A New Approach to Solve Incompressible Navier Stokes Equations Using a Particle Method

Abstract

The goal of this work is to present a new methodology to solve computational fluid dynamics (CFD) problems based on a particle method minimizing the usage of mesh based solvers in order to get a potential computational efficiency to get rid of the new challenges of engineering and science. Thanks to the recent advances in hardware, in particular the possibility of using graphic processors (GPGPU), high performance computing is now available if and only if software development gives an important jump to incorporate this technology. Due to the complexity in programming over such a platform the best way to take advantage of its performance rests on the design of numerical methods able to be viewed as cellular automata. In this sense explicit methods seem to be an attractive choice. However it is well known that explicit methods have a severe stability limitation. On the other hand, the spatial discretization of particle methods offers some advantages against others like finite elements or finite volumes in terms of computational costs. The main reasons of this rest on the low dimensionality of this method (particle methods are a zero dimensional representation of the solution of a given set of PDE’s while finite elements are 3D and finite volume are part 2D and part 3D). In addition particle methods are generally written in Lagrangian formulation avoiding the necessity of defining a spatial stabilization in convection dominated flows. Finite elements and finite volume are generally designed using an Eulerian formulation with some extra diffusion in the solution due to this stabilization requirement. However, some particle methods often require a mesh to interpolate and also to solve the problem losing some of their advantages in terms of efficiency. Even though there are some methods that do not use any mesh in their formulation, the interpolation methods become very complex introducing errors in the computation with noticeable extra diffusion. Having detected two main limitations of particle methods to solve Navier-Stokes equations for viscous incompressible flows, we propose in this work the following: to enhance the time integration using an explicit streamline based scheme computed with the old velocity vector allowing to enlarge the time steps of standard explicit schemes in advection dominated flows, in order to minimize the use of mesh based solvers, the velocity predictor and its correction is formulated purely on the particles as any spatial collocation method using a gradient recovery technique to include the pressure gradient and the viscous terms. This method is written in a Lagrangian formulation in a segregated way like a fractional step method. The computation of the predicted fractional velocity and its correction is done using our proposal explained above, i.e. streamline in time collocation in space scheme. On the other hand the pressure correction (Poisson solver) is carried out using a FEM like method. This method may be implemented in two ways: The mobile mesh version: where the particles represent the mesh nodes and a permanent remeshing is needed in order to avoid the severe restriction in the time steps imposed by the mesh motion. Remember that the mesh is only used to solve the Poisson problem for the pressure correction. The fixed mesh version: where there is a background mesh to do some computations, mainly for the pressure, and a certain amount of particles that move in a Lagrangian way transporting the velocity. Some interpolation between the particles and the fixed mesh is needed but the remeshing is completely avoided. This paper presents this novel approach built with the above mentioned features and shows some results to demonstrate its ability in terms of stability and accuracy with a high potential to be optimized in order to solve the challenging engineering problems of the next decades.of the solution of a given set of PDE’s while finite elements are 3D and finite volume are part 2D and part 3D). In addition particle methods are generally written in Lagrangian formulation avoiding the necessity of defining a spatial stabilization in convection dominated flows. Finite elements and finite volume are generally designed using an Eulerian formulation with some extra diffusion in the solution due to this stabilization requirement. However, some particle methods often require a mesh to interpolate and also to solve the problem losing some of their advantages in terms of efficiency. Even though there are some methods that do not use any mesh in their formulation, the interpolation methods become very complex introducing errors in the computation with noticeable extra diffusion. Having detected two main limitations of particle methods to solve Navier-Stokes equations for viscous incompressible flows, we propose in this work the following: to enhance the time integration using an explicit streamline based scheme computed with the old velocity vector allowing to enlarge the time steps of standard explicit schemes in advection dominated flows, in order to minimize the use of mesh based solvers, the velocity predictor and its correction is formulated purely on the particles as any spatial collocation method using a gradient recovery technique to include the pressure gradient and the viscous terms. This method is written in a Lagrangian formulation in a segregated way like a fractional step method. The computation of the predicted fractional velocity and its correction is done using our proposal explained above, i.e. streamline in time collocation in space scheme. On the other hand the pressure correction (Poisson solver) is carried out using a FEM like method. This method may be implemented in two ways: The mobile mesh version: where the particles represent the mesh nodes and a permanent remeshing is needed in order to avoid the severe restriction in the time steps imposed by the mesh motion. Remember that the mesh is only used to solve the Poisson problem for the pressure correction. The fixed mesh version: where there is a background mesh to do some computations, mainly for the pressure, and a certain amount of particles that move in a Lagrangian way transporting the velocity. Some interpolation between the particles and the fixed mesh is needed but the remeshing is completely avoided. This paper presents this novel approach built with the above mentioned features and shows some results to demonstrate its ability in terms of stability and accuracy with a high potential to be optimized in order to solve the challenging engineering problems of the next decades.