The Role of Near-Bed Turbulence in the Inception of Particle Motion
by A.N. Papanicolaou
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:
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.
1. Introduction
3. Methodology-Results
Next Section: Conclusions