A semi-explicit multi-step method for solving incompressible navier-stokes equations

The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predictor-corrector semi-explicit methods. In the present work, we derive a scheme based on using the fractional step technique in conjunction with explicit multi-step time integration within the framework of Galerkin-type stabilized finite element methods. We show that under certain assumptions, a Runge–Kutta scheme equipped with the fractional step leads to an efficient semi-explicit method, where the pressure Poisson equation is solved only once per time step. Thus, the computational cost of the implicit step of the scheme is minimized. The numerical example solved validates the resulting scheme and provides the insights regarding its accuracy and computational efficiency.Peer ReviewedPostprint (published version

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

An efficient split-step framework for non-Newtonian incompressible flow problems with consistent pressure boundary conditions

Incompressible flow problems with nonlinear viscosity, as they often appear in biomedical and industrial applications, impose several numerical challenges related to regularity requirements, boundary conditions, matrix preconditioning, among other aspects. In particular, standard split-step or projection schemes decoupling velocity and pressure are not as efficient for generalised Newtonian fluids, since the additional terms due to the non-zero viscosity gradient couple all velocity components again. Moreover, classical pressure correction methods are not consistent with the non-Newtonian setting, which can cause numerical artifacts such as spurious pressure boundary layers. Although consistent reformulations have been recently developed, the additional projection steps needed for the viscous stress tensor incur considerable computational overhead. In this work, we present a new time-splitting framework that handles such important issues, leading to an efficient and accurate numerical tool. Two key factors for achieving this are an appropriate explicit–implicit treatment of the viscous and convective nonlinearities, as well as the derivation of a pressure Poisson problem with fully consistent boundary conditions and finite-element-suitable regularity requirements. We present first- and higher-order stepping schemes tailored for this purpose, as well as various numerical examples showcasing the stability, accuracy and efficiency of the proposed framework

High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of L2L^2 and HH(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard $L^2$-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free $H(\operatorname{div})$-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms.Comment: 24 pages, 13 figure

Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

[EN] In numerous engineering applications, such as polymer or blood flow, the dependence of fluid viscosity on the local shear rate plays an important role. Standard techniques using inf-sup stable finite elements lead to saddle-point systems posing a challenge even for state-ofthe-art solvers and preconditioners. Alternatively, projection schemes or time-splitting methods decouple equations for velocity and pressure, resulting in easier to solve linear systems. Although pressure and velocity correction schemes of high-order accuracy are available for Newtonian fluids, the extension to generalised Newtonian fluids is not a trivial task. Herein, we present a split-step scheme based on an explicit-implicit treatment of pressure, viscosity and convection terms, combined with a pressure Poisson equation with fully consistent boundary conditions. Then, using standard equal-order finite elements becomes possible. Stability, flexibility and efficiency of the splitting scheme is showcased in two challenging applications involving aortic aneurysm flow and human phonation.The authors gratefully acknowledge Graz University of Technology for the financial support of the Lead-project: Mechanics, Modeling and Simulation of Aortic Dissection.Schussnig, R.; Pacheco, D.; Kaltenbacher, M.; Fries, T. (2022). Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 335-344. https://doi.org/10.4995/YIC2021.2021.12217OCS33534