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Free-surface flow simulations with a Lagrangian and an Arbitrary Lagrangian–Eulerian methods

Abstract : Free-surface flows have various natures in the environmental and industrial contexts. It may be a gentle mathematically regular surface with waves from which one may want to extract renewable energy, or an also smooth surface of an evaporating pool in case of recirculating pumps deficiency, but also a flow down a spillway with a really disturbed free-surface. Lagrange and/or Euler approaches can be used to solve the discretised Navier-Stokes equations with a free-surface.Among the Lagrangian methods, Smoothed Particle Hydrodynamics (SPH) is a mesh-less numerical method ideal for simulating potentially violent free-surface phenomena such as a wave breaking, or a dam-break for which many Eulerian methods can be difficult to apply.Gentle free-surface flows can also be tackled with the Arbitrary Lagrangian Eulerian (ALE) mesh-based Finite Volumes method, where the free-surface faces of the mesh move so that the kinematic boundary condition is fulfilled.The first Chapter is made to introduce SPH and ALE Finite Volumes and to draw links between the two methods.For SPH, dealing with boundary conditions (walls and open boundaries) is one of the most challenging parts as it is declared as one of the Grand Challenges of the international organisation representing the community of researchers and industrial users of Smoothed Particle Hydrodynamics (SPHERIC). Concerning walls, the proposed methodology introduced in Chapter 2 relies on the semi-analytical approach which consists in renormalising the density field near a solid wall with respect to the missing kernel support area, and intrinsic gradient and divergence operators that ensure conservation properties are employed. The accuracy of the physical field such as the pressure next to walls is considerably improved, and the consistent manner developed to wall-correct operators allows us to perform simulations with turbulence modelsAn axisymmetric formulation with a unified renormalisation factor taking both radial correction and wall renormalisation is proposed as an extension of this work in Appendix.The third Chapter deals with open-boundaries for the SPH approach with the resolution of a Riemann problem associated to the hyperbolic compressible SPH framework used. The discretisation of the boundary in surface elements (segments in 2-D) and vertices is adequate to make particles enter progressively so that no pressure wave are created by the release of new fluid particles. Some details or how to integrate the geometrical renormalisation factor used in the SPH boundary conditions is presented in Appendix. The fourth Chapter presents the ALE Finite Volumes algorithm developed in the massively parallel open-source code code_saturne. An original mixing of cell-based numerical scheme used to get conservation of mass and momentum on each cell control volume and a vertex-based scheme based on the Compatible Discrete Operators (CDO) approach is presented with a particular care on the free-surface condition both on fluid and mesh-displacement. Various verification and validation test cases are presented. Space discretisation of a Poisson equation, used for the mass correction step in the ALE Finite Volumes approach, is presented in Appendix.
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Martin Ferrand. Free-surface flow simulations with a Lagrangian and an Arbitrary Lagrangian–Eulerian methods. Fluids mechanics [physics.class-ph]. École des Ponts ParisTech, 2022. English. ⟨NNT : 2022ENPC0002⟩. ⟨tel-03637275⟩

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