89 Sphercal solutons of an underwater exploson bubble Andrew B. Wardlaw, Jr. Naval Surface Warfare Center, Code 423, Indan Head Dvson, Indan Head, MD 20640-5035, USA E-mal: 423@uwtech.h.navy.ml Hans U. Mar Insttute for Defense Analyses, Operatonal Evaluaton Dvson, 1801 North Beauregard Street, Alexandra, VA 22311-1772, USA E-mal: marh@asme.org Receved 7 March 1997 Revsed 10 February 1998 The evoluton of the 1D exploson bubble flow feld out to the frst bubble mnmum s examned n detal usng four dfferent models. The most detaled s based on the Euler equatons and accounts for the nternal bubble flud moton, whle the smplest lnks a potental water soluton to a statonary, Isentropc bubble model. Comparson of the dfferent models wth expermental data provdes nsght nto the nfluence of compressblty and nternal bubble dynamcs on the behavor of the exploson bubble. 1. Introducton Smulaton of underwater explosons s an essental component of platform vulnerablty and weapon lethalty assessments. Success at ths task requres accurate predcton of target loadng, whch prmarly occurs as a consequence of the ntal shock and subsequent bubble collapse. These damage mechansms occur over a wde range of tme scales, from submllsecond for shock response to hundreds of mllseconds for response to bubble pulses. Unfortunately, three-dmensonal Eular smulatons of such events, partcularly bubble collapse, tax the ablty of supercomputers. Ths has resulted n the use of less complete models for water and exploson products (e.g., Van Tuyl and Collns [11], Szymczak et al. [10] and Chahne [1]). The objectve of ths paper s to examne the consequence of these smplfcatons by studyng the one-dmensonal (1D) exploson bubble. The 1D exploson exhbts two mportant damage mechansms: the ntal shock, and subsequent pressure pulses from bubble collapse and rebound. The lmted sze of ths problem makes t numercally tractable, allowng mesh converged solutons to be acheved to provde nsght nto mesh resoluton requrements n 2D and 3D. Shock and nterface fttng s also easly mplemented n ths sngle-dmensonal settng. Ths problem does not exhbt bubble jettng, a form of collapse that nduces a drected water jet. However, t s reasonable to assume that accurate treatment of the 1D problem s a necessary prerequste for the predcton of jettng. Ths paper examnes four dfferent numercal descrptons of the underwater exploson. The smplest, termed the Incompressble Model, assumes water to be ncompressble and the bubble to be a statonary, homogenous gas, whose pressure changes sentropcally wth bubble volume. The second approach, termed the Isentropc Euler Model, ncludes a bubble descrpton smlar to that of the Incompressble Model, but ntroduces the compressblty of the water. The thrd approach, termed the Unform Euler Model, adds an Euler descrpton of the bubble, and starts the calculaton wth unform bubble propertes. The fourth approach, termed Non-Unform Euler Model, s based on nonunform ntal bubble condtons, derved from a Taylor wave soluton to the gaseous explosve products. The flow feld evoluton predcted by these four models s examned and compared. Also, soluton senstvty to changes n water and exploson product equatons of state s consdered. The paper concludes by comparng calculatons wth expermental data. Comparsons are made only through the frst bubble mnmum; multdmensonal dsspatve mechansms nfluencng bubble evoluton beyond the frst mnmum are outsde to scope of ths paper, but are addressed elsewhere (e.g., Geers and Hunter [5]). Shock and Vbraton 5 (1998) 89 102 ISSN 1070-9622 / $8.00 1998, IOS Press. All rghts reserved
90 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 2. Soluton methodology Two types of materal descrptons are consdered n ths paper: compressble (Euler) and ncompressble. These methods have been assembled nto the four dfferent models mentoned n the Introducton.Ths secton detals the equatons and numercal methods of the compressble and ncompressble descrptons, whle the followng secton defnes the specfcs of the ndvdual models. 2.1. Compressble (Euler) calculatons The water and the gas bubble are descrbed usng the Euler equatons n sphercal coordnates: U t + F = S, (1) r where U = r 2 ρ ρu ρu, F = r 2 ρu 2 + p, ρe ρue + pu S = 0 2rp. ρu To ensure a hgh qualty soluton, the numercal method avods mxed cells that contan gas and water. Ths s acheved usng an Arbtrary Lagrangan/Euleran (ALE) approach whch moves the gas/water nterface node at the computed nterface velocty. A consequence of ths methodology s that the number of grd ponts s constant throughout the calculaton, n both the bubble and the surroundng water. Equatons (1) are solved usng a second order Godunov method based on the technque descrbed by Collela [2]. However, ths method has been modfed to contan both a Lagrangan step and a re-map step, as s descrbed n Appendx A. Fttng the shock to the outer edge of the computatonal mesh durng the shock phase allows the shock jump to be explctly calculated rather than captured by the numercal scheme. The Tat and JWL equatons of state are used to descrbe the water and bubble (gaseous detonaton products), respectvely: where: Tat: p = B[(ρ/ρ) γ 1] + A, (2) JWL: B = 3.31 10 9 d/cm 2, ρ = 1.0; A = 1.0 10 6 d/cm 2, γ = 7.15; ( p = A 1 ωρ R 1 ρ 0 ( + B 1 ωρ R 2 ρ 0 ) e R1ρ0/ρ ) e R2ρ0/ρ + ωρe. (3) Unless stated otherwse, the followng set of JWL coeffcents s used for TNT (Dobratz and Crawford [3]): A = 3.712 10 2 d/cm 2, B = 0.0323 10 12 d/cm 2, R 1 = 4.15, R 2 = 0.95, ω = 0.30, ρ 0 = 1.63 g/cm 3, e 0 = 4.29 10 10 dcm/g. 2.2. Incompressble calculatons In the Incompressble Model, the water s descrbed usng the potental soluton for an expandng sphere (e.g., Karamchet [7]). The pressure on the surface of the bubble s gven by: p(r(t), t) = p + ρ 2 [ 3 ( dr dt ) 2 ] + 2r d2 r dt 2, (4) where r(t) s the bubble radus and p s the ambent pressure. Equaton (4) s coupled to the equaton of state for the bubble, whch n ths study s taken to be of the JWL form. By vrtue of the sentropc assumpton, the equaton of state provdes pressure as a functon of densty, whch n turn s a functon of ntal bubble mass (M) and the bubble volume. Ths reduces the bubble pressure to: p = p ( r(t) ). (5) Combnng Eqs (3) and (4) yelds a dfferental equaton whch can be ntegrated numercally, startng wth a known radus r 0 and dr/dt = 0. 3. Model detals 3.1. Incompressble Model The Incompressble Model descrbes the water usng the ncompressble formulaton of Eq. (4). The bubble
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 91 s assumed to be a statonary, sentropc gas governed by the JWL equaton of state, Eq. (3), and assocated parameter set. Startng an ncompressble calculaton from the detonaton condtons (.e., ρ = 1.63 g/cm 3, e 0 = 4.29 10 10 dcm/g for TNT) results n a perod that s too long; t s therefore common practce to apply emprcal condtons whch effectvely remove the energy assocated wth the shock from the calculaton. Typcally, a set of emprcal formulas derved from experment and talored to a gamma law gas bubble model are used to set the ntal bubble condtons (e.g., Szymczak et al. [10]). In the present study, the ntal condtons are chosen by adjustng the ntal bubble radus and settng the ntal pressure by sentropc expanson from the exploson condtons. Numercal experments ndcated that an ntal bubble radus 1.8 tmes the actual charge radus produces a soluton that closely matches the Non-Unform bubble perod. The motvaton for selectng ths ntal condton s to connect the ntal emprcal bubble pressure to that of the ntal explosve state used n the other models whle also assurng a match between the predcted ncompressble and compressble bubble perod. Ths ntal condton prescrpton s not beng offered as a replacement for tradton emprcal startng condtons, whch have been verfed over a broad range of cases. 3.2. Isentropc Euler Model Here the bubble s smlar to the sentropc, statonary bubble used n the Incompressble model whle the water s treated as compressble va the Euler equatons. Ideally, the bubble gas would be descrbed usng Eq. (5); however, to smplfy model constructon, a sngle Euler cell defnes the bubble. The nner edge of the cell s located at the center of the bubble whle the outer edge s located at the gas/water nterface and moves at the nterface velocty. Ths model produces a homogeneous bubble; however, the velocty of the bubble gas s not zero. Hence, the nternal dynamcs of the bubble are not completely elmnated, and the Isentropc Euler and Incompressble models dffer by more than just the compressblty of the water. However, an examnaton of the soluton ndcates that the gas bubble velocty s small and the bubble knetc energy s neglgble. The ntal bubble condtons are unform and the startng state s that for TNT, the same condtons employed n the Incompressble model as the base state for the sentropc expanson. Fg. 1. Non-Unform Euler Model ntal expanson sub-phase. 3.3. Unform Euler Model The bubble and water are treated as compressble fluds whch are descrbed by the Euler equatons. Ths model dffers from the Isentropc Euler Model only wth regard to the bubble, that s now descrbed by a large number of cells. The same unform, TNT condtons are used as the ntal bubble state n the Unform Euler and Isentropc Euler models. 3.4. Non-Unform Euler Model The Euler soluton descrbed n the Unform Euler Model s appled here and the Unform and Non- Unform Euler models dffer only wth regard to the ntal bubble condtons. The Non-Unform and Unform ntal bubble condtons contan the same total energy, however, for Non-Unform Euler case the dstrbuton of propertes wthn the bubble s determned usng a sphercal analog to the Taylor plane wave soluton based on the work of Gurgus et al. [6]. The resultng bubble property profle features the hghest pressure at the bubble/water nterface wth a constant, mnmum pressure regon at the bubble center (see Fg. 1, t = 0curve). 4. Model comparsons The four bubble models are compared for the followng condtons: detonaton of a 28 kg TNT sphere (radus of 16 cm) at a depth of 91.7 m (ambent pressure of 1.0 10 7 d/cm 2 ). The Euler soluton mesh sze was selected through a convergence study and features 512 ponts n the bubble and 565 pont n the water. The same mesh sze s used for both the shock and collapse phase; however, the shock s only tracked for the shock
92 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble Fg. 2. Non-Unform Euler Model shock formaton sub-phase. Fg. 3. Non-Unform Euler Model shock nteracton sub-phase.
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 93 Fg. 4. Comparson of solutons durng expanson sub-phase, at t = 2.06 10 5. phase. The requred run tme on a Pentum II 266 MHz PC s one hour for a shock problem, and two hours for a bubble collapse calculaton. 4.1. Shock phase Fgures 1 3 trace the evoluton of the Non-Unform Euler soluton through early tmes, where the shock s the domnant feature n the flow feld. Ths shock phase can be dvded nto three sub-phases. The frst s llustrated n Fg. 1 and depcts the ntal nteracton between the hgh pressure bubble and the low pressure water. Ths results n a shock that propagates nto the water and an expanson that travels back nto the bubble. The sub-phase termnates when the nward movng expanson reaches the orgn. Fgure 2 llustrates the reflecton of the expanson from the orgn, whch marks the start of the second sub-phase. As expected, an expanson reflects as an expanson, droppng the mnmum pressure n the bubble several orders of magntude. However, the pressure gradent nsde the bubble and the accompanyng nward flow mpedes the outward progress of the expanson. A sharp pressure gradent develops n the bubble that evolves nto a shock movng toward the center of the bubble. The second sub-phase ends when that shock reaches the orgn. The thrd sub-phase begns when the shock formed n the second sub-phase reflects from the orgn. The reflected shock propagates to the gas/water nterface and nteracts wth t as s shown n Fg. 3. The nteracton generates a reflected shock whch moves back nto the bubble toward the orgn, and a transmtted shock that travels outward from the bubble nto the water. The thrd sub-phase ends when the reflected shock reaches the orgn. The process of shock reflecton from the orgn and nteracton wth the nterface repeats as the bubble expands. Flow feld profles at later tmes bear the legacy of these nteractons; several pressure pulses n the water can be seen movng outward from the bubble. As the bubble expands, pressure levels nsde the bubble decrease, the shock nsde the bubble weakens, and the ntenstes of waves transmtted from the bubble to the water become neglgble. Comparson of the four solutons durng the shock phase s shown n Fgs 4 and 5. The Unform Euler Model soluton s qualtatvely smlar to the Non- Unform Euler Model soluton. The ntal sub-phase 1 expanson reflects from the orgn and subsequently forms an nward movng shock. Ths shock repeatedly reflects from the orgn and nteracts wth the nterface, generatng transmtted pressure pulses n the water, several of whch are shown n Fg. 5. The prncple dfferences between these solutons are the tmes at whch the events occur. The Unform Euler soluton unfolds at a qucker pace, as can be seen n Fg. 5 from the more advanced poston of the transmtted pulse. 4.2. Bubble collapse phase The evoluton of the Non-Unform Euler Model flow feld near the bubble mnmum volume s depcted n Fg. 6. The pressure wthn the bubble s hghly Non- Unform, and a careful examnaton of flow feld data ndcates that the moment of maxmum bubble pressure does not concde wth the tme of mnmum volume. Although the maxmum bubble pressure occurs near the pont of mnmum volume, the peak pressure wll concde wth a tme at whch a shock wave nsde the bubble converges at the bubble center. Fgure 6 also exhbts a water pressure profle wth numerous local peaks, whch are a consequence of the nteracton of the shock wave nsde the bubble wth the gas/water nterface. The solutons for all four models through bubble collapse are compared n Fgs 7, 8 and 9. The bubble radus hstory s shown n Fg. 7, whle Fg. 8 llustrates the bubble-water nterface pressure. The bubble perod s marked by the mnmum volume n Fg. 7 and the pressure peak n Fg. 8. The three Euler models feature sgnfcantly dfferent bubble perods, varyng from 0.127 to 0.136 seconds. The ncompressble soluton perod has been talored to match that of the Non- Unform Euler Model. Note that the maxmum bubble radus dffers between these two cases. Use of the explosve ntal condtons for the ncompressble solu-
94 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble Fg. 5. Comparson of solutons durng the shock nteracton sub-phase. Fg. 6. Non-Unform Euler Model bubble collapse.
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 95 Fg. 7. Comparson of bubble radus hstory. Fg. 8. Comparson of nterface bubble pressure.
96 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble Fg. 9. Comparson of collapse pressure feld. ton rather than the emprcal ones yelds a bubble perod of 0.176 seconds. The Non-Unform and Unform Euler Model nterface pressures oscllate due to the nteractons of the shock waves nsde the bubble wth the gas/water nterface. Ths type of oscllatory nterface behavor has been prevously observed by Mader [8]. The Incompressble and Isentropc Euler solutons, on the other hand, do not have an nternal bubble structure and consequently have smoothly varyng nterface pressures. The pressure feld at bubble collapse s llustrated n Fg. 9 for all four solutons. The three Euler solutons dsplay smlar water pressure levels. However, the Non-Unform Euler Model and Unform Euler Model solutons exhbt local pressure peaks generated by the gas/water nteracton, whle the Isentropc Euler Model and Incompressble Model solutons do not. In addton, the Incompressble Model soluton exhbts a pressure far from the bubble whch exceeds that of the compressble models. One method of examnng the dfferences among the four models s to consder the manner n whch the energy transmtted to the water s parttoned. In the Euler solutons ths energy can be converted to knetc energy or nternal energy va compresson. Incompressble meda do not have nternal energy, but there s a potental energy assocated wth the work requred to move water from the space occuped by the expandng bubble. For the Euler solutons, the energy parttonng must be determned by numercal ntegraton, but for the Incompressble Model an analytc expresson can be derved: E w = 4π = 4π 3 t 0 p s ur 2 (τ)dτ [ r(t) 3 r(0) 3] ( ) 2 dr p + 2πr(t) 3. (6) dt Here the surface pressure, p s s evaluated from Eq. (4) and u = dr/dt. The frst term s the potental energy whle the second term s the knetc. The results are shown n Fg. 10, whch graphs the fracton of the transmtted energy that s converted nto knetc energy. Comparson of the three Euler models ndcates that the bubble perod s drectly related to the water knetc energy level. Examnaton of these solutons also reveals a relatonshp between ntal shock strength and knetc energy, wth the lowest knetc energy assocated wth the strongest ntal shocks. Presumably,
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 97 Fg. 10. Fracton of water energy converted to knetc energy. a hgher shock pressure ncreases the water compresson, leavng a smaller amount of energy for the knetc component that propels the water away from the bubble. As can be seen n Fg. 10, the ncompressble, explosve ntal condton model has a large knetc energy, whch s consstent wth ts lengthened perod. Use of the emprcal ntal condtons reduces ths energy greatly. As the bubble collapses, water rushes toward t to fll the space left by ts dmnshng volume. At the pont of mnmum volume, ths nflow comes to a halt. In the compressble models, some of the water knetc energy s converted to nternal energy by compresson and heatng; however, n the ncompressble case, the only mechansm avalable to remove energy s to transfer t back to the bubble, whch regans ts ntal energy and pressure. Deceleraton of the water n the Incompressble model s accomplshed only through a pressure gradent. Ths accounts for the dfferent functonal form of the ncompressble pressure vsble n Fg. 9, that exhbts hgher pressures farther from the bubble. 5. Euler Model uncertantes Euler bubble models are dependent on the equatons of state for water and the exploson products. These relatons contan a number of parameters that must be assgned. Addtonally, n the case of water, there are several forms of the equaton of state that mght be used. Before comparng wth experment, t s useful to examne the mpact of these factors on the soluton. Wth respect to the water, the HOM equaton of state descrbed n Mader [8] s tested. Ths equaton has the form: p = Γρ(e e h ) + p h, e h = p hµ 2ρ, where: p h = C2 µρ, µ = 1 ρ/ρ, (7) 1 Sµ ρ = 1, C = 0.1483 10 6, S = 2.0, Γ = 1.0. An addtonal equaton of state for water s also appled whch follows the JWL form (.e., Eq. (3)). The JWL coeffcents are A = 15.82 10 12 d/cm 2, B = 4.668 10 10 d/cm 2, R 1 = 8.94088, R 2 = 1.4497, ω = 1.17245, and reference propertes are ρ 0 = 1.00381 g/cm 3, e 0 = 2.51524 10 8 dcm/g. To gage the senstvty of the soluton to the explosve JWL model, a dfferent set of coeffcents, de-
98 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble Table 1 Influence of equaton of states on bubble perod Equaton of state Maxmum Perod water TNT radus (cm) (ms) Tat TNT1 220.9 134.2 HOM TNT1 218.1 132.8 Smth TNT1 221.6 134.4 Tat TNT2 231.4 140.8 veloped for underwater studes by Fessler [4] and labeled TNT2 s appled. Here A = 5.484 10 12 d/cm 2, B = 9.375 10 10 d/cm 2, R 1 = 4.94, R 2 = 1.121, ω = 1.2801, e 0 = 5.183 10 10 dcm/g. The nfluence of these changes n equaton of state on the Unform Euler bubble perod s shown n Table 1. Analogous changes occur n the other Euler solutons. It s evdent that the largest uncertantes are assocated wth the JWL parameters and ntal energy. 6. Comparson wth experment Table 2 shows a comparson among the four models and the expermental data of Swft and Decus [9] for a case featurng 300 g of TNT at 91.4 m. For all cases the Tat equaton of state was used for water, n conjuncton wth the TNT2 set of JWL coeffcents for the detonaton products; ths set was selected because t produced the best agreement between experment and the Non-Unform Euler Model. The results shown n ths table are consstent wth Fgs 7 and 8; the Non-Unform Euler Model exhbts the shortest perod, whle the Isentropc Euler Model features the longest. Two ncompressble results are shown, labeled ncompressble, n whch the ntal explosve condtons are appled to the bubble, and ncompressbleadjusted, n whch the emprcal ntal condtons are used. Comparson of the Incompressble and Isentropc Euler cases demonstrates the nfluence of water compressblty on the soluton. Table 3 compares the Non-Unform Euler Model results to three addtonal test cases taken from Swft and Decus [9] and Yenne and Arons [12]. Close agreement s obtaned n all relevant cases (Case 1 s ncluded only for reference, snce the charge s located close enough to the free surface to be nfluenced by t). In Fg. 11, computed collapse pressures are compared wth the expermental data of Yenne and Arons [12]. The test was conducted usng 227 g of TNT at a depth of 152.4 m and the pressure was measured at a locaton 69.5 cm from the center of the charge. To ad n a comparson of peak pressures, the computed curves have been ndvdually translated by the amount shown n the fgure legend. As can be seen from ths fgure, the ncompressble peak pressure s the hghest, followed by the Isentropc Euler Model. The Unform and Non-Unform Euler solutons are oscllatory n character, reflectng the nteracton between the bubble and the water. The close agreement between the ncompressble and Euler solutons s somewhat fortutous. The peak ncompressble pressure s senstve to changes n the bubble perod; a 5% ncrease n the bubble perod wll double the peak pressure whle a smlar decrease wll halve t. Ths senstvty s a consequence of requrng the emprcal ntal state to be on the ntal TNT sentrope and the mass of the bubble to be that of the ntal explosve. To change the bubble perod requres a change n ntal bubble radus, that alters the ntal pressure va Eq. (5) and consequently the collapse pressure. The standard emprcal methods for generatng ntal condtons do not levy these requrements on the ntal state and thus are free to best adjust the bubble ntal radus and pressure ndependently (e.g., Szymczak et al. [10]). 7. Conclusons The sphercal exploson bubble contans an nternal shock that repeatedly reflects off of the bubble center and nteracts wth the gas/water nterface. Ths nterface nteracton generates a transmtted shock that travels outward nto the water, and a reflected shock that moves nward towards the bubble center. A consequence of ths nternal bubble structure s a fluctuatng gas/water nterface pressure, as well as pressure varaton at fxed ponts throughout the flow feld. All of these features are contaned n the Non- Unform Euler Bubble Model. As the descrpton of the bubble s smplfed, the followng changes n soluton result: 1) Substtutng the unform ntal bubble condtons for the non-unform ones lengthens the bubble perod on the order of 5%. 2) Elmnatng the nternal bubble structure (Isentropc Euler Model) removes the bubble shocknterface nteracton phenomenon whch leads to smooth bubble water nterface pressures and lengthens the bubble perod by about 5%. 3) Elmnatng the compressblty of water ncreases the bubble perod by about 25%.
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 99 Table 2 Predctons and experment for the 91.4 m, 300 g case Method Perod Maxmum bubble radus t (ms) Dfference r (cm) Dfference Experment 29.8 48.1 Predctons Non-unform Euler 29.8 0% 46.4 4% Unform Euler 31.0 4% 48.6 1% Isentropc Euler 32.4 9% 50.6 5% Incompressble 41.3 39% 66.3 38% Incompressble (adjusted) 29.5 1% 49.4 3% Table 3 Comparson of unform Euler predctons and experment Case Depth Weght Perod (ms) Maxmum radus (cm) (m) gms TNT Test Calc. Dfference Test Calc. Dfference 1 1.5 25.7 77.6 79.6 3% 48.6 45.4 7% 2 91.4 300 29.8 29.8 0% 48.1 49.0 2% 3 152.4 227 18.3 18.5 1% 37.9 38.1 1% 4 178.6 300 17.8 17.9 1% 38.7 39.5 2% Fg. 11. Calculated and expermental pressures at 69.5 cm from a 227 g charge at 152.4 m.
100 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble For the Euler solutons, the strength of the ntal exploson shock plays a key role n parttonng the energy transmtted to the water between knetc and nternal. The stronger the shock, the larger the ncrease n water nternal energy and the smaller the knetc energy. Ths dmnshed knetc energy decreases the outward velocty of the water, reduces the maxmum bubble radus, and shortens the bubble perod. Dfferences among three water equatons of state were tested and all produced bubble perods wthn 1 percent. Changes n the coeffcents and ntal energy for the JWL equaton of state yelded varatons of 5 percent. The exploson bubble solutons have been compared to data from four experments, wth depths varyng from 1.5 m to 179 m. Calculated bubble perod and maxmum radus were n reasonable agreement wth experment. The computed pressure at a pont n the flow feld agreed reasonably well wth experment at collaps tme. Acknowledgement The authors wsh to thank Dr. Raafat Gurgus for provdng the Taylor wave soluton for the non-unform bubble ntal condtons, and Mr. Charles Smth for formulatng the JWL water equaton of state. Appendx A. Modfed Godunov numercal method The method descrbed here algns the mesh wth the bubble water nterface and avods mxed cells. Addtonally, when the ntal exploson shock s ftted, the outer edge of the mesh s moved to assure that the shock s located on the nner edge of the outermost cell. These two mesh algnment features preclude the use of ths method n other than 1D applcatons. A dervatve of ths method s avalable for 2D and 3D calculatons, but here the mesh s fxed, mxed cells are ntroduced, and the shock s captured. Equatons (1) are solved usng an algorthm whch s a dvded nto a Lagrangan step and Remap step. The Lagrangan step convects the cell, based on edge condtons. The Remap step rezones the soluton onto the desred mesh, whch s dctated by the moton of the cell nterface and the shock, f t s beng ft. The detals of the algorthm n advancng from step n to n + 1 (.e., t to t + t) are gven below. The subscrpt ndcates cell centers; cells edges are located at +1/2. 1. Lagrangan Step. (a) Predctor: () Compute element slope usng the followng lmter: ( ) df = dx where ( δf + ) sgn δx ( mn k δf + δx, k δf δx, δf c ) δx f δf + δf δx δx > 0, 0 otherwse, (A.1) δf + δx = f +1 f x +1 x ; δf c δx = f +1 f 1 x +1 x 1. δf δx = f f 1 x x 1 ; At both the bubble-water nterface and the shock, the lmter s adjusted to ensure that dfference are not taken across the nterface or shock. () Compute pressure and velocty at the cell edge center (.e., t + t/2, ± 1/2, n + 1/2) usng the modfed method of characterstcs of Collela [2]. Applyng the method of characterstcs n ts unmodfed form would requre tracng the characterstcs back from the cell edge center to the tme plane t n and assgnng ntal characterstc values usng the element slope nformaton from the precedng step (see Fg. 12). The four characterstc relatons whch can be solved for p e, ρ e, u e, e e are: [ pe p ( x ±, t n+1/2)] ± cρ [ u e u ( x ±, t n+1/2)] = 0 along λ ± = u ± c, c 2[ ρ e ρ ( x 0, t n+1/2)] [ p e p ( x 0, t n+1/2)] = 0 along λ 0 = u, c 2[ (ρe) e ρe ( x 0, t n+1/2)] h [ p e p ( x 0, t n+1/2)] = 0 along λ 0 = u. (A.2)
A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble 101 u L = un + [ p(a +1/2 A 1/2 ) + (A 1/2 p 1/2 A +1/2p +1/2 ) t] /(ρ L V L ), (A.4) e L = en + [ (p 1/2 u 1/2 A 1/2 p +1/2 u +1/2 A +1/2 ) t]/ (ρ L V L ). Fg. 12. Cell edge characterstcs. In the above, the subscrpts +,, 0 denote the pont of characterstc ntersecton at t n as shown n Fg. 12. The modfcaton arses on characterstcs orgnatng n the neghborng cell. To preserve the upwnd nature of the scheme, cell center values are appled on these characterstcs. Ths modfed method of characterstcs produces a rght (+)andleftstate( ) at each cell edge center. (b) Corrector: () Solve an approxmate Remann problem at each cell edge to determne cell edge pressure and velocty, p and u, respectvely. The two states defnng the Remann problem are at the left and rght cell edges computed n 1(a)() and denoted by superscrpt and + respectvely. p = [ (ρc) + p + (ρc) p + + (ρc) + (ρc) (u u + ) ] /[ (ρc) + + (ρc) ], u = [ (ρc) u + (ρc) + u + + (p p + ) ] /[ (ρc) + (ρc) +], (A.3) () Convect cell and recompute cell propertes, denoted by superscrpt L, va the followng: V L = V n + ( u +1/2 A +1/2 u 1/2 A 1/2) t, ρ L = ρ n V n /V L, where p = (p 1/2 + p +1/2 )/2 andv = cell volume. 2. Remap step. (a) Construct the new mesh from the nterface and shock locatons computed n the Lagrangan step by lnear stretchng. (b) Compute slopes n each Lagrangan cell usng the lmter of Eq. (A.1). (c) Compute the fluxes, F, that must be transferred between cells to accomplsh the remappng from the Lagrangan mesh to the new mesh. d +1/2 = u +1/2 t D +1/2, F+1/2 1 = ρ d +1/2A +1/2, F+1/2 2 = ûf +1/2 1, F+1/2 3 = (û2 /2 + ê )F+1/2 1. (A.5) Here D +1/2 s the dstance cell edge + 1/2 moves durng step t,and quanttes are evaluated at the locaton x = x +1/2 ± d +1/2 /2, va the slope of step 2(b). The sgn s selected to nsure that the upwnd propertes are used. (d) Compute re-mapped cell propertes usng: ρ n+1 u n+1 e n+1 = ( ρ L V L + F 1 1/2 F 1 +1/2 = ( ρ L u L V L + F 2 1/2 F 2 +1/2 = ( ρ L el V L + F 3 1/2 F 3 +1/2 ) /V n+1, ) ( / ρ n+1 V n+1 ), ) ( / ρ n+1 V n+1 ). Here V n+1 s the new cell volume. In the case n whch the outer shock s tracked, the outermost
102 A.B. Wardlaw, Jr. and H.U. Mar / Sphercal solutons of an underwater exploson bubble References cell s set to the ambent state, whle ts neghbor contans the state behnd the shock. An exact Remann problem s solved across the nner edge of the outer cell; the cell edge velocty s assgned that of the shock. The resultng soluton exhbts the entre shock jump across ths cell edge. [1] G. Chahne, Interacton between an oscllatng bubble and a free surface, Journal of Fluds Engneerng 99 (1977), 709 716. [2] P. Collela, A drect Euleran scheme for gas dynamcs, SIAM Journal on Scentfc and Statstcal Computng 6(1) (1985), 104 117. [3] B.M. Dobratz and P.C. Crawford, Explosve Handbook of Propertes of Chemcal Explosves and Explosve Smulants, Lawrence Lvermore Natonal Laboratory, UCRL-52997, 1985. [4] B. Fessler, DYSMAS/E Input Manual, IABG, Munch, Germany, 1996. [5] T.L. Geers and K.S. Hunter, Dlatatonal dynamcs of an underwater exploson bubble, n: Proceedngs of the 67th Shock and Vbraton Symposum, Vol. I, 1996, pp. 315 324. [6] R. Gurgus, K. Kamel and A.K. Oppenhem, Self-smlar blast waves ncorporatng deflagratons of varable speeds, Progress n Aeronautcs and Astronautcs 87 (1981), 121 156. [7] K. Karamchet, Prncples of Ideal-Flud Aerodynamcs, Wley, 1966, pp. 279 280. [8] C. Mader, Numercal Modelng of Detonatons, Unversty of Calforna Press, Los Angeles, CA, 1979. [9] E. Swft and J. Decus, Measurement of bubble pulse phenomena, Navord Report 97-46, 1946. [10] W. Szymczak, J. Rogers, J. Solomon and A. Berger, Numercal algorthm for hydrodynamc free boundary problems, Journal of Computatonal Physcs 106(2) (1993), 319 336. [11] A. Van Tuyl and P. Collns, Calculaton of the moton of axsymmetrc underwater exploson bubbles by use of source dstrbuton on free and rgd surfaces, Naval Surface Weapons Center Techncal Report NSWC TR 84-314, 1985. [12] D. Yenne and A. Arons, Secondary pressure pulse due to gas globe oscllaton n underwater explosons III. Calculaton of bubble radus tme curve from pressure tme record, Navord Report 429-49, 1946.
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