Microchannels are important components of microelectromechanical systems (MEMSs) that encounter rarefaction effects due to their small-scale characteristics. The influence of rarefaction effects on the flow stability of microchannels should be investigated to improve MEMS performance. Based on kinetic theory, a linear stability analysis approach for low-speed rarefied flows was developed by using the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation with an external force term. This approach was applied to study the linear temporal stability of microchannel flows. A slip flow model was introduced for comparison. The corresponding eigenvalue problem was solved with a Chebyshev collocation method. This novel approach yielded a critical Reynolds number of 5778. Analysis of the validity and accuracy of the slip flow model shows that although this model cannot capture the Knudsen layer structure, this approach effectively improves the prediction accuracy of the growth rate of the least stable mode. However, the prediction accuracy gradually decreases with increasing Knudsen number. Compared with the stability results obtained from the BGK equation, the Navier–Stokes equations-based stability analysis method always underestimates the disturbance growth rate, regardless of whether a slip flow model is used. The stability analysis results show that rarefaction effects stabilize the flow. The degree of rarefaction does not affect the trends of growth rate and phase velocity with wavenumber, nor does it affect the shape of the velocity eigenfunctions. For a rarefied case, increasing the Mach number has a destabilizing effect on low-speed microchannel flows.

Adaptive Cartesian grids, often accompanied by non-uniform hanging grids, result in non-equidistant node distributions, posing numerical challenges for finite difference discretization. This work introduces a novel cell-based adaptive Cartesian grids solver for compressible flows using a finite difference method, combined with an immersed boundary method to effectively simulate various complex flows. The algorithm selects easily locatable and retrievable neighboring cells around the hanging grid as interpolation templates and employs distance-weighted least squares interpolation to achieve high-precision interpolation. Consequently, equidistant finite difference templates are constructed, enabling high-accuracy finite difference discretization in the transition regions between coarse and fine grids. The advantages of this algorithm include ease of integration with commonly used numerical schemes, flexibility, and implementation convenience. Accuracy tests with the Taylor–Green vortex and vortex convection confirm third-order precision at hanging grids. The algorithm's effectiveness is further demonstrated in laminar flat plate flow, Rayleigh–Taylor instability, cylinder flow, and supersonic flow around a high-aspect-ratio aircraft, showcasing its capability to accurately and efficiently simulate complex flow situations.

In the present study, we perform direct numerical simulations to investigate the spatial development and basic flow statistics in the supersonic turbulent boundary layers at the free-stream Mach number of 2.0 over smooth and disturbed walls, the latter of which enforces extra Reynolds shear stress in the streamwise direction to emulate the drag increment and mean streamline curvature effects of rough walls. Such disturbances escalate the growth rate of turbulent boundary layer thickness and the shape factor. It is found that under the rescaled global coordinate, the mean velocity, Reynolds stresses and pressure fluctuation variances manifest outer-layer similarity, whereas the average and fluctuation variances of temperature and density do not share such a property. Compressibility effects are enhanced by the wall disturbances, yet not sufficiently strong to directly impact the turbulent kinetic energy transport under the presently considered flow parameters. The generalized Reynolds analogy that relates the mean velocity and temperature can be satisfied by incorporating the refinement in modifying the generalized recovery coefficient, and that associates the fluctuating velocity and temperature works reasonably well, indicating the passive transport of temperature fluctuations. The dispersive motions are dominant and decay exponentially below the equivalent sand grain roughness height $k_s$, above which the wall disturbances are distorted to form unsteady motions responsible for the intensified density and pressure fluctuations in the free-stream travelling isentropically as the acoustic radiations.

In our originally published paper,1 Eq. (3.1) should read P − P e q e = F ( s ) 2 C f o ( M a 2 − 1 ) 1 / 2 (3.1)and Eq. (3.5) should read S e * = k Δ P q e = 2

The spray formation and breakup process in an open-end swirl injector were studied through experiments and numerical simulations. A high-speed shadowgraph system and a high-speed backlight system were adopted to record the spray. Volume of fluid was used as the interface tracking method to capture the evolution process. The filling process of the liquid film inside the injector was captured. The air core formation process as observed in the experiments differed from that depicted by the numerical simulation results. The results revealed that the spray pattern of the cross-section at the tangential inlets also varied during the filling process. The evolution of the holes on the liquid film and ligaments was observed. It was determined that the liquid sheet repeatedly exhibited thinning, instability, shedding, breakup, and coalescence in the spray formation and breakup process. The spray pattern underwent the distorted pencil stage, onion stage, tulip stage, and fully developed stage with the increased injection pressure drop. The formation process of the open-end swirl injector also underwent these four stages under an injection pressure drop of 0.5 MPa.