Pseudotransient Continuation Method in Extended Period Simulation of Water Distribution Systems

This paper presents and discusses a new static solver that implements the pseudotransient continuation method for the quasi-steady state analysis, or extended-period simulation of water distribution systems. The implementation is based on the concept of virtual tanks and has a clear physical meaning. The steady state solver described in this paper can analyze a pipe network under pressure deficient conditions and is free from some convergence problems that occur in the Newton-Raphson method-based solvers when analyzing a pipe network with control devices. The numerical examples considered in the paper demonstrate the convergence of the proposed method in cases where existing static solvers (e.g., that of the EPANET 2 hydraulic simulator) fail.     

Improved Pressurized Pipe Network Hydraulic Solver for Applications in Irrigation Systems

GESTAR is an advanced computational hydraulic software tool specially adapted for the design, planning, and management of pressurized irrigation networks. A summary is given of the most significant characteristics of GESTAR. The hydraulic solver for quasi-steady scenarios uses specific strategies and incorporates several new features that improve the algorithms for pipe network computation, overcoming some of the problems that arise when attempting to apply drinking water software, using the gradient method, to irrigation systems. It is shown that the gradient method is a nodal method variant, where flow rates are relaxed using head loss formula exponents. Although relaxation produces a damping effect on instabilities, it is still unable to solve some of the numerical problems common to the nodal methods. In this contribution the results of the research on computational strategies capable of dealing with low resistance elements, hydrant modelling, multiple regulation valves, numerous emitters, and pumps with complex curves are presented, obtaining accurate results even in conditions where other software fails to converge. GESTAR incorporates all these computational techniques, achieving a high convergence rate and robustness. Furthermore, GESTAR’s solver algorithm was easily adapted to incorporate inverse analysis options for optimum network control and parameter calibration. Illustrative examples are provided, documenting the improved numerical techniques and examples of GESTAR’s performance in comparison with EPANET2, a widely used gradient method-based hydraulic solver.     

Riemann Solvers with Runge–Kutta Discontinuous Galerkin Schemes for the 1D Shallow Water Equations

The spectrum of this survey turns on the evaluation of some eminent Riemann solvers (or the so-called solver), for the shallow water equations, when employed with high-order Runge–Kutta discontinuous Galerkin (RKDG) methods. Based on the assumption that: The higher is the accuracy order of a numerical method, the less crucial is the choice of Riemann solver; actual literature rather use the Lax-Friedrich solver as it is easy and less costly, whereas many others could be also applied such as the Godunov, Roe, Osher, HLL, HLLC, and HLLE. In practical applications, the flow can be dominated by geometry, and friction effects have to be taken into consideration. With the intention of obtaining a suitable choice of the Riemann solver function for high-order RKDG methods, a one-dimensional numerical investigation was performed. Three traditional hydraulic problems were computed by this collection of solvers cooperated with high-order RKDG methods. A comparison of the performance of the solvers was carried out discussing the issue of L1-errors magnitude, CPU time cost, discontinuity resolution and source term effects.     

Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems

Three issues concerning the iterative solution of the nonlinear equations governing the flows and heads in a water distribution system network are considered. Zero flows cause a computation failure (division by zero) when the Global Gradient Algorithm of Todini and Pilati is used to solve for the steady state of a system in which the head loss is modeled by the Hazen-Williams formula. The proposed regularization technique overcomes this failure as a solution to this first issue. The second issue relates to zero flows in the Darcy-Weisbach formulation. This work explains for the first time why zero flows do not lead to a division by zero where the head loss is modeled by the Darcy-Weisbach formula. In this paper, the authors show how to handle the computation appropriately in the case of laminar flow (the only instance in which zero flows may occur). However, as is shown, a significant loss of accuracy can result if the Jacobian matrix, necessary for the solution process, becomes poorly conditioned, and so it is recommended that the regularization technique be used for the Darcy-Weisbach case also. Only a modest extra computational cost is incurred when the technique is applied. The third issue relates to a new convergence stopping criterion for the iterative process based on the infinity-norm of the vector of nodal head differences between one iteration and the next. This test is recommended because it has a more natural physical interpretation than the relative discharge stopping criterion that is currently used in standard software packages such as EPANET. In addition, it is recommended to check the infinity norms of the residuals once iteration has been stopped. The residuals test ensures that inaccurate solutions are not accepted.     

Advances in Calculation Methods for Supercritical Flow in Spillway Channels

A calculation method is presented for applications to steady supercritical and transcritical flow in spillway channels. The method solves the two-dimensional nonlinear shallow water equations using a cell-centered finite-volume approach. High spatial resolution of shock waves and other steep flow features is achieved by employing MUSCL reconstruction and an approximate Riemann solver for the flux evaluations at each cell interface. The method can be implemented on boundary-conforming meshes to more accurately map the wide range of geometries that may occur in practice. Six analytical test problems are proposed for the validation of calculation methods applied to steady supercritical flow. These problems are used to validate the proposed flow solver, which is then applied to the case of steady supercritical flow in a curved channel transition, and comparisons are made with published data. Despite limitations in the shallow water model, the results show satisfactory agreement with data for the maximum rise in water level through the standing oblique shock waves.     

Analysis of the steady state molten pool obtained by heating a substrate with an electron beam

Surface melting and solidification with high powered beams can be used for enhancing surface properties. The dimensions of the molten zone define the extent of the modified properties and are critical parameters which must be predicted during process design. The flow field in the molten pool has been reported to be one of the key factors which controls the dimensions of the surface layer. However, the calculation of this is only possible through complex numerical schemes and there is a need to look for simple analytical expressions which may be adequate. One approach for this search involves the precise determination of the steady state stationary profiles and then developing a method for extending these values to include the effect of beam motion for predicting the pool dimensions during processing. In this paper, a study of the flow field and its effect on the depth and width of the steady state pool is presented, based on numerical and analytical methods. To validate the predictions, an experimental study is carried out using surface melting of Al-4.5 wt%Cu alloy an electron beam. The pool shapes are presented through optical micrographs and the depth and width of the pool is measured from these micrographs. The experiments are then simulated using a numerical model which includes fluid flow. The flow field is analyzed using streamline plots and the predicted pool shapes are compared with the micrographs. Further, the results are compared to an analytical method based on pure conduction and the pool depth and width are predicted when the liquid thermal conductivity is modified. The numerical and analytical predictions of the pool depth and width are found to be in good agreement with the experimental measurements (obtained from steady state stationary pools and from dimensions inferred on extrapolating moving beam measurements to zero velocity). The reasons for the success of the analytical model is discussed with reference to the two-dimensional flow fields and vortices predicted by the numerical model.     

Network Implementation of the Two-Component Pressure Approach for Transient Flow in Storm Sewers

The two-component pressure approach (TPA) is an alternative to the Preissman slot method (PSM) for modeling highly transient sewer flow, including transitions between free-surface and pressurized conditions. TPA and PSM resolve intralink wave action by discretizing sewers with numerous elements and solving one-dimensional flow equations in contrast to link-node models, such as the popular storm water management model, which resolve only interlink wave action. Here, improvements of TPA are reported to support storm sewer network modeling. These include a source term discretization to preserve stationarity, a wetting and drying scheme, and a local time-stepping scheme to coordinate solution updates across many links and enable coupling to a two-dimensional overland flow model. A unique variant of the Harten, Lax and van Leer (HLL) Riemann solver is also introduced, and a boundary solver is developed to accommodate the wide range of possible flow regimes and transitions. The boundary solver is explicit to facilitate the extension of TPA to large networks and coupling with an overland flow model. Promising results are obtained in a varied set of test problems involving simple sewer networks.     

Robust Symmetric Formulations for Nonassociated Plasticity Problems

Most engineering material behaviors can be described by using nonassociated plasticity models, but their use leads to the formation of unsymmetric matrices that have, to date, been resolved by nonsymmetric solvers, the initial stiffness method, and, more recently, some symmetrization techniques. Some of these methods have proved to be inefficient due to large core storage requirements, excessive iteration, and slow rate of convergence; or the existence of regions of numerical instability within the mapping process. Some robust techniques are outlined in this paper that are capable of providing solutions of acceptable accuracy for nonassociated plasticity problems, by adapting them for banded, symmetric solvers and by returning CPU times comparable to the best available solution techniques. Numerical solutions to problems in both soil and nongranular plasticity are provided to illustrate the capability of the proposed techniques, and the problem of numerical instability inherent in a previous symmetrization technique is addressed.     

Teaching Hydraulic Design Using Equation Solvers

Civil engineering graduates need to be competent in hydraulic theory, as well as in the application of that theory to the solution of practical problems. Teachers of hydraulic design are faced with the dilemma that most realistic hydraulics problems are too complex to solve by hand, while most commercially available software packages obscure the theoretical background for program algorithms. Equation solvers provide a valuable tool for bridging these gaps. Students can develop an appropriate linear or nonlinear mathematical model to depict a realistic system, then use an equation solver package to solve that model for any combination of input data desired. Computer-based studio classrooms further enhance the learning experience by allowing students to solve problems under the instructor's supervision during class periods. This paper will describe how effective equation solvers and the studio classroom can be in teaching hydraulic design for water distribution systems and open-channel flow. The theory is developed in class through the use of printed notes. Students then develop the nonlinear mathematical model for a simple example, solve the model using an equation solver, and check the correctness of the solution. Students are able to investigate the dynamic response and the sensitivity of the model by varying the equation solver input variable values. Next they apply the theory and solution methods to a practical applications exercise. The final step is to complete a comprehensive, realistic design problem. Students are required to present their results to the class at all stages of the process. Course-end evaluation scores have risen significantly since the class has been converted to the studio format. Student comments indicate that they think equation solvers are a valuable engineering design tool, not only for learning, but in professional practice as well. The instructor has observed that students learn and retain the theory much better when they can apply it immediately to realistic problems. Much more realistic and sophisticated quizzes can be given when the students have computers available to assist with the analysis.     

Experimental Analysis of Hydraulic Solver Convergence with Genetic Algorithms

A procedure for the experimental convergence evaluation of a hydraulic-network solver is proposed, based on using genetic algorithms to search for network parameter values that maximize the number of iterations of the hydraulic-network solver under test. The efficiency of the method is demonstrated by the example of convergence evaluation for the EPANET hydraulic simulator. Examples of a pipe network and of combinations of parameter values for which the static solver of the simulator fails to converge in a reasonable number of iterations are given. The features of the EPANET 2.00.12 solver responsible for loss of convergence are discussed. New criteria for the automatic start of solution damping aimed at improving the convergence of the solver are proposed. The better convergence of the EPANET solver modified in accordance with these criteria is confirmed by the random and the proposed search-based testing method.     

Influence of Regulators in Controlling Upstream Water Depth

Control of irrigation canals usually consists of control of water levels upstream from regulators or check structures. Regulators provide the necessary head to offtakes. Generally, influence factor is used to express the extension of the backwater curve effect within the controlled reach. This factor shows how a change in water depth exercised by a regulator can influence the water surface profile along an irrigation canal. No direct equation is available in the technical literature up until now for calculating this factor on the basis of the steady gradually varied flow theorem. In current research, using the steady gradually varied flow equation for a prismatic canal, an elegant algebraic equation for this factor is derived. Control of water levels upstream from regulators is an important application of this equation in irrigation networks.     

Flow Computation and Mass Balance in Galerkin Finite-Element Groundwater Models

In most groundwater modeling studies, quantification of the flow rates at domain and subdomain boundaries is as important as the computation of the groundwater heads. The computation of these flow rates is not a trivial task when a finite-element method is chosen to solve the groundwater equation. Generally, it is believed that finite-element methods do not conserve mass locally. In this paper, a postprocessing technique is developed to compute mass-conserving flow rates at element faces. It postprocesses the groundwater head field obtained by the Galerkin finite-element method, and the calculated flow rates conserve mass locally and globally. The only requirement for the postprocessor to be applicable is the irrotationality of the flow field, i.e., the curl of the Darcy flux should be zero. The accuracy and the mass conservation properties of the new postprocessor are demonstrated using several test problems that include one-, two-, and three-dimensional flow systems in both homogeneous and heterogeneous aquifer conditions.     

Validation of a Large-Eddy Simulation Model to Simulate Flow in Pump Intakes of Realistic Geometry

This paper describes efforts toward developing a reliable numerical model to predict pump intake flow and associated vortices. Numerical prediction of these flows characterized by the formation of unsteady (meandering) intermittent vortices and presence of massive separation is very challenging. Successful prediction of these phenomena and their effects on the mean flow fields requires numerical methods and turbulence models that can accurately capture the dynamics of the main coherent structures in these flows. In the present work, large-eddy simulation (LES) in conjunction with an accurate nondissipative nonhydrostatic Navier-Stokes massively parallel solver is used to predict the flow and vortical structures in a pressurized pump intake of complex geometry. The LES model is validated using particle image velocimetry data recently collected on a laboratory model of a realistic geometry pump intake. To better put in perspective the predictive performance of the LES model, results from steady simulations employing the shear stress transport (SST) Reynolds-averaged-Navier-Stokes (RANS) model are presented and compared with LES. It is shown that even if SST can fairly successfully capture the mean velocity distribution and mean vortical structures in some regions, overall LES can more accurately predict the mean flow and turbulence statistics compared to the steady SST model.     

Method of Fundamental Solutions for Three-Dimensional Stokes Flow in Exterior Field

The main purpose of the present paper is to provide practical and numerical implementations of the method of fundamental solutions for three-dimensional exterior Stokes problems with quiet far-field condition and discuss the issues therein. The solutions of the steady Stokes problems are obtained by utilizing the boundary collocation method as well as the expansion of Stokeslets, which are the fundamental solutions of the steady Stokes equations. To validate the proposed model, numerical results of a lid-driven cavity flow, uniform flow passing a sphere, and a rotating dumbbell-shaped body show good agreement with the numerical and analytical solutions available in the literature. Also, a hypothetical problem with both vorticity and velocity boundary conditions is solved and compared with the analytical solution. The proposed model is then properly exploited to obtain the flow results of uniform flow passing a pair of vertical spheres in tandem and uniform flow passing a pair of horizontal spheres in tandem. Furthermore, the accuracy of the present numerical scheme is addressed and the detail flow characteristics, such as pressure distribution, streamline contour, velocity field, and vorticity fields are sketched.     

Efficient Solution of the Coupled One-Dimensional Surface—Two-Dimensional Subsurface Flow during Furrow Irrigation Advance

Physically based modeling of the coupled one-dimensional surface and two-dimensional (2D) subsurface flow during furrow irrigation advance often causes numerical instabilities and nonconvergence problems. This is particularly the case for low irrigation advance rates when infiltration consumes a predominant part of the inflow volume. The proposed furrow advance phase model (FAPS) further develops the concepts of a previous study. An analytical zero-inertia surface flow model is iteratively coupled with the 2D subsurface water transport model HYDRUS-2. In contrast to the previous study, the flow domain is discretized using fixed space increments and the resulting set of nonlinear flow equations is solved using the Newton method. The complexity of the model was reduced by process adequate simplifications. FAPS exhibited better convergence, numerical stability, and less computational time than the original fixed time interval solution. The new solution converged rapidly for a number of model tests with various inflow rates including runs with very slow irrigation advance. Simulation model predictions agree very well with advance times measured in laboratory and field tests.     

Coagulation on biomaterials in flowing blood: some theoretical considerations

Are truly inert biomaterials feasible? Recent mathematical models of coagulation which are reviewed here suggest that such materials are impossible. This conclusion, which is certainly consistent with our collective experimental evidence, arises from the calculation that conversion of Factor XI to XIa never drops to zero even at the highest flow rates and with virtually no Factor XIIa bound to a surface. Residual amounts of XIa are still formed which can in principle kick-off the coagulation cascade. Furthermore, if the flow rates and corresponding mass transfer coefficients are low and in spite of these near-vanishing levels of the initiating coagulants, the surprising result is that substantial amounts of thrombin are produced. On the contrary, under slightly higher flow conditions, there can be more substantial levels of initiating coagulants, yet paradoxically thrombin production is near zero. This article presents a theoretical understanding of the events which take place during the interaction of biomaterials with flowing blood. We follow these events from the time of first contact to the final production of thrombin. The effect of flow and surface activity on the contact phase reactions is examined in detail and the two are found to be intertwined. The common pathway is also examined and here the main feature is the existence of three flow dependent regions which produce either high or very low levels of thrombin, as well as multiple thrombin steady states. In a final analysis we link the two segments of the cascade and consider the events which result. In addition, we note that multiple steady states arise only in the presence of two (thrombin) feedback loops. Single loops or the bare cascade will produce only single steady states. With some imagination one can attribute to the feedback loops the role of providing the cascade with a mechanism to produce high thrombin levels in case of acute need (e.g. bleeding) or to allow levels to subside to 'stand-by' when there is no need for clotting. We present this as a partial answer to the question: Why is the coagulation cascade so complex and what is the importance of the feedback loops?     

Parameter Estimation in Groundwater Hydrology Using Artificial Neural Networks

The capability of artificial neural networks to act as universal function approximators has been traditionally used to model problems in which the relation between dependent and independent variables is poorly understood. In this paper, the capability of an artificial neural network to provide a data-driven approximation of the explicit relation between transmissivity and hydraulic head as described by the groundwater flow equation is demonstrated. Techniques are applied to determine the optimal number of nodes and training patterns needed for a neural network to approximate groundwater parameters for a simulated groundwater modeling case study. Furthermore, the paper explains how such an approximation can be used for the purpose of parameter estimation in groundwater hydrology.     

Preserving Steady-State in One-Dimensional Finite-Volume Computations of River Flow

When using finite-volume methods and the conservative form of the Saint Venant equations in one-dimensional flow computations, it is important to establish the correct balance between the discretized flux vector and the geometric source terms. Over the last few years various improvements to numerical schemes have been presented to achieve this correct balance, focusing on the capability to simulate water at rest on irregular geometries (C-property). In this paper it is shown that common schemes can lead to energy-violating solutions in the case of steady flow. We present developments based on the Roe TVD finite-volume scheme for one-dimensional Saint Venant equations, which results in a method that not only satisfies the C-property, but also preserves the correct steady flow when stationary boundary conditions are used. We also present a totally irregular channel test case for the verification of the method.     

Transient Fluid Flow in a Continuous Casting Tundish during Ladle Change and Steady‐state Casting

Transient effects occur during both steady‐state casting as well as transient casting, e.g. a ladle change. These effects are caused by transient boundary conditions at the inlet of the tundish. A time‐dependent inlet temperature causes a free convection flow during steady‐state casting. During transient casting, such as during a ladle change, the mass flow at the inlet is time‐dependent and thus a transient flow develops. In general, transient flow is unwanted because transient flow means a change of conditions for the separation of non‐metallic particles. The analysis of the flow in the tundish is carried out by numerical as well as physical simulations. In this case experimental investigations are carried out on a water model. The results of laser optical investigations using Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (DPIV) serve as a validation of the numerical results. The numerical results are then used for the investigation of the thermal melt flow. The effects caused by a changing bath level during transient casting (ladle change) are investigated using the Volume‐of‐Fluid (VoF) model. Beyond this, the interaction between the melt and slag is taken into account, by using the three phase system melt‐slag‐air. In addition to the classical methods a new zonal approach is introduced in this paper. The integral balance localises high turbulence mixing regions as well as the development and intensity of back flows. The levelling of the momentum flux between the inlet and the outlet can also be described.     

Numerical and Experimental Investigations of Curved Fatigue Crack Growth under Proportional Cyclic Loading

The present paper demonstrates that the common computer‐aided two‐dimensional crack path prediction can be considerably improved in accuracy by using a new predictor‐corrector procedure in combination with the modified virtual crack closure integral (MVCCI) method. The numerical crack growth simulation is still based on a step‐by‐step technique, but uses a piecewise curved approximation of the crack path. By this method, both the new locus of the crack tip and the slope of the crack path at the crack tip can be computed simultaneously by the mode I and II stress intensity factors for only one virtual tangential crack extension with respect to the previous step. Furthermore, the paper presents a new finite element technique of calculating stress intensity factors of mixed mode problems by MVCCI Method. The procedure is devised to calculate the separated strain energy release rates by using the convergence of two separate calculations with different element sizes in the neighbourhood of the crack. The method provides robust calculation ability, even in the case of very coarse meshes. In order to evaluate the validity and efficiency of the proposed higher order crack path simulation method in relation to the well‐established basic strategies, experiments of curved fatigue crack growth are carried out with a specially designed specimen under proportional lateral force bending. In all investigated cases the computationally predicted crack trajectories show an excellent agreement with the different types of curved cracks that are obtained in the experiments as a function of the position of crack initiation.