13.021 – Marine Hydrodynamics, Fall 2004
Lecture 21
c 2004 MIT - Department of Ocean Engineering, All rights reserved.
Copyright °
13.021 - Marine Hydrodynamics
Lecture 21
Wave Forces on a Body
UP
U = ωA
Ul
ωAl
R=
=
ν
ν
UT
AωT
A
Kc =
=
= 2π
l
l
l





 A
F

CF =
=
f
 λ ,
ρgAl2
 |{z}

 Wave
steepness




l
h
, R, , roughness, . . .

λ
λ

|{z}


Diffraction
parameter
1
A
h
"
UT
"
Kc
A T
"
2
A
"
Type of Forces
Viscous forces,...
(form drag, viscous drag) :
f R , K c ,1.roughness
f (R, Kc , roughness, . . .).
Form drag (C
primarily
flow separation - normal stresses.
D ). Associated
ag – form (a)
associated
primarily
with
flowwith
separation
wake
Particle vel.
drag
#$
(b) Friction drag (CF ). F~ ∼
RR
τω dS
b.l.
body
S
τω
b.l.
U
forces arising from potential flow wave theory
'
2
$
p forces
gy 2 force, diffraction
% (Froude-Krylov
"
2. Inertial
force, radiation forc): forces arising from potential
t
&
#
flow wave theory.
1
If linear theory
Small amplitude waves
D+




ZZ
 ∂φ
~
Radiated
F = wave potential
pn̂dS where p = −ρ 
R
 ∂t + gy +


body
Diffracted wave

potential
ave
al
p
'
%
%
%
&
(wetted surface)
I
t
(a )
D
t
(b)

1
|∇φ|2
2
| {z }
=0
if linear theory,
small ampitude waves
R
t
(c)
$
"
gy ... "
"
#
2









In general:
φ=
φI
|{z}
+
(a) Incident wave
potential
∂φI
∂t
+
(b.1) Diffracted wave
potential

p = − ρ
φD
|{z}
∂φD
∂t
+
(a)
φR
|{z}
(b.2) Radiated wave
potential
+
(b.1)
∂φR
∂t

+ gy + . . .
(b.2)
(a) Incident wave potential: Froude-Krylov Force approximation, when l << λ, the incident wave
field is not significantly modified by the presence of the body, therefore ignore φD and φR :
φ ≈ φI
µ
Z Z
F~F K =
body
−ρ
|
¶
∂φI
+ gy n̂dS ⇐ can calculate knowing (incident) wave kinematics (and body geometry)
∂t
{z
}
pI
surface
Further mathematical approximationis valid if the body is really small. After applying the divergence theorem, the above integral can be replaced by:
Z Z Z
F~F K = −
∇pI d∀ ≈ −∇pI |
at body center
body volume
∀
|{z}
body volume
(b) Diffraction and Radiation Forces - hydrodynamic coefficients: added mass, wave damping and
wave excitation . . .
(b.1) Diffraction or scattering force: when l not << λ, wave field near body will be affected even if
body is stationary, so that no-flux B.C. is satisfied.
3
(b.1) Diffraction or scattering force: when ! not << ", wave field near body will be
affected even if body is stationary.
φI
φI
Stationary body
φD
φD
φD (b.1) Diffraction or scattering force: when ! not << ", wave field near body will be
∂
(φ I + φ D )
∂n
∂φ D I ∂φ I
=−
or
← givenStationary body
∂n
∂n
∂φ
∂n
φD
affected even if body is stationary. = 0 =
I
D
D
!
FD
!!
body
'
%%
&
$
"nˆdS
t "#
D
D
D
0
n
D
or
µ
Z Z
n
D
I
n
¶
I
n
given
∂φD
!
~D ='% D $"nˆdS
F
−ρ
n̂dS
F
D
added
mass
" damping
% and
∂t coefficient:
t
#
&
body
(b.2) Radiation Force –
!!
body
even in the absence of an incident wave,
body in motion creates waves and hence wave
forces,
and experiences
also
inertial
forces.
(b.2)
Radiation
Force – added
mass and damping
(b.2)
Radiation
Force - added
mass
and damping
coefficient:
even incoefficient:
the absence of an incident
in the
an incident
wave,
in motion creates
waves and
hence wave
wave, body in motion creates even
waves
andabsence
henceofwave
forces,
andbody
experiences
also inertial
forces.
forces, and experiences also inertial forces.
!
φR
φR
φR
!
U
R
n
!
U nˆ
φR
!
U
!!
!
∂φ R
= U ⋅ nˆ
∂n
!
'
$
%
"'% ˆ
!!
&
#&
'
'
Added mass
Added mass
µ
Z Z
F~R =
−ρ
∂φR
∂t
n̂dS = − mij U̇j − dij Uj
|{z}
|{z}
added
wave
mass
radiation
damping
4
Wave radiation damping
Wave radiation damping
¶
body
$
"ˆ
#
Important parameters
(1)Kc =
UT
l
=





2π Al
(2)diffraction parameter
l
λ
interrelated since maximum wave steepness:
 ¡ A ¢ ¡ l ¢ ≤ 0.07


 l λ
A
λ
≤ 0.07
• If Kc ≤ 1: no appreciable flow separation, viscous effect confined to b.l. (hence small), solve
problem via potential theory. In addition, depending on the value of the ratio λl :
– If λl << 1, ignore diffraction , wave effects in radiation problem (i.e., dij ≈ 0, mij ≈ mij infinite
fluid added mass). F-K approximation might be used, calculate F~F K .
– If
l
λ
>> 1/5, must consider wave diffraction, radiation ( Al ≤
0.07
l/λ
≤ 0.035).
• If Kc >> 1: separation important, viscous forces can not be neglected. Further on if:
l
0.07
l
≤
so << 1 ignore diffraction and F-K approximation might be used
λ
A/l
λ
1
F = ρl2 U (t) |U (t)|CD (R)
|{z}
2
relative
velocity
• Intermediate Kc - both viscous and inertial effects important, use Morrison’s formula.
1
F = ρl2 U (t)|U (t)|CD (R) + ρl3 U̇ Cm (R, Kc )
2
5
Summary:
I
Limiting case:
wave breaking occurs
II
III
I. Use: CD and F − K approximation.
II. Use: CF and F − K approximation.
III. CD is not important and F − K approximation. is not valid.
6