Large eddy simulations were performed for a transonic turbine blade on hybrid unstructured meshes. High order accuracy in both space and time discretization is achieved by using a space time extension of flux reconstruction method(STEFR), and in particular, local time-stepping is enabled for time-accurate unsteady simulations. A piecewise integration in-cell method is introduced for shock-capturing within the STEFR method. This method is completely local in both space and time directions, without any explicit addition to the discretisation, which is suitable for the predictor-corrector type space time extension allowing local timestepping. A volume averaged compensation method is used to construct a conservative form of the Chain-Rule scheme for the flux divergence part of the flux reconstruction discretisation, which is proved stable and efficient to solve flow problems with physical discontinuities. This work successfully extends the application of the STEFR method to solve transonic/supersonic unsteady flows. The numerical validation for the 1D burgers equation and its comparison with traditional multi-stage Runge-Kutta method with uniform time-stepping, indicate that this piecewise integration in-cell method is accurate and more robust with STEFR method using local time-stepping.

A significant advantage of this method is the successfully implementation on hybrid unstructured meshes, which makes it very efficient and scalable to solve multi-scale, complex geometries flow problems. The moving shock, shock-boundary layer interaction, shock-wake interaction were crisply resolved and analyzed for the transonic turbine blade, VKI-LS59, and detail post-processing is presented with the comparison with experimental data in this paper.

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