rest, i.e. for the hull surface H , rather than 0 for the hull surface H of the ship at its actual position. Indeed, a main conclusion of [12] is that, for common monohull ships at Froude numbers F ≤ 0.45, the sinkage and the trim can be realistically predicted via computations for the static hull surface H , i.e. 0 without iterative flow computations for several hull positions. This practical simplification stems from the fact that the sinkage and the trim are largely determined by the pressure at the bottom of the ship hull (as is shown further on), and consequently are not very sensitive to the precise position of the ship. The experimental method, based on an analysis of experimental measurements reported in the literature for 22 models of monohull ships, requires no flow computations and is then simpler still than the numerical method. Indeed, the analysis of experimental data considered in [12] yields simple analytical relations that explicitly predict the sinkage and the trim experienced by a monohull ship in terms of the ship speed V and four basic parameters related to the hull geometry: the length L, the beam B, the draft D, and the block coefficient Cb . Both the simple numerical approach and the even simpler experimental approach are found in [12] to yield realistic predictions of sinkage and trim for a wide range of monohull ships at Froude numbers F ≤ 0.45. 1.3. Practical determination of the drag As is required for routine applications to early ship design, and in accordance with the simple methods considered in [12] to determine the sinkage and the trim, a practical method is used here to predict the drag. Specifically, classical semiempirical relations for the friction drag, the viscous pressure drag and the hull-roughness drag are used, and the wave drag is evaluated via a practical linear potential flow method. The wave drag (a major component of the total drag at high Froude numbers) is more sensitive to the hull position than the sinkage and the trim (as is explained further on). Accordingly, the

Fig. 2. Side views (left) and bottom views (right) of the wetted hull surfaces H of the Wigley hull (top), the S60 model (middle) and the DTMB5415 model (bottom) 0 approximated via 7562 (Wigley), 11,542 (S60) and 12,586 (DTMB5415) flat triangular panels, as for the flow computations reported in the study.

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

drag is computed for a dynamic ship hull surface H st that accounts for sinkage and trim effects, as was already noted. However, the hull surface H st does not need to be very precise. Indeed, a notable result of the theoretical predictions reported further on for the Wigley, S60 and DTMB5415 hulls is that the hull H surfaces H a and 1 predicted by the experimental and numerical methods considered in [12] to determine the sinkage and the trim have nearly identical drags. Moreover, the drag of the hull surface H and the (nearly identical) drag of the hull surface H a 1 are significantly higher, and also much closer to experimental measurements, than the drag of the static hull surface H for the Wigley, 0 S60 and DTMB5415 hulls at Froude numbers 0.35 < F for which sinkage and trim are important. These results show that theoretical predictions and experimental measurements of the drag of a typical freely floating ship at F ≤ 0.45 can differ significantly due to sinkage and trim effects, and that these effects can be realistically accounted for in a simple manner that only requires linear potential flow computations without iterative computations for several hull positions. 1.4. The Neumann–Michell theory and basic notations The flow computations around the Wigley, S60 and DTMB hulls that are performed in this study to determine the sinkage, the trim, and the wave drag are based on the simple linear potential flow theory, called Neumann–Michell (NM) theory, expounded in [1,13]. Important aspects of the theory are considered in [14–17] and validation studies are reported in [13,18–21]. Main elements of the theory are summarized in [12]. A useful feature of the NM theory is that it is well suited for routine practical applications to ship design and hull-form optimization, as is amply demonstrated in [22–30]. Indeed, the NM theory yields realistic flow predictions (sufficiently accurate for optimization) in a very practical way. In particular, the flow around a ship hull can be evaluated in about 1 s using a common PC. Hereafter, coordinates and flow variables are made nondimensional in terms of the gravitational acceleration g, the water density , and the length L and the speed V of the ship. The Cartesian system of nondimensional coordinates (x, y, z) ≡ x ≡ X/L is attached to the moving ship. The x axis is chosen along the path of the ship and points toward the ship bow. The undisturbed free surface is taken as the plane z = 0 and the z axis points upward. The ship bow and stern are located at (0.5,0,0) and at (−0.5,0,0). The unit vector n ≡ (nx , ny , nz ) is normal to the hull surface H and points outside the ship (into the water). 2. Sinkage and trim from its position H at rest, at midship, is called ‘midship sinkage’ and denoted 0 as Hm . Similarly, the vertical displacement of H at the ship bow and stern are denoted as Hb and Hs , and called ‘bow sinkage’ and ‘stern sinkage’. Positive values of Hm , Hb or Hs correspond to downward vertical displacements of H at midship, at the bow or at the stern, respectively. The rotation of H from H is defined by the 0 trim angle  ◦ ≡  rad 180/ where the angles  ◦ and  rad are measured in degrees or in radians, or by the equivalent ‘trim sinkage’ H defined as 2H /L ≡ tan(

rad

)≈

numerical method, to determine the sinkage and the trim of a common monohull ship are considered in [12] and used here. 2.1. Experimental method and explicit relations The experimental method considered in [12] is based on an analysis of measurements, given in the literature, of the midship sinkage Hm and the stern sinkage Hs for 22 ship models at Froude numbers within the range 0.1 ≤ F ≤ 0.45. These 22 ship models correspond to a fairly wide range of hull forms characterized by beam/length ratios B/L, draft/length ratios D/L, draft/beam ratios D/B and block coefficients Cb within the ranges 0.066 ≤ B/L ≤ 0.148 , 0.029 ≤ D/L ≤ 0.071 0.276 ≤ D/B ≤ 0.667 , 0.397 ≤ Cb ≤ 0.6 The analysis given in√[12] shows that Hm and Hs are approximately proportional to BD, and that Hm increases as the block coefficient Cb increases. Moreover, Hm increases approximately like F2 as F increases, whereas Hs increases approximately like F2 for F < 0.33 or like F6 for 0.33 < F. Specifically, the analysis of experimental measurements given in [12] yields the analytical approximations √ H m ≈ 0.9 BD (Cb − 0.13) F 2 (3a) √ 2 H ≈ 0.025 BD (F/0.33)



s

1 + (F/0.33)

8

rad

◦

≡  /180

(2)

Positive values of  ◦ ,  rad and H correspond to bow-up rotations. Two practical methods, an experimental method and a

(3b)

√ Experimental measurements of (H m / BD)/(Cb − 0.13) and of √ s H / BD for the 22 ship models considered in [12] are depicted in Fig. 3. This figure shows that the simple explicit relations (3) can be expected to predict the midship sinkage Hm and the stern sinkage Hs for a wide range of monohull ships within ±20% in most cases and within ±30% for Hm , or ±40% for Hs , in nearly all cases. The relations (3a)–(3b) and the two geometrical identities H b = 2H m − H s and H  = H s − H m

(3c) Hm ,

the stern sinkage Hs , explicitly determine the midship sinkage b  the bow sinkage H and the trim sinkage H in terms of the Froude number F and basic parameters (the beam B, the draft D, and the block coefficient Cb ) related to the ship hull geometry. The trim sinkage H and the trim angle  ◦ , measured in degrees, are related via (2). 2.2. Practical numerical method In the numerical method, the midship sinkage Hm and the trim sinkage H are evaluated via the classical relations C0z + ε2 C0zx C0zx + ε0 C0z H m /L 2H  /L ≈ and ≈ (1 − ε0 ε2 ) a0 (1 − ε0 ε2 ) a2 F2 F2 whereε0 ≡ a1 /a0 and ε2 ≡ a1 /a2

The vertical displacement of a ship hull surface H



3

(4a) (4b)

Here, a0 denotes the nondimensional area of the ship water, and a1 and a2 are the related plane W0H of the hull surface H 0 moments, defined as (a0 , a1 , a2 ) ≡

A

0 L2

,

A1 A2 , L3 L4





(1, x, x2 ) dx dy

≡

(4c)

WH 0

Moreover, C0z and C0zx in (4a) are the nondimensional hydrodynamic lift and pitch moment defined as



(C0z

, C0zx )

(nz , nx z − nz x) p da

=

(4d)

H 0

The geometrical identities H s = H m + H  andH b = H m − H 

(4e)

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11 0.25

0.25

0.2

0.2

0.15

0.15

s

H / √BD

m ( H / √BD ) / ( Cb - 0.13 )

4

0.1

0.1

0.05 0.05 0 0 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.1

0.15

0.2

0.25

F

0.3

0.35

0.4

0.45

F

√ √ Fig. 3. Experimental measurements of (H m / BD)/(Cb − 0.13) on the left side, and of H s / BD on the right side, for 22 ship models at Froude numbers 0.1 ≤ F ≤ 0.45. The thick solid (red) line and the two thick dashed (blue) lines correspond to the functions fm or fs and the functions (1 ± 0.3)fm or (1 ± 0.4)fs , and the thin solid (black) lines that bound the shaded region correspond to the functions (1 ± 0.2)fm or (1 ± 0.2)fs . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

can be used to determine the stern sinkage Hs and the bow sinkage Hb from the midship sinkage Hm and the trim sinkage H that are computed via (4a)–(4d). Expressions (4d) show that, except for a ship hull with large flare and rake angles, the upper part of a ship hull surface (where nz ≈ 0) does not contribute appreciably to the hydrodynamic lift C0z and consequently to the midship sinkage Hm , and that the upper hull surface and the parallel midbody (where nx z ≈ 0 and nz x ≈ 0) do not contribute much to the pitch moment C0zx and consequently to the trim H . Thus, the main contributions to the sinkage Hm and the trim H stem from the bottom of the ship hull surface. The relations (4d) then suggest that the sinkage and the trim of a ship are relatively insensitive to the precise position of the ship hull, and can be realistically determined from the pressure distribution around the static hull surface H of the ship at rest, as is illustrated 0 in [12] and below in Fig. 4. The hydrodynamic pressure p in (4d) is given by the Bernoulli relation P p≡ = V 2



(ny )2

+ (nz )2

t +

(nx )2 − (t2 + d2 ) 2

(5)

where t ≡ ∂/∂t and d ≡ ∂/∂d denote the velocity components along the two orthogonal unit vectors t≡

(ny )2 + (nz )2 , −nx ny , −nx nz



(ny )2 + (nz )2

and d ≡

0, −nz , ny



(ny )2 + (nz )2

H . 0

tangent to These tangential velocity components are determined via the Neumann–Michell linear potential flow theory. 2.3. Illustrative applications As was already noted, H denotes the wetted hull surface of the 0 ship at rest. The wetted ship hull surfaces obtained via the vertical displacement Hm and the rotation  ◦ predicted by the experimental approach or the numerical approach explained in Sections 2.1 and H 2.2 are denoted as H a or 1 hereafter. Fig. 4 depicts the midship sinkage Hm /D and the trim sinkage H /D for the Wigley, S60 and DTMB5415 hulls at Froude numbers 0.1 ≤ F ≤ 0.45. The numerical predictions of Hm /D and H /D given by (4) and (5) with the NM theory applied to the static hull surface H 0 or the hull surface H are depicted in Fig. 4 together with exper1 imental measurements and the predictions given by the explicit analytical approximations (3). Fig. 4 shows that the NM predictions for the hull surfaces H 0 or H are very close for the midship sinkage Hm , and do not dif1  fer significantly for the trim sinkage H . Moreover, these numerical

predictions are in relatively good agreement with the experimental measurements. The NM predictions of the trim sinkage H for the hull surface H are not closer to the experimental measure1 ments than the corresponding NM predictions for the hull surface H . These results suggest that it is sufficient to compute the flow 0 around the static ship hull surface H , instead of the dynamic hull 0 surface H , for the purpose of predicting the sinkage and the trim 1 of common monohull ships at Froude numbers F ≤ 0.45. Fig. 4 also shows that the explicit analytical relations (3) yield predictions of the midship sinkage Hm and the trim sinkage H that are in relatively good agreement with the NM predictions, as well as the experimental measurements. Indeed, the predictions given by the explicit relations (3) are found in [12] to be in relatively good agreement with measurements for a broad range of ship models. 3. Drag The drag D is made nondimensional in terms of the water density  and the speed V and the length L of the ship, i.e. C t ≡ D/(V 2 L2 )

(6)

The nondimensional drag coefficient Ct is evaluated in a simple way based on Froude’s basic decomposition into viscous and wave components, as in [18]. Specifically, Ct is expressed as Ct = Cw + Cv + Ca

(7)

where the drag coefficients C w , C v and Ca denote the wave drag, the viscous drag, and the hull-roughness drag. The wave drag coefficient C w is determined via integration of the pressure p at the ship hull surface H , i.e.



Cw =

nx p da

(8)

H

where p is given by the Bernoulli relation (5). As was already noted, the NM theory is used to compute the flow around the ship hull surface H and the related pressure p. Expression (8) shows that the parallel midbody and the bottom of a ship hull surface, where nx ≈ 0, contribute little to the wave drag, which mostly stems from the bow and stern regions where nx = / 0. Fig. 1 then suggests that the wave drag of a ship is more sensitive to the precise position of the ship hull than the sinkage and the trim, which are mostly determined by the pressure at the hull bottom as was noted earlier. The viscous drag C v in (7) is expressed as C v = C f + Cpv = (1 + k) C f

(9a)

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

0.14 0.12

m H /D

0.1 0.08

Exp. Appr. H Σ0 H Σ1

Wigley

5

S60

DTMB5415

0.06 0.04 0.02 0 0.25

Wigley

0.2

S60

DTMB5415

Hτ/D

0.15 0.1

0.05 0 -0.05 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.1

0.15

0.2

0.25

F

0.3

0.35

0.4

0.45 0.1

0.15

0.2

F

0.25

0.3

0.35

0.4

0.45

F

Fig. 4. Midship sinkage Hm /D (top) and trim sinkage H /D (bottom) for the Wigley hull (left), the S60 model (center) and the DTMB5415 model (right). Experimental measurements (Exp.) are shown together with the predictions given by the analytical relations (3) obtained from an analysis of experimental measurements (Appr.) and the or H . numerical predictions given by the Neumann–Michell theory applied to the hull surfaces H 0 1

where Cf corresponds to the friction drag, and Cpv denotes the viscous pressure drag, commonly called form drag, related to the influence of the viscous boundary layer on the pressure distribution at the ship hull surface. The form drag is taken here as Cpv = k C f ,

and AH denotes the mean wetted area of the ship hull surface H . The kinematic viscosity is taken as 1.14 × 10−6 m2 /s. The form factor k is estimated as in [31] via the relation

where k is the usual form factor. The friction drag Cf is evaluated via the ITTC 1957 formula

k = 0.6

AH VL 0.075 where Re ≡ 2L2 (log10 Re − 2)2

0.5

Cr Cr0

0.3 δCr

0.4

δC

r

0.2

0.15 0.1 0.05

Wigley

Wigley

0

0

-0.1

-0.05

0

5

δCr

0.5

δC

0.4

δCt/ Cw 0 0.3 r

δC / C0

0.2

r

r

2

0.1 0

1

δCr/ Cr0

0.4

t

δCw

0.3

3

104 δC

4

Wigley

0.6 Cr Cr0

4 10 Cr , Cr0

w

δC / C0

0.2

w

0.1

0.5

t

r

104 δCr

1

r

δC / Cr0

0.25

t

δC

0.3

1.5

r

104 C , Cr0

2

(9c)

Here, denotes the displacement of the ship. The form factor k is equal to 0.057, 0.074, 0.06 for the Wigley, S60 and DTMB5415 hulls at rest. These values mean that the form drag k Cf is a small

(9b)

2.5

/L3 + 9 /L3 with 0.05 ≤ k ≤ 0.4

δC / C0

Cf =



0.2 0.1

-0.1

S60

S60 S60

-0.2

0

0

-0.3 0.1

0.15

0.2

0.25

0.3 F

0.35

0.4

0.45

0.1

0.15

0.2

0.25

0.3 F

0.35

0.4

0.45

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

F

Fig. 5. The left column depicts experimental measurements, and corresponding smoothing spline fits, of the residuary drag coefficient Cr for the Wigley hull (top row) and the S60 model (bottom) held in position of rest (C0r ) or free to sink and trim, i.e. freely floating, (Cr ). The center column depicts the difference ıC r = C r − C0r together with theoretical predictions of the differences ıC t = C t − C0t and ıC w = C w − C0w related to the total drag Ct and the wave drag C w . The right column depicts the experimental relative difference ıC r /C0r together with the theoretical relative difference ıC t /C0w .

6

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

6 5

3

4

10 C

w

4

2

10

Exp.

Wigley

H

6

S60

8

Σ0

4

H

Σ1

6 3

ΣH a

1 0

4

2

2

1 0

0

-1

-1

6

10

L=2.5m Smooth

v

5

Wigley

8

4

Ct

3

Cv

6

L=4m Smooth

S60

5

6

Ct

4

v

L=5.72m Smooth

DTMB5415 t

4

C

3

v

C

4

t

10 C , C

DTMB5415

5

Cvp 0

6

Cvp

0

10

L=100m Smooth

5 v

1

v

Cp

0

Wigley

8

6

L=100m Smooth

S60

5

4

L=100m Smooth

DTMB5415

4

t

6 3

C

4

Ct 2

t

Cv

2

Cvp

Cv

1 Cvp

v

0

Cp

0

6

0

10

5

C

2

Cv

L=100m Rough

t

3

4

1

104 Ct, Cv, Ca

2

2

1

10 C , C

C

2

Wigley

8

6

L=100m Rough

S60

5

4

L=100m Rough

DTMB5415

4 6

t

C

t

3

C

Ct

3

4 2

Cvp

1 0 0.1

C

2

Cv

v

a

0.15

Cp

2

Cv

1

Ca 0.2

0.25

0.3

0.35

0.4

0.45

0 0.1

0.15

0.2

0.25

F

0.3

0.35

0.4

0.45

0 0.1

Cvp

Cv

a

C

0.15

0.2

0.25

F

0.3

0.35

0.4

0.45

F

Fig. 6. The left, center and right columns correspond to the Wigley, S60 and DTMB5415 hulls. The top row depicts experimental measurements (Exp.) of the residuary drag (blue dashed line) and the related hull surfaces H (red Cr for the freely floating ship models, and Neumann–Michell predictions of the wave drag C w for the hull surface H 0 1 v t w v v f solid line) and H a (black dash dot line). The second and third rows depict the corresponding theoretical total drag C = C + C , viscous drag C = C + Cp and form drag Cpv = k C f for ship models (second row) and corresponding ships of length L = 100 m with smooth hulls (third row). The bottom row depicts the total drag C t = C w + C v + C a and the roughness drag Ca , as well as the viscous drag C v = C f + Cpv and the form drag Cpv = k C f , for ships of length L = 100 m with rough hulls. Experimental values (Exp.) of the total drag Ct are also depicted for freely floating ship models (second row) and for smooth or rough hulls of length L = 100 m (third and bottom rows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

correction to the friction drag Cf for these three ship hulls, as is illustrated further on in Fig. 6. The hull-roughness drag Ca in (7) is determined as in [32] via the relation

C a = 10−4

AH R where 2L2

4 ≤ R ≡ 1050 (ks /L)

1/3

− 6.4 ≤ 8

(10a)

(10b)

Here, ks characterizes the roughness of the hull surface. The standard value ks = 0.00015 m is used. 4. Sinkage and trim effects on the drag Hereafter, AH , AH and AH a denote the wetted areas of the hull sur0 1 face H of the ship at rest or the hull surfaces H or H a determined 0 1 from the sinkage and the trim predicted by the numerical method or the experimental method considered in Section 2. The corresponding hull displacements are similarly denoted as H , H and 0 1

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

H a . Expressions (9b), (9c) and (10a) show that differences among H the wetted areas AH , AH and AH a and the related displacements 0 , 0 1 H H f

1 and a yield differences in the values of the friction drag C , the form factor k, and the roughness drag Ca for the hull surfaces H , 0 f H and H a , although the differences in the form drag kC and the 1 roughness drag Ca are very small and have a negligible influence on the total drag Ct as is illustrated further on in Fig. 6. The total drag coefficients C0t , C1t , Cat (and similarly for the drag components C w , , H or H Cf , Cpv and Ca ) correspond to the hull surfaces H a. 0 1 4.1. Experimental measurements and theoretical predictions Fig. 5 considers the experimental measurements, reported in [33–35], of the residuary drags of the Wigley hull and the S60 model held fixed in their positions at rest (no sinkage or trim allowed) or unrestrained (free to sink and trim). Specifically, the column on the left of Fig. 5 depicts the experimental measurements of the residuary drags C0r and Cr that correspond to the Wigley hull and the S60 model held fixed or freely floating. The (dashed or solid) lines are smoothing spline fits of the experimental measurements (marked as squares and crosses). The experimental data points and the related smoothing spline fits are significantly higher for the residuary drag Cr than for the residuary drag C0r at Froude numbers 0.25 < F. The differences between the experimental measurements of the residuary drags Cr and C0r are further illustrated, and compared to related theoretical predictions, in the columns in the center and the right of Fig. 5. Specifically, the center column depicts the difference ıC r ≡ C r − C0r between the experimental measurements of the residuary drags Cr and C0r together with theoretical predictions of the differences ıC t ≡ C t − C0t and ıC w ≡ C w − C0w between the total drag Ct and the wave drag C w for the ‘free’ or ‘fixed’ Wigley and S60 models. The right column depicts the experimental relative difference ıC r /C0r and the theoretical relative difference ıC t /C0w . The experimental relative difference ıC r /C0r is only depicted for 0.3 ≤ F in the right column of Fig. 5 because Cr is overly affected by considerable scatter among the experimental measurements for F < 0.3. The theoretical predictions of ıCt and ıC w and the experimental measurements of ıCr depicted in the center column of Fig. 5 are consistent and roughly in agreement, and show that the influence of sinkage and trim on the total drag Ct and its components Cr and C w are relatively small for Froude numbers F < 0.25, but increase rapidly and can be quite large for 0.25 < F. The experimental measurements of ıCr stem from both ıC w and ıC v , and are then more closely related to theoretical predictions of ıCt than of ıC w . The right column of Fig. 5 shows that sinkage and trim effects on the drag of a ship can be significant and that theoretical predictions are consistent with experimental measurements, although these unfortunately suffer from a very large scatter (especially for the S60 model). The total drag Ct and the related wave, viscous and roughness components C w , C v and Ca predicted by (7)–(10) are considered in Fig. 6 for the Wigley, S60 and DTMB5415 models (assumed to be smooth) and for corresponding smooth or rough hulls of length L = 100m. This figure depicts theoretical predictions for the hull sur, H and H faces H a , as well as experimental measurements for 0 1 the freely floating hulls. Specifically, the top row of Fig. 6 depicts theoretical predictions of the wave drag C w for the hull surfaces H , H , H a and experi0 1 mental measurements of the residuary drag Cr for the freely floating models. The second and third rows of Fig. 6 depict the theoretical total drag C t = C w + C v , the viscous drag C v = C f + Cpv and the form drag Cpv = k C f for the smooth models (second row) and corresponding smooth hulls of length L = 100 m (third row). The second row

7

also depicts experimental measurements Cet of the total drag for the three freely floating models. The bottom row of the figure depicts the roughness drag Ca , as well as the viscous drag C v = C f + Cpv , the form drag Cpv = k C f and the total drag C t = C w + C v + C a , for rough hulls of length L = 100m. The wave drag C w depicted in the top row is independent of the ship scale (length). The experimental measurements Cet of the total drag Ct depicted in the third and bottom rows of Fig. 6 for freely floating ship hulls of length L = 100m are estimated from measurements of the residuary drag of the freely floating ship models, as in [36], via the relation Cet = C r + C0v + C0a

(11)

where C0v and C0a are the viscous and roughness drags predicted by

(9) and (10) for the hulls H of the ships at rest. 0 Fig. 6 shows that differences among the theoretical drags Ct , C w , C v determined for the static hull surface H or for the dynamic 0 H are fairly small for F < 0.25, but increase hull surfaces H or  a 1 rapidly for 0.25 < F. These results show that sinkage and trim effects can largely be ignored for F < 0.25 but are significant for 0.25 < F, in agreement with the experimental measurements considered in Fig. 5. The form drag Cpv = k C f depicted in the second, third and bottom rows of Fig. 6 is very small, especially in the third and bottom rows for full-scale hulls. The roughness drag Ca depicted in the bottom row for the rough hulls is relatively small, although clearly not insignificant, and nearly constant within the range 0.1 ≤ F ≤ 0.45. The form drags Cpv and the roughness drags Ca that correspond to the hull surfaces H , H or H a are practically indistinguishable for 0 1 0.1 ≤ F ≤ 0.45. Fig. 6 also shows that differences among the theoretical drag coefficients C w , C v and Ct that correspond to the hull surfaces H 1 H and H a are practically negligible. Thus, the hull surface a defined by the explicit analytical relations (3), which require no flow computations, and the hull surface H have nearly identical drags. 1 Moreover, the (nearly identical) drags Ct associated with the hull H surfaces H a and 1 are in relatively good overall agreement with the experimental measurements Cet in Fig. 6 for the Wigley and S60 hulls. Discrepancies are larger for the DTMB5415 hull. The relative differences between experimental measurements Cet of the total drag Ct of a freely floating ship model and the corresponding theoretical predictions C0t or C1t for the static or dynamic ship hull surfaces H or H are defined as 0 1 e0t = (Cet − C0t )/C0t and e1t = (Cet − C1t )/C1t

(12)

The relative differences e0t and e1t provide a basis for validating the simple theoretical method (7)–(10) that accounts for the influence of sinkage and trim on the drag of a freely floating ship. Indeed, the that differences e0t and e1t are associated with the hull surface H 0 ignores the influence of sinkage and trim, or the hull surface H 1 that approximately accounts for this influence. t t Fig. 7 depicts the relative differences e0 and e1 for the Wigley, S60 and DTMB5415 models at Froude numbers within the range 0.25 < F < 0.45, for which sinkage and trim have a significant influence on the drag (the differences e0t and e1t are not considered for F < 0.25 because sinkage and trim effects on the drag are negligible for F < 0.25, as was shown in Fig. 5). The dashed and solid lines in Fig. 7 are smoothing spline fits that correspond to the experimental values of e0t (squares) and e1t (crosses). The difference e0t increases for 0.28 < F and is large for 0.35 < F. Indeed, the difference e0t exceeds 10% for 0.4 < F. The difference e1t is much smaller than the difference e0t for the Wigley hull and the DTMB5415 model at 0.32 < F, and for the S60 model at 0.35 < F (although this conclusion is weaker due to a large scatter in the experimental data). Specifically, the difference e1t for the Wigley

8

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11 0.25

t

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Fig. 7. Relative differences e0t and e1t , and related smoothing spline fits, between experimental measurements Cet of the total drag Ct of the freely floating Wigley (left), S60 of the ship models at rest or the related hull surfaces (center) and DTMB5415 (right) models and corresponding theoretical predictions C0t and C1t for the static hull surfaces H 0 that account for sinkage and trim. H 1

Fig. 8 shows that ıCt and its components ıC w and ıC v are relatively small for F < 0.25 but are significant for 0.25 < F, as was already observed in Fig. 6, and that the variations ıC w and ıCt of the wave and total drag coefficients increase rapidly for 0.25 < F. The bottom row of Fig. 8 shows that the increase ıC w of the wave drag due to sinkage and trim is dominant for the full-scale Wigley and S60 hulls at 0.25 ≤ F and for the full-scale DTMB5415 hull at 0.3 ≤ F. The top row of Fig. 8 shows that the viscous drag component ıC v is significantly larger at model scale than at full scale, as is well known. Moreover, Fig. 8 shows that the wave drag component ıC w is more sensitive to the hull form than the viscous drag ıC v , as is also well known. Fig. 9 depicts the relative increase ıC t /C0t of the total drag due to sinkage and trim for the Wigley, S60 and DTMB5415 hulls at model scale or for a ship with a rough hull of length L = 100 m. Differences between the ratios ıC t /C0t for model scale and full scale are relatively small. This result means that the relative increase ıC t /C0t of the total drag due to sinkage and trim is not appreciably influenced by the ship length, i.e. is largely independent of scale. Fig. 9 also shows that sinkage and trim effects on the total drag of a ship increase rapidly and can be large for 0.25 < F, as was already observed in Fig. 6. In particular, at F = 0.45, Fig. 9 shows that the

and S60 hulls in the Froude number ranges 0.32 < F or 0.35 < F varies within ±2%, and is smaller than 7% for the DTMB5415 model at 0.32 < F. The larger differences for the DTMB5415 hull might be due to the transom stern (ignored here) of this hull. 4.2. Analysis of theoretical predictions Useful insight into the influence of sinkage and trim on the drag of a ship, and on the wave and viscous drag components within the Froude decomposition of the drag that is considered here, can be gained by further analyzing the drag coefficients Ct , C w , C v and and H . This analysis is Ca associated with the hull surfaces H 0 1 considered in Figs. 8–11. Fig. 8 depicts the variations ıC t,w,v,a = C1t,w,v,a − C0t,w,v,a

(13)

of Ct , C w , C v , Ca due to sinkage and trim for the Wigley, S60 and DTMB5415 hulls at model scale (top row) or for a ship with a rough hull of length L = 100 m (bottom row). The drag increase ıCa related to hull roughness is fairly small within the speed range 0.1 ≤ F ≤ 0.45 considered in Fig. 8.

0.7

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Fig. 8. Variations ıCt (red solid lines), ıC w (blue dashed lines), ıC v (black dash dot lines) and ıCa (green dashed lines) of the total drag Ct , the wave drag C w , the viscous drag C v and the roughness drag Ca for the Wigley (left), S60 (center) and DTMB5415 (right) hulls at model scale (top) or at full scale (bottom). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11 0.2

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Fig. 9. Relative increase ıC t /C0t of the total drag Ct for the Wigley (left), S60 (center) and DTMB5415 (right) hulls at model scale (blue dashed lines) or at full scale (red solid lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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Fig. 10. Relative increases ıC w /C0w (left side) and ıC v /C0v (right side) of the wave and viscous drag coefficients for the Wigley (red solid lines), S60 (black dot-dash lines) and DTMB5415 (blue dashed lines) hulls. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

more sensitive to the hull form than the ratio ıC v /C0v related to the viscous drag, which increases monotonically as F increases. This result illustrates the well-known fact that the viscous drag is relatively insensitive to the hull form, whereas the wave drag can be significantly influenced by the hull form. Moreover, Fig. 10 shows that the relative increases ıC w /C0w and ıC v /C0v of the wave and viscous drag components are of the same

relative increase ıC t /C0t of the total drag Ct is approximately equal to 15% for the Wigley and S60 hulls, and to 7% for the DTMB5415 hull. Fig. 10 depicts the ratios ıC w /C0w and ıC v /C0v that define the relative increases of the wave and viscous drag components due to sinkage and trim for the Wigley, S60 and DTMB5415 hulls. This figure shows that the ratio ıC w /C0w related to the wave drag is much

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Fig. 11. Ratios C1w /C1t (top) and ıC w /ıC t (bottom) for the Wigley (left), S60 (center) and DTMB5415 (right) hulls at model scale (blue dashed lines) or at full scale (red solid lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

order of magnitude for the DTMB5415 hull. However, the increase ıC w /C0w of the wave drag is significantly larger than the increase ıC v /C0v of the viscous drag for the S60 hull and (especially) the Wigley hull, as was already noted in Fig. 8. E.g., for F = 0.45, Fig. 10 shows that ıC w /C0w ≈ 20% and ıC v /C0v ≈ 7% for the Wigley hull. The corresponding differences between ıC w /C0w ≈ 16% and ıC v /C0v ≈ 11% are smaller for the S60 hull. The increase ıC w /C0w in the wave drag (determined theoretically via the NM theory) depicted in Fig. 10 for the Wigley and S60 hulls, and the increase ıC r /C0r in the residuary drag (determined via experimental measurements) depicted in Fig. 5, are roughly in agreement, and clearly illustrate the large increase in the wave drag that can occur due to sinkage and trim. The relative importance of the wave component C w is further analyzed in Fig. 11. Specifically, this figure depicts the ratios C1w /C1t and ıC w /ıC t for the Wigley, S60 and DTMB5415 (smooth) models and corresponding rough hulls of length L = 100 m. Both the ratios C1w /C1t and ıC w /ıC t mostly increase as F increases and are large at high Froude numbers, especially for full-scale hulls. E.g., the top row of Fig. 11 shows that one has C1w /C1t ≈ 0.6 for the full-scale Wigley hull at F = 0.45, and the values of C1w /C1t at F = 0.45 are even larger for the full-scale S60 and DTMB5415 hulls. The bottom row of Fig. 11 shows that ıC w /ıC t ≈ 0.8 for the fullscale Wigley and S60 hulls at F = 0.45, and that ıC w /ıC t exceeds 0.5 for F0.5 < F with 0.25 < F0.5 < 0.35 for the full-scale Wigley, S60 and DTMB5415 hulls. Thus, the increase of the total drag Ct due to sinkage and trim largely stems from the increase of the wave drag at high Froude numbers. Fig. 11 also shows that the ratios C1w /C1t and ıC w /ıC t , which provide a measure of the relative importance of the wave drag, are significantly different for the three hull forms. This result agrees with the well-known fact that the wave drag is sensitive to the hull form, as is also clearly illustrated in Fig. 10. 5. Conclusion Experimental measurements and theoretical computations for the Wigley, S60 and DTMB5415 hulls suggest that the sinkage and the trim experienced by a common freely floating monohull ship are small and have limited influence on the drag for Froude numbers F smaller than about 0.25. However, the sinkage and the trim, and their influence on the drag, increase rapidly for 0.25 < F, and are notable for the highest value F = 0.45 of the range 0.1 ≤ F ≤ 0.45 considered here. E.g., at F = 0.45, the Wigley hull and the S60 model are found to experience an increase in drag of about 15%, and the drag of the DTMB5415 model is about 7% higher, due to sinkage and trim. The influence of sinkage and trim on the drag of a ship can then be significant for 0.25 < F, and moreover depends on the hull form. Thus, sinkage and trim should be considered within the design process, arguably even at early design stages and for hull form optimization. Accordingly, practical methods suited for routine applications to ship design have been considered here. Specifically, the two alternative methods to determine the sinkage and the trim considered in [12] are used. These two simple methods yield predictions of sinkage and trim that do not differ greatly, and are in reasonable agreement with experimental measurements for a broad class of monohull ships. The drag is similarly evaluated in a simple manner based on Froude’s basic decomposition into viscous and wave components, as in [18]. One of the two alternative methods considered in [12] and used here to determine the sinkage and the trim is a simple numerical method. This method only involves linear potential flow computations, based on the Neumann-Michell theory [1,13–17], for of the ship at rest. Indeed, numerical the wetted hull surface H 0

predictions of sinkage and trim for the hull surface H of the ship 0 at rest and for the hull surface H , which is determined from flow 1 and thus accounts for the computations for the hull surface H 0 hull sinkage and trim, do not differ significantly for the Wigley, S60 and DTMB5415 hulls. This result suggests that the sinkage and the trim of common monohull ships at F ≤ 0.45 can be predicted without iterative flow computations for a sequence of ship hull surfaces H n , a notable simplification that stems from the fact that the sinkage and the trim are primarily determined by the pressure at the bottom of the ship hull, and thus are not highly sensitive to the precise position of the ship. The other method considered in [12] and used here to determine the sinkage and the trim is based on an analysis of experimental measurements for 22 ship models. This alternative method yields explicit analytical relations for the sinkage and the trim, and thus requires no flow computations. Specifically, the sinkage and the trim of a ship are determined explicitly, via simple analytical relations, in terms of the ship speed V and four basic parameters (the length L, the beam B, the draft D and the block coefficient Cb ) that characterize the hull geometry. As was already noted, the drag is also determined in a simple manner, based on Froude’s basic decomposition of the drag into viscous and wave components. More precisely, well-known semiempirical relations are used for the friction drag, the viscous pressure drag and the drag due to hull roughness, and the wave drag is determined via linear potential flow computations based on the Neumann-Michell theory. This simple approach can readily be used to predict the drag of ship models as well as full-scale ships with smooth or rough hull surfaces, and these three cases indeed are considered in the study. The wave drag is largely determined by the pressure at the bow and the stern of a ship, and is then more sensitive to the precise position of the ship hull than the sinkage and the trim, which are mostly determined by the pressure at the hull bottom. This basic difference explains why the sinkage and the trim of a ship can be realistically estimated from flow computations around the hull surface H of the ship at rest, whereas the drag must be evaluated for 0 a dynamic ship hull surface H st that accounts for the sinkage and the trim experienced by the ship. However, the hull surface H st does not need to be very precise. Indeed, a main result of the theoretical predictions reported here for the Wigley, S60 and DTMB hulls is that the hull surface H a defined deterby the explicit analytical relations (3) and the hull surface H 1 mined from potential flow computations for the hull surface H of 0 the ship at rest have nearly identical total drag coefficients Ct . Another notable result of the study is that the (nearly identical) predictions of the total drag for the ship hull surfaces H a or H , which correspond to the sinkage and the trim predicted by 1 the explicit analytical relations (3) or via flow computations for the hull surface H as was just noted, are notably higher than the drag 0 predicted for the hull surface H of the ship at rest. 0 Moreover, and more importantly for practical applications, the H drag coefficients predicted for the hull surfaces H a or 1 are much closer to experimental measurements than the drag of the hull surface H of the ship at rest for the Wigley, S60 and DTMB5415 hulls 0 at Froude numbers for which sinkage and trim effects are important. Indeed, at Froude numbers greater than about 0.32 to 0.35, the relative differences e1t between experimental measurements of the total drag and theoretical predictions for the dynamic hull surface , which accounts for sinkage and trim, are much smaller than H 1 the differences e0t related to theoretical predictions for the static hull surface H . 0 This finding suggests that the influence of sinkage and trim on the drag of a freely floating monohull ship at F ≤ 0.45 can be determined in a simple way, and indeed partially validates the practical

C. Ma et al. / Applied Ocean Research 65 (2017) 1–11

method considered here. In particular, if the analytical relations (3) are used to estimate the sinkage and the trim, prediction of the drag of a freely floating ship only requires one flow computation per Froude number, namely for the hull H a. As is noted in the introduction, the drag of a freely floating ship is influenced by sinkage and trim, considered in the study, as well as by several complicated flow features that are not considered here. These additional features include flow separation that typically occurs at a ship stern, notably a transom stern, and wavebreaking at a ship bow. For the Wigley and S60 hulls, these additional complications only have a relatively minor influence on the drag. Accordingly, the relative differences e1t between experimental measurements of the total drag and theoretical predictions are small (within ±2%) for these two hulls for the hull surface H 1 at Froude numbers (greater than about 0.32 to 0.35) for which sinkage and trim have a large influence. The differences e1t are appreciably larger (as much as 7%) for the DTMB5415 hull, possibly because this hull has a transom stern (ignored here). In any case, for the three ship hulls considered here, the differences e1t are much smaller than the differences e0t associated with drag predictions for the hull surface H of the ship at rest for Froude numbers 0 (greater than about 0.35) for which sinkage and trim effects are important. References [1] F. Noblesse, F. Huang, C. Yang, The Neumann–Michell theory of ship waves, J. Eng. Math. 79 (1) (2013) 51–71. [2] C.Y. Chen, F. Noblesse, Comparison between theoretical predictions of wave resistance and experimental data for the Wigley hull, J. Ship Res. 27 (1983) 215–226. [3] A.K. Subramani, E.G. Paterson, F. Stern, CFD calculation of sinkage and trim, J. Ship Res. 44 (1) (2000) 59–82. [4] C. Yang, R. Löhner, F. Noblesse, T.T. Huang, Calculation of ship sinkage and trim using unstructured grids, in: European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS, 2000. [5] C. Yang, H.Y. Kim, F. Noblesse, A practical method for evaluating steady flow about a ship, in: 9th Il Conf. on Fast Sea Transportation, Shanghai, China, 2007. [6] C. Yang, R. Löhner, Calculation of ship sinkage and trim using a finite element method and unstructured grids, Int. J. Comput. Fluid Dyn. 16 (3) (2002) 217–227. [7] C.B. Ni, R.C. Zhu, G.P. Miao, J. Fan, Hull gesture and resistance prediction of high-speed vessels, J. Hydrodyn. Ser. B 23 (2) (2011) 234–240. [8] C.B. Yao, W.C. Dong, A method to calculate resistance of ship taking the effect of dynamic sinkage and trim and viscosity of fluid, in: Applied Mechanics and Materials (Taiwan), Trans Tech Publications, 121, 2012, pp. 1849–1857. [9] W. He, T. Castiglione, M. Kandasamy, F. Stern, Numerical analysis of the interference effects on resistance, sinkage and trim of a fast catamaran, J. Mar. Sci. Technol. 20 (2) (2015) 292–308. [10] L.J. Doctors, Hydrodynamics of High-Performance Marine Vessels, 1st ed., CreateSpace Independent Publishing Platform, 2015. [11] X. Chen, R. Zhu, C. Ma, J. Fan, Computations of linear and nonlinear ship waves by higher-order boundary element method, Ocean Eng. 114 (2016) 142–153. [12] C. Ma, C. Zhang, X. Chen, Y. Jiang, F. Noblesse, Practical estimation of sinkage and trim for common generic monohull ships, Ocean Eng. 126 (2016) 203–216.

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