The paper investigates the influence of relative bend curvature on secondary flow, energy losses, and turbulence in sharp open-channel bends. These processes are important in natural streams with respect to sediment transport, the bathymetry and planimetry, mixing and spreading of pollutants, heat, oxygen, nutrients and biological species, and the conveyance capacity. Laboratory experiments were carried out in a configuration with rectangular cross section, consisting of a 193° bend of constant radius of curvature, preceded and followed by straight reaches. This somewhat unnatural configuration allows investigating the adaptation of mean flow and turbulence to curvature changes in open-channel bends, without contamination by other effects such as a mobile bed topography. Experiments were carried out for three different values of the curvature ratio, defined as the ratio of centerline radius of curvature over flow depth, which is the principal curvature parameter for hydrodynamic processes. Commonly used so-called linear models predict secondary flow to increase linearly with the curvature ratio. The reported experiments show that the secondary flow hardly increases in the investigated very sharp bends when the curvature ratio is further increased. This phenomenon is called saturation. Similar saturation is observed for the energy losses and the turbulence. This paper focuses on the analysis and modeling of the saturation of energy losses and turbulence. Secondary flow is found to be the dominant contribution to the curvature-induced increase in turbulence production, which leads to increased energy losses. The curvature-induced turbulence is explained by the fact that the turbulence dissipation lags behind the turbulence production, in agreement with the concept of the turbulence energy cascade. A 1-D model is proposed for the curvature-induced energy losses and turbulence. It could extend 1-D or depth-averaged 2-D models that are commonly used in long-term (scale of a flood event to geological scales) or large-scale (scale of a river basin) investigations on flood propagation, hazard mapping, water quality modeling, and planimetric river evolution.
 The sustainable development and exploitation of water resources is a prerequisite for the welfare of future generations. The wish and need to give more freedom to the river while maintaining its principal economical and sociological functions, calls for ever more subtle measures that intelligently influence the fluvial system rather than forcing it. The development of a generic insight and reliable engineering and management tools for natural river environments is complicated by the wide variety of encountered planimetries and scales, which renders the definition of a representative configuration impossible. To circumvent this problem, a research program is carried out that focuses on river meanders, which are typical and widespread elements in natural streams characterized by a distinctive interaction between hydrodynamic, morphodynamic and ecological processes that may be representative for a wider range configurations in natural streams. This program follows a three-component methodology that combines laboratory experiments, field experiments and numerical modeling.
 The present paper is the part of a series that reports results of laboratory experiments in a sharply curved meander bend. Most foregoing research on open-channel bends was limited to weak and moderate curvatures. At present, models for river dynamics (water quality, hydrodynamics, morphodynamics and planimetry evolution) are unable to explain some relevant processes observed in natural meander bends, especially at high curvature.
 These models predict the secondary flow to grow proportionally to the relative curvature, parameterized by the ratio of flow depth to centerline radius of curvature, H/R. Their validity is limited to weak and moderate curvatures because they do not account for the feedback between streamwise flow and secondary flow, which is known to limit the growth of the secondary flow when the relative curvature increases. Blanckaert and Graf  and Blanckaert and de Vriend  have successfully explained and modeled this feedback in sharp bends. Their model could be used to extend the validity range of the aforementioned models for advective mixing, transverse bed slope, velocity redistribution, and meander migration.
 Existing models for meander dynamics overpredict the meander migration rate in the high curvature range and fail to simulate the reduced rates of bank erosion and meander migration at high curvature, as observed by Hickin , Hickin and Nanson , Nanson and Hickin , Hooke , Biedenharn et al. . Bagnold  suggested that the maximum migration rate corresponds to minimum flow resistance related to flow separation. Parker and Andrews  postulated as possible causes local obstructions in the floodplain, interactions between nonperiodic adjacent bends and higher-order flow interactions. Markham and Thorne  focused on flow separation at the outer bank and the formation of embayments; Sun et al.  attributed it to the interplay between the migrating river and the changing sedimentary environment created by the meandering river itself. Seminara  and Crosato  pointed to the importance of the spatial lag between the local curvature and the location of the maximum velocity excess. The limited growth of the secondary flow has not yet been investigated in detail, but is expected to contribute significantly to the reduced rates of bank erosion and meander migration at high curvature.
 Curvature-induced increase in turbulence leads to the increase in energy losses and results in enhanced mixing and spreading of pollutants, heat, oxygen, nutrients and biological species. Moreover, it increases the flow's capacity to erode and transport sediment as bed load or suspended load. Understanding of these processes is hampered by the almost complete lack of observations and detailed experimental data. Additional shear by the secondary flow and additional shear by horizontal velocity gradients, for example induced by zones of flow separation and recirculation, are known to generate additional turbulence in meander bends (K. Blanckaert, Topographic steering, flow recirculation, velocity redistribution and bed topography in sharp meander bends, submitted to Water Resources Research, 2009).
 These phenomena could possibly be accounted for and resolved by three-dimensional hydrodynamic numerical models with high spatial resolution and including a higher-order turbulence closure. But they are currently not accounted for in 1-D or 2-D depth-averaged morphodynamic models that are required, because of limitations in computational capacity, in long-term (scale of a flood event to geological scales) or large-scale (scale of a river basin) investigations on flood propagation, hazard mapping, water quality modeling, and planimetric river evolution. To date, such models are unable to describe curvature-induced turbulence and energy losses and they are limited to a simplified parameterization of the time-averaged flow based on a model for secondary flow which validity is limited to low and moderate curvatures.
 1. To report experimental data on the adaptation of secondary flow, turbulence, and energy losses to changes in curvature by means of laboratory experiments carried out at three relative curvatures corresponding to sharply curved flow. More particularly, the paper aims at illustrating that curvature-induced secondary flow, turbulence and energy losses tend to saturate at high relative curvature, which means that they hardly grow anymore when the relative curvature is increased, contrary to the growth proportional to the relative curvature that is predicted by “classical” models.
 2. To interpret the data and to analyze the hydrodynamic mechanisms underlying and explaining the reported observations.
 3. To propose 1-D models for the curvature-induced energy losses and turbulence that can be coupled to 1-D or 2-D depth-averaged morphodanymic models.
 4. To provide high-quality data on the mean flow and turbulence that may be useful for model validation.
 The experimental setup and conditions, the applied measuring techniques, estimations of the uncertainty in measured data and definitions of the secondary flow, and the relative curvature are reported and discussed in section 2 and subsequently one section is devoted to each of the objectives.
 The experiments were carried out in the sharply curved laboratory flume shown in Figure 1. A 9 m long straight inflow reach allows the boundary layer to develop fully before the flow enters in the 193° bend of constant radius of curvature, R = 1.7 m, which is followed by a 5 m long straight outflow reach. The width of the flume is constant at B = 1.3 m. Quasi-uniform sand with diameters in the range d = 1.6 to 2.2 mm is glued on the horizontal bed, whereas the vertical banks are made of PVC.
 This laboratory flume has, on purpose, a simplified geometry that is not representative for natural streams:
 1. The experiments were carried out over a horizontal bed, in order to isolate the influence of the relative curvature and avoid contamination by the interaction of the flow with a mobile bed topography. Moreover the horizontal bed topography strongly simplifies the cross-sectional patterns of secondary flow and turbulence variables, which can be represented by their centerline values. Therefore, the present paper will only report data measured on the centerline of the flume. Data measured in various cross sections through the bend that illustrate a more complete picture of the spatial evolution of the hydrodynamics and that demonstrate the representativeness of centerline data have been reported by Zeng et al. . More three-dimensional velocity data can be obtained from the author.
 2. The single-bend geometry is characterized by discontinuities in radius of curvature at the bend entry and exit, which are not representative for natural meander bends, but allow investigating in an isolated way the adaptation of the flow to changes in curvature as well as the recovery of the flow when the curvature vanishes.
 3. The geometric simplicity makes for easier comparison with models.
 The geometric curvature in an open-channel bend depends on the centerline radius of curvature, R, the width of the flow, B and the depth of the flow, H, which can be combined into two independent dimensionless parameters, such as H/R and B/R. In natural streams R, B and H are correlated according to Leopold et al.  and curvature can therefore be parameterized by one of the dimensionless parameters. Field studies typically make use of the dimensionless curvature ratio B/R. The parameters R, B and H are obviously not correlated in laboratory flumes. Since the parameter H/R is the principal curvature parameter in models for secondary flow, advective velocity distribution, transverse bed slope or mixing in open-channel bends (see section 1), it has been adopted in the present paper to parameterize relative curvature. All subsequent references to “curvature” or “relative curvature” mean H/R and not B/R. The reported experiments were designed so as to vary the overall averaged flow depth, and the relative curvature /R, while keeping the bulk flow velocity, = Q/B about constant.
 Data are represented and analyzed in a curvilinear reference system (s, n, z) with curvilinear s axis in streamwise direction along the centerline and taking its origin at the bend entry, transverse n axis to the left and vertically upward z axis (Figure 1). The transverse velocity component, vn, is decomposed as [Blanckaert and de Vriend, 2003, Figure 1]
The depth-averaged transverse velocity Un represents translatory cross flow, whereas (vn − Un) represents the transverse component of the curvature-induced secondary flow, which is often called spiral flow, helical flow or cross-stream circulation. This decomposition of the transverse velocity and definition of the secondary flow are typical for laboratory studies. In field studies, secondary flow is often differently defined as the flow component perpendicular to the local depth-averaged flow.
Table 1 summarizes the main hydraulic and geometric parameters of the experiments. Flow in all experiments is rough turbulent (Re* = u*ks/v > 70 where Nikuradse's equivalent sand roughness ks has been defined according to van Rijn  as three times the sand diameter, Re ≫ 4000), subcritical (Fr < 1), and very sharply curved (R/B = 1.31 and R/ = 8.5 to 15.6). The channel aspect ratio of B/ = 6.1 to 12.1 is rather high for a laboratory flume, but flow is still less shallow than in most natural rivers.
Table 1. Hydraulic and Geometric Conditions in the Series of Experimentsa
Q is the flow discharge, is the flume-averaged flow depth, = Q/B is the flume-averaged velocity, Es,0 is the average energy slope in straight inflow, Cf,0−1/2 = U0/ is a Chézy-type friction coefficient for the straight inflow based on the hydraulic radius Rh, s is the flume-averaged energy slope, f−1/2 = / is a Chézy-type friction coefficient based on flume-averaged flow characteristics, Re = /v is the Reynolds number, Fr = / is the Froude number.
 In order to investigate and quantify the influence of curvature, characteristics of the straight flow in the inflow reach (labeled with the subscript 0) will be adopted as reference values. Table 1 lists the energy slope in the straight inflow reach, Es,0, which is computed from the measured water surface slope, Ss,0, as Es,0 = Ss,0(1 − Fr02). The roughness of the cross section is parameterized by the dimensionless Chézy coefficient defined as Cf,0−1/2 = U0/, where U0 and Rh,0 are the average velocity and hydraulic radius in the straight inflow, respectively.
 Owing to the horizontal bed, the cross-sectional averaged flow depth, H(s), decreases slightly in streamwise direction, causing variations in the aspect ratio B/H(s), the cross-sectional averaged velocity U(s) = Q/BH(s) and the Froude number Fr(s). Table 2 summarized extreme values of these variables at the flume's inlet and outlet, respectively. These variations do not fundamentally modify the hydrodynamics and do not alter the results and interpretations concerning the investigated processes. In order to avoid contamination by this variable flow depth in the data interpretation, mean velocities will be normalized in the present paper with the cross-sectional averaged velocity, U(s) = Q/BH(s), whereas turbulence quantities will be normalized with the characteristic shear velocity for straight flow, defined as
It should be noted that different methods can be applied to estimate the shear velocity in straight uniform flow (energy slope, logarithmic velocity profile, shear stress profile, turbulent kinetic energy). Nezu and Nakagawa  describe these methods and estimate the uncertainty in the estimates as O(30%), which is in agreement with results from the presented experiments. Such an uncertainty is considerable, especially since most turbulence quantities are normalized with u*,02.
Table 2. Extreme Values of the Cross-Sectional Averaged Flow Depth, Channel Aspect Ratio, Flow Velocity, and Froude Number Occurring at the Flume's Inlet and Outleta
H(s), cross-sectional averaged flow depth; B/H(s), channel aspect ratio; U(s) = Q/(BH(s)), the flow velocity; Fr(s), the Froude number.
 Measurements of the water surface topography were made by moving a set of 8 acoustic limnimeters mounted on a carriage along the flume. The limnimeters were mounted in the transverse positions n = [−0.6, −0.5, −0.3, −0.1, 0.1, 0.3, 0.5, 0.6] m. In streamwise direction, the measuring grid was refined near the bend entry and exit where important water surface gradients exist owing to the discontinuity in curvature. Measurements were made in the cross sections in the bend at [0, 5, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 165, 170, 175, 180, 185, 190, 193]° and in the straight outflow at [0.3, 0.5, 0.8, 1, 1.3, 1.5, 1.8, 2, 2.5, 3, 3.5, 4] m downstream of the bend. In order to estimate accurately the small water surface and energy slopes in the straight inflow reach, a refined measuring grid was adopted with 24 transverse measuring positions in cross sections spaced by 0.1 m in the reach from 7.5 m upstream of the bend entrance onto the bend entrance.
 Nonintrusive measurements of velocity profiles were carried out with an Acoustic Doppler Velocity Profiler (ADVP) on the centerline at 2.5 m, 2.0 m, 1.5 m, 1.0 m, 0.5 m upstream of the bend, every 15° in the bend and every 0.5 m in the straight outflow reach (Figure 1). The working principle of the ADVP has been reported by Lemmin and Rolland , Hurther and Lemmin , Blanckaert and Graf , and Blanckaert and Lemmin . The ADVP consists of a central emitter, surrounded by four receivers, placed in a water filled box that touches the water surface by means of an acoustically transparent mylar film (Figure 1). It measures profiles of the quasi-instantaneous velocity vector, (t) = (vs, vn, vz)(t), from which the time-averaged velocity vector, = (s, n, z), the Reynolds stresses, −ρ (i, j = s, n, z) and higher-order turbulent correlations can be computed. Measurements were made with a sampling frequency of 31.25 Hz and a sampling period of 90 s. The series of experiments is designed to make optimal use of the ADVP's strong points, as explained by K. Blanckaert (submitted manuscript, 2009).
 Blanckaert (submitted manuscript, 2009) reports and illustrates the data treatment procedures applied to transform the raw data obtained by the ADVP into a format appropriate for data analysis and presentation. Moreover, he estimates the uncertainty in the experimental data as follows: streamwise velocity vs: 4%, cross-stream velocities (vn, vz): 10%, shear velocity u*: 30%, turbulent shear stresses: 15%, turbulent normal stresses and turbulent kinetic energy: 20% (accounting for noise corruption of high frequency fluctuations by means of the method reported by Blanckaert and Lemmin ), turbulence production rate: 50% and turbulence dissipation rate: 100%.
 The accuracy in the ADVP measurements is reduced near the flow boundaries. At the water surface, the ADVP housing (Figure 1) perturbs the flow in a region of about 2 cm (about 10 to 15% of the flow depth in our experiments). In a flow layer of about 2 cm near solid boundaries, the ADVP seems to underestimate turbulent characteristics, which is tentatively attributed to the high velocity gradients within the measuring volume and/or to parasitical echoes from the solid boundary [Hurther and Lemmin, 2001]. ADVP measurements seem to underestimate systematically the vertical velocity fluctuations.
3. Experimental Observations
3.1. Water Surface Topography
Figure 2 synthesizes the measured water surface topographies by means of the cross-sectional averaged water surface elevation, H (Figure 2a), and the normalized transverse water surface slope (also called superelevation), (Figure 2b). The measured cross-sectional averaged water surface elevation, H, allows estimating the cross-sectional averaged energy level, E (Figure 2a), the cross-sectional averaged streamwise energy gradient, Es and the dimensionless Chézy coefficient, Cf, as detailed in section 2.
Table 1 indicates that the overall averaged streamwise energy gradient s is considerably higher than the streamwise energy gradient in the straight inflow reach Es0: s/Es0 ≈ 1.44, 1.37 and 1.37 in the F11_90_00, F16_90_00 and F21_90_00 experiments, respectively, resulting in a change of the dimensionless Chézy coefficient Cf−1/2 of about 10%. Remarkably, these curvature-induced energy losses do not increase with the curvature ratio /R in the reported experiments but are about identical.
 The transverse tilting of the water surface has a similar behavior in the three experiments. It begins about 1m upstream of the bend entrance, increases about linearly in the first 30° of the bend and reaches its maximum values of ≈ 1.4 in the bend reach between 60° and 90°. Remarkably, the transverse tilting of the water surface slope does subsequently not remain constant in this long bended reach of constant curvature, but decreases and it is reduced at the bend exit to about half of its maximum value in the bend, indicating that the transverse tilting of the water surface overshoots its equilibrium value in the first part of the bend. The transverse tilting decays quickly downstream of the bend and is negligible about 1 m downstream of the bend exit. These observations deviate from the “classical” description of the transversal tilting as adapting very quickly to curvature changes and being constant in a bend of constant radius of curvature.
3.2. Mean Flow Characteristics Along the Centerline
Figure 3 shows the evolution of the streamwise velocity along the centerline in the three experiments. Velocity profiles have been normalized with the local depth-averaged velocity, Us, such that the depth-averaged value of all illustrated profiles is equal to 1.
 The evolution along the centerline of the streamwise velocity profiles is similar in all three experiments. The vs profiles are about logarithmic in the straight inflow reach. In the first part of the bend, vs slightly increases/decreases in the upper/lower part of the water column. Subsequently, the highest velocities plunge toward the bed, causing a decrease/increase of vs in the upper/lower part of the water column, which is expected to cause an increase in bed shear stress. This process seems to be more pronounced and to start earlier in the bend with increasing curvature ratio, /R. The vs profiles recover toward their logarithmic form in the straight outflow reach, whereby the recovery length seems to increase with /R. The depth-averaged velocity Us (Figure 3d) only slightly varies along the centerline, which is due to redistribution of vs in the cross section around the bend.
Figure 4 summarizes the evolution along the centerline of the depth-averaged transverse velocity Un, representing translatory cross flow, and the transverse component of the secondary flow, (vn − Un) (equation (1)). The vertical velocity vz (not shown) is negligible on the centerline, which indicates that the centerline is situated near the center of the secondary flow cell.
 The secondary flow has a similar evolution in all three experiments. It comes into existence at the bend entrance, increases in the first part of the bend, has maximum values in the bend reach between 60° and 135°, decreases in the second part of the bend and decays in the straight outflow (Figures 4a–4c). Figure 4d quantifies the evolution of the strength of the secondary flow by means of the parameter /U (Brackets 〈 〉 indicate depth-averaged values). Remarkably, the maximum secondary flow strength is comparable in all three experiments with maximum outward/inward transverse velocities near the water surface/bed as high as about 40% of the streamwise velocity. Remarkably, values at the bend exit are reduced to about half of these maximum values, which is mainly due to a pronounced decrease in the upper part of the water column. The cross section of maximum secondary flow is located at 135°, 90° and 75° in the bend for the F11_90_00, F16_90_00 and F21_90_00 experiments, respectively, which suggests that its location depends on the flow depth and is related to the aspect ratio B/H.
 Owing to mass conservation, the translatory cross flow Un (Figure 4e) is strongly related to the transverse redistribution of streamwise velocity Us around the bend. Blanckaert and de Vriend [2003, Figure 7] have illustrated the redistribution of streamwise velocity Us around the bend by means of the parameter (αs + 1), which is defined as
It parameterizes the deviation of the Us(n) distribution from a potential vortex distribution. Blanckaert and de Vriend  estimated αs from detailed velocity measurements in various cross sections around the flume for the here reported experiments. Just downstream of the bend entry, (αs + 1) ≈ 0, which implies that Us decays from the inner bank toward the outer bank. Such an inward velocity redistribution is consistent with the observed strong inward cross flow, Un/U(s) ≈ −0.2. Subsequently, the core of maximum Us values gradually shifts outward, as parameterized by a gradual increase of (αs + 1). Such an outward velocity redistribution is consistent with the observed cross flow in the bend of Un/U ≈ 0.05. Near the bend exit, (αs + 1) strongly increases to about 2, which corresponds to a free vortex distribution. Such a strong increase in (αs + 1) is consistent with the observed pronounced outward cross flow Un/U ≈ 0.2. Cross flow is negligible in the straight outflow reach. The pronounced cross flow occurring near the bend entry and exit are due to the discontinuities in curvature and are therefore not representative of natural river bends.
3.3. Turbulence Characteristics Along the Centerline
Figure 5 illustrates the evolution of the turbulent kinetic energy (tke) (Figures 5a–5c and 5e–5g) and the depth-averaged turbulent normal stresses (Figure 5d) along the centerline in the three experiments. Profiles in the straight inflow deviate considerably from the exponential profile proposed by Nezu and Nakagawa  (Figures 5e–5g), which can be attributed to (1) the slightly accelerating flow in the straight inflow reach, which implies lower turbulence activity than in straight uniform flow [Graf and Altinakar, 1998], (2) the reduced accuracy of the ADVP in turbulence measurements near the bed due to large velocity gradients in the measuring volume, (3) the uncertainty in the estimation of the shear velocity which is used as normalization factor, and (4) the systematic underestimation of the vertical fluctuations. In spite of the experimental uncertainty, the measured patterns clearly indicate some relevant curvature-induced processes.
 The tke strongly increases upon entering the bend. Just below middle depth, a core of high tke is observed with maximum values between the cross sections at 75° and 105° that are about four times higher than in the straight inflow. In the second part of the bend, this core of high tke weakens and is reduced to about half of its maximum value at the bend exit. In the straight outflow reach, the tke distribution recovers toward its straight uniform flow pattern. The recovery length seems to increase with flow depth as indicated by the isoline patterns in Figures 5a–5c. Although the streamwise and vertical turbulent normal stresses increase considerably in the bend, the increase in tke is mainly due to the transverse turbulent normal stress (Figure 5d), which becomes the dominant turbulent normal stress in the bend reach.
Table 3 summarizes the normalized depth-averaged turbulent kinetic energy, 〈k〉/u*,02, and turbulent normal stresses, 〈〉/u*,02 (i = s, n, z) in the straight reaches and the bend. The maximum/averaged values of the tke and transverse turbulent normal stress in the bend are about 2.9 ± 0.5/2.2 ± 0.2 times and 4.7 ± 0.7/3.3 ± 0.4 times higher than in the straight inflow reach, respectively.
Table 3. Summary of Normalized Depth-Averaged Turbulent Kinetic Energy and Turbulent Normal Stressesa
Straight (Inflow + Outflow)
Maximum in Bend
Maximum in Bend/Straight
〈k〉/u*,02, turbulent kinetic energy; 〈〉/u*,02, turbulent normal stresses, (i = s, n, z) (Figure 5), respectively. Averaged values in the straight reaches and the bend, maximum values in the bend, ratio of averaged values and maximum values in the bend to averaged values in the straight reaches, respectively.
Figure 6 illustrates the evolution around the bend of the normalized turbulent shear stress, /u*,02. This shear stress is zero in straight flow but of dominant order of magnitude in the bend. The patterns of this shear stress show a core of high positive values in the first half of the bend, followed by a less pronounced core of negative values in the second half. Local extrema in these cores seem to increase slightly with the curvature ratio /R. When considering depth-averaged values (Figure 6d), however, the core of negative values is not discernable anymore and the evolution is similar in all three experiments. This shear stress decays to zero in the straight outflow reach.
Figure 7 illustrates the evolution around the bend of the normalized turbulent shear stress, /u*,02. Profiles in the straight inflow deviate considerably from the theoretical linear profile in the straight inflow reach (Figures 6e–6g), which can be attributed to the same factors as for the tke. In spite of the uncertainty in the experimental data, the measured patterns clearly indicate some relevant curvature-induced processes.
 In the first part of the bend upstream of the cross section at 60°, only weak changes in the /u*,02 profiles are observed. In the second part of the bend, a core of positive /u*,02 values which are of dominant order of magnitude is observed in the upper part of the water column, whereas negative values seem to be concentrated in the near-bed region. The recovery toward straight flow profiles in the straight outflow reach is not yet accomplished at the flume's exit.
 The region with positive /u*,02 values corresponds rather well with the region where ∂vs/∂z < 0 (indicated in Figures 7a–7c based on Figure 3). The core of positive values seems to increase in size and magnitude with increasing curvature ratio /R.
Figure 8 illustrates the evolution around the bend of the normalized turbulent shear stress, /u*,02. This shear stress is zero in straight flow but of dominant order of magnitude in the bend. The behavior of this turbulent shear stress seems to be strongly related to the secondary flow (Figure 4): it increases in the first half of the bend, reaches a maximum value and decays rather strongly in the second half of the bend. Maximum values in the bend seem to increase with the curvature ratio /R. The decay of this shear stress in the straight outflow seems to be considerably faster than that of other turbulent quantities.
4. Data Interpretation and Analysis
 Curvature is found to have a dominant influence on the mean flow and turbulence characteristics in the reported experiments. It generates secondary flow, redistribution of streamwise velocity and an increase in turbulence.
 According to low-amplitude perturbation models, secondary flow and turbulence would be expected to increase linearly with the curvature ratio, /R; therefore they are called “linear models.” The most remarkable feature of the reported experiments is that secondary flow and turbulent normal stresses are of very similar magnitude, whereas turbulent shear stresses only slightly increase in the three experiments, despite the pronounced different /R values. The mechanism underlying these observations will be analyzed hereafter.
 On the basis of the analysis of advective momentum transport in open-channel bends [Blanckaert and Graf, 2004], Blanckaert and de Vriend  have developed a nonlinear model for flow redistribution in open-channel bends that explains and quantifies the interaction between the profiles of streamwise velocity vs and secondary flow (vn − Un), including the observed saturation of secondary flow at high curvature. Their model and findings are briefly summarized hereafter.
 Advective transport of streamwise momentum by the secondary flow modifies the vertical vs profile and thereby also the vertical gradient of the centrifugal force, ∂(vs2/R)/∂z, which is the major driving force for the secondary flow. According to Blanckaert and de Vriend's  model, the profiles of vs and (vn − Un) depend on the dimensionless Chézy coefficient Cf and on a bend parameter which they define as
The interaction between vs and (vn − Un) is negligible when β = 0 but intensifies with increasing values of β. The (αs + 1) contribution (equation (3)) in the definition of β is of particular importance since it accounts for the interaction between the horizontal distribution of the flow (Us) and the vertical structure of the flow (vs and (vn − Un)).
 The evolution of (αs + 1) along the centerline in the reported experiments has been illustrated by Blanckaert and de Vriend [2003, Figure 7] as described in section 3.2. Just downstream of the bend entry, (αs + 1) ≈ 0, which indicates a negligible interaction between vs and (vn − Un) implying that the vs profiles remain close to logarithmic and (vn − Un) tends to grow toward the values predicted by linear models. The parameter (αs + 1) progressively increases around the bend, leading to the observed deformation of the vs profile and decrease of secondary flow in the second part of the bend. The occurrence of the maximum secondary flow strength in the middle of the bend, well downstream of the cross section where (αs + 1) and β are minimal is due to inertia. Blanckaert and de Vriend  describe in detail the interaction between vs and (vn − Un) in the here reported experiments, which explains the saturation of the secondary flow.
 A spectral analysis of the turbulent fluctuations gives clues on the mechanisms underlying the curvature-induced increase in tke and turbulent normal stresses (Figure 5). Figures 9a and 9b show normalized power spectra Snn(f)/u*,02 and normalized cumulative power spectra Snn(f′)df′/u*,02 of the transverse velocity fluctuation in the F16_90_00 experiment, respectively, whereas Figure 9c shows dissipation spectra f2Szz(f)/u*,02 [Nezu and Nakagawa, 1993]; f indicates the frequency in Hz. Since dissipation is dominant in the quasi-isotropic high-frequency domain, the dissipation spectra have been estimated from the vertical velocity fluctuations which are characterized by a lower parasitical noise level than the streamwise and transverse fluctuations [Blanckaert and Lemmin, 2006]. In order to reduce scatter, the computed spectra have been averaged over the water depth, excluding the near bed (z/h < 0.25) and near surface zones (z/h > 0.75).
Figures 9a and 9b show the characteristic −5/3 slope in log-log presentation in the inertial subrange of the power spectra and the typical S shape of the cumulative power spectra [Blanckaert and Lemmin, 2006], although some contamination by parasitical noise occurs in the high frequency range, f > 7 Hz.
 The inertial subrange seems to be situated in the frequency range f > 2 Hz. The increase in transverse turbulent normal stresses mainly occurs in the low frequency range f < 3 Hz, which corresponds to the production range of turbulence. Differences between the spectra in the different cross sections are reduced in the high frequency range. Figure 9c indicates that dissipation also increases considerably in the bend with respect to straight flow. The maximum dissipation seems to occur in the frequency range 3 Hz < f < 7 Hz, whereas the contribution of the frequency range f < 3 Hz seems to be negligible. Figure 10 compares the evolution around the flume of the average values of the spectra of the transversal velocity fluctuations in the production range f < 3 Hz (Figure 9a) and the average values of the dissipation spectra in the range 3 Hz < f < 7 Hz (Figure 9c) in the F11_90_00, F16_90_00 and F21_90_00 experiments. In general, the dissipation variable seems to lag behind the production variable and its variations seem to be attenuated; especially the increase in the first part of the bend seems to be less pronounced.
 These observations are in agreement with the energy cascade concept of turbulence: tke is generated in the low frequency range through the interaction between the time-averaged flow and large turbulent structures. While being advected in streamwise direction by the flow, these large turbulent structures disintegrate in smaller structures accompanied by an increase in frequency and in dissipation rate. On the basis of these observations, the further analysis of the mechanisms underlying the increased tke and turbulent normal stresses (Figure 5) will consider the evolution around the flume of the turbulence production rate (Figure 11) and the turbulence dissipation rate (Figure 12). The turbulence production rate is defined by [Hinze, 1975, chap. 1–13]
The definition of the strain rates, ejk (j, k = s, n, z), according to Batchelor [1967, p. 600] and the applied estimation based on the measured data are given by
 The applied estimations are based on the hypotheses that (1) the centerline is close the center of the secondary flow cell which guarantees that enn ≈ 0 and ezz ≈ 0, (2) ∂vs/∂n = αs(vs/R) according to equation (3). The derivatives of the measured velocity patterns have been computed according to the procedure reported by Blanckaert (submitted manuscript, 2009).
 Agreement with theoretical profiles [Nezu and Nakagawa, 1993] in the straight inflow is satisfactorily (Figures 11e–11g). The accuracy is reduced in the lower part of the water column, z/h < 0.2. The systematic underestimation of the measured profiles may be attributed to the same factors as for tke (Figure 5) and u*,02 (Figure 7).
 The production rate of turbulence strongly increases upon entering the bend. At about middle depth, a core of high is observed with maximum values between the cross sections at 45° and 105° that are about four times higher than in the straight inflow. In the second part of the bend, this core of high weakens and is reduced to less than half of its maximum value at the bend exit. In the straight outflow reach, the distribution recovers rather quickly toward its straight uniform flow pattern. The maximum values of in the bend seem to increase with the curvature ratio /R.
 The turbulent dissipation rate ɛ is estimated from the inertial subrange in the energy spectrum according to the procedure given by Nezu and Nakagawa [1993, section 4.6]. Profiles in the straight inflow reach agree surprisingly well with the theoretical profiles [Nezu and Nakagawa, 1993], given the relatively high uncertainty in the estimates of ɛ.
 Turbulent dissipation seems to be higher in the bend than in the straight reaches, although this tendency is less clear in the straight inflow reach. Just downstream of the bend entry, a core of higher values is discernable in the lower half of the flow depth. When considering depth-averaged values, however, the increase of dissipation seems to be rather uniform over the entire bend. The dissipation rate seems to increase with the curvature ratio /R.
 As suggested by the spectral analysis (Figures 9 and 10), differences between the patterns of turbulence production and dissipation may explain the observed increase in tke and turbulent normal stresses in the bend (Figure 5). Table 4 summarizes normalized turbulence production and dissipation rates, κH/u*,03 (Figure 11) and ɛκH/u*,03 (Figure 12), respectively (κ = 0.4 is the von Karman constant), averaged over the inflow reach, bend reach, outflow reach and bend plus outflow reach. The experimental uncertainties in the turbulence production and dissipation rates are inherently high and have been estimated at 50% and 100%, respectively. Moreover, measurements of turbulence production and dissipation rates are systematically underestimated in the near bed region where these variables attain their highest values (Figures 11e–11g and Figures 12e–12g, respectively), leading to systematic underestimations of the averaged values reported in Table 4. These high experimental uncertainties and systematic errors make the following data interpretation unavoidably somewhat tentative:
Table 4. Summary of Normalized Turbulence Production and Dissipation Rates Averaged Over the Inflow Reach, Bend Reach, Outflow Reach, and Bend Plus Outflow Reacha
 1. Dissipation seems to be considerably overestimated in the straight inflow reach, where it should theoretically be equal to production. It is not clear if these deviations are uniquely due to the experimental uncertainty or if they have a physical basis.
 2. When excluding the straight inflow reach, however, the total production and dissipation differ by less than 10% and can be considered in equilibrium.
 3. Turbulence production seems to react faster to changes in curvature than turbulent dissipation: its growth is more pronounced in the first part of the bend, it reaches higher maximum values, and it recovers faster toward “straight flow” patterns in the straight outflow reach. The slower adaptation of the turbulent dissipation seems to attenuate spatial variations. This different behavior of the production and dissipation rate of tke is in agreement with the concept of the turbulent energy cascade as aforementioned in the spectral analysis.
 4. Averaged over the bend, production seems to be higher than dissipation, which is compensated by the higher dissipation in the straight outflow reach.
 5. The increase of tke and turbulent normal stresses can be attributed to these different spatial distributions of the turbulent production and dissipation. Turbulent production shows more spatial variation than dissipation and reaches considerably higher peak values. The regions where the peak values in production occur correspond rather well with the region of maximum tke and turbulent normal stresses.
 The relation between spatial differences in production and dissipation rate of turbulence at the one hand and the increase in tke at the other can be quantified by means of the transport equation for depth-averaged tke [Hinze, 1975] reduced to its three principal terms:
Figure 13a shows the difference between 〈〉 and 〈ɛ〉 in all three experiments. It is positive in the first part of the bend, and negative toward the bend exit as well as in the straight outflow reach. It deviates from its theoretical zero value in the straight inflow. Figure 13b compares the measured distribution of 〈k〉 to the one obtained by integrating equation (8) based on the measured patterns of 〈〉 and 〈ɛ〉. The order of magnitude and patterns of the curvature-induced tke are satisfactorily reproduced by equation (8), given the high uncertainty in the result due experimental errors and error propagation. The large spatial lag between measured and modeled tke in Figure 13b arises because the measured 〈〉 − 〈ɛ〉 < 0 in the beginning of the bend. Figure 14 investigates the mechanisms underlying the increase in turbulence production rate, by analyzing the dominant individual contributions according to equations (6) and (7).
 In straight uniform flow, sz (Figure 14a) is the only contribution to the turbulence production rate. Profiles of sz are slightly modified in the bend owing to the deformation of the vs profiles (Figure 3), but sz remains of dominant magnitude all along the flume. ss (Figure 14b) is negligible except at the bend entrance. But the core of high values at the bend entrance can be attributed to the discontinuity in curvature and is therefore not representative for natural river configuration. A core of nonnegligible sn (Figure 14c) values, generated by the core of high turbulent shear stress /u*,02, exists in the bend reach between 30° and 75°. nz (Figure 14d) is the dominant curvature-induced contribution to turbulence production. Its close relation to the secondary flow through the strain rate enz ≈ ∂vn/∂z and the turbulence shear stress /u*,02 will be further investigated in section 5.
 According to Hinze [1975, p. 325], nz occurs in the production terms for the transverse turbulent normal stress while ss, sn and sz occur in the production terms for the streamwise turbulent normal stress . This explains why becomes the dominant turbulent stress in the bend reach (Figure 5d).
4.3. Energy Losses and Friction Coefficient
Figure 2a has shown that energy losses, as quantified by the streamwise energy gradient, Es(s), increase by about 40% owing to curvature effects in all three experiments. On the basis of the hypothesis that the dimensionless Chézy coefficient Cf is constant in the cross section, Es can be approximated as
Where 〈〈 〉〉 represents cross-sectional averaged values and ∣b∣ the magnitude of the boundary shear stress vector. The observed increase of the energy gradient can be attributed to the following mechanism: (1) the additional transverse component of the boundary shear stress induced by the secondary flow, (2) the increased near-bed gradient of the velocity vector due to the deformed vs profiles (Figure 3), (3) the nonuniform distribution of the depth-averaged velocity, which implies 〈〈∣2∣〉〉 > U2, and (4) the increased production rate of turbulence, which is by definition an increased loss of mean flow kinetic energy.
Blanckaert and de Vriend's [2003, Figure 10] nonlinear model for curved flow quantifies the first two mechanisms. It defines a correction factor to the friction coefficient ψsecondary flow = Cf/Cf,0 = ∣b,bend∣/∣b,straight∣ as a function of the parameter βCf−0.15. On the basis of the experimental data, the average value of this factor ψ along the centerline in the bend is about
for the F11_90_00, F16_90_00 and F21_90_00 experiments, respectively. Since the secondary flow is maximum on the centerline, the bend averaged value of ψsecondary flow is expected to be lower than the average value along the centerline.
 The relevance of the third mechanism can be estimated on the basis of a linearization of the velocity distribution according to equation (3): Ulinear(s, n) = U(s)(1 + αs(s)n/R). Averaging over the cross section (the average is in fact taken over a bend sector in order to take into account that arc length is longer/shorter in the outer/inner part of the cross section than on the centerline), gives
This effect can only cause a relevant increase in energy gradient of maximum 50% in very sharp bends where a mobile bed topography causes high αs values. In the three reported experiments, however, the negative αs values in a large part of the bend cause a negligible bend-averaged decrease in energy gradient.
 These observations suggest that the increased production rate of tke in the bend of O(100%) (Table 4) is a mechanism of dominant order of magnitude in the experiments. This mechanism will be further analyzed in section 5.
5. Modeling Implications
 As aforementioned, the nonlinear model of Blanckaert and de Vriend  is able to explain the interaction between vertical profiles of the streamwise velocity vs and the secondary flow, (vn − Un). This section will focus on the modeling of the curvature-induced increase in tke and in energy losses.
 According to section 4, the spatial distribution of the production rate of turbulence (Figure 11) is a dominant mechanism with respect to the curvature-induced increase in energy losses and tke (Figure 5). The model proposed hereafter for curvature-induced increase in energy losses and tke will be based on a model for the curvature-induced increase in turbulence production rate .
 The production rate of turbulence in straight uniform flow straight is theoretically known as [Nezu and Nakagawa, 1993]
The contribution ss is probably of minor importance but has been included for completeness. Modeling of the production rate requires modeling of the turbulent stresses. Figure 15 shows the patterns of the turbulent shear stresses for the F16_90_00 experiment estimated from the experimental strain rates according to the parabolic eddy viscosity model:
 The Kronecker delta is indicated by the symbol δjk and νt is the eddy viscosity. The shear stress related to the secondary flow is particularly well reproduced by the eddy viscosity model. The eddy viscosity estimation of the shear stress reproduces satisfactorily the major features of the measured distribution. Contrary to the measured distribution (Figure 7b), it does not suffer from systematic underestimations of the vertical velocity fluctuations which are pronounced near the bed. The eddy viscosity estimation of is less accurate: the estimated core of positive values is too large and too weak. In general, however, the eddy viscosity estimations of the shear stresses are sufficiently accurate for modeling of the production rate of turbulence.
 On the basis of equations (13) and (14) and assuming that sz,bend ≈ sz,straight (Figure 14a) the production rate of turbulence in the bend can be modeled as
This can be further elaborated by approximating the strain rates (equation (7)) based on a linear approximation of the vertical profiles of vn = vn,surface(2z/h − 1), where vn,surface is the transverse component of the secondary flow at the water surface, and an adaptation length λ ∼ H/Cf1/2 [de Vriend, 1981; Johannesson and Parker, 1989a] of the secondary flow into
Figure 16 shows the production rate bend as well as the dominant individual contributions sn and nz in the F16_90_00 experiment according to equation (19), where vn,surf has been estimated by fitting a linear profile to the measured pattern of (vn − Un) (Figure 4) and αs has been based on the experimental data [Blanckaert and de Vriend, 2003, Figure 7]. The total production rate of turbulence bend and the dominant curvature-induced contribution nz estimated according to this simplified model (equation (19)) agree surprisingly well with the experimental data (compare Figures 16a–16c to Figures 11b, 14c, and 14d, respectively). The modeled sn contribution shows the correct tendency but considerably underestimated the experimental values. This contribution is not of dominant order of magnitude, however. The ss contribution (not shown) is negligible.
Since the theoretical profile in straight uniform flow (equation (12)) is not defined at the bed level, a lower integration bound just above the bed has been chosen in equation (20) at z0. When taking z0 at ks/30 ≈ d/30, the denominator in equation (20) varies from about 145 for z0/h = 0.003 to 215 for z0/h = 0.0003.
 This validated model for the curvature-induced increase in production rate allows the following:
 1. Gaining insight in the processes underlying the curvature-induced increase in , which according to equations (19), (20), and (21) is determined by the parameter Cf−1/2(vn,surface/U) that accounts for the dimensionless Chézy coefficient Cf and for the strength of the secondary flow parameterized by vn,surface/U. Further development of the model for the curvature-induced production rate of turbulence requires the description of vn,surface/U as a function of depth-averaged flow parameters. For weakly curved flows a linear model for secondary flow (see discussion in section 4.1) gives the following description: vn,surface/U = (h/R).function(Cf). For moderately to strongly curved flow, however, the use of a nonlinear model is required, such as the one proposed by Blanckaert and de Vriend 
In cases where the parameter αs (equation (3)) cannot be computed from experimental data, it can be estimated from the velocity redistribution model of Johannesson and Parker [1989b]. The elaboration of a 1-D model for the parameter αs and subsequently for the parameter vn,surface/U will be reported elsewhere. This model will allow quantifying the influence of the parameters Cf, h/R and αs.
 2. Further analyzing the relation between curvature-induced energy losses, velocity redistribution, secondary flow, turbulence production rate, and tke in the reported experiments. On the basis of equations (19) and (20), the bend-averaged increase in turbulence production rate has been estimated along the centerline as 〈bend〉/〈straight〉 ≈ 1.4, 1.3 and 1.5 in the F11_90_00, F16_90_00 and F21_90_00 experiments, respectively. These values are considerably lower than the experimental estimations of these ratios according to Table 4, which can entirely be attributed to the reduced accuracy in near-bed turbulence measurements that leads to an underestimation of the measured straight. Since the secondary flow is maximum on the centerline, the bend averaged value of bend/straight is expected to be lower than the average value along the centerline.
 The observations and modeling suggest that the increased energy losses in our experiments can be attributed to two curvature-induced mechanism of comparable importance: the increased boundary shear stress due to the secondary flow and the deformed velocity profiles on the one hand and the increased production rate of turbulence on the other.
 3. Modeling the curvature-induced increase in energy losses in 1-D or 2-D depth-averaged models. When accounting for the turbulence induced increase in energy losses by means of the correction factor
The correction factor ψαs is computed by depth-averaged 2-D models or modeled according to equation (11) in 1-D models.
 4. Modeling the curvature-induced increase in tke in 1-D or 2-D depth-averaged models based on the simplified transport equation for tke (equation 8)
Δkbend, Δbend, and Δɛbend represent curvature-induced increase in tke, turbulence production and turbulence dissipation, respectively. 〈Δbend〉 is modeled by means of equations (19) and (20), whereas 〈Δɛbend〉 is modeled in a first-order approximation as follows:
The modeled streamwise distributions of 〈bend〉 and 〈ɛbend〉 on the centerline in the F11_90_00, F16_90_00 and F21_90_00 experiments are shown in Figure 17a, whereas Figure 17b compares the streamwise distribution of normalized depth-averaged tke 〈k〉/u*,02 from measurements and computed according to model equations (25) and (26).
 Comparison of Figures 11d and 12d to Figure 17a shows that the modeled 〈bend〉 and 〈ɛbend〉 are considerably higher than the measured ones. Comparison of Figures 11b and 16a, however, indicates that these differences can be attributed to the important near-bed contribution to 〈bend〉 and 〈ɛbend〉 which are not resolved by the measurements. The relevance of the proposed model is further confirmed by the satisfactorily reproduction of 〈bend〉_〈ɛbend〉 (Figure 13a versus Figure 17a). The dissipation 〈ɛbend〉 lags behind the production 〈bend〉 and is attenuated. Figure 17b shows that the modeled curvature-increased tke agrees fairly well with the measured one, although model predictions lag somewhat behind measurements. In such a complex 3-D flow, this very simple 1-D model can obviously not be expected to give more than “first-order” estimates of the curvature-induced increase in tke.
 5. Since secondary flow drives the model for curvature-induced increase in tke, this implies that it enhances mixing and spreading of transported quantities (suspended sediment, heat, oxygen, nutrients, biological species) in two ways: directly by what is commonly called advective mixing [Fischer, 1969; Krishnappan and Lau, 1977] but also indirectly by increasing the turbulence level, which seems to be a mechanism of dominant order of magnitude in sharp bends. The model for curvature-induced tke increase can be extended in a straightforward proportional increase in mixing coefficients.
 The further elaboration of the models for the curvature-induced turbulence production rate, energy losses and turbulent kinetic energy, and especially the implementation of Blanckaert and de Vriend's  model for the description of vn,surface/U as a function of depth-averaged flow parameters will be reported elsewhere.
 This paper reported results from laboratory experiments on sharply curved open-channel flow, carried out in the framework of a broader research program that combines laboratory experiments, field experiments and numerical simulations. Although the laboratory experiments were carried out in a single bend configuration with horizontal bed that is not representative for natural rivers, they yielded some generally valid results.
 The paper investigated the adaptation of mean flow and turbulence characteristics to changes in curvature in three experiments with different curvature ratio, H/R. Velocity measurements were made with high spatial and temporal resolution in the vertical plane through the centerline of the flume, allowing to calculate the three time-averaged velocity components, the six Reynolds stresses and the production and dissipation rate of turbulence. These data allowed identifying and visualizing some interesting phenomena, analyzing the underlying hydrodynamic mechanisms and developing models.
6.1. Observed Phenomena
 1. Energy losses increase considerably owing to curvature and are about 40% higher in the investigated sharp bends than in straight flow.
 2. A pronounced secondary flow (vn − Un) develops in the bend, with maximum transverse velocities that are about 40% of the streamwise velocity.
 3. The vertical profiles of the streamwise velocity (vs) deform with an increase of velocities in the lower part of the water column.
 4. The tke increases strongly in the bend. Its maximum/average values in the reported experiments are about 2.9 ± 0.5/2.2 ± 0.2 times higher than in straight flow. The increase is most pronounced for the transverse fluctuations: the maximum/averaged transverse turbulent normal stresses in the bend are about 4.7 ± 0.7/3.3 ± 0.4 times higher than in straight flow. All turbulent shear stresses are of dominant order of magnitude in the bend. This curvature-induced increase in turbulence is relevant with respect to spreading and mixing of suspended matter, oxygen or nutrients, temperature, to sediment transport and scour.
 5. The secondary flow, energy losses, and turbulence do not increase proportionally to the curvature ratio H/R, as could be expected on the basis of a linear perturbation approach. The secondary flow and the energy losses are about identical in the three experiments, and the turbulence characteristics only seem to increase weakly. These observations indicate a saturation of curvature effects in very sharply curved open-channel flow.
 6. The recovery length toward straight flow hydrodynamic characteristics seems to increase with the flow depth H.
6.2. Underlying Flow Mechanisms
 1. Blanckaert and Graf  and Blanckaert and de Vriend  have analyzed and explained the interaction and feedback between streamwise and cross-stream velocities. Advective momentum transport by the secondary flow deforms the vs profiles, which reduces the driving mechanism of the secondary flow. This nonlinear feedback between vs and (vn − Un) ultimately leads to the observed saturation of the secondary flow for very sharp curvatures.
 2. The analysis of spectral dynamics and of streamwise patterns of turbulence production and dissipation rates suggests the following mechanism underlying curvature-induced increased turbulence. Production and dissipation rates are considerably increased by curvature effects. They are about in equilibrium averaged over the bend, but have different spatial distributions. Production is considerably larger/smaller than dissipation in the first/second half of the bend, leading to an increase/decrease of turbulent kinetic energy. This lagging of the dissipation rate behind the production rate is in line with the energy cascade concept of turbulence. The production rate of turbulence is dominated by the contribution due to the cross-sectional turbulence shear stress which is closely related to the secondary flow.
 3. The increased energy losses in the bend can be attributed to four mechanisms: (1) the additional transverse component of the boundary shear stress induced by the secondary flow, (2) the increased near-bed gradient of the velocity vector due to the deformed vs profiles (Figure 3), (3) the increased production rate of turbulence, (4) the nonuniform distribution of the depth-averaged velocity. In the reported experiments, mechanisms 1 and 2 on the one hand and 3 on the other hand were found to be dominant and of comparable magnitude, whereas mechanism 4 was found to be negligible.
 It can be concluded that the curvature-induced secondary flow is the dominant mechanism in the reported experiments: it is directly or indirectly responsible for the deformation of the vs profiles, the curvature-induced increased turbulence production rate, turbulent stresses, tke and energy losses. The role of curvature-induced secondary flow is accentuated in the investigated laboratory configurations with rectangular cross section. In sharply curved real rivers, the mobile bed bathymetry is expected to have also an influence of dominant order of magnitude on the characteristics of the turbulence production rate, turbulent stresses, tke and energy losses.
6.3. Modeling Implications
 1. Blanckaert and de Vriend  have reported a model for the vs and (vn − Un) profiles that accounts for their nonlinear interaction and includes the saturation of the secondary flow in very sharply curved open-channel flow. This model can parameterize curvature effects in 1-D or depth-averaged 2-D hydrodynamic models.
 2. The eddy viscosity concept is shown to represent satisfactorily the turbulent shear stresses, and especially the dominant cross-stream turbulent shear stress which is closely related to the secondary flow.
 3. On the basis of the eddy viscosity concept and a parameterization of the profiles of streamwise velocity and secondary flow, a model for the curvature-induced increase in turbulence production rate has been developed and validated by means of the reported experimental data. This model quantifies the influence of different geometric and hydraulic parameters and identifies (Cf−1/2vn,surf/U) as the dominant scaling parameter.
 4. Correction factors to the friction factor that parameterize curvature-induced increase in energy losses have been derived. They account for curvature-induced turbulence production rate, secondary flow and the nonuniformity of the velocity distribution.
 5. On the basis of the model for curvature-induced increase in turbulence production rate, a 1-D model is developed for the curvature-induced increase in tke and mixing.
 Owing to limitations in computational capacity, long-term (scale of a flood event to geological scales) or large-scale (scale of a river basin) investigations make use of 1-D or 2-D depth-averaged morphodynamic models. As aforementioned, such models are currently unable to describe curvature-induced turbulence and energy losses, and they are limited to a simplified parameterization of the time-averaged flow based on a model for secondary flow which validity is limited to low and moderate curvatures. The here reported results allow enhancing existing 1-D or 2-D depth-averaged morphodynamic models by extending the validity range to sharp curvatures and by including a parameterization for curvature-induced increase in energy losses, tke and mixing. These extensions may improve the models' predictive capacity and reliability in multiple applications. A better understanding and modeling of the investigated processes may lead to more accurate estimations of the boundary shear stress, the sediment transport, the conveyance capacity and the water quality, a better assessment of natural hazards and the development of less expensive and more environmentally friendly bank protection measures as well as river restoration schemes that can guarantee the conservation of biodiversity. The extension of the validity range to sharp curvatures is important with respect to meander dynamics, and especially with respect to the modeling of cutoff events and the accompanying local restraightening. The frequency of occurrence of these events determines the rate of floodplain rejuvenation as well as the sedimentologic structure of the meander belt. The former is an important ecological parameter, whereas knowledge of the latter is required for the sustainable exploitation of drinking water of hydrocarburates.
 Beside enhancing insight in the hydrodynamics of sharply curved open-channel flow and improving modeling tools, the reported data are useful for the validation of numerical codes. More data from the reported experiments and from other experiments (Blanckaert, submitted manuscript, 2009) can be obtained from the author.
 This research was supported by the Swiss National Science Foundation under grants 2100-052257.97 and 2000-059392.99 and by the Netherlands Organization for Scientific Research (NWO) under grant DN66-149. The author wishes to acknowledge W. H. Graf, H. J. de Vriend, and W. S. J. Uijttewaal as well as R. Ferguson, A. Sukhodolov, M. Garcia, and an anonymous reviewer for constructive comments that improved the manuscript.