Prediction of Free Surface Water Plume as a
Barrier to Sea-skimming Aerodynamic
Missiles: Underwater Explosion Bubble
Dynamics
My-Ha D.1, Lim K.M.1, Khoo B.C.1, Willcox K.2
1
1
National University of Singapore,
2
Massachusetts Institute of Technology
Outlines






2
Problem of water barrier formation
Simulation of bubble and free surface interaction
Proper Orthogonal Decomposition (POD)
Water barrier simulation
Numerical results
Conclusions
Problem of water barrier formation
Motivation
3

A set of bubbles are created under the water surface

The free surface is pushed up due to the evolution of the
bubbles to create a water plume

The resultant water plume will act as an effective barrier. It
interferes with flying object skimming just above the free
surface
Water barrier by underwater explosions
S0
z
S
hs
r = 0 r = r01
r = r02
…
ws
4
r = r0No
r = r0i
…
r
Simulation of bubble and free surface
interaction
Mathematical formulation

z
n

FREE SURFACE
UNPERTURBED FREE SURFACE
(0, 0)

r


FLUID
BUBBLE
S

n
Bubble consists of vapor of
surrounding fluid and noncondensing gas
Non-condensing gas is
assumed ideal
Fluid in the domain  is
inviscid, incompressible and
irrotational
Potential flow satisfies Laplace
equation
 2  0
5
Simulation of bubble and free surface
interaction
Mathematical formulation

Integral representation
G  y, x  
  y 
cx x    
G  y, x    y 
dS

n 
 n
Green’s function:

Axisymmetrical formulation as in Wang et al.
G
6
1
G  y, x  
x y

 H
n
Simulation of bubble and free surface
interaction
Governing equations

Kinematic and dynamic boundary conditions
dx
 
dt
d

dt
d

dt

7

1
2
1
2
2
 V0 
2
    z        1
V 
  S
   2 z
  
2
Solving these equations gives the position and the velocity
potential of the nodes on the boundary
Bubble and free surface interaction
1
1
4
3
2
1
0.5
4
5
6 7
0.5
0
0
4
4
3
-0.5
5
-0.5
1
-1
z
z
2
6
-1
7
-1.5
-1.5
-2
-2
-2.5
-2.5
-3
-2
-1.5
-1
-0.5
0
r
0.5
1
1.5
2
-3
-2
-1.5
-1
-0.5
0
r
Evolution of bubble and free surface profile for
8
  1.25,   100,   0.3
0.5
1
1.5
2
Simulation of bubble and free surface
interaction
Linear superposition
Linear superposition of
free surfaces formed by
two bubbles at distance
exact free surface due to two bubbles with dr = 2.0
0.5
0.4
z
0.3
dr  2.0
0.2
0.1
0
-3
-2
-1
0
1
2
r
linear superposition of two surfaces with dr = 2.0
3

h
 1.25
Rm

Pb , 0
0.5
0.4
z
0.3
0.2
0.1
0
-3
9
-2
-1
0
r
1
2
3

Pref
 100
gRm
Pref
 0 .3
Proper Orthogonal Decomposition (POD)
Introduction

POD is also known as
–
–
–

10
Principle Component Analysis (PCA)
Singular Value Decomposition (SVD)
Karhunen-Loève Decomposition (KLD)
POD has been applied to a wide range of disciplines such as
image processing, signal analysis, process identification and
oceanography
Proper Orthogonal Decomposition (POD)
Basic POD formulation


Given a set of snapshots which are solutions of the system at
different instants in time


(
x
)
The basis
are chosen to minimize the truncation error
j
j 1
due to the construction of the snapshots using M basis functions


max


An approximation is given by
u,   2
,  

u,  2
, 
q
u x    a j  j x 
j 1

11
q  M & q  N
Given a number of modes, POD basis is optimal for constructing a
solution
Method of snapshots

The POD basis vectors can be calculated as
M
 j x    bi j ui  x 
i 1

j
The vectors b satisfies the modified eigenproblem
Rb  b where

1
ui , u j 
R ij 
M
Size of R is M x M where M is the number of snapshots
–
–
12
j  1,..., M
M is much smaller than N
N x N eigenproblem is reduced to M x M problem
Parametric POD





ηi
Snapshots u are taken corresponding to the parameter
value ηi i  1,..., M
Basic POD is applied on the set of snapshots
orthonormal basis  M
 i i 1
POD coefficient
to obtain the
a ηj i  ( j , u ηi )
η
a
POD coefficient j for intermediate value η not in the
ηi
a
sample obtained by interpolation of j
η
The prediction of u corresponding to the parameter value η is given by
q
13
u 
ηi M
i 1
u η   a ηj  j
j 1
q
u ( ,  , x)   a j ( ,  ) j ( x)
j 1
Water barrier simulation
Geometric feature
14
The problem is considered as an optimization problem that
minimizes the difference between constructed surface S and
desired one S0
Water barrier simulation
Optimization formulation





15
N0 bubbles are created
Free surface S is approximated by
linear superposition of Sk,
k = 1,…,N
Free surface Sk corresponding to the bubble located at lateral
k
k
k
position r0 , depth  with strength  is calculated using
parametric POD
dr  2.0
Lateral distance between two bubbles is
Strength and depth of the bubbles must be in the ranges of
interest
Offline Phase
Collecting snapshots

Run the described simulation code to collect snapshots

The snapshot is the free surface shape at the time of maximum
height

The ensemble contains 312 snapshots corresponding to 39
values of initial depth in the range [-1.25, -5.05] with interval
step of 0.1, and 8 values of strength in the range [100, 800]
with interval step of 100
16
Online Phase
Optimization formulation
17
Online Phase
Two-stage formulation

BP involves highly nonlinear representations of the mean of
snapshots, POD modes and the interpolation function of POD
coefficient

BP is difficult to solve exactly in a reasonable time even with
small number of bubbles involved

The problem is reformulated as two-stage formulation using the
POD feature that the approximated surface is dependent on the
depth and strength only through the POD coefficient
18
Stage 1: Determine bubble positions
Two methods:
(i) Greedy algorithm
(ii) Approximate Function (AF) algorithm
19
Stage 1: Greedy algorithm
i.
ii.
iii.
iv.
20
Try all possible placement points
Calculate the resultant free surfaces corresponding to each possible
placement points
Choose the point that minimizes the cost function
Repeat process for next bubble
Stage 1: Approximate function (AF) algorithm

The mean and the first POD mode are approximate by exponential functions
in the form of
f (r )  C1e

21
 C2 r 2
The coefficients C1, C2 are obtained by solving the problem
Approximate functions
The mean of snapshots
The first POD mode
0.12
0.05
0
0.1
-0.05
0.08
-0.1
0.06
-0.15
z
z
-0.2
0.04
-0.25
0.02
-0.3
0
-0.35
-0.02
-15
22
-10
-5
0
r
5
10
15
-0.4
-15
-10
-5
0
r
5
10
15
Approximate exponential functions (blue) compare quite well with
exact vectors (red).
Online Phase
Second-stage problem
r 
k N0
0 k 1
 k N 0 ,q
j k , j 1

Let

Stage 2 involves minimizing of a piecewise function and this is given by
the global minimum of the solutions of its piecewise function elements.

The bubble problem is fully solved when the lateral position, the depth and
the strength of the bubbles are determined.
23
and
( a ) 
be the solution of Stage 1
Numerical results
2D water barrier
Example 1
16
0.4
horizontal position
depth
strength /100
inception time 100*t
14
0.35
12
0.3
10
8
0.25
6
S 0.2
4
0.15
2
0.1
0
-2
0.05
-4
0
24
0
2
4
6
8
r
10
12
14
1
2
3
4
5
bubble No.
6
7
8
Numerical results
2D water barrier
Example 2
0.4
16
horizontal position
depth
strength /100
inception time 100*t
14
0.35
12
0.3
10
0.25
8
6
S 0.2
4
0.15
2
0.1
0
-2
0.05
-4
0
0
2
4
6
8
r
25
10
12
14
1
2
3
4
5
bubble No.
6
7
8
Numerical results
2D water barrier
Example 3
26
Numerical results
2D water barrier
Example 4
0.4
16
horizontal position
depth
strength /100
inception time 100*t
14
0.35
12
0.3
10
0.25
8
S 0.2
6
4
0.15
2
0.1
0
-2
0.05
-4
0
0
2
4
6
8
r
27
10
12
14
1
2
3
4
bubble No.
5
6
Numerical results
2D water barrier

Computation time for 2D problems
TABLE I
COMPUTATION TIME FOR 2D PROBLEMS
NO. OF BUBBLES
COMPUTATION TIME
5 bubbles
6 bubbles
7 bubbles
8 bubbles
Full simulation of 1 bubble
14.0s
16.0s
20.0s
25.0s
60.0s
Computation time for solving two-dimensional optimization
problem in the domain of [0,14], 141 grid points (grid size 0.1) using
AF algorithm. Solution is obtained by running a LOQO program on a
Intel Xeon 2.8GHz processor, RAM 2.0Gb

28
Computation time increases fairly linearly with no. bubbles
Numerical results
3D water barrier
Desired surface
0.4
0.3
S
0.2
0.1
0
8
6
8
6
4
y
29
4
2
2
0
0
x
Numerical results
3D water barrier
Constructed free surface corresponding to 6 bubbles
0.4
0.3
S
0.2
0.1
0
8
6
8
6
4
y
30
4
2
2
0
0
x
Numerical results
3D water barrier
Parameters for 6 bubbles
12
9
8
depth
strength /100
inception time 100*t
10
3
7
6
8
6
6
5
y 4
2
4
5
3
2
2
0
1
1
4
-2
0
-4
-1
-1
31
0
1
2
3
4
x
5
6
7
8
9
1
1.5
2
2.5
3
3.5
4
bubble No.
4.5
5
5.5
6
Numerical results
3D water barrier
Constructed free surface corresponding to 8 bubbles
0.4
0.3
S
0.2
0.1
0
8
6
8
6
4
32
y
4
2
2
0
0
x
Numerical results
3D water barrier
Parameters for 8 bubbles
9
12
8
4
depth
strength /100
inception time 100*t
10
8
7
8
6
3
5
6
7
4
y 4
3
2
2
6
2
0
1
-2
1
0
5
-4
-1
-1
33
0
1
2
3
4
x
5
6
7
8
9
1
2
3
4
5
bubble No.
6
7
8
Numerical results
3D water barrier
Constructed free surface corresponding to 10 bubbles
0.4
0.3
S
0.2
0.1
0
8
6
8
6
4
y
34
4
2
2
0
0
x
Numerical results
3D water barrier
Parameters for 10 bubbles
12
9
8
7
4
depth
strength /100
inception time 100*t
10
8
9
8
6
3
5
6
7
4
y 4
2
3
2
6
2
0
1
-2
10
1
0
5
-4
-1
-1
35
0
1
2
3
4
x
5
6
7
8
9
1
2
3
4
5
6
bubble No.
7
8
9
10
Numerical results
3D water barrier

Computation time for 3D problems
TABLE II
COMPUTATION TIME FOR 3D PROBLEMS
NO. BUBBLES
COMPUTATION TIME
6 bubbles
8 bubbles
10 bubbles
12 bubbles
14 bubbles
Full simulation of 1 bubble
(1717 nodes)
24.0s
32.0s
50.0s
83.0s
170.6s
280s
Computation
time
for
solving
three-dimensional
optimization problems with different numbers of bubbles in a
domain of [0 8]x[0 8] grid 17x17 using AF algorithm.
Solution is obtained by running a LOQO program on a Intel
Xeon 2.8GHz processor, RAM 2.0Gb

36
AF algorithm is still efficient for reasonable size problems
Conclusions




37
The problem of water barrier formation can be posed as an
optimization problem coupled with underwater gas bubbles
and free air-water interface
POD with linear interpolation in parametric space is an
excellent tool for constructing a reduced-order model
Solution algorithm is very efficient for 2D problems and
reasonable size 3D problems
The optimization problem may has many local minima. A
robust procedure for finding initial guess may result in a
better final solution