Critical shear stress for incipient motion of a particle on a rough bed

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jf002208, 2012 Critical shear stress for incipient motion of a particle on a rough bed Hyungoo Lee 1 and S. Balachandar 1 Received 31 August 2011; revised 24 January 2012; accepted 27 January 2012; published 17 March [1] The main objectives of the present study are to obtain improved models of hydrodynamic forces and torque on a particle sitting on a bed and to use these models for the investigation of incipient motion and resuspension of particles. The improved models for force and torque are obtained from numerical simulations of a particle sitting on a bed with a turbulent flow of logarithmic mean velocity profile approaching the particle. Since the mean turbulent velocity profile can depart from the logarithmic profile in case of macroscale rough beds or flow down steep slopes, we have also considered forces and torque on a particle due to both linear and uniform mean flow profiles. The computed drag and lift coefficients and the predicted critical shear stress for incipient particle motion and resuspension are compared against available experimental results. The improved force and torque models are also used to evaluate the effect of turbulent velocity fluctuations on the critical shear stress for incipient motion and resuspension. The present results are of direct relevance to cases where the particle is mostly exposed to the ambient flow. In cases where the particle protrusion is small and is submerged mostly within a pocket of other particles, the above formulation can be used with a redefined area of exposure and flow velocity seen by the particle. However, the drag, lift, and torque coefficients and the resisting forces will be influenced by the partial exposure and the details of pocket geometry, which require further investigation. Citation: Lee, H., and S. Balachandar (2012), Critical shear stress for incipient motion of a particle on a rough bed, J. Geophys. Res., 117,, doi: /2011jf Introduction [2] The incipient motion of a particle on a streambed or a rough surface is often predicted by means of a dimensionless critical shear stress, called theshields parameter, which is defined as t B ¼ ~t B = ~r p ~r f ~g d ~, where ~t B is the critical shear stress, ~r p and ~r f are the density of the particle and the surrounding fluid, ~g is the gravitational acceleration and d ~ is the diameter of the particle. Hereinafter the tilde denotes a dimensional variable to distinguish from the dimensionless counterpart. Extensive research in this area performed over the past eight decades has been well reviewed by Buffington and Montgomery [1997], Dey and Papanicolaou [2008], and Garcia [2008]. Most investigators have studied the critical shear stress using experimental or theoretical methods. In particular, Wiberg and Smith [1987], Ling [1995], Wu and Chou, [2003], Vollmer and Kleinhans [2007], Coleman and Nikora [2008], Gregoretti [2008], and Giménez-Curto and Corniero [2009] have conducted theoretical investigations of incipient particle motion. 1 Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida, USA. Copyright 2012 by the American Geophysical Union /12/2011JF [3] The theoretical prediction of the critical condition for incipient motion is based on a force or momentum balance between the destabilizing hydrodynamic drag and lift forces against the resisting gravitational and frictional forces. Apart from geometric details such as bed slope, particle exposure and pocket geometry, accurate prediction of incipient motion requires precise knowledge of the hydrodynamic drag and lift forces acting on the particle. The drag and lift forces are expressed as ~F D ¼ ~r f ~AC D ~u 2 =2 and ~F L ¼ ~r f ~AC L ~u 2 =2, where à is the cross-sectional area of the particle exposed to the oncoming flow, ũ is the velocity of the oncoming flow and C D and C L are the drag and lift coefficients. For a particle fully exposed to the flow the area of cross section can be unambiguously defined and ũ is typically taken to be the undisturbed velocity approaching the center of the particle. [4] The conventional force balance relations have generally taken the drag and lift coefficients to be constants, and this approximation is appropriate in the limit of large particle Reynolds number. At smaller Reynolds numbers (Re), however, both C D and C L are substantially larger and exhibit strong Reynolds number dependence. The Reynolds number dependence of C D is often taken to be given by the standard drag model of Clift and Gauvin [1970] or other similar forms [see Marsh et al., 2004]. Similarly C L can be taken to be the shear lift model of Saffman [1965]. However, these models are valid only for uniform (d~u=d~y = 0) and linear shear (d~u=d~y = constant) flows away from the influence of any 1of19

2 nearby surface, and thus are not appropriate for the case of a particle on a bed. In the limit of a smooth bed it has been theoretically established that the drag coefficient increases 70% over its free fall value due to the effect of the wall [Goldman et al., 1967b]. The analysis of Goldman et al. [1967b] is for Stokes flow, but similar strong influence of the wall on drag force was observed at finite Re as well [Zeng et al., 2009]. In the case of lift force, the influence of the bed is very strong and can increase C L by even an order of magnitude over its value in an unbounded shear flow [Lee and Balachandar, 2010]. [5] The importance of turbulent fluctuations on the incipient motion and resuspension of particles has been well recognized. Since the oncoming flow is turbulent, even if the mean shear stress is below the critical value, instantaneous wall shear stress can exceed the threshold. During intermittent events such as bursts and sweeps, the turbulent flow significantly departs from the time-averaged mean, and such events have been well documented to strongly contribute to incipient particle motion and suspension [Fenton and Abbott, 1977]. There have been recent state-of-the-art timeresolved simultaneous measurements of drag and lift forces on the particle and velocity components of the oncoming flow [Apperley and Raudkivi, 1989; Hofland, 2005; Schmeeckle et al., 2007; Dwivedi et al., 2010a, 2010b, 2011]. These measurements and associated analysis have provided a detailed understanding of mean and fluctuating forces and their relation to the approaching turbulent flow. [6] Real beds are heterogeneous with randomly positioned particles of varying size and as a result the relative protrusion of a particle above the bed (particle exposure) is a random variable. This and additional factors including grain shape, pocket geometry and bed geometries are known to influence the hydrodynamic forces on the particle, but more importantly they strongly alter the resistive forces countering particle motion [Kirchner et al., 1990; Johnston et al., 1998; Schmeeckle and Nelson, 2003]. [7] The main objectives of the present study are to obtain improved models of drag, lift and torque coefficients appropriate for a particle sitting on a bed and use these improved models for the investigation of incipient rolling motion of the particle. We will also use the improved lift model to consider the balance of vertical force in the prediction of critical shear stress for particle resuspension. The improved C D and C L models are obtained from numerical simulations of a particle sitting on a wall subjected to mean turbulent boundary layer flow. The size of the particle relative to the thickness of the boundary layer is varied. The simulations range from a small particle of d + (particle diameter in wall units) equal to 2 that is entirely within the viscous sublayer to a large particle of d + = 500 that extends well into the turbulent log layer. The corresponding particle Reynolds number varies from 2 to about [8] The Reynolds number dependent C D and C L models obtained from the present simulation will be used to evaluate expected drag and lift force fluctuations in terms of velocity fluctuations. The computed mean forces and their estimated fluctuation will be compared against the experimental measurements as appropriate. The Reynolds number dependent drag and lift coefficients to be presented here are most relevant for the case of a particle nearly fully exposed to the oncoming flow. In the case of particles that are partly submerged and shielded within a packet, the present results on C D and C L can be used to evaluate the drag and lift forces with appropriate values for cross-sectional area (Ã) and flow velocity (ũ). This simple approach, although easy to implement, will become less accurate with decreasing protrusion, as the geometric details of the particle and the pocket can be expected to have a direct influence on the drag and lift coefficients. For example, the recent work by Chan-Braun et al. [2011] presents forces on fully submerged particles subjected to a well-resolved turbulent open channel flow. [9] Also, it has been shown that the mean turbulent velocity profile can depart from the logarithmic profile, for example, in the case of macroscale roughness of the bed, or flow down steep slopes [Shvidchenko and Pender, 2001; Nikora et al., 2001; Armanini and Gregoretti, 2005]. In order to study the sensitivity of the drag and lift coefficients to the shape of the mean turbulent velocity profile, we also consider C D and C L on a particle sitting on a bed subjected to (1) a linear velocity profile and (2) a uniform velocity profile. These two additional profiles can also be considered as limiting cases of the logarithmic profile. When the particle is smaller than the thickness of the viscous sublayer of the oncoming turbulent flow, then the wall-bounded linear velocity applies. In the other limit of a very large particle extending well into the log layer of the oncoming turbulent flow, the mean flow seen by the particle can be approximated as a near uniform flow. [10] Finally, the present numerical simulations allow us to obtain a model for the hydrodynamic torque on a particle sitting on a bed. In the moment balance, in addition to the drag and lift forces, we have included the contribution of the hydrodynamic torque in the estimation of the critical shear stress for incipient motion. The importance of the various contributions to overall incipient particle motion and resuspension are analyzed. 2. Theoretical Background [11] A spherical particle sitting on a bed will tend to move (roll and/or slide on the bed) by the combined action of hydrodynamic forces, gravity, frictional and other interaction forces with the bed. Here we assume the rough bed to consist of spherical sediments, shown schematically as the dashed circle in Figure 1. The mobile particle and the bed sediment contacts at a vertex A as shown in the two dimensional projection. The vertical and horizontal distances from the particle center to the pivot A are ~ d V ¼ d=2 ~ ~ and ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d H ¼ ~ d ~ ~, where ~ is the height of the pivot from the base of the particle. The angle of repose f can be expressed as a function of the ratio of the bed roughness to the particle size ( ~ k s = d) ~ from their geometric relationship as! f ¼ cos 1 2~ ¼ cos ~ 1 1 ks 1 þ ~ k s = d ~ : ð1þ [12] Equation (1) is similar to that by Wiberg and Smith [1987] except for the empirical constant z * used in their paper, where z * was the average elevation of the bottom of the almost moving particle from the bed, which depended on the particle s sphericity and roughness. They adopted a small 2of19

3 plane passing through the center of the particle and therefore take the spanwise component of the force and the streamwise and wall normal components of the torque on the particle to be zero. An incipient motion of the particle can be determined by the net moment, ~M A in (6), about the pivot point A : ~M A ¼ ~T þ ~F D ~ dv ~F g ~F ~dh L ; ð6þ where the gravitational force (~F g ) depends on the specific gravity of the particle x ¼ ~r p ~r f =~r f as Figure 1. Schematic diagram of a mobile particle and a bed grain. The diameters of the particle and the grain are ~d and ~ k s, respectively. The hydrodynamic drag (~F D ) and lift (~F L ) forces and torque (~T), as well as the gravitational force (~F g ) acting on the particle, are represented. value of z * = 0.02 in their calculations and here we have ignored z * by setting it zero (see Figure 1). For the case of uniformly sized particle and bed sediment ( ~ k s = ~ d ¼ 1), the angle of repose can be determined from (1) as f =60, which is consistent with Miller and Byrne s [1966] experiment for a nearshore natural sand (f 61.9 ) and Wiberg and Smith s [1987] calculation that f Note that we consider a horizontal bed and as a result the gravitational acceleration directs normal to the wall. Appropriate modifications can be made for the case of a sloping bed. In the above we have used the following geometric relation between the three different length scales ( ~ d, ~ k s and ~): ~k s ~d ¼ 2~=~ d 1 2~= ~ d : ð2þ 2.1. Moment Balance [13] As the flow goes around the particle the distribution of pressure (~p) and shear stress (~t) around the particle results in a net force on the particle. The ambient shear and the presence of the wall breaks the symmetry about the particle center along the wall normal direction and as a result the line of action of the net force in general will not pass through the center of mass (marked O in Figure 1). This hydrodynamic interaction of the surrounding flow with the particle can also be expressed in terms of drag force (~F D ), lift force (~F L ) and torque (~T), which are defined as Z ~F D ¼ ð~pn þ n ~t Þd~S e x ð3þ Z ~F L ¼ ~S ~S Z ~T ¼ ð~pn þ n ~t Þd~S e y ð4þ ~S ð~r ~t Þd~S e z ; ð5þ where ~S denotes the surface of the sphere, n is the unit normal vector to the surface of the sphere, and e x, e y and e z are unit vectors along streamwise, wall normal and spanwise directions. We limit to conditions of symmetry about the x-y ~F g ¼ p 6 ~r f x ~g ~ d 3 : ð7þ Note that the torque can be equivalently replaced by displacing the line of action of the drag force upward by ~T=~F D above the center of mass (O). [14] If ~M A > 0 the particle will start to roll about the vertex A by lifting off the bed, while for ~M A < 0 the particle will remain at the current location in contact with the bed. Therefore, the critical condition of incipient motion can be predicted by solving the following moment balance equation where the distances d V and d H and the gravitational force (7) have been substituted into equation (6): d ~ ~T þ ~F D! 2 ~ p 6 ~r f x ~g d ~ 3 ~F q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ~ d ~ ~ ¼ 0: The above moment balance following Wiberg and Smith [1987] is based on median pocket geometry. In a real bed consisting of particles of different sizes, the pocket geometry, particle exposure and as a result both the hydrodynamic forces promoting particle motion and resistive forces opposing particle motion will vary over the bed. The complexity of the problem and variability of the resisting forces has been well discussed by Schmeeckle and Nelson [2003] [also see Kirchner et al., 1990; Johnston et al., 1998; Shvidchenko and Pender, 2001]. [15] Accurate expressions for the forces ~F D, ~F L and torque ~T on the particle are still required to determine the critical threshold condition from equation (8). The dimensional forces and torque can be nondimensionalized and expressed in terms of dimensionless drag, lift and moment coefficients (C D, C L and C M )as C D ¼ ~F D = p 8 ~r f ~u2 f ~ d 2 ; C L ¼ ~F L = p 8 ~r f ~u2 f ~ d 2 C M ¼ ~T= p 16 ~r f ~u 2 f ~ d 3 ; and where ũ f is the velocity of the oncoming fluid at the particle center and the area of cross section has been chosen to be ~A ¼ p ~ d 2 =4 appropriate for a particle of full exposure. As discussed in particle-laden flow literature [see Crowe et al., 1998] the advantage of the above definitions along with the choice of Reynolds number as Re f ¼ ~u f ~ d=~n (~n is the kinematic viscosity of the fluid) is that C D, C L and C M are now functions of only Re f. For particles that are not fully exposed to the flow, an ideal scenario is to identify appropriate definitions of Ã, ũ f and Re f so that C D, C L and C M are still functions of only Re f. Unfortunately we do not fully ð8þ ð9þ 3of19

4 Figure 2. Mean turbulent velocity profiles in an open channel flow for different bed roughnesses, k + s. Dashed lines show fluid velocity for a rough bed from equation (16). Solid lines show turbulent velocity profiles of viscous sublayer and logarithmic region for a hydraulically smooth bed. understand forces and moments on a partly exposed particle to develop such unique definitions of à and ũ f. Substituting (9) into (8), and dividing by p~r f ~u 2~ f d 3 =8, one can obtain the nondimensional moment balance 1 2 C M þ 1 2 ~ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ C d ~ D ~d 1 ~ d ~ 4 3 x ~g d ~! ~u 2 C L ¼ 0; ð10þ f where again the roughness ratio can be replaced as ~ ~d ¼ Re 2=3 ; ð11þ Re p in terms of particle and roughness height Reynolds numbers, which are defined as qffiffiffiffiffiffiffiffiffiffiffi g x d ~ 3 pffiffiffiffiffiffiffiffiffiffi g x~ 3 Re p ¼ ; and Re ¼ : ð12þ ~n ~n The final dimensionless form of the moment equation which governs the incipient motion of the particle is given as " 1 2 C M þ 1 2 Re # 2=3 C D Re p vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 " Re 2=3 1 Re # u 2=3 t Re 2 p 3 Re p Re p Re f vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " Re 2=3 þ 1 Re # u 2=3 t C L ¼ 0: ð13þ Re p Re p [16] The above approach based on balance of moments about a pivot is somewhat different from the approach pursued in other investigations such as the one by Wiberg and Smith [1987], where a wall friction is used in a streamwise force balance. But a simple comparison presented in Appendix A shows their equivalence Velocity Profile [17] We will first examine the time-averaged turbulent velocity profile in an open channel flow. The classical picture of a turbulent boundary layer applies: there exists a thin viscous sublayer near the bed where the average velocity profile is linear and farther away a logarithmic region satisfying a log law velocity profile exists, with a buffer region connecting the two. In reality, the viscous sublayer is satisfied only when the bottom boundary is sufficiently smooth. It is well established that the time-averaged turbulent velocity ũ normalized qffiffiffiffiffiffiffiffiffiffiffiffi by the friction velocity (or shear velocity, ~u ¼ ~t B =~r f ) can be expressed in terms of distance from the wall in wall units (y + )as u þ ¼ ~u ¼ F y þ ¼ ~u ~y : ð14þ ~u n An empirical expression of the average velocity profile for a hydraulically smooth bed can be written as viscous sublayer u þ ¼ y þ ; logarithmic region u þ ¼ 1 k ln yþ þ 5:5; ð15þ where the von Kármán constant, k, is approximately 0.4. A useful empirical expression for the average velocity profile of a turbulent open channel flow, which covers the entire range from hydraulically smooth to rough as well as transitional beds, was proposed by Swamee [1993]: u þ ¼ ( ðy þ Þ 10 3 þ 1 k ln 1 þ 9y þ ) : ð16þ 1 þ 0:3Re k The above expression continuously applies from the viscous sublayer to the log layer and will be used henceforth for the average turbulent velocity approaching the particle. [18] The time-averaged velocity profile (16) depends on bed roughness Re k ¼ ~ k s ~u =~n and Figure 2 shows the open channel velocity profiles for various values of roughness in the range of 10 1 Re k The velocity profile for a smooth bed (Re k O 10 1 ) is almost independent of the roughness, nearly close to the velocity profile given in (15). While the average velocity of a rough bed (Re k > O 10 1 ) is significantly reduced as the roughness Re k increases. [19] For heterogeneous sediments (Re k d þ ), there exist two length scales; the diameter of the particle d + and the bed roughness Re k [Wiberg and Smith, 1987]. The undisturbed ambient flow velocity (u + f ) at the particle center can be obtained from (16) by substituting y + = d + /2. This resulting average turbulent open channel velocity seen by the particle is shown in Figure 3 as a function of particle diameter in wall units d þ ¼ d~u ~ =n for varying values of bed roughness. A threshold of viscous sublayer thickness d vs ( 11.6) is typically identified. A small particle of size much smaller than the viscous sublayer (d + d vs ) is taken to be fully submerged in the viscous sublayer and u + f d + /2. A particle is considered to be large if it mainly lies in the logarithmic turbulent flow region, (d + d vs ). In this limit if the wall roughness is small (i.e., Re k 1) we have 4of19

5 which are valid over the shear Reynolds number range (Re s 250 ), where shear Reynolds number is defined as Re s ¼ G ~ d ~ 2 = ð2~n Þ. We now use the following definition of uniform shear in a linear shear flow, ~G ¼ d~u d~y ¼ ~t B ~m ¼ t Bx ~g~ d ~n ; ð19þ to rewrite the shear Reynolds number in terms of nondimensional wall shear stress (t B ) and particle Reynolds number as Figure 3. Effect of relative bed roughness k + s /d + on undisturbed fluid velocity u + f evaluated at the particle center. u + f = ln(d + /2)/k, insteadifre k 1thenu + f reaches an asymptotic constant value that depends on Re k =d þ as given below: u þ f ¼ 1 dþ ln 15 : ð17þ k Re k [20] There are instances where the turbulent mean flow significantly departs from the above logarithmic profile. As discussed by Nikora et al. [2001], in cases where the water depth is comparable to roughness height a linear velocity profile may be more appropriate. Such situations arise in cases of either macroscopically rough beds or flow down steep slopes. The influence of such mean flow profiles on incipient particle motion has been discussed by Armanini and Gregoretti [2005]. Here we will investigate the dependence of hydrodynamic forces and torque on the nature of the turbulent mean flow profile. In addition to the logarithmic profile presented above we will also consider the problems of (1) a particle subjected to a linear velocity profile and (2) a particle subjected to a uniform velocity profile. We expect the results for the linear profile to be in agreement with those of the logarithmic profile in the limit of small particles submerged fully in the viscous sublayer. Similarly, the results for the uniform flow will be in better agreement with the logarithmic profile for very large particles which are mostly exposed to the log region and beyond. The Reynolds number dependence of C D, C L and C M for the linear and the uniform mean velocity profiles will be presented below. These results will later be compared with those obtained for the logarithmic profile Linear Mean Velocity Model [21] Here we recast the results of Zeng et al. [2009] on hydrodynamic drag, lift and moment (torque) coefficients acting on a particle sitting on a plane wall, in a wall-bounded linear shear flow. They suggested the following accurate expressions of C D and C L, C D ¼ 40:81 1 þ 0:104Re 0:753 s Re s and 3:663 C L ¼ Re 2 s þ 0:1173 0:22 ; ð18þ Re s ¼ ~ G ~ d 2 2~n ¼ t BRe 2 p 2 ¼ Re f : ð20þ Note that for a linear mean velocity profile Re s is the same as Re f, since the shear flow velocity at the particle center is given by ~u f ¼ ~G d=2. ~ The Stokes flow results of Goldman et al. [1967b] for the torque on a particle subjected to wall-bounded linear shear flow (~T ¼ 0:944p ~m G ~ d ~ 3 =2) can be used to obtain C M ¼ 15:104 Re s : ð21þ The numerical results of Zeng et al. [2009] show the above theoretical result to remain approximately valid up to Re s 200. [22] The above drag, lift and moment coefficients, along with the Reynolds number definition given in (20) can be substituted into equation (13). The resulting expression is an implicit threshold condition for the critical shear stress (t B ) in terms of Re p and Re. We will discuss this expression in greater detail in section Uniform Mean Velocity Model [23] Here we use the results of spectral element simulations of a particle translating parallel to a wall for a range of Re t 250. In this case of particle translating parallel to a wall at velocity ~v t the translational Reynolds number is defined as Re t ¼ ~v t d=~n. ~ Note that in this case there is logarithmic singularity, and as the particle approaches to touch the wall the drag force and the hydrodynamic torque on the particle will increase to infinity logarithmically. In contrast, the lift force will remain bounded. The drag coefficients obtained from the simulation results were expressed as [Zeng et al., 2009] C D ¼ 24 h fðþ1 e þ a t ðþre e b t e t Re t ðþ i ; ð22þ where e ¼ ~e= d ~ is the nondimensional gap between the bottom of the particle and the wall. The dependence of the different functions ( f (e), a t (e) andb t (e)) on the nondimensional gap between the particle and the wall can be expressed as fðþ¼1:028 e 0:07 1 þ 4e ln 270e ; p 135 þ 256e a t ðþ¼0:15 e ½1 expð ffiffi eþš; ð23þ p b t ðþ¼0:687 e þ 0:313expð2 ffiffi eþ: 5of19

6 [24] Note that in the limit e we have f 1, a t 0.15 and b t and thus the drag coefficient reduces to the standard drag law. Here, for the case of a particle moving on a wall we consider the limit of e 0. However, in order to avoid the logarithmic singularity we choose a small gap of e = 0.005, and the different functions reduce to fðe ¼ 0:005Þ ¼ 3:149; a t ðe ¼ 0:005Þ ¼ 0:01024; b t ðe ¼ 0:005Þ ¼ 0:9587: ð24þ [25] Zeng et al. [2009] presented the following expression for the lift coefficient: C L ¼ 0:313 þ 0:812exp 0:125Re 0:77 t : ð25þ Note that the above expression for C L is appropriate for the case of the particle in contact with the wall (i.e., for e = 0). The torque on the particle translating parallel to a wall in a quiescent flow also has the logarithmic singularity and from the Stokes flow results of Goldman et al. [1967a] the moment coefficient for the case of e = can be written as C M ¼ 8:673 Re t : ð26þ Unfortunately, a detailed model for moment coefficient at finite Reynolds numbers for the case of a particle translating parallel to a wall is not available. Here we will employ (26) to be applicable at finite Reynolds number as well. [26] We apply the above model as an approximation for the limiting case of a uniform flow over a very large particle. In the frame of reference attached to the particle ~v t is then the oncoming uniform flow velocity seen by the particle (~u f ), and as a result Re t is the same as Re f. If we take this fluid velocity to be given by the asymptotic constant velocity of a hydraulically rough bed presented in (17), the translational Reynolds number, Re t, that appears in the above equations of drag, lift and moment coefficients can be expressed as pffiffiffiffiffi t BRep Re t ¼ ln 15 ~! d k ~ k s pffiffiffiffiffi t BRep ¼ ln 15 2=3! 1 2Re =Re p k 2 2=3 ¼ Re f : ð27þ Re =Re p When the above relation along with the drag, lift and moment coefficients are substituted into equation (13), the critical shear stress can be expressed as a function of Re p and Re, which will again be discussed in section Numerical Simulation [27] Here we perform numerical simulations of flow around an isolated stationary particle which is embedded in a steady ambient flow of time-averaged turbulent velocity profile over a hydraulically smooth wall. The rigid particle and the wall are fully resolved by an immersed boundary technique [Uhlmann, 2005]. The size of the particle is systematically varied in these simulations. The particle is located almost on the plane wall with a small gap (e = 0.005) between the particle and the wall to prevent logarithmic singularity. This approach is similar to that used in prior investigations [Zeng et al., 2009; Stewart et al., 2010; Lee and Balachandar, 2010; Lee et al., 2011]. [28] We emphasize that the present simulations are approximate and are not direct numerical simulations, as they do not resolve the scales of oncoming turbulent flow. By using the time-averaged mean turbulent profile for the inflow the goal of the present numerical simulations is to obtain approximations for the time-averaged hydrodynamic forces (drag, lift) and torque on a spherical particle for the intermediate regime where 1 Re f O 10 3, corresponding to a particle diameter in the range 1 d + O(10 2 ). Nonlinear interactions in a turbulent flow will influence the time-averaged forces and torque on the particle and even more importantly will contribute to fluctuations about the time averages. These influences will be discussed in section qffiffiffiffiffiffiffiffiffiffiffiffi [29] In the simulations the friction velocity, ~u ¼ ~t b =~r f, and the viscous length scale, ~n=~u, are chosen as the velocity and length scales. The dimensionless governing equations for the incompressible flow reduce to r u ¼ 0 u t þ u ru ¼rp þr2 u þ f i þ f k ; ð28þ where a force field f i is applied within the volume occupied by the particle, which in conjunction with the immersed boundary technique being employed here enforces the correct velocity boundary condition on the surface of the particle. Another external force f k is also applied in order to maintain a steady ambient flow of desired velocity profile. Here we choose this driving force field to be f k ¼ 2 u d y 2 e x; ð29þ where u d (y) is the desired steady streamwise velocity profile and e x is the unit vector along the streamwise direction. Thus, in the absence of the particle, the force field f k will allow u d (y) to be an admissible solution of the governing equations. We choose u d to be the average turbulent open channel flow given in equation (16) and thereby consider the problem of a particle subjected to a time-averaged flow, without the influence of turbulence fluctuations. [30] The key parameter of the simulations is the dimensionless diameter of a particle (d + ) which can be rewritten in terms of particle Reynolds number and t B as d þ d~u ¼ ~ ~n ¼ p ffiffiffiffiffi t BRep : ð30þ Here we perform a sequence of numerical simulations with d + varying in the range 2 d [31] The Cartesian grid employed in the simulations is clustered near the particle to enhance the resolution while optimizing the computational costs. A uniform grid is used in the small cubic computational region around the particle ( d + [0, 1.5d + ] d + ) with a fine resolution of D = d + /40, where D is the uniform grid spacing in all three directions. Table 1 shows the details of the different cases simulated. 6of19

7 Table 1. Cases of Numerical Simulations Case d + Re f L x a L y a L z a N x b N y b N z b d + /D c Dy w +d , , , , , , , , , , , , , , a Size of computational domain: L x L y L z =[15d +,15d + ] [0, 15d + ] [7d +,7d + ]. b Number of grid points: N x N y N z. c Grid resolutions of uniformly distributed mesh around a particle: d + /D. d Grid resolutions near a wall: Dy w +. Additional information on the present immersed boundary technique, numerical algorithms and extensive tests of its accuracy, including grid independence, have been discussed by Lee and Balachandar [2010] and will not be repeated here. [32] The boundary conditions employed in the simulations are Inflow u ¼ eq: ð16þ; v ¼ w ¼ 0 Outflow u= x ¼ 0 Free surface u= y ¼ w= y ¼ 0; v ¼ 0 Lateral u= z ¼ 0: ð31þ No-slip boundary condition is applied in conjunction with the immersed boundary method on the surface of the particle. The force field f i is integrated around the particle to obtain the drag force, lift force and net torque on the particle, which will be used to compute the corresponding hydrodynamic force and moment coefficients. 4. Results 4.1. Flow Features [33] Before we discuss the results on forces and torque we first present pressure distribution and flow structure around the particle for the different cases considered. Figure 4 shows pressure contours and flow streamlines on the symmetry plane (z + = 0) for three distinct particle sizes (d + =5, 20 and 100). The case of (1) d + = 5 represents a small particle that is fully submerged within the viscous sublayer, (2) d + = 20 represents a particle in the buffer region and (3) d + = 100 represents a large grain mostly in the logarithmic region. In Figure 4, the flow is directed left to right. [34] A high-pressure stagnation region (dark gray) is seen at front side of the particle. This high-pressure region is shifted above the particle center (y + d + /2) due to wall effect. But with increasing particle size (and particle Reynolds number) the location of the high-pressure region moves down toward the midsection of the particle. The lowpressure (white) region extends over the top and the leeward side of the particle. In all the three cases shown the difference in pressure between the top and the bottom portion of the particle contributes to a net positive lift force that is directed away from the wall. Unlike in an unbounded uniform ambient flow, the wall-induced symmetry breaking features of the flow are observed in the streamline pattern. Flow separation and recirculation can be seen at the top and behind the particle. The size of the wake behind the particle grows as d + increases. For the case of the large particle (d + = 100), unsteady flow patterns are observed in both Figures 4e and 4f. But, the flow for the two smaller particles of d + = 5 and 20 remains steady state. [35] It must be pointed out that the pressure and flow patterns presented in Figure 4 will be influenced by turbulence in the oncoming flow. Based on the results of Zeng et al. [2008], it can be expected that the effect of oncoming turbulent flow will be to promote self-induced vortex shedding in the wake even for smaller particles and for larger particles the nature of the wake will be somewhat altered by the ambient turbulence. It must also be cautioned that at the highest Reynolds numbers considered the flow around the particle is marginally resolved Hydrodynamic Forces [36] Dimensional drag and lift forces (~F D and ~F L ) and torque (~T ) on the particle are used to define the dimensionless drag, lift and moment coefficients as given in (9). The time history of the drag and lift coefficients for the three cases are shown in Figure 5. For the case of d + = 100 the root mean square (RMS) of drag and lift fluctuations are 1.83% and 5.78% of the mean, respectively (see Table 2). Among the 14 cases simulated (see Table 1) we observe the flow to remain steady in the first 9 cases, for Re f 653:2. We observe that the onset of unsteadiness to occur for d + > 50, also confirmed in Figures 4e and 4f for d + = 100. Beyond this threshold value, time-averaged values of the hydrodynamic forces and torque are used to compute the mean force and moment coefficients. [37] In an unbounded uniform ambient flow, stability analysis and numerical simulations show that the wake becomes unsteady and sheds one-sided vortices above a transition Reynolds number between 270 and 280. The 7of19

8 Figure 4. (left) Pressure contours and (right) streamline patterns around a spherical particle in a turbulent open channel flow: (a, b) d + = 5, (c, d) 20, and (e, f) 100 (instant). effect of the nearby wall is to delay the onset of this transition to unsteadiness. The simulation results of Zeng et al. [2005] for the problem of a particle translating parallel to a flat wall in a quiescent fluid have shown that as the particle to wall gap decreases the transition Reynolds number increases above 300. This is consistent with the transition regime map presented by Stewart et al. [2010]. There is no prior result on transition to unsteadiness in the case of a stationary particle subjected to wall-bounded shear flow. The present results suggest a fairly large transition Reynolds number. However, two additional factors must be taken into account. First, the oncoming velocity field being enforced is not a linear shear flow. In the present simulations a particle of d + > 50 extends well into the log layer and thus the results on the transition Reynolds number are specific to the turbulent mean velocity profile. Furthermore, it has been observed by Bagchi and Balachandar [2003, 2004] and Zeng et al. [2008] that the presence of ambient turbulence in the oncoming flow tends to promote vortex shedding and somewhat lowers the transition Reynolds number, respectively, in both unbounded and wall-bounded flows Drag Coefficient [38] The drag coefficient computed from the present simulations are presented in Figure 6 plotted as a function of the flow Reynolds number (Re f ¼ ~u f ~ d=~n ¼ u þ f d þ ). From the results we can obtain a good fit for the drag coefficient on a particle nearly sitting on a bed in a fully developed turbulent open channel flow as C D ¼ 40:81 1 þ 0:104Re 0:753 f 1 erf 0:002Re f Re f þ 0:54erf 0:002Re f ; ð32þ where erf denotes the error function. On the other hand the above model assumes that the drag coefficient to gradually approaches a constant value of 0.54 at higher Reynolds numbers beyond those considered in the present simulations. [39] Also plotted in Figure 6 as dashed and dash-dotted lines are the drag coefficients obtained for the linear and uniform mean flow profiles discussed in sections 2.3 and 2.4. In the low Reynolds number regime (say Re f 10) the C D from the present simulations with the logarithmic mean velocity are in very good agreement with the corresponding results for the linear mean velocity profile given in (18). At large Reynolds numbers (Re f > O 10 4 ) the drag coefficient for the logarithmic mean velocity approaches a constant value of 0.54 and is quite close to the corresponding result for the uniform mean velocity given in (22). In between these two regimes (10 < Re f < O 10 4 ) the drag for the logarithmic mean velocity profile are smaller than 8of19

9 from the expression given in (32) highlights the sensitivity to the definition of flow velocity seen by the particle. [40] Also shown in Figure 6 are the time-averaged mean C D measured in the experiments of Schmeeckle et al. [2007]. Results from several of their tests using 1.9 cm sphere are shown and the results for the smooth bottom boundary and rough gravel bed are shown as cross and triangle symbols, respectively. Their C D was evaluated using fluid velocity measured directly upstream of the particle and using the full area of cross section of the sphere, and therefore can be directly compared with the simulation results. Their Re f ranged from 4,750 to 13,250. Although the experimental data show some scatter, the simulation results are slightly lower than the experimentally measured C D. Also presented in Figure 6 is C D evaluated from the measurements of Dwivedi et al. [2010a]. Only the result for their largest exposure is presented. Their mean drag coefficient is substantially lower than the present simulation and the experimental results of Schmeeckle et al. [2007]. However, even for the case of Dwivedi et al. [2010a] reported here, the particle exposure was only 25%, thus highlighting the effects of particle exposure and the pocket geometry Lift Coefficient [41] Similar to drag, the computed lift coefficients are plotted in Figure 7. A good fit for the lift coefficient for a particle on a bed in a turbulent open channel flow can be expressed as 3:663 C L ¼ 0:22 1 erf 0:001Re f Re 2 f þ 0:1173 þ 0:223erf 0:001Re f : ð34þ Figure 5. Time variations of (a) drag and (b) lift coefficients on a spherical particle in a turbulent mean velocity profile. Three different particle sizes of d +, 20, 50, and 100, are considered. both the linear and uniform velocity profiles. Nevertheless, the differences in C D between the three different mean velocity profiles are quite modest, considering the fact that the logarithmic, linear and uniform velocity profiles should appear substantially different for a particle of intermediate size. Also shown in Figure 6 is the well-known empirical standard drag coefficient of a sphere in an unbounded uniform flow suggested by Clift and Gauvin [1970]: C D ¼ 24 1 þ 0:15Re 0:687 0:42 f þ Re f 1 þ 4: Re 1:16 f for Re f < : ð33þ It is clear that the standard drag coefficient is consistently smaller over the entire range of Reynolds number considered, thus indicating the influence of the bed. In Figure 6 plotted as star symbol are the present simulation results, but with the fluid velocity in the definitions of Re f and C D taken at 0:15 ~ d above the top of the particle. This results in an increase in Re f and a decrease in C D. The deviation in C D Figure 6. Comparison of drag coefficients. Circles and stars show present numerical simulations for a wall-bounded turbulent mean flow. Solid line represents curve fit in equation (32) for a wall-bounded turbulent mean flow. Dash-dotted line shows the result for linear mean velocity model given in equation (18). Dashed line shows the result for uniform mean velocity model given in equation (22). Dashed double-dotted line shows the standard drag model presented in equation (33) for an unbounded uniform flow [Clift and Gauvin, 1970]. 9of19

10 Figure 7. Comparison of lift coefficients. Circles and stars show present numerical simulations for a wall-bounded turbulent mean flow. Solid line represents the curve fit given in equation (34) for a wall-bounded turbulent mean flow. Dash-dotted line shows the result for linear mean velocity model in equation (18). Dashed line shows the result for uniform mean velocity model given in equation (25) for a wall-bounded uniform flow [Zeng et al., 2009]. Dashed double-dotted line shows low Reynolds number theory [Saffman, 1965]. For Re f 0 the above reduces to (18) appropriate for the limit of a small particle. Wiberg and Smith [1987] assumed the lift coefficient to be a constant, and used C L = 0.2 in their calculation. This constant is consistent with the present numerical simulation results at high Reynolds numbers. But for Re f < 500 there is discrepancy and the difference increases as the Reynolds number decreases. The lift expression proposed by Saffman [1965] is also p widely ffiffiffiffiffiffiffi used. 2 His first-order lift force is ~F L ¼ 1:615~r f ~u f d ~ ~G~n. Since the ambient shear is not uniform, ~G in the above expression must be approximated. A simple approximation assuming a linear velocity change from the wall to the center of the particle as ~G ¼ 2~u f = d ~ yields d In comparison, equation (25) predicts a constant lift coefficient of The difference can be expected since the uniform velocity model completely ignores the approach to zero velocity at the bed and thereby overpredicts the pressure distribution at the bottom of the particle. [43] Experimentally measured time-averaged lift coefficients by Schmeeckle et al. [2007] are also plotted in Figure 7. Unlike for C D, the only cases where the particle is either almost sitting on the bed or within 0:2 ~ d from the bed are shown. It can be seen that the simulation results are in reasonable agreement with the measured lift coefficients. The measured C L for other cases, where there is a larger gap between the particle and the bed, are substantially lower, and even sometimes negative. This is consistent with the observation by Zeng et al. [2009] that the lift coefficient rapidly decreases with increasing separation from the wall. The lift coefficient for the largest exposure case of Dwivedi et al. [2010a] is also shown. The C L is substantially larger again highlighting the possible effects of particle exposure and pocket geometry. [44] Compared to the drag force, since the lift force has not been well investigated, the ratio of the lift to the drag C L /C D is often assumed to obtain an estimate of the lift force. Particularly at large Reynolds numbers, the ratio has been considered to be a constant (C L /C D = ) for a given condition [Bagnold, 1956; Patnaik et al., 1992; Vollmer and Kleinhans, 2007]. Figure 8 shows the ratio of lift to drag coefficient (C L /C D ) obtained from the numerical simulations. Also shown is the ratio computed from the curve fits (32) and (34). The ratio, although remains O(1), is strongly dependent on Reynolds number, showing tendency toward a constant value for Re f > Torque Coefficient [45] Figure 9 shows the computed moment coefficient as a function of the Reynolds number. Also plotted in Figure 9 are the moment coefficients obtained from Zeng et al. s [2009] numerical simulations of a stationary particle subjected to a wall-bounded linear shear flow (as opposed to the C L;Saff ¼ p 5:816 ffiffiffiffiffiffiffi : ð35þ Re f This approximation of ambient shear is quite reasonable for a small particle embedded entirely within the viscous sublayer. However for a larger particle, clearly the shear is greater close to the wall with rapidly decreasing velocity gradient in the top portion of the particle. The lift coefficient for the linear mean velocity model given in equation (18) and for the uniform velocity model given in equation (25) are also presented in Figure 7. Predictions by both Saffman s theory and equation (18) continue to decrease rapidly with increasing Re f. Over the entire regime Saffman s lift is significantly different from the computed C L, since it ignores both the effects of finite Re as well as the bed surface. As expected for Re f 10 equation (18) provides a good approximation. [42] Note that the numerical simulations yield a near constant lift coefficient of about for large particles of Figure 8. The ratio of lift to drag coefficient (or force ratio) on a spherical particle. Solid line shows computed data obtained from the curve fits of drag and lift given in equations (32) and (34). 10 of 19

11 below the critical value as estimated from a steady state analysis. Second, due to nonlinear influence of turbulence, the time-averaged forces and torque on the particle will not be the same as that given by the time-averaged turbulent inflow profile. Here we will get an estimate of these two effects of turbulence. [47] We follow the recent work of Dwivedi et al. [2010a, 2010b, 2011] on the analysis of time-resolved measurements of drag and lift forces and consider the relations given in (9) to be applicable on an instantaneous basis. In other words, instantaneous drag force can be obtained as F D ðþ¼ t p 8 ~r ~ f d 2 C D ~u 2 f ðþ; t ð37þ Figure 9. Comparison of moment coefficients. Circles show present numerical simulations for a wall-bounded turbulent flow. Squares show numerical simulations of Zeng et al. [2009] for a wall-bounded linear shear flow. Solid line shows present curve fit given in equation (36) for a wallbounded turbulent flow. Dashed line shows low Reynolds number theory [Goldman et al., 1967b]. mean logarithmic velocity profile). The moment coefficient for a particle sitting on a plane wall in a linear shear flow was given in (21) as obtained from the low Reynolds number theory [Goldman et al., 1967b]. At finite Reynolds number in a turbulent open channel with logarithmic mean velocity profile, we present the curve fit for the moment coefficient as obtained from present numerical simulations as C M ¼ 15:104 Re f 1 þ 0:0005Re f erf 0:002Re f : ð36þ In the limit of Re f 0 the above curve fit correctly approaches the behavior for a small particle embedded in a linear shear flow. With increasing Reynolds number the moment coefficient continues to decrease as inverse power of Re f and this trend extends to about Re f All of the predictions nearly collapse in this region. With further increase in Reynolds number the decay rate of moment coefficient slows down and the above curve fit results in an asymptotic constant (but very small) value of the moment coefficient for very large Re f. Although the precise asymptotic behavior of C M for large Re f is not known, the above curve fit seems to provide adequate approximation for the numerical results Effect of Turbulence Fluctuations [46] The oncoming wall turbulence will influence incipient particle motion and particle suspension in several significant ways. First and foremost, due to turbulent fluctuations the instantaneous velocity seen by the particle, and therefore the instantaneous forces and torque on the particle can far exceed the time-averaged mean values. Such deviations can result in particle motion and suspension even when the time-averaged velocity or wall shear stress is where C D as a function of Re is taken to be given by (32). The above quasi-steady formulation does not include the effects of spatial and temporal accelerations of the near-bed turbulent flow [Nelson et al., 1995]. A more complete formulation will require additional contributions to force arising from unsteady mechanisms, such as added mass and pressure gradient contributions [see Schmeeckle et al., 2007]. However, due to difficulties in simultaneous measurements of surrounding flow accelerations, the vast majority of investigations of incipient motion and resuspension are limited to the above quasi-steady formulation. [48] Some support for this quasi-steady approximation can be drawn from the results of Zeng et al. [2008], who employed fully resolved direct numerical simulations of flow over a stationary spherical particle subjected to wall turbulence and investigated the role of turbulent fluctuations on drag and lift forces for different sized particles ranging from d + = 3.56 to d + = For a small particle, they observed the time-dependent fluctuations in the drag and lift forces to be well correlated and accurately predicted with the above quasi-steady formulation. For a larger particle, due to self-induced vortex shedding, the instantaneous forces are not well correlated with oncoming turbulence. As a result, time-dependent force evolution cannot be accurately predicted with the quasi-steady formulation. But the prediction does not improve with the inclusion of unsteady force contributions. The difficulty arises from self-induced vortex shedding and the resulting force fluctuations are not correlated with either oncoming velocity fluctuation or with flow acceleration. Fortunately, despite lack of instantaneous correlation, even for a large particle, they observe the RMS force fluctuations to be well predicted by the above quasisteady formulation. [49] In order to fully evaluate the mean and fluctuating forces from (37), we first note that time dependence arises from both ~u f and C D. We first relate fluctuations in C D to velocity fluctuations. To do so, we define C D to be the average drag coefficient corresponding to the time-averaged Reynolds number Re ¼ ~u f d=~n ~. The time-dependent drag coefficient can be Taylor series expanded about this average and when truncated to the first two terms of the expansion can be expressed as C D ðþ t ¼ 1 þ 1 dc D C D C D dre ReðÞRe t ~u f ¼ 1 þ a ; ~u f ð38þ 11 of 19