Numerical Modeling of Bed Evolution in Channel Bends

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data than those obtained by previous numerical models. The model is used to investigate the process of evolution and stability of bed deformation in circular bends.     

3D Numerical Modeling of Flow and Sediment Transport in Open Channels

A 3D numerical model for calculating flow and sediment transport in open channels is presented. The flow is calculated by solving the full Reynolds-averaged Navier-Stokes equations with the k ? ε turbulence model. Special free-surface and roughness treatments are introduced for open-channel flow; in particular the water level is determined from a 2D Poisson equation derived from 2D depth-averaged momentum equations. Suspended-load transport is simulated through the general convection-diffusion equation with an empirical settling-velocity term. This equation and the flow equations are solved numerically with a finite-volume method on an adaptive, nonstaggered grid. Bed-load transport is simulated with a nonequilibrium method and the bed deformation is obtained from an overall mass-balance equation. The suspended-load model is tested for channel flow situations with net entrainment from a loose bed and with net deposition, and the full 3D total-load model is validated by calculating the flow and sediment transport in a 180° channel bend with movable bed. In all cases, the agreement with measurements is generally good.     

Explicit Schemes for Dam-Break Simulations

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.     

Maximum Migration Distance of Meander Channel in Sand Using Hyperbolic Function Approach

A set of large-scale laboratory experiments were conducted to study the migration of meandering channel. Factors affecting the change of banklines, including the ratio of centerline curvature to channel width, bend angle, and Froude number were tested in the experiments. The effect of each factor on the evolution of channel plan form was evaluated and quantified. The channel bankline displacement was modeled by a hyperbolic function with the inclusion of the initial migration rate and the maximum migration distance. A three-dimensional numerical model was also employed to explain some findings in the laboratory tests. It is found that the maximum migration distance along a bend satisfies a Gaussian distribution. A set of equations were developed for predicting the maximum migration distance. With the maximum migration distance being developed as a function of several geometric and flow parameters, a hyperbolic-function model can be applied to estimate the maximum bankline migration distance when the channel reaches equilibrium.     

Effect of Bed Armoring on Bed Topography of Channel Bends

The two-dimensional numerical model previously developed by the writers for modeling the bed variations in a channel bend with uniform sediment is upgraded to incorporate the nonuniformity of sediment particles as well as bed armoring. In this model, the two-dimensional, depth-averaged, unsteady flow equations along with the bed-load mass conservation equation are solved in a body-fitted coordinate system by using the Beam and Warming alternating-direction implicit (ADI) scheme. A one-dimensional bed surface armoring approach is extended herein for application to a two-dimensional domain. The model is applied to a 180° bend with a constant radius under unsteady flow conditions. Numerical simulations are carried out to study the effect of bed armoring on the bed deformations in channel bends. Results show that bed armoring reduces scour in channel bends.     

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.     

Laminar Water Wave and Current Passing Over Porous Bed

The problem of the dynamic interaction of water waves, current, and a hard poroelastic bed is dealt with in this study. Finite-depth homogeneous water with harmonic linear water waves passing over a semi-infinite poroelastic bed is investigated. In order to reveal the importance of viscous effect for different bed forms, viscosity of water is considered herein. In a boundary layer correction approach, the governing equations of the poroelastic material are decoupled without losing physical generality. The contribution of pressure effect and shear effect to the hard poroelastic bed, which is a valuable indication to the mechanism of ripple formation, is clarified in the present study. This approach will be helpful in saving time and storage capacity when it is applied to numerical computation.     

Drainage of Sloping Lands with Variable Recharge: Analytical Formulas and Model Development

Semianalytical transient equations for shallow subsurface transverse drainage systems installed in sloping lands are developed. They provide a general relationship between drain flow rates, water table elevations, and recharge rates. This relationship demonstrates that, depending on the recharge intensity, several drain flow rates can be observed at a given water table elevation. The recharge contribution is shown to depend on a water table shape factor and to decrease when the water table is low or the slope is steep. For very steep slopes, the recharge intensity no longer influences the drain flow rate. These equations can be used to confirm previous results obtained in steady-state conditions and to determine precisely under which conditions slope needs to be considered in drainage design. They have been incorporated into the field drainage model SIDRA, which simulates hourly values of water table elevations and drain flow rates. The model predictions are compared with the predictions of a steady-state equation and a numerical model, which solves the Boussinesq equation (SLOP model).     

Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids

A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge–Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.     

Numerical Modeling of Contaminant Transport through Soils: Case Study

This paper presents the development and validation of a numerical model for simulation of the flow of water and air and contaminant transport through unsaturated soils. The governing differential equations include two mass balance equations for the water phase and air phase together with a balance equation for contaminant transport through the two phases. In the model the nonlinear system of the governing differential equations was solved using a finite-element method in the space domain and a finite difference scheme in the time domain. The governing equation of the miscible contaminant transport including advection, dispersion-diffusion and adsorption effects are presented. The mathematical framework and the numerical implementation of the model are described in detail. The model is validated by application to standard experiments on contaminant transport in unsaturated soils. The application of the model to a case study is then presented and discussed. Finally, the merits and limitations of the model are highlighted.     

Model for Coupled Liquid Water Flow and Heat Transport with Phase Change in a Snowpack

The flow of liquid water in a snowpack is complex because of the coupled processes involved, including the phase change between liquid and solid, and the latent and sensible heat transfer processes. To properly describe the details of spatial and temporal changes in a snowpack it is necessary to include these coupled processes. This paper presents a numerical model of coupled liquid water flow and heat transport in a snowpack. The model is intended to quantify infiltration into a snowpack, and evaluate the potential for the formation of distinct heterogeneities in liquid water and heat transport properties in a snowpack. The numerical model solves the two-dimensional form of the governing coupled equations using a finite difference scheme. The governing equations assume thermodynamic equilibrium between the solid and liquid phases in the snowpack. Equations describing the metamorphosis of ice grains during liquid water flow are applied within the model, and the heat and liquid water transport properties of the snow are treated with relations identical to those used for mineral porous media. Sample solution results for an alternative formulation taken from the literature are used to test the present solution, and it is found that the present model yields similar results but with some distinct differences. The effect of direct coupling of the temperature with the liquid water pressure is presented in a simple horizontal freezing simulation, which is compared with the Stefan problem where liquid water is not redistributed. Overall the direct coupling and water redistribution is found to lead to greater front penetration in comparison to the Stefan formulation. For infiltration with gravity it is shown that grain size growth during infiltration leads to increased wetting front penetration.     

Numerical Simulation of Shallow-Water Flow Using a Modified Cartesian Cut-Cell Approach

The Cartesian cut-cell method can be used to represent irregular and complex computational domains with less computational efforts by cutting the grid cells on the boundary surfaces in a background uniform Cartesian mesh. In this study, a modified Cartesian cut-cell grid technique is proposed to better represent complex physical geometries. A point shifting treatment was employed to determine the start and end points of a line segment in cut-cell grids. This led to an improved representation of sharply-shaped corners in surface polygons. Numerical simulation to solve a set of shallow-water equations was performed by incorporating a finite volume approach into the Cartesian cut-cell mesh. The advective fluxes at intercells were first estimated by a Harten, Lax and van Leer for contact wave approximate Riemann solver. In order to improve the model accuracy to the second order, a total variation diminishing-weighted average flux method was applied to work adaptively with the cut-cell mesh. The numerical model was then employed to simulate dam-break flow propagation in a small channel with a rectangular obstacle or a 45° bend. The numerical results show good agreement with available laboratory measurements.     

Heat Transfer through the Copper Plate of Slab Moulds

In slab moulds the cooling water flows in notches which are machined into the copper body. Therefore, the heat transfer from the hot side of the copper body to the cooling water is two‐ or three‐dimensional. But usually the heat transport is taken to be one‐dimensional and is computed using an average value for the copper wall with the water flowing in a slit parallel to the wall. This simplification is not very satisfactory. In the present study the three‐dimensional heat flow in a slab mould has been computed with an exact numerical model and the resulting heat transfer coefficient for heat flow from the hot copper side to the bulk of the cooling water is compared with that of three simple engineering equations.     

DNA-binding domains of Fos and Jun do not induce DNA curvature: an investigation with solution and gel methods

We demonstrate the use of a DNA minicircle competition binding assay, together with DNA cyclization kinetics and gel-phasing methods, to show that the DNA-binding domains (dbd) of the heterodimeric leucine zipper protein Fos-Jun do not bend the AP-1 target site. Our DNA constructs contain an AP-1 site phased by 1-4 helical turns against an A-tract-directed bend. Competition binding experiments reveal that (dbd)Fos-Jun has a slight preference for binding to linear over circular AP-1 DNAs, independent of whether the site faces in or out on the circle. This result suggests that (dbd)Fos-Jun slightly stiffens rather than bends its DNA target site. A single A-tract bend replacing the AP-1 site is readily detected by its effect on cyclization kinetics, in contrast to the observations for Fos-Jun bound at the AP-1 locus. In contrast, comparative electrophoresis reveals that Fos-Jun-DNA complexes, in which the A-tract bend is positioned close (1-2 helical turns) to the AP-1 site, show phase-dependent variations in gel mobilities that are comparable with those observed when a single A-tract bend replaces the AP-1 site. Whereas gel mobility variations of Fos-Jun-DNA complexes decrease linearly with increasing Mg2+ contained in the gel, the solution binding preference of (dbd)Fos-Jun for linear over circular DNAs is independent of Mg2+ concentration. Hence, gel mobility variations of Fos-Jun-DNA complexes are not indicative of (dbd)Fos-Jun-induced DNA bending (upper limit 5 degrees) in the low salt conditions of gel electrophoresis. Instead, we propose that the gel anomalies depend on the steric relationship of the leucine zipper region with respect to a DNA bend.     

Effects of Lifting Force on Bed Topography and Bed-Surface Sediment Size in Channel Bend

In bed-load sediment transport, the lifting force plays an important role in reducing the friction between sediment particles and the bed surface, and it makes particle transportation by the shear force easier. Because the lifting force is related to vorticity, a three-dimensional (3D) numerical model incorporating large eddy simulations was applied to simulate the vorticity field in a channel bend. The results show that the distribution of vorticity is highly nonuniform, and it can lead to significant variations in lifting force and bed-load sediment transport per unit width in a channel bend. Relevant theories are modified on the basis of physical reasoning and then incorporated into numerical models to investigate the lifting-force effects on the bed topography and bed-surface sediment size gradation in a channel bend. With the lifting-force effects considered, it is shown that the errors in simulated bed topography can be reduced by approximately 40% and in bed-surface sediment size by 50%.     

Dam Break in Channels with 90° Bend

In practice, dam-break modeling is generally performed using a one-dimensional (1D) approach for its limited requirements in data and computation. However, for valleys with multiple sharp bends, such a 1D model may fail for predicting as well the maximum water level as the wave arrival time. This paper presents an experimental study of a dam-break flow in an initially dry channel with a 90° bend, with refined measurements of water level and velocity field. The measured data are compared to some numerical results computed with finite-volume schemes associated with Roe-type flux calculation. The 1D approach reveals the expected limits, while a full two-dimensional (2D) approach provides fine level prediction and rather satisfactory information about the arrival time. A hybrid approach is now proposed, mixing the 1D model for the straight reaches and local 2D models for the bends. The compatibility of the Roe fluxes at the interfaces requires a careful formulation, but the resulting scheme seems able to capture reflection and diffraction processes in such a way that the results are really good in what concerns the water level.     

Solution Method for Beams on Nonuniform Elastic Foundations

This technical note presents an analytical method, accompanied by a numerical scheme, to evaluate the response of beams on nonuniform elastic foundations, namely, when the foundation modulus is k = k(x). The method employs a Green’s function formulation, which results in a system of nonsingular integral equations for the distributed reaction q(x). These equations can be discretized in a straightforward manner to yield a system of linear algebraic equations that can be solved by elementary numerical techniques.     

Model for Predicting Floods due to Earthen Dam Breaching. II: Comparison with Other Methods and Predictive Use

A sensitivity analysis of the model presented in the companion paper is made and, on the basis of the results a criterion is proposed for the choice of values to assign to the side slopes of the breach, in order to use the model for prediction. Moreover, the sensitivity analysis shows that the outflow hydrograph and its peak value depend not only on the dam height and the stored volume, but also on the vertical distribution of the water mass in the reservoir. The model is compared with some previously published methods and the disadvantages, limitations, and errors that can be made using parametric models and predictive equations are pointed out. Finally, easy to use equations interpolating the numerical results of the model are provided that predict not only the peak discharge but the whole outflow hydrograph for overtopping failures.     

Multiquadric Solution for Shallow Water Equations

A computational algorithm based on the multiquadric, which is a continuously differentiable radial basis function, is devised to solve the shallow water equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from partial derivatives of the radial basis function, not by any difference scheme. The method does not require the generation of a grid as in the finite-element method and allows easy editing and refinement of the numerical model. To increase confidence in the multiquadric solution, a sensitivity and convergence analysis is performed using numerical models of a rectangular channel. Applications of the algorithm are made to compute the sea surface elevations and currents in Tolo Harbour, Hong Kong, during a typhoon attack. The numerical solution is shown to be robust and stable. The computed results are compared with measured data and good agreement is indicated.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.