A finite volume discretization method for flow on structured and unstructured anisotropic meshes

This project is concerned with advection discretization technology within the field of Computational Fluid Dynamics (CFD). To this end, two novel methods are proposed which are dubbed the Enhanced Taylor (ET) Schemes. The model equation for this work is the advection-diffusion equation with the industrial application being incompressible ow. The objective of the proposed schemes is to achieve increased accuracy on structured and unstructured anisotropic meshes. One of the schemes focuses on improving advection accuracy, and the other on improving total advection-diffusion accuracy. Fundamental to the design of the ET schemes is the primary focus on face accuracy, with the additional incorporation of the up and downwind mesh stretching factors and ow gradients. Additionally, non-linear blending with the existing NVSF scheme was effected in the interest of robustness and stability, particularly on equispaced meshes. The developed schemes, along with prominent linear ĸ-Upwind schemes were critically assessed and compared. Current methods were shown to be at best 3rd and 1st-order accurate at non-equispaced faces and nodes respectively. In contrast, the developed schemes were shown to be up to 4th and 2nd-order accurate. Numerical experiments followed. This involved applying the prominent and developed schemes to solve the 1D advection-diffusion equation on stretched meshes. The 2D case involved incompressible ow in a lid-driven cavity. Anisotropic structured and unstructured meshes were employed. Significant improvements in accuracy were found with the ET schemes, with average reductions in error measuring up to a 50%. In comparison to existing methods, it is proposed that state-of-the-art technology has been developed

A space-time discontinuous Galerkin finite element method for two-fluid problems

A space-time discontinuous Galerkin finite element method for two fluid flow problems is presented. By using a combination of level set and cut-cell methods the interface between two fluids is tracked in space-time. The movement of the interface in space-time is calculated by solving the level set equation, where the interface geometry is identified with the 0-level set. To enhance the accuracy of the interface approximation the level set function is advected with the interface velocity, which for this purpose is extended into the domain. Close to the interface the mesh is locally refined in such a way that the 0-level set coincides with a set of faces in the mesh. The two fluid flow equations are solved on this refined mesh. The procedure is repeated until both the mesh and the flow solution have converged to a reasonable accuracy.\ud The method is tested on linear advection and Euler shock tube problems involving ideal gas and compressible bubbly magma. Oscillations around the interface are eliminated by choosing a suitable interface flux

An Extended Volume of Fluid Method and its Application to Single Bubbles Rising in a Viscoelastic Liquid

An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume-averaging of the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme is proposed for the second-order accurate finite-volume discretization of the interface stress term. We demonstrate that this scheme allows for a consistent treatment of the interface stress and the surface tension force in the pressure equation of the segregated solution approach. Because of the high Weissenberg number problem, an appropriate stabilization approach is applied to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. Direct numerical simulations of the transient motion of a bubble rising in a quiescent viscoelastic fluid are performed for the purpose of experimental code validation. The well-known jump discontinuity in the terminal bubble rise velocity when the bubble volume exceeds a critical value is captured by the method. The formulation of the interfacial stress together with the novel scheme for its discretization is found crucial for the quantitatively correct prediction of the jump discontinuity in the terminal bubble rise velocity

Stratified shallow flow modelling

Environmental hydraulics covers a very wide range of applications including free surface flows in rivers. estuaries and lakes. To find engineering solutions to environmental hydraulics problems. 3D numerical modelling is nowadays widely used. However. the computation of sharp spatial gradients (such as found in stratified estuaries and lakes. around plumes near outfalls along rivers and coasts or in exchange areas of high shear). and the modelling of these processes along steep bathymetric slopes (such as found at the edge of dredged channels or of the continental shelf) remains a challenge. In addition. crude assumptions (such as the hydrostatic assumption) are often made to the primary differential equations in order to simplify the problem and enable long term prediction of environmental hydraulic changes. In this thesis. a robust adaptive mesh displacement (AMD) method is implemented and validated against the lock exchange case in particular. The AMD method aims at vertically focusing nodes within each water column to capture sharp gradients. while reducing the number of nodes or requiring prior knowledge of the flow structure. Second. a direct computation of dynamic pressure is introduced based on the equation of vertical momentum and validated against the analytical potential flow theory solution of a source-sink pair. Dynamic pressure is necessary to model destratification recirculation devices. or flow over dredge channel. or solitary waves. for instance. This direct computation method makes the hydrostatic assumption redundant. Third. a new advection scheme is implemented. whose main advantage is simplicity averaging over Riemann problems without solving them. while excessive numerical viscosity is compensated for by using high-resolution MUSCL type reconstruction. Recommendations are made in this thesis to extend the advection scheme developed herein for tracer advection to the non-linear shallow water equations. to the diffusion terms and to turbulence closure laws within the same finite element framework

DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models

Atmospheric dynamical cores are a fundamental component of global atmospheric modeling systems and are responsible for capturing the dynamical behavior of the Earth's atmosphere via numerical integration of the Navier-Stokes equations. These systems have existed in one form or another for over half of a century, with the earliest discretizations having now evolved into a complex ecosystem of algorithms and computational strategies. In essence, no two dynamical cores are alike, and their individual successes suggest that no perfect model exists. To better understand modern dynamical cores, this paper aims to provide a comprehensive review of 11 non-hydrostatic dynamical cores, drawn from modeling centers and groups that participated in the 2016 Dynamical Core Model Intercomparison Project (DCMIP) workshop and summer school. This review includes a choice of model grid, variable placement, vertical coordinate, prognostic equations, temporal discretization, and the diffusion, stabilization, filters, and fixers employed by each syste

Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods

Computational science, including computational fluid dynamics (CFD), has become an indispensible tool for scientific discovery and engineering design, yet a key remaining challenge is to simultaneously ensure accuracy, efficiency, and robustness of the calculations. This research focuses on advancing a class of high-order finite element methods and develops a set of algorithms to increase the accuracy, efficiency, and robustness of calculations involving convection and diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses high-order discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for simultaneous mesh and order adaptation to reduce the error in a scalar functional of the approximate solution to the discretized equations. Contributions are made in key aspects of these methods applied to general systems of equations, addressing the scalability and memory requirements, accuracy of HDG methods, and efficiency and robustness with new adaptation methods. First, this work generalizes existing HDG methods to systems of equations, and in so doing creates a new primal formulation by applying DG stabilization methods as the viscous stabilization for HDG. The primal formulation is shown to be even more computationally efficient than the existing methods. Second, by instead keeping existing viscous stabilization methods and developing a new convection stabilization, this work shows that additional accuracy can be obtained, even in the case of purely convective systems. Both HDG methods are compared to DG in the same computational framework and are shown to be more efficient. Finally, the set of adaptation frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options. The second framework intended for simplex meshes extends the mesh optimization via error sampling and synthesis (MOESS) method to incorporate order adaptation. Collectively, the results from this research address a number of key issues that currently are at the forefront of high-order CFD methods, and particularly to output-based hp-adaptation for DG and HDG methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/137150/1/jdahm_1.pd