A computational method for the hydrodynamics of fractured-porous media

A numerical method based on the boundary-fitted finite difference method (BFDM) is presented in this paper. The boundaries are external (the boundary of the physical domain) and internal (which corresponds to the fracture network). The difference between this approach and the usual one lies in the inclusion of discrete fractures in the volume that represents the porous medium. The numerical model has been used in the prediction of the flow pattern in several internationally recognized verification cases and applied to the solution of hypothetical problems of interest to us in the field of nuclear waste repository modelling. The results obtained show that the numerical approach considered gives accurate and reliable predictions of the hydrodynamics of fractured-porous media, thus justifying its use for the above-mentioned studies.     

An operator splitting method with FDM and FEM for the convection- diffusion equation using boundary-fitted coordinate system

An operator splitting method combining finite difference method and finite element method is proposed in this paper by using boundary-fitted coordinate system. The governing equation is split into advection and diffusion equations and solved by finite difference method using boundary-fitted coordinate system and finite element method respectively. An example for which analytic solution is available is used to verified the proposed methods and the agreement is very good. Numerical results show that it is very efficient.     

A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model

In stratified three-dimensional models the use of a boundary-fitted vertical co-ordinate is known to produce errors in the horizontal pressure gradient calculation near steep topography. The error is due to the splitting of the horizontal pressure gradient term in each of the momentum equations into two parts and the subsequent incomplete cancellation of the truncation errors of those parts. In order to minimize these pressure gradient errors, a fourth-order-accurate pressure gradient calculation has been implemented and installed in SPEM, a three-dimensional primitive equation ocean model. The stability and accuracy of the new scheme are compared with those of the original second-order-accurate model in a series of calculations of unforced flow in the vicinity of an isolated seamount. The new scheme is shown to have much smaller pressure gradient errors over a wide range of parameter space as well as a greater parametric domain of numerical stability.     

Application of Boundary-Fitted Coordinate Transformations to groundwater flow modeling

The Boundary-Fitted Coordinate (BFC) Transformation method is a very powerful, efficient and accurate method of modeling heat or fluid flow in two- or three-dimensional domains with complex boundary shapes and abrupt changes in internal properties. Since the late 1970's it has become the modeling method of choice among many aerodynamicists and heat-flow modelers. It is being presented here for the first time as a new approach to modeling groundwater flow, based on successful research results in two dimensions. The BFC transformation method was employed to simulate two hypothetical well-flow scenarios in isotropic and anisotropic domains, and actual groundwater flows in the area of West Lafayette, Indiana. The numerical solutions in those cases were at least as accurate as and/or consistent with those obtained by purely finite difference and finite element methods, but with the added advantage of more accurate representation and implementation of the boundary condition in the region of great sensitivity. The BFC method successfully applied to two-dimensional simulations should be easily extended to simulations of three-dimensional flow and transport and thus, this research is continuing in that direction.     

A shock-capturing model based on flux-vector splitting method in boundary-fitted curvilinear coordinates

In this paper, a robust and accurate high-resolution finite-volume scheme is presented which employs flux-vector splitting (FVS) as the building block for solution of shallow water equations in boundary-fitted curvilinear coordinates. Eddy viscosity approach is used to accommodate shear stresses due to turbulence. Splitting of the convective terms is achieved via flux Jacobians whereas Liou–Steffen Splitting (LSS) technique, but in transformed coordinates, is used to split pressure terms. Limited flux gradients are also used to increase the computational accuracy of evaluation of interface fluxes and decrease the excessive numerical dissipation associated with FVS. This will completely remove spurious oscillations in high-gradient regions without introducing too much numerical dissipations. The method is tested for some classic simulations including hydraulic jump, 1D dam break and 2D dam break problems. The results show very satisfactory agreement with experimental data, analytical solutions and other numerical results.     

An integrated modelling system for coastal area dynamics

Two-dimensional initial-boundary value problems are considered for the shallow water equations and the equation of advection and dispersion of pollutants. The problems are solved in curvilinear boundary fitted co-ordinates. The transformed equations are integrated on a regular grid by the semi-implicit and implicit finite difference methods. Based on the numerical method, the integrated modelling system Cardinal for coastal area dynamics and pollution processes is developed for application on personal computers. Examples of computations are given.     

The numerical solution of the flow in a general bifurcating channel at moderately high Reynolds number using boundary-fitted co-ordinates,primitive variables and Newton iteration

A numerical code has been implemented for the numerical solution of the steady, incompressible Navier–Stokes equations using primitive variables in a bifurcating channel. A boundary-fitted, numerically generated grid is placed onto the domain of the channel which is transformed into either a rectilinear C- or T-shaped region. The differenced equations are solved using Newton's iteration which makes upwinding at high Reynolds number unnecessary. Practical implications of inverting the huge Jacobian matrix of Newton's method are discussed. The results have relative error of 2–3 × 10^?3 at Reynolds number 100, with T-geometry being marginally but significantly more accurate than C-geometry. Results have been obtained for Reynolds numbers up to 1000 for three bifurcations one of which models the carotid arterial bifurcation in the human head. For this latter bifurcation the wall shear stress is calculated in connection with the onset of atherosclerosis. Finally, the results of flows having different daughter tube end pressures are presented.     

River and reservoir flow modelling using the transformed shallow water equations

This paper describes a versatile finite difference scheme for the solution of the two-dimensional shallow water equations on boundary-fitted non-orthogonal curvilinear meshes. It is believed that this is the first non-orthogonal shallow water equation model incorporating the advective acceleration terms to have been developed in the United Kingdom. The numerical scheme has been validated against the severe condition of jet-forced flow in a circular reservoir with vertical side walls, where reflections of the initial free surface waves pose major problems in achieving a stable solution. Furthermore, the validation exercises are designed to test the computer model for artificial diffusion, which may be a consequence of the numerical scheme adopted to stabilize the shallow water equations. The model is shown to be capable of simulating the flow conditions in an irregularly shaped domain typical of the geometries frequently encountered in civil engineering river basin management.     

A scheme for automatic generation of boundary-fitted depth- and depth-gradient- dependent grids in arbitrary two-dimensional regions

In this paper we present a scheme for the numerical generation of boundary-fitted grids that adapt to both water depth and depth gradient. The scheme can be used in arbitrary two-dimensional regions and is based on the application of the well-known control function approach to generate adaptive grids. The method includes the evaluation of water depths at the grid points from a known distribution of depth points and their associated depths plus a procedure for the numerical evaluation of depth gradients. It is demonstrated that the smoothness of the grid can be enhanced by introducing a suitable filtering technique.