Velocity profiles and the structure of turbulence at the outer bank of a compound meander bend

Geomorphology 295 (2017) 191–201

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Velocity profiles and the structure of turbulence at the outer bank of a compound meander bend

MARK

Frank L. Engel⁎, Bruce L. Rhoads Department of Geography and Geographic Information Science, University of Illinois, Urbana, IL 61801, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Meandering Turbulence Secondary flow

Few studies have quantified near-bank turbulence at the field-scale in meander bends. As a result, details of the structure of turbulence at the outer bank of bends are poorly understood, despite recognized linkages among turbulence, bank erosion, and channel migration. This study uses high-frequency measurements of flow velocities to analyze the characteristics of turbulence in close proximity to the outer bank of an actively migrating compound meander bend. Results show that the structure of turbulence in the bend is linked to curvatureinduced effects through the progressive advection of high momentum fluid toward the outer bank as flow moves through the bend. Vertical profiles of streamwise-vertical Reynolds stresses near the outer bank differ considerably from those in wide straight channels because of the effects both of curvature-induced helical motion and of local frictional effects associated with the complex bank morphology. The results of the study provide the basis for a conceptual model of the structure of outer bank turbulence in this meander bend.

1. Introduction Lowland alluvial rivers tend to meander within their floodplains, eroding the outer banks of curving channels (Parker et al., 2011; Engel and Rhoads, 2012). Over the past three decades, the fluvial dynamics of meander bends, including processes related to outer bank erosion, have been the focus of numerous scientific investigations. Despite this effort, a comprehensive understanding of the hydraulic processes of outer bank erosion—especially the role of flow turbulence—remains elusive. Because natural river flows are fully turbulent, the erosive stresses acting on the banks of meandering rivers should be related to turbulent stresses. Current models of outer bank erosion, however, rely on simple parameterization of the flow via excess velocity, excess shear stress at the bank toe, or excess flow depth (Darby and Thorne, 1996; Simon et al., 2000; Langendoen and Simon, 2008; Blanckaert et al., 2012; Motta et al., 2012, 2014). These simplified models do not account directly for the effects of turbulence, which impact bank erosion through bed-material transport at the toe of the bank and through direct shear forces acting on the face of the bank. An enhanced understanding of turbulence near outer banks will help to refine representation of shear stresses and mechanisms of bank erosion in morphodynamic models of meandering rivers. It also is of practical importance. Between 1990 and 2003, over 4000 bank stabilization projects occurred in the United States at a cost of over US$0.5 billion (Bernhardt et al., 2005). Motivations for bank stabilization projects include maintaining a navigable

⁎

channel, protection of property, reduction of fertile soil loss, and increasing flood capacity (Odgaard, 1984; Palmer et al., 2005; Lave et al., 2010). This paper uses unprecedented measurements of near-bank velocities to analyze the characteristics of turbulence close to the outer bank of a compound meander loop along a small meandering river. Based on this analysis, a conceptual model of the structure of turbulence near the outer bank of this bend is developed. The specific objectives of this paper are to (i) determine the structure of turbulence at the outer bank of an actively migrating meander bend, (ii) evaluate the spatial variability of turbulence characteristics in relationship to the curvature-induced mean flow field, and (iii) develop a conceptual model of the nature of turbulence at the outer bank of a meander bends. Currently, a coherent conceptual framework describing the structure of outer bank turbulence in bends does not exist. 2. Turbulence in meandering rivers Natural channel flows typically are fully developed turbulent flows, and thus turbulent fluctuations (and turbulent stresses) are a primary means for redistributing momentum throughout the fluid (Nezu and Nakagawa, 1993). Furthermore, fluid turbulence plays a critical role in the transport and dispersal of sediment, nutrients, and pollutants in the flow (Jamieson et al., 2010; Paiement-Paradis et al., 2010). Meanders produce highly three-dimensional mean flow characteristics through

Corresponding author at: US Geological Survey, Texas Water Science Center, San Antonio, TX 78249, USA. E-mail address: [email protected] (F.L. Engel).

http://dx.doi.org/10.1016/j.geomorph.2017.06.018 Received 18 March 2017; Received in revised form 23 June 2017; Accepted 30 June 2017 Available online 02 July 2017 0169-555X/ Published by Elsevier B.V.

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F.L. Engel, B.L. Rhoads

protecting the outer bank from high momentum flow. Although, Blanckaert et al. (2012) noted that the SOC advects high-momentum near-surface fluid downward toward the base of the outer bank, the net effect of cell development is a reduction in the magnitude of turbulence in the outer bank region. The development of SOCs is especially pronounced in sharp bends and may contribute to the low migration rates of such bends by protecting the outer bank from erosion (Blanckaert, 2011). Other experimental work in curved channels with mobile beds has reported increased outer bank turbulence, in part owing to the interaction of the mean flow with a fully developed bed morphology that topographically forces high momentum fluid into the outer bank region (Abad and Garcia, 2009a, 2009b; Jamieson et al., 2010). The few existing field studies of turbulence in meandering rivers indicate that levels of turbulent kinetic energy k near the outer bank are relatively high because of displacement of the high-velocity core toward the base of the outer bank through advective transport of momentum by helical motion, producing strong velocity gradients near the channel boundary (Anwar, 1986; Engel and Rhoads, 2012; Sukhodolov, 2012). Sukhodolov (2012) noted the presence of a SOC in at least one of his measurement transects while also reporting increased k magnitudes at the bank toe relative to the thalweg. This finding suggests that, contrary to assertions by Blanckaert et al. (2012), high levels of near-bank turbulence can occur even when SOCs develop. Moreover, factors other than SOCs, such as the presence of blocks of failed bank material (Engel and Rhoads, 2012), may impact the strength and structure of turbulence at the outer bank. Further work clearly is needed to determine the structure of near-bank turbulence in meandering rivers.

Fig. 1. Coordinate system definition sketch.

curvature-induced secondary circulation, and therefore the structure and patterns of turbulence and momentum transfer should be different in meandering channels than in straight channels (Blanckaert and de Vriend, 2005). A curved channel can be described by an intrinsic coordinate system with streamwise s, cross-stream n, and vertical z axes (Fig. 1). Whereas in straight channels, the streamwise component of flow dominates the transfer of momentum in the fluid (and by extension, the turbulent stresses), curvature-induced secondary circulation in bends leads to substantial lateral and vertical advection of momentum by the mean flow. This advection may also enhance cross-stream and vertical turbulent shear stresses—including near the channel bed and outer banks where material can be eroded. Table 1 summarizes the findings of relevant laboratory and field experiments that have measured turbulence characteristics in the outer-bank region of curved channels. Existing approaches have examined turbulent characteristics of steady flow in curved flumes with flat-beds, fixed channel characteristics, and bank roughness with mobile sand beds in equilibrium configurations (Blanckaert and Graf, 2001; Abad and Garcia, 2009a; Jamieson et al., 2010; Blanckaert et al., 2012). Of these studies, only Blanckaert and Graf (2001) and Blanckaert et al. (2012) focused on the turbulent characteristics near the outer bank. Abad and Garcia (2009a) and Jamieson et al. (2010) mention patterns of flow structure near the outer bank, but this region was not the focus of their research. Field studies that have explored the characteristics of turbulence in meander bends, especially turbulence near the outer bank, are scarce. Anwar (1986) used electromagnetic current meters (ECMs) to investigate turbulence through a simple bend in the United Kingdom. Additionally, Engel and Rhoads (2012) investigated the patterns of turbulent kinetic energy and planform evolution in a small compound meander loop, but included only a qualitative analysis of near bank shear stresses and outer bank erosion. More recently, Sukhodolov (2012) examined the turbulent structure in a lowland meander bend: however, analysis of patterns of turbulence was restricted to the central region of the channel. The development of a comprehensive understanding of outer bank turbulence in bends is complicated by contradictory findings from experimental studies. On the one hand, some work has shown that turbulent kinetic energy, turbulence intensities, and absolute magnitudes of Reynolds stresses at the outer bank are low relative to magnitudes of these metrics within the adjacent thalweg flow (Blanckaert and Graf, 2001; Blanckaert et al., 2012). Low levels of turbulence near the outer bank have been attributed to the development of a weak outer bank cell of secondary circulation, referred to as a secondary outer bank cell (SOC). This cell, which rotates counter to the main curvature-induced helical cell, develops in a near-bank region with lower streamwise velocities than in the channel thalweg (Bathurst et al., 1979; Markham and Thorne, 1992). Formation of the cell is induced by the effects of roughness of a near-vertical bank on the adjacent flow, which produces reversals in the gradient of streamwise velocity near the water surface along the outer bank. Increases in bank roughness result in expansion of the SOC, which acts as a buffer between the high momentum flow in the main channel and the low momentum flow in the outer bank region,

3. Study area and methods Field measurements of near-bank turbulence were obtained at five cross-sections within an elongated meander loop along Sugar Creek in McLean County, Illinois, USA (Fig. 2). The study bend has a bankfull width of 20–23 m, a bankfull depth of 2–2.5 m, and an average gradient of 0.001 m m− 1. At the time of flow measurements, the flow width was 11.2–13.7 m and the flow depth was 1.4–1.2 m at the measurement cross-sections. The outer bank of the bend is nearly vertical, unvegetated, and consists mainly of basal lag gravels overlain by fine sands and loam soil textures; whereas the inner bank consists of relict point bar deposits and exhibits a typical spatial succession of grasses, reeds, shrubs, and trees with increasing distance from the channel. To characterize material within the vertical outer banks, bulk sediment samples were collected at several locations throughout the study bend. The sampling strategy was to obtain a representative sample at each of the main stratigraphic layers in the cut bank. Sediment samples were dry-sieved in the lab, and grain size distributions by weight were constructed for each sample. Grain size distributions of bank materials finer than 0.063 mm were determined through pipette analysis. The resulting analysis yielded the percent sand, silt, and clay within each bank sediment sample. Past work at the study bend has examined the mean flow structure through the bends for three different flow stages (Engel and Rhoads, 2015). In this paper, time-averaged velocity data for campaign two (26 May 2011)—a moderate flow event—are considered to provide a context for detailed measurements of turbulent characteristics in the outer bank region, which were acquired on the same date. The time-averaged velocity data were obtained with a broadband 1200 kHz Workhorse Rio Grande acoustic Doppler current profiler (ADCP), with a sampling frequency of ~1 Hz, manufactured by Teledyne RD Instruments. Measurements were conducted at cross-sections oriented perpendicular to the local channel direction (Fig. 2). To acquire the measurements, operators on each bank pulled the ADCP, mounted on a catamaran, along the cross-sections using a nylon rope. For each measurement transect, four passes across the channel were completed, ensuring that passes were in reciprocal pairs, where a pair consists of one pass starting on the left bank and another starting on the right bank. Measured 192

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Table 1 Summary of existing literature on outer bank turbulence in meandering channels. Author(s)

Experimental setup

Abad and Garcia (2009a)

Lab: 200° asymmetric compound bend (Kinoshita) flume, rectangular smooth channel upstream & downstream skewed experiments

Anwar (1986)

Field: Sand-bedded 35° simple meander bend

Blanckaert et al. (2012)

Lab: 193° curved flume. Rectangular channel, with varied outer bank roughness (smooth, sand, gravel)

Blanckaert and Graf (2001); Blanckaert and de Vriend (2005)

Lab: 120° constant curvature flume with vertical banks sand bed

Blanckaert (2010); van Balen et al., 2010

Lab: 193° curved flume with vertical banks and sand bed

Engel and Rhoads (2012)

Field: Sand & gravel bedded 200° asymmetrical compound meander bend 2 different measurements in the bend 10 years apart

Jamieson et al. (2010)

Lab: 135° constant curvature flume with sand bed. 2 experiments, on developing and equilibrium beds

Sukhodolov (2012)

Field: Sand-bedded 150° compound meander bend

Outer bank turbulence findings 1. The core of high velocities hugs the inner bank, and in general turbulent metrics find their maxima there 2. k is perfectly correlated with local curvature 3. τss stresses are the major contributor to k, except just downstream of apex, where τnn is dominant 4. τsn are the largest shear stress component, with local maxima at the inner and outer banks 1. Normal stresses have their maxima near the water surface at the outer bank, with the highest values at the bend apex 2. Shear stresses have their maxima at the outer bank near the apex, decreasing downstream 3. Maximum outward directed values of τsn stresses occur at the outer bank near the water surface 1. Outer bank velocities and k are lower than in the main channel. Although τnz magnitudes increase slightly near the bank, they are much lower than in the main channel 2. τnn stresses are the main contributor to peak values of k 3. τnz stresses are related to circulation cells, however correlation in the near-bank region is not good as in the main channel 1. Shear stresses are reduced in the outer bank region relative to the main channel 2. τnz stresses are correlated with circulation cells, and are of the same order of magnitude as the other shear stresses; near-bed τnz is associated with the transversal component of the bed stress 3. τsn stresses are near 0 at the water surface nearbank, and increase linearly to the bed 4. τnn and τzz are smaller than τss near the outer bank. τzz is larger than τnn near the outer bank (Blanckaert and de Vriend, 2005, Fig. 9). 1. There is an increase in depth-averaged k near the outer bank between 30° and 90° in the bend (Blanckaert, 2010, Fig. 16). Increased k is attributed to the flow colliding with the outer bank and to additional shear produced by the strong secondary flow. 2. 3D patterns of k observed suggest important role played by advection of momentum by the secondary flow. 1. Submergence of high velocity fluid below the water surface at bend apices increases velocity gradients near the outer bank and bed 2. k magnitudes are greatest near the bank toe, downstream of bend apices 1. τsz stresses near-bank become less dominant as τsn, and τnz magnitudes increase as flow moves through the bend 2. Magnitudes of τsn at the outer bank are high relative to other stresses, and fluctuations are directed into the bank 3. τnz stresses are correlated with circulation cells (see their Figs. 6 and 9c) 1. Shear stress components are of the same magnitude near the outer bank 2. Generally, shear stress magnitudes increase with curvature, with maxima at or near the apices 3. τnz stresses are directed outward strongest at the bend apices (see cross-sections 5-5 & 7-7, Fig. 10). Outward directed τnz in cross-section 5-5 occurs near the bed. At the water surface τnz is directed inward, perhaps associated with an outer bank cell

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Notes and limitations The channel is rectangular, thus flow patterns are not necessarily representative of self-formed channels (with developed pool and point-bar topography)

Anwar uses different sign conventions. Thus, τsn stresses are positive when directed outward (streamwise and lateral contributions having the same sign)

This study does not present all turbulent stress results, and instead focuses on computing advection and vorticity terms for model verification of the SOC1. Also, as a rectangular channel, flow patterns may not be fully representative of those found in self-formed channels.

1. Parameters were measured at only 1 crosssection (60°) 2. Flume geometry had an artificially sharp outer bank/bank toe interface 3. Velocities and turbulence quantities were extrapolated to the bed due to limitations with measuring near the boundaries

1. The experimental study only reports the depth-averaged k. A validated 3D LES numerical model provides the 3D patterns of k. 2. The flume geometry had an artificially sharp outer bank/bank toe interface.

Only k is quantified, and an empirical relationship between k and boundary shear is used to qualitatively evaluate near-bank erosion patterns

Flume geometry had an artificially sharp outer bank/bank toe interface. Only Reynolds stresses and k evaluated near the bed are presented rather than profiles of quantities over depth

There was no discussion of outer bank turbulence characteristics. All interpretation of outer bank stresses have been derived considering the contour plots of turbulent quantities in the paper

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Fig. 2. Location map of the study site within McLean County, Illinois, USA (A). (B) A ground photo taken in the Fall of 2010 at the location indicated by the camera icon in (C), which shows a topographic map of the channel and the locations of survey transects.

velocities were post-processed using the Velocity Mapping Toolbox (Parsons et al., 2013) to visualize the three-dimensional time- and space-averaged flow field at each channel cross-section. Measured beam depths acquired by the ADCP provided information on the submerged morphology of the outer bank. Engel and Rhoads (2015) provide complete information on the experimental setup, instrumentation, and analysis of the time-averaged velocity and bathymetric data. Measurements of three-dimensional velocities ui (i = s, n, z) (Fig. 1) near the outer bank were conducted during campaign 2 using an acoustic Doppler velocimeter (ADV) deployed on a custom mount and scaffold system (Fig. 3). The ADV mount consists of a metal frame with leveling screws that holds a wading rod to which the ADV sensor is attached. Scaffolds were placed on each side of the mount, which was then aligned with the plane of the ADCP transects to allow access to it by an operator. Sets of measurements were collected in lobe 2 at crosssections 2-5 thru 2-9 (Fig. 2). For each set of measurements, ADV data were sampled at a frequency of 25 Hz within an irregular grid consisting of 2 verticals of 3–10 points spaced approximately 1–2 m apart within 1–3 m of the outer bank, producing a grid of 12–40 points. Each vertical sampling position was geolocated using a robotic total station with centimetric accuracy. ADV sampling was performed for a minimum of 60 s at each grid point, with an average sampling time of 81 s. Near-bed points were sampled for an average of 99 s at each location. Statistical analysis of ADV sampling times in rivers shows that 60–90 s is the optimum sampling length to resolve the mean flow velocities, turbulent intensities, and shear stresses while minimizing the standard error of the measurement (Buffin-Bélanger and Roy, 2005). The ADV data were post-processed and filtered using a phase-space threshold despiking method (Goring and Nikora, 2002; Wahl, 2002). The highly turbulent nature of the flow in the outer bank region made the choice of a phase-space filter preferable over more typical filtering methods (e.g., signal to noise ratio or correlation thresholds), which tend to exclude more data than necessary in low correlation environments. Instantaneous velocities ui (i = s, n, z) were Reynolds

Fig. 3. Acoustic Doppler velocimeter (ADV) custom mount system being used to measure instantaneous velocity near the outer bank.

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Fig. 4. Cross-section plots for campaign 2 showing downstream velocity magnitudes (contours) with superimposed cross-stream/vertical vectors (arrows) for each transect where flow was measured in 2011. Thick black arrows indicate the inferred secondary circulation patterns.

ui (i = s, n, z ) and turbulent ui′ (i = s, n, z) comdecomposed into mean − ponents. Using this decomposition, the turbulent kinetic energy k:

k=

4.2. Profiles of mean velocity and turbulence near the outer bank To examine in detail the relation between mean flow and turbulence near the outer bank, vertical profiles of time-averaged streamwise, cross-stream and vertical velocities, Reynolds stresses, and turbulent kinetic energy in the outer bank region were derived from the ADV measurements at each cross-section where instantaneous velocity data were collected (Fig. 5).

− − −

1 ( us′ 2 + un′ 2 + uz′ 2) 2

(1)

and Reynolds shear stresses in kinematic units τij:

τij = −u′i u′j i , j = s , n, z

(2)

4.2.1. Time-averaged velocity In general, time-averaged velocity profiles throughout the outer bank of the study bend indicate that the maximum streamwise velocities − us occur below the free surface, are greatest toward the channel thalweg, and diminish in magnitude close to the outer bank (Fig. 5). Patterns of secondary flow, as indicated by cross-stream − un and vertical − uz profiles and vectors, vary spatially with proximity to the outer bank and downstream through the study bend. At the exit to lobe 1, nearbank flow is directed toward the bank face with magnitudes of − un about 8% of − us (cross-sections 2-5 and 2-6; Fig. 5). Near the thalweg, the effect of curvature-induced secondary circulation is evident with negative values of − un near the surface and positive values of − un near the bed (e.g. cross-section 2-6). At the entrance to lobe 2 (crosssection 2-7, Fig. 5), flow is directed toward the bank face over the entire profile with values of − un about 17% of − us . Vertical velocities throughout the region near the outer bank are relatively small and do not seem to vary systematically. Immediately upstream of the lobe 2 apex, where the form of the outer bank is strongly influenced by a large block of failed bank material (cross-section 2-8; Fig. 5), the velocity data exhibit a much different pattern than elsewhere in the study bend. This difference reflects interaction of the flow with the submerged bank block. Although an attempt was made to measure two velocity profiles at this location, complications associated with the complex topography of, and vegetation growing on, the bank block compromised efforts to obtain measurements close to the bank; thus only one profile near the edge of the block was measured. Streamwise flow − us near the edge of the bank block approximates a logarithmic profile with the highest velocities at uz indicate that no pronounced secondary the surface. Data for − un and − flow occurs at this location. Downstream of the apex (cross-section 2-9; Fig. 5), the highest values of − us are located below the free surface. Patterns of − uz for both profiles indicate that secondary circuun and − lation, with near-surface flow directed outward and near-bed flow directed inward, occurs near the outer bank at this location.

where i , j are index variables indicating the direction component of the velocities were determined for each sample location. The values of the kinematic Reynolds normal stresses were also computed:

τii = −u′i u′i i = s , n, z

(3)

4. Results 4.1. Time-averaged velocity field Patterns of mean flow through the reach, including near the outer bank, provide a context for interpreting patterns of near-bank turbulence derived from the ADV measurements. The general pattern of streamwise velocities is comprised of a core region of high velocities surrounded by diminishing velocities near the channel bed and banks (Fig. 4). The pattern of cross-stream/vertical (un/uz) velocity vectors indicates that a large curvature-induced cell of helical motion exists in the center region of the channel. This cell is characterized by outwarddirected near-surface flow and inward-directed near-bed flow. Curvature-induced secondary circulation is strongest upstream (cross-section 2-7) and immediately downstream (cross-section 2-9) of the lobe 2 apex. Through the bend the core of highest streamwise velocity is advected toward the outer bank and accelerates near the lobe 1 exit (cross-sections 2-5–2-7) into lobe 2 (cross-sections 2-8–2-9). Near the outer bank streamwise velocities are small within the upstream part of the reach (cross-section 2-5) but increase toward the entrance to lobe 2 (cross-section 2-7) as the high velocity core shifts progressively outward. Just upstream of the lobe 2 apex (cross-section 2-8), a large block of failed bank material rests on the outer bank toe. Interaction between the flow and the bank block results in reduced streamwise velocities in the outer bank region both at the location of the bank block (crosssection 2-8) and downstream in the lee of the block (cross-section 2-9). Interpretation of patterns of secondary flow near the outer bank is limited by the relatively shallow depths and lack of velocity data generated by the ADCP near the surface and the bed. At some locations, flow near the outer bank appears to be directed toward this bank over the entire depth (cross-sections 2-5, 2-7), whereas no systematic pattern of secondary motion is evidence near the failed bank material (crosssections 2-8, 2-9).

4.2.2. Reynolds shear stresses Profiles of Reynolds shear stresses provide insight into the fluxes of momentum produced by turbulent fluctuations (Fig. 5). Patterns of τsz through the exit of lobe 1 (cross-sections 2-5 through 2-7) are complex. Generally, in the upper part of the water column values of τsz are either negative or close to zero and are positive near the bed. An exception is 195

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(caption on next page)

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us ), cross-stream (− un ), and Fig. 5. Vertical profiles of velocity and turbulence components with estimated outer bank profile for each of the survey transects. Time-averaged streamwise (− vertical (− uz ) velocity components with cross-stream/vertical vectors (top); Reynolds shear stress components (τsz , τsn , τnz, in kinematic units) (middle); and Reynolds normal stress components with turbulent kinetic energy (τss , τnn , τzz, in kinematic units) (bottom). Vertical black lines denote zero magnitudes for each measured quantity, dark blue lines denote the water surface elevation, and the shaded area denotes estimated bank geometry. Measurements above ~ 8.45 m (0.6 z/h) for the near-bank profile at cross-section 2-5 were not possible because of a local protrusion of the bank that interfered with operation of the ADV at this location. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the profile over the thalweg at cross-section 2-6 (Fig. 5) where values of τsz are positive over the entire flow depth. In several of the profiles, maximum values of τsz occur near the bed. In lobe 2 (cross-sections 2-8 and 2-9), where flow interacts with failed bank material, the pattern of τsz over depth differs from upstream. Here, values of τsz are positive over depth and exhibit somewhat parabolic profiles with small values at the surface, high values in the middle of the flow, and the smallest values near the bed. Overall, profiles of τsn at the exit of lobe 1 (cross-sections 2-5–2-7) are fairly similar. Values of τsn are negative over the entire depth (i.e., time-averaged values of us′ and un′ have the same sign, where n is positive toward the outer bank), indicating that a net lateral flux of streamwise momentum occurs toward the bank face. These values decrease to a minimum (absolute maximum) that corresponds to the location of the τsz maximum. Again, the exception is the vertical farthest from the bank at cross-section 2-6 where the peak negative value of τsn occurs 7 cm beneath the water surface. Below this level, values increase to zero at mid-depth. At cross-section 2-8, τsn is negative over the entire depth, indicating a net turbulent lateral flux of streamwise momentum toward the surface of the failed bank material at this location. Absolute magnitudes of τsn are similar to those of τsz, in essence mirroring the quasi-parabolic distribution of τsz stresses (Fig. 5). Farther downstream (cross-section 2-9), the distribution of τsn is negative over depth; and peak negative values of τsn are located about 5 cm below the water surface. Values increase linearly below that level. The point at which values of τsn nearly reach zero corresponds roughly with the position in un changes from positive (i.e., directed outward) to the profile where − negative (i.e., directed inward). Thus, patterns of secondary circulation, which redistribute high streamwise momentum outward near the surface and inward lower in the flow column, may influence the strength and distribution of τsn at this location. A local peak in negative values of τsn occurs near the bed, indicating that the transverse turbulent flux of streamwise momentum is directed toward the bank face. The magnitudes of τnz are less than the magnitudes of the other two shear stress components throughout the bend, reflecting small values of uw′. Most of the profiles display a consistent pattern with positive values of τnz near the surface or within the upper portion of the flow profile and negative values near the bed or over the lower portion of the flow profile. This pattern provides evidence for differences in the vertical turbulent flux of transverse momentum between upper and lower portions of the flow.

Fig. 6. Vertical profiles of the contribution of each fluctuation component to the total k upstream (A) and downstream (B) of the failed bank material.

Table 2 Depth-averaged contributions of each fluctuation component to the total k.

−

4.2.3. Reynolds normal stresses and turbulent kinetic energy The patterns of Reynolds normal stresses indicate how various components of turbulent kinetic energy k contribute to the turbulent transport of momentum (Fig. 5). In general, the patterns of normal stresses and turbulent kinetic energy are similar throughout the bend. As flow exits lobe 1 and approaches the lobe 2 apex (cross-sections 25–2-7), local maxima of the normal stresses and turbulent kinetic energy are found near the bed in the bottom 20% of the flow (~ 10–15 cm above the bed). This region corresponds to the portion of the velocity profiles where velocity gradients over depth are the largest. As the mean flow accelerates into lobe 2 (cross-sections 2-8 and 2-9), curvature-induced secondary circulation moves progressively closer to the outer bank, the flow interacts with failed bank material, and velocity gradients near the bank increase. As a result, absolute magnitudes of the normal stresses and k increase over the entire flow depth. At these locations the maximum values of normal stresses and k occur near the water surface and decrease markedly near the bed.

−

−

0.5( us′ 2) /k

0.5( un′ 2) /k

0.5( uz′ 2) /k

0.33 0.82 0.51

0.16 0.64 0.32

0.02 0.31 0.16

Upstream of bank block Minimum 0.33 Maximum 0.82 Average 0.52

0.16 0.64 0.31

0.02 0.31 0.17

Downstream of bank block Minimum 0.33 Maximum 0.66 Average 0.49

0.23 0.52 0.35

0.02 0.23 0.16

All cross-sections Minimum Maximum Average

The contribution of fluctuations of each velocity component to the total k can be quantified from Eq. (1) as 0.5( ui′ 2) k (Fig. 6). Contributions of each component do not differ significantly when averaged

−

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over depth anywhere in the study bend (Table 2). Profiles of relative contributions also are similar for the different locations but vary slightly upstream and downstream of the failed bank material (Fig. 6). Upstream of the bank block in the lobe 1 exit (cross-section 2-5–2-7), us′ accounts for as much as 80% of the total k in a region just above the bed (0.3 ≥ z/h ≥ 0.15). Close to the bed (z/h < 0.15), relative contributions of un′ to k increase but remain less than contributions of us′. In the upper middle portion of the water column (0.5 ≥ z/h ≥ 0.8), un′ and uz′ contribute nearly the same proportion to the total k (~ 25% each). At and downstream of the failed bank block (cross-sections 2-8 and 2-9), total contributions by us′ and un′ in the lower portion of the flow column (0.5 ≥ z/h ≥ 0.1) overlap (us′= 33–53%, and un′= 25–45%), whereas in the upper 50% of the flow contributions by us′ to k clearly exceed those of un′.

Table 3 Relative proportion of ensemble-averaged turbulent normal stresses for each cross-section.

− −

− −

XS

〈 un′ 2〉 〈 us′ 2〉

〈 uz′ 2〉 〈 us′ 2〉

2–5 2–6 2–7 2–8 2–9

0.53 0.61 0.72 0.51 0.81

0.24 0.35 0.35 0.33 0.34

and vertical turbulent fluctuations in straight channels is also well established (Nezu and Nakagawa, 1993):

− − − − 〈 uz′ 2〉 〈 us′ 2〉 = 0.31

〈 un′ 2〉 〈 us′ 2〉 = 0.51

5. Discussion The results of this study show that the structure of turbulence near the outer bank of meander bends is complex. This structure does not seem to be strongly related to the pattern of mean flow through the two lobes of the compound bend, which is dominated by curvature-induced secondary circulation in the mid-channel region, convective acceleration of flow in the downstream direction, and shifting of the locus of the core of highest velocities progressively outward over distance downstream. The ADCP velocity data indicate that streamwise velocity magnitudes increase near the outer bank as the core of high-velocity shifts outward, but velocities near the outer bank are not sufficiently resolved by the ADCP measurements to permit accurate interpretations of secondary flow in that region. The detailed three-dimensional measurements of near-bank mean velocities using ADVs confirm that flow near the outer bank is generally directed toward this bank. Patterns of cross-stream velocities closest to the thalweg appear to capture the margins of the curvature-induced helical motion within the center of the channel. Of the verticals closest to the bank face, only the vertical downstream of the failed bank material (cross-section 2-9) exhibits evidence of helical motion; and this pattern of motion is similar to the sense of rotation associated with the large-scale curvature-induced cell. It is unclear whether this pattern of fluid motion is part of the largescale cell or a local effect induced by the failed bank material immediately upstream. In any case, the ADV data do not provide evidence to support the development of a secondary outer bank cell with a pattern of rotation opposite that of the large curvature-induced helical cell, as has been documented in previous studies (Markham and Thorne, 1992; Blanckaert, 2011). Patterns of Reynolds stresses deviate significantly from those expected in straight, fully-developed open channel flows (Fig. 5). In general the results here suggest that the influence of channel banks on the flow leads to substantial deviations from a linear profile of τsz over depth that is characteristic of wide flows in straight open channels (Dingman, 2009) and that has been confirmed through field measurements of flows in such channels (Nikora and Goring, 2000). Less information is available on profiles of τsn and τnz in natural rivers. In a meander bend with well-developed curvature-induced secondary circulation (i.e., outward directed near-surface flow and inward directed near-bed flow), Blanckaert and Graf (2001) found that streamwise fluctuations were the dominant contributor to the total turbulent momentum flux in their experimental bend. In another experimental study, Jamieson et al. (2010) reported that τsz stresses in the straight channel entrance of their flume were double the values of τsn and four times the values of τnz; whereas in the fully developed curved flow, τsn and τnz actually exceeded values of τsz. Near the outer bank of the study bend, values of τsz and τsn are nearly equal in absolute magnitude, suggesting that turbulent stresses directed toward the bank face (τsn) are as important as stresses directly toward the bed (τsz). With regard to Reynolds normal stresses, the relationship between depth-averaged (denoted by angle brackets) streamwise, cross-stream,

(4)

In the study bend lobe 2, these ratios change from upstream to downstream as the high velocity core moves progressively outward through the bend (Table 3). Relatively equal contributions of crossstream and vertical normal stresses to the turbulent kinetic energy (Table 2) may also be indicative of the likely presence of a vertically oriented shear layer in the lee of the bank block (cross-sections 2-8 and 2-9) (Sukhodolov and Rhoads, 2001). The implications of the near-bank turbulent stresses for bank erosion can be considered in the context of bank material properties, which for the two lobes of the study reach consists of three main stratigraphic layers: an upper cohesive layer about 1.75 m thick that contains abundant roots, about 25–30% clay, and 45–50% silt; an underlying sandy layer about 0.5 m thick that contains about 90% sand; and a basal sand and gravel layer about 0.5 m thick that is similar in size characteristics to bed material in the channel thalweg (Fig. 7). The sandy layer below the cohesive material is noncohesive and highly susceptible to erosion by hydraulic action. Velocity data obtained at the lobe 1 exit (cross-section 2-7) provides information on the structure of near-bank turbulence within the reach for clean, nearly vertical banks (Fig. 4). Values of shear stress and turbulent kinetic energy at this location are highest on the lower part of the outer bank consisting of erodible sand (Figs. 5 and 7C). Erosion of the sand removes material from the bottom part of the outer bank, which destabilizes the upper portion of this bank. Once the weight of material in the upper part of the bank exceeds the tensile strength of cohesion, the overlying material fails (Thorne, 1998), resulting in the production of bank blocks (Fig. 7D). The production of such blocks temporarily protects the base of the bank from turbulent stresses. Downstream of the bank block at cross-section 2-9, near-bank shear stresses are highest in the upper 50% of the profile near the outer bank, corresponding to the cohesive portion of the bank. Thus, high values of Reynolds stresses and turbulent kinetic energy near the base of the outer bank, where exposed and not covered by bank blocks, promotes erosion of basal sandy layers, which in turn leads to subsequent failures of cohesive material into the channel. This primary mechanism of bank erosion within the study bend is supported by field observations of bank failures resting on the bank toe (Fig. 7D) along much of the reach (Fig. 2C). The structure of turbulence along the outer bank region in the study bend can be summarized in the form of a conceptual model that separates this structure into: (i) turbulence associated with interaction between curvature-induced flow paths and the channel boundary (Fig. 8A–C), and (ii) turbulence associated with interaction between near-bank flow and bank blocks resting along the toe of the bank (Fig. 8C2). As curvature effects on the flow act to redistribute high momentum outward progressively through a meander bend, the contributions of cross-stream and vertical velocity fluctuations to turbulent kinetic energy increase. Sharp channel curvature within a lobe, such as in lobe 1 of the study bend, can limit the magnitude of near-bank streamwise velocities, restricting the locus of the high-velocity core and 198

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Fig. 7. Annotated photographs of the outer bank taken 7 December 2011 of the study bend showing characteristic bank material profiles in lobe 2 (A) and in lobe 1 (B). Example of hydraulic erosion of sands in lobe 2 (C). Failed bank material with intact grasses resting along the outer bank of lobe 2 (D).

stream flow and the portion of the outer bank above the bank block, leading to the formation of a shear layer, characterized by moderate to high levels of turbulent kinetic energy and turbulence structure that is quasi-two-dimensional over depth. The bank block resting near the lobe 2 apex (cross-section 2-8) represents an obstacle to flow that results in reduced near-bank velocities and relatively small Reynolds stresses. Additionally, nearly equal values of τss and τnn along with relatively low values of τzz suggest that turbulence near the outer bank is essentially quasi-two-dimensional and represents a shear layer between slow flow in the lee of the block and fast flow in the thalweg. The zone of reduced velocities acts as a buffer that protects the vertical bank above the bank block from high momentum fluid and shear stresses. The dimensions of the zone of reduced velocities above the bank blocks is likely dependent on (i) flow stage and (ii) the dimensions of the bank block. This study examined near-bank turbulence at an intermediate flow stage, and it is unclear whether the zone of reduced velocities would extend to the water surface at stages approaching bank-full flow. Farther downstream from the bank block (cross-section 2-9), the high velocity core again becomes submerged near the outer bank, and strong secondary circulation advected momentum toward the outer bank toe, as reflected in the increase in Reynolds stresses and turbulent kinetic energy at this location compared to upstream (cross-section 2-8). The findings in this study support the assertion by Engel and Rhoads (2012) that local influences on flow in bends, such as persistent bank blocks, may impact patterns of bank erosion in compound bends. Although the curvature-induced structure of the outer bank turbulence can serve as a mechanism to enhance bed and bank shear stresses by increasing the turbulent momentum flux toward the bed and bank, the production of large roughness elements through bank erosion, such as bank blocks, can locally attenuate the influence of curvature on nearbank turbulence. Measurement of turbulence quantities in the outer bank region of an actively migrating meander bend is difficult, requiring instrumentation capable of measuring instantaneous velocities with a high degree of accuracy. Normally, ADVs are employed in labs or wadeable streams. In this study, ADVs were used in adverse flow conditions, under a changing hydrograph (Engel and Rhoads, 2015), requiring a fast-paced

helical motion to the middle of the channel. This situation is exemplified by conditions at the exit to lobe 1 in the study bend (crosssection 2-5) where the highest velocities are located near the center of the channel and the profile of near-bank transverse velocities indicates that helical motion does not penetrate close to the bank at this location. Turbulence is generated as the flow encounters resistance arising from the boundary, with the highest values of turbulent shear stress and kinetic energy corresponding to the near-bed and bank toe region. Although shear and normal stresses here are likely to be dominated by streamwise velocity fluctuations, proximity of the near vertical banks produces nonlinear variation of stress profiles over depth (Fig. 8A). At the apex of the downstream lobe (Fig. 8B), curvature-induced helical motion begins to enhance the contributions of cross-stream and vertical velocity fluctuations to the shear stresses and turbulent kinetic energy as high momentum fluid moves close to the outer bank. In the study bend, the high velocity core moves outward through the entrance to lobe 2 (Fig. 4, cross-sections 2-6 and 2-7) and near-bank shear stresses and turbulent kinetic energy increase as high momentum fluid is advected toward the bank (Fig. 5). Downstream of the apex (Fig. 8C), the core of high velocity is fully submerged and proximate to the outer bank. Turbulent fluctuations of cross-stream, and to a lesser degree vertical velocities, contribute substantially to the overall stresses and turbulent kinetic energy. This location should, in the case of vegetation and debris free, near-vertical banks, comprise the highest near-bed and near-bank toe turbulent stresses in the bend as high momentum fluid directly impinges the outer bank. Helical motion of the time-averaged flow submerges the high velocity core, steepening velocity gradients and enhancing Reynolds stresses and turbulent kinetic energy near the bed and bank toe, creating a mechanism for hydraulic erosion of noncohesive material at the base of the outer bank. If bank material above the toe is cohesive, it will be prone to failure (Soar et al., 1998). Flow encountering large roughness elements along the outer bank—like blocks of failed bank material—can disrupt curvature-induced turbulence structure near the bank (Fig. 8C2). The frictional effects of large bank blocks may shift the submerged high velocity core away from the outer bank (Engel and Rhoads, 2012, 2015). In the study bend, such effects created large velocity gradients between the free 199

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Fig. 8. Conceptual model of turbulence at the outer bank of a meander bend upstream of the apex (A), at the apex (B), and downstream of the apex (C). Also shown is the effect of a bank block on the flow turbulence downstream of the apex (C2). Graphs on the right represent idealized vertical profiles of Reynolds shear stresses and turbulent kinetic energy.

6. Conclusion

measurement campaign. As such, the tradeoff between collecting data at many points within a bend over short sampling durations versus at a few points for long sampling durations is a delicate one. To capture adequately the characteristics of turbulence structure, the number of sampling locations in this study is much less than the density normally obtained in a wadeable stream and/or lab situation. Moreover, obtaining measurements near the complex outer bank topography is problematic, especially when submerged obstacles such as bank blocks are present. Measuring flow near the outer bank when banks are actively failing can be dangerous. Despite these limitations, the measurements and analysis presented herein shed important light on the characteristics of turbulence near the outer banks of meander bends and the relation of these characteristics to bank erosion.

This study has examined in the field at an unprecedented level of detail the structure of the turbulence in the outer bank region of a compound meander loop within the context of the general pattern of mean flow through the loop. Results show that the structure of turbulence is linked to curvature-induced effects through the progressive advection of high momentum fluid toward the outer bank as flow moves through successive lobes of the loop. Vertical profiles of streamwise-vertical Reynolds stresses near the outer bank differ considerably from those in wide straight channels owing to the effects of curvature-induced helical motion and of local frictional effects associated with the complex bank morphology. Persistent outward-directed flow near the outer bank leads to peaks of streamwise-transverse Reynolds stresses that are similar in absolute magnitudes to peaks of 200

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vertical-streamwise Reynolds stresses. When the vertical locations of these peaks in Reynolds stresses, along with peaks in turbulent kinetic energy, coincide with the position of noncohesive sandy layers near the bank toe, removal of this sandy material by hydraulic erosion can trigger failure of overlying cohesive materials. Large roughness elements, such as failed bank material, disrupt the curvature-induced pattern of turbulence. Bank blocks reduce flow velocities and turbulent stresses in the immediate lee of the block. Here turbulence becomes quasi-two-dimensional, and magnitudes of the Reynolds stresses are relatively small compared to locations unaffected by bank blocks. Results of this study support the bank-toe protection hypothesis that large roughness elements can protect the outer bank from fluid forces and reduce bank erosion rates (Carson and Kirkby, 1972). Furthermore, the findings support the assertion that the effects of local topographic features on erosion and deposition within compound bends can be more important than reach-scale effects associated with channel curvature (Engel and Rhoads, 2012). Past work has suggested that the presence of an outer bank circulation cell (SOC) can have a protective effect on outer bank turbulent stresses, reducing the overall turbulence at the outer bank (Blanckaert and Graf, 2001; Blanckaert et al., 2013). No outer bank cell was clearly detected in this study through either the ADCP or ADV measurements. Based on the results of this study, it is not possible to evaluate the role of SOCs on nearbank turbulence structure. Logistical difficulties associated with the relatively flashy hydrological regime at the study site, with setup times for ADV measurements, and with the complex topography of the outer bank, limited the number of locations within the flow near the outer bank at which measurements could be obtained and the proximity of the measurements to the outer bank. More extensive sets of near-bank turbulence measurements may be possible in meandering streams with uniform outer banks and sustained, relatively constant flows. Future work examining the turbulent structure in the outer bank of meander bends should focus on accurate determination of the fluid stresses proximate to the boundary. Also, more research is needed to explore the structure of turbulence around failed blocks of bank material, how curvature-induced turbulence is enhanced or attenuated by the development of bank blocks, and how modification of turbulence by bank blocks influences the persistence of these roughness elements. Acknowledgements This research was funded by the National Science Foundation [BCS1003622] and by the Department of Geography and Geographic Information Science at the University of Illinois at Urbana-Champaign. Thanks to Howard Heatherwick for graciously allowing access to his land. References Abad, J.D., Garcia, M.H., 2009a. Experiments in a high-amplitude Kinoshita meandering channel: 1. Implications of bend orientation on mean and turbulent flow structure. Water Resour. Res. 45 (2), W02401. http://dx.doi.org/10.1029/2008WR007016. Abad, J.D., Garcia, M.H., 2009b. Experiments in a high-amplitude Kinoshita meandering channel: 2. Implications of bend orientation on bed morphodynamics. Water Resour. Res. 45 (2), W02402. http://dx.doi.org/10.1029/2008WR007017. Anwar, H.O., 1986. Turbulent structure in a river bend. J. Hydraul. Eng. 112, 657–669. van Balen, W., Blanckaert, K., Uijttewaal, W.S.J., 2010. Analysis of the role of turbulence in curved open-channel flow at different water depths by means of experiments, LES and RANS. J. Turbul. http://dx.doi.org/10.1080/14685241003789404. Bathurst, J.C., Hey, R.D., Thorne, C.R., 1979. Secondary flow and shear stress at river bends. J. Hydraul. Div. 105, 1277–1295. Bernhardt, E.S., Palmer, M.A., Allan, J.D., Alexander, G., Barnas, K., Brooks, S., Carr, J., Clayton, S., Dahm, C., Follstad-Shah, J., Galat, D., Gloss, S., Goodwin, P., Hart, D., Hassett, B., Jenkinson, R., Katz, S., Kondolf, G.M., Lake, P.S., Lave, R., Meyer, J.L., O'Donnell, T.K., Pagano, L., Powell, B., Sudduth, E., 2005. Synthesizing U.S. river restoration efforts. Science 80 (308), 636–637. http://dx.doi.org/10.1126/science. 1109769. Blanckaert, K., 2010. Topographic steering, flow recirculation, velocity redistribution, and bed topography in sharp meander bends. Water Resour. Res. 46, 1–23. http://dx. doi.org/10.1029/2009WR008303. Blanckaert, K., 2011. Hydrodynamic processes in sharp meander bends and their

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