High-resolution numerical schemes for compressible flows and\ud compressible two-phase flows

Several high-resolution numerical schemes based on the Constrained Interpolation Profile Conservative Semi-Lagrangian (CIP-CSL), Essentially Non-Oscillatory (ENO), Weighted ENO (WENO), Boundary Variation Diminishing (BVD), and Tangent of Hyperbola for INterface Capturing (THINC) schemes have been proposed for compressible flows and compressible two-phase flows. In the first part of the thesis, three high-resolution CIP-CSL schemes are proposed. (i) A fully conservative and less oscillatory multi-moment scheme (CIP-CSL3-ENO) is proposed based on two CIP-CSL3 schemes and the ENO scheme. An ENO indicator is designed to intentionally select non-smooth stencil but can efficiently minimise numerical oscillations. (ii) Motivated by the observation that combining two different types of reconstruction functions can effectively reduce numerical diffusion and oscillations, a better-suited scheme CIP-CSL-ENO5 is proposed based on hybrid-type CIP-CSL reconstruction functions and a newly designed ENO indicator. (iii) To further reduce the numerical diffusion in vicinity of discontinuities, the BVD and THINC schemes are implemented in the CIP-CSL framework. The resulting scheme accurately capture both smooth and discontinuous solutions simultaneously by selecting an appropriate reconstruction function. In the second part of the thesis, the TWENO (Target WENO) scheme is proposed to improve the accuracy of the fifth-order WENO scheme. Unlike conventional WENO schemes, the TWENO scheme is designed to restore the highest possible order interAbstract iv polation when three sub-stencils or two adjacent sub-stencils are smooth. To further minimise the numerical diffusion across discontinuities, the TWENO scheme is implemented with the THINC scheme and the Total Boundary Variation Diminishing (TBVD) algorithm. The resulting scheme TBVD-TWENO-THINC is also applied to solve the five-equation model for compressible two-phase flows. Verified through a wide range of benchmark tests, the proposed numerical schemes are able to obtain accurate and high-resolution numerical solutions for compressible flows and compressible two-phase flows

Sharp interface schemes for multi-material computational fluid dynamics

In this thesis we consider the solution of compressible multi-material flow problems, where each material is governed by the Euler equations and the material interface may be considered to be a perfect discontinuity separating macroscopic pure-material regions. Working in the framework of Godunov-type finite volume methods, we develop numerical algorithms for tracking the material interface and evolving fluid states. For the task of tracking the location of sharp material interfaces, we focus on volume tracking methods due to their ability to conserve mass in highly deformational flows. Three original contributions are presented in this area. First, the accuracy of the volume-of-fluid algorithm is improved through the addition of marker particles. Next, the efficient moment-of-fluid method is presented. This improves the computational efficiency of the moment-of-fluid method by a factor of three by mapping certain quantities during the interface reconstruction step on to a pre-computed data structure. Finally, a general framework for updating volume fractions based on the solution to a quadratic programming problem is presented. The evolution of fluid states in the full multi-material system is an independent problem. We present developments to two numerical approaches for this problem. The first, the ghost fluid method, is widely used due to the ease in which pure-fluid algorithms can be extended to the multi-material case. We investigate the effect of using a number of different interface tracking methods on the solutions, and find that the conservation errors vary by more than an order of magnitude. The ghost fluid method is then altered such that the ghost state extrapolation step is eliminated from the algorithm, allowing volume fraction-based interface tracking methods to be coupled. In the final chapter, we tackle the same problem using a mixture model-based method. The numerical method presented here is based on work by Miller and Puckett in 1996, in which a six-equation system using the assumption of pressure and velocity equilibrium was used to model two-material flow. We have thoroughly overhauled this method, incorporating Riemann solvers developed for the five-equation system, as well as a robust implicit energy update. We present numerical results on a range of one- and two-dimensional shock-interface interaction test problems which demonstrate the ability of the method to match the solutions from the five-equation model while maintaining a perfectly sharp material interface.Funded by AW

Numerical modelling of rapidly varied river flow

A new approach to solve shallow water flow problems over highly irregular geometry both correctly' and efficiently is presented in this thesis. Godunov-type schemes which are widely used with the finite volume technique cannot solve the shallow water equations correctly unless the source terms related to the bed slope and channel width variation are discretized properly, because Godunov-type schemes were developed on the basis of homogeneous governing equations which is not compatible with an inhomogeneous system. The main concept of the new approach is to avoid a fractional step method and transform the shallow water equations into homogeneous form equations. New definitions for the source terms which can be incorporated into the homogeneous form equations are also proposed in this thesis. The modification to the homogeneous form equations combines the source terms with the flux term and solves them by the same solution structure of the numerical scheme. As a result the source terms are automatically discretized to achieve perfect balance with the flux terms without any special treatment and the method does not introduce numerical errors. Another point considered to achieve well-balanced numerical schemes is that the channel geometry should be reconstructed in order to be compatible with the numerical flux term which is computed with piecewise constant initial data. In this thesis, the channel geometry has been changed to have constant state inside each cell and, consequently, each cell interface is considered as a discontinuity. The definition of the new flux related to the source terms has been obtained on the basis of the modified channel geometry. A simple and accurate algorithm to solve the moving boundary problem in two-dimensional modelling case has also been presented in this thesis. To solve the moving boundary condition, the locations of all the cell interfaces between the wet and dry cells have been detected first and the integrated numerical fluxes through the interfaces have been controlled according to the water surface level of the wet cells. The proposed techniques were applied to several well-known conservative schemes including Riemann solver based and verified against benchmark tests and natural river flow problems in the one and two dimensions. The numerical results shows good agreement with the analytical solutions, if available, and recorded data from other literature. The proposed approach features several advantages: 1) it can solve steady problems as well as highly unsteady ones over irregular channel geometry, 2) the numerical discretization of the source terms is always performed as the same way that the flux term is treated, 3) as a result, it shows strong applicability to various conservative numerical schemes, 4) it can solve the moving (wetting/drying) boundary problem correctly. The author believes that this new method can be a good option to simulate natural river flows over highly irregular geometries

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

A Class of upwind methods for Conservation Laws

Various new methods for the solution of hyperbolic systems of conservation laws in one, two and three space dimensions are developed. All are explicit, conservative timemarching methods that are second order accurate in space and time in regions of smooth flow and make use of local Riemann problems at intercell boundaries. In one space dimension, the Weighted Average Flux (w af ) approach of Toro is extended to generate a scheme that is stable with timesteps twice as large as those allowed by the stability conditions of the original scheme. A Riemann problem based extension of the Warming-Beam scheme is considered. Total Variation Diminishing (t v d ) conditions are enforced for both schemes. Numerical results for the Euler Equations of Gas Dynamics are presented. In two and three space dimensions, finite volume versions of the waf scheme on Cartesian grids are derived for the linear advection equation. Two two dimensional schemes are found that are second order accurate in space and time. One of these is extended for the solution of nonlinear systems of hyperbolic conservation laws in two separate ways. The resulting schemes are tested on the Shallow Water equations. The equivalent three dimensional schemes are also discussed. The two dimensional schemes are then extended for use on structured, body-fitted grids of quadrilaterals and one of these extensions is used to demonstrate the phenomena of Mach reflection of shallow water bores.Ph

A dimensionally split Cartesian cut cell method for Computational Fluid Dynamics

We present a novel dimensionally split Cartesian cut cell method to compute inviscid, viscous and turbulent flows around rigid geometries. On a cut cell mesh, the existence of arbitrarily small boundary cells severely restricts the stable time step for an explicit numerical scheme. We solve this `small cell problem' when computing solutions for hyperbolic conservation laws by combining wave speed and geometric information to develop a novel stabilised cut cell flux. The convergence and stability of the developed technique are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). Subsequently, we develop the method further to be able to compute solutions for the compressible Navier-Stokes equations. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a full description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a wide range of test problems ranging from the nearly incompressible to the highly compressible flow regimes. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). It is the first presentation of a dimensionally split cut cell method for the compressible Navier-Stokes equations in the literature. Finally, we also present an extension of the cut cell method to solve high Reynolds number turbulent automotive flows using a wall-modelled Large Eddy Simulation (WMLES) approach. A full description is provided of the coupling between the (implicit) LES solution and an equilibrium wall function on the cut cell mesh. The combined methodology is used to compute results for the turbulent flow over a square cylinder, and for flow over the SAE Notchback and DrivAer reference automotive geometries. We intend to publish the promising results as part of a future publication, which would be the first assessment of a WMLES Cartesian cut cell approach for computing automotive flows to be presented in the literature.My time as a PhD student at the University of Cambridge was funded by a scholarship from the Cambridge Commonwealth, European & International Trust