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Transition from wavelets to ripples in a laboratory flume with a diverging channel
International Journal of Sediment Research 23 (2008) 1-12
Transition from wavelets to ripples in a laboratory flume with a diverging channel William B. RAUEN 1 , Binliang LIN2, and Roger A. FALCONER3
Abstract An experimental investigation on the initiation and development of bed forms on a bed of fine silica sand was conducted under alluvial flow conditions in a laboratory flume with a diverging channel. The main aims of the study were to assess: i) the steepness of bed forms in the transition stage of development; and ii) the threshold height of wavelets (ηt) that triggered the start of ripple development. Detailed bed profile measurements were carried out using an acoustic Doppler probe, traversed longitudinally over the sediment bed at various experimentation times. The bed form dimensions were extracted from such bed profile records and analysed for the wavelet, transition and equilibrium stages. It was found that the steepness of ripples in the transition and equilibrium stages were similar, confirming predictions of previous mathematical model simulations. A lognormal distribution fitted the wavelet length data. The wavelet threshold height was estimated as ηt ≈ 7 mm, or y+ ≈ 80 in wall units. Such a height magnitude suggested that ripple development could be triggered by the wavelets reaching the outer flow zone of a turbulent boundary layer. The ηt value obtained corresponded generally to the intersection point between two predictive equations for bed form dimensions. A formulation was developed to predict ηt as a function of the sediment grain size, which was confirmed for the fine sand used in this study. Key Words: Bed form, Bed ripple, Doppler sonar, Scale model, Wave height, Wavelength
1 Introduction Previous studies of the mechanics of bed form initiation and growth have shown that the current-induced development of bed forms on an initially flat sediment bed of uniform quartz sand can typically be divided into three stages, including: i) wavelet stage; ii) transition stage; and iii) the equilibrium stage. During each of these stages the time variation of the height (η) and the length (λ) occurs generally according to the schematic representation illustrated in Fig. 1a, where η and λ are depicted in Fig. (1b). As shown by Coleman and Melville (1996), wavelets are incipient 2-D cross-flow bed forms from which both ripples and dunes can develop. The wavelet stage can be generally characterised by these incipient bed forms showing a preferred and almost constant spacing, i.e. λw, as depicted in Fig. (1a). Zhou and Mendoza (2005) suggested a possible cause for the occurrence of this phenomenon based on the assumption that the incipient bed forms separated by such an interval would receive the maximum energy transfer from the flow field. This effect would guarantee the fastest growth rate for such bed forms, making them the dominant features locally. The wavelet stage is triggered when the height of random pileups of sediment on the bed reaches a certain critical value, i.e. ηw, at time tw, as depicted in Fig. (1a). Such a threshold height has been related 1
Dr., 3 Prof., Hydro-environmental Research Centre, School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK 2 Prof., Hydro-environmental Research Centre, School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK, E-mail: [email protected] Note: The original manuscript of this paper was received in Sept. 2007. The revised version was received in Dec. 2007. Discussion open until March 2009. International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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to the bed roughness height and is usually expressed as a function of a representative grain diameter (Coleman and Melville, 1996), and also the thickness of the viscous sublayer (Zhou and Mendoza, 2005). According to Baas (2003), for the initiation of wavelets to occur, the height of a sediment pileup should be large enough to induce flow separation in its lee, which suggests that the tip of such a pileup must be located above the laminar sublayer of the near wall zone.
Fig. 1 a) Conceptual diagram of time evolution of dimensions of current-generated sand ripples from a flat bed, depicting typical variations of height (η) and length (λ) during the three stages of development; b) generic representation of η and λ
The wavelet stage can typically last from a few seconds, in conditions of relatively large shear stress excesses, i.e. for θ >> θc, to several minutes, when θ ≈ θc, where θ and θc are given by equations (1) and (2) respectively. τb u2 (1) θ= = * (ρ s − ρ )gd gdΔ 0.30 (2) θc = (1 + 1.2 D* ) + 0.055[1 − e ( −0.02 D ) ] *
where θ is the Shields or mobility parameter, θc is the critical Shields parameter, d is the representative grain size (taken as the median grain size in this study), D* is the dimensionless grain size, τb is the bed shear stress, g is the acceleration due to gravity, ρs and ρ are the densities of the sediment and fluid respectively, s = ρs/ρ and Δ = (ρs-ρ)/ρ (Soulsby, 1997). A review of previous studies suggested that the end of the wavelet stage occurs when λ becomes greater than the λw value. This phenomenon is supposedly triggered by the height of the developing wavelets reaching a second threshold value, i.e. ηt, at time tt, as indicated in Fig. (1a). However, from the literature it is unclear as to what is the critical wavelet height, at which the state of local equilibrium of the wavelets is disturbed and the increase of λ induced. Thus, one of the main aims of this study was to try and determine such threshold height for the further development of ripples on a fine sand bed. 1.1 Estimation of wavelet length Previous investigators have found that the value of λw is mainly a function of the grain size. It does not depend on the flow conditions. The following empirical relationships have been proposed to calculate the wavelet length, with the corresponding authors being indicated in brackets: d λw = 10 2.5 0.2 (Coleman and Melville, 1996) (3) Re*c -2-
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λ w = 150d 0.5 λw = 175d
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(Coleman and Eling, 2000) (5) where Re*c is the critical grain Reynolds number, given as Re*c = u*cd/ν, where u*c is the critical friction velocity, ν is the kinematic viscosity of the fluid, with grain sizes normally given in mm. It can be noted that, with the exception of equation (3), these equations are dimensionally not consistent. Equations (3) and (5) have been derived for the range of particle sizes between 0.2 mm < d < 1.6 mm and have been shown to perform better than equation (4), when the grain size is in that range (Coleman and Eling, 2000). The same authors pointed out that equation (3) is not valid under laminar flow conditions, while equation (5) is valid for both laminar and turbulent flows, and for open-channel as well as closed conduit flows (Coleman et al., 2003). Any comparisons between data sets from different studies have to be made with care, particularly when the scatter level in the data is high. The estimation of representative ripple dimensions from acquired data is affected by the various types of probability distributions used to fit the η and λ data, but statistical studies on this matter seem to be limited to specific cases. The most commonly used parameters to describe ripple dimensions are the median or average η and λ values, as reported by several authors. 1.2 Notes on transition and equilibrium stages The transition stage of ripple development is characterised by the growth of both the bed form height η and length λ following relationships such as those given by Nikora and Hicks (1997) and Coleman et al. (2005). It also typically encompasses a change in the nature of the bed forms from a 2-D to a 3-D rippled field (Venditti et al., 2005). The sediment material required for such growth is provided primarily by the bed load sediment transport under conditions of excess bed shear stress, i.e. when θ ≥ θcr. The transition stage can typically last from several minutes to hundreds of hours, depending on the excess shear stress condition (Baas, 1994; Raudkivi, 1997), after which the bed form dimensions remain constant, characterising the equilibrium stage. Finer sediments tend to produce steeper equilibrium bed forms, as suggested by the following relationship developed to describe the steepness of ripples in equilibrium:
ηe = 0.074 d −0.253 (Raudkivi, 1997) λe which was obtained by combining equations (7) and (8) below. η e = 18.16 d 0.097 (Raudkivi, 1997)
(6) (7)
λe = 245d (Raudkivi, 1997) (8) Yalin (1977) analysed a number of experimental data sets for rippled beds formed with sediment grain sizes spanning from silt to medium sand, for flows of Re* < 10. For such conditions, Yalin found that the corresponding ripple steepnesses would not exceed 0.2, which agrees well with the estimate of ripple steepness evaluated using equation (6) for silt. Table 1 summarises the ripple steepness values calculated using equation (6) for silt and sand beds, where each size class is represented by the corresponding average grain size value d in the Wentworth scale. It can be seen in Table 1 that the equilibrium ripple steepnesses formed on sand beds can be expected to vary from 1:7 to 1:15 approximately, depending on the grain size class under consideration. The values situated in the upper end of this range correspond to the ‘typical’ ripple steepnesses of 1:10 to 1:15, as identified by Soulsby and Whitehouse (2005) from analysing a large data set of current-generated ripples in equilibrium. However, little is known about the variation in steepness of ripples during the wavelet and transition stages. Niño et al. (2002) used mathematical model simulations to infer that, after the wavelet stage, the ripple steepness tends to stabilise at a constant value, which corresponds approximately to the equilibrium steepness. Such an indication does not appear to have been tested experimentally, which raises the question of whether the relative growth of η and λ can be generally described by a relationship similar to equation (6). This aspect has been investigated further in this study, as discussed later. 0.35
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Table 1 Ripple steepness values estimated using equation (6) for silt and sand, where d represents the average grain size value of different classes in the Wentworth scale Grain size class Medium silt Coarse silt Very fine sand Fine sand Medium sand Coarse sand Very coarse sand
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ηe/λe (Eq. 6)
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0.024 0.048 0.095 0.188 0.375 0.750 1.500
0.190 0.160 0.134 0.113 0.095 0.080 0.067
5 6 7 9 11 13 15
2 Experimentation work 2.1 Hydraulic system The experiments reported herein were conducted inside a large flume located in the Hyder Hydraulics Laboratory, at Cardiff School of Engineering. The working part of the flume was 17.0 m long, 1.2 m wide and 1.0 m high. The flume was equipped with a tilting mechanism, and was capable of operating with flows in the range of 10 to 1,000 l/s. The hydraulic system also included an axial flow impeller, powered by an electric engine, which recirculated the flume water in a return pipeline 0.6 m in diameter, located underneath the flume. The flume structure was mostly built of steel, whereas the side walls and bottom panels were made of glass to allow for visualisation of flow and sediment transport processes. A perspective 3-D view of the hydraulics flume and its main components is shown in Fig. 2.
Fig. 2 Perspective 3-D view of laboratory flume
2.2 Hydraulic model As illustrated in Fig. 3, the hydraulic model had three main regions, namely an upstream channel, a diverging channel and a downstream channel. The model was made of translucent PVC and the height of the model walls was 50 cm throughout. The upstream channel was 3.0 m long and 30 cm wide, and corresponded to the region of 2.0 m < x < 5.0 m of the model. The downstream channel was 4.0 m long and 90 cm wide, and was located between 11.0 m < x < 15.0 m. The diverging channel connected these two sections and was 6.0m long, spanning from 5.0 m < x < 11.0 m. The channel width varied linearly in the longitudinal direction with a 1:10 ratio, from 30 cm at x = 5.0 m to 90 cm at x = 11.0 m. Such a threefold increase in the channel width generated a similar increase in the cross-sectional flow area between the upstream and downstream channels. A characterisation of the key hydrodynamic and sediment transport conditions in the model is included in section 2.4. 2.3 Sediment characteristics The sediment used in the experiments was fine quartz sand, obtained commercially in washed and graded condition, with a specified silica content higher than 98%. The grain size characteristics were d = d50 = 130 μm, dg = 120 μm, σg = 2 μm, where d50 is the median grain size, dg is the geometric mean grain size and σg is the geometric standard deviation of sediment sizes. The relatively low value of σg indicated -4-
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that the sand was well sorted and uniform. Some key sediment transport parameters were calculated using the formulations by Soulsby (1997), as ws = 1.2 cm/s, D* = 3.3, θcr = 0.065 and u*cr = 1.2 cm/s, for the settling velocity, dimensionless grain size, critical Shields parameter and critical friction velocity respectively.
Fig. 3 Schematic illustration of the hydraulic model depicting the main regions, type of sediment, initial flat sedimentary bed, dimensions and streamwise flow direction
2.4 Experimentation conditions A typical experiment would start with a flat sediment bed, and as the experiment progressed then bed forms would develop under a fully turbulent flow, at a constant discharge. Several bed profile measurements were undertaken during the experiments to assess the bed form development, from wavelets to the equilibrium stage. The flume was first levelled in the horizontal plane for the experiments described herein. Approximately 1,000 kg of sand was used to fill the model bed. The sand bed was flattened prior to the start of an experiment, so that the initial bed level in the model would be h = 10 cm. Tap water was then introduced through one of the external channels formed between the model and the flume walls, which were connected to the working part of the model during the filling process. Such procedures were performed with care, to minimise the disturbance to the prepared sand bed. The flume was filled to an experimental water depth of H = 30 cm above the initial sand bed level. By taking into account such water depth in the model, the critical mean streamwise velocity for sediment erosion was calculated as U0cr ≈ 25 cm/s, using the friction law of Soulsby (1997). Pebbles with d ≈ 1 cm, as a replacement for the sand, were introduced in the inlet section of the model, in the region of x < 3.0 m, in order to prevent erosion in that region due to a uniform inflow distribution. The pebbles also formed a ramp with a mild slope within 0.0 m < x < 2.0 m, which raised the bed level from the bottom of the flume to the sand surface. The interface between the pebbles and sand occurred at x = 3.0 m. The flow rate used in the experiments reported herein was Q = 36 l/s. Such experimental discharge was selected as it induced a relatively slow rate of development for the bed forms in the model, which allowed for detailed bed profile measurements during the wavelet period to be conducted. For such a flow rate, the distributions of some key flow and sediment transport parameters were estimated for the initial flat bed condition in the model, as shown in Fig. 4. This figure depicts the estimated distributions of normalised parameters along the hydraulic model, where the assessment region of wavelet development corresponded to the upstream channel and the first quarter of the diverging channel, i.e. the region of 3.0 m < x < 6.5 m in the model. Normalisation was made by the corresponding values calculated for the upstream channel, which are shown in Table 2. The bulk cross-sectional velocity was calculated as U0 = Q/A, where A is the cross-sectional area of the flow. Turbulent subcritical flow occurred in the model, as indicated by the distributions of the Reynolds and Froude numbers, calculated as Re = U04Rh/ν and Fr = U0/(gH)0.5 respectively, where Rh is the hydraulic radius. The highest rates of erosion, bed load sediment transport and bed form development were expected to occur in the upstream channel, where θ/θcr ≈ 2.5, International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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which was the highest level estimated for the relative Shields number. The friction velocity distribution was estimated using Soulsby’s friction law. The particle Reynolds and Froude numbers were calculated as Re* = u*d/ν and Fr* = U0/[gd(s-1)]0.5 respectively. The flow was hydrodynamically smooth along the assessment region. 2.5 Bed form data acquisition and processing Bed profile measurements in the model were carried out using an Acoustic Doppler Velocimeter probe, operated in sonar mode. The probe was traversed along the x direction centreline, while the longitudinal positions and bed elevation values were recorded to generate a 2-D profile of the bed. The accuracy of the method when used under the experimentation conditions reported herein was of the order of 1 mm in the horizontal and vertical directions, when compared against results obtained with an HR Wallingford bed profiler equipped with a conductivity probe.
Fig. 4 Estimated distributions of key hydrodynamic and sediment transport parameters along the assessment region, normalised by the corresponding values calculated for the upstream channel, where U0 = mean streamwise velocity, θ /θcr = relative Shields number, Re = Reynolds number, Fr = Froude number, u* = friction velocity, Re* = particle Reynolds number, Fr* = particle Froude number Table 2 Values of key hydrodynamic and sediment transport parameters estimated for the upstream channel (3.0 m < x < 5.0 m) and end of the wavelet assessment region (x = 6.5 m) Parameter 3.0 m < x < 5.0 x = 6.5 m U0 (m/s) 0.40 0.25 Re 137,000 117,000 Fr 0.23 0.16 u* (m/s) 0.019 0.013 Re* 2.2 1.4 Fr* 8.8 5.8 2.5 1.1 θ /θcr Q (L/s) 36.0 36.0 H (m) 0.30 0.30 d = d50 (mm) 0.13 0.13 D* 3.3 3.3 u*cr (m/s) 0.012 0.012 0.065 0.065 θcr
The bed profile assessments were carried out from t = 12 s up to approximately 55 min of experimentation time. A total of 22 profiling passes were carried out during this period, each lasting 90–100 s and with an interval of 50–60 s between passes. This procedure conferred a good temporal -6-
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resolution to the wavelet data, allowing the development of bed forms during the wavelet and early transition stages to be assessed. The processing of bed profiling data involved searching the records for bed surface slopes, so that the slope inflection points (e.g. crests and troughs of the bed forms) in a bed profile were identified. The height difference between a trough and the subsequent crest in the upstream direction was calculated, and height values not lower than d and the corresponding distance along the x direction were recorded. Further analysis of the results compared to the flat bed condition, i.e. prior to the start of an experiment, allowed discerning bed forms from random sand pileups, as further explained below. The calculated median bed form height and length, i.e. η50 and λ50, were the representative bed form dimensions used in the analysis. 3 Results and discussion
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3.1 Determination of wavelet stage The wavelet length estimated for the sand of this study using equation (4) was λw ≈ 55 mm. Figures 5 and 6 show the longitudinal variation of the bed form height and length respectively, as well as their time of evolution during the experiment. The location of x = 5.0 m corresponded to the interface between the upstream and diverging channels.
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Fig. 5 Evolution of bed form height along assessment region with time International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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The bed level results acquired before the start of the test, i.e. corresponding to t = 0 showed that a number of random irregularities were present on the initial bed, as shown in Fig. 5. The random distribution of the spacings between the bed irregularities measured at t = 0 is shown in Fig. 6. Such irregularities were characterised as having a mean height difference and a standard deviation about 0.6 mm, or 5d. These values were used to define the minimum bed form height restriction for the experimental results, which corresponded to the sum of the mean and standard deviation values given above, i.e. ηmin ≈ 10d. A similar height restriction was adopted by Coleman and Melville (1994) in processing their results. Such a value was higher than the flat bed roughness height calculated using most formulations available in the literature – such as 2.5d (Soulsby, 1997), 3.5d (Coleman and Melville, 1996) or 3.5d90 (van Rijn, 1984) – which is thought to be the threshold height of a sand pileup that triggers the initiation of wavelets, i.e. parameter ηw, as depicted in Fig. (1a). The first record of bed forms occurring after the start of the experiment encompassed the interval of 12 s < t < 108 s of experimentation time, as depicted in Fig. 5. This record generally showed that the start of bed form growth was in the upstream channel, i.e. in the region where x < 5.0 m, but not in the divergent channel, where x > 5.0 m. The median initial bed form height calculated for the data in the upstream channel was η50 = 2.4 mm. The corresponding graph depicting the variation of λ is shown in Fig. 6. This graph indicates that the initial wavelet length in the region of x < 5.0 m was typically in the range of -8-
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30 mm < λ < 60 mm. The median bed form length in the upstream channel was λ50 = 47 mm, which was approximately 14% lower than the wavelet length estimated using equation (4), of λw = 55 mm. On the other hand, the distribution pattern of λ in the diverging channel was similar to the corresponding results measured at t = 0, suggesting that bed form development had not yet started in that part of the model for t < 108 s. Further information about the development of λ in the upstream channel, as measured in the first few bed profiling passes, is given in Table 3. The start and end times of each pass are represented by ts and tf respectively, while the calculated medians are given by λ50. The lower and upper limits of the 95% confidence interval (CI) for the median are also shown. An indication of the duration of the wavelet stage in the upstream channel was obtained from an analysis of the 95% CI limits provided in Table 3. The estimated wavelet length of λw = 55 mm was within the CI limits of profile nos. 2, 3 and 4. From profile no. 5 onwards the corresponding λ50 value was lower than the lower limit of the confidence interval, so that the finish time of profile no. 4 (approximately 10 min) was regarded as the time for the end of the wavelet stage in the upstream channel, i.e. tt, as depicted in Fig. (1a). Table 3 Start and finish times, medians and limits of the 95% confidence interval (CI) for the bed form length data acquired in the first six bed profiling passes, as shown in Fig. 6 95% CI limits (mm) Profile no. ts (s) tf (s) λ50 (mm) 1 12 108 47 41 52 2 164 262 52 47 55 3 317 416 54 48 59 4 465 576 57 50 64 5 630 729 68 58 73 6 778 877 70 62 78
Another aspect noted from Table 3 is that λ50 increased with time during the wavelet stage, at a typical rate about 1 mm per 50 s for the experiment. This result suggested that the bed form length is not constant, but increases slightly during the wavelet stage. For the diverging channel Figs. 5 and 6 indicated that there was a gradual increase of η and λ respectively with t. Qualitatively speaking, the trend of bed form growth in this part of the model followed the trend of decreasing values of the hydrodynamic variables depicted in Fig. 4, herein referred to as the diverging channel effects. A key practical implication of such effects for the development of wavelets was a gradual reduction of the growth rate of bed forms, which led to a trend of increase of the duration of the wavelet stage (i.e. parameter tt in Fig. (1a)) along the x axis in the diverging channel. For instance, in the region of 6.0 m < x < 6.5 m – where θ/θcr ≈ 1.1 – and 3,216 s < x < 3,314 s (i.e. even after almost 1 hour of experimentation time), Fig. 6 showed that the bed form lengths were still close to the estimated λw value of 55 mm. 3.2 Bed form steepness The bed form data shown in Figs. 5 and 6 were combined to generate a bed form steepness plot (Fig. 7). The results shown in this graph exclude any bed irregularities with a height lower than ηmin ≈ 10d. Included in this graph are the plots of equations (4), (6), (7) and (8), which gave the estimated wavelet length (λw = 55 mm), ripple steepness (λ/η ≈ 8), and equilibrium ripple height (ηe = 15 mm) and length (λe =121 mm) respectively. Such equilibrium ripple dimensions gave the estimated upper limits of the transition stage of ripple development. The ripple steepness estimated for d = 0.13 mm corresponded to the level expected for very fine to fine sands (Table 1). It can be seen from Fig. 7 that the steepness data of this study followed the trend of equation (4) for η < 7 mm approximately, where most of the ripple length values were around the estimated wavelet length λw = 55 mm. On the other hand, for higher η values the bed form length data were generally higher than the λw value, indicating that the corresponding bed forms were probably no longer in the wavelet stage (Coleman and Melville, 1996). Figure 7 also showed that the relative increase of the bed form length and height after the wavelet stage generally followed the trend of the ripple steepness equation (6). Despite the scatter, it can be noted that International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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such a trend was followed by the ripple data in both the transition and equilibrium stages. It is interesting to note that the steepness data for ripples in the transition stage was described by equation (6), which was originally developed for equilibrium ripples. This fact indicated that such an equation may be applicable to predict the steepness of both transitional and equilibrium ripples.
Fig. 7 Steepness results of bed forms during wavelet, transition and equilibrium stages and plots of predictive equations
Furthermore, the similarity between the ripple steepness in the transition and equilibrium stages confirmed the predictions of the mathematical model simulations of Niño et al. (2002). 3.3 Determination of wavelet threshold height A review of previous studies suggested that the transition from wavelets to ripples normally occurs when the bed form length can no longer be described by a relationship such as equations (3) – (5) (Coleman and Melville, 1996; Raudkivi, 1997; Coleman and Eling, 2000). As discussed above, in this study such a phenomenon occurred when the bed form steepness data followed the trend of equation (6) in Fig. 7. Therefore, an expression for ηt as a function of d was obtained by combining equations (4) and (6) and making λe ≡ λw and ηe ≡ ηt. After rearranging the terms, the equation used to calculate ηt is given by: ηt = 11.10 d 0.247 (for d < 0.2 mm) (9) where the interval of d was determined by the considerations of Coleman and Eling (2000) about the performance of equations (4) and (5) with sands of different grain sizes. The critical wavelet height was defined graphically by locating the intersection point between the curves representing the wavelet (red line) and ripple (green line) stages. It may be pointed out that both equations involved in the derivation of equation (9) consider the median grain size as the representative sediment particle diameter (Raudkivi, 1997). For the fine sand used in this study the threshold height was calculated as being ηt ≈ 7 mm. Conceptually speaking, ηt can be understood as the wavelet height that triggered the start of ripple development. As a result, bed forms with a recorded height lower than ηt were deemed to be in the wavelet stage, while higher height values were associated with transitional ripples, according to the definitions of Fig. (1a). In wall units, this value corresponded to y+ ≈ 80, where y+ = u*cηt/ν, with u*c = 12 mm/s. Such a y+ value is of a similar order of magnitude as the upper limit of the near wall zone of the fully turbulent flow (Rodi, 1993), which suggested that the start of the transition stage of ripple development could be related to the wavelets reaching the outer flow zone of a turbulent boundary layer. Further experimentation on bed form development under different turbulence strength levels, including laminar flow, beyond the wavelet stage would be required to confirm such a hypothesis. - 10 -
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3.4 A probability distribution function for wavelet length Figure 8 depicts a histogram of the bed form length measurements in the interval of ηmin < η < ηt, which corresponds to bed forms in the wavelet stage but excluded bed irregularities. These data were fitted using a lognormal distribution with the median length λ50 shown to be 52 mm, about 5% lower than the estimated wavelet length of λ50 = 55 mm. The fitting of a lognormal distribution to wavelet length data, which has not been reported before, suggested that the median length could also be used to quantify wavelet development, rather than using only the average length.
Fig. 8 Histogram of wavelet length data with fitted lognormal distribution
5 Conclusion Experiments were carried out in a laboratory flume to study the initiation and development of bed forms, from an initially flat sediment bed, and under turbulent and subcritical flow conditions. The model contained a diverging channel, which allowed for varying velocity and sediment transport processes to occur along the developing rippled field. Bed profiles were measured at a relatively high resolution, both spatially and temporally, which allowed for detailed investigations of the bed form development to be made. It was found that the wavelet length measurements can be fitted by a lognormal distribution, while the threshold bed form height that triggered the transition from wavelets to ripples in this study was expressed as ηt = 11.10d0.247. Further experimentation findings reported herein are summarised below: • The duration of the wavelet stage was found to vary from about 10 min (for a cross-sectional mean velocity of 40 cm/s and relative Shields number of 2.5) to more than 1 hour (for 25 cm/s and 1.0 respectively), due to the diverging channel effects in the model. • The ripple steepness measured during the transition stage of development has been shown to be similar to the steepness of equilibrium ripples, providing empirical confirmation to mathematical model simulation results of Niño et al. (2002). As a consequence of this, it has been suggested that the equilibrium ripple steepness equation of Raudkivi (1997) can be used to predict the steepness of transitional ripples as well. • For the fine sand of this study, the threshold wavelet height was determined as ηt ≈ 7 mm. Such a height value corresponded to the intersection point between two predictive equations for bed form dimensions and was deemed as the trigger for the start of the transitional stage of ripple development. Further studies will be carried out to investigate the relationship between the turbulence strength levels and the critical bed form height for the transition from wavelets to ripples. The authors also plan to undertake further experiments to verify the applicability of the findings reported herein to other sediment grain sizes. Acknowledgements Receipt of the equipment grant NE/C513269/1 from the UK’s Natural Environment Research Council (NERC), and the research grant EP/C512316/1 from the Engineering and Physical Sciences Research International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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Council (EPSRC) are greatly acknowledged by the authors. We would also like to thank Mr. Paul Leach for the technical support provided during the experimental work. References Baas J. H. 1994, A flume study on the development and equilibrium morphology of current ripples in very fine sand. Sedimentology, 41, pp. 185–209. Baas J. H. 2003, Ripple, ripple mark, and ripple structure. Encyclopaedia of Sediments and Sedimentary Rocks, Ed. G. V. Middleton. Kluwer Academic Publishers, Dordrecht, pp. 565–568. Coleman S. E. and Melville B. W. 1994, Bed-form development. Journal of Hydraulic Engineering, Vol. 120, No. 4, pp. 544–560. Coleman S. E. and Melville B. W. 1996, Initiation of bed forms on a flat sand bed. Journal of Hydraulic Engineering, Vol. 122, No. 6, pp. 301–310. Coleman S.E. and Eling B. 2000, Sand wavelets in laminar open–channel flows. Journal of Hydraulic Research, Vol. 38, No. 5, pp. 331–338. Coleman S. E., Fedele J. J., and Garcia M. H. 2003, Closed–conduit bed–form initiation and development. Journal of Hydraulic Engineering, Vol. 129, No. 12, pp. 956–965. Coleman S. E., Zhang M. H., and Clunie T. M. 2005, Sediment–wave development in subcritical water flow. Journal of Hydraulic Engineering, Vol. 131, No. 2, pp. 106–111. Nikora V. I. and Hicks D. M. 1997, Scaling relationships for sand wave development in unidirectional flow. Journal of Hydraulic Engineering, Vol. 123, No. 12, pp. 1152–1156. Niño Y., Atala A., Barahona M., and Aracena D. 2002, Discrete particle model for analyzing bedform development. Journal of Hydraulic Engineering, Vol. 128, No. 4, pp. 381–389. Raudkivi A. J. 1997, Ripples on stream bed. Journal of Hydraulic Engineering, Vol. 123, No. 1, pp. 58–64. Rodi W. 1993, Turbulence Models and Their Application in Hydraulics – a State-of-the-art Review, 3rd ed., A.A.Balkema, Rotterdam / Brookfield. Soulsby R. L. 1997, Dynamics of Marine Sands – a Manual for Practical Applications. Thomas Telford Publications, London. Soulsby R. L. and Whitehouse R. J. S. 2005, Prediction of ripple properties in shelf seas. Mark 1 Predictor. HR Wallingford Report TR 150, Release 1.1. van Rijn L. C. 1984, Sediment transport, Part III: Bed forms and alluvial roughness. Journal of Hydraulic Engineering, Vol. 110, No. 12, pp. 1733–1754. Venditti J. G., Church M., and Bennett S. J. 2005, On the transition between 2D and 3D dunes. Sedimentology, 52, pp. 1343–1359. Yalin M. S. 1977, Mechanics of sediment transport. 2nd Ed., Pergamon Press, Guildford. Zhou D. and Mendoza C. 2005, Growth model for sand wavelets. Journal of Hydraulic Engineering, Vol. 131, No. 10, pp. 866–876.
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Transition from wavelets to ripples in a laboratory flume with a diverging channel William B. RAUEN 1 , Binliang LIN2, and Roger A. FALCONER3
Abstract An experimental investigation on the initiation and development of bed forms on a bed of fine silica sand was conducted under alluvial flow conditions in a laboratory flume with a diverging channel. The main aims of the study were to assess: i) the steepness of bed forms in the transition stage of development; and ii) the threshold height of wavelets (ηt) that triggered the start of ripple development. Detailed bed profile measurements were carried out using an acoustic Doppler probe, traversed longitudinally over the sediment bed at various experimentation times. The bed form dimensions were extracted from such bed profile records and analysed for the wavelet, transition and equilibrium stages. It was found that the steepness of ripples in the transition and equilibrium stages were similar, confirming predictions of previous mathematical model simulations. A lognormal distribution fitted the wavelet length data. The wavelet threshold height was estimated as ηt ≈ 7 mm, or y+ ≈ 80 in wall units. Such a height magnitude suggested that ripple development could be triggered by the wavelets reaching the outer flow zone of a turbulent boundary layer. The ηt value obtained corresponded generally to the intersection point between two predictive equations for bed form dimensions. A formulation was developed to predict ηt as a function of the sediment grain size, which was confirmed for the fine sand used in this study. Key Words: Bed form, Bed ripple, Doppler sonar, Scale model, Wave height, Wavelength
1 Introduction Previous studies of the mechanics of bed form initiation and growth have shown that the current-induced development of bed forms on an initially flat sediment bed of uniform quartz sand can typically be divided into three stages, including: i) wavelet stage; ii) transition stage; and iii) the equilibrium stage. During each of these stages the time variation of the height (η) and the length (λ) occurs generally according to the schematic representation illustrated in Fig. 1a, where η and λ are depicted in Fig. (1b). As shown by Coleman and Melville (1996), wavelets are incipient 2-D cross-flow bed forms from which both ripples and dunes can develop. The wavelet stage can be generally characterised by these incipient bed forms showing a preferred and almost constant spacing, i.e. λw, as depicted in Fig. (1a). Zhou and Mendoza (2005) suggested a possible cause for the occurrence of this phenomenon based on the assumption that the incipient bed forms separated by such an interval would receive the maximum energy transfer from the flow field. This effect would guarantee the fastest growth rate for such bed forms, making them the dominant features locally. The wavelet stage is triggered when the height of random pileups of sediment on the bed reaches a certain critical value, i.e. ηw, at time tw, as depicted in Fig. (1a). Such a threshold height has been related 1
Dr., 3 Prof., Hydro-environmental Research Centre, School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK 2 Prof., Hydro-environmental Research Centre, School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK, E-mail: [email protected] Note: The original manuscript of this paper was received in Sept. 2007. The revised version was received in Dec. 2007. Discussion open until March 2009. International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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to the bed roughness height and is usually expressed as a function of a representative grain diameter (Coleman and Melville, 1996), and also the thickness of the viscous sublayer (Zhou and Mendoza, 2005). According to Baas (2003), for the initiation of wavelets to occur, the height of a sediment pileup should be large enough to induce flow separation in its lee, which suggests that the tip of such a pileup must be located above the laminar sublayer of the near wall zone.
Fig. 1 a) Conceptual diagram of time evolution of dimensions of current-generated sand ripples from a flat bed, depicting typical variations of height (η) and length (λ) during the three stages of development; b) generic representation of η and λ
The wavelet stage can typically last from a few seconds, in conditions of relatively large shear stress excesses, i.e. for θ >> θc, to several minutes, when θ ≈ θc, where θ and θc are given by equations (1) and (2) respectively. τb u2 (1) θ= = * (ρ s − ρ )gd gdΔ 0.30 (2) θc = (1 + 1.2 D* ) + 0.055[1 − e ( −0.02 D ) ] *
where θ is the Shields or mobility parameter, θc is the critical Shields parameter, d is the representative grain size (taken as the median grain size in this study), D* is the dimensionless grain size, τb is the bed shear stress, g is the acceleration due to gravity, ρs and ρ are the densities of the sediment and fluid respectively, s = ρs/ρ and Δ = (ρs-ρ)/ρ (Soulsby, 1997). A review of previous studies suggested that the end of the wavelet stage occurs when λ becomes greater than the λw value. This phenomenon is supposedly triggered by the height of the developing wavelets reaching a second threshold value, i.e. ηt, at time tt, as indicated in Fig. (1a). However, from the literature it is unclear as to what is the critical wavelet height, at which the state of local equilibrium of the wavelets is disturbed and the increase of λ induced. Thus, one of the main aims of this study was to try and determine such threshold height for the further development of ripples on a fine sand bed. 1.1 Estimation of wavelet length Previous investigators have found that the value of λw is mainly a function of the grain size. It does not depend on the flow conditions. The following empirical relationships have been proposed to calculate the wavelet length, with the corresponding authors being indicated in brackets: d λw = 10 2.5 0.2 (Coleman and Melville, 1996) (3) Re*c -2-
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λ w = 150d 0.5 λw = 175d
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(Coleman and Eling, 2000) (5) where Re*c is the critical grain Reynolds number, given as Re*c = u*cd/ν, where u*c is the critical friction velocity, ν is the kinematic viscosity of the fluid, with grain sizes normally given in mm. It can be noted that, with the exception of equation (3), these equations are dimensionally not consistent. Equations (3) and (5) have been derived for the range of particle sizes between 0.2 mm < d < 1.6 mm and have been shown to perform better than equation (4), when the grain size is in that range (Coleman and Eling, 2000). The same authors pointed out that equation (3) is not valid under laminar flow conditions, while equation (5) is valid for both laminar and turbulent flows, and for open-channel as well as closed conduit flows (Coleman et al., 2003). Any comparisons between data sets from different studies have to be made with care, particularly when the scatter level in the data is high. The estimation of representative ripple dimensions from acquired data is affected by the various types of probability distributions used to fit the η and λ data, but statistical studies on this matter seem to be limited to specific cases. The most commonly used parameters to describe ripple dimensions are the median or average η and λ values, as reported by several authors. 1.2 Notes on transition and equilibrium stages The transition stage of ripple development is characterised by the growth of both the bed form height η and length λ following relationships such as those given by Nikora and Hicks (1997) and Coleman et al. (2005). It also typically encompasses a change in the nature of the bed forms from a 2-D to a 3-D rippled field (Venditti et al., 2005). The sediment material required for such growth is provided primarily by the bed load sediment transport under conditions of excess bed shear stress, i.e. when θ ≥ θcr. The transition stage can typically last from several minutes to hundreds of hours, depending on the excess shear stress condition (Baas, 1994; Raudkivi, 1997), after which the bed form dimensions remain constant, characterising the equilibrium stage. Finer sediments tend to produce steeper equilibrium bed forms, as suggested by the following relationship developed to describe the steepness of ripples in equilibrium:
ηe = 0.074 d −0.253 (Raudkivi, 1997) λe which was obtained by combining equations (7) and (8) below. η e = 18.16 d 0.097 (Raudkivi, 1997)
(6) (7)
λe = 245d (Raudkivi, 1997) (8) Yalin (1977) analysed a number of experimental data sets for rippled beds formed with sediment grain sizes spanning from silt to medium sand, for flows of Re* < 10. For such conditions, Yalin found that the corresponding ripple steepnesses would not exceed 0.2, which agrees well with the estimate of ripple steepness evaluated using equation (6) for silt. Table 1 summarises the ripple steepness values calculated using equation (6) for silt and sand beds, where each size class is represented by the corresponding average grain size value d in the Wentworth scale. It can be seen in Table 1 that the equilibrium ripple steepnesses formed on sand beds can be expected to vary from 1:7 to 1:15 approximately, depending on the grain size class under consideration. The values situated in the upper end of this range correspond to the ‘typical’ ripple steepnesses of 1:10 to 1:15, as identified by Soulsby and Whitehouse (2005) from analysing a large data set of current-generated ripples in equilibrium. However, little is known about the variation in steepness of ripples during the wavelet and transition stages. Niño et al. (2002) used mathematical model simulations to infer that, after the wavelet stage, the ripple steepness tends to stabilise at a constant value, which corresponds approximately to the equilibrium steepness. Such an indication does not appear to have been tested experimentally, which raises the question of whether the relative growth of η and λ can be generally described by a relationship similar to equation (6). This aspect has been investigated further in this study, as discussed later. 0.35
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Table 1 Ripple steepness values estimated using equation (6) for silt and sand, where d represents the average grain size value of different classes in the Wentworth scale Grain size class Medium silt Coarse silt Very fine sand Fine sand Medium sand Coarse sand Very coarse sand
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0.190 0.160 0.134 0.113 0.095 0.080 0.067
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2 Experimentation work 2.1 Hydraulic system The experiments reported herein were conducted inside a large flume located in the Hyder Hydraulics Laboratory, at Cardiff School of Engineering. The working part of the flume was 17.0 m long, 1.2 m wide and 1.0 m high. The flume was equipped with a tilting mechanism, and was capable of operating with flows in the range of 10 to 1,000 l/s. The hydraulic system also included an axial flow impeller, powered by an electric engine, which recirculated the flume water in a return pipeline 0.6 m in diameter, located underneath the flume. The flume structure was mostly built of steel, whereas the side walls and bottom panels were made of glass to allow for visualisation of flow and sediment transport processes. A perspective 3-D view of the hydraulics flume and its main components is shown in Fig. 2.
Fig. 2 Perspective 3-D view of laboratory flume
2.2 Hydraulic model As illustrated in Fig. 3, the hydraulic model had three main regions, namely an upstream channel, a diverging channel and a downstream channel. The model was made of translucent PVC and the height of the model walls was 50 cm throughout. The upstream channel was 3.0 m long and 30 cm wide, and corresponded to the region of 2.0 m < x < 5.0 m of the model. The downstream channel was 4.0 m long and 90 cm wide, and was located between 11.0 m < x < 15.0 m. The diverging channel connected these two sections and was 6.0m long, spanning from 5.0 m < x < 11.0 m. The channel width varied linearly in the longitudinal direction with a 1:10 ratio, from 30 cm at x = 5.0 m to 90 cm at x = 11.0 m. Such a threefold increase in the channel width generated a similar increase in the cross-sectional flow area between the upstream and downstream channels. A characterisation of the key hydrodynamic and sediment transport conditions in the model is included in section 2.4. 2.3 Sediment characteristics The sediment used in the experiments was fine quartz sand, obtained commercially in washed and graded condition, with a specified silica content higher than 98%. The grain size characteristics were d = d50 = 130 μm, dg = 120 μm, σg = 2 μm, where d50 is the median grain size, dg is the geometric mean grain size and σg is the geometric standard deviation of sediment sizes. The relatively low value of σg indicated -4-
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that the sand was well sorted and uniform. Some key sediment transport parameters were calculated using the formulations by Soulsby (1997), as ws = 1.2 cm/s, D* = 3.3, θcr = 0.065 and u*cr = 1.2 cm/s, for the settling velocity, dimensionless grain size, critical Shields parameter and critical friction velocity respectively.
Fig. 3 Schematic illustration of the hydraulic model depicting the main regions, type of sediment, initial flat sedimentary bed, dimensions and streamwise flow direction
2.4 Experimentation conditions A typical experiment would start with a flat sediment bed, and as the experiment progressed then bed forms would develop under a fully turbulent flow, at a constant discharge. Several bed profile measurements were undertaken during the experiments to assess the bed form development, from wavelets to the equilibrium stage. The flume was first levelled in the horizontal plane for the experiments described herein. Approximately 1,000 kg of sand was used to fill the model bed. The sand bed was flattened prior to the start of an experiment, so that the initial bed level in the model would be h = 10 cm. Tap water was then introduced through one of the external channels formed between the model and the flume walls, which were connected to the working part of the model during the filling process. Such procedures were performed with care, to minimise the disturbance to the prepared sand bed. The flume was filled to an experimental water depth of H = 30 cm above the initial sand bed level. By taking into account such water depth in the model, the critical mean streamwise velocity for sediment erosion was calculated as U0cr ≈ 25 cm/s, using the friction law of Soulsby (1997). Pebbles with d ≈ 1 cm, as a replacement for the sand, were introduced in the inlet section of the model, in the region of x < 3.0 m, in order to prevent erosion in that region due to a uniform inflow distribution. The pebbles also formed a ramp with a mild slope within 0.0 m < x < 2.0 m, which raised the bed level from the bottom of the flume to the sand surface. The interface between the pebbles and sand occurred at x = 3.0 m. The flow rate used in the experiments reported herein was Q = 36 l/s. Such experimental discharge was selected as it induced a relatively slow rate of development for the bed forms in the model, which allowed for detailed bed profile measurements during the wavelet period to be conducted. For such a flow rate, the distributions of some key flow and sediment transport parameters were estimated for the initial flat bed condition in the model, as shown in Fig. 4. This figure depicts the estimated distributions of normalised parameters along the hydraulic model, where the assessment region of wavelet development corresponded to the upstream channel and the first quarter of the diverging channel, i.e. the region of 3.0 m < x < 6.5 m in the model. Normalisation was made by the corresponding values calculated for the upstream channel, which are shown in Table 2. The bulk cross-sectional velocity was calculated as U0 = Q/A, where A is the cross-sectional area of the flow. Turbulent subcritical flow occurred in the model, as indicated by the distributions of the Reynolds and Froude numbers, calculated as Re = U04Rh/ν and Fr = U0/(gH)0.5 respectively, where Rh is the hydraulic radius. The highest rates of erosion, bed load sediment transport and bed form development were expected to occur in the upstream channel, where θ/θcr ≈ 2.5, International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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which was the highest level estimated for the relative Shields number. The friction velocity distribution was estimated using Soulsby’s friction law. The particle Reynolds and Froude numbers were calculated as Re* = u*d/ν and Fr* = U0/[gd(s-1)]0.5 respectively. The flow was hydrodynamically smooth along the assessment region. 2.5 Bed form data acquisition and processing Bed profile measurements in the model were carried out using an Acoustic Doppler Velocimeter probe, operated in sonar mode. The probe was traversed along the x direction centreline, while the longitudinal positions and bed elevation values were recorded to generate a 2-D profile of the bed. The accuracy of the method when used under the experimentation conditions reported herein was of the order of 1 mm in the horizontal and vertical directions, when compared against results obtained with an HR Wallingford bed profiler equipped with a conductivity probe.
Fig. 4 Estimated distributions of key hydrodynamic and sediment transport parameters along the assessment region, normalised by the corresponding values calculated for the upstream channel, where U0 = mean streamwise velocity, θ /θcr = relative Shields number, Re = Reynolds number, Fr = Froude number, u* = friction velocity, Re* = particle Reynolds number, Fr* = particle Froude number Table 2 Values of key hydrodynamic and sediment transport parameters estimated for the upstream channel (3.0 m < x < 5.0 m) and end of the wavelet assessment region (x = 6.5 m) Parameter 3.0 m < x < 5.0 x = 6.5 m U0 (m/s) 0.40 0.25 Re 137,000 117,000 Fr 0.23 0.16 u* (m/s) 0.019 0.013 Re* 2.2 1.4 Fr* 8.8 5.8 2.5 1.1 θ /θcr Q (L/s) 36.0 36.0 H (m) 0.30 0.30 d = d50 (mm) 0.13 0.13 D* 3.3 3.3 u*cr (m/s) 0.012 0.012 0.065 0.065 θcr
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resolution to the wavelet data, allowing the development of bed forms during the wavelet and early transition stages to be assessed. The processing of bed profiling data involved searching the records for bed surface slopes, so that the slope inflection points (e.g. crests and troughs of the bed forms) in a bed profile were identified. The height difference between a trough and the subsequent crest in the upstream direction was calculated, and height values not lower than d and the corresponding distance along the x direction were recorded. Further analysis of the results compared to the flat bed condition, i.e. prior to the start of an experiment, allowed discerning bed forms from random sand pileups, as further explained below. The calculated median bed form height and length, i.e. η50 and λ50, were the representative bed form dimensions used in the analysis. 3 Results and discussion
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Fig. 6 Evolution of bed form length along assessment region with time
The bed level results acquired before the start of the test, i.e. corresponding to t = 0 showed that a number of random irregularities were present on the initial bed, as shown in Fig. 5. The random distribution of the spacings between the bed irregularities measured at t = 0 is shown in Fig. 6. Such irregularities were characterised as having a mean height difference and a standard deviation about 0.6 mm, or 5d. These values were used to define the minimum bed form height restriction for the experimental results, which corresponded to the sum of the mean and standard deviation values given above, i.e. ηmin ≈ 10d. A similar height restriction was adopted by Coleman and Melville (1994) in processing their results. Such a value was higher than the flat bed roughness height calculated using most formulations available in the literature – such as 2.5d (Soulsby, 1997), 3.5d (Coleman and Melville, 1996) or 3.5d90 (van Rijn, 1984) – which is thought to be the threshold height of a sand pileup that triggers the initiation of wavelets, i.e. parameter ηw, as depicted in Fig. (1a). The first record of bed forms occurring after the start of the experiment encompassed the interval of 12 s < t < 108 s of experimentation time, as depicted in Fig. 5. This record generally showed that the start of bed form growth was in the upstream channel, i.e. in the region where x < 5.0 m, but not in the divergent channel, where x > 5.0 m. The median initial bed form height calculated for the data in the upstream channel was η50 = 2.4 mm. The corresponding graph depicting the variation of λ is shown in Fig. 6. This graph indicates that the initial wavelet length in the region of x < 5.0 m was typically in the range of -8-
International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
30 mm < λ < 60 mm. The median bed form length in the upstream channel was λ50 = 47 mm, which was approximately 14% lower than the wavelet length estimated using equation (4), of λw = 55 mm. On the other hand, the distribution pattern of λ in the diverging channel was similar to the corresponding results measured at t = 0, suggesting that bed form development had not yet started in that part of the model for t < 108 s. Further information about the development of λ in the upstream channel, as measured in the first few bed profiling passes, is given in Table 3. The start and end times of each pass are represented by ts and tf respectively, while the calculated medians are given by λ50. The lower and upper limits of the 95% confidence interval (CI) for the median are also shown. An indication of the duration of the wavelet stage in the upstream channel was obtained from an analysis of the 95% CI limits provided in Table 3. The estimated wavelet length of λw = 55 mm was within the CI limits of profile nos. 2, 3 and 4. From profile no. 5 onwards the corresponding λ50 value was lower than the lower limit of the confidence interval, so that the finish time of profile no. 4 (approximately 10 min) was regarded as the time for the end of the wavelet stage in the upstream channel, i.e. tt, as depicted in Fig. (1a). Table 3 Start and finish times, medians and limits of the 95% confidence interval (CI) for the bed form length data acquired in the first six bed profiling passes, as shown in Fig. 6 95% CI limits (mm) Profile no. ts (s) tf (s) λ50 (mm) 1 12 108 47 41 52 2 164 262 52 47 55 3 317 416 54 48 59 4 465 576 57 50 64 5 630 729 68 58 73 6 778 877 70 62 78
Another aspect noted from Table 3 is that λ50 increased with time during the wavelet stage, at a typical rate about 1 mm per 50 s for the experiment. This result suggested that the bed form length is not constant, but increases slightly during the wavelet stage. For the diverging channel Figs. 5 and 6 indicated that there was a gradual increase of η and λ respectively with t. Qualitatively speaking, the trend of bed form growth in this part of the model followed the trend of decreasing values of the hydrodynamic variables depicted in Fig. 4, herein referred to as the diverging channel effects. A key practical implication of such effects for the development of wavelets was a gradual reduction of the growth rate of bed forms, which led to a trend of increase of the duration of the wavelet stage (i.e. parameter tt in Fig. (1a)) along the x axis in the diverging channel. For instance, in the region of 6.0 m < x < 6.5 m – where θ/θcr ≈ 1.1 – and 3,216 s < x < 3,314 s (i.e. even after almost 1 hour of experimentation time), Fig. 6 showed that the bed form lengths were still close to the estimated λw value of 55 mm. 3.2 Bed form steepness The bed form data shown in Figs. 5 and 6 were combined to generate a bed form steepness plot (Fig. 7). The results shown in this graph exclude any bed irregularities with a height lower than ηmin ≈ 10d. Included in this graph are the plots of equations (4), (6), (7) and (8), which gave the estimated wavelet length (λw = 55 mm), ripple steepness (λ/η ≈ 8), and equilibrium ripple height (ηe = 15 mm) and length (λe =121 mm) respectively. Such equilibrium ripple dimensions gave the estimated upper limits of the transition stage of ripple development. The ripple steepness estimated for d = 0.13 mm corresponded to the level expected for very fine to fine sands (Table 1). It can be seen from Fig. 7 that the steepness data of this study followed the trend of equation (4) for η < 7 mm approximately, where most of the ripple length values were around the estimated wavelet length λw = 55 mm. On the other hand, for higher η values the bed form length data were generally higher than the λw value, indicating that the corresponding bed forms were probably no longer in the wavelet stage (Coleman and Melville, 1996). Figure 7 also showed that the relative increase of the bed form length and height after the wavelet stage generally followed the trend of the ripple steepness equation (6). Despite the scatter, it can be noted that International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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such a trend was followed by the ripple data in both the transition and equilibrium stages. It is interesting to note that the steepness data for ripples in the transition stage was described by equation (6), which was originally developed for equilibrium ripples. This fact indicated that such an equation may be applicable to predict the steepness of both transitional and equilibrium ripples.
Fig. 7 Steepness results of bed forms during wavelet, transition and equilibrium stages and plots of predictive equations
Furthermore, the similarity between the ripple steepness in the transition and equilibrium stages confirmed the predictions of the mathematical model simulations of Niño et al. (2002). 3.3 Determination of wavelet threshold height A review of previous studies suggested that the transition from wavelets to ripples normally occurs when the bed form length can no longer be described by a relationship such as equations (3) – (5) (Coleman and Melville, 1996; Raudkivi, 1997; Coleman and Eling, 2000). As discussed above, in this study such a phenomenon occurred when the bed form steepness data followed the trend of equation (6) in Fig. 7. Therefore, an expression for ηt as a function of d was obtained by combining equations (4) and (6) and making λe ≡ λw and ηe ≡ ηt. After rearranging the terms, the equation used to calculate ηt is given by: ηt = 11.10 d 0.247 (for d < 0.2 mm) (9) where the interval of d was determined by the considerations of Coleman and Eling (2000) about the performance of equations (4) and (5) with sands of different grain sizes. The critical wavelet height was defined graphically by locating the intersection point between the curves representing the wavelet (red line) and ripple (green line) stages. It may be pointed out that both equations involved in the derivation of equation (9) consider the median grain size as the representative sediment particle diameter (Raudkivi, 1997). For the fine sand used in this study the threshold height was calculated as being ηt ≈ 7 mm. Conceptually speaking, ηt can be understood as the wavelet height that triggered the start of ripple development. As a result, bed forms with a recorded height lower than ηt were deemed to be in the wavelet stage, while higher height values were associated with transitional ripples, according to the definitions of Fig. (1a). In wall units, this value corresponded to y+ ≈ 80, where y+ = u*cηt/ν, with u*c = 12 mm/s. Such a y+ value is of a similar order of magnitude as the upper limit of the near wall zone of the fully turbulent flow (Rodi, 1993), which suggested that the start of the transition stage of ripple development could be related to the wavelets reaching the outer flow zone of a turbulent boundary layer. Further experimentation on bed form development under different turbulence strength levels, including laminar flow, beyond the wavelet stage would be required to confirm such a hypothesis. - 10 -
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3.4 A probability distribution function for wavelet length Figure 8 depicts a histogram of the bed form length measurements in the interval of ηmin < η < ηt, which corresponds to bed forms in the wavelet stage but excluded bed irregularities. These data were fitted using a lognormal distribution with the median length λ50 shown to be 52 mm, about 5% lower than the estimated wavelet length of λ50 = 55 mm. The fitting of a lognormal distribution to wavelet length data, which has not been reported before, suggested that the median length could also be used to quantify wavelet development, rather than using only the average length.
Fig. 8 Histogram of wavelet length data with fitted lognormal distribution
5 Conclusion Experiments were carried out in a laboratory flume to study the initiation and development of bed forms, from an initially flat sediment bed, and under turbulent and subcritical flow conditions. The model contained a diverging channel, which allowed for varying velocity and sediment transport processes to occur along the developing rippled field. Bed profiles were measured at a relatively high resolution, both spatially and temporally, which allowed for detailed investigations of the bed form development to be made. It was found that the wavelet length measurements can be fitted by a lognormal distribution, while the threshold bed form height that triggered the transition from wavelets to ripples in this study was expressed as ηt = 11.10d0.247. Further experimentation findings reported herein are summarised below: • The duration of the wavelet stage was found to vary from about 10 min (for a cross-sectional mean velocity of 40 cm/s and relative Shields number of 2.5) to more than 1 hour (for 25 cm/s and 1.0 respectively), due to the diverging channel effects in the model. • The ripple steepness measured during the transition stage of development has been shown to be similar to the steepness of equilibrium ripples, providing empirical confirmation to mathematical model simulation results of Niño et al. (2002). As a consequence of this, it has been suggested that the equilibrium ripple steepness equation of Raudkivi (1997) can be used to predict the steepness of transitional ripples as well. • For the fine sand of this study, the threshold wavelet height was determined as ηt ≈ 7 mm. Such a height value corresponded to the intersection point between two predictive equations for bed form dimensions and was deemed as the trigger for the start of the transitional stage of ripple development. Further studies will be carried out to investigate the relationship between the turbulence strength levels and the critical bed form height for the transition from wavelets to ripples. The authors also plan to undertake further experiments to verify the applicability of the findings reported herein to other sediment grain sizes. Acknowledgements Receipt of the equipment grant NE/C513269/1 from the UK’s Natural Environment Research Council (NERC), and the research grant EP/C512316/1 from the Engineering and Physical Sciences Research International Journal of Sediment Research, Vol. 23, No. 1, 2008, pp. 1-12
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Council (EPSRC) are greatly acknowledged by the authors. We would also like to thank Mr. Paul Leach for the technical support provided during the experimental work. References Baas J. H. 1994, A flume study on the development and equilibrium morphology of current ripples in very fine sand. Sedimentology, 41, pp. 185–209. Baas J. H. 2003, Ripple, ripple mark, and ripple structure. Encyclopaedia of Sediments and Sedimentary Rocks, Ed. G. V. Middleton. Kluwer Academic Publishers, Dordrecht, pp. 565–568. Coleman S. E. and Melville B. W. 1994, Bed-form development. Journal of Hydraulic Engineering, Vol. 120, No. 4, pp. 544–560. Coleman S. E. and Melville B. W. 1996, Initiation of bed forms on a flat sand bed. Journal of Hydraulic Engineering, Vol. 122, No. 6, pp. 301–310. Coleman S.E. and Eling B. 2000, Sand wavelets in laminar open–channel flows. Journal of Hydraulic Research, Vol. 38, No. 5, pp. 331–338. Coleman S. E., Fedele J. J., and Garcia M. H. 2003, Closed–conduit bed–form initiation and development. Journal of Hydraulic Engineering, Vol. 129, No. 12, pp. 956–965. Coleman S. E., Zhang M. H., and Clunie T. M. 2005, Sediment–wave development in subcritical water flow. Journal of Hydraulic Engineering, Vol. 131, No. 2, pp. 106–111. Nikora V. I. and Hicks D. M. 1997, Scaling relationships for sand wave development in unidirectional flow. Journal of Hydraulic Engineering, Vol. 123, No. 12, pp. 1152–1156. Niño Y., Atala A., Barahona M., and Aracena D. 2002, Discrete particle model for analyzing bedform development. Journal of Hydraulic Engineering, Vol. 128, No. 4, pp. 381–389. Raudkivi A. J. 1997, Ripples on stream bed. Journal of Hydraulic Engineering, Vol. 123, No. 1, pp. 58–64. Rodi W. 1993, Turbulence Models and Their Application in Hydraulics – a State-of-the-art Review, 3rd ed., A.A.Balkema, Rotterdam / Brookfield. Soulsby R. L. 1997, Dynamics of Marine Sands – a Manual for Practical Applications. Thomas Telford Publications, London. Soulsby R. L. and Whitehouse R. J. S. 2005, Prediction of ripple properties in shelf seas. Mark 1 Predictor. HR Wallingford Report TR 150, Release 1.1. van Rijn L. C. 1984, Sediment transport, Part III: Bed forms and alluvial roughness. Journal of Hydraulic Engineering, Vol. 110, No. 12, pp. 1733–1754. Venditti J. G., Church M., and Bennett S. J. 2005, On the transition between 2D and 3D dunes. Sedimentology, 52, pp. 1343–1359. Yalin M. S. 1977, Mechanics of sediment transport. 2nd Ed., Pergamon Press, Guildford. Zhou D. and Mendoza C. 2005, Growth model for sand wavelets. Journal of Hydraulic Engineering, Vol. 131, No. 10, pp. 866–876.
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