Nonlinear modeling of mean flow redistribution in curved open channels

Authors


Abstract

[1] Phenomena concerning flow, morphology, and water quality in rivers are often investigated by means of a depth-integrated flow model, coupled to a sediment transport model and a water quality model. In such depth−integrated models, the vertical structure of the flow is represented by a closure submodel, which mainly has to account for the secondary circulation, which (1) redistributes by advection the flow, the boundary shear stresses, the sediment transport, and dissolved and suspended matter, (2) causes the direction of the bed shear stress to deviate from the direction of the depth-averaged velocity and thereby influences the bed topography, and (3) gives rise to additional friction losses as compared with straight uniform flow. The commonly used linear closure submodels are shown to fail in reproducing essential features in moderately to strongly curved flow, because they neglect the feedback between the downstream velocity and the secondary circulation. A nonlinear closure submodel taking this feedback into account is proposed and shown to yield results that compare well with experimental data, even for very strongly curved flow. The feedback effects turn out to be controlled almost exclusively by a single parameter, which enables their parameterization in a relatively simple way. This control parameter also helps to objectively distinguish weak, moderate, and strongly curved flows. The proposed closure submodel has the potential of improving the performance of depth-integrated flow-sediment-water quality models without much extra computational effort.

1. Introduction

[2] In the past, numerous river canalization works have been undertaken that confined the river to a prescribed planform, in order to improve navigation or to use the alluvial plane. Recently, there is a tendency to renaturalize rivers, e.g., by giving them more freedom to shape their course in the alluvial plane. Such renaturalized rivers are recognized as a potentially rich biotope and as an important buffer in the flood defense system. Moreover, natural rivers are preferred from a landscape point of view. This new way of dealing with rivers requires more subtle measures than before, aiming at influencing the river in its natural behavior, rather than shaping it according to man's needs. This requires a more thorough understanding of flow, sediment transport, morphology, and mixing processes than before, as well as better predictive capabilities.

[3] Mathematical modeling is one way of predicting a river's behavior. The basis of such models is often a flow model, coupled to a sediment transport model and a sediment balance equation, and/or a dispersion model for dissolved or suspended matter. Although fully three-dimensional flow models are available now, depth-integrated flow models will remain an important complementary tool in research and engineering practice for long to go, especially in morphological studies. They yield a less detailed image of the flow pattern but are computationally at least an order of magnitude less expensive, and thereby allow for covering more complex situations, larger river domains (river reach or basin scale), and longer time periods (historical and geological timescales), or for performing stochastic or probabilistic simulations.

[4] The information related to the vertical structure of the flow field is to a large extent lost by depth-integrating the flow equations. The remaining information has to be introduced via a closure submodel. Johannesson and Parker [1989b], Finnie et al. [1999], Blanckaert [2001], and Blanckaert and Graf [2003] have shown the importance of this closure, which mainly has to account for the secondary circulation. This is a characteristic and dominant feature of curved flow: it redistributes velocity, boundary shear stress, and sediment transport, shapes the bed topography, and mixes dissolved and suspended matter.

[5] It has been a known fact for quite some time now that so-called linear closure submodels that neglect feedback effects between the secondary circulation and the main flow over-predict the effect of the secondary circulation, and have validity limited to weak curvatures [de Vriend, 1981a; Yeh and Kennedy, 1993; Blanckaert, 2001; Blanckaert, 2002, chapter II.3] (see also section 4). To solve the problem, a number of nonlinear models have been proposed in the past [Jin and Steffler, 1993; Yeh and Kennedy, 1993]. However, those models are difficult and computationally time consuming to be implemented. The present paper proposes a nonlinear closure submodel that takes due account of feedback effects. It can be reduced to a semiheuristic form, which requires hardly more computational effort than the linear models. Therefore, from a practice point of view, the current model will be more likely to be adapted by the hydraulic engineering and geologic modeling communities and have wider impact than those previous nonlinear models. Moreover, the nonlinear model provides new insight into the mechanisms and effects of the secondary circulation.

2. Mathematical Framework

[6] The main structure of the three-dimensional (3-D) flow field in a curved open channel is outlined in Figure 1. Furthermore, it defines the varying centerline radius of curvature, R(s), the local flow depth h = zSzb where zS and zb are the elevations of the water surface and the bed above a horizontal datum, and the bed shear stress vector equation image. The curvilinear horizontal s-axis lies along the channel centerline, the horizontal n-axis is perpendicular to the centerline and points to the left, and the z-axis is vertically upward. According to laboratory and field data summarized by Odgaard [1986, 1989], the overall mean water depth H can be confounded with the centerline water depth, h(s, 0).

Figure 1.

Definition sketch of curved open-channel flow and decomposition of transversal velocity according to equation (1).

[7] The local instantaneous velocity components are split into a turbulent and a turbulence-averaged part, and the latter is split into a depth-averaged and a depth-varying part:

display math

in which t is time, the over bars indicate turbulence-averaging, and the angle brackets indicate depth-averaging. Hence, by definition,

display math

in which Uj is the relevant depth-averaged velocity component. Especially for the transversal velocity component, vn, the decomposition according to equation (1) has an important physical meaning: equation image = Un represents the cross flow, whereas v*n represents the transversal component of the secondary circulation (see Figure 1). The former is mainly induced by downstream variations in the bed topography, whereas the latter is a characteristic feature of curved open-channel flow.

[8] The Reynolds and depth-integrated velocity correlations equation image that appear in depth-integrated flow models can be decomposed according to equation (1), to yield

display math

For simplicity, the over bar is omitted henceforth, except in the instantaneous velocity correlations, such as the last term in equation (3), the Reynolds stress.

[9] Introducing these definitions into the 3-D flow equations and averaging the result over turbulence and depth yields a set of equations like that in Figure 2. In principle, these equations can be solved for the depth-averaged velocities equation image = (Us, Un) and the flow depth h after providing appropriate boundary conditions. First, though, the residual nonlinear terms, such as 〈v*iv*j〉 and equation image, need to be expressed in terms of the turbulence-and depth-averaged dependent variables. The depth-averaged Reynolds stresses, −ρequation image, are related to equation image = (Us, Un) and h by means of a turbulence closure submodel. Similarly, a closure submodel has to be provided that relates the so-called dispersion stresses, −ρ 〈v*iv*j〉, to equation image = (Us, Un) and h.

Figure 2.

Depth-integrated flow equations in curvilinear reference system (see also Jin and Steffler [1993] or Dietrich and Whiting [1989]); marked terms, related to vertical structure of flow field, require closure submodel.

[10] Depth-integrated turbulence closure models have amply been published in the literature [Rodi, 1984, chapter 2.6.e; Nezu and Nakagawa, 1993, chapters 6.2.2 and 6.5.2] and will not be discussed herein. The dispersion stresses are typically larger than the depth-averaged Reynolds stresses. Their role has also been investigated extensively [Kalkwijk and de Vriend, 1980; Olesen, 1987; Johannesson and Parker, 1989b; Nezu and Nakagawa, 1993; Yulistiyanto et al., 1998; Finnie et al., 1999; Lien et al., 1999; Blanckaert and Graf, 2003]:

[11] 1. The velocity correlation equation imageU2s, which is related to the shape of the downstream velocity profile, does not play an important role [Olesen, 1987, p. 49], so it will be ignored henceforth.

[12] 2. The correlation 〈v*sv*n〉 is associated with the advective transport of downstream momentum v*s by the secondary circulation v*n; this is a dominant mechanism in the redistribution of the downstream velocity and boundary shear stress and needs to be modeled accurately in depth-integrated flow models [Johannesson and Parker, 1989b; Finnie et al., 1999; Blanckaert and Graf, 2003].

[13] 3. The quantity equation image is a measure of the strength of the secondary circulation.

[14] Apart from the turbulence-and depth-averaged velocity correlations, the bed shear stress vector equation imagebs, τbn) appears as an additional unknown in the depth-integrated flow equations and has to be related to equation image (Us, Un) and h:

[15] 1. The bed shear stress direction τbnbn is determined by the direction of the near-bed velocity vector and thus deviates from the direction Un/Us of the depth-averaged velocity vector if there is a secondary circulation (Figure 1) [Olesen, 1987, equations 2.42 and 2.54]:

display math

The deviation of the bed shear stress due to the secondary circulation, ατ(H/R), affects the direction of the sediment transport vector and thus has an important influence on the bed topography [Olesen, 1987, equations 2.31 and 3.18].

[16] 2. The magnitude of the bed shear stress, ∥equation image∥, is related to the magnitude of the depth-averaged velocity in straight uniform flow through a flow-resistance equation, such as τbsU2s = Cf0 (the subscript 0 refers to straight uniform flow). The friction factor Cf0 relates to the Chezy coefficient C via Cf = g/C2. Given the depth-averaged velocity, the bed shear stress in a curved flow is higher than in the equivalent straight uniform flow. First, there is an additional transversal bed shear stress component, τ*bn, due to the secondary circulation. Second, because of the advective momentum transport by the secondary circulation, the velocity profiles in a curved flow are typically flatter than in a straight flow [Blanckaert and Graf, 2003]. Since the bed shear stress is determined by the near-bed velocity gradients, this will cause an increase of ∥equation image∥ for the same depth-averaged velocity ∥equation image∥. Both effects are expressed by multiplying the flow resistance as in straight uniform flow by the amplification factor ψ, yielding

display math

[17] The remainder of this paper focuses on the closure submodels for the dispersion stresses, the bed shear stress direction, and the bed shear stress amplification factor, which are all closely related to the secondary circulation.

3. Approaches of Model Closure

[18] The flow field in a bend is well described by the three-dimensional hydrostatic flow equations, equations (A1)(A3) in Appendix A. From the 3-D solutions for vs and vn, the normalized dispersion terms could be evaluated:

display math

where U = Q/(BH) is the overall mean velocity, B is the width, and fs = vs/U and fn = v*n/(UH/R) represent the form of the vertical profiles of vs and v*n, respectively. Similarly, the 3-D solution defines the relation between the bed shear stress vector equation image and the dependent variables equation image(Us, Un) and h of the depth-integrated flow model. Consequently a closure model for these quantities can be derived from the 3-D hydrostatic flow equations. This can be done in a number of steps:

[19] 1. The 3-D hydrostatic flow equations are reduced to the simplest form that still represents all essential mechanisms. This simplification is briefly described below, but explained in detail in Appendix A. First, inertia effects are neglected and the flow is considered at the centerline of the river, which is assumed to fall within the central part of the secondary circulation cell, where n = 0, hH, UsU, and vz ≈ 0. Subsequently, an order-of-magnitude analysis is carried out by means of normalization in order to identify the dominant terms/mechanisms in the flow equations (see Figure A1 in Appendix A).

[20] 2. Various approaches can be taken to simplify the normalized flow equation in order to derive the vertical structure of the velocity components in curved flow. A formal first-order perturbation approach with H/R as a perturbation parameter leads to the commonly used linear model approach (see section 4). This will be compared with a more physics-based nonlinear approach that includes the feedback between the secondary circulation and the main flow (see section 5).

[21] 3. Neglecting inertia is equivalent to assuming that the flow field adapts instantaneously to changes in curvature. In reality, the adaptation to curvature changes takes a certain distance to implement itself, and flow in equilibrium with the imposed curvature only occurs in infinite bends of constant curvature. The solutions neglecting inertia will therefore be indicated by the subscript ∞. The inertial adaptation of the vertical flow structure to curvature changes is described by means of a semiheuristic relaxation model:

display math

where Y = 〈fsfn〉, 〈fn2〉, ψ or ατ and ϕ = s/R (bend) or s/B (straight) is the normalized downstream coordinate. This relaxation model expresses that the solution Y lags behind its target value (Y in a bend, 0 in a straight reach), whereby the relaxation factor λ indicates how strong this inertial lag effect is in terms of normalized downstream distance. Various models have been proposed in the literature [Rozovskii, 1957; de Vriend, 1981a; Kitanidis and Kennedy, 1984; Kalkwijk and Booij, 1986; Odgaard, 1986; Ikeda and Nishimura, 1986; Olesen, 1987; Johannesson and Parker, 1989a]. Their main differences lie in the definition of the relaxation factor. Hereinafter the relaxation factor proposed by Johannesson and Parker [1989a] will be adopted for 〈fn2〉 and 〈fsfn〉:

display math

According to de Vriend [1981a] and Kalkwijk and Booij [1986], ατ has a different relaxation factor, which will not be considered in this paper.

[22] 4. The closure model is now defined at the centerline of rivers of varying curvature. However, the closure models need to provide the vertical structure of the flow throughout the flow domain. Ikeda et al. [1990] propose a semiheuristic width-extension. This paper will abstain from any width-extension.

4. Linear Approach of Closure

[23] The linear approach of the closure problem is summarized in Figure 3. It is based on the normalized equations for curved flow at the centerline and a formal perturbation analysis with the curvature ratio, H/R, as a perturbation parameter. The equations shown in Figure 3 correspond to the zero-order system of equations, which includes only the terms in Figure A1 (Appendix A) that remain if H/R ↓0. The solutions of this linear closure submodel will be given the suffix 0.

Figure 3.

Linear model equations for curved flow at centerline. Inertia effects are included by means of equation (7).

[24] The linear model consists of six coupled equations, including the two integral conditions 〈fs0〉 = 1 and 〈fn0〉 = 0. They can be solved for the six variables, fs0, fn0, Ss0, Sn0, ατ0 and ψ0. The downstream momentum equation (first equation in Figure 3) is identical to the one in straight uniform flow and will yield the corresponding profile of the downstream velocity, fs0(η; Cf0), in which Cf0 is the only control parameter. The additional condition 〈fs0〉 = 1 determines the normalized downstream water surface slope as Ss0 = 1, due to the chosen normalization. The transversal momentum equation (second equation in Figure 3) expresses the local imbalance between the centrifugal force and the transversal pressure gradient that is known to give rise to the curvature-induced secondary circulation. The solution of the normalized horizontal component of the secondary circulation, fn, is completely determined by the downstream velocity profile, fs, and the corresponding eddy viscosity profile fν: inserting fs0 and fν0 and applying the condition 〈fn0〉 = 0 yields the solutions for fn0(η;Cf0) and Sn0.

[25] Since fs0 and fν0 depend uniquely on Cf0, this will also be the only control parameter in fn0, Sn0, 〈fs20, and 〈fsfn0. The deviation angle τ*bn = τbs of the bed shear stress is proportional to the curvature ratio, with the factor of proportionality, ατ0, also depending uniquely on Cf0. Furthermore, at zeroth-order, ψ0 = 1.

[26] This closure submodel is called linear, because all resulting (dimensional) variables, i.e., 〈v*sv*n〉, equation image, ατ, increase linearly with the curvature ratio H/R.

[27] Various linear-model solutions have been proposed in the literature [e.g., Engelund, 1974; Kikkawa et al., 1976; de Vriend, 1977; Odgaard, 1986; Johannesson and Parker, 1989a]. They mainly differ in the choice of the eddy viscosity profile, fν0, and the resulting downstream velocity profile, fs0. Here we adopt the model proposed by de Vriend [1977], which is based on a parabolic eddy viscosity profile, fν0 = κν(1 − η) with κ the Karman constant, resulting in a logarithmic fs0; the solutions for equation image, 〈fsfn0, and ατ0 are shown in Figure 4.

Figure 4.

Normalized linear model solutions.

[28] After inclusion of inertia effects according to equation (7), these solutions are compared in Figure 5 with the centerline evolution of 〈fn2〉 and 〈fsfn〉 measured in flume experiments at Ecole Polytechnique Fédérale Lausanne (EPFL) [Blanckaert, 2002, chapter IV.1]. The curvature ratios H/R and estimations of the straight-uniform-flow friction factors Cf0 are tabulated in Figure 5. The flume has a width B of 1.3 m and the curved part a constant centerline radius of curvature of R = 1.7 m.

Figure 5.

Comparison of linear-model results (solid lines) with data (symbols) measured at centerline.

[29] The graphs and the table included in Figure 5 show that the linear model fails to represent the observed behavior at various points: (1) According to the model, the nondimensional quantities only depend on Cf0, not on H/R, whereas the experimental data show a decreasing tendency with H/R; (2) the model generally overestimates the experimental data; and (3) the model results increase monotonically throughout the bend, as they adapt gradually to the imposed constant curvature, only to decrease in the straight reach downstream of the bend. Note that even this 193° bend is too short for the inertial adaptation to get accomplished. Contrary to the linear-model results, the experimental data reach a maximum in the bend and then decrease considerably in the second part of the bend.

[30] Blanckaert [2001] has reported more data that illustrate the shortcomings of the linear closure submodel. As stated above, the linear model solution is completely determined by the assumed eddy viscosity profile and the corresponding downstream velocity profile. De Vriend [1981a] already indicated that the neglected feedback between fs and fn might be responsible of the poor linear-model predictions. His three-dimensional simulations for laminar curved flow indicate that advective momentum transport by the secondary circulation, fsfn, reduces fs near the water surface and increases it near the bed. These flattened fs-profiles lead to a reduction of the force imbalance that drives the secondary circulation. The mechanisms for the flattening of the fs-profile and the subsequent reduction of 〈fn2〉 have been investigated experimentally by Blanckaert and Graf [2003] and Blanckaert and de Vriend [2003], respectively, for turbulent curved flow.

[31] Yet, this feedback mechanism between fs and fn has not been quantified before and it is not clear to what extent it is responsible of the poor performance of the linear model. These questions will be addressed in the next section, where a nonlinear model will be proposed that accounts for this feedback mechanism.

5. Nonlinear Approach of Closure

5.1. Existing Nonlinear Models

[32] Closure submodels that account for the feedback between fs and fn are called nonlinear. Nonlinear models have been proposed by Jin and Steffler [1993] and Yeh and Kennedy [1993]. They adopt predefined profiles of the downstream velocity and the secondary circulation, both with one degree of freedom representing their modification due to curvature influences. This degree of freedom is determined from two equations expressing the depth-integrated conservation of moment-of-momentum, which are added to the system of depth-integrated flow equations. These models simulate the flattening of the fs-profiles and the reduction of the secondary circulation in the second part of the bend. Moreover, they partially account for the bed shear stress amplification factor, ψ.

[33] However, these models are not sufficiently transparent, as they do not clearly indicate the relative importance of the various mechanisms, nor the sensitivity to certain parameters. For example, the influences of the friction factor Cf, the curvature ratio H/R, and the transversal velocity distribution αs (equation (10)) are accounted for in these models but are not discernible in the presented mathematical formulation. Furthermore, the addition of two differential equations makes these models computationally expensive. In the following section, a computationally efficient nonlinear closure model is proposed that aims at improving and extending these models. Instead of using predefined velocity profiles with one degree of freedom, the proposed model will calculate the entire vertical profiles of the downstream velocity and the secondary circulation.

5.2. Nonlinear Model Equations

[34] The proposed nonlinear model is based on the flow equations for curved flow at the centerline, reduced to their simplest form that still includes all essential mechanisms (Figure 6). Whereas in the linear model a formal perturbation approach was taken and all terms multiplied by a power of H/R in Figure A1 (Appendix A) were neglected, the nonlinear model equations are based on a more physical approach. Although formally of the order (H/R)2, the terms representing the advective transport of downstream momentum by the secondary circulation are retained in the downstream momentum equations, for the following reasons:

Figure 6.

Nonlinear model equations for curved flow at centerline. Inertia effects are included by means of equation (7).

[35] 1. The ratio of the advective momentum transport terms and the driving gravity term is of order (H/R)2/Cf0, and thus increasing with increasing curvature and decreasing roughness. Whereas it is negligible for weak curvatures and rough boundaries, (H/R)2/Cf0 ∼ (1/100)2/(0.01) ∼ 0.01, it cannot be neglected anymore for sharp bends and/or smooth boundaries, (H/R)2/Cf0 ∼ (1/20)2/(0.0025) ∼ 1.

[36] 2. Blanckaert and Graf [2003] conducted a term-by-term analysis of the downstream momentum equations (three dimensional as well as depth-integrated), based on experimental data for flow in an open-channel bend with (H/R)2/Cf0 ∼ (0.11/2)2/0.008 ∼ 0.38. They found that when averaged over the flow depth, the advective momentum transport terms were of the same order of magnitude as the driving gravity term. Locally, however, they were an order of magnitude larger, especially near the center of the secondary circulation cell. This shows that the advective-momentum-transport terms are more important than indicated by an order-of-magnitude analysis and that they might even play an important role in weakly curved flows with (H/R)2/Cf0 ≪ 1. Furthermore, Blanckaert and Graf [2003] demonstrated that the advective transport of momentum by the secondary circulation is the main cause of the flattening of the fs-profiles.

[37] The simple eddy-viscosity concept that underlies the linear model is maintained: The shape is kept parabolic throughout the bend, and the magnitude is taken proportional to the local shear velocity equation image and local water depth h(n) (also see equation (A5) in Appendix A). At the centerline, this leads to

display math

[38] Since the nonlinear model equations are applied at the centerline, they cannot solve for the transversal gradients (the shaded ∂/∂ξ-terms in Figure 6). These terms, however, play an essential role and have to be modeled. Adopting a similarity hypothesis, Appendix B shows that these gradients can be expressed in terms of the normalized transversal gradient of the depth-averaged downstream velocity at the centerline

display math

via

display math

[39] Since the nonlinear model will serve as a closure submodel in the depth-integrated flow equations, information on αs becomes available when solving the combined model. This is similar to most turbulence closures, where the turbulent stresses are related to gradients of the to-be-computed mean flow field. The parameter αs has been proposed by Einstein and Harder [1954]. According to Bradshaw [1969], Prandtl [1961] already proposed this parameter in 1930 to investigate the influence of curvature on turbulent flows.

[40] Figure 7 shows that in the EPFL-experiments [Blanckaert, 2002, chapter IV.1] the quantity αs varies from 0 in the straight reach upstream of the bend, via −1 (potential-vortex distribution) in the first part of the bend, to +1 (forced-vortex distribution) just beyond the bend exit, and back to 0 far downstream. Note that αs + 1 can be interpreted as the deviation from the potential-vortex velocity distribution.

Figure 7.

Width-distribution of downstream velocity around bend, parameterized by αs (equation (10)).

[41] This yields the nonlinear closure submodel summarized in Figure 8, where the transversal momentum equation is given in its integral form. As in the linear model, the inertial lag effects in zones of varying curvature are included using a linear relaxation model (equation (7)). Note that by parameterizing the transversal gradients, the original partial differential equations reduce to ordinary differential equations.

Figure 8.

Modeled nonlinear model equations for curved flow at centerline. For H/R ↓0, linear-model equations are found. Inertia effects are included by means of equation (7).

[42] The nonlinear model consists of six coupled equations, including the two integral conditions 〈fs〉 = 1 and 〈fn〉 = 0. They can be solved for the six variables, fs, fn, Ss, Sn, ατ and ψ. The numerical solution procedure is rather intricate, due to the strong coupling of the variables and the sensitivity of the modeled feedback mechanism. It consists of a discretization of the equations, followed by an iterative solution technique, as discussed by Blanckaert [2002, chapter IV.1].

[43] The solution depends on three parameters (those shaded in Figure 8): the friction factor Cf0, the curvature ratio H/R, and the deviation from the potential-vortex distribution αs + 1. Note that for vanishing curvature ratio, H/R ↓0, the dependence on αs + 1 vanishes as well, and the linear model equations emerge. For αs + 1 = 0, the advective momentum transport terms vanish; however, the nonlinear model differs (slightly) from the linear one through the bend friction factor, ψ > 1.

5.3. Validation

[44] Before further discussing the nonlinear model, its results will first be compared with measured data from the EPFL-experiments (Figure 5). Figure 9 shows the evolution of 〈fn2〉 and 〈fsfn〉 along the centerline in the very strongly curved Q89-experiment. The linear model solutions 〈fn2〉 and 〈fsfn0 are shown for comparison.

Figure 9.

Comparison of experimental data measured at centerline (symbols) with linear model solution (0), nonlinear model solution neglecting inertia effects (∞), and nonlinear model solution.

[45] The nonlinear model performs considerably better than the linear one at various points:

[46] 1. The linear-model's overestimations and its failure to account for the decrease of 〈fn2〉 and 〈fsfn〉 with H/R (Figure 5) are explained by the fact that the linear model is an asymptotic solution of the nonlinear one for vanishing curvature. The nonlinear model solution shows a significant decrease with increasing H/R (see Figure 10) and predicts the correct order of magnitude.

Figure 10.

General solution of nonlinear model for curved flow at centerline, normalized with linear-model solution (Figure 4), as a function of bend parameter β. Inertia effects are included by means of equation (7).

[47] 2. The linear model failed to account for the decreasing tendencies of 〈fn2〉 and 〈fsfn〉 in the second part of the bend, since it exclusively depends on Cf0, which does not vary around the bend. The nonlinear model includes the interaction between the main flow and the secondary circulation and therefore reproduces the decrease, via its dependence on αs + 1, the only nonlinear model parameter that varies around the bend (Figure 7). The importance of αs + 1 will be discussed in further depth in section 6. Also note that the experimental data lag behind the nonlinear-model solution neglecting inertia (subscript ∞) in addition to reaching a lower maximum. If inertial lag effects are included, however, the agreement between model results and data is reasonably satisfying (Figure 9). This indicates that the linear relaxation model for the inertial lag effects is adequate.

[48] 3. When averaged over the bend reach, the shear stress amplification factor ψ has a value of 1.56 (not shown), yielding equation image = equation image = 11.55, which compares satisfactorily with the experimental value of 10.8 [Blanckaert, 2002, chapter II.3].

[49] In general, the agreement between the nonlinear-model results and the measured data is good, given that the experiment concerns a very strongly curved channel (H/R = 0.094 will rarely occur in natural open-channel bends) and that no calibration parameters are used in the model.

5.4. Parameter Reduction

[50] In its present form, the nonlinear closure model presented above is not convenient for practical use. The model requires the solution of six coupled equations and the results depend on three parameters, i.e., Cf0, H/R and αs + 1. This section presents approximate solutions in tabular or graphical form that are more convenient to use. The elaboration of these approximate solutions is reported in detail by Blanckaert [2002, chapter IV.1].

[51] The parameters H/R and αs + 1 mostly occur in the combination (H/R)2s + 1) (Figure 8), which is furthermore correlated with the remaining parameter Cf = ψCf0. Figure 10 shows that when plotted against the combined parameter β = (Cf)−0.275 (H/R)0.5s + 1)0.25, the solutions for 〈fn2, 〈fsfn, and ατ∞ each almost collapse on a single curve. The solution for equation image shows a somewhat larger, but still acceptable scatter around a single curve when plotted against the parameter βCf0.15 = (Cf)−0.125 (H/R)0.5s + 1)0.25. The scatter mainly increases at very high H/R values, which are rare in nature. Thus the parameter β = (Cf)−0.275 (H/R)0.5s + 1)0.25 is identified as a control parameter in curved open-channel flow. It will henceforth be named the bend parameter.

[52] The nonlinear model accounts for the feedback between the main and the secondary circulation, which, via the reduced imbalance between the centrifugal force and the transversal pressure gradient, will also affect the strength of the secondary circulation. Since the linear-model profiles depend on Cf0 only, the nonlinear-model profiles will not depend exclusively on β, but also on Cf. Figure 11 shows the simulated flattening of the fs profiles and the reduction of fn for flow over a rough bed (equation image = 10) and a smooth one (equation image = 20) for values of the bend parameter β = 0, 0.5, 0.65, 0.85, 1, and 1.3.

Figure 11.

Nonlinear model solution for fs and fn profiles for values of bend parameter of β = 0, 0.5, 0.65, 0.8, and 1.3 for flow over rough (equation image = 10) and smooth (equation image = 20) bed.

[53] With increasing β, the fs profiles flatten by decreasing/increasing in the upper/lower part of the water column. For high values of β, nonmonotonic fs profiles are found (i.e., with ∂fs/∂η < 0 in the upper part of the water column), leading to a considerable reduction of fn.

[54] Note that Figure 10 represents the general solution of the nonlinear model, neglecting inertia effects. Inertia effects in regions of varying curvature are subsequently accounted for by means of the relaxation equation (equation (7)). The results can be used as a closure submodel (e.g., using look-up tables or analytical functions) in depth-integrated flow models. Moreover, they reflect the relevant flow mechanisms (such as the role of the cross-stream gradient of the downstream velocity, αs) and the sensitivity to the various model parameters, which are lumped into the bend parameter, β.

6. Linear Versus Nonlinear Closure Submodel

[55] De Vriend [1981a] already indicated that the validity of the linear model is limited to weak curvatures. However, neither the range of validity, nor the discrepancies in case of strong curvatures could be quantified. With the nonlinear model, which encompasses the linear model as a limiting case for H/R ↓0, this is possible now. Three regions can be distinguished in the solution domain (see Figure 11), which can be objectively related to the notions of weak, moderate, and strong curvature: (1) For β < 0.4, the linear model solution is acceptable; (2) for 0.4 < β < 0.8, the ratio of the nonlinear to the linear-model solution decreases strongly and almost linearly; and (3) for β > 0.8, the nonlinear model solution for 〈fsfn〉, which is the dominant parameter with respect to the velocity redistribution, is reduced to less than half the linear one and the tail end of the curve is reached. Note that the hypotheses underlying the nonlinear model may cease to be valid at higher β values in this region.

[56] The notions of weak, moderate, and strong curvature used to be defined in a rather arbitrary way, mostly in terms of the H/R or B/R ratio or by means of the turbulent Dean number, De = 13(Cf)−0.5 (H/R)0.5 [de Vriend, 1981b]. The bend parameter now provides an objective criterion to assess the curvature effects, depending on the curvature ratio H/R, the friction factor Cf, and the parameter αs. Note that the arc-length of the bend, the channel width B, and the bed topography implicitly affect the degree of curvature, since they determine the value of αs.

[57] The nonlinear model solution (Figure 10, left panel) clearly indicates that the secondary circulation becomes self-limited in sharp bends. This idea is not completely new, but the nonlinear model clearly shows, underpins and quantifies this phenomenon.

[58] Although the linear and the nonlinear models compute the secondary circulation fn from the same simplified transversal momentum equation (the second equation in Figures 3 and 6), the nonlinear model is far more complete in the representation of the flow mechanism, as explained in detail by Blanckaert [2002, chapter IV.1]. In both models, the downstream fs profile yields the secondary circulation fn; from the fs and fn profiles, the depth-averaged advective momentum transport by the secondary circulation 〈fsfn〉 is computed, which is an important contributor to the cross-stream distribution of the downstream velocity, parameterized by αs. Whereas the straight-uniform-flow profile fs0 is prescribed in the linear model, the curved flow profile fs is deformed through two important feedback mechanisms: (1) fn flattens the fs profiles by advecting main flow momentum, fsfn, thus yielding a decrease of fn, and (2) the vertical distribution of vs, represented by fs, is coupled to its distribution over the width, represented by αs; the larger αs gets, the flatter the fs profile and the smaller fn; this leads to a decrease of 〈fsfn〉 and finally to a reduction of αs.

[59] Note that both feedbacks are negative and thus tend to stabilize their effects. Because of these feedbacks, the nonlinear model depends mainly on the parameter combination β = (Cf)−0.275 (H/R)0.5s + 1)0.25, rather than on Cf only. Especially the factor αs + 1, parameterizing the cross-stream distribution of vs, plays an important role:

[60] 1. It controls the coupling between the vertical (fs) and the horizontal (αs) distributions of vs. Closure models based on the nonlinear approach are therefore dynamic, in that they are dependent on the solution of the depth-averaged flow model. This is equivalent to not imposing the vertical flow structure, but computing it as part of the solution. On the contrary, closure models based on the linear approach impose the vertical flow structure, since their solution uniquely depends on the constant Cf0.

[61] 2. The αs dependence explains the decreasing tendencies of 〈fn2〉 and 〈fsfn〉 in the second part of the bend (Figures 5 and 9).

[62] 3. Bends with a horizontal bed, with αs usually varying between −1 and +1 (Figure 7), are atypical of natural rivers, in which αs can reach significantly higher values [Blanckaert, 2002, chapter IV.1]. The vertical flow structure expressed by the nonlinear model depends on αs and must therefore be different in either case. This implies that laboratory experiments with a horizontal bed may not be representative of the flow in natural river bends.

7. Summary and Conclusions

[63] In depth-integrated flow models, the influence of the vertical structure of the flow field has to be accounted for by a closure submodel. In the case of meandering rivers, this requires taking into account the secondary circulation, which is known to have the following effects: (1) It redistributes the flow, the boundary shear stresses, and the sediment transport by advecting flow momentum, represented by 〈fsfn〉; (2) it causes the direction of the bed shear stress to deviate from the direction of the depth-averaged velocity, given by ατ; and (3) it causes additional friction losses in a bend, parameterized by ψ.

[64] A nonlinear approach of this closure problem is proposed, which is shown to perform better than the commonly used linear one.

[65] Both linear and nonlinear closure submodels are derived from the three-dimensional flow equations, by reducing them to the simplest form that still includes all essential mechanisms. As a first step, a closure model is derived for the river centerline, neglecting inertia effects. In a subsequent step, inertial lag effects in response to curvature variations are included by applying a linear relaxation model. A last step would be to extend the closure model over the entire river width, but this is not treated in the present paper.

[66] The commonly used linear closure models have been derived from the 3-D flow equations via a formal first-order perturbation in the curvature ratio H/R. On the basis of physical arguments, the proposed nonlinear model retains the advective momentum terms in the 3-D flow equations, even though they are of order (H/R)2. Its results depend on the friction factor Cf, the curvature ratio H/R, and the distribution of the downstream velocity over the width, parameterized by αs = [(∂Us/∂n)/(Us/R)]n=0. The linear model is found to be the asymptotic case for vanishing curvature ratio. As opposed to the linear model, the nonlinear model is dynamic, in that the vertical flow structure is not imposed, but computed as part of the solution. The results agree well with experimental data: They are of the right order of magnitude, correctly represent the decrease with increasing H/R, and the αs dependence exhibits the expected maximum in the first part of the bend.

[67] The nonlinear model results can be expressed in a good approximation as functions of a single parameter. For the quantities 〈fsfn〉, 〈fn2〉 and ατ, this is the so-called bend parameter β = (Cf)−0.275 (H/R)0.5s + 1)0.25, whereas ψ shows the least scatter against the parameter β (Cf)0.15. This allows for convenient and computationally inexpensive ways (e.g., using look-up tables or analytical functions) to include this closure submodel into a depth-integrated flow-sediment-water quality model. Furthermore, this identifies β as an important control parameter in curved open-channel flow. It indicates the validity range of the commonly used linear models and helps to distinguish between weak, moderate, and strong curvatures.

[68] As the nonlinear model accounts for the feedback between the secondary circulation fn, the vertical distribution of the downstream velocity fs, and its width-distribution αs, it is able to reproduce the flattening of the fs profiles and the reduction of fn with increasing curvature, which are typical of curved-channel flow. Furthermore, it underpins and quantifies the self-limiting character of the secondary circulation in sharp bends.

[69] The above conclusions indicate that the proposed nonlinear-model closure for the vertical structure of the flow field, and especially for the effects of the secondary circulation, has the potential to effectively and efficiently improve the capabilities of depth-integrated flow-sediment-water quality models.

Appendix A:: Derivation of Linear and Nonlinear-Model Equations

[70] The three-dimensional Reynolds-averaged conservation equations for mass and momentum in curvilinear coordinates are reported by Schlichting and Gersten [2000]. Distinguishing between the cross-flow Un and the secondary circulation v*n according to equation (1), they can be written as

display math
display math
display math

[71] The hydrostatic pressure assumption, p = ρg (zSz), replaces the momentum-conservation equation for the vertical velocity component vz. The Reynolds stresses are expressed in terms of the mean velocity components using the eddy viscosity concept, equation image = 2vteij (i, j = s, n, z), where νt is the eddy viscosity and eij are the strain rates. The horizontal s-axis lies along the channel centerline, the horizontal n-axis is perpendicular to the centerline and points to the left, and the z-axis is vertically upward (Figure 1). R (s) is the varying centerline radius of curvature (positive/negative for bends turning to the right/left); ρ is the fluid density; (1 + n/R) is a metric factor.

[72] Equations (A1)(A3) are now simplified by the following steps:

[73] 1. We consider steady flow and neglect inertia effects, which is equivalent with assuming that the flow field adopts instantaneously to changes in curvature. In reality, the flow field and the bed topography gradually adjust to changes in curvature. The flow in this adaptation zone is called developing curved flow. Only in infinite bends of constant curvature, inertia effects vanish since the flow field and the transversal bed slope adapt completely to the curvature and no longer vary in the downstream direction. In this so-called fully developed curved flow situation, which will be indicated by the subscript ∞, we have

display math

whence all terms between square brackets vanish.

[74] 2. We concentrate on the river centerline, where n = 0. Furthermore, hH and UsU, according to laboratory and field data summarized by Odgaard [1986, 1989], where H and U are the overall mean values of h (s, n) and Us (s, n). This hypothesis is not crucial, but we have adopted it since it simplifies the model presentation.

[75] 3. We assume that the centerline falls within the central part of the secondary circulation cell, where vzvn. We have verified that vz has a negligible effect on the solution of the nonlinear model, even near important transversal bed slopes where vz = −vnzb/∂n, and can therefore be neglected.

[76] 4. We normalize them in order to identify the dominant terms/mechanisms:

display math

Since the solution of the nonlinear model was found to be hardly influenced by the choice of the eddy viscosity, a rather simple formulation is adopted. The parabolic profile shape of the linear model, fν0(η), is retained throughout the model domain, and its magnitude is taken proportional to the local water depth h(n) and the local shear velocity, equation image.

[77] Figure A1 gives the resulting momentum-conservation equations, as well as the corresponding depth-integrated equations [see also Jin and Steffler, 1993; Dietrich and Whiting, 1989].

Figure A1.

Normalized hydrostatic momentum-conservation equations for curved flow at centerline, neglecting inertia effects.

Appendix B:: Transversal Gradients in Nonlinear Model

[78] Since the nonlinear model equations are applied only at the centerline, the transversal gradients (shaded terms in Figure 6) have to be provided. In the following, they will be related to the transversal gradient of the downstream velocity, parameterized by αs (see equation (10)).

B1. Definitions

[79] A similarity hypothesis is adopted for the vertical distributions of fs and fn,

display math

in which equation image represents the shape of the vertical distribution and gj is a gain factor that may vary over the width. Coefficients are now defined as

display math
display math

where αn represents the transversal gradient of the magnitude of the secondary circulation. According to equation (A5), the transversal gradient of the flow depth, which is due to the tilting of the water surface and the bed slope, can be written in a similar form:

display math

If the water depth varies over the width, ξ and η are not mutually orthogonal:

display math

The transversal gradients of fs and fn can now be parameterized as

display math

[80] It has been verified that the depth-varying contribution αj,3-D has negligible effect on the solution of the nonlinear model, even for important transversal bed slopes, whence it is neglected further on. This simplification eliminates the explicit dependence of the nonlinear model on the transversal bed slope ST, which occurs in αH (equation (B4)).

[81] The transversal-gradient term in the depth-integrated momentum equations can now be parameterized as

display math

B2. Determination of αn

[82] The principle of mass-conservation underlies the determination of αn. Since depth-integration of the 3-D mass-conservation equation (equation (A1)) removes fn, the indirect method shown in Figure B1 is proposed.

Figure B1.

Definition sketch of normalized quantities and derivation of αn.

[83] The curvilinear control volume, indicated in Figure B1, covering the upper half of the water column in the central part of the circulation cell is considered. It is assumed that the fn profiles are linear and that the vertical velocity is negligible. The value of αn is now found by expressing the equality of the incoming and outgoing mass flux due to the secondary circulation.

display math

Substituting fn (ξ, 1) = equation image(1)gn (ξ) gives

display math

The transversal gradients in the nonlinear model equations (Figure 6) are now uniquely parameterized by αs:

display math
Notation
B

channel width.

C, Cf = g/C2

Chezy roughness coefficient and friction factor, respectively.

eij

strain rates, i, j = s, n, z.

fj

form of distributions of vs and v*n: fs = vs/U and fn = v*n/(UH/R).

equation image

form of vertical distribution of vs and v*n, equation (B1).

fsfn

normalized advective transport of downstream momentum fs by the secondary circulation fn.

fν0

form of the eddy viscosity distribution, equation (A5).

Fr = U/equation image

Froude number.

g

gravitational acceleration.

gj

gain factor in transversal distribution of vs and v*n, j = s, n, equation (B1).

H

overall mean flow depth, which can be confounded with the centerline water depth h (s, 0) [Odgaard, 1986, 1989].

h

local flow depth.

equation image

normalized local flow depth, equation (A5).

n

transversal reference axis.

p

pressure.

Q

discharge.

R

radius of curvature of channel centerline, varying in downstream direction.

Sj

water surface slope.

ST

normalized transversal bed slope, equation (A5).

s

downstream reference axis.

t

time.

U

Q/(BH) = overall mean velocity.

Uj = 〈vj

depth-averaged velocity component, j = s, n.

vj

time-averaged velocity component, j = s, n, z.

v'j (t)

instantaneous velocity fluctuation, j = s, n, z.

v*n

local deviations from the depth-averaged velocity, j = s, n.

equation image

time–averaged correlation of the velocity fluctuations, i, j = s, n, z.

z

vertical reference axis; elevation above the horizontal (s, n) plane.

Symbols

 

αH

normalized transversal water depth gradient, equation (B4).

αs

normalized transversal velocity gradient, (∂Us/∂n)/(Us/R), equation (10).

ατ

normalized deviation angle between bed shear stress and depth-averaged velocity, equation (4).

β

bend parameter, β = (Cf)−0.275 (H/R)0.5s + 1)0.25.

ϕ

normalized downstream coordinate.

η

normalized vertical coordinate.

κ

Karman constant, κ = 0.4.

λ

relaxation factor that accounts for inertia effects, equation (7).

νt

eddy viscosity.

ρ

fluid density of water.

τb

shear stress at the bed.

τ*bn

part of the transversal bed shear stress due to the secondary circulation, equation (4).

ξ

normalized transversal coordinate, equation (A5).

ψ

flow resistance amplification factor due to bend effects, equation (5).

arrow

vectorial quantity.

over bar

time-averaged values.

〈〉

values averaged over the local flow depth.

O( )

order of magnitude of ( ).

∥…∥

norm of vector.

Subscripts

 

b

bed.

S

water surface.

0

linear model solution (except for η0; see above).

model solution neglecting inertia.

3-D

three-dimensional.

Acknowledgments

[84] This research is sponsored by the Swiss National Science Foundation under grants 2100-052257.97 and 2000-059392.99. The author wishes to acknowledge his Ph.D. supervisor, W. H. Graf, for his support.

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