We have established a novel molecular kinetic model that addresses fundamental challenges in the non-equilibrium transport of nanoscale confined fluids, such as rarefaction and fluid inhomogeneities, which are crucial to a range of scientific and engineering fields. The proposed model explicitly considers fluid–solid molecular interactions in the transport equations, eliminating the reliance on predefined boundary conditions. By consistently accounting for molecular interactions between fluids and solids, the unified model captures both intrinsic and apparent non-hydrodynamic effects, as well as real fluid behaviours. Rigorous comparisons with molecular dynamics simulations demonstrate that the present model accurately predicts unique features of strongly inhomogeneous fluid flows, including fluid adsorption, solvation force, velocity slip and temperature jump. Therefore, this mesoscopic model bridges the gap between molecular-scale dynamics and macroscopic hydrodynamics, enabling a practical simulation tool for nanoscale surface-confined flows. Moreover, it offers valuable insights into the molecular mechanisms underlying anomalous transport phenomena observed in confined flows, such as the disappearance and re-emergence of the Knudsen minimum.

Large-eddy simulations have been conducted to investigate the decay law of homogeneous turbulence influenced by a magnetic field within a cubic domain, employing periodic boundary conditions. The initial integral Reynolds number is approximately 1000, while the initial interaction number $N$ ranges from 0.1–100. The results reveal that the Joule cone angle $\theta$, half of the Joule cone, decays as $\cos \theta \sim t^{-1/2}$ when $N \gg 1$. In the nonlinear stage, small-scale vortices gradually recover and restore three-dimensionality. Moreover, the corresponding critical state at small scales, marking the transition from quasi-two-dimensional structure to the onset of three-dimensionality, has been quantitatively defined. During the linear stage, based on the true magnetic damping number ($\tau _t = \rho / (\sigma {\boldsymbol{B}}^2 \cos ^2 \psi )$, where $\sigma$, $\boldsymbol{B}$ and $\psi$ denote the electrical conductivity, magnetic field and the angle between the wavevector and $\boldsymbol{B}$ in Fourier space, respectively), Moffatt’s decay law, $K \sim t^{-1/2}$, manifests at distinct times and zones in the Fourier space, with $K$ signifying turbulent kinetic energy. In the nonlinear stage, for $N \gg 1$, a $-3$ slope in the energy power spectrum is prominently observed over an extended period. The near-equivalence of the characteristic time scales of inertial and Lorentz forces in the inertial subrange suggests a quasiequilibrium state between energy transfer and Joule dissipation in Fourier space, thereby corroborating the hypothesis proposed by Alemany et al. 1979 Journal de Mecanique 18(2): 277–313. Additionally, it is observed that pressure mediates energy transfer from horizontal kinetic energy ($K_{\parallel }$) to vertical kinetic energy ($K_{\bot }$), accelerating the decay of $K_{\parallel }$. Notably, concurrent inverse and direct energy transfers emerge during the decay process. Our analysis reveals that the ratio $R$ of the maximum inverse to maximum direct energy flux correlates with the dimensionality of the turbulence, following the scaling law $R\sim (\cos \theta )^{-2.2}$.

Flow over bluff bodies encounters instability at supercritical Reynolds numbers, exhibiting the periodic vortex shedding that leads to structural vibrations and acoustic noise. In this paper, a new aerodynamic shape optimisation strategy based on resolvent analysis is proposed to passively control the vortex shedding over two-dimensional cylinders. Firstly, we show that when the flow satisfies the rank-1 approximation, minimizing the maximal resolvent gain enhances flow stability. Secondly, we formulate the geometry-constrained resolvent-based optimisation problem that can be solved by the nonlinear conjugate gradient algorithm. Compared with conventional stability-based optimisation, the proposed approach is more effective as it avoids the cumbersome eigendecomposition of the high-dimensional Jacobian matrix. The efficacy of the proposed resolvent-based optimisation is validated through improving the stability of the one-dimensional Ginzburg–Landau equation. Thirdly, this approach is applied to suppress the vortex shedding of bluff bodies, initialised by a circular cylinder. Although the optimisation is performed at a subcritical state $Re = 40$, reduced vortex shedding and drag forces can be achieved at supercritical Reynolds numbers, while the critical Reynolds number is extended from $47$ to $60$. Dynamic mode decomposition is then performed to reveal that the optimised system becomes more stable and satisfies the rank-1 approximation. Finally, we demonstrate that the combined effects of the flattened surface and the Coanda effect delay flow separation, keeping the separation point nearly unchanged at supercritical Reynolds numbers (e.g. between 80 and 140) for the optimised geometry. This results in a substantial reduction in the strength of vortex shedding, which in turn leads to decreased drag forces. The optimised shape still achieves drag reduction in turbulent flows at a relatively high Reynolds number.

We derive boundary conditions for two-dimensional parallel and non-parallel flows at the interface of a homogeneous and isotropic porous medium and an overlying fluid layer by solving a macroscopic closure problem based on the asymptotic solution to the generalised transport equations (GTE) in the interfacial region. We obtained jump boundary conditions at the effective sharp surface dividing the homogeneous fluid and porous layers for either the Darcy or the Darcy–Brinkman equations. We discuss the choice of the location of the dividing surface and propose choices which reduce the distance with the GTE solutions. We propose an ad hoc expression of the permeability distribution within the interfacial region which enables us to preserve the invariance of the fluid-side-averaged velocity profile with respect to the radius $r_0$ of the averaging volume. Solutions to the GTE, equipped with the proposed permeability distribution, compare favourably with the averaged solutions to the pore-scale simulations when the interfacial thickness $\delta$ is adjusted to $r_0$. Numerical tests for parallel and non-parallel flows using the obtained jump boundary conditions or the generalised transport equations show quantitative agreement with the GTE solutions, with experiments and pore-scale simulations. The proposed model of mass and momentum transport is predictive, requiring solely information on the bulk porosity and permeability and the location of the solid matrix of the porous medium. Our results suggest that the Brinkman corrections may be avoided if the ratio $a=\delta /\delta _B$ of the thickness $\delta$ of the interfacial region to the Brinkman penetration depth $\delta _B$ is large enough, as the Brinkman sub-layer is entirely contained within the interfacial region in that case. Our formulation has been extended to anisotropic porous media and can be easily dealt with for three-dimensional configurations.