Van Rijn Formula Research Articles

The distribution of velocity and sediment concentration under currents in coastal seas

A power law velocity profile for engineering application was obtained based on the field data set of flow velocity and sediment concentration in coastal areas of Jinzhou, Nantong, and Zhoushan. Using theoretical analyses, van Rijn's formula for sediment concentration profile was modified. Tests of the new power-law velocity profile and the modified formula of suspended sediment concentration against a large data set were performed. The results show a good fit to the data and that these two formulae could demonstrate the basic principles of vertical distribution of flow velocity and suspended sediment concentration, respectively.

Field Measurements of Longshore Sediment Transport along Denu Beach, Volta Region, Ghana

Field measurements of longshore sediment transport have been performed in the surf zone along Denu Beach, located in the Volta Region of Ghana, West Africa. This study consisted of measuring sediment transport rates by the deployment of modified versions of Kraus streamer traps on a cross-shore distance, which was about half the average surf zone width. In addition to the measured transport rates, data on waves, longshore currents, and other morphological parameters were collected with simple and low-cost instrumentation. In total, about 22 datasets were obtained through several days of data collection spanning a period of 5 months. Mathematical formulae proposed by CERC (1984), Kamphuis (1991), and the new van Rijn (2014) were used to compute theoretical transports for the entire surf zone, which were validated by measured data. Applying the CERC equation with a K-value of 0.39, the theoretical transports were about one order of magnitude higher than the measured rates. The rates computed by the Kamphuis formula were slightly higher than those yielded by the van Rijn equation. They produced values that were, on average, about 3.5 and 2.9 times higher than the measured rates, respectively. This study then confirmed the capabilities of both the Kamphuis and van Rijn equations to provide closer estimates of longshore sediment transport on Denu Beach. It especially sheds light on the new van Rijn formula, which has not been used extensively in LST quantification across the world. We, therefore, concluded that both the Kamphuis and van Rijn equations could be particularly useful to local engineers in the prediction of LST prior to the design of coastal protection structures. Further long-term studies employing available cutting-edge technologies were also recommended to provide more information on longshore transport not only in Ghana but also in the entire Gulf of Guinea and for the establishment of a reliable sediment budget for the country and the entire region.

On the General Longshore Transport model for resilient beaches

In the present paper, a set of 319 field and laboratory data on longshore transport rate has been gathered, checked, and finally adopted in order to overcome a noticed limitation in the previous version of General Longshore Transport (GLT) procedure for the estimation of Longshore Transport (LT). In particular, the extensive number of additional data, characterized by a wide mobility range, has allowed to obtain a single equation able to determine the longshore transport rate at coastal slopes consisting of loose not-cohesive materials, from reshaping dynamically stable berm breakwaters to sandy beaches. The reliability of the GLT procedure has been assessed by an independent verification with 39 field and laboratory data, for both offshore and breaking wave characteristics as an input. In addition, the GLT procedure has been compared with the van Rijn (2014) formula.

The relative merits of sediment transport formulas applied to the Nakdong Estuary, Korea using a coupled numerical modeling system

Three sediment transport formulas by Lund-CIRP (2012), van Rijn (1998), and Soulsby (1997) were applied for this study with field data by Yoon (2008) and several sedimentary variables of Nakdong Estuary in a coupled numerical modeling system, CMS. CMS is adopted to compare the measured depth change data and the calculated depth change data. As a result, the Lund-CIRP formula with mean grain size (0.25 mm) and multi-bottom friction coefficients based on the mean grain size distribution had higher accuracy (1.6 %) than the other formulas. The Soulsby formula has higher accuracy for multi-grain size and 0.06 of bottom friction coefficients. The van Rijn formula has high accuracy at shallow water areas such as a river. In this study, it is judged that the accuracy of the van Rijn cases was low because the water depth of the Nakdong Estuary and the nearby sea area rapidly deepens.

Observations of near-bed orbital velocities and small-scale bedforms on the Dutch lower shoreface

The lower shoreface, with water depths between about 8 and 20 m, forms the transition between the inner shelf and upper shoreface. Knowledge of the lower shoreface is essential, as it is – in many cases – the sediment source for the upper shoreface and beach. This paper presents new data of near-bed orbital velocities and small-scale bedforms at various depths and locations on the Dutch lower shoreface. Near-bed orbital velocities were beyond 1 m/s during high-energetic wave conditions. They increase with wave height and decrease with water depth, and can be reasonably well described by linear wave theory. Ripple heights range between 0.01−0.03 m and ripple lengths between 0.08−0.20 m. Ripple dimensions are controlled by wave mobility, with lower and shorter ripples for higher waves, and not so much by the currents. The Van Rijn (2007) formula generally overpredicts the ripple heights, and the variation with tidal currents in time. The measurements clearly indicate significant sediment mobility at the lower shoreface under higher wave events. It is yet unclear what this means for the net sand transport. This will depend on the subtle timing of sediment suspension, wave-mean currents and near-bed orbital velocities. It requires detailed modeling to determine lower shoreface net transport rates, and to unravel the controlling sand transport mechanisms.

Sediment Transport and Water Flow Resistance in Alluvial River Channels: Modified Model of Transport of Non-Uniform Grain-Size Sediments

The paper presents recommendations for using the results obtained in sediment transport simulation and modeling of channel deformations in rivers. This work relates to the issues of empirical modeling of the water flow characteristics in natural riverbeds with a movable bottom (alluvial channels) which are extremely complex. The study shows that in the simulation of sediment transport and calculation of channel deformations in the rivers, it is expedient to use the calculation dependences of Chézy’s coefficient for assessing the roughness of the bottom sediment mixture, or the dependences of the form based on the field investigation data. Three models are most commonly used and based on the original formulas of Meyer-Peter and Müller (1948), Einstein (1950) and van Rijn (1984). This work deals with assessing the hydraulic resistance of the channel and improving the river sediment transport model in a simulation of riverbed transformation on the basis of previous research to verify it based on 296 field measurements on the Central-East European lowland rivers. The performed test calculations show that the modified van Rijn formula gives the best results from all the considered variants.

A model for predicting the lateral distribution of sediment transport in river bends

ABSTRACT Distribution of lateral velocity and suspended sediment concentration in river bends is highly non-uniform. Predicting such velocity and sediment distributions is prerequisite for many applications, such as river bank protection, sediment transport, deposition pattern, pollutant transport and flood control. Due to secondary flow development, flow structure in river bends has three-dimensional feature. In this case, one-dimensional mathematical models are generally not suitable, so two- or three-dimensional mathematical models have to be used. In this paper, field measurement of lateral distributions of flow velocity and suspended sediment concentration was conducted in three bends at the Karun River in Iran (namely, Jangieh, Khabineh and Malheh), and the measured velocity data were used for calibrating the Spooner and Shiono model. Based on the lateral velocity distributions by this mathematical model, sediment transport capacities were computed using five commonly used empirical formulas of sediment transport. The results showed that in the three river bends, the sediment transports formula by Yang agrees well with the measured lateral sediment concentration. The Van Rijn formula also predicts the results with suitable accuracy. The other three formulas by Ackers-White, Engelund-Hansen, and Karim-Kennedy significantly over-predict the sediment concentration compared with the measured data.

Transport characteristics of non-cohesive sediment with different hydrological durations and sediment transport formulas

River sediment transport is an important global environmental issue, for it leads to waterway sedimentation, ecological degradation, and river geomorphologic evolution, its quantification is of the highest importance for river and watershed management. However, it remains unclear what the applicability of different sediment transport formulas in sediment transport process, and what the characteristics of non-cohesive sediment transport processes with different hydrological durations. This study developed a two-dimensional (2D) surface water Flow Model (FM) coupled with a Sediment Transport (ST) model in the MIKE 21 modelling system. The characteristic difference on simulated non-cohesive sediment transport under different sediment concentrations, hydrological durations, and sediment transport formulas were quantified by calculating the bed load, suspended load, and bed level change. Results indicated that the bed load and suspended load using the Engelund and Fredsøe formula were significantly lower than the Van Rijn formula for individual rainfall simulation scenarios. With the increase in peak flow, both suspended load and bed load first increased and then decreased. In the long-term duration simulations, high sediment concentrations were accompanied by higher bed load and lower suspension load for the Engelund and Fredsøe formula, but by lower bed load and suspension load for the Van Rijn formula. The high sediment concentrations resulted in a higher erosion rate, and low sediment concentrations resulted in a higher deposition rate. The two formulas had a high degree of consistency in seasonality and the bed level changes in spring and summer were mainly sedimentary, while in autumn and winter they were mainly erosive. The Engelund and Fredsøe formula produced high riverbed erosion capacity at low discharge, but the Van Rijn formula had high erosion capacity at high discharge.

Prediction of the Suspended Transport Rate of River Reach

A new suspended transport rate equation using field measurements that apply to the Euphrates River’s reach in the upstream direction from the Ramadi barrage in Iraq was developed via multiple linear regression. The governing parameters were carefully chosen based on dimensional analysis and grouped into four parameters: shields, transport, mobility, and particles. The ability of the suspended transport rate equations of Karim-Kennedy and van Rijn were tested. A total of 25 series of field data for which the concentration of suspended load, average flow depth, sediment particle size, flow velocity, and the width of the water surface were applied. A comparison of the suspended transfer rates in the river reach and the calculated transport rates in the modified equation showed good coincidence. The evaluation results showed that the new equation performed better than the three equations used. The performance of the new equation was determined utilizing three statistical criteria (the coefficient of determination, discrepancy ratio and relative error). The computed suspended transport rates were found within a discrepancy ratio of 1.02 for the measured values and a relative error of 13.4%. The performance of the Van Rijn formula performed better, with a discrepancy ratio of 1.26 for the measured values and a relative error 39%, than the other two equations. The percentage of data that ranged between 0.5–2.0 of the discrepancy ratio varied between 72–100% for the three equations and the new equation.

Investigation of Flow, Erosion, and Sedimentation Pattern around Varied Groynes under Different Hydraulic and Geometric Conditions: A Numerical Study

Groynes are popular hydraulic structures often used to control the erosion of banks by altering flow and sediment transport. In this paper, the effects of altering groyne orientation and spatial setup (from large to small and vice versa) on flow patterns, bed erosion, and sedimentation are numerically investigated. Studied groynes were parallel to each other, non-submerged, and impermeable. Numerical simulations were conducted in FLOW-3D. A nested mesh configuration combined with Van-Rijn formula on sediment transport yielded more accurate results when comparing numerical results to experiments. Groynes arranged from large to small at an angle of 45° decreased the scour depth by up to 55%, and an arrangement from small to large at an angle of 135° reduced the scour depth by up to 72%. Additionally, it was observed that simulations with an orientation closer to 90 degrees needed more equilibrium time when compared to other simulations.

Modeling bedform development under turbulent flows using Large-Eddy-Simulation and Immersed-Boundary-Method

A numerical model was built to study the mechanism of sedimentary bedform development in hydraulically smooth turbulent flows. The model consisted of a module for flows, a module for sediment transport, and a module for bed surface evolution. The flow was unsteady, three-dimensional and modeled by a Large-Eddy-Simulation (LES) method coupled with an Immersed-Boundary-Method (IBM). Governing equations were discretized on a fixed Cartesian grid by Finite Difference Method. Sediment (bedload) transport was estimated by Van Rijn formula corresponding to bed shear stress distribution obtained from the flow solution. The bed surface evolution was adapted to the sediment flux and described by the Exner–Polya equation. Updated bed surface was then used as the boundary for solving the flow field in the next time step. Time-advancement was discretized by the Adams–Bashforth method. The model was first validated by two test cases of bed shear stress and turbulence statistics over fixed sinusoidal bed surfaces. It was then employed to study initiation and development of bedforms from an initially flat bed to fully developed forms. Formation of newly and successively formed bedforms, growing and downstream propagating process of existing bedforms were very close to experimental observations. Instantaneous bed shear stress and corresponding sediment flux around evolving bedforms, which were difficult to observe in experiments, were also well produced by this model.

Estimating bedload transport in a large sand–gravel bed river from direct sampling, dune tracking and empirical formulas

Three methods used to estimate river bedload discharges were compared in a large sand–gravel bed river. The unit and total bedload transport rates estimated by sediment sampling, dune tracking method and the equations of Van Rijn and Meyer-Peter and Müller were compared. The analysis was based on a large data set obtained from field surveys that combined sediment sampling, acoustic doppler profiler (ADP) measurements, and multibeam echosoundings of the middle reach of the River Loire (France) for contrasting flow conditions ranging from low flows to 2-year floods. For transport stages between 2 and 6 (i.e. between the annual mean discharge and the discharge for a 2-year flood), the tested equations predicted fairly well the unit and total bedload discharges and roughly estimated the temporal variability of the bedload during floods. For these transport conditions, the best agreement with the sediment sampling measurements was observed with the Van Rijn formula. For lower flow conditions, the tested formulas provide lower estimates than those sampled. To improve the accuracy of the equations for low flows, potential solutions include integrating near-bed turbulences and non uniformity of bed particles into the models. The ability of the formulas to predict the temporal variability of the bedload can be improved by using the bedform roughness to adjust the estimates. The bedload transport rates calculated with dune tracking method are lower than the fluxes sampled because a substantial fraction of the bedload does not participate to the dune migration. Finally, a first rating curve that was implemented for the River Loire estimates the annual bedload discharge for 2010 to be 480,000t.

REFINED HYDRODYNAMIC MODELLING OF THE GIRONDE ESTUARY, FRANCE

In the centre of the Gironde estuary, EDF exploits a power plant. A hydrodynamic model is required to prevent different risks. Indeed, accurate water levels and velocity field are needed for flood control, sediment transport and associated deposition and transport of algae or pollutant. In order to enhance the accuracy of the hydrodynamic results and to tend to a fully predictive model, two attempts have been envisaged. The first one focuses on the tide imposed at the border by a combination of different harmonic waves whose constant harmonics are extracted from a tide model. Harmonic constants from different tide models have been compared and the influence of the harmonics number have been studied. The second attempt focuses on the friction coefficient. The values of the friction coefficient have been calibrated and compared to values predicted by the equation of van Rijn 2007.
 With the calibrated values, the differences between the measured and computed water levels are less than 10 cm.
 With the predictive friction formula, limited accuracy losses have been observed, which demonstrates the feasibility of using van Rijn formula for a configuration such as the Gironde.

Three-dimensional (3D) modeling of non-uniform sediment transport in a channel bend with unsteady flow

A fully three-dimensional non-hydrostatic model was applied to compute the water flow and morphodynamic processes in a laboratory flume. The experiments provided data for bed deformation and grain sorting processes in a 180? channel bend. The data were used to investigate the performance of different sediment transport approaches. The original van Rijn formula for uniform sediments was compared with two advanced nonuniform sediment formulas that considered the interaction between individual grain sizes by defining hiding-exposure approaches. The results showed that the default configuration of the model was able to predict bed deformations and grain sorting processes with satisfying magnitude and trend. However, the default approach considerably overestimated the bar evolution. The computed results improved significantly when applying any of the introduced hidingexposure approaches, representing better the physical mechanisms of the hiding-exposure effect. This study shows that the present numerical model is able to predict the morphodynamic bed changes in a flume of nonuniform sediments with good agreement if appropriate sediment formulations are used.

Appraisal of suspended sediment concentration on reference level according to van Rijn's method

Appraisal of suspended sediment concentration on reference level according to van Rijn's method Appraisal of suspended sediment concentration on reference level according to van Rijn's method. The paper presents the results of sediment concentration measurements in the River Dłubnia at differentiated discharges of 0.44 m-3·s-1, 1.96 m-3·s-1 and 7.41 m-3·s-1. The elaborated concentration profiles based on hydrometric measurements permitted determination of Ca concentration on reference level "a" above the bed. The obtained reference concentrations were compared with the values obtained from calculations by use of van Rijn's formula. It was found that the calculated Ca concentration, applying the van Rijn's formula, is over five times higher than determined from concentration profiles. In order to obtain Ca concentration values, calculated by use of the empiric formula, approximate to real values coefficient correction must be determined. The elaborated preliminary estimation of the possibility of van Rijn's formula application for determination of this concentration on the reference level "a" above the bed showed that application of this method for appraisal of Ca concentration, by use van Rijn's formula, performed in the Dłubnia river of the catchment of the Upper Vistula river may be charged by errors. A faulty calculation of suspension sediment transport applying van Rijn's method may also point to it. Adoption of this method needs further verification and adaptation.

Sensitivity analysis of non-cohesive sediment transport formulae

Sand transport models are often based on semi-empirical equilibrium transport formulae that relate sediment fluxes to physical properties such as velocity, depth and characteristic sediment grain sizes. In engineering applications, errors in these physical properties affect the accuracy of the sediment fluxes. The present analysis quantifies error propagation from the input physical properties to the sediment fluxes, determines which ones control the final errors, and provides insight into the relative strengths, weaknesses and limitations of four total load formulae (Ackers and White, Engelund and Hansen, van Rijn, and Karim and Kennedy) and one bed load formulation (van Rijn). The various sources of uncertainty are first investigated individually, in order to pinpoint the key physical properties that control the errors. Since the strong non-linearity of most sand transport formulae precludes analytical approaches, a Monte Carlo method is validated and used in the analysis. Results show that the accuracy in total sediment transport evaluations is mainly determined by errors in the current velocity and in the sediment median grain size. For the bed load transport using the van Rijn formula, errors in the current velocity alone control the final accuracy. In a final set of tests, all physical properties are allowed to vary simultaneously in order to analyze the combined effect of errors. The combined effect of errors in all the physical properties is then compared to an estimate of the errors due to the intrinsic limitations of the formulae. Results show that errors in the physical properties can be dominant for typical uncertainties associated with these properties, particularly for small depths. A comparison between the various formulae reveals that the van Rijn formula is more sensitive to basic physical properties. Hence, it should only be used when physical properties are known with precision.

Dynamics of intertidal gravel dunes

This paper examines the dynamics of intertidal gravel dunes subjected to high‐energy unsteady and reversing tidal currents in the Severn Estuary, UK. The dunes were composed of shale particles with median grain size, D50, around 4 mm and had mean heights and wavelengths of 60 cm and 7 m, respectively. Acoustic instruments were deployed above a dune to measure the streamwise profile and flow turbulence. Radar was used to map a section of the dune field on two occasions. Measurements during the peak spring ebb tides showed that when the measured bed shear stress exceeded the threshold for entrainment of the coarse surface armor, sediments were eroded from the upper surface of the dune. The dune prograded in the direction of the ebb flow a distance O(1 m), and some eroded sediments were deposited on the stoss slope downstream. The dune was then reworked during the following flood tide to reestablish approximately the same pre‐ebb profile shape and location. Radar data showed also that the exposed dunes at low water occupied approximately the same location over a period of 4 days during peak spring tides. The relationship between the depth‐mean current and the net volume of dune sediments moved was found to conform to a power law. The predicted critical bed shear stress, τcrit, for D50 = 4 mm was approximately 1 N/m2. However, owing to a coarse armor layer on the surface of the dune, a measured τcrit value O(4 N/m2) was required to initiate dune erosion. Although the Van Rijn formula was found to overestimate both the height and wavelength of dunes at this site, the dunes appear to be equilibrium bed forms controlled by local hydraulics and sediment supply. Sediment transport predicted by bed load formula were within a factor of two of the observed mass transport of dune sediments for ebb tides.

Longshore sediment transport rate—measurement and estimation, central west coast of India

Measurements of the longshore sediment transport rate (LSTR) along the surf zone at a 4-km-long beach on the central west coast of India were made over a 4-month period. During the study, both the lateral and vertical distributions of the sediment transport rate were measured with traps deployed on a line spanning the surf zone. Sediment transport in the swash zone was not considered in the present study. The longshore current was measured at each trap location. The breaking wave parameters were calculated from a directional wave buoy at 16-m water depth. The measured values were compared with those calculated from three selected empirical formulas. The standard coefficient values in the empirical formulas were used without calibration to the data sets. The measured average gross transport was 726 m 3 /day and that calculated were 1108, 1017, and 781 m 3 /day based on CERC, Walton and Bruno, and Van Rijn formula. During the data collection 69% of the time, the transport was direct towards north, and in the remaining period, it was direct towards south. The correlation coefficient between the longshore sediment transport rates measured and those calculated by CERC, Walton and Bruno, and Van Rijn formula were 0.38, 0.71, and 0.74, respectively. The average RMS error between the measured and the calculated longshore sediment transport rate based on CERC, Walton and Bruno, and Van Rijn formula were 0.91, 0.57, and 0.47.

Cross-shore distribution of longshore sediment transport: comparison between predictive formulas and field measurements

The skill of six well-known formulas developed for calculating the longshore sediment transport rate was evaluated in the present study. Formulas proposed by Bijker [Bijker, E.W., 1967. Some considerations about scales for coastal models with movable bed. Delft Hydraulics Laboratory, Publication 50, Delft, The Netherlands; Journal of the Waterways, Harbors and Coastal Engineering Division, 97 (4) (1971) 687.], Engelund–Hansen [Engelund, F., Hansen, E., 1967. A Monograph On Sediment Transport in Alluvial Streams. Teknisk Forlag, Copenhagen, Denmark], Ackers–White [Journal of Hydraulics Division, 99 (1) (1973) 2041], Bailard–Inman [Journal of Geophysical Research, 86 (C3) (1981) 2035], Van Rijn [Journal of Hydraulic Division, 110 (10) (1984) 1431; 110(11) (1984) 1613; 110(12) (1984) 1733], and Watanabe [Watanabe, A., 1992. Total rate and distribution of longshore sand transport. Proceedings of the 23rd Coastal Engineering Conference, ASCE, 2528–2541] were investigated because they are commonly employed in engineering studies to calculate the time-averaged net sediment transport rate in the surf zone. The predictive capability of these six formulas was examined by comparison to detailed, high-quality data on hydrodynamics and sediment transport from Duck, NC, collected during the DUCK85, SUPERDUCK, and SANDYDUCK field data collection projects. Measured hydrodynamics were employed as much as possible to reduce uncertainties in the calculations, and all formulas were applied with standard coefficient values without calibration to the data sets. Overall, the Van Rijn formula was found to yield the most reliable predictions over the range of swell and storm conditions covered by the field data set. The Engelund–Hansen formula worked reasonably well, although with large scatter for the storm cases, whereas the Bailard–Inman formula systematically overestimated the swell cases and underestimated the storm cases. The formulas by Watanabe and Ackers–White produced satisfactory results for most cases, although the former overestimated the transport rates for swell cases and the latter yielded considerable scatter for storm cases. Finally, the Bijker formula systematically overestimated the transport rates for all cases. It should be pointed out that the coefficient values in most of the employed formulas were based primarily on data from the laboratory or from the river environment. Thus, re-calibration of the coefficient values by reference to field data from the surf zone is expected to improve their predictive capability, although the limited amount of high-quality field data available at present makes it difficult to obtain values that would be applicable to a wide range of wave and beach conditions.

Three-dimensional numerical modelling of bed changes in a sand trap

Water and sediment flow was modelled numerically in three dimensions in a tunnel-type sand trap. The numerical model solved the transient Navier-Stokes equations with the k-ε turbulence model. Simultaneously, the sediment flow was calculated by solving the transient convection-diffusion equation for sediment concentration. Multiple sediment sizes were used. Van Rijn's formula for sediment concentration was used as boundary condition for the bed. Sediment continuity for the cells close to the bed gave the bed changes with time. An adaptive grid was used, adjusting to the bed changes. The processes in the sand trap involved both sedimentation and erosion, as a large volume of sand was formed and moved through the geometry. A physical model test was carried out to verify the numerical model. Bed levels were measured at a longitudinal profile and a cross-section. The results from the physical and numerical model compared well.