gasdynamic model of turbulent combustion in tnt explosions

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Proceedings of the Combustion Institute 33 (2011) 2177–2185

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of the

CombustionInstitute

Gasdynamic model of turbulent combustionin TNT explosions

A.L. Kuhl a,⇑, J.B. Bell b, V.E. Beckner b, H. Reichenbach c

a Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USAb Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, Berkeley, CA 94720, USA

c Ernst Mach Institut, Eckerstraße 4, 79104 Freburg, Germany

Abstract

A model is proposed to simulate turbulent combustion in confined TNT explosions. It is based on: (i)the multi-component gasdynamic conservation laws, (ii) a fast-chemistry model for TNT–air combustion,(iii) a thermodynamic model for frozen reactants and equilibrium products, (iv) a high-order Godunovscheme providing a non-diffusive solution of the governing equations, and (v) an ILES approach wherebyadaptive mesh refinement is used to capture the energy-bearing scales of the turbulence on the grid. Three-dimensional numerical simulations of explosion fields from 1.5-g PETN/TNT charges were performed.Explosions in six different chambers were studied: three calorimeters (volumes of 6.6-L, 21.2-L and40.5-L with L/D = 1), and three tunnels (L/D = 3.8, 4.65 and 12.5 with volumes of 6.3-L)—to investigatethe influence of chamber volume and geometry on the combustion process. Predicted pressures historieswere quite similar to measured pressure histories for all cases studied. Experimentally, mass-fraction ofproducts, Y exp

P , reached a peak value of 88% at an excess air ratio of twice stoichiometric, and then decayedwith increasing air dilution; mass-fractions Y calc

P computed from the numerical simulations followed similartrends. Based on this agreement, we conclude that the dominant effect that controls the rate of TNT com-bustion with air is the turbulent mixing rate; the ILES approach along with the fast-chemistry model usedhere adequately captures this effect.� 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: TNT detonation products; Turbulent combustion; ILES; Barometric calorimeter

1. Introduction

This paper describes the development of a gas-dynamic model of turbulent combustion in con-fined TNT explosions. The particular problemunder consideration is the blast wave formed bythe detonation of a 1.5-g composite chargeconsisting of 0.5-g spherical booster of PETN

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⇑ Corresponding author. Fax: +1 925 424 6764.E-mail address: [email protected] (A.L. Kuhl).

surrounded by a 1.0-g spherical shell of TNT withinitial densities of q0 ¼ 1:0 g/cc. The charge wasplaced in the center of a chamber. Six chamberswere used: cylindrical calorimeters A (V c = 6.6 l,L = 21 cm, D = 20 cm, L/D = 1.), B (V c =21.2 l, L = 30 cm, D = 30 cm, L/D = 1) and C(V c = 40.4 l, L = 37.9 cm, D = 36.9 cm, L/D =1), and circular cross-section tunnel D(V c = 6.3 l, D = 12 cm, L = 55.5 cm, L/D =4.65), and rectangular cross-section tunnels E(V c = 6.3 l, X = Y = 8 cm, L = 100 cm, L/D =12.5) and F (V c = 4 l, X = Y = 10.1 cm, L =

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2178 A.L. Kuhl et al. / Proceedings of the Combustion Institute 33 (2011) 2177–2185

38.6 cm, L/D = 3.8). The detonation of thecharge created a blast wave that reflected off thechamber walls. The main diagnostic was Kistler603B pressure gauges located on the lid of thecalorimeters or end walls of the tunnels. Mixingof hot detonation products with air created aspherical combustion cloud which enhanced thechamber pressure. A typical example of this com-bustion effect is shown in Fig. 1, which depictspressure history measured for 1.5-g compositePETN/TNT charge detonated in air versus N2

(which suppresses combustion) atmospheres.Experimental results for 1.5-g composite PETN/TNT charges and PETN/Al charges were pub-lished in the 32nd Int. Combustion Symposium[1]. Here we describe numerical modeling of thoseexperiments.

Such combustion effects were first studied byOrnellas [2] in detonation calorimeter experi-ments. Twenty-five grams TNT charges were det-onated in a 5.3-L spherical bomb calorimeter;tests done in a vacuum measured the “Heat ofDetonation, while tests done in O2 atmospheremeasured the “Heat of Combustion.” The tempo-ral evolution of the combustion energy wasstudied by Kuhl and Reichenbach using the baro-metric calorimeter technique [3].

Previous numerical simulations include: gas-dynmics of combustion of TNT products in air[4], afterburning of detonation products from 1-kg TNT charges in 16-m3 chamber [5], investiga-tion of effects of confinement on combustion ofTNT explosion products in air [6], and numericalsimulations of thermobaric explosions [7]. In addi-tion, a heterogeneous continuum model of Al par-ticle combustion in explosions has been used tomodel the PETN/Al experiments [8].

Thermodynamic states in such combustionfields can be calculated by the Cheetah code devel-oped by Fried [9]. In addition, Kuhl has found an

Fig. 1. Comparison of pressure histories measured incalorimeter A, for a 1.5-g PETN/TNT charge detonatedin air versus N2 atmospheres.

analytic model to the turbulent combustion ofTNT in a chamber, based on analysis in thermo-dynamic state space [10].

Presented here is a multi-component gasdy-namic model of TNT combustion in explosions.The model details, along with the numerical meth-ods used for integrating the governing equationsare presented in Section 2. Three-dimensionalnumerical simulations were performed for 1.5-gPETN/TNT explosions in six chambers (A–F).Results are compared with experimental pressurehistories in Section 3. This is followed by discus-sion of combustion efficiency in Section 4, andconclusion in Section 5.

2. Model

2.1. Conservation laws

We consider an initial value problem startingat the time the detonation wave reaches the edgeof the charge, so the detonation products aregases. We model the expansion of the detonationproducts and the ensuing blast waves that are cre-ated in the surroundings. We model the evolutionof the combustion fields in the limit of large Rey-nolds and Peclet numbers, where effects of molec-ular diffusion and heat conduction are negligible.The flow field is governed by the following conser-vation laws:

@tqþr � ðquÞ ¼ 0 ð1Þ@tquþr � ðquuþ pÞ ¼ 0 ð2Þ@tqE þr � ðquE þ puÞ ¼ 0 ð3Þ

where q and p represent the gas density and pres-sure, u is the gas velocity vector, E � uþ u � u=2 isthe total energy, and u is the specific internal energy.

2.2. Combustion model

We consider two fuels: PETN detonationproducts (F 1) and TNT detonation products(F 2), along with their corresponding combustionproducts: PETN–air (P 1) and TNT–air (P 2); theircompositions are given in Tables 1 and 2. Wemodel the global combustion of both fuels F k withair (A) producing equilibrium combustion prod-ucts P k :

F k þ A) P k ðk ¼ 1 or 2Þ ð4ÞThe mass-fractions Y k of the components are gov-erned by the component conservation laws:

@tqY Fk þr � qY Fku ¼ �_sk ð5Þ@tqY A þr � qY Au ¼ �

Xk

ak _sk ð6Þ

@tqY Pk þr � qY Pku ¼X

k

ð1þ akÞ_sk ð7Þ

Table 1Reactants composition (mol/kg) at p = 1 bar,T = 298 K.

Fuel H2O CO2 N2 CO CH4 H2 CðsÞ

PETN* 11.3 10.8 6.3 5.0 – 1.3 –TNT* 1.3 2.1 6.6 20.8 3.3 2.7 4.3

* q0 ¼ 1 g/cc.

Table 2Equilibrium products composition (mol/kg) at T ¼ T ad .

Fuel H2O CO2 N2 O2 CO

PETN–air (rs ¼ 0:48) 7.6 6.1 12.8 2.3 4.6TNT–air (rs ¼ 3:35) 2.4 5.1 22.3 1.1 2.0 Fig. 3. Combustion products (CP) loci for stoichiome-

tric mixtures of TNT–air (rs ¼ 3:35) and PETN–air(rs ¼ 0:482). Horizontal lines represent adiabaticcombustion.

A.L. Kuhl et al. / Proceedings of the Combustion Institute 33 (2011) 2177–2185 2179

Fuel and air are consumed in stoichiometric pro-portions: ak ¼ A=F k . In the above, _sk representsthe global kinetics source term. In this work weuse the fast-chemistry limit that is consistent withthe inviscid gasdynamic model (1)–(3), so when-ever fuel and air enter a computational cell, theyare consumed in one time step.

2.3. Equations of state

Our code carries the density and specific inter-nal energy, along with the gas composition in eachcell. These are used to calculate the pressure andtemperature in a computational cell based onEquations of State (EOS). The thermodynamicstates encountered during SDF explosions havebeen analyzed in [11]. Here we summarize onlythe salient features needed for the present numer-ical modeling.

Figures 2 and 3 present the locus of states inthe Le Chatelier diagram of specific internalenergy versus temperature (u� T ). Presented inFig. 2 are curves for reactant components: PETNdetonation products (PETN), TNT detonation

Fig. 2. Detonation products isentropes for TNT andPETN charges (q0 ¼ 1 g=cc), with frozen productscomposition for T < 1800 K.

products (TNT) and air (A); their correspondingequilibrium combustion products (P k) are de-picted in Fig. 3. In this formulation, combustionbecomes:

material transformations from reactants

! products at constant energy

resulting in combustion temperatures of 3175 and2900 K for PETN–air and TNT–air systems,respectively (see Fig. 3). The system is iso-ener-getic—so there is no energy addition (e.g., “heatof combustion”) in the energy conservation Eq.(3). In a reacting cell, before combustion, the reac-tants EOS are used; after combustion, the prod-ucts EOS are used.

In [11] we have shown that these componentsbehave as calorically-perfect gases forT < 3500 K. Thus it is appropriate to fit themsolely as a function of temperature. Piecewise qua-dratic functions were used to define the compo-nents c:

ucðT Þ ¼ acT 2 þ bcT þ cc ð8Þfor c ¼ A; F k ; P k . Given uc, one can solve the qua-dratic to find the temperature. Then for a purecell, the pressure is computed from the perfectgas law:

pc ¼ qcRcT c ð9Þor from the JWL function in the detonation prod-ucts gases [11]:

pJWLðv; T Þ ¼ A 1� x � v0

R1 � v

� �e�R1 �v=v0

þ B 1� x � v0

R2 � v

� �e�R2 �v=v0 þ RT=v ð10Þ

where v is the specific volume (v ¼ 1=q). Formixed cells, the caloric equation of state retainsthe same form as (8), but the coefficients are


weighted by the mass-fraction of components inthe cell. See [11] for more details and fittingconstants.

2.4. Numerical methods

The governing Eqs. (1)–(3) and (5)–(7) wereintegrated with a high-order Godunov schemebased on an efficient Riemann solver for gasdy-namics first developed by Colella and Glaz [12]and extended to generalized conservation lawsby Bell et al. [13]. The solver was modified toaccommodate negative specific internal energies(vid. Figs. 2 and 3) associated with our thermody-namic formulation. In this approach, informationpropagates along characteristics at the correctwave speeds, and it incorporates nonlinear waveinteractions within the cell during the time step.The methodology includes a limiting step (slope

Fig. 4. Evolution of turbulent

flattening) that automatically reduces the orderof approximation in the neighborhood of discon-tinuities, while in smooth regions of the flow thescheme is second order in time and space.

The Godunov scheme has been incorporatedinto an adaptive mesh refinement (AMR) algo-rithm that allows one to focus computationaleffort in complex regions of the flow such as mix-ing layers and reaction zones. Our adaptive meth-ods are based on the block-structured AMRalgorithms of Berger and Colella [14], andextended to three-dimensional hyperbolic systemsby Bell et al. [15]. Embedded boundary methodsare used to represent irregular geometries [16].In this AMR approach, regions to be refined areorganized into rectangular patches, with severalhundred to several thousand grid-points perpatch. One can refine based on discontinuities(shocks and contact surfaces), using Richardson

mixing in a TNT fireball.


error estimates, or for present purposes, on flamesurfaces.

2.5. Turbulent mixing

AMR is also used to refine turbulent mixingregions; by successive refinements we are able tocapture the energy-bearing scales of the turbu-lence on the computational grid. In this way weare able to compute the effects of turbulent mixingwithout resorting to explicit turbulence modeling.This is consistent with the so-called MILES(Monotone Integrated Large-Eddy Simulation)approach of Boris [17]. A comprehensive reviewof Implicit Large-Eddy Simulation (ILES) meth-ods may be found in Grinstein et al. [18]. A veri-fication of the ability of our Godunov scheme toreplicate the Kolmogorov spectrum of turbulentflows has been demonstrated by Aspden et al. [19].

2.6. Initial value problem

We assume that the charge is consumed by aconstant velocity Chapman-Jouguet (CJ) detona-tion wave. Therefore we initialize the computa-tional grid with the self-similar flowfieldcorresponding to a spherical CJ detonation waveof Taylor [20] when the detonation reaches theedge of the charge. Initially we use six levels ofAMR grid refinement—to resolve the initial wavestructure, and then drop refinement levels as theblast wave expands.

Fig. 5. Shadow photograph of a PETN/TNT (0.2 g/0.5 g) detonation products cloud.

Fig. 6. Combustion products surface from PETN/TNTcharge (0.5 g/1 g) in air.

3. Results

Evolution of the flowfield engendered by thedetonation of a 1.5-g PETN/TNT charge wascomputed with the 3-d AMR code. We start withnumerical simulations of the un-confined case,which will be used for flow visualization of themixing (Section 3.1), and then move on to simula-tions of combustion in chambers A–F. Results willbe compared with experimental pressure andimpulse histories in Section 3.2 and Section 3.3.The latter will be used to establish mean chamberpressures and combustion completeness.

3.1. Flow visualization

In Fig. 4 we present a cross-sectional views ofthe evolution of the flow field created by the deto-nation of a 1-g spherical TNT charge in an un-con-fined space. Composition of the flow field isrepresented by colors: fuel is yellow, air is blueand combustion products are red. White dotsdenote the cells that are burning during the currentcomputational cycle. Black lines represent con-tours of r � u thereby making shocks visible, whilegreen contours denote r� u thereby makingvorticity visible. One can see that the detonation

products–air interface is unstable and developsRichtmyer [21]–Meshkov[22] mixing structureswith high vorticity. One can also see the sphericalshock forming the blast wave in air, and animploding shock in the detonation products (DP)gases. By 114 ls, the imploding shock has reachedthe center (r = 0); thereafter it propagates backthrough the DP gases (times 161 and 221 ls),and then emerges from the DP cloud forming asecond blast wave discovered by Brode [23]. Thisimploding shock draws the mixing structures intowards the center (114 ls) forming Taylor cavities[24,25]; Kelvin [26]–Helmholtz [27] structures arevisible on the cavity walls. The vorticity in thesecavities mixes the air with the fuel, forming hot(2900 K) combustion products. Similar mixingstructures are seen in experiments: Fig. 5 presentsa shadow photograph visualization of mixing


inside the fireball from a PETN/TNT (0.5/1 g)charge; Taylor cavities are prominent. An exteriorview of the combustion products interface from anAMR simulation is presented in Fig. 6, whichshows numerous mushroom-capped structures.

Fig. 7. Comparison of pressure histories from numerical simPETN/TNT charges in cylindrical (L/D = 1) chambers withdifferent L/D ratios (Cases D, E and F).

3.2. Pressure histories

Pressure histories from numerical simulationof the 3-d combustion field in chambers A–F arepresented in Fig. 7. These are compared with

ulations with experimental data for explosion of 1.5-gdifferent volumes (Cases A, B and C) and tunnels with


pressure histories measured at r = 5 cm on the lidof cylindrical calorimeters A–C of different vol-umes (L/D = 1), and the end-wall of the tunnelsD–F of different L/D ratios. Computed waveformsare quite similar to measured waveforms—judgedon the basis of initial shock arrival time, peakpressure and waveform, and subsequent shockarrival times, peak pressures and waveforms.Agreement of various shock arrival times indi-cates that the temperature/sound speed fieldscomputed for combustion of PENT/TNT withair are similar to the experimental fields.

3.3. Chamber pressures

Pressure histories from Fig. 7 were integratedto compute the impulse

IkðtÞ ¼Z 3ms

1msDpkðt0Þdt0 ð11Þ

during 1 6 tðmsÞ 6 3. The impulse histories werefit with linear functions of time: IðtÞfit ¼aþ D�pc � t; the slope then represents the meanchamber pressure, D�pc (vid. Table 3). The so

Table 3Chamber pressures and mass-fraction of products.

Case �pCVE*

bars�pexp

*

bars�pcalc

*

barsY P ;exp

(%)Y P ;calc

(%)

A 8.94 8.37 7.65 88.4 73.6B 3.98 3.52 3.24 70.9 53.4C 2.64 2.13 2.17 39.3 45.1D 9.36 7.04 6.49 54.4 43.6E 9.36 5.44 5.84 22.9 30.7F 12.1 10.1 9.12 68.2 51.8

* �pc ¼ pa þ Dp where pa ¼ 0:978 bars.

Fig. 8. Comparison of mean chamber pressures withthermodynamic predictions of Constant-Volume-Explo-sions (CVE) as a function of chamber volume V c

(L/D = 1).

established mean chamber pressures from thecomputations are depicted as a function of cham-ber volume: V c in Fig. 8 (for L/D = constant) andas a function of L/D (at V c = constant) in Fig. 9.They are similar to values inferred from the mea-sured pressure histories [3]. Both are somewhatless than the theoretical values predicted for aConstant Volume Explosion (CVE) in a chamber(Fig. 8). Both diverge more and more from thepCVE as L/D increases (Fig. 10) because the tubeconfinement suppresses mixing and combustion.

4. Discussion

According to theory of combustion in a ther-modynamically-isolated chamber [10], the mass-fraction of combustion products at late times(Y1P ) is related to mean chamber pressures bythe relation:

Fig. 9. Comparison of mean chamber pressures withthermodynamic predictions of Constant-Volume-Explo-sions (CVE) as a function of tunnel L/D (V c = 6.3 L).

Fig. 10. Mass-fraction of combustion products Y p;c as afunction of v.

Fig. 11. Effect of cell size on pressure history (TNT,Case A, gauge 1).


Y1P ¼ ½�p1c � pR�=½pP � pR� ð12Þwhere pR and pP denote the chamber pressurespredicted for frozen reactants R and equilibriumcombustion products P (corresponding to CVEexplosions computed by the Cheetah code). Re-sults are compiled in Table 3 and presented inFig. 10 as a function of excess air ratio:v ¼ achamber=astoich (corresponding to chambersA–C and F). Combustion products mass-fractionreaches a peak value of 88% for air concentra-tions about twice stoichiometric (v � 2), andthen decay as air dilution increases. Presumablythis is due to quenching by the cold excess air.The mass-fraction of combustion products com-puted from the numerical simulations (att = 3 ms) follows similar trends, but their valuesare about 10% lower than the experimental values(see Table 3).

One might naturally wonder: how can an Eulersimulation hope to approximate the correct phys-ical diffusion between species in the gas phase?Admittedly, at the finest scales—where mixing isoccurring—mixing in an ILES simulation isnumerical. However, in the present case, we findcompensating effects with refinement. As we refinethe mesh, the numerical mixing decreases (due todecreased numerical diffusion) but turbulencescales increase (because finer scales becomeresolved). These compensating effects make thesolution insensitive to grid resolution—once onecaptures a significant fraction of the inertial range.Empirically we find that ILES seems to work forexplosion fields—which are characterized byextraordinarily large Reynolds numbers (�108)and extremely strong ignition sources (p � 200kbar and T � 3000 K). But the approach is notuniversal, e.g., in our simulations of turbulentcombustion in pre-mixed hydrogen flames [28]—that are characterized by smaller Reynolds num-bers (�700), high molecular diffusivity and mildignition—it did not accurately predict the behav-ior, even in a large-scale sense. So, the appropri-ateness of the ILES approach must be evaluatedfor each class of problem that one encounters(vid. [18]).

Our previous studies [4] have demonstratedthat one can get convergence of low-dimensionalnorms such as mass-fraction of fuel consumed, asthe mesh is refined. At the beginning of the pres-ent work, grid refinement studies were con-ducted. Typically four levels of refinement wereused (mesh sizes of 3.28, 1.64, 0.8 and 0.4 mm).Next, all grids were refined by a factor of two(1.64, 0.8, 0.4 and 0.2 mm). The reflected pres-sure histories were compared, and we found thatthe 0.4 mm mesh calculation gave virtually iden-tical results to the 0.2 mm calculation (seeFig. 11). So the 0.4 mm mesh was adopted asthe baseline for these studies. Most simulationswere run on an IBM POWER4+ computer using

256 processors. Typical run times varied between6000 CPU hours (Case A) and 60,000 CPUhours (Case C).

5. Conclusions

A gasdynamic model has been proposed fornumerical simulations of turbulent combustionof detonation products gases with air. It is basedon the following elements: (i) the multi-compo-nent gasdynamic conservation laws, (ii) a combus-tion model utilizing the fast-chemistry limit, (iii) athermodynamic model assuming frozen reactantsand equilibrium products, (iv) a high-order Godu-nov scheme providing non-diffusive solutions tothe conservation laws, and (v) adaptive meshrefinement algorithm to capture the energy-bear-ing scales of the turbulent mixing. Pressures histo-ries from the numerical simulations were quitesimilar to measured pressure histories for sixdifferent mixing scenarios (chambers A–F). Themodel was able to capture the influence of cham-ber volume and geometry on the global fuel con-sumption; numerical results followed trendssimilar to the experiments in regard to mass-frac-tion of products versus excess air ratio: Y1P ðvÞ.Based on this agreement, we conclude that thedominant effect that controls the rate of TNTcombustion with air is the turbulent mixing rate;the ILES approach along with the fast-chemistrymodel used here adequately captures this effectin the highly turbulent post blast wave environ-ment studied here.

Acknowledgements

This work performed under the auspices of theUS Department of Energy by Lawrence Liver-more National Laboratory under Contract DE-AC52-07NA27344. LLNL-CONF-422265.


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gasdynamic model of turbulent combustion in tnt explosions

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