Series solutions for steady unsaturated flow in irregular porous domains

For the most part, analytical solutions for steady unsaturated infiltration have been restricted to infinite and semi-infinite seepage geometries, using the quasi-linear approximation for the hydraulic conductivity. We provide analytical series methods to solve the steady quasi-linear flow equations, in finite irregular seepage geometries. Unlike the classical approach, the series method has been modified, to cater for arbitrary boundary geometry and surface recharge distributions. The matrix flux potential and the stream function both satisfy the same governing partial differential equation, and the stream function formulation is used to estimate the series coefficients. For a finite vadose zone, the stream function solution does not uniquely determine the matrix flux potential, when flux boundary conditions are used. Consequently, the stream function solution applies to a range of moisture distributions, for given infiltration and evapotranspiration rates through the surface.     

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

A wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.     

Review of wetting and drying algorithms for numerical tidal flow models

A review of wetting and drying (WD) algorithms used by contemporary numerical models based on the shallow water equations is presented. The numerical models reviewed employ WD algorithms that fall into four general frameworks: (1) Specifying a thin film of fluid over the entire domain; (2) checking if an element or node is wet, dry or potentially one of the two, and subsequently adding or removing elements from the computational domain; (3) linearly extrapolating the fluid depth onto a dry element and its nodes from nearby wet elements and computing the velocities; and (4) allowing the water surface to extend below the topographic ground surface. This review presents the benefits and drawbacks in terms of accuracy, robustness, computational efficiency, and conservation properties. The WD algorithms also tend to be highly tailored to the numerical model they serve and therefore difficult to generalize. Furthermore, the lack of temporally and spatially defined validation data has hampered comparisons of the models in terms of their ability to simulate WD over real domains. A short discussion of this topic is included in the conclusion. Copyright © 2012 John Wiley & Sons, Ltd.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

非规则区域中拉普拉斯方程的有限分析5点格式

建立了非规则区域的有限分析5点格式,增加了有限分析法对不规则边界的适应性。应用所提出的方法对水利工程中常见的有压和无压流动进行了计算,与实验和前人的计算结果相比较,本文的方法都能得到较为满意的结果。本文的计算格式也可以应用到其他非规则区域的计算中。     

A semi‐implicit numerical model for urban drainage systems

In this paper, a semi‐implicit numerical model for one‐dimensional urban drainage networks is formulated in such a fashion as to intrinsically account for arbitrary cross sections, for the occurrence of dry areas, for free surface, and for pressurized flows. The governing differential equations are discretized with a consistent mass conservative scheme that naturally applies to all flow regimes. The resulting mildly nonlinear system, at every time step, is efficiently solved with a converging, properly devised, nested Newton‐type algorithm. It will be shown that with the proposed semi‐implicit model, high accuracy can be achieved at a moderate computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

Turbulence model and numerical scheme assessment for buffet computations

The prediction of shock‐induced oscillations over transonic rigid airfoils is important for a better understanding of the buffeting phenomenon. The unsteady resolution of the Navier–Stokes equations is performed with various transport‐equation turbulence models in which corrections are added for non‐equilibrium flows. The lack of numerical efficiency due to the CFL stability condition is circumvented by the use of a wall law approach and a dual time stepping method. Moreover, various numerical schemes are used to try and be independent of the numerical discretization. Comparisons are made with the experimental results obtained for the supercritical RA16SC1 airfoil. They show the interest in using the SST correction or realizability conditions to get correct predictions of the frequency, amplitude and pressure fluctuations over the airfoil. Copyright © 2004 John Wiley & Sons, Ltd.     

A Cartesian grid method for two‐phase gel dynamics on an irregular domain

We develop a numerical method for simulating models of two‐phase gel dynamics in an irregular domain using a regular Cartesian grid. The models consist of transport equations for the volume fractions of the two phases, polymer network and solvent; coupled momentum equations for the two phases; and a volume‐averaged incompressibility constraint. Multigrid with Vanka‐type box relaxation scheme is used as a preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Ghost points are used to enforce no‐slip boundary conditions for the velocity field of each phase, and no‐flux boundary conditions for the volume fractions. The behavior of the new method, including its rate of convergence, is explored through numerical experiments for a problem in which strong phase separation develops from an initially (almost) homogeneous phase distribution. We also use the method to explore situations, motivated by biology, which show that imposed boundary velocities can cause substantial redistribution of network and solvent. Copyright © 2010 John Wiley & Sons, Ltd.     

Relative permeability relations: A key factor for a drying model

In the modelling of heat, mass and momentum transfer phenomena which occur in a capillary porous medium during drying, the liquid and gas flows are usually described by the generalised Darcy laws. Nevertheless, the question of how to determine experimentally the relative permeability relations remains unanswered for most materials that consist of water and humid air, and as a result, arbitrary functions are used in the drying codes. In this paper, the emphasis is on deducing from both numerical and experimental studies a method for estimating pertinent relations for these key parameters. In the first part, the sensitivity of liquid velocity and, consequently, of drying kinetics in the variation of the relative permeabilities is investigated numerically by testing various forms. It is concluded that in order to predict a realistic liquid velocity behaviour, relative permeabilities can be linked to a measurable quantity: the capillary pressure. An estimation technique, based on simulations coupled with experimental measurements of capillary pressure, together with moisture content kinetics obtained for low or middle temperature convective drying, is deduced. In the second part, the proposed methodology is applied to pine wood. It is shown that the obtained relations provide closer representation of physical reality than those commonly used.     

不规则区域平面弹性问题的正则区域重心插值配点法

将不规则区域嵌入到规则的矩形区域,在矩形区域上将弹性平面问题的控制方程采用重心Lagrange插值离散,得到控制方程矩阵形式的离散表达式。在边界节点上利用重心插值离散边界条件,规则区域采用置换法施加边界条件,不规则区域采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法进行求解,得到整个规则区域上的位移数值解。利用重心插值计算得到不规则区域内任意节点的位移值,计算精度可到10^-14以上。数值算例验证了所建立方法的有效性和计算精度。     

A numerical method for forced convection in porous and homogeneous fluid domains coupled at interface by stress jump

A numerical method was developed for flows involving an interface between a homogeneous fluid and a porous medium. It is based on the finite volume method with body‐fitted and multi‐block grids. The Brinkman–Forcheimmer extended model was used to govern the flow in the porous medium region. At its interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The forced convection through a porous insert over a backward‐facing step is investigated. The results are presented with flow configurations for different Darcy numbers, 10^?2 to 10^?5, porosity from 0.2 to 0.8, Reynolds number from 10 to 800, and the ratio of insert length to channel height from 0.1 to 0.3. The heat transfer is improved by using porous insert. To enhance the heat transfer with minimal frictional losses, it is preferable to have a medium length of insert with medium Darcy number, and larger Reynolds number. The interfacial stress jump coefficients β and β₁ were varied from ?1 to 1, and within this range the average and local lower‐wall Nusselt numbers are not sensitive to the parameters. Copyright © 2007 John Wiley & Sons, Ltd.     

A sequential numerical chemical compositional simulator

Chemical flooding in the petroleum industry has a larger scale of oil recovery efficiency than water flooding. On the other hand, it is far more technical, costly, and risky. Numerical reservoir simulation can be employed to conduct mechanism study, feasibility evaluation, pilot plan optimization, and performance prediction for chemical flooding to improve recovery efficiency and reduce operational costs. In this article, we study numerical simulation of chemical flooding such as alkaline, surfactant, polymer, and foam (ASP+foam) flooding. The main displacement mechanisms in this type of flooding are interfacial tension lowering, capillary desaturation, chemical synergetic effects, and mobility control. The model of chemical flooding involves such physicochemical phenomena as dispersion, diffusion, adsorption, chemical reactions, and in situ generation of surfactant from acidic crude oil. The numerical simulator is based on a sequential solution approach that solves both pressure and compositions implicitly, and is applied to three experiments, a chemical flow without mass transfer between phases, a laboratory sandstone core, and an ASP+foam displacement problem with mass transfer, and to a real oilfield. A comparison with UTCHEM is also performed. These applications and comparison indicate that this numerical simulator is practical, efficient, and accurate for simulating complex chemical flooding processes.      

A variable boundary model of storm surge flooding in generalized curvilinear grids

In this paper, generalized 2D shallow sea dynamic equations in movable curvilinear co-ordinates are derived. Through a differential co-ordinate transformation a self-adaptive grid is proposed to treat a continuously deforming lateral boundary and a kinematical boundary condition is adopted. The self-adaptive grid method (SAM) is used to simulate numerically the storm surge flooding in the Bohai Sea on 23 April 1969, which was one of the largest storm surge inundations in China.     

A numerical study of an unsteady laminar flow in a doubly constricted 3D vessel

Unsteady flow dynamics in doubly constricted 3D vessels have been investigated under pulsatile flow conditions for a full cycle of period T. The coupled non‐linear partial differential equations governing the mass and momentum of a viscous incompressible fluid has been numerically analyzed by a time accurate Finite Volume Scheme in an implicit Euler time marching setting. Roe's flux difference splitting of non‐linear terms and the pseudo‐compressibility technique employed in the current numerical scheme makes it robust both in space and time. Computational experiments are carried out to assess the influence of Reynolds' number and the spacing between two mild constrictions on the pressure drop across the constrictions. The study reveals that the pressure drop across a series of mild constrictions can get physiologically critical and is also found to be sensitive both to the spacing between the constrictions and the oscillatory nature of the inflow profile. The flow separation zone on the downstream constriction is seen to detach from the diverging wall of the constriction leading to vortex shedding with 3D features earlier than that on the wall in the spacing between the two constrictions. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical investigation of fully developed laminar flow in irregular annuli: Square‐circular ducts

Numerical study is carried out on fully developed laminar axial flow of incompressible Newtonian fluid in irregular annuli of different cross‐sectional geometry. Accurate prediction of the friction factor of this flow was obtained using high‐order finite element method. Investigations are performed for the annular flow between rectangular and circular ducts under all possible arrangements. The governing equations are solved by an efficient finite element technique. It was found that a higher annular area‐ratio will lead to a monotonic increase in (f Re) value in the case of regular annuli, and will lead to an increase followed by a decrease in (f Re) value in the case of irregular annuli. Furthermore, this study illustrates that irregular annuli have lower (f Re) value than regular annuli, and that the square‐in‐circle case (SC‐case) has the lowest (f Re) value whereas the circle‐in‐circle case (CC‐case) has the highest (f Re) value. The finite element technique developed in this study was validated against the already reported numerical results in the literature, and the results presented in this work show excellent agreement. Copyright © 2009 John Wiley & Sons, Ltd.     

A Petrov-Galerkin method for the numerical solution of the Bradshaw-Ferriss-Atwell turbulence model

The Bradshaw-Ferriss-Atwell model for 2D constant property turbulent boundary layers is shown to be ill-posed with respect to numerical solution. It is shown that a simple modification to the model equations results in a well-posed system which is hyperbolic in nature. For this modified system a numerical algorithm is constructed by discretizing in space using the Petrov-Galerkin technique (of which the standard Galerkin method is a special case) and stepping in the timelike direction with the trapezoidal (Crank-Nicolson) rule. The algorithm is applied to a selection of test problems. It is found that the solutions produced by the standard Galerkin method exhibit oscillations. It is further shown that these oscillations may be eliminated by employing the Petrov-Galerkin method with the free parameters set to simple functions of the eigenvalues of the modified system.     

A projection method for solving incompressible viscous flows on domains with moving boundaries

In this paper, a projection method is presented for solving the flow problems in domains with moving boundaries. In order to track the movement of the domain boundaries, arbitrary‐Lagrangian–Eulerian (ALE) co‐ordinates are used. The unsteady incompressible Navier–Stokes equations on the ALE co‐ordinates are solved by using a projection method developed in this paper. This projection method is based on the Bell's Godunov‐projection method. However, substantial changes are made so that this algorithm is capable of solving the ALE form of incompressible Navier–Stokes equations. Multi‐block structured grids are used to discretize the flow domains. The grid velocity is not explicitly computed; instead the volume change is used to account for the effect of grid movement. A new method is also proposed to compute the freestream capturing metrics so that the geometric conservation law (GCL) can be satisfied exactly in this algorithm. This projection method is also parallelized so that the state of the art high performance computers can be used to match the computation cost associated with the moving grid calculations. Several test cases are solved to verify the performance of this moving‐grid projection method. Copyright © 2004 John Wiley Sons, Ltd.     

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar–Gross–Krook (BGK) numerical model for one‐ and two‐dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation‐free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one‐ and two‐dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C_r), with the only condition being C_r≤1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks). In addition, comparisons of numerical tests with measured data from dam break laboratory experiments show good agreement for C_r≤0.6. This reduction in the stability domain is due to the explicit integration of the friction term. Furthermore, BGK schemes are easily extended to multidimensional problems and do not require characteristic decomposition. The proposed scheme is second‐order in both space and time when the external forces are zero and second‐order in space but first‐order in time when the external forces are non‐zero. However, since all the test cases presented are either for zero or small values of external forces, the results tend to maintain second‐order accuracy. In problems where the external forces become significant, it is possible to improve the order of accuracy of the scheme in time by, for example, applying the Runge–Kutta method in the integration of the external forces. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical technique for laminar swirling flow at the interface between porous and homogenous fluid domains

There have been a few recent numerical implementations of the stress‐jump condition at the interface of conjugate flows, which couple the governing equations for flows in the porous and homogenous fluid domains. These previous demonstration cases were for two‐dimensional, planar flows with simple geometries, for example, flow over a porous layer or flow through a porous plug. The present study implements the interfacial stress‐jump condition for a non‐planar flow with three velocity components, which is more realistic in terms of practical flow applications. The steady, laminar, Newtonian flow in a stirred micro‐bioreactor with a porous scaffold inside was investigated. It is shown how to implement the interfacial jump condition on the radial, axial, and swirling velocity components. To avoid a full three‐dimensional simulation, the flow is assumed to be independent of the azimuthal direction, which makes it an axisymmetric flow with a swirling velocity. The present interface treatment is suitable for non‐flat surfaces, which is achieved by applying the finite volume method based on body‐fitted and multi‐block grids. The numerical simulations show that a vortex breakdown bubble, attached to the free surface, occurs above a certain Reynolds number. The presence of the porous scaffold delays the onset of vortex breakdown and confines it to a region above the scaffold. Copyright © 2008 John Wiley & Sons, Ltd.