Accuracy of kinematic wave and diffusion wave approximations for space-independent flows with lateral inflow neglected in the momentum equation

Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momentum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included.     

Accuracy of kinematic wave and diffusion wave approximations for space-independent flows on infiltrating surfaces with lateral inflow neglected in the momentum equation

Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momentum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow, infiltration and channel roughness when the initial condition is non-vanishing; it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included.     

ERRORS OF KINEMATIC-WAVE AND DIFFUSION-WAVE APPROXIMATIONS FOR SPACE-INDEPENDENT FLOWS ON INFILTRATING SURFACES

Error equations for the kinematic-wave and diffusion-wave approximations were derived under simplified conditions for space-independent flows occurring on infiltrating planes or channels. These equations specify error as a function of time in the flow hydrograph. The kinematic-wave, diffusion wave and dynamic-wave solutions were parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness, lateral inflow and infiltration when the initial condition is vanishing. The error equations were found to be the Riccati equation.     

Accuracy of the kinematic wave and diffusion wave approximations for time-independent flows with infiltration included

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with infiltration were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1·4% for KF²₀≥7·5, and up to 14% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary, both in terms of the magnitude of the error and the segment of the channel reach for which these approximations would be applicable. © 1998 John Wiley & Sons, Ltd.     

Errors of kinematic wave and diffusion wave approximations for time‐independent flows with infiltration and momentum exchange included

Error equations for kinematic wave and diffusion wave approximations were derived for time‐independent flows on infiltrating planes and channels under one upstream boundary and two downstream boundary conditions: zero flow at the upstream boundary, and critical flow depth and zero depth gradient at the downstream boundary. These equations specify error in the flow hydrograph as a function of space. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors below 2% for values of KF (e.g. KF ≥ 7·5), where K is the kinematic wave number and F is the Froude number. Even for small values of KF (e.g. KF = 2·5), the errors were typically less than 3%. The accuracy of the diffusive approximation was greatly influenced by the downstream boundary condition. For critical flow depth downstream boundary condition, the error of the kinematic wave approximation was found to be less than 10% for KF ≥ 7·5 and greater than 20% for smaller values of KF. This error increased with strong downstream boundary control. The analytical solution of the diffusion wave approximation is adequate only for small values of K. Copyright © 2005 John Wiley & Sons, Ltd.     

A practical method for analysis of river waves and for kinematic wave routing in natural channel networks

The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non‐linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.     

Stability and accuracy of two-dimensional kinematic wave overland flow modeling

A two-dimensional finite element based overland flow model was developed and used to study the accuracy and stability of three numerical schemes and watershed parameter aggregation error. The conventional consistent finite element scheme results in oscillations for certain time step ranges. The lumped and the upwind finite element schemes are tested as alternatives to the consistent scheme. The upwind scheme did not improve on the stability or the accuracy of the solution, while the lumped scheme provided stable and accurate solutions for time steps twice the size of time steps needed for the consistent scheme. A new accuracy based dynamic time step estimate for the two-dimensional overland flow kinematic wave solution is developed for the lumped scheme. The newly developed dynamic time step estimates are functions of the mesh size, and time of concentration of the watershed hydrograph. Due to lack of analytical solutions, the time step was developed by comparing numerical solutions of various levels of discretization to a reference solution using a very fine mesh and a very small time step. The time step criteria were tested on a different set of problems and proved to be adequate for accurate and stable solutions. A sensitivity analysis for the watershed slope, Manning’s roughness coefficient and excess rainfall rate was conducted in order to test the effect of parameter aggregation on the stability and accuracy of the solution. The results of this analysis show that aggregation of the slope data resulted in the highest error. The roughness coefficient had a smaller effect on the solution while the rainfall intensity did not show any significant effect on the flow rate solution for the range of rainfall intensity used. This work pioneers the challenge of providing guidelines for accurate and stable numerical solutions of the two-dimensional kinematic wave equations for overland flow.     

An experimental study on the evolution law of roll wave parameters in overland flow

Roll waves commonly occur in overland flow and have an important influence on the progress of soil erosion on slopes. This study aimed to explore the evolution and mechanism of roll waves on steep slopes. The potential effects of flow rate, rainfall intensity and bed roughness on the laws controlling roll wave parameters were investigated. The flow rates, rainfall intensities and bed roughness varied from 5 to 30 L/min, 0 to 150 mm/h, and 0.061 to 1.700 mm, respectively. The results indicate that roll waves polymerize significantly along the propagation path, and bed roughness and rainfall affect the generation and evolution of roll waves. The wave velocity, length and height decreased with bed roughness, whereas the wave frequency increased with increasing bed roughness under fixed flow rate and rainfall intensity conditions. Rainfall increased the wave velocity and wavelength and decreased the wave frequency. The wave velocity, height and wavelength tended to increase with an increasing flow rate. Rainfall promoted the generation of roll waves, whereas bed roughness had the opposite effect. The generation of roll waves is closely related to the Froude number (Fr) and flow resistance. In this experiment, the range of the Reynolds number for the roll waves generated in the laminar region was 142–416, and the range of the flow resistance coefficient was 0.64–4.85. The critical value of the Fr for flow instability in the laminar region was approximately 0.57. Exploring the generation and evolution law of roll waves is necessary for understanding the processes and dynamic mechanisms of slope soil erosion.     

Accuracy of kinematic wave and diffusion wave approximations for time-independent flows

Errors in the kinematic wave and diffusion wave approximations for time-independent (or steady-state) cases of channel flow were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors in the range 1–2% for values of KF (? 7.5), where K is the kinematic wave number and F₀ is the Froude number. Even for small values of KF (e.g. KF²₀ = 0.75), the errors were typically less than 15%. The accuracy of the diffusion wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic wave approximation was found to be less than 13% in the region 0.1 ? x ? 0.95 for KF = 7.5 and was greater than 30% for smaller values of KF (? 0.75). This error increased with strong downstream boundary control.     

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt→0, the space step size Δx=Q/(2S₀BC]_k) where Q is discharge, S₀ is bed slope, B is channel width and C_k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs. © 1997 John Wiley & Sons, Ltd.     

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE APPROXIMATIONS FOR TIME-INDEPENDENT FLOW WITH MOMENTUM EXCHANGE INCLUDED

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with momentum exchange included were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1% for KF²₀≥7·5 and up to 12% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary condition both in terms of the error magnitude and the segment of the channel reach for which these approximations would be applicable. © 1997 by John Wiley & Sons, Ltd.     

A MATHEMATICAL QUASI-2D OVERLAND FLOW MODEL

1 INTRODUCTIONThe prediction of future impacts on terrestrial ecosystems by atmospheric, climatic and land-usechanges is the aim of watershed management. Meeting these requirements scientists, managers and policymakers try to achieve the sustainable management of the vitally important resources of watersheds due toan integrated ecosystem approach at the catchment scale. As composite landscapes often have a highdegree of contingency between its elements, the transport over these landscape s…     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Experimental analysis of reservoir release wave routing in upland boulder bed rivers

Reservoir release wave routing during 33 controlled reservoir releases, along 15 upland boulder bed river channel reaches, on five different regulated rivers were monitored to assess the importance of river channel roughness and reservoir release magnitude on reservoir release wave speeds. Wave speeds varied between 0.52 and 3.01 m s^?1. Reservoir release wave translation, steepening, and attenuation occurred. With high channel roughness values reservoir release wave arrival speed is retarded in comparison to peak stage and wave steepening occurs, but with a reduction in channel roughness reservoir release wave front arrival is accelerated producing attentuation. The threshold between reservoir release wave front attenuation and steepening occurs at a pre-release discharge/channel width of approximately 0.1, an index of channel roughness. The paper also demonstrates, via comparison of observed and calculated reservoir release wave speeds on the River Washburn, Yorkshire, the difficulty of accurately predicting flood wave movement in upland boulder bed channels using existing prediction equations. The calculated values, however, revealed systematic error with pre-release discharge and reservoir release magnitude. Apparently the equations fail to account for the effects of high channel roughness together with pressure gradient forces, induced by rapid rates of stage change on the rising limb of reservoir releases. In order to accurately predict reservoir release wave movement in regulated rivers, this paper demonstrates that hydraulic studies need to be undertaken and pre-release discharges prescribed to determine desired reservoir release wave routing behaviour. Manipulation of the reservoir release pattern at the dam alone, cannot dictate reservoir release wave front form downstream or wave speed.     

Simulation of summer storms of differing recurrence intervals in a semiarid environment using a kinematic routing scheme

A kinematic flood routing procedure has been devised for a small dendritic headwater gully network on the Western slope of Colorado. the program is spatially-distributed, incorporating lateral inflows from 103 field sites on the network for which channel geometry variables are known. This model, in which a lateral inflow algorithm for the sideslopes between each channel site is convoluted into a Freeze-type (1978) numerical scheme, is fully developed in this paper. Although the field basis of the lateral inflow algorithm has been tested elsewhere (Faulkner, 1990), sensitivity tests were needed for the roughness and hillslope velocity estimates used in the routing procedure. After these successful tests, a suitably precalibrated run of the model was compared with a field-monitored runoff event on the watershed, and results again were encouraging. However, peak attentuation downstream was more pronounced in reality than on the simulation, so the model was also modified by inclusion of allowances for transmission loss. the tendency that the model had displayed for peak size attenuation downstream was considerably enhanced. Using the model, the geomorphic role of the flashfloods which affect the watershed in the summer months is briefly considered by applying the model to existing records of local summer storm rainfall events as a basis for event simulation. These simulations show that downstream attenuation of the flood wave on concave networks in steep semiarid terrain was likely to be a common occurrence, possibly resulting in down-net deposition and differences in geomorphic behaviour between upstream and downstream sites. the discussion is finally broadened to consider the relative importance of ‘common’ as compared to ‘freak’ watershed events in maintaining these differences.     

Influence of upstream inflow on wave celerity and time to equilibrium on an overland plane

Abstract

Based on the kinematic wave equations, formulae for the wave celerity along an overland plane subject to uniform rainfall excess and with a constant upstream inflow together with the corresponding average wave celerity and time to equilibrium for the entire plane are derived. The formulae are further developed in terms of both the Darcy-Weisbach resistance coefficient and the Manning resistance coefficient. By comparing the wave celerities, the average wave celerities and the time to equilibrium for planes with and without upstream inflow show that the upstream inflow causes the wave celerity and the average wave celerity to be faster and the times to equilibrium to be shorter. The effect of upstream inflow is greater with increasing inflow, but the marginal effect decreases with increasing inflow. The effect is greatest for laminar flow and least for turbulent flow. For the wave celerity, the effect is also greatest at the upstream end of the plane and least at the downstream end of the plane.     

A kinematic‐wave‐based distributed watershed model using FEM,GIS and remotely sensed data

For the appropriate management of water resources in a watershed, it is essential to calculate the time distribution of runoff for the given rainfall event. In this paper, a kinematic‐wave‐based distributed watershed model using finite element method (FEM), geographical information systems (GIS) and remote‐sensing‐based approach is presented for the runoff simulation of small watersheds. The kinematic wave equations are solved using FEM for overland and channel flow to generate runoff at the outlet of the watershed concerned. The interception loss is calculated by an empirical model based on leaf area index (LAI). The Green‐Ampt Mein Larson (GAML) model is used for the estimation of infiltration. Remotely sensed data has been used to extract land use (LU)/land cover (LC). GIS have been used to prepare finite element grid and input files such as Manning's roughness and slope. The developed overland flow model has been checked with an analytical solution for a hypothetical watershed. The model has been applied to a gauged watershed and an ungauged watershed. From the results, it is seen that the model is able to simulate the hydrographs reasonably well. A sensitivity analysis of the model is carried out with the calibrated infiltration parameters, overland flow Manning's roughness, channel flow Manning's roughness, time step and grid size. The present model is useful in predicting the hydrograph in small, ungauged watersheds. Copyright © 2007 John Wiley & Sons, Ltd.     

Conditions governing the use of approximations for the Saint-Vénant equations for shallow surface water flow

The solutions of the Saint-Vénant equations are compared with those of the kinematic, diffusion and gravity wave approximations, for a range of constant Froudé and kinematic wave numbers, with two different lower boundary conditions: (1) critical flow; and (2) zero depth gradient. For each lower boundary condition, zones are defined in the F₀,k-field in which either kinematic, diffusion or gravity wave solutions may be used to approximate the full Saint-Vénant solutions.     

基于多波束地形测量的上荆江典型河段沙波形态特征分析

沙波形态影响水流结构、泥沙输移及动床阻力。本研究采用多波束测深系统首次精细测量了上荆江典型河段的床面地形,采用改进后的沙波形态量化算法统计了各类沙波形态参数,分析了不同水流强度下沙波形态的变化特征。计算结果表明:(1)测量河段小型与大型沙波的平均波高分别为0.16～0.81和0.96～2.31 m,波长分别为13～27和16～41m;沙波尺度相较于水深较小,小型与大型沙波的波高分别不超过水深的0.045和0.150倍;(2)沙波背流面坡度基本不超过14°,小于泥沙水下休止角,其与陡度之间的关系可以用线性方程描述;(3)中洪水流量对沙波形态尺度的塑造作用强于枯水流量,且对浅水区大型沙波形态尺度的塑造作用强于深水区。本研究量化了天然河流的沙波形态,较好地反映了沙波形态特征,能为大型冲积河流沙波形态的量化及特征参数的统计分析提供参考。