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.     

Distorted Froude‐scaled flume analysis of large woody debris

This paper presents the results of a movable‐boundary, distorted, Froude‐scaled hydraulic model based on Abiaca Creek, a sand‐bedded channel in northern Mississippi. The model was used to examine the geomorphic and hydraulic impact of simplified large woody debris (LWD) elements. The theory of physical scale models is discussed and the method used to construct the LWD test channel is developed. The channel model had bed and banks moulded from 0·8 mm sand, and flow conditions were just below the threshold of motion so that any sediment transport and channel adjustment were the result of the debris element. Dimensions and positions of LWD elements were determined using a debris jam classification model. Elements were attached to a dynamometer to measure element drag forces, and channel adjustment was determined through detailed topographic surveys. The fluid drag force on the elements decreased asymptotically over time as the channel boundary eroded around the elements due to locally increased boundary shear stress. Total time for geomorphic adjustment computed for the prototype channel at the Q₂ discharge (discharge occurring once every two years on average) was as short as 45 hours. The size, depth and position of scour holes, bank erosion and bars created by flow acceleration past the elements were found to be related to element length and position within the channel cross‐section. Morphologies created by each debris element in the model channel were comparable with similar jams observed in the prototype channel. Published in 2001 John Wiley & Sons, Ltd.     

Macroturbulent coherent structures in an ice‐covered river flow using a pulse‐coherent acoustic Doppler profiler

The aim of this work is to compare macroturbulent coherent structures (MCS) geometry and organization between ice covered and open channel flow conditions. Velocity profiles were obtained using a Pulse‐Coherent Acoustic Doppler Profiler in both open channel and ice‐covered conditions. The friction imposed by the ice cover results in parabolic shaped velocity profiles. Reynolds stresses in the streamwise (u) and vertical (v) components of the flow show positive values near the channel bed and negative values near the ice cover, with two distinctive boundary layers with specific turbulent signatures. Vertically aligned stripes of coherent flow motions were revealed from statistics applied to space‐time matrices of flow velocities. In open channel conditions, the macroturbulent structures extended over the entire depth of the flow whereas they were discontinued and nested close to the boundary walls in ice‐covered conditions. The size of MCS is consequently reduced in scale under an ice cover. The average streamwise length scale is reduced from 2.5 to 0.4Y (u) and from 1.5 to 0.4Y (v) where Y is the flow depth. In open channel conditions, the vertical extent of MCS covers the entire flow depth, whereas the vertical extent was in the range 0.58Y–1Y (u) and 0.81Y–1Y (v) in ice‐covered conditions. Under an ice cover, each boundary wall generates its own set of MCS that compete with each other in the outer region of the flow, enhancing mixing and promoting the dissipation of coherent structures. Copyright © 2012 John Wiley & Sons, Ltd.     

Variable self‐similar sinuosity properties within simulated river networks

River networks have been shown to obey power scaling laws and to follow self‐organization principles. Their self‐similar (fractal) properties open a path to relate small scale and large scale hydrological processes, such as erosion, deposition or geological movements. However, the existence of a self‐similar dimension has only been checked using either the whole channel network or, on the contrary, a single channel link. No study has explicitly addressed the possible spatial variation of the self‐similar properties between these two extreme geomorphologic objects. Here, a new method based on self‐similarity maps (SSM) is proposed to spatially explore the stream length self‐similar dimension D_l within a river network. The mapping principle consists in computing local self‐similar dimensions deduced from a fit of stream length estimations using increasing divider sizes. A local uncertainty related to the fit quality is also computed and localized on every stream. To assess the efficiency of the approach, contrasted river networks are simulated using optimal channel networks (OCN), where each network is characterized by an exponent γ conditioning its overall topology. By building SSM of these networks, it is shown that deviations from uniform self‐similarity across space occur. Depending on the type of network (γ parameter), these deviations are or are not related to Strahler's order structure. Finally, it is found numerically that the structural averaged stream length self‐similar dimension D_l is closely related to the more functional γ parameter. Results form a bridge between the studies on river sinuosity (single channel) and growth of channel networks (watershed). As for every method providing spatial information where they were lacking before, the SSM may soon help to accurately interpret natural networks and help to simulate more realistic channel networks. Copyright © 2011 John Wiley & Sons, Ltd.     

The effects of discharge and slope on hyporheic flow in step‐pool morphologies

This paper focuses on surface–subsurface water exchange in a steep coarse‐bedded stream with a step‐pool morphology. We use both flume experiments and numerical modelling to investigate the influence of stream discharge, channel slope and sediment hydraulic conductivity on hyporheic exchange. The model step‐pool reach, whose topography is scaled from a natural river, consists of three step‐pool units with 0.1‐m step heights, discharges ranging between base and over‐bankfull flows (scaled values of 0.3–4.5 l/s) and slopes of 4% and 8%. Results indicate that the deepest hyporheic flow occurs with the steeper slope and at moderate discharges and that downwelling fluxes at the base of steps are highest at the largest stream discharges. In contrast to findings in a pool‐riffle morphology, those in this study show that steep slopes cause deeper surface–subsurface exchanges than gentle slopes. Numerical simulation results show that the portion of the hyporheic zone influenced by surface water temperature increases with sediment hydraulic conductivity. These experiments and numerical simulations emphasize the importance of topography, sediment permeability and roughness elements along the channel surface in governing the locations and magnitude of downwelling fluxes and hyporheic exchange. Our results show that hyporheic zones in these steep streams are thicker than previously expected by extending the results from streams with pool‐riffle bed forms. Copyright © 2014 John Wiley & Sons, Ltd.     

At‐a‐station hydraulic geometry relations, 1: theoretical development

This paper, the first of two, hypothesizes that: (1) the temporal variation of stream power of a river channel at a given station with varying discharge is accomplished by the temporal variation in channel form (flow depth and channel width) and hydraulic variables, including energy slope, flow velocity and friction; (2) the change in stream power is distributed among the changes in flow depth, channel width, flow velocity, slope, and friction, depending on the boundary conditions that the channels has to satisfy. The second hypothesis is a result of the principle of maximum entropy and the theory of minimum energy dissipation or its simplified minimum stream power. These two hypotheses lead to families of at‐a‐station hydraulic geometry relations. The conditions under which these families of relations can occur in the field are discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Improvement of surface run‐off in the hydrological model ParFlow by a scale‐consistent river parameterization

We propose an improvement of the overland‐flow parameterization in a distributed hydrological model, which uses a constant horizontal grid resolution and employs the kinematic wave approximation for both hillslope and river channel flow. The standard parameterization lacks any channel flow characteristics for rivers, which results in reduced river flow velocities for streams narrower than the horizontal grid resolution. Moreover, the surface areas, through which these wider model rivers may exchange water with the subsurface, are larger than the real river channels potentially leading to unrealistic vertical flows. We propose an approximation of the subscale channel flow by scaling Manning's roughness in the kinematic wave formulation via a relationship between river width and grid cell size, following a simplified version of the Barré de Saint‐Venant equations (Manning–Strickler equations). The too large exchange areas between model rivers and the subsurface are compensated by a grid resolution‐dependent scaling of the infiltration/exfiltration rate across river beds. We test both scaling approaches in the integrated hydrological model ParFlow. An empirical relation is used for estimating the true river width from the mean annual discharge. Our simulations show that the scaling of the roughness coefficient and the hydraulic conductivity effectively corrects overland flow velocities calculated on the coarse grid leading to a better representation of flood waves in the river channels.     

Modelling the bulk flow of a bedrock‐constrained,multi‐channel reach of the Mekong River,Siphandone, southern Laos

The general nature of bulk flow within bedrock single‐channel reaches has been considered by several studies recently. However, the flow structure of a bedrock‐constrained, large river with a multiple channel network has not been investigated previously. The multiple channel network of the Siphandone wetlands in Laos, a section of the Mekong River, was modelled using a steady one‐dimensional hydraulic model. The river network is characterized by a spatially‐varying channel‐form leading to significant changes in the bulk flow properties between and along the channels. The challenge to model the bulk flow in such a remote region was the lack of ideal boundary conditions. The flow models considered both low flow, high inbank and overbank flows and were calibrated using SPOT satellite sensor imagery and limited field data concerning water levels. The application of the model highlighted flow characteristics of a large multi‐channel network and also further indicated the field data that would be required to properly characterize the flow field empirically. Important results included the observation that adjacent channels within the network had different water surface slopes for the same moments in time; thus calibration data for modelling similar systems needs to account for these significant local differences. Further, the in‐channel hydraulic roughness coefficient strongly varied from one cross‐section to the next (Manning's ‘n’ range: 0·01 to 0·10). These differences were amplified during low flow but persisted in muted form during high discharges. Copyright © 2012 John Wiley & Sons, Ltd.     

Time–space fractional governing equations of one‐dimensional unsteady open channel flow process: Numerical solution and exploration

Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time–space fractional governing equations of 1‐dimensional unsteady/non‐uniform open channel flow process. By numerical simulations, results of the proposed fractional governing equations of the open channel flow process were compared with those of the standard Saint‐Venant equations. Numerical simulations showed that flow discharge and water depth can exhibit heavier tails in downstream locations as space and time fractional derivative powers decrease from 1. The fractional governing equations under consideration are generalizations of the well‐known Saint‐Venant equations, which are written in the integer differentiation framework. The new governing equations in the fractional‐order differentiation framework have the capability of modelling nonlocal flow processes both in time and in space by taking the global correlations into consideration. Furthermore, the generalized flow process may possibly shed light on understanding the theory of the anomalous transport processes and observed heavy‐tailed distributions of particle displacements in transport processes.     

Transient analysis of wave propagation in non‐homogeneous elastic unbounded domains by using the scaled boundary finite‐element method

The scaled boundary finite‐element method has been developed for the dynamic analysis of unbounded domains. In this method only the boundary is discretized resulting in a reduction of the spatial dimension by one. Like the finite‐element method no fundamental solution is required. This paper extends the scaled boundary finite‐element method to simulate the transient response of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. To reduce the number of degrees of freedom and the computational cost, the technique of reduced set of base functions is applied. The scaled boundary finite‐element equation for an unbounded domain is reformulated in generalized coordinates. The resulting acceleration unit‐impulse response matrix is obtained and assembled with the equation of motion of standard finite elements. Numerical examples of non‐homogeneous isotropic and transversely isotropic unbounded domains demonstrate the accuracy of the scaled boundary finite‐element method. Copyright © 2006 John Wiley & Sons, Ltd.     

A conceptual–numerical model to simulate hydraulic head in aquifers that are hydraulically connected to surface water bodies

In this paper, we present a conceptual‐numerical model that can be deduced from a calibrated finite difference groundwater‐flow model, which provides a parsimonious approach to simulate and analyze hydraulic heads and surface water body–aquifer interaction for linear aquifers (linear response of head to stresses). The solution of linear groundwater‐flow problems using eigenvalue techniques can be formulated with a simple explicit state equation whose structure shows that the surface water body–aquifer interaction phenomenon can be approached as the drainage of a number of independent linear reservoirs. The hydraulic head field could be also approached by the summation of the head fields, estimated for those reservoirs, defined over the same domain set by the aquifer limits, where the hydraulic head field in each reservoir is proportional to a specific surface (an eigenfunction of an eigenproblem, or an eigenvector in discrete cases). All the parameters and initial conditions of each linear reservoir can be mathematically defined in a univocal way from the calibrated finite difference model, preserving its characteristics (geometry, boundary conditions, hydrodynamic parameters (heterogeneity), and spatial distribution of the stresses). We also demonstrated that, in practical cases, an accurate solution can be obtained with a reduced number of linear reservoirs. The reduced computational cost of these solutions can help to integrate the groundwater component within conjunctive use management models. Conceptual approximation also facilitates understanding of the physical phenomenon and analysis of the factors that influence it. A simple synthetic aquifer has been employed to show how the conceptual model can be built for different spatial discretizations, the parameters required, and their influence on the simulation of hydraulic head fields and stream–aquifer flow exchange variables. A real‐world case was also solved to test the accuracy of the proposed approaches, by comparing its solution with that obtained using finite‐difference MODFLOW code. Copyright © 2011 John Wiley & Sons, Ltd.     

The influence of riparian vegetation on near‐bank turbulence: a flume experiment

Measurements from a fixed‐bed, Froude‐scaled hydraulic model of a stream in northeastern Vermont demonstrate the importance of forested riparian vegetation effects on near‐bank turbulence during overbank flows. Sections of the prototype stream, a tributary to Sleepers River, have increased in channel width within the last 40 years in response to passive reforestation of its riparian zone. Previous research found that reaches of small streams with forested riparian zones are commonly wider than adjacent reaches with non‐forested, or grassy, vegetation; however, driving mechanisms for this morphologic difference are not fully explained. Flume experiments were performed with a 1:5 scale, simplified model of half a channel and its floodplain, mimicking the typical non‐forested channel size. Two types of riparian vegetation were placed on the constructed floodplain: non‐forested, with synthetic grass carpeting; and forested, where rigid, randomly distributed, wooden dowels were added. Three‐dimensional velocities were measured with an acoustic Doppler velocimeter at 41 locations within the channel and floodplain at near‐bed and 0·6‐depth elevations. Observations of velocity components and calculations of turbulent kinetic energy (TKE), Reynolds shear stress and boundary shear stress showed significant differences between forested and non‐forested runs. Generally, forested runs exhibited a narrow band of high turbulence between the floodplain and main channel, where TKE was roughly two times greater than TKE in non‐forested runs. Compared to non‐forested runs, the hydraulic characteristics of forested runs appear to create an environment with higher erosion potential. Given that sediment entrainment and transport can be amplified in flows with high turbulence intensity and given that mature forested stream reaches are wider than comparable non‐forested reaches, our results demonstrated a possible driving mechanism for channel widening during overbank flow events in stream reaches with recently reforested riparian zones. Copyright © 2007 John Wiley & Sons, Ltd.     

Computational study of self‐centering buckling‐restrained braced frame seismic performance

Recent research developed and experimentally validated a self‐centering buckling‐restrained brace (SC‐BRB) that employs a restoring mechanism created using concentric tubes held flush with pretensioned shape memory alloy rods, in conjunction with a buckling‐restrained brace (BRB) that dissipates seismic energy. The present computational study investigated how the SC‐BRB can be implemented in real buildings to improve seismic performance. First, a computational brace model was developed and calibrated against experimental data, including the definition of a new cyclic material model for superelastic NiTi shape memory alloy. A parametric study were then conducted to explore the design space for SC‐BRBs. Finally, a set of prototype buildings was designed and computationally subjected to a suite of ground motions. The effect of the lateral resistance of gravity framing on self‐centering was also examined. From the component study, the SC‐BRB was found to dissipate sufficient energy even with large self‐centering ratios (as large as 4) based on criteria found in the literature for limiting peak drifts. From the prototype building study, a SC‐BRB self‐centering ratio of 0.5 was capable of reliably limiting residual drifts to negligible values, which is consistent with a dynamic form of self‐centering discussed in the literature. Because large self‐centering ratios can create significant overstrength, the most efficient SC‐BRB frame designs had a self‐centering ratio in the range of 0.5–1.5. Ambient building resistance (e.g., gravity framing) was found to reduce peak drifts, but had a negligible effect on residual drifts. Copyright © 2014 John Wiley & Sons, Ltd.     

Rill flow resistance law under equilibrium bed‐load transport conditions

In this paper, a recently deduced flow resistance equation for open channel flow was tested under equilibrium bed‐load transport conditions in a rill. First, the flow resistance equation was deduced applying dimensional analysis and the incomplete self‐similarity condition for the flow velocity distribution. Then, the following steps were carried out for developing the analysis: (a) a relationship (Equation  13 ) between the Γ function of the velocity profile, the rill slope, and the Froude number was calibrated by the available measurements by Jiang et al.; (b) a relationship (Equation  17 ) between the Γ function, the rill slope, the Shields number, and the Froude number was calibrated by the same measurements; and (c) the Darcy–Weisbach friction factor values measured by Jiang et al. were compared with those calculated by the rill flow resistance equation with Γ estimated by Equations  13 and 17 . This last comparison demonstrated that the rill flow resistance equation, in which slope and Shields number, representative of sediment transport effects, are introduced, is characterized by the lowest values of the estimate errors.     

An analytical study on three‐dimensional versus two‐dimensional water table‐induced flow patterns in a Tóthian basin

Although it has been increasingly acknowledged that groundwater flow pattern is complicated in the three‐dimensional (3‐D) domain, two‐dimensional (2‐D) water table‐induced flow models are still widely used to delineate basin‐scale groundwater circulation. However, the validity of 2‐D cross‐sectional flow field induced by water table has been seldom examined. Here, we derive the analytical solution of 3‐D water table‐induced hydraulic head in a Tóthian basin and then examine the validity of 2‐D cross‐sectional models by comparing the flow fields of selected cross sections calculated by the 2‐D cross‐sectional model with those by the 3‐D model, which represents the “true” cases. For cross sections in the recharge or discharge area of the 3‐D basin, even if head difference is not significant, the 2‐D cross‐sectional models result in flow patterns absolutely different from the true ones. For the cross section following the principal direction of groundwater flow, although 2‐D cross‐sectional models would overestimate the penetrating depth of local flow systems and underestimate the recharge/discharge flux, the flow pattern from the cross‐sectional model is similar to the true one and could be close enough to the true one by adjusting the decay exponent and anisotropy ratio of permeability. Consequently, to determine whether a 2‐D cross‐sectional model is applicable, a comparison of hydraulic head difference between 2‐D and 3‐D solutions is not enough. Instead, the similarity of flow pattern should be considered to determine whether a cross‐sectional model is applicable. This study improves understanding of groundwater flow induced by more natural water table undulations in the 3‐D domain and the limitations of 2‐D models accounting for cross‐sectional water table undulation only.     

Variable power‐law scaling of hillslope Hortonian rainfall–runoff processes

Hydrological studies focused on Hortonian rainfall–run‐off scaling have found that the run‐off depth generally declines with the plot length in power‐law scaling. Both the power‐law proportional coefficient and the scaling exponent show great variability for specific conditions, but why and how they vary remain unclear. In the present study, the scaling of hillslope Hortonian rainfall–run‐off processes is investigated for different rainfall, soil infiltration, and hillslope surface characteristics using the physically based cell‐based rainfall‐infiltration‐run‐off model. The results show that both temporally intermittent and steady rainfalls can result in prominent power‐law scaling at the initial stage of run‐off generation. Then, the magnitude of the power‐law scaling decreases gradually due to the decreasing run‐on effect. The power‐law scaling is most sensitive to the rainfall and soil infiltration parameters. When the ratio of rainfall to infiltration exceeds a critical value, the magnitude of the power‐law scaling tends to decrease notably. For different intermittent rainfall patterns, the power‐law exponent varies in the range of ?1.0 to ?0.113, which shows an approximately logarithmic increasing trend for the proportional coefficient as a function of the run‐off coefficient. The scaling is also sensitive to the surface roughness, soil sealing, slope angle, and hillslope geometry because these factors control the run‐off routing and run‐on infiltration processes. These results provide insights into the variable scaling of the Hortonian rainfall–run‐off process, which are expected to benefit modelling of large‐scale hydrological and ecological processes.     

2.5D direct‐current resistivity forward modelling and inversion by finite‐element–infinite‐element coupled method

To reduce the numerical errors arising from the improper enforcement of the artificial boundary conditions on the distant surface that encloses the underground part of the subsurface, we present a finite‐element–infinite‐element coupled method to significantly reduce the computation time and memory cost in the 2.5D direct‐current resistivity inversion. We first present the boundary value problem of the secondary potential. Then, a new type of infinite element is analysed and applied to replace the conventionally used mixed boundary condition on the distant boundary. In the internal domain, a standard finite‐element method is used to derive the final system of linear equations. With a novel shape function for infinite elements at the subsurface boundary, the final system matrix is sparse, symmetric, and independent of source electrodes. Through lower upper decomposition, the multi‐pole potentials can be swiftly obtained by simple back‐substitutions. We embed the newly developed forward solution to the inversion procedure. To compute the sensitivity matrix, we adopt the efficient adjoint equation approach to further reduce the computation cost. Finally, several synthetic examples are tested to show the efficiency of inversion.     

Shaking table test on small‐scale retrofitted model of Sefid‐rud concrete buttress dam

Sefid‐rud concrete buttress dam with a height of 106 m was damaged during the devastating 1990 Manjil earthquake. The dam was repaired and strengthened using epoxy grouting of cracks and the installation of post‐tensioned anchors. In a previous study, nonlinear seismic response of the highest monolith with empty reservoir was investigated experimentally through model testing. A geometric‐scaled model of 1:30 was tested on a shaking table to study dynamic cracking of the model. As a result of the similarity between model and prototype cracking pattern, the model was retrofitted according to prototype retrofitting plan after the Manjil earthquake and re‐tested on shaking table to estimate the current safety of the prototype. Experimental test results showed that the post‐tensioning resulted in a significant decrease in dynamic responses in terms of crest displacement and measured strains of the retrofitted model in comparison with its corresponding responses at the first test. No cracking was observed in the retrofitted model when the base motion peak acceleration exceeded a value that was 22% higher than the one caused cracking in the first model. This can be interpreted as the efficiency of prototype post‐tensioning system in evaluating the seismic safety of Sefid‐rud dam. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of hydrodynamic simulation results for eco‐hydraulic and eco‐hydrological applications: a spatial semivariance approach

Output from a three‐dimensional numerical flow model (SSIIM) is used in conjunction with high‐resolution topographic and velocity data to assess such models for eco‐hydraulic applications in river channel design and habitat appraisal. A new methodology for the comparison between field measurement and model output is detailed. This involves a comparison between conventional goodness‐of‐fit approaches applied to a spatially structured (riffle and pool) sample of model and field data, and a ‘relaxation’ method based upon the spatial semivariance of model/field departures. Conventional assessment indicates that the model predicts point‐by‐point velocity characteristics on a 0·45 m mesh to within ±0·1 m s⁻¹ over 80% of the channel area at low flow, and 50% of the area at high in‐bank flow. When a relative criterion of model fit is used, however, the model appears to perform less well: 60–70% of channel area has predicted velocities that depart from observed velocities by more than 10%. Regression analysis of observed and predicted velocities gives more cause for optimism, but all of these conventional indicators of goodness of fit neglect important spatial characteristics of model performance. Spatial semivariance is a means of supplementing model appraisal in this respect. In particular, using the relaxation approach, results are greatly improved: at a high in‐bank flow, the model results match field measurements to within 0·1 m s⁻¹ for more than 95% of the total channel area, provided that model and field comparisons are allowed within a radius of approximately 1 m from the original point of measurement. It is suggested that this revised form of model assessment is of particular relevance to eco‐hydraulic applications, where some degree of spatial and temporal dynamism (or uncertainty) is a characteristic. The approach may also be generalized to other environmental science modelling applications where the spatial attributes of model fits are of interest. Copyright © 2005 John Wiley & Sons, Ltd.