Analytical inversion of specific energy–depth relationship in channels with parabolic cross-sections

Abstract

The specific energy–depth relationship in open channels with parabolic cross-sections is analytically inverted. Two nondimensional expressions of the specific energy are considered, depending on the prescribed quantity (specific energy or alternate depth). The inversion of such functions consists of finding the roots of cubic and quartic equations. By solving a quartic equation for a given discharge and for each value of the specific energy, a subcritical depth and a supercritical depth are found analytically. In this case, two acceptable roots are recognized and two other roots are discarded, on the basis of their physical meaning. Moreover, by solving a cubic equation for a given discharge and for each value of the depth, the other corresponding alternate depth is found analytically. Then, one acceptable root is recognized and two other complex conjugate roots are discarded. Finally, different examples are presented, to show the efficiency of the proposed solutions. Such analytical solutions can be easily used in natural rivers and parabolic channels.

Citation Vatankhah, A. R. & Valiani, A. (2011) Analytical inversion of specific energy–depth relationship in channels with parabolic cross-sections. Hydrol. Sci. J. 56(5), 834–840.     

Analytical and numerical analysis of tides and salinities in estuaries; part I: tidal wave propagation in convergent estuaries

Analytical solutions of the momentum and energy equations for tidal flow are studied. Analytical solutions are well known for prismatic channels but are less well known for converging channels. As most estuaries have a planform with converging channels, the attention in this paper is fully focused on converging tidal channels. It will be shown that the tidal range along converging channels can be described by relatively simple expressions solving the energy and momentum equations (new approaches). The semi-analytical solution of the energy equation includes quadratic (nonlinear) bottom friction. The analytical solution of the continuity and momentum equations is only possible for linearized bottom friction. The linearized analytical solution is presented for sinusoidal tidal waves with and without reflection in strongly convergent (funnel type) channels. Using these approaches, simple and powerful tools (spreadsheet models) for tidal analysis of amplified and damped tidal wave propagation in converging estuaries have been developed. The analytical solutions are compared with the results of numerical solutions and with measured data of the Western Scheldt Estuary in the Netherlands, the Hooghly Estuary in India and the Delaware Estuary in the USA. The analytical solutions show surprisingly good agreement with measured tidal ranges in these large-scale tidal systems. Convergence is found to be dominant in long and deep-converging channels resulting in an amplified tidal range, whereas bottom friction is generally dominant in shallow converging channels resulting in a damped tidal range. Reflection in closed-end channels is important in the most landward 1/3 length of the total channel length. In strongly convergent channels with a single forward propagating tidal wave, there is a phase lead of the horizontal and vertical tide close to 90^o, mimicking a standing wave system (apparent standing wave).     

Sensitivity analysis of 2D steady-state shallow water flow. Application to free surface flow model calibration

This paper presents the analytical properties of the solutions of the sensitivity equations for steady-state, two-dimensional shallow water flow. These analytical properties are used to provide guidelines for model calibration and validation. The sensitivity of the water depth/level and that of the longitudinal unit discharge are shown to contain redundant information. Under subcritical conditions, the sensitivities of the flow variables are shown to obey an anisotropic elliptic equation. The main directions of the contour lines for water depth and the longitudinal unit discharge sensitivity are parallel and perpendicular to the flow, while they are diagonal to the flow for the transverse unit discharge sensitivity. Moreover, the sensitivity for all three variables extends farther in the transverse direction than in the longitudinal direction, the anisotropy ratio being a function of the sole Froude number. For supercritical flow, the sensitivity obeys an anisotropic hyperbolic equation. These findings are confirmed by application examples on idealized and real-world simulations. The sensitivities to the geometry, friction coefficient or model boundary conditions are shown to behave in different ways, thus providing different types of information for model calibration and validation.     

ASSESSMENT OF MACCORMACK SCHEME IN UNSTEADY SEDIMENT LADEN FLOW COMPUTATION

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Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.     

Variation of specific yield with depth in an alluvial aquifer of the Platte River valley, USA

Estimation of specific yield (Sy) of an aquifer is of great importance in water resource management. In this study, an experimental drainage method was developed to determine specific yield of an alluvial aquifer of the Platte River valley, Nebraska, USA. Sets of sediment cores with continuous interval depth were collected to plastic tubes using the direct push technique and then taken to the laboratory. During the Sy experiment, those sediment cores were re-saturated by placing them vertically in a large water tank. Sy was determined by the water drained from the sediments by force of gravity in a bracket. Our results show that the values of Sy varied largely with depth at each site and the variability for Sy with interval depth between the test sites is also observed. This spatial heterogeneity in Sy might result from the variation of grain size, grain shape, sorting and compaction of sediments in different cores with interval depth. The Sy for all sediment cores ranged from 0.01 to 0.18 and the mean value was 0.08±0.04. Our drainage method can functionally preserve the sedimentary structures in their original state and it is easier to experiment at a lower expense.     

Extension of Vedernikov's graph for seepage from canals

In this investigation, using previously derived equations by Vedernikov and Morel-Seytoux, closed-form solutions have been obtained to compute the seepage from a slit and a strip. Also, a graphical solution as an extension of Vedernikov's graph has been presented for computing quantity of seepage from triangular, rectangular, and trapezoidal canals. The solution replaces approximately the cumbersome evaluation of improper integrals with unknown implicit transformation variables.     

Simulation of dam-break flow with grid adaptation

The unsteady free surface flow caused by sudden collapse of a dam produces discontinuities in the flow variables. As the flow surges downstream, it forms a moving bore front with steep gradients of water height and velocity. In the numerical simulation of this flow, proper grid distribution can play a crucial part in the prediction and resolution of the solutions. The use of presently available numerical schemes to solve this problem on a uniform course grid system fails to resolve the characteristic flow features and hence do a poor job in simulating this flow. In this paper, an adaptive grid which adjusts itself as the solution evolves is used for a better resolution of the flow properties. Rai and Anderson's¹² method is used to determine the grid speed; however, a different partial differential equation based on the conservative principle of grid arc lengths for clustering grids in one-dimensional flow is used along with the St. Venant equations to numerically simulate the flow. Both the subcritical and the supercritical flows under extreme boundary conditions are solved using this technique. With a specified number of grid points, this provides better quality solutions as compared to those obtained with uniformly distributed grids.     

A general model of lateral depth-averaged velocity distributions for open channel flows

This paper reviews a model, developed by Shiono and Knight [Shiono K, Knight DW. Two-dimensional analytical solution for a compound channel. In: Proceedings of the 3rd international symposium on refined flow modelling and turbulence measurements, Tokyo, Japan, July 1988. p. 503–10; Shiono K, Knight DW. Turbulent open channel flows with variable depth across the channel. J Fluid Mech 1991;222:617–46 [231:693]], which yields analytical solutions to the depth-integrated Navier–Stokes equations, and includes the effects of bed friction, lateral turbulence and secondary flows. Some issues about the original model developed by Shiono and Knight (1988, 1991) are highlighted and discussed. Based on the experimental data concerning the secondary flow, two assumptions are proposed to describe the contribution of the streamwise vorticity to the flow. Two new analytical solutions are compared with the conventional solution for three simple channel shapes and one trapezoidal compound channel to highlight their differences and the importance of the secondary flow and planform vorticity term. Comparison of the analytical results with the experimental data shows that the general SKM predicts the lateral distributions of depth-averaged velocity well.     

Streamflow routing in the indian arid zone

A lumped model for streamflow routing in arid ephemeral channels has been developed. The governing equations for movement of flood waves subjected to transmission losses are simplified through a time averaging process to develop an ordinary differential equation describing transmission losses as a function of distance, inflow, channel width, time parameters of flow and effective hydraulic conductivity. The resulting equation has an analytical solution and simulates runoff volume and peak discharge rates for individual storm events. The outflow hydrograph is fairly well approximated with a triangular approximation. The model is simplified and constructed to require a minimum of observed data for calibration. It can also be used for ungauged basins in arid regions through parameterization.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

Obtaining the steady‐state drawdown solutions of constant‐head and constant‐flux tests

The solutions of constant‐head and constant‐flux tests are commonly used to predict the temporal or spatial drawdown distribution or to determine aquifer parameters. Theis and Thiem equations, for instance, are well‐known transient and steady‐state drawdown solutions, respectively, of the constant‐flux test. It is known that the Theis equation is not applicable to the case where the aquifer has a finite boundary or the pumping time tends to infinity. On the other hand, the Thiem equation does not apply to the case where the aquifer boundary is infinite. However, the issue of obtaining the Thiem equation from the transient drawdown solution has not previously been addressed. In this paper, the drawdown solutions for constant‐head and constant‐flux tests conducted in finite or infinite confined aquifers with or without consideration of the effect of the well radius are examined comprehensively. Mathematical verification and physical interpretation of the solutions to these two tests converging or not converging to the Thiem equation are presented. The result shows that there are some finite‐domain solutions for these two tests that can converge to the Thiem equation when the time becomes infinitely large. In addition, the time criteria to give a good approximation to the finite‐domain solution by the infinite‐domain solution and the Thiem equation are investigated and presented. Copyright © 2008 John Wiley & Sons, Ltd.     

An analytical solution to multi-component NAPL dissolution equations

This note presents a novel method for determining the changing composition of a multi-component NAPL body dissolving into moving groundwater, and the consequent changes in the aqueous phase solute concentrations in the surrounding pore water. A canonical system of coupled non-linear governing equations is derived which is suitable for representation of both pooled and residual configurations, and this is solved. Whereas previous authors have handled such problems numerically, it is shown that these governing equations succumb to analytical solution. By a suitable substitution, the equations become decoupled, and the problem collapses to a single first-order equation. The final result is expressed implicitly, with time as a function of the number of moles of the least soluble component, m₁. The number of moles of each other component is expressed explicitly in terms of m₁. It is shown that the time-m₁ relationship has a well behaved inverse. An example is given in which the analytic solution is verified against traditional finite difference analysis, and its computational efficiency is shown.     

Solutions of the heat conduction equation in a non-uniform soil

A class of analytic, periodic solutions of the heat conduction equation in a non-uniform soil is derived. The class may be characterized by the fact that the speed of the temperature wave varies according to the square root of the soil diffusivity (a function of soil depth). In addition it is shown that the constant soil solution is the limiting case when the rate of change with depth of diffusivity and thermal conductivity become very small. The solutions may be regarded as general whenever temperature analysis is restricted to small values of depth or whenever the soil parameters vary slowly. For all other cases the class of solutions possess the additional property that the rate of change of conductive capacity varies directly as the product of the bulk density and specific heat of the soil. A particular temperature profile is given for the case when the diffusivity varies as the nth power of depth.     

Analytic stage–discharge formulae for flow in straight trapezoidal open channels

Analytic stage–discharge formulae are derived for flow in straight trapezoidal channels, based on the 2D analytic velocity distribution in open channels given by Shiono and Knight [Shiono K, Knight DW, Turbulent open-channel flows with variable depth across the channel. J Fluid Mech 1991;222:617–46]. A simple hand-calculation method is provided. Legendre incomplete elliptic integrals of the first and second kinds and a binomial series expansion are used in the derivation of these analytic formulae, together with physically based hydraulic parameters, such as local friction factor (f), dimensionless eddy viscosity (λ) and secondary flow (Γ). The stage–discharge results obtained from the formulae are shown to be in good agreement with experimental data, as are the corresponding analytic velocity and boundary shear stress distributions. The influences of f, λ and Γ on the stage–discharge relationship are also discussed.     

Series solutions for flow in stratified aquifers with natural geometry

An analytical series solution method is presented for modeling regional steady-state groundwater flow in a two-dimensional stratified aquifer cross-section where the water table is well-characterized. The aquifer system may have any number of contiguous or non-contiguous layers and the geometry of each layer is restricted only by the requirement that the elevation of the stratigraphic unconformities between layers is a function of the x-coordinate alone. Various techniques may be used to handle pinching layers, faults, and other discontinuities. The solutions are obtained by minimizing head and flow continuity errors between layers and errors in the Dirichlet surface at a set of control points along these unconformities; the governing equation is met exactly. The solutions are derived and demonstrated on multiple test cases. The errors for some specific, geometrically challenging cases are assessed and discussed.     

Radiation from a Finite Reverse Fault in a Half Space

— Using a set of well-known results for the seismic field radiated by a simple dip-slip dislocation in a half space, we study interesting details of the motion at the surface of the half space. The static solution for a dislocation in a half space was found by Freund and Barnett in 1976. The corresponding elastodynamic solution was solved exactly in the Fourier and Laplace domain by several authors about 20 years ago, however its properties remained unexplored because of analytical difficulties. We remove these difficulties and show that the solution contains three important phenomena: Seismic wave fronts of P, S and SP type; the near-field pulse associated with the propagation of the dislocation front; and the long-time elastic response that converges toward the static solution of Freund and Barnett. Based on these results we show that solutions to all these problems are self-similar and homogeneous in x/h and αt/h so that when the fault depth h approaches 0, the solutions become concentrated near the origin and around the P, S and surface wave travel times. This explains several paradoxes in the radiation from dip-slip faults; among these the most notable are the presence of a point force singularity at the tip of a surface breaking fault and the reduction in high frequency radiation near the surface.     

Characterizing supraglacial meltwater channel hydraulics on the Greenland Ice Sheet from in situ observations

Supraglacial rivers on the Greenland Ice Sheet (GrIS) transport large volumes of surface meltwater toward the ocean, yet have received relatively little direct research. This study presents field observations of channel width, depth, velocity, and water surface slope for nine supraglacial channels on the south‐western GrIS collected between July 23 and August 20, 2012. Field sites are located up to 74 km inland and span 494–1485 m elevation, and contain measured discharges larger than any previous in situ study: from 0.006 to 23.12 m³/s in channels 0.20 to 20.62 m wide. All channels were deeply incised with near vertical banks, and hydraulic geometry results indicate that supraglacial channels primarily accommodate greater discharges by increasing velocity. Smaller streams had steeper water surface slopes (0.74–8.83%) than typical in terrestrial settings, yielding correspondingly high velocities (0.40–2.60 m/s) and Froude numbers (0.45–3.11) with supercritical flow observed in 54% of measurements. Derived Manning's n values were larger and more variable than anticipated from channels of uniform substrate, ranging from 0.009 to 0.154 with a mean value of 0.035 ± 0.027 despite the absence of sediment, debris, or other roughness elements. Ubiquitous micro‐depressions in shallow sections of the channel bed may explain some of these roughness values. However, we find that other, unobserved sources of flow resistance likely contributed to these elevated Manning's n values: future work should explicitly consider additional sources of flow resistance beyond bed roughness in supraglacial channels. We conclude that hydraulic modeling for these channels must allow for both subcritical and supercritical flow, and most importantly must refrain from assuming that all ice‐substrate channels exhibit similar hydraulic behavior, especially for Froude numbers and Manning's n. Finally, this study highlights that further theoretical and empirical work on supraglacial channel hydraulics is necessary before broad scale understanding of ice sheet hydrology can be achieved. Copyright © 2016 John Wiley & Sons, Ltd.     

A new saturated/unsaturated model for stormwater infiltration systems

Infiltration systems are widely used as an effective urban stormwater control measure. Most design methods and models roughly approximate the complex physical flow processes in these systems using empirical equations and fixed infiltration rates to calculate emptying times from full. Sophisticated variably saturated flow models are available, but rarely applied owing to their complexity. This paper describes the development and testing of an integrated one‐dimensional model of flow through the porous storage of a typical infiltration system and surrounding soils. The model accounts for the depth in the storage, surrounding soil moisture conditions and the interaction between the storage and surrounding soil. It is a front‐tracking model that innovatively combines a soil‐moisture‐based solution of Richard's equation for unsaturated flow with piston flow through a saturated zone as well as a reservoir equation for flow through a porous storage. This allows the use of a simple non‐iterative numerical solution that can handle ponded infiltration into dry soils. The model is more rigorous than approximate stormwater infiltration system models and could therefore be valuable in everyday practice. A range of test cases commonly used to test soil water flow models for infiltration in unsaturated conditions, drainage from saturation and infiltration under ponded conditions were used to test the model along with an experiment with variable depth in a porous storage over saturated conditions. Results show that the model produces a good fit to the observed data, analytical solutions and Hydrus. Copyright © 2008 John Wiley & Sons, Ltd.     

The role of molecular diffusion in dispersion theory

The analytic derivations of the equations governing dispersive flow assume that different solutions entering an interstitial flow channel are completely mixed at the exit node of the flow channel. The physical mechanisms which can effect this mixing are turbulence and molecular diffusion between the solutions in the channel. We have examined these mechanisms and find for the interstitial velocities ordinarily encountered in groundwater flow that turbulence is not effective as a mixing mechanism whereas molecular diffusion is.For molecular diffusion to be efficient as a mixing mechanism the interstitial velocities should be less than:

$\text{u < 2.25}\text{ks}\text{h}{}^2$

where k is the diffusion coefficient of the salt solution; s the mean length of the interstitial channels and h the channel diameter.This velocity places an upper limit to the range of validity of the equations of dispersion theory as presently developed.