Sediment Threshold with Upward Seepage

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Sediment Threshold with Upward Seepage

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Subhasish Dey1 and Ulrich C. E. Zanke2 Abstract: An analytical model is presented to determine the threshold bed shear stress for noncohesive sediment motion subject to upward seepage on horizontal sedimentary bed under a stream flow. Hydrodynamic, seepage, and micromechanical forces acting on a solitary sediment particle, resting over a sedimentary bed under slip-spinning condition, are analyzed. The correlation coefficient between the results obtained using the present model and the experimental data of threshold bed shear stress with upward seepage on the horizontal bed is 0.767. It indicates that the model predicts satisfactorily the threshold bed shear stress with upward seepage. DOI: 10.1061/共ASCE兲0733-9399共2004兲130:9共1118兲 CE Database subject headings: Fluvial hydraulics; Hydrodynamics; Open channel flow; Sediment transport; Seepage; Streamflow.

Introduction When the bed shear stress induced by the stream flow over a loose sedimentary bed is just sufficient to begin the sediment motion, it is known as threshold condition. Shields 共1936兲 has been the pioneer to describe the threshold bed shear stress at which the sediment particles on a sedimentary bed are on the verge of motion by a unidirectional stream flow. Since then many investigators studied the beginning of sediment motion experimentally 共Miller et al. 1977; Buffington and Montgomery 1997兲, and a sizable number of mathematical models of sediment threshold were developed 共White 1940; Iwagaki 1956; Mingmin and Qiwei 1982; Wiberg and Smith 1987; Ling 1995; Zanke 1996, 2003; Dey 1999, 2003; McEwan and Heald 2001; Papanicolaou et al. 2002兲. When the stream flow is associated with an upward seepage through the porous sedimentary bed, the sediment particles experience an additional force in the form of seepage force resulting in a reduction of submerged weight of the sediment particles. Martin and Aral 共1971兲 investigated the instability of sediment particles on an inclined bed due to upward and downward seepage conditions to determine the seepage force acting on interfacial sediment particles. Oldenziel and Brink 共1974兲 reported that the submerged weight of sediment particles is reduced by an upward seepage and hence, the Shields parameter is modified. Cheng and Chiew 共1998兲 put forward the modified equation of velocity distribution in open channel subject to upward seepage. Also, Cheng and Chiew 共1999兲 derived the ratio of threshold shear velocity with upward seepage to that without seepage being a function of the ratio of hydraulic gradient of seepage to its threshold value under 1

Associate Professor, Dept. of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India. E-mail: [email protected] 2 Professor, Institut fu¨r Wasserbau und Wasserwirtschaft, Technische Univ. Darmstadt, Rundeturmstrasse 1, D-64283 Darmstadt, Germany. E-mail: [email protected] Note. Associate Editor: Michelle H. Teng. Discussion open until February 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on August 22, 2003; approved on February 10, 2004. This technical note is part of the Journal of Engineering Mechanics, Vol. 130, No. 9, September 1, 2004. ©ASCE, ISSN 0733-9399/2004/9-1118 –1123/$18.00.

quick condition. They conducted experiments for sediment threshold with upward seepage and concluded that the threshold shear velocity reduces with increase in seepage velocity. Cheng 共2003兲 used the generalized Ergun equation 共Ergun 1952兲 to analyze sediment threshold with upward seepage. He derived an alternative relationship between the hydraulic gradient and seepage velocity. This led to a generalized Ergun equation, being used for different flow regimes, and was then applied to the evaluation of the threshold shear velocity for the sediment particles subject to upward seepage. The aim of the paper is to present an analytical model on the threshold of noncohesive sediment motion with upward seepage on horizontal sedimentary beds under a stream flow. To verify the model, the experimental results of Cheng and Chiew 共1999兲 are used for comparisons. In this context, it is pertinent to mention that the upward seepage is important from the point of view of sediment threshold, because it reduces the effective threshold bed shear stress for sediment particles resulting in a motion of sediment under the flow having threshold bed shear stress lower than that obtained from the Shields diagram.

Model Forces Acting on Solitary Particle In a unidirectional stream flow over a sedimentary bed, the most stable three-dimensional configuration of a spherical solitary sediment particle of diameter D resting over three closely packed spherical particles of identical diameter d forming the sedimentary bed is shown in Fig. 1共a兲. Depending on the orientation of the bed particles, the solitary particle has a tendency either to roll over the valley formed by the two particles or to roll over the summit of a single particle due to the hydrodynamic forces. The forces acting on the solitary sediment particle are the downward force due to its submerged weight (F G ), seepage force (F S ), and hydrodynamic forces, which are drag force (F D ) and lift force (F L ), as shown in Fig. 1共a兲. When the solitary particle is about to dislodge downstream from its original position, the equation of moment about the point of contact 共M兲 of the solitary particle downstream is F D Z⫺ 共 F G ⫺F L ⫺F S 兲 X⫽0

(1)

where X and Z⫽horizontal and vertical lever arms, respectively. The submerged weight of the particle is

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For higher Reynolds number 共rough and transitional regimes兲, the solitary particle spins into the groove, formed by the three closely packed bed particles, just before dislodging downstream from its original position due to large velocity gradient 共differential velocity in the vertical direction due to considerable velocity difference between the bottom and top points of the solitary particle兲 at the particle level 共Dey 1999兲. To be more explicit, the hydrodynamic force acting on the upper portion of particle is significantly greater than that acting on the lower portion of particle, resulting in a turning moment to the particle. If the particle is round enough, it spins 共Ling 1995兲. The solitary particle, in a slip–spinning mode, acquires additional lift just before dislodging downstream from its original position. This type of motion may not prevail for all the sediment particles but the particles in a slip–spinning mode move earlier than the other particles 共Debnath 2000兲. This movement is found when the particles are spherical, as the experiments of Halow 共1973兲 revealed that spherical particles spin or roll, and angular particles slide in turbulent flow. Therefore, the inclusion of slip–spinning mode is significant in the analysis of the threshold of sediment motion. The lift force, caused by the spinning mode of particle, is termed lift due to Magnus effect (F Lm ). Rubinow and Keller 共1961兲 formulated it as F Lm ⫽␣ L ␳D 3 u m ␻ Fig. 1. Definition sketch: 共a兲 diagrammatic presentation of forces acting on spherical solitary particle under stream flow with upward seepage; 共b兲 top view of solitary particle resting over three closely packed bed particles; and 共c兲 tetrahedron formed joining centers of four particles

F G⫽

␲ 3 D 共 ␳ s ⫺␳ 兲 g 6

(2)

where ␳ s ⫽mass density of sediment particles; ␳⫽mass density of fluid; and g⫽gravitational acceleration. The drag force developed due to pressure and viscous skin frictional forces is given by F D ⫽C D

c b ⫹ R R2

(4)

where R⫽flow Reynolds number at particle level (⫽u m D/␯); ␯⫽kinematic viscosity of fluid; and a, b and c⫽coefficients dependent on R. The lift force, caused by the velocity gradient, in a shear flow is termed lift due to shear effect (F Ls ). For a sphere in a viscous flow, Saffman 共1968兲 proposed the following equation:

冉 冊

F Ls ⫽␣ L ␳D 2 u m ␯

⳵u ⳵z

F Lm ⫽0.5␣ L ␳D 3 u m

where ␣ L ⫽lift coefficient; ⳵u/⳵z⫽velocity gradient; and u⫽flow velocity at z. The value of ␣ L proposed by Chepil 共1958兲 is 0.85C D .

(7)

冉 冊冋 ⳵u ⳵z

0.5

␯ 0.5⫹0.5D

冉 冊册 ⳵u ⳵z

0.5

(8)

The seepage force F S acting on the solitary particle is given by F S ⫽0.5␳

v 2s

␳ 20

ae

(9)

where v s ⫽seepage velocity; ␳ o ⫽porosity of sediment particles; and a e ⫽effective area of solitary particle receiving the seepage force. The value of porosity ␳ o is assumed 0.4, in this study. In fact, the seepage force F S acting on the bottom portion of the solitary particle at the circular area, whose perimeter is touching the contact points (T 1 , T 2 , and T 3 ) 关see Fig. 1共c兲兴 of the bed particles, is due to the stagnation pressure resulting from an effective seepage velocity of magnitude v s /␳ o through the central space of three bed particles. Hence, the area a e is expressed as ␲(T 1 P) 2 . The expression of T 1 P is given in the succeeding section 关see Eq. 共12兲兴. The seepage velocity v s can be calculated solving Ergun 共1952兲 equation, which is

0.5

(5)

⳵u ⳵z

Saffman 共1965兲 showed analytically that for viscous flow, F Lm might be ignored in the force analysis, as F Ls ⬎F Lm . Nevertheless, the magnitude of ␣ L , later found by Saffman 共1968兲, varies over a wide range. Consequently, F Lm becomes significant in the calculation of total lift force F L . The total lift force F L , a combination of F Ls and F Lm , is expressed as

(3)

where C D ⫽drag coefficient; u m ⫽mean flow velocity received by the effective frontal area 共effective projected area of the particle being right angles to the direction of flow兲 of the solitary particle; and ␰⫽fraction to determine the effective frontal area of the solitary particle. The empirical equation for drag coefficient C D given by Morsi and Alexander 共1972兲 is used here. It is C D ⫽a⫹

where ␻⫽angular velocity of spinning particle. According to Saffman 共1965兲, the maximum angular velocity achieved by a solitary particle equals 0.5⳵u/⳵z. Thus, Eq. 共6兲 is rewritten as

F L ⫽␣ L ␳D 2 u m

␲ 2 2 D ␳u m ␰ 8

(6)

i⫽

共 1⫺␳ o 兲v s

g␳ 3o ␺d



150共 1⫺␳ o 兲

␯ ⫹1.75v s ␺d



(10)

where i⫽hydraulic gradient of seepage; and ␺⫽sphericity of sediment particles.

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Determination of Lever Arms (X and Z) A tetrahedron (OO 1 O 2 O 3 ) is formed joining the centers of the three bed particles and the solitary particle 关Fig. 1共c兲兴. T 1 , T 2 , and T 3 are the contact points of the solitary particle with the three bed particles. Depending on the streamwise orientation of the bed particles, moment is usually taken about T 2 T 3 or T 1 . Accordingly the horizontal lever arm 共X兲 is either PS or T 1 P. Thus, one can write 1

Dd PS⫽ D⫹d 2 冑3

(11)

sider that an equal distribution of the orientations of bed particles within the said two extreme cases prevails. It is important to recognize that the horizontal lever arm for any orientation must lie between PS and T 1 P. Thus, the horizontal lever arm X is averaged as X⫽0.5共 T 1 P⫹ PS 兲 ⫽␭

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Z⫽ Dd T 1 P⫽ 冑3 D⫹d

(13)

where ␭⫽0.433. On the other hand, the vertical lever arm Z (⫽O P), being independent of the orientation of the bed particles, is given by

and 1

Dd D⫹d

1

D 共 3D 2 ⫹2Dd⫺d 2 兲 0.5 2 冑3 D⫹d

(14)

(12)

Since the sedimentary bed is formed by a large number of sediment particles, their orientations with respect to the direction of stream flow are numerous. Therefore, it is appropriate to con-

␶ˆ ⫽ 2 C D ␲uˆ m ␰ 共 3⫹6dˆ ⫺dˆ 2 兲 0.5⫹6␭dˆ



Equation of Sediment Threshold Using Eqs. 共2兲, 共3兲, 共8兲, 共9兲, 共13兲, and 共14兲 into Eq. 共1兲, the equation of sediment threshold in nondimensional form is obtained as

2␲␭dˆ

␣ L uˆ m

冑 冉冑* 冑 ⳵uˆ 2 ⳵zˆ

dˆ ⫹ R



2 ⳵uˆ ␲ Rs • ⫹ ⳵zˆ 3 ␳ 2 R2 o

*

冉 冊册 dˆ

2

(15)

1⫹dˆ

where ␶ˆ ⫽Shields parameter or nondimensional threshold bed shear stress, that is ␶/ 关 (␳ s ⫺␳)gD 兴 ; ␶⫽threshold bed shear stress; uˆ m ⫽u m /u * ; u * ⫽threshold shear velocity (⫽ 冑␶/␳); uˆ ⫽u/u * ; dˆ ⫽d/D; R*⫽particle Reynolds number (⫽u * d/␯); and zˆ ⫽z/D. The accuracy of the results obtained from the model is highly dependent on the accurate determination of dˆ which is not an easy task through field measurement of a sedimentary bed, because the range of variation of dˆ may be from 0.1 to 10. To avoid this difficulty, dˆ is determined from the information on angle-of-repose of sediment particles, using the expression given by Ippen and Eagleson 共1955兲 for the spherical sediments as

⬎1.4), due to its process of deposition or armoring, particles of variable diameters prevail in the upper layers. In this study, ␤⫽0.25 is considered, as was done by van Rijn 共1984兲.

2 tan ␾ 关 6 tan ␾⫹ 共 48 tan2 ␾⫹27兲 0.5兴 dˆ ⫽ 4 tan2 ␾⫹9

where A⫽effective frontal area of the solitary particle exposed to the flow, that is (␲D 2 /4)␰; ␰⫽ 兵 1⫺arccos(1⫺2hˆ)⫹2(1⫺2hˆ)关hˆ(1 ⫺hˆ)兴0.5其 ; hˆ ⫽h/D; h⫽vertical distance between the bottom level of solitary particle and the zero-velocity level, that is ␧-␦; and ␧⫽vertical distance between the bottom level of solitary particle or zero-velocity level and the virtual bed level. The nondimensional mean velocity of flow uˆ m is obtained as

(16)

where ␾⫽angle-of-repose, that is the angle between the gravity force and the radius to the point of contact.

Determination of Position of Virtual Bed Level In the present study, the virtual bed level is considered to be at a depth of ␤d below the top level of the bed particles. Here, ␤ is a factor being less than unity. Therefore, the vertical distance ␦ between the virtual bed level and the bottom level of the solitary particle is ␦⫽

1 2 冑3

共 3D 2 ⫹6Dd⫺d 2 兲 0.5⫺

1 共 D⫹d 兲 ⫹␤d 2

Determination of uˆ m and ⵲ uˆ Õ⵲ zˆ The mean velocity of flow received by the effective frontal area of the solitary particle is given by u m⫽

2 uˆ m ⫽ ˆ A



D⫹␦





u 关共 z⫺␦ 兲共 D⫹␦⫺z 兲兴 0.5dz

ˆ 1⫹␦

ˆ⑀

uˆ 关共 zˆ ⫺␦ˆ 兲共 1⫹␦ˆ ⫺zˆ 兲兴 0.5dzˆ

(18)

(19)

where Aˆ ⫽A/D 2 ; ␦ˆ ⫽␦/D; and ␧ˆ ⫽␧/D. The velocity gradient ⳵u/⳵z can be obtained as follows: ⳵u 1 ⫽ ⳵z D⫹␦⫺␧

(17)

The geometric standard deviation ␴ g of the particle size distribution given by 冑d 84 /d 16 is less than 1.4 for uniform sediments 共Dey et al. 1995兲. Here, it is considered that D equals d 50 for uniformly graded sediments. For nonuniform sediments (␴ g

2 A



D⫹␦



u D⫹␦ ⫺u ␧ ⳵u dz⫽ ⳵z D⫹␦⫺␧

(20)

Thus, the nondimensional velocity gradient ⳵uˆ /⳵zˆ is given by

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⳵uˆ uˆ 1⫹␦ˆ ⫺uˆ ␧ˆ ⫽ ⳵zˆ 1⫹␦ˆ ⫺␧ˆ

(21)

Table 1. Summary of Experimental Data of Cheng and Chiew 共1999兲

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Median sediment size d 50 共mm兲 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63

Seepage velocity v s 共cm/s兲

Threshold shear velocity u * 共cm/s兲

Nondimensional threshold bed shear stress ␶ˆ

0.70 0.73 0.68 0.70 0.98 1.23 1.30 1.38 1.33 0.99 0.37 1.30 1.18 0.62 0.65 0.62 0.57 0.58 0.36 0.25 0.33 0.31 0.57 0.54 0.55 0.41 0.45 0.11 0.12 0.10 0.11 0.12 0.13 0.14 0.13 0.12 0.14 0.14 0.16 0.17 0.11 0.16 0.16 0.16 0.12

2.17 1.81 2.49 2.07 1.52 0.76 0.87 1.09 1.32 1.68 2.50 0.99 1.16 0.65 0.30 0.44 0.56 0.72 1.17 1.49 1.34 1.55 0.98 0.91 0.91 1.33 1.37 0.71 1.12 1.06 0.41 0.92 0.90 0.68 0.50 1.00 0.70 0.60 0.38 0.25 1.05 0.79 0.66 0.48 1.12

0.01492 0.01038 0.01964 0.01358 0.00732 0.00183 0.00240 0.00376 0.00552 0.00894 0.01980 0.00311 0.00426 0.00256 0.00055 0.00117 0.00190 0.00314 0.00829 0.01345 0.01088 0.01455 0.00582 0.00502 0.00502 0.01071 0.01137 0.00494 0.01230 0.01102 0.00165 0.00830 0.00794 0.00453 0.00245 0.00981 0.00481 0.00353 0.00142 0.00061 0.01081 0.00612 0.00427 0.00226 0.01230

Fig. 2. Comparison of values of nondimensional threshold bed shear stress ␶ˆ computed using present model with experimental data of Cheng and Chiew 共1999兲

␩⫽ ␩⫽

冉冊

冉冊



(23a)

*

8.5 8.5␴ ⫺3 exp共 ⫺0.11 ln2.5 R 兲 2.5 ln R ⫹ * 1⫹␴ * 1⫹␴

2 uˆ m ⫽ ˆ kA

R ⬍70

*

(23b)

冕 冋冉 冊 ˆ 1⫹␦

ˆ⑀

冉 冊册

zˆ zˆ ␴ ln ˆ ln ˆ ⫹ z0 4k z0

关共 zˆ ⫺␦ˆ 兲共 1⫹␦ˆ ⫺zˆ 兲兴 0.5dzˆ

(24) where ␧ˆ ⫽zˆ 0 if (zˆ 0 ⫺␦ˆ )⭓0; and ␧ˆ ⫽␦ˆ if (zˆ 0 ⫺␦ˆ )⬍0. The Simpson’s rule is applied to solve Eq. 共24兲. The velocity gradient obtained using Eq. 共21兲 is





⳵uˆ ␴ 2 共 1⫹␦ˆ 兲 ⑀ˆ 1 共 1⫹␦ˆ 兲 1⫹ ln ln ⫽ ⳵zˆ k 共 1⫹␦ˆ ⫺␧ˆ 兲 4k ␧ˆ zˆ 20

(25)

Results

The velocity distribution subject to upward seepage given by Cheng and Chiew 共1998兲 is zˆ zˆ 1 ␴ ln ˆ ⫹ 2 ln ˆ k z0 z0 4k

R ⭓70

In this study, k s is assumed to be d, as was done by Wiberg and Smith 共1987兲. The mean velocity uˆ m determined using Eq. 共19兲 is

Velocity Distribution Subject to Upward Seepage

uˆ ⫽



8.5 1⫹␴

(22)

where k⫽von Karman constant 共⫽0.4兲; zˆ 0 ⫽z 0 /D; z 0 ⫽zerovelocity level, that is k s exp(⫺k␩); k s ⫽equivalent roughness height of Nikuradse; ␴⫽ v s /u ; and *

The equations developed in the preceding section were implemented in a computer program that provided a solution for the nondimensional threshold bed shear stress ␶ˆ for the beginning of sediment motion subject to upward seepage on horizontal sedimentary beds. The experimental data of Cheng and Chiew 共1999兲 given in Table 1 are used to verify the model. It is interesting to note that for d 50⫽1.95, 1.02, and 0.63 mm, the values of ␶ˆ obtained from the Shields diagram 共without upward seepage兲 being 0.04, 0.033, and 0.03, respectively, are greater than ␶ˆ with upward seepage given in Table 1. The values of ␶ˆ are compared using the present model for d 50 and v s given in Table 1. As the sediments

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used by Cheng and Chiew 共1999兲 were uniform, dˆ is assumed unity. The comparison of the values of ␶ˆ obtained using the present model with the experimental data of Cheng and Chiew 共1999兲 is shown in Fig. 2. The value of correlation coefficient between experimentally obtained and computed ␶ˆ is 0.767. It indicates that the model corresponds to the experimental data. The criterion for the threshold of sediment motion depends on how tightly the sedimentary bed is packed, which may vary from nosphere 共flat bed兲 to tightly packed 共compact sedimentary bed兲 conditions. Moreover, it is always difficult to determine the real beginning of sediment movement, owing to the several definitions of sediment threshold 共Kramer 1935; Dey 1999兲. These are perhaps the principal reasons for scatter in the experimental data 共Fig. 2兲. Nevertheless, the present model does a satisfactory job of estimating ␶ˆ with the experimental data of sediment threshold subject to upward seepage on horizontal bed.

Conclusions The threshold of noncohesive sediment motion on a loose horizontal sedimentary bed subject to upward seepage under a stream flow has been modeled analytically using the basic concept of hydrodynamics and micromechanics. Experimental data of sediment threshold with upward seepage reported by Cheng and Chiew 共1999兲 have been used to compare the results obtained from the present model. The model shows a satisfactory agreement with the experimental data.

Acknowledgment This investigation was carried out at the Institut fu¨r Wasserbau und Wasserwirtschaft, Technische Universita¨t Darmstadt, Germany, during the visit of the first writer under the Deutscher Akademischer Austauschdienst 共DAAD兲 program.

R R * u u * uˆ uˆ m um

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

vs X Z zˆ z0 zˆ 0 ␣L ␤,␭,␨ ␦

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

␦ ⫽ ␧ ⫽ ␧ˆ ⫽ ␯ ⫽ ␰ ⫽ ␳ ␳o ␳s ␴ ␶ ␶ˆ

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

␾ ⫽ ␺ ⫽ ␻ ⫽

flow Reynolds number at particle level 关M0 L0 T0兴; particle Reynolds number 关M0 L0 T0兴; flow velocity at z 关 L T⫺1 兴 ; threshold shear velocity 关L T⫺1兴; u/u 关 M0 L0 T0 兴 ; * u m /u 关 M0 L0 T0 兴 ; * mean flow velocity received by effective frontal area of solitary particle 关L T⫺1兴; seepage velocity 关L T⫺1兴; horizontal lever arm 关L兴; vertical lever arm 关L兴; z/D 关 M0 L0 T0 兴 ; zero-velocity level 关L兴; z 0 /D 关 M0 L0 T0 兴 ; lift coefficient 关M0 L0 T0兴; factors 关M0 L0 T0兴; vertical distance between bottom level of solitary particle and virtual bed level 关L兴; ␦/D 关 M0 L0 T0 兴 ; vertical distance between bottom level of solitary particle or zero-velocity level and virtual bed level 关L兴; ␧/D 关 M0 L0 T0 兴 ; kinematic viscosity of fluid 关L2 T⫺1兴; fraction to determine effective frontal area of solitary particle 关M0 L0 T0兴; mass density of fluid 关M L⫺3兴; porosity of sediment particles 关M0 L0 T0兴; mass density of sediment particles 关M L⫺3兴; v s /u 关 M0 L0 T0 兴 ; * threshold bed shear stress 关M L⫺1 T⫺2兴; nondimensional threshold bed shear stress 关M0 L0 T0兴; angle of repose of sediment particles 关M0 L0 T0兴; sphericity of sediment particles 关M0 L0 T0兴; and angular velocity of spinning particle 关T⫺1兴.

Notation References The following symbols are used in this technical note: A ⫽ effective frontal area of solitary particle exposed to flow 关L2兴; Aˆ ⫽ A/D 2 关 M0 L0 T0 兴 ; a e ⫽ effective area of solitary particle receiving seepage force 关L2兴; C D ⫽ drag coefficient 关M0 L0 T0兴; D ⫽ diameter of solitary particle 关L兴; d ⫽ diameter of particles forming sedimentary bed 关L兴; dˆ ⫽ d/D 关 M0 L0 T0 兴 ; F D ⫽ drag force 关M L T⫺2兴; F G ⫽ submerged weight of solitary particle 关M L T⫺2兴; F L ⫽ total lift force 关M L T⫺2兴; F Lm ⫽ lift due to Magnus effect 关M L T⫺2兴; F Ls ⫽ lift due to shear effect 关M L T⫺2兴; F S ⫽ seepage force 关M L T⫺2兴; g ⫽ gravitational acceleration 关L T⫺2兴; h ⫽ vertical distance between bottom level of solitary particle and zero-velocity level 关L兴; hˆ ⫽ h/D 关 M0 L0 T0 兴 ; i ⫽ hydraulic gradient of seepage 关M0 L0 T0兴; k ⫽ von Karman constant 关M0 L0 T0兴; k s ⫽ equivalent roughness height of Nikuradse 关L兴;

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