A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

This is the published version. Copyright 2011 © EDP SciencesThe newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be completely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compressible Navier-Stokes equations to demonstrate its performance

Simulation of steady mixed convection in a lid-driven cavity filled with newtonian fluid by finite volume method

The steady mixed convection flow in a lid-driven cavity was simulated. The cavity was filled with a Newtonian fluid, both vertical walls are adiabatic, while the horizontal walls were either fixed cold and uniformly/oscillatory heated. Firstly, the effect of internal heat generation or absorption on the fluid flow and heat transfer behaviours was studied. The moving upper wall was uniformly heated while the bottom wall was kept cold. The effect of magnetic field on fluid flow and heat transfer was analysed in the second problem. An inclined magnetic field was considered in the third problem. In the fourth problem, the flow inside an inclined cavity was simulated, where the top wall was subjected to an heated oscillating temperature. Finally, the mixed convection within an inclined cavity with the presence of an inclined magnetic field was studied. The dimensionless governing equations were formulated by using appropriate reference variables. These equations were solved using the finite volume method. The convection-diffusion terms were discretized using the power law scheme while the pressure and velocity components were coupled using the SIMPLE algorithms. The resultant matrices were then solved iteratively using the Tri- Diagonal Matrix Algorithm coded in FORTRAN90. The present solutions obtained were then compared with those of previous studies and a good agreement was found. The numerical results were presented in the forms of isotherm and streamline. It was found that the heat transfer rate in an inclined cavity increased mildly for both forced convection dominated and mixed convection dominated regimes. However, for natural convection dominated regime, the heat transfer rate decreased when the inclination angle was 30◦ and increased when the inclination angles reached 60◦. The presence of external forces would affect the local heat transfer and fluid flow behaviours significantly

Space-time hybridized discontinuous Galerkin methods for shallow water equations

The non-linear shallow water equations model the dynamics of a shallow layer of an incompressible fluid; they are obtained by asymptotic analysis and depth-averaging of the Navier-Stokes equations. They are utilized in a wide range of applications, from simulation of geophysical phenomena such as river/oceanic flows and avalanches to the study of hurricane simulation, storm surge modeling, and oil spills. As a hyperbolic system of equations, shocks may develop in finite time and therefore an appropriate numerical discretization of these equations needs to be developed. The purpose of this dissertation is to develop and implement a state of the art numerical method to accurately model these equations. Therefore, a well-balanced space-time hybridized discontinuous Galerkin method was developed for our purpose. The method was implemented and tested for several benchmark problems and very promising results were obtained. An a priori error estimate for the developed method was also obtained with an optimal rate of convergence in an appropriate norm. The estimate obtained is an extension of the existing a priori error estimates in the literature, first to the case of a system of shallow water equations, second to a hybridized mixed DG method, and third to an arbitrary degree of polynomial in time.Computational Science, Engineering, and Mathematic

An Ellam Scheme for Advection-Diffusion Equations in Two Dimensions

We develop an Eulerian{Lagrangian localized adjoint method (ELLAM) to solve two-dimensional advection-diusion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to mass-conservative numerical schemes. The computational advantages of the ELLAM approximation have been demonstrated for a number of one-dimensional transport systems; practical implementations of ELLAM schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper. Extensive numerical results are presented to compare the ELLAM scheme with many widely used numerical methods and to demonstrate the strength of the ELLAM scheme

A space-time high-order implicit shock tracking method for shock-dominated unsteady flows

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. In this work, we extend implicit shock tracking to time-dependent problems using a slab-based space-time approach. This is achieved by reformulating a time-dependent conservation law as a steady conservation law in one higher dimension and applying existing implicit shock tracking techniques. To avoid computations over the entire time domain and unstructured mesh generation in higher dimensions, we introduce a general procedure to generate conforming, simplex-only meshes of space-time slabs in such a way that preserves features (e.g., curved elements, refinement regions) from previous time slabs. The use of space-time slabs also simplifies the shock tracking problem by reducing temporal complexity. Several practical adaptations of the implicit shock tracking solvers are developed for the space-time setting including 1) a self-adjusting temporal boundary, 2) nondimensionalization of a space-time slab, 3) adaptive mesh refinement, and 4) shock boundary conditions, which lead to accurate solutions on coarse space-time grids, even for problem with complex flow features such as curved shocks, shock formation, shock-shock and shock-boundary interaction, and triple points.Comment: 35 pages, 20 figure

Finite element model for three-dimensional compressible turbulent flows

Due to the complexity of the Navier-Stokes equations, numerical methods are widely used to analyze the flows. In this thesis, we establish a finite element model for three-dimensional compressible turbulent flows. We modified an in-house code in order to use several types of elements in a computational domain. We used four types of elements in our mesh: the 8-node hexahedron, the 4-node tetra, the 6-node prism, and the 5-node pyramid. The original code used only the 4-node tetra elements. We used the Streamline Upwind/Petrov-Galerkin stabilization technique with a shock capturing operator. We validated the code with benchmark tests using the 3D Naca0012 model and the DLR F11 model. We used different sets of Reynolds numbers, Mach numbers, and angles of attack to test the code and compare our results with other numerical and experimental results. Because of the strong nonlinearities with the increase of the angle of attack, we need to set up a solution strategy to avoid divergence of the solution. The tests of verification and validation show that the results we obtained are comparable to those of the references

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

A mesh transparent numerical method for large-eddy simulation of compressible turbulent flows

A Large Eddy-Simulation code, based on a mesh transparent algorithm, for hybrid unstructured meshes is presented to deal with complex geometries that are often found in engineering flow problems. While tetrahedral elements are very effective in dealing with complex geometry, excessive numerical diffusion often affects results. Thus, prismatic or hexahedral elements are preferable in regions where turbulence structures are important. A second order reconstruction methodology is used since an investigation of a higher order method based upon Lele's compact scheme has shown this to be impractical on general unstructured meshes. The convective fluxes are treated with the Roe scheme that has been modified by introducing a variable scaling to the dissipation matrix to obtain a nearly second order accurate centred scheme in statistically smooth flow, whilst retaining the high resolution TVD behaviour across a shock discontinuity. The code has been parallelised using MPI to ensure portability. The base numerical scheme has been validated for steady flow computations over complex geometries using inviscid and RANS forms of the governing equations. The extension of the numerical scheme to unsteady turbulent flows and the complete LES code have been validated for the interaction of a shock with a laminar mixing layer, a Mach 0.9 turbulent round jet and a fully developed turbulent pipe flow. The mixing layer and round jet computations indicate that, for similar mesh resolution of the shear layer, the present code exhibits results comparable to previously published work using a higher order scheme on a structured mesh. The unstructured meshes have a significantly smaller total number of nodes since tetrahedral elements are used to fill to the far field region. The pipe flow results show that the present code is capable of producing the correct flow features. Finally, the code has been applied to the LES computation of the impingement of a highly under-expanded jet that produces plate shock oscillation. Comparison with other workers' experiments indicates good qualitative agreement for the major features of the flow. However, in this preliminary computation the computed frequency is somewhat lower than that of experimental measurements.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Spectral hulls: a degree of freedom reducing hp-strategy in space/time

Reducing the degrees of freedom (DOF) of modern finite element methods is investigated using a systematic hp-process. The elements are first agglomerated (h-coarsening) to form convex/concave hulls and then the polynomial degree of the hull basis, is increased (p-refinement). Compared to the conventional continuous/discontinuous FEM, this mechanism yields more accurate solutions with smaller DOF. This methodology is validated throughout the dissertation using various methods including Fourier-Chebyshev collocation, Continuous Galerkin (CG), Discontinuous Galerkin (DG) and Discontinuous Least-Squares (DLS) on structured and/or arbitrary unstructured grids. The feasibility of such procedure is first investigated in time only by letting the spatial discretization to be fixed to an arbitrary spectral/finite element discretization. In this scenario, lower order time steps (elements) are agglomerated into a space-time hull. A general system of Volterra integral equation is then developed which is simultaneously applicable to $\partial^v /\partial t^v$ time dependency of the PDE. The reduction in DOF is demonstrated by validating a one-dimensional periodic convection test case and two-dimensional scattering from engineering geometries. Motivated by these results, the ideas are then generalized to space. This requires special grid generation and general polyhedral basis functions, called spectral hull basis, which are addressed in detail. In particular, a new set of basis functions are derived based on the SVD of the Vandermonde matrix which are proven to have small Lebesgue constant. Various theoretical results are presented including the derivation of a closed form relation for the Lebesgue constant on a polyhedron, derivation of a closed form relation for approximate Fekete points on a polyhedron and a new proof of Weierstrass approximation theorem in a polyhedral subset of d-dimensional space. One application of the proposed hull basis is to reduce the DOF of discontinuous FEM such that it can compete in practice with CG. The accuracy and efficiency of spectral hulls are demonstrated in a linear acoustics test case and a two-dimensional compressible vortex shedding problem