High-order finite-volume TENO schemes with dual ENO-like stencil selection for unstructured meshes

The TENO-family schemes [Fu et al., Journal of Computational Physics 305 (2016): 333-359] have been demonstrated to perform well for compressible gas dynamics and turbulent flow predictions on structured meshes. However, the extension of the TENO schemes to unstructured meshes is non-trivial and challenging, particularly when the multiple design objectives are pursued simultaneously, i.e., restoring the high-order accuracy in smooth regions, retaining the low numerical dissipation for small-scale features, maintaining the sharp shock-capturing property, and featuring the good numerical robustness for high-Mach flows. In this work, a family of very-high-order (up to seventh-order accuracy) robust finite-volume TENO schemes with dual ENO-like stencil selection for unstructured meshes is proposed. The stencils include one large stencil and several small stencils. The novelty originates from a so-called dual ENO-like stencil selection strategy. Following a strong scale separation, the ENO-like stencil selection procedure with a small $C_T$ is first enforced among all the candidates such that the high-order candidate scheme on the large stencil is adopted for the final reconstruction when the local flow is smooth. If the large stencil is judged to be crossed by discontinuities, a second ENO-like stencil selection with a relatively large $C_T$ is applied to all the left small stencils and the ENO property is obtained by selecting the smooth small stencils which are not crossed by discontinuities. The smaller $C_T$ in the first stage ensures that the high-order reconstruction is restored for smooth flow scales with higher wavenumbers. On the other hand, the larger $C_T$ in the second stage can enforce a strong nonlinear adaptation for capturing discontinuities with better robustness. Such a dual ENO-like stencil selection strategy introduces an explicit scale separation and deploys the optimal strategy for different types of flows correspondingly. Without parameter tuning, a set of benchmark simulations has been conducted to validate the performance of the proposed TENO schemes. Numerical results demonstrate the good numerical robustness and the low-dissipation property for highly compressible flows with shockwaves.

A scale-based study of the Reynolds number scaling for the near-wall streamwise turbulence intensity in wall turbulence

Very recently, a defect model which depicts the growth tendency of the near-wall peak of the streamwise turbulence intensity has been developed (Chen $\&$ Sreenivasan, J. Fluid Mech. (2021), vol.908, R3). Based on the finiteness of the near-wall turbulence production, this model predicts that the magnitude of the peak will approach a finite limit as the Reynolds number increases. In the present study, we revisit the basic hypotheses of the model, such as the balance between the turbulence production and the wall dissipation in the region of peak production, the negligible effects of the logarithmic motions on the wall dissipation, and the typical time-scale that the outer-layer flow imposes on the inner layer. Our analyses show that some of them are not consistent with the characteristics of the wall-bounded turbulence. Moreover, based on the spectral stochastic estimation, we develop a framework to assess the wall dissipation contributed by the energy-containing eddies populating the logarithmic region, and uncover the linkage between its magnitude and the local Reynolds number. Our results demonstrate that these multi-scale eddies make a non-negligible contribution to the formation of the wall dissipation. Based on these observations, we verify that the classical logarithmic model, which suggests a logarithmic growth of the near-wall peak of the streamwise turbulence intensity with regard to the friction Reynolds number, is more physically consistent, and still holds even with the latest high-Reynolds-number database.

Linear response analysis of supersonic turbulent channel flows with a large parameter space

In this work, the linear responses of the turbulent mean flow to both harmonic and stochastic forcing are investigated for the supersonic channel flow. Well-established universal relations are utilized to efficiently obtain the mean profiles with a large parameter space, with the bulk Mach number up to 5 and the friction Reynolds number up to \num{e4}, so a systematic parameter study is feasible. The most amplified structure takes the form of streamwise velocity and temperature streaks optimally forced by the streamwise vortices. The outer peak of the pre-multiplied energy amplification corresponds to the large-scale motion, whose spanwise wavelength ($\lambda_z^+$) is very insensitive to compressibility effects. In contrast, the classic inner peak representing the small-scale near-wall motions disappears for the stochastic response with an increasing Mach number. Meanwhile, the small-scale motions become much less coherent. A decomposition of the forcing identifies the different effects of the incompressible counterpart and the thermodynamic components. Wall-cooling effects, arising with the high Mach number, increase the spacing of the most amplified near-wall streaks; the spacing becomes nearly invariant with the Mach number if expressed in the semi-local units. Meanwhile, the coherence of the stochastic response with $\lambda_z^+>90$ is enhanced, but on the opposite, that smaller than 90 is decreased. The geometrical self-similarity of the response in the mid-$\lambda_z$ range is still roughly satisfied, insensitive to the Mach number. Finally, theoretical analyses of the perturbation equations are presented to help understand the scaling of the energy amplification.

A five-point TENO scheme with adaptive dissipation based on a new scale sensor

In this paper, a new five-point targeted essentially non-oscillatory (TENO) scheme with adaptive dissipation is proposed. With the standard TENO weighting strategy, the cut-off parameter $C_T$ determines the nonlinear numerical dissipation of the resultant TENO scheme. Moreover, according to the dissipation-adaptive TENO5-A scheme, the choice of the cut-off parameter $C_T$ highly depends on the effective scale sensor. However, the scale sensor in TENO5-A can only roughly detect the discontinuity locations instead of evaluating the local flow wavenumber as desired. In this work, a new five-point scale sensor, which can estimate the local flow wavenumber accurately, is proposed to further improve the performance of TENO5-A. In combination with a hyperbolic tangent function, the new scale sensor is deployed to the TENO5-A framework for adapting the cut-off parameter $C_T$, i.e., the local nonlinear dissipation, according to the local flow wavenumber. Overall, sufficient numerical dissipation is generated to capture discontinuities, whereas a minimum amount of dissipation is delivered for better resolving the smooth flows. A set of benchmark cases is simulated to demonstrate the performance of the new TENO5-A scheme.