Elsevier

Journal of Computational Physics

Volume 364, 1 July 2018, Pages 111-136
Journal of Computational Physics

Coupled SPH–FV method with net vorticity and mass transfer

https://doi.org/10.1016/j.jcp.2018.02.052Get rights and content

Highlights

  • A SPH/FV coupling strategy has been extended to address complex free-surface flows.

  • SPH is used close to free-surfaces and FV in the bulk of the flow and wall regions.

  • Net mass fluxes and free-surfaces across the coupling interface are addressed.

  • New algorithms for particle creation and a SPH-driven Level-Set approach are proposed.

  • Good results are obtained also when vorticity and free-surface transfer is important

Abstract

Recently, an algorithm for coupling a Finite Volume (FV) method, that discretize the Navier–Stokes equations on block structured Eulerian grids, with the weakly-compressible Lagrangian Smoothed Particle Hydrodynamics (SPH) was presented in [16]. The algorithm takes advantage of the SPH method to discretize flow regions close to free-surfaces and of the FV method to resolve the bulk flow and the wall regions. The continuity between the two solutions is guaranteed by overlapping zones. Here we extend the algorithm by adding the possibility to have: 1) net mass transfer between the SPH and FV sub-domains; 2) free-surface across the overlapping region. In this context, particle generation at common boundaries is required to prevent depletion or clustering of particles. This operation is not trivial, because consistency between the Lagrangian and Eulerian description of the flow must be retained to ensure mass conservation. We propose here a new coupling paradigm that extends the algorithm developed in [16] and renders it suitable to test cases where vorticity and free surface significantly pass from one domain to the other. On the SPH side, a novel technique for the creation/deletion of particle was developed. On the FV side, the information recovered from the SPH solver are exploited to improve free surface prediction in a fashion that resemble the Particle Level-Set algorithms. The combination of the two new features was tested and validated in a number of test cases where both vorticity and front evolution are important. Convergence and robustness of the algorithm are shown.

Introduction

Fluid dynamic design of complex devices and basic research about free surface flow close to rigid boundaries require numerical algorithms able to capture both free surface effects and details of the boundary layers on the walls.

The main difficulties for reliable simulations stem from both the topological variation of the computational domain due to free surface evolution (more easily handled by Lagrangian approaches) and from the need for a correct resolution of vorticity production and evolution, particularly in the boundary layer (for which space discretization is more easily controlled with Eulerian approaches).

In order to exploit the best of the two approaches, in [16] an algorithm that couples the Smoothed Particle Hydrodynamics (SPH) approach with a Finite Volume (FV) scheme was developed and implemented. The results proved that for certain kind of flows (e.g. wave propagation and breaking in deep water) the algorithm is both very accurate and efficient.

The idea of domain decomposition has been largely exploited in fluid dynamics. A classical review of coupling methods between Eulerian solvers can be found in [14], whereas a more recent review of domain decomposition with particle methods can be found in [11]. Coupling methods between mesh-based and particle approaches have been proposed, besides [16], in [20] where an incompressible SPH model has been coupled with an incompressible FV approach with surface fitting. Another example can be found in [3] where both SPH and FV solvers are exploited to improve the quality of the solution.

The algorithm in [16] is here extended to handle both net mass transfer and free surface passage through the coupling boundary. To this end, the interface boundary on the SPH side is supplemented with a procedure to add and remove particles, so that open-boundary conditions can be included to limit the region where the SPH solution is performed. Similarly to what proposed in [1] the seeding/de-seeding region is controlled by the solution exchanged with the adjacent FV solution, and particle production is carried out with particular attention to global mass conservation.

At the same time, in the overlapping zone on the FV side, particle trajectory are exploited to correct the free surface position, in a way that resemble the Particle/Level Set algorithms (see, e.g. [8]), the information obtained from the SPH solution being used to guarantee consistency between the two solutions also in case of violent free–surface motion (e.g. sloshing in a tank).

The new algorithm is checked in a series of 2D test cases where vorticity and free surface move across the overlapping zone, with relevant mass flux. In the first test case (Taylor–Green vortex flow) only mass transfer and relevant vorticity exchange between the two sub-domains takes place. The second test case (jet impinging on a free surface initially at rest) is instead characterized by free-surface passage across the coupling boundary with negligible vorticity production. The last two cases (vortex dipole evolving beneath a free surface and violent sloshing flow in a tank) exhibit both aspects. For all test cases, convergence is carefully checked and verification is performed by comparing the computed solutions with available reference solutions.

Besides accuracy, the mentioned test cases are conceived to check the robustness of the coupling algorithm. Indeed, the adopted domain decomposition between solvers is intentionally not optimal in terms of solver capabilities, but rather mirrors realistic conditions where the complete evolution of the flow is not a priori known. In these cases it can happen that complex free-surface profiles move across the coupling boundaries towards the FV region or that relevant vortex structures enter the SPH domain. By means of these test cases, we want to prove that the proposed coupling algorithm is robust and accurate also when the choice of the domain decomposition is not flawless.

Applications to realistic test cases are finally shown for the flow around a circular cylinder below a free surface and for the sloshing flow in a tank with a corrugated bottom. In both cases the advantage of the coupling strategy is highlighted by comparing the solution with the single solver computations.

Section snippets

Mathematical models

In this section, the mathematical models used in the following will be briefly recalled. For both (Lagrangian and Eulerian) approaches, the weakly compressible flow model was adopted. In this model, the fluid is supposed to be barotropic, i.e. the fluid density ρ depends only on the pressure p; in addition, in the hypothesis of weak compressibilitydρdp=1c2, where c (the sound speed) is supposed to be constant and large with respect to the typical fluid velocity U; generally, this condition is

Finite volume scheme

The discretization of Navier–Stokes equations in their Eulerian formulation is made by mean of a Finite Volume (FV) scheme, implemented on a multi-block structured grid with partial overlapping [5], [19]; with this approach, the computational domain is split into sub-domainsDl,l=1,,L not necessarily disjoint. A curvilinear coordinate system (ξ,η,ζ) is considered on each block, discretized by a structured grid with hexahedral cellsDi,j,kli=1,,Nij=1,,Njk=1,,Nk where the indexes i,j,k runs

Coupling algorithm

By coupling the two approaches, it is possible to obtain an algorithm that retains the best from each single approach when simulating flows with regions characterized by different length and time scales. In particular, we want to exploit the ability of Lagrangian particle methods to simulate free surface flows, possibly with front fragmentation, without mass or momentum losses; at the same time, we would like to retain the ability of mesh-based approaches (chimera-type approaches in particular)

Numerical results

Four test cases are reported to assess convergence properties and to validate the coupled algorithm in comparison with analytical solutions when available or solutions obtained with other solvers.

The first test case consists of a flow (Taylor–Green vortex) where the two solvers exchange mainly vorticity. The second test case, instead, is characterized by free surface passage across the interaction boundary, with limited vorticity transfer. In the third test case, both phenomena are present and

Application to complex configurations

In this section, two examples of application to more realistic situations are reported in order to highlight the effectiveness of the proposed coupling algorithm.

Conclusions

In a previous paper [16], an algorithm that couples a SPH solver with a FV approach was proposed and applied to some test cases where both complex free surface evolution and vorticity were present. The algorithm in its original form was intended for flows where the free surface and the viscous region were clearly separated, easily identifiable and confined in time. In other words, the procedure did not allow relevant mass transfer and/or free surface crossing between the two subdomains. This

Acknowledgements

The work was partially supported by the Flagship Project RITMARE - The Italian Research for the Sea - coordinated by the Italian National Research Council and funded by the Italian Ministry of Education, University and Research within the National Research Program 2015-2016.

As for Ecole Centrale de Nantes and NEXTFLOW Software, the research leading to these results has partially received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No.

References (26)

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