  The recently proposed high-order TENO scheme [Fu et al., Journal of
Computational Physics, 305, pp.333-359] has shown great potential in predicting
complex fluids owing to the novel weighting strategy, which ensures the
high-order accuracy, the low numerical dissipation, and the sharp
shock-capturing capability. However, the applications are still restricted to
simple geometries with Cartesian or curvilinear meshes. In this work, a new
class of high-order shock-capturing TENO schemes for unstructured meshes are
proposed. Similar to the standard TENO schemes and some variants of WENO
schemes, the candidate stencils include one large stencil and several small
third-order stencils. Following a strong scale-separation procedure, a tailored
novel ENO-like stencil selection strategy is proposed such that the high-order
accuracy is restored in smooth regions by selecting the candidate
reconstruction on the large stencil while the ENO property is enforced near
discontinuities by adopting the candidate reconstruction from smooth small
stencils. The nonsmooth stencils containing genuine discontinuities are
explicitly excluded from the final reconstruction, leading to excellent
numerical stability. Different from the WENO concept, such unique sharp stencil
selection retains the low numerical dissipation without sacrificing the
shock-capturing capability. The newly proposed framework enables arbitrarily
high-order TENO reconstructions on unstructured meshes. For conceptual
verification, the TENO schemes with third- to sixth-order accuracy are
constructed. Without parameter tuning case by case, the performance of the
proposed TENO schemes is demonstrated by examining a set of benchmark cases
with broadband flow length scales.
