For compressible flow simulations involving both turbulence and shockwaves, the competing requirements render it challenging to develop high-order numerical methods capable of capturing the discontinuities sharply and resolving the turbulence with high spectral resolution. In particular when deployed with the advanced large-eddy simulation (LES) approach, for which the governing equations are solved with coarse meshes, the solution is extraordinarily sensitive to the numerical dissipation resulting in large uncertainties for cross-code comparisons. Similar sensitivities have also been observed for a wide range of complex fluid predictions, e.g., turbulent reacting flows, two-phase flows, and transitional flows. In this paper, the family of high-order targeted essentially non-oscillatory (TENO) schemes on both the Cartesian and unstructured meshes is reviewed for general hyperbolic conservation laws with an emphasis on the high-speed turbulent flows. As a novel variant of popular weighted ENO (WENO) scheme, the TENO scheme retains the sharp shock-capturing capability of WENO and is suitable for resolving turbulence with controllable low numerical dissipation. The key success of TENO relies on a strong scale-separation procedure and the tailored novel ENO-like stencil selection strategy. In addition, the built-in candidate stencils with incremental width facilitate the construction of arbitrarily high-order (both odd- and even-order) schemes featuring superior robustness. Detailed performance comparisons between WENO and TENO schemes are discussed comprehensively as well as the applications of TENO schemes to challenging compressible fluids. At last, the potential future developments to further boost the performance of TENO schemes from various perspectives are highlighted.
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