International Journal of Fluid Dynamics (2000), Vol. 4, Article 2

The Role of Near-Bed Turbulence in the Inception of Particle Motion

A.N. PAPANICOLAOU

Dept. of Civil and Env. Eng., Albrook Hydraulic Lab,
Washington State University, Pullman, WA 99163-2910;
United States of America

Received May 17 1999 and in Revised Form April 3 2000.  Publication date April 19 2000.


Abstract

In this study, the characteristics of near-bed turbulence were experimentally investigated for three distinct roughness regimes, namely the isolated, wake interference, and skimming. Incipient flow conditions prevailed throughout the experiments. Spherical particles of the same size and density were placed upon a rough particle bed to simulate the three regimes.Experimental runs for the aforementioned regimes were performed in a tilting water-recirculating flume. Flow measurements atop the spherical particles were performed by means of a 3-D Laser Doppler Velocimeter (LDV). The aim of the tests was to determine the contribution of various turbulent stress components to the initial entrainment of spherical particles. Analysis of the constructed joint frequency distributions of  and  reveals a significant variation in the magnitude and duration of sweeps (u>0, w<0), ejections (u<0, w>0), inward (u<0, w<0), and outward interactions (u>0, w>0) for each of the three roughness regimes considered here. Along the same lines, time series plots of the instantaneous normal and shear stresses suggest that the normal stress U2 should be considered as the most dominant stress responsible for particle entrainment. This result is consistent with other reported turbulent measurements at flow conditions well above those corresponding to particle incipient conditions.


Keywords


Copyright Notice

This material is the work of the author listed here. It is original and has not been published previously unless acknowledged. It may be freely copied and distributed provided that the name of the author, the institution and the journal remain attached. 


1. Introduction

The analysis of the turbulence properties of boundary layer flows has been a vibrant area of research for almost a century. Early pioneering work was directed towards the formulation of time and space averaged representation of flow characteristics, culminating in the now familiar boundary layer theory. Nowadays, the discovery of the so-called bursting phenomenon in turbulent flows by Kline et al. (1967)  (i.e., the cycle of sweeps (u>0, w<0), ejections (u<0, w>0), inward (u<0, w<0), and outward interactions (u>0, w>0), where, u is the fluctuating velocity component in the longitudinal direction and w is the fluctuating velocity component in the vertical direction) generated a new interest in further studying the structures of boundary layer turbulence and then applying this new knowledge to the initiation of spherical particles motion problem.

In an attempt to link the characteristics of turbulent episodes with the entrainment of sediment, several researchers (e.g., Keshavarzy and Ball, 1999; Kaftori et al., 1998; Nino and Garcia, 1996; Kirkbride, 1994; Lapointe, 1992; Rashidi et al. (1990); Dyer and Soulsby, 1988; Grass, 1983; Sumer and Deigaard, 1981; Cleaver and Yates, 1976) have considered that the sweeps cause the initiation of bedload motion in a stream bed, while the ejections (or, interchangeably, bursts) are primarily responsible for the particles' suspended motion. The reasoning behind this consideration is that the sweeps and ejections are the only events associated with the bursting process that contribute positively to the fluctuating Reynolds shear stress component (-uw) and, therefore, augment the turbulence production term.

Recently, a second school of thought (e.g., Nelson et al., 1995) has supported the opinion that the sweeps are not the only events responsible for bedload transport of gravel, but thatthe outward interactions are also responsible. Nelson et al., (1995) have clearly shown that when the magnitude of the outward interactions increases comparatively to the other events of a bursting cycle, the sediment flux increases too, although the magnitude of the Reynolds stress decreases. They found a poor correlation between the sediment flux and the Reynolds shear stress component (-uw). Instead, they have indicated a significant positive correlation between the streamwise instantaneous velocity U (where  and is the local time averaged velocity) and the sediment flux for flow conditions well above the sediment incipient motion conditions.

Along these lines, Sterk et al. (1998), Clifford et al. (1991), and Williams et al. (1989)  have suggested that the normal stresses in the longitudinal and vertical direction, u2 and w2, may be more important in sediment transport than the shear stress component (-uw). This suggests that calculations of sediment flux should not be based on shear stress alone, as many of the equations predicting bedload and/or suspension rates generally do. Moreover, it was shown that although the (-uw) term remains the principal shear stress component, its contributions to the total turbulent stress are much less than those of the u2, w2. For the stresses involving mixed products (e.g., ) (Clifford et al., 1991; Corrsin, 1967). According to Corrsin (1967), it is likely that the mixed terms might significantly contribute to the initiation of sediment motion.

These prior studies suggest that the linkage of the sediment motion with the turbulent flow components responsible for the initiation of motion remains an open case despite the substantial progress that has been attained in the sediment-flow interaction research over the last decade. It is questionable as to whether the statistical characteristics of the (-uw) term are of the greatest importance in the prediction of the beginning of sediment entrainment. Nonetheless, the majority of the aforementioned studies have focused on the interaction of turbulence with sediment motion for flow conditions well above the critical sediment motion conditions (Julien, 1995). To the best of the author's knowledge, in none of these laboratory and field studies were the different turbulent events at flow conditions near the inception of sediment motion.

The primary motivation of this study is to identify the flow events that are responsible for the commencement of sediment (spherical particles) motion under different bed configurations. This was accomplished by performing incipient motion tests in a water-recirculating flume for three well defined surface-packing density configurations that simulate the isolated, wake interference, and skimming flow regimes (i.e., the term surface-packing density denotes the inter-particle distance and is defined as the ratio between the projected plan area of all the particles to the total bed section within which the particles are located (Schlichting, 1979)). The instantaneous stress tensor in the vicinity of a particle was measured by using a 3-D Laser Doppler Velocimeter (LDV). The results were analyzed to determine the relative importance of the various stress components, as well as the contributions during the four quadrants of the turbulent bursting cycle.


2. Experimental Facility

The experiments were conducted in a tilting, re-circulating flume with a rectangular cross-section and walls made of plexiglass. The flume is 20.5 m long, 0.6 m wide, and 0.3 m deep. Its useful length is approximately 16 m. The test section, which is 3 m long and 0.4 m wide, was located 13 m from the flume entrance, where fully developed turbulent flow conditions were established during the experiments. Lead spherical particles, 8 mm in diameter, were placed atop a bed of glass beads of identical size (8 mm in diameter) packed four layers deep (with porosity of almost 30%) and distributed uniformly along the flume bed, as shown in figure 1. Detailed flow measurements were obtained by means of a 3-D Laser Doppler Velocimeter (LDV). The LDV employed here is a six-beam, non-orthogonal, color-separated, fringe-mode, off-axis backscatter system. This particular non-intrusive instrument uses three independent optical channels to measure three non-orthogonal components of the velocity. Two of these components are approximately co-planar with a coupling angle of approximately 30 degrees. The third component is approximately orthogonal to the other two. To improve optical access and facilitate near-wall measurements, the LDV system was tilted 4.8 degrees from the horizontal. The LDV measuring volume is roughly an ellipsoid about 0.08 mm in vertical and streamwise extent and 0.3 mm in cross-section extent. Average data rates of about 20 measurements per second were obtained by seeding the flow with silicon carbide.

Figure 1. A sketch of the test section. Test particles atop the well-packed 4-layer bed.

Three packing density tests were performed: the 2% density test, which represents the isolated flow regime, the 50%, which represents the wake interference regime, and the 70%, which corresponds to the skimming flow regime. For the 2% test, the spacing among the particles was almost 6 balls diameter and a total of 530 particles were placed within the test section. In the 50% case, the spacing was about 1 ball diameter and 13,250 particles were employed throughout the run. Finally, for the 70% case, the particles were in contact with their neighboring particles and about 18,600 particles were used. Figures 2(a)-(c) provide a plan view of the test area for the three packing density conditions.


(a)

(b)

(c)

Figures 2(a)-(b)-(c). A top view of the test section for the 2%, 50%, and 70% packing conditions. The flow is from top to bottom.

The flow conditions for the tests conducted here are identical with those defined in an earlier study (Papanicolaou et al., 1999). The flow conditions in that study were determined by performing incipient motion tests for the same packing density configurations that were considered here. The only difference between the current tests and those described in Papanicolaou et al. (1999) is in the choice of roughness elements. In the latter case, the roughness elements were represented with entrainable glass beads of 8 mm diameter and specific gravity 2.54. This allowed the monitoring of the beads' motion. Instead, the focus in this investigation was to examine the near-bed flow characteristics for the three roughness regimes. Subsequently, in order to obtain point measurements without any interference (such as a glass particle rolling and blocking the Laser beam) lead particles (with specific gravity of 12.4) of the same diameter with that of the glass particles were used.

For the 2% and 50% cases, detailed point-velocity measurements were carried out at a vertical distance of 0.8 mm above the particle top surface. For the 70% case, due to presence of noise, measurements were obtained at a distance of 4 mm. Tables 1(a)-(c) summarize the hydraulic conditions for the tests, namely, the slope S of the flume, the depth H, the dimensionless critical shear stress , where  is the density of the spherical particles,  is the density of water, the friction velocity u* , the Reynolds number Re= 4HV/n, where V is the depth-averaged velocity, n denotes the kinematic viscosity , the local mean velocities and turbulent intensities in the stream-wise and vertical directions ,,u', and w' respectively, and the time-averaged Reynolds stress.
 
 


3. Methodology-Results

The analysis of the data performed here aimed to elucidate the flow mechanisms responsible for spherical particle entrainment and to identify the instantaneous stress terms that are most relevant to particle initial motion under various roughness conditions. Time series analysis was performed to evaluate the dominant stress components, while quadrant analysis was employed to determine the contributions of turbulent bursts to the Reynolds stress for the three bed configurations (Lu and Willmarth, 1973).

In an Eulerian coordinate system, the instantaneous stress tensor, at a point in space, is decomposed into the following matrix form, assuming homogeneous turbulent flow conditions:

To date, attention has been directed exclusively towards the shear stress term (-uw), although this term may not be the most relevant to spherical particle motion. The systematic exclusion of the other instantaneous stress components (i.e., the normal and the other shear and mixed terms shown in (1)) from the study of the entrainment problem implies that there is a misunderstanding about the flow mechanisms causing sediment motion. In this study, the role of different instantaneous stress components was evaluated by following a totally different approach. It was presumed that the particles do experience the action of the instantaneous stress components (the particles do not distinguish the contributions corresponding to the mean, fluctuating, or mixed stress components of equation (1) as we conveniently do for mathematical or statistical expediency). This consideration is fully justified considering that in sediment transport theory, the hydrodynamic forces acting on particles, namely the drag and lift forces, are defined as function of the instantaneous velocities U and W squared, respectively (e.g., Yalin, 1992; Naden, 1987). Thus, the time series of the instantaneous normal stresses U2 and W2 and of the shear stress UW were constructed for the three different packing density tests.

The time series plots in figures 3(a)-(c) provide the variation in magnitude of the instantaneous stresses U2, W2, and UW (divided by the fluid density). These measurements, which are on an average 3,072 per measuring point, are taken at close proximity to the top surface of a spherical particle (at a distance of 0.8 mm above the top surface of a particle for the 2% and 50% cases and at a distance of 4 mm for the 70% case). Figures 3(a)-(c) reveal a striking difference in magnitude among the three stress components. Overall, U2 obtains values that are at least 6-7 times higher in magnitude than W2 and UW. This is consistent with recent experimental flow measurements related to coarse sediment movement where it was shown that U2 is a good predictor of sediment (Clifford et al., 1991), while the "total momentum flux" UW (as it is defined by Nelson et al. (1995)) has a poor correlation with sediment entrainment (e.g., Nelson et al., 1995; Williams et al., 1989). The overwhelming importance of U2 reinstates the strong correlation that exists between the drag force and the transport of coarse material (Sterk et al., 1998). Moreover, figures 3(a)-(c) indicate differences in the turbulent structure of flow, especially between the 2% and 70% cases. In the 2% case, the time series plots demonstrate the presence of relatively low frequency events while higher frequency events are recorded for the 70% run. This is partially attributed to the flow separation occurring in the 2% case. In this case, the roughness elements (i.e., the 8 mm diameter spheres) provide sharp breaks in bed elevation, causing flow detachment and slow return of the fluid parcel to the undisturbed boundary. Similar flow boundary layer processes were discussed earlier by Eaton and Johnston (1980).


Figure 3(a). Time series plots of U2, W2, and UW for the 2% packing condition.


Figure 3(b). Time series plots of U2, W2, and UW for the 50% packing condition.


Figure 3(c). Time series plots of U2, W2, and UW for the 70% packing condition.

The time series analysis performed here was complemented with the construction of the joint frequency distributions of  and  for the three roughness regimes. Figures 4(a)-(c) were developed by plotting the normalized - pairs, contouring the density of the points, and normalizing the results to peak values of 100%. The four quadrants in the plots correspond to the four turbulent events (outward interactions, ejections, inward interactions, and sweeps) that characterize the individual turbulent velocity measurements. The ejections (second quadrant) and sweeps (fourth quadrant) contribute positively to the bed shear stress (i.e., UW, the flux of forward momentum to the bed) while the outward (first quadrant) and inward interactions (third quadrant) contribute negatively to the bed shear stress. Figures 4(a)-(c) reveal unique information about the turbulence characteristics under various roughness configurations. In the 2% case (figure 4(a)), the joint frequency distribution clearly demonstrates a positive correlation associated with the bed shear stress at the measuring point and the tilting of the joint frequency distribution into quadrants 1 and 3. Although this is not the anticipated trend, it is not particularly surprising if the nature of the flow for the 2% case is carefully considered (such as fluid detachment occurs at the top of a particle as it was discussed earlier). Moreover, table 2 clearly illustrates that in the 2% case, the inward and outward interactions, on the average, occupy the highest percentage of time within a bursting cycle. The above finding seems to be in agreement with the recent findings of Kaftori et al. (1998). They suggested that for the flow regime in the presence of isolated roughness elements, the joint frequency distribution among the quadrants in the wall region changes dramatically. According to Kaftori et al. (1998), the importance of the second and fourth quadrants diminishes as the roughness increases, while the contributions of the first and third quadrants to the Reynolds stress become more significant. Instead, in the 50% case, the percentage of time that is occupied by each kind of turbulent event is well balanced (table 2). This is well demonstrated in figure 4(b), with the rather circular shape of the joint frequency distribution. This distribution does not have any pronounced peaks or any preferential tilting towards one of the four quadrants and is rather symmetric with respect to the origin of the u-w plane. This suggests that in the wake roughness regime all four events of a bursting cycle contribute equally to the Reynolds stress and therefore to the turbulent production term. Figure 4(c), the 70% case, depicts the anticipated negative correlation associated with the Reynolds stress and the tilting of the distribution into quadrants 2 and 4. This trend is typically encountered in flows over smooth boundaries (Nezu and Nakagawa, 1993) and it is fully justified here considering the fact that fluid motion, for the 70% test, occurs over a well-packed flat bed layer of identical spheres. Table 2 illustrates that the percentage of time occupied from the ejections and sweeps within a bursting cycle in the 70% case is higher than that of the inward and outward interactions. Similar trends of the results shown in figure 4(c) have been reported in the literature by Nezu and Nakagawa (1993) and Balakrishnan and Dancey (1994) (for flows over smooth boundaries) and by Nelson et al. (1995) for flows over roughness (sandy beds-experimental run 7, pp2080).


 

Figure 4(a). Joint frequency distribution of the normalized u/u' and w/w' for the isolated roughness regime (2% case). Events corresponding to quadrants are shown in figure above.
 


Figure 4(b). Joint frequency distribution of the normalized u/u' and w/w' for the wake interference regime (50% case). Events corresponding to quadrants are shown in figure above.

Figure 4(c). Joint frequency distribution of the normalized u/u' and w/w' for the skimming roughness regime (70% case). Events corresponding to quadrants are shown in figure above.

The information shown in figures 3 and 4 raise many questions regarding the significance of the UW term in the entrainment process. If the magnitude of the stress terms is indeed the criterion that is used to decide which terms should be included in the study of the entrainment problem, then the U2 term should be considered as the most relevant stress term to sediment motion (instead of UW). Also, there is a lot of uncertainty regarding the form in which the UW stress applies to a sediment particle and if it does, the area over which it acts. This uncertainty probably explains why most of the entrainment models that focus on the UW term are rather qualitative. Finally, the constructed here joint frequency distributions clearly demonstrate for the first time that all events within a bursting cycle should be considered in the study of the sediment entrainment problem. Their contributions change as the roughness configuration changes as well.


4. Conclusions

In the present investigation, the characteristics of near-bed turbulence were examined for three roughness regimes; the isolated, the wake interference, and the skimming. The joint frequency distributions of  and , (where u' and w' are the turbulent intensities in the horizontal and vertical direction respectively) which were constructed here, clearly demonstrate that the frequency of occurrence and magnitude of turbulent events, under incipient flow conditions, varies significantly with bed roughness. In particular, for the isolated flow regime the outward and inward interactions are the most frequent events. For the skimming regime, the sweeps and ejections are the most dominant, and for the weak interference regime, all events appear to occupy the same percentage of time. Additional analysis was provided by developing the time series of the normal and shear stress components U2, W2, and UW of the instantaneous stress tensor at a very close proximity atop a particle surface. The results deduced here are consistent with the Sterk et al. (1998) and Nelson et al. (1995) measurements where stream-wise velocity was found to correlate well with sediment transport (for conditions well above the critical). The findings, although quite academic at this stage of the investigation, should be capable of extension to real cases to predict incipient motion of sediment in natural streams under various hydrologic and bed roughness conditions.


Acknowledgements

This material is based upon work supported by the U.S. Geological Survey under grant number 14-08-0001-G2271. The author would like to thank Drs. Diplas, Dancey, and Bala for their constructive comments and the anonymous reviewers whose comments have improved the quality of this manuscript.


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