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International Journal of Naval Architecture and Ocean Engineering xx (2016) 1 — 14

http://www.journals.elsevier.com/international-journal-of-naval-architecture-and-ocean-engineering/

Experimental study on hydrodynamic coefficients for high-incidence-angle

maneuver of a submarine

Jong-Yong Parka, Nakwan Kima *, Yong-Ku Shinb

a Department of Naval Architecture and Ocean Engineering, Research Institute of Marine Systems Engineering, Seoul National University, Seoul, South Korea

b Agency for Defense Development, Changwon, South Korea

Received 30 September 2015; revised 8 August 2016; accepted 24 August 2016 Available online ■ ■ ■

Abstract

Snap rolling during hard turning and instability during emergency rising are important features of submarine operation. Hydrodynamics modeling using a high incidence flow angle is required to predict these phenomena. In the present study, a quasi-steady dynamics model of a submarine suitable for high-incidence-angle maneuvering applications is developed. To determine the hydrodynamic coefficients of the model, static tests, dynamic tests, and control surface tests were conducted in a towing tank and wind tunnel. The towing tank test is conducted utilizing a Reynolds number of 3.12 x 106, and the wind tunnel test is performed utilizing a Reynolds number of 5.11 x 106. In addition, least squares, golden section search, and surface fitting using polynomial models were used to analyze the experimental results. The obtained coefficients are presented in tabular form and can be used for various purposes such as hard turning simulation, emergency rising simulation, and controller design.

Copyright © 2016 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: High-incidence-angle maneuver; Towing tank test; Wind tunnel test; Golden section search; High-order polynomial model

1. Introduction

Submarines are required to be stable during certain maneuvers such as steady turning, depth change, and emergency rising. A method for predicting the submarine behavior during such maneuvers is thus required for the design of the vehicle. A high-incidence-angle flow acts on a submarine during hard turning or emergency rising, resulting in excessive motion response of the vehicle. This makes it difficult to use a general dynamics model, which is suitable for linear motion, to predict the behavior of a submarine during such maneuvers. The conduction of a free running test is an accurate method for predicting the motions, but it is time consuming and costly.

* Corresponding author. E-mail address: nwkim@snu.ac.kr (N. Kim).

Peer review under responsibility of Society of Naval Architects of Korea.

Alternative methods include simulation using the equations of motions and the utilization of a data base.

The simulation of submarine maneuver is generally based on Gertler and Hagen (1967)'s equations of motion. The utilized model was revised by Feldman (1979) to provide enhanced, unsteady, and nonlinear modeling for cross flow drag and sail vortex. Watt (2007) proposed an analytical method for estimating the added mass using incompressible potential flow theory and a damping term that is suitable for a high-incidence-angle flow. Many studies have been aimed at obtaining the coefficients of the maneuver model using captive model tests (Seol, 2005; Feldman, 1987, 1995; Nguyen et al., 1995; Quick et al., 2012, 2014; Roddy et al., 1995; Watt and Bohlmann, 2004). Planar Motion Mechanism (PMM) and Rotating Arm (RA) tests were generally used to determine the coefficients. The stability and control characteristics of a submarine have also been determined by RA and PMM tests

http://dx.doi.org/10.1016/j.ijnaoe.2016.08.003

2092-6782/Copyright © 2016 Society of Naval Architects of Korea. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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(Feldman, 1987, 1995; Roddy et al., 1995). Researchers at the Defense Science and Technology Organization (DSTO) experimentally tested a generic submarine model in a wind tunnel (Quick et al., 2012, 2014). The wind tunnel tests were targeted at acquiring steady state aerodynamic force and moment data and investigating the characteristics of the flow field of a submarine. The standard submarine model was developed for a series of systematic hydrodynamic experiments jointly funded by Defense R&D Canada (DRDC) and the Royal Netherlands Navy (RNLN), and has been statically tested at different facilities (Mackay, 2003). A horizontal planar motion mechanism (HPMM) test was performed in a towing tank at the Seoul National University using various depths, and the horizontal dynamic stability of the submarine was analyzed using estimated coefficients (Seol, 2005). Most studies on the design of submarine control were concerned with the maintenance of depth in waves (Dumlu and Istefanopulos, 1995; Tolliver, 1982; Choi, 2006). The adaptive controller was designed by Dumlu and Istefanopulos (1995) to operate under various sea conditions. A mathematical model has been proposed for calculating the wave forces acting on the submarine, and a controller designed by PID has also been used to confirm the possibility of control through simulation (Choi, 2006). However, most previous experimental studies involved only static tests and, to the best of the authors' knowledge, there have been none that considered pitch/yaw. Studies that have considered controller design focused on vertical motions such as heave/pitch, and the developed controllers can therefore not be used to assure performance during 6-DOF motion.

This paper proposes a quasi-steady submarine dynamics model that is suitable for high-incidence-angle maneuvers such as hard turning and emergency rising. The model tests were conducted in the towing tank of Seoul National University and in the wind tunnel at the Agency for Defense Development. The goal of these tests was to obtain the hydrodynamic coefficients of the model. The towing tank test was conducted at a Reynolds number of 3.12 x 106, and the wind tunnel test was performed

at a Reynolds number of 5.11 x 106. Resistance tests, static tests, and dynamic tests were conducted in the towing tank with a 1.3-m model by using the static test device and PMM device. The static test device was used to hold the model at a steady heading angle when resistance tests and static b tests were performed. The static a test was also conducted with the static

test device by rotating the submarine model by 90° onto its side. The static tests were conducted using angles ranging between —20° and +20°. The dynamic tests were comprised of the HPMM and vertical planar motion mechanism (VPMM) tests, which were performed with the PMM device. The PMM device only allowed the model to oscillate in the sway and yaw directions, and the VPMM was conducted by rotating the submarine by 90°. Because of the general complexity of the experiment and alignment issues related to the size of the model, the control surface efficiency tests could not be conducted in the towing tank. The static a test, the static b test, the combined a/b test, and the control surface efficiency tests were all performed in the wind tunnel with a 1.92-m model. The resistance tests were used to measure the surge force at various speeds. Owing to the difficulty of obtaining uniform flow at low speeds in a wind tunnel, the resistance tests were conducted in only the towing tank. A three-axis potentiometer was used for the alignment of the combined a/b tests. Only the wind tunnel was equipped with the angle measurement system, and the combined a/b test could only be performed in the wind tunnel. Angles of attack ranging from —30° to +30° and drift angles ranging from —24° to +24° are used for the static tests in the wind tunnel. The static a tests and static b tests results were used to validate the results of each facility test.

The list of towing tank and wind tunnel tests are given in Table 1.

2. Mathematical model

2.1. Overview

In this section, the equations of motion of a submarine are described. The coordinate system used in this study is shown in Fig. 1.

The origin of the body-fixed coordinates is located at the midship on the centerline. The six degrees of freedom equations of motion can be derived by Newton's second law, and they can be expressed as shown below:

The equations assume that the submarine mass and mass distribution do not change with time. The terms on the right-hand side of Eq. (1) represent the external forces acting on the submarine. In this study, the following modular-type mathematical model is used to represent the external forces.

m[u — vr + wq — xG (q2 + r2) + yG (pq — r) +Zo(pr + q) = X m[v — wp + ur — yG\r2 + p2) + ZG(qr — p) +XG(qp + r) = Y m[W — uq + vp — Zg (p2 + q2) + XG(rp — q) + yG(rq + p)] = Z

Ixp + {h — Iy)qr — Ixz(r + pq)+Iyz( r2 — q2) + Ixy (pr — q) +m[yG (W — uq + vp)—ZG(v — wp + ur)} = K Iyq +(Ix — Iz)rp — Iyx(p + qr)+Izx (p2 — r2 ) + Iyz(qp — r) +m[zG(u — vr + wq) — xg (W — uq + vp)} = M Izr + (Iy — Ix)pq — Izy(q + rp)+Ixy(q2 — p2) + Izx(rq — p) +m[xG (v — wp + ur) —yG(u — vr + wq)} = N

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Fig. 1. Coordinate system.

F =[X Y Z K M N]T = Fhs + FHD + Fp + FC (2) The modular-type mathematical model is expressed in terms of the hydrostatic force, hydrodynamic force acting on the hull, the propeller force, and the control surface force. The submarine control surface system consists of sail planes, stern planes, and rudders. The deflection angles of the sail planes, stern planes, and rudders are denoted by db, ds, and dr, respectively.

angle Q and flow orientation F, which are not limited to a 2D plane. A hydrodynamic coefficient Fuvw(Q,F) is also adopted to address the coupled large angle of attack and large drift angle. The coefficient contains all the effects of the translation motion, and this makes it easier to obtain the coefficient from experimental data compared to using the Feldman (1979) model. An intuitive understanding of (Q,F) is difficult; hence, in the present study, Fuvw(Q,F) was replaced by Fuv-w(a,b) to simplify the modeling. Based on the work of Gertler and Hagen (1967), and using a procedure similar to that of Watt (2007) to consider the effect of the high angle of attack, an external force model appropriate for high-incidence-angle maneuver was established. The model is a quasi-steady dynamics model. The nonlinear cross flow drag effect from rotational motion and unsteady viscous effects are neglected in the model. The mathematical model is as follows: * Surge:

X = Xhs + Xc + Xp + |L2U2[X'(a,b)] + |l4 \x'qqq2 + X'r/2

+ K ' rp

xuu + X'vrvr + X'wqwq] + 2l2 [Xuu'u2]

Xhs = - ( W - B)sin6, Xc = §L2 u2 [X^r^r ¿2 + Xdsdsds2 + X'sbsbô:

2.2. Dynamics modeling for submarine

Advanced analytical studies have been conducted to establish the dynamics model of a submarine for high-incidence-angle maneuver. The external force acting on the submarine was modeled by Gertler and Hagen (1967), Feldman (1979), and Watt (2007). Gertler and Hagen (1967) proposed coefficient-based standard submarine equations of motion. The equations of motion consist of linear and nonlinear coefficients, which are assumed to be constant, and can be obtained from the captive model tests. Although the model is suitable for describing standard mild maneuvering, it is not sufficiently accurate for simulating high-incidence-angle maneuver such as hard turning and emergency rising maneuvers. Feldman (1979) modified the Gertler and Hagen (1967) model by adopting crossflow drag and sail vortex to provide improved unsteady and nonlinear motion characteristics. In a maneuver, the local velocity varies along the hull owing to the angular velocity of the submarine. Feldman (1979) accounts for this effect by using the cross flow drag model. In a turning maneuver, the vorticity shed from the sail induces lift on the hull and appendage, which has significant effects on submarine depth keeping. The unsteady viscous effect is included with the sail vortex model in the maneuvering equations of motion. The modified model is, however, too complex for determining the hydrodynamic coefficients of the crossflow drag and sail vortex using experimental results. Watt (2007) proposed a model of the external force that is suitable for emergency rising. In his work, he defined the flow incidence

Yhs + Yc + |l2U2[Y'(a,b)] + rL4 Y'/ + Y'pP + Yp,p,p|p|

+ Y'pqpq + Y'qrqr

+ Y'U + Y'pup + YV|r|vjrj

2 L r ' p ' *pjpj

YvV + YVqVq + YWpWp + YWrwr

Yhs = +(W - B)cosd sin f, Yc = ^V^ôr + Ydr|dr|¿rjdrj] * Heave:

Z = Zhs + Zc + ^L2U [Z'(a

l+ 2l4

Zq q + Z'ppp2+z'''2

Z'quq+ZWyw|q|+ ZVpvp

Zhs = +(W - B)cos6 cos f, Zc = 2l2u2[Z'Sbôb +

Zdb|<5b|db|db|+ z'st(a ds)]

* Roll:

K = Khs + Kc + Kp + PL3U2 [K'(a, b)] + PL5 [K'_p + K'r

+Kq+K'pqpq+KP|p|Pjpj

+KVqvq+KWpwp+KW'Wr

Kpup + K'ur + K'vv

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Khs = (ygw — yBB)cos q cos f — (zgW — zbB)cos q sin f, Kc = rl\2[k'&rôr + krNdr\ôr|] * Pitch: 2

M = Mhs + Mc + pL3U2 [M'(a, b)] + pL5 \^qfl + M'ppp2

+ K/ + M'rpTp + M'q , q , q I q I + M'vpvp + M'quq + I w I q

V., w + M' tVT

MHS = (xGW - xBB)cos q cos f - (zGW - zBB)sin q Mc = 2L3u2[Mdbdb + Mdb|db Idb I + M'ST(a, ds)]

N = Nhs + Nc + pL3U2 [N(a, b)] + 2L5 [Nr + N'pp + ^pq NVv + N' up + NT ur + N' Iv I r

NHS = — xBB)cos 6 sin f + (yGW — yBB)sin6,

Nc = 2i2"2[Nd r dr + Nd r | dr | dr | dr | ]

Z'ST (a, ds) and M'ST (a, ds) represent the stern plane force and moment, respectively.

3. Experiment

Experiments were performed to determine the values of the hydrodynamic coefficients in Eqs. (3)—(8). The test submarine had a cruciform tail planes and sail planes. The length per diameter, L/D, of the submarine was 10.9, and the leading edge of the sail was located at 0.748 L from the AP. The experiments were performed in a towing tank and wind tunnel. The laboratory of each experiment was equipped with a sting-type balance, which was used as a measurement sensor. It was difficult to conduct a control surface test and combined a/b static test in the towing tank, and these tests were conducted in only the wind tunnel. From the wind tunnel test results, the coefficients related to the control surfaces and Fuvw(a, b) were determined. The resistance tests, which comprised HPMM and VPMM tests, were conducted in the towing tank. It is also important that the Reynolds number be sufficiently high to avoid any significant scale effects. This number is approximately 15 million, based on the overall length of the hull (Feldman, 1995). Because of the allowable ranges of model size and overall speed in the laboratories, the experiments are

Table 2

Comparison of towing tank and wind tunnel test conditions.

Parameter Description Wind tunnel Towing tank

L Model length (m) 1.92 1.30

U Overall speed (m/s) 40 2.4

p Fluid density (kg/m3) 1.225 998.2

V Fluid dynamic viscosity (m2/s) 1.50E-05 1.00E-06

Rn Reynolds number 5.11E+06 3.12E+06

Fn Froude number 9.21 0.67

conducted at a lower Reynolds number than the recommended value. Regarding the wind tunnel tests, the model length was 1.92 m and the wind speed was 40 m/s and the Reynolds number was 5.11 x 106. In the case of the towing tank tests, the model length was 1.30 m and the towing speed was 2.4 m/s and the Reynolds number was 3.12 x 106. The wind tunnel and towing tank tests were conducted for different Reynolds numbers, and it was therefore necessary to validate the results of each test. Additional static b test (static drift test) and static a test (static angle of attack test) were conducted in the towing tank to compare their results with those of the same tests conducted in the wind tunnel. The submarine test conditions are given in Table 2.

3.1. Towing tank test

To measure the forces and moments acting on the submarine model, a sting-type balance was attached to the inside of the model. The origin of the body-fixed coordinates was set at the midship on the centerline. An adapter was fabricated to match the rotation center of the PMM device (or static test device) and the body-fixed coordinate origin and the

Fig. 2. Captive model test system of the towing tank.

Table 1

List of towing tank and wind tunnel tests.

Test list Reynolds number Resistance test Static test Dynamic test Control surface efficiency test

Static a test Static b test Combined test HPMM VPMM Sail plane test Stern plane test Rudder test

Towing tank 3.12 x 106 O O O X O O X X X

Wind tunnel 5.11 x 106 X O O O X X O O O

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measurement center of the balance. The captive model test system of the towing tank is illustrated in Fig. 2.

The shaft of the PMM and static test device could move in the vertical direction, and this enabled its being used to determine the submerged depth of the submarine model. The dimensions of the towing tank are 8 (width) x 3.5 (depth) x 110 m (length). Seol (2005) conducted the PMM tests of a submarine with various depths. According to the experimental results, the effect of depth on the forces and moments is negligible in H/D = 3.5 or more of the depth condition, where H is the depth and D is the diameter of the submarine model. The model was tested in H/D = 6.0 to avoid the free surface effect. The value of distance/D from the towing tank bottom to the model is 22.83. The Reynolds number of the test was less than that of the real submarine maneuver, and a turbulence stimulator was could be fitted to enable a more realistic boundary layer development and pressure distribution along the hull (ITTC, 2002). Studs, wires, and a sand strip were used to form the turbulence stimulator in the towing tank. The employed sand strips were in accordance with ITTC recommendations (ITTC, 2002), comprising adhesive strips of width 10 mm covered with sharp-edged sand of grain size of approximately 0.50 mm. The sand strips were placed about 5% Lpp aft of the FP, and were applied to the control surfaces and the sail.

3.2. Resistance test

The resistance test was used to measure the surge force for various towing speeds, namely, 0.4, 0.6, 0.8, 1, 1.2, 1.44, 1.6, 2, and 2.4 m/s. There was only surge velocity during the test and the hydrodynamic force that acted on the submarine model can therefore be expressed as follows.

Xhd = Xuuu2 (9)

The coefficients Xuu can be determined by least squares. The measurement data and curve fitting results are shown in Fig. 3.

-4 -5 -6

1 1 __ ■

■ E «périment tting

U [m/s]

Fig. 3. Resistance test result.

3.3. Static test

Static a and static b tests using angles ranging between —20° and +20° were conducted in the towing tank to validate the results of the wind tunnel tests. A towing speed of 2.4 m/s was applied. The results of the static b tests are shown in Fig. 4.

The static a tests and VPMM tests were conducted by rotating the submarine by 90°. It was difficult to align the submarine model at 90° using a digital inclinometer because there was no flat surface on the side of the model. A laser alignment equipment was therefore used to set the model vertically. The results of the static a tests are shown in Fig. 5.

It can be observed from Figs. 4 and 5 that the results of the static tests can be fitted to a cubic curve as suggested by Gertler and Hagen (1967). The test results were used to verify those of the towing tank and wind tunnel tests.

3.4. Planar motion mechanism

The PMM is a device developed for the measurement of rotary derivatives and added mass in a long and narrow towing tank (Lewis, 1988). The PMM device at the Seoul National University has one strut that can induce sinusoidal sway and yaw motion. Sway and yaw motions always have a phase shift of 90° relative to the strut. The specifications of the PMM device used in the towing tank are given in Table 3.

If the maximum sway amplitude is denoted by y0, the maximum yaw amplitude by j0, and the frequency by u, the way and yaw motions induced by the PMM device are as follows:

y = y0 sin ut j = j0 cos ut

The sway velocity v, sway acceleration v, yaw rate r, and yaw angular acceleration r can be expressed as follows:

v = (y0u)cos ut, V = — (yoto2)sinut r = — (j0u)sinut, r = — (j0u2)cosut

The HPMM tests consisted of pure sway, pure yaw, and combined sway/yaw tests. The added masses Yv, Kv, and Nv related to the sway acceleration were determined by the pure sway test, and the related yaw coefficients Yr, Yr, ..., Nr, Nr were determined by the pure yaw test. The sway/yaw combined coefficients such as Yv|r|, N|v|r were obtained by the combined sway/yaw test. The VPMM test was conducted by rotating the submarine by 90°. The test conditions are given in Table 4.

The results of the pure sway test for a forced oscillatory frequency of 0.1 Hz are here used as sample PMM test results. The y are shown in Fig. 6.

As can be observed, the measured force and moments are sinusoidal, and they contain structural noise induced by the towing carriage. The nonlinear curve fitting method is thus required. To explain the nonlinear curve fitting method, the sway force measurement data will be used as an example. The sway force data can be fitted to a sinusoidal form as follows:

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Fig. 4. Static b test results in the towing tank.

-10 0 10 20 -20 -10 0 10 20 a [deg] a [deg]

Fig. 5. Static a test results in the towing tank.

Table 3

Specifications of the PMM device.

Parameter

Maximum sway amplitude (m) Maximum yaw amplitude (°) Frequency range (Hz) Motor power (kW) Motor torque (kgm) Maximum weight (kg)

Ymea(t) = Ay sin(uyt + £y)

0.5 40

0.05—2.00 5.5 2.9 650

where AY, uY, and eY are the amplitude, frequency and phase of the sway force data, respectively. Eq. (12) can be expressed as Eq. (13) by application of trigonometric function theory.

Table 4

Test matrix of the PMM.

HPMM Pure sway Maximum sway amplitude (m) 0.475

Maximum yaw amplitude (°) 0.0 0.0 0.0

Frequency (Hz) 0.1 0.15 0.2

Pure yaw Maximum sway amplitude (m) 0.475

Maximum yaw amplitude (°) 7.1 10.7 14.2

Frequency (Hz) 0.1 0.15 0.2

Combined Maximum sway amplitude (m) 0.475

sway/yaw Maximum yaw amplitude (°) 10.0 13.7 17.3

Frequency (Hz) 0.1 0.15 0.2

VPMM Pure heave Maximum heave amplitude (m) 0.475

Maximum pitch amplitude (°) 0.0 0.0 0.0

Frequency (Hz) 0.1 0.15 0.2

Pure pitch Maximum heave amplitude (m) 0.475

Maximum pitch amplitude (°) 7.1 10.7 14.2

Frequency (Hz) 0.1 0.15 0.2

Combined Maximum heave amplitude (m) 0.475

heave/pitch Maximum pitch amplitude (°) 10.0 13.7 17.3

Frequency (Hz) 0.1 0.15 0.2

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50 0 -50

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

time(sec)

5 10 15 20

time(sec)

Fig. 6. Example of the PMM test measurement data (Pure sway, Freq = 0.1 Hz).

raw data

KS**** ¿»■JL. -------fitting X*/

0 5 10 15 20 25

1 ftlhii ! j

0 5 10 15 20 25

time(sec)

0.5 0.4 -0.3 -0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

y(m) v(m/s) v dot(m/s2)

\ i j \ V ,

time(sec)

Fig. 7. Fitting results of the PMM test measurement data.

Ay sin(uYt + £Y) = Ay cos £Y sin(uYt) + Ay sin £Y cos(uYt) = Ci sin(uYt) + C2 cos(uYi), where C1 = Ay cos £Y, C2 = Ay sin £Y

If the frequency u is known, C1 and C2 can be obtained by least squares. The amplitude and phase can then be determined using the following interaction formula:

Ay = C2 + C2, £Y = tan—1(C2/C1) (14)

Generally, sinusoidal data fitting is done by fast Fourier transform (FFT). FFT involves integration over different periods, and its results would not be accurate if the measurement data do not contain sufficient periods. Owing to the limited length of the towing tank, it was difficult to obtain data for more than three periods under the low frequency test conditions. Hence, the golden section search method was used in this study to search the frequency of the data. Fitting results obtained by this procedure are shown in Fig. 7.

The dynamic test results can be analyzed by two approaches: the least square method and the Fourier integral (FI) method. The FI method is widely used in dynamic test analyses of linear models. This method also separates the measured motion and force signals to the in-phase and out-of-phase terms to obtain the added mass and damping term. However, nonlinear coefficients or coupled coefficients cannot be obtained by the FI method. Although the added mass can be deducted by the FI method, the least square method is employed in this research for the consistency of the analysis. The fitted measurement data contains the inertia force induced by the motions of the submarine model, and it is necessary to remove the inertia force in order to determine the hydrody-namic force.

FhD = F mea + FI (15)

The pure sway test only involves u, v, and v in pure sway test. From Eq. (4), Eq. (6), and Eq. (8), the hydrodynamic force acting on the submarine model can be obtained as follows:

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Yhd = Y (a, b) + YvV

Khd = K(a, b)+KvV (16)

Nhd = N(a, b)+Nvv

The added mass related to the sway acceleration can be obtained by subtracting F(a,b) in the left-hand side and applying least squares. In the pure yaw test, there is no sway motion, and the hydrodynamic force can therefore be expressed as

Yhd = Yr r + Yu

KHD = Kr r + Krur (17)

Nhd = Nr r + Nrur

The values of the coefficients in Eq. (17) can be determined by least squares. The coefficients Yvjrj and Njvjr can also be determined from the results of the combined sway/yaw tests. The coefficients determined from the results of the pure sway and pure yaw tests were used to analyze the combined sway/ yaw test results using Eq. (18).

Yv|r|v|r| = Yhd — (Y(a, b) + YvV + Yrr + Yrur) Nv|r|V | r| =Nhd — (N(a,b)+Nvv + Nrr + Nrur) ( 8)

As is generally done in analyzing the results of a HPMM test, the coefficients related to the heave/pitch were obtained from the results of the VPMM test.

3.5. Wind tunnel test

The wind tunnel tests were performed in the low-speed wind tunnel at the Agency for Defense Development. The low-speed wind tunnel is a closed return-type wind tunnel, and the contraction area ratio is 1:9. The dimensions of the test section are 2.7 (width) x 3 (height) x 8.75 m (length). The value of distance/D from the wind tunnel wall to the model is 7.16. Dynamic pressure variation induced by the wind tunnel wall effect is corrected by considering solid and wake blockage. The model is supported by a crescent sting. The

40 r 30 -

-30----------[----------[----------[----------f-------------------f----------}----------}----------f----------i

_4Q_I_I_I_I_I_I_I_I_I_I

-25 -20 -15 -10 -5 0 5 10 15 20 25

p(deg)

Fig. 8. Test case of the combined a/b static test in wind tunnel.

cavity pressure between the model and sting is measured using a Barocell pressure transducer for drag correction. The model used for the wind tunnel tests was 1.91 m long, and the wind speed was 40 m/s. It was difficult to conduct the combined a/b static tests with high-incidence angles and control surface tests in the towing tank; thus, the tests are performed in the wind tunnel. Dynamic tests cannot be conducted in the wind tunnel because oscillation frequencies must be scaled up to match the wind speed.

3.6. Static test

The translation damping coefficients are denoted by Xvv, Yv, and Z„ in the Gertler model. Although the model is simple and suitable for general maneuver, there is a limiting to fitting its experiment data for high incidence angles in a combined a/b area. To address this issue, the coefficients X(a, b), Y(a, b), ■ ■•, N(a, b) were adopted based on the work of Watt (2007), and static tests for angles of attack ranging between —30° and +30° and drift angles ranging between —24° and +24° were conducted to determine their values. The measurement data was fitted by high-order polynomial model. The test cases are shown in Fig. 8.

The submarine was symmetrical in the x—z plane, and the first quadrant data could therefore be expanded to the second quadrant, and the third quadrant data could be expanded to the fourth quadrant. The test results are shown in Fig. 9.

The values of X(a, b), Y(a, b), ■■■, N(a, b) in equations (3)—(8) can be determined from the results of the static tests. Surface fitting is necessary for modeling based on the results of the combined a/b static test. In this paper, the high-order polynomial fitting is used to fit the surface. The structure and order of the model is determined by considering the inherent oddness and evenness of the characteristics it is fitting, and the polynomial model of X(a, b), Y(a, b), ■ ■■, N(a, b) can be expressed as follows:

X'(a, b)=X02b2 + X04b4 + X10a + xnab1 + X14 ab4 + x20a2 + x22a2b2 + x30a3 + x32a3 b2 + x40 a4

Y'(a, b)=y01b + y03b3 + y05b5 + yuab + y^b3 + y21a2 b + y23 a2b3 + y31a3b1

Z'(a, b) = Z00 + Z02b2 + Z04b4 + Z10a + z^ab2 + Z20a2

+ Z22a2b2 + Z30a3 + Z32a3b2 + Z40a4 + Z50a5 (21)

K'(a, b) = k01b + k03b3 + ^05b5 + knab + k13ab3 + k21a2b

+ k23a2b3 + k3ia3b

+ MODEL

J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14

m ; ■ ■

■. ■"„■■ ■ : : .■ ..

■■VBlV" ■■ ■ V

• JWJI■■■ i.f

. , . -20 -20 a(deg) ß(deg)

, . , -20 -20 a(deg) ß(deg)

a(deg)

ß(deg)

V : W'-

. . . -20 -20 a(deg) ß(deg)

» « , -20 -20 a(deg) ß(deg)

! ■ : " ■ ' ■ I

■ . ii j vV^vAi/^.---.-'wL-:1'1 • ►

. -20 -20 a(deg) ß(deg)

Fig. 9. Combined a/ß static test results.

M'(a, b) = m00 + m02b2 + m04b4 + m10a + muab2 + m20a2 + m22a2 b2 + m30a3 + m32a3 b2 + m40a4 + m50a5

N'(a, b) = n01b + n03b3 + n05b5 + nnab + n^afi3 + n2ia2b + n23 a2b3 + n31a3b:

The hydrodynamic coefficients x02, x04, ..., n41 are determined by the least squares method and are listed in Table A.1 and A.2. The fitting results comparisons of Gertler and Hagen (1967) model and modified model are performed to confirm the effectiveness of the own dynamics model. The translation damping of the Gertler and Hagen (1967) model is expressed as terms of translation velocities such as u, v and w. The damping term can be expressed as function of a, b by non-dimensionalize the translation velocities. The surface fitting results of Gertler and Hagen (1967) model and own dynamics model are compared in Fig. 10.

The degree of the fitting level between the two different models is evaluated using coefficient of determination R2. The coefficient of determination of each motion mode is listed in Table 5.

There are only a negligible difference between the two model in sway and roll, but it seems large difference in the other motion mode. It can be known that the Gertler and

Hagen (1967) model is not sufficient to reflect the complex phenomenon occurred in high incidence angle range.

3.7. Control surface efficiency test

The control surface efficiency tests were conducted to measure the force and moment acting on the submarine body in the presence of a control surface deflection angle. The tests were used to evaluate the effectiveness of the control system of the submarine, comprising the sail planes, stern planes, and rudders. The sail plane test was conducted using sail plane deflection angles ranging from —30° to 30° at zero hull incidence. Because a submarine is bilaterally symmetrical, the rudder test was conducted using rudder deflection angles ranging from —5° to 30° at zero hull incidence. The coefficients related to the control surfaces, such as Xdbdb, XgrSi, Ndr, could be determined by the control surfaces tests. Fig. 11 shows the sail plane and rudder efficiency tests results.

The lift of the sail plane was smaller than that of the other control surfaces. It was located near the midship of the submarine, and this caused the pitch moment induced by the sail plane to be small and tending to be rough. The upper rudder was larger than the lower rudder, and this resulted in the inducement of an asymmetrical roll moment by the rudder deflection angle. This effect is reflected by the terms Kdr and Ksrjdri terms. The stern plane test was conducted using stern

+ MODEL

10 J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14

Fig. 10. Fitting results comparison.

Table 5

Comparison of the fitting level between the Gertler and Hagen (1967) model and the modified model.

Motion mode Coefficient of determination

Gertler model Modified model

Surge 0.3053 0.7332

Sway 0.9765 0.9908

Heave 0.9656 0.9937

Roll 0.9913 0.9931

Pitch 0.9200 0.9712

Yaw 0.9050 0.9737

plane deflection angles ranging from —30° to 30° with various angles of attack. The angle of attack range is —30° to 30°.

Part (a) of Fig. 12 shows the stern plane efficiency test results. The hydrodynamic forces induced by the angle of

attack without stern plane deflection is already considered by equations (19)—(24). Data compensation is required to remove the effect of angle of attack and to deduct the hydrodynamic force induced by the control surface only. The compensated data were obtained by subtracting the value at stern plane deflection angle ds = 0 from the test results and they are shown in part (b) of Fig. 12. The third-order polynomial model was chosen for the sail plane force model and is expressed as follows:

Zst'(a, ds) = ZSsds + ZdAd2s + Z&sd3s + ZaS,ads + ZaSAad2s

Zaa5s a ds

+ MODEL

J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14

Rudder test

X -1.5

x 10"3 Sail plane test ■ ■ Experiment - Fitting

-1 ■ + ■

.5 ^_____"""

-30 -20 -10

-30 -20 -10

■30 -20 -10 xlO"3

-30 -20 -10

¿b(deg)

-30 -20 -10 0 10

<5r(deg)

Fig. 11. Control surface efficiency test results for the sail plane and rudder.

Stern plane test

Stem plane test

0.02-1 0.01 N 0$ -0.01 -0.02

T ▼ Y

< a=-30°

■ »=-24°

+ a=-16°

♦ a=-8°

X a=-4°

* a=0°

* a=4°

▼ a=8°

* a=16°

• a=24°

A a=30°

•< a=-30°

■ a=-24°

+ a=-16°

♦ a=-8°

X a—4°

* a=0°

* 0=4"

▼ a=B°

* a=16°

• «-24°

A a=30°

.X10"3

-20 -10

-30 -20 -10 0 10 20 30

„ x103_

-30 -20 -10 0 10 20 30

x * " Y

♦ x * *

. ♦ X *

+ + ♦ x

• + ♦

' I . +

-30 -20 -10 0 10 20 30

(a) Measured data

(b) Compensated data

Fig. 12. Stern plane efficiency test results.

MsT (a, 8s) = MSs8, + M&A8] + M&AAd3 + MaSsa8,

+ MaSsSs a8] + MaaS, a2 8, (26)

The fitting results using equation (25) - (26) are shown in Fig. 13.

3.8. Validation of test results

The dynamics model of a submarine that is suitable for high-angle-of-attack maneuver was developed in this study. Towing tank and wind tunnel tests were conducted to determine the coefficients of the model. The sizes of the model,

+ MODEL

J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14

0.02 0.015 0.01 0.005 N 0 -0.005 -0.01 -0.015

-0.02 -40

Fig. 13. Stern plane model fitting results.

\ — Towing Tank ™ ■ Wind Tunnel

0 20 a (deg)

Fig. 14. Static a test results comparison of towing tank and wind tunnel.

Towing Tank 5 ———■ Wind Tunnel

-0.03L

-40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20

P (deg)

ß (deg)

p (deg)

Fig. 15. Static b test results comparison of towing tank and wind tunnel.

+ MODEL

J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14 13

fluid characteristics, and Reynolds numbers used in the two different test facilities were different. Validation of the results was thus necessary to ensure their continuity for application to simulation. Static a and static b tests were conduct in both facilities and the respective results were thus used for the validation. The results of these tests are compared in Figs. 14 and 15.

From Fig. 14, it can be seen that most parts of the heave forces in both facilities were similar, although there were some differences between the pitch moments for high incidence angles. This indicates that the lifts were comparable but there were differences between the longitudinal centers of pressure for high incidence angles. Nevertheless, good agreement can be observed between the results of the static b tests conducted in the towing tank and wind tunnel.

4. Conclusions

In this study, a submarine dynamics model for high-incidence-angle maneuver was developed using the models previously developed by Gertler and Hagen (1967), Feldman (1979), and Watt (2007). The hydrodynamics coefficients of the model were determined by conducting towing tank and wind tunnel tests. A turbulence stimulator was applied to the submarine model to obtain a more realistic boundary layer development and pressure distribution. Resistance, static, HPMM, and VPMM tests were conducted in the towing tank, and control surface and combined a/b static tests were conducted in the wind tunnel. The PMM test results were fitted to a sinusoidal form using the methods of golden section search and least squares, and the fitted data were used to determine the coefficients related to the angular motion. The results of the combined a/b static tests were fitted by polynomial surface fitting. A high order polynomial model was sufficient for the surface fitting of the other results of the combined a/b static tests. The control force and moment exhibited stall for high deflection angles, and a second order odd function was therefore used to fit the data. The results of the static tests conducted in the two different test facilities were compared and good agreement was observed. The obtained modeling and hydrodynamics coefficients are useful

for the simulation of submarine maneuvers such as hard turning and emergency rising.

Acknowledgments

This research is funded by Agency for Defense Development (UE120033DD).

Appendix 1. (Notation)

u,v,w surge, sway and heave velocities

p, q, r roll, pitch and yaw angular velocities

U = j u2 + v2 + w2 overall speed

a = tan- 1(w/U ) angle of attack

b = tan- 1(-v/U ) drift angle

I moment of inertia

XB,yB,ZB coordinates of center of buoyancy

Xo,yo,ZG coordinates of center of gravity

db, is, dr sail plane, stern plane and rudder deflection

& = tan- 1 (Pv2 + w2 /u) flow incidence, always positive

F = tan- 1 (v/w) flow orientation

X,Y,Z surge, sway and heave forces

K,M,N roll, pitch and yaw moments

<p,6,j Roll, pitch and yaw Euler angles

L Submarine length

D Submarine diameter

p fluid density

V fluid dynamic viscosity

Rn Reynolds number

Fn Froude number

x,y,z body-fixed coordinate

x0y0,z0 space-fixed coordinate

m mass

B buoyancy

W weight

Appendix 2. (Coefficients)

The coefficients of the hydrodynamics and dynamics model of the submarine determined by the tests are presented in Tables A.1 and A.2, where WTT indicates wind tunnel test.

Table A.1

Coefficients of the submarine model (surge, sway and heave)

Method

Method

Method

X'uu -1.264E-03 Resistance Yv -9.079E-03 HPMM zw -1.2471E-02 VPMM

Xibib -1.855E-03 WTT Yv -5.305E-03 HPMM z'i 1.9781E-03 VPMM

Xisis -1.855E-03 WTT Y'r 4.598E-03 HPMM zq -5.6362E-03 VPMM

Xirir -2.438E-03 WTT y'M -4.053E-02 HPMM Z' zwq Z'b -1.8093E-02 VPMM

Y'sr 6.923E-03 WTT -4.152E-03 WTT

Yir 1 ir 1 -5.343E-03 WTT Zib 1 ib 1 6.823E-03 WTT

14 J.-Y. Park et al. / International Journal of Naval Architecture and Ocean Engineering xx (2016) 1—14

Table A.2

Coefficients of the submarine model (roll, pitch and yaw)

Roll Pitch Yaw

Item Value Method Item Value Method Item Value Method

KV —4.366E-04 HPMM MW —8.6710E-04 VPMM N —8.156E-04 HPMM

KV —2.294E-04 HPMM M —2.3490E-03 VPMM N —6.214E-03 HPMM

K 1.707E-04 HPMM M —3.1774E-03 VPMM K —3.435E-03 HPMM

Kr 1.306E-04 WTT M'rr — 1.168E-04 HPMM N'w —7.391E-03 HPMM

KSr\Sr\ — 1.952E-04 WTT Kq — 1.2611E-02 VPMM K —2.727E-03 WTT

Mdb 4.018E-04 WTT N'r\'r\ 1.906E-03 WTT

M'\b\ —4.546E-04 WTT

Table A.3

Coefficients of translation damping (surge, sway and heave)

X{a, ß)' Y {a, ß)' Z{a, ß)'

Item Value Item Value Item Value

X02 1.392E-02 y01 3.166E-02 Z00 —5.132E-04

x04 —6.057E-02 >03 2.522E-01 z02 9.301E-02

X10 —4.478E-05 y 05 —4.938E-01 Z04 —3.716E-01

x12 4.300E-02 y 11 9.876E-02 z10 —2.110E-02

X14 — 1.643E-01 y 13 —3.623E-01 Z12 1.960E-01

x20 1.125E-03 y21 3.778E-01 z14 —4.936E-01

x22 —5.852E-02 y23 — 1.708E+00 Z20 —6.342E-03

x30 — 1.164E-03 y31 —4.792E-01 z22 —5.984E-01

x32 — 1.471E-01 Z30 —7.791E-02

x40 —2.152E-03 z30 —5.052E-01

x50 1.516E-03 Z40 2.611E-02

z50 9.999E-02

Table A.4

Coefficients of translation damping (roll, pitch and yaw)

K(a,ß)' M(a,ß)' N(a,ß)'

Item Value Item Value Item Value

k)1 7.684E-04 m00 —8.566E-05 n01 1.080E-02

k03 5.282E-03 m02 1.329E-02 n03 — 1.090E-02

k05 —2.011E-02 m04 —7.239E-02 n05 2.612E-02

ku —3.498E-03 mm 4.295E-03 n11 —8.682E-03

k13 5.812E-03 m12 6.138E-02 n13 1.256E-02

k21 4.725E-03 m14 —2.889E-01 n21 —7.522E-02

k23 —3.856E-02 m20 —3.629E-03 n23 4.467E-01

k31 7.684E-04 m22 — 1.339E-01 n31 1.428E-01

m30 — 1.625E-02

m32 —2.948E-01

m40 1.175E-02

m50 3.417E-02

Table A.5

Coefficients of the stern plane

Z'CT{a, 5s)' m'ct {a> 5s)'

Item Value Item Value

zl —6.328E-03 M's —2.913E-03

Z0 — 1.197E-04 M's's —5.042E-05

Z0 Zèsèsès 5.625E-03 Misisis 2.760E-03

Z0 —2.159E-04 Ma's —4.641E-05

Za's's 5.260E-03 MaSsSs 2.424E-03

Z0 aa's 1.642E-03 Maa's 4.876E-04

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