De Vriend Research Articles

UNRAVELING MULTIMODAL NEARSHORE WIND-WAVE FIELDS ON THE DUTCH SHOREFACE

Changing (wind) climate might influence the magnitude, direction, and frequency of wave systems (Lobeto et al., 2021). However, in coastal engineering applications, generalized wave parameters are commonly used in climate change assessments with the risk of, for example, misrepresenting the nearshore transformation of wind-driven wave climates (Hegermiller et al., 2017). In consequence, these uncertainties in the nearshore (wind) climate will affect, amongst others, ship navigation, the implementation of marine renewable energy farms, the feasibility of coastal infrastructure and defences, or the efficiency of sandy coastal maintenance, and thus the decision-making of long-term, multidecadal coastal strategies (Rijksoverheid, 2013), especially when they are designed accounting for the Building with Nature concept (de Vriend et al., 2015). This study analyses the importance and application of considering multiple coexisting wave trains on the Dutch shoreface.

Secondary Flow and Flow Redistribution in Two Sharp Bends on the Middle Yangtze River

AbstractIt remains an open question whether the state of knowledge on bend hydrodynamics applies to sharp bends in the largest river systems on Earth. This paper reports a field investigation into the hydrodynamics in two consecutive sharp bends of the Middle Yangtze River (MYR), and demonstrates that the reduced‐order model of Blanckaert and de Vriend is able to resolve the secondary flow strength and continuous flow redistribution in rivers of this size to high accuracy. Consequently, the model has been validated over the entire range of scales. These two bends show similarities to those of smaller size in terms of bathymetry, flow redistribution, and secondary flow evolution. The transverse location of the core of maximum velocity along the bends lags behind the thalweg by approximately half a width. A center‐region cell of secondary flow develops over the outer bank near the apex, and gradually decays downstream until it completely vanishes at the bend exit. Both bends are mildly curved from the perspective of hydrodynamic modeling, with a weak interaction between the secondary flow and primary flow, although they are sharply curved from the perspective of geomorphology (width/radius > 0.5). A term‐by‐term analysis of the model indicates that the flow redistribution is primarily controlled by planimetric changes in curvature and by topographic steering, and is slightly influenced by the secondary flow. However, the secondary flow may indirectly affect the flow redistribution by conditioning the bed topography. Important knowledge gaps remain with respect to the coupling between the hydrodynamic and morphological processes.

Large conservation opportunities exist in >90% of tropic-subtropic coastal habitats adjacent to cities

Large conservation opportunities exist in >90% of tropic-subtropic coastal habitats adjacent to cities

Cixin LiuSupernova Era by Joel Martinsen

in the lowly Directorate-General of Communication , indeed, in its lowliest Department of Culture. While becoming head of this department had been a promotion in position and title for Fenia Xenopoulou, Culture is beneath her ambitions. When her department is tasked with coming up with a “big jubilee” sort of affair to celebrate the commission’s “birthday,” she thus quickly hands this down to subordinate Martin Susman. Such a jubilee seems direly needed because, in sharp contrast to the positive poll ratings of the EU Council, president, and Parliament , the EU Commission has gotten a bad rap. It is, after all, the EU’s powerful (and unelected ) “enforcer.” Susman is an EU idealist (and an Austrian intellectual ) and thus believes that any such “jubilee” should celebrate the commission’s foundational idea(l). (These are among many entertaining evocations of fellow Austrian Robert Musil’s classic The Man without Qualities.) For Susman, the foundational idea(l) of the EU is a lapidary “never again” to world war and especially the Holocaust. But how to celebrate? Round up all the survivors ? Not even Steven Spielberg has such a list! But the obsessive records kept by the Nazis in a nearby deportation center identify all the individuals on the last transport out. Also identified are the handful who escaped thanks to an attack on the train by Belgian resistance fighters. And lo and behold, escapee David de Vriend is right here in Brussels, just retired to an assisted-living home. Meanwhile, Brussels has its own international problem. There’s been an assassination, and, in the havoc wreaked by the pig, the assassin has made his escape. Police Inspector Brunfaut is on the case. But he’s suddenly ordered to drop it—on highest authority. NATO? The CIA? No, the order comes from a much higher organization : no one must know that its operative had shot the wrong man. Then there is Florian Susman, new president and hence head lobbyist of the European Pig Producers, now in Brussels to see how little brother Martin can help him in the commission. En route later to a meeting in Budapest, he’s slowed down by a mass of refugees crossing the border and then almost killed by a cab driver who is racing Books in Review Cixin Liu Supernova Era Trans. Joel Martinsen. New York. Tor Books. 2019. 352 pages. SUPERNOVA ERA is the fifth novel by Chinese sci-fi author Cixin Liu to come out in English translation since 2014, and this doesn’t even include the collection of his short fiction (The Wandering Earth) and his numerous stand-alone stories in US/UK magazines. While this might seem like a lot of Liu in a short amount of time, anglophone readers should be immensely grateful to translators Ken Liu and Joel Martinsen for giving us the opportunity to read so much imaginative and ambitious science fiction. Known for his detailed scientific explanations of phenomena both on the micro and macro level, Liu offers us a complex and fascinating thought experiment in Supernova Era. Written just after the Tiananmen Square massacre in 1989 but only published in China in 2004, this is a novel about how quickly human civilization can come to the verge of collapse in the face of an undetected supernova. Liu devotes seven pages near the beginning of the novel just to a description of exactly how this one particular star was born, how it matured, and then how it collapsed in on itself, sending a blast of intense radiation straight into our solar CIXIN LIU 88 WLT WINTER 2020 SUPERNOVA REMNANT CASSIOPEIA A / NASA/JPL-CALTECH/STSCI/CXC/SAO system, which bathed the Earth for a week. Very quickly, doctors and scientists realize that anyone over the age of thirteen will die within a year due to the radiation’s irreversible physical effects. The adults’ desperate race against time to train their children in their own jobs so that those children can keep the world running leads to a period of “Inertia,” in which the new Children’s World continues functioning seemingly as before. It doesn’t take long, though, for critical systems to shut down and mass panic to break...

SHORELINE RESPONSE TO FUTURE SEA LEVEL RISE

Florida, United States, has shoreline change measurements starting in the 1800s with spacing of about every 300 m. In addition, due to extensive shoreline development and tourism, processes causing shoreline change have been studied extensively. The 1160-km east and 275-km southwest shorelines advanced seaward on average from the 1800s even before widespread beach nourishment and despite sea level rise. Shoreline advance despite sea level rise has been noted along other coasts such as the Netherlands central coast (Stive and de Vriend, 1995). In contrast, the 335-km Florida west coast retreated landward on average almost 30 m from 1867 to 2015.

Parallel Morphodynamic Modelling for the Yangtze Estuary

ABSTRACT Chu A.; Wang Z.B.; de Vriend H.J., and Tai J.A., 2018. Parallel morphodynamic modelling for the Yangtze Estuary. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 641–645. Coconut Creek (Florida), ISSN 0749-0208. Model input reduction and morphological acceleration techniques to bridge the large time scale of morphological evolution and the small time scale of hydrodynamically driven sediment transport is investigated for the Yangtze Estuary morphodynamic model. The discharge series of the river is schematized into a set of multi-discharge levels. Each discharge level coupled with the spring-neap tide forms one morphodynamic model for the estuary with bed stratigraphy approach applied. These models are used to simulate the bathymetry change of the estuary in parallel with the results merged together base on the Mor-Merge (MM) method. The morphological f...

Modeling Flow Pattern and Evolution of Meandering Channels with a Nonlinear Model

Meander dynamics has been the focus of river engineering for decades; however, it remains a challenge for researchers to precisely replicate natural evolution processes of meandering channels with numerical models due to the high nonlinearity of the governing equations. The present study puts forward a nonlinear model to simulate the flow pattern and evolution of meandering channels. The proposed meander model adopts the nonlinear hydrodynamic submodel developed by Blanckaert and de Vriend, which accounts for the nonlinear interactions between secondary flow and main flow and therefore has no curvature restriction. With the computational flow field, the evolution process of the channel centerline is simulated using the Bank Erosion and Retreat Model (BERM) developed by Chen and Duan. Verification against two laboratory flume experiments indicates the proposed meander model yields satisfactory agreement with the measured data. For comparison, the same experimental cases are also simulated with the linear version of the hydrodynamic submodel. Calculated results show that the flow pattern and meander evolution process predicted by the nonlinear and the linear models are similar for mildly curved channels, whereas they exhibit different characteristics when channel sinuosity becomes relatively high. It is indicated that the nonlinear interactions between main flow and secondary flow prevent the growth of the secondary flow and induce a more uniform transverse velocity profile in high-sinuosity channels, which slows down the evolution process of meandering channels.

A parametrical study on secondary flow in sharp open-channel bends: experiments and theoretical modelling

The dependence of curvature-induced secondary flow on the curvature ratio H/R (H is the average flow depth and R is the centreline radius of curvature), the Froude number Fr, and the dimensionless roughness coefficient Cf was systematically investigated in a series of 25 experiments in a laboratory flume. The investigated H/R values were (±0.005) 0.050, 0.073, 0.112 and 0.151, the Froude numbers were (±0.03) 0.10, 0.23, 0.32, 0.40 and 0.50, and roughness heights of the sediment bed were 0.002 m and 0.067 m. With width-to-depth ratios of 4.9 to 17.1, these experiments are more representative of small and medium-size rivers than the largest rivers on earth. Commonly used parameterizations for secondary flow predict the normalized magnitude to grow linearly with H/R, and to decrease with increasing bed roughness. This behaviour was only observed in mildly curved flows. The normalized magnitude was found to grow less than linearly with H/R in moderately curved flows, and to reach a maximum value in strongly curved flow. This confirms observations by Blanckaert (2009), who called this phenomenon the saturation of the secondary flow. He attributed the less than linear growth to nonlinear interactions between the primary and the secondary flows. For high values of H/R, secondary flow was found to be stronger over the very rough gravel bed than over the smoother sand bed. This is explained by the nonlinear interactions between the primary and secondary flows that are less pronounced over the rougher bed. The experiments did not reveal any dependence of the secondary flow on Fr. Predictions of the magnitude of the secondary flow with the nonlinear model of Blanckaert and de Vriend (2003, 2010) agreed very well with the experimental data over the entire range of investigated experimental conditions. Nonlinear effects are always important when the parameter Cf−0.25H/R is larger than an indicative value of 0.1, and negligible when this parameter is smaller than 0.05. These values allow a quantitative and objective distinction between mildly, moderately and sharply curved open-channel flow.

ASSESSMENT OF THE CAPABILITY OF 3D STRATIFIED FLOW FINITE ELEMENT MODEL IN CHARACTERIZING MEANDER DYNAMICS

There have been attempts to simulate meander dynamics (Langbein and Leopold 1966, Oodgard 1989, Campoerale et. al 2007, da Silva and El-Tahawy 2008, Duan and Julien 2010, Blanckaert and de Vriend 2010, Esfahani and Keshavarzi 2011). Meandering geometry is complex phenomena (Chanson 2004, Wu 2008), this would include the dynamics of flow properties and of morphology. Simulating meander flow dynamics is mostly popular using either Finite Element Method (FEM) or Finite Volume Method (FVM) where are based on Eulerian description, and based on stationer grid-based methods (Wormleaton and Ewunetu 2006, Wu 2008, Duan and Julien 2010, Gomez-Gesteira et. al 2010). As such this model is lack of capability in simulating the dynamics of meander morphology; much effort is put through to overcome this issue with such as Smoothed Particle Hydrodynamics (SPH), Boundary Element Methods, Arbitrary Lagrangian Eulerian, etc. This paper has two objectives; to identify meander flow characteristics and sediment transport distribution patterns, and to simulate meander flow characteristics and sediment transport distribution patterns using FEM. This study has identified that the key of dynamics of flow characteristics are helical flow and coherent structures, and the key of dynamics of transport characteristics are erosion-deposition zone patterns. The finite element model using in this study, RMA has shown its capability to simulate the meander key characteristics above, for small deflection angles (30°) location of maximum erosion-deposition zones near the crossover of the sinuosity, for intermediate deflection angles (70°) location of maximum erosion-deposition zones between the crossover and apex of the sinuosity, and for large deflection angles (110°) location of maximum erosion-deposition zones near the apex of the sinuosity, these are agreed with experiments of Odgaard 1989, da Silva 2006, da Silva et. al 2006, and Esfahani and Keshavarzi (2012). These results can be used as a reference to develop a method to model meander morpho-dynamics.

ASSESSMENT OF THE CAPABILITY OF 3D STRATIFIED FLOW FINITE ELEMENT MODEL IN CHARACTERIZING MEANDER DYNAMICS

There have been attempts to simulate meander dynamics (Langbein and Leopold 1966, Oodgard 1989, Campoerale et. al 2007, da Silva and El-Tahawy 2008, Duan and Julien 2010, Blanckaert and de Vriend 2010, Esfahani and Keshavarzi 2011). Meandering geometry is complex phenomena (Chanson 2004, Wu 2008), this would include the dynamics of flow properties and of morphology. Simulating meander flow dynamics is mostly popular using either Finite Element Method (FEM) or Finite Volume Method (FVM) where are based on Eulerian description, and based on stationer grid-based methods (Wormleaton and Ewunetu 2006, Wu 2008, Duan and Julien 2010, Gomez-Gesteira et. al 2010). As such this model is lack of capability in simulating the dynamics of meander morphology; much effort is put through to overcome this issue with such as Smoothed Particle Hydrodynamics (SPH), Boundary Element Methods, Arbitrary Lagrangian Eulerian, etc. This paper has two objectives; to identify meander flow characteristics and sediment transport distribution patterns, and to simulate meander flow characteristics and sediment transport distribution patterns using FEM. This study has identified that the key of dynamics of flow characteristics are helical flow and coherent structures, and the key of dynamics of transport characteristics are erosion-deposition zone patterns. The finite element model using in this study, RMA has shown its capability to simulate the meander key characteristics above, for small deflection angles (30°) location of maximum erosion-deposition zones near the crossover of the sinuosity, for intermediate deflection angles (70°) location of maximum erosion-deposition zones between the crossover and apex of the sinuosity, and for large deflection angles (110°) location of maximum erosion-deposition zones near the apex of the sinuosity, these are agreed with experiments of Odgaard 1989, da Silva 2006, da Silva et. al 2006, and Esfahani and Keshavarzi (2012). These results can be used as a reference to develop a method to model meander morpho-dynamics.

Classification of Tidal Inlets Along the Central West Coast of India

Tidal inlets along the Maharashtra coast on the central west coast of India were classified according to three methods available in the literature. Two classification methods viz., (i) Hydrodynamic classification (Hayes, 1979) and (ii) Classification based on dimensionless parameters (Vu, 2013) used for the classification are compared with the morphological classification of de Vriend et al., (1999). The hydrodynamic classification of Tidal inlets along Maharashtra coast is carried out considering mean annual significant wave height. The classification is also extended considering significant wave heights obtained for South-West monsoon, North-East monsoon and Fair Weather seasons. It has been observed that 74% of the inlets are tide dominated as per morphological classification whereasconsidering annual mean wave heights in Vu (2013) method, 67% of the inlets are wave dominated.

Numerical Simulation of U-Shape Bend Flow Movement Based on CCHE Model

This paper simulates the characteristics of 2D U-shape bend flow using nonlinear k~ε closure turbulence model in the software CCHE2D. Furthermore, we get the pictures about transverse slope of water surface, the transverse velocity distribution and transverse section water level variation. With comparison and analysis, the numerical simulation results are agreement with the experimental data measured by De Vriend. Simulating the movement of 2D bend flow with the software CCHE, an accurate flow basic characteristic can be obtained.

Meander dynamics: A reduced‐order nonlinear model without curvature restrictions for flow and bed morphology

AbstractReduced‐order models remain essential tools for meander modeling, especially for processes at large length scales and long time scales, probabilistic simulations, rapid assessments, or when input data are scarce or uncertain. Present reduced‐order meander models consider their dependent variables either as small‐amplitude variations compared to a basic state (linearity) or as varying gradually in a spatial sense (gradual variation). In a prequel, Blanckaert and de Vriend () derived a nonlinear reduced‐order hydrodynamic model without curvature restrictions and showed that linearity or gradual curvature variations assumptions do not hold in strongly curved channels. Moreover, in strongly curved channels, a nonlinear feedback mechanism causes the secondary flow strength to be smaller than its linear mild‐curvature equivalent. In the limit of mild‐amplitude variations and mild curvature, their nonlinear meander flow model simplifies to a well‐known linear formulation. The present paper extends this nonlinear modeling to the bed morphology in strongly curved bends, making use of Exner's sediment conservation principle. Furthermore, the model quantifying the relative influence of the downslope gravitational force is refined by considering nonlinear effects. The coupled nonlinear flow and bed morphology model yields satisfactory results for the bed topography, whereas the corresponding linear model strongly overpredicts the magnitude of the transverse bed slope. Analysis of the forcing mechanisms indicate that this erroneous behavior is caused by an overestimation of the upslope drag force due to the secondary flow.

Processes governing the flow redistribution in sharp river bends

Insight is provided in hydrodynamic processes governing the velocity redistribution in sharp river bends based on simulations of three recent experiments by means of Blanckaert and de Vriend's (2003, 2010) reduced-order nonlinear model without curvature restrictions. This model successfully simulated the flow redistribution and the secondary flow in all three experiments. The results indicate that the flow redistribution is primarily governed by topographic steering, curvature variations and secondary flow, in a broad range of different configurations, including mildly to sharply curved bends, narrow to shallow bends, smooth to rough bends, bends with additional complexities such as horizontal recirculation zones or patches of riverbed vegetation. The relative importance of these three dominant processes is case dependent, and controlled by the parameters Cf−1H/B, R/B and streamwise curvature variations. The first parameter characterizes a river reach, whereas the second and third parameters are characteristics of individual bends. Major differences exist between the hydrodynamic processes in mildly and sharply curved bends. First, velocity redistribution induced by curvature variations is negligible in mildly curved bends, but the dominant process in sharp bends. This result is relevant, because most meander models are based on the assumption of weak-curvature variations. Second, nonlinear hydrodynamic interactions play a dominant role in sharp bends, where mild-curvature models overpredict the secondary flow and in some cases even falsely identify it as the dominant process governing the velocity redistribution, which leads to unsatisfactory flow predictions. The reduction in secondary flow strength provoked by the nonlinear hydrodynamic interactions is accompanied by a reduction in the transverse bed slope, which reduces the effect of topographic steering.

Discussion of “Modeling the Evolution of Incised Streams. II: Streambank Erosion” by Eddy J. Langendoen and Andrew Simon

The authors are to be congratulated on having successfully implemented a conceptual model for two bank failure mechanisms into their numerical model CONservational Channel Evolution and Pollutant Transport System CONCEPTS and on having validated it against long-term field data of streambank erosion. A comparison of measured and simulated cross sections at the study site indicates reasonable behavior of their numerical model of bank stability for streambank failure. As noted by the authors, CONCEPTS does not predict the increased shear stress along an outer riverbank caused by the helical flow pattern within a bend. Curved channel flow is associated with a helical motion—often termed “secondary flow”— generating a transverse velocity component in addition to the main streamwise velocity. Yalin and da Silva 2001 mention the prominence of the secondary flow that progressively decreases as the width-to-depth ratio B / h of a curve increases. The authors’ treatment of the missing helical motion within their field validation is, however, questionable. The increased shear stresses at the bend apex due to secondary flow are accounted for by a reduction of the critical shear stress. However, the range of critical shear stresses used, between 1 Pa up- and downstream of the bend and 8 Pa at the bend apex, is surprising. This is nearly a one-order-of-magnitude difference, while the magnitude of the secondary flow velocity is typically some 10% of the streamwise velocity magnitude Blanckaert and de Vriend 2004. The range of the selected critical shear stresses appears therefore to be extremely large. This is also reflected by the results presented. Cross sections with a small critical shear stress of 1 Pa underlie stronger and temporally increased streambank retreat than those considered using a higher critical shear stress of 4 Pa. The discusser therefore questions the sensitivity of the authors’ approach relative to the selected model to account for secondary flow. Were simulations also made with a larger or smaller ratio of critical shear stresses at cross sections along a typical bend? Finally, the discusser would like to ask the authors how the critical shear stresses can be selected in terms of bend hydraulics, topography and geotechnical data?

Comments on ‘Modelling the morphodynamic impact of offshore sandpit geometries’ by Roos et al. (2008)

This brief comment of Roos et al.'s [Roos, P.C., Hulscher, S.J.M.H. & de Vriend, H.J., 2008. Modelling the morphodynamic impact of offshore sandpit geometries. Coast. Eng. doi:10.1016/j.coastaleng.2008.02.019] paper is aimed at supporting the validity of idealized process-based models as a valuable tool for the design of sand pits. The discussion moves from a brief description of the capabilities of an already existing idealized process-based model [Blondeaux, P. and Vittori, G. 2005a Morphological development of shallow sand pits. Proceedings 29th International Conference Coastal Engineering 2004, World Scientific (Jane McKee Smith ed) ISBN 981-256-995-2 pp. 2581–2593; Blondeaux, P. and Vittori, G., 2005b. Flow and sediment transport induced by tide propagation: part 2: the wavy bottom case. J. Geophys. Res., Vol. 110, No. C8, C08003] which can predict the morphological time development of sand pits. Then, the differences between Blondeaux & Vittori's model and Roos et al.'s model are analysed to give to the potential user an idea of the range of applicability of the different models, also in relation to practical applications.

Three-dimensional structure of wave-induced momentum flux in irrotational waves in combined shoaling-refraction conditions

In the paper, the three-dimensional structure of the wave-induced momentum flux in irrotational waves propagating over a two-dimensional, irregular bathymetry is analyzed. The expansion method developed by de Vriend and Kitou [de Vriend, H.J., Kitou, N., 1990a. Incorporation of wave effects in a 3D hydrostatic mean current model. Delft Hydraulics Report H-1295. de Vriend, H.J., Kitou, N., 1990b. Incorporation of wave effects in a 3D hydrostatic mean current model. Proc. 22nd Int. Coast. Eng. Conf. ASCE, 1005–1018.] for unidirectional waves has been extended to derive expressions for velocity components in three-dimensional waves over sloping bottom. The vertical wave-induced momentum flux resulting from this solution has been shown to be vertically-varying (contrary to the 2D-V case) and to act as a counterbalance for the vertical variability of the other wave forcing terms in the momentum equations. Thus, the total wave forcing remains depth-invariant, but—contrary to the ‘traditional’ solution based on the radiation stress concept—it does not depend explicitly on the direction of wave propagation and is a simple function of gradients of wave energy and water depth only. One of the most important consequences of this fact is the lack of the longshore-current-generating force in the case of non-dissipative waves approaching a shore with a bottom profile uniform in the along-shore direction. To illustrate the meaning of the new solution, the wave forcing due to waves approaching a barred beach has been analysed in detail. Also, the present solution has been shown to give the same results as the one obtained by extending of the approach by Rivero and Arcilla [Rivero, F.J., Arcilla, A.S., 1995. On the vertical distribution of 〈 ũw˜〉. Coast. Eng. 25, 137–152.] to three dimensions.

Reply to comment by Z. B. Wang and H. J. De Vriend on “Depth integrated modeling of suspended sediment transport”

[1] Let us first thankWang and De Vriend [2004] for their attention and their careful scrutiny of our paper. Let us also clarify that the motivation of our work was not to show that Galappatti’s [1983] model (hereinafter referred to as G83) was incorrect, but rather to develop a depth-averaged formulation of suspended sediment transport suitable for long-term morphological predictions, able to take into account weak non equilibrium effects of the kind often encountered in tidal as well as fluvial environments. Unfortunately, the mathematical framework appropriate to perform such investigation is precisely that proposed in G83, which forced us to revisit critically the latter work. [2] Let us come to the points raised by the Wang and De Vriend, which do concern G83’s approach rather than our formulation, whose validity does not seem to be questioned. The issue underlined in the discussion is whether or not G83’s perturbation scheme is correct and the basis for the Wang and De Vriend’s defense are the works of Wang [1992] and Wang and Ribberink [1986] (hereinafter referred to as W92 and WR86, respectively). We then also need to revisit the latter works.

Depth Averaged Model of Open Channel Flows with Lag between Main Flows and Secondary Currents in a Generalized Curvilinear Coordinate System

It is important to consider the secondary currents induced by the stream line curvature to predict the river channel processes. In view of the refinements of the model proposed by Kalkwijk·De Vriend, the depth averaged 2-D flow equations including the development and attenuation of secondary currents along a stream line, which are considered by Ikeda et al. and Parker et al. analytically, are studied in the generalized curvilinear coordinate system. The depth averaged equation of the strength of secondary currents along a stream line is derived by using the Engelund model, and then is transformed into the curvilinear coordinates. It is shown that the experiments by Rozovskii can be predicted well by using the model equations

Erratum: “Main Flow Velocity in Short River Bends” by H. J. de Vriend and H. J. Geldof (July, 1983)

Erratum: “Main Flow Velocity in Short River Bends” by H. J. de Vriend and H. J. Geldof (July, 1983)