Integral surface tension model for 2-dimensional flows simulated using smoothed particle hydrodynamics

  • Majda, Marian (FIFTY2 Technology)
  • Rastelli, Pietro (FIFTY2 Technology)

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Smoothed particle hydrodynamics (SPH) is a numerical method that is well-suited for simulating free boundary problems of time-dependent PDEs like the Navier-Stokes equations. In free-surface applications, robust modelling of the surface tension is necessary. Two classes of describing surface tension can be identified: curvature-based and the so-called integral formulation [3]. Before discretization, they are equivalent. Numerically they differ, since the integral formulation only involves low-order derivatives of the geometry in contrast to curvature which needs higher-order derivatives. Low-order derivatives can be evaluated numerically with higher accuracy. In modelling surface tension using curvature in particle-based methods, accuracy limitations are amplified since quantities are interpolated to obtain the curvature, and the interpolation becomes less accurate for geometries resolved with a low number of particles. In the present work, a novel approach to compute surface tension forces using the integral formulation is introduced and applied to 2d fluids. The method utilizes the SPH number density estimate as a function defining a level set, that is, a level curve. On this curve, which represents the physical free surface, locations of points and their respective tangents are approximated numerically. No other information about the curve is required. These points and tangents are computed using the information in a local SPH neighborhood only and do not need to be stored for the next time-step or for use by another particle, thus lowering computational effort. The integral formulation allows for a high precision and a restriction of surface tension forces to the SPH particles closest to the free surface. Typical test cases like an oscillating 2d droplet are performed, where the fluid flow is simulated using incompressible SPH. Results are compared with a continuous surface force method [2], a particular case of curvature-based surface tension model and interparticle interaction force approaches as presented in [4] and [1].