Lateral transfer of streamwise momentum caused by a roughness transition across a shallow channel


  • D. A. Vermaas,

    1. Hydrology and Quantitative Water Management Group, Wageningen University, Wageningen, Netherlands
    2. Environmental Fluid Mechanics Section, Faculty of Civil Engineering and Geosciences, Delft University, Delft, Netherlands
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  • W. S. J. Uijttewaal,

    1. Environmental Fluid Mechanics Section, Faculty of Civil Engineering and Geosciences, Delft University, Delft, Netherlands
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  • A. J. F. Hoitink

    1. Hydrology and Quantitative Water Management Group, Wageningen University, Wageningen, Netherlands
    2. Institute for Marine and Atmospheric Research Utrecht, Department of Physical Geography, Utrecht University, Utrecht, Netherlands
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[1] Research on lateral exchange of streamwise momentum between parallel flows in open channels has mainly been focused on compound channels composed of a main channel and a floodplain, on mixing layer development downstream of river confluences, and on partially vegetated channels. This study aims to establish the mechanisms responsible for streamwise momentum exchange between concurrent parallel flows subject to different bed roughnesses. The contribution of momentum exchange of each mechanism is determined on the basis of flume experiments. For an initially uniform flow that experiences a bed with two parallel lanes of different roughnesses, three mechanisms for exchange of streamwise momentum can be distinguished: cross-channel secondary circulations, turbulent mixing resulting from vortices acting in the horizontal plane, and mass transfer from the decelerating flow over the rough-bottomed lane to the accelerating flow in the parallel smooth-bottomed lane. The mass transfer and associated momentum transfer across the channel cause a gain in longitudinal momentum. The secondary circulations are driven by turbulence anisotropy and feature a main cell that extends over the full water depth, which is centered at the smooth side of the smooth-to-rough transition. The gain of momentum corresponding to the mass transfer in the developing reach is on the same order of magnitude as the momentum exchange by turbulent mixing and of that by secondary circulations in the most downstream position, where the flow is nearly developed. The contribution of the secondary circulations to the exchange of streamwise momentum between the parallel flows gradually becomes dominant over the contribution of turbulent mixing when depth increases.

1. Introduction

[2] Streamwise velocities in straight, open channels are controlled by water depth, bed composition, and vegetation density. Channels with an abrupt lateral difference in these properties result in a compound flow, in which parallel flows are controlled by local equilibrium conditions outside the vicinity of the lateral change in bed properties. In the vicinity of this lateral change the parallel flows cannot be studied in isolation, since they interact by means of secondary circulations and turbulence-induced vortices acting in the horizontal plane, causing exchange of momentum between the different flow domains.

[3] Knowledge about the origin and magnitude of this momentum exchange is required to predict lateral profiles in flow velocity and their associated properties. Shallow flows, such as floodplain flows, often deal with spatial variation in bed level, bed composition, and vegetation density. Most evident is the interface between the main channel and the floodplain, featuring a change in bed level and a change in bed roughness. Changes in only bed level or only bed roughness can be found within a main channel or within the floodplain domain. A variation in these properties not only redistributes the flow locally but also can influence the flow as a whole [e.g., Smart, 1992] because of lateral momentum transfer. Quantifying the contributions of alternative mechanisms of lateral momentum transfer provides a support to design an appropriate setup for a numerical flow model by providing a means to evaluate the importance of local heterogeneity in bed properties.

[4] By restricting ourselves to an idealized case of two parallel flow domains, the transverse velocity difference will result in a single mixing layer. Mixing layers in parallel flows can be subdivided in three categories, according to the cause of the velocity difference: compound channels, confluences of flows with different velocities, and channels with compound bed roughness. This paper focuses on the flow subject to compound bed roughness. Hereafter, related research on each type of mixing layer is reviewed briefly.

[5] A significant amount of research has been devoted to turbulence-induced momentum exchange in compound channels, representing main channels and adjacent floodplains with a different bed level [Shiono and Knight, 1991; Van Prooijen et al., 2005]. Smart [1992] showed that the turbulence-induced interaction between the floodplain and the main channel can account for an increase in the water level by about 15%. Myers et al. [2001] inferred relationships between discharge and water level, presuming two different values of floodplain roughness. Myers et al. [2001] show the effect of floodplain roughness on the velocity in the main channel. The exchange of momentum between the floodplain and the main channel is due both to secondary circulation, in a vertical plane perpendicular to the main flow direction, and to large-scale vortices moving in the horizontal plain. These causes of momentum exchange retard the flow in the main channel. The contribution of secondary flow to the lateral momentum exchange in compound channels depends very much on the depth of the floodplains relative to the depth of the main channel and on the geometrical details of the interface [Shiono and Knight, 1991; Van Prooijen et al., 2005]. It is therefore important to analyze the strength and shape of the secondary circulation. Proust et al. [2009] recently recognized the importance of cross-channel mass transfer for transverse momentum exchange in compound channels, which needs to be taken into account when the flow adapts to nonuniform cross-section geometry.

[6] Prandtl [1925] distinguished between secondary currents of the first kind, which are induced by flow curvature, and secondary currents of the second kind, induced by anisotropy of turbulence. The coarse features of secondary currents induced by flow curvature are well understood [Rozovskii, 1957; De Vriend, 1977] and can be reproduced without detailed modeling of turbulence [Rodi, 1986; Nezu and Nakagawa, 1993]. A more detailed analysis of curved flows [Van Balen et al., 2009] shows that secondary currents of the second kind can also be found in curved flow, near the outer bank of a bend. For straight flows, lateral momentum exchange by shear-induced large-scale vortices and secondary currents of Prandtl's second kind remain contemporary interrelated research subjects [Nezu, 2005]. Common turbulence models, such as those assuming an isotropic eddy viscosity [Nezu and Nakagawa, 1993], cannot properly reproduce the features governing momentum exchange, thereby limiting the accuracy of predicted profiles of flow velocity and bed shear stress.

[7] An adjoining field of research is focusing on mixing layers at confluences. Shallow mixing layers developing downstream from a splitter plate feature lateral momentum exchange dominated by large eddies emerging from Kelvin-Helmholtz instabilities [Uijttewaal and Booij, 2000]. With the idealized laboratory conditions of shallowness, in combination with a flat and smooth bottom, the secondary currents do not easily develop. Confluences in deeper flows can feature a secondary current [Sukhodolov et al., 2010]. Rhoads and Sukhodolov [2001] enumerate that the structure of flow within the confluence hydrodynamic zone is influenced by the degree of symmetry of the confluence [Mosley, 1976], the junction angle of the confluence, the momentum flux ratio of the confluent streams [Best, 1987], and the degree of concordance of the channel beds at the apex [Biron et al., 1996]. The variety of hydrodynamic processes at stream confluences is further enhanced by the mobility of the bed [Szupiany et al., 2009], often featuring deep scour holes where flow converges. Rhoads and Sukhodolov [2008] invoked an analogy between river confluence hydrodynamics and wake flow, associated with a stagnation zone near the apex [Best, 1987]. The dependence of river confluence hydrodynamics on the bed morphology and wake effects leads to the conclusion that the mixing layers downstream of a confluence cannot be schematized as simple parallel flows with different initial velocities.

[8] Studies on the flow subject to a lateral change in bed roughness are sparse. Recently, White and Nepf [2008] analyzed the flow in a channel that was partially covered with emergent vegetation. They observed coherent vortices spanning both flow layers, which were the dominant contributors to lateral momentum fluxes. The absence of a significant contribution of secondary circulations allowed for the construction of an analytical model for the fully developed profiles of velocity and shear stress. The configuration in the study by White and Nepf [2008] can be perceived as a special case of parallel flows subject to different bed roughness, in which the roughness in one of the lanes is at its upper extreme, inhibiting vertical circulations. Studerus [1982] and Tominaga and Nezu [1991] described the flow structure in a developed mixing layer over a bed with a more subtle difference in lateral roughness, which revealed the secondary flow driven by turbulence anisotropy and the structure of mixing layer vortices. Neither Studerus [1982] nor Tominaga and Nezu [1991] quantified the contribution of secondary flows and the vortices rotating about a vertical axis to the exchange of streamwise momentum, nor did they elaborate on the dynamics in the developing range of the mixing layer. This paper presents laboratory experiments on the mechanisms causing exchange of streamwise momentum between parallel flows with different bed roughness, focusing both on the downstream adaptation of the flow and on the developed situation starting with a laterally uniform inflow condition.

[9] The remainder of this paper is structured as follows. In section 2 the theoretical framework is explained, which includes a new formulation for momentum exchange between parallel lanes of different roughness due to mass flux redistribution. Section 3 presents the setup of the laboratory experiments. In section 4 the results are presented, including the quantification of the transfer of streamwise momentum by the three mechanisms. In section 5, conclusions are drawn.

2. Theoretical Framework

[10] A synoptic schematization of an archetypical mixing layer that develops between two parallel flows subject to different bed roughnesses is presented in Figure 1. A uniform flow approaches the start of two parallel lanes of equal width, with the roughness of the smooth bed lane being the same as the roughness of the bed upstream of the starting line of the split bed channel section. Upstream and downstream of the start of the rough lane, the flow decelerates, whereas the reversed situation occurs around the start of the smooth lane. The acceleration and deceleration imply a mass transfer from the rough lane to the smooth lane, satisfying continuity. Both the acceleration and deceleration and the associated mass transfer will decrease asymptotically to zero with distance from the starting line. Given the nonlinear relation between velocity and advected momentum, the redistribution of flow from a uniform to a nonuniform velocity profile requires additional longitudinal momentum. Consequently, the redistribution of discharge over the flume affects locally the longitudinal water level gradient.

Figure 1.

Schematization of a mixing layer between two parallel flows subject to different bed roughnesses, as it develops after an initially uniform situation.

[11] The longitudinal change in momentum due to mass transfer, with adjoining exchange of longitudinal momentum associated with secondary circulation and turbulent mixing, can be expressed in two coupled along-channel momentum balances, pertaining to the smooth and the rough side of a stationary channel flow. The control volume of a momentum balance covers the full cross section of the flow above either the rough bed or the smooth bed and is infinitesimally small in the longitudinal direction. Regarding the smooth side of the flume, such a momentum balance reads

equation image

and regarding the rough side of the flume, it reads

equation image

where D is water depth, B is width of the smooth or rough section, M is advection of longitudinal momentum, TTM and TSF denote the interfacial shear stress due to momentum fluxes by turbulent mixing and secondary flow, respectively, equation image is mass density, g is the gravitational constant, h is water level relative to a reference level, and equation image is bed shear stress. The coordinates x, y, and z are defined for the respective longitudinal, transverse, and vertical directions, with the origin at the right bank at the beginning of the rough bed.

[12] The left-hand sides of equations (1) and (2) reflect the change in advected longitudinal momentum, caused by flow redistribution near the start of the parallel flow lanes. For a cross-sectional area equation image, the longitudinal momentum flux can be expressed as

equation image

where u is velocity in the x direction and equation image and equation image are the width and depth of the domain, respectively. Neglecting small water level differences between the two lanes, mass conservation requires that the deceleration above the rough bed equals the acceleration above the smooth bed at the same x position. Because of the nonlinear relation between velocity and advected momentum, equation image does not equal equation image. The redistribution of water flow, resulting from the starting compound roughness, is therefore not merely a redistribution of longitudinal momentum. As a result of the quadratic relation between longitudinal momentum and longitudinal velocity in (3), the overall longitudinal momentum is increased when the longitudinal velocities adapt to the distinct roughness. A locally steeper slope of the free surface is required to satisfy the momentum balance.

[13] The transverse momentum exchange, quantified by the terms DTTM and DTSF in equations (1) and (2), is expected to be most pronounced in the mixing layer and will act to smooth the cross-channel velocity profile. The friction coefficients in a cross-channel profile, however, will vary as abruptly as the roughness changes. The retarding effect the mixing layer has on the flow over the smooth bed and the accelerating effect it has on the flow over the rough bed will increase the channel-averaged bed shear stress compared to a situation where the two flow lanes do not interact. Hence, the mixing layer enhances the total drag imposed on the flow as a whole. Translating this effect to a channel with compound roughness means that mixing layers induced by lateral roughness changes will reduce the conveyance capacity of the channel. The increase of the total drag by momentum exchanges in the mixing layer in the two-lane experiment is a combined result of momentum transfer caused by mass transfer, secondary circulation, and turbulent mixing. The remainder of this section elaborates on each of these aspects of mixing layers.

2.1. Mass Transfer

[14] Except for the region close to the sidewalls, the flow is considered uniform over the width at equation image m. The upstream distance from x = 0 over which the flow is affected by the roughness change will decrease with increasing Froude number (Fr), up to Fr = 1 where the flow becomes supercritical. Our focus is on flow regimes characterized by equation image, as in floodplain flows. By approximation, the longitudinal rate of change of Ms and Mr as a result of mass transfer from the decelerating flow over the rough lane to the accelerating flow over the smooth lane can be calculated according to

equation image
equation image

where V100 is the depth-averaged transverse velocity at the parallel smooth to rough interface, equation image is the cross-section-averaged longitudinal velocity, and L is the downstream distance from x = 0. Appendix A offers derivations of equations (4) and (5), which rely on the following assumptions: (1) the shapes of the vertical profiles of streamwise velocity do not affect the depth-averaged longitudinal momentum; (2) equation image, where U100 is the depth-averaged longitudinal velocity at the parallel smooth to rough interface; (3) there are no systematic changes in sidewall effects along the flow lanes; and (4) the width of the smooth section equals the width of the rough section B.

[15] The gradient in longitudinal momentum averaged over the total cross section can be obtained by summing equations (4) and (5), which yields

equation image

where Mt = Ms + Mr. Equation (6) describes the gain of longitudinal momentum over the total flume width as a result of the velocity redistribution. As in the case of a compound channel with sections of different bed levels [Bousmar et al., 2005], the width of the channel influences the rate of change of the longitudinal momentum related to mass exchange. Equation (6) depends on the fact that the width of the two sections is equal, which limits the generality of the case under study.

2.2. Turbulent Mixing

[16] The velocity gradient in the mixing layer induces vortices in a horizontal plane, quantified by the Reynolds stress equation image, where the primes denote the fluctuations of the local longitudinal velocity (u) and local transverse velocity (v), and an overbar denotes averaging over an interval that encompasses several times the period of the largest turbulent motions. This turbulent mixing transports momentum from the smooth to the rough side of the channel. In a depth-averaged approach, the transverse momentum exchange by turbulent mixing is defined by

equation image

2.3. Secondary Flow

[17] Studerus [1982] and Wang and Cheng [2006] observed a secondary current of the second kind in flow over a bed with longitudinal smooth and rough strips. These secondary currents had an upward velocity above the smooth bed and a downward velocity above the rough bed. The secondary flow can cause a depth-averaged momentum transfer in the transverse direction. The transverse component of the secondary flow transports high-momentum water from the smooth side to the rough side in the mixing layer and low-momentum water to the smooth side. This exchange can be quantified by

equation image

Studerus [1982] expected this mechanism for transverse momentum exchange to be more efficient than that of turbulent mixing, without providing quantitative arguments.

[18] Having identified the three mechanisms of momentum transfer, an experiment was set up where all mechanisms could be measured to provide better insight into the mechanisms controlling the relative importance of each mechanism.

3. Experimental Setup

3.1. Flume Setup

[19] Laboratory experiments were conducted in a horizontal (i.e., nontilting) flume of the Fluid Mechanics Laboratory at Delft University of Technology. This flume is 2 m wide and has a length of about 30 m (Figure 2). The inflow section of the flume is equipped with flow straighteners and a foam plate to establish homogeneity of the flow and reduce surface waves. The flume is equipped with a weir at the far end to adjust the water level. Looking in the downstream direction, the right half of the bed was covered with a homogeneous layer of stones over a length of 15 m in the central part of the flume. The leading and trailing sections were sufficiently long to avoid interference with the inflow section and the downstream boundary. The stones were characterized by a D50 of 7.6 mm and a D90 of 9.3 mm, where D50 and D90 denote the median grain size and the grain size that is exceeded by 10% of the stones in a sample. The left half of the bed is covered with 6 mm thick polished wooden plates to create a smooth bed. The effective levels of the smooth and rough beds were equal.

Figure 2.

Flume configuration.

3.2. Velocity Measurements

[20] Velocity measurements were taken using an acoustic Doppler velocimeter (ADV), manufactured by Nortek (type Vectrino). The ADV probe is composed of a sound emitter (10 MHz) and four receiver beams, tilted 30° from the vertical to minimize the part near the surface that remained unmeasured. The ADV measurements ranged from 0.045 m beneath the surface to 0.01 m above the bottom. The length of the sampling volume was set to 4.0 mm. The velocity data were recorded at 25 Hz for at least 3 min at every position. To obtain velocities near the surface, the velocity profiles were extrapolated using cubic splines for the vertical and streamwise velocity components and linear splines for the cross-stream velocity components. The ADV was mounted in a frame on a movable carriage.

[21] Acoustic measurements introduce an uncertainty in velocity measurements because of Doppler noise, a finite measurement volume, and limited accuracy in solving the Doppler phase shift [Voulgaris and Trowbridge, 1998]. Voulgaris and Trowbridge [1998] and McLelland and Nicholas [2000] showed that the error in time mean velocity and turbulent shear stress as measured by an ADV is very small (within 1%) and close to values derived by laser Doppler velocimetry. In the present experiments, four ADV receivers were available, whereas three beams are sufficient to resolve the velocity vector. The information from the redundant beam can be used for error estimation, which is also common in research with acoustic Doppler current profilers [Lu and Lueck, 1999]. The average magnitude of the error established accordingly is 3.5 × 10−3 m s−1 for the individual samples. This is slightly larger than the value mentioned in the instrument specifications, which claim an accuracy of 1 × 10−3 m s−1 plus 0.5% of the measured value. Hence, the time mean values, calculated over 4500 samples, have a negligible sampling error. The sampling errors in the covariance can be expected to be similarly small [Voulgaris and Trowbridge, 1998].

[22] Another source of error can be misalignment of the ADV [Roy et al., 1996]. The pitch, roll, and heave angles are calibrated by defining the x direction according to the direction of the time- and space-averaged velocity vector in a cross section with uniform flow. The alignment of the ADV was calibrated for all data sets and differed at most by 0.3° between different cases. Adopting an angle variation of 0.3°, an imprecision of the velocity data can be estimated. This angle variation implies an imprecision of 0.4% in the time-averaged longitudinal velocity, and the Reynolds shear stress equation image may be biased by 4 × 10−3 Pa (7% of the average magnitude).

3.3. Experimental Cases

[23] The range of settings of the experiments was restricted by the requirements of (1) a nearly developed mixing layer in the downstream part of the 30 m long flume, (2) the condition of Re > 4000 to ensure a fully turbulent flow, (3) Froude numbers smaller than around 0.35 to suppress surface waves, and (4) the water depth being an order of magnitude smaller than the width of the flume to ensure that the mixing layers were unconfined. Given these restrictions, four experimental settings were chosen and are listed in Table 1. Depths were between 0.08 and 0.22 m, and discharges were 0.040 or 0.100 m3 s−1. The Reynolds numbers were about Re = 18 × 103 for the two shallow cases and about Re = 40 ×103 for the two deepest cases.

Table 1. List of Experiment Cases With Corresponding Settings
CaseMethodDischarge (m3 s−1)Water Depth (m)equation image
E08Flume experiment0.0400.08117 × 103
E11Flume experiment0.0400.11018 × 103
E15Flume experiment0.1000.15141 × 103
E22Flume experiment0.1000.22239 × 103

3.4. Water Level Slopes

[24] The longitudinal section over which mixing layer development can be analyzed is restricted by nonuniformity at the inflow boundary and backwater effects from the weir at the downstream boundary. The inflow was equipped with flow straighteners and a flat plate on the surface to homogenize the flow. The split bed started 11.5 m behind this plate to obtain uniform flow at the start. Water level measurements show a backwater effect starting at x =15 m, about 6 m before the end of the flume. Hence, the region between x = −1.5 and 12.5 m can be considered free from effects of nonuniformity at the inflow and backwater effects.

[25] Water level measurements were performed by using an electronic water level gauge (manufactured by Delft Hydraulics Laboratory) at y = 1.75 m that could be moved in a streamwise direction. Above the smooth bed section, measurements with a manual gauge were performed, which only differ from the measurements at the rough side around x = 0 m. Water level measurements are used for calculating the overall bed friction. The discharge was measured by an acoustic discharge meter (Endress+Hauser Prosonic Flow 91W) for that purpose. The error in the overall bed friction is estimated to be 5%, using the specifications of the devices and confidence intervals of the fitted water level slopes.

[26] Since the flume bed is horizontal and the water level is sloping, the water depth slowly decreases in the downstream direction. Consequently, a fully developed flow cannot be reached, even if the length of the measuring section were to be much longer. The results presented herein focus on mixing layer development and on the nearly developed flow.

[27] A minor part of the flow variation can be attributed to the flow accelerations resulting from the decrease in water depth. The water level profile is relatively steepest in cases E08 and E15. These water levels, as measured by the electronic gauge, are shown in Figure 3. The water depth that is used in the calculations and in Table 1 refers to the water depth in the middle of the flume (x = 6 m). The water depth differs by at most 5% with respect to the value at x = 6 m. The same holds for the change in U since equation imageU/equation imagex is by approximation (for small changes in h, in a steady state) proportional to equation imageh/equation imagex:

equation image
Figure 3.

Water level profiles at y = 1.75 m for cases (left) E08 and (right) E15.

[28] Given the low Froude numbers in the experiments, this justifies considering U and h constant over the relatively short experimental section.

4. Results

4.1. Streamwise Development of Velocity Profiles

[29] Downstream of the start of the rough lane a mixing layer develops in each of the experiments (Figure 4). The development is slowest and extends over larger longitudinal distances for the deeper case. This can be explained by the fact that the development of a boundary layer downstream of a roughness change starts at the bottom [Antonia and Luxton, 1971] and requires a length proportional to the depth. In shallower water the velocity difference develops over a shorter distance, and the mixing layer appears more pronounced. Figure 5 shows the development of the depth-averaged velocities at the center of the rough-bottomed section (y = 0.50 m) and at the center of the smooth-bottomed section (y = 1.50 m), which are referred to as U50 and U150, respectively. The longitudinal velocities do not attain their asymptotic values over the 12.5 m length under study, and the velocity difference is still increasing between x = 6 m and x = 12.5 m. Hence, the mixing layer cannot be assumed to be completely developed even after a length of one hundred times the water depth.

Figure 4.

Depth mean velocity profiles across the mixing layer in the four flume experiments. Streamwise velocity is divided by the mean upstream velocity at x = −1.5 m.

Figure 5.

Downstream development of the depth-averaged longitudinal velocities measured at y = 0.5 m and y = 1.5 m.

[30] The transverse velocity at the smooth-to-rough interface (y = 1 m), denoted by V100 (x), is larger for shallower cases when scaled by the mean streamwise velocity equation image (Figure 6). The amount of water that has to be transported laterally is proportional to the width B of the sections and to the streamwise velocity difference U150U50. When scaled by the velocity difference U150U50, the profiles for V100 (x) collapse on a single curve (Figure 6b). In contrast to the findings for the streamwise velocities of the mixing layer, the development of the transverse exchange appears largely independent of the water depth.

Figure 6.

Depth-averaged transverse velocity at the smooth-to-rough interface, (a) scaled by equation image and (b) scaled by U150U50.

4.2. Momentum Exchange Between Two Flows in Developing Flow

[31] The gain in longitudinal momentum due to the flow redistribution in the individual smooth and rough lanes is dominated by the term equation image in equations (4) and (5). Consequently, the individual profiles of equation image and equation image are expected to follow the same pattern as that shown in Figure 6. The gradient in longitudinal momentum for the total flume width, as described in equation (6), does not include this term. The profile of equation image is shown in Figure 7, where equation image is scaled by the pressure gradient term equation image, defined by

equation image
Figure 7.

Longitudinal momentum gradient, scaled by equation image.

[32] Values of equation image are dependent on the local cross-interface flow, represented by V100, as well as the cumulative upstream flow across the interface, represented by the integral in equation (6). Therefore, its maximum value occurs between x = 1 m and x = 2.5 m, whereas the cross-interface velocity peaks slightly before or at x = 0 m. The relative momentum increase due to flow redistribution appears largest for the shallowest case E08, and a weak trend is visible of decreasing values with increasing water depth. In this dimensionless form the magnitude of V100 (x) and its downstream distribution accounts for the observed variations.

[33] Figure 8 presents the midchannel development of momentum transfer by horizontal turbulent mixing across the smooth-to-rough interface equation image. The magnitude gradually increases from 0% to around 2.5% for all but the shallowest case, labeled E08. The further increase over the interval x = 6.0 m to x = 12.5 m emphasizes, together with earlier observations of the development of velocity profiles and streamwise momentum, that the system needs a much larger downstream distance to adapt to the new boundary conditions over the split bed configuration than expected on the basis of a transversely uniform roughness distribution.

Figure 8.

Development of equation image at y = 1 m.

4.3. Momentum Exchange Between Nearly Developed Flows

[34] Although the flow appeared not to be fully developed at x = 12.5 m, the processes governing the momentum exchange are expected to be indicative for developed conditions. Figure 9 presents the secondary flow patterns observed in the flume experiments at x = 12.5 m. A secondary flow driven by lateral difference in bed roughness was observed earlier by, e.g., Studerus [1982] and Tominaga and Nezu [1991], associated with stresses due to turbulence anisotropy at the smooth-to-rough interface [Nezu and Nakagawa, 1993]. Figure 9 shows a secondary flow, featuring a main circulation cell that is asymmetrical about the smooth-to-rough interface (y = 1 m). The center of this main circulation cell in the mixing layer is located at the smooth side. An analysis of this observation is given by Hinze [1967], who attributes the secondary circulation to the local inequality of production and dissipation of turbulent kinetic energy. In the shallow, largely unidirectional flow under study, the turbulence energy balance equation can be simplified by applying the boundary layer approximation and neglecting small terms, yielding

equation image

where u = (u, v, w) is the velocity vector and equation image is turbulence energy dissipation. At the rough side of the interface, turbulence production exceeds the dissipation, resulting in a positive value for the sum of the terms at the right-hand side of equation (11). It can be assumed that

equation image

and that

equation image
Figure 9.

Secondary circulation pattern for all four cases, at x = 12.5 m. Black vectors are measured, and gray vectors are extrapolated. The background is shaded according to the longitudinal velocity u (m s−1).

[35] This implies that equation image above the rough bed and equation image above the smooth bed, forming a circulation cell. Zooming in to the smooth-to-rough interface, the rough bed stimulates turbulence production rather than the smooth bed stimulating turbulence dissipation. Hence, the excess in turbulence production at the rough side can be expected to be more concentrated close to the smooth-to-rough interface compared to the excess in turbulence dissipation, which is spread more over the smooth side. The pronounced excess in turbulence production close to the interface causes downwelling in this region, in agreement with the observations in Figure 9.

[36] Figure 10 shows transverse profiles of the depth-integrated interfacial stress DTSF associated with secondary circulation and scaled with equation image, at x = 12.5 m. The highest contribution of the secondary circulation to the longitudinal momentum balance is found between y = 0.98 m and y = 1.15 m, which coincides with the location of the main circulation cell in Figure 9. For cases with a larger water depth, the local maximum in equation image above the smooth lane increases, and moves away from the smooth-to-rough interface.

Figure 10.

Transverse profiles of DTSF at x = 12.5 m, scaled by equation image.

[37] Zooming in on the details of the secondary flow pattern observed in the laboratory experiments in the near-wall region at y = 0.96 m and y = 0.98 m (Figure 9), the near-bed velocity is relatively strong. The results suggest the presence of a small counterrotating circulation cell adjacent to, and possibly driven by, the main circulation cell. The spatial resolution of the ADV measurements is insufficient to obtain a clear view on this small-scale circulation cell. The dips at y = 0.96 m and peaks at y = 0.98 m in Figure 10 are also an indication of the presence of a counterrotating microcell, contributing to the momentum transfer locally.

[38] Figure 11 shows transverse profiles of equation image. The highest values of equation image can be found around y = 1 m and y = 0.92 m. Low values occur at the smooth side of the interface region. The profiles become more asymmetric for cases characterized by larger water depths. The distribution of the Reynolds stress tensor component equation image, capturing momentum exchange by horizontal vortices, is shown in Figure 12 for midchannel sections across the interface. The maximum Reynolds shear stresses are located above the smooth-to-rough interface, with a band tilting upward toward the rough side, which may be caused by interaction of the vortices with the secondary circulation. Low values of equation image, and even negative values, can be found at the smooth side of the mixing layer, around y = 1.10 m. The transverse momentum exchange due to the secondary flow is strongest in this region (Figure 9), reducing the cross-channel gradient in longitudinal velocity. The secondary circulation cell may block the near-bottom momentum exchange by turbulent mixing between the parallel channel sections by disrupting the near-bed development of coherent vortex structures.

Figure 11.

Transverse profiles of DTTM at x = 12.5 m, scaled by equation image.

Figure 12.

Distribution of equation image at x = 12.5 m, quantifying momentum exchange by vortices acting in the horizontal plane.

[39] Comparing equation image (Figure 10) and equation image (Figure 11) at y = 1 m, the contributions of secondary circulations and turbulent mixing have the same order of magnitude, just as the peak values of equation image in the developing region do. In more detail, the ratio between the contributions of secondary circulations and turbulent mixing at y = 1 m depends on the water depth, as shown in Figure 13. For the case with the smallest depth, TTM is slightly larger than TSF according to the ADV measurements. For larger water depths, TSF becomes significantly larger than TTM, up to about 50% in cases E15 and E22. The ratio between TSF and TTM is expected to further increase up to fully developed conditions of the secondary circulation. This is especially relevant for experiment E22, where conditions at x = 12.5 m cannot be considered fully developed.

Figure 13.

Ratio of TSF to TTM deduced from ADV measurements at y = 1 m and x = 12.5 m. The data are connected with spline fits (dashed lines).

5. Conclusions

[40] Three mechanisms control the momentum exchange between two lanes of different bed roughness, namely, mass transfer from the decelerating flow over the rough-bottomed lane to the accelerating flow over the smooth-bottomed lane, secondary circulations driven by turbulence anisotropy, and vortices moving in the horizontal plane. As a result of the associated momentum exchange, a mixing layer is established at the transition of the parallel flows over the smooth and rough beds, featuring stronger horizontal shears for shallower cases.

[41] The longitudinal gradient in momentum, associated with transverse depth-averaged mass transfer, is roughly proportional to the depth and to the depth mean transverse velocity. It was shown that the depth mean transverse velocity at the smooth-to-rough interface, when scaled by the difference in velocity along the centerlines through the two lanes, is independent of depth or discharge and reduces by approximately 90% over the 12.5 m range of measurements. The mass transfer does not merely imply a redistribution of momentum but also results in an overall increase of momentum, which is associated with a steeper free surface slope.

[42] At larger depths, the mixing layer at the smooth-to-rough interface was not fully developed in the experimental cases. Nevertheless, a secondary current established, visible in the cross section perpendicular to the main flow direction. The circulation cell was observed with its center above the smooth lane side of the mixing layer. The momentum transfer from the smooth-bottomed lane to the rough-bottomed lane by secondary circulation increases with depth, largely independent of the discharge. Close to the smooth-to-rough interface, an anomalous circulation pattern was observed from ADV measurements, which suggests the presence of a counterrotating microcell close to the bed. This cell seems to impact the Reynolds stress patterns, possibly confining the spatial extent of vortex structures.

[43] Momentum transfer by vortices acting in the horizontal plane is inhomogeneous over the depth, which can partly be attributed to the interlinked mechanisms underlying secondary circulations. The vortex-induced momentum transfer is concentrated near the bed at the smooth-to-rough interface.

[44] The momentum exchange contributions from the secondary circulation and from mixing layer vortices are, in the nearly developed mixing layer, on the same order of magnitude and so is the peak in momentum gradient associated with transverse depth-averaged mass transfer in the developing mixing layer. The contribution to momentum exchange by the secondary flow exceeds the contribution by the mixing layer vortices as the water depth increases.

Appendix A

[45] Aiming to quantify the rate of change of streamwise momentum integrated over the cross section of a split channel as sketched in Figure 1, we consider the two control volumes (CV1 and CV2 as defined in Figure A1). The control volumes extend in the longitudinal direction from equation image (where flow is uniform) up to x = L and in the transverse direction over the full width of either of the two flow lanes. The respective mass balances for CV1 and CV2 read

equation image
equation image

where equation image and equation image denote averaging over the width of the smooth and rough parts of the cross section, respectively. The gain or loss of momentum with respect to the upstream boundary Mi, integrated over the downstream cross section of CV1 can be obtained from

equation image

which assumes that equation image over the full length of the mixing layer. For the second line, use is made of mass conservation. Analogously, a momentum balance for CV2 can be formulated, which reads

equation image
Figure A1.

Definition of control volumes for which a mass and a momentum balance are set up. Dashed lines border the control volumes; arrows indicate the mass and momentum fluxes.

[46] Differentiating equations (A3) and (A4) with respect to x yields

equation image
equation image

and consequently, for the rate of change of streamwise momentum integrated over the entire cross section,

equation image


[47] The Rijkswaterstaat Water Service of the Directorate General for Public Works and Water Management in the Netherlands provided financial support for the present study. Arjan Sieben (Rijkswaterstaat Water Service) is acknowledged for fruitful discussions during this research.