p. colella et al- mach reflection from he-driven blast waves

5/12/2018 P. Colella et al- Mach Reflection from HE-Driven Blast Waves

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e ecl,lonfrom HE-Driven Blast WaveP. Colella, ~.E. Ferguson, H.M. Glaz, and A.L. Kuhlr/.,'

Reprinted from o,.namin of fdplMl edi ted by J.R. Bowen. J.-C.Leyer, and R.I. Soloukhin, Vo regress in Astronautics andAeronautics series. Published n 1986 b the American Ins titute ofAeronautics and Astronautic Inc 33 Broadway, N.Y. 10019,644pp., 6 )(9, iUus. , ISBN ()' 9304031S~, $39 .50 Membe r, $79.50 Lis t.


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. . . .

Mach Reflection from an HE-Driven Blast WaveP. Colella-

Lawrence Berkeley Laboratory. Berkeley. CaliforniaR.E. Ferguson'[ an d H.M. GlutNalJOlSur/ace Weapons Center, Silver Spring, MarylandandA.L. K u h l ~

R&D Associates, Marina del Rey, California

AbstractP lo wf ie ld s a ss oc ia te d w it h o ne -d im en si on al f re e- ai rexplosions are well known both for the point-source caseand for the case of s blast wave driven by the detonationof a high-explosive (HE) charge. Considered here is thet wo -d ime ns ion al ca se of t he re fle ct ion o f a sp he ric al . H E-driven blast wave from an ideal plane surface. The evolu-tion of the flowfield was calculated with 8 nondiffusivenu mer ic al a lg or ith m f or ac cu rat el y s ol vi ng th e E ule r e qua -tions. This algorithm is based on a second-order Godunovscheme and a monotonicity algorithm that is designed togive sharp shocks and contact surfaces while smoothregions of the flow remain smooth yet free of numericaldiffusion. The incident HE-driven blast wave was accu-r at el y c ap tu re d b y a f in e- zo ne d o ne -d im en si on al c al cu la -t io n th at wa s c on tin uo usl y fe d i nt o t he tw o-d im ens io nalmesh. The latter incorporated a fine-zoned mesh thatf ol lo wed t he r ef lec ti on r egi on an d ac cu rat el y re sol ve d t he

c om pl ic at ed f lo w s tr uc tu re o cc ur ri ng o n m ul ti pl e l en gt hscales. Major findings in the regular reflection regionwere as follows. Portions of the main reflected shockreflected within the channel formed by the wall and thed en se H E pr odu ct s. t hu s c rea ti ng ad dit io nal pr ess ur epulses on the wall. Coherent vortex structures formed onthe fireball as a result of the interaction of the re-flected shock with this contact surface. The flow did! >r es en te d a t t he 1 0t h I CD ER S, B er ke le y, C al if or ni a, A ug us t 4 9 .1985. C o p y r i g h t e M er lc an I ns ti tu te o f A er on au ti cs a nd A st ro na ut ic s.I n c . , 1985. A ll r ig ht s r es er ve d.

* H at h ea a tl cl 8 n. H at h e. a ti c 8 D ep a rt . en t .t H at h e. . t l ci an . A pp l le d H at h em a tl c a B r an c h.t se nl or S ta ff S ci en ti at . N uc le ar E ff ec ta D ep ar t . . nt .

38 8


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MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 389

indeed make a transition to a double-Mach structure. butthis transition was delayed 1.5 to 3.8 deg beyond the two-shock limit of regular reflection because the nascent Machstem was less than one cell high in this region. Thedouble-Mach structure with its two moving stagnationpoints was similar (but not identical) to an equivalentshock-on-wedge case. A key feature of this flow was asupersonic wall jet (velocity of 3.5 to 4.3 km/s) consist-ing of a free shear layer and a wall boundary layer. Thewall jet was laminar in these calculations. but shouldactually be turbulent due to Reynolds number considera-tions. Nevertheless. calculated peak pressures were foundto be in excellent agreement with experimental data at allg ro un d r an ge s.

I . I nt ro du ct io nConsidered here is the two-dimensional axisymmetric

reflection of a spherical (high-explosives-driven) blastwave from a plane surface. The temporal evolution of theflowfield was calculated with a second-order EulerianGodunov scheme that accurately solves such inviscid com-pressible flow problems on a very fine computational mesh.The accuracy of the solution was confirmed by experimentalpressure data for the same problem.

The details of flowfields associated with one-dimen-sional free-air explosions are well established. Con-sider. for example. the similarity solutions for sphericalblast waves: the point explosion solution of Taylor (1941)and Sedov (1946). and all classes of blast waves boundedby strong shocks (Oppenheim et al. 1972a) and by strongChapman-Jouguet detonations (Oppenheim et al. 1972b).Other'examples are the non-self-similar solutions of thedecay of a point-aource explosion: the original finitedifference calculation (Von Neumann and Goldstine 1955),the method of integral relations solution (Korobeinikovand Chushkin 1966). the method of characteristics solution(Okhotsimskii et ale 1957). and the Lagrangian finite-difference calculations (Brode 1955). Also well estab-lished are non-self-similar solutions of the decay ofspherical blast waves driven by a solid, high-explosives(HE) charge (Brode 1959).However, when one considers the reflection of suchspherical blast waves from a plane surface. a detaileddescription of the flowfields is not generally available.Such flows are inherently two dimensional. They are driv-en by decaying blast waves. and hence they are intrinsi-cally non-.elf-.imilar. They depend parametrically on the

39 0 P. COLELLA ET Al.scaled height of burst (HOB) of the explOSion, the blastsource. and the equation of state (EOS) of the medium(e.g., y varies for real air). Hence. such flowfields arenot amenable to general solution; each represents a par-t ic ul ar c as e.

Much of our knowledge of such reflections comes fromconsidering the flowfield in the near viCinity of thereflection point. By neglecting the rarefaction wavebehind the incident shock, one can equate the flow to thatproduced by a plane, square wave shock reflecting from aplane surface. This is. of course. a good approximationwhen the flow behind the reflected shock is supersonic(relative to the reflection point). Many tools thenbecome available. For example. one can use the shockpolar technique (Courant and Friedrichs 1948) with anappropriate equation of state to predict peak pressures inthe regular reflection regime; whereas in the Mach reflec-tion regime. one must resort to experimental data of shockreflections from wedges (e.g Bertrand 1972). One canuse experimental shock-an-wedge results and their asso-ciated empirical theories to predict the transition toMach reflection and the approximate shock structure.-Indeed. such analysis predicts that for strong shocks.transition will proceed from regular to double-Machreflection. One can even view the height-of-burst problemas a continuous sequence of shock-on-wedge configurationsfor which the wedge angle varies from 90 deg at groundzero to 0 deg at an infinite ground range. Nevertheless.such techniques have a limited utility. They are alwaysapproximations to a truly non-self-similar problem. andthey do not describe the entire flowfield. To overcomesuch limitations. one must resort to height-of-burste xp er im en t~ a nd t wo -d im en si on al n um er ic al s im ul at io ns .

Height-of-burst experiments utilizing HE blast wavesources have been conducted (Baker 1973). Typically.flowfield measurements are limited to near-surface staticand total pressure histories at a s.all number of groundranges. and high-speed photography. Often there is muchscatter in the data due to nonrepeatabillty of the HEcharges; this scatter limits the scientific usefulness ofthe data. Some of the most repeatable data come fromtests performed with 8-1b spheres of PBx-9404 (Carpenter1974). Nevertheless. such measurements are not sufficient

See, for example, Ben-Cor and Glass (1978, 1979). Ando andGlass (1981). Shirouzu and Gla (1982). Lee and Gla (1984).neacha.bault and Gla (1981). Bazhenova et al. (1984). Hu andGl.. a (1986). Rornunl (198S), .and Rornul\I an d Taylor (&982).


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MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 391to allow one to reconstruct the entire flowfield. Forthat. a numerical simulation of the flow is required.Today, one can simulate the reflection of a sphericalblast wave from a plane surface with numerical codes thats ol ve t he i nvi sci d t wo- dim ens ion al E ul er e qu ati ons of g as -dynamics: for example. a simulation of the Tunguska mete-orite explosion at an H08 - 305 m/kt3 ( Sh ur sh al ov 1 97 8)and the calculation of a point-source case detonated at anHOB - 31.7 m/kt3 (Fry et al. 1981). How accurate are suchcalculations? One of the difficulties in numerical simu-lation of such flows is the disparity of length scales inthe problemi for example. the height-of-burst scsle vs theMach stem height (typically less than 1/10 the height-of-burst scale) vs the boundary layer scale (which is muchsmaller than the Mach atem height). One must take specialcare to design the computational mesh to take such dis-parate length scales into account. With the memory sizeand speed of class VI computers such as the CRAY I. suchl ar ge -s ca le c om pu ta ti on s a re n ow p os si bl e ( al th ou gh e xp en -sive). Of course. one needs a minimal-diffusion numericalalgorithm to maximize the information per grid point. Ano tew ort hy ex amp le i s t he se con d-o rde r Eu ler ian G odu novscheme of Colella and Glaz (1984. 1985). This code hasb een us ed t o s im ula te sh ock -on -we dge ex per ime nts in th eregular reflection regime and in the simple. complex, andd oub le- Mac h r efl ect ion re gim es. Ex cel len t ag ree men t wi thdata was obtained for those cases for which viscous andn on eq ui li br iu m e ff ec ts w er e n eg li gi bl e i n t he e xp er im en ts(Glaz et al. 1985a, 1985b. 1986). In Some of the double-Mach reflection cases for which such effects were nots ma ll. qu ali tat ive ag re em ent w as st ill f oun d f or fl owf iel df ea tu re s s uc h 88 c on ta ct s ur fa ce ls ec on d M ac h s te m i nt er ac -tion and subsequent vortex rollup. Nevertheless. theq ues tio n re mai ns: Ho w ac cur ate ly c an on e nu mer ica llys im ul at e t he t ru ly n on st ea dy h ei gh t- of -b u' rs t c as e?The objective of this work was then to perform ah ig hly r eso lve d n um eri cal s im ula tio n of t he tw o-d ime n-sional reflection of an HE-driven blast wave with theabovementioned Godunov acheme and to check the accuracy oft he so lut ion b y c om par ing i t w it h p re cis ion ex per ime nta ldata. An 8- lb P 8X -94 04 c har ge e xp eri men t de ton ate d a tHOB - 51.66 cs (Carpenter 1974) was selected for that pur-pose. A zoning convergence study (with a fine grid meshspacing of 1.2. 0.6, and, finally. 0.3 mm) was performedto demonstrate that the results were independent of cellsize.T he c om pu ta ti on al t ec hn iq ue i nc lu di ng t he s ec on d- or de rGodunov scheme. the equations of state. the initial condi-tions, and the grid dynamics are.described in Sec. 11.

" . . . .

392 P. COLELLA ET AL.Th e n ume ric al re sul ts, s uch a s t he i nci den t o ne- dim en-s ion al b las t w av e. th e re gul ar re fle cti on r eg ime . tr ans i-ti on, th e d ou ble -Ma ch r efl ect ion r eg ime , c omp ari son s o fs urf ace d ata . a nd c omp ari son s wi th an eq uiv ale nt s hoc k-o n-wedge case, are presented in Sec. Ill. Conclusions andre com men ded i mpr ove men ts a re o ffe red i n S ec . IV .

I I. C om pu ta ti on al M et hO dT he eq uat ion s o f c omp res sib le h ydr ody nam ics i n o nes pa ce va ria ble . w ri tte n i n c on ser vat ion f or m, a re

a a aat !! + aV A ! + ar'!! 0 (la)where

(lb)

. 2 2Here, P is the density. E - (1/2)(u + v ) + e is the totalenergy per unit mas8, where e is the internal energy perunit mass. u is the component of velocity in the r-direc-tion, and v is the transverse component of velocity; p isth e p res sur e; X re pre sen ts a n ar bit rar y a dv ect ed s cal arquantity. and V ; VCr) - ra+1/(a + I) Is a volume coordi-nate, A ; A(r) - dV/dr - ra. The values u 0,1.2 corres-po nd t o C art esi an. cy lin dri cal . a nd sp her ica l s ymm etr y,r es pec tiv ely . T his p ar tic ula r r ep res ent ati on o f t he e qua -t ion s f ol low s C ole lla a nd W oo dwa rd (1 984 ) a nd co rre spo ndsc lo sel y t o th e fi nit e-d iff ere nce e qua tio ns th at fo llo w.The pressure is given by an equation of state:(2)

f or si ngl e-f lui d h yd rod yna mic s. F or t he ca lc ul ati ons p re -sented here. it is necessary to use a two-fluid model.where the two fluids are the detonation product gases andair. Each of these materials has associated with it anequat i on of state of the form of Eq. (2). We let X denotethe volume fraction of high explosives (HE). so that in amixed cell, 0 ( X (1. Then our two-fluid treatment isdefined by the last equation in Eq. (1) and by setting

p XPHE + (l-X)Pair (3)


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MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 39 3wherever a pressure is needed by the numerical method.This relatively crude treatment (in particular. our reli-ance on the mixture density and internal energy precludesreferring to the model as a true two-fluid model) turnsout to be sufficient for the present problem. This islargely due to the fact that the dynamics of the materialinterface are not of major interest and they do notdirectly interact with the Mach stem region flowfield.which is the focal point of this study. Our treatmenthere will be superseded by a true multimaterial algorithmbased on the simple line interface calculation (SLIC)algorithm of Noh and Woodward (1976) and the Euleriansecond-order Godunov scheme for single-fluid hydrodynamics(Colella et al. 1986).The numerical method used in this study is the versionof the second-order Eulerian Godunov scheme described inColella and Glaz (1985). This version was espeCiallydesigned to handle general equations of state of the typeencountered here. The modifications necessary for non-Cartesian symmetries (i.e a - 1.2) are described inColella and Woodward (1984). Operator splitting is usedto solve multidimensional problems; in the axisymmetriccalculation of Sec. III. this means that Eqs. (1) witha-I are solved in the radial direction with u set to theradial component of velocity; and then Eqs. (1) with a - 0are solved in the axial direction with u set to the axialc omp on en t o f v el oc it y. A brief overview of the method forsolving Eqs. (1) is presented below.

Let Un {uj} represent the cell-averaged solution attime level t tn. i.e

nVj+I/2f U(r.tn)dVnVj-l/2

(4)

n+ 1The computational objective is to define U in termS ofUn. The conservative. second-order-in-tiae. finite-dif-ference representation of Eqs. (I) is

AVj+l Un+1 AV n Un _ Atn[An+l/2 ,n+l/2 _ An+l /2 Fn+l/2j j j j+l/2 j+l/2 j-l/2 j-l/2(n+I/2 n+l/2) 2. 0 - ] (5 )Hj+l/2 - Hj-l/2 n A n+lAr j + rj

39 4 P. COLELLA ET AL.where

AVj Vj+I/2 - Vj-l/2.n+ 1/2 n -I f 1 [ nAj+I/2 (At) 0 A rj+1/2 + n+l n ]8 (rj+l/2 - rj+l/2) dB

n+l/2 n+l/2 n+l/2Here. typical F1+1/2 - F(Uj+l/2). and Uj+l/2 representsthe average of D along the (j.j+l) interface. i.e

ftn+1Un +1/2 _ (Atn)-1 u[ rn n+l/2 n]j+l/2 j 1/ + s (t - t ).t dte" + 2 j+l/2where

n+l/2 (n+l/2 n \/ nSj+l/2 - rj+1/2 - rj+1/2/1 AtEvidently. a computational scheme in the form Eq. (5) isn+l/2 ndefined by specifying Uj+l/2 as a function of U The first-order Godunov scheme is defined by settingn+l/2 n nUj+l/2 to the solution of the Riemann problem (Uj' Uj+l)n+l/2evaluated along the line r/t Sj+l/2' The high-orderscheme is conceptually similar in that a Riemann problemn n(Uj+I/2.L. Uj+l/2.R) is constructed and solved in the S8meway. However. the left and right states are now functionsn no f ( Uj -2 U j+ 2) an d (U j- l U j+ 3) . re spe ct iv el y.These additional data are used to create aonotonizedpiecewi'Be-linear profiles in each computational zone. fromwhich a version of the .ethod of characteristics is basedto get new values centered on the interface. The overallconstruction. including the solution of the Riemann prob-lem. is eqUivalent to the method of characteristics (up tosecond order) for smooth flow in determ1niRJ the interfacefluxes. Further details. such as a on ot on ic it y c on st ra in tsand additional constructions necessary near strong discon-tinuities. asy be found in the references mentioned.

An important aspect of our numerical .ethod is that wedo not require equation-ot-state evaluations at each stepin the Rie.ann problem. iterative solution. it is onlynecessary tor the approximate .ethod that the equation ofstate be evaluated for each u j . T he i nf or ma ti on r eq ui re dby the algorithm is the dimensionless quantities 1 =y(p.e). f = f(p.e) such that

p (y-l)pe (6)


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M AC H R EF I..!C TIO N F RO M A N H ED RIV EN B I.A ST W AV E 395

an d (7)f d Note that Y # r for a non-where c is the speed 0 soun.polytr opic equation of sta te.

The equations of state used in the calculations ofSec. III are the equilibrium air EOS of Gilmore (1955) andHansen (1959). and the Jones-Wilkins-Lee (JWL) EOS forPBX-9404 detonation product gases (Dobratz 1974). Thecaloric JWL equation of state takes the form-R P t -RZP t (8 )p A{l-~Po/RlP)e 1 0 + B(l-wPo/RZp)e 0 + wp e

whereas the isentrope is given by-RIPo/P + B e - R2 Po/ pPs As . (9)

where p is the initial charge density and the JWL param-eters f~r PBX-9404 are A - 8.545 Mbar9; B 0.Z049 Mbars;C _ 0.00754 Hbars; R1 - 4.60; R2 - 1.35; w - 0.25. Thebehavi or of y for the JWL EOS may be found b y f ittingEq. (6) to Eq. (8). and r(p,e) can be calculated from theisentrope using Eq. (9); in this case. c2 is obtained inclosed form (Glaz 1979) and Eq. (7) may be used to calcu-late r.The calculation was run in two stages: first as aone-dimensional free air burst until ground strike. andthen as a two-dimensional reflection problem. The one-dimensional calcula tion was initialized when the detona-tion wave reached the charge radius Rc. The flowfieldinside the charge at that time was assumed to b e that ofan ideal Chapman-Jou guet (CJ) detona tion (Tayl or 195 0;Kuhl aod Seizew 1978) with no afterburning. Using the JWLparameters for a PBX-9404 charge with an initial densityof Po - 1.84 g/ca3 the C J state is PCJ - 370 kbars; PCJ -2 .4 85 g /c m3; ecJ - S.142x1010 erg/go WCJ - 8.8 km/s; UCJ -2.28 km/s; qCJ - 5.543x1010 erg/go r - 2.85; X-I.For an 8-1b sphere the charge radius was R c 7.76 em.The ambient atmoaphere was initialized as Pa - 1.00 bar;P _ 1.1687x10-3 g/cm3; e8 - 2.1390x109 erg/go u - 0;X8_ O. A fine-zoned grid (6r - 0.3 am) was dynamicallymoved with the shock to accurately capture the complexflow in that region. Coarse zones (6r - 3 mm) were usednear r - 0 and for large r; and a transition region con-nected these cells with the fine grid. After initializa-tion. the evolution of the one-dimensional blaat wave wascalculated by solving Eqs. (1) with a - 2 until the shockradius was equal to the height of burst (51.66 cm. t -

39 8 P . C O LE LL A E T A I...97.44 ~s). This solution was then conservatively interpo-lated onto a two-dimen sional mesh.The two-dimensional mesh covered a region 0 < r


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MACHREFLECTIONFROMAN HEDRIVENBLASTWAVE 397 39 8 P. COLELLAETAL.caused the detonation products to overexpand to a velocitylarger than that induced by the air shock. This incom-patibility was resolved by an inward-facing shock. whicheventually imploded and created a series of secondarypulses at late times.Our calculation was performed for a spherical PBX-9404charge (initial charge density of 1.84 g!cm3 detonationpressure of 370 kbars). The resulting blast wave wasqualitatively similar to Brode's results; hence, theresults will not be reported here In detail. Quantitativedifferences were as follows. Peak velocities reachedabout 17 km!s. whereas the maximum peak air-shock pressurereached about 1 kbar. owing to the larger detonation pres-sure of the PBX charge. The blast wave approached thepoint-source solution at a shock overpressure of about

20. ~Nr~~~~tN(Rj; '. -I!O . . 9 E~G/C,'__,n'\r'-"'r:~ ,- -'-h'", ""--".....!lllI.:~ 1

Fig. ,. T r . f t. i t i on f r o . r e l u l.r t o d ou bl e- Ma ch r ef le ct io n f or . ph er ie al . H E- dr iv en b I. at . av e r ef le ct in g fraan ideal p la ne . ur -race. De_ity c:ontoun (10-)lie.).

40.l 110 .5 80.8 .00.'r .cm~20.4

~II~

r fQn' 100.'

P ig. Ib T ra n. it io n f r~ re gu la r t o d ou bl e-M ac h r ef lee ti on f or a' ph er ie at . H E- dr iv en b la .t V lv e r ef le et in g f ro . a nflee 1 t 1 ideal plane aur- n eena energy contour. (109 e r g /g ) 13 bars vs 7 bars for TNT. The air shock arrived at~~~u~d5~e~~ (i.e at a shock radius corresponding to the cm. or 6.78 charge radii) with an incidentover-pressure of 98.86 bars; hence. the flowfield corre-sponded to a piston-driven wave throughout the entireregime of the two-dimensional calculation. This led toshock interactions that are unique to the HE case.B. Overall View of the Two-D imensional Reflection

An overall view of the two-dimensional reflection ofthe spherical HE-driven blast wave from an ideal plane~u~face is depicted in Fig. 1 in terms of isodensity i80-n ernal energy. and isopressure contours at differe~t

t~mes. Thirty equally spaced contour values were usedw th the minimum and maximum value and step size identi-


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MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 39 9 400 P. COLELLA ET AL.

I101 1008

ground ranges, the reflected shock propagates upward veryslowly because of the large, downward-direct~d dynamicpressure of the detonation products gases of the incidentwave. This is markedly different from the case of thereflection of a point eKplosion blast wave in which thereflected shock propagates very rapidly through the low-density, high-sound-speed region near the blast center(Fry et a1. 1981)

Interactions of the reflected shock R with the contactsurface CS and the shock l' create additional shocks nearthe ground and generate vortex structures on CS and SL',

1 10.2~2

2O.4P

1 10.2L0 40l

.~.al SfE~1.C6f'!ll----._ - T - ~ - l124___ ______ ,__J_ _ r - - t _ j_

eo.1 108 100.'

JU . . . . . ,6.n(.00 Til 3 .25( .02 STEPI. IOE-OI

. /! t > " ~

. 1 1 53.2

, l a W '- , - - . , - " ' " 1'0.2

. . . . . , 1001 5l.2i10.2

1001f 'M I .P il. le Tra~ ition fro - r elul ar t o dou ble- Mach r_f leeti on f or _e ph _r ie al . H E- dr iv en b le et . .. . r .f le et in , f ro a e n i de al p le ne . ur -f ac e. P re ee ur e c oa to ur e ( ba re ).

. 1 1 13.2

fied on the plot. This technique gives a concise displayof the major features of the two-dimensional flowfield:D iscontinuities appear as heavy dark lines (where manycontours group together), rarefaction waves appear as afan of contour lines, while plateau regions are contour-free. Contact .urfaces may be i de nt if ie d a s d is co nt in ui -ties in density and internal energy. without any jump inpressure or velocity. slip lines may be d is ti ng uis he d a scontact surfaces with a discontinuous change in velocity.shocks are denoted by discontinuities With sharp jumps inpressure.In Fig. I, the incident shock (I), the contact surface(CS) separating the detonation products and air. and theinward-facing shock (I') of the incident blast wave areclearly visible. Reflection of the incident Rhock 1 fromthe plane surface creates the main reflecte6 shock R.which effectively 8tops the contact 8urface CS. At small

o ~~ '; ~ _ : ; _ " _ , J r304 43.'. . . . . . 1

ri,. 2a In tera ction of t he r eflec ted ...e 1 wit h th_ con tacte ur fe ce C S a nd e ho ek I ' i n t he r _, ul ar r ef le ct io n r e, i. .(t 1 7 . ~ ei refl actl oa po lat at SO ea ).


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MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 401 402 P. COLELLA ET AL.

~8.D 532 " " " I1.1

! ]3.14U

4Dr h:rnl ~32 [. .. . 1

[an'rl,. 2e. Interaction of the reflect ed wava l.tth the contacte ur fa ca C S a nd e ho ck I f i n t he r e, ul ar r ef le ct io n r e, i. ..Sch e.. tic ahowin, wave interectione: (a) t 124 ~e, r J O c - .(b) t 14 S ~e. r 40 e - . (e) t 17 1 ~a. r 50 ea.

ri,. 2b Interaction of the reflected wava I with the c ontactaurface CS and ahoek I' ln the re,ular reflection regi..(t 171 ~.; reflection point at ~ o cal.

tinuities are somewhat difficult to pick out when they arelocated in the coarae-zoned region; hence they have beende pi ct ed s che mat ic al ly in F ig . 2 c.Th e r efl ect ed s hoc k R in ter act s w ith t he i nc ide nt -w avecontact surface es at point A, creating a reflected shockRCt and deflecting es. Shock Ret reflects off the wall atpoint B as a regular reflection, thus creating a secondpeak pressure on the wall. The reflected portion of RCIreflects off contact surface CS at point C, creatingre fl ec ted sh ock R C2 , a nd f urt her d efl ect s co nt ac t su rf ac eCS. Shock RC2 reflects off the wall at point D as a regu-lar reflection, thus creating a third peak pressure on thewall. The reflected portion of shock RC2 reflects offcontact surface CS at point E, c re at in g a t hi rd r ef le ct eds ho ck R C) .

T he t ra ns mi tt ed p or ti on of shock R e ma na ti ng f ro .p o i n t A i n t e r a c t s w i t h t h e i n c i d e n t s b o c k I' ( o b l i q u e. h o c k i n t e r a c t i o n ) a t p o i n t ' . c r e . t i n e a _ l i p l i n . IL'.

as shown in Figs. 1a and lb. T he r ef le cte d s hoc k a ls odeflects the contact surface away from the Mach stemregion so that in this calculation, the detonation prod-ucts are not entrained in the Mach stem flow.C . Th e R eg ul ar R efl ec ti on R egi on

A detailed view of the flowfield near the end of ther eg ul ar r ef le ct io n r eg io n (t 1 71 ~ s. r ef le ct io n pointIt ' 0 c. ) il I h o w n i n P l . s . 2. a n d 2b . T h e w e a k e r d i s c o n -" ; ; ii . _ (, . . , , ,, . , ~ r .. ' . " ' I If , ~


10/18

MACHREFLECTIONFROMANHEDRIVENBLASTWAVE 403 404 P. COLELLAETAl.The transmitted portion of shock I' emanating from point Finteracts obliquely with the transmitted portion of shockRCI emanating from C at point G; transmitted shocks fromthis reflection interact with the contact surface CS atpoint H. and with slip line SL ' at J . At earlier times, atr an sm itt ed s ho ck f rom p oi nt J in te ra cte d w it h t he ma inreflected shock at point L. D en si ty /i nt er na l e ne rg y g ra -dients in the incident wave cause kinks in the mainreflected wave at points K an d K' .In summary. the following features were found in theregular reflection region. The main reflected wave Rreflects within the channel formed by the wall and thed en se de to na tio n p ro du ct s (C S) . c au si ng a ddi ti on al p re s-sure pulses on the wall. Shock interactions with contactsurfaces at points A and F inviscidly generate positiveand negative vorticity, respectively, which rolls up intovortex structures shown in Figs. la and lb. Finally, themain contact surface CS is idealized in this calculationas a discontinuity. We know experimentally. however. thatt his s ur fac e i s i rr egu la r a nd d if fu sed -- pe rtu rb at ion s o nthis surface grow owing to a Rayleigh-Taylor mechanism andthese lead to local turbulent mixing during the evolutionof the incident blast wave (Anisiaov et al. 1983). Th est re ng th o f r ef le cte d s ho ck s RC I an d RC 2 will depend onthe mixing across the contact surface CS. These inviscidcalculations. which do not take into account such turbu-lent mixing, no doubt overestimate the strength of shocksRC I and RC2.

O. T he Tr an si tio n Re gi meA detailed view of the shock structure In the tran-sition region is given in Fig. 3. I n t hi s c al cu la ti ont ra ns it io n f ro m r eg ul ar r ef le ct io n (RR) t o d o ub l e- M ac h

reflection (OMR) occurred at a ground range of greaterthan 52.5 cm and less than 55 cm, with corresponding inci-dent shock angles of 44.5 and 46.8 deg, respectively.C om pa ri so n w it h th e l im it o f e xi st en ce r egu la r r ef le ct ion(i.e., the So-called deflection criterion) for real air 1nTable 1 i nd ica te s t ha t th e ca lc ul at ed r eg ul ar r ef lec ti onr eg io n pe rs is ted i n t hi s he ig ht -of -b ur st c al cu la ti on f or1 . 5 to 3 . B deg beyond the theoretical limit. Note that a ~s im il ar p er si st en ce o f r eg ul ar r ef le ct io n (PRR) has been

Table 1 Co .par hon of r eg u la r d ou b le - Ma c h t r an s it i on sIncident

Wedge shock Transitionangle, a angle, G g ro un d r an geSouree (deg) (deg) (elll)

'1""1

,1"'1L l. tt o f r eg ul ar r ef le et lo nJ.uhl,1982 48.2Gilmore'e Air EOS 4 7 43IllllOre 1955])

,I-IGla.... 1 9 8 2 49.9H an se n's A ir BO S 46 44(Hanlen 19590HO B ! !aleulation

Po . 45.S 44.5 52.5P i, . 3 )

k :m lP re aa ur e c on to ur a a ho vi na t ra ns it io n f ra . r eg ul ar t oIJlI. 43.2 46.8 5S double-Mach refleetion: Ca) t IS8 ~., r 4S ca (II). (b ) t 179 ~a. r 52.5 ca ( PR R) j ( e) t 18 7 Pa, r 55 c. (DKR).- ' . ,"

. . r 'IIII:

~1riii


11/18

M A O H R E FI. EO T IO N F ROM A N H ED RIV EN B LA ST W A VE 40 5observed for shock rafla etlon. I ro m w ed S' ., b ot h e .p er l-mentall y (B1eackney and Taub 1949. H enderson and Lo zzi1975) a nd numer ically, with t he same hydrocode used here(Glaz e t all 19 85a. 198 5b. 1986).One can identify three potentia l reaso ns for persis-tence of reg ular reflection: 1) real viscosity. 2) numeri -cal viscosity. an d 3) inadequ ate zoning. Re al viscositymust be r ejected f or this case because it was not includedin the calculat ion. The second-o rder Godunov schem e usedhere leav es essen tially no numerical viscosity in thesmooth regi ons of the flow. However . all sh ock-capturingsch emes i ntroduce numerical dissipation at shock fr onts toallow a s mooth transit ion b etween pres hock and pos tshocksta tes. When such a lgorithms are used to calculate shoc kwaves near a wall. a "nume rical wall boundary l ayer" i sfor med (Noh 1976) . Th e primar y effect of this 1s tocreate an artif icial "wall hea ting"--typical ly a few per-cent. This effec t can be seen 1n the density and radi alvelocity con tours of Fig. 2. wh ich exhibit a kink at aboutthe 3-mm height (about ten cells).

A co ncerted effort was made to m inimize comput ationalcel l-size effect s. The 617 by 214 g rid used ess entiallyall of the one-megaword fa st core spac e av ailable on aCRA Y 1 comp uter. The fine-zoned grid (267 by 140 cells)that slid w ith the r eflection region used ce lls of 0.3 by0.3 mm. Thi s resul ted in 83 radial cel ls between thereflecti on poi nt at 52.5 em and the 55-cm point, and onewould think that would cons titute adequate zoni ng. How-ever, the M ach st em grows from a po int (in the inv iscidthe ory) and is never captu red compu tationally u ntil thesho ck structure grows large enough to be resolved on t hemesh. Note that a t the 55-cm location. the Mach stem was 'onl y about four cell s high. If a Mach stem e xisted at the52.S-cm gr ound ra nge. it woul d be less than one cell high.hen ce, it would not hav e been resolved. A more detailedinvi acid calculation of tr ansition using a local adaptiv egrid refinement (e.g., Berger and Cole lla 1986) isrequired to concl usively reso lve this zoning question. Wespeculate t hat such inv iscid calculat ions wi ll ind eed con-firm that double-Mach reflect ion will exi st im mediatelyafter passing the RR limit. Therefo re, we believe thatthe persistence of r egular refl ection in these calcula-tions is caus ed by inadeq uate zoning a nd the numericalwal l boundary layer . while the persistence beha viorobse rved in experiment s Is due to a vis cous wall boundar yeffect. To concl usively prove the latter , a viscous cal-culation of t he oblique sh ock structure at the wall isrequired.

406 P. COLELLA ET AL.

,I

' . c u l t a , . a R d y n a m t o . " . O t l w t r l a b r v . d t n t h t .calcu lation in the PRR region. It 18'well k nown that a8t he incident shock angle increa ses in the regular re flec-tion region, one encounters the so nic criterion (w heresound waves can r each all the way t o the re flection point)about 1 deg be fore one reaches the RR limit (He nderson andLozzi 1975). In such a case. the reflected shock is nolonger s trsight but continuou sly curv ed near the reflec-tion point. The present calculatio ns also exhibit sucheffects. Fi gure 3 shows that the r eflected sho ck isstrai ght at a ground range of 45 cm (RR) but curved nearthe reflection point at 52.5 cm. A s shown in Table 2. theangle of the main portion of the reflecte d shock incr easessmoothly through transiti on; howe ver. at the wall it jum psfrom about 24 d eg at the 50-cm range to about 37 deg atthe 52.S-cm ra nge.

The pressure an d velocity profiles on the wa ll als ochanged dramati cally in the PRR r egion. When th e reflec -tion p oint was at 45 em, the pres sure and vel ocity gra -dients were w ell behaved. Howe ver, in the PRR region(e.g., with t he refle ction point at .52.5 cm). the p ressureand velocity gradients on th e wall become very larg e asone approach es the re flection point from t he left.

In summary. the limi t of regular reflection for thiscase i s 43 to 44 deg (depen ding on th e particular equat ionof state used f or air) with a c orresponding gr ound rangeto t ransition of 48.2 to 49.9 cm. In this calculati on.regu lar re flection seemed to pers ist to a gr ound ran ge ofabout 52.5 cm (a - 44.5 de g), but the reflected shock

IfIIII

1tI!iI!

T a b l e 2 S ho c k a n gl e . n e ar t ra n. l tl onI n c i d e n t R e f l e c t e d s ho ck a nr tl eC r o u n d . h o c kr a n g e a n g l e , u O ff w a ll . N ea r w al l,(e_) ( d e g ) 8 ( d e g ) 8 o ( d e g ) R e g i _

t4 5 4 2 . 5 19 19 R I5 0 44 2 4 2 4 P I R5 2 . ' ) 4 6 . 5 2 6 - 3 1 - P R R5 5 4 8 2 8 . ' ) ..37 [ P I R5 1 . 5 4 9 3 3 - 3 1 I J o I R6 0 51 34 -37 [ P I R


12/18

MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 407 408 p, COLELLA ET AL,angle near the surface at this range was consistent Witht ha t o f d ou bl e- Ma ch r ef le ct io n (a ~ 37 deg). Hence, w ebe li ev e t ha t i t w as i nd ee d a na sc en t do ub le -M ac h s tr uc tu rethat was not computationally resolved on the mesh. As weshall see in the next section, both local adaptive meshre fi ne me nt an d t ur bu le nc e m ode li ng ar e re qu ir ed t o pr op -erly model certain details of the flow in the double Machr eg io n a nd , by im pl ic at io n. t o a cc ur at el y p re di ct tr an si -tion.E. Th e D ou bl e- Ma ch R eg io n

A detailed view of the complex flowfield in thedouble-Mach region is presented in Fig. 4 (t - 270 vs,Mach stem at 80 em). The domain of these figures repre-aents the fine-zoned region of the calculation (267 r b y

140 z cells with a cell size of 0.3 nm). The incidentshock (I), the reflected shock (R'). and the main Machstem (HI) meet at the main triple point (TPI), generatinga slip line (SLI) that haa positive vorticity. Air flowsalong SLI and impacts on the wall, thus creating a largelocal pressure. This point actually corresponds to amo vi ng st ag na ti on po in t ( SP 1) , wh ic h i s p ar ti cu la rl y e vi -dent in the relative vector velocity plot of Fig. 4c (thecoordinate system is moving with the velocity of SP1 at2.308 km/s). The flow overexpands from SPI (about 145bars) by means of a strong rarefaction wave ( R W ) a nd f or asa l ow -p re ss ur e (- IO -b ar m in im um ) su pe rs on ic w al l j et (B eethe Mach number contours, relative to SPl, of Fig. 4c).The gas velocity in the wall jet (3.5 to 4.3 ka/s) islarger than the wave velocity of the Mach stem (-2.75km/s) , so the jet rams into the rear of the Mach stem . .,

.2UU

_ 2.'tI 1.7

0.1 M'~4 11.2f I m R I 74.' 7'.4 . 1 . 2

f t-).2Fig. 4a S h oc k . t ru ct u re

in t he d ou bl e- Ma ch r ef le c' 3.4 Fig. 4b S ho ck t ru e t u ret lo n r eg t_ ~t .27 0 118, in t h e d ou b le -H a ch reflec-Mac h a te. at 80 ca) jU t io n r eg im e (t 270 ",a,H 1.7 Ma ch .te ~ a t 80 c~).i'0.. III'

a' . . . . 13.2 7.. 7... .1.2. . . . ,4.J .2 'U

I _uIH U

I.0.. I ~I I. III'II' , .,, i ' ,.u a

, ... 1 71 2 ". 71.0 71.1 . I . J.t-II , _. - . ~ "I . . . . . .


13/18

MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE

This interaction pushes ~t the foot of the Mach stem andforces the jet to expand upward two dimensionally. thusforming a rotational flow and a main vortex VI. which haspositive rotation. The rotational flow near VI is locallysupersonic, and embedded shocks (5'. 5' '. and 5"') can beseen.The toeing-out of the Mach stem creates a secondtriple point, TP2. This is actually an inverted Mach stemstructure with an incident shock HI'. a reflected shock5", a Mach ste. HI. and a slip line SL2 that has negativevorticity. This slip line flows up and over the main vor-tex VI, approaches the wall and stagnates, thus creating asecond moving stagnation point (SP2). which is also evi-dent in the relative velocity vector plot of Pig. 4c. Ata range of 80 em. SP2 has shocked-up on the wall. All ofslip line SL2 and some of SLI are entrained in a secondv or te x s tr uc tu re V 2. w hi ch b aa n eKa ti ve r ot ati on ( 8e e

r 1 0 0 > 1

409 P. COLELLA ET AL.10Fig. 4c). All of the fluid entering the main Mach stem HIbetween triple points TPl and TP2 is entrained in vortexV2 (see the vorticity contour plot of Pig. 4c).The gas velocity is supersonic above slip line SLl andsubsonic below it. Pressure waves from SPI coalesce.forming the second Mach stem H2. The latter interactswith the reflected shock forming a third triple point(TP3). This also appears to be a n i nv er te d M ac h s tr uc tu rewith an incident.wave R'. a reflected wave H2, a Mach stemR, and a slip line (SL3). A fourth triple point (TP4) canbe seen on shock H2. It appears to be a remnant of theinteraction of the embedded shock 5'" with SL1. It hasan incident shock H2. a reflected shock R", a Mach stemH2'. and a slip line SL4. Shock R" terminates on slipline SL3. while shock H2' terminates on slip line SLI.Secondary vortex structures are evident on slip lineSLI (caused by shock H2 and local rarefaction waves) andslip line SL2 (induced by shock 5" I}, and near vortex VL(the entrained part of SLI that was shocked by 5' ' ).T he r are fa cti on wa ve be hi nd th e i nc id ent s hoc k p ro pa -gates through the DHR structure (see, for example, thed en si ty . in te rna l e ner gy . a nd p re ssu re co nt our p lot s o fFig. 4a) just as in the regular 'reflection casei but thisap pe ars t o be a w eak e ff ec t, si nc e t he m ai n d is co nti nu i-ties (R'. SLl. and M2) are basically straight lines.Even at the 80-cm range. the wall jet was quite thin(1.8 mm) and not well resolved (about six cells high).although very fine zoning was used here. it was still toocoarse for adequate numerical resolution. The slip lineSLI on top of the jet is a free shear layer subject toK el vi n- He lm ho lt z i ns ta bi li ti es . H er e t he R ey no ld s n um be rof the jet was about 3xl04 based on jet height. It willn o d ou bt d ev elo p vo rt ex s tr uc tu res . l ea din g t o t urb ul entmixing. Also. the wall boundary layer will reduce theradial momentum of the jet. Hence. turbulence effectswill influence the entrainment of the main vortex VI andthe toeing-out of MI' (i.e., there will be l e ss p u sh i ng ) .These effects were not modeled in this calculation andm ay. i n f ac t. i nf lu enc e t ra ns it ion .

Fi g. 4 c Sh ock stru ctu rei n t he d ou bl e- Ma ch r ef le c-t io n r eg 1~ (t 27 0 ~s.Ma ch ste m at 80 em).

P . Su rf ace D at aPigure 5 gives a detailed snapshot of the completeflowfield on the surface at the end of regular reflection(t - L71 ~s, reflection point at 50 Cd) and in the fully

reaolved double-Mach reglon (t - 270 ~s. Mach ste. toe at80 cm). Such plots augment the interpretatlon of the con-tour plots In Figs. 2 and 4. The reflected 'hocks R. Ret.

rI

ItII~IIi


14/18

M AC H R EF LE CT IO N F RO M A N H E-D RIV EN B LA ST W AV E

300 I II I C '

o .. ... . '-----o 20 40 eo eo ,00r .cm~~ 1 I ~ l ~ ~.- - l

I I c ~ N li , o ) I iA


15/18

M AC H R EFL EC TIO N F RO M A N H ED RIV EN B LA ST W AV EA.. . I., A.floet ion (I 111 ,..,

~~~~~~~~~.

10~I; SO" ' s ! 60

...o . .. . .. . .. . .. . .. . .. . ..o ro 4D 60 ~ 100r I cml

1600 ~.,." . , ... ", .. ,1400

- 1200i000~e ~I!. 600oit 400200

ro 40 ~ ~ 100r le ml

Ooubl. Moc: h A .f le ctio n I t 2 10 ~. ,413 414 p, C OL EL LA E T A L.

e o 6S 70 7S ~,Icm,

1200 - . " .. ,., ..... ,.

1000

200o .110 66 10 15 110

r lo ml

1 II. 1-- :~ ~- . .. . w - rr T - '-_ . ' 1 91_~_ . . . . .~:---. . . . ,OCI~ .CI1-1ia.K."~ ""..1...-.. I c . r " " " , , , 191... . . . . " " .. .....n 0

Fig. ~c Co~pariaon of surface-level flowfields tn the reRularreflectlon regime (t 171 ~s) and double-Mach reflection regime(t - 270 III).

in Fig. 6. The incident overpressure at ground zero of98.86 bars reflects to a peak value of 880 bars. The cor-r es po nd in g t he or et ic al r ef le ct io n f ac to r f or r ea l a ir i s9.43, which results in a theoretical re flected pressure of932 bars (vs 880 bars for the two-dimensional calculation,or 6 percent low due to zoning). The reflected pressurecurve R agrees very well with the experimental data ofC ar pe nt er ( 19 74 ), t hu s i nd ir ec tl y c on fi rm in g t ha t t he c al -c ul at ed i nc id en t b la st w av e c lo se ly s im ul at ed t he e xp er i-mental blast wave. Near ground zero, the shocks RCi andRC2 are much stronger than the reflected shock R, but theyd ec ay m or e r ap id ly . A s m en ti on ed b ef or e, c al cu la te dvalues for RCi and RC2 are expected to be too largebecause of the sharp contact surface in this calculation.Note in particular that the pressure range curve forshock R suffers a jolt at 49.5 cm (i.e., near the RRlimit) and locally increases at 53 cm--this behavior per-

OR 1 c m 'Flg. 6) Co.parisons of calculated peak pressurea on the surfacew lt h e xp er im en ta l d at s ( Ca rp en te r (974).

haps being a c onsequence of the arrival of the sonic pointsingularity and the formation of a nascent Mach stem.I n t he d ou bl e- Ma ch r eg io n. t he . ai n 8 ta gn at io n p oi ntSPI decays from 290 bars at transition to a value of about100 bars at 80 cm. Stagnation point SP2 and shock MItdecay rather slowly from about 100 bars to 75 bars. Ing en er al . t he c al cu la te d p ea ks i n t he d ou bl e- Ma ch r eg io na re i n e xc el le nt a gr ee me nt w it h t he e xp er im en ta l d at a(Carpenter 1974) even at ~3 em, where the grid points werei na de qu at e t o r es ol vi ng t he d ou bl e- Ma ch a te

rI


16/18

M A CH R EF LE CT IO N F RO M A N H E-O RIV EN B LA ST W A VE 415 416 P . C O LE LL A E T ALG. Comparis ons with D MR on Wedg es

Considerab ly more is known about the det ails ofd ou bl e- Ma ch s ho ck s tr uc tu re s c re at ed b y p la ne s ho ckref lections fro m wedges . Such f lows have many usefu lfeatures:1) T he f lo ws a re s el f- si mi la r ( tw o- di me ns io na lCar tesian), and hen ce are mo re amen able to analysIs.2 ) E xp er im en ta l p ho to gr ap hi c r es ul ts ( e. g. , S ch li er en ,s ha do wg ra ph , a nd i nt er fe ro me tr ic d at a) a re a va il ab le t ov er if y c o de c al c ul at io ns .3) The complica ting effects of a rarefac tion wavebeh ind t he i ncident shoc k are absent.I n a dd it io n, w ed ge r es ul ts ( e. g r ef le ct io n f ac to rs )are often u sed to ap proximate the heigh t-ot-burst case.Hence, it 1s useful to explore the equivalence of the DMRflowfie ld for th e wedge case correspond ing to the height-o f- bu rs t c as e.The double-Mach structure at a time of 270 ~s (Machsted at 80 em) from the present calc ulation was sele ctedfor comparison. At this time the incident shock Mach num-be r (HI) was 5.46 with an incident shock angle of about57 deg. The equivalent wedge case was constructed as fol-lows. A 500xlO O y-eell two-dimensi onal Cartesia n mesh wa sc ho se n w it h s qu ar e z on es (Ax Ay 1 unit). The shockproperties corresponding to an MI 5.46 real air shockwere continuously fed into the left side of the grId at aahock angle of 57 deg (wedge angle of 33 deg). Thes ec on d- or de r G od un ov s ch em e ( wi th G il mo re 's e qu at io n o fstate for real air) was then used to calculate ther e fl e ct e d f l ow f i el d .The results of the wedge case are shown in Fig. 7. Bydesign. the calculation s are ide ntical a t the ma in tripl epoint TPl. The overall features of the wedge flowfieldare qui te similar to the height-of- burst case (Fig. 4).considering that the wed~e case was about 2.4 times morecoarsely zoned. Peak pressures on the wall (SP1) were13 3 bars (in stantaneou s value at 75.2 ca) for the height-o f- bu rs t c as e a nd 1 2 2 bars f or t he w edge. yIelding" re fl ec ti on f ac to rs " o f 3 .8 a nd 3 .5 , r es pe ct iv el y.

T he p ri nc ip al d if fe re nc es a re t he r ef le ct ed s ho ckangle and the location of triple point TP3. T he r ef l ec te dshock angle of 49 deg for the height-of-burst case is con-sid erably s teeper than the 22-deg angle for the wedg ecase. In the h eight-of-bu rst ca se. the rare faction wavebehind t he i ncident shock alloW S the ref lected shock R.tomove upward .are easIly into the incident wave. Thiscauses t he seco nd Mach s tem (HZ) to be m or e v er ti ca l a nd

. . . . . .

i100L 10

20G_u0 U 0.4 0.1 0.1 IAI~

Fig. 7) Double-Mach reflection fro. s rs.p (HI ~ 5.46. 8v ~ 1 1 d es.G il .o re 's a ir EO S [Gilmore 1955): { a) 'O ve ra ll d en si ty c on to ur s;(b) density contours in the double-Mach region (dashed lines cor-respond to the HO J r esu lt s at 27 0 ~)i (c ) wall p r es s ur e d i st ri b u-tion.

.,. __ . . . . . '"

the length of the reflected shock R' to be about half thevalue found for the wedge case. (To elucidate thesepoints. shocks M2 and R for the helght-of-burs~ case aredepicted as dashed lInes on the wedge results.) Conse-quently. the dista nce betwe en points SPl an d HI ' (i.e.,the D HR durati on) 1s somewh at s horter in the he ight-of-b ur st c as e.I n s um ma ry . w e m ay c on cl ud e t ha t t he " he ig ht -o f- bu rs tcase is truly nonsteady. and hence not amenable to simi-la rity analysi8. The rarefact ion w ave behind the i ncidentshock modifies the reflected shock angle at TP3 and there-

by influences the location and shape of the second Machste. HZ. co.pa red to the equivalen t wedge case . Becau se


17/18

MACH REFLECTION FROM AN HEDRIVEN BLAST WAVE 417 418 P. COLELLA ET AL.peak refle cted pressur es and "r eflectio n fa ctors" a re aconsequence of the gasdynamic state at the main triplepoint, they are similar for the two cases. However, theh e ig ht -o t- b ur st " re fl e ct io n f ac to r " (R-lIPSPl/llpl) m us t bebased on instantaneous values (with 8Pt. TPt. and Ml' allbeing at diff erent rad iI from the explo sion c enter).

due to viscous wall boundary layer effects. Adaptivegrid ding and a visc ous wall bou ndary layer ca pability areag ain ne eded t o accura tely calcul ate such f lows.The do uble-Mac h shock structu re dir ects some of thebl ast en ergy t oward the surf ace. and the reby extends thehigh enthalpy flow to larger ground ranges. The calcu-l at ed s ur fa ce -l ev el p ea k p re ss ur es a re i n e xc el le nt a gr ee -me nt with exp erimenta l da ta at all grou nd ranges.A shock-o n-wedge calc ulation was also per formed tos im ul at e t he d ou bl e- Ma ch f lo wf ie ld f ro m t he h ei gh t- of -burst case at t 270 ~s (Mach stem at 80 em). Overallfeatures of the flow were quite similar in both cases.The principa l differe nces were the refl ected shock an glewhich was larger in the height-of-burst case; and the 'location of triple point TP3, which was closer to TPI int he h el gh t- of -b ur st c as e. T he se e ff ec ts w er e a tt ri bu te dt o the inc ident wave rarefa ction effec ts and tr ue non-s te ad In es s o f t he h ei gh t- of -b ur st c as e.

I V . C on c lu si on sT he p re se nt c al cu la ti on d em on st ra te s t ha t t he r ef le c-tion of a spherical HE-driven blast wave from a plane sur-face cre ates c omplex flow structu res on multi ple le ngth8cales. In the regular Teflection region. portions ofshock R refle ct wit hin th e chan nel fo rmed by the wall andt he d en se d et on at io n p ro du ct s, t hu 8 p ro du ci ng a dd it io na lpressure pulses on the wall. The interaction of shock Rwith contact surface CS and slip lIne SL' inviscidly gen-erates vor ticitYt which leads t o the formatio n of l arge-

scale vortex struct ures (i.e turbulen t Mixing) on th einterface between the detonation products and the air. Inth e double -Mach flo w s tructure . slip lines emana ting f romtriple pOints TPl and TP2 are directed downward. The flowis forced to turn parallel at the wall. thereby convertingsome of the flow kinetic energy into pressure and creatingstagnation points SPl and 8P2 that move with the DHRstructure. This also creates a supersonic wall jet con-Sisting of a free shear layer and a wall boundary layer.The ReynOlds number of the jet is quite large, rangingf ro m 3 xl 04 for this case to 107 f or l ar ge -s ca le e xp lo -sions. Hence, one would expect strong turbulent mixing atthe free shear layer; however, the wall jet in these cal-c ul at io ns w as l am in ar . T he s ec on d- or de r G od un ov a lg or it hmused here is nondiffusive enough to be a bl e t o c al cu la tethe evolut ion of discr ete vortex struct ures started fromi nv is cl d K el vi n- He lm ho lt z i ns ta bi li ti es ( Gl ow ac ki e t a l.1986) if adequate zoning is used in the jet (about fivetimes finer than that used here). The wall boundary layerwas not modeled. Both effects will influence the horizon-tal momentum of the jet. the toeing-out of the Mach 8tem,and the rotational flow of the asin vortices Vl and V2.Adaptiv e griddin g and a viscou s wa ll bound ary la yer capa-bilit y a re neede d t o accura tely mo del th ese details .A do uble-Mac h shock s tructure appeared In th is calcu-lation at a ground range between 52.5 and 55 em, which wasl.~ to 3.8 deg beyond the limit of regular reflection. WebeHev e that the so -called p ersisten ce of reg ular refl ec-ti on in thi s c alculati on was ca used by ina dequate comp uta-ti onal zoni ng, Whereas the pers istence i n experi ~nts i8

ReferencesAndo, S. and Glasa, 1.1. (1981) D o.alna and boundarles of ~aeUdo-I ta tl on ar y o bl iq ue -s ho ck -w av e r ef le ct io ns I n c ar bo n d io xi de .Proceedinge of the 7th International Sy.poslu. on the MilitaryApplicationa of Blast Simulation, pp. 3.6-1 to 3.6-24. D efenceResearch Establish~nt, Suffield Ralston Alberta CanadaV.I. "" ,A ni sl mo v, S .l ., Z e l' do vi ch , Y.I., I no ga ao v, N .A ., a nd I va no v, ".r.(1983) The Taylor inltability of contact boundary between deto-nation products and a surrounding gaa. Shock Waves, Exploaionsand Detonations, edited by J.R. Iowen et al. Progresa in

Astronautics and Aeronautica, 87, ~, New York.Baker, W.E. (1973) ExplOSions in Air. University of Texa. Pre ,A u. ti n, T ex as , p p. 11 8- 14 9.B az he no vs , T .V ., C vo ld ev a, L .G ., a nd N et tl et on , M .A . ( 19 84)unsteady interactiona of ahock vavea. Pro. Aerosp S~i 21,249-331. !:!' -' ~.B en -D or , G . a nd G la .a , I .t . ( 19 78 ) N on at at io na ry o bl iq ue -a ho ck -w av er ef le ct io ns : A ct ua l i ao py cn ic a a nd n u. er ic al " ex pe ri me nt s.~. 16{ll), 1146-ll~3.Ben-Dor, G. and Glasa, t.t. (1979) Domains and boundaries of non-s ta ti on ar y o bl iq ue -a ho ck -w av e r efl ex io ns . I . D ia to mi c g as .J. Fluid Mach. 92 (12 June), 4S9-496.Berger, M. and Colella, P. (1986) Local adaptive Gesb refineMentf or a bo Ck h yd ro dy na .i ca ( in p re s. ).

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Hornung. H.G. end T.ylor, J.R. (1982) Tr.n.ttlon fro. regular toM.ch reflection of .hock waves. Part 1. The effect of via-cosity in the p.eudo.teady ca.e. J. Fluid Hech. 123(OCt.),143-153.

Hu, J.T.C. and GI 1.1. (1986) An i nt er fe ro . . t ri c a nd n u .. r ic .lat ud y o f pa eu do ,t. tl on .ry obl iq ue -sh oc k- wa ve r efl ec ti ons i nlulfur hex.fluorid. (5'6)' 'roc. R. Soc. London (to be pub-Hahed).

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and .pheric.l bla.t wave. in g with counterpre ure. Pro-cee di ng St ekl ov In ati tu te of M ath e . . tlc .. Ak ade .i y. HaU~Noacow, SSSR. pp. 4-33.G l 1.1. (1982) private co..unic.tton. UTIAS, Uniyer,ity ofT or on to . T or on to . C an ad a.

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