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CHAPTER 3

Environment Mathematical

Models and Resulting

Aero/Hydro Forces Effects

Copyright Transas Marine Ltd. 2003

Wind Forces

According to the forces classification shown on Fig. 14, external aeroand hydrodynamic forces include:

•forces due to wind influence (wind forces);

•forces due to current influence (current forces);

•forces due to waves influence (wave forces);

•forces due to shallow waters and wall influence(ship to bottom and ship to wall interaction force);

•forces due to channel configuration influence;

•forces due to another ship influence (ship to ship interaction force).

It is convenient to divide the forces mentioned above in two groups: forces relating to the external and weather conditions (wind, waves, current) and forces relating to hydrodynamic interaction process (shallow waters, wall, another stationary or moving object).

External forces are obtained as the additional component of forces on deep calm water.

The mathematical description of the external forces is shown below.

WIND FORCES

The air flow around the ship is considered as uniform flow of constant direction and velocity. The true wind velocity at given Boutfort number (according to the Boutfort scale) is obtained as average value of wind velocity at 6m height above the sea level.

Fig. 14. Wind Forces. Water Plane

Formulas for calculating the aerodynamic forces and moments are as follows:

aerodynamic longitudinal force

Fx A = [CxAH (ϕwk ) + dCxA (ϕwk ,AxN )]0.5 ρ AVA 2 (Ax + AxN )

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

51

Wind Forces

aerodynamic lateral force

Fy A = [CyAH (ϕwk ) + dCyA (ϕwk ,AyN )]0.5 ρ AVA 2 (Ay + AyN )

aerodynamic vertical force

Fz A = [CzAH (ϕwk ) + dCzA(ϕwk ,AyN )]0.5 ρ AVA 2 (Az + AyN )

aerodynamic yaw moment

Mz A = [CmzAH (ϕwk ) + dCmzA (ϕwk ,AyN )]0.5 ρ AVA 2 (Ax + AyN )L

aerodynamic roll moment

Mx A = [CmxAH(ϕwk ) + dCmxA (ϕwk ,AxN )]0.5 ρ AVA 2 (Ax + AxN )L

where

Cx AH , Cy AH , Cz AH – non-dimensional aerodynamic force components (longitudinal, lateral, vertical), Cmx A , Cmx A – non-dimensional aerodynamic moment components in body axis (yaw and roll aerodynamic moments);

dCx A , dCy A , dCz A – the additional values of non-dimensional aerodynamic forces due to the superstructures, dCmx A , dCmx A – the additional values of nondimensional aerodynamic moment due to the superstructures;

VA , ρ A – relative wind velocity and relative wind angle velocity;

Ay , Ax the above water hull area projected to the central plane and to the midship plane;

AxN, AyN the superstructure area projected to the central plane and to the midship plane.

The value of relative wind velocity is defined as the sum of two components: wind velocity in fixed axis and ship velocity (see Fig. 14). The calculation formula is as follows:

VA = Vx A 2 +VyA 2 ,

where Vx A = Vx w − Vx and Vy w = Vw cos(ϕ − ϕw ) – the longitudinal and lateral wind velocity components in body axis, ϕwk is the relative wind angle. Relative wind

angle is calculated by formula ϕwk = arctg(Vy + V y w ) .

Vx + Vx w

The value of the aerodynamic characteristics can be calculated using the tests results or approximately, using the database.

Structural formulas for complete aerodynamic characteristics calculating (characteristics of the hull and superstructures) are defined by functions represented by partial sums of Fourier series:

non-dimensional longitudinal aerodynamic force

52 NAVI-TRAINER 4000. Mathematical Models. Technical Description.

Wind Forces

5

5

∑

axk cos(kϕwk ) + 0.5 ax6 cos(6 ϕwk ) +

∑

Cx A (ϕwk ,L ) = 0.5ax0 +

bxk sin(kϕwk ) kA ,

k =1

k =1

~0

~1

~2

2

,

where axk = axk

+ axk L

+ axk L

~0

~1

~2

2

,

bxk = bxk

+ bxk L

+ bxk L

non-dimensional lateral aerodynamic force

5

5

∑

ayk cos(kϕwk ) + 0.5ay 6 cos(6 ϕwk ) +

∑

byk

Cy A (ϕwk ,L ) = 0.5ay0 +

sin(kϕwk ) kA +

k =1

k=1

3

∑

(q0 + q1

L + q2 L 2 )sin(kϕ

,

yk

yk

yk

wk

AN

k=1

~0

~1

~2

2

,

ayk = ayk

+ ayk L

+ ayk L

~0

~1

~2

2

,

byk = byk

+ byk L

+ byk L

non-dimensional vertical aerodynamic force

~0

~1

~2

2

,

azk = azk

+ azk L

+ azk L

~0

~1

~2

2

,

bzk = bzk

+ bzk L

+ bzk L

non-dimensional yaw aerodynamic moment

5

5

∑

amzk cos(kϕwk ) +

0.5amz6 cos(6 ϕwk ) +

∑

Cm zA (ϕwk ,L ) = 0.5amz0 +

bmzk sin(kϕwk ) k1A +

k=1

k =1

3

∑

(qmzk0 + qmzk1 L + qmzk2

k1AN ,

L 2 )sin(kϕwk )

k =1

~0

~1

~2

2

L

,

amzk = amzk + amzk L + amzk

~0

~1

~2

2

L

,

bmzk = bmzk + bmzk L + bmzk

non-dimensional roll aerodynamic moment

Cm xA

amxk

bmxk

5

(ϕwk ,

) = [0.5amx0 + ∑amxk cos(kϕwk ) + 0.5amx6 cos(6 ϕwk ) +

L

k =1

5

3

∑

b

sin(kϕ

)]k1

+

∑

(q0

+ q1

L + q2

L 2 )sin(kϕ

mxk

wk

A

mxk

mxk

mxk

wk

AN

k =1

k=1

~0

~1

~2

2

,

= amxk + amxk L

+ amxk L

= b~0

+ b~1

+ b~2

2

,

L

L

mxk

mxk

mxk

where L = L / H is the hull aspect ratio.

Coefficients ax0, ax1, K , ax5, bx1, K , bx5, ay 0, ay1, K , ay 5, K are depended on the superstructures shape and dimensions.

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

53

Current Forces

Coefficients kA, kAN , k1A, k1AN are depended on the superstructures area and above-water hull area. Coefficients are provided in the database.

CURRENT FORCES

The current is modeled as a stationary flow with given velocity distribution. Velocity distribution in depth is not considered.

Fig. 15. Current Forces. Water Plane

The current is determined as the map (see Fig. 15).

Forces and moments due to the current are defined as a sum of two components: current the forces and moments in steady uniform flow and the forces and moments due to current irregularity.

The calculations can be fulfilled using the formulas:

longitudinal current force

F

= C

yBH

(β

Ck

)0.5 ρV

2LT + F

,

y C

Ck

yC

current yaw moment

M

zC

= C

mzBH

(β

Ck

)0.5 ρV

2L2T +

M

zC

,

Ck

current roll moment

Mx C

=

Mx C

where CxBH , CyBH , CmzBH

– the above mentioned non-dimensional components of

the hydro-dynamic force at bare hull when moving in uniform current; Vck – the ship relative velocity, β ck – the relative drift angle, FyC , MzC , MxC – additional

lateral force and yaw and roll moments due to the current irregularity.

54 NAVI-TRAINER 4000. Mathematical Models. Technical Description.

Current Forces

Additional forces and moments due to the current irregularity can be calculated using the flat section method depending on the hull frame shape and depending on the current speed in the considered area (local velocity). The formulas are as follows:

additional lateral force

N

FyC = ∑ FyC i ( Vk i ) , i =1

additional yaw moment

N

MzC = ∑ FyC i ( Vk i )xi , i =1

additional roll moment

N

MxC = ∑ FyC i ( Vk i ) zi , i =1

where i – section number, FyC i – the force acting in the i-section, Vk i – the velocity component normal to the ship central plane; xi , zi – the X-coordinate of the section and distance from the free surface to the point situated at half frame height.

Force	FyC i can be calculated using the value of non-dimensional lateral force
CyC i . The formula is as follows:
F	= C	yC i	(B T )0.5ρ( V	k i	)2 .
yCi

The ship speed value in true axis can be obtained by formula:

VCk = (Vx C + Vx )2 +(Vy C + Vy )2 ,

where Vx C = VC sin(βCk − ϕC ) – the longitudinal component of current velocity in body axis;

Vy C = VC cos(βCk − ϕC ) – the lateral component current velocity in body axis, βCk – the relative drift angle.

Relative drift angle is calculated by the formula:

β		Vy + VyC
	Ck	= arctg			.
		Vx + V xC

Local velocity value is calculated by formulas:

Vx C i = VxC i + Vx C ,

Vy C i = VyC i + Vy C ,

where VxC i and VyC i , VxC and Vy C – the local current velocity components and the ship velocity.

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

55

Wave Forces

WAVE FORCES

Sea roughness is modeled as a stationary process with spectral characteristics corresponding to the real sea waves characteristics (at point – Pirson – Moskovits spectrum; at direction – Artur spectrum).

A 3D polyharmonic non-regular wave model is used. The roughness is defined by the height H1 3 and general sea direction ϕwv .

Wave surface is obtained as a sum of harmonics. It can be defined by formula:

N	N
ζ (xg , yg ,t) = ∑ζ i (x, y,t) =∑ Ai cos(kxi x + kyi y − ω i t + φi ) ,
i =1	i =1
where i – the harmonic’s number, ζ – the wave surface Z-coordinate, N – the
quantity of harmonics, Ai	– the amplitude of i-harmonic, ki – the wave number, ωi

– the frequency of i-harmonic, ψ i – the direction of i-harmonic, φi is the phase of i-

harmonic. The wave surface shape are defined according to the sea roughness spectral characteristics.

The sea roughness model consists of 20 harmonics.

When the ship motion in shallow waters obtained, the wave height, wave length, wave crest velocity are changed in compared with the corresponding values at deep waters.

Forces and moments due to the waves influence are calculated according to the following formulas:

longitudinal force

FxWAVE = F1xWAVE (t) + F2xWAVE + XWAVE

lateral force

Fy WAVE = F1y WAVE (t) + F2y WAVE

vertical force

FzWAVE = F1zWAVE (t)

roll moment

Mx WAVE = M1xWAVE (t)

trim moment

My WAVE = M1y WAVE (t)

yaw moment

MzWAVE = M1zWAVE (t) + M2zWAVE

56 NAVI-TRAINER 4000. Mathematical Models. Technical Description.

Wave Forces

Fig. 16. Wave Forces at Rough Sea

Wave forces F1xWAVE , F1y WAVE , F1zWAVE , and wave moments

M1xWAVE , M1y WAVE , M1zWAVE are the function of time and can be calculated using formulas:

wave longitudinal force

n

F1xWAVE = −m ∑ Fx(λi, ξi) γi(t),

i =1

wave lateral force

n

F1y WAVE = −m ∑ Fy(λI, ξi) γi(t),

i =1

wave vertical force

			n
F1zWAVE = −m ∑ Fz(λi, ξi) ξi(t),
			i =1
wave roll moment
						n
M	= (J	x	+ λ	44	)n2	∑ Fmy(λi, ξi) γi(t),
1x WAVE					θ

i =1

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

57

Wave Forces

wave trim moment

			n		γ i (t)
	M1y WAVE = −m ∑ Fmy(λi, ξi)					,
					ki
			=
		i		1
	wave yaw moment
				n
	M	= −g J	z	∑ Fmz(λi, ξi) ζi(t) ki2,
	1zWAVE

i =1

where Fx(λi, ξi), Fy(λi, ξi), Fz(λi, ξi), Fmx(λi, ξi), Fmy(λi, ξi), Fmz(λi, ξi) are the reducing coefficients depending on the ship draught, on the ratio of ship length to wave length

of i-harmonic, ξi – wave encounter angle, λi – the wave length of i-harmonic, γ i – slope of surface.

Additional resistance in wave depends on the values of average parameters XWAVE = f (ζ ,H13,V ) , where V is the ship velocity.

Slope of surface is calculated using the formula:

N

γ i (xg ,yg ,t) = ∑Ai ki sin(kxi x + kyi y − ωi t + φi ) i =1

Forces and moments components designated as F2x wave,F2y wave, M2zwave (wave

drift forces) define the time-independent wave forces and are calculated using formulas:

longitudinal force

F 2x WAVE = ρgL2H123 Ix (ξ ,H1 3 )

lateral force

F 2y WAVE = ρgL2H123 Iy (ξ,H1 3 )

yaw moment

M 2zWAVE = ρgL2H123 Iz (ξ ,H1 3 )

where Ix (ξ,H13 ) , Iy (ξ,H13 ) , Imz (ξ ,H13 ) are non-dimensional coefficients depending on the ship hull form and size, H1 3 – significant height.

58 NAVI-TRAINER 4000. Mathematical Models. Technical Description.