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The present work elucidates the boundary behaviors of the velocity gradient tensor ($\bm{A}\equiv\bm{\nabla}\bm{u}$) and its principal invariants ($P,Q,R$) for compressible flow interacting with a stationary rigid wall. Firstly, it is found that the well-known Caswell formula exhibits an inherent physical structure being compatible with the normal-nilpotent decomposition, where both the strain-rate and rotation-rate tensors contain the physical effects from the spin component of the vorticity. Secondly, we derive the kinematic and dynamic forms of the boundary $\bm{A}$-flux from which the known boundary fluxes can be recovered by applying the symmetric-antisymmetric decomposition. Then, we obtain the explicit expression of the boundary $Q$ flux as a result of the competition among the boundary fluxes of squared dilatation, enstrophy and squared strain-rate. Importantly, we emphasize that both the coupling between the spin and surface pressure gradient, and the spin-curvature quadratic interaction, are \textit{not} responsible for the generation of the boundary $Q$ flux, although they contribute to both the boundary fluxes of enstrophy and squared strain-rate. Moreover, we prove that the boundary $R$ flux must vanish on a stationary rigid wall. Finally, the boundary fluxes of the invariants of the strain-rate and rotation-rate tensors are also discussed. It is revealed that the boundary flux of the third invariant of the strain-rate tensor is proportional to the wall-normal derivative of the vortex stretching term, which serves as a source term accounting for the the spatiotemporal evolution rate of the wall-normal enstrophy flux. These theoretical results provide a unified description of boundary vorticity and vortex dynamics, which could be valuable in understanding the formation mechanisms of complex near-wall coherent structures and the boundary sources of flow noise.

Traditionally, deriving aerodynamic parameters for an airfoil via Computational Fluid Dynamics requires significant time and effort. However, recent approaches employ neural networks to replace this process, it still grapples with challenges like lack of end-to-end training and interpretability. A novel and more efficient neural network is proposed in this paper, called AirfoilNet. AirfoilNet seamlessly merges mathematical computations with neural networks, thereby augmenting interpretability. It encodes grey-scale airfoil images into a lower-dimensional space for computation with Reynolds number, angle of attack, and geometric coordinates of airfoils. The calculated features are then fed into prediction heads for aerodynamic coefficient predictions, and the entire process is end-to-end. Furthermore, two different prediction heads, Gated Recurrent Unit Net(GRUNet) and Residual Multi-Layer Perceptron(ResMLP), designed to support our iteratively refined prediction scheme. Comprehensive analysis of experimental results underscores AirfoilNet's robust prediction accuracy, generalization capability, and swift inference.

From the Navier-Stokes-Korteweg (NSK) equations, the exact relations between the fundamental surface physical quantities for two-phase viscous flow with diffuse interface are derived, including density gradient, shear stress, vorticity, pressure, enstrophy flux and surface curvature. These theoretical results provide a solid foundation of the boundary/interfacial vorticity dynamics and a new tool for analysis of complex interfacial phenomena in two-phase viscous flows. To demonstrate the application of the developed results, simulation of a droplet impacting and spreading on a solid wall is conducted by using a recently developed well-balanced discrete unified gas kinetic scheme (WB-DUGKS), focusing on spreading process when the separation bubbles form inside the droplet. The distributions of shear stress, pressure and enstrophy flux at the interface and wall are analyzed, particularly near the moving contact points and other characteristic points. This example gives an unique perspective to the physics of droplet impingement on a wall.

In this paper, we introduce a new physical concept referred to as the wall-normal Lamb dilatation flux (WNLDF), which is defined as the wall-normal derivative of the Lamb dilatation (namely, the divergence of the Lamb vector) multiplied by the dynamic viscosity for incompressible viscous flows. It is proved that the boundary Lamb dilatation flux (BLDF, namely the wall-normal Lamb dilatation flux at the wall) is determined by the boundary enstrophy flux (BEF) and the surface curvature-induced contribution. As the first step to explore this new concept, the present study only considers flow past a stationary no-slip flat wall without curvature-fluid dynamic coupling effects. It is found that the temporal-spatial evolution rate of the WNLDF is contributed by four source terms, which can be explicitly expressed using the fundamental surface quantities including skin friction (or surface vorticity) and surface pressure. Therefore, the instantaneous near-wall flow structures are directly related to the WNLDF in both laminar and turbulent flows. As an example, for the turbulent channel flow at $Re_{\tau}=180$, the intensification of the temporal-spatial evolution rate of the WNLDF is caused by the strong wall-normal velocity event (SWNVE) associated with quasi-streamwise vortices and high intermittency of the viscous sublayer. In addition, near the SWNVE, this evolution rate is mainly contributed by the coupling between skin friction and the boundary enstrophy gradient, as well as the coupling between the skin friction divergence and the boundary enstrophy. The exact results presented in this paper could provide new physical insights into complex near-wall flows.