  High-order gas-kinetic scheme (HGKS) has become a workable tool for the
direct numerical simulation (DNS) of turbulence. In this paper, to accelerate
the computation, HGKS is implemented with the graphical processing unit (GPU)
using the compute unified device architecture (CUDA). To conduct the much
large-scale DNS of turbulence, HGKS also be further upgraded with multiple GPUs
using message passing interface (MPI) and CUDA architecture. The benchmark
cases for compressible turbulence, including Taylor-Green vortex and turbulent
channel flows, are presented to assess the numerical performance of HGKS with
Nvidia TITAN RTX and Tesla V100 GPUs. For single-GPU computation, compared with
the parallel central processing unit (CPU) code running on the Intel Core
i7-9700 with open multi-processing (OpenMP) directives, 7x speedup is achieved
by TITAN RTX and 16x speedup is achieved by Tesla V100. For multiple-GPU
computation, the computational time of parallel CPU code running on 1024 Intel
Xeon E5-2692 cores with MPI is approximately 3 times longer than that of GPU
code using 8 Tesla V100 GPUs with MPI and CUDA. Numerical results confirm the
excellent performance of multiple-GPU accelerated HGKS for large-scale DNS of
turbulence. HGKS in GPU is also compiled with FP32 precision to evaluate the
effect of number formats precision. Reasonably, compared to the computation
with FP64 precision, the efficiency is improved and the memory cost is reduced
with FP32 precision. For turbulent channel flows, difference in long-time
statistical turbulent quantities is acceptable between FP32 and FP64 precision
solutions. While the obvious discrepancy in instantaneous turbulent quantities
can be observed, which shows that FP32 precision is not safe for DNS in
compressible turbulence. The choice of precision should depended on the
requirement of accuracy and the available computational resources.
