# A Heterogeneous Parallel Algorithm for Euler-Lagrange Simulations of Liquid in Supersonic Flow
## 1 Introduction
[29] ported and optimized a Fortran-based code and performed an accurate simulation of turbulent flow on a NVIDIA Tesla S2070 (NVIDIA Corporation, Santa Clara, CA, USA). The acceleration is about 22 times faster than one AMD Opteron 2352 Barcelona chip (AMD, Sunnyvale, CA, USA) and about 11 times faster than one Intel Xeon X5650 Westmere core (Intel Corporation, Palo Alto, CA, USA). [PERSON] et al. [30] ported existing code to multiple GPUs and achieved massive acceleration for the large eddy simulation (LES) of the planetary boundary layer. [PERSON] et al. [31] proposed a heterogeneous parallel computational fluid dynamics (CFD) solver and conducted a multi-parallel computation. The application of GPU acceleration in the numerical simulation of multiphase flow has also been studied. [PERSON] et al. [32] used four NVIDIA GPU devices to accelerate an existing particle-filled turbulence simulation code based on message passing interface (MPI). For 10\({}^{7}\) particles, calculation using GPU can achieve about 14 speedups compared to the original CPU execution. However, the method is one-way coupled, and the gas phase information needs to be transmitted to the devices for each iterative step, rendering the method impractical for large-scale simulations. [PERSON] et al. [33] developed a CPU/GPU portable software library Grit and combined it with a direct numerical simulation solver to simulate the particle-laden turbulent flow and the dilute turbulent jet evaporation. [PERSON] et al. [34] used more than 200 GPUs to achieve a quasi-real-time simulation of an industrial scale rotating drum. [PERSON] et al. [35] developed a CPU-GPU mixed computing mode, in which the gas phase was solved by CPUs and the particle was calculated by GPUs, and they applied this method to three cases with 2500, 5000, 1000 cells, and 32,000, 64,000, 128,000 parcels. Compared with the CPU method, respective speedups of 1.05, 1.44, and 2.37 were achieved. [PERSON] et al. [36] developed a GPU-accelerated model to accomplish the actual simulation of the entire toilet bowl system. At present, GPU acceleration technology has been widely used in multiphase flow, but mostly for certain expensive and parallelizable sections. To the authors' knowledge, the GPU technology is not applied to accelerate both the gas and liquid phase simultaneously. The current work is executed as a portion of the establishment of a universal CFD code for fast simulation of the full-field process of liquid in supersonic flow. To achieve it, a CPU/GPU heterogeneous parallel algorithm is built, in which a novel method for droplet dynamic management is proposed to solve the droplets' fluctuation in the flow field, and a droplet-locating approach is developed to conform with the weakened three-dimensional relationships of the cells. The purpose of this paper is to evaluate the internal two-phase CFD code, demonstrate the successful design of a GPU-based computing system, and compare the acceleration ratios compared to the CPUs. Although the two-phase CPU/GPU heterogeneous parallel computation mode based on the Euler-Lagrange approach proposed in this paper is used for liquid in supersonic flow, it is also applicable to other Euler-Lagrange simulations. This paper is organized as follows. The relevant physical models for the gas phase, liquid phase, phase-to-phase exchange terms, and code structure are introduced in Section 2. The computational condition, physical parameters, and performance of the algorithm are presented in Section 3, where we also present the key findings of this study. Conclusions are drawn in Section 4.