Journal of Geophysical Research: Earth Surface

A flume experiment on the development of subaqueous fine-gravel dunes from a lower-stage plane bed

Authors


Abstract

[1] Determining the hydraulic conditions whereby gravel dunes first develop in subaqueous environments is fundamental as their presence may influence engineering solutions designed to maintain bed stability. In addition, estimates of the flow conditions associated with preserved gravel bed forms in sedimentary sequences are useful for reconstructing the depositional environments and geometries of, for example, oil- and gas-bearing geological strata. Consequently, a series of experiments considered dune initiation. In these experiments, defects and latterly incipient dunes developed from lower-stage plane gravel beds during near-threshold conditions of motion (θ/θcrit = 1.0–1.016) and long periods of marginal bed load transport rates. The three-dimensional defects were almost imperceptible positive ovoid features with heights of one or two grain diameters and lengths and spans of a few decimeters. After 17 hours of flow, incipient, low-amplitude, simple two-dimensional dunes developed from the defects, with heights ranging between 0.029 and 0.055 m, nonequilibrium wavelengths of 1–4 m and spans of 0.6–0.9 m. Continued development over several days, with θ/θcrit ratios of around 1.3, resulted in near-equilibrium two-dimensional dunes with wavelengths averaging 2.6–3.5 m and spans equal to the flume width (4 m). The inception of incipient dunes could be predicted using bulk flow models; however, this approach was not suitable for the prediction of defect development. Near-bed turbulence, in the form of small-scale sweep events of limited breadth, controls the initiation of defects, but larger-scale, coherent turbulent structures in the outer flow are related to dune development. Significantly, both defects and incipient dunes can exist at the same time, which indicates that the effects of sweeps on the bed morphology persist at the same time as larger-scale turbulent structures are beginning to effect sediment transport.

1. Introduction

[2] Defining the initial hydraulic conditions whereby bed forms develop in gravel is fundamental to aspects of river and shallow marine fluid dynamics because bed forms affect flow resistance, water levels and navigation depths and so their presence may influence engineering solutions designed to maintain bed stability. In addition, geologists frequently need to estimate the flow conditions over bed forms preserved within gravel-rich sedimentary sequences to provide robust reconstructions of the depositional environments and, for example, geometries of oil- and gas-bearing geological strata. Despite this, there are few controlled experiments concerning the initial development of bed forms in gravel from mobile lower stage plane beds (LSPBs) [Carling, 1999]. This paper primarily considers this issue but also reports briefly on the evolution of steeper gravel dunes. Here, structured groupings of a few grains as positive bed features are termed “clusters” whilst incipient, strongly three-dimensional bed forms appearing as vague humps on an otherwise planar bed are termed “defects.” Various cluster geometries consisting of discrete groups of clasts one or two grains high, or networks of grouped particles, have been reported on LSPBs [see Strom et al., 2004]. However, it is not clear if these are precursors to bed forms of greater amplitude, specifically dunes. Alternatively, sometimes it is argued that rapidly migrating gravel sheets on LSPBs are precursors to dune development [see Carling, 1999, and references therein]. The two mechanisms are not mutually exclusive but the incipient conditions for gravel dune development have never been defined. Broad flumes are required to negate sidewall distortions of defect geometry and subsequent dune height and planform [e.g., Crickmore, 1970; Menduni and Paris, 1986]. Defects initially are without steep leeside faces, have heights <20 mm and are the precursors to incipient dunes. The latter are small, flow transverse bed forms with heights >20 mm and steep lee sides. For simplicity, bed forms developed with long, straight, or slightly sinuous crest lines that are approximately transverse to the flow are termed two-dimensional (2-D) dunes. Similarly, dunes with long, highly sinuous crest lines, or short, recurved crest lines are termed three-dimensional (3-D) dunes [after Allen, 1968, 1969; Costello and Southard, 1980].

[3] The experiments were conducted in the 160 m long flume (Figure 1) within the Water Resources Center of Tsukuba University, Japan [Ikeda, 1983]. The flume width is 4 m, water is recirculated and there is an option to recirculate sediment. In these experiments, the sediment recirculation option was only used in run numbers 7 and 8. At the end of each run the flume was slowly back flooded by closing an undershot tailgate whilst first maintaining, and then progressively reducing, the inflow until ponded water filled the flume. The flume was then drained slowly by slightly raising the tailgate. In this manner, bed form development could be arrested, and the bed forms exposed, without rapid drawdown modifying the dune shape. Subsequently bed form migration could be reinitiated, without any disturbance, by slowly back flooding the flume using a hosepipe with the tailgate closed, and then incrementally increasing the inflow whilst slowly opening the tailgate. The slope of the steel channel is fixed at 0.01 but the slope of bed sediments is adjusted by altering the height of an exit weir that retains the sediment filament within the flume. The water depth was controlled, by adjusting both the discharge, weir height and bed sediment gradient.

Figure 1.

Illustration of the flume, which is 2 m deep, 4 m wide, and 160 m long. Water can be supplied at a maximum rate of 1.5 m3 s−1. Water from the lower reservoir (A) is pumped up (B) to flow through the main flume; the water then returns to the lower reservoir. Sediment discharged with water from the outlet is returned upstream by conveyors after its weight is measured in the sump chamber (C). The sediment is introduced to the flume either directly (D) or from the sediment feeder system (F) after the original sediments are sieved (E).

[4] The observations on bed defects and incipient dunes reported herein were part of a broader study of the plan view evolution of dunes from predominately 2-D to 3-D morphologies through time [cf. Baas, 1994; Baas et al., 1993] and from shallow to deep waters. Although it would be preferable to have near-bed turbulence data to relate to defect development, bulk flow data relate well to large-scale changes in planform and this was the nature of most of the flow data collected. The defects and incipient dunes, reported below, developed within near-uniform flow that pertained within a short length of the flume between 85 and 115 m. Eight experimental runs were conducted, including the four for which defects and incipient dunes were observed. Turbulence data are only available for incipient dunes during run 5 and for well-developed bed forms in runs 6 and 7. The test results for run 3 are not reported here, as a buildup of leaves (for that test alone) on the intake screen meant that the flow data were suspect.

2. Method

[5] The flume was filled with 5 mm gravel (Figure 2 and Table 1) from 50 m downstream of the entry section (0 m) as far as the tailgate (160 m) thus providing a bed sediment filament varying from zero thickness upstream to 1 m thick downstream. For runs 1 through 4, during which defects were observed, the bed surface was screeded flat to produce a bed of uniform grain size distribution both vertically and spatially. For runs 5 through 8, the bed gradient was allowed to develop through the marginal sediment transport process. Judging from the representative grain size curves (Figure 2) significant bed armoring by surface segregation processes did not occur during runs 1 through 4. Instead, the coarsest fractions (16 mm) tended not to be transported but remained in the troughs of evolving bed forms. The fractions between 2 mm and 12 mm were transported and contributed to bed form evolution. The lee sides of incipient dunes were deficient in the coarsest fractions (>12 mm) and the finest fractions (<4 mm), this partitioning is explained by differential transport, whereby the coarsest grains form a lag on the stoss side and entrap the finest fractions. The study reach, in which incipient bed forms were observed, was characterized by a gentle bed gradient, shallow flow depth (h = ∼0.367 m; standard deviation = 0.011) and the presence of a LSPB with or without defects and incipient low-aspect dunes.

Figure 2.

Probability plot of representative grain size distributions of bed material sampled from an incipient dune. More curves are not shown as they would be coincidental.

Table 1. Settling Velocities of Selected Grain Size Fractions
Grain Size Fraction equation imageDensity, gm cm−3Settling Velocity, m s−1
−3.0 to −2.52.570.3758
−2.5 to −2.02.570.3264
−2.0 to −1.52.580.2889
Bulk sample2.53-

[6] To ensure two-dimensional flow, the width/depth ratio (B/h) was greater than 5 [Nezu and Nakagawa, 1993], the ratio (h/D) of the average flow depth (h) to particle size (D) was greater than 3, such that no water surface effects would interfere with the bed processes and the critical shear stress is independent of the h/D ratio [Bettess, 1984]. Wall effects clearly distorted the crest lines of transverse straight-crested dunes for a distance of ∼100 mm from the flume sides. This is typical of shallow flows [Vanoni and Brooks, 1957] and so no defect or incipient dune data were collected from these marginal areas. A carriageway was equipped with a bed surface indicator, a water surface point gauge and a 10 MHz 5 cm Sontek acoustic Doppler velocimeter (ADV) [Kraus et al., 1994]. The height and lateral position of these instruments could be selected to within 10 mm using digital position monitoring. Sediment and water surface elevations were recorded at 5 m downstream intervals (i.e., from x = 70 to 160 m) at 0.5 m intervals across the flume width. For completeness, manometer readings were recorded at 69, 89, 122 and 154 m along the length of the flume. Together, these data provided water and bed surface slope information, but the measurements of the bed topography were not adequate (10 mm accuracy) to define the dimensions of defects and the smaller incipient dunes. Fortunately, the outlines of the defects were distinctive; differential drying of the bed caused raised areas to appear lighter in color than depressed areas. Consequently, a spirit level and millimeter graduated rule were used to measure carefully the geometry of defects and incipient dunes. A digital inclinometer was used to estimate the bed form slopes. However, for the very “flat” features, with heights of one or two grain diameters, values of the bed form slopes should be regarded as indicative rather than absolute.

[7] Unrotated average (5 Hz) downstream ADV flow speeds at 0.6 of the depth below the water surface were filtered and integrated over 120 seconds. Vertical photographs of the full 4 m width of the drained flume, including bed forms, were obtained using a digital camera mounted 3.92 m above the gravel surface on a mobile gantry with illumination of the test section from a halogen lamp. An ERC bed load sampler (with mouth width of 100 mm [Ikeda, 1983]) was deployed occasionally in the defect/incipient dune region at 105 m.

[8] With flowing water in the large flume, it was not possible to observe the bed as the water was discolored, being supplied from an outdoor pond. The critical shear velocity (u*crit) for initial motion was determined using a range of methods [Thompson et al., 2003] that consider the energy slope (u*E = 0.0559 m s−1), the Reynolds stresses (u*RE = 0.0192 m s−1), the turbulent kinetic energy (u*TKE = 0.0274 m s−1) and Prandtl's seventh power law (u*P = 0.0339 m s−1). In this manner, and given τcrit = ρu*crit2, the critical shear stress (τcrit) was determined to be >0.4 < 3.1 Pa (u*crit < 0.0559 m s−1; nondimensional critical shear stress, θcrit < τcrit/(ρs − ρ)gD50 = 0.0386) for the stable bed:LSPB transition. This range of threshold values is less than the average (3.6 Pa) reported for gravel bed load in fully turbulent flow (see reviews by Komar [1996] and Buffington and Montgomery [1997]) and so is consistent with marginal transport rates. Nevertheless, it was observed (when draining the flume) that, for these conditions of initial motion, where just a few grains per second were being entrained from any 1 m2 flat bed area, defects and incipient dunes had developed after several hours. To better constrain the stable bed:LSPB:dune transition, the same bed material was placed within a 0.3 m wide flume with a working section of 4 m. Here slightly higher, but still marginal, transport rates were observed in uniform clear water flow. This latter condition was associated with a critical shear stress of 4.69 Pa (u*crit = 0.068 ms−1; θcrit = 0.057) for marginal bed load transport rates developed within a few seconds; thus it was concluded that within the large flume, the LSPB:dune transition occurred within the θcrit –range: 0.0386–0.057. This range is consistent with a number of studies of initial motion of gravel on LSPBs where the general θcrit value for incipient motion is typically 0.045 [Komar, 1996] but incipient dunes are possible [Carling, 1999, Figure 4]. For brevity, later in this paper only energy slope-derived shear stresses are reported. The settling velocities (ω) of three discrete size fractions (Table 1) were measured in still clear water at 12°, for a 3 m drop within a 10 m high setting tube of 0.3 m diameter. The purpose was to satisfy the Rouse criterion, such that no suspended motion should occur in the tests (i.e., z = ω/κu* = >5 [see Strom et al., 2004] wherein κ denotes the von Kármán constant ≈ 0.4). The Rouse numbers in these tests were z > 12. Subsequent to individual runs, the general dimensions of vague bed defects were noted and the height (H), length (L) and stoss and lee side slope angles were measured for a total of 429 incipient and well-formed dunes.

3. Summary Bed Form Development

3.1. Incipient Bed Forms

[9] Data reported in this section are for near-uniform flow between 85 and 115 m (Figure 3) wherein, for each test run, flow was fully turbulent (Reynolds number, Re ∼ 105) and subcritical (Froude number, Fr ∼ 0.54; standard deviation, 0.008; see Table 2). Run 1 established threshold sediment transport conditions for the stable bed:LSPB:incipient dune transition. The nondimensional shear stress during run 1 was 0.0386 throughout the section, such that incipient motion on a plane bed occurred (Fr = 0.52; h = 0.375 m). Nevertheless, small strongly three-dimensional defects and latterly a few short wavelength (1–1.5 m) low-amplitude dunes were evident after the 17 hour run (Figures 4 and 5) . These were of limited span (typically 1–2 m) with straight, flow-normal, or slightly skewed crest lines. During the 16.5 hour run 2, the nondimensional shear stress increased in a downstream direction from 0.0386 to 0.055, whilst during the 22.75 hour run 4 the nondimensional shear stress was sustained throughout the reach at 0.055. For both of these latter runs, the Froude number was 0.5 and the water depth was ∼0.375 and 0.382 m respectively. In both cases, long wavelength incipient straight-crested transverse dunes with distinctive spacing developed (Figure 6). These bed forms had lengths of between 0.95 m and 5.67 m whilst heights averaged 0.022 m (range: 0.011 m to 0.047 m). Stoss slopes remained slight, averaging 0.63° (range: 0.1° to 1.3°) but lee sides rapidly became steep ∼20° (range: 15° to 32°). The steepest lee slope measured was 49°, well in excess of the measured angle of repose: 32° to 34°, and this requires some comment. Usually steep lee side angles are associated with angular material and widely graded sediments [Julien, 1995; Nakagawa and Miyata, 2002]. The bed material in the present experiments was subangular but narrowly sorted (Figure 2). It appears that the angularity of the sediment [Makse et al., 1998] coupled with the absence of highly turbulent, fully separated flow in the leeside of these gravel dunes occasionally precluded the avalanching of sediments that usually reduces leeside slopes to angle of repose.

Figure 3.

Variation in bed and water surface elevation during runs. Symbols represent measurement points for upstream and downstream reaches.

Figure 4.

(a) Strongly three-dimensional ovoid defects of limited span. Transverse shadowing defines downstream fronts of defects as shown in sketch. (b) Irregular defect. (c) Two-dimensional incipient dune of limited span evolved from defect. (d) Incipient 2-D dune evolved from defect with defects (arrowed) on stoss. In Figures 4a and 4d, longitudinal stripes are due to condensation dripping from roof struts. Flow is top to bottom of images. Scale bar is in 20 cm divisions. All images are from run 2 and effectively represent a time series of bed form development.

Figure 5.

(a) Plot showing defects develop distinctive leeside slopes (shadowed). (b) Distinct ovoid defects with steep (shadowed) lee sides. (c) Two-dimensional incipient dune of limited span evolved from defect. Note secondary crest line developing on the stoss (arrowed). In Figures 5b and 5c, longitudinal stripes are due to condensation dripping from roof struts. Flow is top to bottom in the images. Scale bar is in 20 cm divisions. All images are from run 4 and effectively represent a time series of bed form development.

Figure 6.

(a) Long wavelength incipient straight-crested transverse dunes with distinctive spacing. Longitudinal stripes are due to condensation dripping from roof struts. (b) Defects present between 2-D dune crest lines. Some that appear lighter in the image are arrowed. Note that the wavelengths of the transverse dunes (∼3–4 m) are too large for flow separation at the crest of one dune to affect the evolution of the next crest line downstream. Flow is top to bottom. Scale bar is in 20 cm divisions. All images are from run 2.

Table 2. Summary Data for the Test Section 70–100 m Where Defects and Incipient Dunes Were Observed During Runs 1–5 and Small Dunes During Runs 6–8
RunDischarge,a m3 s−1Cumulative Duration, hoursTemperature, °CBed SlopeWater Surface SlopeWater Depth, mMean Velocity, m s−1Froude Number
  • a

    Nominal discharge from in-line meter.

  • b

    Nominal values.

  • c

    Flow was nonuniform, increasing in depth down section, so data are exemplary values only.

11.517161/3184b1/678b0.3751.000.52
21.533.5161/3184b1/678b0.3751.000.52
41.522.75161/31841/6780.3820.980.51
51.528.216.51/5261/6960.4200.890.44
61.523.5161/526b1/8360.3501.010.57
7c1.552141/10871/11110.2111.060.74
8c1.573141/5881/6250.3501.100.60

[10] The bed was screeded flat after run 2 and by the end of run 5 (a total of 28.2 hours duration) the nondimensional shear stress was increased further (θ = 0.067 to 0.074). For θ = 0.067, a single bed load sample provided a transport rate of 0.052 kg ms−1. For these latter conditions, dunes were of similar wavelength (3–4 m) as noted during previous runs, but the heights of the dunes degraded slightly. Many of these dunes had gentle lee sides, but others had steep lee sides, such that flow separation should have occurred at each crest line [Dyer, 1986]. However, in each case there was no visual evident of scour pits [e.g., Dalrymple et al., 1978; Costello and Southard, 1980] in the dune troughs, nor erosion of the bed at a distance downstream where flow reattachment might be anticipated for isolated bed forms (e.g., 8 to 10H [Allen, 1984a; Best, 1996; Wilbers, 2004]). Instead, the bed downstream of each dune lee-toe was planar. The stoss toe of each dune was not defined by a break of slope, rather, the stoss sides gradually increased in gradient to a maximum of 1°, or so.

[11] The manner of initial defect development upon a LSPB, giving rise to incipient dunes in this narrowly graded fine gravel, was not evident visually, although from studies of broadly graded sand and gravel beds one would expect the formation of small particle clusters [see Best, 1996] as precursors to development of defects [Williams and Kemp, 1971; Gyr and Schmid, 1989]. Formerly, it has been assumed that broadly graded sediment is required for cluster development such that larger obstacle clasts form stationary nuclei around which smaller particles cluster [see Strom et al., 2004, and references therein]. Recently, Papanicolaou et al. [2003] and Strom et al. [2004] demonstrated that broadly graded sediment is not required, as simple clusters developed on beds of 8 mm diameter spheres for conditions of marginal transport. In contrast to beds of spheres, the present test bed material is more natural. Once drained, and as the bed dried, topographic highs became distinct as lighter-colored sediment in contrast to topographic lows in which sediments appeared darker in color, being still damp. Close inspection of the bed showed slight positive bed features had developed with heights of only a couple of grains, negligible stoss and lee slopes, wavelengths of decimeters up to 0.4 m, and crest spans of some decimeters. These bed defects often were noticeably ovoid with spans typically two to three times greater than the flow-parallel lengths (Figure 7). With further exposure to flow, steep lee faces with breadths of a few centimeters and heights of one to three grains developed transverse to the flow (Figures 4a, 5a, and 5b). In time the distinctive lee sides propagated laterally forming more readily identifiable 2-D transverse bed forms; precursors of low-amplitude 2-D dunes (Figures 4b, 4c, 4d, 5b, and 5c).

Figure 7.

Defect span and length relationships.

[12] Although initial groupings of a few grains must occur, no initial small-scale (e.g., tens of mm2) distinct, prominent and stable clusters of grains were noted in this uniformly sized sediments, in contrast to observations of clusters in broadly graded sediments. Rather defect development appeared to occur owing to vertical accretion over a relatively large bed area (e.g., tens of cm2, see Figures 4 and 5). In addition, once incipient dunes with distinctive crest lines were present, the morphologies of both defects and dunes were remarkably similar to those incipient bed forms reported from well-sorted sand beds by Gyr and Schmid [1989, Figures 3 and 4]. Thus narrow grading in both sand and gravel might preclude cluster formation and instead promote rapid defect development. However, whereas the defects reported by Gyr and Schmid had lengths and breadths of around 20 mm, the gravel defects noted herein were an order of magnitude larger. Additional distinct bed defects (a couple of grains high with crest line lengths and flow-parallel lengths measuring a few centimeters) could readily be discerned, usually high on the stoss sides of existing dunes (Figures 4d and 5c) and rarely near the stoss toes. These distinct defects, which eventually developed into new bed forms, formed on the stoss slope of existing bed forms of any length. For example, the crest line of one short wavelength incipient dune (L = 0.95 m) began to degrade, in height and lee side angle, as a new bed form began to develop immediately upstream of the crest line of the parent dune. More usually, bed form wavelengths exceeded about 4 m before minor topographic defects appeared and began to form new bed forms in intermediate positions (Figure 6b).

3.2. Two-Dimensional Dunes

[13] During runs 1–5, defined 2-D bed forms, with steep lee sides, developed downstream of 115 m. These had regular, straight or slightly sinuous crest lines that were essentially flow-normal or slightly skewed, exhibiting spans of 2–4 m, which thus often extended across the full width of the flume (Figure 8). The 2-D dunes of run 1 lengthened from 0.70 to 0.85 m and grew in height from 0.039 to 0.1 m during run 2 as θ increased in a downstream direction from 0.056 to 0.077. As a result, average stoss slopes increased from 5.2° to 6.7° and lee slopes from 27° to 37.5° (Figure 9). However during runs 4 and 5 the nondimensional shear stress fell to 0.05 or less, and the 2-D dunes were prograding downstream into water of increasing depth (0.42–0.74 m), concomitantly the Froude number fell to less than 0.44. Given the downstream reduction in shear stress and flow speed, the dunes in run 5 degraded somewhat in height. However, throughout these runs it was observed that as dunes migrated into deeper, slower-flowing water very steep dunes (H:L ∼ 0.12 to 0.14) with steep lee slopes could develop. The steepest lee slope measured was 49°, well in excess of the measured angle of repose: 32° to 34°. Comment made above for incipient bed forms also applies here. For these two-dimensional dunes, migrating slowly into relatively quiescent water, over steepening may be sustained by the absence of highly turbulent flow. However, these dunes tended to degrade, as they slowly migrated still further downstream (beyond 130 m) into deeper waters (Fr < 0.19; θ < 0.05) where the dune:LSPB transition pertained.

Figure 8.

Strongly two-dimensional transverse dunes with ∼0.5 m wavelength and 0.05–0.10 m high. Flow is top to bottom. Note wall drag effect extends to 100 mm from channel margins. Scale bar is in 20 cm divisions. Image is from run 6.

Figure 9.

Stoss and lee slope angles for incipient 2-D dunes.

[14] The incipient low-amplitude 2-D dunes were not depth limited. Allen [1984a] for example demonstrates that fully developed dunes usually exhibit H:h ratios of 0.12 before flow blocking and accelerated flow speed above the crest induces a depth limitation, causing crestal flattening. In contrast, the incipient and well-developed 2-D dunes (for runs 2, 4 and 5) had relative heights ranging between 0.025 and 0.12, averaging 0.06. Thus these dunes, without obvious toe scour, appear to have been equilibrium responses to the imposed flows, the bed form morphology being maintained over several hours in each run. Costello [1974] and Costello and Southard [1980] describe similar long wavelength, low-amplitude, 2-D dunes in coarse sand. Best [1996] has argued that this class of bed form is distinct from fully developed 2-D and 3-D dunes, in as much as erosion at the point of flow reattachment (downstream of steep lee sides) plays no effective role in the dune-building process. This supposition appears to be confirmed in the majority of the present examples of incipient dunes. Specifically, the increase in the stoss slope scour that drives sediment transport must be large relative to the value in the vicinity of flow reattachment. This conclusion can be inferred because the incipient and low-amplitude 2-D dunes clearly migrated downstream. Thus gravel particles must be entrained from the stoss sides, and deposited on the lee sides, facilitating downstream migration. However, no scour hollows could be detected in the expected region of flow reattachment, rather the bed remained plane. Thus it must be inferred that either scour was present but is not measurable (i.e., < ∼1D50), or sediment for dune building is sourced from upstream of the dune field and moved over a LSPB to develop dunes. Any slight irregularities, developing in the lee of these dunes (which could form the bed defects leading to further dune development) would be overrun by the advancing lee side of existing dunes. It should be noted that a few steeper and higher dunes developed within the field of incipient dunes at the end of run 5 as the shear stress (θ > 0.074) and flow speed were increased. Additionally steep flow-transverse 2-D and 3-D dunes developed when the shear stress was further increased and these exhibited leeside angles commensurate with the presence of flow separation (e.g., 10°–15° [Best and Kostaschuk, 2002]). Thus, although it was not possible to discern the presence or absence of flow separation in any of the runs, it would seem that the development of separated flow is instrumental in controlling the phase transition from incipient 2-D dunes to well-developed 2-D dunes exhibiting increased crest line sinuosity and bifurcation, leading to the development of 3-D dunes.

3.3. Three-Dimensional Dunes

[15] Prior to run 6 the bed was screeded flat. On starting the flume, standing waves (due to initial unsteady flow) were observed at 112–130 m above the initially flat bed, but these reduced in height and spread upstream and downstream as flow became steady and more uniform. However, nonbreaking (and a few slightly breaking) standing waves were locally persistent throughout the rest of the run during which flow was steady. During run 6 the bed load transport rate was ∼0.046 kg m−1s−1, Froude number was 0.61 at x = 70 m increasing to 0.69 at x = 145 m. On draining the channel, dunes extended the full length of the flume to 155 m, but locally (where the standing waves had been observed) graded into bed forms which appeared to be symmetrical to slightly asymmetrical transitional bed forms; the latter steepest on the down-flume side.

[16] By the end of run 6 the difference between the slopes within the upstream and downstream sections was much reduced compared to run 4 (Figure 3). A clear change in bed form character was evident. In the upstream reach, incipient dunes had been replaced by steep 2-D dunes but (unlike the 2-D dunes in earlier runs) there were broad smooth, longitudinal spurs [after Allen, 1968] associated with the dunes but with a height less than that of the dunes. Typically, spurs extended downstream from lobes developed in the dune crest lines. At the downstream end of this section, crests begin to bifurcate and trend diagonally across the flume. This change in planform may represent a transition to more 3-D dunes. Dune crest lines here were more sinuous, and tended to run diagonally across the flume, with relatively sharp changes in direction, producing distinct lobes and embayments. The 3-D dune crests had shorter spans compared with the more 2-D dunes in the upstream section. They had smaller mean wavelength, but mean height and slope angles were essentially the same. Variations in measured wavelengths were owing to appearance of 3-D bed forms of shorter spans and particularly an increase in crest bifurcations. Locally, short span, low dunes with parabolic or lobate forms also developed initially with negligible amplitude. These poorly developed, feint bed forms indicate that new dune crests sometimes were initiated in a highly 3-D state. Those 2-D dunes with longitudinal low-amplitude broad spurs may be transitional between 2-D and 3-D, the spurs being forerunners of the lobate forms of 3-D crest lines.

[17] During run 7 Froude numbers increased steadily from 0.66 at x = 75 m to 0.75 at x = 100 m. The average Froude number for the downstream portion of the channel between x = 100 m and x = 145 m was 0.73. In this latter reach, a strongly 3-D, wavy regular planform was evident in the bed form crest lines where rooster tail standing waves had been observed breaking gently upstream. These bed forms were categorized as transitional dunes/antidunes.

4. Models of Bed Form Initiation From Lower-Stage Plane Bed

[18] Costello and Southard [1980] argued that the existence field for low-amplitude, straight-crested transverse 2-D dunes “pinched out” for grain sizes >2 mm. Given that 2-D fine-gravel dunes were reported by Ikeda [1983] and are described here in 5 mm gravel demonstrates that, the existence field in certain conditions should be extended to include coarser bed stocks [Carling, 1999]. However, the controls on defect development and the initial spacing and planform of incipient bed forms are not known. Note, in respect of the points made below, that as strongly 2-D incipient dunes first appear on the bed, vague 3-D defects still persist between 2-D dune crest lines.

[19] A number of theories have been proposed to account for the initiation of bed forms from a LSPB. Aspects of these models may not be mutually exclusive and some elements of each are considered later in this paper to provide a fuller explanation for defect initiation and growth into incipient dunes. For the purposes of this paper, these theories may be defined as falling within five groups.

[20] 1. Inherent mesoscale coherent flow structures acting over large bed areas define the initial spacing of bed forms [e.g., Liu, 1957; Folk, 1976; Richards, 1980; Sumer and Bakioglu, 1984]. Such models are derived from the potential flow analysis of bulk flow first presented by Anderson [1953] and Kennedy [1963] and latterly developed by others [see Carling and Shvidchenko, 2002]. This approach can explain the seemingly spontaneous appearance of self-similar 2-D bed forms over extended bed areas but otherwise provide only weak explanation for the small-scale initial 3-D bed forms spacings and, utilizing bulk flow parameters, provide no explanation of the fluid–grain interactions giving rise to bed form growth.

[21] 2. The development of a sweep-induced random isolated clustering of particles results in flow separation and the generation of a single isolated bed form [Gyr and Schmid, 1989]. Reattachment downstream of the bed form then causes a further bed form to develop and, in this manner, bed defects are generated progressively from a single point over increasing bed areas [Raudkivi, 1963; Southard and Dingler, 1971; Williams and Kemp, 1972; Leeder, 1980]. This mechanism is evidently initially local in effect.

[22] 3. A similar model to group 2 accounts for spontaneous bed form development over extended bed areas. Grass [1970] and Williams and Kemp [1971] proposed that spatially pervasive sweep impacts on a flat bed, once the threshold for particle motion is exceeded, induce bed defects to appear over broad areas. However, initial 2-D bed form spacings often are large which is not consistent with the close spacing of coherent energy dissipative structures [Pal et al., 2001].

[23] 4. Random, isolated, oversized or protruding particles serve as local nuclei for the formation of particle clusters. The literature does not agree on whether these clusters form and breakup randomly such that no equilibrium spacing exists across the bed, or that equilibrium spacing rapidly evolves through the bed load transport process (discussed by Strom et al. [2004]). Significantly, no studies have demonstrated the development of dunes from clusters.

[24] 5. Inherent transport instability develops so bed load flux becomes unsteady, leading to particle segregation and congestion, such that incipient bed forms occur, usually termed “bed load sheets” [Best, 1996]. These sheets [Carling, 1999, and references therein] may be the precursors to simple 2-D dunes [Best, 1996], but each sheet tends to migrate quickly as a coherent particle grouping, in contrast to the slower evolution of 2-D dunes by stoss side erosion and lee side deposition. There has been little comment on this celerity issue in the literature, and these sheets have not been shown unequivocally to evolve into 2-D dunes in water. However, similar particle groupings readily evolve into transverse and lunate bed forms in experimental subaerial [Prasad et al., 2000; Pal et al., 2001] and subaqueous conditions [Schmidt, 1989; Gyr and Schmid, 1989].

[25] Presently, there are no quantitative predictors for initial bed form characteristics in gravel and, to shape arguments presented below, it is worthwhile considering some predictors for sandy bed forms. Considering bulk flow rather than turbulence, Coleman and Melville [1996] and Coleman and Cornelius [1999] suggest that in steady unidirectional flow above LSPBs the initial wavelength (Li) of incipient bed forms in sand (D50 = 0.2 to 1.55 mm) is relatively insensitive to the flow, the latter characterized by the grain Reynold's number, Re*:

equation image

but instead is proportional to the median grain size of the bed stock [Coleman and Eling, 2000; Coleman et al., 2003];

equation image

which may be compared with a function for incipient sand ripples presented by Raudkivi [1997];

equation image

In equations (2) and (3) the constants and coefficients pertain to millimeter units. These three equations are not applicable to gravel as is explained below but, for the purposes of argument, the gravel data from the present study are shown in Figure 10 in relation to predictions obtained using Equations (1) and (3).

Figure 10.

Representation of bed form data in terms of the bulk flow parameters proposed by Coleman and Cornelius [1999]. Measured parameters from this study: circles, 2-D transverse dunes; horizontal bars, defects. Predicted normalized wavelengths: triangle, equation (1); diamond, equation (3).

[26] Although developed for sand ripples in flows of low Reynolds numbers, the form of equations (1), (2), and (3) might provide clues as to the physical mechanism underpinning the relationship between the grain size of the bed and the wavelength of incipient bed forms. Equation (1) indicates a weak inverse dependency of bed form wavelength on the Reynolds number of the flow such that, for a given grain size, as the Reynolds number increases then the incipient bed form wavelength decreases. Equations (2) and (3) demonstrate that as grain size increases then the relative length of the incipient bed form decreases. Engel [1981] and Yoon and Patel [1996] showed that if the bed stock is coarsened, then the length of the flow separation cell in the lee of bed forms decreases because increased turbulence is generated above larger grains and this inhibits flow separation. Thus, if a vertical length scale (H) is provided by a multiple of the grain size, the length of a flow separation zone (Ls) developed on the downstream side of an individual grain might control initial bed form wavelength. For example, Engel [1981] and Kadota and Nezu [1999] demonstrated that for solitary bed defects, Ls/H decreases as the Reynolds number increases to reach a constant value of about 6.5 when Re = 20,000. Consequently, as incipient gravel dunes form in fully turbulent flow (Re > ∼ 104) a constant relationship between incipient bed form wavelength and defect height would be anticipated if grain size is the appropriate scalar. Note that equations (1), (2) and (3) predict Li/D50 ratios that are too large by a factor of 10 to 20 when compared with the separation lengths widely reported as typically five to ten times the height of isolated bed defects (see review in the work of Wilbers [2004]). Thus flow separation associated with individual grains does not seem to be the controlling factor mediating incipient bed form wavelengths in either sand or gravel. Separation ratios of Ls/H = 8 to 10 as reported in the literature are equivalent to separation angles of around 30 degrees. In contrast, the results of equations (1) and (2) (Ls/D50 ∼ 100) would indicate separation angles of less than 6 degrees. For such low angles, it is difficult to envisage coherent separation zones, emanating from single grains, continuing to exist downstream above multiple queues of gravel-sized grains as the turbulent flow structure immediately above the bed integrates the effects of multiple roughness elements and cannot be related to the local influence of any one bed element [Grass et al., 1991]. Thus there is no clear grain-related length scale to associate with an incipient bed form wavelength.

[27] Fries [2004] addressed the eddy coalescence problem by arguing that when the frequency of eddy formation is greater than the frequency of eddy coalescence (ff > fs ≈ Re*H > 1400equation image) then increased turbulence will occur downstream of a given obstacle of height H, drag coefficient Cd and obstacle Reynolds number Re*H. For this condition he argued incipient bed forms would appear when Re*HT ≥ 70, where T = τ/τcrit. This approach, although similar to that of Coleman and associates is not restricted to initiation of sand ripples, and applies reasonably well to the present 2-D dune data when H is considered as a multiple of D (Figure 11).

Figure 11.

Representation of bed form data in terms of the bulk flow parameters proposed by Fries [2004]. The drag coefficient in the present experiments increased as the nondimensional shear stress was increased. Diamonds represent calculated points to define: Re*H = 77T. Two-dimensional transverse dunes (circles) from this study occur when Re*T = 70 and H is defined as 4D ≈ 20 mm. Defects (H < 4D) in the present study occur at R*H ∼ 275–280 for T ∼ 1–1.02 but are not represented.

[28] The wavelengths of incipient 2-D dunes, when observed for conditions of incipient motion, were less than 1.0. m and further, as shear stress increased the bed form spacings increased rapidly to around 1.2 m (see (Re0.2)L/D = 740 in Figure 10). Consequently, these data permit an estimate of the minimum wavelength at inception. An extrapolated of the polynomial function relating the nondimensional excess shear stress with the nondimensional wavelength (Figure 10) indicates that, for incipient motion (T ≈ 0.98), wavelengths would be ∼ 0.29 m (see (Re0.2)L/D = 179 in Figure 10). Although the data in Figure 10 are few, the scarcity of individual measures of incipient 2-D dune wavelengths less than 1.0 m (see (Re0.2)L/D = 617 in Figure 10) might indicate that, in fine gravel as opposed to sand, wavelengths at inception are relatively long. Further, for conditions of low excess shear stress, it was evident that one solitary dune did not first appear and then, by its presence induce the development of a train of additional dunes. Indeed initial crest spacings are too large and bed form heights to small to generate downstream evolving turbulent flow structures capable of remaining coherent and initiating a further downstream bed form. Rather, as several, “irregularly spaced” and/or “uniformly spaced” low-amplitude 2-D incipient dunes appeared “spontaneously” it may be deduced that an inherent resonance between the fluid and the plane mobile bed material dictates the initial average spacing of 2-D dunes. This method of initiation is not consistent with local turbulence at bed defects being responsible for engendering the 2-D bed form development. Rather it is consistent with the application of larger-scale flow instability mechanisms [Coleman and Melville, 1996; Coleman and Fenton, 1996].

5. A Model for Initiation of Gravel Dunes From LSPB

[29] When the size of individual grains on a LSPB exceeds 4(ν/u*), where ν is the viscosity, then the viscous sublayer is disrupted and flow perturbations develop downstream of each grain [Müller and Gyr, 1996]. Flow perturbations interrupt sediment transport, induce deposition and there is a propensity for bed defects to develop. As the vertical expression of any defect increases, there is a propensity for lateral and longitudinal development of the defect. According to Gyr and Müller [1996], for a rough turbulent boundary, quasi-circular evenly spaced defects with height-breadth ratios of 0.3 develop when the nondimensional grain size D+ = D/(ν/u*) is within the range 12 < D+ < 70. In test 1, D+ = 25, and for this condition the lateral sweep dimension (λ) should be of the order of 10 mm according to Gyr and Müller [1996]. Such a dimension equates to 2 to 3 grain diameters and this scaling reasonably should characterize the lateral dimension of initial bed defects in the present tests. It was not possible to see initial bed defects developing at this scale in the present series of tests but initial isolated grouping of a few particles, proportionate in size to the length-scale of sweeps, is a logical precursor to the (rapid) development of the low-amplitude mounds of greater span observed in the present tests. Significantly, the development of small-scale irregularly spaced 3-D bed forms cannot be reconciled with a formative 2-D flow disturbance as associated with stability analysis, rather interacting higher-frequency smaller-scale near-bed sweeps seem appropriate. Grass et al. [1991] argued that the initial value of λ leading to defect development is constant and proportional to the roughness length ks, where ks ≈ 1D. Grass and Mansour-Tehrani [1996] and Defina [1996] showed that for 3–12 mm spheres and small pebbles, λ ≈ 3–4 D.Best [1992] argued that sweep impacts are grouped, such that entrainment and deposition occurs in “patches” that have dimensions greater than the individual sweep events. Williams [1990] showed that centimeter-scale sweep-induced plane bed disturbances in gravel propagate laterally over breadths of several decimeters. Thus for grouped sweeps, or laterally propagating sweeps, λ ≫ 3–4 D is consistent with the rapid lateral propagation of individual, or grouped, sweep-induced gravel transport [Williams, 1990]. This is in accord with the rapid lateral development of the defects observed in these experiments and in the field observations of patch development in 4 mm river gravel [Drake et al., 1988] and coarser marine gravels [Thorne et al., 1989]. Once defects exist, topographic forcing of the flow will induce spanwise variation in the bed shear stress distribution enhancing bed form development.

[30] For poorly understood flow conditions, but at increased turbulence intensities, Gyr and Schmid [1989] and Gyr and Müller [1996] commented that 3-D defects can develop into 2-D transverse bed forms. For this latter development, the relatively short length-scale of sweeps is replaced with a longer length-scale proportional to the intermittency of larger flow structures that scale with the boundary layer thickness or, in shallow flow, with the flow depth [Allen, 1984b]. At this scale, it is possible for the pressure field, produced by muted water surface waves or outer layer large-scale periodic turbulence [Gyr and Schmid, 1989; Ferguson et al., 1996], to generate preferred 2-D bed form spacings. The length-scale of the incipient dunes that result might reasonably be obtained by a consideration of the time and length scales of larger turbulent flow structures: TU = 5–6 h, where T is the time between events and U is the free stream velocity [Gyr and Müller, 1996; Raudkivi, 1997]. For test 1 this corresponds to lengths of 1.8–2.2 m which is consistent with the visual observation that 3-D defects scaled in decimeters rapidly were augmented or replaced by transverse incipient dunes with wave lengths of 1–2 m; this being consistent with Allen's [1984b] general scaling relationship: L = 3.35h for incipient 2-D bed forms. This approximate scaling was addressed further by consideration of time series of ADV turbulence data for run 5. These data are not ideal for understanding processes active at the bed level; being obtained at 0.15 m above the bed in a 0.42 m flow depth. However, outer layer periodic turbulence penetrating to the bed produces similar scale near-bed turbulence [Ferguson et al., 1996]. Thus, assuming a constant stress layer extends upward throughout the majority of the shallow flow, the data are used to estimate the scale of eddies shed into the outer flow in association with the growth of incipient 2-D bed forms. Twenty-one independent FFT power spectra of the u′ flow component exhibited a spectral slope approximating to −5/3 over the frequency (f) range 10−1f〉 100. Being therefore in accord with other laboratory measurements of turbulence and with theory [Tennekes and Lumley, 1972], the present measurements are considered to accurately represent the turbulent flow field above the incipient dunes. Nineteen of the 21 spectra exhibited peaks at a frequency of between 0.08 and 0.2 (Figure 12). By converting the power spectra to wave number spectra (not shown), and as frequency is the reciprocal of period, typical eddy length scales may be derived [see Carling et al., 2000]. Using this approach, the peak values reflect a characteristic eddy length scale of 0.56 to 1.42 m, which is consistent with the initial (but somewhat variable) spacings of the crest lines of incipient 2-D bed forms (Figure 6). Consequently, it is at this phase of bed form development, and latterly for fully developed 2-D dunes (Figure 7), that larger-scale “wedge” flow models [Yalin, 1992; Ferguson et al., 1996, Roy et al., 1996] might prove appropriate and potential flow models [after Kennedy, 1963] prove applicable.

Figure 12.

Example of one of 19 self-similar spectral signatures obtained from 21 independent turbulent time series of flow data above gravel bed for conditions of incipient bed forms.

6. Conclusions

[31] On 5 mm gravel LSPBs, defects developed from small flow-induced disturbances. The scale of these defects is commensurate with the disturbance being caused by small-scale turbulent sweeps on the flat bed. These defects sometimes developed laterally into incipient 2-D dunes of short span. However, and importantly, low-amplitude transverse dunes of broad span (4 m), and of a preferred wavelength of one to several meters, developed after several hours. The preferred dune spacing indicates a fluid control induced by large-scale coherent turbulence structures. For this latter condition, defects still occurred between dune crest lines. Thus, for these prescribed conditions, the two scales of small- and large-scale coherent flow structures are related to two scales of coexisting bed forms.

[32] These experiments show that in some situations time is the controlling factor dictating whether a bed is viewed as LSPB or dune phase. Both defects and dunes developed for shear stress conditions below, or marginally above, nondimensional θ values typical for incipient motion on LSPBs. Defects initially were strongly 3-D and lacked distinctive crest lines or slopes but rapidly developed into 2-D straight-crested transverse dunes with negligible amplitude, and distinctive lee slopes marked by the absence of visible bed scour in the lee sides. These simple bed forms appear to be a stable state over the time scales of this investigation. However, increase in shear stress and/or development into deeper water resulted in 2-D transverse dunes steepening, and then evolving into firstly sinuous long-crested transverse bed forms with distinctive long flow-parallel spurs emanating from the crests and latterly these sinuous forms were replaced by short-crested low-amplitude lunate 3-D dunes or transitional wavy-crested bed forms. Equilibrium can occur for both 2-D and 3-D dunes over a range of θ values, the value tending to constrain time to equilibrium. Bed forms transitional between lower stage and upper stage flow develop at a given θcrit; usually this is around >0.25 to <0.3 [e.g., Carling, 1999]. However, in the literature this is considered simply as a transition from “dune” to “upper stage plane bed” or “antidune” and in terms of bulk flow parameters only. No mechanistic descriptions are available to explain the development of seemingly complex transitional bed forms, at high Froude numbers and shallow flow depths, such as the wavy-crested, and short-span lunate bed forms noted here and the rhomboid gravel bed forms reported by Ikeda [1983].

Acknowledgments

[33] The British Council is thanked for funding this collaborative venture.

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