A GPU-accelerated δ+-LES-SPH framework for simulating moderate-to-high-Reynolds-number turbulent flows with complex boundaries

Ocean engineering involves numerous moderate-to-high-Reynolds-number turbulent flows. To facilitate the applications of Smoothed Particle Hydrodynamics (SPH) to efficiently simulate complex turbulent flows in engineering problems, a Taichi-based GPU-accelerated SPH code based on the δ + -LES-SPH model is developed in this work. By simulating Taylor-Green flows in 2D and 3D (closed isotropic turbulent flow with Reynolds number Re up to 10 6 ), flow past a sphere (adverse-pressure-gradient flow with Re up to 10 4 ), and open channel flow (wall-bound turbulence with Re up to approximately 19620), the computational accuracy and efficiency of the present δ + -LES-SPH framework are quantitatively validated. In particular, turbulent flows in practical engineering applications involving violently deforming free-surface (flow past a vertical free-surface-piercing cylinder with Re up to 25640) and complex moving/deforming boundaries (manta ray swimming with Re up to 14800) are investigated. Results show that turbulent first-order and second-order moments (e.g., cascade slope, mean velocity, velocity RMS, Reynolds shear stress, and resolved turbulence kinetic energy) and turbulence characteristics (e.g., coherent vortex structures, hairpin vortices, and wake evolution) are effectively predicted by the present δ + -LES-SPH. The efficiency of the GPU implementation, which is highly enhanced and reaches a speedup of approximately 19-20 times compared to the CPU implementation (NVIDIA A100 compared with Intel Xeon Platinum 8358P, double-precision), is similar to that of the mesh-based Finite Volume Method (FVM) solver. In addition, the selection and effects of critical coefficients (e.g., the Smagorinsky coefficient C s and the density diffusion coefficient C δ ) of the δ + -LES-SPH model are discussed. Inspired by the practice of varying C s for different turbulent flows, we propose that the value of C δ be adjusted according to the specific turbulent flow under consideration, ranging from 1 to 6. Specifically, C δ = 6 is recommended for fully developed and isotropic turbulent flows, while C δ should be reduced to avoid excessive numerical dissipation for anisotropic turbulent flows (e.g., mean shear flows). Furthermore, the effect of the discrete scheme of the pressure gradient is discussed. In flows with a gravity field, the hydrostatic pressure induces spurious vorticity noise when using the classical scheme (a sum form: p j + p i ). A difference form ( p j - p i ) of the pressure gradient is necessary to avoid this issue.