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High dissipative second-order schemes are widely used in Reynolds Averaged NavierStokes (RANS) simulations owing to the robustness, while they are harmful to the numerical accuracy for large eddy simulation (LES). It is well known that high dissipative numerical schemes can be as influential as the sub-grid scale (SGS) models . In the present paper, a new high-order numerical scheme for unstructured grids is proposed by the authors. The scheme is based on a central-type reconstruction, which is non-dissipative. A circulative gradient correction is proposed to extend the stencils of reconstruction without a large CPU/memory requirement. To achieve high-order accuracy, the free parameter is determined by expending a Taylor expression. The dispersion property of the scheme is optimized by minimizing the numerical errors in large eddy simulations. Furthermore, a hybrid method combining the proposed scheme with a weighted essentially non-oscillatory (WENO) scheme is constructed to make it possible for the discontinuity-capturing. Several benchmark test problems have been solved to validate the properties of the present schemes. The results of a linear wave advection problem indicate that the numerical error of the present schemes is reduced by 1∼2 orders of magnitude compared with the traditional second-order schemes. The better dispersion properties are shown by solving the transportation of inviscid vortex, with no lagging or advancing phase error observed. The high resolution for turbulent flow is demonstrated by LES of turbulent channel flow. At last, the ability for shock-wave capturing of the hybrid WENO scheme is validated by a forward step problem, with a clear thrice reflecting structure present.