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Laboratory for Scientific Computing


In mining or defence applications, condensed-phase explosives interact with one or more compliant inert materials in the ignition or detonation process. Examples include confined rate-sticks and explosives that include gas cavities and solid beads. To model these, a mathematical formulation that can simultaneously model the explosive multi-phase mixture and its immiscible interaction with inert materials is necessary.

The challenge

MiNi16 combines the advantages of simpler augmented Euler and complex multi-phase models, while avoiding their shortcomings.

When standard conservative shock-capturing schemes are employed for the solution of multi-component versions of the Euler equations, they face several challenges. These include the solution of complex equations of state for describing the materials involved in the simulation, large (1000:1) density differences across the material interface boundary, maintaining oscillation-free interfaces (in terms of pressures, velocities, and temperatures), recovering accurate temperature fields and heavy computations required for well-resolved, three-dimensional simulations. Several of these points have been addressed in a piece-meal fashion the past, but a formulation that addresses them all at the same time, was not available. Of particular significance for predictive combustion modelling is the accurate and oscillation-free temperature field recovery.

The research

To this end, we have produced a mathematical formulation (MiNi16) which is based on an augmented Euler approach, to account for the mixture of the explosive and its products, and a multi-phase diffuse interface approach to solve for the immiscible interaction between the mixture and the inert materials. 

This formulation is in essence a hybrid (augmented Euler and multi-phase) model. With careful selection of its components, we have manage to retain desirable features of the two approaches, while losing their shortcomings. Critically for our applications of interest, MiNi16 (Michael and Nikiforakis, 2016) allows the accurate recovery of temperature fields across all components. Moreover, it conveys a lot more physical information than augmented Euler, without the complexity of the full multi-phase Baer–Nunziato-type models, and avoids robustness issues associated with augmented Euler models in the presence of more than two components. The model can sustain large density differences across material interfaces without the presence of spurious oscillations in velocity and pressure, and it can accommodate realistic equations of state and arbitrary (pressure- or temperature-based) reaction-rate laws.

Two examples are shown in the figures below. On the left is the small-scale interaction of a shock wave with an air cavity and on the right a rate-stick example.

The formulation can take one explosive material (material 2) that is composed of reactants (material α) and products (material β) and one or more inert materials (material 1). Each material is described by its own equation of state which can be simple (e.g. ideal gas) or complex (e.g. Mie-Grüneisen). As this is a diffuse-interface formulation, we apply mixing rules between materials α and β to obtain material 2. Mixing rules are also applied across the interface between materials 1 and 2, keeping in mind that this is an interface between two physically immiscible materials. The equations look like:

where ρ denotes density, p pressure, E specific total energy, u the velocity vector, z the volume fraction of a material to the total volume, λ the mass fraction of material α with respect to material 2 and K is the reaction rate of the explosive material.

We use high-resolution shock-capturing methods and adaptive mesh refinement to generate our numerical solutions.

The following figure illustrates how the full MiNi16 model can be simplified to reduced models, limiting cases and simpler formulations. Examples of applications for each case are also given. This also means that once the full model is coded, the initial data set up can be arranged such that one of the reduced models is in fact used.


We applied the MiNi16 model to study, among others, detonation propagation in compliantly confined rate-sticks (Ioannou et al., 2017) and shock-induced cavity collapse in nitromethane  (Michael and Nikiforakis 2018,2019). An example of the detonation propagation in a compliantly confined rate-stick is seen below.


  • A hybrid formulation for the numerical simulation of condensed phase explosives, Michael L. and Nikiforakis N., Journal of Computational Physics, vol. 316, (2016), 193-217,
  • The evolution of the temperature field during cavity collapse in liquid nitromethane. Part II: Reactive case, Michael L. and Nikiforakis N. Shock Waves, 29(1), 173-191 (2019),
  • The evolution of the temperature field during cavity collapse in liquid nitromethane. Part I: Inert case, Michael L. and Nikiforakis N. Shock Waves, 29(1), 153-172 (2019)
  • Control of condensed-phase explosive behaviour by means of voids and solid particles, Michael L. and Nikiforakis N. Active Flow and Combustion Control (2018), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer,
  • Detonation propagation in annular arcs of condensed phase explosives, Ioannou E., Schoch S., Nikiforakis N. and Michael L. Physics of Fluids, 29, 116102 (2017),