The velocity gradient tensor (VGT) has been extensively employed to describe many fundamental and intrinsic spatial characteristics of turbulence. However, the temporal evolution characteristics following fluid particles are crucial for understanding the dynamics of turbulence. Therefore, we utilize the acceleration gradient tensor (AGT) including both temporal evolution and spatial characteristics of flow as a tool to probe turbulence. Our focus is directed towards the flow characteristics described by three principal invariants of the AGT. It is confirmed that the AGT principal invariants include not only the spatial characteristics described by VGT principal invariants, but also the evolution features such as the temporal evolutions of dilatation, strain rate, and rotation rate. Moreover, the statistical analysis of AGT principal invariants is conducted in compressible channel flows to discern the flow characteristics affected by the compressibility. The distributions of normalized AGT principal invariants with different bulk Mach numbers, $Ma_b$, collapse together far from the viscous sublayer but exhibit significant differences in the viscous sublayer. These differences are attributed to the intense temporal evolutions of dilatation, strain rate, and rotation rate at high $Ma_b$, induced by the generation of alternating positive and negative structures (APNSs) near the wall in high-speed flows with strong compressibility. By examining the principal invariants of filtered AGT, it is confirmed that the compressibility effect on flow characteristics near the wall becomes negligible as the flow scale increases. We further prove that AGT is determined by the spatial non-uniformities of the density and the force acting on the fluid particles. Hence, the compelling spatial non-uniformity of the force in the viscous sublayer has been identified as the primary dynamics driving the intense temporal evolutions and the generation of APNSs at high $Ma_b$.

The skin-friction coefficient is a dimensionless quantity defined by the wall shear stress exerted on an object moving in a fluid, and it decreases as the Reynolds number increases for wall-bounded turbulent flows over flat plate. In this work, a novel transformation, based on physical and asymptotic analyses, is proposed to map the skin-friction relation of high-speed turbulent boundary layers (TBLs) to the incompressible skin-friction relation. Through this proposed approach, it has been theoretically confirmed that the transformed skin-friction coefficient $C_{f,i}$ and transformed momentum-thickness Reynolds number $Re_{\theta,i}$ for compressible TBLs with and without heat transfer, follow a general scaling law that aligns precisely with the incompressible skin-friction scaling law, expressed as $\left(2/C_{f,i}\right)^{1/2}\propto\ln Re_{\theta,i}$. Furthermore, the reliability of the skin-friction scaling law is validated by compressible TBLs with free-stream Mach number ranging from $0.5$ to $14$, friction Reynolds number ranging from $100$ to $2400$, and the wall-to-recovery temperature ratio ranging from $0.15$ to $1.9$. In all of these data, $\left(2/C_{f,i}\right)^{1/2}$ and $\ln Re_{\theta,i}$ based on the present theory collapse to the incompressible relation, with a squared Pearson correlation coefficient reaching an impressive value of $0.99$, significantly exceeding $0.85$ and $0.86$ based on the established van Driest II and the Spalding-Chi transformations, respectively.

For compressible flow simulations involving both shock waves and turbulence, the competing requirements render it challenging to develop high-order numerical methods capable of capturing the discontinuities sharply and resolving the turbulence with high spectral resolution. In this paper, an efficient class of high-order TENO schemes with local adaptive dissipation for compressible flow simulation on unstructured meshes is proposed based on three new concepts: (1) a novel reliable troubled-cell indicator is proposed for the unstructured finite-volume method without case-sensitive parameter to tune; (2) different from the classical shock-capturing schemes for unstructured meshes, which conduct characteristic decomposition at each cell interface, an efficient hybrid weighting strategy is proposed by recasting the high-order linear scheme based on conserved variables for smooth flow scales and invoking the nonlinear TENO weighting process in characteristic space for non-smooth flow scales; (3) noticing that the low-order undivided difference deployed in the calculation of the new indicator is more effective in terms of separating the high-wavenumber fluctuations from the genuine discontinuities than the high-order difference, a new adaptive dissipation control strategy is introduced to combine the good numerical robustness for shock waves with the low-dissipation property for broadband physical fluctuations. Without the necessity of parameter tuning case by case, a set of benchmark simulations reveals that the proposed TENO-E scheme features robust shock-capturing capability and state-of-the-art high-resolution properties for highly compressible flows involving strong shock waves and a wide range of flow scales. Moreover, the proposed scheme is substantially less computationally expensive than the straightforward deployment of classical shock-capturing schemes, and thus is promising for high-fidelity DNS/LES simulation of more complex practical engineering flows.

The well-known quadratic temperature--velocity (TV) relation is significant for physical understanding and modelling of compressible wall-bounded turbulence. Meanwhile, there is an increasing interest in employing the TV relation for laminar modelling. In this work, we revisit the TV relation for both laminar and turbulent flows, aiming to explain the success of the TV relation where it works, improve its accuracy where it deviates, and relax its limitation as a wall model for accurate temperature prediction. We show that the general recovery factor defined by Zhang et al. (J. Fluid. Mech., vol.739, pp.392--440, 2014) is not a wall-normal constant in most laminar and turbulent cases. The effective Prandtl number $\Prd_e$ is more critical in determining the shape of temperature profiles. The quadratic TV relation systematically deviates for laminar boundary layers irrespective of Mach number and wall boundary conditions. We find a universal distribution of $\Prd_e$, based on which the TV relation can be notably improved, especially for cold-wall cases. For turbulent flows, the TV relation as the wall model can effectively improve the near-wall temperature prediction for cold-wall boundary layer cases, but it involves boundary-layer-edge quantities used in the Reynolds analogy scaling, which hinders the application of the wall model in complex flows. We propose a transformation-based temperature wall model by solving inversely the newly-developed temperature transformation of Cheng and Fu (Phy. Rev. Fluids, vol.9, no.054610, 2024). The dependence on edge quantities is thus removed in the new model, and the high accuracy in turbulent temperature prediction is maintained for boundary layer flows.

In this work, an implicit moving-least-squares immersed boundary method (MLS-IBM) is proposed to accurately enforce the velocity boundary condition on immersed objects. This method effectively eliminates errors induced by the inequality between the interpolation and spreading operations, while preserving the conservation of the force and torque. The instantaneous discretization errors for the velocity boundary conditions are negligible, approaching machine round-off levels, which render the proposed method much more accurate than previous MLS-IBMs. In terms of computational efficiency, the proposed implicit MLS-IBM outperforms the explicit variant MLS-IBM for stationary problems and shows comparable performance for moving-boundary problems. Additionally, the assembly of the correlation matrix in the implicit MLS-IBM is optimized to improve the computational efficiency, making it superior to previous implicit IBMs. The proposed implicit MLS-IBM integrated with the lattice Boltzmann flux solver can achieve second-order spatial accuracy through a mesh-refinement study. The robustness and accuracy of the proposed implicit MLS-IBM are validated through several complex fluid-structure interaction (FSI) problems involving complex geometries, moving boundaries, and large deformations.