lawrence m. hull- mach reflection of spherical detonation waves

8/3/2019 Lawrence M. Hull- Mach Reflection of Spherical Detonation Waves

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L ~ siarnos Nat ional Lmoratorv is ooeratea OY tne univers iw of Cali lornia tor me unite0 States Oeoartment of Ecerov unoer z:-!ract %'.:d05-ENG-36

TITLE: Mach Reflection of Spherical Detonation Waves

AUTHORIS): Lawrence M. Hull, M-8

SUBMITTED TO: Tenth International Detonation Symposium

DECLAIMERThis report was prepared as an account of work sponsored by an agency of the United Statesemployees, makes any warranty, express or im plied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or

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Hull, # 97

' When two detonation waves collide, the shape of the wave front at their inter-section can be used to categorize the flow as regular or irregular reflection. In thecase of regular reflection, the intersection of the waves forms a cusp. In the caseof irregular reflection, the cusp is replaced by a leading shock locus that bridgesthe incident waves. Many workers have studied irregular or Mach reflection ofdetonation waves,l4 but most of their experimental work has focused on the inter-action of plane detonation waves. Reflection of spherical detonation waves has re-ceived less attention. This study also differs from previous work in that the focusis to measure the relationship between the detonation velocity and the local wavecurvature for irregular reflection of spherical detonation waves. Two explosiveswith different detonation urouerties. PBX 9501 and PBX 9502. are comuared.

1

MACH REFLECTION OF SPHERICAL DETONATION WAVES

L. M. HullLosAlamos National hboratoryLosAlamos,NM 7544

INTRODUCTIONRecent work in this country and in the UK indicatesthat divergent detonation wave propagation can be de-scribed n terms of a relationship between the local wavecurvature and the local normal detonation velocity (seeRefs. 5-7). In the USA. , the theory is called Deto-nation Shock Dynamics @SD). The application of DSDto convergent waves has a less solid theoretical foun-dation because the flow behind the convergent wave issubsonic andcan thusbeinfluenced by disturbances prop-agated forward to the wave front. Whitham's8 methodfor treating shock diffraction in gas dynamics suffersfrom the same difficulty; nevertheless, it has been suc-cessful. The results presented here, along with those ofBdzil and Davis? show that the detonation velocity is re-lated to the local wave curvature in a continuous mannerfrom convergent to divergent configurations. Thus itmay be possible to apply wave-tracking methods, basedon DSD, to arbitrary convergent and divergent geome-tries, with success similar to that enjoyed by Whitham'smethod.When two divergent spherical detonation waves col-lide, regular reflection is first observed and, subse-quently, irregular reflection develops. The Mach stem isthe bridge that develops between the spheres, (Herein,any irregular reflection is called a Mach reflection.) Thesaddle surface generated by the collision (Figure 1) is asurface of revolution that is truncated because of thefinite dimensions of the explosive. Therefore, the inter-section of th e surface with a plane results in a record of

entire surface generated when two "divergent sphkricthe temporal evolution of the surface, To describe twaves collide, all the usual parameters that describewave reflection must be measured or estimated, as

as the parameters that describe the expanding sphericalwaves. Experimental wave-arrival-timedata are modeledwith analytic functions to describe the wave shape andpropagation for both the convergent and divergentwaves. We also use metal plate acceleration experimentsand assumed equations of state to provide independentmeasurements of the pressure in the Mach reflectionregion.A surface has two principal radii of curvature. Wehave chosen the mean curvature as the descriptive pa-rameter. The sign of the mean curvature K is chosen sothat a divergent wave has positive curvature. The meancurvature of the saddle point (Mach stem) is negative andapproaches zero as the spheres expand and the Machstem grows. The data analysis technique assumes thatthe D(K) elationship is linear, primarily because this isthe simplest form. The convergent wave results are onlyslightly dependent on this assumption. The linear rela-tionship can be written as

D = Dj (1 - VK),where D is the localdetonation velocity, D. is the CT det-onation velocity, v is a coefficient dependent on the ex-plosive, and K is the mean cuvature of the detonationwave. For a spherical wave, D = drldt, so substitutionof Equation (1) and integrationwith respect to time gives

r - vDj(t - to ) = r - ro +vln-. ro - vEquation 1 applies to both convergent and divergentflow, but Equation 2 applies only to the spherically ex-panding waves.


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FIGURE 1. MACH REFLECTION OF SPHERICALDETONATION WAVES. DETONATIONS WERERIVAL RECORDED BY A SMEAR CAMERA. THELATED TO OBTAIN AN APPROXIMATIONOF THESHAPEOFTHE COMPLETE SURFACE

INITIATED AT (OfJ0,O) AND WAVE FRONT AR-RECORDS WERE ANALYZED AND INTERPO-

EXPERIMENTALThe divergent-wave detonation-velocity-curvature e-lationship for PBX 9501 was determined with the ex-periment shown schematically in Figure2. Two detona-

tor-booster systems were placed so as to initiate the ex-plosive under two thicknessesof the test explosive. Thearrival time of the detonation wave at the observationsurface over the center of each detonator was recordedthrough a single slit with th e smear camera (Figure 3).Camera data corresponding to each wave were modeledas a sphere located at some arbitrary origin; the sphereexpanded at a constant velocity only for the duration ofthe breakout at the free surface. Position-time datafromthe region directly over th e detonators were fit to themodel equation to determine the radius of each wave.When the measured radiiand time differencesare used tosolve Equation (2) with Dj = 8.802 d p s , = 0.639mm. Then Equation (1) gives the corresonding values ofD so that we have two D(K)points.To determine the variationof the detonation velocitywith the local mean surface curvature for convergent det-onations, the Mach reflections of two propagating spher-ical detonation waves aremeasured. The mean curvatureof the saddle surface, and thus the Mach stem, is themean of the two principal curvatures. These principalcurvatures are the inverse of the radius of curvature ofthe intersection of the saddle with thex-z plane (rm), andthe inverse of the radius of curvature of the intersectionof the saddle with th e y-z plane at y = 0 (rs) (Figure 4).The radius of curvature rs is much smaller than the radius

of curvature rm, so r, dominates the estimated meanMach stem curvature from the data. On the other hand,the velocity of the Mach stem is the rate of increase ofrm. Therefore, accurate measurement of both rm and r,is necessary to determine the D(K) elationship for theconvergent region./ IGHTSHIELD

/ IIN I9.05 mm

INITIATOR INITIATORFIGURE 2. DIVERGENT WAVE CHARACTER-IZATION EXPERIMENT

FIGURE 3. SMEAR CAMERA RECORD OF THEBREAKOUT O F THE TWO DIVERGING SPHER-ICAL WAVES GENERATED BY THE TEST AS-SEMBLY SHOWN IN FIGURE 2. CLEAR PAINTWAS USED TOPROVIDE A SHARP TRACE. TIMEINCREASES UPWARDTest pieces used for the convergent wave studies areright-circular cylinders with one end cut at an angle, withrespect to the charge axis (see Figure 5). Two detona-tors are placed on the y-axis at y =eo and are initiatedsimultaneously. The arrivalof the detonation wave at theangled observation surface is recorded through 11 slitswith a smear camera. Slits are arranged parallel to they-axis with the center slit at x =0. A typical camera recordis shown in Figure 6. The curved Mach stem can beclearly seen at the intersection of the two spheres.


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z

A

EINFLECTION POINT

MACH STEM\, \FIGURE 4. THE GEOMETRY OF THE MACH RE-FLECTION OF SPHERICAL WAVES

curve fi t is given in Figure 7. The value of rm for eachslit is given by

The corresponding arrival time of the Mach stem at y = 0is known from the data; differentiation of this (rm,t) datagives the normal velocity of the Mach stem, D =dr,,,/dt.

FIGURE 5. SHOT GEOMETRY FOR MACH STEMEXPERIMENTSFIGURE 6 . A MULTISLIT SMEAR CAMERARECORD OF SPHERICAL DETONATION WAVE IN-TERACTION IN PBX 9501. TIME INCREASESUPWARD

The arrival time data,well away from the Mach stem,are fit to the equation of a sphere centered at (xo,yo,q)and expanding as given by Equations 1and 2. At eachslit, the values of x = a and z= c are known from the ge-ometry. Therefore, the model function for the expandingwave becomes2 2(a-xo) +(Y-Yo) +tc-zO)2=r2, (3)

the data set is t = t(y), and the parameters determined bythe fit y e_g ,yo ,q, and to. The ro value in Equation (1)is taken to be Iql. The use of Equations (1) and (2),rather than a sphere expanding at the CJ velocity, resultin a small correction because the radius of the sphere islarge, compared with v. A fixed value of v = 0.6 mmwas used for PBX 9501 and a value of v = 2.0 mm wasused for PBX 9502. An example of the result of this

The arrival time data in th e saddle region defines theshape of the Mach stem and allows the estimation of theMach stem radius of curvaturers Mach stem data are fitto the equation of a circle lying in the y-iiplane (ii is thenormal direction of the Mach stem). The center of thecircle is moving in theiidirection at the velocity for thatparticular slit, and the radius of the circle is rs.Therefore, the model equation we use to determine rs is

where data to be fit are t = t(y) and the parameters areyo,@, and r, (note that this is a different origin than in theprevious discussion). The result of this curve-fitting pro-cedure for the center slit is shown in Figure 8; judgment


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was used to select the number of data points to be in-cluded in the curve fit. The rs data for PBX 9501 are inthe approximate interval4 as27 mmand for PBX 9502in the approximate interval 9 cr,c34 mm. Thus we areapproaching the region where the radius of curvature ofthe wave front is comparable to the reaction zonethickness (the distance from the leading shock to thesonic locus), a region of questionable applicability ofDSD. The theoretical basis of DSD and the applicabilityof Equation (1)are dependent on the condition that theinverse of the wave curvature is much greater than thereaction zone thickness.

6.000

5.000

P '.-wIF 3.0002.000

1.000

0Y(m)

FIGURE 7. DATA AND CURVE FIT USED TOPANDING PART OF THE DETONATION WAVEFOR THERECORDSHOWN NFIGURE 6FIND THE ORIGIN OF THE SPHERICALLY EX-

The mean curvature, K = 1/2 (l/rm - l/rs), can nowbe calculated and the D(K) relationship plotted (seeFigure 9). Results for PBX 9501 indicate that, to withina goodapproximation over the measured range, the D(K)relationship is continuous and linear for both K >0 and K0 isnonliiear.9The data from Bdzil and Davis's study9 and this studypatch together: the D(K) elationship appears tobe con-tinuous in both slope and magnitudeacross K =0. Theserelations can now beused with a detonation-front track-ing routine (see Reference 10) to predict wave profilesfor both convergent and divergent waves. -

Properly defied, the Mach stem width is the trans-verse distance from the planeof symmetry to the positionof the intersection of the incident detonation and the dis-turbance reflected into the products. The position of thereflected disturbance is not readily availablefromarrivaltime data. Therefore, we choose to assume that thetransverse position of th e inflection point of the leading

wave front defie s the stem width. Figure 10shows theMach stem width data as a function of position rm . Theestimated Mach stem width w completes the informationnecessary to calculate the interaction angle a from r cos a= yo - w (see Figure 4) . By this calculation, we showthat the measurementsare for interaction angles that varyfrom about 70" to 75" for both PBX 9501 and PBX 9502.Although we have made measurements over a limitedrange of interaction angle, the Mach stem dimensions de-tected are initially only slightly larger then the reactionzone thickness and they are observed until nearly planewave conditionsare achieved. Over the measured range,the growth rate of the Mach stem is approximately con-stant. From the linear fit shown in Figure 10, thegrowth angle for PBX 9501 is x = 2.4" and for PBX9502,x = 7.4". Extrapolating the linear relationship to w= 0 allows the estimation of the critical angle (the valueQ, at which an irregular reflection first appears). ForPBX 9501e 56", and for PBX 9502 Q= 63". Thisextrapolationisexpected to overestimate thecritical anglebecause of the neglected variation of the growth rate ofthe stem.

28Wc

W W=I1wFIGURE 8. DATA AND CURVE FIT TO THECENTER SLIT MACH STEM REGION OF THERECORD SHOWN IN FIGURE 6. THE RADIUSVALUE OBTAINED IS 16.4 mm

In the reflectance-change flash gap (RCFG) test, a2-m-t hi ck aluminum plate is placed on top of a right-circular cyliider of explosive, along with a piece ofLucite@ n which a gap ismachined and polished.* Theplate surface is illuminated with an argon lamp, and theslit of a smear camera is aligned parallel with the line de-fined by the detonators and the gap. When a shock wavearrives at the metal-air interface, the diffuse reflectivity of* Lucite is a registered trademark for acrylic resins, E. I.du Pont de Nemours & Co.


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Hull, t 9 7 5

the surface is reduced. This is recorded by the smearcamera as reduced exposure on the film. The metal platearrives at the Lucite surface and causes the trappedair oflash, and th e subsequent shock induced in Lucite causesth e Lucite to become opaque. This is recorded on thesmear camera as a thin line of bright light. From theknown gap depth and time difference, we calculate theaverage free-surfacevelocity of the aluminum plate.

l 0 .W7

2 . W'EM x lPBX 9501: D = 8.8q1-0.79~)PBX 9502: D = 7.71(1- 2.1~)

I I I I I I I I I I4.10 4.08 4.06 404 4.02 0.00 0.02 00 4 0.06

K mm-1)FIGURE 9. THE DETONATION VELOCITY-CURVATURE RELATIONSHIP FOR PBX 9501(THE OPEN CIRCLES) AND PBX 9502 (THE SOLIDPOINT TAKEN FROM REFERENCE 14CIRCLES). THE BLACK SQUARE IS A PBX-9404

FIGURE 10. THE MACH STEM WIDTH AS AFUNCTION OF DISTANCE. THE SLOPE OF THELINES GIVES THE GROWTH ANGLE OF THESTEM

Figure 11reproduces records obtained in reflectance-change flash gap experiments. The absence of a cusp inthe reflectance-change trace in Figure 1a indicates ir-regular or Mach reflection. The presence of a cusp inFigure 1 b indicates regular reflection. Dips toward ear-lier time at the center line of the flash trace imply thatgreater velocity is inducedin the plate by higher pressurein the interaction region. We assumed that the particlevelocity of the aluminum is one-half the measured free-surface velocity and that the free-surface velocity vectorbisects the original surface normal and the shock frontnormal. To find the detonation wave pressure, the pres-sure and particle velocity of the explosive products andaluminum are shock matched (assuming the productsfollow the JWL equation of state). A curve-fitting pro-cedure similar to the multistreak technique outlined aboveis used to find the interaction anglea

FIGURE 11. RECORD OBTAINED FROM RE-MENTS IN PBX 9501. (A) IRREGULAR OR MACHREFLECTION. (B) REGULAR REFLECTION. TIMEINCREASES UPWARDFLECTANCE-CHANGE FLASH-GAP EXPERI-


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Hull, 897 6ANALYSIS

The data are best discussed in the context of an anal-ysis. The most basic analysis is to approximate the flowas the intersectionof two plane waves that are tangents tothe spheres. Flow is separated into regions by the deto-nation and shock waves, the flow in each region is as-sumed to beone-dimensional, and all waves are assumedto follow the appropriate jump conditions. This is theclassical shock polar theory of gas dynamics, a toolfound useful over the decades and still in use today.ll-l3The quasi-steady assumption now encompasses the ef-fects of interaction angle variation with time.

Figure 12 represents the geometry of the classicalMach reflection analysis. The plane of symmetry of thecolliding spherical waves is represented as a perfectlyrigid wall. Figure 12 shows the Mach stemas a straightline. However, other assumptions can be made about, the shape of the Mach stem and the resulting trajectory ofth e triple point. The triple point follows a path thatmakes an angle x with the wall. In a stem that grows,the velocity of the point P is given by

+Dj csc(a - x)sin$. (6 )In general, x is variable and the path can have curvature.Additionally, for interacting spheres,a s variable.

vp= -D j csc(a - x ) cos

I R

The system of equations is closed by requiring theflow velocity in regions 3 and 4 to be parallel and thepressures in regions 3 and4 to be equal. Regions 3 and4 are separated by a contact discontinuity (slip stream);the velocities differ, but the pressures are equal.Our experimental observation is that the Mach stemsare curved, they grow, and they are normal to the planeof symmetry. Thus the flow is at least two-dimensionalin the region behind the stem and the triple point maydegenerate into a region of interaction between the inci-dent detonation and the reflected disturbance. In general,the rate of growth of the Mach stem will not be constant.On the other hand, our data (Figure 10) show that thegrowth rate is fairly constant over the interval observed.Because the observed stems are curved, we also expectthat the straight-stem assumption will overpredict thegrowth rate. Therefore, we compare our pressure data tothe results of the shock polar analysis using both thestraight-stem assumption and the constant growth rate(measured) assumption. From Figure 12, if the growthrate and interaction angle are both specified, the pressure

behind the stem is easily calculated from the jump condi-tions.The explosive is assumed to follow the JW L equa-tion of state. Figure 13shows the pressure produced bythe wave interaction for both regular and Mach reflectionas a function of interaction angle. For low values of a,regular reflection occurs. As the critical angle Q is ap-proached, the pressure increases. When Mach reflectionfmtappears, the pressure jumps discontinuously to veryhigh values. As a becomes larger in the Mach reflectionregime, the pressure decreases and eventually appmach-es the CJ value. Mach stems near the critical anglewould be the size of the reaction zone or smaller. Thusthe classical theory, which assumes a jump from the ini-tial state to the fully detonated state, may not apply.For regular reflection (x = 0, 03 = O), the pressure

I . ...................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-X

(a)

calculated from the theory and the data from the RCFGexperiments agree reasonably well. For Mach reflection,pressure calculated from the theory using the straight-stem assumption is significantly greater than the datafrom the RCFG experiments. Use of the measured valueof x , which forces the measured Mach stem velocity,brings the calculation into agreement with the pressuremeasured in the RCFG experiments. The values of xcalculated from the straight-stem assumption are consid-erably larger than the measured values. This results inoverprediction of the phase velocity along the plane ofsymmetry, and so also overpredicts the pressure behindthe stem.CONCLUSION

I R

The data show that, over the measured interval, thedetonation velocity is a continuous function of curvaturefor convergent flow, in spite of the presence of subsonicflow behind the Mach stem The D(K) elationships pro-vided by this study combined with that of Reference 9allow the calculation of convergent and divergent deto-nation wave propagation in PBX 9501 and PBX 9502

@)

FIGURE 12. MACH OF A PLANEFROM A RIGID WALL. (A) LABORATORYC O O R D I N A ~ ~ .B) C O O ~ ~ A ~ S~ A C ~ DTOP


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Hull, #97 7using DSD wave trackers. Comparisons of such calcu-lations with experiments are required to determine theextent of applicability of DSD to convergent detonationwaves.

160

140130

403020E0L-

a

I I I I I I I I I10 20 30 50 50 60 70 80 90

aC)FIGURE 13. PRESSURE BEHIND THE REFLECT-ED WAVE AS A FUNCTION OF INTERACTIONANGLE. THE OPEN CIRCLES INDICATE PBX9501 AND THE OPEN SQUARES INDICATE PBX9502. THE BLACK SQUARES INDICATE THESURED GROWTH ANGLE AND THE BLACKDOTS INDICATE THE PRESSURE MEASUREDWITH THE RCFG TECHNIQUE, BOTH FOR PBX950 1

PRESSURE CALCULATED WITH THE MEA-

The classical shock polar theory fails to accuratelyreplicate the parameters of the flow when the straightMach stem assumption was used, but use of the mea-sured stem growth rate brings the theory into agreementwith the experiment. Therefore, the analytic treatmentrequires proper handling of the curved-stem growth ratedynamics. Equation (1) used with Whithams methodsuggests a nonlinear diffusive Mach stem growth pro-cess.7 For application to spherical wave interactions, theprocess must be convoluted with the time-dependentboundary condition introduced by the spherically ex-panding wave. This modeling approach seems valid andis a current topicofour esearch.Both the classical shock polar theory and the use ofDSD wave trackers are expected to be applicable whenthe features suchas the Mach stem width and the inverseof the surface curvature are relatively large, comparedwith the reaction zone thickness. The study of the flowwhere the features of interest are smaller than the reactionzone thickness remains unprobed by this study. Recentcomputer simulations15 by Stewart suggest that reflec-tion configurations familiar from gas dynamics may oc-cur entirely within the reaction zone or span across thereaction zone.

REFERENCES1.

2.

3.

4.

5 .

6.

7.

8.9.

10.

11.

12.

Gardner, S. D., and Wackerle, J., Interactions ofDetonation Waves in Condensed Explosives,Proceedings of the Fourth Symposium (Inter-national) on D etonation, October 1965, pp. 154-155.Lambourn, B. D., and Wright, P. W., Mach In-teraction of Two Plane Detonation Waves, Pro-ceedings of the Fourth Symposium (Internationa l)on Detona tion,October 1965,pp. 142-152.J. P. Argous, C. Peyre, and J. Thouvenin, Ob-servation and Study of the Conditions for Formationof Mach Detonation Waves, Proceedings of theFourth Symposium (Internationa l) on De tonation,October 1965, pp. 135-141.Mader, C. L., Detonation Wave Interactions,Pro-ceedings of the Seventh Symposium (International)on Detonation,June 1981,pp. 669-677.Stewart, D. S., and Bdzil, J. B., The ShockDynamics of Stable Multidimensional Detonation,in Combustion and Flame, 72,311 -323 (1988).Bdzil, J. B., Fickett, W., and Stewart, D. S.,Detonation Shock Dynamics: A New Approach toModeling Multi-dimensional Detonation Waves,Proceedings of the Ninth Symposium (International)on Detonation,August 1989, pp. 730-742.Lambourn,B. D. and Swift, D. C. Application ofWhithams Shock Dynamics Theory to the Prop-agation of Divergent Detonation Waves, Proceed-ings of the Ninth Symposium (International) on Det-onation,August 1989, pp. 784-797.Whitham, G. B., Linear and NonlinearWaves, John Wiley & Sons, New York, 1974.Bdzil, J. B. DSD Calibration for PBX 9502, Pro-ceedings of theTenth Symposium (International)on Detona tion,July 1993.Fickett, W. and Bdzil, J., DSD Technology - ADetonation Reactive Huygens Code Los AlamosNational Laboratory report LA-12235-M S, (1992).Courant, R. and Friedrichs, K. O., SupersonicFlow and Shock Waves, Interscience, New York,1948.Hornug, H., Regular and Mach Reflection ofShock Waves, Ann. Rev. Fluid Mech. Vol. 18,No. 33, (1986) pp. 58.

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13. Ben-Dor ,G. and Takayama, K., The Phenomenaof Shock Wave Reflection - A Review ofUnsolved Problems and Future Research Needs,Shock Waves Vol. 2, 1992, pp. 211-223.


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14. Campbell, A. W. and Engelke, Ray, The DiameterEffect in High-Density Heterogeneous Explosives,Proceedingsof the Sixth Symposium (International)on Detonation, August 1976, pp. 642-652.15. Stewart, D. S.CMHOG Calculations of a ReactiveWave Interactionpersonal communication, 1993.

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lawrence m. hull- mach reflection of spherical detonation waves

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