The present study is concerned with a one-dimensional problem in explosive welding that pertains to the collision of lead and steel plates. The metal plates and the surrounding air are represented as separate immiscible phases governed by independent equations of state. A multifluid Godunov-type (finite-volume) computational algorithm, based on the mechanical-equilibrium Euler equations and incorporating pressure relaxation, is used to numerically describe the evolution of the waves resulting from the collision. The position of the interface (contact discontinuity) between immiscible phases is tracked by means of the volume-of-fluid (VOF) method. The numerical model allows one to account for the existence of tensile stresses in metal and registers them as regions of negative pressure. The computed arrival time of the unloading wave at the interface between the plates agrees with the experimental data and with simulation results obtained via different methods.
Explosive welding is an established method for joining dissimilar metals and alloys. It is a process of metal bonding by way of high-speed impact due to energy released by an explosive. This process is accompanied by complex dynamic effects, which form the subject matter for many experimental and numerical studies. Classical scientific research in this field is associated with the names of A. A. Deribas [1] and S. K. Godunov [2]. In O. M. Belotserkovsky's work [3], it is proven numerically that one of the reasons for the joining of metals may be the Rayleigh-Taylor instability, which arises at the interface of colliding metal plates.
Experimental research conducted for the scientific articles [4,5] shows that welded metals remain in a pseudo-fluid condensed state in the first several microseconds after their collision. Therefore, the numerical simulation of the collision process may be carried out via a multifluid hydrodynamic approach, based on the solution of the Euler equations. In this case, every material (phase) is considered a compressible continuum in the Eulerian approximation.
Many fields of science and technology require modelling immiscible compressible continua while accounting for the presence of condensed phases [6]. Several approaches to the construction of the requisite multiphase models exist in the literature. Some of them, such as the nonequilibrium multi-velocity Rakhmatulin-Nigmatulin model [6,7], account for phase transitions and the non-locality of interphase interaction. Others, like the Baer-Nunziato model [8,9], account for the disequilibrium of the process with respect to velocities, pressures and temperatures, assuming local relaxation. The homogenised description of compressible media based on the Euler equations with equilibrium pressures and velocities has become widely used [6,10,11]. To numerically solve the Euler equation system, Godunov-type [12], discontinuous-Galerkin-type [13], finite-element-type [14] and other methods are actively employed.
The main task in numerical simulation of the collision process is to track the phase boundaries (interfaces) of the colliding materials. Various methods [15,16] can be used to capture or track sharp interfaces between immiscible materials, including level-set methods, arbitrary Eulerian-Lagrangian (ALE) methods and volume-of-fluid (VOF) methods. Among these methodologies, we highlight the VOF method [17] with material-interface reconstruction, which has a number of advantages, such as relative ease of use, execution speed, global mass conservation and conservation of volume fraction at the interfacial contact discontinuity. This method is used to track phase interfaces for both incompressible [17] and compressible [18][19][20] flows.
In compressible media, the choice of the method for density and internal-energy determination for each particular phase in computational cells containing several such phases -so-called mixed cells -presents some difficulty [21]. Our approach to simulating multiphase flows is based on the idea suggested in an unpublished manuscript by P. Colella, H. M. Glaz and R. E. Ferguson (CGF). In their algorithm, interfaces between phases are tracked using the VOF method with the equations of motion for the multiphase medium complemented by the evolution equations for the volume fractions, energies and densities of each phase in mixed cells. This formulation accommodates using separate and arbitrary equations of state for the phases. The mathematical model of CGF in the case of two phases is equivalent to Kapila et al.’s mechanical-equilibrium five-equation 𝑉 𝑝-model [10]. The latter model emerges as a reduction of the non-equilibrium Baer-Nunziato model for describing arbitrary two-phase flows under the assumption that the exchange of momentum and energy at the phase interfaces runs its course at orders-of-magnitude-shorter timescales than advection processes.
In the works [18,19,22], the CGF method, originally developed for gases, is improved upon and generalised to condensed-medium problems, subject to various constraints on the utilised equations of state.
In the present work, we suggest a modification of the method due to CGF for multifluid modelling and use different equations of state for each phase. The developed method is demonstrated on a one-dimensional problem of high-speed collision of metal plates. It is shown that this approach has low numerical diffusion, thereby yielding higher solution accuracy. A comparative analysis of the results obtained here with similar computation results from the paper [9] is performed.
We represent the state inside a mixed cell by a single phase with its own density, pressure, internal energy and speed of sound, all consistent with the multiphase continuum. In accordance with the works [11,[22][23][24], this phase is called effective. Its density is calculated from the conservation of mass together with phase immiscibility via the following formula:
where 𝜌 (𝛼)
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