Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been
developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful
for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for
these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values
from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

Mean temperature scalings in compressible wall turbulence

In the present study, we report a new Mach number invariant function ($\phi_{S}$) for the mean temperature field in compressible wall turbulence. We demonstrate its validation by comparing it with the invariant functions derived in the previous studies, i.e., the semi-local-type ($\phi_{SL}$) and van-Driest-type ($\phi_{VD}$) scalings, case by case. To be specific, $\phi_{SL}$ works well in the inner layer of compressible channel flows with isothermal walls; $\phi_{S}$ works well in the inner layer of compressible channel flows with adiabatic walls, and supersonic/hypersonic turbulent boundary layers with cold walls; $\phi_{VD}$ does not work the best among all three functions in the flows under consideration. The newly proposed temperature transformations based on $\phi_{S}$ show an improvement in channel flows over adiabatic walls and supersonic/hypersonic turbulent boundary layers with cold walls. The effects of the generated high-order terms during derivation are also clarified. These findings may be revealing for the development of the near-wall model in high-speed aerodynamics.

A new high-order shock-capturing TENO scheme combined with skew-symmetric-splitting method for compressible gas dynamics and turbulence simulation

The high-order shock-capturing scheme is one of the main building blocks for the simulation of the compressible fluid characterized by strong shockwaves and broadband length scales. However, the classical shock-capturing scheme fails to perform long-time stable and non-dissipative simulations since the quadratic invariants associated with the conservation equations cannot be conserved as a result of the inherent numerical dissipation. Additionally, the overall computational cost for classical shock-capturing schemes is quite expensive as a result of the time-consuming local characteristic decomposition and the nonlinear-weights computing process. In this work, based on a new efficient discontinuity indicator, which distinguishes the non-smooth high-wavenumber fluctuations and discontinuities from smooth scales in the wavenumber space, a paradigm of high-order shock-capturing scheme by recasting the non-dissipative skew-symmetric-splitting method with newly optimized dispersion property for smooth flow scales and invoking the nonlinear targeted ENO (TENO) schemes for non-smooth discontinuities is proposed. The resulting TENO-S scheme not only successfully performs long-time stable computations for smooth flows without numerical dissipation, but also recovers the robust shock-capturing capabilities with adaptive numerical dissipation. Without the necessity of parameter tuning case by case, extensive benchmark simulations involving a wide range of flow length scales and strong discontinuities demonstrate that the proposed TENO-S scheme performs significantly better than the straightforward deployment of WENO/TENO-family schemes with better spectral property and higher computational efficiency.