Shock wave interaction with a transcritical fuel droplet

Shock-droplet interactions occur in a spectrum of high-speed propulsion systems involving liquid fuels. When the combustion chamber pressure is above the critical pressure of the fuel, transcritical behaviour involving the transition from liquid-like to gas-like states is observed. Our understanding of multiphase-shock interaction is significantly less developed than its gasphase counterpart (i.e., shock-bubble interaction) and is particularly limited at transcritical conditions. We consider the interaction of a shockwave with a liquid n-dodecane droplet exposed to a nitrogen environment at a supercritical pressure. A fully-conservative diffuseinterface framework coupled with the Peng-Robinson equation of state is developed to accurately determine the state of the fluid as the shock propagates through the droplet. The shock-droplet interaction results show the development of interfacial instabilities and an axial

Shock-bubble and shock-droplet interactions have been the subject of many studies over the past decades [30][31][32]. The seminal work of Haas and Strutevant [30] revealed very complex phenomena occurring during shock interaction with helium and R22 refrigerant bubble. Later, these cases were numerically modelled highlighting the feature of the shock-bubble interactions [31]. Since then, many additional cases have been considered, including the bubble composition of SF6 and Krypton [33][34][35][36]. However, these studies are all conducted at atmospheric pressure and room temperature where the fluids are in a gaseous state. Understanding the disintegration of droplets impacted by shocks at supercritical conditions is relevant to liquid-fuelled scramjet engines during low hypersonic, i.e., start-up operations, and will inform the mixing and combustion behaviour of liquid fuel sprays in a spectrum of supercritical and high-speed propulsion systems, particularly in hypersonic flights.
Although a better understanding of the sub-and supercritical conditions has been obtained by numerical and experimental efforts, there is still a critical gap in the knowledge on the transcritical conditions where the transition from liquid-like to gas-like behaviour occurs by crossing the pseudo-boiling line [37,38]. At transcritical conditions, both subcritical (twophase) and supercritical (diffusion-controlled) behaviours might emerge [15,20]. Classical transcritical cases involve fuel injected at a pressure above the critical pressure. In the case of fuel injection into a combustion chamber, the fuel with temperature initially below the critical temperature is injected at supercritical pressure, mixes with the hot ambient air, and increases the temperature of the fuel. As the fuel is heated, it transitions from a liquid-like to gas-like supercritical fluid and crosses the Nishikawa Widom line (termed pseudo-boiling). At a highertemperature, the fuel will behave as an ideal gas when the compressibility factor is equal to unityi.e., n-dodecane at 1100 K and 6 MPa. For the design of high-pressure and high-speed propulsion systems, it is of great interest to understand the behaviours of fuel droplets impacted by shockwave at pressures above the critical pressure. The lack of knowledge on multiphase shock-driven instabilities due to the dearth of detailed experimental data at such high pressure (temperature) is the main motivation behind this computational study that focuses on the shock interaction with fuel droplets at supercritical pressures.
There are many reports on simulation of droplet/shock interactions using sharp-interface and diffuse-interface approaches at subcritical conditions [39][40][41][42][43][44]. However, there are only two reports that simulate the fuel droplet-shock interaction at transcritical conditions [45,46]. To the authors' knowledge, the underlying physics behind this problem have not been previously investigated. In the study of our previous work [46], we briefly considered the shock-interaction with an n-dodecane droplet in a nitrogen environment at near-critical conditions. It is important to note that the physics associated with a fluid near the critical point is significantly different from conventional liquid droplets or ideal-gas bubbles. For the shock-bubble interaction problem, the sphere of fluid is referred to as a bubble because it is in a gaseous state, typically an ideal gas. The term droplet is typically associated with a liquid sphere with significant surface tension effects. However, this differentiation breaks down for a supercritical fluid where gas-like and liquid-like properties better represent the state of the fluid [47]. Additionally, at supercritical conditions, surface tension is typically neglected. For this study, however, we use the term 'droplet' for the fuel sphere. In the present paper, we present the shock interaction with the supercritical fuel droplet by visualizing the shock characteristics and instability development.

Methodology
We consider a fully compressible (FC), inviscid, multiphase model. The governing equations for a diffuse-interface two-species system are given below: where D Y is the mass fraction of the bubble/droplet fluid species,  is the density, u is the velocity vector, p is the pressure, E is the total energy ( ) 2 2 Ee  =+ u , e is the internal energy, and I is the identity matrix. The mass fraction of the second species of the twocomponent system is given by the mixture rule 1 SD YY =− . The system of equations, Eqs. (1) -(4), is closed using the PR-EoS [48] as follows where T is the temperature, M is the molar mass, and a and b are coefficients that depend on the state and composition of the fluid. The PR-EoS is modified [46] to approximate the state of the fluid in the vapor dome region to increase the robustness of the numerical method. The parameters for the NASA polynomials that are used to determine the internal energy, enthalpy, and entropy were taken from Ref [49] This study uses an in-house, density-based, finite volume solver in C++ parallelized using domain decomposition and message passing interface (MPI) which is presented in [46]. In the same study, we introduced a hybrid (HY) numerical method, which hybridizes the FC procedure with the DF method [46]. HY method is chosen for this study as our earlier study proved, the HY method can significantly reduce the magnitude of pressure oscillation while limiting the loss of energy conservation when a shock impacts the nitrogen-n-dodecane interface. To avoid the interaction of the characteristic fields and increase the robustness and stability of the solver, we reconstruct the primitive variables in the characteristic space -Characteristic-wise reconstruction (CW) [46,[50][51][52][53].
WENO5-CW reconstruction is performed in the axisymmetric cylindrical coordinatesr, z (see Appendix C of [58] for more details). We also apply the artificial interface thickening method as discussed in our earlier study [46] to the initial condition of the droplet interface in axisymmetric cylindrical coordinates with 2  = , where  is the number of finite volume cells the interface is initially smeared over. The interface thickening reduces the jump in properties between cells, reducing pressure oscillations and increasing the numerical stability [46]. Additionally, interface thickening reduces the loss in energy conservation when used with a quasi-conservative method like the HY method [46]. Because the droplet is defined on a uniform Cartesian mesh, this interfacial thickening also helps provide the smooth curvature of the initial droplet shape.

Results and Discussion
We consider the interaction of a shock wave in a nitrogen environment with a droplet of ndodecane at supercritical pressure in a liquid-like state to showcase the transcritical shock interaction in 2D axisymmetric cylindrical coordinates. In this study, we focus on the ndodecane/nitrogen interface as n-dodecane has been extensively used as a diesel fuel surrogate in the combustion community.
The shock wave travels from right to left impacting the droplet, where the droplet diameter is 5 cm, see Figure 1. The computational domain is shown in Figure 1 where the axial length is 50 cm (from z=-25 to z=25cm) and with a radius of 30 cm (which is approximately 0 6D as is the suggested dimension in Ref. [59]). The size of the computational domain is sufficiently large so that potential erroneous reflections from the non-reflective boundaries will not interact with the droplet throughout the simulation. The computational domain consists of a uniform mesh from 16 to 16 cm z =− and from the axis of symmetry to 5 cm r = to increase the computational speed of the simulation where the cell dimensions are 0.23 mm zr     . The mesh then grows at a rate of 10% in the z-direction to the outer boundaries. Due to the axis of symmetry boundary condition, only the top half of the depicted computational domain is required for the simulation (Figure 1). A transmissive BC is used at the other boundaries (left, right, and outer boundaries). Here we show the key flow features of the shock interaction with a fuel droplet where the droplet is initially at a pressure of 6 MPa and a temperature is 650 K. The critical temperature of n-dodecane is 658.1 K and the critical pressure is 1.82 MPa. In this case, the n-dodecane is at a supercritical pressure near the pseudo-boiling line (close to the critical temperature). For a Mach 1.2 shock wave, the initial state of the three regions, (1) pre-shock, (2) post-shock, and (3) Note that z u is the velocity component in the z-direction and the initial velocity in the r-direction ( ) r u is zero everywhere. The shock wave, which is initially at z=10 cm, travels to the left towards the droplet and t begins when the shock first impacts the droplet interface (at z=7.5 cm).
Once the shock impacts the droplet, the vertical incident shock above and below the droplet is joined by two shock waves: (1) the refracted shock that lies within the droplet and (2) the reflected shock that is outside the droplet (Figure 2(a)). As time progresses, the refracted shock travels through the undisturbed region and converges towards the downstream interface (Figure 2(b)). The passage of the refracted shock across the downstream interface results in a transmitted shock and the reflection of a rarefaction wave, Figure 2(c). The acoustic impedance mismatch experienced by the refracted shock at the downstream interface results in the internal reflection of a rarefaction wave [32]. As time progresses, interfacial instabilities begin to grow at the interface between the fuel and the surrounding nitrogen, Figure 2(d). After the shock-droplet interaction, an axial jet develops outwards at the downstream interface, Figure 2  The axial jet is a result of the refracted shock wave convergence (Figure 3). The refracted shock convergence reaches a focal point just inside the droplet near the downstream interface. Figure 3(a) shows the point in time where the refracted shock has fully wrapped around the undisturbed region and is near the focal point. Figure 3(b) shows the axial pressure distribution before shock convergence, where the leftward traveling shock has a significantly higher pressure of about 18 MPa. There is no more undisturbed region once the shock converges ( Figure 3(c)) and the peak pressure at the axis near the downstream interface reaches about 30 MPa (Figure 3(d)). After the shock convergence, the shock wave is reflected as a strong rarefaction wave as seen in Figure 3(e)-(f) where the pressure drop is large (about 5 MPa).

Conclusions
In this paper, we developed a model to simulate the shock wave interaction with a fuel droplet at supercritical pressure. The shock-droplet resulted in various interesting flow features, including axial jetting and interfacial instability. The model developed will be used in future studies to further investigate the physics of the transcritical shock-droplet interaction.