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- ItemOptimal control of buoyancy-driven liquid steel stirring modeled with single-phase Navier–Stokes equations(Berlin ; Heidelberg : Springer, 2021) Wilbrandt, Ulrich; Alia, Najib; John, VolkerGas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible Navier–Stokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.
- ItemOn the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows(Amsterdam [u.a.] : Elsevier Science, 2021) GarcĂa-Archilla, Bosco; John, Volker; Novo, JuliaThe 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.
- ItemOn the feasibility of using open source solvers for the simulation of a turbulent air flow in a dairy barn(Amsterdam [u.a.] : Elsevier, 2020) Janke, David; Caiazzo, Alfonso; Ahmed, Naveed; Alia, Najib; Knoth, Oswald; Moreau, Baptiste; Wilbrandt, Ulrich; Willink, Dilya; Amon, Thomas; John, VolkerTwo transient open source solvers, OpenFOAM and ParMooN, and the commercial solver Ansys Fluent are assessed with respect to the simulation of the turbulent air flow inside and around a dairy barn. For this purpose, data were obtained in an experimental campaign at a 1:100 scaled wind tunnel model. All solvers used different meshes, discretization schemes, and turbulence models. The experimental data and numerical results agree well for time-averaged stream-wise and vertical-wise velocities. In particular, the air exchange was predicted with high accuracy by both open source solvers with relative differences less than 4% and by the commercial solver with a relative difference of 9% compared to the experimental results. With respect to the turbulent quantities, good agreements at the second (downwind) half of the barn inside and especially outside the barn could be achieved, where all codes accurately predicted the flow separation and, in many cases, the root-mean-square velocities. Deviations between simulations and experimental results regarding turbulent quantities could be observed in the first part of the barn. These deviations can be attributed to the utilization of roughness elements between inlet and barn in the experiment that were not modeled in the numerical simulations. Both open source solvers proved to be promising tools for the accurate prediction of time-dependent phenomena in an agricultural context, e.g., like the transport of particulate matter or pathogen-laden aerosols in and around agricultural buildings. © 2020 The Authors
- ItemExistence of solutions of a finite element flux-corrected-transport scheme(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2020) John, Volker; Knobloch, PetrThe existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection-diffusion-reaction equations and transport equations.
- ItemOptimal control of a buoyancy-driven liquid steel stirring modeled with single-phase Navier--Stokes equations(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2020) Wilbrandt, Ulrich; Alia, Najib; John, VolkerGas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible NavierStokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.
- ItemAn assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2020) Ahmed, Naveed; John, VolkerA numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the the residual-based VMS methods, in particular when used with a pair of inf-sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver.
- ItemOn reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion-reaction equations(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2020) Frerichs, Derk; John, VolkerA standard discontinuous Galerkin (DG) finite element method for discretizing steady-state convection-diffusion-reaction equations is known to be stable and to compute sharp layers in the convection-dominated regime, but also to show large spurious oscillations. This paper studies post-processing methods for reducing the spurious oscillations, which replace the DG solution in a vicinity of layers by a constant or linear approximation. Three methods from the literature are considered and several generalizations and modifications are proposed. Numerical studies with the post-processing methods are performed at two-dimensional examples.
- ItemError analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2021) John, Volker; Moreau, Baptiste; Novo, JuliaA reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov--Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. Numerical studies illustrate the performance of the SUPG-ROM method.
- ItemOn a technique for reducing spurious oscillations in DG solutions of convection-diffusion equations(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2022) Frerichs-Mihov, Derk; John, VolkerThis note studies a generalization of a post-processing technique and a novel method inspired by the same technique which significantly reduce spurious oscillations in discontinuous Galerkin solutions of convection-diffusion equations in the convection-dominated regime.