# A Heterogeneous Parallel Algorithm for Euler-Lagrange Simulations of Liquid in Supersonic Flow
## 1 Introduction  
The scramjet engine is a hot topic in the research field of hypersonic vehicles [1; 2; 3; 4; 5]. Compared with gaseous fuel, liquid fuel has advantageous characteristics of high volumetric energy density, easy handling and pumping, convenient transportation, and a high degree of safety [6]. Thus, the applications of liquid fuel in scramjet engines are drawing increasing attention. Liquid fuel atomization and mixing have significant impact on the performance of scramjet engines. Crossflow jet is one way to improve the atomization and mixing characteristic [7; 8; 9]. However, such a process is highly complex. Under the action of the supersonic transverse airflow, the liquid fuel will go through multiple sub-processes such as fragmentation, atomization, mixing, etc. [10]. Therefore, a liquid jet in supersonic flow is a temporally and spatially multi-scale problem, which requires tremendous amounts of computational resources to solve numerically [11]. Numerous research has been conducted to investigate the spray structure [12; 13; 14], atomization properties [15; 16], and spray distribution [12; 13; 14] of the two-phase flow. [PERSON] et al. [17] employed a computation domain with 2.79 \(\times\) 10\({}^{6}\) grid points, divided into 132 blocks, to study the breakup and atomization of a water jet in a subsonic crossflow at different liquid-gas momentum flux ratios and cross-flow temperatures. [PERSON] et al. used direct numerical simulation via the Euler-Lagrange point-source approach to investigate the turbulent modulation of particles in compressible isotropic turbulence [18], the particle dispersion in a three-dimensional spatially developing compressible mixing layer [19]. To capture the small-scale structure, each case was computationally intensive. The experiments of [PERSON] et al. [20] and [PERSON] et al. [21] were recreated numerically by [PERSON] et al. [13; 22] using their in-house code to investigate the gas-liquid interaction and elucidate the mixing process. The number of cells in both cases was 19.4 million, while the number of computational droplets was about 1 million and 0.55 million, respectively. Thereafter, an investigation of a liquid jet in a cavity-based supersonic combustor was carried out both experimentally [23] and numerically [14]. The numerical simulation with 26.46 million grids and 5 million droplets was performed on 756 central processing unit (CPU) cores of the Tianhe-1 for 10 flow-throughs, running for approximately 31 days. Recently, [PERSON] et al. [24] added an evaporation model to further study the spray characteristics of kerosene jet in the cavity-based supersonic combustor. In the study, the number of cells and droplets was 18 million and 0.115 million, respectively. The simulation was also implemented on 600 CPU cores of the Tianhe-1 for 10 flow-throughs, running for 36 days. In conclusion, most of the simulations of jet in a crossflow were conducted using CPU-based high-performance computing, employing enormous numbers of cores and very long computational time. Therefore, a fast, efficient, and accurate computational method to simulate a liquid jet in supersonic flow is highly sought after. With the overall hardware performance and knowledge base increasing, GPU has become more prevalent for computational acceleration in recent years [25]. Fast computing speed, easy maintenance, and high-power efficiency make using graphics processing units (GPUs) for scientific computing an attractive option in many fields [26; 27]. [PERSON] et al. [28] proposed an improved hybrid Euler-Lagrange method for solving the Navier Stokes equation via GPU and performed a simulation of lid-driven cavity problem with 8 million grids. Speedups of 7, 14, 21, and 27 were obtained, respectively, for one to four GPUs, with respect to two Intel Xeon E5-2630 v2 CPUs (Intel Corporation, Palo Alto, CA, USA) with 12 cores. [PERSON] et al.