1
Low Speed Cou se-Keeping and Zigzag Manoeu es o he
KSUPRAMAX Bulk Ca ie in Wa es
Dong-Jin Kim
*
, Kunhang Yun, Chang-Seop Kwon, Yeon-Gyu Kim, Seung-Hyun Hwang
Ko ea Resea ch Ins i u e o Ships and Ocean Enginee ing, Daejeon, Republic o KOREA
Abs ac . Manoeu ing pe o mances o ships in wa es a e signi ican ly di e en om hose in calm wa e .
Reliable p edic ion echniques on he ship’s manoeu abili ies in wa es a e equi ed a he ea ly design s age
o i s sa e ope a ions in ac ual seas. In pa icula , a en ions should be paid o he ship’s cou se-keeping and
changing abili ies which become mo e impo an in ough wa es. In his s udy, cou se-keeping and zigzag
manoeu es o a ‘KSUPRAMAX’ model ship which has a sup amax class bulk ca ie hull o m a e
expe imen ally in es iga ed. The ull-scale ship leng h is 192 m, a 1/64-scaled model ship is cons uc ed o
ee- unning model es s in Ocean Enginee ing Basin o Ko ea Resea ch Ins i u e o Ships and Ocean
Enginee ing (KRISO). A i s , u ning ci cle es s a e epea edly conduc ed in long-c es ed i egula wa es
wi h he model p opulsion poin co esponded o he ull-scale design speed o 14.5 kno s. Equi alen egula
wa es ha show simila u ning ci cle ajec o ies and app oach speeds compa ed o i egula wa es a e
sea ched. Nex , cou se-keeping es s a e pe o med in i egula and equi alen egula wa es wi h wa e
inciden angles o 0 o 180 deg ees a speeds lowe han 5 kno s. Low speed cou se-keeping pa hs, check helms,
and d i angles in equi alen egula wa es a e close o hose in i egula wa es. 3-DoF manoeu ing
simula ion models a e cons uc ed based on cap i e model es s and empi ical me hods. Since he wa e d i
o ces a e equal o sums o hyd odynmaic o ces ac ing on he hull, he p opelle , and he udde du ing cou se-
keeping manoeu es, wa e d i o ces a e iden i ied om cou se-keeping simula ions. Zigzag es s a e also
conduc ed in equi alen egula wa es wi h app oaching condi ions which a e he same as hose in cou se-
keeping es s, zigzag beha io s a e simula ed by he p esen models including wa e d i o ces iden i ied.
Cou se-keeping and changing abili ies a e discussed wi h conside a ions o wa e d i o ces which a e a ied
wi h wa e inciden angles.
Keywo ds: KSPURAMAX bulk ca ie , F ee- unning model es , Cou se-keeping abili y, Zigzag manoeu e,
Wa e d i o ce
1 In oduc ion
Ship’s manoeu abili ies a e con en ionally e alua ed in calm wa e , which a e di e en om hose in a
seaway due o en i onmen al loads. A an ea ly design s age, i is necessa y o p edic manoeu ing pe o mance
o a ship in wa es as well as in calm wa e o i s sa e ope a ions in seas. Mo eo e , ins alled engine powe s o
new-buil ships a e ecen ly dec eased o mee EEDI equi emen s which aim o educe g eenhouse gas emissions.
A en ions should be paid o check whe he ships ha e su icien manoeu abili ies o ope a ing in ad e se
wea he condi ions.
In ad e se condi ions, ship ad ance speeds a e dec eased, manoeu abili ies o ships a e ema kably di e en
om hose in calm wa e . Since ships should o e come iolen en i onmen al loads and keep o change hei
cou ses by hei own s ee ing sys ems, cou se-keeping and changing abili ies o ships a e o pa icula impo ance
in such si ua ions. As in con en ional IMO s anda d manoeu ing es s, cou se-keeping and changing abili ies o
ships can be e alua ed hough zigzag manoeu ing pe o mances. In addi ion, in ad e se wea he condi ion unde
s ong en i onmen al loads, asymme ic beha io s o ships such as hull d i s and check helms o keeping s aigh
cou ses need o be in es iga ed in de ail.
The e a e some p e ious s udies on cou se–keeping o zigzag abili ies o ships in wa es h ough expe imen al
o nume ical echniques. Hi ayama and Kim (1994) pe o med zigzag model es s o a anke in long and sho
c es i egula wa es gene a ed in a owing ank, simula ions by manoeu ing models including wa e d i o ces
a e compa ed wi h model es esul s [1]. Yasukawa and Adnan (2006) expe imen ally in es iga ed mo ions and
wa e d i o ces o a con aine model in egula head and beam wa es, e ec s o d i angles on wa e d i o ces
*
Co espondence o: djkim@k iso. e.k
16 h In e na ional Symposium on P ac ical Design o Ships and O he Floa ing S uc u es PRADS 202 5
Ann A bo , MI, USA, Oc obe 19 h – 23 d 2025
2
we e discussed, Yasukawa (2008) alida ed zigzag and s opping manoeu ing simula ions h ough compa ison
wi h ee- unning model es s [2][3]. In a esea ch by Sp enge e al. (2017), ee- unning u n and zigzag model
es s o a KVLCC2 anke and a DTC con aine we e pe o med in ou Eu opean ins i u es and we e analyzed
wi hin he amewo k o SHOPERA p ojec [4]. Zhang e al. (2017) ca ied ou wo- ime scale simula ions on u n
and zigzag manoeu es o a S175 con aine in egula wa es and compa ed hem wi h model es esul s [5]. Wang
e al. (2018) in es iga ed s anda d zigzag manoeu ing cha ac e is ics o an ONRT model in egula wa es by
using CFD sol e [6]. Sanada e al. (2018) ca ied ou cou se-keeping and manoeu ing model es s o an ONRT
in a ious egula wa es, in pa icula , ime-a e aged and ha monic ampli udes o ship mo ions and udde angles
du ing cou se-keeping es s a e analyzed in a ious wa e inciden angles [7]. Kim e al. (2022) pe o med ee-
unning CFD calcula ions o a KCS model and analyzed low-speed cou se-keeping and u ning abili ies o he
model in di e en wa e inciden angles [8]. Suzuki e al. (2023, 2024) ca ied ou ee- unning cou se-keeping
model es s o a KVLCC1 anke in egula sho wa es and compa ed hem wi h nume ical simula ions
conside ing s eady wa e d i o ces and momen s ob ained by cap i e model es s and CFD [9][10].
In his s udy, cou se-keeping and zigzag abili ies o a ‘KSUPRAMAX’ bulk ca ie in wa es a e in es iga ed
by ee- unning model es s. A i s , 35° u ning ci cle es s a e ca ied ou in long-c es ed i egula and a ious
egula wa es. The signi ican heigh and he peak pe iod o long-c es ed i egula wa es a e selec ed wi h
conside a ion o ad e se wea he condi ions de ined in an IMO minimum p opulsion powe guideline [11].
Equi alen egula wa es a e de e mined so ha model ship app oach speeds and ajec o ies in i egula wa es
a e simila o hose in egula wa es. I is also con i med ha low speed cou se-keeping abili ies depending on
wa e inciden angles in i egula wa es a e simila o hose in equi alen egula wa es. By using s eady s a e
a iables o he hull, he p opelle , and he udde in cou se-keeping es s, wa e d i o ces and momen s
depending on inciden angles a e iden i ied h ough modula ype manoeu ing models. Low speed 10°/10°
s a boa d and po zigzag es s a e conduc ed in equi alen egula wa es wi h a ious inciden angles. Zigzag
abili ies in wa es a e signi ican ly di e en om hose in calm wa e , and a e a ied depending on wa e inciden
angles, i.e. wa e d i o ces and momen s. Zigzag beha io s depending on wa e inciden angles a e analyzed wi h
conside a ions o wa e d i o ces iden i ied.
2 F ee- unning model es
2.1 Model ship and es basin
F ee- unning model es s a e ca ied ou wi h a 1/64-scaled model ship o ‘KSUPRAMAX’ in his s udy. The
ull-scale ship is a 66,000 DWT sup amax bulk ca ie , which we e used in ecen CFD o model es s udies on
mainly i s added esis ances in wa es [12][13][14]. Main pa icula s o ull-scale and model ships a e summa ized
in Table 1. Figu e 1 shows he body plan and he scaled model ship cons uc ed. The ship geome y and es da a
can be downloaded om KRISO websi e [15], o he wise, can be p o ided ia he co esponding au ho ’s email.
Table 1. Main pa icula s o ull-scale and model ships (design d a )
Pa icula
Full-scale
Model
Scale a io (λ)
1
64
Leng h be ween pe pendicula s (L) [m]
192.0
3.000
B ead h (B) [m]
36.0
0.563
Fo e / a d a ( TF / TA) [m]
11.2 / 11.2
0.175 / 0.175
Displacemen (∇) [m3]
65028.2
0.248
LCB ( wd, +) [m]
5.773
0.090
Yaw adius o gy a ion (kzz) [L]
0.250
0.238
Me acen ic heigh (GM) [m]
-
0.080
P opelle diame e (DP) [m]
6.0
0.094
Rudde la e al a ea (AR) [m2]
45.91
0.0112
3
Figu e 1. Body plan (le ) and 1/64-scaled model ship ( igh ) o KSUPRAMAX bulk ca ie
F ee- unning model es s a e conduc ed in Ocean Enginee ing Basin o Ko ea Resea ch Ins i u e o Ships and
Ocean Enginee ing (KRISO) as desc ibed in Figu e 2. The basin is 56 m in leng h, 30 m in beam, and 4.5 m in
dep h. Wa e make s a e ins alled along sho and long ends o he basin. Fo p esen model es s, egula and long-
c es ed i egula wa es a e gene a ed by sho end wa e make s. Ship mo ions, p opulsion and s ee ing signals a e
measu ed by onboa d PC moun ed on he model, and all synch onized signals a e sen o he g ound con ol PC
wi elessly. Th ee p isms a e ixed on he model deck. They a e aced by h ee o al s a ion de ices on he g ound
so ha he 3-dimensional ship posi ions can be measu ed au oma ically, which ha e been upda ed compa ed o he
p e ious 2-dimensional measu emen sys em [16][17]. Figu e 3 shows he coo dina e sys em used in he p esen
s udy. O-XY is a space- ixed coo dina e, X-coo dina e is se owa d he model ship heading when each es s a s.
o-xy is a body- ixed coo dina e, whose o igin is loca ed on he midship o he ship.
Figu e 2. F ee- unning model es in KRISO Ocean Enginee ing Basin
Figu e 3. Coo dina e sys em
4
2.2 Tes p og am
F ee- unning u ning ci cle and zigzag es s a e ca ied ou in calm wa e a he model p opulsion poin s o 19.0
and 8.0 RPS which co espond o he ull-scale speeds o 14.5 (design speed) and 5.0 kno s, espec i ely. 35°
po side u ning ci cle es s a e conduc ed in long-c es ed i egula wa es and se e al kinds o egula wa es a
19.0 RPS. Based on abo e u ning ci cle es esul s, one speci ic egula wa e condi ion is selec ed. A he
p opulsion poin o 8.0 RPS, cou se-keeping and 10/10 zigzag es s a e pe o med in i egula and selec ed egula
wa es wi h a ying wa e inciden angles. To al p og am is summa ized in Table 2.
Table 2. F ee- unning model es p og am (SB: s a boa d, PS: po side)
3 Sea ch o ‘equi alen egula wa es’ co esponding o he ad e se condi ion
3.1 Tu ning ci cle es s a 19.0 RPS (co . ull-scale 14.5 kno s in calm wa e )
To main ain he manoeu abili y o ships in ad e se wea he condi ions, IMO in e im guideline o de e mining
minimum p opulsion powe o ships was p oposed [11]. In he guideline, ad e se wea he condi ion is de ined as
Table 3. Fo he p esen ull-scale KSUPRAMAX bulk ca ie , he signi ican wa e heigh and peak pe iod can be
de e mined as Table 4. Wind loads a e no conside ed in his s udy.
Table 3. Ad e se wea he condi ion (IMO, 2021)
Tes
Wa e
RPS [/s]
Wa e inciden angle [°]
(e.g. 180°: head wa e)
35° SB, PS Tu n
Calm wa e
19.0
(co . 14.5 kno s)
-
35° SB, PS Tu n
10°/10° SB, PS Zigzag
8.0
(co . 5.0 kno s)
35° PS Tu n
I egula
(Hs= 4.5m, Tp=12.0s)
19.0
180
Regula
(H= 2.589~4.500 m,
T= 8.520~12.000 s)
Cou se-keeping
I egula
(Hs= 4.5m, Tp=12.0s)
8.0
180, 150, 120, 90, 60, 30, 0
Regula
(H= 2.878 m, T= 9.780 s)
10°/10° SB, PS Zigzag
I egula
(Hs= 4.5m, Tp=12.0s)
8.0
180, 150, 90, 30, 0
Regula
(H= 2.878 m, T= 9.780 s)
Ship leng h [m]
Signi ican wa e heigh [m]
Peak wa e pe iod [s]
Mean wind speed [m/s]
L > 250
6.0
7.0 ~ 15.0
22.6
200 ≤ L ≤ 250
Linea ly in e pola ed depending on ship leng h
L < 200
4.5
7.0 ~ 15.0
19.0
5
Table 4. Long-c es ed i egula wa e condi ion in he p esen s udy
No.
App oach
speed [kno s]
𝐻𝐷 720−360
[L]
𝜇𝐷 720−360
[°]
1
11.5
2.19
30
2
11.1
2.11
37
3
11.2
1.78
19
4
11.1
1.94
27
5
11.4
1.84
30
Mean
11.3
1.97
29
S d. de .
0.2
0.18
6
Figu e 4. 35° PS u ns in i egula wa es (le ) and de ini ion o d i ing dis ance and angle (SIMMAN,2023) ( igh )
A i s , he model p opulsion poin is ixed o 19.0 RPS which co esponds o he ull-scale 14.5 kno s in calm
wa e , 35° po side u ning ci cle es s a e ca ied ou in long-c es ed i egula wa es de e mined in Table 4. Fi e
epe i ion es esul s a e analyzed in Figu e 4. Due o he wa e d i o ces and momen s, u ning ci cle ajec o ies
a e d i ed in wa es. De ini ion o ajec o y d i ing dis ance and angle a e desc ibed on he igh o Figu e 4 [18].
S a is ically, he p esen model ship is d i ed app oxima ely wo imes ship leng h du ing a 360 ° u n in i egula
wa es. The app oach speed is 14.5 kno s in calm wa e , whe eas he mean alue o app oach speeds is educed by
11.3 kno s in i egula wa es.
Wa e ene gy lux pe uni c es leng h (𝑃𝑊 [W/m]) a e heo e ically o mula ed in bo h egula and long-
c es ed i egula wa es, espec i ely, which a e shown in Eqs. (1) and (2) [19]. 𝜌 is he densi y o wa e , 𝑔 is he
g a i a ional accele a ion, 𝜁𝑎 is he ampli ude o he inciden wa e, 𝑐𝐺 is he g oup eloci y. 𝑓 is he wa e
equency, 𝑆(𝑓) is he spec al densi y unc ion o inciden i egula wa es. 𝐻𝑆 and 𝑇𝐸 a e he signi ican wa e
heigh and he ene gy pe iod, espec i ely. 𝑇𝐸 is de ined as Eq. (3) [20].
𝑃𝑊(𝑟𝑒𝑔𝑢𝑙𝑎𝑟)=1
2𝜌𝑔𝜁𝑎2𝑐𝐺=1
32𝜋𝜌𝑔2𝐻2𝑇
(1)
𝑃𝑊(𝑖𝑟𝑟𝑒𝑔𝑢𝑙𝑎𝑟)=𝜌𝑔∫𝑐𝐺(𝑓)𝑆𝑖(𝑓)𝑑𝑓
∞
0=1
64𝜌𝑔2𝐻𝑆2𝑇𝐸
(2)
Signi ican wa e heigh [m]
Peak wa e pe iod [s]
4.5
12.0
6
𝑇𝐸=𝑚−1/𝑚0 , 𝑚𝑛= ∫𝑓𝑛𝑆𝑖(𝑓)𝑑𝑓
∞
0
(3)
When Eq. (1) equals o Eq. (2), ‘equi alen egula wa es’ whose ene gy lux is iden ical o ha o long-c es ed
i egula wa es a e es ima ed as Table 5.
Table 5. Equi alen egula wa e condi ion
In o de o in es iga e he model ship ope a ed in he equi alen egula wa es in Table 5 shows manoeu ing
pe o mances which a e simila o hose in long-c es ed i egula wa es, 35° u ning ci cle es s a e ca ied ou in
a ious egula wa es as desc ibed in Table 6. ‘REG03’ is he es ima ed condi ion in Table 5. Regula wa e heigh s
a e a ied om 𝐻𝑆/2 o 𝐻𝑆 , pe iods a e changed o 𝑇𝑍, 𝑇𝑀, and 𝑇𝑃, espec i ely. 𝑇𝑍, 𝑇𝑀, and 𝑇𝑃 a e ze o-c ossing,
mean, and peak pe iods on i egula wa es ene gy spec um.
Resul s o 35° u ning ci cle es s a 19.0 RPS in egula wa es a e analyzed in Figu e 5. In he iewpoin o
wa e ene gy, ship app oach speeds and ajec o y d i ing dis ances a e ocused on. Bo h a e deeply ela ed o
wa e d i o ces and momen s in longi udinal and la e al di ec ions. REG03 shows u n ajec o y d i ing dis ance
which is simila o ha in i egula wa es as an icipa ed, bu he ship app oach speed in REG03 is lowe han ha
in i egula wa es. The added esis ance in REG03 is expec ed o be la ge han ha in i egula wa es due o
la ge ela i e ship mo ions, al hough wa e ene gies in bo h wa es a e heo e ically iden ical o each o he .
Table 6. Regula wa e condi ions (REG01 o REG06)
Figu e 5. App oach speeds (le ) and ajec o y d i ing dis ances ( igh ) in REG01 o REG06
In e es ingly, ajec o y d i ing dis ances in egula wa es a e p opo ional o wa e slopes. To main ain he
ajec o y d i ing dis ance, and o inc ease he ship app oach speed, REG07 and REG08 a e addi ionally gene a ed
as shown in Table 7. Wa e slopes o REG07 and REG08 a e he same as ha o REG03.
Regula wa e heigh , 𝐻
Regula wa e pe iod, 𝑇
𝐻=𝐻𝑆/√2
𝑇=𝑇𝐸
Pe iod
Heigh
𝑇𝑍 (8.520s)
𝑇𝑀 (9.276s)
𝑇𝐸 (10.284s)
𝑇𝑃 (12.000s)
𝐻𝑆 / 2
(2.250m)
-
-
REG05
-
𝐻𝑆 / √2
(3.182m)
REG01
REG02
REG03
REG04
𝐻𝑆
(4.500m)
-
-
REG06
-
7
Table 7. Addi ional egula wa e condi ions (REG07 and REG08)
Figu e 6. App oach speeds (le ) and ajec o y d i ing dis ances ( igh ) in REG07 and REG08 as well as REG03
Resul s o app oach speeds and ajec o y d i ing dis ances a e desc ibed in Figu e 6. Bo h he app oach speed
and he ajec o y d i ing dis ance in ‘REG07’ a e simila o hose in i egula wa es. The e o e, ‘REG07’ is
de e mined as he equi alen egula wa es. Tu ning ajec o ies in i egula and egula (REG07) wa es a e shown
in Figu e 7.
I egula wa es
(𝐻𝑆= 4.5m, 𝑇𝑃= 12.0s)
Regula wa es, REG07
(H= 2.878m, T= 9.780s)
Figu e 7. 35° PS u ns in i egula wa es (le ) and in egula wa es REG07 ( igh )
Pe iod
Heigh
𝑇𝑀 (9.276s)
(𝑇𝐸+𝑇𝑀)/2 (9.780s)
𝑇𝐸 (10.284s)
(2.589m)
REG08
-
-
(2.878m)
-
REG07
-
𝐻𝑆 / √2
(3.182m)
-
-
REG03
8
3.2 Cou se-keeping manoeu es a 8.0 RPS in REG07 (co . ull-scale 5.0 kno s in calm wa e )
Figu e 8. T ajec o ies, check helms, d i angles, and ship speeds o cou se-keeping es s in i egula wa es (le ) and in egula
wa es REG07 ( igh )
9
In i egula wa es and equi alen egula wa es p e iously de e mined, low speed cou se-keeping es s a e
conduc ed wi h a ying wa e inciden angles. Model p opelle o a ion a e is ixed o 8.0 RPS, which co esponds
o he ull-scale 5.0 kno s in calm wa e . Wa e inciden angles a e a ied om 180° o 0° wi h he in e al o 30°.
Figu e 8 shows cou se-keeping ajec o ies, ime a e aged check helms, d i angles, and ship speeds, which a e
also in oduced in he p e ious s udy by Kim e al. (2024) [21]. Low speed cou se-keeping manoeu es in
equi alen egula wa es a e conside ably simila o hose in i egula wa es, al hough ime a e aged alues in
i egula wa es a e qui e a ied depending on he encoun e ed wa e ele a ions.
4 Iden i ica ion o wa e d i o ces and momen s
4.1 3-DoF manoeu ing simula ion model
To iden i y wa e d i la e al o ces and momen s depending on wa e inciden angles, 3-DoF modula ype
manoeu ing models a e used. Ho izon al plane manoeu es o he ship a e ep esen ed by Eq. (1). Subsc ip H,
P, and R deno e o ce and momen componen s ac ing on he hull, he p opelle , and he udde .
𝑚(𝑢−𝑣𝑟−𝑥𝐺𝑟2) =𝑋𝐻+𝑋𝑃+𝑋𝑅
𝑚(𝑣+𝑢𝑟+𝑥𝐺𝑟) =𝑌𝐻+𝑌𝑅
𝐼𝑧𝑧𝑟+𝑚𝑥𝐺(𝑣+𝑢𝑟)=𝑁𝐻+𝑁𝑅
(1)
Hull, p opelle , and udde o ces and momen s can be o mula ed as Eqs. (2) o (4).
𝑋𝐻=𝑋𝑢𝑢+𝑋𝑢𝑢𝑢2+𝑋𝑣𝑣𝑣2+𝑋𝑣𝑟𝑣𝑟+𝑋𝑟𝑟𝑟2+𝑋𝑣𝑣𝑣𝑣𝑣4
𝑌𝐻=𝑌𝑣𝑣+𝑌𝑟𝑟+𝑌𝑣𝑣+𝑌𝑟𝑟+𝑌𝑣𝑣𝑣𝑣3+𝑌𝑟𝑟𝑟𝑟3+𝑌𝑣𝑣𝑟𝑣2𝑟+𝑌𝑣𝑟𝑟𝑣𝑟2
𝑁𝐻=𝑁𝑣𝑣+𝑁𝑟𝑟+𝑁𝑣𝑣+𝑁𝑟𝑟+𝑁𝑣𝑣𝑣𝑣3+𝑁𝑟𝑟𝑟𝑟3+𝑁𝑣𝑣𝑟𝑣2𝑟+𝑁𝑣𝑟𝑟𝑣𝑟2
(2)
𝑋𝑃=(1−𝑡)𝜌𝑛2𝐷𝑃
4∙𝐾𝑇
(3)
𝑋𝑅=−(1−𝑡𝑅)𝐹𝑁sin𝛿
𝑌𝑅= −(1+𝑎𝐻)𝐹𝑁cos𝛿
𝑁𝑅=−(𝑥𝑅+𝑎𝐻𝑥𝐻)𝐹𝑁cos𝛿
(4)
In Eqs. (3) and (4), de ailed cha ac e is ics o he p opelle and he udde a e ob ained by Eqs. (5) and (6).
𝐾𝑇=𝑓(𝐽)
𝐽= 𝑢𝑃
𝑛𝐷𝑃
𝑢𝑃=(1−𝑤𝑃)𝑢 , 𝑤𝑃=𝑤·exp (−𝐶𝑃𝛽𝑃
2) , 𝛽𝑃=𝛽−𝑥𝑃′𝑟′
( 𝐶𝑃= 𝐶𝑃+ 𝑤ℎ𝑒𝑛 𝛽𝑃>0, 𝐶𝑃= 𝐶𝑃− 𝑤ℎ𝑒𝑛 𝛽𝑃<0 )
(5)
𝐹𝑁=1
2𝜌𝐴𝑅𝑈𝑅𝑓𝛼sin𝛼𝑅
UR=√𝑢𝑅+𝑣𝑅, αR=𝛿− an−1(𝑣𝑅/𝑢𝑅)
𝑢𝑅=𝜖𝑢𝑃√𝜂[1+𝜅(√1+8𝐾𝑇
𝜋𝐽2−1)]2+(1−𝜂)
𝑣𝑅=𝛾𝑅(𝑣+𝑙𝑅𝑟) , 𝛽𝑅=𝛽−𝑙𝑅′𝑟′
( 𝛾𝑅= 𝛾𝑅+ 𝑤ℎ𝑒𝑛 𝛽𝑅>0, 𝛾𝑅= 𝛾𝑅− 𝑤ℎ𝑒𝑛 𝛽𝑅<0 )
(7)
Hyd odynamic coe icien s in Eqs. (2) o (7) a e ob ained based on 1/26.087-scaled s a ic HPMM es s [22]
and empi ical o mulas. P esen simula ion models can p edic calm wa e u n and zigzag beha io s o he model
ship wi h high accu acy as shown in Figu e 9.
16
Subs i u ing iden i ied wa e d i o ces and momen s in o manoeu ing models, zigzag manoeu es
a e simula ed in di e en wa e inciden angles. Zigzag simula ions a e in good ag eemen s wi h ee-
unning model es s.
Acknowledgmen s
This esea ch was suppo ed by a g an om Ko ea Resea ch Ins i u e o Ships and Ocean Enginee ing
Endowmen P ojec o “De elopmen o e alua ion echnology o ship’s pe o mance in ex eme en i onmen ”
unded by Minis y o Oceans and Fishe ies, Republic o Ko ea (PES5461).
Re e ences
[1] T. Hi ayama and S.E. Kim. Manoeu abili y o a ull ship in di ec ional spec um wa es o sho wa e leng h. Jou nal o
he Socie y o Na al A chi ec s o Japan, 176:129-136, 1994.
[2] H. Yasukawa and F.A. Adnan. Expe imen al S udy on Wa e-Induced Mo ions and S eady D i Fo ces o an Obliquely
Mo ing Ship. Jou nal o he Japan Socie y o Na al A chi ec s and Ocean Enginee s, 3:133-138, 2006.
[3] H. Yasukawa. Simula ions o Ship Maneu e ing in Wa es (2nd epo : zig-zag and s opping maneu e s). Jou nal o he
Japan Socie y o Na al A chi ec s and Ocean Enginee s, 7:163-170, 2008.
[4] F. Sp enge , A. Ma on, G. Dele o ie, T. an Zwijns oo de, A. Cu a-Hochbaum, A. Lengwina and A. Papanikolaou.
Expe imen al S udies on Seakeeping and Maneu e abili y o Ships in Ad e se Wea he Condi ions. Jou nal o Ship
Resea ch, 61(3):131-152, 2017.
[5] W. Zhang, Z.J. Zou and D.H. Deng. A s udy on p edic ion o ship maneu e ing in egula wa es. Ocean Enginee ing,
137:367-381, 2017.
[6] J. Wang, L. Zou and D. Wan. Nume ical simula ions o zigzag maneu e o ee unning ship in wa es by RANS-O e se
g id me hod. Ocean Enginee ing, 162:55-79, 2018.
[7] Y. Sanada, H. Elshiekh, Y. Toda and F. S e n. ONR Tumblehome cou se keeping and maneu e ing in calm wa e and
wa es. Jou nal o Ma ine Science and Technology, 24(3):948-967, 2018.
[8] D. Kim, J. Yim, S. Song, Y.K. Demi el and T. Tezdogan. A sys ema ic in es iga ion on he manoeu ing pe o mance o
a ship pe o ming low-speed manoeu es in ad e se wea he condi ions using CFD. Ocean Enginee ing, 263, 112364,
2022.
[9] R. Suzuki, Y. Tsukada and M. Ueno. E ec s o s eady wa es o ces on cou se-keeping manoeu es o ull- and model-
scale ships mo ing obliquely in sho wa es. Ship Technology Resea ch, 70(3):190-208, 2023.
[10] R. Suzuki, K. Ohashi and M. Ueno. Es ima ion o s eady wa e o ces and momen ac ing on an obliquely mo ing ship in
sho wa es, and i s applica ion in a manoeu ing simula ion. Applied Ma hema ical Modelling, 125:261-292, 2024.
[11] IMO. Guidelines o de e mining minimum p opulsion powe o main ain he manoeu abili y o ships in ad e se
condi ions. MEPC.1/Ci c.850/Re .3, 2021.
[12] C.M. Lee, S.C. Pa k, J.W. Yu, J.E. Choi and I. Lee. E ec s o di ac ion in egula head wa es on added esis ance and
wake using CFD. In e na ional Jou nal o Na al A chi ec u e and Ocean Enginee ing, 11:736-749, 2019.
[13] J.H. Lee, Y. Kim, B.S. Kim and F. Ge ha d . Compa a i e s udy on analysis me hods o added esis ance o ou ships in
head and oblique wa es. Ocean Enginee ing, 236, 109552, 2021.
[14] G.H. Kim, S. Hwang, S.H. Lee, J.H. Lee, J. Hwangbo, K.S. Kim and K.J. Paik. A Nume ical S udy on he Pe o mance
o he 66k DWT Bulk Ca ie in Regula and I egula Wa es. Jou nal o Ma ine Science and Enginee ing, 11, 1913,
2023.
[15] KRISO websi e. Resea ch - KRISO s anda d ships da a. h ps://www.k iso. e.k /menu.es?mid=a20206000000
[16] D.J. Kim, K. Yun, D.J. Yeo and Y.G. Kim. Ini ial and s eady u ning cha ac e is ics o KCS in egula wa es. Applied
Ocean Resea ch, 105, 102421, 2020.
[17] D.J. Kim, H. Choi, K. Yun, D.J. Yeo and Y.G. Kim. Expe imen al s udy on u ning cha ac e is ics o KVLCC2 anke in
long-c es ed i egula wa es. Ocean Enginee ing, 244, 110362, 2022.
[18] SIMMAN. Final epo o SIMMAN2020 wo kshop – Volume III: Analysis o ajec o y submissions.
h p://simman2020.k , 2023.
[19] ITTC. Wa e Ene gy Con e e Model Tes Expe imen s. Recommended P ocedu es and Guidelines, 7.5-02-07-03.7, 2021.
[20] ITTC. The Specialis Commi ee on Wa es – Final Repo and Recommenda ions o he 23 d ITTC. P oceedings o he
23 d ITTC, 2003.
[21] D.J. Kim, K. Yun, C.S. Kwon, Y.G. Kim and S.H. Hwang. Fo wa d Speeds and Tu ning T ajec o ies o a KSUPRAMAX
Model Ship in Long-C es ed I egula and Equi alen Regula Wa es. Jou nal o he Socie y o Na al A chi ec s o Ko ea,
61(4):258-266, 2024. (in Ko ean)
[22] ITTC. Repo o he Wind Powe ed and Wind Assis ed Ships Commi ee. P oceedings o he 30 h ITTC, 50-53, 2024.
