Water Resources Research

Mean flow, turbulence structure, and bed form superimposition across the ripple-dune transition

Authors


Abstract

[1] This paper describes a laboratory study of the dynamics of flow associated with three different stages of bed form amalgamation across the ripple-dune transition. Measurements of flow velocity were obtained over simplified fixed bed forms, designed to simulate conditions at the ripple:dune transition, using a 2D laser Doppler anemometer. This yielded information detailing the mean velocity field, turbulence statistics, local turbulence production, local deviations from the mean pressure caused by dynamic effects, turbulent kinetic energy and the contributions to the total Reynolds stresses from different coherent turbulent events. The results broadly confirm previous hypotheses that as bed form amalgamation proceeds across the ripple-dune transition, a superimposed bedstate can induce a series of critical changes to the flow structure, with higher Reynolds stresses being produced near and downstream of flow reattachment. In particular, quadrant 4 events (turbulent flow structures with a downstream velocity greater than average, and directed towards the bed) dominate the vertical turbulent diffusion of longitudinal momentum in near-bed regions close to the crest of the next downstream ripple, thus providing the potential for increased erosion and sediment transport. These experiments using simple fixed beds also provide support for recent measurements that document increased suspended sediment concentrations across the ripple-dune transition (Schindler and Robert, 2004, 2005), and that at the transition the largest bed forms are not necessarily those associated with the most intense shear layer activity (Schindler and Robert, 2005). Bed form superimposition and amalgamation may thus significantly alter the mean and turbulent flow field as compared to bed forms without superimposition.

1. Introduction

[2] The nature of the mean and turbulent flow structure over transverse, two-dimensional ripples and dunes is now well-documented (see reviews, and references therein, by Kennedy [1969], McLean [1990], ASCE Task Force [2002], and Best [2005]), and recent work has begun to tackle the more complex, but more commonly occurring, flow fields of dunes that have either a spanwise curvature in their vertical crestline height [Maddux, 2002; Maddux et al., 2003a, 2003b] or a planform sinuosity [Allen, 1968; Venditti, 2003; Parsons et al., 2005]. Due to the importance of large-scale bed forms in controlling bedload transport and greatly influencing flow resistance [e.g., Davies, 1982; Karim, 1995; Fedele and García, 2001; Julien and Klaassen, 1995; Julien et al., 2002], much effort has been devoted to understanding the evolution and growth of bed forms within natural alluvial channels [e.g., Lyn, 1993; Mendoza and Shen, 1990; McLean, 1990; McLean and Smith, 1979, 1986; McLean et al., 1994, 1996, 1999a, 1999b; Nelson and Smith, 1989; Nelson et al., 1993, 1995, 2001; Bennett and Best, 1995; Mohrig and Smith, 1996]. However, the transitional areas between bed forms have received relatively little attention, despite their critical influence on changing flow resistance [Karim, 1995], although several papers have sought to examine: i) the transition between current ripples and dunes, using a range of approaches from linear stability analysis [Fredsøe, 1974; Richards, 1980] to laboratory experimentation [Bennett and Best, 1996; Robert and Uhlman, 2001; Schindler and Robert, 2004, 2005], and ii) the transition from dunes to upper-stage plane bed [Engelund and Fredsøe, 1974; Fredsøe and Engelund, 1975; Bridge and Best, 1988].

[3] Dunes are generated in a range of bed grain sizes, from coarse silts to gravels [Ashley, 1990; Dinehart, 1992; Best, 1996, 2005; Carling, 1999; Kleinhans, 2001, 2004; Bridge, 2003] and their origin has been linked to increasing bedload sediment transport as bed shear stress is increased over bed forms that are either ripples (in sands <0.6 mm diameter) or low-relief bedload sheets/low 2D bed forms in coarser sediments [Southard and Boguchwal, 1990; Best, 1996; Bridge, 2003]. Several features distinguish dunes from the ripples and low-relief bed forms from which they are generated: i) dune height has been found to scale with the flow depth [Jackson, 1976; Yalin, 1964, 1977], whereas ripple height is independent of flow depth, although both bed forms have a similar asymmetric cross-sectional form; ii) dunes appear to have a far more regular wavelength than bedload sheets/low 2D bed forms [Southard and Boguchwal, 1990; Best, 1996]; iii) ripples have a larger height:wavelength ratio than dunes [Allen, 1968; Robert and Uhlman, 2001]; iv) dunes may interact strongly with the water surface with an out-of-phase relationship, whereas ripples do not [McLean, 1990; Bridge, 2003; Best, 2005]; and v) dunes have been shown to influence the flow structure throughout the entire flow depth, often generating large-scale turbulence that erupts on the water surface as ‘boils’ or a ‘kolk-boil vortex’ [Coleman, 1969; Jackson, 1976; Müller and Gyr, 1983, 1986, 1996; Kadota and Nezu, 1999; Best, 2005]. These large-scale turbulent structures may transport appreciable sediment in temporary suspension [Kostaschuk and Church, 1993; Lapointe, 1992, 1996; Babakaiff and Hickin, 1996; Schmeeckle et al., 1999; Venditti and Bennett, 2000]. The formation and maintenance of a dune field thus requires that the sediment transport rate across these bed forms is greater than those from which they have initiated, with this increased transport rate being speculated to result in greater bed form heights and generation of larger flow separation cells in the leeside of the dune [Bennett and Best, 1996]. In this context, the initiation of a dune bedstate has been proposed to require a bed form of sufficient size that will significantly alter the entire mean flow and turbulence field to one that becomes dominated throughout its depth by the bed form [Bennett and Best, 1996]. The formation of this ‘larger-than-average bed form’ may revolve around production of a ‘rogue ripple’ [Leeder, 1983] or amalgamation of bed forms to give a larger than average form, although it is clear that this must be accompanied by an increase in bed shear stress and bedload transport rate over the ripple-dune transition. If it is accepted that an initial dune results from amalgamation of smaller bed forms, then it is the subsequent modification of the downstream turbulent flow field and bedload transport rate that appear critical, with the flux of suspended sediment also becoming increasingly important in finer sands and silts.

[4] Bennett and Best [1996] presented a conceptual model of flow across the ripple-dune transition (Figure 1a) that has recently been examined and modified by Schindler and Robert [2004, 2005]. Bennett and Best [1996] speculated that the transition from ripples to dunes is linked to bed form amalgamation that induces critical changes to the flow field (Figure 1a). Flow over both ripples and dunes is dominated by shear layer instability associated with separation zones in the bed form leeside, but Bennett and Best [1996] argue that, as amalgamation proceeds, the rogue ripple is able to generate larger-scale turbulence that begins to advect throughout the entire flow depth. As turbulent coherent structures (quadrant 2 events, Figure 1a) are able to penetrate further into the outer flow, they induce return flows (quadrant 4 events) that are of greater magnitude [Best, 2005] and that are thus able to exert greater shear stresses as they impact the bed (Figure 1a). Bennett and Best [1996] reason that this change in flow structure is responsible for generating increasing sediment transport across this bed form transition, and producing higher bed forms that evolve into dunes. The required increase in bedload transport at the transition thus occurs through the influence of quadrant 2 events advecting further upwards within the flow, which lead to higher-magnitude instantaneous Reynolds stresses acting upon the bed (e.g. quadrant 4 events). This contention finds qualitative support in the observation that the transition from bedload sheets (also see the two-dimensional dunes of Southard and Boguchwal, 1990) to dunes is marked by both larger bed form troughs and greater erosion at reattachment [Best, 1996]. Additionally, recent work by Schindler and Robert [2004] has demonstrated the increase in suspended sediment transport across this bed form transition, which they attribute to the increasing importance of the wake region and shear layer generated turbulence, with their study reporting that turbulence intensity is at its maximum in the transitional region between ripples and dunes.

Figure 1.

(a) A conceptual model of flow across the ripple dune transition [from Bennett and Best, 1996]. τ and τo represent the bed shear stress and threshold bed shear stress for sediment entrainment, respectively. Q2 and Q4 denote quadrant 2 and 4 events, respectively (Q2: −u′, +w′; Q4: +u′, −w′). (b) Schematic model of bed morphology and flow at one stage of the transition from ripples to dunes, which is characterized by both the development of superimposed bed forms and 3D linguoid ripples [after Schindler and Robert, 2005]. See text and Notation for explanation.

[5] The increased influence of the shear layer may be due to the greater absolute velocity differential across the shear layer over the ripple-dune transition [Bennett and Best, 1995, 1996], and therefore both higher instantaneous velocities/shear stresses at reattachment and higher magnitude ejections from along the shear layer. Robert and Uhlman [2001] and Schindler and Robert [2004] both report large changes in turbulence intensity in the transition region, and attribute this to both more active shear layers and the increased bed form height. Schindler and Robert [2005] also report that the development of more three-dimensional, linguoid ripples at the transition, together with bed form superimposition, is critical (Figure 1b). Schindler and Robert [2005] argue that these 3D bed forms can create greater levels of shear layer turbulence than the higher 2D dunes, thus highlighting that bed form height may not be the sole determinant of shear layer turbulence intensity.

[6] However, the role of bed form superimposition in this process has not been addressed, and there is a need for research on both the flow fields of interacting bed forms and the nature of turbulence across the ripple-dune transition. The present paper seeks to investigate the mean and turbulent structure of flow across the ripple-dune transition by examining a triangular, fixed bed form of ripple-size which amalgamates with a smaller, triangular, bed form to generate a final bed form approximately the size of a dune. The laboratory experiments achieved a high-resolution quantification of flow using laser Doppler anemometry to examine mean flow and turbulence at three stages of amalgamation: i) flow solely over a ripple (flow depth:bed form height (equation image:hs) = 6.67; RUN I), ii) flow over a bed form state where a smaller bed form is superimposed on the primary bed form just upstream from its crest (equation image:hs = 4.35; RUN II), and iii) flow over the combined bed form that is representative of a dune (equation image:hs = 4.00; RUN III).

2. Experimental Setup

2.1. Flow Channel

[7] The experiments were performed in a glass-sided flume in the Sorby Fluid Dynamics Laboratory, School of Earth and Environment, University of Leeds. The channel is 0.30 m wide, 0.30 m deep and 12 m in length. All experiments were conducted under a uniform, steady unidirectional flow, with a flow depth of 0.10 m and with the flow surface being parallel with the bed. The overall average velocity, equation image, was held constant at ∼0.40 m s−1 by maintaining a constant flow discharge during the experiments. The flow Reynolds and Froude numbers, based on average velocity and flow depth, thus resulted in values of ∼37 000 and 0.40, respectively (Table 1).

Table 1. Summary of Experimental Conditions for RUNS I, II, and III
 RUN IRUN IIRUN III
  • a

    Flow depth expressed as depth from ripple trough to flow surface.

  • b

    Shear velocity was determined using four methods using double-averaged values incorporating all profiles [see Fernandez, 2002], and the average is given here.

Bed form height, hs [m]0.0150.0230.025
Flow depth, equation image [m]a0.100.100.10
equation image:hs6.674.354
Mean velocity, equation image [m s−1]0.3720.3740.363
Mean shear velocity, u* [m s−1]b0.0330.0400.039
Reynolds number, Re37 25037 44036 790
Froude number, Fr0.380.380.37

2.2. Bed Configurations

[8] Three bed roughness conditions were investigated, herein termed RUN I, RUN II and RUN III (see Figure 2), whose geometry was based on previous work detailing the bed form height:flow depth and wavelength:bed form height ratios of ripples and dunes [Allen, 1968, 1982; Bennett and Best, 1995, 1996]. During RUN I, the flume bed was completely covered by a periodic train of simulated ripples, consisting of two-dimensional metal triangles with a leeside slope angle of 25°, a wavelength, λL, of 0.30 m and a crest height, hs, of 0.015 m, this producing an aspect ratio (λL/hs) of 20 and a flow depth: bed form height (equation image:hs) ratio of 6.67. The geometrical characteristics of these bed forms were chosen to be at the larger end of the ripple field. In order to characterize evolution of the mean flow field and turbulent coherent structures induced by bed form superimposition and amalgamation across the ripple-dune transition, a smaller triangular metal bed form, λs = 0.10 m and hs = 0.01 m (λs/hs = 10), was then superimposed upon one of the above train of ripples in the test section (4.5 m downstream of the flume inlet), at two different positions in the amalgamation process: i) in RUN II, the smaller bed form was placed on the stoss-side of the larger bed form, approximately 0.03 m (0.33λs) upstream of its crest (Figure 2b), corresponding to an intermediate stage of the amalgamation process (equation image:hs = 4.35); ii) in RUN III, the smaller bed form was positioned so that it had amalgamated with the ripple (Figure 2c), with the two lee-sides forming one continuous larger leeface (equation image:hs = 4.00), and generating a larger bed form representative of a dune.

Figure 2.

Schematic diagram showing the three bed configurations studied in RUNS I, II and III.

2.3. Measurement Techniques

[9] Velocity measurements were collected using a DANTEC, two-dimensional laser Doppler anemometer (LDA) that was operated in backscatter mode with a 161 mm focal length lens (see Bennett and Best [1995] for details of this LDA). The flow was seeded with small (∼<5 μm) white water-based paint particles, which provided excellent light-scattering sources, and permitted sampling frequencies in the range 50–300 Hz. A high resolution positioning carriage allowed the measuring volume (0.07 mm3) to be located with a precision of ±0.1 mm. The above setup thus allowed establishment of a very accurate measuring grid that, dependent on the bed configuration, had between 26 to 30 vertical profiles with a varying streamwise (x) spacing, and with each vertical (z) profile having ∼20 measuring points (see later vertical profiles). All measurements were located along the centerline of the flume to minimize any sidewall effects. In this way, each experimental configuration had a measurement grid with approximately 600 points, at which the longitudinal and vertical components of velocity (u, w) were measured. The sampling time for the LDA measurements was defined, from initial sampling tests, as that required to allow accurate resolution of the fourth-order velocity moments, and was set as 180 seconds [Fernandez, 2002]. The free surface profile over each bed configuration was measured after each run with a manual point gauge accurate to ±0.1 mm.

3. Results

3.1. Mean Flow and Turbulence Intensities

[10] Velocity data was processed at each point of the measuring grid to calculate the mean downstream and vertical components of flow, equation image and equation image, respectively, as:

equation image

where n is the number of the observations and ui, wi are the instantaneous velocities in the horizontal and vertical directions, respectively. The second moments of the velocity distribution (root-mean-square) were also estimated according to:

equation image

in which the fluctuations are u′ = uiequation image and w′ = wiequation image for the downstream and vertical components of velocity, respectively. The resulting contour plots (Figures 3a and 3b) illustrate the changing mean flow structure at the three stages of amalgamation. The greatest downstream velocities (∼1.2equation image; Figure 3a) are present just over the crest of the first bed form, with RUN II showing highest velocities over the smaller ripple, reflecting flow acceleration due to the enhanced topographic forcing effects over the higher regions of the bed. These regions of flow acceleration are followed by a region of separated flow immediately downstream of each crest, characterized by negative (i.e. upstream) horizontal velocities reaching −0.2equation image. Flow in the leeside of the smaller ripple in RUN II shows a negative equation image, but at this distance upstream of the crest of the larger bed form, the region of flow separation frequently combined with that of the downstream form (see below), thus generating a larger unified region of recirculating flow. Downstream of the crest of the larger bed form, the flow shows a gradual recovery downstream, with velocities gradually increasing up the stoss side of the next bed form, illustrating the recovery of the boundary layer after separation and growth of a new internal boundary layer, as shown in many past studies of flow over dunes [e.g., McLean, 1990; Nelson et al., 1993, 1995; Nelson and Smith, 1989; McLean et al., 1994, 1996].

Figure 3a.

Distribution of mean streamwise velocity, equation image, for RUNs I, II and III.

Figure 3b.

Distribution of mean vertical velocity, equation image, for RUNs I, II and III.

[11] The patterns of vertical velocity (Figure 3b) show positive vertical velocities over the stoss side of the bed forms illustrating the topographic forcing of flow, and flow towards the bed in the leeside that shows the influence of flow separation. Positive vertical velocities in the immediate leeside of the bed forms are due to the recirculating flow in this region. It is noticeable that the region of flow directed towards the bed in the leeside of the larger bed form becomes farther away from the bed in RUNs I–III, reflecting the growth of the separation zone. However, it is also apparent that the region of downward flow becomes closer to the crest in RUN II, reflecting the flow separation and topographic forcing of flow over the smaller superimposed bed form. The data presented herein on the mean flow fields for RUNs I and III is very similar to that obtained in many previous studies of flow over 2D dunes and ripples [e.g., van Mierlo and de Ruiter, 1988; Mendoza and Shen, 1990; Lyn, 1993; Nelson et al., 1993, 1995; McLean et al., 1994, 1999b; Bennett and Best, 1995, 1996; Kadota and Nezu, 1999]. During RUN I (Figure 3a), the mean flow structure over ripples is the result of flow separation downstream of the crest, reattachment of flow and growth of a new boundary layer downstream, and convective accelerations due to the bed topography. This pattern is identical over the larger combined form in RUN III but the effects of separation are felt both further downstream and higher in the flow [Bennett and Best, 1996; Robert and Uhlman, 2001; Schindler and Robert, 2004]. However, RUN II clearly shows that a superimposed bed form significantly alters the original flow field, inducing a larger recirculation region and a different spatial distribution of low horizontal velocities and vertical flow velocities associated with flow separation.

[12] The length of the flow separation zone, defined as the distance from the separation point to flow reattachment, i.e. xcrestxreattachment (where the brinkpoint of the crest, xcrest = 0), is shown in Figure 4 for the three runs. The length of the recirculation cell, xr, was estimated using the negative intermittent velocity method described by Kadota and Nezu [1999]; briefly, this method allows estimates of the location of the reattachment point by computing, at each point, the total time, Ir (also called the negative velocity probability), when negative streamwise velocities were measured. Then, for Ir = 0.5, positive and negative downstream velocities have the same probability, the mean velocity is thus zero and indicates the point of flow reattachment. This criterion allowed determination of the location of the reattachment point across the bed form transition, with xrvalues being 3.7, 5.0 and 5.9hs as amalgamation proceeded between RUNs I, II and III respectively. It is also clearly evident that in RUN II (Figure 4) a more unified recirculation region was occasionally present at the intermediate stage of bed form amalgamation, and the separation zone of the upstream smaller ripple was interacting with that of the downstream larger form. However, a region with Ir = 0.50 is present downstream of both the smaller and larger bed forms, indicating that, at this bed form spacing, a time-averaged point of reattachment was associated with both. However, the contours for 0 < Ir < 0.50 occur across the two bed forms, and show that the separation zone of the superimposed ripple was interacting with that of the larger downstream bed form.

Figure 4.

Length of the flow separation zone for RUNs I, II and III as defined using Ir (the negative velocity probability) where Ir = 0.50 defines an equal probability of upstream and downstream directed streamwise velocities (see text for explanation). The point at which Ir = 0.50 at the bed is used to define the reattachment point and length of the flow separation zone, xr. Downstream and vertical distances are expressed in relation to the bed form height, hs.

[13] Comparison of the distributions of Ir (Figure 5) plotted against equation image/urms shows that RUNs I and III follow a Gaussian distribution, similar to that reported by Kadota and Nezu [1999]. However, RUN II departs from this Gaussian distribution and shows lower values of equation image/urms for a particular value of Ir: as will be shown below, this is associated with higher turbulence (increased urms) at this stage of bed form amalgamation that lowers the value of equation image/urms. Additionally, the recirculation cell present in RUN III is 38% larger than values reported in the literature for trains of two-dimensional regular dunes [Kadota and Nezu, 1999], and may be attributed to the smaller topographic accelerations induced by the particular bed configuration used herein.

Figure 5.

Distribution of Ir plotted against equation image/urms for RUNs I, II and III, together with a Gaussian distribution.

[14] Figures 6a and 6b shows the vertical profiles of streamwise (Figure 6a) and vertical (Figure 6b) turbulence intensity over the three configurations investigated. The turbulence intensities for all three RUNs remain unaltered in the flow field upstream of the bed configurations. In RUNs I and III, the distribution of turbulence intensity, as with the patterns of mean flow, is very similar to results from past studies of flow over 2D bed forms [van Mierlo and de Ruiter, 1988; Nelson et al., 1993, 1995; McLean et al., 1994, 1996; Bennett and Best, 1995; Kadota and Nezu, 1999], with turbulence highest along the separation zone shear layer and in the region near reattachment. However, RUN II shows higher absolute levels of horizontal and vertical turbulence intensity in the leeside of the bed forms, and also levels of turbulence intensity that extend into the outer flow as far as that over the amalgamated bed form (RUN III). Flow is thus strongly influenced by the superimposition process, with peaks in turbulence intensity present close to the first crest and in the recirculation region where the flow separation zones have been shown to interact (Figure 4).

Figure 6a.

Distribution of root-mean-square values of streamwise velocity, u′, for RUNs I, II and III. The solid circles in the top diagram indicate the points on the profiles that are used in the quadrant analysis of Figure 11.

Figure 6b.

Distribution of root-mean-square values of vertical velocity, w′, for RUNs I, II and III.

3.2. Reynolds Stresses, Turbulent Kinetic Energy, and Turbulence Production

[15] The Reynolds stress was determined from:

equation image
equation image

where τR is the Reynolds stress and ρ is the fluid density, and is shown in contour maps (Figure 7) for all three experiments. This equation image Reynolds stress was examined as it will be important in the transport of sediment over mobile beds [Nelson et al., 1993, 1995; McLean et al., 1994, 1996; Bennett and Best, 1995, 1996]. These results illustrate how the field of the vertical diffusion of longitudinal momentum becomes greatly affected by the amalgamation process, with higher absolute values of τR being present in RUN II and with these higher values extending further downstream of the region of amalgamation than in either RUNs I or III.

Figure 7.

Distribution of Reynolds stress, −ρequation image, for RUNs I, II and III.

[16] In this context, the spatial variation of the vertical flux of turbulent kinetic energy, FTKE, also reflects the transport processes that govern turbulence within the flow. In the absence of measurements of the spanwise component of turbulence, v′, FTKE may be given by the expression [Antonia and Luxton, 1971]:

equation image

[17] The distribution of FTKE over the bed forms in RUNs I–III (Figure 8) shows a change in the sign of FTKE (positive is upwards) at the location of the shear layer bounding the flow separation recirculation region. Positive vertical fluxes of turbulent kinetic energy are seen to dominate the regions of flow above the shear layer, while negative fluxes prevail below this layer. The magnitude of FTKE, similar to τR, is higher in the stage of bed form superimposition (RUN II) and with the downstream extent of these regions of high FTKE reaching further downstream in RUNs II and III. This turbulence structure thus suggests the existence of localized sources of turbulence production along the shear layer, and that this production is enhanced in the stage of bed form superimposition (RUN II), possibly due to the interaction of the regions of flow separation noted above.

Figure 8.

Distribution of the vertical flux of turbulent kinetic energy, FTKE, for RUNs I, II and III.

[18] Estimates of the production of turbulence, P, can be obtained from:

equation image

that results from the non-linear interactions between the Reynolds stress (−equation image) and the local mean velocity gradient equation image [Raupach et al., 1991; Nelson et al., 1993]. Figure 9 illustrates that increased rates of turbulence production are clearly observed in the intermediate stages of the amalgamation process (RUN II), with peaks in turbulence production in RUN II being larger by a factor of three as compared to the initial and final conditions in RUNs I and III.

Figure 9.

Distribution of turbulence production, P, for RUNs I, II and III.

3.3. Dynamic Pressure Field

[19] The alterations to the turbulence field induced by the superimposed bed form and amalgamation are also detectable in the dynamic pressure field, where induced flow accelerations and decelerations affect the deviations of the pressure field, denoted in this study by ΔP, from the hydrostatic pressure field. These deviations were evaluated herein following Kadota and Nezu [1999], where the 2D momentum equations are used to compute ΔP as a function of the velocity field, as given by:

equation image

where P is the total pressure. Figure 10 shows that the deviations from the dynamic pressure field, ΔP, normalized by ρurms2/2, are similar to those observed over a regular field of 2D dunes [see Kadota and Nezu, 1999]. Both positive and negative values of ΔP can be clearly identified, with the greatest positive values reached in the reattachment zone, and the largest negative values being located along the shear layer generated immediately downstream of the first crest. Regions of high and low ΔP are thus associated with flow separation and reattachment. The bed forms in RUN II show the largest values of −ΔP that are associated with the shear layer of the interacting separation zones, and a large region +ΔP on the stoss side of the downstream bed form. It is likely that under mobile bed conditions, this region of higher +ΔP, together with increased values of FTKE and P, may be expected to lead to greater erosion and sediment transport in this region as compared to the larger, amalgamated bed form in RUN III.

Figure 10.

Distribution of deviations in pressure from the hydrostatic pressure field, ΔP, for RUNs I, II and III (see text for explanation of derivation of ΔP).

3.4. Turbulent Coherent Structures

[20] Analysis of the type and occurrence of turbulent coherent structures over these three bed configurations was achieved using the classical discrimination technique of quadrant analysis [e.g., Lu and Willmarth, 1973; Nezu and Nakagawa, 1993; Bennett and Best, 1995; Nelson et al., 1995] where the instantaneous u and w velocities are used to classify the uw′ events within the uw′-plane. Four quadrants (i = 1–4) can be examined: quadrant 1 [+u′ and +w′]; quadrant 2 [−u′ and +w′]; quadrant 3 [−u′ and −w′] and quadrant 4 [+u′ and −w′]. Comparison of the quadrant contributions to the Reynolds stress (S1–S4 for quadrants 1–4 respectively, and the total Reynolds stress, S5; Figure 11) at two heights above the bed (4 and 30 mm) at two downstream positions (x = 589 and 864 mm, positions chosen to depict flow in the leeside and on the upper stoss slope of the downstream bed form, see Figure 6a for location) illustrates some of the principal changes in the occurrence of turbulent coherent flow structures. In the leeside of the bed form (x = 589 mm, z = 4 mm; Figure 11i), quadrant 2 events are seen to become more important from RUN I–III, with larger magnitude ejections being produced in the leeside. Quadrant 1 and 3 events are also seen to increase in their magnitude and contribution between RUN I and III. At 30 mm above the bed at x = 589 mm (Figure 11iii), quadrant 2 events also increase in importance between RUN II and III, as do quadrant 4 events, although it is evident that at this location quadrant 4 events are of lesser importance in RUN II than in the ripple bedstate (RUN I). At x = 864 mm near the bed (z = 4 mm, Figure 11ii), quadrant 4 events become more important in contributing to the Reynolds stress production between RUN I and III, whereas quadrant 2 events clearly dominate the flow further from the bed (z = 30 mm, Figure 11iv). Furthermore, quadrant 4 events become more dominant near the bed (Figures 11i and 11ii) as bed form superimposition and amalgamation occur, with quadrant 4 events contributing more to the Reynolds stresses, and at higher threshold hole sizes, between RUN I and III.

Figure 11.

The contributions to the Reynolds stress, Si, for quadrants 1–4 (S1–S4, respectively), and the total Reynolds stress (S5) for different threshold values, H. These distributions are shown for points at two heights above the bed (4 and 30 mm) at two downstream positions (x = 589 and 864 mm, which are positions in the leeside and on the upper stoss slope of the downstream bed form, see Figure 6a for location).

[21] In order to examine the relative differences between quadrant 4 and 2 events in contributing to the Reynolds stress, for a certain threshold or hole size H, a variable ΔSH can be defined as:

equation image
equation image

in which Ii,pHis the detection function depending on H, defined as ∣uw′∣ = H(urmswrms) [see Nezu and Nakagawa, 1993] and p(u′,w′) is the joint probability density between u′ and w′. Negative ΔS values indicate when flow is dominated by Q2 motions, whereas positive ΔS shows where the flow is dominated by Q4 events. Figure 12 shows the resulting contour plots for H = 2 for RUNs I–III and illustrates that quadrant 4 events become dominant in the lower flow near the bed for RUNs II and III. In RUN I, Quadrant 4 events are restricted in their area of dominance to the immediate leeside of the bed form, but as amalgamation proceeds (RUN II) quadrant 4 events become more important both at, and downstream of, reattachment. Once amalgamation has occurred (RUN III), the region of quadrant 4 dominance near the bed extends over the entire length of the stoss side, illustrating the higher potential for increased bedload transport across the ripple-dune transition [Bennett and Best, 1996; Robert and Uhlman, 2001].

Figure 12.

Distribution of the relative differences between quadrant 4 and 2 events in contributing to the Reynolds stress, for a certain threshold or hole size H, expressed by ΔSH for RUNs I, II and III (see text for explanation of derivation of ΔSH).

[22] The quadrant signatures and ΔSH profiles (see Figure 13) can also be used to quantify the vertical distance from the bed, zc, at which the structure of turbulence begins to be dominated by quadrant 2 rather than quadrant 4 events (see also Figure 11). The line that connects these points of changing quadrant 4:2 dominance can be used to define the upper boundary of a roughness layer [Raupach et al., 1991] (see Figure 14), that develops downstream of the reattachment point and reaches a maximum thickness, δ (Figure 14), equivalent to ∼1.2–1.4hs, over the crest of the downstream bed form. Figures 13 and 14 also confirm that, whereas quadrant 2 events dominate the production of Reynolds stresses in regions z > 4 mm over the crest of the second ripple in RUN I, quadrant 4 events dominate these regions up to ∼z = 20 mm during superimposition (RUN II) and after amalgamation has occurred (RUN III).

Figure 13.

Vertical profiles of ΔSH for RUNs I and III.

Figure 14.

Thickness of the roughness layer, δ, that develops on the stoss side of the downstream bed form, and defines the point of changing quadrant 4:2 dominance over the downstream bed form.

4. Discussion

[23] The results from the present experiments show significant changes to the spatial structure of turbulence over a ripple as a smaller bed form is superimposed on the stoss side of the bed form and then amalgamates to form a dune: this mimics, albeit in a simplified geometry over fixed forms and in the absence of sediment transport, the process of bed form amalgamation, in which a larger than average bed form (a ‘rogue ripple’) grows in size to form a dune [Leeder, 1983; Bennett and Best, 1996; Schindler and Robert, 2004]. However, it is worthy of note that the ripple-dune transition is also accompanied by an increase in the three-dimensionality of ripples at higher shear stresses [Robert and Uhlman, 2001; Schindler and Robert, 2005] which can also be expected to significantly influence the flow structure [Maddux, 2002; Maddux et al., 2003a, 2003b; Venditti, 2003]. The results presented herein broadly confirm previous hypotheses [Bennett and Best, 1996, Figure 1] that the production of a larger bed form induces a series of critical changes to the flow structure, where higher Reynolds stresses near reattachment may be expected to generate increased bed erosion. Significantly, the superimposition of a smaller bed form on the stoss side of a larger bed form is shown to alter the leeside flow structure through interaction of the two shear layers associated with flow separation. At the bed form spacing investigated herein (0.33λs), this interaction periodically created a larger unified flow separation zone, which appears critical in generating increased turbulence production and turbulent kinetic energy at this stage of bed form superimposition than either over the ripple or the amalgamated form. Increased turbulence over the superimposed bed form also confirms that the absolute height of the bed form may not be the sole determinant of leeside shear layer activity, as also recently suggested by Schindler and Robert [2005]. The presence of a ‘rogue ripple’ may thus not be a prerequisite to produce a transition from ripples to dunes, and bed form superimposition may provide a source for the higher levels of turbulence required for this change.

[24] In particular, it has been shown how quadrant-four events dominate the vertical turbulent diffusion of longitudinal momentum in near-bed regions close to the crest of the next downstream ripple, thus providing the potential for increased erosion and sediment transport by suspension. As amalgamation proceeds, the magnitude of ejections associated with vortex shedding along the free shear layer increases and these events are able to penetrate further into the outer flow, eventually interacting with the water surface as macroturbulent events (kolks or boils), which are noticeably absent in flow over ripples. These observations clearly demonstrate that during the simulated amalgamation process, conditions exist that may explain the mechanisms by which the ripple-dune transition is triggered and how evolution towards a new state of regular bed forms (dunes) is maintained. Additionally, the higher bed shear stresses caused by flow separation from the upstream bed form during superimposition (RUN II), provide an explanation for why crestal erosion of a larger form is common just before amalgamation is complete [cf. Ditchfield and Best, 1992; McCabe and Jones, 1977]: these higher stresses may cause increased erosion and lowering of the height of the larger downstream form.

[25] The superimposition and amalgamation of bed forms studied herein is also representative of one of the two types of bed form coalescence noted by Raudkivi and Witte [1990] and Coleman and Melville [1994], where a smaller bed form approaches from upstream, and merges with, a larger downstream bed form. In this case, the approach of the smaller bed form may restrict sediment supply to the downstream form and thus decrease its celerity. As long as erosion of the downstream form is not severe (see above [Ditchfield and Best, 1992]), this may then lead to coalescence of the two forms. However, Coleman and Melville [1994] also note cases where the downstream bed form diminishes in size as the upstream form approaches, and the present study indicates that this may be tied to both restriction of sediment supply to the downstream bed form during unification as well as downstream erosion caused by enhanced turbulence generated over the superimposed form. The issue of which of these two scenarios occurs will depend on the relative height of the superimposed bed forms, as well as their spacing and relationship to flow depth (which will determine the maximum bed shear stress exerted near the crests), and clearly is an area of research that demands further study.

[26] These present results, together with recent studies of dune three-dimensionality, may be used to provide a revised account of flow and bed form change across the ripple-dune transition. As ripples develop and adopt a more three-dimensional form [Baas, 1999; Robert and Uhlman, 2001; Schindler and Robert, 2005], it is likely that regions of higher bed shear stress will develop over both nodal points in the crestline and associated with spanwise secondary flows [Maddux, 2002; Maddux et al., 2003a, 2003b], with this vorticity also potentially generating more longitudinal vortices [Zedler and Street, 2001]. Additionally, the developing spanwise three-dimensionality of the ripples as they evolve [Baas, 1999; Schindler and Robert, 2004, 2005] may favor production of regions of higher turbulence, with Venditti [2003] reporting that turbulence is increased over a lobe-shaped crestline (i.e. convex downstream) when compared to either a straight-crested or saddle-shaped (i.e. concave-downstream) bed form. Thus, as ripples develop an increasingly three-dimensional morphology, the spatial variability of turbulence and bed shear stresses becomes larger [Schindler and Robert, 2005] and this increases the range of ripple sizes present. As bed shear stress is increased, the superimposition of bed forms, of different size and celerity [Raudkivi and Witte, 1990; Coleman and Melville, 1994], further enhances turbulence production due to the interaction of the separation zones associated with each bed form. Such increased turbulence generates ejections of fluid that penetrate further into the outer flow, and critically induces larger-magnitude quadrant 4 events near the bed, and over a greater percentage of the stoss side of the downstream bed form. These changes will increase bedload transport, thus aiding the growth in the downstream bed form height, until a dune is produced that is able to interact with the entire flow depth, generate turbulence that advects to the flow surface, and produce a wake region that dominates the downstream flow and sets the wavelength for the dunes. As such internal boundary layers are generated on these developing dunes, smaller bed forms grow in response and yield a complex flow field of stacked, interacting wakes.

[27] The significance of bed form superimposition may have been underplayed in past work detailing the fluid dynamics of dunes [Best, 2005], as records of bed morphology from many natural environments show the frequent occurrence of dunes of different scale [e.g., Allen and Collinson, 1974; Allen, 1968; Rubin and McCulloch, 1980; Harbor, 1998; ten Brinke et al., 1999; Wilbers and ten Brinke, 2003; Wilbers, 2004; Parsons et al., 2005], that may be caused by a variety of factors including non-uniform and unsteady flow, hysteresis effects within a flood hydrograph, or the developing internal boundary layer on the stoss side of large dunes. Additionally, recent work on aeolian dunes [Hersen et al., 2004; Endo et al., 2004; Katsuki et al., 2005] has illustrated the clear importance of superimposition and the interactions between bed forms. Endo et al. [2004] identify three types of behavior of interacting barchanoid dunes: i) absorption, where a small barchan catches up a larger dune and these merge to form one larger barchan; ii) ejection, where the approaching bed form causes the detachment of sediment from the downstream dune that subsequently forms a new barchan downstream, and iii) splitting, where there is no physical contact between the two dunes, but the vortices generated by the upstream barchan are responsible for deformation, erosion and, if the upstream form is large enough, splitting of the downstream dune. Endo et al. [2004] note that larger upstream dunes are needed to cause ejection and splitting behaviors than absorption, a finding also reached in the numerical model of Katsuki et al. [2005], and this highlights the possible increased significance of flow separation over these larger forms. These three modes of barchan behavior can also be viewed in the light of the results from the present study which suggest that superimposition may provide the downstream turbulence and potential for erosion/sediment transport that can lead to ejection and splitting. In these cases, the higher upstream bed forms are likely to increase turbulence associated with leeside separation enough to cause significant downstream erosion. Additionally, the limitation of sediment supply by the upstream form to the downstream barchan may also provide a mechanism by which the downstream dune may become temporarily stabilized in its position and may favor later absorption.

5. Conclusions

[28] This paper has presented experimental data detailing the changes to the mean and turbulent flow fields of simplified, fixed, triangular roughness elements that are representative of the ripple-dune bed form transition. Four principal conclusions can be drawn from this work:

[29] 1. The presence of a smaller bed form superimposed on the stoss side of a ripple at the ripple-dune transition produces higher levels of turbulence intensity, turbulent kinetic energy and Reynolds stresses than either over ripples or the amalgamated dune bedstate.

[30] 2. Increases in the production of turbulence during superimposition appear due to interaction of the separation zone shear layers associated with flow separation in the leeside of each bed form.

[31] 3. Bed form superimposition and amalgamation increases the height of the near-bed region in which quadrant 4 events are dominant on the downstream bed form, and also increases the downstream area over which this dominance persists. The importance of near-bed quadrant 4 events is likely to increase the transport of bed and suspended load sediment across the ripple-dune transition, as has been recently reported by Schindler and Robert [2004].

[32] 4. These data support the previous contentions of Bennett and Best [1996] and Schindler and Robert [2004, 2005] that the ripple-dune transition is linked to significant changes in the mean and turbulent flow field that causes large changes in the sediment transport over these bed forms.

[33] The present study has illustrated that bed form superimposition significantly influences the mean and turbulent flow field, and suggests further research is required to fully investigate the effects of superimposition on turbulence production, periodicity, flow resistance, sediment transport and developing bed morphology.

Notation
The following symbols are used in this paper:

 

equation image

flow depth.

Fr

Froude number equation image.

FTKE

vertical flux of turbulent kinetic energy.

g

gravitational acceleration.

h

bed form height.

hs

height of bed form crest.

H

threshold value in quadrant analysis.

Ir

negative velocity probability.

p

pressure.

P

turbulence production.

pdf

joint probability density distribution.

q

turbulent kinetic energy.

Re

Reynolds number equation image.

u*

mean shear velocity.

equation image, equation image

mean streamwise and vertical components of velocity, respectively.

u′, w

instantaneous fluctuation from the mean value of u and w, respectively.

urms,wrms

root mean square values of turbulent fluctuations corresponding to u and w, respectively.

x, z

streamwise and vertical coordinates (with velocity components u and w, respectively).

xr

downstream length of flow separation zone.

zc

height above bed at which ΔSH ≤ 1 (i.e. quadrant 2 events become dominant over quadrant 4 events).

ΔSi,H

difference between contributions to Reynolds stress due to i event for size hole H.

ΔP

deviations in pressure from hydrostatic pressure.

δ

thickness of roughness layer.

κ

von Kárman constant.

λL

wavelength of larger bed form.

λs

wavelength of smaller bed form.

μ

molecular viscosity of water.

ρ

density of water.

τ

bed shear stress.

τo

threshold bed shear stress for sediment entrainment.

τR

Reynolds stress (=−ρequation image).

Acknowledgments

[34] We are grateful for the financial support of CONICOR-British Council and SECyT-UNC that provided funding for FL to visit Leeds to undertake this collaborative project, and for JB to pay a reciprocal visit to Córdoba. RF would like to thank the National Institute for Water and Environment, Argentina, for a scholarship under which much of this analysis was undertaken. JB is grateful to the UK Natural Environment Research Council for continuing support of the Sorby Fluid Dynamics Laboratory through grants GR3/8235 and GR3/10015 that facilitated this work. The writing of this paper was enabled whilst JB was in receipt of a Leverhulme Trust Research Fellowship, and that was partly conducted at the Ven Te Chow Hydrosystems Laboratory, University of Illinois at Urbana-Champaign. JB is indebted to Marcelo García and UIUC for hosting him and making facilities available during this time. Our thanks also to Rob Schindler and André Robert for provision of Figure 1b. Lastly, we are extremely grateful for the thoughtful and constructive reviews of Jeremy Venditti, André Robert and an anonymous Water Resources Research reviewer, together with the editorial advice of Marc Parlange.

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