DETERMINATION OF THE VELOCITY-CURVATURERELATIONSHIP FOR UNKNOWN FRONT SHAPES

Scott I. Jackson and Mark Short

Shock and Detonation Physics Group, WX-9, LANL, Los Alamos, NM 87545

Abstract. Detonation Shock Dynamics (DSD) is a detonation propagation methodology that replaces thedetonation shock and reaction zone with a surface that evolves according to a specified normal-velocityevolution law. DSD is able to model detonation propagation when supplied with two components: the normal-detonation-velocity variation versus detonation surface curvature and the surface edge angle at the explosive-confiner interface. The velocity-curvature relationship is typically derived from experimental rate-stick data.Experimental front shapes can be fit to an analytic equation with an appropriate characteristic shape toexamine detonation velocity-curvature variation computed from that analytic expression. However, in somecomplex explosive-confiner configurations, an appropriate functional form for the detonation front shape maybe difficult to construct. To address such situations, we numerically compute the velocity-curvature variationdirectly from discrete experimental front-shape data using local rather than global fitting forms. The resultsare then compared to the global method for determining the velocity-curvature variation. The possibilitiesand limitations of such an approach are discussed.Keywords: DSD, detonation shock dynamics, curvature, detonationPACS: 47.40.Rs, 47.40.Nm, 82.40.Fp

INTRODUCTION

Measurement of the relationship between a detona-tion’s normal velocity and its curvature is a key basisfor Detonation Shock Dynamics theory (DSD), a sur-face propagation concept that replaces the detonationshock and reaction zone with a surface that evolvesaccording to a specified normal-velocity evolutionlaw [1]. The velocity-curvature (Dn-κ) relationshipis found by fitting assumed κ(Dn) forms to digitizedexperimental front shape data, usually obtained fromrate-stick tests.

Steady rate-stick configurations tend to exhibitgradual variations of Dn and κ near the charge centerand rapid variations near the charge periphery, whereedge effects are significant. This can be analyzed byfitting the front shape to the functional form,

z(r) =−n

∑i=1

Ai ln[cos

[η

π

2rR

]]i(1)

where z is the front height, r is the local radius, and Ris the charge radius [1, 2]. Equation 1 is fit globallyacross an entire experimental rate-stick front shapeby varying parameters Ai and η where 0 < η < 1.Parameter n is typically chosen as 1 or 3.

Once a global analytic fit z(r) has been obtained,the normal velocity Dn can then be found from

Dn =D√

1+ z′(2)

where z′ = dz/dr and D is the axial detonation phasevelocity. For a cylindrical rate stick, κ can be ex-pressed as

κ =z′′

[1+(z′)2]3/2 +z′

r√

1+(z′)2(3)

with z′′ = d2z/dr2. For a one-dimensional slab ge-ometry, the second term of Eq. 3 would be zero.

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For complex explosive-confiner configurations, anappropriate functional form z(r) to analyze detona-tion front shape may be difficult to construct. In thisstudy we address such situations by locally comput-ing the Dn-κ relationship directly from discrete ex-perimental front-shape data with no global assump-tions of wave shape. This is done by locally fitting aparabolic form to local portions of each front shape.

The effectiveness of this technique is then ex-plored by applying it to three experimental config-urations: (1) a TATB-based explosive cylindrical ratestick for verification purposes, (2) a highly heteroge-nous ANFO-based cylindrical rate stick to resolve lo-cal and complex variations in wave shape, and (3) aTATB-based explosive slab whose wave shape can-not be fit by Eq. 1.

METHOD

Experimental front record profiles consist of a list Zcomposed of n data points representing the detona-tion shape as it emerges from an explosive charge.Each point (xi,zi) has an x and z coordinate, corre-sponding to its radial and vertical measurement lo-cation as shown in Fig. 1. All xi points lie inside thecharge radius such that −R≤ xi ≤ R.

0 5 100

2

4

6

Radius (mm)

z (m

m) Circle fit

to r = 9 mmFront points

Global fit

Circle fit to r = 0

FIGURE 1. Front shape from a PBX 9502 rate stick.

The local fitting process starts by extracting a sub-set Si of points from Z that lie within ∆r

2 of a givenfront radius position xc. A parabolic analytic form

zp(r) = a(r−Xp)2 +b(r−Xp)+Yp (4)

is then fit to the front segment Si by varying fitparameters a, b, Xp, and Yp . Using this local fit,along with Eqs. 2 and 3, then allows Dn and κ valuesto be calculated for the point xc. This process is

iterated for each point in Z, with xc taking the placeof each value of xi along the front shape.

The parameter ∆r is an adjustable variable. Largervalues fit across a greater portion of Z and yielda higher quality fit, but do not detect rapid localvariations. Smaller values recover these features, butare more sensitive to experimental and digitizationerrors due to a decreased number of fit points in Si.

Near the edge of each front profile, it is not possi-ble to center Si on xc as no experimental front shapedata exists past the charge periphery. This results indecreased fit quality near r = R; for this reason wedo not fit when xc is not within the center of Si. Thus,an annular region of Z where R− ∆r

2 ≤ xi ≤ R is notfit with this process.

PBX 9502 RATE STICK VERIFICATION

PBX 9502 is a plastic bonded explosive composedof 95% TATB and 5% Kel-F binder. The explo-sive’s morphology is very fine grained and yieldsvery smooth front shape measurements. Data froman 18-mm-diameter unconfined PBX 9502 rate stickfrom Ref. [3] is shown in Fig. 1. Digitized frontshape points are green and the corresponding globalfit is black. Red and purple circles highlight the largevariation in local wavefront curvature at the chargecenter and outer radius, respectively.

Figures 2 and 3 illustrate the variations in Dnand κ along the front as determined by the globalfit method (black line) and the local fit methodology(data circles). Different color data points correspondto different values of ∆r, as listed in the Fig. 3 legend.The local fit data agree with the global fit over most

7.4

7.2

7.0

6.8

Dn (

mm

/µs

)

86420

Radius (mm)

FIGURE 2. Dn versus r for the front from Fig. 1 data.

the front. The Dn fit quality is excellent for all val-ues of ∆r. The κ fit experiences increased variationwith decreasing ∆r due to the increased sensitivity ofthe second derivative to experimental noise. Near thecharge edge however, lower values of ∆r are seen to

Preprint of http://dx.doi.org/10.1063/1.3686290

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

κ (1

/mm

)

86420

Radius (mm)

∆r=5mm ∆r=4mm ∆r=3mm ∆r=2mm ∆r=1mm ∆r=0.75mm ∆r=0.5mm ∆r=0.25mm Global

FIGURE 3. κ versus r for Fig. 1 data.

7.5

7.4

7.3

7.2

7.1

7.0

6.9

Dn (

mm

/µs

)

0.40.30.20.10.0

κ (1/mm)

FIGURE 4. Dn-κ data for Fig. 1 data.

more accurately match the global fit down to a break-down limit, in this case ∆r = 0.5mm, after which thelocal-fit κ dissolves into scatter. Approximately 380points (42 points/mm) were available for this frontshape, yielding approximately 21 points in the localfit at breakdown. This highlights the importance ofobtaining high-resolution front shape data.

ANFO RATE STICK TEST

Data from a cylindrical rate stick composed ofANFO confined by aluminum (test 12-305 fromRef. [4]) is shown in Figs. 5 and 6. The front break-out reflects the heterogenous nature of explosive,which is composed of approximately 2-mm-diameterammonium-nitrate prills soaked with fuel oil. As in

3.12

3.08

3.04

3.00

2.96

Dn (

mm

/µs

)

403020100

Radius (mm)

Global ∆r = 5mm ∆r = 10mm ∆r = 15mm ∆r = 20mm ∆r = 25mm ∆r = 30mm

FIGURE 5. ANFO (a) front shape, (b) Dn-r, and (c) κ-rdata.

-0.10

-0.05

0.00

0.05

0.10

κ (1

/mm

)403020100

Radius (mm)

FIGURE 6. ANFO (a) front shape, (b) Dn-r, and (c) κ-rdata.

the previous example, larger values of ∆r (≥ 15 mm)approximate the global fit well, while decreased val-ues of ∆r (5 and 10 mm) recover the prill morphol-ogy.

PBX 9502 SLAB TEST

The rectangular slab test or "sandwich-style test,"consists of an unconfined rectangular slab of explo-sive. The charge aspect ratio and boosting methodare intended to isolate long-axis edge effects fromthe charge center during the test time [5].

An image of an unconfined slab test and detona-tion front shape is shown in Fig. 7 for a 151× 127×8 mm thick slab of PBX 9502. Normally, detonationbreakout is observed across the 8-mm-thick dimen-sion [5]. This test was intended to verify the one-dimensional nature of the geometry and measuredthe front shape across the 151-mm-long dimension asindicated by the black line denoting the streak mea-surement location. As the front shape is still evolv-

Preprint of http://dx.doi.org/10.1063/1.3686290

Experimental image:

Streak image:

streak location

t

FIGURE 7. Slab (a) experiment and (b) front shape.

7.45

7.40

7.35

7.30

7.25

7.20

Dn (

mm

/µs

)

-50 0 50

Width (mm)

∆x = 1 ∆x = 2 ∆x = 5 ∆x = 10 ∆x = 20

FIGURE 8. PBX 9502 slab (a) Dn-r, and (c) κ-r data.

0.10

0.05

0.00

-0.05

κ (1

/mm

)

-50 0 50

Width (mm)

FIGURE 9. PBX 9502 slab (a) Dn-r, and (c) κ-r data.

ing over much of this axis, such front data is unsteadyin nature, not well-represented by Eq. 1, and requiresanalysis by other means, such as the local fit method.

Dn and κ data from the local-fit method are shownin Fig. 9. The data indicate that minimal curvature isindeed present near the charge center and show thatlong-axis edge effects do not affect the flow in thisregion.

CONCLUSIONS

A numerical method was presented to measure lo-cal variations in the normal detonation velocity Dnand surface curvature κ from detonation front-shapedata. Specifically, the method operates by locally fit-ting a parabolic equation to small sections of a front.Such an approach allows variation of detonation ve-locity and curvature to be examined for explosiveconfigurations with no assumptions of global waveshape.

The methodology was verified and compared to aprior technique for Dn-κ variation determination thatinvolves globally fitting an analytic form across theentire front shape. Front shape data from a hetero-geneous ANFO rate stick and a non-standard slab-geometry test were also analyzed. In general, datafrom the local-fit method agreed with the global fitmethod. Reduction of the fitting section size wasfound to increase the method’s sensitivity to localwave shape variations and also to experimental noise.

ACKNOWLEDGEMENTS

This effort was funded by the Department of Energy.Support was provided through programs adminis-tered by Dan Hooks, Jim Koster, Rick Martineau,and David Robbins.

REFERENCES

1. Bdzil, J., J. Fluid Mech., 108, 195–226 (1981).2. Catanach, R., and Hill, L., “Diameter Effect Curve

and Detonation Front Curvature Measurements forANFO,” in Shock Compression of Condensed Matter,American Institute of Physics, 2001, pp. 906–909.

3. Hill, L., Bdzil, J., Davis, W., and Critchfield, R., “PBX9502 Front Curvature Rate Stick Data: Repeatabilityand the Effects of Temperature and Material Variation,”in Ninth Symposium (Int.) on Detonation, Office ofNaval Research, 2006, pp. 331–341.

4. Jackson, S., Kiyanda, C., and Short, M., “ExperimentalObservations of Detonation in Ammonium-Nitrate-Fuel-Oil (ANFO) Surrounded by a High-Sound-Speed,Shockless, Aluminum Confiner,” in Proceedings ofthe 33rd International Combustion Symposium, TheCombustion Institute, 2010, pp. 2219–2226.

5. Hill, L., and Aslam, T., “The LANL Detonation-Confinement Test: Prototype Development and SampleResults,” in Shock Compression of Condensed Matter,American Institute of Physics, 2003, pp. 847–850.

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