Particle Method for Simulation of Free Surface Flows
In this paper we present a particle method for simulations of free surface flows. This is a fully Lagrangian and grid free method. A fluid domain is first replaced by a discrete number of points, which are referred to particles. Each particle carries all
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Fraunhofer Institut Techno- und Wirtschaftmathematik Gottlieb-Daimler-Strasse, Geb . 49 D-67663 Kais erslautern, Germany tiwari@itwm·fhg·de [email protected]
1 Introduction In this paper we present a particle method for simulations of free surface flows. This is a fully Lagrangian and grid free method. A fluid domain is first replaced by a discrete number of points, which are referred to particles. Each particle carries all fluid informations, like density, velocity, temperature etc and moves with fluid velocity. Therefore, particles themselves can be considered as geometrical grids of the fluid domain. This method has some advantages over grid based techniques, for example, it can handle fluid domains which change naturally, whereas grid based techniques require additional computational effort. Numerical simulations of free surface flows have many industrial applications like casting, tank filling and others. Many methods have been developed to simulate free surface flows[5 , 6, 9]. A classical grid free Lagrangian method is the Smoothed Particle Hydrodynamics (SPH) method, which was originally introduced to solve problems in astrophysics [8, 3]. It has since been extended to simulate the compressible Euler equations in fluid dynamics and applied to a wide range of problems, see [11, 12]. The implementation of the boundary conditions is the main problem of the SPH method. Our approach differs from the method of SPH in approximations of derivatives and the treatment of boundary conditions. We approximate the derivatives of a function by the weighted least squares method. This is similar to the moving least squares method, see [1, 2, 7] for detail references. In [7] the boundary conditions are well treated by the moving least squares technique. Like in [7] we have replaced the boundary by particles and prescribe the boundary condition on the boundary particles. These approaches are similar to the central difference rule in the finite difference discretization and have well known problems of stability. Some viscosity is required to stabilize the numerical scheme. In [15] a weighted least squares method is proposed to solve the full Navier-Stokes equations. The artificial viscosity is not required to stabilize the scheme but the natural Navier-Stokes viscous term is used. In this approach the solutions ofthe compressible Euler equations can be obtained from the Navier-Stokes equations
T.Y. Hou et al.(eds.), Hyperbolic Problems: Theory, Numerics, Applications © Springer-Verlag Berlin Heidelberg 2003
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S. Tiwari and J. Kuhnert
by letting viscosity and heat conductivity tend to zero [15]. Furthermore, in [16] the method is extended to solve incompressible viscous Navier-Stokes equations and shown that the Navier-Stokes equations can be solved quite accurately. In this paper and in [16] we have simulated the incompressible free surface flows as a limit of the compressible, viscous Navier-Stokes equations with the equation of state such that the flow is weakly incompressible. This type of equation of state was
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