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

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

Computability and Adaptivity in CFD

We give a brief introduction to research on adaptive computational methods for laminar compressible and incompressible flow, and then focus on computability and adaptivity for turbulent incompressible flow, where we present a framework for adaptive finite element methods with duality- based a posteriori error control for chosen output quantities of interest. We show in concrete examples that outputs such as mean values in time of drag and lift of a bluff body in a turbulent flow are computable to a tolerance of a few percent, for a simple geometry using some hundred thousand mesh points, and for complex geometries some million mesh points

Acceleration Methods for Nonlinear Solvers and Application to Fluid Flow Simulations

This thesis studies nonlinear iterative solvers for the simulation of Newtonian and non- Newtonian fluid models with two different approaches: Anderson acceleration (AA), an extrapolation technique that accelerates the convergence rate and improves the robustness of fixed-point iterations schemes, and continuous data assimilation (CDA) which drives the approximate solution towards coarse data measurements or observables by adding a penalty term. We analyze the properties of nonlinear solvers to apply the AA technique. We consider the Picard iteration for the Bingham equation which models the motion of viscoplastic materials, and the classical iterated penalty Picard and Arrow-Hurwicz iterations for the incompressible Navier–Stokes equations (NSE) which model the Newtonian fluid flows. All these nonlinear solvers have some drawbacks. They lack robustness, and the required number of iterations for convergence could be large. In this thesis, we show that AA significantly improves the convergence properties of these nonlinear solvers, makes them more robust, and significantly reduces the number of iterations for convergence. We support our accelerated convergence analysis with various numerical tests. We also consider CDA, which is used to improve the convergence of Picard iterations for the first time in literature. We analyze the improved contraction property of CDA applied to the Picard scheme for steady NSE. We give results for several numerical experiments of CDA applied to the Picard iteration to solve 1D, 2D and 3D nonlinear partial differential equations and show that significant reduction in the required number of iterations thanks to CDA

Stabilized Isogeometric Collocation Methods For Scalar Transport and Incompressible Fluid Flow

In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation schemes address spurious oscillations that can arise from advection and pressure instabilities. Numerical examples for the advection-diffusion equation, Stokes equations, and incompressible Navier-Stokes equations show the effectiveness of the proposed stabilized schemes while maintaining the high-order convergence rates and accuracy of standard isogeometric collocation on smooth problems