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Incipient sediment motion based on turbulent fluctuations
Journal Pre-proof Incipient sediment motion based on turbulent fluctuations Wan Hanna Melini Wan Mohtar, Lee Ji Wang, Najwa Izzaty Mohammad Azha, NianSheng Cheng PII:
S1001-6279(18)30345-7
DOI:
https://doi.org/10.1016/j.ijsrc.2019.10.008
Reference:
IJSRC 261
To appear in:
International Journal of Sediment Research
Received Date: 25 October 2018 Revised Date:
27 August 2019
Accepted Date: 29 October 2019
Please cite this article as: Melini Wan Mohtar W.H., Wang L.J., Mohammad Azha N.I. & Cheng N.-S., Incipient sediment motion based on turbulent fluctuations, International Journal of Sediment Research, https://doi.org/10.1016/j.ijsrc.2019.10.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V. on behalf of International Research and Training Centre on Erosion and Sedimentation/the World Association for Sedimentation and Erosion Research.
Incipient sediment motion based on turbulent fluctuations Wan Hanna Melini Wan Mohtara, Lee Ji Wanga, Najwa Izzaty Mohammad Azhaa, NianSheng Chengb a
Department of Civil & Structural Engineering, Faculty of Engineering and Built
Environment, Universiti Kebangsaan Malaysia, Bandar Baru Bangi, 43600, Malaysia b
School of Civil and Environmental Engineering, College of Engineering, Nanyang Technical University, 50 Nanyang Avenue, 639798 Singapore
Incipient sediment motion based on turbulent fluctuations
The current study modifies the representation of the Shields parameter using turbulent strength, i.e. the root-mean-square (rms) fluid velocity. Experiments were done under a steady, uniform flow using eight sediment sizes with particle Reynolds numbers ( ) ranging between
1.0 (fine sediment) and 183.4 (coarse sediment). Utilising the peak rms horizontal () values, the
critical Shields parameter, , was calculated and a trend similar to the well-established Shields
curve was developed. The analysis was extended to the Shields curve obtained based on the critical shear velocity, Reynolds shear stress, and data extracted from the oscillating grid-turbulence experiments. Results show that turbulent fluctuations are crucial for the incipient sediment motion and are essentially better predictors than the commonly used critical shear velocity. A quadrant analysis to identify the role of turbulent bursting events in incipient sediment motion also was done where sweeps and ejections are dominant for finer and coarser sediment sizes, respectively.
Keywords: Incipient sediment motion, Root-mean-square fluid velocity, Grid-turbulence, Quadrant analysis, Turbulent fluctuations.
1. Introduction Information on the mobility state of a particular particle grain in comparison to its previous stationary position in a water stream is crucial. This condition often is described as incipient sediment motion or the threshold criteria of sediment movement and is represented by the critical Shields parameter, . This parameter was famously developed by Shields (1936) and is expressed as =
( )
,
(1)
where = ∗ is the critical bed shear stress, ∗ is the critical shear velocity (characteristic
velocity defined in the near-bed region), is the grain diameter (commonly described as median grain size, ), and and are sediment and fluid densities, respectively.
One of the most common and well established approaches in determining the value of ∗ to represent the velocity scale in the near-bed region is by using the logarithmic layer profile,
which is defined as
∗
!
= ln ,
!"
(2)
where # is the mean average fluid velocity, $ is the von Karman constant which is commonly
applied as 0.4. The symbols % and % represent distance above the bed and surface roughness,
respectively. In incipient sediment motion experiments, under the condition for which the threshold movement was observed, the calculated shear velocity (using Eq. 2) is defined as the critical shear velocity ∗ . Note that the shear velocity is a pseudo velocity scale which represents shear strength and cannot be measured directly.
In places where the measurement of turbulence is possible, the critical near-bed kinematic Reynolds shear stress, ,& , and turbulent kinetic energy (TKE), '( , can be used to define incipient
sediment motion (i.e. can be substituted as in Eq. (1)). The critical shear stress is obtained using Eq. (3) as follows, ,& = −′+′,
(3)
,,- = .'( = .(′ + 0′ + +′ ),
(4)
where ′, 0′, and +′ are horizontal, transversal, and vertical velocity fluctuations, respectively, obtained through Reynolds decomposition. The symbol . is a coefficient, usually taken as 0.19 or 0.2 (Kim et al., 2000).
In the boundary layer, the instability generated by shear stress creates turbulence, and this is significant. As the incoming flow is turbulent, the instantaneous forces and torque are dominant and are highly likely to exceed the critical shear stress value ( ) although the mean shear stress () acting on the particles is much less than = ∗ (Lee & Balachandar, 2012). This turbulence
effect accounts for the observation made by Paintal (1971), where bedload sediment transport occurred despite the fact that the (time-averaged) shear stress was much less than the (calculated) critical shear stress.
Turbulence behaviour in the near bed region fluctuates and is dependent on the bed formation (McLean, 1994; Nelson et al., 1995), obstructions on the bed (Cheng & Emadzadeh, 2016; Sumer et al., 2003), the presence of coherent vortex structures in the near-bed region (Nino & Garcia, 1996), and the impulse acting on the particles (Diplas & Dancey, 2013; Valyrakis et al, 2010). To account for the spike in fluid velocity, it is best to describe the representation of sediment entrainment using the maximum forces acting on the particles rather than the mean bed shear stress (Giménez-Curto & Corniero, 2009). On a micro-scale level close to the boundary layer, the self-induced vortex shedding in the wake promotes higher turbulent fluctuation for larger-sized particles within the hydraulically rough regions. This consequently influences the strength of the fluctuating lift force on the incipient sediment motion, providing approximately 40% of the effect on sediment movement whilst the remaining effect is due to drag force (Lee & Balachandar, 2012; Schmeeckle et al., 2007).
Several studies have been done to observe the influence of isolated turbulent fluctuations (with negligible contribution by the mean flow) on the threshold of motion criteria, by adapting the statistically stationary flow behaviour in oscillating-grid chambers (Bellinsky et al., 2005; Wan Mohtar & Munro, 2013). A similar Shields pattern was observed where was monotonically increased for sediment within the hydraulically smooth area and reached a relatively constant value of ≈ 0.05 within the hydraulically rough regions. The findings from the grid turbulence experiments provide the basic and fundamental concept on the effects of isolated turbulent fluctuations (without the influence from the mean flow) on incipient sediment motion. The current study empirically analyses the effect of turbulent fluctuations in a (more realistic) steady open channel flow. The behaviour of particles subjected to turbulent fluctuations (defined as root mean square (rms) fluid velocity), the shear velocity calculated from the logarithmic profile, and the Reynolds shear stress and TKE based shear stress were investigated to evaluate which parameter significantly affects incipient sediment motion. Analysis also is extended to a quadrant analysis to evaluate the turbulence structure responsible for incipient sediment motion for varying sediment sizes.
2. Materials and methods
Experiments were done in a 0.5 × 0.5 × 10 m laboratory flume, and a schematic diagram of the setup is shown in Fig. 1. The working area, i.e. the sediment bed spans approximately 0.6 × 0.5 m and is located about 5.5 m from the inlet where pre-measurement of the velocity profile showed that, at this location, the flow is well developed and uniform. To minimise the wall effect, the observation of threshold criteria is made at the mid-plane region covering about 0.6 × 0.15 m. A self-made, systematically stacked group of 0.025 × 0.5 m PVC pipes which acts as a flow strengthener, was placed at the inlet to ensure a smooth, uniform flow and to minimise the turbulence effects. The flow velocity and water depth were controlled both by a tail gate (located at the end of flume) and a flow valve located before the flume inlet. Water depth was kept constant at 15 cm and the level measurement was kept constant for each set of experiments by using a vernier scale.
Figure 1 The sediment used in the experiments is homogeneous and ranges between 40 to 1300 9m
in size. If the roughness length scale was taken as ' ≈ , the ratio of /;< (where ;< is the depth of viscous sublayer) showed that the range of sediment size discussed here is within both hydraulically smooth and rough regions (see Table 1). The sediment gradation parameter = = >?@ / A is less than 1.6, obeying the uniformly distributed standard rule (Blott & Pye,
2001; Vanoni, 2006). The parameters ?@ and A denote sediment sizes at 84% and 16%, respectively on the grain distribution curve. The particle Reynolds number ( ) is defined as Table 1
=
D B( )C"
E
,
(5)
where F = / is the specific gravity of sediment, G is gravitational acceleration, and H is water kinematic viscosity. Sediment density is taken as 2650 kg/m3.
The bed was prepared by placing sediment in the working area, with the bed depth set at 5
cm with false floors installed upstream and downstream of the working area. The top layer was then levelled using a bed scraper to produce a horizontal smooth bed. To ensure a consistent surface roughness throughout the channel, layers of the same sediment (for each experiment) were placed upstream and downstream of the working area. Each experiment was initiated by filling in the flume with a flow at the lowest velocity to minimise disturbance to the prepared sediment bed. The flow rate was kept constant until a specified water level had been reached before increasing the flow velocity. The initial velocity was set at 15 cm/s, followed by a systematic increase of approximately 1 cm/s until incipient sediment motion was observed.
The critical velocity was measured after the threshold movement was established. The definition of the threshold motion varied between researchers where most of the studies utilised observational measurements. Here, the critical velocity is defined when a small number of the smallest particles are in motion in isolated zones (within 10 min of the observation period), which is described as weak transport in Kramer (1935). In the current study, the definition of "a small number" denotes about 10 - 20 particles moving with a relatively similar velocity in the longitudinal direction. An Acoustic Doppler Velocimeter (ADV) was used to measure the instantaneous fluid velocity (captured at 200 Hz frequency) at distances of 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, and 8.5 cm above the bed. The ADV was positioned exactly at the center of working area for each experiment to ensure a consistent measurement in all experiments. Throughout the data measurement, the Signal to Noise Ratio (SNR) is constantly between 22-25, which is deemed appropriate as specified in the ADV manual. The ADV allows for a three dimensional velocity to be measured. Let #, I, and J be the horizontal (in the flow direction), transverse (normal direction of the flow), and vertical velocity components, respectively. Taking the horizontal flow velocity as an example, the Reynolds decomposition separates the turbulent fluctuating components as # = # + ′ where # denotes
the instantaneous flow velocity, and # and ′ are the mean flow velocity and turbulent
fluctuations, respectively. The turbulent strength, based on the rms velocities are obtained as = B′(K) . As such, following the same procedure, the rms velocity for transversal velocity, 0,
and vertical velocity, +, were calculated at each point based on the statistical parameters.
Figure 2 Figures 2a and 2b show the cross sectional vertical turbulence intensity profile for /#
and 0/#, respectively against the distance above bed L for varying ranges of Reynolds number,
= #ℎ/H, where ℎ is the flow depth. The data clearly shows that and 0 are amplified near
the bed and have peak values, which are denoted as and 0 for horizontal and transversal
components, respectively, at approximately L = 1.5 cm above the bed (≈ L/ℎ = 0.1. The peak value can reach up to 0.1 ∼ 0.13#, which is in agreement with the findings of Lamb et al. (2008). Similar amplification of also was observed in a steady uniform flow, where peak values
can reach up to 10% of the mean flow velocity, # (Cheng & Emadzadeh, 2016) at L ≈ 0.5 cm,
where L/ℎ is about 0.025, much closer to the bed layer. A corresponding rise in the rms
transversal fluid velocity 0 at L/ℎ = 0.1 cm also was observed. The amplification is within a similar range as that for , i.e. 0.1 ∼ 0.12# for the range of evaluated here.
It is noted that the peak value recorded in the current study is slightly lower than the achievable maximum (as discussed in Nakagawa et al. (1975)); however, measurement of a smaller range of (consistently fixed) vertical distances was not possible in the current study. As coherent velocity profiles were observed for the range of considered in this study, the distance
above bed at % =1.5 cm is believed to give a fair representation of the peak value. To assess for the variation, the discrepancy of obtained at dimensional distance L = 1.5 cm and at L ≈ 0.5
cm were compared based on the vertical velocity profile reported by Cheng and Emadzadeh (2016). The /# for both studies were found comparable between 0.9-0.98, and, thus, it is deemed acceptable for the peak value of to be defined at such a distance.
Note that the current study focuses on the horizontal movement of sediment. By taking into account the streamwise (i.e. horizontal) peak value, , the threshold criteria for each sediment size is redefined as =
PQ
( )C"
.
(6)
To facilitate the correlation of based on , the current study compares the presentation
of using other shear-based variables in incipient sediment motion including the commonly used critical shear velocity, ∗, , Reynolds shear stress, ,& , and TKE, ,,- as described in Eq. (1).
3. Results and discussion 3.1. Redefining the Shields diagram The critical Shields parameter obtained for each sediment size was plotted against and is shown in Fig. 3. The well-established Shields curve also also plotted on the same graph, as a solid line and was calculated based on the empirical function of = 0.22.A + 0.06exp(−17.77 .A ),
(7)
as described by Brownlie (1982).
Figure 3 Parameter Shields was found to monotonically increase with decreasing for the
region < 100 before reaching a constant value of ≈ 0.08 for larger particle Reynolds number. A qualitatively similar profile with the empirical Shields curve is evident. This curve was developed based on the critical shear velocity measurements taken in steady open channel flow. Despite that, the magnitude of obtained through is consistently (up to 4 times) higher than
the Shields curve, particularly for the lower region. Figure 4
The analysis was extended to the characterisation of , which was calculated using the
critical shear velocity, ∗ , and was obtained through the logarithmic law (as described in Eq. (2))
applied to the measured velocity profile at the near-bed region, about 20% of the flow depth. The Shields profile was evidently fell closer to the well- established Shields curve (the solid line in Fig. 4) and follows the same entrainment behaviour. The representation of the Shields parameter using has a higher magnitude of X(10 ) and was consistent throughout the range of
considered in the current study. However, the discrepancy is notably evident for lower values although this discrepancy begins to diminish at > 80 as the particle size increases. Although
increases as decreases, the slope of the increase is less steep compared to the well
established Shields curve. Since the ADV is able to measure instantaneous fluid velocity, it is possible to represent
using the critical Reynolds shear stress, ,& , and the turbulent kinetic energy, ,,- . The discussion starts with the ,& based . The values fall much closer to the Shields curve and
produce a similar slope pattern with the using . It was found that the Reynolds shear
stress-based Shields parameters approximately fall on the same curve as the ∗, based .
Apparently, momentum flux terms proved to be consistent and dependable elements for estimating the critical shear velocity in describing incipient sediment motion.
Presenting the incipient sediment motion in terms of TKE is permissible as the magnitude of fluctuating drag and lift forces is well correlated with the near-bed turbulent kinetic energy (Yang et al., 2016). In a bare channel, the near-bed TKE is proportional to the bed shear stress (as described in Eq. 4) and may be used as a predictor for the incipient motion condition (Stapleton & Huntley, 1995). The curve obtained slightly deviates from other shear-based variables and the established Shields curve, in particular at > 200 where the values are consistently significantly higher (Fig. 4). Even so, based on the similarity of the profiles, it can be said that , ∗ , and Reynolds shear stress, & , and ,,- functioned in the same way to promote sediment
transport.
Surface roughness (i.e. the sediment size) apparently plays an important role in the near-bed critical turbulence intensity, /∗ . The ratio of /∗ is ∼ 1 for > 40, however,
the value of increases for finer sediment and can essentially be /∗ ≈ 2 at < 10 (i.e.
within the hydraulically smooth region). Note that at = 1 (the smallest in the current
study), /∗ = 2.34. The observed ratio of /∗ for smaller sediment sizes lies within the
range of the hydraulically smooth limit (of 2.3 to 2.7) as described by Nezu and Rodi (1986). Regression analysis was done on the near-bed critical turbulence intensity /∗ , in terms
of particle Reynolds number . The data yielded the relation /∗ = 2.22. @ . The
measured ∗ monotonically decreased for finer sediment sizes (as it is inversely proportional to
), and is very likely to fall within the laminar viscous sublayer, where viscous force dominates and reduces the shear strength. This explains the higher calculated when using (compared to
when ∗ was used) for lower .
3.2. Comparison with oscillating grid turbulence data
The oscillating grid approach permits the representation of incipient sediment motion based on turbulent fluctuations, with no or negligible effect from the mean fluid velocity. The generated flow is nearly-isotropic (that is invariant in rotation), quasi-homogeneous (i.e. invariant in translation) turbulence and allows for a characterisation of incipient sediment motion based solely on the effects of turbulence. Although several studies are available for comparison, here the works of Bellinsky et al. (2005), Redondo et al. (2001), and Wan Mohtar and Munro (2013) were selected as these studies used a range of varying sediment sizes and are comparable to the ones discussed in the current study. The Shields parameter, , was obtained based on the approximation of ∼ , which
allows for to be determined using Eq. (1). Note that use of , denoted as the critical in the grid turbulence experiments, differed in each study. Bellinsky et al. (2005) and Redondo et al. (2001) used the calculated based on the empirical equation developed by Hopfinger and Toly (1976). The behaviour of grid generated turbulence at the boundary layer posed similar characteristics as the ones obtained in open channel flow. The velocities of and 0, which were amplified in the near-bed region due to the kinematic boundary condition, are known as splat-antisplat events (Perot & Moin, 1995). Because of this, the work of Wan Mohtar and Munro (2013) used the peak value of in the near bed region, where the dimensional distance above bed
% ≈ 1.8 cm.
Figure 5 Figure 5 shows the plots of experimental data with from the grid turbulence
experiments. To avoid confusion in the discussion, the Shields parameter from grid turbulence experiments is denoted as . In general, the Shields profiles fall close to the envelope of the well-established Shields curve, where as decreased (for < 40) the is increased
monotonically while for higher > 100, reached a relatively constant value. It should be noted that the Shields curve line based on the data obtained by Redondo et al.(2001) is significantly lower, and this is believed to be due to variations in the definition of incipient sediment motion and the lack of homogeneity in the distribution of Redondo et al.’s (2001) marine sediment. The Shields parameter obviously has a higher magnitude than , approximately 3 higher than the Shields parameter characterised by the near-isotropic turbulence. To account for the variation of the obtained profiles, a power law regression = [\ was applied to each
data set. Coefficient ] denotes the slope in the log scale plot. Coefficients ] in the data obtained by Wan Mohtar and Munro (2013) and Bellinsky et al. (2005) as well as the current data fall within a similar range, i.e. 0.39, 0.35, and 0.44, respectively. A higher ] value of 0.51 was obtained
based on the data Redondo et al. (2001). A steeper slope for calculated using indicates that as sediment size decreases, higher turbulent strength is required and is responsible for incipient sediment motion, which, therefore, suggests higher mean fluid velocity.
The ratio of peak rms and shear velocity /∗ was expressed by Lamb et al. (2008) as P
∗
= ^ _5.62log
b
,
+ 4c,
(8)
where the coefficient ^ is = 0.2 and ' is the roughness length-scale of the bed. Using this representation, the data from the grid turbulence experiments also were redefined based on critical shear velocity, ∗ . Figure 6
Figure 6 shows the representation of the modified Shields parameters (denoted here d )
based on the recalculated and ∗ from the current grid turbulence experiments. Both velocity
parameters were represented using the modification factor e =
fghij fghk l(fghk)Q
, where mn is the
particle’s angle of repose and o is the bed slope. The modified particle Reynolds number (d ) is defined as >e and d = /e. Note that, in this plot, the modification of the Shields
parameters were only done for data from Wan Mohtar and Munro (2013), considering the robustness approach of how the Shields parameter was defined. Recall that Wan Mohtar and Munro (2013) used the peak value of root-mean-square horizontal fluid velocity . Nevertheless, a similar trend is expected if the same modification procedure was applied to the data obtained by Bellinsky et al. (2005) and Redondo et al. (2001) due to the similarity of the Shields profiles presented in Fig. 5. Despite the similarity in the obtained profiles, the difference in the magnitude of d
varies up to X(10 ). The Shields curve obtained through turbulent fluctuations is consistently higher than the estimated values when calculated using shear velocity. This analysis supports the idea that turbulent fluctuations are consistently higher than shear velocity for estimating incipient sediment motion.
3.3. Analysis of the quadrant of Reynolds shear stress plane during incipient sediment motion The availability of instantaneous ′ and +′ allows for a quadrature analysis of the
′ − +′ plane to identify the dominant turbulence structure during the critical velocity identified
as the threshold criteria for sediment motion. Most of the studies measuring simultaneous particle movement and fluid velocities were done to elucidate the dominant fractions, thus, the flow velocity was much higher than the threshold value (/ ⋙ 1) to permit obvious sediment movement (Wu & Jiang, 2007; Wu & Yang, 2004). Here, the authors opted to investigate the turbulence characteristics at the paradoxical demarcation of immobility-mobility of a particle, for which the flow velocity is much the lower considering lower observational threshold criteria used compared to previous studies.
The vertical distance plays an important role in determining the turbulence structure, where the sweep and ejection dominance varies within the wall shear layer (Dey et al., 2011). However, the distance where the highest rms horizontal velocity is a representative coherent height at which the turbulence intensity is maximal and the contributions of sweeps and ejections are equal (Wu & Yang, 2004). As such, it is believed that the same dataset at where was obtained provides an accurate representation for the quadrant analysis.
Four quadrants (Qq) provide the types of bursting event as: (1) outward interactions Q1 (q=1, ′ > 0, +′ > 0), 2) ejections Q2 (q=2, ′ < 0, +′ > 0); 3) inward interactions Q3 (q=3,
′ < 0, +′ < 0); and 4) sweep Q4 (q=4, ′ > 0, +′ < 0). The frequency of each bursting event is
calculated based on the significant contributions to −′+′, through filtration using a hyperbolic region, referred to as the hole size parameter, r. The parameter r ignores the smaller ′ and +′
velocity components which correspond to more quiescent periods, whereby only the dominant events will be accounted for (Lu & Willmarth, 1973; Nezu & Nakagawa, 1993). The detection function, os (K) , is calculated through the decomposition of velocity
fluctuations into four quadrant domains of q = 1, 2, 3, and 4 as . . os,t (K) ={ 1, if |′+′| ≥ r(′′) (+′+′) , 0, otherwise,
(9)
where the overbar denotes time averaging. In the current study, as the velocity threshold for sediment motion is much lower (which translates into lower fluid velocity) than that for medium or general sediment (transport) movement, the hole size only varied between 1-5 in the conditional analysis. Figure 7 The variations of fractional contribution s,t of each types of event were investigated with
hole size r for all the range considered here. Note that s,t = s,t /s,t , where s,t
denotes the frequency of an event and s,t is the total number of all four quadrant events. The current study analysis showed that the events decreased as the r value increased, in agreement
with the findings of Dey et al. (2011) and Yue et al. (2007). Interestingly, there is a distinctive pattern of frequency of event occurrence for fine and coarse sediment sizes. To facilitate a comprehensible plot, Fig. 7 shows the influence of hole size on the values of s for = 0.99
and = 7.19 to represent fine sediment and = 142.71 and = 186.36 for coarser sediment. At r = 1, both ejection and sweep are dominant in entraining the sediment, but as r increases, the turbulence structure responsible to initiate the movement for fine sediment are
sweep events, whereby is seen to monotonically decrease. On the contrary for coarser sediment sizes, the influence of ejection became prominent and consequential and dominated the scenario up to 80%. This indicates that the movement of fine sediment is through sweeps of high momentum onto the particles, whereas the coarser sediment moved due to the lifting forces from the bed. However, Wu and Jiang (2007) found that the entrainment for approximately similar = 142.71 is through lifting and is dominated by sweep (Q4), which is contrary to the ejection mode found in the current study. The contradictory findings are believed to be the result of variation in the vertical distance of where the flow velocity measurements were taken. The frequency and occurrence of events varied along the boundary layer, by which ejections monopolise the turbulence structure at the top of the boundary layer (Dey et al. 2011).
In general, sweep and ejection events are responsible for the incipient sediment motion, which echoes the findings of Kesharvazi et al. (2012) and Thorne et al. (1989), to name a few. During entrainment of particles, both sweep and ejection events have a higher frequency of occurrence than outward and inward interactions.
Figure 8
To provide a qualitative overview on the detailed fluctuation contribution on each type of event, Fig. 8 shows the quadrant analysis for each case when the conditional analysis (i.e.
hole size) r = 1. The data show that the momentum and kinetic energy transfer needed to initiate
sediment movement is mostly the result of ejection (′ < 0, +′ > 0) and sweep (′ > 0, +′ < 0) events based on the consistent top-left (Q2) slanting towards right-down (Q4), although both
inward and outward interactions do make a bit of contribution. At the lowest particle Reynolds number = 0.99, the turbulence intensity is considerably lower than occurs as the particle Reynolds number increases. Both ejection and sweep events composed more than 80% of the dominant turbulence structure, and interestingly the fraction between the two is comparably similar, i.e. between 36-46% each (Table 2).
Table 2
To assess on the quantitative influence of sediment size on the types of bursting events, Table 2 lists the fractional contribution (based on r = 1) for the ranges considered here. The inward and outward interactions consistently have low frequencies. There is no distinct pattern of motions types based on the , except sweep and ejection events are significant during incipient sediment motion. Whereas for higher flow velocities which promote sediment entrainment (and weak sediment transport), the sediment movement also can be influenced by outward interactions (Wu & Jiang, 2007). Wu and Jiang (2007) studied at much higher flow velocity to observe energetic sediment movement. Thus, both outward and inward interactions become more developed and distinctive at higher flow velocities. Based on this, it is speculated that both outward and inward interactions are not significant for incipient sediment motion and only display influence entering the phase of sediment entrainment, suspension, and transport.
Bed roughness changes the dominant turbulence structure, whereby in a gravel bed river the sediment mobility is influenced by the ejection-sweep events for densely packed and low-density bed geometries. Whereas outward-inward interactions are the most frequent events for isolated flow regimes (such boulders or clasts) (Papanicolaou et al. 2001). Note that the threshold condition in the current study is based on a (flat) smooth surface, thus, the bedform wake (due to the changes in surface roughness) is not significant. The pressure difference between grains during the dominant sweep (Q4) induced sediment entrainment for both exposed or shielded grains (Dwivedi et al., 2010).
4. Conclusions
The effect of turbulence is inherent in the threshold criteria for sediment movement. The representation of based on turbulent fluctuations, presented here as the root-mean-square horizontal velocity, gives similar profiles as that obtained when the Shields parameter was calculated using the critical shear velocity, ∗ , and Reynolds shear stress, & . Both turbulent fluctuations and shear velocity function in the same way to promote incipient sediment movement. These results indicate that the second order moments of velocity fluctuation (i.e. turbulence intensity) may reduce the ambiguity in the determination of incipient sediment motion. It is believed that by incorporating the peak value of turbulence intensity, a more accurate threshold criterion is feasible.
It is interesting to note that ejections and sweeps are the dominant turbulence structures and they initiate the incipient sediment motion even at low flow velocities, spanning from hydraulically smooth to rough regions. Significant turbulence structure influences on the incipient sediment motion for coarser and finer sediment sizes are ejection and sweep types, respectively.
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Figure 1: A schematic diagram of the experimental setup (not to scale). A false floor is installed both upstream and downstream of the working area. (Note: ADV is an Acoustic Doppler Velocimeter)
Figure 2: The vertical turbulence intensity profiles for varying Reynolds number Re (a) ݑ/ܷ and (b) ݒ/ܷ. Both fluid velocities are made dimensionless with the mean horizontal velocity, ܷ. Each symbol represents associated Re, with Re = 25000 (∘), Re = 29000 (×), Re = 31000 (□), Re = 36140 (⋄), Re = 36300 (△), Re = 40000 (∗), Re = 46000 (⊳), and Re = 48000 (⊲).
Figure 3: The Shields diagram for experimental data when the critical Shields stress is calculated using ݑො. The solid line represents the Shields curve from the Brownlie (1982) equation.
Figure 4: The Shields diagram for experimental data (•), ߠ calculated based on ∗ݑ, (□), ߬, (⋄), ߬,ோ (△), and ∗ݑ (×). The solid line represents Shields curve calculated using the Brownlie (1982) equation.
Figure 5: Representation of Shields diagram with the current experimental data (•), obtained with ݑ and ߠ from selected grid turbulence experiments. Data was extracted from Wan Mohtar & Munro (2013) (□), Bellinsky et al. (2005) (△) and Redondo et al. (2001) (⋄). The solid line represents Shields curve from the Brownlie (1982) equation.
Figure 6: Replot of Shields curve based on the modified ߠ and particle Reynolds number, ܴ݁ , based on ݑො (∘) and ∗ݑ (□) obtained from the current grid turbulence data.
Figure 7: Fractional contribution ܨ for (a) outward interaction (Q1), (b) ejection (Q2), (c) inward interaction (Q3), and (d) sweep (Q4). Symbols denote ܴ݁ = 0.99 (∘), ܴ݁ = 7.19 (□), ܴ݁ = 142.71 (△) and ܴ݁ = 186.36 (×).
Figure 8: The quadrant analysis for (a) ܴ݁ = 0.99, (b) ܴ݁ = 7.19 , (c) ܴ݁ = 20.33, (d) ܴ݁ = 40.50, (e) ܴ݁ = 57.50, (f) ܴ݁ = 88.52, (g) ܴ݁ = 142.71, and (h) ܴ݁ = 186.36 based on = ܪ 1.
Table 1: Sediment and flow characteristics used in the current study ݀ହ (ߤm) 40 150 300 475 600 800 1,100 1,300
ܴ݁ 0.99 7.19 20.33 40.50 57.50 88.52 142.72 183.36
ܷ (m/s) 0.150 0.209 0.194 0.202 0.221 0.279 0.293 0.300
( ∗ݑm/s) 0.0087 0.0150 0.0159 0.0182 0.0226 0.0238 0.0326 0.0351
ߜ௩ (mm) 0.577 0.334 0.315 0.275 0.221 0.210 0.153 0.143
݀ହ /ߜ௩ 0.0693 0.449 0.951 1.729 2.711 3.811 7.180 9.102
Table 2: Fractional contributions ܨ of types of bursting event for varying ܴ݁ , when = ܪ1.
Event Outward (Q1) Ejection (Q2) Inward (Q3) Sweep (Q4)
0.99 11.0 40.2 4.8 44.0
7.19 11.3 43.0 8.8 36.9
20.33 9.1 44.1 7.2 39.6
40.50 10.0 40.7 6.9 42.4
ܴ݁ 57.50 5.6 46.0 8.0 40.4
88.52 8.6 37.7 8.9 44.8
142.72 6.1 41.9 5.2 46.7
186.36 9.5 44.2 9.3 37.0
The authors have no conflict of interest with any people or organisations.
S1001-6279(18)30345-7
DOI:
https://doi.org/10.1016/j.ijsrc.2019.10.008
Reference:
IJSRC 261
To appear in:
International Journal of Sediment Research
Received Date: 25 October 2018 Revised Date:
27 August 2019
Accepted Date: 29 October 2019
Please cite this article as: Melini Wan Mohtar W.H., Wang L.J., Mohammad Azha N.I. & Cheng N.-S., Incipient sediment motion based on turbulent fluctuations, International Journal of Sediment Research, https://doi.org/10.1016/j.ijsrc.2019.10.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V. on behalf of International Research and Training Centre on Erosion and Sedimentation/the World Association for Sedimentation and Erosion Research.
Incipient sediment motion based on turbulent fluctuations Wan Hanna Melini Wan Mohtara, Lee Ji Wanga, Najwa Izzaty Mohammad Azhaa, NianSheng Chengb a
Department of Civil & Structural Engineering, Faculty of Engineering and Built
Environment, Universiti Kebangsaan Malaysia, Bandar Baru Bangi, 43600, Malaysia b
School of Civil and Environmental Engineering, College of Engineering, Nanyang Technical University, 50 Nanyang Avenue, 639798 Singapore
Incipient sediment motion based on turbulent fluctuations
The current study modifies the representation of the Shields parameter using turbulent strength, i.e. the root-mean-square (rms) fluid velocity. Experiments were done under a steady, uniform flow using eight sediment sizes with particle Reynolds numbers ( ) ranging between
1.0 (fine sediment) and 183.4 (coarse sediment). Utilising the peak rms horizontal () values, the
critical Shields parameter, , was calculated and a trend similar to the well-established Shields
curve was developed. The analysis was extended to the Shields curve obtained based on the critical shear velocity, Reynolds shear stress, and data extracted from the oscillating grid-turbulence experiments. Results show that turbulent fluctuations are crucial for the incipient sediment motion and are essentially better predictors than the commonly used critical shear velocity. A quadrant analysis to identify the role of turbulent bursting events in incipient sediment motion also was done where sweeps and ejections are dominant for finer and coarser sediment sizes, respectively.
Keywords: Incipient sediment motion, Root-mean-square fluid velocity, Grid-turbulence, Quadrant analysis, Turbulent fluctuations.
1. Introduction Information on the mobility state of a particular particle grain in comparison to its previous stationary position in a water stream is crucial. This condition often is described as incipient sediment motion or the threshold criteria of sediment movement and is represented by the critical Shields parameter, . This parameter was famously developed by Shields (1936) and is expressed as =
( )
,
(1)
where = ∗ is the critical bed shear stress, ∗ is the critical shear velocity (characteristic
velocity defined in the near-bed region), is the grain diameter (commonly described as median grain size, ), and and are sediment and fluid densities, respectively.
One of the most common and well established approaches in determining the value of ∗ to represent the velocity scale in the near-bed region is by using the logarithmic layer profile,
which is defined as
∗
!
= ln ,
!"
(2)
where # is the mean average fluid velocity, $ is the von Karman constant which is commonly
applied as 0.4. The symbols % and % represent distance above the bed and surface roughness,
respectively. In incipient sediment motion experiments, under the condition for which the threshold movement was observed, the calculated shear velocity (using Eq. 2) is defined as the critical shear velocity ∗ . Note that the shear velocity is a pseudo velocity scale which represents shear strength and cannot be measured directly.
In places where the measurement of turbulence is possible, the critical near-bed kinematic Reynolds shear stress, ,& , and turbulent kinetic energy (TKE), '( , can be used to define incipient
sediment motion (i.e. can be substituted as in Eq. (1)). The critical shear stress is obtained using Eq. (3) as follows, ,& = −′+′,
(3)
,,- = .'( = .(′ + 0′ + +′ ),
(4)
where ′, 0′, and +′ are horizontal, transversal, and vertical velocity fluctuations, respectively, obtained through Reynolds decomposition. The symbol . is a coefficient, usually taken as 0.19 or 0.2 (Kim et al., 2000).
In the boundary layer, the instability generated by shear stress creates turbulence, and this is significant. As the incoming flow is turbulent, the instantaneous forces and torque are dominant and are highly likely to exceed the critical shear stress value ( ) although the mean shear stress () acting on the particles is much less than = ∗ (Lee & Balachandar, 2012). This turbulence
effect accounts for the observation made by Paintal (1971), where bedload sediment transport occurred despite the fact that the (time-averaged) shear stress was much less than the (calculated) critical shear stress.
Turbulence behaviour in the near bed region fluctuates and is dependent on the bed formation (McLean, 1994; Nelson et al., 1995), obstructions on the bed (Cheng & Emadzadeh, 2016; Sumer et al., 2003), the presence of coherent vortex structures in the near-bed region (Nino & Garcia, 1996), and the impulse acting on the particles (Diplas & Dancey, 2013; Valyrakis et al, 2010). To account for the spike in fluid velocity, it is best to describe the representation of sediment entrainment using the maximum forces acting on the particles rather than the mean bed shear stress (Giménez-Curto & Corniero, 2009). On a micro-scale level close to the boundary layer, the self-induced vortex shedding in the wake promotes higher turbulent fluctuation for larger-sized particles within the hydraulically rough regions. This consequently influences the strength of the fluctuating lift force on the incipient sediment motion, providing approximately 40% of the effect on sediment movement whilst the remaining effect is due to drag force (Lee & Balachandar, 2012; Schmeeckle et al., 2007).
Several studies have been done to observe the influence of isolated turbulent fluctuations (with negligible contribution by the mean flow) on the threshold of motion criteria, by adapting the statistically stationary flow behaviour in oscillating-grid chambers (Bellinsky et al., 2005; Wan Mohtar & Munro, 2013). A similar Shields pattern was observed where was monotonically increased for sediment within the hydraulically smooth area and reached a relatively constant value of ≈ 0.05 within the hydraulically rough regions. The findings from the grid turbulence experiments provide the basic and fundamental concept on the effects of isolated turbulent fluctuations (without the influence from the mean flow) on incipient sediment motion. The current study empirically analyses the effect of turbulent fluctuations in a (more realistic) steady open channel flow. The behaviour of particles subjected to turbulent fluctuations (defined as root mean square (rms) fluid velocity), the shear velocity calculated from the logarithmic profile, and the Reynolds shear stress and TKE based shear stress were investigated to evaluate which parameter significantly affects incipient sediment motion. Analysis also is extended to a quadrant analysis to evaluate the turbulence structure responsible for incipient sediment motion for varying sediment sizes.
2. Materials and methods
Experiments were done in a 0.5 × 0.5 × 10 m laboratory flume, and a schematic diagram of the setup is shown in Fig. 1. The working area, i.e. the sediment bed spans approximately 0.6 × 0.5 m and is located about 5.5 m from the inlet where pre-measurement of the velocity profile showed that, at this location, the flow is well developed and uniform. To minimise the wall effect, the observation of threshold criteria is made at the mid-plane region covering about 0.6 × 0.15 m. A self-made, systematically stacked group of 0.025 × 0.5 m PVC pipes which acts as a flow strengthener, was placed at the inlet to ensure a smooth, uniform flow and to minimise the turbulence effects. The flow velocity and water depth were controlled both by a tail gate (located at the end of flume) and a flow valve located before the flume inlet. Water depth was kept constant at 15 cm and the level measurement was kept constant for each set of experiments by using a vernier scale.
Figure 1 The sediment used in the experiments is homogeneous and ranges between 40 to 1300 9m
in size. If the roughness length scale was taken as ' ≈ , the ratio of /;< (where ;< is the depth of viscous sublayer) showed that the range of sediment size discussed here is within both hydraulically smooth and rough regions (see Table 1). The sediment gradation parameter = = >?@ / A is less than 1.6, obeying the uniformly distributed standard rule (Blott & Pye,
2001; Vanoni, 2006). The parameters ?@ and A denote sediment sizes at 84% and 16%, respectively on the grain distribution curve. The particle Reynolds number ( ) is defined as Table 1
=
D B( )C"
E
,
(5)
where F = / is the specific gravity of sediment, G is gravitational acceleration, and H is water kinematic viscosity. Sediment density is taken as 2650 kg/m3.
The bed was prepared by placing sediment in the working area, with the bed depth set at 5
cm with false floors installed upstream and downstream of the working area. The top layer was then levelled using a bed scraper to produce a horizontal smooth bed. To ensure a consistent surface roughness throughout the channel, layers of the same sediment (for each experiment) were placed upstream and downstream of the working area. Each experiment was initiated by filling in the flume with a flow at the lowest velocity to minimise disturbance to the prepared sediment bed. The flow rate was kept constant until a specified water level had been reached before increasing the flow velocity. The initial velocity was set at 15 cm/s, followed by a systematic increase of approximately 1 cm/s until incipient sediment motion was observed.
The critical velocity was measured after the threshold movement was established. The definition of the threshold motion varied between researchers where most of the studies utilised observational measurements. Here, the critical velocity is defined when a small number of the smallest particles are in motion in isolated zones (within 10 min of the observation period), which is described as weak transport in Kramer (1935). In the current study, the definition of "a small number" denotes about 10 - 20 particles moving with a relatively similar velocity in the longitudinal direction. An Acoustic Doppler Velocimeter (ADV) was used to measure the instantaneous fluid velocity (captured at 200 Hz frequency) at distances of 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, and 8.5 cm above the bed. The ADV was positioned exactly at the center of working area for each experiment to ensure a consistent measurement in all experiments. Throughout the data measurement, the Signal to Noise Ratio (SNR) is constantly between 22-25, which is deemed appropriate as specified in the ADV manual. The ADV allows for a three dimensional velocity to be measured. Let #, I, and J be the horizontal (in the flow direction), transverse (normal direction of the flow), and vertical velocity components, respectively. Taking the horizontal flow velocity as an example, the Reynolds decomposition separates the turbulent fluctuating components as # = # + ′ where # denotes
the instantaneous flow velocity, and # and ′ are the mean flow velocity and turbulent
fluctuations, respectively. The turbulent strength, based on the rms velocities are obtained as = B′(K) . As such, following the same procedure, the rms velocity for transversal velocity, 0,
and vertical velocity, +, were calculated at each point based on the statistical parameters.
Figure 2 Figures 2a and 2b show the cross sectional vertical turbulence intensity profile for /#
and 0/#, respectively against the distance above bed L for varying ranges of Reynolds number,
= #ℎ/H, where ℎ is the flow depth. The data clearly shows that and 0 are amplified near
the bed and have peak values, which are denoted as and 0 for horizontal and transversal
components, respectively, at approximately L = 1.5 cm above the bed (≈ L/ℎ = 0.1. The peak value can reach up to 0.1 ∼ 0.13#, which is in agreement with the findings of Lamb et al. (2008). Similar amplification of also was observed in a steady uniform flow, where peak values
can reach up to 10% of the mean flow velocity, # (Cheng & Emadzadeh, 2016) at L ≈ 0.5 cm,
where L/ℎ is about 0.025, much closer to the bed layer. A corresponding rise in the rms
transversal fluid velocity 0 at L/ℎ = 0.1 cm also was observed. The amplification is within a similar range as that for , i.e. 0.1 ∼ 0.12# for the range of evaluated here.
It is noted that the peak value recorded in the current study is slightly lower than the achievable maximum (as discussed in Nakagawa et al. (1975)); however, measurement of a smaller range of (consistently fixed) vertical distances was not possible in the current study. As coherent velocity profiles were observed for the range of considered in this study, the distance
above bed at % =1.5 cm is believed to give a fair representation of the peak value. To assess for the variation, the discrepancy of obtained at dimensional distance L = 1.5 cm and at L ≈ 0.5
cm were compared based on the vertical velocity profile reported by Cheng and Emadzadeh (2016). The /# for both studies were found comparable between 0.9-0.98, and, thus, it is deemed acceptable for the peak value of to be defined at such a distance.
Note that the current study focuses on the horizontal movement of sediment. By taking into account the streamwise (i.e. horizontal) peak value, , the threshold criteria for each sediment size is redefined as =
PQ
( )C"
.
(6)
To facilitate the correlation of based on , the current study compares the presentation
of using other shear-based variables in incipient sediment motion including the commonly used critical shear velocity, ∗, , Reynolds shear stress, ,& , and TKE, ,,- as described in Eq. (1).
3. Results and discussion 3.1. Redefining the Shields diagram The critical Shields parameter obtained for each sediment size was plotted against and is shown in Fig. 3. The well-established Shields curve also also plotted on the same graph, as a solid line and was calculated based on the empirical function of = 0.22.A + 0.06exp(−17.77 .A ),
(7)
as described by Brownlie (1982).
Figure 3 Parameter Shields was found to monotonically increase with decreasing for the
region < 100 before reaching a constant value of ≈ 0.08 for larger particle Reynolds number. A qualitatively similar profile with the empirical Shields curve is evident. This curve was developed based on the critical shear velocity measurements taken in steady open channel flow. Despite that, the magnitude of obtained through is consistently (up to 4 times) higher than
the Shields curve, particularly for the lower region. Figure 4
The analysis was extended to the characterisation of , which was calculated using the
critical shear velocity, ∗ , and was obtained through the logarithmic law (as described in Eq. (2))
applied to the measured velocity profile at the near-bed region, about 20% of the flow depth. The Shields profile was evidently fell closer to the well- established Shields curve (the solid line in Fig. 4) and follows the same entrainment behaviour. The representation of the Shields parameter using has a higher magnitude of X(10 ) and was consistent throughout the range of
considered in the current study. However, the discrepancy is notably evident for lower values although this discrepancy begins to diminish at > 80 as the particle size increases. Although
increases as decreases, the slope of the increase is less steep compared to the well
established Shields curve. Since the ADV is able to measure instantaneous fluid velocity, it is possible to represent
using the critical Reynolds shear stress, ,& , and the turbulent kinetic energy, ,,- . The discussion starts with the ,& based . The values fall much closer to the Shields curve and
produce a similar slope pattern with the using . It was found that the Reynolds shear
stress-based Shields parameters approximately fall on the same curve as the ∗, based .
Apparently, momentum flux terms proved to be consistent and dependable elements for estimating the critical shear velocity in describing incipient sediment motion.
Presenting the incipient sediment motion in terms of TKE is permissible as the magnitude of fluctuating drag and lift forces is well correlated with the near-bed turbulent kinetic energy (Yang et al., 2016). In a bare channel, the near-bed TKE is proportional to the bed shear stress (as described in Eq. 4) and may be used as a predictor for the incipient motion condition (Stapleton & Huntley, 1995). The curve obtained slightly deviates from other shear-based variables and the established Shields curve, in particular at > 200 where the values are consistently significantly higher (Fig. 4). Even so, based on the similarity of the profiles, it can be said that , ∗ , and Reynolds shear stress, & , and ,,- functioned in the same way to promote sediment
transport.
Surface roughness (i.e. the sediment size) apparently plays an important role in the near-bed critical turbulence intensity, /∗ . The ratio of /∗ is ∼ 1 for > 40, however,
the value of increases for finer sediment and can essentially be /∗ ≈ 2 at < 10 (i.e.
within the hydraulically smooth region). Note that at = 1 (the smallest in the current
study), /∗ = 2.34. The observed ratio of /∗ for smaller sediment sizes lies within the
range of the hydraulically smooth limit (of 2.3 to 2.7) as described by Nezu and Rodi (1986). Regression analysis was done on the near-bed critical turbulence intensity /∗ , in terms
of particle Reynolds number . The data yielded the relation /∗ = 2.22. @ . The
measured ∗ monotonically decreased for finer sediment sizes (as it is inversely proportional to
), and is very likely to fall within the laminar viscous sublayer, where viscous force dominates and reduces the shear strength. This explains the higher calculated when using (compared to
when ∗ was used) for lower .
3.2. Comparison with oscillating grid turbulence data
The oscillating grid approach permits the representation of incipient sediment motion based on turbulent fluctuations, with no or negligible effect from the mean fluid velocity. The generated flow is nearly-isotropic (that is invariant in rotation), quasi-homogeneous (i.e. invariant in translation) turbulence and allows for a characterisation of incipient sediment motion based solely on the effects of turbulence. Although several studies are available for comparison, here the works of Bellinsky et al. (2005), Redondo et al. (2001), and Wan Mohtar and Munro (2013) were selected as these studies used a range of varying sediment sizes and are comparable to the ones discussed in the current study. The Shields parameter, , was obtained based on the approximation of ∼ , which
allows for to be determined using Eq. (1). Note that use of , denoted as the critical in the grid turbulence experiments, differed in each study. Bellinsky et al. (2005) and Redondo et al. (2001) used the calculated based on the empirical equation developed by Hopfinger and Toly (1976). The behaviour of grid generated turbulence at the boundary layer posed similar characteristics as the ones obtained in open channel flow. The velocities of and 0, which were amplified in the near-bed region due to the kinematic boundary condition, are known as splat-antisplat events (Perot & Moin, 1995). Because of this, the work of Wan Mohtar and Munro (2013) used the peak value of in the near bed region, where the dimensional distance above bed
% ≈ 1.8 cm.
Figure 5 Figure 5 shows the plots of experimental data with from the grid turbulence
experiments. To avoid confusion in the discussion, the Shields parameter from grid turbulence experiments is denoted as . In general, the Shields profiles fall close to the envelope of the well-established Shields curve, where as decreased (for < 40) the is increased
monotonically while for higher > 100, reached a relatively constant value. It should be noted that the Shields curve line based on the data obtained by Redondo et al.(2001) is significantly lower, and this is believed to be due to variations in the definition of incipient sediment motion and the lack of homogeneity in the distribution of Redondo et al.’s (2001) marine sediment. The Shields parameter obviously has a higher magnitude than , approximately 3 higher than the Shields parameter characterised by the near-isotropic turbulence. To account for the variation of the obtained profiles, a power law regression = [\ was applied to each
data set. Coefficient ] denotes the slope in the log scale plot. Coefficients ] in the data obtained by Wan Mohtar and Munro (2013) and Bellinsky et al. (2005) as well as the current data fall within a similar range, i.e. 0.39, 0.35, and 0.44, respectively. A higher ] value of 0.51 was obtained
based on the data Redondo et al. (2001). A steeper slope for calculated using indicates that as sediment size decreases, higher turbulent strength is required and is responsible for incipient sediment motion, which, therefore, suggests higher mean fluid velocity.
The ratio of peak rms and shear velocity /∗ was expressed by Lamb et al. (2008) as P
∗
= ^ _5.62log
b
,
+ 4c,
(8)
where the coefficient ^ is = 0.2 and ' is the roughness length-scale of the bed. Using this representation, the data from the grid turbulence experiments also were redefined based on critical shear velocity, ∗ . Figure 6
Figure 6 shows the representation of the modified Shields parameters (denoted here d )
based on the recalculated and ∗ from the current grid turbulence experiments. Both velocity
parameters were represented using the modification factor e =
fghij fghk l(fghk)Q
, where mn is the
particle’s angle of repose and o is the bed slope. The modified particle Reynolds number (d ) is defined as >e and d = /e. Note that, in this plot, the modification of the Shields
parameters were only done for data from Wan Mohtar and Munro (2013), considering the robustness approach of how the Shields parameter was defined. Recall that Wan Mohtar and Munro (2013) used the peak value of root-mean-square horizontal fluid velocity . Nevertheless, a similar trend is expected if the same modification procedure was applied to the data obtained by Bellinsky et al. (2005) and Redondo et al. (2001) due to the similarity of the Shields profiles presented in Fig. 5. Despite the similarity in the obtained profiles, the difference in the magnitude of d
varies up to X(10 ). The Shields curve obtained through turbulent fluctuations is consistently higher than the estimated values when calculated using shear velocity. This analysis supports the idea that turbulent fluctuations are consistently higher than shear velocity for estimating incipient sediment motion.
3.3. Analysis of the quadrant of Reynolds shear stress plane during incipient sediment motion The availability of instantaneous ′ and +′ allows for a quadrature analysis of the
′ − +′ plane to identify the dominant turbulence structure during the critical velocity identified
as the threshold criteria for sediment motion. Most of the studies measuring simultaneous particle movement and fluid velocities were done to elucidate the dominant fractions, thus, the flow velocity was much higher than the threshold value (/ ⋙ 1) to permit obvious sediment movement (Wu & Jiang, 2007; Wu & Yang, 2004). Here, the authors opted to investigate the turbulence characteristics at the paradoxical demarcation of immobility-mobility of a particle, for which the flow velocity is much the lower considering lower observational threshold criteria used compared to previous studies.
The vertical distance plays an important role in determining the turbulence structure, where the sweep and ejection dominance varies within the wall shear layer (Dey et al., 2011). However, the distance where the highest rms horizontal velocity is a representative coherent height at which the turbulence intensity is maximal and the contributions of sweeps and ejections are equal (Wu & Yang, 2004). As such, it is believed that the same dataset at where was obtained provides an accurate representation for the quadrant analysis.
Four quadrants (Qq) provide the types of bursting event as: (1) outward interactions Q1 (q=1, ′ > 0, +′ > 0), 2) ejections Q2 (q=2, ′ < 0, +′ > 0); 3) inward interactions Q3 (q=3,
′ < 0, +′ < 0); and 4) sweep Q4 (q=4, ′ > 0, +′ < 0). The frequency of each bursting event is
calculated based on the significant contributions to −′+′, through filtration using a hyperbolic region, referred to as the hole size parameter, r. The parameter r ignores the smaller ′ and +′
velocity components which correspond to more quiescent periods, whereby only the dominant events will be accounted for (Lu & Willmarth, 1973; Nezu & Nakagawa, 1993). The detection function, os (K) , is calculated through the decomposition of velocity
fluctuations into four quadrant domains of q = 1, 2, 3, and 4 as . . os,t (K) ={ 1, if |′+′| ≥ r(′′) (+′+′) , 0, otherwise,
(9)
where the overbar denotes time averaging. In the current study, as the velocity threshold for sediment motion is much lower (which translates into lower fluid velocity) than that for medium or general sediment (transport) movement, the hole size only varied between 1-5 in the conditional analysis. Figure 7 The variations of fractional contribution s,t of each types of event were investigated with
hole size r for all the range considered here. Note that s,t = s,t /s,t , where s,t
denotes the frequency of an event and s,t is the total number of all four quadrant events. The current study analysis showed that the events decreased as the r value increased, in agreement
with the findings of Dey et al. (2011) and Yue et al. (2007). Interestingly, there is a distinctive pattern of frequency of event occurrence for fine and coarse sediment sizes. To facilitate a comprehensible plot, Fig. 7 shows the influence of hole size on the values of s for = 0.99
and = 7.19 to represent fine sediment and = 142.71 and = 186.36 for coarser sediment. At r = 1, both ejection and sweep are dominant in entraining the sediment, but as r increases, the turbulence structure responsible to initiate the movement for fine sediment are
sweep events, whereby is seen to monotonically decrease. On the contrary for coarser sediment sizes, the influence of ejection became prominent and consequential and dominated the scenario up to 80%. This indicates that the movement of fine sediment is through sweeps of high momentum onto the particles, whereas the coarser sediment moved due to the lifting forces from the bed. However, Wu and Jiang (2007) found that the entrainment for approximately similar = 142.71 is through lifting and is dominated by sweep (Q4), which is contrary to the ejection mode found in the current study. The contradictory findings are believed to be the result of variation in the vertical distance of where the flow velocity measurements were taken. The frequency and occurrence of events varied along the boundary layer, by which ejections monopolise the turbulence structure at the top of the boundary layer (Dey et al. 2011).
In general, sweep and ejection events are responsible for the incipient sediment motion, which echoes the findings of Kesharvazi et al. (2012) and Thorne et al. (1989), to name a few. During entrainment of particles, both sweep and ejection events have a higher frequency of occurrence than outward and inward interactions.
Figure 8
To provide a qualitative overview on the detailed fluctuation contribution on each type of event, Fig. 8 shows the quadrant analysis for each case when the conditional analysis (i.e.
hole size) r = 1. The data show that the momentum and kinetic energy transfer needed to initiate
sediment movement is mostly the result of ejection (′ < 0, +′ > 0) and sweep (′ > 0, +′ < 0) events based on the consistent top-left (Q2) slanting towards right-down (Q4), although both
inward and outward interactions do make a bit of contribution. At the lowest particle Reynolds number = 0.99, the turbulence intensity is considerably lower than occurs as the particle Reynolds number increases. Both ejection and sweep events composed more than 80% of the dominant turbulence structure, and interestingly the fraction between the two is comparably similar, i.e. between 36-46% each (Table 2).
Table 2
To assess on the quantitative influence of sediment size on the types of bursting events, Table 2 lists the fractional contribution (based on r = 1) for the ranges considered here. The inward and outward interactions consistently have low frequencies. There is no distinct pattern of motions types based on the , except sweep and ejection events are significant during incipient sediment motion. Whereas for higher flow velocities which promote sediment entrainment (and weak sediment transport), the sediment movement also can be influenced by outward interactions (Wu & Jiang, 2007). Wu and Jiang (2007) studied at much higher flow velocity to observe energetic sediment movement. Thus, both outward and inward interactions become more developed and distinctive at higher flow velocities. Based on this, it is speculated that both outward and inward interactions are not significant for incipient sediment motion and only display influence entering the phase of sediment entrainment, suspension, and transport.
Bed roughness changes the dominant turbulence structure, whereby in a gravel bed river the sediment mobility is influenced by the ejection-sweep events for densely packed and low-density bed geometries. Whereas outward-inward interactions are the most frequent events for isolated flow regimes (such boulders or clasts) (Papanicolaou et al. 2001). Note that the threshold condition in the current study is based on a (flat) smooth surface, thus, the bedform wake (due to the changes in surface roughness) is not significant. The pressure difference between grains during the dominant sweep (Q4) induced sediment entrainment for both exposed or shielded grains (Dwivedi et al., 2010).
4. Conclusions
The effect of turbulence is inherent in the threshold criteria for sediment movement. The representation of based on turbulent fluctuations, presented here as the root-mean-square horizontal velocity, gives similar profiles as that obtained when the Shields parameter was calculated using the critical shear velocity, ∗ , and Reynolds shear stress, & . Both turbulent fluctuations and shear velocity function in the same way to promote incipient sediment movement. These results indicate that the second order moments of velocity fluctuation (i.e. turbulence intensity) may reduce the ambiguity in the determination of incipient sediment motion. It is believed that by incorporating the peak value of turbulence intensity, a more accurate threshold criterion is feasible.
It is interesting to note that ejections and sweeps are the dominant turbulence structures and they initiate the incipient sediment motion even at low flow velocities, spanning from hydraulically smooth to rough regions. Significant turbulence structure influences on the incipient sediment motion for coarser and finer sediment sizes are ejection and sweep types, respectively.
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Figure 1: A schematic diagram of the experimental setup (not to scale). A false floor is installed both upstream and downstream of the working area. (Note: ADV is an Acoustic Doppler Velocimeter)
Figure 2: The vertical turbulence intensity profiles for varying Reynolds number Re (a) ݑ/ܷ and (b) ݒ/ܷ. Both fluid velocities are made dimensionless with the mean horizontal velocity, ܷ. Each symbol represents associated Re, with Re = 25000 (∘), Re = 29000 (×), Re = 31000 (□), Re = 36140 (⋄), Re = 36300 (△), Re = 40000 (∗), Re = 46000 (⊳), and Re = 48000 (⊲).
Figure 3: The Shields diagram for experimental data when the critical Shields stress is calculated using ݑො. The solid line represents the Shields curve from the Brownlie (1982) equation.
Figure 4: The Shields diagram for experimental data (•), ߠ calculated based on ∗ݑ, (□), ߬, (⋄), ߬,ோ (△), and ∗ݑ (×). The solid line represents Shields curve calculated using the Brownlie (1982) equation.
Figure 5: Representation of Shields diagram with the current experimental data (•), obtained with ݑ and ߠ from selected grid turbulence experiments. Data was extracted from Wan Mohtar & Munro (2013) (□), Bellinsky et al. (2005) (△) and Redondo et al. (2001) (⋄). The solid line represents Shields curve from the Brownlie (1982) equation.
Figure 6: Replot of Shields curve based on the modified ߠ and particle Reynolds number, ܴ݁ , based on ݑො (∘) and ∗ݑ (□) obtained from the current grid turbulence data.
Figure 7: Fractional contribution ܨ for (a) outward interaction (Q1), (b) ejection (Q2), (c) inward interaction (Q3), and (d) sweep (Q4). Symbols denote ܴ݁ = 0.99 (∘), ܴ݁ = 7.19 (□), ܴ݁ = 142.71 (△) and ܴ݁ = 186.36 (×).
Figure 8: The quadrant analysis for (a) ܴ݁ = 0.99, (b) ܴ݁ = 7.19 , (c) ܴ݁ = 20.33, (d) ܴ݁ = 40.50, (e) ܴ݁ = 57.50, (f) ܴ݁ = 88.52, (g) ܴ݁ = 142.71, and (h) ܴ݁ = 186.36 based on = ܪ 1.
Table 1: Sediment and flow characteristics used in the current study ݀ହ (ߤm) 40 150 300 475 600 800 1,100 1,300
ܴ݁ 0.99 7.19 20.33 40.50 57.50 88.52 142.72 183.36
ܷ (m/s) 0.150 0.209 0.194 0.202 0.221 0.279 0.293 0.300
( ∗ݑm/s) 0.0087 0.0150 0.0159 0.0182 0.0226 0.0238 0.0326 0.0351
ߜ௩ (mm) 0.577 0.334 0.315 0.275 0.221 0.210 0.153 0.143
݀ହ /ߜ௩ 0.0693 0.449 0.951 1.729 2.711 3.811 7.180 9.102
Table 2: Fractional contributions ܨ of types of bursting event for varying ܴ݁ , when = ܪ1.
Event Outward (Q1) Ejection (Q2) Inward (Q3) Sweep (Q4)
0.99 11.0 40.2 4.8 44.0
7.19 11.3 43.0 8.8 36.9
20.33 9.1 44.1 7.2 39.6
40.50 10.0 40.7 6.9 42.4
ܴ݁ 57.50 5.6 46.0 8.0 40.4
88.52 8.6 37.7 8.9 44.8
142.72 6.1 41.9 5.2 46.7
186.36 9.5 44.2 9.3 37.0
The authors have no conflict of interest with any people or organisations.