Local pressure‐correction for the Navier‐Stokes equations

This article presents a novel local pressure correction method for incompressible fluid flows and documents a numerical study of this method. Pressure correction methods decouple the velocity and pressure components of the time-dependent Navier-Stokes equations and lead to a sequence of elliptic partial differential equations for both components instead of a saddle point problem. In some sit- uations, the equations for the velocity components are solved explicitly (with time step restrictions) and thus the elliptic pressure problem remains to be the most expensive step. Here, we employ a multiscale procedure for the solution of the Poisson problem related to pressure. The procedure replaces the global Poisson problem by local Poisson problems on subregions. We propose a new Robin-type boundary condition design for the local Poisson problems, which contains a coarse approximation of the global Poisson problem. Accordingly, no further communication between subregions is necessary and the method is perfectly adapted for parallel computations. Numerical experiments regarding a known analytical solution and flow around cylinder benchmarks show the effectivity of this new local pressure correction method

Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1, Reference solutions

There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of 0.6, (constant property and variable property cases) and (variable property case). These reference solutions were produced after a first international workshop organized by CEA and LIMSI in January 2000, in which the above authors volunteered to produce accurate numerical solutions from which the present reference solutions could be established

Footprint and detectability of a well leaking CO2 in the Central North Sea: Implications from a field experiment and numerical modelling

Highlights • CO2 gas bubbles are completely dissolved within 2 m above the seabed. • CO2 is not emitted into the atmosphere but retained in the North Sea. • Dissolved CO2 is rapidly dispersed by tidal currents in the North Sea. • Harmful effects on benthic biota occur in the direct vicinity of the leak. • Monitoring has to be performed at the seabed and close to the leak. Abstract Existing wells pose a risk for the loss of carbon dioxide (CO2) from storage sites, which might compromise the suitability of carbon dioxide removal (CDR) and carbon capture and storage (CCS) technologies as climate change mitigation options. Here, we show results of a controlled CO2 release experiment at the Sleipner CO2 storage site and numerical simulations that evaluate the detectability and environmental consequences of a well leaking CO2 into the Central North Sea (CNS). Our field measurements and numerical results demonstrate that the detectability and impact of a leakage of <55 t yr−1 of CO2 would be limited to bottom waters and a small area around the leak, due to rapid CO2 bubble dissolution in seawater within the lower 2 m of the water column and quick dispersion of the dissolved CO2 plume by strong tidal currents. As such, the consequences of a single well leaking CO2 are found to be insignificant in terms of storage performance. Only prolonged leakage along numerous wells might compromise long-term CO2 storage and may adversely affect the local marine ecosystem. Since many abandoned wells leak natural gas into the marine environment, hydrocarbon provinces with a high density of wells may not always be the most suitable areas for CO2 storage

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instance streamline upwind Petrov-Galerkin (SUPG), are optimal on isotropic meshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. We describe the new method and prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropic stabilization methods. The capability of the method is illustrated by means of two numerical test problems

Linear-quadratic optimal control for the Oseen equations with stabilized finite elements

For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilized Petrov-Galerkin (PSPG). In this work we study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. In particular, we address the question whether a possible commutation error in optimal control problems lead to a decline of convergence order. Therefore, we give a priori estimates for SUPG/PSPG. In a numerical study for a flow with boundary layers, we illustrate to which extend the commutation error affects the accuracy

Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of capillarity effects close to phase boundaries. Standard numerical discretizations are known to violate discrete versions of inherent energy inequalities, thus leading to spurious dynamics of computed solutions close to static equilibria (e.g., parasitic currents). In this work, we propose a time-implicit discretization of the problem, and use piecewise linear (or bilinear), globally continuous finite element spaces for both, velocity and density fields, and two regularizing terms where corresponding parameters tend to zero as the mesh-size h > 0 tends to zero. Solvability, non-negativity of computed densities, as well as conservation of mass, and a discrete energy law to control dynamics are shown. Computational experiments are provided to study interesting regimes of coefficients for viscosity and capillarity

Computing and Information HIERARCHICAL A POSTERIORI RESIDUAL BASED ERROR ESTIMATORS FOR BILINEAR FINITE ELEMENTS

Abstract. We present techniques of a posteriori error estimation for Q1 finite element discretizations based on residual evaluations with respect to test functions of higher-order. This technique is designed for quadrilateral (or hexahedral) triangulations and gives local error indicators in terms of nodal contributions. We show reliability and efficiency of the estimator. Moreover, we present a simplification which is attractive from computational point of view as well. Key words. error estimates, adaptivity, finite elements 1