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Numerical model of spherical particle saltation in a channel with a transversely tilted rough bed

Numerical model of spherical particle saltation in a channel with a transversely tilted rough bed J. Hydrol. Hydromech., 57, 2009, 3, 182­190 DOI: 10.2478/v10098-009-0017-x NIKOLAY LUKERCHENKO, SIARHEI PIATSEVICH, ZDENEK CHARA, PAVEL VLASAK* Institute of Hydrodynamics of Academy of Sciences of the Czech Republic, v. v. i., Pod Patankou 30/5, 166 12 Prague 6, Czech Republic; *Corresponding author, mailto: vlasak@ih.cas.cz, phone: +420 233109019, fax: +420 233324361 This paper deals with the numerical simulation of spherical particle saltation in a channel with a rough transversely tilted bed. The numerical model presented is based on the 3D model of spherical particle saltation developed by the authors, which takes into account the translational and rotational particle motion. The stochastic method and the concept of a contact zone were used for the calculation of a particle trajectory and its dependence on the bed lateral slope, particle diameter, and shear velocity. The effect of the bed lateral slope results in a deviation of the particle trajectory from the downstream direction. Some examples of the calculation are presented. The trajectories of the saltating particles starting their movements from one point were calculated and it was shown that they are of random character and together create a bundle or fascicle of trajectories. It was found that the centrelines of the bundles can be approximated by the straight lines for low and moderate values of the bed transverse slope, i.e. slopes less than 20°. The angle of deviation of the centreline from the downstream direction increases when the bed lateral slope and/or the particle diameters increase. However, with increasing shear velocity, the deviation angle decreases. Due to the lateral bed slope the particles are sorted according to their size, and the criteria for sorting particles were defined. An example of the particle sorting was calculated and the separable and nonseparable regions were determined. KEY WORDS: Saltation, Transversely Tilted Bed, Particle-Bed Collision, Particle Sorting. Nikolaj Lukerchenko, Siarhei Piatsevich, Zdenk Chára, Pavel Vlasák: NUMERICKÝ MODEL SALTACE KULOVITÉ CÁSTICE V KORYT S PÍCN SKLONNÝM DRSNÝM DNEM. J. Hydrol. Hydromech., 57, 2009, 3; 16 lit., 5 obr. Studie popisuje numerickou simulaci saltacního pohybu kulovité cástice v koryt s pícn sklonným drsným dnem. Pedlozený numerický model je zalozen na autory vyvinutém 3D modelu saltacního pohybu kulovité cástice, který pocítá s translacním i rotacním pohybem cástice. Pro výpocet trajektorie cástice a její závislosti na pícném sklonu dna koryta, prmru cástice a smykové rychlosti nosné kapaliny byla pouzita stochastická metoda a koncept kontaktní zóny. Vlivem pícného sklonu dna koryta dochází k odchylce trajektorie cástice od smru proudu. Trajektorie cástic zacínajících svj pohyb v jednom bod byly vypocteny a bylo ukázáno, ze trajektorie jsou náhodného charakteru a tvoí spolecn svazek trajektorií, jehoz osa mze být pro nízké a stední hodnoty pícného sklonu dna koryta aproximována pímkou. Vlivem pícného sklonu dna koryta mze dojít k roztídní cástic podle velikosti. Bylo spocteno nkolik píklad tídní, definováno kriterium tídní a urceny oblasti tídní podle velikosti cástic a sklonu dna koryta. KLÍCOVÁ SLOVA: saltace, pícn sklonné dno, kolise cástice-dno, tídní cástic. 1. Introduction The modelling of the saltation process in the case of a transversely tilted bed is important for the understanding of natural phenomena such as bank erosion, river bed topography, sediment sorting, and stream braiding (Sekine and Parker, 1992). According to Talmon et al. (1995), in this case "correct modelling is very important because it, for 182 instance, affect transverse bed slopes in bends and the wave length and damping of spatial river-bed deformations". Most of the numerical models of saltation particle movement are 2D (e.g. Wiberg and Smith, 1985; Nino and Garcia, 1994; Lee et al., 2000; Lukerchenko et al., 2006). However, a description of the saltation process in the case of a transversely tilted bed can be realised properly only if the 3D pattern of particle saltation is used (Lukerchenko et al., 2009). Sekine and Kikkawa (1992) and Lee et al. (2006) developed 3D saltation models which were deterministic for the particle motion in fluid but stochastic for the particle­bed collision stage. Neither model took into account the particle rotation and both assumed a spherical shape and uniform size of the saltating particles and a bed formed by equal particles. As a result of the particle­bed collisions, the particle gains angular velocity, which can reach a few tens of revolutions per second (Nino and Garcia, 1998). Therefore, to describe the saltation process reliably the model of saltation must take into account not only the translational but also the rotational particle motion. The present paper uses as a basis the 3D numerical model of spherical particle saltation (Lukerchenko et al., 2003, 2004, 2009), in which the particle rotation is taken into account and the problem of a particle­bed collision is solved using the concept of a contact zone. The model allows the calculation of the particle trajectory and expression of the bed lateral slope and other parameters of the effect of the saltation process on the lateral deviation of the particle trajectory. 2. The mathematical model of the saltation The principal assumptions of the model are as follows. The saltating particle is spherical and its translational and rotational movements are taken into account. The effect of turbulence fluctuations on a saltating particle is neglected. The concentration of the conveyed particles is sufficiently low to neglect mutual collisions between moving particles and the influence of the particles on fluid flow. Therefore the motion of a set of particles can be represented by the motion of a single particle. The effects of the drag force, submerged gravitational force, Basset history force, force due to added mass, Magnus force, and drag moment acting on the particle are taken into account in the model. Let us define the channel bed as a plane spaced over the top of the bed's roughness. The fluid flow is steady and uniform, and the velocity profile can be described by the logarithmic law. The bed level is inclined to the horizontal plane in the downstream direction and also in the lateral direction. The longitudinal and lateral slopes can be defined as follows. The coordinate axis Oy'' is vertical and the coordinate axis Ox'' is horizontal in the down- stream direction, that is, in the direction of the fluid velocity vector. The angle s is the angle of the channel bed slope in the downstream direction and the angle l is the angle of the channel bed slope in the lateral direction (see Fig. 1). The coordinate plane Oxz associated with the bed level can be obtained from the coordinate's plane Ox''z'' using two transformations of the coordinate system. The first is the rotation of the coordinate system Ox''y''z'' by the angle s around the axis Oz'' (see Fig. 1). The new coordinate system is Ox'y'z'. The second transformation is the rotation of the coordinate system Ox'y'z' by the angle l around the coordinate axis Ox'. The system of equations governing particle motion is written in the resulting coordinate system, Oxyz. The gravitational acceleration vector in the coordinate system Oxyz can be written as g = (g cos s, ­ g cos s cos l, g cos s sin l). (1) Fig. 1. Definition of the angle of the bed slope to the horizontal plane. Obr. 1. Definice úhl sklonu dna koryta k vodorovné rovin (particle trajectory ­ trajektorie cástice). Let us consider a spherical homogeneous particle of diameter dp and density p moving in a fluid of density and dynamic viscosity . The system of governing equations for the particle saltation motion can be written as d rO p dt =v, (2) (3) 183 dv = FD + Fm + Fg + FB + FM , dt d =M, dt (4) where t is the time, rOp (xp, yp, zp) ­ the radius-vector of the particle centre of mass, v (vx, vy, vz) ­ the vector of the velocity of the particle centre of mass, (x, y, z) ­ the vector of angular velocity of the particle rotation about its centre of mass, J ­ the particle moment of inertia, FD, Fm, Fg, FB, and FM are the drag force, force due to added mass, submerged gravitational force, Basset history force, and Magnus force per unit volume, respectively, and M is the drag torque of viscous forces acting on the particle. The particle motion is determined by two dimensionless parameters: the translational Reynolds number (or Reynolds number) Re = | vR | dp/ and the rotational Reynolds number Re = | R | r2p/, where vR = v ­ vf is the vector of particle­fluid relative velocity, vf (vfx, vfy, vfz) ­ the vector of fluid velocity, R = ­ 0.5 rot vf ­ the particle relative angular velocity, and rp = 0.5 d p is the particle radius. The flow velocity distribution can be described by the logarithmic law Eq. (8) is valid for the Reynolds number range relevant to the saltation mode of sediment transport. For the 3D case, the expression for the force due to added mass can be written as dv Fm = Cm ( v ) v f - , dt (9) where Cm =1/2 is the added mass coefficient. The submerged gravitational force is the difference between the gravitational force and the Archimedean force Fg = (p ­ ) g. (10) The Basset history force is FB = - 9 dp t dv R d . 0 d t - (11) The translational movement of the solid sphere with simultaneous rotation in the viscous fluid induced the lateral force known as the Magnus force: FM = CM ( R × v R ) , (12) where CM is the dimensionless Magnus force coefficient (Oesterle and Dinh, 1998): CM = 0.0844 Re + Re y u v fx ( y ) = * ln , v fy = v fz = 0 , k y0 (5) where k = 0.4 ­ the Karman constant; y0 = = 0.11(/u*) + 0.033 ks (Nikuradze, 1933), u* ­ the fluid shear velocity, and ks ­ the bed roughness. From Nino and Garcia (1994) the drag force term can be written as Re 0.3 0.4 + 0.75 - 0.0844 exp -0.1Re Re Re The drag moment can be written as . (13) FD = ­ 0.75 CD | vR | vR / dp, (6) M = - C 5 R R rp , (14) where dimensionless drag force coefficient CD of a rotating spherical particle moving in fluid can be calculated from the equation (Lukerchenko et al., 2008): 0.3 CD = CD 0 1 + 0.065 Re , where C is the dimensionless drag rotation coefficient of the rotating particle moving in fluid, which can be calculated according to Lukerchenko et al. (2008) as C = C 0 1 + 0.0044 Re , (7) (15) where CD 0 = 1 24 (1 + 0.15(Re) 2 + Re , 0.208 + 0.017 Re) - 1 + 104 Re -0.5 (8) is the drag force coefficient of a particle moving in fluid without rotation (Nino and Garcia, 1994). where C0 is the dimensionless drag rotation coefficient given by the experimental investigation by Sawatzki (1970) for the rotating sphere in calm water. The system of equations describing the particle motion in the channel is solved numerically using a fourth-order Runge-Kutta scheme (Lukerchenko et al., 2004). The numerical model of the particle­bed collision is described in detail by Lukerchenko et al. (2004). 3. Result of the calculations The effect of the transversely tilted bed on the particle saltation mainly results in the lateral deviation of the particles' trajectories from the downstream direction in agreement with the lateral slope of the channel bed. For the purpose of particle movement simulation, the fluid density, = 103 kg m-3, and the fluid dynamic viscosity, = 10 -3 Pa s-1 (both corresponding to water), the particle density, p = 2650 kg m-3 (corresponding to sand), and the gravitational acceleration, g = 9.81 m s-2, were used. 3.1 The bundle of saltating particle trajectories The particle trajectory has a random character and thus the bundle or fascicles of the particles' trajectories, which start from a point named the bundle origin, were studied. Some examples of the bundle of saltating particle trajectories projected onto the coordinate plane Oxz are shown in Fig. 2 for the case of the channel bed without the lateral slope (i.e. for l = 0) and for the cases of the channel bed with the lateral slope, that is, for l is equal to 5 and 10 degrees. Let us consider the bundle of particle trajectories zi = zi (x) in the projection onto the coordinate plane Oxz. The bundle axis is the line za = za (x), where za (x) is the average value of the coordinates zi of all bundle trajectories for the same coordinate x. In this case the bundle boundary is the pair of lines zb = za (x) ± z (x), where z (x) is the standard deviation of the coordinates zi (x) of the particle trajectories from the average value za (x) for the same coordinate x. The calculations showed that the bundle axis can be approximated by a nearly straight line and, similarly, the bundle boundaries can be approximated by straight lines. Let us also define a bundle region as the region between the bundle boundaries. The deviation angle p is the angle between the bundle axis and the downstream direction, and the disperse angle p is the angle between the bundle boundary and the bundle axis. The bundle boundary is determined by the standard deviation of the particle trajectories from the bundle axis. Both p and p are functions of the saltation parameters p = p(l, dp, u*, ks, p, , , g), and p = p(l, dp, u*, ks, p, , , g). (17) (16) The effects of the shear velocity u*, the particle diameter dp, and the channel bed lateral slope l are shown in Fig. 3 for three values of the shear velocity and three different particle diameters. The value of the deviation angle p increases when the value of the bed lateral slope l increases. In the studied range of l , that is from zero to 20 degrees, the dependence is close to linear. With increases in shear velocity, the deviation angle decreases, while in contrast the deviation angle increases with increases in particle diameter. 3.2 Particle sorting The effect of the particle diameter dp on the deviation angle p is shown in Fig. 4 for the lateral slope of the channel bed l = 20° and the bed roughness ks = 1 mm. The dependence of the deviation angle on particle size is a monotonously increasing function; it suggests the possibility of the selection of particles according to the diameter of the particle in a channel with a laterally sloped bed. This process can be called particle sorting. However, particle sorting is a relatively complex process which depends not only on the particle diameter but also on the lateral slope of the channel bed, liquid velocity, bed roughness, and other parameters. However, if the particle diameter varies from 0.5 mm to 5 mm, the deviation angle changes only slightly, in the range of a few degrees. Hence the problem of particle sorting due to the transversely tilted bed is connected to the question: Particles of what diameters can be sorted in this way? The answer is not simple and requires special investigation of the particle sorting criteria. Let us consider two bundles of particle trajectories of the same origin. The particles have different diameters, dp0 and dp, while all other parameters of the particles and channel, that is, their density p, shear velocity u*, bed roughness ks, and lateral slope l, are the same. It is reasonable to suppose that the particles will be separated if their bundle regions do not intersect, that is, do not have common points except the common origin. This condition can be written as (p0 + p0 < p ­ p) (p0 ­ p0 < p + p). (18) Condition (18) is the first approximation and can be called the criterion for the rough rate of sorting, because about 32% of the particle trajectories of the given bundle can exceed the bundle boundary. The criteria for the middle, fine, and very fine rates 185 Fig. 2. The bundles of saltating particle trajectories for different values of the bed level lateral slope l (dp = 1 mm, p/ = 2.65, ks = = 1 mm, u* = 0.025 m s-1). Obr. 2. Svazky trajektorií cástic pohybujících se saltací pro rzné pícné sklony dna koryta l (dp = 1 mm, p/ =2,65, ks = 1 mm, u* = 0,025 m s-1; bundle axis ­ osa svazku trajektorií, bundle boundaries ­ hranice svazku trajektorií , downstream direction ­ smr proudu). Fig. 3. The deviation angle p versus the lateral slope of the channel bed l. Obr. 3. Závislost deviacního úhlu p na pícném sklonu dna koryta l. of sorting can be determined in a similar way, and hence the bundle boundaries can be defined as 2, 2.6, and 3 standard deviations of the particle trajectories from the bundle axis, respectively. In these cases, only about 5%, 1%, and 0.3% of the particle trajectories of the given bundle can exceed the bundle boundary, respectively. Furthermore, the rough sorting criterion is used. mass, Magnus force, and drag moment act on the particle. The effect of the transversely tilted channel bed mainly results in the lateral deviation of the particle trajectories from the downstream direction; the deviation is in agreement with the lateral slope of the channel bed. Fig. 4. The deviation angle p of the bundle axis from the downstream direction versus the particle diameter dp (l = 20°, p/ = 2.65, ks = 1 mm, u* = 0.07 m s-1). Obr. 4. Závislost deviacního úhlu p osy svazku na prmru cástice dp (l = 200, p/ = 2.65, ks = 1 mm, u* = 0.07 m s-1). Fig. 5. The separable and non-separable regions (dp0 = 1 mm, ks = 1 mm, p/ = 2.65, u* = 0.058 m s-1). Obr. 5. Zóny separace a neseparace (dp0 = 1 mm, ks = 1 mm, p/ = 2,65, u* = 0,058 m s-1). The calculated results of sorting the particles with diameter dp0 = 1 mm and the particles with diameters varying from 0.2 mm to 4 mm are shown in Fig. 5. The dotted line is the boundary of the separation, which indicates the minimal value of the channel bed lateral slope l for which the separation of particles with the given diameter dp and diameter dp0 = 1 mm is realised. Above this value of the lateral slope the separation occurs and below this value the separation does not occur. The higher the value of the channel bed lateral slope l, the more efficient the particle separation. 4. Conclusions A 3D numerical model of spherical particle saltation in a channel with a transversely tilted rough bed was developed. The model takes into account the translational and rotational movement of the particle and the particle-bed collisions. During the saltation, the drag force, submerged gravitational force, Basset history force, force due to added Trajectories of particles starting their movements from one point are of a random character and together create a bundle or fascicle of trajectories. Definitions of the bundle origin, bundle axis, bundle boundaries, and bundle region were introduced. Based on the particle trajectory simulation it was found that the bundle axis and the bundle boundaries can be approximated by straight lines. The angle of deviation p of the bundle axis from the downstream direction is nearly linearly dependent on the lateral channel bed slope l, varying between zero and 20 degrees. The deviation angle p increases when the channel bed lateral slope l or particle diameter dp increases and the shear velocity u* decreases. The possibility of particle sorting according to particle size occurring during their saltation movement in a channel with a transversely tilted rough bed was studied and the criteria for particle separation were defined. The separable and non-separable regions depending on the values of the particle diameter and the channel bed lateral slope were determined. Acknowledgements. The authors gratefully acknowledge the support of the Grant Agency of the Czech Republic, through project no. GA103/06/1487, and the Academy of Sciences of the Czech Republic, through Institutional Research Plan AV0Z20600510. List of symbols CD CD0 Cm CM C C0 ­ drag force coefficient [­], ­ drag force coefficient in the case R = 0 [­], ­ force due to added mass coefficient [­], ­ Magnus force coefficient [­], ­ drag rotation coefficient [­], ­ drag rotation coefficient in the case vR = 0 [­], ­ diameter of the moving particle [m ], ­ Basset history force per unit volume [N m-3], ­ drag force per unit volume [N m-3], ­ submerged gravitational force per unit volume [N m-3], ­ force due to added mass per unit volume [N m-3], ­ Magnus force per unit volume [N m-3], ­ vector of gravitational acceleration [m s-2], ­ particle moment of inertia [kg m2], ­ Karman constant [­], ­ bed roughness [m], ­ drag moment of viscous forces acting on a rotating particle in fluid [N m], ­ radius of the moving particle [m], ­ radius-vector of the particle centre of mass [m], ­ translational Reynolds number [­], ­ rotational Reynolds number [­], ­ time [s], ­ fluid shear velocity [m s-1], ­ vector of the fluid velocity [m s-1], ­ vector of velocity of the particle centre of mass [m s-1], ­ vector of the particle relative velocity [m s-1], ­ elevation of zero fluid velocity [m ], ­ deviation angle ­ angle of the bundle axis deviation from the downstream direction, [degrees], ­ disperse angle ­ angle between the bundle boundary and the bundle axis [degrees], ­ dynamic viscosity [Pa s], ­ angle of the bed level lateral slope [degrees], ­ angle of the bed level downstream slope [degrees], ­ fluid density [kg m-3], ­ density of the moving particle [kg m-3], http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Numerical model of spherical particle saltation in a channel with a transversely tilted rough bed

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de Gruyter
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10.2478/v10098-009-0017-x
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J. Hydrol. Hydromech., 57, 2009, 3, 182­190 DOI: 10.2478/v10098-009-0017-x NIKOLAY LUKERCHENKO, SIARHEI PIATSEVICH, ZDENEK CHARA, PAVEL VLASAK* Institute of Hydrodynamics of Academy of Sciences of the Czech Republic, v. v. i., Pod Patankou 30/5, 166 12 Prague 6, Czech Republic; *Corresponding author, mailto: vlasak@ih.cas.cz, phone: +420 233109019, fax: +420 233324361 This paper deals with the numerical simulation of spherical particle saltation in a channel with a rough transversely tilted bed. The numerical model presented is based on the 3D model of spherical particle saltation developed by the authors, which takes into account the translational and rotational particle motion. The stochastic method and the concept of a contact zone were used for the calculation of a particle trajectory and its dependence on the bed lateral slope, particle diameter, and shear velocity. The effect of the bed lateral slope results in a deviation of the particle trajectory from the downstream direction. Some examples of the calculation are presented. The trajectories of the saltating particles starting their movements from one point were calculated and it was shown that they are of random character and together create a bundle or fascicle of trajectories. It was found that the centrelines of the bundles can be approximated by the straight lines for low and moderate values of the bed transverse slope, i.e. slopes less than 20°. The angle of deviation of the centreline from the downstream direction increases when the bed lateral slope and/or the particle diameters increase. However, with increasing shear velocity, the deviation angle decreases. Due to the lateral bed slope the particles are sorted according to their size, and the criteria for sorting particles were defined. An example of the particle sorting was calculated and the separable and nonseparable regions were determined. KEY WORDS: Saltation, Transversely Tilted Bed, Particle-Bed Collision, Particle Sorting. Nikolaj Lukerchenko, Siarhei Piatsevich, Zdenk Chára, Pavel Vlasák: NUMERICKÝ MODEL SALTACE KULOVITÉ CÁSTICE V KORYT S PÍCN SKLONNÝM DRSNÝM DNEM. J. Hydrol. Hydromech., 57, 2009, 3; 16 lit., 5 obr. Studie popisuje numerickou simulaci saltacního pohybu kulovité cástice v koryt s pícn sklonným drsným dnem. Pedlozený numerický model je zalozen na autory vyvinutém 3D modelu saltacního pohybu kulovité cástice, který pocítá s translacním i rotacním pohybem cástice. Pro výpocet trajektorie cástice a její závislosti na pícném sklonu dna koryta, prmru cástice a smykové rychlosti nosné kapaliny byla pouzita stochastická metoda a koncept kontaktní zóny. Vlivem pícného sklonu dna koryta dochází k odchylce trajektorie cástice od smru proudu. Trajektorie cástic zacínajících svj pohyb v jednom bod byly vypocteny a bylo ukázáno, ze trajektorie jsou náhodného charakteru a tvoí spolecn svazek trajektorií, jehoz osa mze být pro nízké a stední hodnoty pícného sklonu dna koryta aproximována pímkou. Vlivem pícného sklonu dna koryta mze dojít k roztídní cástic podle velikosti. Bylo spocteno nkolik píklad tídní, definováno kriterium tídní a urceny oblasti tídní podle velikosti cástic a sklonu dna koryta. KLÍCOVÁ SLOVA: saltace, pícn sklonné dno, kolise cástice-dno, tídní cástic. 1. Introduction The modelling of the saltation process in the case of a transversely tilted bed is important for the understanding of natural phenomena such as bank erosion, river bed topography, sediment sorting, and stream braiding (Sekine and Parker, 1992). According to Talmon et al. (1995), in this case "correct modelling is very important because it, for 182 instance, affect transverse bed slopes in bends and the wave length and damping of spatial river-bed deformations". Most of the numerical models of saltation particle movement are 2D (e.g. Wiberg and Smith, 1985; Nino and Garcia, 1994; Lee et al., 2000; Lukerchenko et al., 2006). However, a description of the saltation process in the case of a transversely tilted bed can be realised properly only if the 3D pattern of particle saltation is used (Lukerchenko et al., 2009). Sekine and Kikkawa (1992) and Lee et al. (2006) developed 3D saltation models which were deterministic for the particle motion in fluid but stochastic for the particle­bed collision stage. Neither model took into account the particle rotation and both assumed a spherical shape and uniform size of the saltating particles and a bed formed by equal particles. As a result of the particle­bed collisions, the particle gains angular velocity, which can reach a few tens of revolutions per second (Nino and Garcia, 1998). Therefore, to describe the saltation process reliably the model of saltation must take into account not only the translational but also the rotational particle motion. The present paper uses as a basis the 3D numerical model of spherical particle saltation (Lukerchenko et al., 2003, 2004, 2009), in which the particle rotation is taken into account and the problem of a particle­bed collision is solved using the concept of a contact zone. The model allows the calculation of the particle trajectory and expression of the bed lateral slope and other parameters of the effect of the saltation process on the lateral deviation of the particle trajectory. 2. The mathematical model of the saltation The principal assumptions of the model are as follows. The saltating particle is spherical and its translational and rotational movements are taken into account. The effect of turbulence fluctuations on a saltating particle is neglected. The concentration of the conveyed particles is sufficiently low to neglect mutual collisions between moving particles and the influence of the particles on fluid flow. Therefore the motion of a set of particles can be represented by the motion of a single particle. The effects of the drag force, submerged gravitational force, Basset history force, force due to added mass, Magnus force, and drag moment acting on the particle are taken into account in the model. Let us define the channel bed as a plane spaced over the top of the bed's roughness. The fluid flow is steady and uniform, and the velocity profile can be described by the logarithmic law. The bed level is inclined to the horizontal plane in the downstream direction and also in the lateral direction. The longitudinal and lateral slopes can be defined as follows. The coordinate axis Oy'' is vertical and the coordinate axis Ox'' is horizontal in the down- stream direction, that is, in the direction of the fluid velocity vector. The angle s is the angle of the channel bed slope in the downstream direction and the angle l is the angle of the channel bed slope in the lateral direction (see Fig. 1). The coordinate plane Oxz associated with the bed level can be obtained from the coordinate's plane Ox''z'' using two transformations of the coordinate system. The first is the rotation of the coordinate system Ox''y''z'' by the angle s around the axis Oz'' (see Fig. 1). The new coordinate system is Ox'y'z'. The second transformation is the rotation of the coordinate system Ox'y'z' by the angle l around the coordinate axis Ox'. The system of equations governing particle motion is written in the resulting coordinate system, Oxyz. The gravitational acceleration vector in the coordinate system Oxyz can be written as g = (g cos s, ­ g cos s cos l, g cos s sin l). (1) Fig. 1. Definition of the angle of the bed slope to the horizontal plane. Obr. 1. Definice úhl sklonu dna koryta k vodorovné rovin (particle trajectory ­ trajektorie cástice). Let us consider a spherical homogeneous particle of diameter dp and density p moving in a fluid of density and dynamic viscosity . The system of governing equations for the particle saltation motion can be written as d rO p dt =v, (2) (3) 183 dv = FD + Fm + Fg + FB + FM , dt d =M, dt (4) where t is the time, rOp (xp, yp, zp) ­ the radius-vector of the particle centre of mass, v (vx, vy, vz) ­ the vector of the velocity of the particle centre of mass, (x, y, z) ­ the vector of angular velocity of the particle rotation about its centre of mass, J ­ the particle moment of inertia, FD, Fm, Fg, FB, and FM are the drag force, force due to added mass, submerged gravitational force, Basset history force, and Magnus force per unit volume, respectively, and M is the drag torque of viscous forces acting on the particle. The particle motion is determined by two dimensionless parameters: the translational Reynolds number (or Reynolds number) Re = | vR | dp/ and the rotational Reynolds number Re = | R | r2p/, where vR = v ­ vf is the vector of particle­fluid relative velocity, vf (vfx, vfy, vfz) ­ the vector of fluid velocity, R = ­ 0.5 rot vf ­ the particle relative angular velocity, and rp = 0.5 d p is the particle radius. The flow velocity distribution can be described by the logarithmic law Eq. (8) is valid for the Reynolds number range relevant to the saltation mode of sediment transport. For the 3D case, the expression for the force due to added mass can be written as dv Fm = Cm ( v ) v f - , dt (9) where Cm =1/2 is the added mass coefficient. The submerged gravitational force is the difference between the gravitational force and the Archimedean force Fg = (p ­ ) g. (10) The Basset history force is FB = - 9 dp t dv R d . 0 d t - (11) The translational movement of the solid sphere with simultaneous rotation in the viscous fluid induced the lateral force known as the Magnus force: FM = CM ( R × v R ) , (12) where CM is the dimensionless Magnus force coefficient (Oesterle and Dinh, 1998): CM = 0.0844 Re + Re y u v fx ( y ) = * ln , v fy = v fz = 0 , k y0 (5) where k = 0.4 ­ the Karman constant; y0 = = 0.11(/u*) + 0.033 ks (Nikuradze, 1933), u* ­ the fluid shear velocity, and ks ­ the bed roughness. From Nino and Garcia (1994) the drag force term can be written as Re 0.3 0.4 + 0.75 - 0.0844 exp -0.1Re Re Re The drag moment can be written as . (13) FD = ­ 0.75 CD | vR | vR / dp, (6) M = - C 5 R R rp , (14) where dimensionless drag force coefficient CD of a rotating spherical particle moving in fluid can be calculated from the equation (Lukerchenko et al., 2008): 0.3 CD = CD 0 1 + 0.065 Re , where C is the dimensionless drag rotation coefficient of the rotating particle moving in fluid, which can be calculated according to Lukerchenko et al. (2008) as C = C 0 1 + 0.0044 Re , (7) (15) where CD 0 = 1 24 (1 + 0.15(Re) 2 + Re , 0.208 + 0.017 Re) - 1 + 104 Re -0.5 (8) is the drag force coefficient of a particle moving in fluid without rotation (Nino and Garcia, 1994). where C0 is the dimensionless drag rotation coefficient given by the experimental investigation by Sawatzki (1970) for the rotating sphere in calm water. The system of equations describing the particle motion in the channel is solved numerically using a fourth-order Runge-Kutta scheme (Lukerchenko et al., 2004). The numerical model of the particle­bed collision is described in detail by Lukerchenko et al. (2004). 3. Result of the calculations The effect of the transversely tilted bed on the particle saltation mainly results in the lateral deviation of the particles' trajectories from the downstream direction in agreement with the lateral slope of the channel bed. For the purpose of particle movement simulation, the fluid density, = 103 kg m-3, and the fluid dynamic viscosity, = 10 -3 Pa s-1 (both corresponding to water), the particle density, p = 2650 kg m-3 (corresponding to sand), and the gravitational acceleration, g = 9.81 m s-2, were used. 3.1 The bundle of saltating particle trajectories The particle trajectory has a random character and thus the bundle or fascicles of the particles' trajectories, which start from a point named the bundle origin, were studied. Some examples of the bundle of saltating particle trajectories projected onto the coordinate plane Oxz are shown in Fig. 2 for the case of the channel bed without the lateral slope (i.e. for l = 0) and for the cases of the channel bed with the lateral slope, that is, for l is equal to 5 and 10 degrees. Let us consider the bundle of particle trajectories zi = zi (x) in the projection onto the coordinate plane Oxz. The bundle axis is the line za = za (x), where za (x) is the average value of the coordinates zi of all bundle trajectories for the same coordinate x. In this case the bundle boundary is the pair of lines zb = za (x) ± z (x), where z (x) is the standard deviation of the coordinates zi (x) of the particle trajectories from the average value za (x) for the same coordinate x. The calculations showed that the bundle axis can be approximated by a nearly straight line and, similarly, the bundle boundaries can be approximated by straight lines. Let us also define a bundle region as the region between the bundle boundaries. The deviation angle p is the angle between the bundle axis and the downstream direction, and the disperse angle p is the angle between the bundle boundary and the bundle axis. The bundle boundary is determined by the standard deviation of the particle trajectories from the bundle axis. Both p and p are functions of the saltation parameters p = p(l, dp, u*, ks, p, , , g), and p = p(l, dp, u*, ks, p, , , g). (17) (16) The effects of the shear velocity u*, the particle diameter dp, and the channel bed lateral slope l are shown in Fig. 3 for three values of the shear velocity and three different particle diameters. The value of the deviation angle p increases when the value of the bed lateral slope l increases. In the studied range of l , that is from zero to 20 degrees, the dependence is close to linear. With increases in shear velocity, the deviation angle decreases, while in contrast the deviation angle increases with increases in particle diameter. 3.2 Particle sorting The effect of the particle diameter dp on the deviation angle p is shown in Fig. 4 for the lateral slope of the channel bed l = 20° and the bed roughness ks = 1 mm. The dependence of the deviation angle on particle size is a monotonously increasing function; it suggests the possibility of the selection of particles according to the diameter of the particle in a channel with a laterally sloped bed. This process can be called particle sorting. However, particle sorting is a relatively complex process which depends not only on the particle diameter but also on the lateral slope of the channel bed, liquid velocity, bed roughness, and other parameters. However, if the particle diameter varies from 0.5 mm to 5 mm, the deviation angle changes only slightly, in the range of a few degrees. Hence the problem of particle sorting due to the transversely tilted bed is connected to the question: Particles of what diameters can be sorted in this way? The answer is not simple and requires special investigation of the particle sorting criteria. Let us consider two bundles of particle trajectories of the same origin. The particles have different diameters, dp0 and dp, while all other parameters of the particles and channel, that is, their density p, shear velocity u*, bed roughness ks, and lateral slope l, are the same. It is reasonable to suppose that the particles will be separated if their bundle regions do not intersect, that is, do not have common points except the common origin. This condition can be written as (p0 + p0 < p ­ p) (p0 ­ p0 < p + p). (18) Condition (18) is the first approximation and can be called the criterion for the rough rate of sorting, because about 32% of the particle trajectories of the given bundle can exceed the bundle boundary. The criteria for the middle, fine, and very fine rates 185 Fig. 2. The bundles of saltating particle trajectories for different values of the bed level lateral slope l (dp = 1 mm, p/ = 2.65, ks = = 1 mm, u* = 0.025 m s-1). Obr. 2. Svazky trajektorií cástic pohybujících se saltací pro rzné pícné sklony dna koryta l (dp = 1 mm, p/ =2,65, ks = 1 mm, u* = 0,025 m s-1; bundle axis ­ osa svazku trajektorií, bundle boundaries ­ hranice svazku trajektorií , downstream direction ­ smr proudu). Fig. 3. The deviation angle p versus the lateral slope of the channel bed l. Obr. 3. Závislost deviacního úhlu p na pícném sklonu dna koryta l. of sorting can be determined in a similar way, and hence the bundle boundaries can be defined as 2, 2.6, and 3 standard deviations of the particle trajectories from the bundle axis, respectively. In these cases, only about 5%, 1%, and 0.3% of the particle trajectories of the given bundle can exceed the bundle boundary, respectively. Furthermore, the rough sorting criterion is used. mass, Magnus force, and drag moment act on the particle. The effect of the transversely tilted channel bed mainly results in the lateral deviation of the particle trajectories from the downstream direction; the deviation is in agreement with the lateral slope of the channel bed. Fig. 4. The deviation angle p of the bundle axis from the downstream direction versus the particle diameter dp (l = 20°, p/ = 2.65, ks = 1 mm, u* = 0.07 m s-1). Obr. 4. Závislost deviacního úhlu p osy svazku na prmru cástice dp (l = 200, p/ = 2.65, ks = 1 mm, u* = 0.07 m s-1). Fig. 5. The separable and non-separable regions (dp0 = 1 mm, ks = 1 mm, p/ = 2.65, u* = 0.058 m s-1). Obr. 5. Zóny separace a neseparace (dp0 = 1 mm, ks = 1 mm, p/ = 2,65, u* = 0,058 m s-1). The calculated results of sorting the particles with diameter dp0 = 1 mm and the particles with diameters varying from 0.2 mm to 4 mm are shown in Fig. 5. The dotted line is the boundary of the separation, which indicates the minimal value of the channel bed lateral slope l for which the separation of particles with the given diameter dp and diameter dp0 = 1 mm is realised. Above this value of the lateral slope the separation occurs and below this value the separation does not occur. The higher the value of the channel bed lateral slope l, the more efficient the particle separation. 4. Conclusions A 3D numerical model of spherical particle saltation in a channel with a transversely tilted rough bed was developed. The model takes into account the translational and rotational movement of the particle and the particle-bed collisions. During the saltation, the drag force, submerged gravitational force, Basset history force, force due to added Trajectories of particles starting their movements from one point are of a random character and together create a bundle or fascicle of trajectories. Definitions of the bundle origin, bundle axis, bundle boundaries, and bundle region were introduced. Based on the particle trajectory simulation it was found that the bundle axis and the bundle boundaries can be approximated by straight lines. The angle of deviation p of the bundle axis from the downstream direction is nearly linearly dependent on the lateral channel bed slope l, varying between zero and 20 degrees. The deviation angle p increases when the channel bed lateral slope l or particle diameter dp increases and the shear velocity u* decreases. The possibility of particle sorting according to particle size occurring during their saltation movement in a channel with a transversely tilted rough bed was studied and the criteria for particle separation were defined. The separable and non-separable regions depending on the values of the particle diameter and the channel bed lateral slope were determined. Acknowledgements. The authors gratefully acknowledge the support of the Grant Agency of the Czech Republic, through project no. GA103/06/1487, and the Academy of Sciences of the Czech Republic, through Institutional Research Plan AV0Z20600510. List of symbols CD CD0 Cm CM C C0 ­ drag force coefficient [­], ­ drag force coefficient in the case R = 0 [­], ­ force due to added mass coefficient [­], ­ Magnus force coefficient [­], ­ drag rotation coefficient [­], ­ drag rotation coefficient in the case vR = 0 [­], ­ diameter of the moving particle [m ], ­ Basset history force per unit volume [N m-3], ­ drag force per unit volume [N m-3], ­ submerged gravitational force per unit volume [N m-3], ­ force due to added mass per unit volume [N m-3], ­ Magnus force per unit volume [N m-3], ­ vector of gravitational acceleration [m s-2], ­ particle moment of inertia [kg m2], ­ Karman constant [­], ­ bed roughness [m], ­ drag moment of viscous forces acting on a rotating particle in fluid [N m], ­ radius of the moving particle [m], ­ radius-vector of the particle centre of mass [m], ­ translational Reynolds number [­], ­ rotational Reynolds number [­], ­ time [s], ­ fluid shear velocity [m s-1], ­ vector of the fluid velocity [m s-1], ­ vector of velocity of the particle centre of mass [m s-1], ­ vector of the particle relative velocity [m s-1], ­ elevation of zero fluid velocity [m ], ­ deviation angle ­ angle of the bundle axis deviation from the downstream direction, [degrees], ­ disperse angle ­ angle between the bundle boundary and the bundle axis [degrees], ­ dynamic viscosity [Pa s], ­ angle of the bed level lateral slope [degrees], ­ angle of the bed level downstream slope [degrees], ­ fluid density [kg m-3], ­ density of the moving particle [kg m-3],

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Sep 1, 2009

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