Jun Zhang

Accurate modeling of vibration–dissociation coupling remains a primary challenge in the direct simulation Monte Carlo framework for hypersonic reacting flows because conventional phenomenological models often rely on equilibrium assumptions or empirical calibration. To address this, a novel nonequilibrium coupled vibration–dissociation (NCVD) model is proposed. Built within a physically constrained deconvolution framework based on a two-temperature energy distribution, the NCVD model explicitly incorporates two critical nonequilibrium indicators (the nonequilibrium rate factor and the average vibrational energy of dissociating molecules) into the microscopic collision probability. Unlike traditional models, the present formulation ensures a consistent mapping between macroscopic nonequilibrium behaviors and microscopic kinetics. Comprehensive zero-dimensional heat-bath simulations demonstrate that the NCVD model quantitatively reproduces the target dissociation characteristics in both vibrationally cold and hot conditions, significantly outperforming the standard total collision energy (TCE) and vibrationally favored dissociation models. Furthermore, in one-dimensional simulations of high-enthalpy shock-heated oxygen flows, the NCVD model shows superior agreement with experimental measurements and state-to-state calculations for vibrational temperature and species concentration profiles. The proposed NCVD model for simulating thermochemical nonequilibrium flows proves to offer a robust compromise between physical fidelity and computational cost, maintaining efficiency comparable to the TCE model for engineering applications.

Hypersonic gas flows encountered in aerospace engineering often exhibit multiscale characteristics together with translational, rotational, and vibrational nonequilibrium. The direct simulation Monte Carlo (DSMC) method provides reliable particle-based simulations in rarefied flows, but its computational cost increases substantially near the continuum regime because of strict spatiotemporal resolution constraints. In this work, the multiscale simulation Monte Carlo (MSMC) method is extended and reformulated for polyatomic gas flows with rotational and vibrational nonequilibrium. The proposed method is built upon the standard stochastic particle framework and reduces spatiotemporal discretization errors through a modified collision scheme and pair-selection approach. The temporal error is reduced through an exponential Runge-Kutta collision scheme with an anti-dissipation distribution designed to reproduce the target midpoint collision effect, thereby compensating for the splitting error caused by the decoupling of transport and collision. The spatial error is mitigated by a local sampling collision strategy that suppresses nonlocal cell-based collision pairing. The method is validated for several representative polyatomic gas flows, including homogeneous relaxation, normal shock structure, hypersonic cylinder flows at different rarefaction levels, low-density nozzle flow, and hypersonic flow around a blunted cone. The results show that the present MSMC method agrees well with DSMC reference solutions, while allowing coarser grids and larger time steps, demonstrating improved computational efficiency at comparable accuracy for nonequilibrium polyatomic gas flows.

In recent years, significant progress has been made in Fokker-Planck (FP) approximations of the Boltzmann equation, where binary collisions are modeled as drift and diffusion processes in velocity space. To address the discrepancy in the Prandtl number for the original linear FP model, several modified formulations have been developed. Among these, one strategy introduces additional degrees of freedom into the drift term. However, this leads to an increase in model complexity, as the presence of higher-order velocity moments complicates the solution of the equations with underdetermined parameters. An alternative strategy primarily modifies the diffusion term, but the anisotropy introduced in the diffusion term may compromise its positive definiteness. In this work, a quadratic-FP (quad-FP) model with correct Prandtl number is proposed. In the quad-FP model, a quadratic nonlinear modification is introduced into the drift term and corresponding anisotropy modifications are also implemented in the diffusion term. This design simplifies the model structure and facilitates numerical implementation. Moreover, it mitigates the anisotropy introduced by diffusion-term modification, better preserving the positive definiteness of the diffusion term even under strongly nonequilibrium conditions. Numerical tests demonstrate that the quad-FP model provides predictions consistent with direct simulation Monte Carlo results in a variety of flows and achieves improved computational efficiency.

Operator splitting methods are widely used for efficient simulation of multiscale systems with disparate time scales. Although classical schemes such as Strang splitting are formally second order, they often suffer accuracy degradation in stiff relaxation systems when the fast dynamics are temporally under-resolved. In this work, we revisit Strang splitting in stiff regimes from a geometric perspective, leveraging the slow-manifold structure induced by rapid relaxation. Our analysis reveals that in resolved regimes, the substep composition exhibits a hidden symmetry that enables error cancellation and thereby underpins Strang’s second-order accuracy; in under-resolved stiff regimes, the relaxation substep induces a stiff-limit attraction toward quasi-equilibrium, which disrupts the stage-wise cancellation mechanism and leads to order reduction. Motivated by this mechanism, we introduce a symmetry-preserving modification that retains the Strang composition and replaces only the stiff relaxation substep with a time-reversible implicit-midpoint update. The resulting scheme preserves the cancellation mechanism and achieves uniform second-order accuracy for the slow dynamics, even in strongly under-resolved stiff regimes. Numerical experiments on a linear damped oscillator, a nonlinear reaction--diffusion system, and the Bhatnagar--Gross--Krook model corroborate the analysis: classical Strang splittings exhibit order reduction when under-resolved, whereas the proposed method maintains robust second-order accuracy for the slow dynamics and captures the correct first-order stiff-limit behavior of the fast dynamics over a wider range of time steps, allowing larger steps for a prescribed accuracy.

Recently, data-driven methods have shown great promise for discovering governing equations from simulation or experimental data. However, most existing approaches are limited to scalar equations, with few capable of identifying tensor relationships. In this work, we propose a general data-driven framework for identifying tensor equations, referred to as symbolic identification of tensor equations (SITE). The core idea of SITE – representing tensor equations using a host–plasmid structure – is inspired by the multidimensional gene expression programming approach. To improve the robustness of the evolutionary process, SITE adopts a genetic information retention strategy. Moreover, SITE introduces two key innovations beyond conventional evolutionary algorithms. First, it incorporates a dimensional homogeneity check to restrict the search space and eliminate physically invalid expressions. Second, it replaces traditional linear scaling with a tensor linear regression technique, greatly enhancing the efficiency of numerical coefficient optimization. We validate SITE using two benchmark scenarios, where it accurately recovers target equations from synthetic data, showing robustness to noise and flexible expressive capability. Furthermore, SITE is applied to identify constitutive relations directly from molecular simulation data, which are generated without reliance on macroscopic constitutive models. It adapts to both compressible and incompressible flow conditions and successfully identifies the corresponding macroscopic forms, highlighting its potential for data-driven discovery of tensor equation.

Vehicles operating over wide altitude ranges, such as reusable spaceplanes and atmospheric reentry vehicles, face complex aerodynamic environments spanning from continuum to rarefied flow regimes. Conventional aerodynamic designs, typically optimized for a single regime, tend to struggle to maintain consistent performance across such a wide range of flow conditions. This challenge motivates the development of design frameworks capable of accommodating variable flow physics throughout the mission envelope. To this end, we propose CrossAero, a deep reinforcement learning-based framework for aerodynamic optimization that operates directly on a continuous design space of the Knudsen number (Kn), thus removing the need to manually select discrete design points. This capability is facilitated by introducing a probability density function of Kn, which encodes the relative importance of different flow conditions and can be tailored to mission requirements or operation time fractions. Leveraging deep reinforcement learning, CrossAero learns directly from the entire Kn distribution, effectively capturing cross-regime aerodynamic effects within a unified framework. The framework's capabilities are first demonstrated through two single-regime optimization tasks, thereby confirming its reliability under distinct aerodynamic conditions. Building upon this, CrossAero optimizes a fixed-geometry airfoil to maximize average performance over a prescribed Kn distribution, achieving a 5.9% drag reduction relative to the baseline airfoil. Finally, the third case trains a flow-conditioned airfoil generator capable of rapidly producing near-optimal designs for given Kn values without retraining, demonstrating the framework's efficiency in condition-adaptive design and its potential for application in morphing platforms. These results demonstrate CrossAero's robustness, efficiency, and adaptability for the cross-regime aerodynamic design.