An effective multigrid method for high-speed flows

The use is considered of a multigrid method with central differencing to solve the Navier-Stokes equations for high speed flows. The time dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for 2-D laminar flow over a circular cylinder and a 15 deg compression ramp

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

Improving the accuracy of central difference schemes

General difference approximations to the fluid dynamic equations require an artificial viscosity in order to converge to a steady state. This artificial viscosity serves two purposes. One is to suppress high frequency noise which is not damped by the central differences. The second purpose is to introduce an entropy-like condition so that shocks can be captured. These viscosities need a coefficient to measure the amount of viscosity to be added. In the standard scheme, a scalar coefficient is used based on the spectral radius of the Jacobian of the convective flux. However, this can add too much viscosity to the slower waves. Hence, it is suggested that a matrix viscosity be used. This gives an appropriate viscosity for each wave component. With this matrix valued coefficient, the central difference scheme becomes closer to upwind biased methods

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

High-order, linearly stable, partitioned solvers for general multiphysics problems based on implicit–explicit Runge–Kutta schemes

This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit–explicit Runge–Kutta (IMEX-RK) discretization of the semi-discrete equations. The generic multiphysics problem is modeled as a system of n systems of partial differential equations where the ith subsystem is coupled to the other subsystems through a coupling term that can depend on the state of all the other subsystems. This coupled system of partial differential equations reduces to a coupled system of ordinary differential equations via the method of lines where an appropriate spatial discretization is applied to each subsystem. The coupled system of ordinary differential equations is taken as a monolithic system and discretized using an IMEX-RK discretization with a specific implicit–explicit decomposition that introduces the concept of a predictor for the coupling term. We propose four coupling predictors that enable the monolithic system to be solved in a partitioned manner, i.e., subsystem-by-subsystem, and preserve the IMEX-RK structure and therefore the design order of accuracy of the monolithic scheme. The four partitioned solvers that result from these predictors are high-order accurate, allow for maximum re-use of existing single-physics software, and two of the four solvers allow the subsystems to be solved in parallel at a given stage and time step. We also analyze the stability of a coupled, linear model problem with a specific coupling structure and show that one of the partitioned solvers achieves unconditional linear stability for this problem, while the others are unconditionally stable only for certain values of the coupling strength. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems including a simple linear system of ODEs, advection–diffusion–reaction systems, fluid–structure interaction problems, and particle-laden flows, where we verify the design order of the IMEX schemes and study various stability properties

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a non-destructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well-balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme -- even on unstructured grids -- and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, super-linear convergence, mass-conservation, well-balancedness, and stability are verified