This study presents a hybrid reduced-order modeling (ROM) framework for incompressible flows on collocated finite-volume grids. The method combines a discretize-then-project consistent-flux formulation for velocity and pressure with a non-intrusive neural-network closure for the turbulent-viscosity coefficients. The intrusive formulation preserves the discrete mass-conservation constraint and pressure–velocity coupling, while a reduced pressure reference-cell constraint fixes the pressure gauge ambiguity. Three neural-network architectures are evaluated for the closure: a Multilayer Perceptron (MLP), a Transformer, and a Long Short-Term Memory (LSTM) network. The framework is first assessed on a three-dimensional lid-driven cavity simulation at Re=100, where the LSTM-based ROM achieves relative errors of 0.7% for velocity and 4% for turbulent viscosity. A more demanding case at Re=3200 is then considered. A systematic mode-sensitivity study identifies N=15 POD modes as the best overall configuration, balancing velocity and pressure accuracy, reduced dimension, closure robustness, and computational cost. For this configuration, the LSTM-based ROM achieves a final relative velocity error of approximately 12.3% and an online-phase wall-clock speedup of approximately 50× over the full-order model. Energy and enstrophy errors remain below 11% across all three evaluated architectures. Application to this second regime requires case-specific neural-network retraining and pressure reference-cell parameter retuning. A time-extrapolation test, using training data limited to t∈[0,3] s and a ROM rollout to t=6 s, shows that the ROM remains bounded over the investigated extrapolation interval. Velocity and pressure errors increase beyond the training window, while the LSTM turbulent-viscosity closure remains robust, identifying long-horizon pressure accuracy as the primary current limitation. These results demonstrate the potential of combining a consistent projection-based formulation with a data-driven turbulence closure for efficient reduced-order simulation.
An improved discrete unified gas kinetic scheme (DUGKS) with implicit flux reconstruction based on the right-rectangular rule, referred to as RR-DUGKS, is developed for continuum compressible flows. The RR-DUGKS is formulated within the total-energy double-distribution-function framework, where the velocity and total-energy distribution functions involve distinct relaxation times and equilibrium representations, with an additional coupling term for Prandtl-number adjustment. The resulting formulation preserves the discrete conservation of mass, momentum, and total energy. Compared with the conventional trapezoidal treatment, the RR-DUGKS maintains non-negative reconstruction coefficients while retaining overall second-order accuracy, despite introducing an additional second-order temporal truncation term. A series of benchmark tests covering smooth, discontinuous, multidimensional, and high-Mach-number compressible flows is performed to assess the method. The results confirm that the RR-DUGKS retains the accuracy of the conventional trapezoidal formulation for smooth flows, while providing improved numerical stability and reduced nonphysical oscillations in problems involving strong gradients and discontinuities. For three-dimensional compressible vortex flows, the method also yields dissipation characteristics in good agreement with reference data. These results demonstrate that the RR-DUGKS provides a simple and conservative approach for improving the numerical stability of the total-energy DUGKS for continuum compressible flows.
A new lattice Boltzmann collision model is proposed for high-Reynolds-number flows and large Courant–Friedrichs–Lewy (CFL) numbers. The model, referred to as the Filtered Recursive Regularized (FRR) collision model, is based on a modification of previously developed recursive regularized collision models within the Bhatnagar–Gross–Krook (BGK) lattice Boltzmann framework. A guided non-equilibrium distribution function is constructed from the Chapman–Enskog expansion at Navier–Stokes order, thereby effectively filtering out unstable modes associated with higher-order non-equilibrium terms. Its contribution can be explicitly adjusted through a weighting coefficient. The numerical performance of the proposed model is evaluated using four benchmarks: (i) 2D double shear layer, (ii) 2D lid-driven cavity flow, (iii) 3D Taylor–Green vortex flow, and (iv) periodic hill flow, demonstrating a significant increase in numerical stability for under-resolved simulations of high-Reynolds-number flows. The double shear layer and Taylor–Green vortex cases further demonstrate stable simulations at relatively large characteristic velocities compared with the lattice speed of sound. In addition, the cubic-velocity error associated with the defects of Galilean invariance is effectively mitigated through the introduction of a correction term and the filtered recursive regularization procedure. These results indicate that the FRR collision model provides a robust framework for extending the applicability of the lattice Boltzmann method (LBM) to high-Reynolds-number flows.
