The high-speed impact of a droplet onto a flexible substrate is a highly non-linear process of practical importance, which poses formidable modelling challenges in the context of fluid–structure interaction. We present two approaches aimed at investigating the canonical system of a droplet impacting onto a rigid plate supported by a spring and a dashpot: matched asymptotic expansions and direct numerical simulation (DNS). In the former, we derive a generalisation of inviscid Wagner theory to approximate the flow behaviour during the early stages of the impact. In the latter, we perform detailed DNS designed to validate the analytical framework, as well as provide insight into later times beyond the reach of the proposed analytical model. We draw from both methods to observe the influence the material properties of the plate-spring-dashpot have on the dynamics of the droplet, predominantly through altering its internal hydrodynamic pressure distribution. We build on the interplay between these techniques, demonstrating that a hybrid approach leads to improved model and computational development, as well as result interpretation, across multiple length and time scales.
The DNS for this paper uses Basilisk, which is a free software program written in C to solve partial differential equations, most commonly using the volume-of-fluid method to solve the Navier-Stokes equations. The driver code is therefore pure C code, with outputs in plain text files in order to be compatible with any post-processing software. The analytical solutions are reduced to a set of second-order non-linear ODEs, so are readily solved with off-the-shelf software packages such as MATLAB or SciPy.
The direct numerical simulations (DNS) for this paper were conducted using Basilisk (http://basilisk.fr/). As Basilisk is a free software program written in C, it can be readily installed on any Linux machine, and it should be straightforward to then run the driver code to re-produce the DNS from this paper. Given this, the numerical solutions presented in this paper are a result of many high-fidelity simulations, which each took approximately 24 CPU hours running between 4 to 8 cores. Hence the difficulty in reproducing the results should mainly be in the amount of computational resources it would take, so HPC resources will be required.
The DNS in this paper were used to validate the presented analytical solutions, as well as extend the results to a longer timescale. Reproducing these numerical results will build confidence in these results, ensuring that they are independent of the system architecture they were produced on.
The reviewers should first make sure they understand the problem formulation (Section 2). They can skip the analytical model in Section 3 if they wish, as this gets deep into Wagner theory and is not required knowledge to understand the DNS. Section 4 details how the direct numerical simulations (DNS) are set-up, and understanding this section is essential before looking at the code. The DNS validation in Appendix B may then be a great place to start off before re-producing the main results. The results to re-produce are given in Section 5, in particular Figures 5 and 6. You can also attempt to re-produce the analytical solutions given in these figures and Figure 7, as the equations they are plotting are referenced in the captions.