Several 2D compressible vortex models, proposed in the literature to investigate vorticity dynamics or to assess the accuracy of numerical methods, are reviewed and reformulated with a common non-dimensional notation. Two vortex types may be distinguished, namely the zero-global circulation vortex and the finite-global circulation vortex, where the global circulation amounts to the circulation value around a far-field path enclosing the vortex core. Numerical experiments carried out on the convected and stationary vortex benchmark tests are analyzed to clarify the error dynamics during the flow evolution in time. It is shown that both classes of compressible vortex models are affected, on the long term, by a flow instability of acoustic type. It is then concluded that this classical problem may be used to assess the spatial accuracy of a numerical scheme, through a standard h-refinement approach, only with the necessary caution.
This paper develops a two-relaxation-time lattice Boltzmann method (TRT-LBM) framework for simulating electro-thermo-convection (ETC) driven by conduction-based charge transport in a two-dimensional sidewall-heated square cavity. The coupled flow, temperature, and charge density fields are resolved using the TRT-LBM, while the electric potential is computed via a fast Fourier transform (FFT) based Poisson solver with domain extension. Numerical validations under hydrostatic and convective conditions demonstrate strong agreement with analytical and literature results. Parametric studies on the conduction number (), Rayleigh number (), and ion mobility ratio () reveal that the electric field suppresses natural convection by modifying the structure and thickness of heterogeneous charge layers. In the ohmic regime, a lower leads to stronger suppression of flow and heat transfer. At low Rayleigh numbers, the coupling between the electric body force and buoyancy within the charge layer can even induce symmetric secondary vortices. Furthermore, heat transfer suppression initially becomes more pronounced but gradually stabilizes as increases, as indicated by the decreasing relative Nusselt number . Within the investigated parameter range, the strongest suppression occurs at and , where . When the ion mobility ratio deviates from unity, the asymmetry introduced by unequal ion mobilities significantly alters the flow structure and heat transfer performance. As increases from 0.5 to 2.0, the average Nusselt number decreases from 2.49 to 2.21, corresponding to a reduction of approximately .
When solving partial differential equations using finite-difference methods, near-boundary accuracy is often limited by truncation of finite-difference stencils and limitations of boundary treatments. These limitations often lead to local order reduction, making it challenging to maintain globally consistent second-order accuracy. To address these challenges, this paper proposed a constrained matrix reconstruction optimization (CMRO) method with an implicit wall constraint. The method employs a matrix-based simultaneous solution strategy that integrates physical boundary constraints and interior discrete equations into a unified coupled system while incorporating a ghost-point technique to preserve stencil compactness. Results indicate that, within a non-iterative framework, the proposed method helps maintain second-order accuracy near boundaries in the tested cases. Timing analysis shows that the additional cost is confined to wall nodes and remains modest relative to the overall time-marching cost. Numerical examples demonstrate the method's accuracy consistency, robustness, and bounded computational overhead, offering a practical approach for high-precision simulations with complex boundaries and demonstrating its value for engineering applications.
ABSTRACT The impact dynamics of droplets on a single rectangular superhydrophobic ridge are systematically investigated using the lattice Boltzmann method. The effects of ridge width, droplet deviation, and Weber number on droplet deformation and breakup are examined through extensive numerical simulations. The results show that kinetic and surface energies undergo continuous interconversion, while the vertical momentum exhibits a characteristic three‐stage evolution and vorticity is primarily concentrated near the gas‐liquid interface and ridge corners. Depending on the impact conditions, droplets exhibit symmetric or asymmetric deformation and undergo either non‐breakup or breakup into two or three secondary droplets. The breakup behavior is governed by the competition between inertial effect and surface tension, together with the resulting stress concentration near the ridge corners. In general, lower Weber numbers and larger ridge widths favor non‐breakup, whereas higher Weber numbers and larger droplet deviations promote breakup, with deviation favoring breakup into two secondary droplets. Phase diagrams further reveal narrow transition regions between different impact outcomes, enabling the corresponding critical boundaries to be quantitatively described by linear relationships. These findings provide a mechanistic framework for understanding and predicting droplet breakup on structured surface.
This study develops invariance-embedded machine learning (ML) sub-grid-scale (SGS) stress models admitting turbulence kinetic energy (TKE) backscatter toward more accurate large eddy simulation (LES) of meso-scale turbulent hurricane boundary layer flows. The new ML SGS model consists of two parts: a classification model used to distinguish regions with either strong energy cascade or energy backscatter from those with mild TKE transfer and a regression model used to calculate SGS stresses in regions with strong TKE transfer. To facilitate model implementation in computational fluid dynamics (CFD) solvers, the Smagorinsky model with a signed coefficient , where a positive value indicates energy cascade and a negative one indicates energy backscatter, is employed as the carrier for the ML model. To improve its robustness and generality, both physical and geometric invariance features of turbulent flows are embedded in the model input for classification and regression, and the signed Smagorinsky model coefficient is used as the regression model's output. All model assessments in this work are carried out in an a priori setting against filtered high-fidelity data. Specifically, various machine-learning methods and input configurations have been used to evaluate the classification model's performance. The F1-scores, which measure balanced precision and recall, of classification models with embedded physical and geometric invariance can be improved by about over those without geometric invariance. Regression models based on ensemble neural networks have demonstrated superior performance in predicting the signed Smagorinsky model coefficient, exceeding that of the dynamic Smagorinsky model (DSM) in a priori tests.
