An unstructured finite volume technique for the 3D Poisson equation on arbitrary geometry using a σ‐coordinate system

We present a solver for a three‐dimensional Poisson equation issued from the Navier–Stokes equations applied to model rivers, estuaries, and coastal flows. The three‐dimensional physical domain is composed of an arbitrary domain in the horizontal direction and is bounded by an irregular free surface and bottom in the vertical direction. The equations are transformed vertically to the σ‐coordinate system to obtain an accurate representation of top and bottom topographies. The method is based on a second‐order finite volume technique on prisms consisting of triangular grids in the horizontal direction. The algorithm is accompanied by an analysis of different linear system solvers in order to achieve fast solutions. Numerical experiments are conducted to test the numerical accuracy and the computational efficiency of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Dilation‐based shock capturing for high‐order methods

In this paper, we introduce a shock‐capturing artificial viscosity technique for high‐order unstructured mesh methods. This artificial viscosity model is based on a non‐dimensional form of the divergence of the velocity. The technique is an extension and improvement of the dilation‐based artificial viscosity methods introduced in Premasuthan et al., 15 and further extended in Nguyen and Peraire 27 . The approach presented has a number attractive properties including non‐dimensional analytical form, sub‐cell resolution, and robustness for complex shock flows on anisotropic meshes. We present extensive numerical results to demonstrate the performance of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Central finite volume methods for ideal and shallow water magnetohydrodynamics

We propose two‐dimensional central finite volume methods based on our multidimensional extensions of Nessyahu and Tadmor's one‐dimensional non‐oscillatory central scheme and a constrained transport‐type method to solve ideal magnetohydrodynamic problems (MHD) and shallow water magnetohydrodynamic problems (SMHD). The main numerical scheme is second‐order accurate both in space and time and uses an original Cartesian grid coupled to a Cartesian‐ or diamond‐staggered dual grid to by‐pass the resolution of the Riemann problems at the cell interfaces. To treat the non‐vanishing magnetic field/flux divergence we have constructed an adaptation of Evans and Hawley's constrained transport method specifically designed for central schemes. Our numerical results show the efficiency and the potential of the scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

A numerical model for run‐up of breaking waves

In the present paper, the numerical method for the three‐dimensional run‐up, given in Johnsgard and Pedersen [‘A numerical model for three‐dimensional run‐up’, Int. J. Numer. Methods Fluids, 24 , 913–931 (1997)], is extended to include wave breaking. In the fundamental problem of run‐up of a uniform bore, the present model is compared with analytical solutions from the literature. The numerical solutions converge, but very slowly. This is not due to the numerical model, but rather to the structure of the solutions themselves. Numerical results for two realistic but simplified tsunami cases are also presented. In the first case, two‐dimensional simulations are performed concerning the run‐up of a tsunami in Portugal, in the second case, the three dimensional wave pattern generated after a slide in Tafjord, Norway in 1931, is studied. A discussion of different aspects of the model is summarized at the end of the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite element method for the two‐dimensional extended Boussinesq equations

A new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119 :618)two‐dimensional extended Boussinesq equations is presented using a linear triangular finite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a finite element method for the one‐dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two‐dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright © 2002 John Wiley & Sons, Ltd.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Applications of the artificial compressibility method for turbulent open channel flows

A three‐dimensional (3‐D) numerical method for solving the Navier–Stokes equations with a standard k–ε turbulence model is presented. In order to couple pressure with velocity directly, the pressure is divided into hydrostatic and hydrodynamic parts and the artificial compressibility method (ACM) is employed for the hydrodynamic pressure. By introducing a pseudo‐time derivative of the hydrodynamic pressure into the continuity equation, the incompressible Navier–Stokes equations are changed from elliptic‐parabolic to hyperbolic‐parabolic equations. In this paper, a third‐order monotone upstream‐centred scheme for conservation laws (MUSCL) method is used for the hyperbolic equations. A system of discrete equations is solved implicitly using the lower–upper symmetric Gauss–Seidel (LU‐SGS) method. This newly developed numerical method is validated against experimental data with good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.