On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open

A sensitivity study of artificial viscosity in a defect-deferred correction method for the coupled Stokes/Darcy model

This paper analyzes the sensitivity of the artificial viscosity in the defect deferred correction method for the non-stationary coupled Stokes/Darcy model. For the defect step and the deferred correction step of the defect deferred correction method, we respectively give the corresponding sensitivity systems related to the change of artificial viscosity. Finite element schemes are devised for computing solutions to the sensitivity systems. Finally, we will verify the theoretical analysis results through numerical experiments. Our results reveal that the solution is sensitive for small values of the artificial viscosity, and when the viscosity/hydraulic conductivity coefficients are small

Parallel Multiphase Navier-Stokes Solver

We study and implement methods to solve the variable density Navier-Stokes equations. More specifically, we study the transport equation with the level set method and the momentum equation using two methods: the projection method and the artificial compressibility method. This is done with the aim of numerically simulating multiphase fluid flow in gravity oil-water-gas separator vessels. The result of the implementation is the parallel Aspen software framework based on the massively parallel deal.II . For the transport equation, we briefly discuss the theory behind it and several techniques to stabilize it, especially the graph laplacian artificial viscosity with higher order elements. Also, we introduce the level set method to model the multiphase flow and study ways to maintain a sharp surface in between phases. For the momentum equation, we give an overview of the two methods and discuss a new projection method with variable time stepping that is second order in time. Then we discuss the new third order in time artificial compressiblity method and present variable density version of it. We also provide a stability proof for the discrete implicit variable density artificial compressibility method. For all the methods we introduce, we conduct numerical experiments for verification, convergence rates, as well as realistic models

Fast, Adaptive Algorithms for Flow Problems

Time-accurate simulations of physical phenomena (e.g., ocean dynamics, weather, and combustion) are essential to economic development and the well-being of humanity. For example, the economic toll hurricanes wrought on the United States in 2017 exceeded $\$200$ billon dollars. To mitigate the damage, the accurate and timely forecasting of hurricane paths are essential. Ensemble simulations, used to calculate mean paths via multiple realizations, are an invaluable tool in estimating uncertainty, understanding rare events, and improving forecasting. The main challenge in the simulation of fluid flow is the complexity (runtime, memory requirements, and efficiency) of each realization. This work confronts each of these challenges with several novel ensemble algorithms that allow for the fast, efficient computation of flow problems, all while reducing memory requirements. The schemes in question exploit the saddle-point structure of the incompressible Navier-Stokes (NSE) and Boussinesq equations by relaxing incompressibility appropriately via artificial compressibility (AC), yielding algorithms that require far fewer resources to solve while retaining time-accuracy. Paired with an implicit-explicit (IMEX) ensemble method that employs a shared coefficient matrix, we develop, analyze, and validate novel schemes that reduce runtime and memory requirements. Using these methods as building blocks, we then consider schemes that are time-adaptive, i.e., schemes that utilize varying timestep sizes. The consideration of time-adaptive artficial compressibility methods, used in the algorithms mentioned above, also leads to the study of a new slightly-compressible fluid flow continuum model. This work demonstrates stability and weak convergence of the model to the incompressible NSE, and examines two associated time-adaptive AC methods. We show that these methods are unconditionally, nonlinearly, long-time stable and demonstrate numerically their accuracy and efficiency. The methods described above are designed for laminar flow; turbulent flow is addressed with the introduction of a novel one-equation unsteady Reynolds-averaged Navier-Stokes (URANS) model with multiple improvements over the original model of Prandtl. This work demonstrates analytically and numerically the advantages of the model over the original

Ensemble time-stepping algorithms for natural convection

Predictability of fluid flow via natural convection is a fundamental issue with implications for, e.g., weather predictions including global climate change assessment and nuclear reactor cooling. In this work, we study numerical methods for natural convection and utilize them to study predictability. Eight new algorithms are devised which are far more efficient than existing ones for ensemble calculations. They allow for either increased ensemble sizes or denser meshes on current computing systems. The artificial compressibility ensemble (ACE) family produce accurate velocity and temperature approximations and are fastest. The speed of second-order ACE degrades as $\epsilon \rightarrow 0$ or $\Delta t \rightarrow 0$ due to the iterative solver. However, first-order ACE has a uniform solve time since $\gamma = \mathcal{O}(1)$. The ensemble backward differentiation formula (eBDF) family are most accurate and reliable. The penalty ensemble algorithm (PEA) family are strongly affected by the timestep and are least accurate. In particular, $\gamma = \mathcal{O}(1/ \Delta t^2)$ for second-order PEA leads to solver breakdown. We also propose an ACE turbulence (ACE-T) family of methods for turbulence modeling which are both fast and accurate. A complete numerical analysis is performed which establishes full-reliability. The analysis involves techniques that are novel and results that subsume, elucidate, and expand previous results in closely related fields, e.g., iso-thermal fluid flow. Numerical tests show predicted accuracy is consistent with theory. Predictability is a highly complex and problem-dependent phenomenon. Predictability studies are performed utilizing the new second-order ACE algorithm. We perform a numerical test where the flow reaches a steady state. It is found that increasing the size of the domain increases predictability. Also, spatial averages increase predictability with increasing filter radius. We also study a problem with a manufactured solution. Sufficiently large rotations increase the predictability of a flow. Further, spatial averages decrease predictability with increasing filter radius

Partitioned methods for coupled fluid flow problems

Many flow problems in engineering and technology are coupled in their nature. Plenty of turbulent flows are solved by legacy codes or by ones written by a team of programmers with great complexity. As knowledge of turbulent flows expands and new models are introduced, implementation of modern approaches in legacy codes and increasing their accuracy are of great concern. On the other hand, industrial flow models normally involve multi-physical process or multi-domains. Given the different nature of the physical processes of each subproblem, they may require different meshes, time steps and methods. There is a natural desire to uncouple and solve such systems by solving each subphysics problem, to reduce the technical complexity and allow the use of optimized legacy sub-problems' codes. The objective of this work is the development, analysis and validation of new modular, uncoupling algorithms for some coupled flow models, addressing both of the above problems. Particularly, this thesis studies: i) explicitly uncoupling algorithm for implementation of variational multiscale approach in legacy turbulence codes, ii) partitioned time stepping methods for magnetohydrodynamics flows, and iii) partitioned time stepping methods for groundwater-surface water flows. For each direction, we give comprehensive analysis of stability and derive optimal error estimates of our proposed methods. We discuss the advantages and limitations of uncoupling methods compared with monolithic methods, where the globally coupled problems are assembled and solved in one step. Numerical experiments are performed to verify the theoretical results