A flume experiment on the effect of channel width on the perturbation and recovery of flow in straight pools and riffles with smooth boundaries

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1 JOURNAL OF GEOPHYSICAL RESEARCH: EARTH SURFACE, VOL. 118, , doi: /jgrf.20133, 2013 A flume experiment on the effect of channel width on the perturbation and recovery of flow in straight pools and riffles with smooth boundaries Bruce MacVicar 1 and Jim Best 2 Received 7 March 2013; revised 4 August 2013; accepted 19 August 2013; published 16 September [1] The scaling relation between channel width and the spacing of macroscale bed forms has long interested earth scientists and engineers. The current paper conceptualizes flow over such macro bed forms using perturbation theory. The objectives are to characterize the response of flow to pressure gradients that occur in convectively accelerating flow and convectively decelerating flow (CDF), as occurs in pools and riffles, and to determine how the response is modified by the width of the channel. Flume experiments are described that use idealized two-dimensional bed forms and an inner movable wall to isolate the effect of channel width. Ultrasonic Doppler velocimetry profilers operating at 40 Hz are used to measure velocity. Results show that the recovery of the shear velocity (u * ) and Coles wake parameter (Π) follows a simple relaxation response toward uniform flow conditions that is insensitive to channel width, while the lateral concentration of flow (Ψ) and the principal Reynolds stress ( u w ) occur as two-stage spreading and relaxation responses that follow a scaling relation on the order of 3 4 times the channel width, or approximately one half of the typical distance between pools. The u w increases during CDF, precisely in the location where mean bed velocity is at a minimum. It thus appears that hydrodynamic recovery from perturbation helps to explain the sensitivity of the scale of macro bed forms in rivers to channel width. Mobile beds and 3-D geometries should be tested to verify how mean flow and turbulent scales evolve as linked aspects of a complex response to perturbation. Citation: MacVicar, B., and J. Best (2013), A flume experiment on the effect of channel width on the perturbation and recovery of flow in straight pools and riffles with smooth boundaries, J. Geophys. Res. Earth Surf., 118, , doi: /jgrf Introduction [2] The macroscale variation of bed topography in rivers has significant implications for our ability to predict roughness, the distributions of velocity and turbulence, and sediment transport. The frequency of macro bed forms such as riffles and pools is thought to scale with the width of the channel [Keller and Melhorn, 1978; Leopold and Wolman, 1957]. Yalin [1971] postulated that their frequency is limited by the size of the largest eddies that can form from geometrical irregularities in the channel. In support of this idea, Yalin and da Silva [2001] described qualitatively the presence of horizontal bursts with a tornado action at their origin that 1 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario, Canada. 2 Ven Te Chow Hydrosystems Laboratory and Departments of Geology, Geography and Geographic Information Science and Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. Corresponding author: B. MacVicar, Department of Civil and Environmental Engineering, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1, Canada. (bmacvicar@uwaterloo.ca) American Geophysical Union. All Rights Reserved /13/ /jgrf generates scour and then deposition downstream, thus leading to formation of alternate bars. In contrast, other research has ascribed no role for turbulence, but rather stressed the importance of flow convergence in the maintenance of macro bed forms [MacWilliams et al., 2006; Sawyer et al., 2010]. This lateral convergence of mean flow vectors over a relatively narrow band of the channel width is thought to route sediment over the channel bar and away from the deep part of the macro bed form. However, given that mean velocity and turbulence are linked components of the governing Navier-Stokes momentum equation, a complete explanation of the scaling relation between channel width and bed form dimensions should consider both aspects of the flow hydrodynamics. Yet few high quality data sets over bed forms exist that could be used to test how mean flow and turbulence over bed forms are influenced by the channel width. [3] If changes in bed slope are conceptualized as perturbations from which flow recovers to some normal self-similar state, then flow over macro bed forms can be viewed as responding to a series of perturbations. In their review of perturbations to boundary layer flows, Smits and Wood [1985] delineate two stages in the recovery process. The first stage, called spreading, can be understood by considering the perturbation generated by a change in the streamwise pressure gradient. While initially confined to the inner flow 1850

2 transitions recovery to CDF recovery to CAF uniform flow uniform flow mean velocity turbulence intensity Figure 1. Conceptualization of the scale of flow recovery due to transitions in bed slope in a pool in relation to the overall bed form length. Increasing water depth results in convectively decelerating flow (CDF) due to an adverse pressure gradient, while decreasing water depth results in convectively accelerating flow (CAF) due to a favorable pressure gradient. Conceptual velocity (solid line) and turbulence intensity (dotted line) profiles from Kironoto and Graf [1995] are shown for comparison. region, the changes to the velocity gradient and turbulence production that result from the perturbation eventually spread over the whole boundary layer. After the effects of the perturbation have reached a given location, i.e., the spreading is complete, the second stage of the recovery process, called relaxation, commences. In this second phase, the distributions of turbulence and mean velocity relax toward those observed in uniform flow. The combination of spreading and relaxation as flow adjusts to changes in streamwise pressure gradient over a bed form should thus produce a series of measurable recovery distances as conceptualized in Figure 1. [4] While the gravitational and frictional forces in the channel are balanced in uniform flow [Nezu and Nakagawa, 1993], this situation is perturbed over bed forms with changing bed slopes (Figure 1) due to an imbalance in the pressure forces on the flow. An increase in flow depth results in convectively decelerating flow (CDF), which requires an adverse (positive) pressure gradient in the direction of the flow. Conversely, a decrease in channel depth results in convectively accelerating flow (CAF), which requires a favorable (negative) pressure gradient. The effect of pressure gradients on the streamwise velocity profile is commonly characterized by the law of the wake [Coles, 1956], with open channel velocity profiles in CAF being more convex in the downstream direction than uniform flow profiles, and CDF profiles being nearly concave [Kironoto and Graf, 1995; Song and Chiew, 2001] (Figure 1). Nonuniform flow also perturbs the local balance between the production and dissipation of turbulent energy [Smits and Wood, 1985], with relatively high levels of turbulence being generated where velocity gradients are steeper than in uniform flow [Boussinesq, 1877]. CDF results in increased turbulence generation and a convex turbulence intensity profile, while CAF results in reduced turbulence generation and a concave profile [Kironoto and Graf, 1995; Song and Chiew, 2001] (Figure 1). Following Yang and Chow [2008] and Thompson [2004], MacVicar and Rennie [2012] showed that CDF and CAF also affect the lateral distribution of flow in an open channel due to their effect on turbulence and the patterns of secondary flow. MacVicar et al. [2013] illustrated the changes to the strength and character of coherent flow structures that are generated over idealized macroscale bed forms. [5] The distance of recovery downstream from a perturbation varies depending on the hydrodynamic parameter being considered. Smits and Wood [1985] used the momentum equation to show that turbulence production in the outer layer responds more slowly to a change in the pressure gradient because the velocity gradient is less in the outer flow. As a result, the velocity profile is shifted without altering the gradient, and turbulence production is not affected immediately. In contrast, in the inner layer, the velocity gradients are stronger and a change in pressure tends to immediately affect the turbulent stresses and velocity gradient. In CDF, Lee and Sung [2008] found that skin friction decreases rapidly and that the balance between the pressure and skin friction recovered more quickly than the shape of the velocity profile. Onitsuka et al. [2009] reported that the velocity profile in CDF required a streamwise distance about 12 times the flow depth to reach an equilibrium state. Lateral perturbations and recovery have been examined by Ettema and Muste [2004] in their study of flow recovery distances downstream of a spur dike in a flatbed channel, as well as by Chen et al. [2012] in their study of flow downstream of vegetation patches. Ettema and Muste [2004] found that the thalweg alignment, which is pushed to one side by the spur dike, requires a longer distance to recover than the length of flow separation behind the dike. Ettema and Muste [2004] reasoned that the increased vorticity downstream of the dike entrains slower moving water into the central part of the channel, which subsequently delays the recovery of a typical flow distribution across the channel. Chen et al. [2012] observed a recovery distance that scaled with the wake region behind the vegetation patch and described how even a small positive velocity through the patch was sufficient to delay the formation of the vortex street where turbulence intensities were too high for sediment deposition to occur. [6] The aim of the current paper is to present further experimental results on the response of open channel flow to perturbations in the form of convective acceleration and deceleration. An innovative laboratory apparatus was designed to measure velocity and turbulence at a high spatial density in a channel with modular bed forms and a variable width. This apparatus was previously used by MacVicar et al. [2013] to investigate coherent flow structures over macro bed forms. The objectives here are to characterize the response of the mean flow and Reynolds stress to convective acceleration and deceleration and to determine how the response is modified by the width of the channel. The scope of the present study is limited to simplified two-dimensional smooth bed forms without mobile sediment. 2. Methods 2.1. Apparatus [7] Experiments were conducted in a 17 m long, 0.6 m wide recirculating flume. The flume walls were made of smooth molded fiberglass with inset acrylic windows, and the channel slope was fixed at m m 1. The bed forms were constructed from PVC sheets that were heated and bent to shape to form uniform and nonuniform modules so that different channel and bed form geometries could be tested. Uniform depth modules were 0.4 m long and were either low (0.025 m) or high (0.085 m) to create deep or shallow uniform sections. The nonuniform depth modules were 0.51 m long and fixed at an angle of 7.2 from the horizontal. This low angle was selected to ensure that permanent flow separation did not occur in the nonuniform sections. The 1851

3 Figure 2. Photos of experimental setup showing (a) low angle transitions, (b) moveable sidewall in flume that allowed channel width (Y) to be varied, and (c) the UDVP probes and holders. nonuniform modules could be turned 180 to create either CAF or CDF sections (Figure 2a). A modular inner wall was also constructed from PVC that followed the bed form profile, and could be moved laterally to vary the width of the channel (Figure 2b). [8] An array of 4 MHz ultrasonic Doppler velocity profilers (UDVPs) manufactured by Metflow was used to measure flow velocities (Figure 2c). UDVPs use measurement principles that are similar to those employed in other acoustic-type instruments such as acoustic Doppler velocimeters, relying on detection of the Doppler shift in sound velocity from neutrally buoyant tracers particles in the flow [Best et al., 2001]. UDVPs are suited to laboratory measurements due to their high sampling frequency (up to 100 Hz), small size (8 mm diameter), which means that they are easily deployed in different configurations and locations, and short blanking distance (~1.7 cm), which means they can measure relatively close to the water surface. UDVPs are a singlebeam instrument that measure velocity vectors in a quasisimultaneous manner in a series of bins along the beam axis, with the recorded velocity vectors oriented in the direction of the beam. Beams cannot be measured simultaneously. [9] In the current experiments, the UDVP parameters were set to record 2 min time series at 40 Hz in sampling bins that were 1 to 2 mm long in the beam direction and less than 5 mm in diameter. In this manner, between 50 and 100 bins were defined along the beam axis in water depths between 6 and 12 cm. Four to 12 profiles were measured across one half of the channel width, and cross sections were spaced between 0.12 and 0.3 m in the streamwise direction, with closer sections being measured over bed forms and wider spacings in the longer uniform sections (Figure 3) Error Analysis [10] The data series error analysis procedure comprised four steps: (i) dephase wrapping of data points; (ii) the detection and replacement of errors due to inadequate seeding; (iii) the detection and replacement of data spikes, following Goring and Nikora [2002]; and (iv) the application of a third-order Butterworth filter to remove all variability above the Nyquist frequency (Figure 4). Some of these errors, such as high-frequency white noise in the data series and the occurrence of occasional data spikes, are similar to what has been observed using other velocimeters, such as electromagnetic current meters and acoustic Doppler velocimeters [MacVicar et al., 2007]. However, no measure of the quality of the return signal is provided with the Metflow software, and an additional algorithm to detect and replace errors where seeding was inadequate was found to be necessary. [11] The seeding error, so-called because preliminary experiments determined that it occurred when levels of seeding material in the flow were low, was characterized by a highly skewed velocity distribution. The error occurred because the seeding material tended to settle out in slack-water areas of the flume, most notably the tailwater tank, and this became more severe with run time. The error could not be removed using a despiking algorithm because the poor data tended to be grouped in time, while spikes are assumed to Figure 3. Measurement locations for runs (a) Y20, (b) Y30, (c) Y40, and (d) Y60. Flow direction is from left to right. Measurement section is 6 m downstream from the flume entrance. 1852

4 and 3, respectively). Reynolds decomposition was then used to calculate the mean (u i ) and fluctuating parts (u i ) of each beam velocity. The mean and variance of each beam velocity were then interpolated through the sampling volume on a three-dimensional grid. Grid spacing was at 1/100 of the measurement section length (0.073 m), 1/20 of the channel width (0.03 m), and 1/50 of the flow depth (variable from 1 to 2 mm depending on local depth). Following Pedocchi and Garcia [2009], the streamwise (u) and vertical (w) mean velocities were calculated at each grid node as u ¼ u 1 u 3 2 sinα (3) w ¼ u 1 þ u 3 2 cosα ¼ u 2 (4) [14] The principal Reynolds stress ( u w ) was calculated as Figure 4. Boxplots of data quality treatment stages: 1 raw data; 2 dewrapped for phase aliasing; 3 zero-skewed data removed; 4 despiked using Nikora-Goring algorithm, and 5 filtered using third-order Butterworth low-pass filter. be randomly distributed [Goring and Nikora, 2002]. The highly skewed data points were identified by first plotting a histogram of the data and finding the mode (m). An initial estimate of the standard deviation (s * ) was then calculated using only the data from the unskewed side of the velocity distribution as follows: where s ¼ ðjj u jmj H Þ2 H H ¼ 1 if jj> u jmj 0 if jj u jmj [12] Data points farther than seven standard deviations away from the mode were assumed to be erroneous. This extreme value was chosen after trial and error tests using repeat measurements with different levels of seeding. Subsequent tests in areas around shear zones with high skewness showed that the selected threshold was sufficient to avoid the misclassification of good data Analysis [13] The streamwise and vertical components of flow and the principal Reynolds stress were calculated from measured beam velocities. Beam velocities (u i ) were recorded at orientations of 30, 0, and 30 to the vertical (components 1, 2, (1) (2) u w ¼ u 2 1 þ u sin 2 α where α is the beam angle measured from the vertical. [15] The shear velocity (u * ) and the Coles wake parameter (Π) were calculated to describe the effect of CDF and CAF on streamwise velocity profiles. The velocity gradient in the inner zone (i.e., close to the channel bed) responds quickly to changes in skin friction as flow accelerates and decelerates while the velocity gradient in the outer zone reflects inertial effects of nonuniform flow. Following MacVicar and Rennie [2012], the inner zone was defined by calculating the best fit for a two-segment u -ln(z) relation, where the breakpoint between the segments was allowed to vary within 0.10 z/z 0.20; u * was calculated by fitting the law-ofthe-wall to measured velocity values in the inner zone. The streamwise velocity profile log law can be written as (5) u þ ¼ 1 κ lnzþ þ A (6) where u + and z + are the normalized time-averaged velocity and height above the bed, respectively, κ is the von Karman constant, and A is a constant of integration. Over smooth beds, u þ ¼ u=u and z + = zu*/ν, where ν is the kinematic viscosity. Both κ (~0.41) and A (~5.3) can be considered universal constants [Nezu and Rodi, 1986]. Π was calculated by fitting the law-of-the-wake to the entire flow depth as follows: u c u u ¼ 1 κ lnzþ þ 2Π π z κ sin2 (7) 2 where u c is the maximum velocity in the profile at elevation z c and Π is the Coles wake parameter. Π is approximately 0 in uniform flows, greater than 0 in CDF, and less than 0 in CAF [Coles, 1956; Kironoto and Graf, 1995; Song and Chiew, 2001]. [16] The lateral distribution of flow was characterized using a specific discharge ratio termed the lateral concentration of flow (Ψ): z c Ψ ¼ q p =q b (8) where q p is the profile-specific discharge measured at a lateral location (y) in the channel and q b is the bulk-specific discharge for the channel. q p is determined as the product of 1853

5 Table 1. Channel Geometry and Flow Parameters for Experimental Runs a Run ID MACVICAR AND BEST: FLOW PERTURBATION AND RECOVERY IN POOLS Geometry Y (m) Z s (m) Y/Z X s (m) X s /Y X d (m) X d /Y X b (m) X b /Y Q (L/s) Re Fr u * o (m/s) Π o Ψ o Y Y Y Y a Channel geometry includes the width (Y), depth over shallow sections (Z s ), length of shallow section (X s ), length of deep section (X d ), and total bed form length including one deep, one shallow, and two transitional sections (X b ). Flow parameters include the discharge (Q), bulk velocity (U b ), flow Reynolds number (Re), Froude number (Fr), shear velocity, wake parameter, and the lateral concentration of flow as measured upstream of the bed form (u * o, Π o, and Ψ o, respectively). Flow U p and Z, where U p is the depth-averaged velocity at y and Z is the corresponding flow depth. q b is determined as Q/Y, where Q is the total discharge and Y is the channel width. By definition, Ψ = 1 if the flow has a uniform lateral distribution. Past work [MacVicar and Rennie, 2012] has found that in a straight channel with a rough bed, Ψ ~ 1.1 in uniform flow (aspect ratio Y/Z = 6.0) but increased to ~1.35 following an increase in flow depth (Y/Z = 3.0 at deepest location), and decreased back to ~1.1 following a decrease in flow depth (Y/Z = 6.0). 3. Experiments [17] Four channel widths were investigated in the current experimental program (Table 1). As described earlier, the experimental apparatus was designed so that the inner wall of the channel could be shifted laterally to change the channel width (Y). Flow depth, represented by the depth in the shallow section (Z s ), the bed form profile geometry, represented by the length of the shallow (X s ) and deep (X d ) subunits, and the overall bed form length (X b ) were held constant. Bed form geometry ratios thus only varied as a function of Y. In all experiments, the total bed form length-width ratio (X b /Y) exceeded the commonly cited relation of X b /Y ~6, which allowed sufficient distance to examine the relevant hydrodynamic effects. A second, shorter bed form was installed downstream of the deep uniform section to create a negative pressure gradient in this area as occurs in a natural pool that was used to roughly scale the experiments [MacVicar and Roy, 2007]. The flow was fully turbulent and was roughly scaled using the Froude number at bankfull flow for a forced riffle pool(fr = 0.59 at bankfull discharge from MacVicar and Roy [2007]. The ratio of flow depths in the deep and shallow uniform sections (~2:1) was also set to match the field example. Shear velocities (u * ), as estimated from near-bed velocity profiles at the channel centerline, were ~0.010 m s 1 for all four runs. The Coles wake parameter was low and in a similar range to that observed in other straight channels [Kironoto and Graf, 1994; Nezu and Rodi, 1986]. The lateral concentration of flow (Ψ) was always greater than 1 and slightly higher in the narrower channels, as expected due to the higher relative significance of the sidewall friction. 4. Results 4.1. Streamwise Velocity Distribution at Channel Centerline [18] The streamwise velocity (u) profile at the channel centerline is perturbed by changes in flow depth over the bed forms. For all channel widths, u is more evenly distributed throughout the flow depth as the bulk velocity (U b ) increases in convectively accelerating flow (CAF), a feature shown in all cases (Figures 5a 5d) by the u contours becoming increasingly vertical between x = 0.5 and 1.0 m. A similar pattern was observed in previous studies of CAF in open channels [Kironoto and Graf, 1995; Song and Chiew, 2001]. In the shallow uniform flow section (x = m), a stronger vertical gradient of u gradually redevelops, as shown by more horizontal velocity contours. As U b decreases in convectively accelerating flow (CDF), the vertical gradient of u is stretched so that there is an increasing area of low, but still positive, u in the lee of the bed form with relatively Figure 5. Centerline mean velocity for runs (a) Y20, (b) Y30, (c) Y40, and (d) Y

6 Figure 6. Lateral concentration of flow (Ψ) calculated as the specific discharge (q) divided by the bulk discharge (Q) for runs (a) Y20, (b) Y30, (c) Y40, and (d) Y60. These figures depict one half the width of the channel. Bed and water surface profiles are shown in the lower inset for reference. high u near the water surface from x = 3.5 to 4.0 m. A similar pattern has also been observed in previous studies of CDF in open channels [Kironoto and Graf, 1995; Song and Chiew, 2001]. Intermittent flow separation may occur near the bed in CDF, although it was not possible to assess this with the current apparatus due to the beam orientation of the UDVP probes. In the deep uniform section (x = 4.0 and 6.8 m), the gradient in u decreases, a trend shown by the u contours becoming angled toward the water surface and the channel. At x = 6.8 m, the distribution of u matches the pattern observed upstream of the bed form at x = 0.5 m, for all channel widths. [19] The vertical distribution of u at the channel centerline is particularly sensitive to the channel width (Y ) within the deep uniform flow section (x = 4.0 to 6.8 m). In this zone, high velocity flow near the water surface decelerates relatively rapidly in the narrower channels (Figures 5a and 5b) in comparison with the wider pools (Figures 5c and 5d). The u = 0.25 m s 1 contour line, for example, extends only to x = 5.25 m in Y20 (Figure 5a), but all the way to x = 6.5 m for Y40 and Y60 (Figure 5c and 5d). The fastest velocities do not occur at the highest measured elevation in the narrower channels as shown by the u =0.20and0.25ms 1 contours between x = 4.5 and 6.0 m in Figures 5a and 5b. This phenomenon whereby the fastest velocities occur below the water surface is known as the velocity dip and is related to weak secondary currents [Nezu and Nakagawa, 1993;Yang et al., 2004] Lateral Distribution of Streamwise Velocity [20] The lateral distribution of flow across the channel width is sensitive to changes in channel depth over the modeled bed forms (Figure 6). Flow enters the test section with lateral concentration (Ψ) values between 0.90 at the channel sidewall and 1.10 at the channel centerline at x = 0.5 m, although instrumentation limitations prevented measurements at y < m (white regions in Figures 6a 6d). In CAF, the flow is more evenly distributed across the width of the channel for all experimental runs, as shown by the Ψ contours that are angled toward the sidewalls and the centerline from x = 0.5 to 1.0 m (Figures 6a 6d). In the shallow uniform section (x = 1.0 to 3.5 m), this trend is reversed so that the contours are angled away from the sidewalls and higher value Ψ contours appear at the channel centerline. In the CDF section, the flow is increasingly concentrated in the center of the channel, as shown by the rapid increase of Ψ at the channel centerline and the steeper angle of the Ψ < 1.0 contours relative to the sidewall of the channel from x = 3.5 to 4.0 m. Ψ tends to peak in the deep uniform section, a trend shown by the appearance and subsequent disappearance of the highest (e.g., Ψ max ~ 1.30 in Y30 and 1.15 in Y60) and lowest values (e.g., Ψ min = 0.80 in Y30 and 0.60 in Y60) near the channel centerline and sidewalls, respectively, between x = 4.0 and 6.8 m. This peaking behavior is characteristic of a combined spreading and relaxation response to perturbation [Smits and Wood, 1985]. The calculated values of Ψ are similar to those reported by MacVicar and Rennie [2012] in a rough-bed experiment. [21] Similar to the vertical distribution of u, the lateral distribution of flow over the bed forms as represented by Ψ is sensitive to channel width, particularly in the deep uniform section (x = 4 to 6.8 m). Through the CAF and shallow uniform sections (x = 0.5 to 3.5 m), Ψ is generally higher in the narrow channels (Figures 6a and 6b) than in the wider channels (Figures 6c and 6d). This result is expected due to the increasing relative roughness of the walls in narrow channels due to the lower aspect ratio. The maximum value of Ψ at the transition from CDF to the deep uniform section occurs at the channel centerline in the narrowest channel (Figure 6a), but about halfway between the sidewall and the channel centerline in the widest channel (Figure 6d). An area of relatively high Ψ values is angled away from the sidewall in the deep uniform section of the widest channel so that the maximum Ψ at the channel centerline occurs farther downstream 1855

7 a) u/u b u/u b u/u b u/u b b) c) d) z (m) z/z x (m) CAF Shallow uniform CDF Deep uniform CAF 0 Figure 7. Streamwise velocity profiles in run Y60 at different vertical and lateral positions (a) y/y = 0.5 (centerline), (b) y/y = 0.35, (c) y/y = 0.20, and (d) y/y = 0.05 (near sidewall). Note that z/z = 0.1 and z/z = 0.8 are not plotted in all locations due to measurement limitations. Bed and water surface profiles are shown in the lower inset for reference. (x = 5 to 6.5 m; Figure 6d). In addition, the spatial extent of the area with Ψ < 1 close to the sidewall is greater when the channel width is larger. The area with Ψ < 1.0 does not continue to expand laterally within the deep uniform section in narrow channels, contrary to the trend in the widest channel. Additionally, the area with Ψ < 1.0 does not contract laterally within the test section in the widest channel (Figure 6d). At the downstream end of the deep uniform section (x ~ 6.8 m), the lateral distributions of flow remain more asymmetric and laterally concentrated than the distributions observed upstream of the bed form (x ~ 0.5 m) for all runs except the narrowest channel Longitudinal Profiles of Streamwise Velocity [22] A set of longitudinal profiles for run Y60 (Figure 7) help visualize the response of the streamwise velocity (u) over the bed forms. These velocity profiles are defined as the relative streamwise velocity (u=u b, where U b is the bulk velocity) versus streamwise position (x) at given relative depth (z/z) and width (y/y). If the flow were to uniformly accelerate and decelerate over the bed form as a function of the flow area, u/u b would appear as a flat line. Instead, as shown for run Y60, u/u b profiles tend to converge during CAF (x = 0.55 to 1.05 m) and diverge during CDF (x = 3.5 to 4.0 m). For example, at the downstream limit of the CAF section, u=u b ~ 1.0 on all the profiles, which means that all the mean velocities are more or less equal throughout the cross section. In contrast, u=u b is as high as 1.65 closer to the water surface and channel centerline at the downstream limit of the CDF section (x = 4.0 m; Figure 7c), but as low as ~0.12 near the bed and at the channel sidewall (Figure 7d). This wide range of mean velocities shows that the flow is highly concentrated in the middle of this section with low velocities near the margins during CDF. [23] In the zones of uniform depth, u=u b tends to diverge following CAF (x = 1.05 to 3.50 m) and converge following CDF (x = 4.0 to 6.8 m). In both cases, the flow tends to return to uniform flow conditions. After reaching a minimum at the end of the CDF, u=u b profiles near the channel bed increase through the deep uniform section. This acceleration occurs across the full width of the channel despite the fact that near the water surface, the flow is still decelerating. Lateral convergence occurs because the deceleration of flow near the water surface occurs more rapidly near the sidewalls. Near the water surface (z/z = 0.6 and 0.8), u=u b reaches a maximum within the CDF zone rather than at the transition to the deep uniform flow. Similar to that observed near the bed, minimum values with u=u b < 1 occur early in the deep uniform flow section and then increase downstream. A velocity dip occurs close to the sidewall at all channel widths and is characterized by a higher u=u b at z/z = 0.60 than at z/z = The slow and accelerating flow near the sidewalls and the bed, combined with fast and decelerating flow away from the sidewalls and the bed, results in a central core of relatively high velocity flow that gradually dissipates. In narrower channels, this high velocity core may be located below the water surface due to the velocity dip at the channel centerline Recovery of Velocity Profile Parameters [24] The shear velocity (u * ) responds rapidly to changes in the pressure gradient. When normalized by the shear velocity measured upstream of the bed form (u * o), u * /u * o at the channel centerline (Figure 8a) increases in shallower flow and decreases when the flow deepens. Downstream of transitions in both the shallow uniform section (x = 1.1 to 3.5) and the deep uniform section (x = 4.0 to 6.8), the recovery of u * /u * o can be characterized as a simple relaxation response because it tends toward an equilibrium value once the perturbation caused by the pressure gradient is removed. The equilibrium value for u * /u * o in the shallow section is approximately twice than that in the deep section due to the higher bulk velocity. In the shallow uniform section following CAF, more variability between the runs is visible, and in some cases u * /u * o appears to peak just downstream of the transition (x ~ 1.3 m), while in other cases u * /u * o appears to peak at the transition to CDF. The shallow flow in this area makes accurate determination of u * /u * o difficult and the observed variability with x and 1856

8 / a) u*/u* o b) c) o ΨΨ z (m) MACVICAR AND BEST: FLOW PERTURBATION AND RECOVERY IN POOLS Y20 Y30 Y40 Y x (m) CAF Shallow uniform CDF Deep uniform CAF Figure 8. Response of mean velocity distribution parameters to flow over the bed form: (a) shear velocity (u * ) normalized by shear velocity upstream of bed form (u * o), (b) Coles wake parameter (Π), and (c) lateral flow concentration (Ψ) normalized by lateral flow concentration upstream of bed form (Ψ o ). See Table 1 for values measured upstream of bed form. Solid lines have been added at u * /u * o =1,Π = 0.20, and Ψ/Ψ o =1 to aid the visual interpolation of recovery lengths. Bed and water surface profiles are shown in the lower inset for reference. between the runs may be due to measurement error. In the deep uniform section, there is little difference in u * /u * o between the runs, indicating that u * /u * o is insensitive to channel width; u * /u * o reaches a minimum at the transition from CDF to deep uniform section, and within the deep uniform section u * /u * o relaxes toward the uniform flow value, reaching the uniform flow value of u * /u * o = 1.0 between 0.8 and 1.1 m downstream of the transition. [25] The Coles wake parameter (Figure 8b) responds more slowly to changes in pressure gradient than u * but is similarly insensitive to channel width. In CAF, the convergence of streamwise velocity values (Figure 7) translates to Π decreasing from Π ~ 0 to a minimum of Π ~ 0.50 (Figure 8b). This direction of change follows what is expected from Coles [1956]. In the shallow uniform section, the divergence of u=u b profiles (Figure 7) translates to the relaxation of Π from its minimum toward Π = 0 from x = 1.1 to 2.5 m. The flow field is again perturbed in CDF, with Π increasing rapidly through the area of flow expansion before relaxing toward Π = 0 in the deep uniform section. The direction of change again follows what is expected from Coles [1956]. The highest value of Π (~6.7) was observed in the widest channel (Y60), while the lowest peak value (Π ~ 4.5) was observed in the narrowest channel. At other locations, the differences between runs are within the variability of the measured results. [26] Unlike the profile parameters u * and Π, the lateral concentration of flow (Ψ) is sensitive to channel width, particularly in the deep uniform section (Figure 8c). In this Figure 9. Normalized centerline Reynolds stress (τ þ R ¼ u w u o * 2 ) for runs (a) Y20, (b) Y30, (c) Y40, and (d) Y

9 Figure 10. Planform view of the normalized maximum Reynolds stress in a profile (τ + R max ) for runs (a) Y20, (b) Y30, (c) Y40, and (d) Y60. These figures depict one half the width of the channel. Bed and water surface profiles are shown in the lower inset for reference. section, the flow recovery is characterized by a spreading response prior to relaxation, as shown previously in Figure 6. What Figure 8c makes clear, however, is that the spreading distance of Ψ increases with channel width. For example, Ψ/Ψ o begins to relax toward the uniform flow value as soon as the pressure gradient is removed at the transition from CDF to the deep uniform section in the narrowest channel, such as observed for u * and Π, but Ψ/Ψ o continues to increase for some distance within the deep uniform section for the other three channel widths. When measured from the beginning of the CDF section, the distance to peak (i.e., the spreading distance) of Ψ/Ψ o is 1.1, 1.5, and 2.2 m for the 0.30, 0.40, and 0.60 m wide channels, respectively. The maximum values of Ψ/Ψ o are between 1.12 and 1.17, with the highest value measured for run Y Reynolds Stress Distribution at Channel Centerline [27] Similar to the centerline streamwise velocity, the distribution of the centerline normalized Reynolds stress (τ þ R ¼ u w u o 2) is sensitive to changes in flow depth over the bed forms (Figure 9). The Reynolds stress is relatively low upstream of the bed form. During CAF, despite occasional negative stresses observed near the water surface in run Y40 (Figure 9c) and slightly higher values observed in run Y30 (Figure 9b), the shear stress consistently increases so that τ + R ~ 2.0 for the four runs in this area. Through the shallow uniform section, Reynolds stress continues to increase near the bed, generally reaching τ + R ~ 7.0 at x = 3.5 m. While slightly higher values were again observed in run Y30, the distributions of Reynolds stress are consistent for the four runs in the shallow uniform section. [28] Within CDF and the deep uniform section, Reynolds stress increases rapidly, particularly in the lee of the bed form as the flow depth increases. For example, τ + R is greater than 14 at x = 4.1 to 4.9 m for run Y20 (Figure 9a). The peak Reynolds stress is higher in run Y30 (Figure 9b) than in the other runs (peaks occur with τ R + > 20), but similar at other channel widths. In all experimental runs, the maximum Reynolds stress values occur in the middle of the flow profile rather than very close to the channel bed as is typical of uniform flows. The streamwise location of the highest τ R + values is sensitive to channel width. The highest τ R + values occur immediately downstream of the transition from CDF to the deep uniform section in both the narrowest (Figures 9a and 9b) and the widest (Figure 9d) channels, but at about x = 5.25 to 5.5 m for run Y40 (Figure 9c). Other differences in the spatial distribution of τ R + include areas with negative τ R + values near the water surface of the narrower channels Lateral Distribution of Reynolds Stress [29] Turbulence intensity contours at the channel centerline cannot be reliably interpreted without considering the lateral distribution of Reynolds stress, particularly in CDF and the deep uniform section. Observed in a plan view and considering only the maximum Reynolds stress values in a profile (τ R max + ), the highest turbulence intensities occur at the transition from CDF to deep uniform flow and decrease overall as flow moves downstream (Figure 10). However, the plan view plot also shows that τ R max + is highest, not at the channel centerline, but relatively close to the sidewall. Local peaks of τ R max + > 15 occur at x ~ 4.0 m and y ~ m at all four channel widths. Secondary peaks are visible closer to the channel centerline, particularly for the widest channel. No secondary peak of τ R max + is visible at the channel centerline for the Y30 or Y20 runs. [30] A second interesting observation from Figure 10 is that even as the overall magnitude of τ R max + is decreasing, the area with high τ R max + tends to narrow and shift toward the channel centerline as the flow moves downstream through the deep uniform section. Contours are thus generally angled relative to the sidewall and the principal flow 1858

10 [31] A final observation from Figure 10 is that τ R max + is highest in an intermediate width channel (Y30). The contour line for τ R max + > 22.5, for example, extends from the sidewall all the way to channel centerline for this run (Figure 10b) whereas only the contour line τ R max + > 12.5 is continuous from the sidewall to the centerline for the other runs (Figures 10a, 10c, and 10d). The high turbulence zone is relatively short in run Y30, as shown by the tightly spaced contours between x = 5.0 and 5.5 m at the channel centerline, which indicates a rapid decrease in turbulence intensity in this area (Figure 10b). Figure 11. Principal Reynolds stress profiles in run Y60 at different vertical and lateral positions: (a) y/y = 0.5, (b) y/y = 0.35, (c) y/y = 0.20, and (d) y/y = Note that z/z = 0.1 and z/z = 0.8 are not plotted in all locations due to measurement limitations. Bed and water surface profiles are shown in the lower inset for reference. direction. Lower contours such as τ R max + > 5.0 are angled downstream and across the deep uniform section for all channel widths. The angle of this contour line is similar for all channel widths, which means the contour occurs farthest downstream at the channel centerline in the widest channel. Higher contours such as τ R max + > 10.0 tend to peak between the sidewall and centerline in the widest channel so that while they occur farther downstream in the widest channel (Figure 10d), they occur farther downstream at the channel centerline in an intermediate width channel (Figure 10c) Longitudinal Profiles of Reynolds Stress [32] Longitudinal profiles of Reynolds stress (τ R + ) help to quantify the turbulent response to changing flow depths over the bed forms (Figure 11). The τ R + profiles for run Y60 are relatively flat upstream of the bed form, with higher values near the bed showing turbulence is generated due to shear at the wall. Flow acceleration does not have a strong effect on τ R + at any relative depth, as the profiles are essentially flat from x = 0.55 to 1.1 m. In the section of shallow uniform flow (x = 1.1 to 3.5 m), the Reynolds stress increases near the bed so that τ R + ~ 4.0 at z/z = 0.1 across the full width of the channel by x = 3.5 m. Higher up in the flow in the shallow uniform section (z/z = 0.60), τ R + increases from 0.5 to 2.0 at the channel centerline (Figure 11a) but decreases from 0.2 to 2.0 near the sidewall (Figure 11d). These observations are consistent with the development of a boundary layer in the shallow, relatively fast, flow with higher Reynolds stress due to turbulence generated at the bed. Negative τ R + values occur in the sidewall region and correspond with the area where a velocity dip was observed (Figure 7). [33] Within CDF and the deep uniform section, τ R + profiles are characterized by a rapid and high magnitude response near the bed and sidewalls to the change in pressure gradient and a lagged and lower magnitude response higher in the water column and the middle of the channel. For example, profiles at z/z = 0.1 to 0.2 peak within CDF (x = 3.5 to 4.0 m) whereas profiles at z/z = 0.6 to 0.8 and y/y = 0.35 to 0.5 peak closer to the downstream end of the deep uniform section (x > 5.0 m). The magnitudes of the peaks are generally higher closer to the sidewall, with the global maximum of τ R + ~ 10.5 occurring at z/z = 0.2, y/y = 0.05, and x = 4.0 m. Higher values of τ R + may have been present at z/z = 0.1 if reliable measurements could have been obtained in this area. Negative τ R + values near the sidewalls again correspond with areas where a velocity dip occurs (Figure 7). The global minimum of τ R + ~ 8 occurs at z/z = 0.8, y/y = 0.05, and x = 3.7 m. [34] Spreading distances for the recovery of Reynolds stress vary as a function of the distance from the channel boundaries. A smoothing spline was fitted to the τ R + profiles to determine the streamwise location of the peaks, which allowed a Reynolds stress spreading distance (l sr ) to be defined from the beginning of the CDF section to the peak value of τ R +. As a note, it was difficult to obtain accurate estimates of l sr higher in the water column, and points were excluded if they explained less than 90% of the variability. Occasional l sr estimates near the water surface that met this criterion are difficult to interpret and should be considered outliers. Cross-sectional plots show that l sr increases with distance from both the channel bed and sidewalls in all experimental 1859

11 Figure 12. Spreading distance in the pool (l sr ) estimated from a smoothing spline fitted to the Reynolds stress distribution in runs (a) Y20, (b) Y30, (c) Y40, and (d) Y60. Only values with R 2 > 0.90 are shown. These figures depict one half the width of the channel. runs (Figure 12). As a result, the largest l sr values are shorter in the narrower runs (Figures 12a and 12b) than in the wider channels (Figures 12c and 12d). For example, excluding the surface outliers, l sr > 2.0 m at z/z ~ 0.7 and y =0.24 m in Y60, but l sr < 1.25 m at z/z ~ 0.6 and y = 0.10 m in Y20. A secondary effect is that l sr appears to be longer near the bed at the channel centerline for narrower channels. For example, l sr < 0.5 m at z/z ~ 0.1 and y = 0.24 m in Y60, but l sr > 1.0 matz/z ~ 0.1 and y = 0.10 m in Y20. This effect can be understood by reexamining Figure 9, where it can be seen that the high τ R zones in Y20 and Y30 remain relatively close to the bed, while those of Y40 and Y60 tend to angle away from the bed from x = 4.0 to 6.0 m. The movement of the high shear zone away from the bed thus results in shorter spreading distances at the bed for wider channels. 5. Discussion [35] The results presented herein demonstrate that the hydrodynamics of straight riffle-pool bed forms can be understood using the concepts from perturbation theory. Flow in open channels is perturbed where the channel geometry is nonuniform and recovers toward uniform flow conditions where the channel geometry is uniform. Recovery of some variables, such as shear velocity and Coles wake parameter, occurs simply as a relaxation toward an equilibrium value. For other variables, such as the principal Reynolds stress and the lateral distribution of velocity, the perturbation appears to spread across the channel prior to relaxation. In this discussion we argue that despite the simple 2-D arrangement of the experiment, this spreading behavior may be the link between channel width and macro bed form spacing that leads to the scaling relation between these two key geometrical parameters. [36] The perturbation and recovery of flow strongly affect the spatial distribution of velocity over the bed form. According to the boundary layer theory, the velocity profile responds to unbalanced pressure forces that act on the entire water column [Smits and Wood, 1985; Townsend, 1976], with an adverse pressure gradient acting to decelerate flow over the entire cross section. What is particularly relevant for the near-bed hydrodynamics is that the rate of response is not the same at all locations, with velocity responding more quickly in the inner zone where flow is bounded by the noslip condition at the channel boundary [Lee and Sung, 2008; Smits and Wood, 1985]. Therefore, although flow deceleration and acceleration occur across the entire cross section in CDF and CAF, respectively, the gradient of velocity between the inner and outer zones is also increasing in CDF and decreasing in CAF, as shown by the convergent and divergent velocity profiles (Figure 7). [37] Lateral velocity gradients respond to changes in the pressure gradient in a similar manner to the gradients in vertical velocity. The current study both confirms that vertical contractions and expansions of flow are sufficient to induce lateral flow convergence and divergence, as previously observed by MacVicar and Rennie [2012] over a rough bed, and extends those results to smooth beds. The occurrence of these lateral flow effects in both smooth and rough beds confirms that they are related to the overall acceleration and deceleration of flow rather than bed roughness. Lateral flow convergence can be interpreted in a similar manner to the changes in the streamwise velocity profile. The flow closer to the sidewall reacts more quickly to changes in pressure gradient than the flow in the middle of the channel because it is bounded by the no-slip condition. Both the vertical and lateral velocity gradients are thus steeper and milder in adverse and favorable pressure gradients, respectively. [38] The steep velocity gradients present in CDF are the most important source of turbulence generation over the modeled bed forms. To examine the correlation of velocity gradients with turbulence generation, cross sections of Reynolds stress ( u w ) and the strain rate (S uw ) were plotted for visual comparison for run Y60 at several cross sections within the deep uniform section (Figure 13). The strain rate is defined as the sum of the streamwise and vertical velocity gradients following Boussinesq [1877]: u u w ¼ ν t S uw ¼ ν t z þ w (9) x where ν t is the eddy viscosity. Immediately downstream of the CDF section, u w and S uw are highly correlated (R ij ~ 0.90), with high strain rates in the middle of the profile closely matched by high Reynolds stress values across the full width of the section (Figure 13a). The eddy viscosity is relatively constant over the entire flow section in this area. Both the vertical and lateral velocity gradients are highest near the sidewalls in wider channels where the global maximum in Reynolds stress occurs (Figure 10). [39] The observed turbulence in the deep uniform section decays as it is advected downstream and toward the middle of the channel after being generated close to the bed and sidewalls in the CDF section. Within the deep uniform section, the strain rate decreases in the outer layer as it increases close 1860

12 Figure 13. (left) Strain rate and (right) Reynolds stress in the deep uniform section of run Y60 at x = (a) 4.0, (b) 4.8, (c) 5.3, and (d) 6.0 m. These figures depict one half the width of the channel. to the channel bed (Figures 13b 13d). This observation is consistent with recovery toward uniform flow exhibited by a relaxation of both the inner zone parameter (u * ) and the outer zone parameter (Π) (Figure 8). As streamwise velocity vectors recover, the core of high u w moves laterally toward the centerline of the channel while decreasing in magnitude near the sidewall (Figures 13b 13d). At the end of the uniform section, a larger triangular-shaped zone pointing downward toward the corner formed by the bed and the sidewall is characterized by negative u w and S uw values, patterns that are consistent with the formation of a corner region in straight uniform rectangular sections studies [Tominaga et al., 1989; Yang et al., 2004]. However, the core of high u w in the center of the channel does not match with the high strain rate near the bed, and the overall correlation between the two parameters is lower (R ij ~ 0.65). A parabolic eddy viscosity distribution is more suitable in this area, as is characteristic of uniform flows [Nezu and Rodi, 1986]. The u w remains above the values observed upstream of the bed form (Figure 11), which indicates that the turbulence would likely continue to decay given a longer pool length. [40] The lateral movement generated during CDF explains the lag between the removal of the pressure gradient and the peak in Reynolds stress at the channel centerline. The streamwise locations of τ R max + show that turbulence generation is amplified by the effect of the corner between the sidewall and the bed during CDF (Figure 10). Downstream of the transition todeepuniform flow, flow convergence means that velocity vectors are angled toward the center of the channel, which also advects the highly turbulent flow toward the center. The rapid decrease of τ R max + near the sidewall in the deep uniform section fits with this interpretation (Figure 10), as do the locations of the τ R + peaks (Figure 9). The τ R max + decreases rapidly near the sidewall because the highly turbulent flow is advected away from this area; τ R max + appears to spread (i.e., peak) within the deep uniform section at the channel centerline because highly turbulent flow is moving toward the center from the sidewall areas. The locations of the peaks are thus determined by the distance it takes for the highly turbulent flow advecting from the sidewalls to arrive. [41] The recovery of the lateral concentration of flow (Ψ) and the Reynolds stress ( u w ) is sensitive to the channel width, while the velocity profile parameters (τ and Π) are not. Observed spreading and relaxation lengths for different channel widths include the relaxation length of u * (l ru* ) and Π (l rπ ) (from Figures 8b and 8c), the spreading length of Ψ (l sψ ) (from Figure 8d), and the spreading length of τ R + (l sr ) (from Figure 12) (summarized in Table 2). Of the four measured lengths, l ru* is the most consistent in absolute terms (18% relative standard deviation), which demonstrates that it is the least sensitive to channel width; l rπ is also relatively consistent for the four channel widths (21% relative standard deviation); l sr is similarly constant with channel width, despite the visual appearance of longer spreading distances in wider channels. It is thought that the inability to accurately estimate l sr near the water surface, combined with the compensating effects of the velocity dip phenomena, makes this estimate the least reliable of the four. The visual impression from the planform views of τ R max is that the Reynolds stress responds in a similar manner to the lateral flow concentration through CDF and the deep uniform section (Figures 6 and 10), and this is supported by the spreading length estimates for Ψ and u w for the two wider channels. When normalized by the channel width, l sψ /Y is the most consistent variable (10% relative standard deviation). [42] The combination of two factors, namely, the velocity dip phenomenon and the decay of turbulence as it is advected through the deep uniform section, explains the higher magnitude u w in run Y30 in comparison with the other runs. Simply due to geometry, the high turbulence generated near the sidewalls has decayed more by the time it reaches the channel centerline in the wider channels than in the narrower channels. We would thus expect turbulence to be greatest in narrow channels. In the narrowest channel, however, a large zone with negative u w occurs near the water surface (Figure 9a) where the velocity gradient is inverted (Figure 5a) as a result of the velocity dip. It is possible that this zone is suppressing turbulence generation. The combination of the velocity dip and turbulence decay appears to result in an optimum width for high turbulence magnitudes. In the experiments presented here, this optimum occurs at an aspect ratio of 5.0 at the riffle crest (Table 1), which is equal to 1861