eISSN 2658–5782

DOI 10.21662/mfs

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2026. Vol. 21. Issue 1, Pp. 25–31
URL: https://multiphasesystems.online/mfs2026.1.005,en
DOI: https://doi.org/10.21662/mfs2026.1.005
Modeling the interaction of a shock wave with a dispersed medium in OpenFOAM
O.V. Yakovlev 🖂
Ufa University of Science and Technology, Ufa, Russia

Abstract

The study investigates the applicability of a kinetic-theory-based model for granular flows to simulate the interaction of a shock wave with a dispersed medium using the OpenFOAM solver blastEulerFoam. The kinetic model was benchmarked against experimental measurements and the Baer–Nunziato (BN) model, which is widely employed for granular dynamics in shock-wave propagation. Numerical simulations revealed significant discrepancies between the kinetic theory model and both experimental observations and BN results. The primary source of the divergence is the omission in the kinetic model of the dependence of granular pressure on particle volume fraction and inter-particle friction, leading to inaccurate predictions of compaction effects. In contrast, the BN model incorporates a strong coupling between particle volume fraction and frictional interactions, providing results that better agree with experimental data.In the numerical experiments, blastEulerFoam was used to solve the Eulerian–Eulerian equations with kinetic-theory granular closures. Gas properties (γ = 1.4, µ = 1.81×10−5 Pa·s) and dispersed-phase parameters (particle diameter is 0.05 mm, αmax = 0.63) were specified. The radial distribution function was taken from the Sinclair–Jackson model, while inter-particle friction was modeled using the Johnson–Jackson formulation. Simulations were performed on a 2-D grid with a 0.5 mm cell size, imposing a constant air inflow over the wall. Compaction angles ψ and ϕ were computed and compared with experimental data and BN predictions. The kinetic-theory model yielded ψ values between 2.3 and 2.7, higher than the experimental range of 1.5–1.8, an over-prediction of compaction. These findings highlight the limitations of model for shock-wave compaction and call for investigation under conditions, including inter-particle interaction and packing density.

Received: 27.02.2026
Accepted: 17.03.2026
Published: 31.03.2026
Citation

Yakovlev OV. Modeling the interaction of a shock wave with a dispersed medium in OpenFOAM. Multiphase Systems. 2026;21(1):25–31 (in Russian).

Keywords

KTGF;
shock wave;
dispersed media;
blastEulerFoam;
compaction

Article outline

Introduction

Shock‑wave propagation in dusty environments is a critical issue in propulsion, defense, and industrial safety (e.g., fuel‑air mixtures, dust explosions). Accurate prediction of the interaction between a shock front and a packed bed of particles is essential for design and safety assessments. Lagrangian particles embedded in an Eulerian gas field; accurate for dilute suspensions but computationally expensive for dense beds. Two interpenetrating continua; the choice of closure relations determines the model’s fidelity. Kinetic‑theory models assume a granular temperature and derive a pressure from particle collisions and a radial‑distribution function (Sinclair–Jackson). Baer–Nunziato model incorporates inter‑phase momentum, energy, and volume‑fraction coupling, and has been shown to capture compaction phenomena in high‑energy shocks. While kinetic‑theory models have been used in CFD codes such as OpenFOAM, their applicability to dense shock–granular‑flow problems has not been rigorously validated against experiments and BN predictions. This paper systematically compares the kinetic‑theory model with experimental compaction angles and BN results for a canonical shock–bed configuration.

Mathematical Model

Governing equations follow the Euler–Euler framework: conservation of mass, momentum, and energy for both gas and solid phases. Granular pressure is derived from the kinetic‑theory closure (based on the works by Lun), includes a granular temperature term and a radial‑distribution function (Sinclair–Jackson). Frictional stresses uses Johnson–Jackson model that introduces a frictional pressure that depends on the solid volume fraction and a friction coefficient. Ideal‑gas equation of state, with standard thermodynamic properties (γ = 1.4, μ = 1.81×10⁻⁵ Pa·s, etc.). Momentum transfer via inter‑phase drag (Gidaspow  model). Energy exchange through heat conduction and collisional dissipation.

Numerical Method

blastEulerFoam (OpenFOAM‑based BlastFOAM) implements the PIMPLE algorithm, combining PISO and SIMPLE for pressure–velocity coupling. Finite‑volume discretization with second‑order accuracy in space and time. 2‑D domain with a solid wall at the bed base (AG) and an inlet channel with prescribed air velocity. Outlet and lateral walls treated as slip walls; the top boundary is open. The gaseous phase is ideal gas, γ =v1.4, μ = 1.81×10⁻⁵ Pa·s, Pr = 1.0. The Granular phase has particle diameter fixed, specific heat of 987 J·kg⁻¹·K⁻¹, thermal conductivity 0.36 W·m⁻¹·K⁻¹, maximum solid volume fraction 0.63. Radial‑distribution function: Sinclair–Jackson. Friction parameters: αmin = 0.4, Fr = 0.05, p = 5 Pa, η = 2, φf = 30° (Johnson–Jackson). Courant number limited to 0.5 for stability; time step adjusted to capture shock propagation accurately.

Results

At 122 µs the pressure field shows a clear deformation of the shock front as it interacts with the packed bed. The pressure distribution matches qualitatively the experimental observations reported by Chuprov  and Utkin & Chuprov. The granular volume fraction reaches the maximum allowed value (0.63) more rapidly than in the experimental data, indicating an over‑prediction of compaction. The spatial distribution of α_s differs from that reported in Chuprov’s works, with a broader compressed region. A series of simulations were performed for inlet Mach numbers 2.5, 3.0, 3.5, and 4.0. The kinetic‑theory model predicts an increase in the thickness of the compressed layer with increasing Mach number (Fig. 3 in the paper). Experimental data and BN simulations (Utkin & Chuprov ) show the opposite trend: the compressed layer becomes thinner as the shock strengthens. Two angles, φ (compression) and ψ (expansion), were extracted from the pressure field. Table 1 (in the paper) compares the kinetic‑theory predictions with BN results: For Ma = 2.5, φ ≈ 0.53° (kinetic) vs 0.81° (BN). For Ma = 4.0, φ ≈ 0.68° (kinetic) vs 0.98° (BN). Table 2 (in the paper) presents a direct comparison with experimental measurements (Fan et al. ): For a shock speed of 1049 m·s⁻¹, the kinetic model gives φ ≈ 2.01°, ψ ≈ 2.28°, whereas experiments report φ ≈ 1.6°, ψ ≈ 3.8°. The kinetic model overestimates φ by up to30 % and underestimates ψ by 30–50%. The kinetic‑theory pressure term (Lun ) lacks an explicit dependence on solid volume fraction and inter‑particle friction, leading to an over‑prediction of granular pressure at high concentrations. The BN model, by contrast, includes a strong coupling between volume fraction and frictional stresses, capturing the compaction dynamics more accurately. The over‑prediction of compaction thickness and angles in the kinetic‑theory model is consistent across all Mach numbers examined.

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