Multifluid Hydrodynamic Simulation of Metallic-Plate Collision Using the VOF Method

TL;DR

The study addresses 1D high-speed collisions in explosive welding among immiscible phases (lead, steel, and air) using a multifluid Euler framework with a single common pressure and velocity in mixed cells. It introduces a VOF-based Godunov finite-volume method with pressure relaxation, separate phase energies and densities, and phase-specific stiffened-gas EOSs, coupled by Wood-like averaging for the effective compressibility: $\bar{K}_{S}=\left(\sum_{\alpha} f^{(\alpha)}/K_{S}^{(\alpha)}\right)^{-1}$ and the advection of volumes $\partial_{t}f^{(\alpha)}+\partial_{x}(f^{(\alpha)}u)=\dfrac{\bar{K}_{S}}{K_{S}^{(\alpha)}}\partial_{x}u$. The final closed system includes $\partial_{t}f^{(\alpha)}+\partial_{x}(f^{(\alpha)}u)=\dfrac{f^{(\alpha)}\bar{K}_{S}}{K_{S}^{(\alpha)}}\partial_{x}u$, $\partial_{t}(f^{(\alpha)}\rho^{(\alpha)})+\partial_{x}(f^{(\alpha)}\rho^{(\alpha)}u)=0$, $\partial_{t}(\bar{\rho}u)+\partial_{x}(\bar{\rho}u^{2}+p)=0$, and $\partial_{t}(f^{(\alpha)}\rho^{(\alpha)}E_{total}^{(\alpha)})+\partial_{x}(f^{(\alpha)}\rho^{(\alpha)}E_{total}^{(\alpha)}u)+u\dfrac{f^{(\alpha)}\rho^{(\alpha)}}{\bar{\rho}}\partial_{x}p=-p\dfrac{f^{(\alpha)}\bar{K}_{S}}{K_{S}^{(\alpha)}}\partial_{x}u$, with $p^{(\alpha)}=P^{(\alpha)}(\rho^{(\alpha)},E^{(\alpha)})$ relaxing to $p$. Results show unloading-wave arrival times consistent with experimental data and prior simulations, while the method maintains sharp interfaces and handles tensile stresses without additional fixes; extending to 2D is proposed to study interface instabilities in explosive welding.

Abstract

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.

Figures (5)

• Figure 1: A schematic for the distribution of densities, velocities and pressures at time $t = 0$. The left-pointing arrow indicates the direction of movement of the lead plate towards the steel plate.
• Figure 2: Advection of Lagrangian volumes (drawn as thin shaded rectangles) through a mixed cell (a cell centred at $x_i$) for two phases in a constant velocity field directed to the right. The phase interface is located at the point $x_{i-1} + 0.5 \Delta x + f_{i}^{\left(1\right)}\Delta x$ inside the Eulerian cell, of size $\Delta x$ with coordinate $x_{i}$
• Figure 3: Pressure profiles at time $t=\unit[1.3]{\upmu s}$, obtained by various methods. The thin solid line corresponds to data from Chuprov:2021. The dashed black line corresponds to data calculated by HLLC with MUSCL reconstruction on a grid of $10\,000$ cells. The thick solid and pale dashed lines correspond to data calculated by the current method with and without reconstruction, with the same spatial resolution as the method in Chuprov:2021. The interfaces between phases are indicated by dotted grey vertical lines. The inset in the upper right corner shows an enlarged plot of the pressure field on the $x$-axis segment with $x \in \left[ \unit[5]{mm} \mathop{..} \unit[6]{mm} \right]$
• Figure 4: Plot of the metallic-plate interface velocity versus time, obtained by various methods. The solid black line corresponds to the data from Chuprov:2021. The thickest line --- made up by sparsely spaced dashes --- corresponds to data obtained with HLLC and MUSCL reconstruction on a grid of $10\,001$ cells. The less thick dashed and thin dash-dot lines correspond to data obtained with the current method without and with reconstruction with the same spatial resolution as the method in Chuprov:2021
• Figure 5: Profiles for pressure $p$ (a), density $\rho$ (b), velocity $u$ (c) and internal specific energy $e$ (d) at time $t=\unit[1.5]{\upmu s}$, obtained with the current method, with and without reconstruction. The plots calculated using MUSCL at $10\,001$ cells with CFL$=0.28$ are shown by a thick solid black line. The palest densely dashed line represents the solution obtained using MUSCL at $3\,601$ cells; the dashed line with larger dash spacing is the solution obtained without MUSCL at the same resolution, and the dash-dot line is the solution without MUSCL at $10\,001$ cells with CFL$=0.2$. The bold dashed grey vertical lines correspond to the phase interfaces