Experimental investigation of the wetted surfaces of stepped planing hulls

Ocean Engineering 187 (2019) 106164

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Experimental investigation of the wetted surfaces of stepped planing hulls Amin Najafi a, Hashem Nowruzi b, 1, *, Mohammad Karami b, Hosein Javanmardi c a

Mechanical Engineering Department, Imam Hossein University, Tehran, Iran Department of Maritime Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran c Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran b

A R T I C L E I N F O

A B S T R A C T

Keywords: Experimental study Planing hull Transverse step Wetted surface

In the current study, experimental tests are conducted to evaluate the hydrodynamic characteristics and the bottom wetted surfaces of the planing hulls equipped by one transverse step. For this purpose, total resistance, dynamic trim angle, dynamic sinkage, reattachment lengths of the separated flow and forebody and aftbody wetted areas are quantitatively measured at different geometrical parameters of the step and hull velocities. Moreover, the bottom wetted surfaces are qualitatively investigated by analyzing the underwater photographs. Our experimental results are also compared by archival data with good accordance. Based on our experimental observations, dynamic trim angle and dynamic sinkage are enhanced by increasing in the height of the step and decreasing the longitudinal position of step from transom. However, lower total resistance is achieved for greater height of the step. In addition, total wetted area is enhanced by decreasing the step height and hull velocity. However, larger total wetted area is detected by an increase in the longitudinal distance of the step from transom. Finally, we formulated the wetted areas of one step planing hull according to the reattached flow patterns.

1. Introduction Hydrodynamics lift force of the high speed craft is a determinative parameter to attain high speeds by overcoming their weight (Doctors, 1985). By increasing the speed of the high speed craft, their drag to lift ratio will be increased. One of the effective strategic solutions to improve the drag to lift ratio of high speed craft is using transverse step in their hulls (stepped planing craft). Using step on the bottom of planing hulls will cause the flow separation at the step region (re-attach at aft­ body location) and it will accomplish a decrease in the wetted area. This happen may lead to decrease in drag to lift ratio of high speed craft. Moreover, transverse steps bring about more uniform pressure distri­ bution on the bottom of planing hull which may provide more longi­ tudinally stability (Savitsky and Morabito, 2010). Hydrodynamics performance of stepped planing craft is dependent on the wetted bottom pattern and area which may be related to the height and longitudinal position of the step, Froude number and etc. Therefore, study of the wetted surfaces of stepped planing hulls is a precondition for achieving an efficient stepped planing craft. Several experimental and numerical studies are done on

hydrodynamic behavior of planing craft. Pioneering investigation on drag and flow pattern around the high speed planing hull is related to Blount and Clement experimental study (Blount and Clement, 1963). Then, Savitsky (1964), conducted a reference experimental work on wedge-like hulls. He proposed semi-empirical formulation to predict of drag and lift forces for without step planing craft. Brizzolara and Serra (Brizzolara and Serra, 2007) studied the drag and lift forces of planing surface by CFD analysis. Their numerical results were in good accor­ dance with experimental data of Savitsky (1964) and Shuford (1958). Effect of whisker spray on total drag of planing craft was studied experimentally and analytically by Savitsky et al. (2007). In 2010, Savitsky and Morabito (2010), experimentally investigated the aftbody surface wake profiles of simple prismatic planing hull. They suggested empirical formulation to predict the centerline and ¼ beam wake pro­ files. They also presented some details about the bottom wetted surface for one step planing craft. Recently, Seo et al. (2016) investigated total resistance and sea keeping of a planing hull using wave-piercing and spray rails, experi­ mentally. Hydrodynamics characteristic of trimaran planing craft under different Froude number were studied experimentally and numerically

* Corresponding author. National Iranian Marine Laboratory (NIMALA), Tehran, Iran. E-mail address: [email protected] (H. Nowruzi). 1 Postal Address: Department of Maritime Engineering, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave, No 424, P.O. Box 15875–4413, Tehran, Iran. https://doi.org/10.1016/j.oceaneng.2019.106164 Received 13 February 2019; Received in revised form 22 May 2019; Accepted 29 June 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Table 1 Summary of some archival works related to investigate of geometrical parameters of the step in stepped planing hull. Author

Methoda

Water condition

Clement and Pope (Clement and Pope, 1961) Gassman, and Kartinen (Gassman and Kartinen, 1994) Svahn (Svahn, 2009) Makasyeyev (Makasyeyev, 2009) Savitsky and Morabito (Savitsky and Morabito, 2010) White and Beaver (White and Beaver, 2010) Taunton et al. (Taunton et al., 2010) Taunton et al. (Taunton et al., 2011) Lee et al. (Lee et al., 2014) Lotfi et al. (Lotfi et al., 2015) De Marco et al. (De Marco et al., 2017)

Exp Exp Analytic & Num Analytic Exp Exp Exp Exp Exp Num Exp & Num

Calm water Calm water Calm water Wavy condition Calm water Calm water Calm water Wavy condition Calm water Calm water Calm water

a b c

Evaluated geometrical parameter of the step Deadrise (β) [deg]

Height (Hs) [m]

Distance from transom (Ls) [m]

5� � β � 15� 15� 10� � β � 30� NP 10� � β � 30� 15� 22.5� 22.5� 15� 10� � β � 30� 23�

NPb 0.013 0.05 NP FQ 0.003 � Hs � 0.025 NP NP 0.0032 � Hs � 0.0096 FQ 0.006

NP 0.495� Ls � 0.799 2.5 FQc NP 0.542 NP NP 0.381� Ls � 1.016 FQ NPEqn (1)-(4)

Exp ¼ experimental method; Num ¼ numerical method; and Analytic ¼ analytical method. NP¼Not Reported. FQ¼Formulated in Equations.

models is experimentally studied by analyzing the reattachment length of the separated flow and forebody and aftbody wetted areas. The remainder of present paper is organized as follows: in section 2, experimental setup and physical model are presented. Experimental results are discussed in Section 3. Section 4 is also provided for conclusions.

Table 2 Specification of NIMALA towing tank. Length (m)

392

Width (m) Water depth (m) Maximum carriage speed (m/s) Maximum capacity of the force gauge (N) Accuracy of force gauge (FS) Maximum measurement range of potentiometer (degree) Accuracy of potentiometer (degree)

6 4 19 600 0.02% (maximum force) �30 0.01

2. Experimental setup In the current study, according to ITTC guidelines (ITTC, 2002) (Committee, 2002), all experimental tests are done in the towing tank of National Iranian Marine Laboratory (NIMALA), Tehran, Iran. Specifi­ cation of NIMALA towing tank is shown in Table 2. Based on our pre­ vious studies (Najafi et al., 2018a, 2018b; Nowruzi and Najafi, 2019), calm water condition with no wave is used for water fluid, where water temperature, density and kinematic viscosity have fixed value of 293.15 K, 1002 kg/m3 and 1.19E-6 m2/s, respectively. According to dynamic and geometrical similarities criteria, Fridsma planing hull model is chosen. In order to determine the dimensions of hull model, the remarks of maximum Froude number and blockage factor are considered. To assessment of the blockage factor, we evalu­ ated the maximal cross-sectional area of the hull model (Ax) per the sectional area of the towing tank (A) (geometrical scale factor is λ ¼ 8) as follows:

by Jiang et al. (2016). De Marco et al. (De Marco et al., 2017), studied the flow pattern and hydrodynamic behavior of one step planing craft, both experimentally and numerically. Cucinotta et al. (2017), experi­ mentally indicated the positive effects of artificial air cavity on drag reduction of planing craft. For more detail, some archival works related to study on geometrical parameters of the step in stepped planing hull are presented in Table 1. According to the cited works, the lack of study on detail of bottom wetted surfaces of the stepped planing hulls is evident. In addition, the effect of different geometrical parameters of the step on the bottom wetted pattern is not well-known so far. Therefore, the main purpose of the current paper is to study the effects of different height of the step and longitudinal distance of the step from transom and hull velocity on the detail of the wetted surface of stepped planing hulls. To this accom­ plishment, bottom wetted surface of stepped Fridsma planing hull

m¼

Ax 0:156 ¼ 0:0065≪0:1 ¼ 24 A

(1)

Table 3 Main geometrical dimensions of full scale and hull model without any transverse step and considered test cases for one-step planing craft. cases

Without transverse step hull model Cases of one-step planing craft

full scale model 1 2 3 4 5 6 7 8 9 10 11

Deadrise angle

Longitudinal position of step from transom

Height of the step

Scale factor

Length of overall

Beam

Draft from keel

Length per beam

Displacement

LCG from AP (m)

В (deg)

Ls (mm)

Hs (mm)

λ

LoA (m)

B (m)

TFrom keel (m)

L/B [-]

Δ (kg)

LCG (m)

20

–

–

1

20

4

2.5

5

25000

7.2

20 20 20 20 20 20 20 20 20 20 30 30

– 600 600 600 600 600 600 800 800 1100 800 800

– 10 10 20 30 30 30 10 10 30 20 20

1:8 1:8

2.5 2.5

0.5 0.5

0.312 0.312

5 5

48 48

0.9 0.9

2

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Fig. 1. Schematic of (a) considered Fridsma model (dimensions are in mm) as prototype of without transverse step and (b) fabricated one-step planing craft.

Fig. 2. Teflon connecting block and its connection to the forebody module.

As may be seen in Eq. (1), where the blockage factor is remarkably lower than the 0.1, the effects of towing tank walls on the model is negligible. Table 3 is presented for the main geometrical dimensions of full scale and hull model without any transverse step (i.e., which has been used as the basic model for embedding the transverse step on it) and considered test cases for one-step planing craft. As may be seen in Table 3, we designed experimental tests in such a way that different investigation on the geometrical parameters of the step can be provided with minimum number of lab tests. Fig. 1 shows the schematic of intended Fridsma model (i.e., prototype of without transverse step model) and fabricated one-step planing craft. Conducted towing tests are tabulated in Table 3.

Hull model is fabricated by three separate parts of forebody module, aftbody module and connecting block. Fore body module as integrated structure is fabricated by fiberglass composite material. Connecting block is fabricated by high strength Teflon and it is used to connect the forebody module to aftbody module (see Fig. 2). Aftbody module has multi movable blocks in order to change the longitudinal position and height of the step. One of these blocks is CG block with six Polyvinyl chloride (PVC) sections to adjust the center of gravity (see Fig. 3 (a)). Another block as transom block is also embedded at the end of the aft­ body module for placement of ballast weights (see Fig. 3 (b)). Other blocks of aftbody module have middle section made of PVC and outer section made of Teflon. A block of aftbody after the assembly is 3

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Fig. 3. Tow block of aftbody module: (a) CG block and (b) transom block.

Fig. 4. (a) Block of aftbody after the assembly and (b) fabricated blocks before the assembly.

Fig. 5. Schematic of longitudinal location of the step: (a) 600 mm from AP, (b) 800 mm from AP and (c) 1100 mm from AP.

presented in Fig. 4. All parts of the hull model are made based on the ITTC tolerances (ProceduresITTC Recommended, 2011) and then, displacement weight of the constructed model is adjusted. Maximum of measured geometrical fabrication error of the hull model for deadrise

20� and 30� are equal to 1.19 mm and 0.92 mm. Three different location of the step is also depicted in Fig. 5. Continued the experimental results of the reattached length of the separated flow and the bottom wetted surfaces areas are presented and 4

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3.2. Reattachment length and wetted areas

Table 4 Experimental results of dynamic trim angle, total trim angle, non-dimensional forms of total resistance and dynamic sinkage. Measured criteria

V [m/ s]

dynamic trim angle

total trim angle

total Resistance

τDTA [deg]

τTTA

[deg]

RT Δ

3.92 3.07 3.81 5.04 4.83 4.27 3.15 2.93 4.01 4.14 3.85

0.176 0.244 0.200 0.163 0.168 0.188 0.194 0.227 0.173 0.229 0.280

7 10 10 7 8 10 7 8 7 8 10

Frr [-]

3.71 5.30 5.30 3.71 4.24 5.30 3.71 4.24 3.71 4.24 5.30

2.33 1.48 1.89 2.79 2.78 2.02 1.51 1.29 1.56 1.79 1.51

½ �

Schematic of the bottom wetted surfaces of the one step planing hull and the main geometrical parameters of the reattachment length of the separated flow are shown in Fig. 6. As may be seen in Fig. 6, the Ls is the longitudinal location of the step from transom and, Lk1 and Lk2 are forebody and aftbody wetted keel lengths, respectively. Moreover, Lc is the wetted chine length and xcl is the distance of reattached length of the separated flow from the step on centerline and x14 is the distance of reattached length of separated flow from the step on the line of ¼ hull beam. The parameters of Swfore and Swaft are representative of forebody and aftbody wetted area, respectively. As may be seen in Fig. 6, two main wetted regions are recognizable at the bottom of stepped planing hulls with one transverse step. The first triangular shape region is located before the step that initiates from the stagnation points of water surface with bottom of the planing hull and two stagnation lines of this region will be ended to the fore body spray line. Another wetted region is located after the step and it is started from reattachment point of the separated flow on the step. The pattern of this region corresponds to the step location, step height, hull velocity, deadrise angle and etc. In the present study, the reattachment length and wetted area are precisely calculated by measuring our recorded experimental underwater images via SolidWorks package. It is also notable that the used camera has 720 � 1280 resolution with frame rate of 30 FPS. Pictograph of some experimental underwater images measured and analyzed by SolidWorks is also presented in Fig. 7. Reattachment length, pattern of wetted surfaces and wetted areas are illustrated in Fig. 7. As may be seen in Fig. 7 (a), the stagnation lines of forebody wetted surface are crossed from chine sides and smaller aftbody wetted area is achieved at higher hull velocity. Fig. 7 (b) shows remarkable effects of Ls on the pattern of aftbody wetted surface. So that, as the Ls increases, local re­ gion of aftbody wetted surface is distributed across the hull beam and it is intersected the side chine. Pattern of forebody and aftbody wetted area under different height of the step is also depicted in Fig. 7(c). Forebody angles of projected stagnation line (αSLðforeÞ ) and spray edge (αWSðforeÞ ), and aftbody angles of projected stagnation line (αSLðaftÞ ) and spray edge (αWSðaftÞ ) are measured as defined in Begovic and Bertorello (2012) (for more details, please see Figs. 6 and 7 in Ref (Begovic and Bertorello, 2012)). Experimental results of forebody and aftbody angles of projected stagnation line and spray edge are presented in Table 5. As may be seen in Table 5, αSLðforeÞ and αWSðforeÞ are increased by an enhancement in Hs and Ls. Values of αSLðaftÞ and αWSðaftÞ are also decreased

dynamic sinkage at LCG Zv

13

r

=

1 2 3 4 5 6 7 8 9 10 11

Testing conditions

=

case

½ �

0.084 0.053 0.068 0.102 0.101 0.073 0.054 0.046 0.056 0.065 0.054

discussed. 3. Results and discussion 3.1. Resistance, dynamic trim and sinkage First, experimental results of dynamic trim angle, total trim angle, non-dimensional forms of total resistance (RT) and dynamic sinkage (ZV) are presented in Table 4. As may be seen in Table 4, dynamic trim angle, total trim angle and dynamic sinkage are decreased by an enhancement in hull velocity. However, total resistance is enhanced by an increase in hull velocity. Moreover, dynamic trim angle, total trim angle and dy­ namic sinkage are increased by an enhancement in the height of the step, while, lower total resistance is achieved for greater height of the step. In addition, dynamic trim angle, total trim angle and dynamic sinkage are decreased and total resistance is increased by increasing the longitudinal position of step from transom. According to our obtained experimental results (please see Table 4), increase of the Hs and decrease of the Ls is one of the strategies to reduce the total resistance. However, the interval of appropriate changes on these geometrical parameters should be determined in proportion to any particular type of stepped planing craft.

Fig. 6. Schematic of bottom wetted surfaces along with defined parameters. 5

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Fig. 7. Pictograph of experimental underwater images measured by SolidWorks under different (a) hull velocities, (b) longitudinal distance of the step from transom (Ls), and (c) heights of the step (Hs).

6

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Table 5 Experimental results of forebody and aftbody angles of projected stagnation line and spray edge. case

Input of experimental tests

1 2 3 4 5 6 7 8 9 10 11

Measured angles

β [deg]

Ls [m]

Hs [m]

V (m/s)

Frr [-]

αSLðforeÞ [deg]

αWSðforeÞ [deg]

αSLðaftÞ [deg]

αWSðaftÞ [deg]

20 20 20 20 20 20 20 20 20 30 30

0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 1.1 0.8 0.8

0.01 0.01 0.02 0.03 0.03 0.03 0.01 0.01 0.03 0.02 0.02

7 10 10 7 8 10 7 8 7 8 10

3.71 5.30 5.30 3.71 4.24 5.30 3.71 4.24 3.71 4.24 5.30

21.99 16.84 18.3 22.95 22.41 20.22 23.28 21.68 26.56 13.28 12.62

27.98 19.26 23.45 30.42 28.31 25.83 33.92 26.52 37.24 16.12 14.97

29.47 22.51 19.78 18.92 21.13 12.52 61.27 64.09 24.72 31.1 21.82

38.35 31.78 26.67 24.38 29.69 17.31 71.8 77.13 32.36 42.65 30.73

Table 6 Experimental results of non-dimensional form of reattachment length, forebody wetted area, aftbody wetted area and total wetted area. Measured criteria Hs [m]

V (m/s)

Frr [-]

xcl

13

r 1 2 3 4 5 6 7 8 9 10 11

20 20 20 20 20 20 20 20 20 30 30

0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 1.1 0.8 0.8

0.01 0.01 0.02 0.03 0.03 0.03 0.01 0.01 0.03 0.02 0.02

7 10 10 7 8 10 7 8 7 8 10

3.71 5.30 5.30 3.71 4.24 5.30 3.71 4.24 3.71 4.24 5.30

=

Ls [m]

½ �

0.551 0.799 1.308 0.972 1.201 1.514 0.330 0.413 1.101 0.826 1.056

Swfore 2

r 3 2.42 1.98 1.52 2.42 2.02 1.39 2.05 1.61 1.40 3.35 3.31 =

β [deg]

Swaft ½ � 2 r 3 0.977 0.324 0.046 0.178 0.084 0.005 2.439 2.396 1.997 1.180 0.744

½ �

Swtotal 2

r 3 3.40 2.30 1.57 2.60 2.10 1.40 4.48 4.01 3.39 4.53 4.05

=

Input of experimental tests

=

case

½ �

Table 7 Comparison between our experimental results of reattachment length and aftbody wetted area by results extracted from Savitsky and Morabito (2010) empirical formulation. case

1 2 3 4 5 6 7 8 9 10 11

Input of experimental tests

Present study

Results extracted from Ref. (Savitsky and Morabito, 2010) formulation

Difference on xcl

Difference on Swaft

β (deg)

Ls (mm)

Hs (mm)

V (m/s)

xcl (mm)

Swaft (mm2)

xcl (mm)

Swaft (mm2)

(%)

(%)

20 20 20 20 20 20 20 20 20 30 30

600 600 600 600 600 600 800 800 1100 800 800

10 10 20 30 30 30 10 10 30 20 20

7 10 10 7 8 10 7 8 7 8 10

199.4 290.2 476.6 352.8 435.1 550.1 120.4 150.3 400.1 300.5 383.5

128907 42791 6071 23500 11050 674 321615 315973 263365 155670 98150

200 290 475 390 436 550 120 150 400 328 386

138000 45839 5662 26859 13800 801 327990 305530 237440 148320 100190

0.301 0.069 0.336 10.544 0.207 0.018 0.332 0.200 0.025 9.151 0.652

7.054 7.123 6.737 14.294 24.887 18.843 1.982 3.305 9.844 4.722 2.078

by increasing the Hs and decreasing the Ls. Experimental results of non-dimensional form (i.e., based on displacement volumer) of reattachment length, forebody wetted area, aftbody wetted area and total wetted areas for conducted test cases are tabulated in Table 6. In addition, our experimental results of reattach­ ment length and aftbody wetted area are compared by results extracted from Savitsky and Morabito (2010) empirical formulation and we ob­ tained the differences. As may be seen in Table 7, the maximum dif­ ference on reattachment length (xcl ) and aftbody wetted area (Swaft ) are obtained 10.544% and 24.887%, respectively. For more detail, experimental results of reattachment length (xcl ) at different hull velocities, longitudinal locations of the step from transom (Ls) and heights of the step (Hs) is presented in Fig. 8. As may be seen in Fig. 8, for all the tested cases, xcl is enhanced by an increase in the hull velocity and heights of step. However, there isn’t any regular trend on

the growth of xcl by increasing the Ls. Indeed, Fig. 8(b) shows the different effects of Ls on the reattachment length of separated flow on step under various heights of the step. Fig. 9 shows the effects of different hull velocities, longitudinal locations of the step from transom (Ls) and heights of step (Hs) on the parameter of forebody wetted area (Swfore ). Based on Fig. 9, as hull velocity increases, Swfore is decreased. Indeed, the planing hull more exit from the water by increasing the hull velocity and it will be resulted to decrease of forebody wetted area. An enhancement in Ls also brings about the forebody wetted area decre­ ment, while, an irregular trend on the value of Swfore is detected by an increase in the Hs. As may be seen in Fig. 10, Swaft is decreased by increasing the hull velocity that is related to higher reattachment length of the separated flow at upper hull velocity. Based on Fig. 10 (b), as the Ls increase, Swaft is remarkably increased and Fig. 10 (c) shows a decrease in Swaft by increasing the height of step from 10 mm to 20 mm, 7

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Fig. 8. Reattachment length (xcl ) at different (a) hull velocity, (b) Ls and (c) Hs.

Fig. 9. Forebody wetted area (Swfore ) at different (a) hull velocity, (b) Ls and (c) Hs.

Fig. 10. Aftbody wetted area (Swaft ) at different (a) hull velocity, (b) Ls and (c) Hs.

Fig. 11. Total wetted area (Swtotal ) at different (a) hull velocity, (b) Ls and (c) Hs. 8

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Fig. 12. Underwater images of formed wetted surfaces on the bottom of stepped planing craft using one transverse step.

while, Swaft is enhanced by increasing the height of step from 20 mm to 30 mm. Fig. 11 shows that the total wetted area (Swtotal ) is increased by an enhancement in the Ls. Moreover, Swtotal is decreased by increasing the hull velocity and Hs. Indeed, as the hull velocity and Hs are decreased and the Ls is increased, an increase in the Swtotal as representative of stepped planing hull resistance is achieved. It is notable that, in all tested cases, trim angle is obtained lower than 4.5� .

and step dimensions, wetted length of the keel, trim angle, hull velocity and etc. Variation of the wetted surfaces is remarkably effective on the lift force and stability of planing craft. Total wetted areas after planing of one-step planing craft is the summation of aftbody and forebody wetted areas as follows: Swtotal ¼ Swaft þ Swfore

(2)

where, we have two scenarios for Swfore as follows:

3.3. Morphology, classification and formulation of wetted surfaces

- If flow spray line (stagnation line) crossing from the corners of the step, the forebody wetted area is:

Fig. 12 shows some underwater images of formed wetted surfaces on the bottom of stepped planing hull using one transverse step. As illus­ trated in Fig. 12, two main wetted regions are formed on the bottom of stepped planing craft using one transverse step. Lateral spray sheets separated from side chine is also detectable for both forebody and aft­ body wetted surfaces. Typical top-view of aftbody wetted area in case of β ¼ 30� , Ls ¼ 800 mm, Hs ¼ 20 mm, 8 ¼ 7 m/s is also presented in Fig. 13. However, necessity of comprehensive experimental tests on different planing hull forms is evident for determination of these spec­ ified intervals. As stated before, the wetted area is significantly depended on the hull

Swfore ¼ 0:5BLK

(3)

- If flow spray line (stagnation line) crossing from side chine before the step, the forebody wetted area is: Swfore ¼

9

B ðLc þ LK Þ 2

(4)

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Fig. 13. Typical formation of aftbody wetted area in case of β ¼ 30� , Ls ¼ 800 mm, Hs ¼ 20 mm, 8 ¼ 7 m/s (Image taken from top-view).

step from transom (Ls). However, lower total resistance is obtained for greater Hs and smaller Ls. Moreover, for all tested cases, total wetted areas are reduced by decreasing the Ls. In addition, increase of the Hs and hull velocity is resulted to reduce of total wetted areas. Appropri­ ately accordance is also achieved between our experimental results of reattachment length and aftbody wetted area against the extracted re­ sults from Savitsky and Morabito (2010) empirical formulation. Finally, we categorized and formulated the bottom wetted areas of planing hulls with one transverse step according to different reattached patterns of the separated flow. The present paper merits future works. Study on the wetted surfaces of two steps planing craft at different geometrical pa­ rameters of the steps can be regarded as future investigation. In addition, implementation of comprehensive experimental tests on different geometrical properties of transverse step for presentation a formulation to correlate the measure wetted surfaces with the properties of the step can be considered in future works.

here, Lc is the wetted chine length. On the other hands, there are four scenarios for Swaft according to x14 . In Table 8, in case of one step planing craft, possible scenarios of formation of aftbody wetted surfaces are categorized. In addition, aftbody wetted areas are formulated ac­ cording to the distance of reattached length of the separated flow from th e step on the line of ¼ (i.e., x14 ). =

=

4. Conclusion Stepped planing craft are considered as interesting planing hull in maritime industries. Proper knowledge on the bottom wetted surface of these craft has remarkable role to advance their design. In this study, hydrodynamic characteristics and wetted surfaces of stepped planing craft are evaluated via experimental towing tests. For this purpose, total resistance, dynamic trim angle, dynamic sinkage, quantitative parame­ ters of reattachment length of the separated flow and wetted area are measured and qualitative criteria of wetted surface pattern is investi­ gated via underwater photographs under three step heights, three dis­ tance of the step from transom, three hull velocities and two deadrise angles. According to our experimental results, dynamic trim angle, total trim angle and dynamic sinkage are increased by an enhancement in the height of the step (Hs) and by decreasing the longitudinal distance of

Conflicts of interest The authors declare that there is no conflict of interests regarding the publication of this paper.

Appendix 1 In this appendix, an uncertainty analysis is presented for all obtained experimental results according to study of Coleman and Steele (1999). To this accomplishment, uncertainty analysis is conducted for all test variables by analysis of systematic, precision and total uncertainties. Systematic un­ certainty (i.e., bias Bi that “i” is the considered parameter) is according to root sum of square (RSS) of primary error sources such as information achievement and calibration. Precision uncertainty (i.e., Pi that “i” is the considered parameter) is proportion toPj ðsÞ ¼ K:SDevj (i.e., according to Ref (Coleman and Steele, 1999), SDevj is the standard deviation of the jth run where K ¼ 2). In addition, RSS of the total precision uncertainty (PT) and the total bias uncertainty (BT) is total uncertainty (UT). As may be seen in Table A.1, systematic, precision and total uncertainties are presented for dynamic trim angle (τDTA ) and non-dimensional form of Froude number, total resistance, dynamic sinkage, reattachment length and total wetted area 1

3

2

and Swtotal =r 3 ). Based on Table A.1, proper confidential interval of about 92% is achieved. =

=

1

(i.e., Frr, RT =Δ, Zv =r 3 , xcl =r

10

=

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Table 8 Classification and formulation of aftbody wetted surfaces according to the x1

.

4

=

Corresponding Figure

Condition x1

4

=

8 > > > > > < > > > > > :

x1

4

BB @x1 4 4

B < Ls b � 2

BB B Ls b ¼ B 4 @x1 4

x1

4

< Ls b >

B 2

C xcl A

8 1> > > B > > B > > b< > > : 2

B ðLs 2

xcl Þ

Swaft ¼ bðLs

xcl Þ

Swaft ¼

1

0

=

> > > > > :

Swaft ¼

0

=

8 > > > > > <

Aftbody wetted area Formulation 1 0

¼ Ls

BB B Ls b ¼ B 4 @x1 4 1

1

0 xcl C C B CL1 ¼ 2@x1 xcl A 4

0

C xcl AL2 ¼ Ls

B 2x1 þ xcl Swaft ¼ B@Ls 4

C x1 A 4

x1

4

1

0

> Ls

BB B Ls b ¼ B 4 @x1

xcl C C CSwaft ¼ bðLs xcl A

xcl Þ

=

4

Table A.1 Uncertainty analysis on experimental data. Term

Froude number (Frr)

Dynamic trim angle (τDTA )

Frr B Frr P Frr UT %UT

τDTA B τDTA P τDTA

Case number (see Table 3) 1

2

3

4

5

6

7

8

9

10

11

3.71 0.0093 0.0200 0.0221 0.595% 2.33 0.0186 0.0291

5.30 0.0175 0.0238 0.0296 0.558% 1.48 0.0222 0.0163

5.30 0.0148 0.0355 0.0385 0.726% 1.89 0.0253 0.0529

3.71 0.0156 0.0282 0.0322 0.868% 2.79 0.0591 0.0678

4.24 0.0318 0.0280 0.0423 0.999% 2.78 0.0634 0.0828

5.30 0.0440 0.0249 0.0505 0.954% 2.02 0.0394 0.0644

3.71 0.0119 0.0197 0.0230 0.619% 1.51 0.0483 0.0491

4.24 0.0089 0.0097 0.0132 0.311% 1.29 0.0445 0.0430

3.71 0.0200 0.0282 0.0346 0.932% 1.56 0.0401 0.0328

4.24 0.0233 0.0047 0.0238 0.561% 1.79 0.0240 0.0240

5.30 0.0334 0.0154 0.0367 0.694% 1.51 0.0445 0.0483

(continued on next page)

11

A. Najafi et al.

Ocean Engineering 187 (2019) 106164

Table A.1 (continued ) Term

2

3

4

5

6

7

8

9

10

11

=

0.0346 1.484% 0.176 0.0040 0.0056 0.0069 3.941% 0.084

0.0275 1.860% 0.244 0.0060 0.0057 0.0083 3.388% 0.053

0.0587 3.104% 0.200 0.0053 0.0085 0.0100 5.002% 0.068

0.0900 3.224% 0.163 0.0051 0.0036 0.0062 3.827% 0.102

0.1043 3.752% 0.168 0.0071 0.0044 0.0084 4.986% 0.101

0.0755 3.738% 0.188 0.0081 0.0067 0.0105 5.598% 0.073

0.0689 4.560% 0.194 0.0071 0.0080 0.0107 5.503% 0.054

0.0619 4.794% 0.227 0.0076 0.0079 0.0110 4.823% 0.046

0.0518 3.318% 0.173 0.0048 0.0067 0.0083 4.789% 0.056

0.0339 1.895% 0.229 0.0051 0.0064 0.0082 3.580% 0.065

0.0657 4.352% 0.280 0.0115 0.0072 0.0135 4.839% 0.054

=

13

0.0010

0.0005

0.0016

0.0027

0.0015

0.0013

0.0017

0.0010

0.0011

0.0009

0.0007

13

0.0025

0.0014

0.0011

0.0015

0.0021

0.0023

0.0008

0.0008

0.0012

0.0014

0.0008

=

0.0027 3.194% 0.551

0.0014 2.695% 0.799

0.0019 2.835% 1.308

0.0031 3.040% 0.972

0.0026 2.582% 1.201

0.0026 3.583% 1.514

0.0019 3.453% 0.330

0.0013 2.749% 0.413

0.0016 2.843% 1.101

0.0017 2.582% 0.826

0.0011 2.061% 1.056

=

13

0.0072

0.0128

0.0190

0.0252

0.0450

0.0391

0.0154

0.0148

0.0232

0.0135

0.0282

13

0.0114

0.0236

0.0407

0.0270

0.0252

0.0283

0.0065

0.0120

0.0337

0.0267

0.0306

=

0.0135 2.444% 3.397

0.0269 3.365% 2.302

0.0449 3.431% 1.570

0.0369 3.800% 2.598

0.0516 4.298% 2.101

0.0483 3.186% 1.396

0.0167 5.050% 4.485

0.0190 4.607% 4.008

0.0409 3.717% 3.394

0.0299 3.618% 4.530

0.0416 3.942% 4.049

=

Zv =r

23

0.1107

0.1377

0.0703

0.1130

0.0557

0.0497

0.1049

0.1207

0.1324

0.2315

0.2118

23

0.1654

0.0539

0.0333

0.1553

0.0912

0.0725

0.1749

0.2445

0.1171

0.1871

0.2389

0.1991 5.860%

0.1478 6.422%

0.0778 4.956%

0.1921 7.395%

0.1068 5.085%

0.0879 6.294%

0.2040 4.548%

0.2727 6.802%

0.1767 5.207%

0.2976 6.570%

0.3193 7.884%

13

=

BZv =r

13

)

xcl =r

=

=

Reattachment length (xcl =r

PZv =r UT %UT

13

Bxcl =r

=

Total wetted area (Swtotal =r

23

Pxcl =r UT %UT )

Swtotal =r

23

BSwtotal =r PSwtotal =r UT %UT

=

)

=

13

1 UT %UT RT =Δ BRT =Δ PRT =Δ UT %UT

Total Resistance (RT =Δ)

Dynamic sinkage (Zv =r

Case number (see Table 3)

References

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