my-ha d. 1 , lim k.m. 1 , khoo b.c. 1 , willcox k. 2

1

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 National University of Singapore, 2 Massachusetts Institute of Technology

2

Problem of water barrier formation Simulation of bubble and free surface interaction Proper Orthogonal Decomposition (POD) Water barrier simulation Numerical results Conclusions

Outlines

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

Motivation

Problem of water barrier formation

4

Water barrier by underwater explosions

rr = 0 r = r01 r = r0

2 r = r0No

ws

S0

S

r = r0i

… …

hs

z

5

Bubble consists of vapor of surrounding fluid and non-condensing gas

Non-condensing gas is assumed ideal

Fluid in the domain is inviscid, incompressible and irrotational

Potential flow satisfies Laplace equation

FREE SURFACE

UNPERTURBED FREE SURFACE

(0, 0)

FLUID

SBUBBLE

n

n

r

z

02

Simulation of bubble and free surface interaction

Mathematical formulation

6

Integral representation

Green’s function:

Axisymmetrical formulation as in Wang et al.

dSxyG

yxyGy

xxcnn

,,

Hn

G

yx

xyG

1

,


Mathematical formulation

7

Kinematic and dynamic boundary conditions

Solving these equations gives the position and the velocity potential of the nodes on the boundary


Governing equations

dt

dx

1022

21

V

Vz

dt

d

zdt

d 22

21

S

8

Bubble and free surface interaction

Evolution of bubble and free surface profile for

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

r

z

432 1

1

2

3

4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

rz

4

4

5

5

6

6

7

7

3.0,100,25.1

9

Linear superposition

Linear superposition of free surfaces formed by two bubbles at distance 0.2dr

-3 -2 -1 0 1 2 3

0

0.1

0.2

0.3

0.4

0.5

linear superposition of two surfaces with dr = 2.0

r

z

-3 -2 -1 0 1 2 3

0

0.1

0.2

0.3

0.4

0.5

r

z

exact free surface due to two bubbles with dr = 2.0

3.0

100

25.1

0,

ref

m

ref

b

m

P

gR

P

P

R

h


10

POD is also known as– 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)Introduction

11

Given a set of snapshots which are solutions of the system at different instants in time

The basis are chosen to minimize the truncation error due to the construction of the snapshots using M basis functions

An approximation is given by

Given a number of modes, POD basis is optimal for constructing a solution

1

)(jj x

,

,

,

,max

22uu

Proper Orthogonal Decomposition (POD)Basic POD formulation

q

jjj xaxu

1NqMq &

12

The POD basis vectors can be calculated as

The vectors satisfies the modified eigenproblem

Size of R is M x M where M is the number of snapshots– M is much smaller than N– N x N eigenproblem is reduced to M x M problem

M

ii

jij xubx

1

jb

Mj ,...,1

bb R jiij uuM

,1

Rwhere

Method of snapshots

13

Snapshots are taken corresponding to the parameter

value

Basic POD is applied on the set of snapshots to obtain the

orthonormal basis

POD coefficient

POD coefficient for intermediate value not in the

sample obtained by interpolation of The prediction of corresponding to the parameter value is given by

iηiηu

Parametric POD

Mi ,...,1

Miηiu 1 Mii 1

),( ii ηj

ηj ua

ηiηja

ηja

ηu

q

jj

ηj

η au1

η

q

jjj xaxu

1

)(),(),,(

14

Water barrier simulationGeometric feature

The problem is considered as an optimization problem that minimizes the difference between constructed surface S and desired one S0

15

N0 bubbles are created Free surface S is approximated by linear

superposition of Sk, k = 1,…,N

Water barrier simulation

Free surface Sk corresponding to the bubble located at lateral position , depth with strength is calculated using parametric POD

Lateral distance between two bubbles is Strength and depth of the bubbles must be in the ranges of

interest

kr0k k

0.2dr

Optimization formulation

16

Offline PhaseCollecting 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

17

Online PhaseOptimization formulation

18

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

Online Phase

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Stage 1: Determine bubble positions

Two methods: (i) Greedy algorithm (ii) Approximate Function (AF) algorithm

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Stage 1: Greedy algorithm

i. Try all possible placement points

ii. Calculate the resultant free surfaces corresponding to each possible placement points

iii. Choose the point that minimizes the cost function

iv. Repeat process for next bubble

21

Stage 1: Approximate function (AF) algorithm

The mean and the first POD mode are approximate by exponential functions in the form of

The coefficients C1, C2 are obtained by solving the problem

22

1)( rCeCrf

22

Approximate functions

-15 -10 -5 0 5 10 15-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

r

z

-15 -10 -5 0 5 10 15-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

r

z

The mean of snapshots The first POD mode

Approximate exponential functions (blue) compare quite well with exact vectors (red).

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Second-stage problem

Let and be the solution of Stage 1

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.

0

10

N

kkr

qN

jk

kja

,

1,

0)(

Online Phase

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Numerical results

Example 1

2D water barrier

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r

S

1 2 3 4 5 6 7 8

-4

-2

0

2

4

6

8

10

12

14

16

bubble No.

horizontal positiondepthstrength /100inception time 100*t

25

Numerical results2D water barrier

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r

S

1 2 3 4 5 6 7 8

-4

-2

0

2

4

6

8

10

12

14

16

bubble No.


Example 2

26


Example 3

27

Numerical results

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r

S

Example 4

2D water barrier

1 2 3 4 5 6

-4

-2

0

2

4

6

8

10

12

14

16

bubble No.


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Computation time for 2D problems

Computation time increases fairly linearly with no. bubbles

TABLE I COMPUTATION TIME FOR 2D PROBLEMS

NO. OF BUBBLES COMPUTATION TIME

5 bubbles 14.0s 6 bubbles 16.0s


Full simulation of 1 bubble 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

29


Desired surface

02

46

8

0

2

4

6

80

0.1

0.2

0.3

0.4

xy

S

30


Constructed free surface corresponding to 6 bubbles

02

46

8

0

2

4

6

80

0.1

0.2

0.3

0.4

xy

S

31


Parameters for 6 bubbles

-1 0 1 2 3 4 5 6 7 8 9-1

0

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

x

y

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-4

-2

0

2

4

6

8

10

12

bubble No.

depthstrength /100inception time 100*t

32 02

46

8

0

2

4

6

80

0.1

0.2

0.3

0.4

xy

S



33



-1 0 1 2 3 4 5 6 7 8 9-1

0

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

x

y

1 2 3 4 5 6 7 8

-4

-2

0

2

4

6

8

10

12

bubble No.


34



02

46

8

0

2

4

6

80

0.1

0.2

0.3

0.4

xy

S

35



-1 0 1 2 3 4 5 6 7 8 9-1

0

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

10

x

y

1 2 3 4 5 6 7 8 9 10

-4

-2

0

2

4

6

8

10

12

bubble No.


36


Computation time for 3D problems

AF algorithm is still efficient for reasonable size problems

TABLE II COMPUTATION TIME FOR 3D PROBLEMS

NO. BUBBLES COMPUTATION TIME


10 bubbles 50.0s 12 bubbles 83.0s 14 bubbles 170.6s

Full simulation of 1 bubble (1717 nodes)

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

37

Conclusions

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

my-ha d. 1 , lim k.m. 1 , khoo b.c. 1 , willcox k. 2

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