1. Dynamics of a body falling in waves
A. Friedman
G. Zilman
T. Miloh
Tel Aviv University
Faculty of Engineering - Israel

2. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive2
Background and motivations
Motion dynamics of 3D bodies falling through water
Y. Kim, Y. Liu & D. Yue. 17-th IWWWFB (2002).
Representative patterns of falling body trajectories observed in tank tests.

3. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive3
Equations of motion
2 2
[ ( )]Gm u vr wq x q r X− + − + =&
[ ( )]Gm v wp ur x qp r Y− + + + =& &
KqrIIpI yyzzxx =−+ )(&
( ), ,V u v w
r
( ), ,p q rω
r
( ) ( )yy xx zz GI q I I rp mx w uq vp M+ − − − + =& &
( ) ( )zz yy xx GI r I I pq mx v wp ur N+ − + − + =& &
[ ( )]Gm w uq vp x rp q Z− + + − =& &

4. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive4
Inertial forces
( ) ( ) 1
0 0 2
kl
ijk j i
B
Ed
M V U V U d x x dB
dt t t
ω
ω ω ω ε
∂∂
= − − × + − × − + × − +
∂ ∂ ∫
E
R R T J
r rr r r r rr r r
( ) ( ) ( )
1y
d
z L
F
C x a x v rx v rx dx
M x
ρ
   
= + +   
  
∫
( ) ( ) [ ]( )0
0 01 ,
DUdV
F B V U V U
dt Dt
ρ ω= − + + − × − + −T T T T E
rr
r r r r rr
Viscous effects
1
,
2
ji
ij
j i
uu
E
x x
 ∂∂
= + ÷ ÷∂ ∂ 
( )3 ,ijl i j k
s
J x x ds
n
φ +
∂
=
∂∫
1
2
i kli kl ikl kld J E J E= +
Non linear terms
( ) 0
0 0
0 0 0
0
B
y y
x s
y
D
F n dl dy dz
Dt x x y
n dl
y n
φ φ
ρ φ ρ
φ φ
ρ φ φ
∞
Σ
Σ
∂ ∂ ∂
′ = − −
∂ ∂ ∂
 ∂ ∂
− − ∇ ×∇ ÷
∂ ∂ 
∫ ∫∫
∫
Ñ

5. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive5
Results of numerical simulations
(calm water)
(a) (b.1) (b.2) (c) (d)

6. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive6
Steady current effects
0
2
4
6
8
10
12
14
16
18
20
0 10 20 30 40 50
x c /L
zc/L
Xg=0L
Xg=0.05L
Xg=0.2L
1.15
/ 6
m B
L a
ρ =
=

7. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive7
Progressive wave effects
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12 14
x c /L
zc/L
1.15
/ 6
m B
L a
ρ =
=
case no. 2
case no. 1

8. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive8
Currents and waves
0
2
4
6
8
10
12
14
16
18
20
-80 -60 -40 -20 0 20 40 60 80
x c /L
zc/L
-->
<--
1.15
/ 6
m B
L a
ρ =
=

9. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive9
Phase effects
-5
0
5
10
15
20
-10 -5 0 5 10 15
x c /L
zc/L
(kx-ωt)=0
(kx- ω t)= π
1.15
/ 6
m B
L a
ρ =
=

10. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive10
Bodies dropped off from a ship
1.15
/ 6
m B
L a
ρ =
=
2 knots
-5
0
5
10
15
20
-30 -20 -10 0 10 20 30 40 50 60
x c /L
zc/L

11. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive11
3D motion of a falling body in the
presence of a progressive wave
1.15m Bρ =
3D motion of a body falling in a
progressive wave. The body starts its
motion with a small velocity in the
transverse direction. Yaw and pitch angles of the body

12. 05/17/15 Dynamics of a body falling in waves Tel Aviv Unive12
Real time simulation
3D motion of a body falling in waves