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Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column

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Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column applied sciences Article Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column Viola Rossano and Giuliano De Stefano * Engineering Department, University of Campania Luigi Vanvitelli, Via Roma 29, 81031 Aversa, Italy; [email protected] * Correspondence: [email protected] Abstract: Computational fluid dynamics was employed to predict the early stages of the aerody- namic breakup of a cylindrical water column, due to the impact of a traveling plane shock wave. The unsteady Reynolds-averaged Navier–Stokes approach was used to simulate the mean turbulent flow in a virtual shock tube device. The compressible flow governing equations were solved by means of a finite volume-based numerical method, where the volume of fluid technique was employed to track the air–water interface on the fixed numerical mesh. The present computational modeling ap- proach for industrial gas dynamics applications was verified by making a comparison with reference experimental and numerical results for the same flow configuration. The engineering analysis of the shock–column interaction was performed in the shear-stripping regime, where an acceptably accurate prediction of the interface deformation was achieved. Both column flattening and sheet shearing at the column equator were correctly reproduced, along with the water body drift. Keywords: computational fluid dynamics; droplet breakup; industrial gas dynamics; shock tube Citation: Rossano, V.; De Stefano, G. Computational Evaluation of Shock 1. Introduction Wave Interaction with a Cylindrical The aerodynamic breakup process of liquid droplets induced by the interaction with Water Column. Appl. Sci. 2021, 11, 4934. https://doi.org/10.3390/ plane shock waves is of crucial importance for many industrial gas dynamics applica- app11114934 tions. The latter include, just to mention a couple of them, supersonic combustion airbreath- ing jet engines (scramjets) and rain impact damage to supersonic aircraft [1,2]. The shock Academic Editor: Sergio Hoyas wave interaction with a water droplet actually represents the initial stage of the breakup process induced by the high-speed gas stream. Since this process plays an important Received: 30 April 2021 role in the droplet breakup, a number of studies have been conducted to investigate the Accepted: 25 May 2021 shock/droplet interaction for correctly interpreting the mechanism of aerobreakup. In Published: 27 May 2021 particular, this process has been the subject of several experimental studies employing shock tube devices, wherein a traveling planar shock wave is reproduced, with uniform Publisher’s Note: MDPI stays neutral gaseous flow conditions being established, e.g., [3,4]. In these experiments, the shock front with regard to jurisdictional claims in passes over the droplet and causes its deformation and successive breakup. published maps and institutional affil- The physical mechanism of aerobreakup is controlled by the Ohnesorge number (Oh), iations. which compares liquid viscosity to surface tension effects, as well as the Weber num- ber (We), which compares disruptive aerodynamic forces to restorative surface tension ones. However, at low Ohnesorge numbers (Oh < 0.1), the influence of liquid viscosity can be neglected, and the process is governed essentially by the Weber number [5]. The Copyright: © 2021 by the authors. interpretation and identification of breakup mechanisms have been recently reviewed, Licensee MDPI, Basel, Switzerland. based on experimental studies using highly resolved images of liquid droplets suddenly This article is an open access article exposed to supersonic gas streams [6]. According to this reclassification, at low Weber distributed under the terms and numbers (10 < We < 102), Rayleigh–Taylor instability has to be considered the main driv- conditions of the Creative Commons ing mechanism for the aerobreakup, which is referred to as Rayleigh–Taylor piercing (RTP). Attribution (CC BY) license (https:// Differently, at high Weber numbers (We > 103), significant shear-induced motions exist, creativecommons.org/licenses/by/ and the breakup process is governed by the instability of the stretched liquid sheets that 4.0/). Appl. Sci. 2021, 11, 4934. https://doi.org/10.3390/app11114934 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 4934 2 of 15 are formed at the periphery of the deforming droplet. This regime is referred to as shear- induced entrainment (SIE) [7]. Another non-dimensional parameter that is sometimes considered when examining this process is the ratio between liquid and gas densities. As for other challenging engineering applications, computational fluid dynamics (CFD) can be effectively employed to predict the early stages of the aerobreakup of a water droplet due to the impact of a traveling shock wave. In the context of preliminary analyses, the shock–droplet interaction can be evaluated by considering a water cylinder with a circular cross-section in the high-speed flow behind a planar shock front. The cylindrical geometry of the water body can be efficiently modeled using a two-dimensional compu- tational domain that provides shorter turnaround times compared to three-dimensional simulations. Several numerical studies exist in the literature dealing with two-dimensional shock–column interactions, e.g., [8–10], where the sheet-thinning process was simulated and the column deformation and drift were measured. Usually, such works focus on the early stages of aerobreakup and neglect the effects of both surface tension and molecular viscosity. The two-dimensional approach is also justified based on experimental investiga- tions of liquid columns that allow for easier visualization of the wave structures [11,12]. Indeed, in the early stages of the interaction process, the deformation of two-dimensional liquid columns has been found to be very similar to that of three-dimensional spherical droplets [13]. Compressible multi-phase solvers based on a five-equation model [14] are often developed to computationally study shock–water column interactions, where the interface is modeled using volume fractions. The governing equations consist of two continuity equations for the two different phases, the mixture momentum and energy equations, and the volume fraction advection equation, e.g., [8,15]. As far as flow turbulence modeling is concerned, both under-resolved direct numerical simulation (DNS) and large eddy simulation (LES) approaches are commonly followed, e.g., [4]. The main goal of the present work was the computational evaluation of the initial stages in the interaction process between the air shock front and the cylindrical water column following a relatively light approach, which was the unsteady Reynolds-averaged Navier–Stokes (RANS) modeling. The mean turbulent flow in a virtual shock tube device was simulated by supplying the calculation with a suitable turbulence closure model. The interaction process was simulated in the shear-stripping regime that was at relatively high Weber and low Ohnesorge numbers. Numerical calculations were conducted by employing one of the CFD solvers that are commonly and successfully used for building virtual wind tunnels in industrial gas dynamics research, e.g., [16,17]. The compressible flow governing equations were solved by means of a finite volume (FV)-based numerical technique, while the transient tracking of the air–water interface was approximated on the fixed FV mesh using the volume of fluid (VOF) method [18]. The present CFD modeling procedure was validated against reference data that were provided by both high-resolution numerical solutions and experiments. The rest of this manuscript is organized as follows. After describing the overall computational model in Section2, the results of the numerical simulations are presented and discussed in Section3. Finally, in Section4, some concluding remarks are drawn. 2. Computational Model In this section, after introducing the industrial gas dynamics application under study, the details of the CFD model are provided, including the method used for tracking the liquid–gas interface and the CFD solver settings. 2.1. Case Study According to the two-dimensional approach, the simplified geometry of the shock tube device is represented by the rectangular domain sketched in Figure1 , where the planar shock, and thus the post-shock air flow, moves from left to right. The reference coordinates system (x, y) was chosen with the x-axis aligned with the streamwise direction, while Appl. Sci. 2021, 11, 4934 3 of 15 the origin corresponded to the leading-edge of the water cylinder in its initial position. Based on previous numerical simulations conducted for the same flow configuration, the spatial domain size was chosen such that 8 < x/d < 16 and y /d < 10, where d − 0 j j 0 0 stands for the initial diameter of the cylindrical water column, which represents the natural reference length. 10 5 0 -5 -10 -8 0 8 16 Figure 1. Sketch of the physical model with traveling shock wave. The shock tube was divided into two parts by means of a virtual cross diaphragm, which was initially located at x/d = 2 (upstream of the column). In the following, the 0 − gas conditions at the right and left sides of the tube are indicated by the subscripts 1 and 4, respectively. The two tube sections were initialized with the same temperature (T4/T1 = 1) and very different pressure and density levels (p /p = r /r 1), with the ideal gas 4 1 4 1 air being at rest on either side (V4 = V1 = 0). Starting from these initial conditions, a planar
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