– The purpose of this paper is to highlight the possibilities of a novel Lagrangian formulation in dealing with the solution of the incompressible Navier‐Stokes equations with very large time steps.
Design/methodology/approach
– The design of the paper is based on introducing the origin of this novel numerical method, originally inspired on the Particle Finite Element Method (PFEM), summarizing the previously published theory in its moving mesh version. Afterwards its extension to fixed mesh version is introduced, showing some details about the implementation.
Findings
– The authors have found that even though this method was originally designed to deal with heterogeneous or free‐surface flows, it can be competitive with Eulerian alternatives, even in their range of optimal application in terms of accuracy, with an interesting robustness allowing to use large time steps in a stable way.
Originality/value
– With this objective in mind, the authors have chosen a number of benchmark examples and have proved that the proposed algorithm provides results which compare favourably, both in terms of solution time and accuracy achieved, with alternative approaches, implemented in in‐house and commercial codes.
Abstract
Purpose
– The purpose of this paper is to highlight the possibilities of a novel Lagrangian formulation in dealing with the solution of the incompressible Navier‐Stokes equations [...]
In a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier‐Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite‐element code is extremely easy with the version presented here because the new shape functions are based on the usual finite‐element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.
Abstract
In a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using [...]
Addresses two difficulties which arise when using a compressible code with equal order interpolation (non‐staggered grids in the finite‐difference nomenclature) to capture a steady‐state solution in the incompressible limit, i.e. at low Mach numbers. Explains that, first, numerical instabilities in the form of spurious oscillations in pressure pollute the solution and, second, the convergence to the steady state becomes extremely slow owing to bad conditioning of the different speeds of propagation. By using a stabilized method, allows the use of equal‐order interpolations in a consistent (weighted‐residual) formulation which stabilizes both the convection and the continuity terms at the same time. On the other hand, by using specially devised preconditioning, assures a rate of convergence independent of Mach number.
Abstract
Addresses two difficulties which arise when using a compressible code with equal order interpolation (non‐staggered grids in the finite‐difference nomenclature) to capture a steady‐state [...]