Forces on stationary particles in near-bed turbulent flows



[1] In natural flows, bed sediment particles are entrained and moved by the fluctuating forces, such as lift and drag, exerted by the overlying flow on the particles. To develop a better understanding of these forces and the relation of the forces to the local flow, the downstream and vertical components of force on near-bed fixed particles and of fluid velocity above or in front of them were measured synchronously at turbulence-resolving frequencies (200 or 500 Hz) in a laboratory flume. Measurements were made for a spherical test particle fixed at various heights above a smooth bed, above a smooth bed downstream of a downstream-facing step, and in a gravel bed of similarly sized particles as well as for a cubical test particle and 7 natural particles above a smooth bed. Horizontal force was well correlated with downstream velocity and not correlated with vertical velocity or vertical momentum flux. The standard drag formula worked well to predict the horizontal force, but the required value of the drag coefficient was significantly higher than generally used to model bed load motion. For the spheres, cubes, and natural particles, average drag coefficients were found to be 0.76, 1.36, and 0.91, respectively. For comparison, the drag coefficient for a sphere settling in still water at similar particle Reynolds numbers is only about 0.4. The variability of the horizontal force relative to its mean was strongly increased by the presence of the step and the gravel bed. Peak deviations were about 30% of the mean force for the sphere over the smooth bed, about twice the mean with the step, and 4 times it for the sphere protruding roughly half its diameter above the gravel bed. Vertical force correlated poorly with downstream velocity, vertical velocity, and vertical momentum flux whether measured over or ahead of the test particle. Typical formulas for shear-induced lift based on Bernoulli's principle poorly predict the vertical forces on near-bed particles. The measurements suggest that particle-scale pressure variations associated with turbulence are significant in the particle momentum balance.

1. Introduction

[2] Predicting the initiation and evolution of bed forms produced by water flowing over loose sediment beds requires specifying a relation giving the transport of the sediment in terms of the motion of the fluid. The relationships typically proposed are empirical or semiempirical, with the fluid motion generally characterized by spatially or temporally averaged quantities, such as mean fluid velocity or bed shear stress. In most cases these relations also incorporate a condition for initiation of sediment motion, such as a critical bed shear stress below which no significant movement occurs. In this way, the motion of sediment particles is tied to parameters that are relatively easy to measure or predict. What entrains and moves sediment, however, is neither bed shear stress nor any other average characteristic of the flow, but instead is the fluctuating forces, such as lift and drag, exerted directly by the flow on the particles. Average flow parameters can at best provide only a parameterization of the underlying physical processes that actually cause sediment erosion, transport, and deposition.

[3] Recently, a number of researchers studying a variety of sediment transport problems have noted that parametric relations for sediment entrainment and transport by steady uniform flows yield poor results when applied to more complex flows that involve spatial or temporal accelerations [Nelson et al., 1995; Sumer et al., 2003; McLean, 2004]. Nelson et al. [1995] suggested that this discrepancy may arise because in complex flows the scaling of turbulence quantities with bed shear stress that characterizes steady uniform flows breaks down. The distributions of turbulence quantities, such as fluctuations in streamwise velocity near the bed, can significantly differ between flows having identical values of average parameters like bed shear stress. Sumer et al. [2003], for example, experimentally induced up to fiftyfold changes in the transport rate with no change in the mean bed shear stress simply by altering the turbulence fluctuations in a variety of ways. Schmeeckle and Nelson [2003] found that increasing the amplitude of turbulent fluctuations in three-dimensional numerical simulations of bed load transport significantly increased bed load concentrations and transport rates. Thus predicting local sediment flux in a wide range of natural flows with sufficient accuracy to characterize the shapes and dynamics of bed forms requires a detailed understanding of the physics of the interaction of turbulent flows with sediment particles.

[4] This is not a new idea. Many authors have developed theoretical models for the motion of a sediment particle in a turbulent flow [Wiberg and Smith, 1985; Anderson and Hallet, 1986; Sekine and Kikkawa, 1992; Niño and Garcia, 1994]. The models, however, all proceed from assumptions about how the forces on sediment particles originate and, more importantly, how they should be treated. Unfortunately, because very little actual data exist on the details of these forces and their relation to the flow, many of the pieces of a physically accurate model for bed load motion remain conjectural. Removing the barriers to better understanding does not require further models. Rather, it requires careful experimental measurement to define the flow-particle interaction accurately, so that models may be developed that are more realistic.

[5] This paper reports measurements at turbulence-resolving frequencies (200 and 500 Hz) of the fluctuating forces on a stationary near-bed sediment particle mounted on a thin adjustable-height stalk made synchronously with measurements of the nearby flow velocity either close to the top of the particle or one particle diameter in front of it. The measurements were made for a diverse suite of bed configurations and particle heights and shapes and show that the forces and their generation differ significantly from the conventional models.

2. Experimental Setup

[6] The experiments were conducted in a racetrack-type flume (described by Nelson and Smith [1989] and Nowell et al. [1989]) with a false wall added to provide a test section 700 cm long by 25 cm wide by 31 cm deep. Flow depth was about 25 cm, and mean velocity ranged from 20 to 90 cm s−1. Thus channel Reynolds numbers ranged from 5 × 105 to 2 × 106. The drag and lift on a fixed, near-bed test particle and the nearby fluid velocity were synchronously measured at turbulence frequencies in a series of 175 runs using a variety of test particles and bed configurations. Each run involved a single particle position and flow condition and produced 14,700 or 17,640 paired measurements of force and velocity. Particle Reynolds numbers ranged from 4000 to 20,000.

2.1. Measurement of Forces

[7] The heart of the experiments was the force transducer used to measure the horizontal and vertical components of force, herein called the nominal drag and the nominal lift, which are not the same as the hydrodynamic drag and lift (Appendix A). It was built by William M. Bruner of Hylozoic Products, Seattle, Washington. It was suspended under a false floor in the flume on spring-loaded mounting screws projecting downward through the floor (Figure 1). The forces on the test particle were transmitted to the transducer via a thin stalk rising through a small hole in the flume floor. The test particle was adjusted up and down by turning the mounting screws from above to raise or lower the whole assembly.

Figure 1.

Diagram of experimental apparatus for force transducer experiments. Water depth is not drawn to the scale of the test particle. The right inset diagram defines the particle height.

[8] The internal mechanism of the transducer permits only streamwise and vertical translation of the test particle in response to the corresponding applied forces and minimizes cross coupling between them. The displacements of the test particle are less than about 10 μm and are sensed by an arrangement of light beams, knife edges, and photocells inside the transducer.

[9] The analog output of the transducer is converted to digital and stored in a pair of 14-bit output registers. When the output registers are subsequently read, a new conversion is performed and stored within about 20 μs. Thus the output values reflect the actual forces at the time of the preceding readout.

[10] The static calibration of the transducer was done in air by recording the digital output while a series of known weights were hung from it in the horizontal and vertical directions. In addition, the calibration was verified with the transducer mounted in position in the water-filled flume by pushing on the test particle with three carefully calibrated springs. The tests covered the whole range of possible applied forces and showed (Figure 2) that the digital output is directly proportional to the applied static forces.

Figure 2.

Static calibration curves for the force transducer. Squares and inverted triangles indicate measurements of downstream and upward components of force in air. Crosses indicate measurements of downstream component in water.

[11] The frequency response of the transducer was measured in water using a special magnetic test particle and streamlined support stalk, which also were used in many of the experimental runs. The particle consisted of a 1.9-cm-diameter foam core sphere with a rare earth bar magnet centered inside and the stalk consisted of a brass rod with a 0.6 × 0.3 cm teardrop-shaped cross section. A sinusoidally varying electric current passing through a pair of coils arranged symmetrically on either side of the particle produced a sinusoidally varying force acting on the magnetic particle in a known direction. Curves of frequency response for both the horizontal and vertical directions (Figure 3) were generated by applying appropriately oriented forces to the particle at constant amplitude and numerous single frequencies. The frequency response of the transducer with the magnetic test particle and streamlined support stalk is essentially flat up to about 40 Hz; and the resonant frequency of the assembly is around 65–75 Hz. Cross coupling (i.e., vertical force measured when only horizontal force was applied and the converse) is < 3.5% at 50 Hz and considerably less at lower frequencies. The power spectrum of an uncorrected time series of drag as recorded at 500 Hz in an otherwise typical experimental run (Figure 4) shows that the output is virtually all at frequencies below about 20 Hz and in the vicinity of the resonance near 70 Hz. Inasmuch as the test particle diameter was 1.9 cm and the mean flow velocity was about 30 cm s−1, Taylor's hypothesis [Tennekes and Lumley, 1972, p. 253] implies that turbulence frequencies higher than 20 Hz correspond to structures smaller than about 1.5 cm. Because the forces exerted by structures smaller than the particle tend to cancel, the strong output near 70 Hz, which corresponds to structures only 0.4 cm in size, must be due to excitation of vibrations near the resonant frequency of the assembly. For all runs, therefore, the transducer output was converted to force units and corrected for resonant response by (1) multiplying each measurement by the appropriate force constant from the static calibration, (2) transforming the resulting time series to the frequency domain by means of the Fast Fourier Transform, or FFT, (3) leaving the amplitudes for frequencies up to 40 Hz unchanged and setting those for the higher frequencies to zero, and finally (4) transforming the result back to the time domain by means of the inverse FFT.

Figure 3.

Frequency response curves for the force transducer. Amplitudes are relative to amplitude at 1.2 Hz. Squares and circles indicate measurements of downstream and upward components of force.

Figure 4.

Amplitude versus frequency for nominal drag Fx on a sphere in a gravel bed (run b60). The peak at the right is due to resonance near the natural frequency of the force transducer and test particle assembly. Amplitudes are obtained as the magnitude of the complex fast Fourier transform coefficients.

[12] The simple recipe in step 3 does not directly utilize the frequency response curves. For cases involving higher fluid velocity or smaller particle size, the low-pass cutoff frequency might have to be adjusted upward, and each complex amplitude below the cutoff divided by the appropriate complex value from the response curves.

[13] Immediately before every run the output of the force transducer in still water was adjusted to zero. Thus the measurements did not include the force due to buoyancy. Immediately after each run the output in still water was checked again, in order to detect drift, which in all cases was negligible.

2.2. Measurement of Velocities

[14] The streamwise and vertical components of fluid velocity were measured using a two-component laser Doppler velocimeter (LDV), which measures the velocity of suspended light-scattering particles at the intersection of two pairs of laser beams. The measurement volume is approximately an ellipsoid whose axial diameters are 0.01, 0.05, and 0.01 cm in the streamwise, cross stream, and vertical directions. To obtain a measurement rate suitable to the desired sampling frequency, which was typically 500 Hz but was 200 Hz for the final set of experiments, the flow was seeded with silver-coated alumina spheres having a mean diameter of 4 μm, which is small enough to assure that they closely followed the flow at the highest turbulence frequencies measured. Overseeding was avoided, inasmuch as it can reduce the number of usable measurements.

[15] Because the light-scattering particles pass through the measurement volume intermittently, the actual velocity measurements are necessarily spaced at unequal time intervals and are therefore digitally stored in output registers for subsequent readout. The LDV was programmed to send the contents of the velocity and force output registers to a personal computer at predetermined equal time intervals (every 2 ms or 5 ms at the measurement rate of 500 or 200 Hz, respectively). With the seeding used in the experiments, roughly 50% of the readout velocities reflected the actual instantaneous velocities at a time in the immediately preceding interval and 75% in the immediately preceding two intervals. Because the highest-frequency component of the filtered time series was typically one fifth to one tenth the frequency of the observations, the filtered force and velocity measurements were taken to be synchronous.

[16] The velocity measurements were made at one of two positions relative to the test particle. They were made either one particle diameter directly ahead of the front of the particle or 0.2 cm directly above the top of it. Measurement ahead of the particle was intended to provide the best estimate of the reference velocity at the particle (defined in Appendix A), and only these experiments were used to estimate drag coefficients. The distance chosen was a compromise between getting the best correspondence between the measured velocity and the reference velocity and minimizing the effect of the deceleration due to the presence of the particle. Because the particle Reynolds numbers were much greater than unity, the magnitude of the effect was estimated assuming irrotational flow around the sphere [Batchelor, 1967, pp. 450, 452] and found to be a decrease of about 4%. For a 1.9-cm-diameter test particle and a 30 cm s−1 fluid velocity, which are typical of the experiments, changes in velocity at the measurement volume take about 100 ms to propagate to the position of the particle center. Measurement above the particle was used when placement of the LDV measurement volume ahead of the particle was impossible, as when the center of the test particle was below the top of the gravel bed, or when comparison between the two positions was desired. Although this position minimizes the time delay, it does not give the best estimate of the reference velocity.

[17] For all runs in which velocity was measured upstream of the test particle, the association of force measurements with velocity measurements was adjusted for the time lag between when the velocity measurements were made and when they became representative of the reference velocity at the particle. In addition, measurements for intervals for which the LDV did not provide new information were ignored, thus assuring that all force and velocity measurements were made at known times. For each run the adjustment was taken to be the integer multiple of the measurement time step that produced the strongest correlation between Fx and Ux2. As expected, it was always approximately the particle diameter divided by Ux.

2.3. Bed Materials and Arrangements

[18] Three types of bed material were used in five plane bed arrangements. They were acrylic with and without a step, randomly arranged gravel, and aligned and staggered spheres.

2.3.1. Acrylic With and Without Step

[19] A series of 92 runs were made with a bed consisting of the false floor of the flume, which was constructed of smooth, flat, acrylic sheet. Their purpose was to investigate drag and lift on a particle in a uniform, well-developed turbulent boundary layer over a smooth floor, in which the turbulent structure is well known and considerably less variable than in nonuniform and rough turbulent boundary layers. For comparison, 24 runs were made with a 4-cm-high downstream-facing step placed on the acrylic floor at various distances ranging from 40 to 100 cm upstream of the test particle. Their purpose was to elucidate the effect of adding strong, low-frequency turbulence to a boundary layer over a smooth wall.

2.3.2. Randomly Arranged Gravel

[20] A group of 39 runs were made with a single layer of well-sorted, naturally rounded, 2.2-cm gravel laid on the false floor. The approximate height of the tops of surrounding grains above the false floor of the flume was about 2.3 cm. A water-worked gravel bed was not possible because a single layer of grains was necessary to minimize the length of the stalk connecting the test particle to the force transducer. The gravel was rearranged several times in order to ascertain the effects of variations in local bed particle arrangement.

2.3.3. Aligned and Staggered Spheres

[21] Another 20 runs were made with the gravel near the test particle replaced by seven transverse rows of 1.9-cm spheres like the test particle spaced 2.54 cm apart both transversely and longitudinally. The spheres were attached to a thin plastic sheet which was itself attached to the acrylic false bed of the flume. One row was positioned at the particle, three upstream, and three downstream. In order to test the importance of bed particles located directly upstream of the test particle, half the runs were made with all the rows aligned to form a square grid and half were made with alternate rows offset half the 2.54-cm spacing to form a triangular grid. The height of the tops of surrounding spheres above the false floor of the flume was about 2.3 cm.

2.4. Test Particles and Support Stalks

[22] Runs were made with a variety of test particles and support stalks. A total of 141 were made with a 1.9-cm sphere, six were made with a 2.54-cm cube mounted with edges in the downstream, cross-stream, and vertical directions, and 28 were made with natural gravel particles (described in Table 1). The stalk used in 72 of the runs was the original 0.6 × 0.3 cm streamlined version; and in 65 it was a simple 0.18-cm stainless steel rod with threaded ends. In 24 others the rod was enclosed up to a point 0.2 cm below the test particle in a fixed 0.6-cm outer diameter cylindrical shroud. Use of the shroud was discontinued, however, because it caused a spurious increase in lift with no detectable change in drag, even when the particle was at maximum height, so that the maximal amount of stalk was exposed to the flow.

Table 1. Angularity and Dimensions of Natural Test Particles
PebbleAngularityLengths of AxesAreas
a1, cma2, cma3, cmA, cm2A, cm2
6well rounded1.

[23] In all runs particle height was measured from the false floor of the flume to the bottom of the test particle (Figure 1). Zero particle height means that the particle was positioned as close as possible to the floor without actually touching it.

3. Results

[24] The basic data from the experiments consist of 175 time series of paired measurements of the horizontal and vertical components of velocity and force. Examples of two typical time series are shown in Figure 5. Figure 5a is an experimental run over a smooth bed and Figure 5b is an experimental run over a rough bed. Inspection of these time series data shows clear coupling between the horizontal components of velocity and force. The peaks and troughs of the nominal drag lag slightly behind those of the horizontal velocity because the velocity was measured 2.8 cm upstream of the center of the test particle. Coupling between the vertical force and the horizontal velocity is less obvious, but the vertical force appears to undergo strong positive and negative fluctuations when the horizontal force and velocity are strongly positive. The relationship between the vertical velocity and the other components of force and velocity is not readily apparent in Figure 5.

Figure 5.

Synchronous time series of nominal drag and lift Fx and Fz on a sphere and of downstream and upward components of velocity Ux and Uz in front of it (a) over the smooth bed (run b11) and (b) in a gravel bed (run f28). Horizontal lines indicate the mean value of each force and velocity.

[25] It is important to note that the scales are not the same in Figures 5a and 5b. The variance of all force and velocity components is much greater for the gravel bed case (Figure 5b) than the smooth bed case (Figure 5a). For example, the range of the vertical force variations is nearly twice that over the smooth bed, despite the fact that the mean horizontal force is greater for the smooth bed case. Low-frequency fluctuations, with periods of roughly a second, dominate the horizontal components of force and velocity in the gravel bed case (Figure 5b) whereas the horizontal components over the smooth bed (Figure 5a) appear to have a combination of high- and low-frequency fluctuations.

[26] For each of the time series collected for various configurations, mean values, standard deviations, skewness, and various correlation coefficients were computed for use in evaluating the results. Tables 2 and 3 list each experiment along with a number of calculated statistics for each run. The reported values of the forces contain neither the gravitational nor the buoyant parts of the actual force. The streamwise force (nominal drag) Fx is taken positive in the direction of flow Ux ; and the vertical force (nominal lift) Fz and the velocity component Uz are taken positive upward. The cross-stream components of force Fy and velocity Uy were not measured. Correlation coefficients between Fx and the product UxUz′ and between Fz and the product UxUz′were also calculated but are not reported in Table 3 because of their uniformly low values. The mean correlation coefficient between Fx and the product UxUz′ for all of the runs with calculated drag coefficients was found to be 0.00, and the mean correlation coefficient between Fz and the product UxUz′ was −0.02. Thus, even though there may be a strong relationship between the near bed Reynolds stress (−ρfequation image and the mean drag force, there is not a relationship between the instantaneous velocity cross product, UxUz′, and the instantaneous drag or lift force.

Table 2. Table of Force and Velocity Dataa
RunTest GrainmUx, cm/sUz, cm/sFx, dynFz, dyn
  • a

    Test numbers beginning with the letter b were conducted with the aerofoil-shaped support stalk as described in the text. The other experiments used a 0.18 cm threaded rod as a support stalk. The stalk in runs d23–d66 were shrouded as explained in the text. All other experiments were not shrouded. The test grain column refers to either a 1.9-cm-diameter sphere, a 2.54-cm cube, or a natural particle number whose dimensions are specified in Table 1. Particle height is the distance between the lowest point on the particle and the acrylic bed (Figure 1). For the gravel bed runs the height of the tops of the grains making up the bed are approximately 2.3 cm above the acrylic bed. The mean, standard deviation (σs), and skewness are calculated from 14,700 points (29.4 s at 500 Hz) in runs b through e and 17,640 points (88.2 s at 200 Hz) in runs f.

Velocity Measurements Made Upstream of Particle
d71part. 11.665.24.3−0.801.02.30.5028581840.14282390.18
d74part. 21.671.54.1−1.451.62.10.292084156−0.361966181−0.12
d77part. 31.669.63.5−−0.191172344−0.08
d80part. 41.671.83.4−−0.463869242−0.16
d84part. 51.671.63.2−0.961.71.80.252600178−0.2510661610.03
d87part. 61.670.83.4−−0.262169302−0.02
d90part. 71.668.23.5−1.351.82.10.767343409−0.695765050.09
d92part. 7053.96.4−0.335.03.20.0266595090.01−41941900.21
d95part. 6051.77.2−
d98part. 5056.46.1−−0.08503770.05
e02part. 4056.26.7−−0.20914148−0.07
e05part. 3055.95.8−0.562.13.10.2036463460.115571130.24
e08part. 2055.86.8−
e11part. 1054.36.2−
Velocity Measurements Made 0.2 cm Above Particle
d93part. 71.472.36.4−0.578.23.2−1.1267675180.11−41401970.45
d96part. 61.475.15.7−0.453.32.8−0.1425192830.0010231230.31
d99part. 51.470.25.0−0.602.72.70.2214611690.01526760.14
e03part. 41.479.25.6−0.49−
e06part. 31.473.14.4−0.738.92.2−0.2036863540.015811160.13
e09part. 21.474.16.6−−0.01637640.10
e12part. 11.472.55.0−0.626.52.4−0.3926282870.16611880.22
Table 3. Table of Calculated Valuesa
RunBed Typeequation image, cm2 s−1Drag CoefficientCorrelation Coefficients r
CDIequation imageCDUUxUzUxFxFxFzUxFzUzFxUzFz
  • a

    Four types of bed are indicated in the table; acrylic and gravel beds are referred to as “smooth” and “gravel,” respectively. For measurements downstream of a backward step the number of step heights downstream is indicated. For measurements in an arrangement of spheres, “stag.” and “unstag.” indicates whether the arrangement was staggered or unstaggered, as explained in the text. Velocity measurements were made one and a half particle diameters upstream of the test grain for runs in which drag coefficients were calculated, and velocity was measured 0.2 cm above the grain for runs in which the drag coefficients were not calculated. -The average drag coefficient CDI is computed by averaging the instantaneous values of the drag coefficient. The average drag coefficients CDU2 and CDU are computed from equation image and equation image and equation image, respectively. The correlation coefficients (Pearson's r) are for the indicated pair of measured quantities. All values calculated from Ux or Uz include only updated values of Ux and Uz.

Velocity Measurements Made Upstream of Particle
Velocity Measurements Made 0.2 cm Above Particle

3.1. Sampling Errors

[27] Any empirical study should consider the importance of sampling errors. It is assumed that the processes sampled in these experiments (force and velocity components) were stationary because the flume speed was held constant, the bed was not moving, and we confirmed that there was no drift in our instruments. The critical question is whether or not the stationary processes were sampled over long enough periods to reduce the sampling error to an acceptable level. Excessively long sampling periods were not possible in this study because the number of data points collected in an experimental run was limited by the size of the memory in the computer dedicated to the LDV system. The velocity and force data were collected at 500 Hz for 29.4 seconds for the majority of experiments (run numbers starting with b, c, d, and e). A minority of experiments (run numbers starting with f) were colleted at 200 Hz for 88.2 s. The time series plotted in Figure 5 are typical of all the runs in that we found no trend in the data over the period of each run, and there were a significant number of excursions above and below the mean. To quantitatively test these observations, each component of force and velocity was linearly regressed against time for several of the experimental runs. The coefficients of determination (R2 values) of all of these regressions were less than 1%, thus lending confidence to the assertion that there were no trends over the period of any of the experimental runs. Also, the autocorrelation times were calculated for all of the collected velocity and force time series as the lag time in which the autocorrelation first fell below e−1. The gravel bed experiments have the largest autocorrelation times. The mean autocorrelation times for the gravel bed experiments are 0.08 s for downstream velocity, 0.02 s for vertical velocity, 0.2 s for nominal drag, and 0.1 s for nominal lift. Autocorrelation times for velocity components are significantly shorter than the force components in all experiments, and the autocorrelation times of the downstream components of force and velocity are longer than the vertical components in all experiments. Because the autocorrelation times are many times smaller than those of the sampling periods, and because the time series have no trend over the sampling period, the sample statistics presented in Tables 2 and 3 should be very close to the population statistics that would be obtained for very long sampling periods.

3.2. Temporal and Spatial Distribution of Drag and Lift

[28] Entrainment of grains from a sediment bed is more likely to occur when the instantaneous drag and lift are large relative to mean values. Thus a key goal of this study was to measure the temporal distribution of drag and lift on particles under systematically varied turbulence conditions and at systematically varied positions relative to either surrounding bed grains or the smooth acrylic bed.

[29] The temporal variability of the drag is greatly affected by the turbulence structure. Shown in Figure 6 are histograms of the ratio of the instantaneous drag to the mean drag Fx/equation image for three cases: the 1.9-cm test sphere with its bottom positioned 0.5 cm above a smooth, plane acrylic bed; the same sphere at the same height above the same bed but 80 cm downstream of 4-cm-high downstream-facing step; and the same sphere with its top positioned approximately 0.7 cm above the tops of the highest grains of a bed of gravel of similar size. The range of the variability of the drag is striking, increasing from less than a third of the mean for the isolated particle over the smooth bed to more than 3 times the mean for the particle in the gravel bed.

Figure 6.

Histograms of instantaneous relative nominal drag Fx/equation image on the sphere above (a) the smooth bed (run b11), (b) the smooth bed downstream of the step (run b52), and (c) the acrylic floor in a gravel bed (run b60). The top of the test particle in Figure 6c was approximately 0.3 diameters above the tops of surrounding natural gravel particles. Some 14,700 counts are represented in each histogram.

[30] The standard deviation of the drag increases only slightly with increasing height of the test particle within the gravel bed, whereas the mean increases rapidly (Figure 7a). Schmeeckle and Nelson [2003] measured velocities within and above a bed of spheres. They showed that when a particle is partially below the level of upstream particles, the mean velocity of the fluid impinging on it is significantly reduced, whereas the variation of the velocity hardly changes. Similarly, Figure 7a shows that the variability of drag on a particle remains relatively constant with particle height, even though the mean force is significantly reduced by the degree of sheltering by surrounding upstream particles. Thus the large variance of drag relative to the mean on a particle in a gravel bed (Figure 6c) is caused by both the increased relative variance in downstream velocity in turbulence over a rough bed as well as a decrease in the mean velocity without a concomitant decrease in the variance of velocity due to upstream sheltering. This observation suggests that variability becomes increasingly important for the motion of a particle as the vertical position of the particle is decreased, and models that attempt to predict entrainment of grains from the sediment bed using only mean quantities are likely to give incorrect results without some empirical calibration to account for the variability. In addition, this suggests that a physically based model of particle entrainment should explicitly incorporate variability.

Figure 7.

Mean nominal force (squares) and standard deviations (triangles) for four gravel bed arrangements, with the test sphere in the gravel bed versus height of the bottom of the sphere above the acrylic bed relative to the diameter of the sphere, as shown in the insets. (a) Nominal drag Fx and (b) nominal lift Fz. The predicted profile of nominal lift (solid line with no symbols) in Figure 7b was calculated from (12) using CL = 0.2 and the measured mean horizontal velocity profile.

[31] Just as for drag, the variability of the lift is greatly affected by the turbulence structure. When more than about half of the test particle is above the top of the gravel bed, the standard deviation of the lift exceeds the mean (Figure 7b). Shown in Figure 8 are histograms of the ratio of instantaneous lift to the mean lift Fz/equation image for the same three cases as Figure 6. The downstream-facing step dramatically increases the magnitude and frequency of large lift events relative to a smooth bed (Figure 8a and 8b). This finding supports the conclusion of Nelson et al. [1995] and McLean et al. [1994] that turbulence structures caused by upstream separation can markedly increase the sediment transport rate. For a particle in a pocket in a gravel bed (Figure 8c) the excursions in lift are even greater, with some upward events almost 6 times the mean. Inasmuch as these lift events occurred in a record only 29.4 seconds in length, longer measurement periods probably would include even larger ones. Moreover, adding a downstream-facing step, such as a dune slip face, to the gravel bed doubtless would generate lift events of even greater magnitude.

Figure 8.

Histograms of instantaneous relative nominal lift Fz/equation image for the same runs as in Figure 6.

4. Discussion

[32] To provide a framework for discussion of the coupling between the synchronously measured velocity components and the nominal drag and lift, a derivation of the equation of motion for a rigid particle in a fluid flow is presented in Appendix A. This derivation also serves to emphasize the approximations and assumptions that typically are employed in developing such a relation.

4.1. Drag

[33] According to equation (A10) derived in Appendix A, the nominal drag on a stationary particle is given by

equation image

where x, y, z are the coordinates x1, x2, x3. Full experimental evaluation of any of the terms on the right is not possible using a single two-component LDV. For turbulent simple shear flow in the x direction parallel to a plane bed, however, the hydrodynamic drag term appears to be dominant, inasmuch as the dominant velocity component Ux and dominant derivative ∂Ux/∂ z in the other terms are multiplied by quantities that typically are small in comparison to Ux.

[34] The acceleration force is produced primarily by pressure gradients in the flow, which can arise from externally imposed unsteadiness or spatial accelerations or from turbulence fluctuations. These latter forces are present even in steady, horizontally uniform turbulent flows and are neglected in virtually all models for sediment particle motion. In the following analysis, they are initially neglected (primarily because they cannot be accurately estimated from the data) and then qualitatively revisited in light of the experimental observations.

[35] Because the LDV measured only Ux and Uz and the error made in omitting Uy in the drag term in equation (1) generally is greater than that made in omitting Uz, the approximation is made that ∣U∣ ≈ Ux. Note that, even for Uz as great as 20% of Ux, which would be unusually large, the error made by neglecting it in estimating the force is only about 2%. Hence the measured nominal drag Fx was modeled simply by

equation image

A time-averaged version of (2) is generally assumed in models of sediment transport. A primary goal of the experimental runs in which velocities were measured upstream of the test particle was to examine the applicability of (2) to predict the downstream force both instantaneously and in a time-averaged sense near the bed of a turbulent flow. Drag coefficients were determined only for experiments in which the fluid velocity was measured in front of the test particle, which means that drag coefficients were only calculated when the test particle was at least half a grain diameter above the tops of surrounding bed grains. Also, since the test particle could not be in contact with surrounding particles, A was calculated as the full projected area of the test particle.

4.1.1. Instantaneous Drag Coefficients

[36] Measured values of instantaneous Fx versus instantaneous equation imageρfAUx2 are plotted in Figure 9 for three representative experimental runs with the spherical test particle over a smooth bed, downstream of a backward step, and in a gravel bed.

Figure 9.

Instantaneous nominal drag Fx versus instantaneous equation imageρf AUx2 for the 1.9-cm sphere (a) over the smooth bed (run b11), (b) over the smooth bed downstream of a step (run b52), and (c) in a gravel bed (run f35). The lines in each graph indicate a drag coefficient of 0.7. The first 1000 points of each run are plotted in the graphs.

[37] Figure 9 shows that using a single value of CD (2) predicts instantaneous Fx for given instantaneous Ux2 reasonably well. The nonrandom distribution of the residual errors, however, indicates the existence of systematic features not included in this simple formulation of the drag. Figure 10, for instance, shows instantaneous CD plotted against instantaneous Ux for the three runs shown in Figure 9. In each case CD varies nonlinearly and inversely with Ux and its range increases with the variability of the flow. The effect is that, for all tests on spheres where velocity was measured in front of the test grain, the measured standard deviation of the instantaneous values of Fx is only about 50–65% of that predicted using (2) with constant CD = 0.7 (Figure 11).

Figure 10.

Instantaneous drag coefficient versus instantaneous downstream velocity for the sphere (a) over the smooth bed (run b11), (b) over the smooth bed downstream of a step (run b52), and (c) in a gravel bed (run f35). The first 1000 points of each run are plotted in the graphs.

Figure 11.

Measured standard deviation of nominal drag Fx versus that predicted using fixed CD = 0.7. The dashed line is the locus of perfect agreement.

[38] The range of particle Reynolds numbers of the experiments in Figures 9 and 10 based on instantaneous velocities was between 2000 and 9000. For settling spheres in still fluid CD is found to be relatively constant between Reynolds numbers of 1000 and 200,000 [Schlichting, 1979, p. 17]. Thus the systematic variation of CD in Figure 10 cannot be explained by variation of Reynolds number. One possible cause of this behavior is that the turbulent fluctuations smaller in scale than the test particle effectively average out over the surface of the particle and therefore contribute little to the drag. To test this hypothesis, the time series of Ux for several of the runs were low-pass filtered with a cutoff at 5 Hz. Although the remaining length scales, assuming Taylor's hypothesis [Tennekes and Lumley, 1972, p. 253], are at least several times the size of the test grain, the calculated standard deviations and instantaneous drag coefficients were nearly identical to those obtained from the unfiltered velocity data. Particle-scale turbulent structures therefore apparently do not cause the discrepancy between the calculated and the measured variability in the drag.

[39] The discrepancy may be the effect of particle-scale pressure fluctuations in the flow. In the derivation (Appendix A) leading to (1) this effect was excluded when the drag term in (A5) was replaced with the empirical expression (A6), although it clearly is present in the full equation of motion (A1). The role of particle-scale pressure gradients in producing a correlation between the instantaneous drag coefficient and the velocity is not completely understood, but at least two scenarios appear likely.

[40] First, reconsider the force caused by the turbulent pressure fluctuations in the flow, or its surrogate in equations (A10) and (1), mf(1 + Cm) 〈Duox/Dtequation image. A fluid parcel that has a high instantaneous Ux at a particular height above the bed has generally been transported there from above. As it enters the lower-momentum layer, a pressure gradient in the upstream direction develops, decelerating the fluid parcel to a velocity closer to that of nearby parcels. This adverse pressure gradient also will act on a rigid particle in the flow, such as the test particle, and will cause the measured force when Ux is higher than average to be smaller than calculated from a constant drag coefficient. The inverse also is true. A low-momentum fluid parcel will cause a pressure gradient in the downstream direction, thus increasing the force on the particle and hence the instantaneous CD. The result is that the variability in drag is less than would be expected based on the variability of Ux2 assuming constant CD, even though the mean drag for each test run is well modeled by a single drag coefficient.

[41] Second, pressure fluctuations may also significantly affect the locus of boundary layer separation on the particle, which in turn is important for determining the instantaneous distribution of pressure over it. At this point, however, the relation between the locus of separation and the pressure fluctuations is not precisely known. From studies in aerodynamics, it is clear that its position dramatically affects the lift and drag.

[42] The extreme values of drag within individual runs are not well modeled by a constant CD, because the turbulent pressure fluctuations or the streamwise locus of separation or both are most important for velocity extremes. Most of the decrease in variability of the measured drag compared to the variability of the calculated drag, however, is due to fluctuations of velocity below the mean, whereas the fluctuations that generally are more important for determining sediment transport are those above the mean, especially near threshold conditions. For the highest fluctuations above the mean, the instantaneous CD asymptotically approaches about 0.4–0.5 (Figure 10). Thus, for the crucial positive velocity fluctuations the error made by using a constant drag coefficient is no more than about 30%.

4.1.2. Time-Averaged Drag Coefficients

[43] For steady turbulent flow an average CD can be computed in several ways. For instance, a coefficient CDI, listed in Table 3 for each run, may be computed by averaging the instantaneous values of the drag coefficient. Its global average for all runs with the spherical test particle is 0.72. Or, a coefficient equation image(Table 3), whose global average for the spherical particle is 0.66, may be computed from equation image and the average of the velocity squared. In Figure 12 the drag coefficient, CDR, is found as the slope of a linear regression of equation image and equation imageρfAequation image. Finally, a coefficient CDU (Table 3), whose global average for the spherical particle is 0.73, may be computed from equation image and the square of the average velocity.

Figure 12.

Mean nominal drag equation image versus equation imageρfAequation image for the 1.9-cm sphere over the smooth bed, the smooth bed downstream of a step, and the gravel bed (46 squares); for the 2.54-cm cube over the smooth bed (6 circles); and for the natural gravel particles over the smooth bed (14 triangles). The linear regressions for the three groups of measurements give drag coefficients of 0.76 (dotted line) for the spheres, 1.36 (dashed line) for the cubes, and 0.91 (solid line) for the natural gravel particles. The dot-dashed line indicates the expected drag coefficient of 0.4 for spheres falling in a still fluid for the range of particle Reynolds numbers in these experiments (Rp = 4000–20,000).

[44] The average nominal drag equation image as a function of equation imageρfAequation image, where equation image is the average downstream velocity, is plotted in Figure 12 for runs with various test particles, particle heights, and bed configurations. The points for the sphere all lie nearly on a single straight line independent of flow variability or particle height, implying a constant CDR = 0.76. The drag coefficient CDR = 1.36 for the cube test particle is considerably greater because the sharp edges cause flow separation around the perimeter of the front face and hence a large drag. The drag on the natural gravel particles is quite variable. The regressed line for the natural particles in Figure 12 gives CDR = 0.91. A couple of the more streamlined gravel particles have drag coefficients slightly less than that for spheres, whereas others with edges near their upstream faces have drag coefficients approaching that of the cube.

[45] The drag coefficients obtained in this study are much greater than those typically found for spheres falling through still water (Figure 12). At the particle Reynolds numbers of this study CDU for a sphere settling in still fluid is only about 0.4 [Schlichting, 1979, p. 17]. For particles resting on a smooth bed in turbulent flow Roberson and Chen [1970] reported drag coefficients of 0.65–0.70 for spheres and 1.1–1.2 for cubes in the same range of particle Reynolds numbers.

[46] Apperley and Raudkivi [1989] made high-frequency measurements of nominal drag and lift on a 0.6-cm sphere above a bed of similar spheres using a force transducer but did not simultaneously measure velocity. Although they therefore did not report a drag coefficient, it can be calculated from their data as follows. They graphically reported that at one grain diameter above the top of the bed

equation image

in which u* is the shear velocity, given as 7.22 cm s−1. Using the log law of the wall velocity profile, taking z0 to be 1/30th the particle diameter, and noting that the von Karman constant κ = 0.41 leads to

equation image

Finally, CDU is calculated from

equation image

giving nearly the same value for CDU as this investigation and the study by Roberson and Chen [1970].

[47] Using a static strain gauge Coleman [1967] measured average drag force on a sphere resting on a bed of other spheres and reported CDU about 0.5 for particle Reynolds number 103–104 but did not give values of A. Because the test sphere was resting in a pocket on a bed of other spheres, its front face must have been partially blocked. Not taking this into account would result in overestimating A and hence underestimating CDU.

4.2. Lift

[48] According to equation (A10) derived in Appendix A the nominal lift on a stationary particle in any flow is given by

equation image

As with the streamwise force, full experimental evaluation of any of the terms on the right is not possible using a single two-component LDV. For turbulent simple shear flow in the x direction parallel to a plane bed, however, the hydrodynamic lift term appears dominant, inasmuch as the dominant velocity component Ux and dominant derivative ∂Ux/∂z in the other terms are multiplied by quantities small in comparison to Ux and ∂Ux/∂z. As will emerge, however, this is not in fact the case, as previously pointed out by Auton et al. [1988] on the basis of theoretical calculation.

[49] Replacing the hydrodynamic lift term by the expression equation imageρfACL(Uxtop2Uxbot2) for uniform simple shear flow from (A8) suggests that the lift ought to correlate strongly with the velocity at the top of the particle, inasmuch as the velocity at the bottom generally will be much smaller and less variable, especially when the bottom is at or below the general level of the bed. In fact, for the 109 runs in which velocity was measured above the particle, the correlation, if any, is extremely weak and the correlation with the downstream velocity measured at the front of the particle, which also should be positive, is weakly negative. Hence the conclusion is inescapable that the lift is not adequately predicted by (A8) and doubtless also not by (A7), and thus that it is pointless to compute a lift coefficient in the same manner as the drag coefficient.

[50] In addition, (A8) predicts the wrong variation of mean lift with particle height (Figure 7b). The predicted lift was computed taking CL = 0.3, using the measured profile of mean velocities above the top of the gravel bed, and assuming zero velocity below it. This assumption probably is close to accurate because the measured mean drag was slightly negative and the standard deviation was very small when the test grain was below the top of the gravel bed. The measured mean lift was maximum when the top of the test particle was actually below the top of the gravel bed and decreased as the particle was moved higher, whereas the formula predicts it should have increased.

[51] The lift appears to be influenced somewhat by the support stalk. The observations suggest that when the particle was close to the bed the lift was consistently smaller with the 0.18-cm cylindrical stalk than with the 0.3 × 0.6 cm teardrop-shaped one. Experiments by Willets and Murray [1981] have shown that lift is upward on a sphere touching a smooth plane bed, whereas it is downward on one whose bottom is only 1/20th its diameter above it. Apparently, such small differences in position significantly affect the distribution of pressure on the bottom of the sphere. Thus the difference in cross-sectional area of the support stalks may be enough to have a significant effect, even when the bottom of the particle is a grain diameter below the top of a gravel bed, where the velocity must be very close to zero.

[52] When the test particle is wholly above the top of the gravel bed, the measured mean lift is downward, whereas the predicted lift is positive but, owing to the decrease in shear rate above the bed, relatively small. A possible explanation is that the vertical component of the mean hydrodynamic drag is downward and not negligible. According to (6), taking the dominant term, it is equation imageρfAequation image, which is always negative, and hence downward, because equation image is small and the fluctuations in Uz and Ux are negatively correlated near the bed. In addition, because the vertical hydrodynamic drag is nonlinearly dependent on Uz, it can have an additional mean vertical component having the same sign as that of the skewness of the distribution of Uz. Recomputing runs f28, f30, f33, f35, and f37 with these corrections, assuming a constant CD = 0.7, resulted in a negative mean lift, but in all cases it was less than 7% of the measured values. Thus the vertical component of the hydrodynamic drag seems unlikely to account for the negative mean lift, although it does account for nearly 70% of the standard deviation of the measurements.

[53] The mean lifts on the 7 natural particles over the smooth acrylic bed differed strongly under otherwise similar conditions (Figure 13). One was strongly downward, although all the others were upward; and their magnitudes in some cases were much larger than those for the spheres. No rationale to explain either the signs or magnitudes of the measurements is apparent from inspection of the particles, although the discrepancies must be due to the form lift produced by the shapes of the particles. Although it was ignored in the derivation of (6) and is zero for spheres, form lift appears to be quite important for natural particles.

Figure 13.

Mean nominal lift equation image on natural particles above the smooth bed versus equation imageρfAequation image measured in front of the particle. Solid squares are measurements at particle heights of 0.0 cm, and open squares are measurements at particle heights of 1.6 cm.

[54] Overall, the measurements suggest that the processes producing lift on particles in near-bed turbulent flow are not captured by the simple models for lift in uniform flows, such as (A7) or (A8), that were adopted in the derivation of (A9) and (A10). The hydrodynamic forces acting on a particle are the buoyancy, which arises from the gravitational acceleration, the dynamic pressure distribution over the particle surface, which arises from the fluid acceleration field, and the viscous stresses acting over it. For sufficiently high particle Reynolds number, the viscous stresses acting directly on the particle are negligible in comparison to the dynamic pressure. The viscous stresses are important, however, because they lead to the formation of a boundary layer and wake, which in turn dominantly determines the streamwise force on the particle. The vertical force, on the other hand, apparently is not dominantly determined by the wake or by Bernoulli pressure differences across the particle, as embodied in the first and second terms of (6). Thus the source of the vertical force must be sought either in the acceleration force, as found on theoretical grounds by Auton et al. [1988], or in other pressure differences caused by the presence of the particle.

[55] If a local turbulence structure produces high pressure on the bottom of the particle and a low pressure on the top, the vertical acceleration force is large and a high-lift event will occur. The term ∂Uz/∂t in the vertical acceleration has to average zero in steady flows and hence cannot produce nonzero mean lift, although it changes rapidly (Figure 5) and could produce strong lift events both upward and downward. The term UzUz/∂z, or, equivalently, equation imageUz2/∂z, is mathematically nonnegative at the bed and hence will be positive near it during both upward and downward turbulence fluctuations; thus it will produce positive mean lift and lift events. The other two terms cannot be discussed even this crudely. Thus further experiments are required to establish the role and significance of the acceleration force in producing vertical force.

[56] The pressure differences between the top and bottom of the particle that are generated by the presence of the particle also could be important. In the runs with the downstream-facing step and the ones with the gravel bed the peak lifts were comparable to the mean drag. Thus, at least for brief periods, the pressure differences across the grain from top to bottom must have been comparable to those from front to rear. These differences were far greater than can be explained by application of Bernoulli's principle, because the measured velocities over the top of the particle are too small and do not correlate with the lift. Instantaneous lifts comparable to mean drag require pressures beneath the particle comparable to the average stagnation pressure at its front. Thus the key to understanding the high variability in the vertical force appears to be the flow and pressure distribution at the bottom of the particle.

5. Conclusions

[57] 1. Instantaneous streamwise force (nominal drag) on a near-bed particle in a turbulent boundary layer is not correlated with instantaneous Reynolds stress but strongly correlates with instantaneous streamwise velocity. Consequently, the standard drag coefficient formulation predicts reasonably well the instantaneous drag from the instantaneous velocity.

[58] 2. The value of the drag coefficient for natural gravel is a factor of 2 to 3 greater than for a sphere settling in still fluid. This implies a factor of 4 to 9 error in predicting the velocity required to produce a given drag, or a factor of approximately 2–3 in the bed stress required to reach some critical value. Drag coefficients of spheres in still fluid are not adequate for use in sediment transport models.

[59] 3. The instantaneous drag coefficient is lower for higher instantaneous velocities and the converse. The cause may be particle-scale pressure gradients caused by the turbulence that affect the particle both directly and through the streamwise locus of boundary layer separation on it.

[60] 4. Instantaneous drag commonly deviates substantially from the mean. Maximum drag on gravel-sized particles protruding from a gravel bed by about half their diameter is as much as 4 times the mean. Hence parameterization of particle entrainment or motion based on mean conditions is not realistic.

[61] 5. Instantaneous vertical force (nominal lift) correlates poorly with the flow velocity either upstream or over the particle. Furthermore, it is not well approximated by any expression for lift, such as the shear-lift formula, that depends on Bernoulli pressure differences.

[62] 6. Mean lift on spheres appears to result from a complex superposition of shear lift, vertical hydrodynamic drag, near-bed vertical fluid acceleration, and other unidentified pressure effects. Mean lift on natural particles additionally is strongly influenced by particle shape.

[63] 7. Excursions in instantaneous lift can far exceed mean lift, especially on a rough bed or downstream of an obstacle. Mean lift therefore can be relatively unimportant in particle entrainment during sediment transport.

[64] 8. Pressure differences acting at the scale of the particle play the dominant role in determining particle motion, except for the very smallest particles, as shown by the fundamental equation of particle motion relating the force on a particle to the integral of pressure over the particle surface. These pressure differences arise not only from the wake formation and Bernoulli effects embodied in the usual formulations of drag and lift but also from other processes associated with the turbulence of the flow and the presence of the particle that require further study.

Appendix A:: Forces on a Sediment Particle in a Fluid Flow

[65] Although existing models for the forces on particles in turbulent flows disagree in various particulars, all proceed from the momentum balance of a sediment particle in a fluid flow, make similar assumptions and approximations, and arrive at similar though usually not identical results. Most derivations contain errors, principally in connection with the fluid acceleration, and many overstate the validity of assumptions or fail even to identify them. The following is intended to correct the errors, to state plainly the assumptions and approximations, to give key references, and to present the complete derivation within the limitations of current knowledge. It provides in one place the indispensable basis for interpreting the experimental results.

A1. Equation of Motion

[66] Adopting a right-handed Cartesian coordinate system xi in which the subscript i denotes the three coordinate directions, and assuming that, because the particle Reynolds number Rp is large, the flow is effectively inviscid and hence forces due to fluid viscosity can be ignored, the momentum balance of a particle of mass mp is given by

equation image

where dVi/dt is the time derivative of the particle velocity Vi, taken following the particle, FSi is the solid contact force exerted by other particles or, in the case of the experimental test particle, by the stalk holding it stationary, gi is the acceleration due to gravity, P is the fluid pressure, and ni is the outward unit vector perpendicular to the element dequation image of the particle surface equation image.

[67] If the flow is irrotational as well as inviscid, the fluid velocity ui can be written as the gradient of a potential Φ, that is, ui = ∂Φ/∂xi. Substituting this into the Navier-Stokes equation for inviscid flow [Batchelor, 1967, p. 380], letting ρfu2 = ρfuiui, where ρf is the density of the fluid and terms with repeated subscripts follow the usual summation convention, writing gi in terms of the gravitational potential Γ, that is, gi = ∂Γ/∂xi, and moving the gradient operator ∂/∂xi out of all the terms shows that the expression P + equation imageρfu2 + ρf∂Φ/∂t + ρfΓ has the same value everywhere in the flow [Batchelor, 1967, pp. 382–383]. If, as usually assumed, the particle affects only the nearby flow field, this expression would therefore also have the same value were the particle absent. Hence

equation image

where the subscript o denotes quantities in the flow with the particle absent, hereafter termed the reference flow.

[68] Substituting (A2) into (A1), letting 〈∂Po/∂xiequation image signify the average of ∂Po/∂xi over the particle volume equation image in the reference flow, transforming the surface integral involving Po into a volume integral by means of

equation image

which is a form of Gauss's divergence theorem, eliminating ∂Po/∂xi by means of the Navier-Stokes equation for inviscid flow [Batchelor, 1967, p. 380], and rearranging the result leads to

equation image

where mf is the mass of fluid displaced by the particle. The second term on the right-hand side of (A4) is the buoyancy force FBi on the sediment particle.

[69] The quantity averaged over the volume equation image in the third term is the acceleration following a fluid particle [Batchelor, 1967, pp. 72–73] in the reference flow. If the smallest length scale of the pressure gradient fluctuations is large compared to the size of the volume equation image, which is not true of turbulent near-bed flows, in which bed particle wakes are important, the average can be approximated by DUi/Dt, which is Duoi/Dt evaluated at the center of the sediment particle. In the sediment transport literature, except, notably, the paper by Madsen [1991], this term usually has been ambiguously, if not incorrectly, given as dUi/dt. Maxey and Riley [1983] and Auton [1987] have pointed out the importance of distinguishing between the derivative following the geometric center of the sediment particle and the derivative following the fluid at the position of the geometric center in the reference flow. Although for low particle Reynolds number Rp the error is not large, for high Rp it is substantial [Maxey and Riley, 1983; Mei et al., 1991].

[70] The fourth term on the right-hand side of (A4) is often called the added mass (or virtual mass) force FMi. It arises from the difference in acceleration of the reference fluid and the sediment particle. For weakly shearing flow, in which the reference velocity difference across the particle is small in comparison to the reference velocity relative to the particle, most authors have given for it the formula mfCMd(UiVi)/dt. Auton et al. [1988] noted, however, that the formula instead should be mfCM(DUi/Dt − dVi/dt). For spheres in weakly shearing flow the added mass coefficient CM = equation image. For the strongly shearing turbulent near-bed flows studied in this investigation the formula equation image, with CM empirically determined, will be adopted instead.

[71] Evaluation of the last term in (A4) requires solution for the flow around the sediment particle. The approach generally taken, and the one followed in this investigation, is to decompose the term into two forces that are treated separately: the drag FDi, which is in the direction of the reference velocity at the center of the particle, and the lift FLi, which is perpendicular to it.

[72] Inserting all these formulas into (A4) then finally leads to the schematic equation

equation image

which apart from the term FSi resembles virtually all equations of motion that have been used in the computation of saltation trajectories of sediment particles.

[73] Application of (A5) requires calculation of the drag and lift. Solution for the drag on a spherical particle using assumptions consistent with the development of (A5), namely inviscid, irrotational flow with pressure gradients only on scales large compared to the particle size, leads to the physically incorrect result FDi = 0 [Batchelor, 1967, p. 453]. A correct solution for large Rp requires treating the particle boundary layer and separation bubble, which means that the forces due to viscosity cannot be disregarded in the momentum balance for the fluid. The experiments reported in this paper show that a similar conclusion applies to the lift. Thus the process leading to (A4) and (A5) is faulty.

[74] To obtain closure, empirical formulas for drag and lift are used. These formulas, whose forms for simple flows can be obtained by dimensional analysis, take account of the viscous forces somewhat arbitrarily.

A2. Drag

[75] An example derived from dimensional analysis that works adequately for numerous particle shapes is the formula

equation image

for the drag, where FD, U, and V are the vectors whose components are the previously defined FDi, Ui, and Vi, Ais the area of the projection of the particle on a plane perpendicular to UV, and the drag coefficient CD is an empirically derived function of Rp. Coleman [1967] has shown that this formula works well for a sphere resting on a bed of spheres when time-averaged FD and U are used.

A3. Lift

[76] The lift on natural sediment particles near the wall in a turbulent boundary layer is less well understood than the drag. Some authors [Davies and Samad, 1978; Willets and Naddeh, 1986], for example, dispute even the direction of lift on particles near the bed, raising doubt as to whether a formula analogous to (A6) is appropriate.

[77] Lift arises in shear flow when the fluid velocity on one side of the particle is faster than on the other. For inviscid, weakly rotational flow around the particle, Auton et al. [1988] obtained the formula

equation image

in which Ω = ∇ × U is evaluated at the position of the particle center and CLA is the Auton lift coefficient. This solution requires that the difference in velocity across the particle be much smaller than ∣UV∣, whereas the two commonly are roughly equal for a fixed particle near a sediment bed. Unfortunately, it is the only solution available for large Rp.

[78] If for a uniform simple shear flow the particle and flow velocities are both in the x1 direction, taken horizontal, and the direction of increasing flow velocity and the perpendicular to the shear planes are both in the x3 direction, taken vertically upward, (A7) reduces to

equation image

where A is the area of projection of the particle on the x1x2 plane, and uot1 and uob1 are the reference flow velocities at the top and bottom of the particle. For a spherical particle the analysis of Auton et al. [1988] gives CLA = equation image.

[79] Wiberg and Smith [1985] adopted the same form for the lift but set the numerical coefficient to equation image instead of equation image, thereby defining a different lift coefficient CLVS. The analysis of Auton et al. [1988] for a spherical particle in a uniform simple shear flow thus gives CLVS = equation image. From the data of Chepil [1958] for a hemisphere in a bed of evenly spaced hemispheres of the same size, Wiberg and Smith [1985] obtained the empirical result CLVS = 0.2; reanalysis gives the range as 0.18–0.34. Analysis of the data of Chepil [1961] for a sphere in a bed of evenly spaced spheres of the same size gives about 0.4–0.5. Anderson and Hallet [1986] used Wiberg and Smith's [1985] formula and Chepil's [1958] data but set CLVS = 0.85CDMA, where CDMA is the drag coefficient read from the empirical curve of Morsi and Alexander [1972]; for Chepil's [1958, 1961] data the range of the proportionality factor is 0.53–1.32.

[80] Other types of lift, such as spin lift and form lift will not be explored here. Spin lift has been investigated by Niño et al. [1994], Niño and Garcia [1994], and Lee and Hsu [1994], who showed that it has a small but perceptible effect on the length and height of saltation trajectories. Form lift due to asymmetry in particle shape remains largely unstudied.

A4. Equation of Motion of Sediment Particle

[81] Using (A6), (A7), and the adopted formula for FMi, (A5) can be expanded to

equation image

for large particle Reynolds number Rp, where, by analogy with the drag coefficient CD, the lift coefficient CL is an empirically derived function of RP. This equation of motion applies whether or not the particle is fixed in position, touching the bed, or contacting other particles.

A5. Forces on Fixed Particle

[82] In the experimental setup the particle velocity V is fixed at zero and the force exerted by the stalk FS is (FB + FH), where FB is the buoyant force and FH is the hydrodynamic force exerted on the particle by the fluid. Substituting these relationships for V and FS in (A9) and rearranging gives the formula

equation image

which, despite the dubious assumptions and approximations in its derivation, of necessity serves as the starting point for the discussions in this paper. The horizontal and vertical components of the term on the left are the nominal drag and lift measured by the force transducer. The three terms on the right are the hydrodynamic drag, the hydrodynamic lift, and a force analogous to buoyancy, herein called the acceleration force, that arises from acceleration of the reference flow. In turbulent flows none of these three forces is always negligible in comparison to the others, nor are the hydrodynamic drag and lift usually in the directions of the nominal drag and lift; hence explicit distinction between the nominal and hydrodynamic forces is necessary.


Projected area of particle for hydrodynamic drag, equation (A6).


Projected area of particle for hydrodynamic lift, (A8).


Added mass coefficient, (A4).


Hydrodynamic or nominal drag coefficient, (A6).

CDI, CDU, CDR;equation image

Various averages of nominal drag coefficient, (3).


Hydrodynamic drag coefficient from Morsi and Alexander [1972], (A8).


Hydrodynamic or nominal lift coefficient, (A9).


Hydrodynamic lift coefficient of Auton et al. [1988], (A8).


Hydrodynamic lift coefficient of Wiberg and Smith [1985], (A8).


Added mass coefficient, (A4).

Duo/Dt, Duoi/Dt

Material derivative of velocity in reference flow, (A4).

DU/Dt, DUi/Dt

Material derivative of velocity in reference flow at center of particle, (A4).


Hydrodynamic drag force, (A5) and (A6).


Calculated nonbuoyant hydrodynamic force on particle, (A10).


Hydrodynamic lift force, (A5) and (A7).


Added mass force, (A5).


Force due to solid contacts, (A1).


Nominal drag and lift forces, (1) and (6).


Acceleration due to gravity, (A1).


Subscript indicating coordinate direction, (A1).


Mass of particle, (A1).


Mass of fluid displaced by particle, (A4).


Outward unit vector perpendicular to particle surface, (A1).


Pressure of fluid in actual flow, (A1).


Pressure of fluid in reference flow, (A2).


Particle Reynolds number, (A1).

equation image

Surface of particle, (A1).


Velocity of fluid in actual flow, (A2).


Magnitude of fluid velocity in actual flow, (A2).

uo, uoi or uox,uoy,uoz

Velocity of fluid in reference flow, (A2), (1), and (6).

uot1, uob1

Reference flow velocity at the top and bottom of a particle, (A8).

U, Ui or Ux, Uy, Uz

Velocity of fluid in reference flow at center of particle, (A4) and (1).


Magnitude of fluid velocity in reference flow, (A2).

V, Vi

Velocity of particle, (A1).

equation image

Volume of particle, (A3).

xi or x,y,z

Cartesian coordinates, (A1) and (1).


Gravitational potential, (A2).


Density of fluid, (A2).


Velocity potential in actual flow, (A2).


Velocity potential in reference flow, (A2).


Curl of reference velocity at center of particle, (A7).


[83] Gratefully acknowledged are financial support from the National Science Foundation (awards 9506415, 0125525, 0352079, and 0353205) and space, logistical, and other support from the U. S. Geological Survey, the UCLA Department of Earth and Space Sciences, the University of Washington School of Oceanography and Friday Harbor Laboratories, the Florida State University Department of Geological Sciences, and the Arizona State University School of Geographical Sciences.