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Spherical combustion clouds in explosions

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Abstract

This study explores the properties of spherical combustion clouds in explosions. Two cases are investigated: (1) detonation of a TNT charge and combustion of its detonation products with air, and (2) shock dispersion of aluminum powder and its combustion with air. The evolution of the blast wave and ensuing combustion cloud dynamics are studied via numerical simulations with our adaptive mesh refinement combustion code. The code solves the multi-phase conservation laws for a dilute heterogeneous continuum as formulated by Nigmatulin. Single-phase combustion (e.g., TNT with air) is modeled in the fast-chemistry limit. Two-phase combustion (e.g., Al powder with air) uses an induction time model based on Arrhenius fits to Boiko’s shock tube data, along with an ignition temperature criterion based on fits to Gurevich’s data, and an ignition probability model that accounts for multi-particle effects on cloud ignition. Equations of state are based on polynomial fits to thermodynamic calculations with the Cheetah code, assuming frozen reactants and equilibrium products. Adaptive mesh refinement is used to resolve thin reaction zones and capture the energy-bearing scales of turbulence on the computational mesh (ILES approach). Taking advantage of the symmetry of the problem, azimuthal averaging was used to extract the mean and rms fluctuations from the numerical solution, including: thermodynamic profiles, kinematic profiles, and reaction-zone profiles across the combustion cloud. Fuel consumption was limited to \(\sim \)60–70 %, due to the limited amount of air a spherical combustion cloud can entrain before the turbulent velocity field decays away. Turbulent kinetic energy spectra of the solution were found to have both rotational and dilatational components, due to compressibility effects. The dilatational component was typically about 1 % of the rotational component; both seemed to preserve their spectra as they decayed. Kinetic energy of the blast wave decayed due to the pressure field. Turbulent kinetic energy of the combustion cloud decayed due to enstrophy \(\overline{\omega ^{2}} \) and dilatation \(\overline{\Delta ^{2}} \).

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Notes

  1. Implicit large eddy simulations (ILES), see reference [32].

  2. \(\text{ Kroneker} \text{ delta}: \delta _{k_{3}} =1 \text{ if} \ k=3 \ and \ \delta _{k_{3}} =0\ \text{ if} \ k\ne 3\).

  3. We found that the \(u(T)_\mathrm{isobar} =u(T)_\mathrm{isochor} \) for \(T < 4,\!000~\text{ K}\), hence it was sufficient to fit the internal energy as solely a function of temperature; above 4,000 K this is an approximation, with an error less than 10 %.

  4. See Ree et al. [18] for justification.

  5. We found that the perfect-gas law accurately describes constant volume explosions for \(p\le 1\;\mathrm{kbar}\) [2].

  6. This Al powder density corresponds to a porosity of 80 % (i.e., a volume fraction of 20 %); it was a consequence of the inefficient packing density of the flake Al particles used in the experiments. It is also why the dilute heterogeneous continuum model used here is successful.

  7. Mean enstrophy is computed from: \(\overline{\omega (R_n ,t)^{2}} \equiv \frac{1}{\delta V}\int \int \!\!\!\int {\omega (R_n ,\theta ,\varphi )\cdot \omega (R_n ,\theta ,\varphi )\mathrm{d}V} \) according to A3.

  8. Since \(\int {\overline{u_r } \mathbf{{u}^{\prime }}} dV=0\).

  9. By orthogonality: \(\int \mathbf{u}^{\prime }_\omega \cdot \mathbf{u}^{\prime }_\Delta \mathrm{d}V=0\) (see Eqs. 26 and 27).

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Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. It was sponsored by the Defense Threat Reduction Agency under IACRO # 11-43821. UCRL-CONF 231319 and LLNL-JRNL-417022.

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Correspondence to A. L. Kuhl.

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Communicated by L. Bauwens.

This paper is based on work that was presented at the 23rd International Colloquium on the Dynamics of Explosions and Reactive Systems, Irvine, California, USA, July 24–29, 2011.

Appendices

Appendix A

1.1 Azimuthal averaging

We take advantage of the point symmetry inherent in spherical blast waves and azimuthally average the flow field to extract the mean and rms profiles from the numerical solution. Recall that the flow field \(\Phi (x,y,z,t)\) is computed and stored at cell-centered points \(P(x,y,z)\) on a Eulerian grid. The points are transformed to points \(P(r,\theta ,\phi )\) on a corresponding spherical grid, by the trigonometric formulas:

$$\begin{aligned} \left| \mathbf{r} \right|&= \sqrt{x^{2}+y^{2}+z^{2}}\quad and\quad \theta =\text{ arccos} (z/\left| \mathbf{r} \right|) \quad and \quad \phi =\text{ arctan}(x/y) \end{aligned}$$
(30)

We consider spherical shell volume \(\delta V\) at radius \(R_n \)

$$\begin{aligned} \delta V=\left[ {\int \int {(R_n \mathrm{d}\theta )(R_n \sin \theta \mathrm{d}\phi )} } \right]\delta r=4\pi R_n^2 \delta r. \end{aligned}$$
(31)

A shell thickness equal to the mesh size is assumed \((\delta r=\Delta )\). We denote computational cells located within this shell volume by \(P_n (R_n ,\theta ,\phi )\) and the corresponding flow field values by \(\Phi _n (R_n ,\theta ,\phi ,t)\). We then average the flow field at fixed \(R_n \) to evaluate the mean field:

$$\begin{aligned} \text{ Mean}: \overline{\Phi (R_n ,t)}&= \frac{1}{\delta V(R_n )}\int \int \!\!\!\int {\Phi (R_n ,\theta ,\phi ,t)\mathrm{d}V}\nonumber \\&= \frac{1}{N}\sum _N {\Phi _n }. \end{aligned}$$
(32)

The ensemble size: \(N=4\pi (R_n /\Delta )^{2}\), is approximately \(10^{5}\), except near the origin (see Table 1). Given the mean, one can then compute fluctuations about the mean:

$$\begin{aligned}&\overline{{\Phi }^{\prime }(R_n ,t)^{2}} =\frac{1}{\delta V(R_n )}\int \int \!\!\!\int {[\Phi (R_n ,t)-\overline{\Phi (R_n ,t)} ]^{2}\mathrm{d}V}\nonumber \\&\quad =\frac{1}{N}\sum _N {[\Phi _n -\Phi (R_n ,t)]^{2}} \end{aligned}$$
(33)

and root mean-squared (rms) fluctuations

$$\begin{aligned} \text{ rms}: {\Phi }^{\prime }(R_n ,t)_\mathrm{rms} =\sqrt{\overline{{\Phi }^{\prime }(R_n ,t)^{2}} } \end{aligned}$$
(34)

These were used to construct the evolution of the mean and rms profiles of the combustion cloud. We note in passing that this azimuthal-averaging technique was first used by Kuhl [36] to analyze spherical mixing layers in TNT explosions.

Table 1 Ensemble size versus radius \((\Delta _2 =0.8)\) mm

Appendix B

1.1 Derivation of the global kinetic energy equation for viscous compressible flow

The compressible Navier–Stokes equations may be written in the form:

$$\begin{aligned} \partial _t \rho \mathbf{u}+\nabla \rho \mathbf{uu}=-\nabla p+\mu \nabla ^{2}\mathbf{u}+\frac{1}{3}\mu \nabla (\Delta ) \end{aligned}$$
(35)

where \(\Delta \equiv \nabla \cdot \mathbf{u}=\mathrm{div}\mathbf{u}, \nabla ^{2}\mathbf{u}=\mathrm{div}(\mathrm{grad}(\mathbf{u})), \mu \) represents the coefficient of viscosity and assumes the Stokes hypothesis. Taking the inner product with u gives the compressible kinetic energy equation:

$$\begin{aligned} \partial _t \rho \mathrm{KE}+\nabla \rho \mathrm{KE}\mathbf{u}=-\mathbf{u}\cdot \nabla p+\mu \mathbf{u}\cdot \nabla ^{2}\mathbf{u}+\frac{1}{3}\mu \mathbf{u}\cdot \nabla (\Delta )\nonumber \\ \end{aligned}$$
(36)

where \(\mathrm{KE}=\mathbf{u}\cdot \mathbf{u}/2\). Using vector calculus relations, one finds

$$\begin{aligned}&\text{ Dilatation} \text{ function}: \mathbf{u}\cdot \nabla (\Delta )=-\Delta ^{2}+\nabla \cdot (\mathbf{u}\Delta )\end{aligned}$$
(37)
$$\begin{aligned}&\text{ Dissipation} \text{ function}: \mathbf{u}\cdot \nabla ^{2}\mathbf{u}=-\omega ^{2}-\Delta ^{2}+\nabla \cdot (\mathbf{u}\Delta )\nonumber \\&\quad +\nabla \cdot (\mathbf{u}\times \omega ). \end{aligned}$$
(38)

Inserting these into (36), one finds the kinetic energy equation for compressible viscous flow:

$$\begin{aligned} \partial _t \rho \mathrm{KE}+\nabla \cdot \rho \mathrm{KE}\mathbf{u}&= -\mathbf{u}\cdot \nabla p-\mu \omega ^{2}-\frac{4}{3}\mu \Delta ^{2}\nonumber \\&+\mu \{\nabla \cdot (\mathbf{u}\times \omega )+\frac{4}{3}\nabla \cdot (\mathbf{u}\Delta )\}.\nonumber \\ \end{aligned}$$
(39)

Integrating over a spherical volume \(V(R) \) whose radius \(R\) is greater than the spherical shock radius \(R_{s}(t)\), all divergence terms are zero. Thus we find the evolution equation for the global (volume-averaged) kinetic energy:

$$\begin{aligned} \frac{d}{\mathrm{d}t}\overline{\rho \mathrm{KE}} =-\overline{\mathbf{u}\cdot \nabla p} -\mu \overline{\omega ^{2}} -\frac{4}{3}\mu \overline{\Delta ^{2}}. \end{aligned}$$
(40)

Decompose the kinetic energy, based on the mean and fluctuating velocity components \(u_{r}\) and \(\mathbf{u}^{\prime }\):

$$\begin{aligned} \mathrm{KE}\equiv \frac{1}{2}\mathbf{u}\cdot \mathbf{u}=\frac{1}{2}\overline{u_r } +\overline{u_r } \mathbf{{u}^{\prime }}+\frac{1}{2}\mathbf{{u}^{\prime }}\cdot \mathbf{{u}^{\prime }}. \end{aligned}$$
(41)

Integrating over the same control volume \(V\) gives Footnote 8:

$$\begin{aligned}&\overline{\rho \mathrm{KE}} =\frac{1}{V}\int {\frac{1}{2}} \rho u_r^2 \mathrm{d}V+\frac{1}{V}\int {\frac{1}{2}\rho \mathbf{{u}^{\prime }}\cdot \mathbf{{u}^{\prime }} \mathrm{d}V}\nonumber \\&\quad =\overline{\rho \mathrm{KE}} _\mathrm{BW} +\overline{\rho \mathrm{FKE}}. \end{aligned}$$
(42)

Decompose the fluctuating kinetic energy into rotational and dilatational components Footnote 9:

$$\begin{aligned} \overline{\rho \mathrm{FKE}} =\frac{1}{V}\int {\frac{1}{2}} \rho \mathbf{u}^{\prime }\cdot \mathbf{u}^{\prime }dV=\overline{\rho \mathrm{KE}} _\omega +\overline{\rho \mathrm{KE}} _\Delta \end{aligned}$$
(43)

where \(\overline{\rho \mathrm{KE}} _\omega =\frac{1}{V}\int {\frac{1}{2}\rho \mathbf{{u}^{\prime }}_\omega \cdot \mathbf{{u}^{\prime }}_\omega \mathrm{d}V} \) and \(\overline{\rho \mathrm{KE}} _\Delta =\frac{1}{V}\int {\frac{1}{2}\rho \mathbf{{u}^{\prime }}_\Delta \cdot \mathbf{{u}^{\prime }}_\Delta \mathrm{d}V} \). Thus, the global kinetic energy equation for compressible viscous flow may be written as:

$$\begin{aligned}&\frac{d}{\mathrm{d}t}\overline{\rho \mathrm{KE}} _{\mathrm{BW}} \!+\!\frac{d}{\mathrm{d}t}\overline{\rho \mathrm{KE}} _\omega \!+\!\frac{d}{\mathrm{d}t}\overline{\rho \mathrm{KE}} _\Delta \!=\!-\!\overline{\mathbf{u}\cdot \nabla p} \!-\!\mu \overline{\omega ^{2}} \!-\!\frac{4}{3}\mu \overline{\Delta ^{2}}.\quad \nonumber \\ \end{aligned}$$
(44)

This provides insight into the decay of turbulent kinetic energy in compressible viscous flows.

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Kuhl, A.L., Bell, J.B., Beckner, V.E. et al. Spherical combustion clouds in explosions. Shock Waves 23, 233–249 (2013). https://doi.org/10.1007/s00193-012-0410-y

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