Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft

HIGH ACCURACY METHODS AND REGULARIZATION TECHNIQUES FOR FLUID FLOWS AND FLUID-FLUID INTERACTION

This dissertation contains several approaches to resolve irregularity issues of CFD problems, including a decoupling of non-linearly coupled fluid-fluid interaction, due to high Reynolds number. New models present not only regularize the linear systems but also produce high accurate solutions both in space and time. To achieve this goal, methods solve a computationally attractive artificial viscosity approximation of the target problem, and then utilize a correction approach to make it high order accurate. This way, they all allow the usage of legacy code | a frequent requirement in the simulation of fluid flows in complex geometries. In addition, they all pave the way for parallelization of the correction step, which roughly halves the computational time for each method, i.e. solves at about the same time that is required for DNS with artificial viscosity. Also, methods present do not requires all over function evaluations as one can store them, and reuse for the correction steps. All of the chapters in this dissertation are self-contained, and introduce model first, and then present both theoretical and computational findings of the corresponding method

Efficient split-step schemes for fluid–structure interaction involving incompressible generalised Newtonian flows

Blood flow, dam or ship construction and numerous other problems in biomedical and general engineering involve incompressible flows interacting with elastic structures. Such interactions heavily influence the deformation and stress states which, in turn, affect the engineering design process. Therefore, any reliable model of such physical processes must consider the coupling of fluids and solids. However, complexity increases for non-Newtonian fluid models, as used, e.g., for blood or polymer flows. In these fluids, subtle differences in the local shear rate can have a drastic impact on the flow and hence on the coupled problem. There, existing (semi-) implicit solution strategies based on split-step or projection schemes for Newtonian fluids are not applicable, while extensions to non-Newtonian fluids can lead to substantial numerical overhead depending on the chosen fluid solver. To address these shortcomings, we present here a higher-order accurate, added-mass-stable fluid–structure interaction scheme centered around a split-step fluid solver. We compare several implicit and semi-implicit variants of the algorithm and verify convergence in space and time. Numerical examples show good performance in both benchmarks and an idealised setting of blood flow through an abdominal aortic aneurysm considering physiological parameters

Variable Time Step Method of DAHLQUIST, LINIGER and NEVANLINNA (DLN) for a Corrected Smagorinsky Model

Turbulent flows strain resources, both memory and CPU speed. The DLN method has greater accuracy and allows larger time steps, requiring less memory and fewer FLOPS. The DLN method can also be implemented adaptively. The classical Smagorinsky model, as an effective way to approximate a (resolved) mean velocity, has recently been corrected to represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. In this paper, we apply a family of second-order, G-stable time-stepping methods proposed by Dahlquist, Liniger, and Nevanlinna (the DLN method) to one corrected Smagorinsky model and provide the detailed numerical analysis of the stability and consistency. We prove that the numerical solutions under any arbitrary time step sequences are unconditionally stable in the long term and converge at second order. We also provide error estimate under certain time step condition. Numerical tests are given to confirm the rate of convergence and also to show that the adaptive DLN algorithm helps to control numerical dissipation so that backscatter is visible

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

Approximating fast, viscous fluid flow in complicated domains

Typical industrial and biological flows often occur in complicated domains that are either infeasible or impossible to resolve. Alternatives to solving the Navier-Stokes equations (NSE) for the fluid velocity in the pores of these problems must be considered. We propose and analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity tends to infinity and permeability K tends to 0 in solid obstacles, and K tends to infinity in fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider inhomogeneous Dirichlet boundary conditions and non-solenoidal velocity (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish conditions for the well-posedness of the continuous and discrete problem. We also establish convergence as Brinkman viscosity tends to infinity and K tends to 0 in solid obstacles, as K tends to infinity in fluid region, and as the mesh width vanishes. We prove similar results for time-dependent Brinkman equations for backward-Euler (BE) time-stepping. We provide numerical examples confirming theory including convergence of velocity, pressure, and drag/lift.We also investigate the stability and convergence of the fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We show herein that no such restriction is required. Moreover, we propose a new linear extrapolation of the convecting velocity for CNLE so that the approximating velocities converge without without time-step restriction in l^{infty}(H^1) along with the discrete time derivative of the velocity in l^2(L^2). The new extrapolation ensures energetic stability of CNLE in the case of inhomogeneous boundary data. Such a result is unknown for conventional CNLE (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE

Variable Stepsize, Variable Order Methods for Partial Differential Equations

Variable stepsize, variable order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in complex applications due to their computational complexity and the difficulty to implement them in legacy code. The goal of this dissertation is to develop, analyze, and test new VSVO methods that have the same computational complexity as their nonadaptive counterparts per step. Adaptivity allows these methods to take fewer steps, which makes them globally less complex. Herein, we show how to use any backward differentiation formula (BDF) method as the basis for a VSVO method. Order adaptivity is achieved using an inexpensive post-processing technique known as time filtering. Time filters do not add to the asymptotic complexity of these methods, and allow for every possible order in the VSVO family to be computed for the same cost as one BDF solve. This approach yields new, nonstandard timestepping methods that are not in the literature, and we analyze their stability and accuracy herein. Backward Euler (BDF1) and BDF2 are extremely ubiquitous methods, and this research demonstrates how they can be converted to order adaptive codes with only a few additional lines of code. We also develop a solver called Multiple Order One Solve Embedded 2,3,4 (MOOSE234). MOOSE234 is a VSVO method based on BDF3 that computes approximations of order two, three and four each step. All three approximations in MOOSE234 are at least A(alpha) stable, and the second order approximation is A stable. While these methods are generally applicable to any system that is first order in time, we focus on issues pertaining to the Navier-Stokes equations. Our methods have been optimized for Navier-Stokes solvers, and we include linearly implicit and implicit-explicit (IMEX) versions