A hybrid pressure formulation of the face-centred finite volume method for viscous laminar incompressible flows (2501.04864v2)

Abstract: This work presents a hybrid pressure face-centred finite volume (FCFV) solver to simulate steady-state incompressible Navier-Stokes flows. The method leverages the robustness, in the incompressible limit, of the hybridisable discontinuous Galerkin paradigm for compressible and weakly compressible flows to derive the formulation of a novel, low-order face-based discretisation. The incompressibility constraint is enforced in a weak sense, by introducing an inter-cell mass flux defined in terms of a new, hybrid variable, representing the pressure at the cell faces. This results in a new hybridisation strategy where cell variables (velocity, pressure and deviatoric strain rate tensor) are expressed as a function of velocity and pressure at the barycentre of the cell faces. The hybrid pressure formulation provides first-order convergence of all variables, including the stress, without the need for gradient reconstruction, thus being less sensitive to cell type, stretching, distortion, and skewness than traditional low-order finite volume solvers. Numerical benchmarks of Navier-Stokes flows at low and moderate Reynolds numbers, in two and three dimensions, are presented to evaluate accuracy and robustness of the method. In particular, the hybrid pressure formulation outperforms the FCFV method when convective effects are relevant, achieving accurate predictions on significantly coarser meshes.

• The paper formulates a novel hybrid pressure-based Face-Centred Finite Volume (FCFV) method for simulating viscous laminar incompressible flows.
• This method introduces a hybrid variable for face pressure and weakly imposes the incompressibility constraint, enabling analytical expressions for variables within a cell.
• Numerical results demonstrate enhanced accuracy and robustness, particularly on coarse meshes, outperforming existing methods and offering a unique global matrix structure beneficial for solver development.

On a Hybrid Pressure Formulation of the Face-Centred Finite Volume Method for Viscous Laminar Incompressible Flows

The paper outlines a novel hybrid pressure approach for face-centred finite volume (FCFV) methods targeting the simulation of steady-state incompressible Navier-Stokes flows. This method leverages the adaptable nature of hybridisable discontinuous Galerkin (HDG) methodologies to deal with the incompressibility limit effectively. Through introducing a hybrid variable that defines pressure at the cell faces, the paper addresses both theoretical and practical implications in CFD computations.

Key Contributions and Methodology

The primary contribution of the research is the formulation of a hybrid pressure-based FCFV method that assures first-order convergence across all principal variables, such as velocity, pressure, and the deviatoric stress tensor. This convergence is achieved irrespective of cell topology, stretching, or distortion, which underscores the robustness of the proposed approach.

The methodology involved conceptualizing a distinct hybridisation technique where velocity, pressure, and deviatoric strain rate tensors within a cell are analytically expressed in terms of velocity and pressure at the barycentre of the cell faces. The incompressibility constraint is imposed weakly, wherein an inter-cell mass flux is defined by an introduced hybrid variable representing the face pressure.

Numerical Results

The research highlights strong numerical exemplifications through various test cases such as low and moderate Reynolds number flows in both two and three-dimensional contexts. A notable observation was the enhanced accuracy and robustness that the hybrid pressure methodology exhibits particularly where convective effects are dominant. This enhanced performance allows for accurate simulations even on significantly coarser meshes as compared to the traditional methods.

For incompressible flows within a cavity and flow past a sphere, the proposed approach demonstrated a more precise depiction of the physical phenomena, validating its efficacy over existing FCFV methods. The numerical results repeatedly show that the hybrid pressure formulation achieves superior accuracy, with results closely agreeing with high-fidelity reference solutions found in literature.

Computational Implications

From a computational standpoint, the method introduces a reimagined structure of the global system matrix, characterized by a negative definite matrix devoid of any saddle point structure. This theoretical insight not only differentiates it from classical FCFV methods but also suggests potential developments in designing efficient iterative solvers and preconditioning techniques beneficial to solving large-scale CFD problems.

Prospects and Future Directions

The paper asserts that its hybrid pressure methodology, combined with the FCFV principles, sets a direction for future investigations into high-fidelity CFD simulations for a variety of complex flow phenomena. Future work could be oriented towards refining iterative solutions and preconditioner designs tailored for such hybrid methods, as well as exploring higher Reynolds number turbulent flows.

This research marks a relevant step in computational fluid dynamics, presenting significant theoretical developments backed by rigorous numerical validation, all while opening avenues for the next generation of adaptive, robust, and efficient computational solvers.