**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# An octree-based adaptive semi-Lagrangian free surface flow solver

Abstract

A numerical method based on an adaptive octree space discretization for the simulation of 3D free-surface fluid flows is proposed. The Navier-Stokes equations are solved with a time-splitting scheme, which decouples advection from diffusion/incompressibility. The advection step is solved with a semi-Lagrangian VOF-based scheme on the octree. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The octree scheme allows anisotropy, refinement of interfacial cells to an arbitrary level and supports arbitrary complex domains. It does not require a 2:1 cell size ratio condition between adjacent cells. The octree is then coupled with a tetrahedral mesh on which we solve the second step of the splitting algorithm, the Stokes' equations. Numerical validation is done on both advection benchmark test cases and results are compared with the uniform cell grid scheme. Paddle-generated water waves are also simulated and results are compared with experimental water wave profile measurements. \bigskip First order finite element stabilization schemes for the time-dependent Stokes' equations are studied. A unified proof of stability and convergence of velocity and pressure for consistent and non-consistent PSPG schemes for the time-dependent Stokes' equations is given with explicit dependence on viscosity and stabilization parameter. The link between bubble enrichment and Pressure Stabilized Petrov-Galerkin (PSPG) schemes in the context of time-dependent Stokes' equations is discussed and two bubble-based PSPG-type schemes are studied. Different possibilities for stabilization parameters are discussed. Numerical comparisons are done to determine stability, convergence and conditioning issues associated with different PSPG schemes, bubble-based schemes and local pressure projection schemes in different settings.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related publications (37)

Loading

Loading

Loading

Related concepts (16)

Navier–Stokes equations

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier an

Numerical method

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming l

Free surface

In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids wo

Viljami Henrikki Laurmaa, Marco Picasso, Gilles Steiner

A numerical method based on an adaptive octree space discretization for the simulation of displacement of free surfaces is proposed and applied to 3D free surface flow problems. A VOF approach is combined with a mass-conserving semi-Lagrangian time-stepping scheme. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The scheme is unconditionally stable with respect to the CFL number and does not require solving of a linear system. The octree scheme allows anisotropy and refinement of interfacial cells to an arbitrary level. It does not require a 2:1 cell size ratio condition between neighbouring cells. Numerical validation is done on benchmark test cases and results are compared with the structured analog. The scheme is coupled with a Stokes solver on a tetrahedral grid for solving of time-dependent Navier-Stokes equations and numerical results are compared with experimental water wave profile measurements. (C) 2016 The Authors. Published by Elsevier Ltd.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

We present a numerical model for the simulation of 3D mono-dispersed sediment dynamics in a Newtonian flow with free surfaces. The physical model is a macroscopic model for the transport of sediment based on a sediment concentration with a single momentum balance equation for the mixture (fluid and sediments).
The model proposed here couples the Navier-Stokes equations, with a
volume-of-fluid (VOF) approach for the tracking of the free surfaces between the liquid
and the air, plus a nonlinear advection equation for the sediments (for the transport, deposition, and resuspension of sediments).
The numerical algorithm relies on a splitting approach to decouple diffusion and advection phenomena such that we are left with a Stokes operator, an advection operator, and deposition/resuspension operators.
For the space discretization, a two-grid method couples a finite element discretization for the resolution of the Stokes problem, and a finer structured grid of small cells for the discretization of the advection operator and the sediment deposition/resuspension operator.
SLIC, redistribution, and decompression algorithms are used for post-processing to limit numerical diffusion and correct the numerical compression of the volume fraction of liquid.
The numerical model is validated through numerical experiments.
We validate and benchmark the model with deposition effects only for some specific experiments, in particular erosion experiments. Then, we validate and benchmark the model in which we introduce resuspension effects. After that, we discuss the limitations of the underlying physical models.
Finally, we consider a one-dimensional diffusion-convection equation and study an error indicator for the design of adaptive algorithms. First, we consider a finite element backward scheme, and then, a splitting scheme that separates the diffusion and the convection parts of the equation.