Sediment transport in high‐speed flows over a fixed bed: 1. Particle dynamics

Particle dynamics are investigated experimentally in supercritical high‐speed open channel flow over a fixed planar bed of low relative roughness height simulating flows in high‐gradient non‐alluvial mountain streams and hydraulic structures. Non‐dimensional equations were developed for transport mode, particle velocity, hop length and hop height accounting for a wide range of literature data encompassing sub‐ and supercritical flow conditions as well as planar and alluvial bed configurations. Particles were dominantly transported in saltation and particle trajectories on planar beds were rather flat and long compared with alluvial bed data due to (1) increased lift forces by spinning motion, (2) strongly downward directed secondary currents, and (3) a planar flume bed where variation in particle reflection and damping effects were minor. The analysis of particle saltation trajectories revealed that the rising and falling limbs were almost symmetrical contradicting alluvial bed data. Furthermore, no or negligible effect of particle size and shape on particle dynamics were found. Implications of experimental findings for mechanistic saltation‐abrasion models are briefly discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

A wind tunnel investigation of particle segregation,ripple formation and armouring within sand beds of systematically varied texture

This paper reports on a wind tunnel investigation of particle segregation, ripple formation and surface armouring within sand beds of systematically varied particle size distribution, from coarsely skewed to bimodal. By design, the system was closed with no external inputs of mass from an external particle feed. Particles too coarse to travel in saltation for the given range in wind speed were dyed red in order to distinguish them in optical images from finer sand particles, which could be entrained into the unidirectional airflow. A 3D laser scanner measured the changing bed topography at regular time intervals during 18 experiments involving varied combinations of wind speed and bed texture. Image classification techniques were used to investigate the coincident self‐organization of the two populations of particles, as distinguished by their colour. As soon as saltation commenced, some of the red particles segregated into thin discontinuous patches. Particle trapping and sheltering on these rough patches was strongly favoured, causing them to grow preferentially. During the earliest stages of formation, bedform growth coincided with: (i) rapid coarsening of the surface texture; and (ii) the merging of proto‐ripple ‘crests’ to generate larger rhythmic bedforms of lower frequency. Consistent with previous work, ripple size was observed to increase under stronger winds when not exceeding the threshold for entrainment of the coarse‐mode or red particles from the crest. With declining rates of mass transport and particle segregation as the bed surface armoured, and the consequent deceleration of ripple propagation through to the end of each experiment, all surfaces eventually attained a steady‐state morphometry. At saturation, the largest ripples developed on beds having the lowest initial concentration of red particles. Copyright © 2016 John Wiley & Sons, Ltd.     

Transport of moderately sorted gravel at low bed shear stresses: The role of fine sediment infiltration

A reliable estimation of sediment transport in gravel‐bed streams is important for various practical engineering and biological studies (e.g., channel stability design, bed degradation/aggradation, restoration of spawning habitat). In the present work, we report original laboratory experiments investigating the transport of gravel particles at low bed shear stresses. The laboratory tests were conducted under unsteady flow conditions inducing low bed shear stresses, with detailed monitoring of the bed topography using a laser scanner. Effects of bed surface arrangements were documented by testing loose and packed bed configurations. Effects of fine sediments were examined by testing beds with sand, artificial fine sand or cohesive silt infiltrated in the gravel matrix. Analysis of the experimental data revealed that the transport of gravel particles depends upon the bed arrangement, the bed material properties (e.g., size and shape, consolidation index, permeability) and the concentration of fine sediments within the surface layer of moving grains. This concentration is directly related to the distribution of fine particles within the gravel matrix (i.e., bottom‐up infiltration or bridging) and their transport mode (i.e., bedload or suspended load). Compared to loose beds, the mobility of gravel is reduced for packed beds and for beds clogged from the bottom up with cohesive fine sediments; in both cases, the bed shear stress for gravel entrainment increases by about 12%. On the other hand, the mobility of gravel increases significantly (bed shear stress for particle motion decreasing up to 40%) for beds clogged at the surface by non‐cohesive sand particles. Copyright © 2017 John Wiley & Sons, Ltd.     

Virtual velocity of sand transport in water

Using the 160‐m‐long flume at Tsukuba University we undertook an experiment to provide a first estimate of the virtual velocity of sand in the size range 0.5–2.0 mm. For the flow velocity used in our experiment this sediment‐size range would conventionally be regarded as suspended sediment. The virtual velocity was found to be 37–41% of the flow velocity. Paradoxically, virtual velocity decreases as particle size decreases. Such a lower virtual velocity of finer sediment is not inconceivable. First, trapping of the sediment appears to be a function of bed roughness, and there is a probable relationship between bed roughness and trapping efficiency for particles of different sizes. Second, finer particles are more likely to find sheltered positions on a rough bed and thus experience lower mobility, relative to the more exposed coarser grains, as observed for bedload transport. Third, the virtual velocity of particles undergoing bedload transport has been found, in some instances, to be lower for finer clasts. We combine our data with previous studies of virtual velocity of bedload to develop, for the first time, a hypothesis for a holistic analysis of sediment movement in rivers. Copyright © 2017 John Wiley & Sons, Ltd.     

Wind tunnel tests of the dynamic processes that control wind erosion of a sand bed

Aeolian sand transport is a complicated process that is affected by many factors (e.g. wind velocity, sand particle size, surface microtopography). Under different experimental conditions, erosion processes will therefore produce different results. In this study, we conducted a series of wind tunnel experiments across a range of wind velocities capable of entraining sand particles (8.0, 10.0, 12.0, and 14.0 m s^-1) to study the dynamic changes of the shear velocity, aerodynamic roughness length, and sand transport. We found that the shear velocity and aerodynamic roughness length are not constant; rather, they change dynamically over time, and the rules that describe their changes depend on the free-stream air velocity. For wind tunnel experiments without feeding sand into the airflow, the sand bed elevation decreases with increasing erosion time, and this change significantly affected the values of shear velocity and aerodynamic roughness length. A Gaussian distribution function described the relationships between the sand transport rate (q_T) and the duration of wind erosion (T). It is therefore necessary for modelers to consider both deflation of the bed and the time scale used when calculating sand transport or erosion rates. © 2018 John Wiley & Sons, Ltd.     

Effect of saltating sediment on flow resistance and bed roughness in overland flow

The acceleration of saltating grains by overland flow causes momentum to be transferred from the flow to the grains, thereby increasing flow resistance and bed roughness. To assess the impact of saltating sediment on overland flow hydraulics, velocity profiles in transitional and turbulent flows on a fixed sand-covered bed were measured using hot-film anemometry. Five discharges were studied. At each discharge, three flows were measured: one free of sediment, one with a relatively low sediment load, and one with a relatively high sediment load. In these flows from 83 to 90 per cent of the sediment was travelling by saltation. As a result, in the sediment-laden flows the near-bed velocities were smaller and the velocity profiles steeper than those in the equivalent sediment-free flows. Sediment loads ranged up to 87·0 per cent of transport capacity and accounted for as much as 20·8 per cent of flow resistance (measured by the friction factor) and 89·7 per cent of bed roughness (measured by the ratio of the roughness length to median grain diameter). It is concluded that saltating sediment has a considerable impact on overland flow hydraulics, at least on fixed granular beds. Saltation is likely to have a relatively smaller effect on overland flow on natural hillslopes and agricultural fields where form and wave resistance dominate. Still, saltation is generally of greater significance in overland flow than in river flow, and for this reason its effect on overland flow hydraulics is deserving of further study. © 1998 John Wiley & Sons, Ltd.     

A prediction method for the size distribution of saltating particles above a mixed-size bed

In aeolian saltation, the sand bed is a mixture of sand particle with a wide range of particle sizes. Generally, the particle size distribution (PSD) of saltating particles is ignored by previous aeolian transport models, which will result in differences between predictions and observations. To better understand the saltation process, a prediction method of the PSD of saltating particles was proposed in this article. The probability of contact between incident sand and bed sand was introduced into the particle-bed collision process. An artificial PSD of the incident saltating particles was set as the initial condition. A stochastic particle-bed collision model considering contact probability was then used in each iteration step to calculate a new PSD of saltating particles. Finally, the PSD of saltating particles can be determined when aeolian saltation reaches a steady state (saltation is in a steady state when its primary characteristics, such as horizontal mass flux and the concentration of saltating particles, remain approximately constant over time and distance). Meanwhile, according to the experimental results, a calculation formula for the contact parameter n is given, which characterizes the shielding effect of particles on each other. That is, if soil PSD and friction velocity were given, the PSD of saltating particles can be determined. Our results do not depend on the initial conditions, and the predicted results are consistent with the experimental results. It indicated that our method can be used to determine the PSD of saltating particles. © 2020 John Wiley & Sons, Ltd.     

Representing the bed roughness of coarse‐grained streams in computational fluid dynamics

Computational fluid dynamics (CFD) applications are increasingly utilized for modelling complex flow patterns in natural streams and rivers. Although CFD has been successfully implemented to model many complex flow situations in natural stream settings, adequately characterizing the effects of gravel and cobble beds on flow hydraulics in CFD is a difficult challenge due to the scale of roughness lengths and the inadequacy of traditional roughness representations to characterize flow profiles in situations with large roughness elements. An alternative method of representing gravel and cobble beds is presented. Appropriate drag forces associated with different grain sizes are computed and included in the momentum equations to account for the influence of a hydraulically rough bed. Comparisons with field measurements reveal reasonable agreement between measured and modelled profiles of spatially averaged velocity and turbulent kinetic energy, and model fidelity to the non‐logarithmic behaviour of the velocity profiles. The novel method of representing coarse beds expands the utility of CFD for investigating physical processes in natural channels with large bed roughness. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of the saltation system under controlled environmental conditions

The transport of sand by the wind occurs predominantly by the process of saltation. Following the entrainment of sand by an above threshold wind, the saltation system is regulated by the mutual interaction of the atmospheric boundary‐layer, the sand cloud and the sand bed. Despite existing data on the spatial and temporal development of the sand transport system, very little is known about the development of the saltation system towards equilibrium. Results are presented from wind‐tunnel experiments that were designed to address the simultaneous spatial and temporal development of the saltation system, with and without artificial sand feed. The development of the saltation system was monitored over a streamwise length of 8 m during a period of 3600 s. Mass flux data were measured simultaneously at 1 m intervals by the downwind deployment of seven Aarhus sand traps. Wind velocity data were collected throughout the experiments. The downwind spatial development of the saltation system is manifested by an overshoot in mass flux and friction velocity prior to declining towards a quasi‐equilibrium. Mass flux overshoots at approximately 4 m downwind, in remarkable agreement with existing data of a comparable scale. Friction velocity overshoots at approximately 6 m downwind, a result not previously witnessed in saltation studies. The overshoot of mass flux prior to the overshoot in friction velocity is a spatial manifestation of the time lag between the entrainment of grains and the deceleration of the wind by the grains in transport. Temporally, the development of the saltation system is controlled by the availability of entrainable grains from the sand bed. Through time the saltation system develops from a transport‐limited to a supply‐limited system. The depletion of the sand bed through time limits the appropriateness of the assumption of ‘equilibrium’ for the universal prediction of mass flux. Copyright © 2002 John Wiley & Sons, Ltd.     

Particle velocity and concentration profiles in bedload experiments on a steep slope

Depth profiles of particle streamwise velocity, concentration and bedload sediment transport rate were measured in a turbulent and supercritical water flow. One‐size 6 mm diameter spherical glass beads were transported at equilibrium in a two‐dimensional 10% steep channel with a mobile bed. Flows were filmed from the side by a high‐speed camera. Particle tracking algorithms made it possible to determine the position, velocity and trajectory of a very large number of particles. Approximately half of the sediment transport rate was composed by rolling grains, and the other half by saltation. This revealed a complex structure, with several concentration and flux peaks due to rolling, and one peak due to saltation. With an increase of the sediment transport rate, the depth structure remained the same at the water/granular interface, with peak value increases but with no shift in elevations. The saltation region expanded towards higher elevations with an increase of the particle velocity commensurate to the water velocity. The proportion of the sediment transport rate in saltation did not vary significantly. The particle streamwise velocity profiles exhibited three segments: an exponential decay in the bed, a linear increase where rolling and saltation co‐existed, and above this, a logarithmic‐like shape due to saltating particles. These results are comparable to profiles measured and modelled in dry granular free surface flows and in more intense bedload such as sheet flows. Copyright © 2014 John Wiley & Sons, Ltd.     

Experimental study on the effects of artificial bed roughness on turbidity currents over abrupt bed slope change

A series of experimental observations are presented in the current study to discuss the effects of artificial bed roughness on the turbidity current flowing in a rectangular channel with an abrupt change in bed slope.For this purpose,two different types of elements,sinusoidal and trapezoidal,with various heights and arrangements are considered as artificial bed roughness.A Vectrino velocity meter was used to measure the velocity and sediment concentration profiles.The effects of inlet sediment concentration on front velocity,body velocity,unit discharge,sediment concentration,and suspended load transport rate also were investigated.Accurate equations were developed for estimation of the velocity of a turbidity current over smooth and rough beds.The unexpected experimental results showed that unlike the effect of roughness height,a change in the roughness arrangement has no significant influence on the velocity of a turbidity current.Also,the effect of bed roughness on the front velocity of a denser current is more significant.     

Three-dimensional saltating processes of multiple sediment particles

The purpose of this study was to investigate the interacting mechanism between the saltating particles near a channel bed. A three-dimensional real-time flow visualization technique was developed to measure the interparticle collision behaviors during the saltating process. Based on the experimental data, the distribution of the collision points was found to be symmetric. This confirms the assumption that the projections of the collision points onto the reasonable plane are uniformly distributed. A three-dimensional saltating model was also developed. This model produced satisfactory results. The model is able to simulate the continuous saltating trajectories of several particles. The simulated dimensionless saltating height, longitudinal and vertical saltation velocity components were found to increase as the dimensionless particle diameter and the dimensionless flow transport capacity parameter increase, while the simulated lateral saltation velocity component varies inversely with the dimensionless flow transport capacity parameter. A regression equation for the bed load transport rate was also obtained.     

Controls on the aerodynamic roughness length and the grain‐size dependence of aeolian sediment transport

Estimates of the wind shear stress exerted on Earth's surface using the fully rough form of the law‐of‐the‐wall are a function of the aerodynamic roughness length, z₀. Accurate prediction of aeolian sediment transport rates, therefore, often requires accurate estimates of z₀. The value of z₀ is determined by the surface roughness and the saltation intensity, both of which can be highly dynamic. Here we report field measurements of z₀ values derived from velocity profiles measured over an evolving topography (i.e. sand ripples). The topography was measured by terrestrial laser scanning and the saltation intensity was measured using a disdrometer. By measuring the topographic evolution and saltation intensity simultaneously and using available formulae to estimate the topographic contribution to z₀, we isolated the contribution of saltation intensity to z₀ and document that this component dominates over the topographic component for all but the lowest shear velocities. Our measurements indicate that the increase in z₀ during periods of saltation is approximately one to two orders of magnitude greater than the increase attributed to microtopography (i.e. evolving sand ripples). Our results also reveal differences in transport as a function of grain size. Each grain‐size fraction exhibited a different dependence on shear velocity, with the saltation intensity of fine particles (diameters ranging from 0.125 to 0.25 mm) saturating and eventually decreasing at high shear velocities, which we interpret to be the result of a limitation in the supply of fine particles from the bed at high shear velocities due to bed armoring. Our findings improve knowledge of the controls on the aerodynamic roughness length and the grain‐size dependence of aeolian sediment transport. The results should contribute to the development of improved sediment transport and dust emission models. © 2018 John Wiley & Sons, Ltd.     

Aeolian sediment flux decay: Non-linear behaviour on developing deflation lag surfaces

Wind tunnel simulations of the effect of non-erodible roughness elements on sediment transport show that the flux ratio q/q_s, shear velocity U*, and roughness density λ are co-dependent variables. Initially, the sediment flux is enhanced by kinetic energy retention in relatively elastic collisions that occur at the roughness element surfaces, but at the same time, the rising surface coverage of the immobile elements reduces the probability of grain ejection. A zone of strong shearing stress develops within 0·03 to 0·04 m of the rough bed because of a relative straightening of velocity profiles which are normally convex with saltation drag. This positive influence on fluid entrainment is opposed by declining shear stress partitioned to the sand bed. Similarly, because the free stream velocity U_f is fixed while U_* increases, velocity at height z and particle momentum gain from the airstream decline, leading eventually to lower numbers of particles ejected on average at each impact. When the ratio of the element basal area to frontal area σ is approximately equal to 3·5, secondary flow effects appear to become significant, so that the dimensionless aerodynamic roughness parameter Z₀/h and shear stress on the exposed sand bed T_s decrease. It is at this point that grain supply to the airstream and saltation drag appear to be significantly reduced, thereby intensifying the reduction in U_*. The zone of strong fluid shear near the bed dissipates.     

Influence of bed clusters and size gradation on operational time distribution for non‐uniform bed‐load transport

The operational time distribution (OTD) defines the time for bed‐load sediment spent in motion, which is needed to characterize the random nature of sediment transport. This study explores the influence of bed clusters and size gradation on OTD for non‐uniform bed‐loads. First, both static and mobile bed armouring experiments were conducted in laboratorial flumes to monitor the transport of mixed sand/gravel sediments. Only in the mobile armouring experiment did apparent bed clusters develop, because of stable feeding and a longer transport period. Second, a generalized subordinated advection (GSA) model was applied to quantify the observed dynamics of tracer particles. Results show that for the static armour layer (without sediment feed), the best‐fit OTD assigns more weight to the large displacement of small particles, likely because of the size‐selective entrainment process. The capacity coefficient in the GSA model, which affects the width of the OTD, is space dependent only for small particles whose dynamics can be significantly affected by larger particles and whose distribution is more likely to be space dependent in a mixed sand and gravel system. However, the OTD for the mobile armour layer (with sediment recirculation) exhibited longer tails for larger particles. This is because the trailing edge of larger particles is more resistant to erosion, and their leading front may not be easily trapped by self‐organized bed clusters. The strong interaction between particle–bed may cause the capacity coefficient to be space‐dependent for bed‐load transport along mobile armour layers. Therefore, the combined laboratory experiments and stochastic model analysis show that the OTD may be affected more by particle–bed interactions (such as clusters) than by particle–particle interactions (e.g. hiding and exposing), and that the GSA model can quantify mixed‐size sand/gravel transport along river beds within either static or mobile armour layers. Copyright © 2016 John Wiley & Sons, Ltd.     

Bed roughness and grain sorting-an experimental study over fine to medium sand beds

The relation between grain-size distribution of the bed and in suspension was critically examined under a uniform flow velocity of 50 cm/s over two beds: one of mainly fine sands and the other of medium sands. Two sections – one 2.85 m downstream and the other 6.35 m downstream in the experimental channel-were selected for sampling to study the grain-sorting pattern in the vertical direction along the direction of transport. The shape and type of the grain-size distribution pattern were critically studied with height above the bed. The change in the distribution pattern has been attributed to the change of local bed roughness causing scouring against the protruded relatively coarse grains on the bed. Such trends are important to predict the nature of river bed topography. The sand of Bed-1 initially exhibits a log-skew-Laplace distribution at different heights of suspension. The distribution pattern, however, changes but this changing pattern is not consistent along the upstream side. For Bed-2, which initially exhibits a log-normal distribution, the same pattern persists from the height of suspension at 5 cm up to 20 cm. Such consistency in log-normality is also observed at the downstream points of measurement. It is generally expected that the mean grain-size would reduce with increases of suspension height but the results of the experiments, in some occasions, differ significantly from the gradual fining upward trend. This result has been attributed to local changes of bed roughness arising from the protruded relatively coarse grains causing eddies, scouring, and turbulent phenomena which moves coarse particles higher in suspension adding a coarse tail to the distribution increasing the mean grain-size.     

Particle travel distances and bed and sediment compositions associated with rain‐impacted flows

Laboratory experiments were performed with rain of uniform drop size (2·7 mm, 5·1 mm) impacting flows over non‐cohesive beds of uniform sized sand (0·11–0·9 mm) and coal (0·2–0·9 mm) particles with flow velocities (20 mm s^?1, 40 mm s^?1) that were insufficient for the flow to entrain the particles without the aid of raindrop impact. Measurement of particle travel distance under rain made up of 2·7 mm drops confirmed a theoretical relationship between settling velocity and the distance particles travel after being disturbed by drop impact. Although, in theory, a relationship between settling velocity and particle travel distance exists, settling velocity by itself was unable to account for the effect of changes in both particle size and density on sediment discharge from beds of uniform non‐cohesive material. Particle density was also a factor. Further study of how particle characteristics influence sediment discharge will aid modelling of the impact of the soil in process‐based models of erosion by rain‐impacted flow. Copyright © 2001 John Wiley & Sons, Ltd.     

Collisions of quartz grains with a sand bed: The influence of incident angle

The velocities with which grains were observed to emerge from a sand bed after an intersaltation collision at u_* = 40 cm s^?1 are reported for four bed attitudes, from horizontal bed to adverse bed slope 15°. The principal effect of bed angle is to alter the magnitude and direction of the ricochet velocity. However, emergent velocities of dislodged grains are consistent with reptation path lengths comparable to the length of the upwind face of ripples in the corresponding wind. Calculations of the loss of forward momentum at collision, using the data for the range of bed attitudes studied suggest that creep is most vigorous on the sloping upwind face of the ripple and diminishes at the crest. As a result, the crest would be expected to accumulate the coarse material which moves predominantly by creep. The saltations originating in ricochet from the sloping back of the ripple are more vigorous and more concentrated in plan than are those originating at the crest. However, the saltation path length is at least an order of magnitude greater than the ripple wavelength and the probability distribution of path lengths is quite dispersed. Consequently it is very unlikely that these spatial patterns of ricochet are preserved sufficiently distinctly in the saltation cloud and subsequent collision distribution to be the agent of ripple development. This study therefore supports a view of moving grain interaction with the bed in which saltation provides the power to mobilize grains but ripple growth is associated with reptation and particularly with a pattern of impact which develops with the bed relief. Creep is more active on upwind facing slopes than at the crest, which therefore is a zone of net creep grain deposition.     

Graded and uniform bed load sediment transport in a degrading channel

Twenty runs of experiments are carried out to investigate non-equilibrium transport of graded and uniform bed load sediment in a degrading channel. Well-sorted gravel and sand are employed to compose four kinds of sediment beds with different gravel/sand contents, i.e., uniform 100%gravel bed, uniform 100% sand bed, and two graded sediment beds respectively with 53% gravel and 47% sand as well as 22%gravel and 78%sand. For different sediment beds, the experiments are conducted under the same discharges, thereby allowing for the role of sediment composition in dictating the bed load transport rate to be identified. A new observed dataset is generated concerning the flow, sediment transport and evolution of bed elevation and composition, which can be exploited to underpin devel-opments of mathematical river models. The data shows that in a degrading channel, the sand greatly promotes the transport of gravel, whilst the gravel considerably hinders the transport of sand. The promoting and hindering effects are evaluated by means of impact factors defined based on sediment transport rates. The impact factors are shown to vary with flow discharge by orders of magnitude, being most pronounced at the lowest discharge. It is characterized that variations in sand or gravel inputs as a result of human activities and climate change may lead to severe morphological changes in degrading channels.     

Computational fluid dynamics modelling of boundary roughness in gravel‐bed rivers: an investigation of the effects of random variability in bed elevation

Results from a series of numerical simulations of two‐dimensional open‐channel flow, conducted using the computational fluid dynamics (CFD) code FLUENT, are compared with data quantifying the mean and turbulent characteristics of open‐channel flow over two contrasting gravel beds. Boundary roughness effects are represented using both the conventional wall function approach and a random elevation model that simulates the effects of supra‐grid‐scale roughness elements (e.g. particle clusters and small bedforms). Results obtained using the random elevation model are characterized by a peak in turbulent kinetic energy located well above the bed (typically at y/h = 0·1–0·3). This is consistent with the field data and in contrast to the results obtained using the wall function approach for which maximum turbulent kinetic energy levels occur at the bed. Use of the random elevation model to represent supra‐grid‐scale roughness also allows a reduction in the height of the near‐bed mesh cell and therefore offers some potential to overcome problems experienced by the wall function approach in flows characterized by high relative roughness. Despite these benefits, the results of simulations conducted using the random elevation model are sensitive to the horizontal and vertical mesh resolution. Increasing the horizontal mesh resolution results in an increase in the near‐bed velocity gradient and turbulent kinetic energy, effectively roughening the bed. Varying the vertical resolution of the mesh has little effect on simulated mean velocity profiles, but results in substantial changes to the shape of the turbulent kinetic energy profile. These findings have significant implications for the application of CFD within natural gravel‐bed channels, particularly with regard to issues of topographic data collection, roughness parameterization and the derivation of mesh‐independent solutions. Copyright © 2001 John Wiley & Sons, Ltd.