A probability density function of liftoff velocities in mixed-size wind sand flux

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1 Science in China Series G: Physics, Mechanics & Astronomy 008 SCIENCE IN CHINA PRESS Springer-Verlag phys.scichina.com A probability density function of liftoff velocities in mixed-size wind sand flux ZHENG XiaoJing, ZHU Wei & XIE Li Key Laboratory of Mechanics on the Western Disaster and Environment, Ministry of Education, Lanzhou University, Lanzhou , China With the discrete element method (DEM), employing the diameter distribution of natural sands sampled from the Tengger Desert, a mixed-size sand bed was produced and the particle-bed collision was simulated in the mixed-size wind sand movement. In the simulation, the shear wind velocity, particle diameter, incident velocity and incident angle of the impact sand particle were given the same values as the experimental results. After the particle-bed collision, we collected all the initial velocities of rising sand particles, including the liftoff angular velocities, liftoff linear velocities and their horizontal and vertical components. By the statistical analysis on the velocity sample for each velocity component, its probability density functions were obtained, and they are the functions of the shear wind velocity. The liftoff velocities and their horizontal and vertical components are distributed as an exponential density function, while the angular velocities are distributed as a normal density function. wind sand flux, mixed-size particle diameter, liftoff velocity, probability density function (PDF), discrete element method (DEM) It is obvious that the liftoff velocities of sand particles with different diameters are of crucial importance for the prediction of wind sand flux. Therefore, a key issue in the research of wind erosion is to clarify the dynamical characteristic of sand particles lifting from the natural sand bed surface due to particle-bed collision [1], that is, with a given wind shear velocity, to obtain the probability density function (PDF) of liftoff velocities of sand particles rising from the mixed-size sand bed. There have already been many studies on the PDF of liftoff velocities of sand particles since the mid and later 1980s, e.g., deducing from the experimental photographs of 100 glass grains trajectories with diameters ranging from 0.35 mm to 0.71 mm. Anderson and Hallet [] thought that the probability densities of liftoff velocities are subject to an exponential function or a Gamma function with regard to shear wind velocity; Willetts and Rice [3] had direct observations on particle-bed collisions with 100 spherical and planar sand particles using a cine-camera. The diameters of sand Received December 15, 007; accepted March 11, 008 doi: /s Corresponding author ( xjzheng@lzu.edu.cn) Supported by the Key Project of the National Natural Science Foundation of China (Grant No ) Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

2 particles lie within the range of mm, and their data suggest that the densities of sand velocities follow a Gauss function for rebound sands and an exponential function for ejection sands. Whereas, some other experimental results can be concluded that the probability densities are subject to the log-normal density function [4], PearsonVII function [5] or Weibull function [6], and so on. These experimental studies have made considerable progress on measurement of the liftoff velocities, and hence on understanding of wind sand movement; however, liftoff velocities in these studies are generally derived from the trajectory photographs captured by high-speed cine-film, or taken in a certain height above the sand surface as a surrogate approximately. Therefore, the accuracy of distinguishing sand particles trajectories, the reasonableness of inverse deduction and the degree of approximation using the surrogate all have significant effects on the resulting PDF of liftoff velocities. To overcome the inaccuracy produced by direct observation on the velocities and trajectories of saltation particles, some researchers attempted to investigate the PDF of liftoff velocities based on the measurement of sand flux and wind speed profile. They firstly gave an initial PDF including several undetermined parameters, and numerically calculated the sand transport rate, then let all numerical results be equal to the experimental results, so an optimization PDF could be obtained. For example, Huang et al. [7] obtained an exponential density function for vertical liftoff velocities of sand particles with respect to shear velocity; Namikas [8] concluded that both Gamma and exponential density functions can fit the densities of liftoff; Zhu et al. [9] thought the probability densities of liftoff velocities follow the normal distribution. Though the measurement of sand flux is of better accuracy than directly tracking sand motion close to the sand surface, as Anderson et al. [1] suggested, the existing models which quantify the conceptual model of Aeolian sand transport are still far from the point at which reliable predictions of transport rates be made. Therefore, it may be one of the main reasons for the large discrepancy between the existing PDFs of liftoff velocities (see Figure 1). Recently, Zheng et al. [10] established a model based on the stochastic particle-bed collisions to Figure 1 Existing probability density function (PDF) of vertical liftoff velocities with shear wind velocity u * = 0.68 m/s. ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

3 obtain the PDF of liftoff velocities. In this model, liftoff velocities of ejected and rebound particles can be presented by analytical formulas with some random parameters. Through introducing arithmetic methodology for determining the density function of multi-random variables, the theoretical prediction on the PDF of rebound and ejected velocities can be presented, including the vertical and horizontal component, as well as the distribution of angular velocity. However, until now, this model cannot give a description of multiple-size particle-bed collision, and therefore cannot present the natural PDF of liftoff velocities. With the development of numerical technology, some researchers introduced the DEM method to describe Aeolian sand movement, such as the simulation of sand ripples shapes, sizes and even evolution [11]. However, few studies completely achieve the PDF of liftoff velocities with multiple particle sizes. In order to obtain a PDF of liftoff velocities in the mixed-size wind sand movement, the particle-bed collision with regard to mixed-size sand particles was presented by employing the DEM. By statistical analysis on the liftoff velocities of sand particles, an empirical expression of PDF for liftoff velocities at different wind speed is obtained. 1 Simulation of mixed-size particle-bed collisions In this paper, the cooling method [1] is required to produce an initial sand bed for particle-bed collision simulation. In order to emulate natural sand particle-bed collision, natural sand particles are sampled from sand dune in southern area of the Tengger desert in China, and their particle diameters obey a lognormal distribution (see Figure ) [13]. In the simulation, particle diameters are divided into 10 representative sets: 0.09 mm, 0.1 mm, 0.14 mm, 0.16 mm, 0.18 mm, 0.19 mm, 0. mm, 0.1 mm, 0.8 mm and 0.45 mm. According to the volume percentage of each diameter set, that is 0.095%, 0.13%, 1.58%, 31.74%, 7.69%, 15.83%, 7.47%, 3.%, 1.33% and 0.11%, respectively. The mean diameters of sand particles in the sand bed and on the top layer are 0.03 mm and 0. mm, and the length and height of the sand bed is 4.5 cm and 0.45 cm (see Figure 3). In order to get the liftoff velocities of sand particles after the particle-bed collision, we analyze the collision process of every sand particle including impact sand particle and sand particle in the Figure Particle diameter distribution of sand particles in the mixed-size sand bed employed in this paper. 978 ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

4 Figure 3 Schematic view of mixed-size sand bed employed for particle-bed collisions in the numerical simulation. sand-bed. Here, firstly let we consider any two neighboring particles i and j, and the linear spring-damper model is employed to exhibit the contact force between the two particles. The particle i is the objective particle (impact particle or particle in sand bed), whose colliding sand particle is particle j. The amount of overlapping between the two particles during contact is δ ij = i / j / ( i j) ( i j) D + D x x + y y, where ( x, y ) and ( x, y ) represent the coordinates of the particles i and j, respectively. For each particle as the particle i, there are several forces applied at its different points or within the particle. The gravity G i acts at the center of mass of sand particle i. The contact force and the damping force act at the contact point between the two particles. The contact force which is also a damping force is of two types, the normal and tangential contact forces. The normal contact force can be expressed as Fn, ij = Kδijn n, ij ( δij > 0) (1) and the tangential contact force as follows: μ Fnij, ntij, with slipping Ft, ij= K t rext, ij without slipping () where K, K t and μ are normal spring constant, tangential spring constant and friction coefficient, respectively, which are taken values as i i 3 j j K = K t = 10 kg/s, μ = 0.5 in this paper; n nij, and n t, ij are unit vectors parallel to the normal line at the contact point and the relative velocity of particle i to j; r ext, ij is the tangential vector at the contact point, the length of which is equal to the accumulated relative tangential displacement during all slip episodes affecting the current contact event. Owing to heat energy dissipation in collision between the two particles, normal and tangential damping forces are introduced to reflect inelastic collision. Assuming that δ and ( = drext, d ) ξ are the change rates of both the normal and tangential distances between centers ij ij t of the contact particles. So the normal and tangential damping forces can be written as Dnij, = b nδ ij and Dtij, = b tξ ij, respectively, with an assumption that the normal and tangential damping constant b n and b t are equal. According to the regulation of linear spring-damper model, the damping constant b n can be written as: bn = mi ω1 ω0, in which m i is the mass of particle i determined by the quartz density ρ = 650 kg/m 3 and its diameter. ω1 π T and ω 0 = K/ mi are the damping vibration frequency and natural frequency, where T is the contact period. By introducing the coefficient of restitutionε, which represents the ratio of the relative velocity after the collision to the one before the collision, it can conclude that the relation between ij ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

5 b and ε is ε ( b mω ) n = exp π. Noting that Anderson and Haff [14] investigated the sensitivity n i 1 of the nominal values of particle properties to simulation results, such as spring constant, damping constant and friction coefficient with incorporation of experimental results, and their appropriate values or distributed regions are given. We adopt ε = 0.7 and b n = kg/s in this paper, which satisfy the confine in ref. [14]. ω i and I i are the angular velocity (taken positive for anticlockwise rotations) and the moment of inertia. Consider the total force acting on particle i by its all of l neighboring particles, therefore, the motion of particle i is governed by the following differential equations: l l l d δ l d ij ξij d xi ( Kδijnn, ij ) x + ( Ft, ij ) x + b + bt = mi, j j j dt (3) j dt dt and where d δ d ξ d ( ) ( ), l l l l ij ij yi Kδijnn, ij y + Ft, ij y + b + bt Gi = m i j j j dt j dt dt y y l j r ij ij,con Ft, ij bt = Ii x d ξ dt dωi, dt r ij,con is a vector running from the centre of the particle i to the contact point of particle j, which can be derived from ( x, y ) and ( x, y ). The initial condition for particle i can be written i i j j as t = 0: xi = xi0, yi = yi0, x i = x i0, y i = y i0, ωi = 0, (6) in which ( x i0, y i0) and ( xi0, y i0) are initial velocities and displacement of particle i (including its horizontal and vertical components), and all of the sand particles are initially assumed staying without rotational motion. The simulation starts at the moment the incident grain contacting the sand bed; and here we must input parameters and initial values, shear wind velocity, particle diameters, and initial locations velocities of sand particles, into the simulation system. If the particle i lies in the sand bed, its initial location ( xi0, y i0) and velocity ( x i0, y i0) are obtained when producing sand bed; else if the particle i is the incident sand particle, its initial location ( xi0, y i0) is constrained within the region which is 5 times the mean diameter away from the boundary to eliminate the effect of boundary on the particle-bed collision. In order to reflect the stochastic property of diameter of the impacted sand particle and impact location ( xi0, y i0), ( xi0, y i0) is generated randomly by computer. The initial velocity ( x i0, y i0) of incident particle is selected from the experimental results measured in wind tunnel [6]. The selection process is demonstrated below: firstly employing the equal distance histogram method, the experimental results are fitted into an analytical function with regard to shear wind velocity and sand particle diameter (see ref. [15] in detail), that is: λ ( λ 1) f( vim ) = λ1 vim exp( λ1vim )/ Γ( λ ), (7) in which λ 1 and λ are parameters, and their expressions relative to shear velocity and sand particles diameters are given in Appendix A; v im is the incident velocity of the particle i, and in simulation its values distribute with eq. (7). According to the suggestion from experimental meas- x (4) (5) 980 ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

6 measurement [16], the incident angels in our simulation are chosen randomly ranging from 8 to 15 or each collision event. With the Runge-Kutta method, the location and velocity of each sand particle for arbitrary moment in the simulation can be obtained by solving eqs. (3)-(5), which can also be used to describe the splash process [17,18]. In order to give a real description for particle-bed collision occurring in natural sand movement, the lognormal distribution of sand particle diameters is employed not only for producing simulation sand bed but also for the incident sand particles. In this paper, for a given shear velocity, 10 groups of sand diameters (0.09 mm, 0.1 mm, 0.14 mm, 0.16 mm, 0.18 mm, 0.19 mm, 0. mm, 0.1 mm, 0.8 mm and 0.45 mm) are firstly chosen. Then for each diameter group, 10 sets of incident velocities are selected according to the PDF given by eq. (7), and finally for each combination of sand diameter and incident velocity, 5 different impact locations are randomly chosen. Repeating the above procedure, our numerical mixed-size particle-bed collision experiments are completed for a given shear wind velocity. Statistical calculations on the PDF of liftoff velocities After each particle-collision, we collect all the liftoff velocities (including the horizontal, vertical, and resultant liftoff velocities and angular velocities). For a given shear velocity, 500 particle-bed collisions are simulated. Denote V 1, V,, V n as a simple stochastic sample of liftoff velocities, and its sample size is n, therefore the corresponding sample observation values are v 1, v,, v n. Let the maximum and the minimum value be a( = min{ v1, v,, v n }) and b( = max{ v1, v,, v n }), re- a δ, b+ δ can be divided into m subregions equally where each region spectively. The region ( ] is denoted by [ vi, v i + 1) ( i = 1,, ). δ is an arbitrary small quantity which is set as δ =0.005 in this paper. The relation of m and n is determined by the distribution, which satisfies m = 1.87 (n 1) 0.4 given by Jeffrey [19]. Denote t i as the number of velocities belonging to region [ vi, v i + 1) and u i as the arithmetic mean of this velocity region ( ui = ( vi + v i + 1)/). The probability for velocity u i can be expressed approximately as P( ui) = ti / n, (8) and the probability density is fi = Pu ( i)/ Δ v= ti /( nδ v) (9) with relation of fidv= 1, where Δ v= vi vi 1. On the basis of the statistical calculations, the PDF of liftoff velocities for a given shear velocity can be obtained. Fitting all sets of the numerical results of particle-bed collision with the least squares method, the PDF of vertical, downwind and upwind horizontal and resultant velocities follow an exponential function as A( v vcr ) f(, v u* ) = Ae, (10) where v represents vertical, downwind and upwind horizontal or resultant velocities; v cr is the threshold starting velocity of a sand particle which is proved to be a constant approximately for any shear wind velocity; A ( > 0) is a function of shear wind velocity u * and it represents the possibility of a sand particle launching with low velocity. Ten groups of wind velocities used in ref. [6] are employed to provide the incorporation of shear wind velocity in our simulation, and eight ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

7 groups (0.15 m/s, m/s, 0.5 m/s, 0.68 m/s, 0.84 m/s, 1.0 m/s, 1. m/s, and 1.36 m/s) are selected to conclude the relations of A and v cr to the shear wind velocity. Results fitted by the least squares method are shown below: u* Av= e, ( vcr ) v = 0.034, π/ A A A v v y x v x+ u* = e, ( vcr ) v = 0.37, y π / = , ( v ) = , = e, ( vcr ) v = 0.536, x π / u* where v, v, v and v x+ are resultant, vertical, upwind horizontal and downwind horizontal y x liftoff velocities of sand particles, respectively. Substituting eq. (11) into eq. (10), we get the PDFs of resultant, horizontal and vertical liftoff velocities in mixed-size wind sand flux, where the sand particle diameters distribute with log-normal distribution shown as Figure. The PDF for angular velocities of sand particles is formulated as * ω+ cr v x u* f( ω, u* ) = e. ( u ) π / It is worth noticing that there was not any results published on the PDF for angular velocities of sand particles till now. 3 Results and discussion The resulting PDFs of liftoff velocities obtained from eqs. (10)-(1) are shown in Figure 4, where, for differentiability, only the distributions under the shear velocity 0.68 m/s are given. From Figure 4, it is indicated that all of the PDFs for absolute values of upwind and downwind horizontal velocities, vertical velocities and resultant velocities follow an exponential density function, whereas, the angular velocities which lie within rev/s distribute with a normal density function, and its symmetrical axis is ω = rev/s. To verify our density functions, we firstly simulate particle-bed collisions using our numerical program under the same confined parameters and initial conditions as the experimental ones in ref. [0]. Sand particles in sand bed are divided into three fractions, called fine ( mm), medium ( mm) and coarse ( mm) fractions, and the corresponding ratio is 17.7:70.6:11.7. Incident velocities are 375 cm/s for the diameter of 0.15 mm, 335 cm/s for the diameter of 0.45 mm and 75 cm/s for the diameter of 0.45 mm, respectively. For each combination of incident velocity and particle diameter, 10 random collisions are simulated. Our simulation rebound/ejected velocities and the number of liftoff sand particles are compared with experimental results obtained by Rice and Willetts [0] shown in Figure 5 and Table 1. From Table 1, we can find that the rebound velocities agree well with the experimental results in ref. [0], and the mean value of ejected velocities recorded in ref. [0] lies in the range of simulation results (see Figure 5). Therefore, it indicates that our method and programs used in this paper are valid. Worthy of note, as Table 1 shows, the mean number of liftoff sand particles is larger than the experimental values, (11) (1) 98 ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

8 Figure 4 The resulting probability densities of (a) angular velocities and (b) horizontal velocity, vertical velocity, resultant velocity with u * = 0.68 m/s. Figure 5 The comparison of simulation results of ratio of ejected velocities to the incident velocity with experimental data. Table 1 Comparison of the mean velocity and mean number of ejected particles in this paper with the results in ref. [18] Coarse impactor Medium impactor Fine impactor Particle diameter (mm) Impact velocity (cm s 1 ) Ejection velocity (cm s 1 ) Number of ejecta per collision Ref. [0] This paper Ref. [0]/mean This paper Ref. [0]/mean This paper Ref. [0]/mean This paper while the mean value of ejected velocities in ref. [0] is higher than the result in this paper, and we think it mainly results from the difficulty of recording ejected launching sand particles with much ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

9 lower velocities than rebound ones. Then, we use our numerical program to simulate the particle-bed collision with single-size sand particles (particle diameter 0.35 mm). The liftoff velocities are analyzed and the PDFs of liftoff velocities and their vertical components are given under the single-size wind sand flux. Figure 6(a) displays the comparisons of our simulation PDF of vertical liftoff velocities with experimental results in ref. [15]. Figure 6(b) is the comparison of our simulation PDFs of rebound/ejected liftoff velocities with theoretical ones predicted by Zheng et al. [10]. From Figure 6, we can find our simulation results accord with both the experimental and theoretical results for the uniform wind sand flux. As Figure 6(b) reveals, the possibility of launching sand particle with low speed is higher than its predictions in ref. [10], which may be derived from the choice of threshold start velocity. Because our threshold start velocity in the simulation is higher than the oneselected in ref. [10], which hence leads to a fewer number of sand particles recorded. Figure 6 The comparisons of the probability densities of (a) liftoff velocity with ref. [14] and (b) vertical liftoff velocity with ref. [10] for the case of uniform particles with particle diameter of 0.35 mm and shear wind velocity u * = m/s. 984 ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no

10 Figure 7 The comparison of simulation results with the prediction results given by eqs. (10) and (11) for shear velocity of 0.93 m/s and 1.54 m/s. In order to show that our simulation PDFs are correct and credible for any shear wind velocity, we compare the numerical results under u * = 0.93 m/s and u * = 1.54 m/s with the results calculated by formulas (10)-(1), shown in Figure 7. To save space, only the probability density of resultant liftoff velocities is plotted. The comparison implies that the prediction on probability densities of liftoff velocities given by eq. (10) is reliable for other shear velocities, and our PDFs of liftoff velocities also can characterize the statistical regularity of liftoff velocities of sand particles in mixed-size wind sand flux. These PDFs of liftoff velocities are valid under the log-normal diameter distribution given in ref. [13]. The authors express their thanks to the people who helped with this work, and acknowledge the valuable suggestions from the peer reviewers. 1 Anderson R S, Sorensen M, Willettes B B. A review of recent progress in our understanding of Aeolian sediment transport. Acta Mech, 1991, 1(Suppl 1): 1-19 Anderson R S, Hallet B. Sediment transport by wind: Toward a general model. Bull Geol Soc Am, 1986, 97: [DOI] 3 Willetts B B, Rice M A. Collisions in Aeolian saltation. Acta Mech, 1986, 63: 55-65[DOI] 4 Nalpanis P, Hunt J C R, Barrett C F. Saltating particles over flat beds. J Fluid Mech, 1993, 51: [DOI] 5 Zou X Y, Wang Z L, Hao Q Z, et al. The distribution of velocity and energy of saltating sand grains in a wind tunnel. Geomorphology, 001, 36: [DOI] 6 Dong Z B, Liu X P, Li F, et al. Impact-entrainment relationship in a saltating cloud. Earth Sur Proc Land, 00, 7: [DOI] 7 Huang N, Zheng X J, Zhou Y H, et al. Simulation of wind-blown sand movement and probability density function of liftoff velocities of sand particles. J Geophys Res, 006, 111: D001 [DOI] 8 Namikas S L. Field measurement and numerical modeling of Aeolian mass flux distributions on a sandy beach. Sedimentology, 003, 50: [DOI] 9 Zhu J J, Qi L X, Kuang Z B. Velocity distribution of particle phase in saltating layer of wind-blown sand two phase flows. Acta Mech Sin, 001, 13: Zheng X J, Xie L, Zhou Y H. Exploration of probability distribution of velocities of saltating sand particles based on the stochastic particle-bed collisions. Phys Lett A, 005, 341: [DOI] 11 Werner B T. A physical model of wind-blown sand transport. Thesis of Ph. D. California: California Institute of Technology, Fortin J, Millet O, Saxce G. Numerical simulation of granular materials by an improved discrete element method. Int J Num Meth Eng, 005, 6: [DOI] 13 Zhou Y H, Guo X, Zheng X J. Experimental measurement of wind-sand flux and sand transport for naturally mixed sands. Phys Rev E, 00, 66: 01305[DOI] 14 Haff P K, Anderson R S. Grain scale simulation of loose sedimentary beds: The example of grain-bed impacts in aeolian saltation. Sedimentology, 1993, 40: [DOI] 15 Xie L, Dong Z B, Zheng X J. Experimental analysis of sand particles liff-off and incident velocities in wind-blown sand flux. Acta Mech Sin, 005, 1: [DOI] 16 Mitha S, Tran M Q, Werner B T, et al. The grain-bed impact process in Aeolian saltation. Acta Mech, 1986, 63: [DOI] 17 Anderson R S, Haff P K. Wind modification and bed response during saltation of sand in air. Acta Mech, 1991, 1(Suppl 1): Zheng X J, Bo T L, Xie L. DPTM simulation of aeolian sand ripple. Sci China Ser G-Phys Mech Astron, 008, 51(3): Jeffrey S. Simonoff Smoothing Methods in Statistics. Berlin: Springer-Verlag, Rice M A, Willetts B B, McEwan I K. An experimental study of multiple grain-size ejecta produced by collisions of saltating grains with flat bed. Sedimentology, 1995, 4: [DOI] ZHENG XiaoJing et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no