2 019 D
2.019
Desiign off O
Ocean Systems
t
Lecture 11
Drift and Slowly
y-Varying
y g Loads (I)
)
March 11, 2010
Drift (or Mean) Forces/Moments
• Wind: Steady (and unsteady) drag forces/moments
• Current: Steady drag forces/moments
• Waves: Nonlinear wave loads including steady and unsteady loads
Imp
portant for desig
gn of mooring
g syystem for station keep
ping
g !!
Wind Drag Forces/Moments
Image by MIT OpenCourseWare.
Fwind =
1
2 ρair Cs CH V10 |V10 |S
Mwind =
P
Fwind × h
Cs: shape coefficient (or Drag coefficient), Cs=1.0 for large flat surface
CH: height coefficient, CH=1.0 for FPSO
V10: wind velocity at 10m above sea surface
S: Project area of the exposed surfaces in the vertical or the heeled condition
h: vertical distance between center of wind force and center of resistance (by
mooring lines, etc)
Current Drag Forces and Moments
For normal incidence,
incidence
Uf
Fcurrent
• Body is fixed:
Fcurrentt = 12 ρw CD Uf |Uf |S
CD: Drag coefficient
Uf: Current velocity
S: Project area of the exposed surfaces
ρw: water density
• Body moves at Ub:
CD of cylinder
1.4
1.0
0.6
0.2
1/10000
Fcurrent = 12 ρw CD (Uf − Ub )|Uf − Ub |S
• Frictional drag coefficient on a ship hull:
Cf =
1/1000
1/100
Roughness k/D
1/10
Image by MIT OpenCourseWare.
0.075
(log10 (Rn)−2)2
Rn: Reynolds number in
terms of ship length L
Current Drag Forces and Moments
I oblique
In
bli
iincident,
id t
Uf
Ft
Uf
β
Un
Ut
Fn
Normal component Un causes flow separation drag, tangential component Ut
causes frictional drag
Fn = 12 ρCD Un |Un |S
Ft = 12 ρCf Ut |Ut |St
Valid for β in the range of 30o to 150o.
Wave Drift (Mean) Force/Moment
Incident wave:
z
ηI = a cos(ωt − kx)
y
x
Wave drift force/moment comes from:
(1) 2nd-ord
der h
hyd
drod
dynamiic pressure due tto tth
he fifirstt ord
der wave
(2) Interaction between the first-order motion and the first-order wave
2nd-order Hydrodynamic Mean Pressure
Consider a simple plane progressive wave in deep water:
−kz
Φ(x z,
Φ(x,
z t) = gA
e
sin(ωt − kx)
kx)
ω
η(x, t) = a cos(ωt − kx)
We look at the pressure field of the wavefield:
P (x,z,t)
ρ
1
= − ∂Φ
∂t − 2 ∇Φ · ∇Φ − gz
∇Φ · ∇Φ = Φ2x + Φ2y + Φ2z
n
o2
gAk kz
2
Φx = − ω e cos(ωt − kx) = (ωAekz )2 cos2 (ωt − kx)
n
o2
gAk kz
2
Φz =
sin(ω t − kx) = (ωAe
(ωAekz )2 sin2 (ωt − kx)
ω e
∇Φ · ∇Φ = Φ2x + Φ2z = (ωAekz )2
Steady, independent of time,
and ~A2.
2nd-order mean pressure component: ρA2 (ωekz )2
Integration this mean pressure component over body surface to give mean force/moment
Interaction Between Body Motion and First-Order Wave
F~ (t) =
R
S(t)
P (t)~nds =
Source: Faltinsen, O. M. Sea Loads on Ships and Offshore Structures.
Cambridge University Press, 1993. © Cambridge University Press.
All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
R
S0
R
P (t)~nds +
S0
R
∆S(t)
P (t)~nds
P ~nds
2nd-order mean pressure effect
Direct Pressure Integration
Drift
D
ift force
f
can be obtai
bt ined
db
by di
directt iinttegratition off th
the pressure over th
the body
d
surface and then taking time average:
F~ (t) =
R
P (t)~n
nds
ds =
S(t)
R
S(t)
©
ª
−ρgz − ρ ∂∂Φt − ρ2 (Φ2x + Φ2y + Φ2z )2 ~nds = · · · · · ·
For example,
Φ((x, z, t)) =
Z
2gA kz
ω e
cos ωt cos kx
standing wave
η(x, t) = 2A sin ωt cos kx
The first two terms’
terms contribution:
x
−ρg
Rη
0
zdz − ρ
¯
∂Φ ¯
∂t z=0
η = ρgA2
The third term’s
term s contribution:
ρ
−
2
Diagram showing drift forces and incident
waves on a vertical wall, Z. Image by MIT
OpenCourseWare.
2
The total horizontal drift force on the wall is: 1
ρgA
2
Z
0
−∞
(Φ2x + Φ2y + Φ2z )2 ds
Z
ρ 0 1 4g 2 A2 2 2kz
k e dz
=−
2 −∞ 2 ω 2
1
= − ρgA2
2
Far-Field Formula Mean force or moment on a floating body can also be obtained using the soso
called far-field formula developed from the momentum theorem.
Momentum theorem:
z
η_
η
8
8
x
S_
S
8
8
Control Volume
Reflected Waves
Transmitted Waves
y
AR
A
AT
Image by MIT OpenCourseWare.
Fx =
ρg
2
4 (A
+ A2R − A2T )
P
x
Fx (t) + F M = dM
dt
P
P
x
Fx (t) + F M = dM
dt
Fx is th
the force actiting on conttroll
volume, FM is total momentum flux
into the control volume, Mx is linear
momentum in control volume
dMx
dt
z
Incident Waves
P
=0
Fx contains force on body, force
on S-∞ and S+∞
FM contains momentum flux
into control volume from
boundaries S- and
∞ S+∞
Since A2 = A2R + A2T , we finally have Fx =
ρg 2
2 AR
•Mean force/moment is 2nd-order in wave amplitude
_
F2
ρgζa2
F2 is the horizontal
f
force
in
i wave
direction on the 2D
body
ω
_
D
g
F2 is the horizontal force in wave direction on the 2D body;
D = draught, ζa = wave amplitude of the incident waves,
ω = circular frequency of oscillations.
Image by MIT OpenCourseWare.
Mean Force/Moment in Irregular Sea
F̄j = HF̄j (ω)A2 , j = 1, . . . , 6
¯
where HF
F̄j is the transfer function for mean force Fj
In irregular sea,
F̄j =
N
X
HF̄j (ωi )A2i
j = 1, . . . , 6
i=1
Ai is the wave amplitude of the i-th wave component.
p
Ai =
F̄j
F
=
N
X
2 S ( ω i ) ∆ω
HF¯j (ωi )2S ((ωi )∆ω
j = 1,
1 ...,6
i=1
=
2
Z
0
∞
S (ω )HF̄¯j (ω )dω
j = 1,
1 ...,6
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2.019 Design of Ocean Systems
Spring 2011
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