HybridFlow: Eulerian–Lagrangian CFD Framework

• HybridFlow is a hybrid Eulerian–Lagrangian computational framework designed to simulate incompressible viscous flows by combining sharp near-wall resolution with low numerical diffusion in wakes.
• It leverages finite element methods in the near-wall region and vortex particle methods in the wake to achieve efficient domain decomposition and conserve circulation to machine precision.
• Validated against canonical benchmarks, HybridFlow accurately predicts flow phenomena such as dipole collisions and cylinder wake dynamics, demonstrating robust performance in CFD simulations.

HybridFlow is a hybrid Eulerian–Lagrangian computational framework designed to efficiently and accurately simulate incompressible viscous flows. The method leverages the complementary strengths of Eulerian solvers (sharp near-wall resolution, effective at handling solid boundaries) and Lagrangian vortex particle methods (low numerical diffusion, superior wake-capturing capability) by spatially decomposing the fluid domain into near-wall and wake regions, each handled with its optimal numerical approach. The hybrid solver advances both domains with a tightly coupled, circulation-conserving scheme, validated against a suite of canonical two-dimensional benchmarks.

1. Domain Decomposition and Solver Orchestration

The core principle of HybridFlow is the spatial domain decomposition into two overlapping subdomains:

• Near-wall region (Eulerian): An unstructured finite element discretization solves the full Navier–Stokes equations in velocity–pressure (𝒖, p) form, enforcing boundary conditions and faithfully capturing viscous and boundary layer phenomena.
• Away from boundaries (Lagrangian): The remainder of the domain is modeled using a regularized vortex particle method (VPM), wherein the vorticity field is discretized by particles carrying circulation and advected according to the local velocity field.

A defining feature is that the Lagrangian solver formally covers the entire fluid domain, while the Eulerian patch acts as a corrective overlay near boundaries. The coupling eschews Schwarz–type global iteration: at every macro time step, the Lagrangian solution is evolved first (neglecting wall vorticity generation), after which the Eulerian solver is advanced using velocity boundary conditions derived from the updated Lagrangian state at its outer interface. The critical “correction” step occurs by deleting and redeploying all vortex particles in the overlap region, with assigned circulations exactly matching the updated Eulerian vorticity—a process which preserves total circulation to machine precision.

2. Mathematical Formulation

Lagrangian VPM: The vorticity–velocity formulation is used: $$\frac{\partial \omega}{\partial t} + (\mathbf{u} \cdot \nabla)\omega - \nu\,\nabla^2\omega = 0$$ where advected vortex particles $p$ with position $\mathbf{x}_p$ and circulation $\Gamma_p$ reconstruct the velocity via

$$\mathbf{u}(\mathbf{x}) = -\frac{1}{2\pi} \sum_p \frac{g_\sigma(\|\mathbf{x} - \mathbf{x}_p\|)}{\|\mathbf{x} - \mathbf{x}_p\|^2}\,(\mathbf{x} - \mathbf{x}_p) \times \mathbf{e}_z\,\Gamma_p + \mathbf{U}_\infty$$

with $g_\sigma$ denoting a regularized core function (e.g., Gaussian).

Eulerian Subdomain: The finite element formulation for the momentum equation, with the stress tensor $$\sigma = 2\nu\varepsilon(\mathbf{u}) - p\,\mathbf{I}$$, is: $$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla\cdot \sigma = 0\,, \qquad \nabla \cdot \mathbf{u} = 0$$ Pressure–velocity coupling is handled with an incremental pressure correction scheme (IPCS), which successively updates tentative velocities, corrects pressure, and enforces solenoidal constraints at each substep.

Coupling Algorithm: For each total time step $\Delta t_L$, the following sequence is executed:

1. VPM: Evolve all vortex particles over $\Delta t_L$, ignoring wall vorticity generation.
2. Map the Lagrangian field to the outer boundary of the Eulerian patch for Dirichlet velocity conditions.
3. Eulerian: Evolve the near-wall region over $\Delta t_L$, using possibly smaller substeps $\Delta t_E$, with boundary conditions provided by the Lagrangian field.
4. Correction: In the overlap region $\Omega_I$, remove existing particles, remesh, and reinitialize, assigning circulation according to the Eulerian-computed vorticity.

All remeshing operations are parameter-dependent (location and width of $\Omega_I$), with careful attention required for accuracy and conservation.

3. Numerical Evaluation and Benchmarks

The HybridFlow scheme is extensively validated against reference finite element (FE) solutions and published data. Major benchmarks include:

Test Case	Main Result/Observation
Unbounded vortex dipole	For small particle spacing, hybrid evolution of maximum vorticity and dipole position agrees with FE; larger particles yield vorticity loss and lag.
Bounded dipole–wall collision	Near-wall vortex generation captured accurately; quantitative agreements in vorticity, energy, enstrophy, and palinstrophy with FE.
Impulsively started cylinder (Re=550)	Correct recovery of drag/lift coefficients and wake structure for appropriate overlap parameters (offset distance $d_{bdry}$), matching prior reference data.
Stalled ellipsoid, high Re (Re=5000)	Strong agreement at early times; discrepancies grow as vortex shedding complexity increases (consistent with nonlinear regime sensitivity).

These cases demonstrate not only high-fidelity force and flow-field prediction, but also pinpoint the necessity of tuning overlap and remeshing parameters for optimal accuracy.

4. Advantages, Limitations, and Conservation Properties

Strengths:

• Spatially adaptive cost: The Eulerian solver is limited to near-wall regions, reducing the global degrees of freedom and associated computational cost without loss of wall resolution.
• Low numerical diffusion in wakes: VPM particles only exist where vorticity is present, preserving fine wake structures over long distances.
• No iterative domain-matching: All Lagrangian–Eulerian coupling is done in one correction per time step, conserving total fluid circulation exactly—a crucial invariant for long-time integration.
• Scalability: Naturally parallelizable/multi-body friendly, where each solid body in a complex environment can host an individual Eulerian patch within a global Lagrangian domain.

Limitations:

• Sensitivity in overlap region: The fidelity of coupling depends on the precise definition and discretization of the interpolation region $\Omega_I$. Remeshing/interpolation errors may arise when transferring vorticity between grids.
• Handling of complex or turbulent flows: Parameter tuning (e.g., interface location) becomes nontrivial for time-dependent geometries or strong turbulence; additional model sophistication or adaptivity may be required.
• Force transients: Slight deviations in computed force coefficients are observed during rapid transients or the onset of vortex shedding, highlighting the need for parameter refinement or improved matching schemes.

5. Implementation and Computational Considerations

HybridFlow’s workflow is built for efficient modular deployment:

• Finite element Eulerian solver uses arbitrary turbulence models or DNS, subject to computational resource availability and flow Reynolds number.
• Vortex particle method is grid-free beyond the overlap zone and is computationally efficient due to the natural adaptivity to regions of nonzero vorticity.
• Conservation: The method enforces exact global circulation conservation at each correction by matching total vortex strength in the overlap region between the two representations.
• Coupling and remeshing: To minimize errors, the transfer from the unstructured Eulerian grid to the structured Lagrangian remesh grid is performed via carefully constructed interpolation.

Further computational benefits arise from the natural parallelizability: multiple objects/bodies can be assigned independent Eulerian regions, enabling distributed-memory strategies for high-performance computing.

6. Future Directions and Research Outlook

The HybridFlow paradigm identifies several promising research and application avenues:

• Moving/deforming objects: Direct extension to problems with time-dependent boundaries, such as rotating or morphing aerofoils, wind turbines, or biological flows.
• High-Reynolds number and turbulence modeling: Seamless coupling to advanced turbulence closures in the Eulerian patch and possible downstream extension to large-eddy simulation (LES) or direct numerical simulation (DNS).
• Wind farms and multi-object interactions: Each component body handled with its own local Eulerian subsystem and coupled globally via the long-range, low-diffusion particle method for wake dynamics.
• Full three-dimensional generalization: The authors note that the framework extends to 3D problems, which remains a natural avenue for future high-fidelity CFD paper.

In conclusion, HybridFlow bridges the gap between computationally intensive, highly resolved Eulerian simulations and efficient—but sometimes less accurate—Lagrangian approaches. By deploying each in the spatial regime where it excels and orchestrating their interaction with strict physical conservation, the framework supplies a robust and efficient alternative for engineering and scientific CFD applications, validated by both canonical benchmarks and realistic flow scenarios (Palha et al., 2015).