# Analysis of hydrodynamic lift on a bed sediment particle

## Abstract

[1] An experimental study of lift force on a spherical sediment particle at different exposures was conducted in a laboratory flume. It is found that the functional form of the theoretical probability density function (pdf) of lift force similar to that of drag force is inadequate. An improved expression for the pdf based on the normal error law is provided which matches the measured data up to ±3 times the standard deviation of lift force for all exposures. The skewness of the measured lift force is found to increase with increase in exposure. In addition, the kurtosis of the measured lift force is found to be higher than that for a normal distribution for all exposures, resulting in higher probability of occurrence of extreme events (beyond ±3 times standard deviation). A spectral density function of lift force as a function of streamwise and stream-normal velocity is developed and validated using the measured lift force data. The spectrum predicted by the proposed model closely follows the measured spectrum for exposures 7.5 mm and 10 mm while it over predicts at frequencies below 1 Hz for exposures 0 mm and 2.5 mm because of poor coherence between lift force and streamwise velocity. The time history plots of measured lift at lower exposures showed that the high-frequency positive and negative fluctuations of lift correspond to strongly positive streamwise velocity fluctuations, a phenomenon not captured by the predictive model. Predominance of fluctuating Bernoulli's lift over lift due to stream-normal velocity fluctuations is observed at all exposures, more so in the case of lower exposures.

## Key Points

- Hydrodynamic forces on bed sediment particle
- Probability density function of lift force
- Spectral density function of lift force

## 1. Introduction

[2] In alluvial streams, hydrodynamic forces are exerted on the sediment particles at the bed surface. An increase in flow velocity induces an increased magnitude of hydrodynamic forces, that is, drag and lift force. Knowledge of drag and lift is therefore important in predicting the threshold or critical condition of incipient motion for the sediment bed particles.

[3] While the effect of drag force has received considerable attention in sediment entrainment research, the role of lift in sediment entrainment is less well understood. There are no reliable quantitative relationships for lift force, which could be used for the determination of the sediment threshold condition. However, it is evident that both drag and lift forces always exist and both contribute to the threshold movement of the bed sediment [*Dey and Papanicolaou*, 2008; *Hofland*, 2005; *Coleman and Nikora*, 2008; *Gregoretti*, 2008; *Lamb et al.* 2008; *Giménez-Curto and Corniero*, 2009; *Recking*, 2009].

[4] *Einstein and El-Samni* [1949] measured the lift force directly as a pressure difference. They carried out experiments using plastic spherical balls of diameter *D* = 68.6 mm and graded gravel with the median size *D*_{50} = 68.6 mm. The lift coefficient *C*_{L} was found to be constant for different flow velocities measured at 0.15*D* above the top of the particle. In addition, the turbulent fluctuations of lift force superimposed on the mean lift force were found to follow the normal error law. *Aksoy* [1973] measured forces on a 20 mm diameter sphere at roughness Reynolds number [*R*_{*} = *k _{s}u*

_{*}/

*υ*,

*k*

_{s}is the equivalent sand grain roughness,

*u*

_{*}is the shear velocity and

*υ*is the kinematic viscosity of the fluid] of 300 and found a ratio of lift to drag of about 0.1.

*Bagnold*[1974] investigated lift and drag on a 16 mm diameter sphere for

*R*

_{*}values of about 800 and obtained a lift to drag ratio of about 0.5.

*Brayshaw et al.*[1983] measured lift and drag on a 115 mm diameter hemisphere at

*R*

_{*}= 52000 and found a lift to drag ratio of about 1.8.

*Saffman*[1965] studied the lift on a sphere for

*R*

_{*}< 5 and presented an expression for lift that

*Yalin*[1977] found to be compatible with the Shields criterion for that range.

*Apperley*[1968] measured hydrodynamic forces on a 6.4 mm diameter sphere laid on 6.4 mm diameter gravels for

*R*

_{*}= 70 and found the lift to drag ratio of 0.5, with the ratio increasing to 0.78 if the sphere was raised by 0.25 times its diameter. Higher exposures of the sphere lead to reduced values of the lift to drag ratio.

*Coleman*[1967] studied the lift forces acting on a sphere placed on a hypothetical streambed using plastic and steel spheres to obtain the variation of lift coefficient with Reynolds number.

*Watters and Rao*[1971] measured lift and drag forces on a 95.3 mm diameter sphere, placed in different bed configurations, and observed negative lift for 20 <

*R*

_{*}< 100.

*Davies and Samad*[1978] reported that the lift force on a sphere adjacent to a boundary becomes negative if both significant underflows occur beneath the sphere and

*R*

_{*}< 5. For

*R*

_{*}≥ 5, however, the lift was found to be positive.

*Schmeeckle et al.*[2007] measured hydrodynamic lift on a bed sediment particle and found that vertical force correlated poorly with downstream velocity, vertical velocity, and vertical momentum flux whether measured over or ahead of the test particle.

[5] The studies highlight limitations in knowledge regarding the magnitude and variation of lift with *R*_{*} and considerable variation in the ratio of lift and drag. This might be attributed to the different functional forms of the theoretical lift models used as well as different flow characteristics and particle configuration used in previous experimental studies. The aim of this paper is to report an experimental study of lift force on a spherical bed particle at different exposures in a laboratory flume. Statistical analysis of the measured lift force on a bed particle is carried out to investigate the probability distribution and spectral distribution at different particle exposures. The fluctuating lift force model proposed by *van Radecke and Schulz-DuBois* [1988] was used to develop the probability and spectral density functions of lift force as a function of streamwise and stream-normal point velocity at 1*D* upstream and 0.15*D* above the top of the particle and validated using the measured lift force data, as discussed below.

## 2. Experimental Setup

[6] Experiments were carried out using spherical particle beds as roughness elements, with PIV used to measure the instantaneous flow velocity field. The 0.43 m wide by 0.3 m deep glass-sided recirculating flume used in the present experiments is 11.8 m long. The flume contains a 0.26 m deep and 0.90 m long recess section situated 7.0 m downstream from the inlet of the open channel section. At the upstream inlet to the open channel section, a series of guide vanes straighten the flow and a wave skimmer reduces water surface fluctuations created at the pipe/open-channel transition. The flume tilts about a single pivot point, with slope adjusted using a built-in jack. A 5.5 m length of the water flume was covered by a layer of 39.7 mm average diameter ping-pong balls glued to aluminum sheets in a hexagonal close-packed structure. Upstream of the ping-pong ball section, rounded river gravel (approximately 20 mm diameter) was placed to match the level and approximate roughness of the ping-pong balls in order to prevent any midflume rapid flow transitions. The river gravel was also placed down the sides of the flume to maintain bed roughness where whole ping-pong balls would not fit. The particle bed, comprising one layer of fixed spheres over most of the bed, was increased to three layers of spheres over a 1 m length in the flume recess (see Figure 1).

[7] The six component force sensor (JR3 force-torque sensor specially built for this study) used to measure the load imposed on it is a monolithic aluminum device instrumented with metal foil strain gauges and sealed with Silicone gel sealant (Dow Corning 3-4222) to make it water proof. The strain gauge signals were connected to the external amplifier and signal conditioning equipment through the sensor cable. In the external electronic system the strain gauge signals were amplified and combined to produce signals representing the forces. The unfiltered response of the amplifier is greater than 10 KHz enabling high-frequency sampling. The sensitivity of the sensor is 0.000980N (0.1 g) in the x direction and 0.00196N (0.2 g) in the z direction. The top surface of the sensor is the sensing face and is precisely machined to be extremely flat. To transfer the loads from the sphere to the force sensor, through the cylindrical supporting rod, 6 mm thick circular steel plate was machined perfectly flat and attached to the top face of the sensor. The circular plate was tightened to the sensitive face of the sensor (Figure 2, left). Particular care was taken while tightening the captive button-head bolts. The bolts were tightened with a hex key through the bolt holes. Each bolt was turned a few turns at a time, and then the next bolt was turned until the sensor mounting surface was flat. The 6 mm diameter supporting hollow brass rod was then attached to the 5 mm thread on the steel plate. The target sphere was attached at the other end of the 6 mm hollow supporting brass rod (Figure 2, right).

[8] To water proof the top sensitive surface of the sensor, a canister was cut from high-grade aluminum to cover the sensor body with a 7 mm diameter hole at its center (Figure 2, right). A clearance of 1.5 mm all around was kept between the sensor body and the aluminum canister. With a 5 mm diameter thread on the steel plate, the hole (on the canister) provided a clearance of 1 mm all around the thread. Except for the circular hole at the center of the canister, all the open joints and edges were made water proof using high-performance silastic 732 offering excellent adhesion, elasticity, wide temperature resistance and long life reliability. The major challenge was to prevent the inflow of water through the gap between the circular hole at the center of the canister and the supporting rod (see Figure 2, right) since inflow of water through this gap could impart false force reading from the sensor. A small amount of white petroleum jelly (shell snow) was applied in the gap, thus sealing the gap perfectly and allowing flexibility for small movement of the threaded rod. The supporting rod and the target sphere attached to it were placed in such a way that they were not touching the surrounding spheres. With this setup, the force sensor recorded the hydrodynamic forces on the sphere due only to flow of the water.

[9] The performance of sensor when installed in the flume was checked by applying known weights to the sensor in the x and z directions. Figure 3 shows the response of the sensor in x and z direction on application of a weight of 0.00981N (1 g) in the negative z direction. Peaks observed between 22 and 25 s are due to the manual loading of the weight on the force sensor.

[10] Due to temperature drift in one of the channels only fluctuating components of forces were recorded. The voltage data obtained were multiplied by the calibration matrix provided by the manufacturer, and the resulting time series was transformed into the frequency domain using fast Fourier transformation (FFT). The real part of the Fourier magnitude of an uncorrected time series of drag sampled at 1000 Hz (Figure 4) shows that virtually all of the output is at frequencies below 20 Hz. For the test particle diameter (*D*) of 38.3 mm and the mean flow velocity (measured at 1*D* upstream and 0.15*D* above the test particle) of about 40 cm/s, Taylor's hypothesis implies that turbulence frequencies higher than 20 Hz correspond to eddies smaller than about 20 mm. Peaks in the vicinity of 70 Hz and 150 Hz were observed in the unfiltered signal of drag and lift force, respectively. Because the net force caused by pressure fluctuations created by smaller eddies is negligible due to phase cancellations when integrated over the surface of the particle, the output near 70 Hz (which corresponds to eddies only 5.7 mm in size) was suspected to be due to excitation of vibrations near the natural frequency of the assembly. The peak at 150 Hz in lift force signal is possibly due to higher modes of vibration of the assembly.

[11] Vibration tests of the assembly were then performed using an electro-dynamic shaker (consisting of a coil and a magnet to produce magnetic fields causing vibration of the assembly) and accelerometer attached to the assembly to establish its frequency response curve as shown in Figure 5. As can be seen in Figure 5, the gain of the assembly is flat up to 55 Hz and shows a sharp peak at about 70 Hz corresponding to its natural frequency.

[12] To remove the frequency component due to the natural frequency of the assembly as well as those contributed by the higher modes of vibration, frequency domain analysis involved setting the amplitudes for frequencies above 50 Hz to zero and then transforming the result back to the time domain by means of the inverse fast Fourier transformation (IFFT). Figure 6 shows the filtered and unfiltered power spectra curve. Ten consecutive filtered force recording were box car averaged for the purpose of low-frequency noise reduction resulting in a net sampling frequency of 100 Hz.

[13] The PIV system of The University of Auckland uses the scanning beam technique to generate up to 200 PIV images per second (i.e., sampling frequency of 100 Hz). The light source is a frequency-doubled 5W CW Nd: YVO4 laser (Spectra-Physics Millenia), which has a wavelength of 532 nm. The water is seeded with 50 *μ*m Polyamide particles. An iterative continuous-window-shift, cross-correlation algorithm with a linear velocity-gradient correction was used. Cross correlation was performed using the Fast Fourier Transform method and the correlation peak location was estimated with a three-point Gaussian function. The PIV light sheet entered the water through a streamlined window (see Figure 7), which skimmed the water surface to prevent scattering of the laser beam due to water surface undulations. The shape of the window is streamlined to minimize potential impact on flow behavior. The similar velocity profile shown in Figure 8 confirms that there is no effect of window on the flow. The dimensions of PIV image area measured were 140 mm × 140 mm. The complete PIV setup and flume is shown in Figure 9.

[14] The software for PIV image analysis calculates approximately 2.5 vector maps per second using a two-pass discrete window-shift (DWS) algorithm, and 0.11 vector maps per second using the four-pass continuous-window-shift shear-corrected (CWSSC) algorithm used herein (based on 59 × 59 vectors per frame and a 2.0 GHz Athlon64 processor). The present PIV velocity estimates have a random error component of < ±0.2 pixels (≡ ±0.014 m/s based on the typical pixel/mm calibrations used in this study). The bias error associated with nonuniform beam scan velocity, lens distortion, and refraction at the fluid-air interface velocity component is less than ±1% [*Cameron*, 2006].

## 3. Flow Configuration

[15] This study considers rough-boundary flows of relative submergence, *H*/*D* ≈ 2.7 − 5.2 (where *H* is the depth of flow with origin at 0.2D below top of spherical particles). The experiments were undertaken for a range of mean velocity Reynolds number (*R* = 4*m*/*υ*) from 124 × 10^{3} and 342 × 10^{3} and are hydraulically rough with roughness Reynolds number (*R*_{k*} = *k _{s}u*

_{*}/

*υ*) ranging from 1983 to 3240 (calculated assuming

*k*

_{s}=

*D*). All flows have Froude numbers (

*Fr*= /) less than unity, indicating a subcritical flow environment. The experiments were performed under uniform flow conditions. Four different particle exposures (

*e*) and three flow depths (

*H*) were tested. The exposure (

*e*) is the distance between the top of the target particle and the top of the surrounding particles while

*S*

_{b}is channel bed slope. The flow properties for the exposures and flow depths tested are shown in Table 1.

e (mm) |
H (mm) |
(m/s) | u_{*} (m/s) |
Re × 10^{3} |
R_{k*} |
θ_{c} |
Fr |
---|---|---|---|---|---|---|---|

0.0 | 108 | 0.69 | 0.0826 | 198.4 | 2844 | 0.14 | 0.67 |

0.0 | 158 | 0.79 | 0.083 | 285.9 | 3055 | 0.15 | 0.63 |

0.0 | 208 | 0.81 | 0.083 | 342.1 | 3240 | 0.15 | 0.57 |

2.5 | 108 | 0.61 | 0.0745 | 176.1 | 2522 | 0.12 | 0.59 |

2.5 | 158 | 0.74 | 0.074 | 268.1 | 2739 | 0.12 | 0.59 |

2.5 | 208 | 0.75 | 0.0673 | 316.3 | 3137 | 0.11 | 0.52 |

7.5 | 108 | 0.51 | 0.060 | 145.3 | 2207 | 0.08 | 0.49 |

7.5 | 158 | 0.60 | 0.061 | 218.5 | 2480 | 0.08 | 0.48 |

7.5 | 208 | 0.61 | 0.057 | 257.9 | 2642 | 0.07 | 0.43 |

10.0 | 108 | 0.43 | 0.053 | 124.1 | 1983 | 0.06 | 0.42 |

10.0 | 158 | 0.53 | 0.054 | 189.4 | 2160 | 0.06 | 0.41 |

10.0 | 208 | 0.55 | 0.053 | 233.8 | 2406 | 0.06 | 0.38 |

[16] Table 2 shows the shear velocity estimated by all three methods, namely using channel slope (*u*_{*slope}), using the slope of the logarithmic velocity profile (*u*_{*log}) and by extrapolating the Reynolds shear stress profile to the bed (*u*_{*uw}). A close inspection of Table 2 reveals that *u*_{*uw} is smaller than *u*_{*log} for all the flows. If secondary currents are present, then there are additional terms in the Reynolds stress which cannot be neglected [*Nezu and Nakagawa*, 1993]. That the shear velocity estimates obtained from the Reynolds stress method are smaller than those obtained from the log-fit method suggests that secondary currents were present in the experiments. However, the difference in their magnitude is of the order of only a few millimeters per second.

e (mm) |
H (mm) |
S_{b} (×10^{−3}) |
u_{*slope} (m/s) |
u_{*log} (m/s) |
u_{*uw} (m/s) |
---|---|---|---|---|---|

0.0 | 108 | 6.479 | 0.083 | 0.0826 | 0.080 |

0.0 | 158 | 4.284 | 0.081 | 0.083 | 0.081 |

0.0 | 208 | 3.448 | 0.084 | 0.083 | 0.080 |

2.5 | 108 | 5.329 | 0.075 | 0.0745 | 0.073 |

2.5 | 158 | 3.971 | 0.078 | 0.074 | 0.070 |

2.5 | 208 | 3.030 | 0.079 | 0.073 | 0.072 |

7.5 | 108 | 3.239 | 0.059 | 0.060 | 0.059 |

7.5 | 158 | 2.508 | 0.062 | 0.061 | 0.058 |

7.5 | 208 | 1.985 | 0.064 | 0.057 | 0.056 |

10.0 | 108 | 3.334 | 0.060 | 0.053 | 0.052 |

10.0 | 158 | 2.299 | 0.060 | 0.054 | 0.053 |

10.0 | 208 | 1.985 | 0.064 | 0.053 | 0.057 |

[17] There are uncertainties involved in the precise measurement of water surface elevation based on pointer gauge flow depth estimates. These uncertainties are due to fluctuating water surface, relatively close spacing of depth measurements used in assessment of flow uniformity and effects of secondary current on shear stress profile. Hence, *u*_{*log} obtained from log-law fit is most reliable estimate of shear velocity for present experiments.

[18] Initially the flume slope, pump speed and the overflow water level were incrementally adjusted to determine the flume settings that caused moveable particle (Specific gravity of the particle tested was 1.12.) at the same location (for each exposure) as the fixed particle to entrain approximately once every two minutes for uniform flow at the required flow depth. Flow depth was measured using a pointer gauge attached to a carriage on top of the flume and the flume slope was measured from a gauge on the side of the flume. The uniformity of the flow was ascertained by measuring the flow depth at two positions along the flume 2.5 m apart. The uniformity of flow was further checked by measuring the flow velocity at three locations (two at the end and one at the center of the light sheet) along the width of PIV light sheet. Figure 8 shows velocity profile at the three locations along the flume. It can be seen that all the three velocity profiles collapsed perfectly well with very small deviations close to the particle top (due to spherical particle roughness effects) indicating that flow was uniform. With the flume settings determined, the particle was attached to the force sensor. The PIV system was aligned with the flume centerline and the center of the target particle to capture streamwise and vertical velocity components, and left running for 5 min. Force recordings were synchronized with PIV recordings.

## 4. Double-Averaged Streamwise Velocity

*H*/

*D*) between 2.5 and 5.2 are classified as Type II according to

*Nikora et al.*[2001], where the logarithmic region of the flow may not be present. On the basis of the location of the roughness layer it is possible to evaluate the existence and thickness of log layer. In general the log law is written as

*d*= displacement height,

*u*

_{*}= shear velocity, and

*z*

_{0}= roughness length which depends on roughness characteristics. For flows over sand glued to flat,

*Nikuradse*[1933] found

*z*

_{0}=

*D*/30.

[20] Equation (1) can be applied in a region between the upper boundary of the roughness layer and 0.2*δ*, where *δ* is boundary layer thickness [*Jiménez*, 2004]. The top of roughness layer can be evaluated from the Reynolds stress profile [*Nakagawa et al.*, 1991; *Nikora et al.*, 2001]. For the experiments reported herein, adopting the maximum in the Reynolds stress profile (0.15D from the top of the particle) and 0.2*δ* as the inner and outer limit of the log layer, respectively, indicates the presence of log layer of height 15.6, 25.6 and 35.6 mm for flow depth of 108, 158 and 208 mm, respectively. Assuming a value of κ = 0.41, the best fit of the data was obtained for *d* = 8 mm (0.2D) and *z*_{0} = *D*/33.

[21] Figure 10 shows the streamwise double-averaged velocity profile normalized with shear velocity. The velocity profiles collapsed reasonably well in the region between *z*/*D* = 0.6 and *z*/*D* = 1.25.

## 5. Total Stress Profiles

*t*= 0), uniform (∂/∂

*x*= 0), two dimensional @(〈〉 = 〈〉 = ∂/∂

*y*= 0), no secondary currents, flow above roughness tops, the momentum conservation equation in the principal flow direction is given by [

*Nikora*, 2004]

*S*

_{b}is mean bed slope and

*H*is flow depth. The right-hand side of equation (2) is the double-averaged fluid stress (〉(

*z*)), while the left-hand side is the gravity term. The vertical distribution of fluid stress is linear, from zero at the water surface to a maximum at the theoretical wall level, as balanced by the right-hand side term. In a 2D approach where no sidewall friction exists

*τ*

_{o}= 〈〉(

*z*= 0). Consequently, the stress that acts on the boundary follows from equation (2) to be

[23] For this study, near-bed velocities (*u*) at point 1*D* upstream and 0.15*D* above the top of the target particle were used. The location 1*D* upstream of the target particle was chosen so that the instantaneous velocity measurements are representative of velocities in the vicinity of the particle, but not so close that the velocities are affected by the presence of the particle. The velocity measuring location at 0.15*D* above the top of the particle was chosen because *Xingkui and Fontijn* [1993] found maximum correlation between the instantaneous velocity at this height and the drag force. *Einstein and El-Samni* [1949] also found a constant *C*_{L} (lift coefficient) for different flow velocities at this height. Sensitivity analysis of the velocity measuring location above the target particle was carried out by estimating the correlation coefficient (Figure 12) at a time lag (*τ*) of 1*D*/ between drag force and velocity at that location. The correlation coefficient is found to maximum at 0.15D above the top of the target particle.

## 6. Lift Force Formulation

[24] The lift force on a bed sediment particle may be induced by the pressure difference due to the vertical velocity gradient over the height of the target particle. The sediment particle might also experience lift due to the upward velocity component as a result of near-bed turbulence fluctuations.

*White*[1940] carried out a single experiment and found that the lift on an individual particle is very small compared to its weight.

*Einstein and El-Samni*[1949] used the following equation for lift force (

*F*

_{L}):

*C*

_{L}is the lift coefficient,

*ρ*is density of fluid,

*A*is the frontal area of the particle, and

*u*is the measured instantaneous streamwise velocity at 0.35

*D*above the theoretical bed (0.15

*D*above the particle top). The measured fluctuating lift (

*F*′

_{L}) due to turbulent velocity fluctuations was found to follow the normal-error law.

*Wiberg and Smith*[1987] applied Bernoulli's equation to the bed particles, giving

*u*

_{t}and

*u*

_{b}are instantaneous velocities at the top and bottom of the particle, respectively, the flow velocity at the top of the particle being higher than that at the bottom.

*t*, the following relation proposed by

*van Radecke and Schulz-DuBois*[1988] was used:

*a*=

*ρAC*/

_{L}D d*dz*,

*d*/

*dz*is streamwise velocity gradient over a height of 1

*D*above the particle top,

*b*=

*ρAC*

_{D},

*C*

_{D}is coefficient of drag and is mean velocity in the streamwise direction 1

*D*upstream and 0.15

*D*above the particle top. The parameters

*u*′ and

*w*′ are the fluctuating streamwise and stream-normal velocities at the same location where is estimated. The first term on the right-hand side of equation (6) represents the Bernoulli's lift due to streamline contraction while the second term represents quasi-steady lift force due to vertical velocity fluctuations.

## 7. Statistical Distribution of Lift Force

*p*(

*F*

_{L})], as per equation (4), can be written as [

*Hofland*, 2005]

*σ*

_{u}is the standard deviation of the streamwise velocity at 0.15

*D*above the particle top. The streamwise and stream-normal fluctuating velocities

*u*′ and

*w*′ in equation (6) can be assumed to be normally distributed as shown in Figures 13a and 13b, respectively, for a flow depth

*H*= 208 mm and all exposures. Table 3 shows that the third- and fourth-order moments are nearly (but not exactly) Gaussian for all exposures. For a point in the boundary layer,

*u*′ and

*w*′ are usually anticorrelated [

*ρ*

_{uw}(

*τ*)] as seen in Figure 13c with maximum value at zero time lag (

*τ*= 0). The weaker anticorrelation at lower exposures are possibly due to the local flow effects and disturbances caused by the pocket geometry.

e (mm) |
/σ(u′)^{3} |
/σ(u′)^{4} |
/σ(w′)^{3} |
/σ(w′)^{4} |
---|---|---|---|---|

0 | −0.12 | 2.53 | 0.03 | 3.15 |

2.5 | −0.04 | 2.47 | 0.24 | 3.61 |

7.5 | −0.01 | 2.49 | −0.04 | 3.44 |

10 | −0.01 | 2.65 | 0.06 | 3.33 |

*au*′ and

*bw*′. The pdf of the lift force can be written as

*δ*is the Delta-Dirac function,

*σ*

_{w}is the standard deviation of stream-normal velocity, and

*ρ*

_{uw}is the correlation coefficient between

*u*′ and

*w*′. Applying the convolution theorem given by the following equation:

*H*= 208 mm and all exposures. Figures 14a–14d represent the pdf plots on normal scale, and Figures 14e–14g represent the same in log scale for exposure

*e*= 0, 2.5, 7.5, and 10 mm, respectively. It can be seen from Figure 14 that equation (7) does not match the measured lift for any of the exposures used in the experiments. However, equation (11), which predicts the lift force to be Gaussian, gives a good match with the measured data up to ±3

*σ*

_{F}

_{L}. Beyond this, the influence of higher-order nonlinear and non-Gaussian terms possibly becomes significant beyond the predictive ability of the proposed model.

*Hofland*[2005],

*Einstein and El-Samni*[1949], and

*Xingkui and Fontijn*[1993] also reported the experimentally measured lift force to be normally distributed between ±2

*σ*

_{F}

_{L}and found a positive deviation beyond that. Table 4 (/

*σ*(

*F*′

_{L})

^{4}column) shows that the kurtosis of the measured lift force is slightly higher than that for a normal distribution for all exposures, resulting in higher probability of occurrence of extreme events (beyond ±3

*σ*

_{F}

_{L}). It is also found that the skewness of the measured lift force increases with increase in exposure. This could mean that the quasi-steady lift force starts to act with increase in exposure yielding a skewed distribution in accordance with equation (7). The statistical analysis of lift force shows that equation (6) provides a better estimate of the fluctuating lift force for the bed particles, as compared to equation (4).

e (mm) |
/σ(F′_{L})^{3} |
/σ(F′_{L})^{4} |
---|---|---|

0 | 0.05 | 3.48 |

2.5 | 0.11 | 3.45 |

7.5 | 0.17 | 3.50 |

10 | 0.22 | 3.40 |

## 8. Spectral Analysis of Lift Force

*t*+

*τ*) is given by equation (6) as

*R*

_{F}

_{L}(

*τ*)] of lift force at a time lag

*τ*can be written using equations (6) and (12) as

*R*

_{uu}and

*R*

_{ww}are the autocorrelation coefficients of streamwise and stream-normal velocity and

*R*

_{uw}is the cross-correlation coefficient between them.

*S*

_{F}

_{L}(

*f*) is the autospectral density function of lift force at the frequency

*f*. Substituting equation (14) in equation (15), the autospectral density function of lift force is obtained as

*S*

_{uu}(

*f*) is the power spectral density of

*u*velocity,

*S*

_{ww}(

*f*) is the power spectral density of

*w*velocity and

*C*

_{uw}(

*f*) is the cospectrum (real part of the cross spectrum) of the

*u*and

*w*velocity.

*b*(hence

*C*) has been calculated assuming a quasi-steady behavior of the fluctuating drag force as [

_{D}A*Dwivedi et al.*, 2010]

*std*is standard deviation and

*F*

_{D}and

*C*

_{D}are drag force and drag coefficient, respectively. The numerator and denominator in equation (17) were estimated from force and velocity measurements, respectively. Table 5 shows the variation of

*C*of a bed particle as a function of mean exposure (

_{D}A*e*) and flow depth (

*H*). Owing to the complexity and practical limitations in determining the exposed area of a bed particle as well as its drag coefficient, the quantity

*C*is practically more useful than its individual constituents (

_{D}A*C*

_{D}and

*A*) since it avoids the difficulty associated with the determination of the area of a bed particle exposed to the flow.

*a*and

*C*for Different Exposures

_{L}Ae (mm) |
H (mm) |
z (mm) | u_{*} (m/s) |
(m/s) | C × 10_{D}A^{−4} (m^{2}) |
b (N s/m) |
d/dz (1/s) |
ρD d/dz (N s/m^{3}) |
a (N s/m) |
C × 10_{L}A^{−4} (m^{2}) |
C/_{D}AC_{L}A |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 208 | 11.4 | 0.081 | 0.429 | 1.232 | 0.027 | 17.30 | 692.0 | 0.103 | 1.50 | 1.217 |

2.5 | 208 | 16.5 | 0.065 | 0.403 | 1.363 | 0.028 | 9.61 | 384.4 | 0.077 | 2.0 | 1.467 |

7.5 | 208 | 21.5 | 0.056 | 0.383 | 1.702 | 0.033 | 6.35 | 254.0 | 0.076 | 3.0 | 1.763 |

10 | 208 | 24.5 | 0.046 | 0.329 | 1.468 | 0.024 | 4.58 | 183.2 | 0.064 | 3.5 | 2.380 |

0 | 158 | 11.4 | 0.076 | 0.451 | 1.218 | 0.027 | 16.26 | 650.4 | 0.096 | 1.48 | 1.215 |

2.5 | 158 | 16.5 | 0.068 | 0.465 | 1.234 | 0.029 | 10.05 | 402.0 | 0.072 | 1.80 | 1.458 |

7.5 | 158 | 21.5 | 0.052 | 0.389 | 1.555 | 0.030 | 5.90 | 236.0 | 0.065 | 2.74 | 1.762 |

10 | 158 | 24.5 | 0.040 | 0.312 | 1.488 | 0.023 | 3.98 | 159.2 | 0.057 | 3.55 | 2.390 |

0 | 108 | 11.4 | 0.072 | 0.471 | 1.309 | 0.031 | 15.40 | 616.0 | 0.098 | 1.59 | 1.214 |

2.5 | 108 | 16.5 | 0.063 | 0.469 | 1.361 | 0.032 | 9.31 | 372.4 | 0.074 | 1.99 | 1.462 |

7.5 | 108 | 21.5 | 0.055 | 0.445 | 1.671 | 0.038 | 6.24 | 249.6 | 0.073 | 2.94 | 1.759 |

10 | 108 | 24.5 | 0.042 | 0.353 | 1.338 | 0.024 | 4.18 | 167.2 | 0.054 | 3.20 | 2.390 |

[33] The unknown quantity *a* (hence *C _{L}A*) was thus calculated using equations (16) and (17) as the mean over the frequency range 1 to 10 Hz using the measured stream-normal gradient [

*d*/

*dz*] of the streamwise velocity over a distance of 1

*D*above the particle top. The calculated values of

*a*and

*C*are shown in Table 5 for different exposures and flow depth investigated in the study. It can be seen that the quantities

_{L}A*C*and

_{D}A*C*are nearly constant for a given exposure irrespective of the flow depth. The mean values of

_{L}A*C*,

_{D}A*C*and the ratio

_{L}A*C*/

_{L}A*C*shown in Table 5 for different exposures can be seen to increase with exposure. Similar observation has been reported by

_{D}A*Apperley*[1968] for the ratio

*C*

_{L}/

*C*

_{D}of spherical particles.

[34] While the quantity *C _{L}A* of a bed particle is not linearly related to its exposure, it can be inferred from Table 5 (z and

*u*

_{*}columns) that the variability of lift force fluctuations [

*std*(

*F*

_{L})] which decrease with increase in exposure at entrainment condition is less than the vertical gradient of streamwise velocity (

*ρD d*/

*dz*) over a distance of 1

*D*above the particle top. The decrease in velocity gradient with increase in exposure is almost double than that of lift force fluctuations such that the quantity

*C*compensates for this effect and hence increases with increase in exposure. It is to be noted that reduction in the quantity

_{L}A*C*at lower exposures does not imply a reduction in lift force; it indicates that the velocity gradient is much higher so that a particle at low exposures can only be entrained by lift force (i.e., Bernoulli's lift).

_{L}A[35] Table 6 shows the values of *C*_{D} and *C*_{L}. It is to be noted that *C _{D}A* for

*e*= 7.5 appears higher than that for

*e*= 10 meaning that there is a threshold protrusion level (

*e*= 7.5 in this study) beyond which the fluctuating drag coefficient falls off due to apparent dominance of the mean flow and hence mean drag. Also to be noted in Table 6 are that the values of

*C*

_{L}are greater than

*C*

_{D}values making ratio

*C*

_{L}/

*C*

_{D}greater than 1. This can be explained in terms of the predominance of Bernoulli's lift induced by a large-scale sweep-burst event that causes a strong vertical velocity gradient (

*du/dz*). Figure 15 shows flow structure just before particle entrainment. It can be seen from Figure 15 that large-scale sweep structure occurring just before particle entrainment.

*C*

_{L}/

*C*

_{D}of a Bed Particle for Different Exposures

e (mm) |
mean [C] × 10_{D}A^{−4} (m^{2}) |
mean [C] × 10_{L}A^{−4} (m^{2}) |
A(e) × 10^{−4} (m^{2}) |
mean [C]/_{D}AA(e) |
mean [C]/_{L}AA(e) |
C_{L}/C_{D} |
---|---|---|---|---|---|---|

10.0 | 1.43 | 3.42 | 5.321 | 0.2687 | 0.6427 | 2.392 |

7.5 | 1.64 | 2.89 | 4.297 | 0.3815 | 0.6742 | 1.762 |

2.5 | 1.32 | 1.93 | 2.547 | 0.5181 | 0.7577 | 1.462 |

0.0 | 1.25 | 1.52 | 1.721 | 0.7260 | 0.8848 | 1.219 |

[36] Figure 16 shows the relative contributions of the different terms on the right-hand side of equation (15) to the lift force fluctuations at different frequencies for all exposures. Figure 16 (left) shows the reduction in Bernoulli lift force fluctuations (Term 1) due to *u*′ with increase in exposure. Figure 16 (right) shows the relative magnitudes of the sum of Term 2 (quasi-steady lift force due to vertical velocity fluctuations) and Term 3 (cross correlation between Bernoulli and quasi-steady lift velocities) for all exposures at different frequencies. The magnitude of the sum of Term 2 and Term 3 is found to increase with decrease in exposure implying increased suppression of the lift force fluctuations below 10 Hz; thus counteracting the effect of Bernoulli's lift. However, Bernoulli's lift (Term 1) being the dominant term governs the spectral distribution of energy at different exposures.

[37] Figures 17a–17d show the comparison between the measured and predicted lift force spectra for exposures, *e* = 0, 2.5, 7.5, and 10 mm, respectively. It can be seen that the spectra predicted by the model closely follows the measured spectra for *e* = 7.5 and 10 mm in the range of frequencies 1–10 Hz with reasonable agreement below 1 Hz. However, the agreement is not so good at frequencies below 1 Hz for *e* = 0 and 2.5 mm. While the predicted spectra have been matched with the measured lift force spectra by adjusting the value of *a* in the frequency range 1–10 Hz for all exposures, the greater departure of the two spectra at frequencies below 1 Hz for particles at lower exposures could be explained by observing the frequency dependant coherence plots of lift force with *u* (*γ*_{uF}_{L}) and *w* (*γ*_{wF}_{L}) in Figure 18 for *e* = 0 and 10 mm. While the coherence function *γ*_{wF}_{L} is marginally higher for the highest exposure (10 mm) compared to the completely shielded particle for frequencies below 1 Hz, *γ*_{uF}_{L} for *e* = 10 mm is 2.5–3 times greater than the corresponding coherence function for the case of completely shielded particle (*e* = 0 mm) at lower frequencies. In other words, the larger eddies capable of producing extreme hydrodynamic forces are constrained by the presence of the wall. For particles at lower exposure such that the lift force phenomena completely departs from the quasi-steady process resulting in poor model predictions. The coherence is in fact better at midfrequency range (1–10 Hz) for *e* = 0 mm.

[38] This is unlike the drag force, for which the coherence equaled one for the no particle exposure case (not shown). This implies not only the non-quasi-steadiness of the lift phenomena to fluctuations in velocity, but also the possibility of the presence of more complicated higher order (or nonlinear) processes. Similar observations were made by *Schmeeckle et al.* [2007] that reported a poor coupling between the streamwise velocities and lift force unlike that between streamwise velocities and drag force for both smooth and gravel beds.

[39] Figure 19 shows the significant coherence plots of the velocity components (*u* and *w*) with the measured lift force. Significant coherence is estimated using equation proposed by *Shumway and Stoffer* [2000]. The significant coherence level of the two random signals (say *u* and *F*_{L}) at each frequency is calculated as follows:

[40] The sequence of signal *F*_{L} has been randomized, that is, the mean and RMS of the signal remains unchanged but the phase information is randomized (reshuffled/reordered). The coherence of this randomized signal with the undisturbed signal (*u*) is now calculated using the usual Fourier transform method. Using the 95%, confidence level, the probability density (using histogram) corresponding to the given value of confidence level gives the desired significant coherence value at that frequency

[41] Significant coherence values (*sig* [*u* (*γ*_{uF}_{L})] and *sig* [*w* (*γ*_{wF}_{L})]) for *e* = 0 are much smaller (around 0.02–0.05) than that for *e* = 10 mm (around 0.1–0.2) over most of the turbulence spectrum and thus helps to explain the departure of the lift force from quasi-steady effects (as normally observed with drag force). This low coherence between the lift force and the u and w velocity signals is also responsible for the relatively poor performance of the proposed velocity based model at lower exposures. Higher-order phenomena such as negligence of higher-order terms in the streamwise and stream-normal velocity, wake effects from upstream roughness blocks (balls in this case) as well as bursting events (such as sweep flow structure, shown in Figure 15) are supposedly responsible for this departure.

[42] In view of the poor predictions of the velocity based lift force model at lower exposures, a model based on different parameter such as pressure might be more appropriate but has not been pursued in this study. For a completely shielded particle, *Hofland* [2005] and *Schmeeckle et al*. [2007] have reported the concurrence of high-frequency positive and negative lift force fluctuations during times of low frequency increased positive streamwise velocity. Similar observations were made in this study, something that the velocity based linear model fails to capture as shown in Figures 20a and 20b. At higher exposures, the phenomena tend toward quasi-steadiness leading to improved model predictions.

[43] Figures 21a and 21b show the time series record for 45 s of the measured lift force as well as that predicted by equation (6) for *e* = 0 mm, while Figures 21c and 21d show the same for *e* = 10 mm. It can be seen that the magnitude of the Bernoulli's lift is orders of magnitude higher than that of the lift due to vertical velocity fluctuations for both exposures (Figures 21a and 21c) with better match obtained for *e* = 10 mm than for *e* = 0 mm.

## 9. Conclusion

[44] An experimental study of lift force on a spherical sediment particle at different exposures was carried out in a laboratory flume. The functional form of the theoretical pdf of lift similar to that of drag is inadequate and an improved expression for pdf based on normal error law is provided which matches the measured data up to ±3*σ*_{F}_{L} for all exposures. The skewness of the measured lift force is found to increase with increase in exposure. This could mean that the quasi-steady lift force starts to act with increase in exposure yielding a skewed distribution. In addition the kurtosis of the measured lift is found to be higher than that for a normal distribution for all exposures resulting in higher probability of occurrence of extreme events (beyond ±3*σ*_{F}_{L}). It is speculated that beyond ±3*σ*_{F}_{L}, the influence of nonlinear and non-Gaussian effects become significant beyond the capability of the proposed model.

[45] The fluctuating lift force model reported in the literature is used to develop the spectral density of lift force as a function of streamwise and stream-normal velocity and validated using the measured lift force data. In particular, the ratio of *C _{L}A*/

*C*is found to increase with increase in exposure. The fluctuating Bernoulli's lift is found to play a dominant role in generating high lift on bed sediment particles at lower exposures due to high-velocity gradient over the particle top.

_{D}A[46] The spectrum predicted by the proposed model is found to closely follow the measured spectrum for exposures *e* = 7.5 and 10 mm, while it over predicts at frequencies less than 1 Hz for *e* = 0 and 2.5 mm. The poor correlation between lift force and streamwise velocity at lower exposures is attributed to this apparent mismatch. The low correlations between lift force and streamwise velocity at lower exposures is caused by the concurrence of high-frequency lift force fluctuations, both positive and negative along with low frequency increased streamwise velocity; a phenomena observed in the measured lift force data. As a result, the time history plots of the velocity based predictive lift force model show a poorer match with the measured lift force at lower exposures compared to the particle at higher particle exposures. While predominance of fluctuating Bernoulli's lift over lift due to stream-normal velocity fluctuations is observed at all exposures, the highly nonlinear process of lift force generation at lower exposures gradually tends toward quasi-steady phenomena at higher exposures leading to better model predictions of lift force.