THE REFLECTED IMPULSE ON A CURVED WALL PRODUCED BY A SPHERICAL EXPLOSION I N A I R

Y. K I V I ' I Y RAFAEL B a l l i s t i c s Center, Haifa, Israel,

25th DoD Explosives Safety Seminar, 18-20 August 1992, Anaheim, CA.

ABSTRACT

The paper deals with the impulse on a concave w a l l r e su l t i ng from the r e f l - ect ion of the spherical b l a s t wave of an explosion i n air. This s i t ua t ion arises when a spherical b l a s t wave h i t s a non-planar surface, such as the w a l l of a containment vessel . Two types of surfaces are considered, cylin- d r i c a l and spherical , i n addition t o the standard plane w a l l . The work is based on computations with the PISCES two-dimensional flow code, employing second order numerical schemes. The r e s u l t s f o r the plane surface are compared with the compiled experimental da ta i n the l i t e r a t u r e . The study w a s ca r r ied out f o r a 2 K g TNT charge a t a distance of 1 m from the surface, assuming t h a t the r a d i i of the cylinder or sphere are a l so 1 meter. It was found t h a t the re f lec ted impulse on a cy l indr ica l surface is =18% over the plane w a l l case. For a spherical w a l l the re f lec ted impulse ( f o r the first shock reverberation) is about 50% over the plane w a l l impulse.

INTRODUCTION

I n designing s t ruc tures t o withstand the effects of explosions, the relevant b l a s t parameter is usually the impulse contained i n the pressure pulse, and not the pressure i tself . The reason f o r t h i s is t h a t most s t ruc tures have a long response t i m e compared t o the b l a s t wave duration.

The b l a s t wave parameters f o r spherical explosions i n air w a s d e a l t with i n many publications. The r e s u l t s w e r e compiled by Baker [ l ] , and Baker e t a l . [2]. The da ta i n these references comprise a l so the re f lec ted impulse on a plane w a l l . I n p rac t i ca l problems, however, there is a need f o r estimating the re f lec ted impulse on a curved w a l l , such as i n designing cy l indr ica l or spherical containment vessels. The present paper a i m s t o obtain an estimate of the re f lec ted impulse on concave w a l l s . Two types of w a l l s are considered i n the present paper: cy l indr ica l and spherical . I n both cases the charge is assumed t o be located a t a distance equal to the radius of curvature of the surface. This means tha t the charge lies along the axis of symmetry, i n the case of the cy l indr ica l surface, and on the center , f o r the spherical surface. (See Fig.1).

The present study is limited t o one value of the shock parameter,

467

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4. TITLE AND SUBTITLE The Reflected Impulse on a Curved Wall Produced by a Spherical ExplosioninAir

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13. SUPPLEMENTARY NOTES See also ADA260984, Volume I. Minutes of the Twenty-Fifth Explosives Safety Seminar Held in Anaheim,CA on 18-20 August 1992.

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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

R/W1l3 2 ft/Lb1'3 (4.8 m/Kg 1/3 1 where R is the distance from the center of the charge and W is the charge weight. Actually, the calculat ions w e r e carr ied ~- out with the following values :

R = l m w = 2 Kg (TNT)

Using the s i m i l a r i t y ru les f o r explosions i n air (a t sea l eve l w e m a g apply the r e s u l t s of the present calculat ions t o of R and W according t o the r e l a t ion

conditions), other combinations

i / w 1 l 3 = f( R/W 1/3 1

i = J (Pw - Po)dt

where i is the impulse per un i t area of the re f lec ted wave, Po and Pw are the atmospheric pressure and the ref lected wave pressure, respect ively, and t is the t i m e . It should be s t ressed , however, t ha t t h i s r e l a t ion s t r i c t l y holds f o r the s i t ua t ion described above, i.e. tha t the charge is located a t the center of the surface, so tha t R = R c , where R c is the radius of curva- tu re . Otherwise, the s i m i l a r i t y ru l e should bg modified to take i n t o account the wall curvature. This modified rule would have the form:

i/W1I3 = f (R/W1'3, Rc/R)

- - THE eBMPUTATIONAL, MODEL -

The solut ion t o the problem is obtained numerTcally, using the PISCES 2DELK hydrocode [ 3 ] . This code has a second order Eulerian processor with a multi- material capabi l i ty , so tha t a differFslt equation of state may be employed t o describe the ambient air and the explosion products.

The computational model consists of a quadr i la te ra l g r id for the cy l indr ica l surface case, and a wedge-like gr id f o r the ssherical. surface case. (F ig .1 ) . The axis of symmetry and the surface on which- the wave is re f lec ted are d e f i e d by a "r ig id w a l l " boundary condition. For the plane surface case, a 'fcontfnuative flow1' boundary conrfftion w a s employed at the open s ides of' the g r id , t o minimize edge effects.

The a5r i s described by an idea l gas equa t ions f state, with a spec i f i c heat r a t i o ~=1.4. The explosive charge is approximted by a sphere of dense $as, with ~ = 1 . 3 . Although t h i s representation of the detonation products is very crude, especial ly at the high pressure r e g i m e , i t produces the correct b l a s t wave Zmpulse. This r e s u l t w a s ver i f ied by varying the equation of state of the detonation products and examining the r e s a t i n g impulse. Such s tudies show that although some d e t a i l s of the flow f i e l d depend on the equation of state; the impulse fs insens i t ive t o it. A sfinilar approach w a s employed i n treat2ng air b l a s t s induced by high explosives i n previous works C4.51.

4 6 8

I n accordance with the above descr ipt ion, the i n i t i a l conditions are uniform pressure and density i n the e n t i r e mesh, except for a s m a l l spherical region which represents the high explosive. The i n i t i a l conditions i n the ambient a i r are:

p = 1.25 Kg/m3

p = 0.1 Wa

and the i n i t i a l conditions f o r the detonation products are:

p = 1600 Kg/m3

e = 4.1 M J / K g

where e , the spec i f i c i n t e rna l energy of the explosive, w a s taken from [6].

THE NUMERICAL SOLUTION

The r e s u l t s of the flow simulation w i l l be described i n some d e t a i l f o r the plane surface case. The computational gr ids are shown i n Fig.2. I n these g r ids the cell s i z e var ies as a geometrical series i n both direct ions. In t h i s way the flow f i e l d d e t a i l s are preserved i n the ea r ly stages of the b l a s t wave formation, when the flow gradients are very large.. The gr id shown i n Fig.2a has a geometric r a t i o of 1.032, which produces a very f i n e g r id i n the v i c i n i t y of the or ig in , so t h a t the charge occupies about seven mesh cells rad ia l ly . This gr id w a s used up t o t = 0.2 m s , when rezoning t o the coarser g r id of Fig.2b w a s ca r r ied out. The new g r i d had a geometric r a t i o of 1.02, and w a s used t o the end of the calculat ion without fur ther changes.

Fig.3 shows the flow f i e l d at t=0.2 m s . The symmetry of the f l o w is not per fec t , due t o numerical effects near the axis of symmetry. This asymmetry increases a t later times, but the solut ion is still acceptable s ince the main object ive of the work is the integrated impulse, and the d e t a i l s of the pressure t i m e h i s tory do not affect the impulse s ign i f icant ly .

The flow f i e l d at t=0.6 m s is shown i n Fig.4. A t t h i s t i m e the main shock wave has already h i t the w a l l and a re f lec ted wave w a s created. A t later t i m e s , (Figs.5 and 6 ) , the re f lec ted wave moves f a r the r from the w a l l . A t t = l . O m s the pressure near the w a l l has dropped t o about 0.2 MPa (twice the atmospheric pressure) , and at t=1.35 m s the pressure is below atmospheric, and no fu r the r pos i t ive contribution to the impulse is made beyond t h i s t i m e .

The calculat ions f o r the other cases were car r ied out similarly.. I n f a c t , the first phase of the b l a s t wave calculat ion ( to 0.2 m s ) described above was used t o create a ' s t a r t i n g state f o r the other cases using appropriate rezoning.

469

RESULTS AND DISCUSSION

The impulse per un i t area on the w a l l is defined by: t

i f t ) = f [ P f t ' ) - Po] d t ' n -

The b l a s t wave is charactarized by a high peak followed by a quick decay below atmospheric pressure. most eases is the peak value, which corresponds t o pos i t ive overpressure.

t o The relevant impulse f o r s t ruc tu ra l response i n

Fig.7 shows the t i m e h i s tory of the impulse for the three cases. The peak pressure is ident ica l f o r the three types of sur faces , and therefore the curves coincide f o r early t i m e s . A t later t i m e s , however, the differences i n w a l l curvature affect the loca l flow, w i t h a corresponding e f f e c t on the w a l l impulse.

For the plane surface, the peak impulse is a t ta ined a t t-1.1 m s , about 0.6ms after the shock f ron t a r r iva l . The peak value is 0.80 MPa-ms. This value should be compared with the compiled data of r2], which predict an impulse of 0.87 MPa-ms. The difference ( - 8 % ) is reasonable i n view of the scatter i n the experimental r e su l t s ,

For the cy l indr ica l surface, the peak impulse i s attained somewhat later, a t tx1.6 m s . However, about 98% of the peak is obtained a t t ~ 1 . 2 ms. The peak impulse is 0.94 MPa-ms, 17.5% over the plane surface case.

The impulse curve fo r the spherical case exhib i t s a d i f f e ren t behavior. In t h i s C a s e , due to the spherical symmetry, shock wave reverberations occur between the w a l l and the or igin. The timing is such t h a t the re f lec ted wave from the center h i t s the w a l l and contributes fur ther t o the impulse, before its "saturation" due t o the local flow is completed. I n t h i s case i t is d i f f i c u l t t o define the maximum impulse. For the purpose of comparing with the other cases, one may take the value just before a r r i v a l of the re f lec ted wave from the or igin. This value is 1.22 MPa-as, about 53% over the plane surface case, and about 303; over the cyl indr ica l surface case.

CONCLUSIONS

The impulse produced on a curved w a l l by a :spherical explosion i n a i r depends on the loca l w a l l curvature. For the spec ia l case of explosion a t the center of a cy l indr ica l o r spherical surfarse, the following values were obtained using hydrocode calculat ions,

Plane Surface: 0.80 MPa-ms Cylindrical Surface: 0.94 M P a - m s . Spherical Surface: 1.22 MPa-ms.

These values were obtained f o r a "T charge of 2 Kg, a t a distance of m . They may be extrapolated t o other combinations of charge weight and distance according t o the w e l l known s i m i l a r i t y ru l e f o r sca l ing impulse da ta , a s explained i n the introduction.

1

470

REFERENCES

[l] B a k e r , W.E.:: "Explosions i n Air", University of Texas Press., Austin, Texas (1973).

[2] Baker, W.E. e t al: "A Manual f o r the Prediction of B l a s t and Fragment Loading on Structures", DOE report , DOE/TIC-11268, Nov. 1980.

[3] Hancock, S.L.: "PISCES ZDELK Theoretical Manual", Physics Internat ional Company, San Leandro, CA., August 1985.

[4] Kivity, Y. and Feller, S.: " B l a s t Venting From a Cubicle", 22nd DoD Explosives Safety Seminar, Anaheim, CA., August 1986.

[5] Kivity, Y. and Kalkstein, A . : " B l a s t Wave Penetration i n t o Cubicles", 23rd DoD Explosives Safety Seminar, A t l a n t a , GA., August 1988.

[6] Johansson, C.H. and Persson, P.A.: "Detonics of High Explosives, Academic P res s , (1970). Table 1.1.1. (p.9) .

471

continuative flow boundary /---

-I

rigid wall boundary

I/ charge

(b)

J

(a) charge

/- f

axis of symmetry

4

axis of symmetry

b) 3

rigid wall boundary

axis of symmetry

FIGURE I: COMPUTATIONAL SEX-Up FOR THE PROBLEM (a) P l a n e Surface (b) Cylindrical Surface (c) Spherical. Surface

f3 location of impulse computation

4 72

FIGURE 2: THE COMPUTATIONAL GRID FOR THE PLANE SURFACE CASE Top: The grid for 0 < t < 0.2 ms. Botttom: The grid for t > 0.2 ms.

Geometric ratio = 1.032 Geometric ratio = 1.020

4 7 3

CONTClUR LEVELS

R 0.0 B 0.3

E 1.2 F 1.5 G 1.8 H 2.1 I 2.4 J 2.7 K 3.0 t 3.3

FIcBlpRe 3: wB;LoIcm VECTOR PUrr AND PRESSURE CONTOWFES AT t = 0.2 m.

f UI

MPa

' CONTOUR LEVELS

F1 0.0 B 0 . 3 C 0.G D 0.9 E 1.2 F 1.5 G 1.8 H 2.1 I 2.4 J 2.7 K 3.0 L 3 . 3 M 3.6 N 3.9

FIGURE 4: VELOCITY VECTOR PLOT AND PRESSURE CONTOURS AT t = 0.6 ms.

.. -, .... . . . . . . . . . _. - .. 1 ., ...

--.!. .. .' -c .'. ...

CONTOUR LEVELS

R 0.0 B 0.1

D E 0.q F Q.5 C 0.6 H 0.7

J 0.9 K 1.0 L 1 . 1

r c ' j:g I

I 0.8

. . . . . . 1 . . . . . . . . . . . . . .

. . . . y. . . -

. . -

_ . . * . I . . . . . ( - . - * . , . . . . . . . . . . . . _ . . , . I . . . . . . _ _ _ . . , . . . . . . . . . . . ..... - . ~. - . - . ! . . . . . - - ~ ., . .... ., - . - . . - . - . . - . - . . . . . . . - _ _ . . I . . . . . . . . . . . ...... . . . . . . - - - . . , . . . . . . - - - . . , - - - . . , - - - . . ,

1,

. . . . . . ....... . . . . . . . . . . . . - - - . . , . . . . . . . . . . . . MPa

CONTOUR LEVELS

FI 0.0 B - 0 3 C .OG D .09 E - 1 2 F "15 G "18 H " 2 1 I "211 J "27 K 0 .3

FIGURE 6: VELOCITY VECTOR PLOT AND PRESSURE CONTOURS AT t = 1.35 ms .

1.5

1.0

0.5

0 . 10.5 1.0 1.5

FIGURE 7: THE REFLECTED WAVE IMPULSE VS. TIME

2 .

t, ms