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Hydrodynamic Interaction Forces Mathematical Description

HYDRODYNAMIC INTERACTION FORCES MATHEMATICAL DESCRIPTION

The influence of shallow waters, channel walls, stationary or moving objects is obtain by additional components to the corresponding values of hydrodynamic forces and moments. The brief description of several particular cases is shown below.

Ship-Bottom Interaction

Additional force FBOT influenced the ship at shallow waters depends on the

kinematics parameters, propeller operation, ship trim and bottom inclination angle (see Fig. 17).

Fig. 17. Forces Due To Shallow Waters Influence

Structural formulas to obtain the hydrodynamic force components due to the bottom influence and their moment are as follows:

longitudinal force due to bottom influence

F

x BOT

= C

x BOT

( χ, H

BOT

/T, V

x

,V

y

,ω

z

)f ( n, P

/ D

,u

THR

)0.5ρV 2LT

P

P

lateral force due to bottom influence

F

= C

y BOT

(χ, H

BOT

/T, V

x

,V

y

,ω

z

)f ( n, P

/ D

,u

)0.5ρV 2LT

y BOT

P

P

THR

vertical force due to bottom influence

F

= C

z BOT

(χ, H

BOT

/T, V

x

,V

y

,ω

z

)f ( n, P

/ D

,u

)0.5ρV 2LT

,

zBOT

P

P

THR

hydrodynamic roll moment due to bottom influence

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where CxWALL, Cy WALL, CmzWALL

Hydrodynamic Interaction Forces Mathematical Description

Mx BOT = Cmx BOT (χ, HBOT /T, Vx ,Vy ,ω z )f ( n, PP / DP ,uTHR )0.5ρV 2L2T

hydrodynamic trim moment due to bottom influence

My BOT = Cmy BOT (χ, HBOT /T, Vx ,Vy ,ωz )f ( n, PP / DP ,uTHR )0.5ρV 2L2T

hydrodynamic yaw moment due to bottom influence

Mz BOT = Cmz BOT (χ, HBOT /T, Vx ,Vy ,ω z )f ( n, PP / DP ,uTHR )0.5ρV 2L2T ,

where Cx BOT , Cy BOT , Cz BOT , Cmy BOT , Cmz BOT – ship hull-bottom interaction non-

dimensional hydrodynamic force (longitudinal, lateral and vertical) and their moments, HBOT is the sea depth at the ship gravity center, χ is the bottom

inclination, Vx,Vy ,ωz are the corresponding ship velocity components, n is the propeller RPM, PP/DP is the propeller pitch, uTHR is the thruster control signal.

The mathematical model of ship motion at shallow waters considers the depth influence on added masses as additional components λ11, λ22 и λ66 , on

propeller hull interaction coefficients (thrust deduction factor, wake scaling factor), propeller thrust and rudder lateral force.

The influence of the shallow water onto the propeller force is considered indirectly by changing the thrust deduction factor and wake scaling factor. The change of the lateral force on rudder is considered to be dependent on the propeller operating conditions and, consequently, on the flow speed behind the propeller.

The mathematical model allows to consider the ship manoeuvring at shallow waters taking into account the squat phenomenon.

Ship to Wall and Ship to Ledge Hydrodynamic Interaction Forces

When ship manoeuvring close to the quay the additional force FWALL appears on

her hull. To calculate the force the quay is modelled as smooth vertical wall. The value of the force depends on kinematics parameters, propellers and thruster operating conditions as well as parameters defining the ship position near the wall.

The last are the distance from midship to the wall (Y0) and angle between the wall and the central plane ξWALL (Fig. 18).

Structural formulas to additional force in body axes define are as follows:

longitudinal force due to wall influence

FxWALL =CxWALL (YWALL / B,HBOT /T,ξWALL , Vx ,Vy ,ω z )f ( n,PP / DP ,uTHR )0.5ρV 2LT

lateral force due to wall influence

Fy WALL = Cy WALL (YWALL / B,HBOT /T,ξWALL , Vx ,Vy ,ω z ) f ( n,PP / DP ,uTHR )0.5ρV 2LT ,

yaw moment due to wall influence

MzWALL = CmzWALL (YWALL / B,HBOT /T,ξWALL , Vx ,Vy ,ω z )f ( n,PP / DP ,uTHR )0.5ρV 2LT ,

– non-dimensional longitudinal and lateral force components and non-dimensional moment due to wall influence, YWALL / B – the

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Hydrodynamic Interaction Forces Mathematical Description

distance from the wall, HBOT/T – the relative depth, Vx,Vy ,ωz the ship velocity component, n – the propeller RPM, PP DP is the propeller pitch, uTHR – the thruster control signal.

It is possible to obtain the ship wall interaction if the wall has arbitrary shape. Then wall with ledge is modeled as isometric trapezium with bases h1, h2 and height l

(see Fig. 19).

The interaction force depends on ship motion kinematics parameters, propeller operating conditions, lateral distance between the ship gravity center and the ledge (Y0/B) and height (m0/L), parameters defining the ship position relatively to the wall, and the ledge dimensions – l (length), h1, h2 (height in the beginning and in the end of ledge).

Fig. 18. Forces Due To Vertical Wall Influence

Structural formulas for FWALL−1 components in body axis are as follows:

longitudinal force due to wall influence

FxWALL−1 = CxWALL−1(Y0 / B,				m0 / L, HBOT /T, ξWALL ,			l / L,	h1 / B,	h2 / B,Vx ,Vy, ω z ) -
f1(n,P	/ D	,u			,) 0.5ρV 2LT;
P	P		THR
lateral force due to wall influence
Fy WALL−1 = Cy WALL−1(Y0 / B,				m0 / L, HBOT /T, ξWALL ,			l / L,	h1 / B,	h2 / B,Vx ,Vy, ω z )
f1(n,P	/ D	,u			,) 0.5ρV 2LT;
P	P		THR
yaw moment due to wall influence
MzWALL−1 = CmzWALL−1(Y0 / B, m0 / L, HBOT /T, ξWALL , l / L, h1 / B, h2 / B,Vx ,Vy, ω z )
f1(n,P	/ D		,u	THR		,)0.5ρV 2L2T;
P	P

where CxWALL−1, Cy WALL−1, CmzWALL−1 – the non-dimensional longitudinal and lateral

components of force and non-dimensional moment of ship-ledge wall interaction, yWALL / B, m0 / L – the distance between the ship center of gravity and the ledge beginning in width and height, ξWALL – the angle between ship central plane and

wall, l / L, h1 / B, h2 / B – the relative ledge dimensions.

In equations system besides the additional hydrodynamic forces due to ship-wall interaction, the corresponding corrections of added masses λ11, λ22 и λ66 is

fulfilled.

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Hydrodynamic Interaction Forces Mathematical Description

The correcting coefficients are stored in the database.

Forces and moments due to the ledge effect on the bottom are considered in the same way.

Fig. 19. Forces Due To Vertical Ledged Wall Influence

Ship maneuvering close to the quay may be performed using the thrusters. Current mathematical model allows to consider the wall influence to the thrust value developed.

Ship to Wall and Ship to Bottom Interaction Force

The mathematical model of ship maneuvering in the channel considers the effect of kinematics parameters, of ship dimensions ratio, channel shape and dimensions, ship position relatively to the channel axis. Some additional parameters to channel form and ship position obtained (width, depth, wall inclination, distance between the ship and channel axis or between the ship and the wall and i.e.) areused.

Structural formulas to FCHAN components calculate:

longitudinal force due to channel form influence

FxCHAN = CxCHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z )

f	( n,P		/ D ,u		THR	)0.5ρV 2LT;
		P		P
lateral force due to channel form influence
Fy CHAN = Cy CHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z )
f ( n,P / D	P	,u		)0.5ρV 2LT;
P		THR
vertical force due to channel form influence
FzCHAN = CzCHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z ) ,
f ( n,P / D	P	,u		)0.5ρV 2LT;
P		THR

hydrodynamic roll moment due to channel form influence

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Hydrodynamic Interaction Forces Mathematical Description

Fig. 20. Forces Due To Channel Configuration Influence.

MxCHAN = CmxCHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z )

f ( n,P	/ D ,u				)0.5ρV 2L2T;
P	P		THR
hydrodynamic trim moment due to channel form influence
My CHAN = Cmy CHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z )
	f ( n,P			/ D ,u			THR	)0.5ρV 2L2T;
			P			P
hydrodynamic yaw moment due to channel form influence
MzCHAN = CmzCHAN (HCHAN /T, BCHAN / B, mCHAN / B, αCHAN , ξCHAN ,Vx ,Vy ,ω z )
f ( n,P	/ D	P	,u			) 0.5ρV 2L2T;
P			THR

where CxCHAN , Cy CHAN , CzCHAN , Cmx CHAN , Cmy CHAN , CmzCHAN – non-dimensional

components of hull-channel hydrodynamic interaction force (longitudinal, lateral and vertical) and their moments; BCHAN, HCHAN and αCHAN – channel width, depth and wall inclination; ξCHAN – the angle between ship and channel axes, mCHAN – the ship shift relatively to the channel axis; n , PP DP – propeller RPM and pitch; uTHR

– the thruster control signal.

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Hydrodynamic Interaction Forces Mathematical Description

The equations include the added masses’ changes λ11, λ22 и λ66 due to the wall

and channel bottom influence. Corresponding correcting coefficients are stored in the database.

Ship to Ship Interaction Force

Mathematical model considers additional force due to ship to ship interaction at meeting and passing. This additional force (hull interaction force) appears on the own ship hull and on the hull of passing ship as well. The forces values are depended on the own ship motion kinematics parameters (ship 1 at Fig. 21), on the passing ship motion kinematics parameters (ship 2 at Fig. 21) and on the relative positions of ships.

Ships’ relative position is defined by value YSHIP , which characterizes the lateral distance between ships hulls at midships and mSHIP , which characterizes longitudinal distance between the ships gravity centers.

Fig. 21. Forces Due To Ship to Ship Interaction

Structural formulas to additional values of force and moment calculate for one of the ship2: is as follows

longitudinal force due to ship to ship interaction
				~		~	ϕ, β1 ,V1	V2 ) f1(u1THR , u2THR ,P1 / D1,n1,
Fx SHIP1 = Cx SHIP1(xSHIP / L,ySHIP / B,
P / D	2	,n	2	) 0.5 ρV 2	L T ;
2				1	1	1

lateral force due to ship to ship interaction

2 – 1st vessel parameters are marked with index “1”, 2nd vessel parameters are marked with index “2”.

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Hydrodynamic Interaction Forces Mathematical Description

~

~

ϕ,

β1

,V1 V2 ) f1(u1THR , u2THR ,P1 / D1,n1,

Fy SHIP1 = Cy SHIP1(xSHIP / L,ySHIP / B,

P / D

2

,n

2

)0.5 ρV 2 L T ;

2

1

1

1

yaw moment due to ship to ship interaction

~

~

ϕ,

β1 ,V1 V2 ) f1(u1THR , u2THR ,P1 / D1,n1,

MzSHIP1 = CmzSHIP1(xSHIP / L,ySHIP / B,

P / D

2

,n

2

)0.5 ρ V 2

L2 T ;

2

1

1

1

whereCx SHIP 1,2 , Cy SHIP 1,2 , Cmz SHIP 1,2

– non-dimensional longitudinal and lateral

components and non-dimensional moment of the hulls interaction force;

~

~

3

,

xSHIP / L,ySHIP

/ B

– longitudinal and lateral distance between ships gravity centers

ϕ – the angle between central planes of ships, V1 V2 – the relative speed of

ships, u1THR, u2THR

– ships thrusters control signals, n1, n 2 – propellers RPM; β1

– the drift angle of the own ship; PP1 / DP1, PP2 / DP2 – the propeller pitch.

When two or more ships maneuvering in shallow water the influence of Freude depth number and ship principal dimensions (block coefficient, L/B ratio, and trim) are taken into account.

When more than two ships maneuvering nearby the interaction forces are calculated in pairs.

3	~	~	~	~	+ B2 ) ).
	– Parameters L and B		are calculated by formulas: L	= 0.5(L1 + L2 ),B= 0.5(B1

Chapter 3. Environment Mathematical Models and Resulting Aero/Hydro Forces Effects.

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CHAPTER 4

External Mechanic Forces

Mathematical Models

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