Incompressible lagrangian fluid flow with thermal coupling

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Lagrangian FE methods for coupled problems in fluid mechanics

This work aims at developing formulations and algorithms where maximum advantage of using Lagrangian finite element fluid formulations can be taken. In particular we concentrate our attention at fluid-structure interaction and thermally coupled applications, most of which originate from practical “real-life” problems. Two fundamental options are investigated - coupling two Lagrangian formulations (e.g. Lagrangian fluid and Lagrangian structure) and coupling the Lagrangian and Eulerian fluid formulations. In the first part of this work the basic concepts of the Lagrangian fluids, the so-called Particle Finite Element Method (PFEM) [1], [2] are presented. These include nodal variable storage, mesh re-construction using Delaunay triangulation/tetrahedralization and alpha shape-based method for identification of the computational domain boundaries. This shall serve as a general basis for all the further developments of this work.Postprint (published version

A finite Element model for incompressible flow problems

Postprint (published version

Lagrangian FE methods for coupled problems in fluid mechanics

Lagrangian finite element methods emerged in fluid dynamics when the deficiencies of the Eulerian methods in treating free surface flows (or generally domains undergoing large shape deformations) were faced. Their advantage relies upon natural tracking of boundaries and interfaces, a feature particularly important for interaction problems. Another attractive feature is the absence of the convective term in the fluid momentum equations written in the Lagrangian framework resulting in a symmetric discrete system matrix, an important feature in case iterative solvers are utilized. Unfortunately, the lack of the control over the mesh distortions is a major drawback of Lagrangian methods. In order to overcome this, a Lagrangian method must be equipped with an efficient re-meshing tool. This work aims at developing formulations and algorithms where maximum advantage of using Lagrangian finite element fluid formulations can be taken. In particular we concentrate our attention at fluid-structure interaction and thermally coupled applications, most of which originate from practical “real-life” problems. Two fundamental options are investigated - coupling two Lagrangian formulations (e.g. Lagrangian fluid and Lagrangian structure) and coupling the Lagrangian and Eulerian fluid formulations. In the first part of this work the basic concepts of the Lagrangian fluids, the so-called Particle Finite Element Method (PFEM) [1], [2] are presented. These include nodal variable storage, mesh re-construction using Delaunay triangulation/tetrahedralization and alpha shape-based method for identification of the computational domain boundaries. This shall serve as a general basis for all the further developments of this work. Next we show how an incompressible Lagrangian fluid can be used in a partitioned fluid-structure interaction context. We present an improved Dirichlet-Neumann strategy for coupling the incompressible Lagrangian fluid with a rigid body. This is finally applied to an industrial problem dealing with the sea-landing of a satellite capsule. In the following, an extension of the method is proposed to allow dealing with fluid-structure problems involving general flexible structures. The method developed takes advantage of the symmetry of the discrete system matrix and by introducing a slight fluid compressibility allows to treat the fluid-structure interaction problem efficiently in a monolithic way. Thus, maximum benefit from using a similar description for both the fluid (updated Lagrangian) and the solid (total Lagrangian) is taken. We show next that the developed monolithic approach is particularly useful for modeling the interaction with light-weight structures. The validation of the method is done by means of comparison with experimental results and with a number of different methods found in literature. The second part of this work aims at coupling Lagrangian and Eulerian fluid formulations. The application area is the modeling of polymers under fire conditions. This kind of problem consists of modeling the two subsystems (namely the polymer and the surrounding air) and their thermomechanical interaction. A compressible fluid formulation based on the Eulerian description is used for modeling the air, whereas a Lagrangian description is used for the polymer. For the surrounding air we develop a model based upon the compressible Navier-Stokes equations. Such choice is dictated by the presence of high temperature gradients in the problem of interest, which precludes the utilization of the Boussinesq approximation. The formulation is restricted to the sub-sonic flow regime, meeting the requirement of the problem of interest. The mechanical interaction of the subsystems is modeled by means of a one-way coupling, where the polymer velocities are imposed on the interface elements of the Eulerian mesh in a weak way. Thermal interaction is treated by means of the energy equation solved on the Eulerian mesh, containing thermal properties of both the subsystems, namely air and polymer. The developments of the second part of this work do not pretend to be by any means exhaustive; for instance, radiation and chemical reaction phenomena are not considered. Rather we make the first step in the direction of modeling the complicated thermo-mechanical problem and provide a general framework that in the future can be enriched with a more detailed and sophisticated models. However this would affect only the individual modules, preserving the overall architecture of the solution procedure unchanged. Each chapter concludes with the example section that includes both the validation tests and/or applications to the real-life problems. The final chapter highlights the achievements of the work and defines the future lines of research that naturally evolve from the results of this work

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $P_2$ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results

Development of CAE tools for fluid-structure interaction and erosion in turbomachinery virtual prototyping

The work presented in this thesis is based on the development of advanced computer aided engineering tools dedicated to multi-physics coupled problems. Starting from the state of the art of numerical tools used in virtual prototyping and testing of turbomachinery systems, we found two interesting and actual possible developments focused on the improved implementation of fluid-structure interaction and material wearing solvers. For both the topics we will present a brief overview with the contextualization on the industrial and research state of the art, the detailed description of mathematical models (Chapter 2), discretized (FEM) stabilized formulations, time integration schemes and coupling algorithms used in the implementation (Chapter 3). The second part of the thesis (Chapter 4-7) will report some application of the developed tools on some latest challenges in turbomachinery field as rain erosion and load control in wind turbines and non-linear aeroelasticity in large axial fans

Discretisations and Preconditioners for Magnetohydrodynamics Models

The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. We present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at a high coupling number $S=1{,}000$ by choosing the Rayleigh number in the range between 0 and $100{,}000$ as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174 page

Advancements in Fluid Simulation Through Enhanced Conservation Schemes

To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study new methods in solving the Navier-Stokes equations using models which have enhanced conservation qualities, in particular, the energy, momentum, and angular momentum conserving (EMAC) scheme. The EMAC scheme has gained popularity in the mathematics community over the past few years as a desirable method to model fluid flow. It has been proven to conserve energy, momentum, angular momentum, helicity, and others. EMAC has also been shown to perform better and maintain accuracy over long periods of time compared to other schemes. We investigate a fully discrete error analysis of EMAC and SKEW. We show that a problematic dependency on the Reynolds number is present in the analysis for SKEW, but not in EMAC under certain conditions. To further explore this concept, we include some numerical experiments designed to highlight these differences in the error analysis. Additionally, we include other projection methods to measure performance. Following this, we introduce a new EMAC variant which applies a differential spatial filter to the EMAC scheme, named EMAC-Reg. Standard models, including EMAC, require especially fine meshes with high Reynold\u27s numbers. This is problematic because the linear systems for 3D flows will be far too large and take an extraordinary amount of time to compute. EMAC-Reg not only performs better on a coarser mesh, but maintains conservation properties as well. Another topic in fluid flow computing that has been gaining recognition is reduced order models. This method uses experimental data to create new models of reduced computational complexity. We introduce the concept of consistency between a full order and a reduced order model, i.e., using the same numerical scheme for the full order and reduced order model. For inconsistency, one could use SKEW in the full order model and then EMAC for the reduced order model. We explore the repercussions of having inconsistency between these two models analytically and experimentally. To obtain a proper linear system from the Navier-Stokes equations, we must solve the nonlinear problem first. We will explore a method used to reduce iteration counts of nonlinear problems, known as Anderson acceleration. We will discuss how we implemented this using the finite element library deal.II \cite{dealII94}, measure the iteration counts and time, and compare against Newton and Picard iterations