Showing papers in "International Journal for Numerical Methods in Fluids in 2006"
TL;DR: The aim of this paper is to assist the special issue on MPDATA methods for fluids by providing an up to date comprehensive review of the approach, including the underlying concepts, principles of implementation, and guidance to the accumulated literature.
Abstract: Multidimensional positive definite advection transport algorithm (MPDATA) was proposed in the early eighties as a simple positive-definite advection scheme with small implicit diffusion, for evaluating the advection of water-substance constituents in atmospheric cloud models. Over the two decades, MPDATA evolved from an advection scheme into a class of generalized transport algorithms that expand beyond advective transport to alternate PDEs and complete fluid models with a wide range of underlying governing equations. Recently, MPDATA has attracted attention in the context of several mutually-beneficial developments such as (i) quantification of MPDATA implicit turbulence modelling capability in the spirit of monotonically integrated large eddy simulations (MILES), (ii) extensions to flow solvers cast in generalized time-dependent curvilinear coordinates, and (iii) unstructured-grid formulations. The aim of this paper is to assist the special issue on MPDATA methods for fluids by providing an up to date comprehensive review of the approach, including the underlying concepts, principles of implementation, and guidance to the accumulated literature. Copyright © 2005 John Wiley & Sons, Ltd.
222 citations
TL;DR: In this paper, a framework is described and demonstrated for CFD analysis of helicopter rotors in hover and forward flight, where the rotor blades are assumed to be rigid and the rotor to be fully articulated with separate hinges for each blade.
Abstract: A framework is described and demonstrated for CFD analysis of helicopter rotors in hover and forward flight. Starting from the Navier–Stokes equations, the paper describes the periodic rotor blade motions required to trim the rotor in forward flight (blade flapping, blade lead-lag and blade pitching) as well as the required mesh deformation. Throughout, the rotor blades are assumed to be rigid and the rotor to be fully articulated with separate hinges for each blade. The employed method allows for rotors with different numbers of blades and with various rotor hub layouts to be analysed. This method is then combined with a novel grid deformation strategy which preserves the quality of multi-block structured, body-fitted grids around the blades. The coupling of the CFD method with a rotor trimming approach is also described and implemented. The complete framework is validated for hovering and forward flying rotors and comparisons are made against available experimental data. Finally, suggestions for further development are put forward. For all cases, results were in good agreement with experiments and rapid convergence has been obtained. Comparisons between the present grid deformation method and transfinite interpolation were made highlighting the advantages of the current approach. Copyright © 2006 John Wiley & Sons, Ltd.
220 citations
TL;DR: In this paper, the authors provide definitions and equations for the quantitative assessment of numerical (verification) and modelling (validation) errors and uncertainties for CFD simulations and of intervals of certification for RANS codes.
Abstract: Definitions and equations are provided for the quantitative assessment of numerical (verification) and modelling (validation) errors and uncertainties for CFD simulations and of intervals of certification for CFD codes. Verification, validation, and certification methodology and procedures are described. Examples of application of quantitative certification of RANS codes are presented for ship hydrodynamics. Opportunities and challenges for achieving consensus and standard V&V and certification methodology and procedures are discussed.
214 citations
TL;DR: In this article, the analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Butler et al. proposed a method to interpolate boundary conditions from the solid body to the Cartesian mesh on which the computation is performed.
Abstract: The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high-order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two-dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three-dimensional flow in an air-conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.
165 citations
TL;DR: A new implementation of surface tension effects in interfacial flow codes is proposed which is both fully implicit in space, that is the interface never has to be reconstructed, and also semi-implicit in time, with semi‐implicit referring to the time integration of the surface tension forces.
Abstract: A new implementation of surface tension effects in interfacial flow codes is proposed which is both fully implicit in space, that is the interface never has to be reconstructed, and also semi-implicit in time, with semi-implicit referring to the time integration of the surface tension forces. The main idea is to combine two previously separate techniques to yield a new expression for the capillary forces. The first is the continuum surface force (CSF) method, which is used to regularize the discontinuous surface tension force term. The regularization can be elegantly implemented with the use of distance functions, which makes the level set method a suitable choice for the interface-tracking algorithm. The second is to use a finite element discretization together with the Laplace–Beltrami operator, which enables simple reformulation of the surface tension term into its semi-implicit equivalent. The performance of the new method is benchmarked against standard explicit methods, where it is shown that the new method is significantly more robust for the chosen test problems when the time steps exceed the numerical capillary time step restriction. Some improvements are also found in the average number of nonlinear iterations and linear multigrid steps taken while solving the momentum equations. Copyright © 2005 John Wiley & Sons, Ltd.
154 citations
TL;DR: In this paper, a high-order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq-type equations (the so-called Serre equations).
Abstract: A high-order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq-type equations (the so-called Serre equations) (J. Fluid Mech. 1987; 176:117–134; Surveys Geophys. 2004; 25(3–4):315–337). The choice of this discretization strategy is motivated by the fact that this particular set of equations is recasted in a convenient quasi-conservative form. Cell face values are reconstructed using implicit compact schemes (J. Comput. Phys. 1999; 156:137–180; J. Comput. Phys. 2004; 198:535–566) and time integration is performed with the help of a four-stage Runge–Kutta method. Numerical properties of the proposed scheme are investigated both, analytically using linear spectral analysis, and numerically for highly nonlinear cases. The numerical analysis indicates that the newly developed scheme has wider stability regions and better spectral resolution than most of the previously published numerical methods used to handle equivalent set of equations. Moreover, it was also noticed that the use of mixed-order strategies to discretize convective and dispersive terms may result in an important overall reduction of the spectral resolution of the scheme. Additionally, there is some numerical evidence, which seems to indicate that the incorporation of a high-order dispersion correction term as given by Madsen et al. (Coastal Eng. 1991; 15:371–388) may introduce instability in the system. Copyright © 2006 John Wiley & Sons, Ltd.
146 citations
TL;DR: Based on this discretization, a posteriori error estimates for the error measured in terms of arbitrary target functionals, like, e.g. the drag and lift coefficients of an airfoil immersed in a viscous or inviscid fluid are derived.
Abstract: We present the Interior Penalty discontinuous Galerkin method for the compressible Navier-Stokes equations. Shock-capturing is used to reduce over-shoots at discontinuities and sharp gradients. This stabilization introduces artificial viscosity at places of large local residuals, but preserves conservation and Galerkin orthogonality of the DG method. Based on this discretization we derive a posteriori error estimates for the error measured in terms of arbitrary target functionals, like, e.g. the drag and lift coefficients of an airfoil immersed in a viscous or inviscid fluid. The performance of the nonlinear solution process, the a posteriori error estimation and an adaptive mesh refinement specially tailored for the accurate computation of the force coefficients are demonstrated for supersonic laminar flows around the NACA0012 airfoil.
145 citations
TL;DR: In this article, a multigrid finite element solver is used to simulate particle-particle and particle-wall interactions and a new collision model based on Glowinski, Joseph, Singh and coauthors is examined to handle particle particle and particle wall interactions.
Abstract: Direct numerical simulation techniques for particulate flow by the fictitious boundary method (FBM) are presented. The flow is computed by a multigrid finite element solver and the solid particles are allowed to move freely through the computational mesh which can be chosen independently from the particles of arbitrary shape, size and number. The interaction between the fluid and the particles is taken into account by the FBM in which an explicit volume based calculation for the hydrodynamic forces is integrated. A new collision model based on papers by Glowinski, Joseph, Singh and coauthors is examined to handle particle-particle and particle-wall interactions. Numerical tests show that the present method provides a very efficient approach to directly simulate particulate flows with many particles.
142 citations
TL;DR: Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy.
Abstract: SUMMARY A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601×601. Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.
136 citations
TL;DR: In this article, a finite volume method, second-order accurate in space and time, employing non-staggered arrangement of the variables with momentum interpolation for the pressure-velocity coupling is developed.
Abstract: Numerical simulations of two-dimensional laminar flow past a triangular cylinder placed in free-stream at low Reynolds number (10⩽Re⩽250) are performed. A finite volume method, second-order accurate in space and time, employing non-staggered arrangement of the variables with momentum interpolation for the pressure–velocity coupling is developed. Global mode analysis predicts Recr=39.9 which confirms the results of earlier studies. Vortex shedding phenomena is found to be similar to the square cylinder with no second bifurcation in the range of Re studied. A discussion on the time-averaged drag coefficient, rms of lift coefficient and Strouhal number is presented. Particle tracking and the instantaneous streaklines provide an excellent means of visualizing the von Karman vortex street. Copyright © 2006 John Wiley & Sons, Ltd.
125 citations
TL;DR: In this article, a modified HLL Riemann solver was used to discretize the modified fluxes of a two-dimensional shallow water model with porosity and an extra source term was added in the momentum equation.
Abstract: Two-dimensional shallow water models with porosity appear as an interesting path for the large-scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two-dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity-based shallow water models for large scale floodplain simulations. Copyright (c) 2005 John Wiley & Sons, Ltd.
TL;DR: An Eulerian framework for modelling fluid–structure interaction (FSI) of incompressible fluids and elastic structures by tracking the movement of the initial positions of all ‘material’ points is proposed.
Abstract: We propose an Eulerian framework for modelling fluid-structure interaction (FSI) of incompressible fluids and elastic structures. The model is based on an Eulerian approach for describing structural dynamics. This is achieved by tracking the movement of the initial positions of all 'material' points. In this approach the displacement appears as a primary variable in an Eulerian framework. Our approach uses a technique which is similar to the level set method in so far that it also tracks initial data, in our case the set of initial positions (IP), and from this determines to which 'phase' a point belongs. To avoid the occasional reinitialization of the initial position set we employ the harmonic continuation of the structure velocity field into the fluid domain. By using the IP set for tracking the structure displacement, we can ensure that corners and edges of the fluid-structure interface are preserved well. Based on this monolythic model of the FSI we apply the Dual Weighted Residual (DWR) method for goal-oriented a posteriori error estimation to stationary FSI problems. Examples are presented based on the model and for the goal-oriented local mesh adaptation.
TL;DR: Gresho and Sani as discussed by the authors showed that the Navier-Stokes momentum equation can be solved simultaneously with the pressure Poisson equation (PPE) whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal components of velocity is specified as a Dirichlet BC.
Abstract: In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111–1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a so-called equivalence theorem (assertion) that stated/asserted that if the Navier–Stokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) would be exactly the same as if the ‘primitive’ equations, in which the PPE plus Neumann BC is replaced by the usual divergence-free constraint (∇ · u = 0), were solved instead.
This issue is explored in sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the original/primitive equations that require no BC for the pressure, the new results establish the same thing when the PPE approach is employed. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: A generic, mass conservative local grid refinement technique for the lattice-Boltzmann method (LBM) is proposed in this article, where mass conservation can be imposed by allowing the LBM particles to move from coarse grid cells to fine grid cells and vice versa in the propagation step.
Abstract: A generic, mass conservative local grid refinement technique for the lattice-Boltzmann method (LBM) is proposed As a volumetric description of the lattice-Boltzmann equation is applied, mass conservation can be imposed by allowing the lattice-Boltzmann particles to move from coarse grid cells to fine grid cells and vice versa in the propagation step In contrast to most existing techniques, no spatial and temporal interpolation of particle densities is applied Moreover, since the communication between the coarse and the fine grids is independent on the collision step, the method can be used for any LBM scheme
It was found that the method is second-order accurate in space for 2-D Poiseuille flow and different grid setups The method was also applied to the case of 2-D lid driven cavity flow at Re=1000, where half of the cavity was locally refined It was found that the locations of the two lower vortices could be captured accurately Finally, a direct numerical simulation (DNS) of turbulent channel flow at Reτ=360 was performed where the grid was locally refined near the walls of the channel Good first- and second-order turbulence statistics were obtained, showing the applicability of the local grid refinement technique for complex flows Copyright © 2005 John Wiley & Sons, Ltd
TL;DR: In this article, the authors consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) finite element method as well as two discontinuous Galerkins (DG) approaches.
Abstract: We consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) finite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, flux-continuous on each element edge, and is suitable for both smooth and highly advective flows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous–continuous Galerkin method illustrates advantages and disadvantages in accuracy and efficiency. Copyright © 2006 John Wiley & Sons, Ltd.
TL;DR: In this article, a truly incompressible version of the smoothed particle hydrodynamics (SPH) method is presented to investigate the surface wave over-topping.
Abstract: In this paper a truly incompressible version of the smoothed particle hydrodynamics (SPH) method is presented to investigate the surface wave overtopping. SPH is a pure Lagrangian approach which can handle large deformations of the free surface with high accuracy. The governing equations are solved based on the SPH particle interaction models and the incompressible algorithm of pressure projection is implemented by enforcing the constant particle density. The two-equation k–e model is an effective way of dealing with the turbulence and vortices during wave breaking and overtopping and it is coupled with the incompressible SPH numerical scheme. The SPH model is employed to reproduce the experiment and computations of wave overtopping of a sloping sea wall. The computations are validated against the experimental and numerical data found in the literatures and good agreement is observed. Besides, the convergence behaviour of the numerical scheme and the effects of particle spacing refinement and turbulence modelling on the simulation results are also investigated in further detail. The sensitivity of the computed wave breaking and overtopping on these issues is discussed and clarified. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors describe flow dynamics in a patient-specific model of carotid artery with a saccular aneurysm under Newtonian and non-Newtonian fluid assumptions.
Abstract: Flow dynamics plays an important role in the pathogenesis and treatment of cerebral aneurysms. The temporal and spatial variations of wall shear stress in the aneurysm are hypothesized to be correlated with its growth and rupture. In addition, the assessment of the velocity field in the aneurysm dome and neck is important for the correct placement of endovascular coils. This work describes the flow dynamics in a patient-specific model of carotid artery with a saccular aneurysm under Newtonian and non-Newtonian fluid assumptions. The model was obtained from three-dimensional rotational angiography image data and blood flow dynamics was studied under physiologically representative waveform of inflow. The three-dimensional continuity and momentum equations for incompressible and unsteady laminar flow were solved with a commercial software using non-structured fine grid with 283 115 tetrahedral elements. The intra-aneurysmal flow shows complex vortex structure that change during one pulsatile cycle. The effect of the non-Newtonian properties of blood on the wall shear stress was important only in the arterial regions with high velocity gradients, on the aneurysmal wall the predictions with the Newtonian and non-Newtonian blood models were similar. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, an improved moving-particle semi-implicit (MPS) method was developed for numerical simulations of convective heat transfer problems, which is a fully Lagrangian particle method for incompressible flows.
Abstract: An improved moving-particle semi-implicit (MPS) method was developed for numerical simulations of convective heat transfer problems. The MPS method, which is based on particles and their interactions, is a fully Lagrangian particle method for incompressible flows. A new Laplacian model and a new method for treating boundary conditions were proposed to solve numerical difficulties resulting from the original MPS method. Results of several numerical tests show the validity of the improved MPS method with the proposed model and method.
The application of the present MPS method to Rayleigh–Benard convection phenomena demonstrated the effectiveness of the proposed model and method on the numerical simulation of convective heat transfer problems. The dependence of the Nusselt number on the Rayleigh number was in good agreement with an empirical formula. The temperature contour and velocity distribution also agree well with the simulation results obtained with other methods. The roll pattern developed in the horizontal fluid layer as well as the convective heat transfer was successfully simulated with three-dimensional MPS calculations. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented and is proved analytically and ascertained by analysing both local and global errors in a numerical example.
Abstract: A semi-implicit three-step Runge-Kutta scheme for the unsteady incompressible Navier-Stokes equations with third-order accuracy in time is presented. The higher order of accuracy as compared to the existing semi-implicit Runge-Kutta schemes is achieved due to one additional inversion of the implicit operator / - τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure-Poisson equation or evaluation of Navier-Stokes operator is needed. The scheme is supplied with a local error estimation and time-step control algorithm. The temporal third-order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example.
TL;DR: In this article, a computational fluid dynamics (CFD)-based multi-objective optimization method for ship design is presented, where the aim is to simultaneously minimize opposite hydrodynamic performances in design tradeoff.
Abstract: This paper concerns development and demonstration of a computational fluid dynamics (CFD)-based multi-objective optimization method for ship design. Three main components of the method, i.e. computer-aided design (CAD), CFD, and optimizer modules are functionally independent and replaceable. The CAD used in the present study is NAPA system, which is one of the leading CAD systems in ship design. The CFD method is FLOWPACK version 2004d, a Reynolds-averaged Navier-Stokes (RaNS) solver developed by the present authors. The CFD method is implemented into a self-propulsion simulator, where the RaNS solver is coupled with a propeller-performance program. In addition, a maneuvering simulation model is developed and applied to predict ship maneuverability performance. Two nonlinear optimization algorithms are used in the present study, i.e. the successive quadratic programming and the multi-objective genetic algorithm, while the former is mainly used to verify the results from the latter. For demonstration of the present method, a multi-objective optimization problem is formulated where ship propulsion and maneuverability performances are considered. That is, the aim is to simultaneously minimize opposite hydrodynamic performances in design tradeoff. In the following, an overview of the present method is given, and results are presented and discussed for tanker stem optimization problem including detailed verification work on the present numerical schemes.
TL;DR: In this paper, a parabolic spline based algorithm is developed and tested to ensure mass conservation, which solves an integral form of the transport equation rather than the differential form, and a truncation error analysis is presented and discussed in terms of results from test cases.
Abstract: A new and efficient parabolic spline based remapping algorithm is developed and tested herein. To ensure mass conservation, the scheme solves an integral form of the transport equation rather than the differential form. The integrals are computed from reconstructed parabolic splines with mass conservation constraints. For higher dimensions, this remapping can be used within a standard directional splitting methodology or within the flow-dependent cascade splitting approach. A grid and sub-grid based monotonic filter is also incorporated into the overall scheme. A truncation error analysis of the scheme is presented and discussed in terms of results from test cases. The analysis shows that although it has a similar truncation error in the converged limit as that of the widely used Piecewise Parabolic Method (PPM) for infinitely differentiable functions, PSM is more accurate than PPM for problems with slow spectral decay. Additionally, an operation count of the scheme is given which demonstrates the computational advantage of PSM compared to PPM. © Crown copyright 2005. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.
TL;DR: In this article, a linear stability analysis of the governing equations for incompressible flows is carried out via a stabilized finite element method to predict the primary instability of the wake, and the generalized eigenvalue problem is solved using a subspace iteration method to get the most unstable eigenmode.
Abstract: The effect of location of the lateral boundaries, of the computational domain, on the critical parameters for the instability of the flow past a circular cylinder is investigated. Linear stability analysis of the governing equations for incompressible flows is carried out via a stabilized finite element method to predict the primary instability of the wake. The generalized eigenvalue problem resulting from the finite element discretization of the equations is solved using a subspace iteration method to get the most unstable eigenmode. Computations are carried out for a large range of blockage, 0.005 ≤ D/H ≤ 0.125, where D is the diameter of the cylinder and H is the lateral width of the domain. A non-monotonic variation of the critical Re with the blockage is observed. It is found that as the blockage increases, the critical Re for the onset of the instability first decreases and then increases. However, a monotonic increase in the non-dimensional shedding frequency at the onset of instability, with increase in blockage, is observed. The increased blockage damps out the low-frequency modes giving way to higher frequency modes. The blockage is found to play an important role in the scatter in the data for the non-dimensional vortex shedding frequency at the onset of the instability, from various researchers in the past.
TL;DR: In this paper, the authors revisited the derivation of the ALE form of the incompressible Navier-Stokes equations in order to retain insight into the nature of geometric conservation.
Abstract: This note revisits the derivation of the ALE form of the incompressible Navier-Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context.
TL;DR: In this paper, numerically the slip with friction boundary condition in the time-dependent incompressible Navier-Stokes equations is studied and the influence of the friction parameter on the flow field is studied.
Abstract: The paper studies numerically the slip with friction boundary condition in the time-dependent incompressible Navier–Stokes equations. Numerical tests on two- and three-dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three-dimensional results differ in many respects from the two-dimensional ones. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, an adaptive hierarchical grid based method for predicting complex free surface flows is used to simulate collapse of a water column and its subsequent interaction with an obstacle in a lid driven cavity at Re=1000.
Abstract: An adaptive hierarchical grid based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high-resolution interface capturing approach and pressure based coupling of the Navier Stokes equations. The Navier-Stokes flow solution scheme is verified for simulation of flow in a lid driven cavity at Re=1000. Two approaches to the coupling of the Navier-Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree-based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids.
TL;DR: In this article, the authors present a discontinuous Galerkin formulation of the shallow water equations and use an orthogonal basis for spatial discretization and an explicit Runge-Kutta scheme for time discretisation.
Abstract: In this paper, we present a discontinuous Galerkin formulation of the shallow-water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge-Kutta scheme is used for time discretization. Some results of second-order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps. Copyright (c) 2006 John Wiley & Sons, Ltd.
TL;DR: This work presents a method obtained by the combination of a high‐order shock capturing scheme, built from Shu–Osher's conservative formulation, with a fifth‐order weighted essentially non‐oscillatory (WENO) interpolatory technique and the adaptive mesh refinement (AMR) technique.
Abstract: The numerical simulation of physical phenomena represented by non-linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a method obtained by the combination of a high-order shock capturing scheme, built from Shu–Osher's conservative formulation (J. Comput. Phys. 1988; 77:439–471; 1989; 83:32–78), a fifth-order weighted essentially non-oscillatory (WENO) interpolatory technique (J. Comput. Phys. 1996; 126:202–228) and Donat–Marquina's flux-splitting method (J. Comput. Phys. 1996; 125:42–58), with the adaptive mesh refinement (AMR) technique of Berger and collaborators (Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. Thesis, Computer Science Department, Stanford University, 1982; J. Comput. Phys. 1989; 82:64–84; 1984; 53:484–512). Copyright © 2006 John Wiley & Sons, Ltd.
TL;DR: In this paper, a computational procedure is presented for solving complex variably saturated flows in porous media, that may easily be implemented into existing conventional finite-volume-based computational fluid dynamics codes, so that their functionality might be geared upon to readily enable the modelling of a complex suite of interacting fluid, thermal and chemical reaction process physics.
Abstract: A computational procedure is presented for solving complex variably saturated flows in porous media, that may easily be implemented into existing conventional finite-volume-based computational fluid dynamics codes, so that their functionality might be geared upon to readily enable the modelling of a complex suite of interacting fluid, thermal and chemical reaction process physics. This procedure has been integrated within a multi-physics finite volume unstructured mesh framework, allowing arbitrarily complex three-dimensional geometries to be modelled. The model is particularly targeted at ore heap-leaching processes, which encounter complex flow problems, such as infiltration into dry soil, drainage, perched water tables and flow through heterogeneous materials, but is equally applicable to any process involving flow through porous media, such as in environmental recovery processes. The computational procedure is based on the mixed form of the classical Richards equation, employing an adaptive transformed mixed algorithm that is numerically robust and significantly reduces compute (or CPU) time. The computational procedure is accurate (compares well with other methods and analytical data), comprehensive (representing any kind of porous flow model), and is computationally efficient. As such, this procedure provides a suitable basis for the implementation of large-scale industrial heap-leach models. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this article, a 2D coupled vertically averaged flow/transport model is presented, which is based on an upwind finite volume method, using Roe's approximate Riemann solver.
Abstract: In this paper, a transient 2D coupled vertically averaged flow/transport model is presented. The model deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. The model is based on an upwind finite volume method, using Roe's approximate Riemann solver. A specific modification of the Riemann solver is proposed to overcome the generation of negative values of depth and concentration, that can appear as a consequence of existing wetting/drying and solute advance fronts over variable bed levels, or by the generation of new ones when dry areas appear. The numerical stability constraints of the explicit model are stated incorporating the influence of the flow velocity, the bed variations and the possible appearance of dry cells. Faced to the important restriction that this new stability condition can impose on the time step size, a different strategy to allow stability using a maximum time step, and in consequence a minimum computational cost is presented.
TL;DR: In this paper, a numerical convergence study of a family of flux-continuous schemes is presented, where the local position of continuity defines the quadrature point and hence the family.
Abstract: In this paper, a numerical convergence study of family of flux-continuous schemes is presented. The family of flux-continuous schemes is characterized in terms of quadrature parameterization, where the local position of continuity defines the quadrature point and hence the family. A convergence study is carried out for the discretization in physical space and the effect of a range of quadrature points on convergence is explored. Structured cell-centred and unstructured cell-vertex schemes are considered. Homogeneous and heterogeneous cases are tested, and convergence is established for a number of examples with discontinuous permeability tensor including a velocity field with singularity. Such cases frequently arise in subsurface flow modelling. A convergence comparison with CVFE is also presented. Copyright © 2006 John Wiley & Sons, Ltd.