novelmodelformanoeuvrabilityofshipsadvancingin landslide...

Research ArticleNovel Model for Manoeuvrability of Ships Advancing inLandslide-Generated Tsunamis

Peiyin Yuan 1 Pingyi Wang2 and Yu Zhao23

1College of Shipping and Naval Architecture Chongqing Jiaotong University Chongqing 400074 China2Collegeh of River and Ocean Engineering Chongqing Jiaotong University Chongqing 400074 China3College of Architecture and Urban Planning Chongqing Jiaotong University Chongqing 400074 China

Correspondence should be addressed to Peiyin Yuan yuan_pei_yin163com

Received 1 June 2020 Revised 29 July 2020 Accepted 14 August 2020 Published 28 August 2020

Academic Editor Chiara Bedon

Copyright copy 2020 Peiyin Yuan et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)e rock and soil on the shore of the bank are unsteady and slide in a poor environment affecting the water body in the riverchannel and forming landslide-generated tsunamis )is directly impacts the navigation of vessels in the river In this studythe river course and sailing ships in the Wanzhou section of the )ree Gorges Reservoir area were taken as the researchobjects )rough a physical model test with a large scale ratio the variation of the water level at the monitoring points in thechannel was determined and the variation law of the water level in the whole channel was derived and converted into aprototype through the scale ratio A model of the shiprsquos manoeuvring motion was established and the shiprsquos manoeuvringmotion characteristics in still water were verified )e correlations between the maximum roll angle and the navigationposition sailing speed and rudder angle were investigated in detail A safety risk response theory of navigation in the area oflandslide-generated tsunamis was proposed and a scientific basis was provided for the safe navigation of ships in the )reeGorges Reservoir area

1 Introduction

Under the influence of strong earthquakes heavy rainand changes in the reservoir water level the rock and soilof reservoirs are prone to landslides and the movementof landslides into river channels at high speeds causeslandslide-generated tsunamis which propagate along theupper and lower reaches of the river affecting hydraulicstructures passing ships and residents of coastal areas

Scholars have used theoretical analyses model testmethods and numerical simulation methods to study thepropagation of landslide-generated tsunamis and thecharacteristics of ship manoeuvring motions To studylandslide-generated tsunamis via model tests experts haveconducted two-dimensional (2D) test methods Fritz [1]performed preliminary experiments involving a falling blockand a wedge block )e wave generation between the twomethods differs because the sliding wedge does not produce

the reverse separation vortex Watts et al [2] performedexperiments on submarine landslides using a semiellipsoidand compared the experimental results with the depth-av-eraged nonlinear shallow water wave equations Expertshave also used three-dimensional (3D) testing methodsPanizzo et al [3] performed 3D block slide experiments in a6m-wide 12m-long and 08m-deep wave basin )erectangular block slide was released at the end of the basinadjacent to the side wall assuming the symmetry of the wavepropagation Di Risio et al [4] used half an ellipsoid-shapedblock to model a landslide in a 55m-wide 18m-deep and108m-long wave basin Experts have compared the char-acteristics of landslide-generated tsunami propagation in 2Dand 3D tests Heller et al [5] Bruggemann [6] and Hellerand Spinneken [7] obtained the initial high-speed calcula-tion formula and the law of landslide-generated tsunamis

)e uncertainty and propagation characteristics oflandslide-generated tsunamis are difficult to measure and

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8897202 15 pageshttpsdoiorg10115520208897202

scholars have used different numerical methods to study thecharacteristics of landslide-generated tsunamis Yavari-Ramshe and Ataie-Ashtiani [8] presented a new landslide-generated model based on incompressible Euler equationsA two-layer model was developed that included a layer ofgranular-type flow beneath a layer of inviscid flow Ataie-Ashtiani and Yavari-Ramshe [9] estimated the impact oflandslide-generated waves using a 2D fourth-order Bous-sinesq-type numerical model Ruffini et al [10] focused onnumerical landslide tsunami propagation in the far field toquantify the effect of the water-body geometry Kelfoun et al[11] used a new two-fluid (seawater and landslide) numericalmodel to estimate the wave amplitudes and the propagationof tsunamis associated with landslide events on ReunionIsland Farhadi [12] performed a numerical simulation of thegravity currents of non-Newtonian fluids via the incom-pressible smoothed particle hydrodynamics approachCecioni and Bellotti [13] presented a depth-integrated nu-merical model for the simulation of the generation andpropagation of tsunamis due to submerged landslides

)emovement of ships on waves has been investigated vianumerical simulations and model tests Hirdaris et al [14]reviewed some of the recent advances in the assessment ofloads for ships and offshore structures with the aim of pre-senting the overall technological landscape for further in-vestigation validation and implementation by the academicand industrial communities Sasa et al [15] performed anumerical simulation of ship motions by using a coastalnetwork wave database Some experts focused on themovement characteristics of the ship in its six degrees offreedom Kianejad et al [16] performed numerical and ex-perimental simulations to examine the effects of differentwave heights and wave frequencies on the ship motioncharacteristics Piscopo et al [17] examined the heave andpitch motion time histories via a time-domain simulationaccording to theoretical wave spectra Scholars have alsostudied the manipulation of ships Seo and Kim [18] per-formed a numerical analysis of the ship manoeuvring per-formance in the presence of incident waves and the resultantshipmotion responses Szlapczynski et al [19] used amodel ofthe ship dynamics to assess the time and distance necessaryfor amanoeuvre to avoid domain violations Lee andKim [20]considered the effects of the steady flow approximation in theanalysis of ship manoeuvring in waves Owing to the strongnonlinear effect of landslide surges the traditional linearpotential flow theory is not accurate Scholars such as Yuanet al [21 22] employed engineering examples and the or-thogonalmodel test method to study the navigational safety ofships in landslide-generated tsunami waters

In summary scholars have performed considerable researchon the propagation characteristics of landslide-generated tsu-namis the laws of shipmovement and the shipmanoeuvres viavarious methods such as theoretical analysis numerical sim-ulation field observation and indoor experiments Howeverfew studies have been performed on the direct development oflandslide-generated tsunamis and ship motion and there hasbeen even less theoretical research on the effect of landslide-generated tsunamis on the ship )erefore according to thefour-degree-of-freedom motion equation of ships and the

characteristics of landslide-generated tsunami propagation thispaper presents simulation procedures for studying the effects oflandslide-generated tsunamis on ship manoeuvring along withscientific suggestions and theoretical support for the naviga-tional safety of ships in landslide-generated tsunami waters

2 Mathematical Model of ShipManipulation Motion

21 Coordinate Systems To facilitate the study of themovement of the hull in the water two right-handed coor-dinate systems are selected )e first is the inertial coordinatesystem fixed to the Earthrsquos surface (o0 minus x0y0z0) and z0 )eshiprsquos transverse velocity along the Y-axis is denoted as v theforward angular velocity around the z-axis is denoted as r andthe roll angular velocity around theX-axis is denoted asp)eexternal force and external torque suffered by the ship in themanoeuvring motion coordinate system can be expressed interms of the longitudinal force x along the X-axis transverseforce y along the Y-axis foreword moment N around the z-axis and roll moment K around the X-axis

)e main characteristic of the MMG model is applied tothe hydrodynamic force and moment on the ship in accor-dance with the physical meaning and is decomposed into thework on the naked hull the propeller open water and thehydrodynamic force and moment on the open-water rudderas well as the mutual interference between the hydrodynamicforce and moment )e MMG model is based on a deep andextensive experimental study together with theoretical anal-ysis and is a popular international mathematical model forship motion It has high reliability (Figures 1 and 2)

Relationship between the physical quantities in theprocess of ship manipulation

_xGO μ cos ϕ minus ] sinϕ

_yGO μ sinϕ + ] cosϕ

_ϕ r

_φ p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where ϕ represents the direction angle and φ represents theroll angle

22 Equation of Ship Manipulation Motion )e shiprsquosmovement in the water is rigid-body movement )e mass isdenoted as m According to the law of rigid weight heartmovement Newtonrsquos law and the momentum theorem thefollowing can be obtained

m euroxGO X0

m euroyGO Y0

Izzeuroϕ N0

Ixxeuroφ K0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

where Izz represents the moment of inertia of the hullagainst the X-axis and Ixx represents the moment of inertiaof the hull against the Y-axis )e derivatives of both sides ofequation (2) can be obtained as follows

2 Advances in Civil Engineering

euroxGO _μ cos ϕ minus _] sinϕ minus (μ sinϕ + ] cos _ϕ) _ϕ

yGO _μ sinϕ + _] cos ϕ +(μ cosϕ minus ] sin _ϕ) _ϕ

euroϕ _r

euroφ _p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)

Substituting equations (4) and (1) into equation (3)yields

X m( _μ minus ]r)

Y m( _] + μr)

N Izz _r

K Izz_p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)

According to the MMGmodel a mathematical model ofthe shiprsquos movement in four degrees of freedom can beobtained by dividing the external force and external torqueof the hull by the bare ship strength propeller force andrudder force

X m( _μ minus ]r) XHO + XP + XR

Y m( _] + μr) YHO + YP + YR

N Izz _r NHO + NP + NR

K Ixx_p KHO + KP + KR

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

where the subscripts H0 P and R correspond to the hullpropeller and rudder respectively If the effects of wind andwaves are considered the wind interference and wave in-terference are added to the various backs

23 Model of Fluid Dynamic Stream and Torque CalculationforBareHull )e fluid force and torque on the bare hull canbe divided into two components the inertial fluid dynamicscaused by inertia and the viscous fluid dynamics caused byviscosity In this study the interaction between the two fluiddynamics is not considered in calculating the forces andtorques )e fluid dynamics HO on the bare hull can beexpressed as follows

XHO X1 + XH

YHO Y1 + YH

NHO N1 + NH

KHO K1 + KH

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

γ

φδ

X0

O0 y0

X

G

u

vy

Figure 1 Coordinate systems

Z0

P

Φ

Figure 2 State of ship motion

Advances in Civil Engineering 3

where the subscripts I and H correspond to the inertial andviscous fluid dynamics respectively

24Model ofPropeller andCalculationofHostCharacteristicsIn this study the shiprsquos main speed and vertical speed areconsidered to be gt0 in the course of navigation and thepropeller thrust and torque model is established as follows

XP 1 minus tp1113872 1113873T

T ρn2D

4PkT JP( 1113857

QP ρn2D

5PkQ JP( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(7)

where tP represents the thrust deduction factor T representsthe propeller thrust ρ represents the fluid density n rep-resents the propeller revolution DP represents the propellerdiameter kT(JP) represents the propeller thrust coefficientand kQ(JP) represents the propeller torque coefficient

In this study the thrust coefficient and the torque co-efficient are both functions of the propellerrsquos advance speedcoefficient which can be expressed as follows

kT a0 + a1JP + a2J2P

kQ b0 + b1JP + b2J2P

⎧⎨

⎩ (8)

where a0 a1 and a2 and b0 b1 and b2 are the regressioncoefficients for the open-water characteristics of thepropeller

25 FluidDynamics and Torque CalculationModel of RudderFor rotational motion the ship is mainly steered using therudder )e calculation model for the rudder is as follows

XR 1 minus tR( 1113857FN sin δ

YR 1 + aH( 1113857FN cos δ

NR xR + aHxH( 1113857FN cos δ

KR minus 1 + aH( 1113857zRFN cos δ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)

where FN represents the positive pressure perpendicular tothe rudder surface tR represents the rudder force deratefactor δ represents the rudder angle aH represents thesteering-induced lateral force correction factor for the hullxH represents the distance between the point of steering toinduce the hull lateral force and the centre of gravity xR

represents the longitudinal distance from the centre of therudder to the centre of gravity of the hull and zR representsthe vertical distance from the centre of the rudder to thecentre of gravity of the hull

)e formula for calculating the positive pressure of therudder is as follows

FN minus12ρARfαU

2R sin α

2R

fα 613λ

225 + λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(10)

where AR represents the rudder area UR represents theeffective incoming velocity αR represents the effective angleof the incoming flow and λ represents the thickness ratio ofthe rudder

3 Model Test

Because of the large-scale landslide surge occurred in thenature it is difficult to capture the landslide surge of originaldata )rough the present research of the )ree GorgesReservoir area river by the model test the model scale isdetermined according to the Froude number In this studythe physical model mainly includes the chute carriage riverlandslide ultrasonic wave acquisition analyser etc )rougha large-scale model test the spread characteristic of thelandslide surge is examined

31 Chute )e chutes are made of high-strength steel )eyhave overall lengths of 2 and 06m and their widths can beadjusted to 05 10 and 15m )e chutes are equipped witha side board floor internal demand placed landslide thechute through ascension device fixed on the sliding shelvesthe chute and chute plate inside have been burnished beforethe test in order to total frictional resistance of equal chuteand carriage as shown in Figure 3

32 Channel Model )e length of the straight section is28m the centre line radius of the bending section is 7mthe length of the straight section of the bending section is13m the width of the upper edge of the river is 8 m andthe river section has the shape of an inverted trapezoid)e main construction process of the river is as follows (1)by the method of model of generalised channel model (2)cross section design and processing in accordance withthe actual survey data (3) the channel model of each 3Dlayout (ensuring the accuracy of the data for every crosssection) (4) the lofting process of the boards (ensuring theaccuracy of each section after conversion in accordancewith the original data) (5) decomposition of the wholeriver (at the bottom) into a linear area a landslide areaand a river bend area for generalised processing (6)casting of the river and building based on the model afterthe lofting (7) waterproofing the constructed model toprevent leakage and ensure its effectiveness as shown inFigure 4

33 Measurement Methods )e measuring instrument wasa self-developed ultrasonic wave acquisition analyserwhich can determine the maximumwave height maximumperiod meaningful wave height and meaningful period atdifferent monitoring points In the test the acquisition timeof the equipment was 50 s and the acquisition frequencywas 50Hz Instruments were arranged in the river tomeasure the swell characteristics )e instruments weretested and had good adaptability and accuracy )e ul-trasonic wave analyser had 16 channels To facilitate thesorting of the test data each channel corresponded to a


measurement point Channel 8 was a ground wire placed atthe bottom of the channel model In the test a 1M ul-trasonic probe with an accuracy of lt01mm was used tocalibrate the sensor on an electric displacement platform of5 microm Additionally the ultrasonic wave meter was installedon the measuring bridge )e most generous slope wasselected for the sliding down the chute and the ultrasonicwave meter was placed 30 cm below the water surfaceaccording to the maximummeasured value of the first waveheight of the landslide surge )e measuring instrumentwas calibrated

34 Verification of Results After the sliding of the slideralong the chute the changes of the liquid level at differenttimes were compared with the calculation results of Heinrich[23] As shown in Figure 5 the test results were consistentwith Heinrichrsquos calculations

)e calculated results of this experiment were slightlysmaller than those of Heinrich mainly because thelandslide in this paper was 3D and scattered and thecollision of the landslide during the sliding process wasaccompanied by energy dissipation )e second is theinverted trapezoid of the test channel )e landslide surgeis accompanied by reflection and superposition in theprocess of propagation which affect the water level at themonitoring point

4 Simulation of Ship Manipulation Motion inStill Water

41 Roll Analysis of Operating Conditions for PropellerRevolution In this study a Yangtze River cruise ship wastaken as the research object A roll test of the ship wassimulated and the results were compared with the experi-mental data )e shiprsquos manoeuvring performance waspredicted and the mathematical model for the shipmanoeuvring was verified)emain parameters of the shiprsquosmain scale and rudder and oars are presented in Table 1

Under hydrostatic conditions the initial state of the pilotshiprsquos roll test is presented in Table 2

By establishing a model of the ship-operated motion theturning circle motion of the ship in still water was analysed)e shiprsquos roll trajectories for different propeller speeds areshown in Figures 6 and 7

)e roll characteristic parameters solved via numericalsimulation were compared with the ship test results and theerror was evaluated as shown in Table 3

For the ship in static water the following conclusionscan be drawn from the gyration trajectory and characteristicparameters of the ship at different main engine speeds

(1) )e propeller revolution was timed and the roll ringsizes differed slightly between the port and starboard)is was due to the ldquozero positive rudder anglerdquo forthe ship in the operating corner of the rudder theactual rudder did not have an angle of zero but wasslightly skewed in a certain direction leading to asmall difference between the port and starboardunder the same roll diameter

(2) )e propeller speed was 122 rmin corresponding toan initial ship speed of 26 kmh )e rotationcharacteristic parameters compared to the propellerspeed of 154 rmin corresponding to the initialspeed of 32 kmh Ship propeller speed increased the

Figure 3 Chute and carriage

Figure 4 Solid model diagram of the channel

Wat

er le

vel e

leva

tion

(m)

ndash065

ndash060

ndash055

ndash050

ndash045

ndash040

ndash035

25 30 35 40 4520Horizontal distance (m)

Model testHeinrich (1992)

Figure 5 Comparison between the model test results and thecalculations of Heinrich for t 15 s


ship speed increased but the roll performance wasdegraded

(3) )e simulation results were compared with the shiptest data )e error of each roll characteristic pa-rameter of the ship was within plusmn20 and far smallerthan 20 satisfying the precision requirements forengineering applications )us the mathematical

model of ship manipulation motion used in thisstudy is correct and reasonable and further simu-lations can be performed using this model

(4) )e simplified manoeuvring motion model of shipsin landslide surge waters such as the hydrodynamicperformance of propellers and rudders may causeerrors in the calculation results However duringship rotation the variation trends of various pa-rameters of ship rotation were consistent with thetheoretical research results for the ship manoeuvringmotion indicating the reliability and correctness ofthe mathematical model

)e rolling change curves as the starboard rolls the shipat different propeller speeds are shown in Figure 8 )efollowing observations were made

(1) )e shiprsquos roll process can be divided into threestages In the initial stage of rotation the ship tiltedtowards the inside of the circle and reached the initialangle of rotation owing to the transverse force of thehull and rudder In the transition stage it graduallyinclined towards the outside of the circle exceededthe roll angle and reached the maximum dynamicroll angle In the steady turning stage the steadyturning angle was stabilised

(2) When the propeller speed was 154 rmin the simu-lation indicated that the initial tilt angle of the roll was074deg the maximum power roll angle was 875deg andthe roll angle of steady rotation was approximately257deg When the propeller speed was 122 rmin themaximum power roll cross-roll angle was 579deg andthe roll angle of steady rotation was approximately162deg While propeller speed affected the initial speedof the ship the ship roll had a greater impact Whenthe speed of the main engine increased the maximumdynamic roll angle of the shiprsquos hydrostatic rotationincreased along with the stable roll angle of the steadyrotation stage

Table 1 Target ship master scale and main parameters of therudder and pulp

Overall length L (m) 14180Length on the waterline Lwl (m) 1320Breadth moulded B (m) 1940Depth moulded D (m) 1140Designed draft d (m) 680Block coefficient Cb 0821Prismatic coefficient CP 0863Displacement nabla (t) 12000Area of rudder AR (m2) 2515Rudder height HR (m) 62Aspect ratio of rudder λ 102Number of blades Z 4Diameter of propeller DP (m) 48Screw pitch P (m) 32

Table 2 Initial state of the target ship in a static water turningcircle

Order rudder angle δE (deg) 350 ndash350Draft d (m) 440 440Distance from midship to centre of buoyancy xB

(m) 080 080

Displacement nabla (t) 52340 52340Engine speed n (rmin) 154 122Initial speed V (kmh) 32 26Initial set rudder angle δ (deg) 00 00Height of CG (m) 56 56Initial transverse metacentric height (m) 68 68

ndash100

0

100

200

300

400

500

ndash500 ndash400 ndash300 ndash200 ndash100 0 100 200 300 400 500 600ndash600

y-displacement (m)

x-di

spla

cem

ent (

m)

Rudder 35 degRudder ndash35 deg

Figure 6 Rotation trajectory for a rudder angle of approximately35deg (main engine speed of 154 rmin)

ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)

0ndash300 ndash200 100 400ndash400 200 300ndash100 500 600ndash500

y-displacement (m)




(3) According to the calculation book of the loadingcondition of the selected ship when the main enginespeed was 154 and 122 rmin the steady rotary rollangle of the ship was 26deg and 18deg respectively Asindicated by the curve of the roll angle the simu-lation results fit well )erefore the mathematicalmodel of manoeuvring movement developed in thisstudy is reasonable

42 Rotation at Different Rudder Angles )e initial speed ofthe ship was set as 32 kmh with starboard roll angles of 10deg20deg and 35deg To simulate the roll motion at different rudderangles of the ship in still water the shiprsquos roll trajectory wasexamined as shown in Figure 9 )e roll angle change curveis presented in Figure 10

As indicated by the shiprsquos roll trajectory the shiprsquos rolldiameter was 21729m at the steering angle of 10deg 9908m atthe rudder angle of 20deg and 4878m at the rudder angle of35deg With the increasing rudder angle the shiprsquos gyration instill water decreased gradually and the ship exhibited goodgyration when sailing with a large rudder angle

From the analysis of the ship at different rudder anglesthe transverse variation curve was obtained

(1) With the increasing rudder angle the time takenfor the shiprsquos roll angle to stabilise increasedWhen the ship sailed with a larger rudder angle itneeded to go through more rolls to reach the stableroll angle whereas when the ship sailed with asmaller rudder angle it entered the steady turningstage earlier

(2) In the roll transition stage the maximum dynamicroll angle of the ship was 234deg 524deg and 875deg whenthe rudder angle was 10deg 20deg and 35deg respectivelyAs the rudder angle increased the maximum dy-namic roll angle of the ship increased )ereforewhen the ship turned in static water the rotary rollangle increased with the rudder angle When therudder angle increased to 20deg the rotary stable phaseof the roll angle is no longer a significant changealways maintain relatively stable

Table 3 Ship turning characteristic parameters

Rotary elementsShip test data Simulation results Error ()

δ 35deg δ ndash35deg δ 35deg δ ndash35deg δ 35deg δ ndash35deg

Engine speed 154 rmin

Advance (m) 4306 4132 4197 3989 ndash253 ndash346Transfer (m) 2540 2398 2498 2340 ndash165 ndash242

Tactical diameter (m) 5582 5333 5510 5228 ndash129 ndash197Steady turning diameter (m) 4706 4584 4878 4727 +365 +312


Advance (m) 4122 4003 3911 3724 ndash512 ndash697Transfer (m) 2268 2115 2340 2206 +317 +430

Tactical diameter (m) 5187 4892 5199 4924 +023 +065Steady turning diameter (m) 4539 4346 4473 4336 ndash145 ndash023

100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)

n = 154rminn = 122rmin

Figure 8 Roll angle for the shiprsquos starboard side at different mainengine speeds

0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)

Rudder 10 degRudder 20 degRudder 35 deg

Figure 9 Ship roll tracks for different rudder angles


5 Analysis of Ship Roll Characteristics inLandslide-Generated Tsunamis

)e water entry point of the landslide body is mostly locatedon both sides of the channel and the heeling moment is thelargest and the risk is the highest when the transverse waveoccurs )erefore in this study the wave direction of thelandslide-generated tsunamis is selected to be 90deg that is theship suffers from the action of the right transverse wave atthe initial moment )e coupling action between thelandslide surge and the ship is shown in Figure 11

)e working conditions and wave-height test results forthe landslide example are presented in Table 4

)e landslide-genertated wave was highly nonlinear)rough the model test results they show the landslide-genertated wave as stokes wave elliptical cosine wavesolitary wave superposition therefore in this paper thewave forces acting on the ideal landslide-genertated waveinto irregular wave second-order drift force by the rules offormula of irregular wave spectrum and wave interferenceformula to calculate

51 Maximum Roll Angle at Different Sailing PositionsTo study the rolling situation of the ship at different directsailing positions five groups of repeated tests should beconducted when the ship is sailing )e first group involvesthe measurement of the shiprsquos speed in still water the secondgroup involves themeasurement of the time taken for the shipto sail to a fixedmonitoring point the third group involves theverification of the shiprsquos passage position and the fourth andfifth groups involve the extraction of test data If there is a 5difference between the results of the fourth and fifth groupsthe next group of tests is conducted If the error value of thetwo groups is lt5 the average value of the two groups is

taken as the test result )e ship sails in a straight line at theselected position with an initial speed of 26 kmh )emaximum wave height plot for different locations is shown inFigure 12 and the maximum roll angle calculations fordifferent locations are presented in Figure 13

As indicated by the analysis results in Figures 12 and 13the height of landslide-generated tsunamis decreasedsharply at a position close to the water entry point With anincrease in the distance from the water entry point themaximum wave height of the surge gradually decreased andthe energy carried by the surge was continuously consumed)e maximum roll angle is an important index for the safetyof ships sailing in waves )e International Maritime Or-ganization stipulates that the limit roll angle of ships sailingsafely in waves is 40degWhen the roll angle is gt40deg the stabilitycharacteristics of ships change which can lead to disastersTo prevent cargo movement and waves on deck the waterinlet angle and the maximum dynamic inclination angle ofthe ship should be considered )e minimum values of thethree should be taken as the standard tomeasure the safety ofthe ship )e engineering experience value is 15deg that is thelimit roll angle for the shiprsquos safe navigation is 15deg

According to the regulations of the InternationalMaritime Organization the maximum safe roll angle forstraight sailing in the wind and waves is 15deg that is whenthe roll angle exceeds 15deg the ship is in considerabledanger of capsizing As shown in Figure 13 the maximumroll angle of the ship reaches 2174deg 1630deg and 1562deg at100 200 and 280m respectively when the ship is proneto capsizing When the ship is 400 and 500m from thelandslide entry point the maximum roll angle is 898deg and764deg respectively (both less than 15deg) thus the ship isrelatively safe to navigate In order to guarantee safety ofship by the form of traffic control appropriate to allowships can drift along in the appropriate scope away fromthe landslide area decreased risk capsizing )roughconsiderable manoeuvring motion analysis the rela-tionship between the wave height and the maximum rollangle was determined as shown in Table 5

When the ship sails at an initial speed of 26 kmh and thedistance from the landslide entry point is gt400m the area isrelatively safe At this time the ship is safe during thenavigation According to the results of this study for sailingto be safe the ship should be 400ndash560m from the landslideentry point under this condition

100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)


Figure 10 Shiprsquos transverse change curves for different rudderangles

Bank

Ship Chute90

Figure 11 Schematic of the landslide surge and ship action


52 Maximum Roll Angle at Different Speeds To study themaximum roll angle of the ship at different speeds the pro-peller propulsion speedwas controlled by changing the rotatingspeed of the main engine and the average speed of this sectionwas taken as the monitoring speed for fixed-distance naviga-tion monitoring Initial speeds of 15 18 and 21kmh wereselected Due to ship is located at 100m and 200m and 280mthe maximum roll angle is more than maximum securityheeling angle this period belongs to the dangerous areatherefore this section of the ship in the 400 and 500m fromlandslides in a relatively safe location as shown in Figure 14

As shown in Figure 14 when the ship was 500m fromthe landslide entry point the maximum roll angle of theship at all speeds was smaller than the roll angle at 400mthat is the maximum roll angle of the ship decreasedgradually with the increasing distance from the landslideentry point )e largest value of the maximum roll anglewas 1416deg from the water entry point of the landslide to400m which occurred at the maximum speed Under thisworking condition the shiprsquos roll angle was close to theultimate roll angle of 15deg which was a dangerous workingcondition that is the wave height was 268m and thesailing speed was 38 kmh )e minimum roll angle was898deg which occurred at the lowest speed When the shipwas 500m from the landslide entry point its maximumroll angle was 1244deg at a speed of 38 kmh and 764deg at a

Table 4 Properties of landslide-generated tsunamisDepth (m) 755Actual landslide volume (m3) 103times105

Landslide angle (deg) 40degSlope width of landslide section (m) 560Distance between model and water entry point (m) 143 286 400 571 714Distance between original model and water entry point (m) 100 200 280 400 500Maximum wave height of model surge (cm) 7284 5026 4591 3829 3142Maximum wave height of original model surge (m) 5099 3518 3214 2680 2199

400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)

Figure 12 Maximum wave height of surges at different locations

4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)

Figure 13 Maximum roll angle of a ship at different locations

Table 5 Relationship between the wave height and the maximumroll angle

Wave height(m)

Location(m)

Maximum ship roll angle(deg) Safety

5099 100 2174 Unsafe3518 200 1630 Unsafe3214 280 1562 Unsafe2680 400 898 Safe2199 500 764 Safe

16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)

Position = 400mPosition = 500m

Figure 14 Maximum roll angle at different speeds


speed of 26 kmh As shown in Figure 14 when the dis-tance from the landslide entry point was fixed the shiprsquosmaximum roll angle gradually increased with the speed)e maximum roll angle of the ship did not exceed 15deg inany of the working conditions )erefore when the shipsails in the area of 400ndash500m 26ndash38 kmh is a safe speedrange that does not present a danger of capsizing )erelationship between the shiprsquos initial speed and themaximum roll angle is presented in Table 6

In summary in the process of ship navigation the shiprsquosspeed is too fast resulting in a large roll angle which caneasily cause the hull to overturn thus the pilot should fullyconsider the state of the ship according to the actual situ-ation to weigh the advantages and disadvantages of thechoice

53 Maximum Roll Angle at Different Rudder AnglesSteering angles of 10deg 15deg 20deg 25deg and 35deg were selected andthe ship sailed at relatively safe positions of the 400m and500m landslide water inlet points at initial speeds of 26 32and 38 kmh )e maximum roll angle was recorded asshown in Figures 15 and 16

As shown in Figures 15 and 16 the maximum roll anglefor each rudder angle always increased with the speed Forships under the action of landslide-generated tsunamiswhen the initial speed was constant the maximum roll angleincreased with the rudder angle As shown in Figure 15when the shiprsquos sailing speed was 26 kmh the shiprsquos sailingrudder angle was 15deg and the shiprsquos maximum rolling anglewas 1245deg When the shiprsquos sailing speed was 32 kmh theshiprsquos sailing rudder angle was 10deg and the shiprsquos maximumrolling angle was 1321deg When the shiprsquos sailing speed was26 kmh the shiprsquos sailing rudder angle was 15deg and theshiprsquos maximum rolling angle was 1245deg When the shiprsquossailing speed was 38 kmh the shiprsquos rudder angle was 10degthe shiprsquos maximum roll angle was 1769deg and the shiprsquosnavigation was in an unsafe state

As shown in Figure 16 when the shiprsquos sailing speed was26 kmh the shiprsquos sailing rudder angle was 20deg and theshiprsquos maximum rolling angle was 1364deg When the shiprsquossailing speed was 38 kmh the shiprsquos rudder angle was 10degthe shiprsquos maximum roll angle was 1443deg 15deg and 1812degand the shiprsquos navigation was in an unsafe state )e rela-tionship between the steering angle and the roll angle ispresented in Tables 7ndash9

As shown in Figures 15 and 16 the maximum roll anglefor each rudder angle always increased with the speed Asindicated by Tables 7ndash9 for ships under the action oflandslide-generated tsunamis when the initial speed wasconstant the maximum roll angle increased sharply with thesteering A larger rudder angle corresponded to a higher riskof capsizing )erefore it is inappropriate to conduct large-rudder angle steering in surges and even less appropriate toconduct emergency steering With the increasing distancefrom the water entry point of the landslide the range of thesteering angle of the ship increased but the safe steeringangle remained small (lt20deg) When the ship is far from thewater entry point of the landslide a small rudder angle can

be used to ensure the navigation safety after the state of theship and the wind and wave conditions are confirmed

6 Ship Track Analysis in Landslide-Generated Tsunamis

To analyse the ship-shore collision risk one should firstensure that the vessels under landslide-generated tsunamis isnot happened beforehand therefore to study the trackconditions of ship in the landslide-generated tsunamis thelargest rolling of ship is the basis of the analysis resultsselection of navigation position the initial speed andsteering rudder angle are no overturning danger relativesafety conditions the simulation are in the initial wave totransverse wave under the condition of specific conditionsas shown in Table 10

61 Shiprsquos Navigation Trajectory at Zero Rudder AngleWorking conditions 1ndash3 and 4ndash6 were selected to study theshiprsquos motion trajectory during steering from the water entrypoint of the landslide to distances of 400 and 500m re-spectively as shown in Figures 17 and 18

As shown in Figures 17 and 18 under the action oflandslide-generated tsunamis the ship did not conductsteering Additionally the shiprsquos trajectory graduallyshifted from the initial straight course to the direction ofwave propagation and the deviation became increasinglyobvious )is is because the wave direction angle changedduring the process of movement An abscissa diagramindicated the ship motion offsets and the variation of thevertical displacement for the ship sailing direction At thebeginning the ship was only affected by the transversewave and the transverse oscillation amplitude change ofthe vessel was partly a result of the landslide surge on themain body and partly due to the landslide surge over thesurface of the rudder which caused the turning of therudder and made the ship deviate significantly from theinitial position

)e ship was eventually affected by the component forcein the wave direction When the landslide-generated tsu-namis disappear if no steering is performed the ship will sailin a straight line along the deflected heading direction thusthe ship will eventually be far from the original course andcollide with the shore wall When the distance from thewater entry point of the landslide was fixed the shiprsquosmovement trajectories at different sailing speeds wereanalysed and compared A higher initial sailing speedyielded a greater track offset of the ship under the action oflandslide-generated tsunamis )erefore to maintain goodcourse stability the speed of the ship should be appropriatelyreduced in a landslide-generated tsunami At a given speedship trajectories at different positions were analysed com-paratively With an increase in the distance from the waterentry point of the landslide the wave declined and the forceon the ship decreased Additionally the offset along the Y-axis caused by the landslide-generated tsunamis in the shiprsquostrajectory decreased significantly


Table 6 Relationship between the initial speed and the maximum roll angle

Wave height (m) Initial speed (kmh) Maximum ship roll angle (deg) Safety

268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)

Velocity = 26kmhVelocity = 32kmhVelocity = 38kmh

Figure 15 Maximum roll angle at 400m with respect to the rudderangle

468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



Table 7 Relationship between the steering angle and themaximumroll angle at the initial speed of 26 kmh

Wave height(m)

Rudder angle(deg)

Maximum ship rollangle (deg) Safety

2680

10 1064 Safe15 1245 Safe20 1509 Unsafe25 1694 Unsafe35 1843 Unsafe

2199

10 904 Safe15 1037 Safe20 1364 Safe25 1552 Unsafe35 1748 Unsafe


Wave height(m)

Rudder angle(deg)


2680

10 1321 Safe15 1534 Unsafe20 1632 Unsafe25 1873 Unsafe35 2017 Unsafe

2199



Wave height(m)

Ruder angle(deg)


2680

10 1769 Unsafe15 1962 Unsafe20 2226 Unsafe25 2511 Unsafe35 285 Unsafe

2199



62 Shiprsquos Navigation Trajectory during Steering )e shipencountered landslide-generated tsunamis during sailingand the shiprsquos track gradually deviated without steering )ewidth of the river course in the landslide area selected in thisstudy was 560m As the ship continued to sail its offset inthe Y direction at 400 and 500m exceeded 160 and 60mrespectively that is the ship hit the bank slope and acollision occurred )erefore to avoid ship collision acci-dents and ensure the safety of the ship navigation steeringshould be performed in advance

To ensure the navigation safety of the ship it was as-sumed that the ship started steering after sailing 100m underthe action of landslide-generated tsunamis and the shiprsquostrack at the limit rudder angle was predicted under eachworking condition

When the shiprsquos initial speed reached 38 kmh at theposition 400m from the landslide entry point the ship waseasily capsized with the rudder whereas a ship-shore

collision occurred without the rudder )erefore in thecalculation example a ship with a speed of 38 kmh cannotsail safely with a rudder thus it is in a dangerous sailingstate When the initial sailing speed is 26 kmh the ultimatesafe rudder angle is selected to be 15deg and the safe rudderangle is selected to be 10deg when the sailing speed is 32 kmh)e ship movement trajectories in the two foregoing con-ditions (working conditions 1 and 2) were simulated )eresults are presented in Figures 19 and 20

As shown in Figures 19 and 20 before the steering theshiprsquos sailing path was shifted in the direction of landslide-generated tsunami propagation When the ship startedsteering it moved in the direction of migration for a certaindistance but the track offset increased and the ship sloweduntil the maximum offset was reached

When the speed was 26 kmh and the ultimate safetyrudder angle was 15deg the maximum lateral offset of the shiprsquostrajectory was 533m which is significantly smaller than

500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)

160ndash120 ndash80 1200 40 80 200ndash200 ndash160 ndash40y-displacement (m)


Figure 17 Shiprsquos trajectory without steering at 400m

500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)

ndash80 0ndash40 ndash20ndash60 20 40 60 80 100ndash100y-displacement (m)



Table 10 Ship track forecast operating conditions

Case Distance (m) Speed (kmh) Rudder angle (deg)1 400 26 lt15deg2 400 32 lt10deg3 400 38 mdash4 500 26 lt20deg5 500 32 lt15deg6 500 38 lt10deg


160m )us the ship could avoid a collision with the bankslope by turning the rudder without capsizing When thespeed was 32 kmh and the ultimate safety rudder angle was10deg the maximum lateral offset of the shiprsquos trajectory was1081m which is smaller than 160m At this time the shipcould still avoid a collision with the bank slope

For the position 500m from the water inlet point of thelandslide the shiprsquos movement trajectory under workingconditions 4ndash6 was simulated and the influence of the steeringangle was considered )e results are shown in Figure 21

As shown in Figure 21 the maximum track lateral offsetwas 304m for condition 4 564m for condition 5 and

1041m for condition 6 To avoid the collision between theship and the bank slope the maximum offset should be lt60mat this time )us working conditions 4 and 5 satisfied therequirements whereas working condition 6 did not

In summary according to the example of the cruise shipof Yangtze river in the)ree Gorges Reservoir area sailing inthe Tuokou area of Yiwan Yangtze river bridge to ensure thenavigation safety of the ship it should sail at a distance of400ndash500m from the water point of the landslide Whencourse deviation occurs a ship-shore collision can beavoided via proper steering and the ship can return to theinitial route At a distance of 400m when the initial sailingspeed was 26 kmh the ship steering angle was 15deg and whenthe initial sailing speed was 32 kmh the ship steering anglewas 10deg for ensuring the navigation safety of the ship Inthese two cases the maximum roll angle was 1245deg and1321deg respectively and the maximum yaw distance was 533and 1081m respectively At a distance of 500m a steeringangle of 20deg at a speed of 26 kmh ensured the navigationsafety of the ship In this case the maximum roll angle was1364deg and the maximum yaw distance was 304m the speedof the ship at 26 kmh and the steering is 20deg to ensure thesafety of the shiprsquos navigation

7 Conclusions

)e geological environment of the)ree Gorges Reservoir iscomplex and prone to landslides )e falling of the landslidebody into the water results in landslide-generated tsunamiswith a large amount of energy which pose a considerablethreat to the navigation safety of ships )erefore it is ofpractical significance to investigate the shiprsquos handlingcharacteristics in landslide-generated tsunamis In thisstudy according to the empirical formula method a

ndash160 ndash40ndash80ndash120 16040 80 120 200ndash200 0y-displacement (m)

500

1000

1500

2000

x-di

spla

cem

ent (

m)

Figure 19 Trajectory for a rudder angle of 15deg and a ship speed of26 kmh

500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)

ndash100 ndash80 ndash60 ndash40 ndash20 0 20 40 60 80 100 120ndash120y-displacement (m)

20deg rudder angle at 26kmh15deg rudder angle at 32kmh10deg rudder angle at 38kmh

Figure 21 Trajectory of the ship at different speeds and rudderangles


mathematical model of four-degree-of-freedom shipmanoeuvring movement was established and the maximumroll angle and ship track line changes in static water andlandslide-generated tsunamis were investigated through theestablished simulation program According to the resultsthe following conclusions are drawn

(1) At a shorter distance to the water inlet landslide-generated tsunamis are higher the shiprsquos roll angle islarger and navigation is more dangerous for largerships )erefore the ship sailing in the range of con-ditions permit in case of sudden landslide-generatedtsunamis according to ship the location quickly awayfrom the area )e ship should avoid a large rudderangle particularly the emergency steering

(2) If no steering is performed in landslide-generatedtsunamis the shiprsquos trajectory gradually shifts from theinitial straight course to the propagation direction ofthe landslide-generated tsunami In the same externalenvironment a higher initial speed results in a largerhorizontal deviation distance of the ship When theship starts to steer the increase in the shiprsquos outwardmigration slows and after the maximum offset isreached the direction of the track line can be restoredunder the action of the rudder force

(3) )e developed simulation program for ship controlmovement can be used to study the effects oflandslide-generated tsunamis on the ship controlcharacteristics )e examples selected in this studywere rudder angles of 15deg 10deg and 20deg at initial shipspeeds of 26 32 and 26 kmh respectively from thelandslide point of 400m )ese conditions are rec-ommended for safe navigation of the ship

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is study was financially supported by the National NaturalScience Foundation of China (51479015) the Social Scienceand Technology Innovation Program for Peoplersquos Livelihoodin Chongqing (cstc2019ngzx0017 cstc2019jscx-msxmX0302cstc2018jscx-msybX0233 and cstc2019jscx-lyjsAX0008) andthe Science and Technology Program of Chongqing Educa-tion Commission (KJQN201900735)

References

[1] H M Fritz Initial phase of landslide generated impulse wavesPhD thesis Eidgenossische Technischen Hochschule ZurichZurich Switzerland 2002

[2] J S Walder P Watts O E Sorensen and K JanssenldquoTsunamis generated by subaerial mass flowsrdquo Journal ofGeophysical Research vol 108 p 2236 2003

[3] A Panizzo P De Girolamo and A Petaccia ldquoForecastingimpulse waves generated by subaerial landslidesrdquo Journal ofGeophysical Research vol 110 2005

[4] M Di Risio G Bellotti A Panizzo and P De Girolamoldquo)ree-dimensional experiments on landslide generatedwaves at a sloping coastrdquo Coastal Engineering vol 56 no 5-6pp 659ndash671 2009

[5] V Heller M Moalemi R D Kinnear and R A AdamsldquoGeometrical effects on landslide-generated tsunamisrdquoJournal of Waterway Port Coastal and Ocean Engineeringvol 138 no 4 pp 286ndash298 2012

[6] M Bruggemann ldquoComposite modelling of the influence ofgeometry on landslide generated impulse wavesrdquo in Pro-ceedings of the 1st Civil and Environmental EngineeringStudent Conference Imperial College London London UKJune 2012

[7] V Heller and J Spinneken ldquoOn the effect of the water bodygeometry on landslide-tsunamis physical insight from lab-oratory tests and 2D to 3D wave parameter transformationrdquoCoastal Engineering vol 104 no 10 pp 113ndash134 2015

[8] S Yavari-Ramshe and B Ataie-Ashtiani ldquoA rigorous finitevolume model to simulate subaerial and submarine landslide-generated wavesrdquo Landslides vol 14 no 1 pp 203ndash221 2017

[9] B Ataie-Ashtiani and S Yavari-Ramshe ldquoNumerical simu-lation of wave generated by landslide incidents in dam res-ervoirsrdquo Landslides vol 8 no 4 pp 417ndash432 2011

[10] G Ruffini V Heller and R Briganti ldquoNumerical modelling oflandslide-tsunami propagation in a wide range of idealizedwater body geometriesrdquo Coastal Engineering vol 153 ArticleID 103518 2019

[11] K Kelfoun T Glachetti and P Labazuy ldquolandslide-generatedtsunamis at reunion islandrdquo Journal of Geophysical Research-Earth Surface vol 115 2010

[12] A Farhadi ldquoISPH numerical simulation of tsunami genera-tion by submarine landslidesrdquoArabian Journal of Geosciencesvol 11 no 12 2018

[13] C Cecioni and G Bellotti ldquoModeling tsunamis generated bysubmerged landslides using depth integrated equationsrdquoApplied Ocean Research vol 32 no 3 pp 343ndash350 2010

[14] S E Hirdaris W Bai D Dessi et al ldquoLoads for use in thedesign of ships and offshore structuresrdquo Ocean Engineeringvol 78 pp 131ndash174 2014

[15] K Sasa D Terada S Shiotani et al ldquoA study of numericalsimulation of ship motions while underway using a coastalwave databaserdquo International Journal of Offshore and PolarEngineering vol 22 no 1 pp 38ndash45 2012

[16] S Kianejad H Enshaei J Duffy and N Ansarifard ldquoCal-culation of ship roll hydrodynamic coefficients in regularbeam wavesrdquo Ocean Engineering vol 203 Article ID 1072252020

[17] V Piscopo A Scamardella and S Gaglione ldquoA new wavespectrum resembling procedure based on ship motion anal-ysisrdquo Ocean Engineering vol 201 Article ID 107137 2020

[18] M-G Seo and Y Kim ldquoNumerical analysis on ship ma-neuvering coupled with ship motion in wavesrdquo Ocean En-gineering vol 38 no 17-18 pp 1934ndash1945 2011

[19] R Szlapczynski P Krata and J Szlapczynska ldquoShip domainapplied to determining distances for collision avoidancemanoeuvres in give-way situationsrdquo Ocean Engineeringvol 165 pp 43ndash54 2018


[20] J-H Lee and Y Kim ldquoStudy on steady flow approximation inturning simulation of ship in wavesrdquo Ocean Engineeringvol 195 Article ID 106645 2020

[21] P Yuan P Wang and Y Zhao ldquoModel test research on thepropagation of tsunamis and their interaction with navigatingshipsrdquo Applied Sciences vol 9 no 3 p 475 2019

[22] P-Y Yuan P-Y Wang Y Zhao and M-L Wang ldquoEx-perimental study on the nonlinear behavior of a sailingcontainer ship under landslide-induced surgesrdquo Advances inCivil Engineering vol 2019 Article ID 3240812 10 pages2019

[23] P Heinrich ldquoNonlinear water waves generated by submarineand aerial landslidesrdquo Journal of Waterway Port Coastal andOcean Engineering vol 118 no 3 pp 249ndash266 1992


scholars have used different numerical methods to study thecharacteristics of landslide-generated tsunamis Yavari-Ramshe and Ataie-Ashtiani [8] presented a new landslide-generated model based on incompressible Euler equationsA two-layer model was developed that included a layer ofgranular-type flow beneath a layer of inviscid flow Ataie-Ashtiani and Yavari-Ramshe [9] estimated the impact oflandslide-generated waves using a 2D fourth-order Bous-sinesq-type numerical model Ruffini et al [10] focused onnumerical landslide tsunami propagation in the far field toquantify the effect of the water-body geometry Kelfoun et al[11] used a new two-fluid (seawater and landslide) numericalmodel to estimate the wave amplitudes and the propagationof tsunamis associated with landslide events on ReunionIsland Farhadi [12] performed a numerical simulation of thegravity currents of non-Newtonian fluids via the incom-pressible smoothed particle hydrodynamics approachCecioni and Bellotti [13] presented a depth-integrated nu-merical model for the simulation of the generation andpropagation of tsunamis due to submerged landslides

)emovement of ships on waves has been investigated vianumerical simulations and model tests Hirdaris et al [14]reviewed some of the recent advances in the assessment ofloads for ships and offshore structures with the aim of pre-senting the overall technological landscape for further in-vestigation validation and implementation by the academicand industrial communities Sasa et al [15] performed anumerical simulation of ship motions by using a coastalnetwork wave database Some experts focused on themovement characteristics of the ship in its six degrees offreedom Kianejad et al [16] performed numerical and ex-perimental simulations to examine the effects of differentwave heights and wave frequencies on the ship motioncharacteristics Piscopo et al [17] examined the heave andpitch motion time histories via a time-domain simulationaccording to theoretical wave spectra Scholars have alsostudied the manipulation of ships Seo and Kim [18] per-formed a numerical analysis of the ship manoeuvring per-formance in the presence of incident waves and the resultantshipmotion responses Szlapczynski et al [19] used amodel ofthe ship dynamics to assess the time and distance necessaryfor amanoeuvre to avoid domain violations Lee andKim [20]considered the effects of the steady flow approximation in theanalysis of ship manoeuvring in waves Owing to the strongnonlinear effect of landslide surges the traditional linearpotential flow theory is not accurate Scholars such as Yuanet al [21 22] employed engineering examples and the or-thogonalmodel test method to study the navigational safety ofships in landslide-generated tsunami waters

In summary scholars have performed considerable researchon the propagation characteristics of landslide-generated tsu-namis the laws of shipmovement and the shipmanoeuvres viavarious methods such as theoretical analysis numerical sim-ulation field observation and indoor experiments Howeverfew studies have been performed on the direct development oflandslide-generated tsunamis and ship motion and there hasbeen even less theoretical research on the effect of landslide-generated tsunamis on the ship )erefore according to thefour-degree-of-freedom motion equation of ships and the

characteristics of landslide-generated tsunami propagation thispaper presents simulation procedures for studying the effects oflandslide-generated tsunamis on ship manoeuvring along withscientific suggestions and theoretical support for the naviga-tional safety of ships in landslide-generated tsunami waters

2 Mathematical Model of ShipManipulation Motion

21 Coordinate Systems To facilitate the study of themovement of the hull in the water two right-handed coor-dinate systems are selected )e first is the inertial coordinatesystem fixed to the Earthrsquos surface (o0 minus x0y0z0) and z0 )eshiprsquos transverse velocity along the Y-axis is denoted as v theforward angular velocity around the z-axis is denoted as r andthe roll angular velocity around theX-axis is denoted asp)eexternal force and external torque suffered by the ship in themanoeuvring motion coordinate system can be expressed interms of the longitudinal force x along the X-axis transverseforce y along the Y-axis foreword moment N around the z-axis and roll moment K around the X-axis

)e main characteristic of the MMG model is applied tothe hydrodynamic force and moment on the ship in accor-dance with the physical meaning and is decomposed into thework on the naked hull the propeller open water and thehydrodynamic force and moment on the open-water rudderas well as the mutual interference between the hydrodynamicforce and moment )e MMG model is based on a deep andextensive experimental study together with theoretical anal-ysis and is a popular international mathematical model forship motion It has high reliability (Figures 1 and 2)

Relationship between the physical quantities in theprocess of ship manipulation

_xGO μ cos ϕ minus ] sinϕ

_yGO μ sinϕ + ] cosϕ

_ϕ r

_φ p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where ϕ represents the direction angle and φ represents theroll angle

22 Equation of Ship Manipulation Motion )e shiprsquosmovement in the water is rigid-body movement )e mass isdenoted as m According to the law of rigid weight heartmovement Newtonrsquos law and the momentum theorem thefollowing can be obtained

m euroxGO X0

m euroyGO Y0

Izzeuroϕ N0

Ixxeuroφ K0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

where Izz represents the moment of inertia of the hullagainst the X-axis and Ixx represents the moment of inertiaof the hull against the Y-axis )e derivatives of both sides ofequation (2) can be obtained as follows




euroϕ _r

euroφ _p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)


X m( _μ minus ]r)

Y m( _] + μr)

N Izz _r

K Izz_p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)



XHO X1 + XH

YHO Y1 + YH

NHO N1 + NH

KHO K1 + KH

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

γ

φδ

X0

O0 y0

X

G

u

vy


Z0

P

Φ





XP 1 minus tp1113872 1113873T

T ρn2D

4PkT JP( 1113857

QP ρn2D

5PkQ JP( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(7)





⎧⎨

⎩ (8)




YR 1 + aH( 1113857FN cos δ



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)




FN minus12ρARfαU

2R sin α

2R

fα 613λ

225 + λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(10)


3 Model Test



















Wat

er le

vel e

leva

tion

(m)

ndash065

ndash060

ndash055

ndash050

ndash045

ndash040

ndash035
















(m) 080 080


ndash100

0

100

200

300

400

500


y-displacement (m)

x-di

spla

cem

ent (

m)



ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)


y-displacement (m)



















100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References




























euroϕ _r

euroφ _p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)


X m( _μ minus ]r)

Y m( _] + μr)

N Izz _r

K Izz_p

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)



XHO X1 + XH

YHO Y1 + YH

NHO N1 + NH

KHO K1 + KH

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

γ

φδ

X0

O0 y0

X

G

u

vy


Z0

P

Φ





XP 1 minus tp1113872 1113873T

T ρn2D

4PkT JP( 1113857

QP ρn2D

5PkQ JP( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(7)





⎧⎨

⎩ (8)




YR 1 + aH( 1113857FN cos δ



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)




FN minus12ρARfαU

2R sin α

2R

fα 613λ

225 + λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(10)


3 Model Test



















Wat

er le

vel e

leva

tion

(m)

ndash065

ndash060

ndash055

ndash050

ndash045

ndash040

ndash035
















(m) 080 080


ndash100

0

100

200

300

400

500


y-displacement (m)

x-di

spla

cem

ent (

m)



ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)


y-displacement (m)



















100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References




























XP 1 minus tp1113872 1113873T

T ρn2D

4PkT JP( 1113857

QP ρn2D

5PkQ JP( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(7)





⎧⎨

⎩ (8)




YR 1 + aH( 1113857FN cos δ



⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)




FN minus12ρARfαU

2R sin α

2R

fα 613λ

225 + λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(10)


3 Model Test



















Wat

er le

vel e

leva

tion

(m)

ndash065

ndash060

ndash055

ndash050

ndash045

ndash040

ndash035
















(m) 080 080


ndash100

0

100

200

300

400

500


y-displacement (m)

x-di

spla

cem

ent (

m)



ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)


y-displacement (m)



















100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References







































Wat

er le

vel e

leva

tion

(m)

ndash065

ndash060

ndash055

ndash050

ndash045

ndash040

ndash035
















(m) 080 080


ndash100

0

100

200

300

400

500


y-displacement (m)

x-di

spla

cem

ent (

m)



ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)


y-displacement (m)



















100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References





































(m) 080 080


ndash100

0

100

200

300

400

500


y-displacement (m)

x-di

spla

cem

ent (

m)



ndash100

0

100

200

300

400

x-di

spla

cem

ent (

m)


y-displacement (m)



















100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References









































100 200 300 4000Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



0 500 1000 1500 20002500ndash500

y-displacement (m)

ndash1000

ndash500

0

500

1000

1500

x-di

spla

cem

ent (

m)













100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References



































100 200 300 4000

Time (s)

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

Roll

(deg)



Bank

Ship Chute90







400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References






























400300 500100 200

Position (m)

15

20

25

30

35

40

45

50

55

Max

surg

e hei

ght (

m)


4

6

8

10

12

14

16

18

20

22

24

Roll

max

(deg)

400300 500100 200

Position (m)



Wave height(m)

Location(m)



16 17 18 19 20 2115

Velocity (kmh)

5

6

7

8

9

10

11

12

13

14

15Ro

ll m

ax (deg

)



















268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References









































268026 898 Safe32 1021 Safe38 1416 Unsafe

219926 764 Safe32 883 Safe38 1244 Safe

68

1012141618202224262830

Roll

max

(deg)

2520 3510 15 30Rudder (deg)



468

10121416182022242628

Roll

max

(deg)

2520 3510 15 30Rudder (deg)




Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Rudder angle(deg)


2680


2199



Wave height(m)

Ruder angle(deg)


2680


2199









500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References
































500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)




500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)












7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References































7 Conclusions



500

1000

1500

2000

x-di

spla

cem

ent (

m)


500

1000

1500

2000

2500

x-di

spla

cem

ent (

m)



500

1000

1500

2000

2500

3000

3500

4000

x-di

spla

cem

ent (

m)









Data Availability




Acknowledgments


References






























Data Availability




Acknowledgments


References


























novelmodelformanoeuvrabilityofshipsadvancingin landslide...

Documents