Hybrid computational solvers that integrate Eulerian and Lagrangian methods are emerging as powerful tools in computational fluid dynamics, particularly for external aerodynamics. These solvers rely on the strengths of both approaches: Eulerian methods efficiently handle boundary layers, while Lagrangian methods excel in reducing numerical diffusion in flow convection. Building on our prior development of a two-dimensional hybrid solver that combines OpenFOAM with vortex particle method, this paper extends its application to the complex phenomena of airfoil stall at low Reynolds numbers. Specifically, we examine both static and dynamic stall conditions of a National Advisory Committee for Aeronautics (NACA) airfoil series 0012 (NACA0012) across a wide range of attack angles and oscillation frequencies, comparing our results with established data. The findings demonstrate the accuracy of hybrid Eulerian–Lagrangian solvers in replicating known stall behaviors, underscoring their potential for advanced aerodynamic studies. This work not only confirms the capability of hybrid solvers in accurately modeling challenging flows but also paves the way for their increased involvement in the field of external aerodynamics.@en

The field of external aerodynamics encompasses various engineering disciplines with a significant impact on wind energy technology. Aerodynamic investigations provide insights not only into the characteristics of individual blades or standalone wind turbines but also into entire wind farms. As advancements in wind turbine design continue, understanding the interactions between turbines in close proximity becomes crucial, presenting a multi-body problem. Researchers require efficient and accurate tools to comprehensively study such dynamics. This paper presents a hybrid Eulerian-Lagrangian solver designed to leverage the strengths of Eulerian solvers in resolving boundary layers and Lagrangian solvers in convecting wakes downstream without introducing significant numerical diffusion. The solver adeptly handles multi-body simulations, allowing the construction of independent Eulerian meshes that communicate seamlessly through Lagrangian particles. In this way, the computational study of multibody problems does not require very large and dense meshes. Validation in single-body cases has already been conducted, with this paper demonstrating the solver's application to a pair of cylinders in different configurations. A comparative performance analysis is carried out against pure Eulerian solvers. The results highlight that the hybrid solver efficiently reproduces the accuracy of the Eulerian solver, demonstrating its effectiveness in handling complex aerodynamic simulations.@en

Hybrid Eulerian–Lagrangian solvers have gained increasing attention in the field of external aerodynamics, particularly when dealing with strong body–vortex interactions. This approach effectively combines the strengths of the Eulerian component, which accurately resolves boundary layer phenomena, and the Lagrangian component, which efficiently evolves the wake downstream. This study builds on our team's previous work by enhancing the capabilities of a two-dimensional hybrid Eulerian–Lagrangian solver. We aim to upgrade our solver which was initially designed for static cases, to now also simulate cases involving moving objects. To ensure the reliability and applicability of a new solver, it is essential to validate its performance in complex cases. Here, the solver is validated across the case of a traveling cylinder and the case of a rotating cylinder in two different rotational speeds at low Reynolds numbers. In the realm of Eulerian solvers, such as OpenFOAM (utilized for the Eulerian component of this hybrid approach), traditional techniques include the use of morphing meshes, overset meshes, and Arbitrary Mesh Interfaces (AMI) to model body motion. The proposed methodology involves extending the Eulerian mesh up to a short distance from the solid boundary and moving it entirely as a solid entity. Then the Lagrangian solver is responsible for calculating the updated boundary conditions, thereby completing the hybrid solver's functionality. This approach is very similar to the overset mesh technique. However, unlike the traditional method where an Eulerian mesh moves on top of a static one, our method involves the motion of an Eulerian mesh over a Lagrangian grid. We compared the results from our hybrid solver with those from a purely Eulerian solver, specifically OpenFOAM. The comparison demonstrates that our solver can replicate OpenFOAM's results with high accuracy. Another interesting point highlighted in this study is the presence of high-frequency oscillations in the body forces in hybrid solvers that incorporate the redistribution of Lagrangian particles and do not utilize surface elements such as vortex panels, specifically when dealing with dynamic mesh simulations. When the Eulerian mesh travels on top of the Lagrangian grid of particles, the positions of the particles with respect to the Eulerian mesh continuously change. This results in a constant shift of particles near the solid body, where the highest vorticity is observed. Particles that are close to the solid boundary at one time step may find themselves inside the boundary at the next time step, leading to their removal. This pattern continuously changes during the simulation, causing fluctuations in the boundary conditions of the Eulerian solver and manifesting as oscillations in the forces acting on the body. It is shown that this issue can be alleviated either by increasing the spatial resolution of the Lagrangian solver or by synchronizing the movement of the Lagrangian grid with the motion of the Eulerian mesh. The results of the study make the solver trustworthy and pave the way for more demanding external aerodynamic simulations.

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The past few decades have witnessed a growing popularity in Eulerian–Lagrangian solvers due to their significant potential for simulating aerodynamic flows, particularly in cases involving strong body–vortex interactions. In this hybrid approach, the two component solvers are mutually coupled in a two-way fashion. Initially, the Lagrangian solver can supply boundary conditions to the Eulerian solver, while the Eulerian solver functions as a corrector for the Lagrangian solution in regions where the latter cannot achieve high accuracy. To utilize such tools effectively, it is vital for them to be capable of handling dynamic mesh movements. This study builds upon the previous research conducted by our team and extends the capabilities of the hybrid solver to handle dynamic meshes. While OpenFOAM, the Eulerian component of this hybrid code, incorporates built-in dynamic mesh properties, certain modifications are necessary to ensure its compatibility with the Lagrangian solver. More specifically, the evolution algorithm of the pimpleFOAM solver needs to be divided into two discrete steps: first, updating the mesh, and later, evolving the solution. This division enables a proper coupling between pimpleFOAM and the Lagrangian solver as an intermediate step. Therefore, the primary objective of this specific paper is to adapt the OpenFOAM solver to meet the demands of the hybrid solver and subsequently validate that the hybrid solver can effectively address dynamic mesh challenges using this approach. This approach introduces a pioneering method for conducting dynamic mesh simulations within the OpenFOAM framework, showcasing its potential for broader applications. To validate the approach, various test cases involving dynamic mesh movements are employed. Specifically, all these cases employ the Lamb–Oseen diffusing vortex, but each case incorporates different types of mesh movements, including translational, rotational, oscillational, and combinations thereof. The results from these cases demonstrate the effectiveness of the proposed OpenFOAM algorithm, with the maximum relative errors —when compared to the analytical solution across all presented cases—capped at (Formula presented.) for the worst-case scenario. This affirms the algorithm’s capability to successfully handle dynamic mesh simulations with the proposed solver.

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In the field of computational aerodynamics, it is vital to develop tools that can accurately, but also efficiently, simulate the flow around bluff objects and calculate the aerodynamic forces acting on them. When strong body–vortex interactions take place, the simulations become more demanding, since complex phenomena appear. To address this issue, hybrid Eulerian–Lagrangian solvers have been developed and are increasingly used in the field. In this paper, a Vortex Particle Method (VPM) is coupled with the OpenFOAM software. The Eulerian solver (OpenFOAM) resolves the regions close to the solid boundaries, while the vortex particles evolve the wake downstream, significantly reducing artificial diffusion. The coupling strategy and the validation results of a hybrid code based on the domain decomposition technique are presented. This work is the first to couple OpenFOAM with a Lagrangian solver in the framework of a hybrid solver. Our objective is twofold: to verify the capability of OpenFOAM to run with a VPM and to validate the hybrid solver using benchmark cases. We demonstrate the validation of the solver on the Lamb–Oseen vortex case, the dipole case in the unbounded domain, and the flow around a cylinder at Re = 550. Our results show that coupling OpenFOAM with a VPM can be achieved without complications and efficiently reproduces the results of pure Eulerian simulations.@en