a. l. kuhl- mixing in explosions

8/3/2019 A. L. Kuhl- Mixing in Explosions

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UCRL-JC-115690PREPRINT

Mixing in Explosions

A. L. Kuhl

This paper was prepared for submittal to theTechnical Meeting at the Russian Academy of SciencesMoscow, Russia

December 1-3,1993

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DISCLAIMERThis document was prepared as an account of work sponsored by an agency of theUnited States Government. Neither the United States Government nor the Universityof California nor any of their employees , makes any wm-rJmty, express or implied, orassumes my legal liability or respemibility forthe Rccuracy, completeness, er usefulnessof any information, apparatus, product, orprocess disclosed, or represents that its useweald not infringe privately owned rights. Referenceherein to any specific commercialproduc/s, process, or service by trade name, trademark, mmufacturer, or otherwise,does net necessarily constitute or imply its endorsement, _mmendatiomb or favoringby the United States Government or (he University of Californ_ The views andopinions of authors expressed herein do not necessarily state or reflect lhose of _heUnited States Government or the University of California, .and shall net be used foradvertising or product endorsement purposes.

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-,.42.7,Mixing in Explosions

A. L. KuhlLawrence Livermore National LaboratoryEl Segundo, California

Abstract

Explosions always contain embedded turbulent mixing regions, for example:boundary layers, shear layers, wall jets, and unstable interfaces. Described here is oneparticular example of the latter, namely, the turbulent mixing occurring in the fh'eball ofan HE-driven blast wave. The evolution of the turbulent mixing was studied via two-dimensional numerical simulations of the convective mixing processes on an adaptivemesh. Vorticity was generated on the fireball interface by baroclinic effects. Theinterface was unstable, and rapidly evolved into a turbulent mixing layer. Four phases ofmixing were observed: (i) a strong blast wave phase; (ii) and implosion phase; (iii) a re-shocking phase; and (iv) an asymptotic mixing phase. The flowfield was azimuthallyaveraged to evaluate the mean and r.m.s, fluctuation profiles across the mixing layer. Thevorticity decayed due to a cascade process. This caused the corresponding enstrophyparameter to increase linearly with time -- in agreement with homogeneous turbulencecalculations of G.K. Batchelor.

KEY WORDS: fireball instabilities, baroclinic effects, turbulent mixing layer profiles,scaling, enstrophy, cascade of vorticity.

INTRODUCTIONOne can say that the science of explosion dynamics began with the seminal

publication of similarity solutions to the point-explosion problem by G. I. Taylor (1941), and L.I. Sedov (1946). In Russia, this phase plane analysis was used to solve a broadspectrum of blast wave problems, as described in the monographs by Sedov (1959),Zel'dovich and Raizer (1966), Stanyukovich (1955), Barenblatt (1979), and Korobeinikov(1985). In the United States, A. K. Oppenheim (1972 a,b) also utilized the phase planemethod to systematically investigate explosions (e.g., all blast wave solutions bounded byMASTEB

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a strong shock, all blast wave solutions bounded by a strong C. J. detonation, etc.), andthereby discovered whole flew classes of blast wave solutions (Barenblatt et al., 1980).

The next phase of research led to investigations of non-self-similar blast waves,which by necessity, required finite difference solutions of the gas dynamics equations.Initially, the research focused on one-dimensional problems, for example, the decay ofpoint explosions by Goldstine and von Neumann (1955), and Brode (1955), and HE-driven blast waves by Brode (1959). Later, such calculations were extended to two-dimensional problems such as the reflection of blast waves from planar surfaces.Eventually, the numerical algorithms were improved to the point where accuratesolutions to very complex flows such as double-Mach reflection (DMR) could beobtained (Colella et al., 1986). However, such numerical simulations only producedlaminar solutions.

Obviously, real explosions always contain turbulent mixing regions, for example:boundary layers, slip lines and shear layers, wall jets, and unstable interfaces. Due torecent advances in computational power and improvements in numerical algorithms, it isnow feasible to perform direct numerical simulations of three-dimensional, turbulentmixing. This paper offers an example of such mixing in explosions.

The tool used for this purpose is a Convective Mixing Model (CMM) forsimulations of compressible turbulent flows. It is based on the idealization thatconvective mixing--rather than diffusive _'ansportmis the dominant physics for high-Reynolds-number flows. It can be viewed as a compressible-flow extension of the ZeroMach Number Model proposed by Oppenheim (1986) and Chorin (1986) for directsimulations of turbulent combustion. Specifically, CMM solves the inviscid conservationlaws of gasdynamics by means of a high-order Godunov algorithm (Colella and Glaz,1985). Adaptive Mesh Refinement (AMR) is used to follow the turbulent regions of theexplosion, and calculate the convective mixing processes on the computational mesh.

CMM avoids the perils and limitations of turbulent models which are, bynecessity, based on integration of mean-field equations. As Chorin (1975) has so aptlypointed out, the laws of fluid dynamics are nonlinear, so the integration of the averagedequations does not equal the average of the integrated equations. CMM is based on thelatter approach, namely, we integrate the equationsmby calculating the convectivemixing on an adaptive grid_and then average the solution. As demonstrated here, this

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approach provides a more faithful numerical simulation to a wide spectrum of fluid-dynamic mixing problems. Oppenheim himself was one of the first and most forcefulproponents of this approach for explosion dynamics, hence it is natural that this reviewshould appear in a volume dedicated in his honor.

THE PROBLEM

The problem considered here is the turbulent mixing in the fireball, created by thedetonation of a spherical high-explosives (HE) charge. The fireball gases expand at ahigh velocity, initially around 9 km/s, which drives a strong blast wave into thesurrounding atmosphere. The detonation products from solid explosives charges aretypically very dense (--2.5 g/cm 3) compared to the shock-compressed air (-0.01 g/cm3),hence the density ratio across the interface is very large (around 250). Such densityinterfaces are unstable to constant accelerations such as Rayleigh-Taylor instabilities(Youngs, 1984), impulsive accelerations as in Richtmyer (1960)-Meshkov (1960)instabilities, and misaligned pressure gradients (i.e., baroclinic effects). Smallperturbations on the interface, arising either from granular irregularities on the chargesurface or from molecular fluctuations, rapidly grow into a turbulent mixing region--as isquite evident from high-speed photography of explosions (Figure 1).

Turbulent mixing in fireballs is of scientific interest for a variety of reasons. First,mixing on a f'k,'eball interface can lead to afterburning for explosives that are not oxygenbalanced (e.g., TNT). Afterburning can add considerable energy to the blast wave (asmuch as 50 percent) that is exceedingly difficult to calculate, since it depends on theturbulent mixing rate which is not known. On a more general note, turbulent fireballsprovide an example of nonsteady compressible, turbulent mixing on density interfaces inspherical geometry. As will be shown here, such problems contain a wealth of fluidmechanic phenomena that are certainly worthy of detailed scientific investigation.

The first finite difference calculations of HE-driven blast waves (e.g., Brode.1959) were one-dimensional, hence mixing effects were neglected. The growth ofinstabilities on spherical HE-fireball interfaces was f'n'st mentioned by Anisimov andZel'dovich (1977). This was followed by experimental studies for cylindrical explosionsby Davydov et al. (1978, 1979). Some of the f'n-st analysis of such mixing layers wasreported by Anisimov et al. in 1983. They identified two regimes: (i) a strong blast waveregime where the mixing width was thin compared to the distance between the inner and


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outer shock (they called this free Rayleigh-Taylor turbulence); and (ii) an asymptoticregime, where the mixing width was of the order of the fireball radius. Much of therecent progress on interface instabilities is summarized in a review article by Anisimovand Inogamov (1993).

While others (Marcus et al., 1991; Ruppert, 1991) have studied the growth ofinstabilities on planar density interfaces accelerated by shock waves, this paperinvestigates instabilities on a spherical density interface imbedded in an HE-driven blastwave. In particular, we shall study the dynamics of mixing in f'u'eballs by followingconvective mixing processes on an adaptive mesh. This same methodology has beenused to study spherical interface instabilities in high-altitude explosions (Kuhl et al.,1993) and implosions (Klein et al., 1993).

The formulation of the calculations is described in the next section. The resultssection presents: visualization of the mixing process, analyses of the mixing layergrowth, mean and r.m.s, profiles across the layer, and evolution of the turbulent kineticenergy. This is followed by conclusions and recommendations for further work.

FORMULATION

The problem considered here is the blast wave created by the detonation of aspherical charge of PBX-9404, with an initial charge density of P0 = 1.84 g/cm 3 and acorresponding detonation velocity of WCj = 8.8 krn/s. For convenience, the chargeradius was assumed to be Ro = 1.0 m; this results in a charge mass of 7,700 kg. Theambient gas was air at sea level conditions:

p**= 1.01325 bars; p**= 1.2254 x 10.3 g/cm3; e_,--- 1.94x10 ) erg/g;a**- 334 m/s; u = 0; y = 1.41.

In order to focus on the convective mixing effects, a number of idealizingapproximations were made. In particular, we assumed that: (1) the explosion is initiallyspherically symmetric; (2) The flow is inviscid (i.e., an infinite Reynolds numberapproximation, where molecular diffusion effects are neglected); (3) both the air and thedetonation products are in local thermodynamic equilibrium; (4) the explosive is oxygenbalanced (i.e., no afterburning), so the blast wave has constant energy; (5) buoyancy


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effects are neglected (in order to focus on baroclinic effe:cs); (6) the flow is two-dimensional (2-D).

Based on the preceding assumptions, the flow is governed by the inviscidconservation laws of gas dynamics:

O_ttP+ V. (pu) = 0 (1)

a_ttPu + V. (puu) = -Vp (2)

3x,pE + V .(pEu)=-V.(pu) (3)at

Where U denotes the velocity and E represents the total energy E = e + 0.5 usu. Thepressure p is related to the density Oand internal energy e by the equation of state. Forair, it takes the well-known form:

p=(y,-1)pe (4)

where the effective gamma _te = ye(O,e) comes from the equilibrium air calculations ofGilmore (1955). A JWL equation of state was used for the detonation products:

p= A(1-wPo/PR1)e -R'p*/p+ B(1- WPo/PR2)e -R'p*/p+ (yd - 1)pe (5)

where according to Dobratz (1974): A = 8.545 Mbars, B. = 0.2049 Mbars, RI= 4.60, R2= 1.35 and Yd= 1.25 for PBX-9404. In mixed cells, the pressure was defined as a mass-weighted average of the air and HE pressures.

In order to follow the mixing processes, an additional transport equation was usedfor concentration C:

-_3C =(us V)C=O (6)at

where C was initialized an C = 1 inside the charge (r < r0) and C = 0 in the air (r > r0).


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The above set of equations (1, 2, 3 and 6) were integrated by means of a second-order Godunov algorithm (Colella and Glaz, 1985). Adaptive Mesh Refinement (AMR)by Berger and Colella (1989) was used to follow the convective dynamics in the turbulentmixing region of the fireball.

In order to initialize the problem, we assumed that the chemical energy wasreleased in the charge by an ideal Chapman-Jouguet (CJ) detonation wave (Taylor,1950). Hence, the flowfield in the charge was assumed to be that of a spherical CJdetonation at the time when the wave reached the edge of the charge. The CJ parametersfor a PBX-9404 charge are (Dobratz, 1974)

PCJ = 370 kbars; PCJ = 2.485 g/cm3; ecj = 8.142 x 1010 erg/g;qcJ = 5.543 x 1010erg/g; ucj = 2.28 krn/s; F = 2.85

To de-singularize the problem, and to qualitatively model the effects of HE gains on thecharge surface, a linear gradient region was added to the outside of the charge totransition from the CJ state to the ambient state. In this region (1.0 < r/R o < 1.10),Gaussianly distributed random fluctuations were, added to the density and energy profiles.This also served as a "white noise" seed for instabilities on the f'treball interface

An r-z cylindrical coordinate grid was used. To save computational time, only aquarter-space region of the spherical blast wave was actually calculated. Three levels ofAMR grid refinement were used: a base grid, with a mesh size A1 that extended to 100Ro, was used to capture the shock; a transitional grid A2; and fine grid A3 to follow themixing. The refinement strategy was based on following the detonation products region(i.e., where C _


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The calculations were run out to a time of 62 ms or 125 ms (i.e., 2 to 4 times the positivephase duration of the blast wave). This took between 5 and 30 hours of CPU time on aCray Y-MP computer, depending on grid resolution. The results are described in thenext section.

RESULTS

A. Flow Visualization

The next four figures present a sequence of frames that show the evolution of theblast wave in cross-section. In particular, density contours are used to visualize theshocks and mixing processes.

Figure 2 depicts the evolution of the flow during the strong blast wave phase. Itshows that the detonation products rapidly expand, forming the main blast wave shock Sand a secondary backward facing shock I that is swept outward by the blast wave flow.The f'u'eball interface is located in the region between these two shocks.

The fireball interface is impulsively accelerated when the detonation wave reachesthe edge of the charge. Thus, the interface is flu'st subjected to impulsive accelerationsthat can lead to Richtmyer-Meshkov (RM) instabilities. As the fireball expands, it isdecelerated by the shock-compressed air, and it is thus subjected to Rayleigh-Taylor(RT) instabilities. By a time of 2.4 ms, perturbations are clearly visible on the fireballinterface (Figure 2a), which has expanded to about 8 charge radii.

Perturbations in this calculation originate from discretization errors resulting fromthe projection of the spherical interface onto the square r-z mesh. In the physicalexperiment, perturbations arise either from granular irregularities on the charge surface,or from molecular fluctuations. According to linear stability analysis, such perturbationswill grow exponentially with time. Eventually they will be resolvable on the finitedifference grid. Here we are interested in the late-time phase where there are stronginteractions between various modes. This leads to a similarity spectrum of frequencies,and a concomitant loss of memory of the initial perturbations. Hence, for our purposesthe cause of the initial perturbations is not particularly important.


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By a time of 3.9 ms (or about 12 charge radii), the perturbations have evolved intomushroom-shaped structures that are characteristic of RM and RT-driven mixing layers(Figure 2b). These structures rapidly evolve into the nonlinear regime. By a time of 9.8ms (or about 17.5 charge radii), they have generated new fine-scale structures through anonlinear cascade process (Figure 2d). Also, Kelvin-Helmholtz (KH) instabilitiesdevelop due to shear flow along the bubble walls Note that pressure gradients in theblast wave are essentially radial. Perturbations on the fireball interface cause amisalignment of pressure and density gradients, that creates vorticity by the baroclinicmechanism:

=-V(l/p) vp (7)This leads to a self-induced production of vorricity in the mixing region.

Figure 3 depicts the evolution of the flow during the implosion phase. At a timeof about 12 ms, the secondary shock I falls into the negative phase of the blast wave, andstarts propagating back toward the origin. It implodes at a time of about 24 ms. Thisdraws the inner boundary of the mixing layer back toward the origin, thus greatlyincreasing the mixing width. During this phase, the flow exhibits many of featurescharacteristic of imploding spherical shells (e.g., elongated bubbles with KI-I structureson the bubble walls, perturbations on the imploding shock shape, etc; see Klein et al.,1993).

The next two figures depict the evolution of the mixing in the f'trebaU during eachof the four characteristic phases of the mixing. Colors are used to visualize variousaspects of the mixing processes. For example, color bar (Figure 4c) is used for airinternal energy--where shock-compressed air is red and the ambient air is blue, whilef'treball gases are yellow. The green and white lines represent vorticity contours (positiveand negative, respectively) which help to visualize the mixing, while black lines representdilatation contours which make the shocks visible. Figure 4a depicts a cross-sectionalview of the flow at the end of the strong blast wave phase (t = 12 ms). Figure 4b depictsthe flow field at the time shock I implodes (t = 24 ms). Clearly, the mixing extends allthe way to the origin.

The implosion of shock I creates a second explosion shock S' that expandsoutward from the origin. This re-shocks the mixing layer, and creates additional


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vorticity through baroclinic effects. At a time of 63 ms, shock S' emerges from thefireball (Figure 5a). The intensification of fine-scale mixing is evident. Shock S'continues to expand, and leaves behind a turbulent fh'eball. This leads to an asymptoticmixing phase that is shown in Figure 5b (t = 126 ms). It is characterized by a continuedcascade process among the various scales of vonicity, and a continued folding of thefireball interface. Since the mean radial expansion of the explosion has ceased, thefireball size remains essentially constant. However, the mixing processes continue togenerate acoustic waves within the f'treball which are quite evident as fine black lines inFigure 5b.

B. Growth of th,e Mixing Laver

In order to study the growth of the mixing region, we found it useful to follow theevolution of certain mean concentration lines. Three lines were selected '

m1. The line R0.9(t) where C = 0.9, which characterizes the inner boundary ofthe mixing region;

2. The line R0.5(t) where C = 0.5, which represents the center of the mixingregion; and

3. The line R0.1(t) where C = 0.1, which characterizes the outer boundary ofthe mixing region.

These are depicted in the wave diagram for an HE-driven blast wave (Figure 6). Fourdistinct phases are evident: (i) a blast wave phase--where the mixing region is sweptoutward by the shock-induced flow; (ii) an implosion phase--that stretches the innerboundary of the mixing region back toward the origin; (iii) a re-shocking phasemthat re-energizes the mixing layer by RM effects; and (iv) an asymptotic mixing phase--wherethe fireball size remains essentially constant.

Let us define the mixing width by the relation:

3(t) / Ro = [R0., (t)- Ro.9(t)] / R0 (8)

The evolution of the mixing width is shown in Figure 7. During the blast wave phase, themixing width grows linearly in time:


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_(t) / R0 = 0.396t (0 < t < 10 ms) (9)

with a velocity of 396 rn/s. This is consistent with an impulsive acceleration of thefireball interface (i.e., a RM growth). During the implosion phase, the power-lawexponent increases to a value of 1.44:

/ R0= 0.151t TM (10ms < t < 32 ms) (10)

as a result of the inward stretching of the interface back toward the origin. Mixing occursin a strong gradient of velocity (i.e., acceleration is not constant), so the quadratic growthassociated with RT mixing is not achieved. During the re-shock phase, the mixing layeris compressed from 8/Ro - 22 to 8/Ro --- 16 by the shock S'. During the asymptoticphase, the mixing width is essentially constant : 8/Ro -- 21.

C. Azimuthal Averaging

In order to study the profiles in the mixing region, it is convenient to define anaveraging procedure. In this problem, azimuthal averaging is most appropriate. The flowfield variables _, were stored as a function of r, z, and t, that is _(r,z,t). One can define apolar-coordinate system in the r-z plane:

R = _Jr2 + z2 (11)

0 = Tan-_(z / r) (12)

and convert the flowfield tG the ilew variables, i.e., O(R,0,t). It is then possible to definean azimuthal average of the flowfield by integrating over 0 :

(R,t) = 2 f,r,_ _(R,O,t)dO (13),10

Given a mean, one can then calculate r.m.s, fluctuations about the mean from the relation:

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(_')2 2 _/2=--J0 [_(R'O't)--_(R't)12dO (14)

and

_' (R,t)= __ (15)

D. Averaged Profiles

In order to check for similarity, the profiles were scaled with the mixing width 8:

r/= (R- Ro.5) / _ (16)

In addition, the flowfield was nondimensionalized by the ambient atmosphericconditions, denoted by subscript .o.

Figure 8 presents the azimuthally-averaged mean flow profiles across the mixinglayer, at the end of each of the four phases of mixing. The mean concentration profilestend to collapse to a single curve, independent of time. This provides indirect verificationof the validity of the scaling procedure. The mean density profiles peak at the fireballinterface at a value of ~ 7/9. at early times, and then decay with time as that mass isspread over a larger and larger volume. Figure 8c indicates that there is a mean radialpressure gradient at the fireball interface during the blast wave phase. Figure 8ddemonstrates that the azimuthal averaging procedure is adequate to produce smooth,well-behaved mean radial velocity profiles. Shock S and I axe indicated on the figure. Ofcourse, the magnitude of the velocity decays with time. The mean azimuthal velocities

mare typically less than 0.1 u, and they oscillate about a mean value of zero--as one mightexpect. Figure 8f displays the azimuthally-averaged vorticity profiles which decay withtime (Figure 8f), by means of a cascade process.

The corresponding r.m.s, profiles across the mixing layer are presented inFigure 9. Again note that the azimuthal-averaging procedure is adequate to give smooth,well-behaved profiles of the second moments of the flow variables. Again theconcentration profiles are approximately self-similar, with peak fluctuation around 40percent. Again, density fluctuations decay with time, as do the pressure fluctuations.The radial velocity fluctuation profiles are surprisingly smooth, and have a trimodal

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character, with peaks at the inner and outer edges of the rrrxing layer and a peak at thecenter of the layer. Even the azimuthal velocity profiles are relatively smooth and well-behaved. Hence, we conclude that the azimuthal-averaging procedure is adequate to givesmooth, well-behaved profiles. The following profiles are approximately self-similar andindependent of time: c, p, p, c', p', p', u', and v'. Profiles exhibiting a marked decayare: the mean radial velocities u (due to blast wave decay), and vorticity _ and co' (as acensequence of a cascade process).

MIXING PROPERTIES

In order to check the global properties of the mixing processes, the variouscomponents of the kinetic energy were integrated over the mixing volume, in particular,the kinetic energy of the mean flow k (t), the kinetic energy of the fluctuation flow k"(t),and the total kinetic energy K(t), were calculated from the following relations:

3-- f ""u(R,t) +2R. 3 J0 (17)

, 3.._fR, v' t)2]RZdR (18)k' (t)= 2t?,3,do [u'(R't)2 + (R,

._3 fR.K(/)-'- d_j fo _ [u(R,O,t)2+v(g,o,l)2]R2dOdR (19)

=k(t)+k"(t) (20)where a value of R, = 40 Ro was used. The results are presented in Figure 10. It showsthat the kinetic energy reaches a peak value of 6.5 x 108 (cm/s) 2 at t _.=6 ms, or abouthalfway through the blast wave phase. The mean kinetic energy then decays as theinterface does work on the ambient atmosphere, and eventually approaches zero at latetimes. This is, of course, consistent with the finite energy of the explosion, which impliesthat the velocity field of the blast wave must eventually decay to zero. The fluctuatingkinetic energy k" reaches a peak value of 3.1 x 107 (era/s) 2 (or about 10 percent of theUlEinstantaneous k) during the implosion phase. It decays slightly during the re-shockphase, and finally reaches a constant asymptotic value of k"** = 2.2 x 107 (crrds)2; this is

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consistent with a residual fluctuating velocity of about 66 m/s that continues to mix thefluid in the fireball. The total kinetic energy, which represents the sum of the two

Bcomponents, tracks k at early times and asymptotically approaches k" at late times.

The fluctuating kinetic energy corresponds to the rotational energy of mixinginduced by the vorticity distribution. The persistence of fluctuating kinetic energy at latetimes may be explained by considering the mechanical energy equation:

d K -u (Vp/p +vu. (Wudr (21)

where the overbars denote averaging over the computational volume. The pressuregradients rapidly disappear and can no longer influence the kinetic energy. Since thepresent calculations are inviscid, shear work cannot affect the kinetic energy. Hence,conservation of mechanical energy implies that the rotational kinetic energy approaches aconstant for inviscid blast waves. In the present calculations, the asymptotic value of thisconstant is K**- 2.3 x 107 (cm/s)2 m or about 3.5 percent of its peak value at early times.From a numerical point of view, this persistence of fluctuating kinetic energydemonstrates that Godunov-AMR algorithm is non-dissipative. From a physical point ofview, additional dissipation is required so that the kinetic energy approaches zero at latetimes. Three-dimensional (3-D) calculations have shown that vorticity amplification inhairpin structures leads to the continued decay of kinetic energy (Bell and Marcus, 1992).

Since the velocity or kinetic energy fluctuations are driven by the vorticity field, itis useful to consider the global properties of the vorticity, namely:

==== 6 _0_.,,/2 w(R,O,t) R2dOdR (22)co(t) = _R3 Jo

I_l 6 R.f,,/2= _-z-mo leoR,O,t)iR2aOdR (23)_R. Jo

==_=co(t)2'_-6 f0_00['_/2o2(R'O't)R2 dOdR (24)

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where the overbars denote integration over the mixing volume, and the vorticity wascalculated as a finite difference approximation to the curl of the velocity field.

mm

Figure 11 presents the evolution of the vorticity co(t) averaged over the mixingregion 0


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His results were based on 2-D simulations of incompressible, isotropic, homogeneousturbulence in a box, as performed by R.W. Bray. Batchelor showed that this lineargrowth in time was associated with a cascade process for to2. Although the kineticenergy was constant, w2 decayed as 1/t2. He found that the spectrum of t02 had a slopeof k"1 in the inertial range 2, and that this spectrum leads to an overall decay in t02.Evidently there is also a cascade process in the present calculations. Vorticity starts atsmall scales which grow with time. Additional fine-scale structures are continuously

created by a folding mechanism in 2-D flows, analogous to the re-generation of fine-scalestructures by hairpin vorticies in three-dimensional (3-D) flows. This leads to a net decayin volume-averaged vorticity--even though the kinetic energy remains constant.

SUMMARY AND CONCLUSIONS

The fireball interface was unstable, and rapidly evolved into a convective mixinglayer. During the blast wave phase the mixing width grew linearly with time8/Ro = 0.396 t, while during the implosion phase it grew with a power-law function oftime 8lKo = 0.151 t 1.44 as a result of the inward stretching of the interface by theimplosion shock. At late times it attained an asymptotically-constant width of 8/Ro = 21.

The azimuthal-averaging procedure was adequate to produce smooth, well-behaved profiles in the mixing layer. The profiles scale with the mixing layer thickness,i.e., according to rl = (R-R0.5)/8; some of the profiles were self-similar (i.e., independentof time).

The mean kinetic energy of the blast wave rapidly decayed to zero, but thefluctuating kinetic energy approached a constant (about 0.035 Kmax) at late times. Thissmall residual value represents the rotational energy associated with turbulent mixing,and is driven by the vorticity field. Three-dimensional calculations are required todissipate this small residual energy. Vorticity is rapidly generated by baroclinic effects,and then evolves through a cascade process by convective mixing. For example, vorticityis created at fine scales that grow to larger and larger scales by vortex merging. At thesame time, fine-scale structures are regenerated in 2-D by stretching of material lines,whereby vorticity gradients are amplified by velocity gradientstsimilar to vortex-linestretching in 3-D flows (see Batchelor, 1969). This leads to a vorticity distribution thatdecays with time because of a cascade process. These results are in agreement with other

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2-D simulations of turbulent mixing for incompressible flows (Batchelor, 1969) andcompressible shear layers (Vuillermoz and Oran, 1992).

The HE-driven blast wave problem considered here provides an opportunity forinvestigating a variety of fluid-mechanic effects related to interface instabilities inspherical geometry. Initially the interface is accelerated impulsively as the detonationwave breaks out of the charge and instabilities grow by the Richtmyer-Meshkovmechanism during the blast wave phase. During the implosion phase, the mixing layer isdrawn back toward the origin, and the mixing structures resemble those found inimploding spherical shells. During the re-shocking phase, the secondary shock S'interacts with the density structures in the f'n'eball thus generating additional fine-scalestructures by the baroclinic mechanism. What remains after shock S' leaves the f'n-ebaUisa vorticity distribution that evolves according to convective mixing dynamics. This canbe used to study the decay of turbulence in spherical mixing layers to arbitrarily latetimes.

Turbulence is, or course, a three-dimensional phenomenon. We plan torecalculate this problem in 3-D to study the evolution of turbulence in spherical mixinglayers. In addition, more detailed experimental data are needed to check the accuracy ofthe numerical simulations.

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ACKNOWLEDGMENTSWork performed under ihe auspices of the U. S. Department of Energy by theLawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Alsosponsored by the Defense Nuclear Agency under DNAIACRO#93-824, and Work Unit0000I.

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REFERENCES

Anisimov, S.I., and Zel'dovich, Y.B. (1977). Rayleigh-Taylor instability ofboundary between detonation products and gas in spherical explosion. Pis'ma Zh.Eksp. Teor. Fiz. 3, 1081-1084.

Anisimov, S.I., Zel'dovich, Ya.B., Inogamov, M.A. and Ivanov, M.F. (1983). TheTaylor instability of contact boundary between expanding detonation products and asurrounding gas. Shock Waves, Explosions and Detonations, J.R. Bowen, J.-C.Leyer, and R.I. Soloukhin, eds., Progress in Astronautics and Aeronautics Series,87, ( M. Summerfield, ed.), AIAA, Washington, D.C., pp. 218-227.

Anisimov, S.I., and Inogamov, N.A. (March 1993). Studies in Physics of HighDensity Energy, Landau Institute, Moscow.Batchelor, G.K., (1969). Computation of the energy spectrum in homogeneoustwo-dimensional, turbulence, Phys. of Fluids (Supplement II), pp. I1233 to II 239.

Barenblatt, G.I. (1979). Similarity, Self-Similarity and Intermediate Asymptotics(N. Stein, translator), M. Van Dyke, ed., Consultants Bureau of Plenum Publishing,New York.

Barenblatt, G.I., Gurguis, R.H., Kamel, M.M., Kuhl, A.L., and Oppenheim, A.K.(1980). Self-similar explosion waves of variable energy at the front. J. Fluid Mech.,99 (4), 841-858.

Bell, J.B. and Marcus, D.L. (1992). Vorticity intensification and transition toturbulence in the 3-D Euler equations, Comm. Math. Phys., 147, 374-394.

Berger, M.J. and Colella, P. (1989) Local Adaptive Mesh Refinement for shockhydrodynamics. J. Comput Phys. 82, 64-84.

Brode H.L. (1955). Numerical solutions of spherical blast waves, Journal ofApplied Physics, 26 (6), 766-775.

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Brode, H.L. (1959). Blast wave from a spherical charge, Physics of Fluids, 2 (2),217-229.

Chorin, A.J. (1975). Lectures on Turbulence Theory, Mathematics Lecture Series5, Publish or Perish, Inc., Boston (vid. esp. Chapter I-3: "An example of the perilsof dishonesty", pp. 21-23).

Chorin, A.J. (1986). Vortex methods for the study of turbulent combustion at lowmath number, Dynamics of Reactive Systems, (Part 1: Flames andConfigurations), J.R. Bowen, J.-C. Leyer, and R.I. Soloukhin, eds., Progress inAstronautics and Aeronautics Series, 105 (M. Summerfield, ed.), AIAA,Washington, D.C., pp. 14-21.

Colella, P. and Glaz, H.M. (1985) Efficient solution algorithms for the riemannproblem for real gases, J. Comput. Phys, 59, 264-289.

Colella, P., Ferguson, R.E., Glaz, H.M., and Kuhl, A.L., (1986). Mach reflectionfrom an HE-driven blast wave, Dynamics of Explosions and Reactive Systems,Progress in Astronautics and Aeronautics-Series, 106, ( M. Summerfield, ed.),AIAA, Washington, D.C., pp. 388-421.

Davydov, A.N., Lebedev, E.F., and Perkov, S.A. (1978). Gasdynamic instability inpropagation of cylindical shock waves. Detonation. Critical Phenomena: Physico-Chemical Transformations in Shock Waves, Chemogolovka, OIKhF AN SSSR, pp.76-79.

Davydov, A.N., Lebedev, E.F., and Perkov, S.A. (1979). Experimentalinvestigation of gas-dynamic instability in the a plasma flow following thecylindrical shock wave. Preprint N 1-40, Institute of High Temperatures, USSR, p.26.

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Klein, R.I., Bell, J. Pember, R., and Kelleher, T., Numerical simulations usingAdaptive Mesh Refinement hydrodynamics of a shock-driven mixing layer, 4th Int.Workshop: Physics of Compressible Turbulent Mixing, 29 March- 1 April, 1993,Cambridge, England, (in press).Korobeinikov, V.P. (1985). Problems of Point-Blast Theory, (Translated by G.Adashko), American Institute of Physics, New York.

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English translation: Hayes, W.D., and Probstein, R.F., eds., Academic Press, NewYork, Vols. I and 2, 1967.

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FOOTNOTES1. I.e., zero, to within a few percent of [co[. Nonzero values arise due to asymmetriesin the vortex pairings, finite-difference errors in computing V x u, and anomoliesarising for the r - z form of the governing equations.2. Note: The corresponding slope for 3-D turbulent is k "5/3.

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CAPTIONSFigure I. Shadow photograph showing a cross-sectional view of an HE-driven blastwave at t = 0.2 ms. A 0.45-g hemispherical charge of Nitropenta wasmounted on the explosion chamber window, and centrally-detonated by anintense electrical discharge. (Courtesy of Dr. H. Reichenbach of the ErnstMach Institut, Freiburg, Germany; 1993).Figure 2. Growth of instabilities on the fireball interface during the blast waveexpansion phase (log p contours): (a) t = 0.31 ms (b) t = 1.1 ms; (c) t = 2.4ms; (d) t = 3.9 ms; (e) t = 5.8 ms; (f) t = 7.8 ms; (g) t = 9.8 ms.Figure 3. Inward stretching of the mixing region during the implosion phase (log pcontours): (a) t -- 12 ms (b) t ,,- 14.5 ms; (c) t ---17 ms; (d) t ---20 ms; (e) t = 22ms; (t3 t = 23 ms.Figure 4. Cross-sectional view of the mixing in a HE-driven blast wave: (a) at the endof the strong blast wave phase (t = 12 ms); (b) at the end of the implosionphase (t = 24 ms); (c) color scale for air internal energy. Note that yellowdenotes the detonation products gases in the fireball, shock-compressed air isred, while the ambient air is blue. The green and white lines representvorticity contours (positive and negative, respectively) which help visualizethe mixing, while the black lines represent dilatation contours which make theshocks visible.Figure 5. Cross-sectional view of the flow field at later times: (a) at the end of the re-shocking phase (t = 63 ms); (b) in the asymptotic-mixing phase (t ---126 ms).Color scheme is the same as the previous figure.Figure 6. Wave diagram for a HE-driven blast wave.Figure 7. Growth of the mixing region.Figure 8. Evolution of the mean-flow profiles in the mixing region: (a) concentration;(b) density; (c) pressure; (d) radial velocity: (e) azimuthal velocity;(f') vorticity.Figure 9. Evolution of the fluctuation profiles in the mixing region: (a) concentration;(b) density; (c) pressure; (d) radial velocity; (e) azimuthal velocity;(f) vorticity.

111Figure 10. Evolution of the kinetic energy in the region R < 40 Ro: mean component k,fluctuating component k", and total K = k + k".Figure 11. Evolution of volume-averaged vortieity in the fn-eball.Figure 12. Evolution of volume-averaged _ in the f'trebaU.Figure 13. Evolution of the enstrophy parameter I_ in the mixing region. Data pointsdenote the present calculation and their fit: E (t) = 0.005 + 0.0345 t.

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Figure 1. Shadow photograph showing a cross-sectional view of an HE-driven blastwave at t = 0.2 ms. A 0.45-g hemispherical charge of Niwopenta wasmountedon theexplosionhamberwindow,andcenn-ally-


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Figure 2. Growth of instabilities on the fireball interface during the blast waveexpansion phase (log p contours): (a) t = 0.31 ms (b) t = I.I ms; (c) t = 2.4ms; (d) t = 3.9 ms; (e) t = 5.8 ms; (f) t = 7.8 ms; (g) t = 9.8 ms.


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Figure3. Inwardstretchingfthemixingregionduringtheimplosiorihase(logpcontours):a)t= 12ms (b)t= 14.5ms; (c)t= 17ms;(d)t= 20ms; (e)t= 22ms;(t')= 23ms.


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"tU.

_

20.

10.

Figure 4. Cross-sectional view of the mixing in a HE-driven blast wave: (a) at the endof the strong blast wave phase (t - 12 ms); (b) at the end of the implosionphase (t - 24 ms); (c) color scale for air internal energy. Note that yellowdenotes the detonation productsgases in the fireball, shock-compressed airisred, while the ambient air is blue. The green and white lines representvordcity contours (positive and negative, respectively) which help visualizethe mixing, while the blacklines representdilatation contourswhich make theshocks visible.


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i

Figure 5. Cross-sectional view of the flow field at latertimes: (a) at the end of the re-shockingphase (t = 6 3 ms); (b) in the asymptotic-mixingphase (t = 126 ms).Color schemeis the sameas the previousfigure.e


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Figure 6. Wave diagram for a HE-driven blast wave.


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uo._ Su_t_ oq_.;o_o.z D "L_!I!.4

oot o_ c_, c_ ol, 9 tri ! ' '_ r _ .tL i-,_ i _i _!'_[+:t !__!.!_4-/,.'-,-!----_-................-_i,,+,,,u_wm,r>,:__l[-_2_J:m,_F-I+^_,u.._:-_ _ l_.-.,:..-t-::--_......... i" -t'i' -- ..... : ,,-- ,, I : ' ,.,: ..... "+':.. ._ , t I ..


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v_ m

4t_,4 "l 1L '"* Jl0- a, i i

w-- __ 1 ,,., i' '_24._/t-|.I "l,O LI t.O i._. 'i,i ,l.| O,O I,I . *'I "L

d i iiii _ ii _ iii!ii __ i i i i __ iii iii iQ'q-- _

.s,.:._ Ib} iI ...........e)e, }-" 0 Iz

0 Z4 i!"" 6.,.I ,..; 12_ !I e, *l !: - ' :'- ,,,, I j ..... Aa.e *_,e Ike _.e 1,: "a.o ._.e o.o _.o- a.:

i O qlO "l.O "l,O O.O 1,0 1.,:

' Figure 9. Evolution of the r.rn.s, fluctuation profiles in the mixing region:

(a)concentration;b)density;c.)_ssm-c: (d)radialeloci_;(=)azimu_ velocity;f)vomc2_.


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Figure10.Evolutionof thekinetic energyin theregon R _ 40 Ro: meancomponent_,nucmaungcomponentk, andtotalK= _:+ k".


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Figure 13.Evolution of the cnsu'ophyparameterg in themixing r_&,ion.Data pointsdenote thepresentcalculationand theirfit: g (t) --0.005 . 0.0345 t.


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oo

Wql ,JP -

o

o


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a. l. kuhl- mixing in explosions

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