A second-order in time and space particle-based method to solve flow problems on arbitrary meshes

Abstract This work presents a novel proposal of a second-order accurate (in time and space) particle-based method for solving transport equations including incompressible flows problems within a mixed Lagrangian-Eulerian formulation. This methodology consists of a symmetrical operator splitting, the use of high-order operators to transfer data between the particles and the background mesh, and an improved version of the eXplicit Integration Following the Streamlines (X-IVS) method. In the case of incompressible flows, a large time-step iterative solver is employed where the momentum equation is split to improve the numerical approximation of the convective term. New interpolation and projection operators are evaluated and quadratically accurate solutions of scalar transport tests are presented. Then, incompressible flow problems are solved where the rate of convergence of the method is assessed using both structured and unstructured background grids. The method is implemented in the open source platform OpenFOAM ® allowing employing arbitrary meshes and obtaining reliable computing time comparisons with standardized solvers. The results obtained reveal that the current method is able to obtain a lower level of error than a fast Eulerian alternative, without increasing the total computing time.

[1]  Sergio Idelsohn,et al.  A fast and accurate method to solve the incompressible Navier‐Stokes equations , 2013 .

[2]  P. Nadukandi.  Numerically stable formulas for a particle-based explicit exponential integrator , 2015 .

[3]  A. Gosman,et al.  Solution of the implicitly discretised reacting flow equations by operator-splitting , 1986 .

[4]  Hrvoje Jasak,et al.  Enhanced coupling of solid body motion and fluid flow in finite volume framework , 2017 .

[5]  Francis H. Harlow,et al.  PIC and its progeny , 1987 .

[6]  Eugenio Oñate,et al.  The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves , 2004 .

[7]  Matthias Teschner,et al.  Implicit Incompressible SPH , 2014, IEEE Transactions on Visualization and Computer Graphics.

[8]  Rodolfo Bermejo,et al.  Finite element modified method of characteristics for the Navier–Stokes equations , 2000 .

[9]  Tony W. H. Sheu,et al.  High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework , 2017 .

[10]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[11]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[12]  T. Driscoll,et al.  Observations on the behavior of radial basis function approximations near boundaries , 2002 .

[13]  S. Koshizuka,et al.  Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid , 1996 .

[14]  Matthias Möller,et al.  A hybridized discontinuous Galerkin framework for high-order particle-mesh operator splitting of the incompressible Navier-Stokes equations , 2018, J. Comput. Phys..

[15]  Eugenio Oñate,et al.  The meshless finite element method , 2003 .

[16]  Yao-Hsin Hwang,et al.  A Moving Particle Method with Embedded Pressure Mesh (MPPM) for Incompressible Flow Calculations , 2011 .

[17]  Borja Servan-Camas,et al.  Seakeeping with the semi-Lagrangian particle finite element method , 2017, CPM 2017.

[18]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[19]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[20]  Eugenio Oñate,et al.  Large time-step explicit integration method for solving problems with dominant convection , 2012 .

[21]  Nathan J. Quinlan,et al.  Extension of the finite volume particle method to viscous flow , 2009, J. Comput. Phys..

[22]  Hrvoje Jasak,et al.  A tensorial approach to computational continuum mechanics using object-oriented techniques , 1998 .

[23]  S. Sarra,et al.  Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations , 2009 .

[24]  Eugenio Oñate,et al.  Lagrangian versus Eulerian integration errors , 2015 .

[25]  J. M. Giménez.  Enlarging time-steps for solving one and two phase flows using the particle finite element method , 2015 .

[26]  G. Taylor,et al.  Mechanism of the production of small eddies from large ones , 1937 .

[27]  Gaofeng Wang,et al.  An overset grid method for large eddy simulation of turbomachinery stages , 2014, J. Comput. Phys..

[28]  Jos Stam,et al.  Stable fluids , 1999, SIGGRAPH.

[29]  Leo M. González,et al.  An extended validation of the last generation of particle finite element method for free surface flows , 2015, J. Comput. Phys..

[30]  A. D. Gosman,et al.  Two calculation procedures for steady, three-dimensional flows with recirculation , 1973 .

[31]  Norberto M. Nigro,et al.  Conservative handling of arbitrary non-conformal interfaces using an efficient supermesh , 2017, J. Comput. Phys..

[32]  Leonardo Di G. Sigalotti,et al.  On the kernel and particle consistency in smoothed particle hydrodynamics , 2016, 1605.05245.

[33]  Juan M. Gimenez,et al.  Evaluating the performance of the particle finite element method in parallel architectures , 2014, Computational Particle Mechanics.

[34]  Rémi Abgrall,et al.  High‐order CFD methods: current status and perspective , 2013 .

[35]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[36]  Rainald Löhner,et al.  A vectorized particle tracer for unstructured grids , 1990 .