Coupling between shallow water and solute flow equations: analysis and management of source terms in 2D

A two‐dimensional model for the simulation of solute transport by convection and diffusion into shallow water flow over variable bottom is presented. It is based on a finite volume method over triangular unstructured grids. A first order upwind technique is applied to solve the flux terms in both the flow and solute equations and the bed slope source terms and a centred discretization is applied to the diffusion and friction terms. The convenience of considering the fully coupled system of equations is indicated and the methodology is well explained. Three options are suggested and compared in order to deal with the diffusion terms. Some comparisons are carried out in order to show the performance in terms of accuracy and computational effort of the different options. Copyright © 2005 John Wiley & Sons, Ltd.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

Preserving bounded and conservative solutions of transport in one‐dimensional shallow‐water flow with upwind numerical schemes: Application to fertigation and solute transport in rivers

This work intends to show that conservative upwind schemes based on a separate discretization of the scalar solute transport from the shallow‐water equations are unable to preserve uniform solute profiles in situations of one‐dimensional unsteady subcritical flow. However, the coupled discretization of the system is proved to lead to the correct solution in first‐order approximations. This work is also devoted to show that, when using a coupled discretization, a careful definition of the flux limiter function in second‐order TVD schemes is required in order to preserve uniform solute profiles. The work shows that, in cases of subcritical irregular flow, the coupled discretization is necessary but nevertheless not sufficient to ensure concentration distributions free from oscillations and a method to avoid these oscillations is proposed. Examples of steady and unsteady flows in test cases, river and irrigation are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Depth‐averaged two‐dimensional curvilinear explicit finite analytic model for open‐channel flows

A depth‐averaged two‐dimensional model has been developed in the curvilinear co‐ordinate system for free‐surface flow problems. The non‐linear convective terms of the momentum equations are discretized based on the explicit–finite–analytic method with second‐order accuracy in space and first‐order accuracy in time. The other terms of the momentum equations, as well as the mass conservation equation, are discretized by the finite difference method. The discretized governing equations are solved in turn, and iteration in each time step is adopted to guarantee the numerical convergence. The new model has been applied to various flow situations, even for the cases with the presence of sub‐critical and supercritical flows simultaneously or sequentially. Comparisons between the numerical results and the experimental data show that the proposed model is robust with satisfactory accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

A Taylor series‐based finite volume method for the Navier–Stokes equations

A Taylor series‐based finite volume formulation has been developed to solve the Navier–Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible. Copyright © 2008 John Wiley & Sons, Ltd.     

Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing

Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented. The reaction rate of the solute is considered inversely proportional along the plate. Adopting the similarity transformation technique the governing equations are converted into the self-similar ordinary differential equations which are solved by shooting procedure using Runge-Kutta method. For increase of the Schmidt number the solute boundary layer thickness is reduced. Most importantly, the effects of reaction rate and order of reaction on concentration field are of conflicting natures, due to increasing reaction rate parameter the concentration decreases, but for the increase in order of reaction it increases. In presence of chemical reaction, the concentration profiles attain negative value when Schmidt number is large.     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

Non‐linear turbulence models can be seen as an improvement of the classical eddy‐viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non‐linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non‐linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non‐linear constitutive equation using a new semi‐implicit numerical treatment for the non‐linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward‐facing step was computed using the control volume method applied to a boundary‐fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi‐implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non‐orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non‐linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

The design of improved smoothing operators for finite volume flow solvers on unstructured meshes

Spatial operators used in unstructured finite volume flow solvers are analysed for accuracy using Taylor series expansion and Fourier analysis. While approaching second‐order accuracy on very regular grids, operators in common use are shown to have errors resulting in accuracy of only first‐, zeroth‐ or even negative‐order on three‐dimensional tetrahedral meshes. A technique using least‐squares optimization is developed to design improved operators on arbitrary meshes. This is applied to the fourth‐order edge sum smoothing operator. The improved numerical dissipation leads to a much more accurate prediction of the Strouhal number for two‐dimensional flow around a cylinder and a reduction of a factor of three in the loss coefficient for inviscid flow over a three‐dimensional hump. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

Development of level set method with good area preservation to predict interface in two‐phase flows

A two‐step conservative level set method is proposed in this study to simulate the gas/water two‐phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the coupled compact scheme. For accurately predicting the modified level set function, the dispersion‐relation‐preserving advection scheme is developed to preserve the theoretical dispersion relation for the first‐order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long‐time accurate Casimir functionals and Hamiltonian in the transport equation for the level set function, the time derivative term is discretized by the sixth‐order accurate symplectic Runge–Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second‐step equation of the modified level set method. For the verification of the proposed dispersion‐relation‐preserving scheme applied in non‐staggered grids for solving the incompressible flow equations, three benchmark problems have been chosen in this study. The conservative level set method with area‐preserving property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam‐break, Rayleigh–Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreements with the referenced solutions are demonstrated in all the investigated problems. Copyright © 2010 John Wiley & Sons, Ltd.     

An attempt to improve accuracy of higher‐order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms

We attempt to improve accuracy in the high‐wavenumber region in DNS of incompressible wall turbulence such as found in fully developed turbulent channel flow. In particular, it is shown that the improvement of accuracy of viscous terms in the Navier–Stokes equations leads to the improvement of accuracy of higher‐order statistics and various spectra. It is emphasized that increase in required computational cost will not be crucial when incompressible flow is simulated, because the introduction of a higher‐order scheme into the viscous terms does not increase computational cost for solving the Poisson equation. We introduced fourth‐order and eighth‐order central compact schemes for discretizing the viscous terms in DNS of a fully developed turbulent channel flow. The results are compared with those using second‐order and fourth‐order central‐difference schemes applied to the viscous terms and those obtained by the spectral method. The results show that accuracy improvement of the viscous terms improve accuracy of higher‐order statistics (i.e., skewness and flatness factors of streamwise velocity fluctuation) and various spectra of velocity and pressure fluctuations in the high‐wavenumber region. Copyright © 2013 John Wiley & Sons, Ltd.     

Efficient Lagrangian scalar tracking method for reactive local mass transport simulation through porous media

Mass transfer in the presence of chemical reactions for flows through porous media is of interest to many disciplines. The Lattice Boltzmann method (LBM) is particularly attractive in such cases due to the ease with which it handles complicated boundary conditions. However, useful Lagrangian information (such as solute survival distance, effective diffusivity, collision frequency) is challenging to obtain from the LBM. In this paper, we present a straightforward and efficient Lagrangian methodology (Lagrangian scalar tracking, LST) for performing solute transport simulations in the presence of heterogeneous, first‐order, irreversible reactions, based on a velocity field obtained from LBM. The hybrid LST/LBM technique tracks passive mass markers that have two contributions to their movement: convective (obtained through interpolation of a previously obtained velocity field) and Brownian. Various Schmidt number solutes and different solute release modes can be modeled with a single solvent flow field using this method. Moreover, the mass markers can have a range of reaction rate coefficients. This allows for the exploration of the whole spectrum of first‐order heterogeneous reaction rates with just a single simulation. In order to show the applicability of the LST/LBM scheme, results from a case study are presented in which the consumption of oxygen and/or nutrients within a porous bone tissue engineering scaffold is modeled under flow perfusion culturing conditions. Although the reactive LST methodology described in this paper compliments the LBM, it can also be used with any other flow simulation that can generate the velocity field. Copyright © 2010 John Wiley & Sons, Ltd.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.