Numerical simulation of shock-induced motion of a cuboidal solid body using the overset grid functionality of OpenFOAM

Abstract

Numerical simulation of the motion of a cuboidal solid body induced by a propagating shock wave was conducted by using the overset grid functionality of OpenFOAM (v1712). A body-fitted grid associated with the moving body was overlapped with a background Cartesian grid fixed in space, and transient, three-dimensional, inviscid flows around the moving body were simulated coupled with the six-degrees-of-freedom motion of the solid body. The results from the present simulation were compared with those from shock tube experiments, in which the shock-induced motion of a cuboidal solid body floating in air and the flow field around the solid body were visualized using the shadowgraph technique and recorded by a high-speed video camera with a frame rate of 20,000 frames per second. The good agreement found between the numerical and experimental results for the translational and angular displacements of the shocked solid body indicates the capability of OpenFOAM to predict the shock-induced motion of a solid body.

Appendices

Validation of the numerical solver

This appendix describes the validation test for the numerical solver used in the present work. The numerical solver was validated by the standard shock tube (or one-dimensional Riemann) problem, which was first suggested by Sod [24]. The initial conditions are the following:

$$\begin{aligned} \left\{ \begin{array}{llll} u =0~\mathrm {m/s}, &{} p=100~\mathrm {kPa}, &{} \rho =1~\mathrm {kg/m}^3 &{} \text{ for } x < 5~\mathrm {m}\\ u = 0~\mathrm {m/s}, &{} p=10~\mathrm {kPa}, &{} \rho =0.125~\mathrm {kg/m}^3 &{} \text{ for } x >5~\mathrm {m} \end{array} \right. \end{aligned}$$

(9)

where x is the one-dimensional coordinate along the longitudinal direction of a shock tube of 10 m in length, in which a diaphragm is located at $x=5~\mathrm {m}$, and u is the flow velocity in the x-direction. In the present validation test, the shock tube was divided by 250 cells in the x-direction.

Figure 13a–c illustrates the distributions of density, pressure, and velocity along the shock tube, respectively, at 6 ms after the rupture of the diaphragm. The analytical solutions for the shock tube problem [33] are also shown in these figures by solid lines. It is seen from these figures that these typical flow properties obtained from the present numerical solvers well agree with those from the analytical solutions. Although there are overshoot and oscillation in the velocity distribution from the numerical solution, they are limited just behind the shock front, and the overall flow velocity is well captured. From the comparison between the numerical and the analytical solutions, it can be said the present numerical solvers can be used for the simulation of unsteady flows with shock waves.

Fig. 13

Comparisons of typical flow properties at 6 ms after the rupture of the diaphragm obtained by the numerical solvers used in the present work with those from the analytical solutions (open circles: numerical solutions, solid lines: analytical solutions)

Grid convergence test

Preliminary numerical simulations were conducted to determine the cell size required to reproduce the shock-induced motion of the cuboidal solid body. The cell numbers of the grid systems used for the present and preliminary simulations are summarized in Table 2.

In Fig. 14, the time histories of the translational and angular displacements of the shocked solid body such as shown in Fig. 11 are compared with each other. It is found that these results clearly show the converging trend for the grid refinement. In particular, the translational displacements of the solid body obtained with Grid 1 are practically the same as those with Grid 2 (see Fig. 14a, b).

On the other hand, the discrepancy in the angular displacement between the two results becomes noticeable with time (Fig. 14c). In order to estimate how much the solutions were converged, we determined the observed order of convergence P for the angular displacement at $t=5$ ms according to Roache’s methodology (Eq. (5.10.6.3) in [34]) and found that $P = 1.71$ based on the cell numbers of the body-fitted grids. The grid convergence index (GCI) for Grid 1 and Grid 2 and that for Grid 2 and Grid 3 were then evaluated as GCI$_{12}=11.0$% and GCI$_{23}=18.1$%, respectively. Since the ratio $\mathrm {GCI}_{23}/(r_{12}^P\mathrm {GCI}_{12})$, where $r_{12}$ is the effective grid refinement ratio [34] for Grid 1 and Grid 2, is approximately one, it can be said that the solutions are within the asymptotic range of convergence for the angular displacement of the solid body. We have eventually decided to use Grid 1 for the present work.

Table 2 Cell numbers of the three different grid systems

Fig. 14

Comparisons of the time histories of the displacements of the shocked solid body obtained with three different grid systems shown in Table 2