Environ Fluid Mech (2017) 17:449–472 DOI 10.1007/s10652-016-9498-4 ORIGINAL ARTICLE

A mathematical model for type II profile of concentration distribution in turbulent flows Snehasis Kundu1 • Koeli Ghoshal2

Received: 10 March 2016 / Accepted: 1 December 2016 / Published online: 27 December 2016 Ó Springer Science+Business Media Dordrecht 2016

Abstract This paper presents a mathematical model to investigate type II profile of suspension concentration distribution (i.e., the concentration profile where the maximum concentration appears at some distance above the bed surface) in a steady, uniform turbulent flow through open-channels. Starting from the mass and momentum conservation equations of two-phase flow, a theoretical model has been derived. The distribution equation is derived considering the effects of fluid lift force, drag force, particle inertia, particle–particle interactions, particle velocity fluctuations and drift diffusion. The equation is solved numerically and is compared with available experimental data as well as with other models existing in the literature. Good agreement between the observed value and computed result, and minimum error in comparison to other models indicate that the present model can be applied in predicting particle concentration distribution for type II profile for a wide range of flow conditions. The proposed model is also able to show the transition from type I profile to type II profile. Keywords Suspension distribution  Type II profile  Particle inertia  Granular temperature  Drift diffusion Mathematics Subject Classification 76F25  76R50  76T20

& Snehasis Kundu [email protected] Koeli Ghoshal [email protected] 1

Department of Basic Sciences and Humanities, IIIT Bhubaneswar, Bhubaneswar 751003, India

2

Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

123

450

Environ Fluid Mech (2017) 17:449–472

1 Introduction In a sediment-laden turbulent flow through open channels, particles are partly transported as bed-load and suspended-load. In suspended-load, sediment particles occasionally come in contact with bed. The transport of suspended sediment is an important area in fluvial hydraulics and has wide applications in industry and geophysical research. The knowledge of particle concentration profile helps us to compute sediment transport rate in rivers and thus to model transport process in rivers. In the last several years, diffusion theory has been largely used to model the profile of suspension distribution. Despite its applications in several practical models, it has serious limitations [43]. Firstly, it is based on a single composite media and does not provide clear picture for dynamics of particle suspension. In reality, sediment-laden flows should be considered as a two-phase system consisting of liquid phase and solid phase. Secondly, the effect of the presence of particles is not counted. Lastly, the effects of fluid lift force on particles, particle inertia, particle velocity fluctuations and effect of fluid vertical velocity induced by secondary vortex have not been incorporated in the diffusion theory. Another drawback of diffusion theory is that it always results dC=dy\0 (where C denotes mean volumetric sediment concentration and y denotes vertical coordinate) throughout the flow depth as sediment diffusion coefficient es [ 0 and particle settling velocity x0 [ 0 for particles heavier than fluid. Several experimental observations [4, 24, 52] show that another type of concentration distribution profile (called as type II profile) exists in which it may happen that dC=dy [ 0 in a thin region near the bed. This phenomenon cannot be described by diffusion theory. The existence of two different types of concentration profiles was experimentally observed both in pipe flow and open-channel flow [4, 36, 42, 44, 52]. These two types of concentration profiles differ by the location of maximum sediment concentration from channel bed surface. The type I profile is defined to be that concentration profile where the maximum concentration appears at the channel bed and sediment concentration gradually decreases from bed surface to the free surface; whereas the type II profile is defined to be that concentration profile where the maximum concentration appears at a height which is significantly above the channel bed or in other words, sediment concentration increases first and then begins to decrease when the height from the bed reaches a certain value. A schematic diagram of type II profile is shown in Fig. 1. Though several scientists have developed models for type I and type II profile using several theoretical approaches, but the understanding of the mechanism for type II profile is still not clear. Also a condition which can distinguish these two types of profiles, is still lacking in literature. As several

Fig. 1 Schematic diagram of sediment-laden flow in open channels (side view)

y

C x

gcos

nearbed lift force

123

g

Environ Fluid Mech (2017) 17:449–472

451

progresses have been done for type I profile, this study focuses only on type II profile of suspension distribution. As type II profile can not be described by diffusion theory, several researchers adopted kinetic theory of two-phase flow to describe it. Ni and Wang [42] proposed that two types of concentration distribution can be explained from the effects of fluctuating characteristics of fluid and sediment particles. Similarly, Wang and Ni [52, 53] and Ni et al. [44] used kinetic theory of two-phase flow incorporating the effect of fluid induced lift force on concentration distribution. They proposed kinetic theory based models to describe the distribution of sediment concentration and results showed that a comprehensive lift force (effect of all the vertical forces acting on particles by fluid) is important for unusual distribution of particles in the near bed region. Bouvard and Petkovic [4] analyzed from their experimental study of heavy particles that fluid lift force and drag force have dominant influence on particles diffusion and dispersion. Fu et al. [14] studied average properties of particles using kinetic theory of two-phase flow in sediment-laden turbulent flow. They derived mathematical model for particle concentration distribution and from their analysis they found that fluid induced lift force and granular temperature (measure of the strength of particle velocity fluctuation) are effective factors for type II distribution of concentration. Wilson and Sellgren [56] studied the effect of near wall lift force on the basis of the Kutta-Zhukovski equation in slurry pipelines. In their analysis they found that particles significantly larger than the thickness of viscous sublayer tend to move upward from the near-bed region by fluid lift force and creates type II profile. They also added that for smaller particles with diameter d\0:15 mm, the lift force in viscous sublayer is not strong enough to remove particles from near-bed region. Later on, Kaushal and Tomita [28] experimentally investigated the near-bed fluid lift force for coarse particles in slurry pipelines using c-ray densitometer. Their results supported the analysis of Wilson and Sellgren [56]. They found that near-bed fluid lift force appears as a result of impact of viscous-turbulent interface at the bottom and on the top of coarse particles. As the particle size is larger than the thickness of viscous sublayer, the interface effect causes the particle to move upward. Wang and Fu [50] studied the diffusion mechanism in sediment-laden flow and found that effects of lift force and particle velocity fluctuation change the gravitational settlement of particles in the near-bed region. Later on, Ni et al. [41] studied the characteristics of hyperconcentrated flow using kinetic theory. They observed that when average volumetric concentration C [ 0:4, vertical profile of particle concentration alters and becomes unusual. In their study, Liu et al. [33] found that particle concentration profile is significantly influenced by particle fluctuation intensity. Apart from the kinetic theory, two-phase flow theory was also adopted by several researchers [19, 20, 27, 51, 57]. In addition to the gravitational force, the advantages of using this approach are that the effects of particle inertia, lift force, drag force, particle velocity fluctuation are also included. In the two phase flow theory, all the forces are included in the phase interaction force between solid and fluid phases. Although several advances are made, the effect of secondary current on suspension distribution are still not investigated by researchers. The secondary current is defined as the flow of fluid along vertical and transversal direction. It is the vertical component of velocity that affects the suspension of particles along vertical direction. Kundu and Ghoshal [30] and Kundu [29] investigated the effect of secondary current on suspension distribution and from the analysis they found that it has significant effect on two types of suspension distribution. This effect of secondary current can be taken as a phase interaction effect of fluid phase on solid phase. Cao et al. [5] derived predictive equations for vertical profile of concentration

123

452

Environ Fluid Mech (2017) 17:449–472

distribution and mixture velocity with an assumed sediment diffusivity and sediment velocity similar to the assumption in conventional diffusion theory. Later on, Greimann et al. [20] developed analytical expressions for vertical suspension distribution profile; Greimann and Holly [19] extended this formulation to hyperconcentrated flows. In these two studies, analytical models are derived assuming sediment particle velocity fluctuation to be constant and no effect of secondary current has been considered in governing equation. However in their study, Liu et al. [33] showed that particle velocity fluctuation is an important factor that significantly influences the patterns of concentration profiles in the near-bed region. Similar conclusion was obtained by Fu et al. [14] and Wang and Fu [50]. Zhong et al. [57] developed a model using two-fluid model and pointed out from theoretical analysis that proposed model can be applied to describe type II profile of concentration distribution; but they did not provide any experimental verification. Recently, Kundu and Ghoshal [31] proposed a model for type II profile of suspension distribution using Almedeij’s [1] asymptotic matching method which provides a good agreement with experimental data. From the above discussion it can be found that in most of the studies different authors have assumed different type of forces acting on particle phase. It is also clear that fluid lift force and particle fluctuating intensity are influencing factors for type II profile. In all of these aforementioned studies authors did not consider the effect of secondary current in their models though it has significant effect. Therefore in the present study we include this effect in the governing equation. The main objectives of this study are: (i) to develop a theoretical model for type II concentration distribution based on the description of two-fluid model; (ii) to check the validity of the proposed model for a wide range of experimental data; (iii) to compare the model with previous models to get a quantitative idea about the accuracy of the model; (iv) to find a condition which can distinguish the type I and type II profile of suspension distribution.

2 Governing equation for sediment suspension In most of the cases, transport of sediment particles occurs in water environment and the motion of sediment particles in water flow is treated as group flow. Therefore, in this study we will analyze the distribution of suspended sediment particles from the point of view of the momentum equilibrium of group particle motion. We consider a two-dimensional (2D) incompressible flow over sediment bed (Fig. 1), with uniform channel slope. Thus the sediment and fluid inertia terms vanish and sediment particles are distributed only along vertical direction (considered as y-direction). Further we consider that the flow is fully developed turbulent flow and sediment concentration is low and therefore the collision stresses of solid phase is negligible. The equation of conservation of mass for solid phase and the motion for sediment particles in the vertical direction can be expressed by balancing all the forces as oC o o þ ðCus Þ þ ðCvs Þ ¼ 0 ot ox oy and

123

ð1Þ

Environ Fluid Mech (2017) 17:449–472

 ovs ovs ovs C op 1 ors 1 ¼ Cgy  C þ þ us þ vs þ Fsy qs oy qs oy qs ot ox oy

453



ð2Þ

where x and y denotes the longitudinal and vertical co-ordinates respectively, t is the time, us and vs are the instantaneous particle velocity components in x and y direction respectively, qs is the density of sediment and C is the volume fraction of sediment particles, gy is the component of gravity acceleration in y-direction, p is the pressure of water flow, rs is the particulate stress arising from particle-particle interactions and Fsy is the phase interaction forces in vertical direction. Generally the slope of channel h is small so gy ¼ g cos h  g. Combining Eqs. (1) and (2), one obtains o o o C op 1 ors 1 ðCvs Þ þ ðCus vs Þ þ ðCvs vs Þ ¼ Cg  þ þ Fsy ot ox oy qs oy qs oy qs

ð3Þ

Equation (3) gives the governing equation for vertical distribution of sediment particles on open channel flows. Similar type of governing equation was also adopted by Liu et al. [33], Liu and Singh [34] and Jiang et al. [27]. Liu et al. [33] decomposed the term Fsy into two parts as flow resistance along vertical direction and lift force of flow acting on sediment particles. Applying Reynolds decomposition to each instantaneous quantities (defined as u ¼ u þ u0 where u is the time average part and u0 is the fluctuation part) and taking time R t þT average (which is defined as u ¼ T1 t00 u dt0 where t0 is some initial time and T is taken sufficiently long time interval such that time average of fluctuating quantities become zero i.e. u0 ¼ 0) on Eq. 3 it can be expressed after rearranging the terms as o o ðCvs þ C0 v0s Þ þ ðCus vs þ Cu0s v0s þ us C0 v0s þ vs C0 u0s ot ox o 0 0 0 þ C us vs Þ þ ðCvs vs þ Cv0s v0s þ vs C0 v0s þ vs C0 v0s þ C0 v0s v0s Þ oy ¼ Cg 

ð4Þ

C op 1 0 op0 1 ors 1 oRs 1  C þ þ þ F sy qs oy qs oy qs oy qs oy qs

where Rs denotes the Reynolds like shear stress which appears due to velocity fluctuations of solid particles which is defined as Rs ¼ qs v0s v0s [57] and overbars denote average part and primes denote fluctuating part of corresponding instantaneous variables. After rearranging the terms, Eq. 4 can be written as   oC o o ovs þ ðCus þ C0 u0s Þ þ ðCvs þ C0 v0s Þ þ C vs ot ox oy ot ovs ovs oðC0 v0s Þ þ ðCvs þ C0 v0s Þ þ ot ox oy o o ðCu0s v0s þ us C0 v0s þ C0 u0s v0s Þ þ ðCv0s v0s þ vs C0 v0s þ C0 v0s v0s Þ þ ox oy

þ ðCus þ C 0 u0s Þ

¼ Cg 

ð5Þ

C op 1 0 op0 1 ors 1 oRs 1  C þ þ þ F sy qs oy qs oy qs oy qs oy qs

Combination of the conservation of mass equation (i.e. Eq. 1) of the solid phase and Eq. 5 yields

123

454

Environ Fluid Mech (2017) 17:449–472

C

ovs ovs ovs oðC0 v0s Þ þ ðCus þ C0 u0s Þ þ ðCvs þ C 0 v0s Þ þ ot ot ox oy o o ðCu0s v0s þ us C0 v0s þ C0 u0s v0s Þ þ ðCv0s v0s þ vs C0 v0s þ C0 v0s v0s Þ þ ox oy ¼ Cg 

ð6Þ

C op 1 0 op0 1 ors 1 oRs 1  C þ þ þ F sy qs oy qs oy qs oy qs oy qs

For a steady and uniform open-channel flow we have oðÞ=ot ¼ oðÞ=ox ¼ 0. Therefore Eq. 6 can be simplified as ðCvs þ C0 v0s Þ

ovs o þ ðCv0s v0s þ vs C0 v0s þ C0 v0s v0s Þ oy oy

¼ Cg 

C op 1 0 op0 1 ors 1 oRs 1  C þ þ þ F sy qs oy qs oy qs oy qs oy qs

ð7Þ

Similarly, for a steady and uniform flow, the mass conservation equation i.e. Eq. 1 can be written after taking time average as o ðCvs þ C0 v0s Þ ¼ 0 oy

ð8Þ

For an equilibrium sediment transport, the net mass flux is zero. Therefore, integration of Eq. 8 gives Cvs þ C0 v0s ¼ 0

ð9Þ

Therefore combining Eqs. 7 and 9, the sediment transport equation can be expressed as o C op 1 0 op0 ðCv0s v0s þ vs C 0 v0s þ C 0 v0s v0s Þ ¼ Cg   C oy qs oy qs oy 1 oRs 1 þ F sy qs oy qs

þ

1 ors þ qs oy

ð10Þ

The second term on the left hand side (LHS) of Eq. 10 can be approximated as vs C0 v0s  vs CUsT , where UsT is the turbulent diffusion velocity of sediment particle [33]. qffiffiffiffiffi Generally for open channel flows, UsT is much smaller than the fluctuating velocity v02 s of particles. Therefore, the second term on the LHS of Eq. 10 is less effective compared to the first term on the LHS of Eq. 10 and can be neglected. In order to simplify Eq. 10, higher order correlation term C0 v0s v0s needs to be defined. But generally higher order correlated terms are neglected due to the complicated expressions of the resultant equations. In fact, in some previous studies [33, 34] these terms have been neglected. In turbulent flows, pressure fluctuation is generated by the turbulent intensities. Batchelor [3] and Uberoi [48] found that the root mean square of pressure fluctuation in an isotopic and homogeneous turbulence depends on the turbulent intensity. Hetsroni [22, 23] concluded that the particle Reynolds number Rep ð¼ x0 d=mf , where x0 is the settling velocity of particle, d is particle diameter and mf is kinematic viscosity of fluid) determines the weakening or strengthening of the flow turbulence intensity. When the particle Reynolds number exceeds a certain value (Rep [ 400), trailing vortices detach from particles and as a result turbulence intensity increases. On the other hand, when the particle Reynolds number is small

123

Environ Fluid Mech (2017) 17:449–472

455

(Rep \100), the trailing vortices will not detach from particles and the turbulence intensity will not increase. We have computed the particle Reynolds number for all selected data sets (shown in Table 2) and found for such type of suspension profiles, Rep / 100. Thus the pressure fluctuation term is ignored as turbulent intensity would not increase. Therefore neglecting the smaller quantities and higher order correlation terms, Eq. 10 is finally simplified after omitting the overbars as o C op 1 ors 1 oRs 1 ðCv0s v0s Þ ¼ Cg  þ þ þ Fsy oy qs oy qs oy qs oy qs Equation 11 can be set in the following form   qf o 1 ors 1 oRs 1 ðCv02 þ þ Fsy þ Þ ¼ Cg 1  s oy qs oy qs oy qs qs

ð11Þ

ð12Þ

where op=oy ¼ qf g in which qf is the density of fluid phase. The first term in the lefthand side of Eq. 12 represents the rate of momentum transport by fluctuations of sediment particles. The first term on the right-hand side of Eq. 12 represents gravity of particles in water. Generally, the mean velocity of sediment particles is assumed to be equal to settling velocity x0 [10]. The mean vertical velocity of sediments can be neglected for neutrally buoyant or very small sediment particles. In this case, Eq. 12 is converted to a simple form considered by Liu et al. [33]. The distribution of sediment particles can be calculated from Eq. 12 if the external forces acting on sediment particles are provided and the closures of the Reynolds like shear stress and particulate stress are provided.

3 Particle forces To estimate forces acting on particles, the equation of motion of a single particle is adopted following Maxey and Riley [35]. In this case sediment particles and water within a mixture are considered as two sets of mass points with different densities. The diameter of sediment particles is assumed to be uniform with diameter d. For mixture of heterogeneous diameter of sediment particles, a mean diameter can be used. The external force Fsy is comprised of drag force, lift force (such as Saffman force and the Magnus force), gravitational force, virtual mass, pressure gradient force and Basset history force [35]. It is clear from the literature that among all these forces drag force and the fluid lift force are important for an inverse distribution of suspended sediment concentration near the bed region [14, 15, 33, 52]. Therefore in this study, only drag force and fluid lift force are considered. Accordingly, the phase interaction force Fsy can be expressed as Fsy ¼ FD þ Msy

ð13Þ

where FD denotes the drag force and Msy denotes the resultant force of other external forces apart from drag force.

3.1 Drag force The resistance of the particle to motion is called the drag force. When a particle is moving along vertical direction (upward or downward), the drag force is also acting along vertical direction with opposite to the direction of movement of the particle. If we assume the shape of the particle is spherical having diameter d, the drag force is expressed as

123

456

Environ Fluid Mech (2017) 17:449–472

 2 1 pd v2r FD ¼ qf CD 2 4

ð14Þ

where CD is the drag coefficient, vr ¼ vs  vf denotes the relative velocity of the particle to fluid and vf denotes the vertical velocity of the fluid phase. Under the action of gravitational force, the vertical velocity vs of sediment particles is the sum of vertical velocity vf of fluid together with the fall velocity x0 of sediment particles [5, 17, 18]. Thus vs ¼ vf  x0 . Therefore the drag force is expressed as  2 1 pd x20 FD ¼ qf CD ð15Þ 2 4 Equation (16) was adopted by several researchers [19, 20, 33]. The drag coefficient is a function of particle Reynolds number. For Stokes flow when Rep \1, flow around sphere is laminar and drag coefficient follows the linear relation CD ¼ 24=Rep and for Rep [ 105 , CD  0:5. For any intermediate value of particle Reynolds number between one and 105 , Cheng [9] suggested that h

i 0:43 24  1 þ 0:27Rep þ 0:47 1  exp 0:04Rep0:38 CD ¼ ð16Þ Rep Equation 16 shows that as particle Reynolds number increases, the Stokes relation would be modified. From the experimental data, we have seen that for such type of suspension profiles Rep [ 1 in most of the test cases except SF2 and SF5 of Wang and Qian [54]. Therefore, it is reasonable to express Eq. 16 into the following form CD ¼

24 þ UðRep Þ Rep

ð17Þ

where U is a function of particle Reynolds number Rep which counts the effect of high particle Reynolds number and U  0 when Rep \1. The first term on the right hand side of Eq. 17 represents the effect of laminar flow around the particle and second term incorporates the effect of fluctuations of particle due to flow turbulence and interaction between particles. Inserting Eq. 17 into Eq. 14 one obtains FD ¼

pd 2 24 2 pd 2 qf q UðRep Þx20 x þ Rep 0 8 8 f

ð18Þ

The first term on the right hand side of Eq. 18 denotes the Stokes drag force when particle is falling in a laminar flow. Second term defines the effect of fluctuations of particles. Therefore Eq. 18 can be expressed as linear combination of laminar drag force and a correction due to the effect of fluctuation of particles due to flow turbulence and collisions. Consequently one can write the total drag force as FD ¼ FD;L þ FD;T

ð19Þ

where FD;L is called the ‘laminar drag force’ and FD;T is called as ‘turbulent drag force’ in this study. In turbulence sediment-laden  flows, two time scales are used. One is the sediment or particle time scale sp ½¼ x0 = gð1  qf =qs Þ  which characterizes the time scale of fluid momentum transfer to the particles in the two-phase system as well as the time scale

123

Environ Fluid Mech (2017) 17:449–472

457

for the fluid-particle interactions. The drag force is usually expressed as the ratio of the relative particle velocity to a particle timescale sp which is taken from Stokes drag [47]. The other one is the integral turbulence timescale st for the net drag force due to turbulent motion. This is equivalent to sp =ðStb  1Þ, and reduces to sp for small Stb where Stb is the bulk Stoke’s number [20]. When a particle moves in a turbulent flow, it produces random fluctuations due to fluid velocity fluctuation and correlation between instantaneous particle distribution. As a result a diffusive flux is generated which affects the movement of particle by generating a drift velocity. The net drift resulting from fluctuating vertical movements indeed requires a considerable different time scale which is generally considered as integral turbulence timescale [47]. Therefore the time averaged drag force must be decomposed into two parts. Such type of consideration was adopted by the previous researchers Liu [32], Greimann et al. [20], Jiang et al. [27] and Toorman [47]. Generally, laminar drag force is approximated using Stokes law. When a spherical sediment particle falls under the action of gravitational force in still fluid, the laminar drag force is same as the submerged weight of the particle which is also known as the buoyancy force. Therefore the laminar drag force is expressed as   qf ð20Þ FD;L ¼ Cqs g 1  qs The time-averaged turbulent drag force is proportional to the ratio of difference of drift velocities of two phases to integral turbulence timescale [27, 32, 47]. Therefore it is expressed as [33] FD;T ¼ Cqs

DUD x0 ¼ Cqs ðStb  1Þ st sp

ð21Þ

where DUD denotes the difference of drift velocities between two phases which is considered as proportional to the settling velocity of particles [19, 33]. Consequently the total drag force is expressed as   qf x0 þ Cqs ðStb  1Þ ð22Þ FD ¼ Cqs g 1  qs sp

3.2 Lift force Apart from the drag force, the lift force of fluid plays an important part in the motion of sediment particles in near wall regions [44, 52]. There are two causes for which lift force may occur. One is due to the rotation of a particle moving in a fluid, and another is due to the translation of particles in the presence of shear of the fluid or due to flow velocity gradient. In the core flow region, suspension of particles is met by the turbulent diffusion. But near the wall, turbulence is diminished or eliminated in the viscous sublayer zone. Here the mirror effect (defined as upward transfer of suspended particles due to turbulence is in equilibrium with downward exchange due to gravitational force) cannot be provided by turbulent diffusion, and the second contribution to suspension, hydrodynamic lift force, comes into prominence. Kaushal and Tomita [28] suggested that, apart from these two well known reasons lift force may occur as a result of impact of viscous–turbulent interface on the bottom most layer of particles. When the particle size is larger than the viscous sublayer thickness, the interface impacts on the particles to lift upward. However they

123

458

Environ Fluid Mech (2017) 17:449–472

concluded that it is not possible to express it mathematically. Therefore in this study, we assume that lift force acting on a spherical particle along the main flow direction includes contributions of fluid shear and particle rotation. Therefore if only this lift force is considered among other external forces, then the time-averaged force Msy on solid phase can be expressed as Msy ¼ Cqs Ly

ð23Þ

where Ly denotes the lift force per unit mass on a particle. According to Auton [2], lift force FL acting on a single spherical particle of diameter d along main flow direction can be expressed as FL ¼

4p duf CL qf d3 ur dy 3

ð24Þ

where CL is the lift coefficient and CL  0:5, uf is the velocity of the fluid phase in main flow direction, ur is the relative velocity of sediment with respect to fluid surrounding it, which is usually defined according to [19] as ur ¼ uf  us þ ud where us is the mean velocity of particles along main flow direction, uf is the mean flow velocity along main flow direction and ud is the drift velocity. The drift velocity occurs due to correlation between the instantaneous particle distribution and fluid velocity fluctuation which can be modeled as [19, 20, 57]   1 oC 1 oð1  CÞ ð25Þ  ud ¼ D C oy 1  C oy where D is the drift diffusion. Most often it is assumed to be related to the fluid eddy viscosity. Simonin [46] suggested its relation to the correlation between vertical fluctuation of fluid and particle multiplied by the integral turbulence timescale as D ¼ st v0f v0s ¼ 

sp v0 v0 Stb  1 f s

ð26Þ

When sediment particles moves along with fluid, there exists a velocity-lag between particle phase and fluid phase and it changes with flow depth from channel bed. The lag velocity ul , is defined as ul ¼ uf  us . The lag-velocity can be modeled as [8] 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31:5     ul 4 y 1=1:5 1 mf 2=1:5 1 mf 1=1:5 5 32 32 ¼ 22 þ ð27Þ  h 4 2 u u d u d where mf is the the kinematic viscosity of fluid. Then the relative velocity ur can be written from Eqs. 25–27 as ðx0 =u Þv0f v0s ur ul oC ¼ þ u u gCð1  CÞðStb  1Þð1  qf =qs Þ oy

ð28Þ

The lift force Ly per unit mass on a spherical particle is then expressed as Ly ¼

123

6FL pqs d 3

ð29Þ

Environ Fluid Mech (2017) 17:449–472

459

The log-wake velocity distribution in sediment-laden flow through open channels with the assumption that sin2 ðpn=2Þ  3n2  2n3 for the fluid phase is expressed as [30]  uf 1 y 2P  2 3n  2n3 ¼ ln þ j u j h

ð30Þ

where P is Coles’ wake parameter, u is the shear velocity and y is the vertical distance from channel bed and h is the flow depth. The second term on the RHS of Eq. (30) denotes the effect of wake in the flow. Inserting Eqs. (28), (29) and (30) into Eq. (24), the lift force can be expressed as   y u2 h y ul / oC þ 12P ð31Þ þ 1 Ly ¼ k2 h h h y u Cð1  CÞ oy where k2 ½¼ 8CL qf =ðjqs Þ is a parameter and / is a function of y which is given as /¼

x0 =u v0 v0 gðStb  1Þð1  qf =qs Þ f s

ð32Þ

It can be observed from Eq. 31 that it consists of two terms. If P ¼ 0 and effect of drift velocity is neglected, Eq. 31 reduces to a simple form which was considered in several previous studies [14, 44, 50, 52, 57]. The value of P may be positive or negative [21, 31]. Generally in open channel flows, large cellular secondary vortices are generated due to variation of bed roughness or bed elevation along lateral direction which has height equal to flow depth and span wise spacing of twice of the flow depth. Figure 2 shows a schematic diagram of such type of secondary vortices. From the figure it can be observed that circulation is in upward direction over sand ridges consisting of fine sands and it is downward over sand troughs consisting of coarse sands. Kundu and Ghoshal [30] and Kundu [29] have studied extensively the effects of secondary circulations on suspension profiles. They mentioned that secondary current has significant effect on the types of suspension concentration profile and therefore it cannot be neglected. They also found that particle suspension increases over sand ridges and decreases over sand troughs. In the Fig. 2, the line of boil denotes high sediment zone. Their study supports the findings of previous studies by Coleman [11] who observed boil lines on arial photographs of the Brahmaputra River, India. These suggests that large secondary vortices has significant

h

Coarse sand

Sang ridge (fine sand)

River bed 2h

Fig. 2 Schematic diagram of cellular secondary vortices in open channel flows

123

460

Environ Fluid Mech (2017) 17:449–472

impact on suspension of particles in vertical direction. The effect of secondary vortices on concentration profile has not been considered till now. This effect is considered in this study by the consideration of the wake effect in the velocity profile in Eq. 30. Consideration of this effect in the present study makes a significant difference in the proposed model from previous studies on concentration distribution. This makes the present study more effective than previous studies. Consequently, the external force acting on sediment particles can be expressed as   qf x0 þ ðStb  1ÞCqs Fsy ¼ Cqs g 1  qs sp   2  ð33Þ y u h y ul / oC þ 12P þk2 Cqs þ 1 h h h y u Cð1  CÞ oy

4 Dimensionless form of concentration distribution equation Substitution of Eq. 33 into Eq. 12 gives the momentum equation of the solid phase as ov02 oC x0 s ¼ ðStb  1ÞC þ þ v02 s oy oy sp   2  y u h y ul / oC 1 ors 1 oRs þ 12P 1 þ þ þ k2 C h h qs oy qs oy h y u Cð1  CÞ oy C

ð34Þ

In our study we emphasize to give a mathematical model for concentration distribution of type II profile. There are several closures available for sediment diffusion coefficient in literature. Using particle turbulent intensity, the sediment diffusion coefficient is expressed as es ¼ sp v02 s . Therefore Eq. 34 can be rewritten as   y ov02 oC h y

¼ Cx0 ðStb  1Þ þ k2 Stb Cu þ 12P 1 : sp C s þ es oy y h h oy  ð35Þ ul / oC sp ors sp oRs þ þ þ u Cð1  CÞ oy qs oy qs oy where Stb ð¼ sp u =hÞ is the bulk Stokes’ number introduced by [20]. The distribution of suspended particles is investigated by introducing the following dimensionless variables as y n¼ ; h

eþ s ¼

es ; u h

xþ 0 ¼

x0 ; u

v0þ2 ¼ s

v02 s ; u2

rþ s ¼

rs ; qs u2

Rþ s ¼

Rs qs u2

ð36Þ

Equation 35 then becomes   ov0þ2 1 þ oC þ s ¼ Cx0 ðStb  1Þ þ k2 Stb C þ 12Pnð1  nÞ : þe Stb C n on  s on ul ð/=hÞ oC orþ oRþ þ Stb s þ Stb s þ u Cð1  CÞ on on on Equation 37 can further be written as

123

ð37Þ

Environ Fluid Mech (2017) 17:449–472



eþ s

461

"  ovþ2 k2 Stb wðnÞ/þ oC orþ oRþ þ s s s0   C ¼ Cx0 ðStb  1Þ þ Stb k2 CwðnÞuþ þ þ l on 1C on on on ð38Þ

þ where uþ l ¼ ul =u and functions w and / are given as

1 wðnÞ ¼ þ 12Pnð1  nÞ; n

/þ ¼

/ x0 =u ¼ v0 v0 h ghðStb  1Þð1  qf =qs Þ f s

ð39Þ

The term on the left-hand side of Eq. 38 represents the diffusion caused by the vertical gradient of sediment concentration and drift velocity of particles. The bulk stokes number Stb is related to particle relaxation time sp . For fine sediment particles, relaxation time is smaller compared to the medium or coarse particle. Therefore for fine sediment particles, Stb  1 can be considered and k2 is introduced here as the fluid lift coefficient. If in the flow only fine sediments are transported and if there is no significant amount of lift force; drift velocity of particles, particle-particle interactions and Reynolds like stress are neglected, then it can be seen from Eq. 38 that for Stb  1, k2  1 and for very small particles, Eq. 38 reduces to eþ s

oC ¼ Cxþ 0 on

ð40Þ

which is the widely used diffusion equation in previous studies. In other words, the present study shows that the description based on diffusion theory for suspended sediment distribution is applicable only in the case when Stb  1, fluid lift force and effect of drift velocity, particle-particle interactions and Reynolds like stresses are negligible, and if we consider particles are very small. Various results also show that Rouse type profile is well applicable only in dilute flow conditions [25, 26, 49]. Concentration distribution of suspended particles can be determined from Eq. 37 if the turbulent intensity of solid phase in vertical direction can be determined.

4.1 Closure equations þ þ 0 0 0þ2 From Eq. 38 one can observe that variables eþ s , vf vs , vs , rs and Rs require modeling before determining concentration profile of suspended load in turbulent open-channel flows. The closure of these variables can be determined by using the kinetic theory of granular flows and by solving turbulence models, such as the k  e model proposed by Celik and Rodi [6]. In this study instead of applying turbulence models simple analytical or empirical relations are used which was also adopted by Greimann and Holly [19], Jiang et al. [27] and Fu et al. [15]. Empirical equation for the eddy viscosity ef of the fluid is used and an polynomial approximation of the turbulence intensity v02 s of sediment particles has been introduced. One can estimate sediment diffusivity from es ¼ spt v02 s . But it contains spt which is difficult to determine. Therefore in this study we estimate sediment diffusivity from Reynolds analogy. According to Reynolds analogy, the sediment diffusivity coefficient es is usually taken to be proportional to the fluid eddy viscosity ef as

123

462

Environ Fluid Mech (2017) 17:449–472

es ¼ cef

ð41Þ

where c is the proportionality constant which is inverse of the turbulent Schmidt number. Several investigations regarding the variation of parameter c with various flow characteristics and therefore in the present study the value of this parameter is calculated from experimental data Kundu and Ghoshal [30]. Experimental measurements of the eddy viscosity in open channels by Nezu and Nakagawa [39, p. 66], show that the eddy viscosity ef of the fluid phase can be modeled by standard parabolic distribution as ef y y

ð42Þ ¼j 1 h h u h in which jð¼ 0:4Þ is the von Karman coefficient in clear water. Several authors [45, 55] used Eq. 42 to describe velocity profile in sediment-laden open-channel flows. Following the studies of Greimann and Holly [19] and Greimann et al. [20], it is assumed that the covariance of the turbulent intensities of fluid and particle is equal to the covariance of the fluid fluctuation velocities; i.e., v0f v0s ¼ v02 f

ð43Þ

This assumption may overestimate the fluid-sediment correlations, As the sediment does not respond instantaneously to changes in the surrounding fluids velocity, this assumption may overestimate the fluid-particle correlation. The particulate stress arising from particle– particle interactions is described as [57] 2 rs ¼ qs Csy

ð44Þ

2 denotes the peculiar velocity of sediment particles. Similar to the kinetic theory where Csy 2 as of granular flows, Zhong et al. [57] proposed a closer for the term Csy

4 2 ¼ Cg ð1 þ eÞk Csy 0 p 3

ð45Þ

where kp ¼ ð3=2Þv02 s is turbulent kinetic energy of particles; e is the restitution coefficient of particle collision with the value e ¼ 0:95; and g0 is radial distribution function, which is given by Ding and Gidaspow [13] as g0 ¼

1 1  ðC=Cm Þ1=3

ð46Þ

where Cm denotes the maximum volumetric concentration in particle packing. Following [37], Cm ¼ 0:74 is considered in this study. Similarly the Reynolds like shear stress is modeled as [57] Rs ¼ qs v0s v0s ¼ qs v02 s

ð47Þ

The relationship between vertical turbulent intensity of particles and that of fluid is not clearly known. Experimental measurements by Muste and Patel [38] and Cellino and Graf [7] show that particle turbulent intensity is lower than fluid turbulent intensity far from the near-bed region. On the other hand, experimental results of Garcya et al. [16] and Liu and Singh [34] suggest that in the near-bed region vertical turbulence intensity of particles is

123

Environ Fluid Mech (2017) 17:449–472

463

higher than that of the fluid. Generally, in the turbulent flow over sediment bed, due to the high concentration in the near bed region turbulent intensity of fluid is damped by the presence of particles; whereas far from bed, the sediment concentration of particles is small and therefore turbulent intensity of fluid is higher than that of sediment particles. Overall consideration says that the turbulent intensity of particles can be considered to be proportional to the turbulent intensity of fluid. Since in this study we concentrate on the deviation of concentration profile in the near-bed region. Therefore following Greimann and Holly [19], vertical turbulent intensity of particle is assumed as 02 v02 s ¼ Ds vf

ð48Þ

where v02 f is the turbulent intensity of fluid phase and Ds denotes the damping of particle turbulence relative to fluid and the value is considered as 1.2 as proposed by Greimann and Holly [19] considering that the sediment turbulent intensity is slightly larger than fluid in the near-bed region. The fluid turbulence intensity is obtained by modifying the clear water expression of Nezu and Rodi [40] by an empirical coefficient Df as [19] 2 v02 f ¼ Df Cv u expð1:34nÞ

ð49Þ

where Cv ¼ 1:51 as proposed by Nezu and Rodi [40] and Df is the damping of turbulence due to presence of sediment particles and is expressed as [19] rffiffiffiffiffiffi jm Df ¼ ð50Þ j where jm is the von Karman coefficient of sediment-water mixture and jð¼ 0:4Þ is the von Karman coefficient in clear water. Substitution of Eqs. 49–50 into Eq. 48 gives the vertical turbulent intensity of particles as rffiffiffiffiffiffi jm 0þ2 vs ¼ Ds Cv expð1:34nÞ ð51Þ j has been plotted in Fig. 3 for jm ¼ 0:35. where jm is a free parameter. The profile of v0þ2 s From Fig. 3 one can observe that intensity of particles due to turbulence gradually decreases from bed surface towards the free surface. The form of Eq. 51 is not simple and therefore leads to more complicated equation when substituted into Eq. 37 of concentration Fig. 3 Polynomial approximation of v0sþ2

123

464

Environ Fluid Mech (2017) 17:449–472

distribution. Therefore a simple polynomial approximation to Eq. 51 is required. Figure 3 suggests that a second order polynomial could be well fitted to the profile which is given as v0þ2 ¼ l0  nðl1  nÞ s

ð52Þ

where l0 and l1 are constants to be determined. Comparing Eqs. 51 and 52 and using boundary conditions at n ¼ 0 and n ¼ 1, the value of constants are obtained as pffiffiffiffiffiffiffiffiffiffiffi ð53Þ l0 ¼ Ds Cv jm =j and l1 ¼ 1 þ l 0  D s C v

pffiffiffiffiffiffiffiffiffiffiffi jm =j expð1:34Þ

ð54Þ

Equation 52 is also plotted in Fig. 3 with the value of parameters l0 and l1 from Eqs. 53 and 54 respectively. From the figure one can observe that the proposed approximation Eq. 52 agree well with Eq. 51 in the region 0:01  n  0:3 and slightly deviates from Eq. 51 in the region 0:3\n  1. The coefficient of determination is R2 ¼ 0:995. To simplify the calculation, Eq. 52 is used in this study instead of Eq. 51.

5 Final governing equation Substituting Eqs. 44 and 47 into Eq. 38 one get the differential equation for the sediment distribution as " #  þ2 oCsy k2 Stb wðnÞ/þ oC ov0þ2 þ þ þ s ¼ Cx0 ðStb  1Þ þ Stb k2 CwðnÞul  ð1 þ CÞ  es  on 1C on on ð55Þ þ2 ¼ C 2 =u2 . It can be observed from Eq. 45 that C þ2 depends on suspension where Csy sy sy  concentration. Therefore using Eqs. 45 and 55 can be expressed as  k2 Stb wðnÞ/þ oC 2 þ þ A0 Stb ðl0  l1 n þ n Þg1 ðCÞ es  on 1C

 þ ¼ Cxþ ðSt  1Þ þ St k CwðnÞu  ð2n  l Þ f 1 þ C þ A0 f1 ðCÞg ð56Þ b b 2 0 0 l

where A0 ¼ 2ð1 þ eÞCm1=3 and functions f1 ðCÞ and g1 ðCÞ are given as f1 ðCÞ ¼

C 1=3

Cm  C1=3

and

g1 ðCÞ ¼

3Cm1=3  2C 1=3 1=3

3ðCm  C1=3 Þ2

ð57Þ

Equation 56 gives the differential equation for suspension distribution. It is solved numerically using a standard 4th-order Runge–Kutta method with the initial condition C ¼ Ca at the reference level n ¼ na . The values of reference level and reference concentration is taken from experimental data in this study. From Eq. 56 it can be seen that sediment concentration depends on parameters Stb and k2 . The effect of these parameters on suspension profile is discussed next. To show the effects of bulk Stokes’ number and fluid lift force on suspension concentration profile, examples are considered. Figure 4 shows the variation of suspension

123

Environ Fluid Mech (2017) 17:449–472

465

0

10

St = 0.0005 b

ξ(= y/h)

St = 0.0007 b

Type II profile

St = 0.001 b

−1

10

Type I profile

St = 0.003 b

−2

10

−2

10

−1

0

10

10

1

10

C/C

Fig. 4 Variation of suspension concentration with bulk Stokes’ number Stb

concentration for different values of Stokes’ number where the value of the parameters are kept as: na ¼ 0:03, Ca ¼ 4 103 , d ¼ 2:4 mm, k2 ¼ 1, l1 ¼ 2:32, c ¼ 0:1, x0 ¼ 3:51 cm/s and P ¼ 0:2. Also in this example, dilute sediment concentration is assumed. In Fig. 3 concentration profiles are plotted for five different values of Stokes’ number Stb ¼ 5 104 ; 7 104 ; 1 103 and 3 103 . From the figure one can observe that when Stokes number is comparatively small, Type I profile is obtained or Rouse type profile is obtained. This indicates that Rouse equation is valid only for small Stokes’ number. Also from the example it is observed that when k2 ¼ 1, Type I profile is obtained for Stb \0:0006 and when Stb exceeds this value Type II profile is obtained. This indicates that in dilute flow the pattern of suspension distribution depends on bulk Stokes’ number. When Stokes’ number is large, particles experience additional diffusion due to the turbulent fluctuations of particles. Further it is also found that the suspension profile is less sensitive with the change of the parameter k2 .

6 Comparison with experimental data To test the applicability of the model presented in this study for open channel flows, experimental data of Bouvard and Petkovic [4] and Wang and Qian [54] are selected. These data sets satisfy the criteria for the present study i.e., maximum volumetric concentration Cmax appears at some distance significantly above the channel bed, spanned a significant range of Stokes’ number and have been widely cited in literatures [41, 44, 51, 52]. The present model is also compared with other existing models for type II profile of Wang and Ni [52], Ni et al. [44] and Zhong et al. [57]. The detail of the models are shown in Table 1. To test the accuracy of the proposed model Eq. ??, Mean Absolute Standard Error (MASE) is calculated as MASE ¼

n 1X Mi n i¼1

ð58Þ

where n is the total number of observed data points and

123

466

Environ Fluid Mech (2017) 17:449–472

Table 1 Summary of some analytical models with applicability Author

Equations

Wang and Ni [52]

C C

Applicability

¼ ZC nf1 expfZ ng

Dilute flow

0 p6d ffiffiffiffiffiffiffiffi for pipe flow; where Z ¼ 5x u and f ¼ 1 þ

h

qs =qf

0 p18d ffiffiffiffiffiffiffiffi Z ¼ 15x u and f ¼ 1 þ h qs =qf for open channel flow and R 1 f1 C ¼ 0 n expfZ ngdn.

Ni et al. [44]

C C

0

 ng:½1þc expfðA LÞng ¼ cexpfA 0 ð1expfLgÞþ 1 ð1expfA gÞ L

where c0 ¼ Zhong et al. [57]



A

1t1 t3 t1 t2 1; t1

¼

Ca ;t C 2

¼ L1 ð1  eL Þ; t3 ¼ A1 ð1  eA Þ; A ¼

a0 x0 u .

C 1Ca Ca 1C

¼ F1 ðStb ; Z; b; nÞ:F2 ðStb ; Z; b; nÞ h1 iZ ðvþ1Þn 1ðv1Þþn cv where F1 ðStb ; Z; b; nÞ ¼ 21ðv1Þþn : 21ðvþ1Þna ; a 2 2  h i

1=2 k2 cbStb Z n2 na b Stb Z F2 ðStb ; Z; b; nÞ ¼ nna na2 nb : exp  k2cbv fF3 ðnÞ  F3 ðna Þg ;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D D C St , Z ¼ jxm u0  ; v ¼ 1 þ 4b; b ¼ s cjf mv b and F3 ðnÞ ¼ tanh1 2n1 v

Dilute and dense flow Dilute and dense flow

8k C q

Present model

k2 ¼ 1jqL f . s n o k2 Stb wðnÞ/þ eþ þ A0 Stb ðl0  l1 n þ n2 Þg1 ðCÞ oC ¼ Cxþ s  0 ðStb  1Þþ 1C

on þ Stb k2 CwðnÞul  ð2n  l0 Þf1 þ C þ A0 f1 ðCÞg where f1 ðCÞ ¼

C , 1=3 Cm C 1=3

and g1 ðCÞ ¼

8 Cmeasured > < observed Mi ¼ C > : Cobserved Cmeasured

Dilute and dense flow

1=3

3Cm 2C1=3 1=3 3ðCm C 1=3 Þ2

if Cmeasured [ Cobserved ð59Þ if Cmeasured \Cobserved

It can be observed from Eq. 58 that MASE 1 and a model is considered to be more accurate if MASE value is closer to one and MASE ¼ 1 represents the condition of perfect agreement. MASEs are calculated for the present model as well as for other selected models. Table 2 shows MASEs for selected test cases. In Table 2, EWN , EN , EZ and EPM denote MASEs corresponding to the models of Wang and Ni [52], Ni et al. [44], Zhong et al. [57] and the present model respectively.

6.1 Estimation of parameters To compare between various models, correct estimation of the parameters present in the models is needed. The parameters Z and f present in Wang and Ni’s model are calculated using the qffiffiffiffiffiffiffiffiffiffiffi formula Z ¼ ða3 x0 Þ=u and f ¼ 1 þ ða4 dÞ=ðh qs =qf Þ respectively as proposed by Wang and Ni [52] where a3 and a4 are parameters. The value of other parameters d, h, x0 , u , qs and qf are taken from experimental data. In this study, a3 ¼ 15 and a4 ¼ 18 are considered as suggested by Wang and Ni [52]. Model of Ni et al. [44] contains three parameters A , L and c0 . Estimation of c0 requires the value of reference concentration Ca at the bed level. In this study, the value of reference

123

0.96

0.96

1.42

1.42

Run SM6

Run SM7

Run SC5

Run SC7

0.268

Run SF5

9

Run 4

0.268

5

Run 3

Run SF2

2

Run 2

Wang and Qian [54]

2.4

Run 1

Bouvard and Petkovic [4]

Mean diameter d (mm)

Run no.

Authors

1.052

1.052

1.052

1.052

1.052

1.052

1.006

1.011

1.04

1.04

Particle density qs (g/cm3 )

2.29

2.29

1.59

1.59

0.197

0.197

1.81

1.86

2.01

2.27

Settling velocity x0 (cm/s)

7.37

7.37

7.37

7.37

7.16

7.74

2.67

2.67

2.67

5.41

Shear velocity u (cm/s)

12.25

6.51

13.72

7.54

9.06

1.02

0.45

0.37

0.21

0.32

Average con. C ( 102 )

Table 2 Summary of experimental data with MASEs ( corresponds to minimum error)

32.52

32.52

15.26

15.26

0.53

0.53

100.9

93

40.2

54.48

Particle Reynolds no. Rep

0.0348

0.0348

0.0242

0.0242

0.0036

0.0031

0.1101

0.0620

0.0190

0.0434

Stokes’ no. Stb

0.85

0.82

0.97

0.74

0.68

1.07

0.76

0.74

0.70

0.69

k2

1.04

1.28

1.10

1.30

1.24

0.62

0.78

0.62

0.90

0.68

c

–

–

–

–

–

–

1.7049

1.3014

1.6393

1.4735

EWN

1.0881

1.0311

1.0543

1.0642

1.0529

1.0226

1.0499



1.1261

1.1003

1.0513

1.0746

1.0147

1.0289

1.6723 1.0317

1.5370

1.3004 

1.1339

1.3194

EPM

1.5333

1.6780

1.3495

EZ



1.0521

1.0486

1.0397

1.0582

1.7276

1.6457

1.8549

1.3204

EN

Environ Fluid Mech (2017) 17:449–472 467

123

468

Environ Fluid Mech (2017) 17:449–472

concentration is taken from experimental data. The value of other parameters A and L are calculated using the formula A ¼ ða0 x0 Þ=u and L ¼ 3 respectively where a0 is a parameter which is determined from experimental data. The value of average volumetric concentration Cavg is taken from experimental data and for the data set of Bouvard and Petkovic [4] it is computed from the formula proposed by Wang and Ni [52] which is given R1 expfZ na gC where C ¼ 0 nf1 expfZ ngdn. as Cavg ¼ Ca n1f a In the model of Zhong et al. [57], the value of all parameters are calculated as mentioned by them except the parameters c and jm which are treated as free parameters and these are determined from experimental data. The estimated value of jm from experimental data shows that von Karman coefficient of the mixture is not a universal constant and jm  0:4. For the computation purpose, the characteristic reference level na is taken from the available experimental data above the bed surface. The proposed model contains several parameters (e.g. na , Ca , k2 , l1 , c and P). The value of parameters na and Ca are taken from experimental data as no such formula for type II profile is available in the literature to the best of the authors’ knowledge. According to [12], the value of parameter P increases with the increase of Reynolds number R ¼ ðu hÞ=mf and tends to a constant for large Reynolds number. Laser Doppler Anemometry (LDA) velocity measurements in 2D fully-developed open-channel flow over smooth beds indicated that P  0:2 for R [ 2000 [40]. Other parameters k2 and c are considered as free parameters whose values are calculated by using the least-square fitting technique. A MATLAB programme has been written to compute these values. Validity of the proposed model under different flow conditions through open-channels are discussed separately in the following sections.

6.2 Verification with experimental data Experimental data set of Bouvard and Petkovic [4] has been used to verify the present model through open-channels. The experiments were carried out in a laboratory flume of 0.35 m wide and 11 m long with adjustable slope. Throughout the experiment flow depth was kept as 0.075 m. Spherical plastic particles with diameter having the range from 2 to 9 mm have been used in the experiment. As size of particles are large enough, particles can be considered as gravel particle. Other flow conditions are given in Table 2.

0.4

0.6

Run 1

0.8

Run 2 0.3

ξ

ξ

ξ

0.2

0.4

0

0

0.2

0.4

C/Ca

Fig. 5 denote denote denote

0.6

0

0.6

0.3

0.4

0.2

0.1

0.2

0.1 0

1

2

C/Ca

3

0

Run 4

0.8

0.4

0.6

0.2

1

Run 3

0.5

ξ

1

0

0.5

C/Ca

1

0

0

0.2

0.4

0.6

C/C

Relative concentration profiles for particle suspension in dilute flow through open-channels. Squares data of Bouvard and Petkovic [4], dashed lines denote model of Wang and Ni [52], dotted lines model of Ni et al. [44], dash dotted lines denote model of Zhong et al. [57] and continuous lines the proposed model

123

Environ Fluid Mech (2017) 17:449–472

469

In Fig. 5 comparison of relative concentration profiles obtained from the present model together with the experimental data of Bouvard and Petkovic [4] is presented. Selected previous models proposed by Wang and Ni [52], Ni et al. [44] and Zhong et al. [57] are also plotted in this figure. From the figure it can be observed that the proposed model agrees well with the experimental data throughout the flow depth and well predicts the near-bed deficiency of particles. To measure the accuracy of models with the experimental data, MASEs are calculated and shown in Table 2 for all test cases. In most of the test cases, the present model gives minimum MASE compared to other models which indicates the efficiency of the present model. From Fig. 5 one can observe that the location of maximum concentration point shifts upward with the increase of particle diameter as well as Stokes’ number. The present model predicts this phenomenon well. Due to significantly lareg Stokes’ number, particles obtain additional diffusion due to turbulent fluctuation. These effects help gravel particles to move upward and as a consequence maximum concentration point significantly rises above the bed level. From the Table 2 one can observe that value of k2 slightly decreases with the increase of particle density; which is consistent with the assumption of the parameter as k2 v 1=qs . This experimental results indicate that type II profile occurs for gravel particles and proposed model can be applied to predict near-bed deficiency of particle concentration in dilute open channel flows. Refined data set of Wang and Qian [54] are also used to check the validity of the present model through open-channels. The experiment was conducted in recirculating flume of length 20 m, width 30 cm with a wide range of sediment concentrations from 0.62 to 21%. Three types [fine (SF), medium (SM) and coarse (SC)] of plastic particles having density 1.05 gm/cm3 and sand particles having density 2.64 gm/cm3 have been used to examine the effects of light and heavy particles. In this experiment type II profile is observed for plastic particles only. The Stokes’ number has a range from 0.003 to 0.035. Detail flow conditions are shown in Table 2. Figure 6 compares the present model with the experimental data of Wang and Qian [54] together with previous models of Ni et al. [44] and Zhong et al. [57]. Average concentration in selected test cases has the range from 1.02 to 13.72%. From Fig. 6 one can observe that present model can predict well the type II profile of suspension concentration throughout the flow depth. It is important to mention here that in the present model, the effect of particle-particle interaction and drift diffusion is included in the modele. It can be observed from Fig. 6 that due to the inclusion of these effects, the obtained results agree well with observation. This indicates that interact between particles, particle inertia, drift diffusion are important factors for this type of suspension profile. Also in the present study, the particle turbulent intensity is considered as function of flow depth rather than a constant. This also makes the obtained results more appropriate. The MASEs corresponding to selected test cases are given in Table 2. In average number of test cases, minimum error corresponds to the present model which indicates that particle deficiency in the near bed region is reasonably predicted by the present model. It can be observed also that the location of maximum concentration point is well predicted by the present model for open channel flow.

123

470

Environ Fluid Mech (2017) 17:449–472 1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

SF2

0.2 0

ξ

ξ

ξ

1

0

0.4

SF5

0.2

0.5

1

1.5

0

0

0.5

C/C

1

1.5

0

0.8

0.8

0.6

0.6

0.6

0.4

0.2 1

C/C

2

0

0.4

SC5

0.2 0

2

ξ

ξ

0.8

ξ

1

SM7

1

C/C

1

0

0

C/C

1

0.4

SM6

0.2

0

SC7

0.2 1

C/C

2

0

0

1

2

C/C

Fig. 6 Relative concentration profiles for particle suspension in dense flow through open-channels. Solid circles denote data ofWang and Qian [54], dotted lines denote model of Ni et al. [44], dash dotted lines denote model of Zhong et al. [57] and continuous lines denote the proposed model

7 Conclusions In this paper, a distribution equation for type II profile of suspension concentration is derived based on two-fluid model. The effects of lift force on sediment particles, particle inertia and particle velocity fluctuation are considered in the governing equation. Finally the equation is solved numerically. The model is validated with a wide range of experimental data and a good agreement is found between computed and observed values. The model is also compared with other models existing in the literature. To get a quantitative idea about the goodness of fit, the MASE is calculated. Minimum error of the present model in comparison to the other models indicates that the present model can be employed to find suspension profile of type II. From the analysis it is found that particle inertia, particle-particle interactions, drift diffusion and lift force are influential factors for type II profile of concentration distribution. It is observed that when both Stokes’ number Stb and lift force coefficient k2 are small, Rouse type profile or type I profile is obtained. For a fixed value of k2 , if the value of Stb is increased, pattern of concentration distribution changes from type I profile to type II profile.

123

Environ Fluid Mech (2017) 17:449–472

471

References 1. Almedeij J (2009) Asymptotic matching with a case study from hydraulic engineering. In: Recent Advances In Water Resources, Hydraulics and Hydrology, pp. 71–76. Cambridge 2. Auton TR (1987) The lift force on a spherical body in a rotational flow. J Fluid Mech 183:199–218 3. Batchelor GK (1951) Pressure fluctuations in isotropic turbulence. Proc Cam Phil Soc 47(2):359–374 4. Bouvard M, Petkovic S (1985) Vertical dispersion of spherical, heavy particles in turbulent open channel flow. J Hydraul Res 23(1):5–20 5. Cao Z, Wei L, Xie J (1995) Sediment-laden flow in open channels from two-phase flow viewpoint. J Hydraul Eng 121(10):725–735 6. Celik I, Rodi W (1988) Modeling suspended sediment transport in nonequilibrium situations. J Hydraul Eng 114(10):1157–1191 7. Cellino M, Graf WH (1997) Measurements on suspension flow in open channels. In: Proceedings of 27th IAHR Conference on Environment and Coastal Hydraulics, vol 1, pp 179–186. IAHR, Delft 8. Cheng NS (2004) Analysis of velocity lag in sediment-laden open channel flows. J Hydraul Eng 130(7):657–666 9. Cheng NS (2009) Comparison of formulas for drag coefficient and settling velocity of spherical particles. Powder Technol 189:395–398 10. Chien N, Wan Z (1999) Mechanics of sediment transport. ASCE, New York 11. Coleman JM (1969) Brahmaputra river; channel process and sedimentation. Sedimen Geol 3:129–239 12. Coles DE (1956) The law of the wake in the turbulent boundary layer. J Fluid Mech 1:191–226 13. Ding J, Gidaspow D (1990) A bubbling fluidization model using kinetic theory of granular flow. AIChE J 36(4):523–538 14. Fu XD, Wang GQ, Dong Z (2001) Theoretical analysis and numerical computation of dilute solid/liquid two-phase pipe flow. Sci China Ser E 44(3):298–308 15. Fu XD, Wang GQ, Saho XJ (2005) Vertical dispersion of fine and coarse sediments in turbulent openchannel flows. J Hydraul Eng 131(10):877–888 16. Garcya M, Nino Y, Lopez F (1996) Coherent flow structures in open channels In: Laboratory observations of particle entrainment into suspension by turbulent bursting, pp 63–86. Wiley, New York 17. Ghoshal K (2004) On velocity and suspension concentration in a sediment-laden flow: experimental and theoretical studies. Ph.D. thesis, Indian Statistical Institute, Kolkata 18. Glenn SM (1983) A contenental shelf bottom boundary layer model: the effects of waves, currents and a moveable bed. Ph.D. thesis, Massachusetts Institute of Technology, Massachusetts 19. Greimann BP, Holly FM (2001) Two-phase analysis of concentration profiles. J Hydraul Eng 127(9):753–762 20. Greimann BP, Muste M, Holly FM (1999) Two-phase formulation of suspended sediment transport. J Hydraul Res 37(4):479–500 21. Guo J, Julien PY (2008) Application of the modified log-wake law in open-channels. J Appl Fluid Mech 1(2):17–23 22. Hetsroni G (1989) Particles-turbulence interaction. Int J Multiph Flow 15(2):735–746 23. Hetsroni G (1993) The effect of particles on the turbulence in boundary layer In: Rocco M.C.(ed.) Particulate two phase flow, Butterworth Heinemann, Wobum 24. Hung TC, Hung TK, Bugliarello G (1980) Blood flow in capillary tubes: curvature and gravity effects. Biorehology 17:331–342 25. Hunt JN (1954) The turbulent transport of suspended sediment in open channels. Proc R Soc Lond A 224:322–335 26. Jha SK, Bombardelli FA (2010) Toward two-phase flow modeling of nondilute sediment transport in open channels. J Geophys Res 115(F03015) 27. Jiang JS, Law AWK, Cheng NS (2004) Two-phase modeling of suspended sediment distribution in open channel flows. J Hydraul Res 42(3):273–281 28. Kaushal DR, Tomita Y (2007) Experimental investigation for near-wall lift of coarse particles in slurry pipeline using c-ray densitometer. Powder Technol 172:177–187 29. Kundu S (2016) Effect of lateral bed roughness variation on particle suspension in open channels. Environ Earth Sci 75(8):1–18 30. Kundu S, Ghoshal K (2014) Effects of secondary current and stratification on suspension concentration in an open channel flow. Environ Fluid Mech 14(6):1357–1380 31. Kundu S, Ghoshal K (2014) Explicit formulation for suspended concentration distribution with near-bed particle deficiency. Powder Tech 253:429–437 32. Liu D (1993) Fluid dynamics of two-phase systems. Chinese Higher Education Press, Beijing

123

472

Environ Fluid Mech (2017) 17:449–472

33. Liu QQ, Shu AP, Singh VP (2007) Analysis of the vertical profile of concentration in sediment-laden flows. J Eng Mech 133(6):601–607 34. Liu QQ, Singh VP (2004) Fluid-solid interaction in particle-laden flows. J Eng Mech 130(12):1476–1485 35. Maxey MR, Riley JJ (1983) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26:883–889 36. Michalik A (1973) Density patterns of the inhomogeneous liquids in the industrial pipeline measured by means of radiometric scanning. La Houille Blanche 1:53–59 37. Mih WC (1999) High concentration granular shear flow. J Hydraul Res 37(2):229–248 38. Muste M, Patel VC (1997) Velocity profiles for particles and liquid in open-channel flow with suspended sediment. J Hydraul Eng 123(9):742–751 39. Nezu I, Nakagawa H (1993) Turbulence in open-channel flows. IAHR Monograph, Balkema, Rotterdam 40. Nezu I, Rodi W (1986) Open-channel flow measurements with a laser dropper anemometer. J Hydraul Eng 112(5):335–355 41. Ni JR, Huang XJ, Borthwick AGL (2003) Characteristics of hyperconcentrated sediment-laden flows. J Hydraul Eng Mech 129(12):1489–1493 42. Ni JR, Wang GQ (1987) On the two patterns of vertical distribution of sediment concentration and their formation causes. J Hydraul Eng 7:60–68 43. Ni JR, Wang GQ (1991) Vertical sediment distribution. J Hydraul Eng 117(9):1184–1194 44. Ni JR, Wang GQ, Borthwick AGL (2000) Kinetic theory for particles in dilute and dense solid-liquid flows. J Hydraul Eng 126(12):893–903 45. Odgaard A (1986) Meander flow model i. J Hydraul Eng 112(12):1117–1150 46. Simonin O (1991) Prediction of the dispersed phase turbulence in particle-laden jets. Gas-solid flows, American Society of Mechanical Engineers-Fluid Engineering Division 121:197206 47. Toorman A (2008) Vertical mixing in the fully developed turbulent layer of sediment-laden openchannel flow. J Hydraul Eng 134(9):1225–1235 48. Uberoi MS (1953) Quadruple velocity correlations and pressure fluctuations in isotropic turbulence. J Aero Sci 20(3):197–204 49. Umeyama M, Gerritsen F (1992) Velocity distribution in uniform sediment-laden flow. J Hydraul Eng 118(2):229–245 50. Wang GQ, Fu X (2004) Mechanics of particle vertical diffusion in sediment-laden flows. Chin Sci Bull 49(10):1086–1090 51. Wang GQ, Fu XD, Huang YF, Huang G (2008) Analysis of suspended sediment transport in openchannel flows: kinetic-model-based simulation. J Hydraul Eng 134(3):328–339 52. Wang GQ, Ni JR (1990) Kinetic theory for particle concentration distribution in two-phase flows. J Eng Mech 116(12):2738–2748 53. Wang GQ, Ni JR (1991) The kinetic theory for dilute solid liquid two-phase flow. Int J Multiph Flow 17(2):273–281 54. Wang X, Qian N (1989) Turbulence characteristics of sediment-laden flows. J Hydraul Eng 115(6):781–799 55. White S, Parker G (2004) Density stratification effects in sand-bed rivers. J Hydraul Eng 130(8):783–795 56. Wilson KC, Sellgren A (2003) Interaction of particles and near-wall lift in slurry pipelines. J Hydraul Eng 129(1):73–76 57. Zhong D, Wang G, Sun Q (2011) Transport equation for suspended sediment based on two-fluid model of solid/liquid two-phase flows. J Hydraul Eng 137(5):530–542

123