Optimization of the G&H bubble modeldownloads.hindawi.com/journals/sv/2005/430767.pdfto the case of no bubble distortion, thereby limiting the motion to dilation plus translation

Shock and Vibration 12 (2005) 3–8 3IOS Press

Optimization of the G&H bubble model

Thomas L. Geers∗ and Chung-Kyu ParkDepartment of Mechanical Engineering University of Colorado, Boulder, CO, USA

Abstract. A spherical model for the bubble created by an underwater explosion is enhanced to account, in approximate fashion,for the effect of bubble distortion on translation and dilation. The enhancement consists of introducing artificial drag in the formC|ν|P , whereν is translation velocity, andC, P produce an optimum fit to empirical formulas for the second dilation maximumand first two translation jumps. The recommended values areC = 0.4 andP = 1 for charge weights between 100 lb–1000 lband and depths exceeding 200 ft.

Keywords: Underwater explosions, bubble dynamics

1. Introduction

Recently, Geers and Hunter developed a bubble bubble model for a very deep to moderately deep underwaterexplosion that integrates the shockwave and oscillation phases of the motion [2]. The model was then specializedto the case of no bubble distortion, thereby limiting the motion to dilation plus translation. The specialized modelproduced bubble-motion histories exhibiting predictive capability superior to that of earlier bubble models.

The principal limitation of the specialized model is that its neglect of bubble distortion results in the over-predictionof bubble translation, which in turn affects bubble dilation. Hicks reviously attempted to remedy such deficiencyby introducing the hydrodynamic drag forceFD(t) = πa2(t) · 1

2ρ�D(t), in whicha(t) is the bubble radius,ρ� isthe density of the liquid surrounding the bubble, andD(t) = CDu2(t), whereu(t) is the translation velocity of thebubble [3]. Hicks recommendedCD = 2.25, which is an order of magnitude larger than the typical value for a rigidsphere at high Reynolds number. Hunter and Geers found that neither this nor any otherC D-value simultaneouslyyields accurate dilation and accurate translation at moderate depths [4].

In principle, the above limitation of the specialized model can be relieved by using the boundary element methodto implement the more general model. However, this increases the number of degrees of freedom by one or twoorders of magnitude. Furthermore, the resulting non-spherical collapse process is difficult to treat [1]. Hence, itis worthwhile to seek an approximate technique that accurately accounts for the effects of bubble distortion ontranslation and dilation within the confines of the specialized model. This is the motivation for the work reportedhere.

2. Equations of motion for the specialized bubble model

With a(t) and u(t) as the spherical bubble’s radius and vertical displacement, withφD(t) and φT (t) as thedilation and translation velocity potentials for the external liquid at the bubble surface, and withϕ T (t) as thetranslation velocity potential for the internal gas at the bubble surface, the oscillation-phase equations of motion forthe specialized bubble model, including the drag functionD(t), are [2]

∗Corresponding author. E-mail: [email protected].

ISSN 1070-9622/05/$17.00 2005 – IOS Press and the authors. All rights reserved

4 T.L. Geers and C.-K. Park / Optimization of the G&H bubble model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

time(s)

bubb

le r

adiu

s(ft)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-250

-200

-150

-100

-50

0

time(s)

bubb

le tr

ansl

atio

n (ft

)

Fig. 1. Dilation and translation histories for a 1000-lb charge of TNT detonated at a depth of 250 ft, calculated with two different hydrodynamicdrag coefficients.CD = 0.0 (dashed),CD = 2.25 (solid).

a = a−1φD − c−1� (φD − a2 − 1

3u2 − 2

3ua−1φT ),

u = −2a−1φT − c−1� (φT − 2au),

φD = (1 + ζ)−1

{[12

+12(ρg/ρ�) + ζ

] (a2 +

13u2

)− (ρg/ρ�)cga

−1φD +23(1 + ζ)ua−1φT

−ρ−1� [Pg − Patm − ρ�g(d − u)] − 1

3[(a−1φT )2 − (ρg/ρ�)(a−1ϕT )2

]}, (1)

φT = (1 + ζ)−1

{[1 + (ρg/ρ�) + 2ζ]au − (ρg/ρ�)cg(2a−1φT + a−1ϕT ) − (1 − ρg/ρ�)ga +

38D(t)

},

ϕT = (1 + ζ)−1

{[2 + (cg/c�) + ζ]au − cg(2a−1φT + a−1ϕT ) + (cg/c�)(1 − ρg/ρ�)ga − 3

8(cg/cl)D(t)

}.

In these equations,c� andcg(t) are the speeds of sound in the liquid and the gas,ρg(t) is the density of the gas,ζ(t) = ρg(t)cg(t)/ρ�c�, Pg(t), is the uniform component of pressure in the gas defined by its equation of state,P atm

is atmospheric pressure,g is the acceleration due to gravity, andd is the initial depth of the explosive charge. In theexplicit numerical integration of Eq. (1), the right sides of (1c) and (1d) are used forφD andφT on the right sides of(1a) and (1b).

If c�(t) andρg(t) are both set to zero, andD(t) is taken asCDu2(t), Eq. (1) reduce to the equations of Hicks [3]:

aa +32a2 − 1

4u2 − gu = ρ−1

� [Pg − (Patm + ρ�gd)],(2)

au +34CDu2 + 3au − 2ga = 0.

T.L. Geers and C.-K. Park / Optimization of the G&H bubble model 5

These equations clearly exhibit the coupling between dilation and translation. Of particular interest are the termsgu (in the first equation) and2ga (in the second). The first accounts for the change in ambient pressure as the bubblerises, and the second accounts for the buoyancy force that causes the bubble to rise. Although insights provided byEq. (2) are valuable, the equations are not accurate predictors of bubble motion [2].

Shown in Fig. 1 are response histories for bubble dilation and translation produced by Eq. (1) for a 1000-lb chargeof TNT detonated at a depth of 250 ft; one pair of histories pertains toC D = 0 and the other pair toCD = 2.25. Thehorizontal and vertical lines in Fig. 1(a) mark empirical values for maximum bubble radius and time of minimumbubble radius, respectively. The asterisks and circles in Fig. 1(b) mark empirical values for bubble depth at the timesof bubble maximum and bubble minimum, respectively. These values are based on extensive experimental data [3,5,6]. In [5] Snay characterizes the data beyond the third bubble minimum as “sparse and uncertain”.

The histories in Fig. 1 demonstrate that the use ofCD = 0 over-predicts both dilation and translation, while the useof CD = 2.25 under-predicts these responses. This implies that values between these extremes might work better.In fact, a drag formula that admits velocity to a power other than two might be better still. These considerations areonly pertinent to motion during and after the first bubble minimum, as the bubble undergoes negligible translationbefore that time.

3. Modification of the drag formula

Seeking to improve model performance without increasing computational effort, we replaced the drag formulaD(t) = CDu2(t) with the ad hoc relation

D(t) = C|u(t)|P . (3)

Then we searched for optimum values ofC andP over practical ranges of charge weightW and detonation depthd. We based our search on accurate computation of the wave field generated by an explosion bubble rather than onaccurate prediction of bubble motionper se. Specifically, emphasis was placed on accurately predicting the first andsecond pressure pulses generated by bubble collapse/rebound. Bubble dilation is the primary contributor to thesepulses, translation is a secondary contributor, and distortion is a minor contributor. The pulse characteristics aresignificantly affected by wave effects in both the external liquid and internal gas, as well as by bubble depth [4].

In accordance with the above criterion, the search employed an error measure computed with the followingprocedure:

1. For specified values ofW , d andC, P , perform a bubble simulation with the G&H equations (Eq. (1)) andrecord the valuesa(t2), u(t1), u(t2), andu(t3), wheretn is the time of thenth bubble maximum.

2. For the same values ofW andd, calculatea(t2), u(t2)−u(t1), andu(t3)−u(t2) from the empirical formulasof Snay [5].

3. With the results produced in the preceding steps, compute the errors

εD2 = {[a(t2)]G&H − [a(t2)]Snay}/a,

εT1 = {[u(t2) − u(t1)]G&H − [u(t2) − u(t1)]Snay}/h, (4)

εT2 = {[u(t3) − u(t2)]G&H − [u(t3) − u(t2)]Snay}/h.

Here,a andh are the equilibrium radius and initial head, respectively, given by

a = ac(Kc/ρgh)1/3γ , h = d + Patm/ρ�g, (5)

whereac is the charge radius,Kc is the energetic constant for the charge material [2], andγ is the ratio ofspecific heats for the bubble gas.

4. With Eq. (4), compute the error magnitude

ε =√

(ε2D2 + ε2

T1 + ε2T2), (6)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-450

-400

-350

-300

-250

-200

time(s)

bubb

le tr

ansl

atio

n (ft

)

P=0.8P=0.6P=0.4P=0.2

Fig. 2. Translation histories for a 1000-lb charge of TNT detonated at a depth of 250 ft, calculated withC = 1.0 and various values ofP .

Thirty-six hundred G&H simulations and corresponding Snay calculations were performed forW = 100, 400,700, 1000 lb;d = 200, 250, 300, 400, 500 ft;C = 0.2 (0.2) 3.0;P = 0.8 (0.2) 3.0.

Three different optimization procedures were pursued, as follows:

A. Copt andPopt were considered functions ofW andd. With the 20W, d pairs as points on a horizontalW -d plane, theC- and P -values were sought for eachW, d pair that minimizedε. Then the surfacesCopt(W, d) andPopt(W, d) were determined by performing least-squares fits of the quadratic representationc0 + c10W + c01d + c20W

2 + c11Wd + c02d2. Fits with polynomials of higher order were also carried out,

with little improvement.B. Copt andPopt were considered functions of the parameterη = h/a, where, from Eq. (5),h is a function ofd

anda is a function ofW andd. With the 20η-values as points on a horizontalη-axis,C- andP -values weresought for eachη-value that minimizedε. Then the curvesC opt(η) andPopt(η) were determined by performingleast-squares fits of the quadratic representationc0 + c1η + c2η

2. Fits with higher-order polynomials werealso carried out, to little effect.

C. Copt andPopt were considered constants. Theε-values were grouped according to theirC, P pairs, and theaverage and maximum values ofε for eachC, P pair were recorded in Table 1.

4. Computational results

The most successful of the optimization procedures was the last. Table 1 produced by that procedure is shownbelow. We see that the errors in the vicinity ofC = 0.4, P = 1.0 are the smallest. As the maximum error forC = 0.4, P = 0.8 is slightly smaller than that forC = 0.4, P = 1.0, we performed additional simulations forC = 0.4 andP < 0.8. Figure 2 shows representative translation histories generated by these simulations. Wesee that a sufficiently small value ofP causes largenegative translation. For all of the simulations performed, aminimumP -value of unity was safe in this regard. Hence, the recommended drag formula for all cases in the ranges

T.L. Geers and C.-K. Park / Optimization of the G&H bubble model 7

Tabl

e1

Ave

rage

and

max

imum

erro

r-m

agni

tude

valu

esfo

r20

8co

mbi

natio

nsof

Can

dP

PC

0.2

0.4

0.5

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0.8

Ave

rage

0.04

80.

047

0.05

30.

059

0.07

00.

077

0.08

20.

086

0.08

90.

091

0.09

30.

094

0.09

60.

096

0.09

70.

098

Max

0.15

00.

118

0.12

70.

144

0.18

30.

212

0.23

30.

247

0.25

80.

266

0.27

30.

278

0.28

20.

286

0.28

80.

291

1.0

Ave

rage

0.05

20.

046

0.04

90.

054

0.06

30.

070

0.07

50.

079

0.08

20.

084

0.08

60.

087

0.08

90.

090

0.09

10.

092

Max

0.15

90.

127

0.13

00.

141

0.17

00.

196

0.21

60.

230

0.24

00.

248

0.25

40.

259

0.26

30.

267

0.27

00.

273

1.1

Ave

rage

0.05

30.

048

0.05

00.

054

0.06

30.

069

0.07

40.

077

0.08

00.

082

0.08

30.

085

0.08

60.

087

0.08

80.

089

Max

0.16

20.

133

0.13

60.

146

0.17

30.

198

0.21

60.

229

0.23

80.

245

0.25

00.

254

0.25

80.

261

0.26

40.

267

1.2

Ave

rage

0.05

40.

050

0.05

20.

056

0.06

40.

070

0.07

50.

077

0.08

00.

081

0.08

20.

083

0.08

40.

085

0.08

60.

087

Max

0.16

40.

139

0.14

50.

156

0.18

20.

205

0.22

20.

233

0.24

10.

246

0.25

00.

253

0.25

60.

259

0.26

10.

263

1.4

Ave

rage

0.05

60.

056

0.06

00.

064

0.07

10.

076

0.08

00.

081

0.08

20.

083

0.08

30.

084

0.08

40.

084

0.08

50.

085

Max

0.16

50.

158

0.16

90.

184

0.21

20.

232

0.24

50.

252

0.25

50.

257

0.25

80.

259

0.25

90.

260

0.26

00.

261

1.6

Ave

rage

0.05

80.

065

0.07

00.

075

0.08

20.

085

0.08

70.

087

0.08

70.

086

0.08

60.

086

0.08

50.

085

0.08

50.

085

Max

0.16

80.

186

0.20

50.

224

0.25

10.

265

0.27

10.

272

0.27

10.

269

0.26

70.

266

0.26

40.

263

0.26

30.

263

1.8

Ave

rage

0.06

20.

076

0.08

30.

087

0.09

20.

093

0.09

20.

091

0.09

00.

089

0.08

80.

087

0.08

70.

086

0.08

60.

086

Max

0.17

90.

226

0.24

90.

267

0.28

50.

290

0.28

90.

284

0.28

00.

276

0.27

20.

269

0.26

70.

266

0.26

40.

264

2.0

Ave

rage

0.07

00.

089

0.09

40.

097

0.09

80.

097

0.09

50.

093

0.09

10.

090

0.08

90.

088

0.08

70.

087

0.08

60.

086

Max

0.20

40.

271

0.29

10.

301

0.30

60.

302

0.29

40.

287

0.28

10.

277

0.27

30.

270

0.26

80.

266

0.26

50.

264

2.2

Ave

rage

0.08

20.

100

0.10

20.

102

0.10

00.

098

0.09

50.

093

0.09

10.

090

0.08

90.

088

0.08

70.

087

0.08

60.

086

Max

0.24

30.

309

0.31

80.

319

0.31

20.

302

0.29

20.

285

0.27

90.

275

0.27

10.

269

0.26

70.

265

0.26

40.

263

2.4

Ave

rage

0.09

50.

100

0.10

20.

102

0.10

00.

098

0.09

50.

093

0.09

10.

090

0.08

90.

088

0.08

70.

087

0.08

60.

086

Max

0.28

60.

309

0.31

80.

319

0.31

20.

302

0.29

20.

285

0.27

90.

275

0.27

10.

269

0.26

70.

265

0.26

40.

263

2.6

Ave

rage

0.10

40.

107

0.10

50.

103

0.09

90.

095

0.09

30.

091

0.09

00.

089

0.08

80.

087

0.08

70.

086

0.08

60.

086

Max

0.32

10.

332

0.32

40.

315

0.30

00.

290

0.28

20.

277

0.27

30.

270

0.26

70.

266

0.26

40.

263

0.26

20.

262

2.8

Ave

rage

0.11

00.

106

0.10

30.

101

0.09

70.

094

0.09

20.

090

0.08

90.

088

0.08

70.

087

0.08

60.

086

0.08

60.

085

Max

0.34

00.

326

0.31

60.

306

0.29

30.

284

0.27

80.

273

0.27

00.

268

0.26

60.

264

0.26

30.

262

0.26

20.

261

3.0

Ave

rage

0.11

10.

104

0.10

10.

098

0.09

50.

092

0.09

10.

089

0.08

80.

088

0.08

70.

087

0.08

60.

086

0.08

60.

085

Max

0.34

30.

317

0.30

80.

298

0.28

70.

279

0.27

40.

271

0.26

80.

266

0.26

40.

263

0.26

20.

262

0.26

10.

261


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

time(s)

bubb

le r

adiu

s(ft)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-250

-200

-150

-100

-50

0

time(s)

bubb

le tr

ansl

atio

n (ft

)

Fig. 3. Dilation and Translation histories for a 1000-lb charge of TNT detonated at a depth of 250 ft, calculated withC = 0.4 andP = 1.0.

100 lb� W � 1000 lb,d � 200 ft is

D(t) = 0.4|u(t)|. (7)

Shown in Fig. 3 are response histories for bubble radius and translation produced by the G&H equations withEq. (7) for a 1000-lb charge of TNT detonated at a depth of 250 ft. We see that overall agreement with experimentaldata is better than that seen in Fig. 1.

Acknowledgements

The authors express their appreciation to Dr. Kendall S. Hunter for providing valuable help in the use of hisbubble-simulation and empirical-data software.

References

[1] J.P. Best, The formation of toroidal bubbles upon the collapse of transient cavities,J. FI. Mech. 251 (1993), 79–107.[2] T.L. Geers and K.S. Hunter, An integratedwave-effects model for an underwater explosion bubble,J. Acoust. Soc. Am. 111 (2002),

1584–1601.[3] A.N. Hicks, Effect of bubble migration on explosion-induced whipping in ships, Technical Report 3301, Naval Ship Research and Develop-

ment Center, Bethesda, MD, 1970.[4] K.S. Hunter and T.L. Geers, Pressure and velocity fields produced by an underwater explosion,J. Acoust. Soc. Am. 115 (2004), 1483–1496.[5] H.G. Snay,Underwater explosion phenomena: The parameters of migrating bubbles, Technical Report NAVORD 4185, Naval Ordnance

Laboratory, White Oak, MD, 1962.[6] E. Swift and J.C. Decius,Measurements of bubble pulse phenomena: III. Radius and period studies, in Underwater Explosion Research

(Office of Naval Research, Washington, D.C., 1950), Vol. 2, 1948, pp. 553–599.

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Optimization of the G&H bubble modeldownloads.hindawi.com/journals/sv/2005/430767.pdfto the case of no bubble distortion, thereby limiting the motion to dilation plus translation