Direct numerical simulation (DNS) is a technique that solves the fluid Navier-Stokes equations directly without further modeling. The high spatial and temporal resolution makes DNS one of the most effective methods to advance turbulence research. However, it also hinders the application of DNS in high Reynolds number (Re) turbulence of particular interest, because the memory and computation requirements scale rapidly with Re^4. Recent studies have shown that developing efficient parallel methods for heterogeneous many-core systems is promising to solve this computational challenge. We propose four parallel methods and optimizations to develop a high-performance and scalable implicit finite difference solver called PowerLLEL, aiming to accelerate extreme-scale DNS of incompressible turbulence on the heterogeneous many-core system. Firstly, an adaptive multi-level parallelization strategy is proposed to fully exploit the multi-level parallelism and computing power of heterogeneous many-core systems. Secondly, hierarchical-memory-adapted data reuse/tiling strategy and kernel fusion are adopted to improve the performance of memory-bounded stencil-like operations. Thirdly, a parallel tridiagonal solver based on the parallel diagonal dominant (PDD) algorithm is developed to minimize the number of global data transposes. Fourthly, three effective communication optimizations are implemented by Remote Direct Memory Access (RDMA) to maximize the performance of the remaining global transposes and halo exchange. Results show that the solver exploits the heterogeneous computing power of the new Tianhe supercomputer, and achieves up to 10.6x speedup (against the CPU-only performance). Linear strong scaling is obtained up to a 25.8 billion problem size.