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Launch velocity characteristics of non-uniform sand in aeolian saltation
Physica A 388 (2009) 1367–1374
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Launch velocity characteristics of non-uniform sand in aeolian saltation D.J. Feng, Z.S. Li, J.R. Ni ∗ Department of Environmental Engineering, Peking University, The Key Laboratory of Water and Sediment Sciences, Ministry of Education, Beijing 100871, China
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Article history: Received 13 November 2008 Received in revised form 19 December 2008 Available online 9 January 2009 PACS: 92.40.-t 02.60.-x 45.70.-n Keywords: Aeolian saltation Numerical modelling Mass flux Launch velocity
a b s t r a c t A numerical model for the vertical profile of aeolian mass flux was established based on the launch velocity probability distribution of saltating sand particles. The mean launch velocity (v¯ 0 ) was used as a primary variable in the simulation, from which the average saltation height and length could be further calculated. The results showed that the mean launch velocity, the average saltation height, and the average saltation length increased with increasing wind friction velocity (u∗), although the increases for the first two were not significant. On the other hand, the relative mean launch velocity (v¯ 0 /u∗) was found to decrease with increasing wind friction velocity. Other interesting results were obtained from the simulation: both v¯ 0 and v¯ 0 /u∗ decreased with an increase in the sand grain size, which agreed with a previous report claiming that the launch velocity was limited by the inertia of bed grains. Furthermore, decreasing dependence of the average saltation height and length was also found with increasing sand grain size. Overall, the sand grain size was of significance to the determination of probability distribution of the launch velocity and thus the saltation characteristics. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Aeolian sand transport causes major environmental problems such as wind erosion, sand storms, and desertification in arid and semi-arid regions [1]. Saltation is one of the three major modes of particle motion during wind erosion, along with suspension and creep, and it plays a key role in the erosion process, accounting for 75% of the total sand transport flux [2]. Most particles on the sand bed surface are mainly initiated by aerodynamic forces or by the impact of other descending particles, and then accelerated during their flight in air. After they impact the surface, the rebounding particles are ejected. The impact entrainment is dominated in the equilibration of aeolian saltation (e.g., Refs. [3–5]). The grain–bed collision process is an important and perhaps the most complex component of saltation, but substantial uncertainty remains regarding many aspects of this process. The initial or launch velocity of saltating sand particles is a key issue for the investigation of the grain–bed collision process. The numerous studies on the grain–bed collision process have generally been treated in two ways: numerical and theoretical modelling (in Refs. [3–13]), and experimental measurements (in Refs. [14–27]). The mean launch angle has variously been found to be 90◦ , 50◦ , and in the range of 21◦ –33◦ , 43◦ –46◦ , 34◦ –41◦ , 27◦ –30◦ (in Refs. [2,14,16–18, 26], respectively). Many studies have revealed that the mean launch speed was positively proportional to the wind friction velocity (e.g., Refs. [3,7,9]), but recently Namikas [28] reported that the mean launch speed was essentially constant and independent of the wind friction velocity. The launch velocity probability distribution of saltating sand particles is a bridge to link microscopic and macroscopic aeolian research. Various modes of the launch velocity distribution have been found, corresponding to the normal function [17,24], log-normal function [18,25,27], Gaussian function [4,5,25], exponential
∗
Corresponding author. Tel.: +86 10 62751185; fax: +86 10 62756526. E-mail addresses: [email protected] (D.J. Feng), [email protected] (J.R. Ni).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.01.002
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function [4,5,27], modified exponential function [10], gamma function [14,16,24], and Weibull function [23]. Thus, these previous results are inconclusive, and uncertainties still exist. To date, much of the insight regarding the grain–bed collision process has been concentrated on a macroscopic representation of total sand, and it was assumed that saltating particles were uniform in size and followed identical trajectories (e.g., Refs. [2,29]). In practice, however, a natural sand bed is generally composed of differently sized and shaped grains, and has many variable geometric configurations. Saltating particles follow different trajectories, and then their characteristics have different functional dependences on size when colliding with the bed. Few investigations have been conducted on the separate launch velocity characteristics of different sizes of sand particles, even though they are fundamentally important for understanding the microscopic process of non-uniform saltation [30]. In this paper, the launch velocity characteristics of non-uniform sand in aeolian saltation were investigated by simulating the vertical profile of the sand mass flux combined with experimental data. Section 2 briefly introduces a numerical model of aeolian sand transport. The simulated mass flux was compared with both the experimental data of total sand particles studied by Ni et al. [31], shown in Section 3, and that of different-sized sand particles studied by Li et al. [32], shown in Section 4. The influences of wind velocity and grain size on the mean launch velocity were estimated in the simulation, and then the average saltation height and length were further calculated as a function of wind velocity and grain size. Section 5 summarizes the main conclusions. 2. Numerical modelling Most numerical models of aeolian sand transport are generally subdivided into four component processes: aerodynamic entrainment, grain trajectories, grain–bed collision, and wind field modification (e.g., Refs. [3,5,6]). These components form a self-limited feedback process between air motion, particle motion, and particle–air and particle–surface interactions. This model allows simulation of the whole process of aeolian saltation from inception by aerodynamic entrainment to a steady state when each sub-model is combined [33]. The proposed model in this study was used to simulate the relative proportion of the vertical profile of the mass flux, rather than the absolute magnitude of the mass flux. Thus, aerodynamic entrainment was not needed. Furthermore, the goal of the present model was to investigate the fully developed flow of wind-blown sand, rather than to study the temporal development of aeolian sand transport. The simulated sand mass flux herein was saturated. In this state, no more particles can enter into the flow since the amount of sand entrained by the wind has a maximum value [34]. Hence, it was assumed that the equilibrium of aeolian saltation was achieved throughout simulations, and no wind field modification component was required. 2.1. Grain trajectories It is widely accepted that there are four forces that govern the flight of saltating particles: force due to gravity, aerodynamic drag, the Magnus effect, and aerodynamic lift. Following the precedent set (e.g., Refs. [6,17,28]), gravity and aerodynamic drag are only considered for the calculation of grain trajectories. The equations of motion used in the model are dvx /dt = −0.75ρa /(ρp d)Cd Ur (vx − U )
(1)
dvz /dt = −0.75ρa /(ρp d)Cd Ur vz − g
(2)
where vx and vz are the sand velocities (v ) in the horizontal (x) and vertical (z) directions, respectively; ρa is the density of air (1.23 kg m−3 ); ρp is density of sand (2650 kg m−3 ); d is sand diameter; g is gravity; and U is time-averaged horizontal wind velocity at vertical height. The drag coefficient Cd is specified using the function proposed for a spherical grain by Morsi and Alexander [35]. Ur is the relative velocity between sand particle and wind flow, which is written as Ur = ((vx − U )2 + vz2 )1/2 .
(3)
2.2. Wind velocity profile In order to solve the trajectory equations, it is necessary to specify the vertical profile of the horizontal wind velocity and the initial or launch velocity of the sand particles. The mean stream-wise wind velocity profile without particle motion is well governed by the standard logarithmic model. Particle motion significantly alters the wind velocity profiles. In accord with the corresponding wind velocity data of Ni et al. [31], the wind velocity profile represented here, as proposed by Li et al. [36], is as follows: U = u ∗ /κ ln(2gz /(0.04u ∗2 )) U = u ∗ /κ
Z
z
1/z
q
z > u ∗2 /(6.7g )
(4)
1 − (1 − u ∗t 2 /u ∗2 )(1 − 6.7gz /u ∗2 )e−6.7gz /u ∗ dz 2
z ≤ u ∗2 /(6.7g )
(5)
z0
where u∗ is the wind friction velocity; u ∗t is the threshold friction velocity; k is the von Karman constant (taken as 0.41), and z0 is the roughness length of the surface or the height above the bed at which flow velocity tends to zero. The values of
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Fig. 1. Comparison of the experimental data of Ni et al. [31] and the simulated results using Eqs. (6) and (7) with a mean launch angle of 35◦ .
u∗ and z0 are generally derived from experimental measurements, using the simple slope method. Bagnold [2] suggested z0 = 1/30d50 (where d50 is the mean grain diameter). 2.3. Grain–bed collision The grain–bed collision process determines the initial or launch velocity of saltating particles, which further exerts a strong effect on the particle trajectories and sand mass flux distribution. On striking the surface, saltating particles impart momentum to stationary particles. This impact may result in the rebound of the original particles as well as the ejection of other stationary particles into the air stream. Grain–bed collision is an extremely complex process, so it is difficult to observe and measure it directly under natural conditions. Great effort has been made to study the empirical and statistical characteristics of the launch speed and angle, commonly described by a probability distribution function. Different patterns of the launch velocity distribution have been proposed by many investigators, but no general agreement has been achieved according to the published literature (e.g., Refs. [5,25, 27]). In general, there are two major modes of the launch velocity distribution: the exponential and log-normal functions, which could be expressed as follows: p(v0 ) = 1/¯v0 exp(−v0 /¯v0 )
(6)
p(α0 ) = 1/α¯ 0 exp(−α0 /α¯ 0 )
√
(7)
p(v0 ) = 1/( 2π σv0 v0 ) exp(−(ln v0 − ln v¯ 0 ) /(2σv0 )) 2
2
√ p(α0 ) = 1/( 2π σα0 α0 ) exp(−(ln α0 − ln α¯ 0 )2 /(2σα20 ))
(8) (9)
where p(v0 ) and p(α0 ) are the probability distribution functions of the launch speed (v0 ) and angle (α0 ), respectively; v¯ 0 and α¯ 0 are the mean values of launch speed and angle, respectively; σv0 and σα0 are the standard deviations of the launch speed and the angle’s logarithm, respectively. Using the probability distribution functions of the launch speed and angle (Eqs. (6)–(9)) and integrating the Eqs. (1)–(5), the vertical profile of sand mass flux can be expressed as q(z ) = nm
Z 0
∞
p(v0 )
π
Z
p(α0 )(vx / |vz + | + vx / |vz − |)dv0 dα0
(10)
0
where n is the dislodgment rate; m is the mass of a single sand grain; |vz + | represents the average of |vz | over ascending particles, and |vz − | represents its average over descending particles in the slice z ± 1/2dz. 3. Launch velocity characteristics of total sand particles Ni et al. [31] measured the vertical profiles of the stream-wise wind velocity and the saturated mass flux in a wind tunnel containing a bed of naturally mixed sand. The present simulated sand mass flux was compared with the wind tunnel data of Ni et al. [31]. The setting parameters were as follows. The launch speed and angle distribution used Eqs. (6) and (7), respectively. The sand diameter was 0.35 mm (mean grain size of the test sands in Ni et al. [31]), and its density was 2650 kg m−3 . The mean launch angle was 35◦ . The mean launch velocity was used as a primary variable in the simulation. For comparison with the tunnel data of Ni et al. [31], the simulated mass flux was scaled by the q1exp /q1sim (q1exp and q1sim were the mass fluxes at 1 cm above the sand bed surface for the experimental data of Ni et al. [31] and for the model-simulated results, respectively). These were plotted in a log-linear coordinate system, as shown in Fig. 1. The variation of the mean
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Fig. 2. Variations of the mean launch velocity of total sand particles as a function of the wind friction velocity obtained from the simulated results in Fig. 1.
Fig. 3. Variations of the average saltation length and height of total sand particles with the wind friction velocity based on the estimated mean launch velocity in Fig. 2. Table 1 Comparison of the vertical slopes between the simulated and the experimental (Ni et al. [31]) sand mass flux distribution in Fig. 1. u∗ (m s−1 )
Vertical slopes
0.47 0.77 1.11 1.53 2.31
Experiment
Simulation
Difference (%)
−0.113 −0.101 −0.097 −0.090 −0.081
−0.108 −0.110 −0.107 −0.102 −0.091
−4.8 8.8 10.5 13.2 12.6
launch velocity obtained from the simulation as a function of the wind friction velocity is shown in Fig. 2. After estimating the mean launch velocity, we calculated the average saltation length (Ls0 ) and height (Hs0 ) by the following formulas: ∞
Z
p(v0i )
Ls0 = 0
Z Hs0 = 0
π
Z
p(α0i )Lsi dv0 dα0
(11)
p(α0i )Hsi dv0 dα0
(12)
0
∞
p(v0i )
π
Z 0
where Lsi and Hsi are the maximum jump length and height of the i-th particle with a launch speed of v0i and a launch angle of α0i , respectively. Fig. 3 illustrates the variations of average saltation length and height with wind friction velocity. Fig. 1 showed that reasonable agreement between the simulated and experimental results has been achieved. The vertical slopes of the best-fit exponential function for the experimental and simulated mass flux above the saltation layer (z > 4 cm) were obtained following Namikas [28], and the comparison results are shown in Table 1. From Table 1, the difference
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between the simulated and the experimental slopes increased with increasing wind velocity, which implies that the errors of simulation would increase as the wind velocity becomes greater. Fig. 2 shows a mild increase of the mean launch velocity (v¯ 0 ) from 1.2 m s−1 to 1.5 m s−1 as the wind friction velocity increases from 0.47 m s−1 to 2.31 m s−1 , while an apparent decrease of the relative mean launch velocity (v¯ 0 /u∗) occurs correspondingly from 2.7 to 0.5. Note that the mean launch velocities of 1.22 m s−1 and 1.23 m s−1 corresponded to the two runs with the lowest wind velocities u∗ = 0.47 m s−1 and u∗ = 0.77 m s−1 (see Fig. 2), respectively. Thus, the mean launch velocity remained approximately constant under the two lowest u∗ conditions (within experimental error). This finding agreed well with the results reported by Namikas [28] that v¯ 0 was largely independent of u∗. In Namikas’ [28] data set, u∗ ranged from about 0.27 m s−1 to 0.63 m s−1 , roughly corresponding to the two lowest values mentioned above. The simulated mean launch velocities reported for the three highest u∗ experiments were different from those for the two smallest u∗ experiments. This may be explained as follows: sand would not saltate under these conditions, and it would fly into suspension. Thus, modelling saltation under such conditions seems inappropriate. While the wind speeds of these conditions were relatively rare and represented unusual extremes rather than typical conditions, they certainly occurred in the natural environment, and especially in wind tunnel experiments with a saltation cloud. The results in Fig. 2, under the two lowest u∗ conditions, disagreed with those of Bagnold [2], who suggested that the mean launch velocity scaled with the wind friction velocity. The mean launch velocities have previously been found to be 1.8u∗, 3.3u∗, 5.8u∗, and 3.5(4.4)u∗ in Refs. [14–16,18], respectively. Anderson and Hallet [3] proposed a mean launch velocity of 0.63u∗. McDonald and Anderson [9] obtained good results with a value of 1.0u∗. Nishimura and Hunt [21] found that the relative mean launch velocity of ice particles decreased from 5.4 to 2.5 with increasing wind friction velocity (0.35–0.65 m s−1 ). However, the three largest u∗ experiments showed a clear trend of an increase in v¯ 0 with u∗. As the wind friction velocity increases, saltating sand particles obtain more momentum from the wind flow, and impact the sand surface with a larger velocity. Both the rebound velocity of impact particles and the ejected velocity of surface particles increase accordingly. This physical explanation may account for this study’s results, that the mean launch velocity was positively proportional to the wind friction velocity. By considering the negative dependence of the relative mean launch velocity on the wind friction velocity, we may infer that more factors in addition to wind friction velocity are involved in influencing the mean launch velocity. Fig. 3 showed that the average saltation length (Ls0 ) increased from about 0.25 m to 0.65 m with increasing wind friction velocity. Many experiments have been carried out to measure the saltation trajectory characteristics of sand particles [14– 18,26]. Willetts and Rice [16] reported that Ls0 was about 0.64–0.8 m under the wind friction velocity of 0.39 m s−1 . Nalpanis et al. [18] found that Ls0 varied between 0.064 m and 0.08 m for the given u∗ values ranging from 0.18 m s−1 to 0.2 m s−1 . Zhang et al. [26] obtained Ls0 values of 0.036–0.06 m under a u∗ of 0.21–0.39 m s−1 . Sørensen’s [17] results revealed that there was a clear increase of Ls0 with increasing u∗, which was similar to the present results. We can explain this by the fact that the sand saltation length was determined by the flight time and the particle horizontal velocity. Saltating particles in air are dragged by the wind to the downstream position. They will absorb more momentum if the wind flow velocity increases; the particle horizontal velocity increases accordingly, and this leads to an increase in the saltation length. Fig. 3 also shows that the average saltation height (Hs0 ) was almost constant (about 0.026 m) and independent of wind friction velocity, with the exception of high wind speed conditions of a larger value (about 0.031 m). The values of Hs0 have been experimentally reported to be about 0.048–0.056 m, 0.005–0.006 m, 0.002–0.004 m in Refs. [16,18,26], respectively. In contrast to the present study, some investigators have proposed that the saltation height is positively proportional to the wind friction velocity (e.g., Refs. [2,29]). The disagreement may be attributed to the fact that the saltation height increases with an increase in the launch velocity of sand particles. In general, the vertical wind velocity is irrelevant to steady aeolian sand transport, and the vertical motion of sand particles is mainly determined by the launch velocity. Fig. 2 reveals that the difference in the magnitude of the mean launch velocity was small under different wind velocities, except for the largest one (u∗ = 2.31 m s−1 ). Thus, the variation of the estimated mean launch velocities resulted in the effects on the average saltation height. There were some uncertainties in the present simulation study. Although it is generally accepted that the mean grain size of the surface sand should be used as the input parameter to simulate the vertical profile of the mass flux of total sand, and that this treatment gives the overall characteristics of total sand transport, this simplification does not represent the actual conditions, and a more appropriate grain size for the simulation needs to be better specified. The mean launch velocity in this section represented the average launch velocity of total sand particles. 4. Launch velocity characteristics of different-sized sand particles Li et al. [32] investigated the vertical profiles of the sand mass flux of different size gradings by re-analyzing the wind tunnel data of Ni et al. [31], and suggested that the vertical profile of the sand mass flux depended on the size grading. The data for the four size gradings of Li et al. [32] were compared with the simulated results of the present model: 0.18–0.25 mm, 0.25–0.355 mm, 0.355–0.5 mm, and 0.5–0.71 mm. The input parameters are as follows. Eqs. (6) and (9) were used in the size gradings of 0.18–0.25 mm and 0.25–0.355 mm. Eqs. (6) and (7) were used in the size gradings of 0.355–0.5 mm and 0.5–0.71 mm. The sand diameter used the mean grain diameter of each size grading, and its density was 2650 kg m−3 . The mean launch angle was 35◦ , and its standard deviation was 0.5. The mean launch velocity was also used as a primary variable
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Table 2 Comparison of the vertical slopes between the simulated and the experimental (Li et al. [32]) sand mass flux distributions in Fig. 4. Grain size grading (mm)
Vertical slopes u∗ = 1.11 (m s−1 )
0.18–0.25 0.25–0.355 0.355–0.5 0.5–0.71
u∗ = 1.53 (m s−1 )
u∗ = 2.31 (m s−1 )
Exp.
Sim.
Dif. (%)
Exp.
Sim.
Dif. (%)
Exp.
Sim.
−0.025 −0.075 −0.166 −0.353
−0.030 −0.068 −0.172 −0.377
16.6 −8.4 3.7 6.9
−0.033 −0.061 −0.136 −0.198
−0.030 −0.060 −0.146 −0.228
−8.9 −0.8
−0.016 −0.048 −0.105 −0.169
−0.017 −0.045 −0.113 −0.170
7.9 14.9
Dif. (%) 3.8
−5.1 7.6 0.9
Fig. 4. Comparison of the experimental data of Li et al. [32] and the simulated results using Eqs. (6) and (7) with a mean launch angle of 35◦ for the grain size grading of 0.355–0.5 mm and 0.5–0.71 mm, and using Eqs. (6) and (9) with a mean launch angle of 35◦ and its standard deviation of 0.5 for the grain size grading of 0.18–0.25 mm and 0.25–0.355 mm.
in the simulation. After estimating the mean launch velocity, the average saltation length and height were calculated as a function of wind friction velocity and grain size. A comparison of the simulated sand flux with the experimental data of Li et al. [32] for different size gradings is shown in Fig. 4 (also following Namikas [28]). The discrepancy of the vertical slopes of the sand mass flux between the experimental data of Li et al. [32] and the simulated results for the outer saltation layer are shown in Table 2. The comparison results showed that the agreement between the simulated results and the experimental data was good. Fig. 4 shows that both the simulated and experimental sand mass fluxes of the grain size grading ≤0.355 mm decayed exponentially with height whereas those of the grain size grading >0.355 mm represented a remarkably positive deviation in the near-bed region. This result may support the idea that the creep of coarse sand particles causes the deviation of the sand mass flux from the exponential distribution near the sand surface. The dependence of the mean launch velocity on both the wind friction velocity and the sand grain size is shown in Fig. 5. The variation of mean launch velocity with the wind friction velocity in Fig. 5 was similar to that of Fig. 2. This may indicate that the wind friction velocity exerts a similar effect on the launch velocity of different-sized sand particles. Fig. 5 also shows that the (relative) mean launch velocity decreased with an increase in grain size. Rice et al. [19] used highspeed photographic techniques to observe the launch velocity of different-sized particles on a sand surface of multiple grain sizes that was divided into 0.15–0.25 mm, 0.3–0.35 mm, and 0.425–0.6 mm gradings. Rice et al. [19] found that the mean rebound velocities of coarse sand particles were larger than those of fine particles, and the mean ejected velocities of all particles were almost the same. Therefore, the experimental data of Rice et al. were similar to the present simulated results: the mean launch velocities of coarse sand particles were larger than those of fine grains. In addition, the present results agreed with the results presented by Namikas [37], that the launch velocity was limited by the inertia of bed grains. This may be explained by the fact that the number of sand particles ejected from the bed after impact with coarse particles is larger than that caused by fine particles [8]. So coarse particles lose more momentum in the grain–bed collision process and then leave the bed at lower launch velocities than fine grains, but with approximately the same kinetic energy level (i.e., the launch velocity is inversely proportional to the grain mass). If the impact energy exceeds the inertia of the bed grains, the bed grain is set into motion and the impactor slams into more grains and transfers more energy, until it finally hits a grain with sufficient inertia to allow rebound [37]. However, the grain–bed collision process is too complex to be understood only by using such theoretical models. A great deal of effort has been made to investigate the launch velocity of total sand particles on a surface containing multiple grain sizes, but few research studies have been done on the launch velocities of different-sized sand particles, which is the particular focus of this section.
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Fig. 5. Variations of the mean launch velocity of the different-sized sand particles as a function of the wind friction velocity and the sand grain size obtained from the simulated results in Fig. 4.
Fig. 6. Variations of the average saltation length and height of the different-sized sand particles with the wind friction velocity and the sand grain size based on the estimated mean launch velocity in Fig. 5.
Fig. 6 illustrates the variations of average saltation length and height with the wind friction velocity and sand grain size. It shows that the average saltation length increased with increasing wind friction velocity, whereas the average saltation heights are almost the same value for u∗ = 1.11 m s−1 and 1.53 m s−1 , which are smaller than that for u∗ = 2.31 m s−1 . This result was similar to that of Fig. 3. Fig. 6 also shows that all of the saltation lengths and heights decreased with increases in grain size. We may infer that the motion of coarse sand particles is more easily influenced by other particles than that of fine grains. Nalpanis et al. [18] obtained similar results in wind tunnel experiments containing different-sized sand beds. Usually, the height representing 50% of the cumulative percentage of the total sand transport rate is defined as the average saltation height [1]. Using this definition, Dong et al. [38] analyzed the experimental data of mass flux and found that the average saltation height increased with increasing grain size, which was different from the present results. The discrepancy between the present results and those of Dong et al.’s may be explained by two factors. First, the definition of average saltation height for the present study was different from that in Dong et al. [38], although the two different definitions may reflect similar characteristics of saltation trajectory. Second, Dong et al.’s conclusion merely reflected the saltation height characteristics obtained assuming uniform bed sand when the particle size varied in a series of separate tests. Whether the characteristics remained similar for non-uniform bed sand is another matter. Thus, it can be seen that the behavior of average saltation length and height shown in Fig. 6 reflected the results shown in Fig. 5. The simulated results further revealed that the launch velocity distribution was determined by the sand particle size, which supports the assumption of Shao and Mikami [30] that the profiles of saltation fluxes of particles of different sizes differed, but these profiles can be universal if the initial velocity probability density function is given. However, the present results only reflected the launch velocity characteristics obtained using the experimental data from high wind velocity (u∗ ≥ 1.11 m s−1 ). Whether the characteristics would remain similar for low wind velocity was another matter. In addition, we assumed that the motion in air of different-sized sand particles was independent, and midair inter-particle collisions were ignored. Actually, there should be interaction among saltating sand particles [39]. Thus, future studies need to be conducted on the characteristics of non-uniform sand in aeolian saltation.
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5. Conclusions Based on the two typical probability distribution functions of the launch velocity of saltating sand particles, a numerical model was established to simulate the vertical profile of the sand mass flux. The mean launch velocity (v¯ 0 ) was used as an important parameter in the simulation, which was also of significance in the calculation of the average saltation length and height. The simulated sand mass fluxes agreed well with both the experimental results on total sand particles by Ni et al. [31] and those on different-sized sand particles by Li et al. [32]. The mean launch velocity and average saltation height of total sand particles was essentially constant and independent of the shear velocity under lower wind velocity conditions, although they increased with increasing wind friction velocity (u∗) under higher wind velocity conditions. The relative mean launch velocity (v¯ 0 /u∗) decreased with increasing wind friction velocity but the average saltation length increased. The simulated results on launch velocity depending on particle size agreed with Namikas’ claim in [37] on the launch velocity being limited by the inertia of bed grains. Both the absolute and the relative mean launch velocities of different-sized sand particles varied with the wind friction velocity; so did the average saltation length and height. On the other hand, the above four parameters decreased with increasing particle size. Acknowledgements The project was supported by the National Natural Science Foundation of China under Grant No. 49625101, 40371011. The reviewers’ valuable comments for improving the manuscript are especially acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
Z. Wu, Geomorphology of Wind-Drift Sands and Their Controlled Engineering, Science Press, Beijing, 2003 (in Chinese). R.A. Bagnold, The Physics of Blown Sand and Desert Dunes, Methuen, London, 1941. R.S. Anderson, B. Hallet, Geol. Soc. Am. Bull. 97 (1986) 523. R.S. Anderson, P.K. Haff, Science 241 (1988) 820. R.S. Anderson, P.K. Haff, Acta Mech. Suppl. 1 (1991) 21. I.K. McEwan, B.B. Willetts, Acta Mech. Suppl. 1 (1991) 53. I.K. McEwan, B.B. Willetts, J. Fluid Mech. 252 (1993) 99. P.K. Haff, R.S. Anderson, Sedimentology 40 (1993) 175. R.R. McDonald, R.S. Anderson, Sedimentology 42 (1995) 39. M. Xing, L.J. Guo, Phys. Lett. A 332 (2004) 389. X.J. Zheng, L. Xie, Y.H. Zhou, Phys. Lett. A 341 (2005) 107. Y.H. Zhou, W.Q. Li, X.J. Zheng, J. Geophys. Res. - Atmos. 111 (2006) D15108. W.Q. Ta, Z.B. Dong, Geomorphology 89 (2007) 348. B.R. White, J.C. Schulz, J. Fluid Mech. 81 (1977) 497. K. Araoka, N. Maeno, in: 3rd Symposium on Polar Meteorology and Glaciology, Tokyo, National Institute of Polar Research, Technical Memorandum, 1981. B.B. Willetts, M.A. Rice, in: O.E. Barndorff-Nielsen, J.T. Møller, K.R. Rasmussen, B.B.Willetts, (Eds.), Proceedings of International Workshop on the Physics of Blown Sand, Aarhus University, Denmark, 1985. M. Sørensen, in: O.E. Barndorff-Nielsen, J.T. Møller, K.R. Rasmussen, B.B. Willetts, (Eds.), Proceedings of International Workshop on the Physics of Blown Sand, Aarhus University, Denmark, 1985. P. Nalpanis, J.C.R. Hunt, C.F. Barrett, J. Fluid Mech. 251 (1993) 661. M.A. Rice, B.B. Willetts, I.K. McEwan, Sedimentology 42 (1995) 695. M.A. Rice, B.B. Willetts, I.K. McEwan, Sedimentology 43 (1996) 21. K. Nishimura, J.C.R. Hunt, J. Fluid Mech. 417 (2000) 77. F. Rioual, A. Valance, D. Bideau, Phys. Rev. E 62 (2000) 2450. Z.B. Dong, X.P. Liu, F. Li, H.T. Wang, A.G. Zhao, Earth Surf. Processes Landf. 27 (2002) 641. L. Xie, Z.B. Dong, X.J. Zheng, Acta Mech. Sin. 21 (2005) 564. H. Cheng, X.Y. Zou, C.L. Zhang, J. Geophys. Res. - Atmos. 111 (2006) D22205. W. Zhang, J.H. Kang, S.J. Lee, Geomorphology 86 (2007) 320. L.Q. Kang, L.J. Guo, D.Y. Liu, Geomorphology 96 (2008) 1. S.L. Namikas, Sedimentology 50 (2003) 303. P.R. Owen, J. Fluid Mech. 20 (1964) 225. Y.P. Shao, M. Mikami, Bound. - Lay. Meteorol. 115 (2005) 359. J.R. Ni, Z.S. Li, C. Mendoza, Geomorphology 49 (2003) 205. Z.S. Li, D.J. Feng, S.L. Wu, A.G.L. Borthwick, J.R. Ni, Geomorphology 100 (2008) 484. G. Sauermann, K. Kroy, H.J. Herrmann, Phys. Rev. E 64 (2001) 031305. O. Duran, H. Herrmann, J. Stat. Mech. 7 (2006) P07011. S.A. Morsi, A.J. Alexander, J. Fluid Mech. 55 (1972) 193. Z.S. Li, J.R. Ni, C. Mendoza, Geomorphology 60 (2004) 359. S.L. Namikas, J. Coastal Res. 22 (2006) 1250. Z.B. Dong, X.P. Liu, H.T. Wang, A.G. Zhao, X.M. Wang, Geomorphology 49 (2002) 219. Z.B. Dong, N. Huang, X.P. Liu, J. Geophys. Res. - Atmos. 110 (2005) D24113.
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Launch velocity characteristics of non-uniform sand in aeolian saltation D.J. Feng, Z.S. Li, J.R. Ni ∗ Department of Environmental Engineering, Peking University, The Key Laboratory of Water and Sediment Sciences, Ministry of Education, Beijing 100871, China
article
info
Article history: Received 13 November 2008 Received in revised form 19 December 2008 Available online 9 January 2009 PACS: 92.40.-t 02.60.-x 45.70.-n Keywords: Aeolian saltation Numerical modelling Mass flux Launch velocity
a b s t r a c t A numerical model for the vertical profile of aeolian mass flux was established based on the launch velocity probability distribution of saltating sand particles. The mean launch velocity (v¯ 0 ) was used as a primary variable in the simulation, from which the average saltation height and length could be further calculated. The results showed that the mean launch velocity, the average saltation height, and the average saltation length increased with increasing wind friction velocity (u∗), although the increases for the first two were not significant. On the other hand, the relative mean launch velocity (v¯ 0 /u∗) was found to decrease with increasing wind friction velocity. Other interesting results were obtained from the simulation: both v¯ 0 and v¯ 0 /u∗ decreased with an increase in the sand grain size, which agreed with a previous report claiming that the launch velocity was limited by the inertia of bed grains. Furthermore, decreasing dependence of the average saltation height and length was also found with increasing sand grain size. Overall, the sand grain size was of significance to the determination of probability distribution of the launch velocity and thus the saltation characteristics. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Aeolian sand transport causes major environmental problems such as wind erosion, sand storms, and desertification in arid and semi-arid regions [1]. Saltation is one of the three major modes of particle motion during wind erosion, along with suspension and creep, and it plays a key role in the erosion process, accounting for 75% of the total sand transport flux [2]. Most particles on the sand bed surface are mainly initiated by aerodynamic forces or by the impact of other descending particles, and then accelerated during their flight in air. After they impact the surface, the rebounding particles are ejected. The impact entrainment is dominated in the equilibration of aeolian saltation (e.g., Refs. [3–5]). The grain–bed collision process is an important and perhaps the most complex component of saltation, but substantial uncertainty remains regarding many aspects of this process. The initial or launch velocity of saltating sand particles is a key issue for the investigation of the grain–bed collision process. The numerous studies on the grain–bed collision process have generally been treated in two ways: numerical and theoretical modelling (in Refs. [3–13]), and experimental measurements (in Refs. [14–27]). The mean launch angle has variously been found to be 90◦ , 50◦ , and in the range of 21◦ –33◦ , 43◦ –46◦ , 34◦ –41◦ , 27◦ –30◦ (in Refs. [2,14,16–18, 26], respectively). Many studies have revealed that the mean launch speed was positively proportional to the wind friction velocity (e.g., Refs. [3,7,9]), but recently Namikas [28] reported that the mean launch speed was essentially constant and independent of the wind friction velocity. The launch velocity probability distribution of saltating sand particles is a bridge to link microscopic and macroscopic aeolian research. Various modes of the launch velocity distribution have been found, corresponding to the normal function [17,24], log-normal function [18,25,27], Gaussian function [4,5,25], exponential
∗
Corresponding author. Tel.: +86 10 62751185; fax: +86 10 62756526. E-mail addresses: [email protected] (D.J. Feng), [email protected] (J.R. Ni).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.01.002
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function [4,5,27], modified exponential function [10], gamma function [14,16,24], and Weibull function [23]. Thus, these previous results are inconclusive, and uncertainties still exist. To date, much of the insight regarding the grain–bed collision process has been concentrated on a macroscopic representation of total sand, and it was assumed that saltating particles were uniform in size and followed identical trajectories (e.g., Refs. [2,29]). In practice, however, a natural sand bed is generally composed of differently sized and shaped grains, and has many variable geometric configurations. Saltating particles follow different trajectories, and then their characteristics have different functional dependences on size when colliding with the bed. Few investigations have been conducted on the separate launch velocity characteristics of different sizes of sand particles, even though they are fundamentally important for understanding the microscopic process of non-uniform saltation [30]. In this paper, the launch velocity characteristics of non-uniform sand in aeolian saltation were investigated by simulating the vertical profile of the sand mass flux combined with experimental data. Section 2 briefly introduces a numerical model of aeolian sand transport. The simulated mass flux was compared with both the experimental data of total sand particles studied by Ni et al. [31], shown in Section 3, and that of different-sized sand particles studied by Li et al. [32], shown in Section 4. The influences of wind velocity and grain size on the mean launch velocity were estimated in the simulation, and then the average saltation height and length were further calculated as a function of wind velocity and grain size. Section 5 summarizes the main conclusions. 2. Numerical modelling Most numerical models of aeolian sand transport are generally subdivided into four component processes: aerodynamic entrainment, grain trajectories, grain–bed collision, and wind field modification (e.g., Refs. [3,5,6]). These components form a self-limited feedback process between air motion, particle motion, and particle–air and particle–surface interactions. This model allows simulation of the whole process of aeolian saltation from inception by aerodynamic entrainment to a steady state when each sub-model is combined [33]. The proposed model in this study was used to simulate the relative proportion of the vertical profile of the mass flux, rather than the absolute magnitude of the mass flux. Thus, aerodynamic entrainment was not needed. Furthermore, the goal of the present model was to investigate the fully developed flow of wind-blown sand, rather than to study the temporal development of aeolian sand transport. The simulated sand mass flux herein was saturated. In this state, no more particles can enter into the flow since the amount of sand entrained by the wind has a maximum value [34]. Hence, it was assumed that the equilibrium of aeolian saltation was achieved throughout simulations, and no wind field modification component was required. 2.1. Grain trajectories It is widely accepted that there are four forces that govern the flight of saltating particles: force due to gravity, aerodynamic drag, the Magnus effect, and aerodynamic lift. Following the precedent set (e.g., Refs. [6,17,28]), gravity and aerodynamic drag are only considered for the calculation of grain trajectories. The equations of motion used in the model are dvx /dt = −0.75ρa /(ρp d)Cd Ur (vx − U )
(1)
dvz /dt = −0.75ρa /(ρp d)Cd Ur vz − g
(2)
where vx and vz are the sand velocities (v ) in the horizontal (x) and vertical (z) directions, respectively; ρa is the density of air (1.23 kg m−3 ); ρp is density of sand (2650 kg m−3 ); d is sand diameter; g is gravity; and U is time-averaged horizontal wind velocity at vertical height. The drag coefficient Cd is specified using the function proposed for a spherical grain by Morsi and Alexander [35]. Ur is the relative velocity between sand particle and wind flow, which is written as Ur = ((vx − U )2 + vz2 )1/2 .
(3)
2.2. Wind velocity profile In order to solve the trajectory equations, it is necessary to specify the vertical profile of the horizontal wind velocity and the initial or launch velocity of the sand particles. The mean stream-wise wind velocity profile without particle motion is well governed by the standard logarithmic model. Particle motion significantly alters the wind velocity profiles. In accord with the corresponding wind velocity data of Ni et al. [31], the wind velocity profile represented here, as proposed by Li et al. [36], is as follows: U = u ∗ /κ ln(2gz /(0.04u ∗2 )) U = u ∗ /κ
Z
z
1/z
q
z > u ∗2 /(6.7g )
(4)
1 − (1 − u ∗t 2 /u ∗2 )(1 − 6.7gz /u ∗2 )e−6.7gz /u ∗ dz 2
z ≤ u ∗2 /(6.7g )
(5)
z0
where u∗ is the wind friction velocity; u ∗t is the threshold friction velocity; k is the von Karman constant (taken as 0.41), and z0 is the roughness length of the surface or the height above the bed at which flow velocity tends to zero. The values of
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Fig. 1. Comparison of the experimental data of Ni et al. [31] and the simulated results using Eqs. (6) and (7) with a mean launch angle of 35◦ .
u∗ and z0 are generally derived from experimental measurements, using the simple slope method. Bagnold [2] suggested z0 = 1/30d50 (where d50 is the mean grain diameter). 2.3. Grain–bed collision The grain–bed collision process determines the initial or launch velocity of saltating particles, which further exerts a strong effect on the particle trajectories and sand mass flux distribution. On striking the surface, saltating particles impart momentum to stationary particles. This impact may result in the rebound of the original particles as well as the ejection of other stationary particles into the air stream. Grain–bed collision is an extremely complex process, so it is difficult to observe and measure it directly under natural conditions. Great effort has been made to study the empirical and statistical characteristics of the launch speed and angle, commonly described by a probability distribution function. Different patterns of the launch velocity distribution have been proposed by many investigators, but no general agreement has been achieved according to the published literature (e.g., Refs. [5,25, 27]). In general, there are two major modes of the launch velocity distribution: the exponential and log-normal functions, which could be expressed as follows: p(v0 ) = 1/¯v0 exp(−v0 /¯v0 )
(6)
p(α0 ) = 1/α¯ 0 exp(−α0 /α¯ 0 )
√
(7)
p(v0 ) = 1/( 2π σv0 v0 ) exp(−(ln v0 − ln v¯ 0 ) /(2σv0 )) 2
2
√ p(α0 ) = 1/( 2π σα0 α0 ) exp(−(ln α0 − ln α¯ 0 )2 /(2σα20 ))
(8) (9)
where p(v0 ) and p(α0 ) are the probability distribution functions of the launch speed (v0 ) and angle (α0 ), respectively; v¯ 0 and α¯ 0 are the mean values of launch speed and angle, respectively; σv0 and σα0 are the standard deviations of the launch speed and the angle’s logarithm, respectively. Using the probability distribution functions of the launch speed and angle (Eqs. (6)–(9)) and integrating the Eqs. (1)–(5), the vertical profile of sand mass flux can be expressed as q(z ) = nm
Z 0
∞
p(v0 )
π
Z
p(α0 )(vx / |vz + | + vx / |vz − |)dv0 dα0
(10)
0
where n is the dislodgment rate; m is the mass of a single sand grain; |vz + | represents the average of |vz | over ascending particles, and |vz − | represents its average over descending particles in the slice z ± 1/2dz. 3. Launch velocity characteristics of total sand particles Ni et al. [31] measured the vertical profiles of the stream-wise wind velocity and the saturated mass flux in a wind tunnel containing a bed of naturally mixed sand. The present simulated sand mass flux was compared with the wind tunnel data of Ni et al. [31]. The setting parameters were as follows. The launch speed and angle distribution used Eqs. (6) and (7), respectively. The sand diameter was 0.35 mm (mean grain size of the test sands in Ni et al. [31]), and its density was 2650 kg m−3 . The mean launch angle was 35◦ . The mean launch velocity was used as a primary variable in the simulation. For comparison with the tunnel data of Ni et al. [31], the simulated mass flux was scaled by the q1exp /q1sim (q1exp and q1sim were the mass fluxes at 1 cm above the sand bed surface for the experimental data of Ni et al. [31] and for the model-simulated results, respectively). These were plotted in a log-linear coordinate system, as shown in Fig. 1. The variation of the mean
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Fig. 2. Variations of the mean launch velocity of total sand particles as a function of the wind friction velocity obtained from the simulated results in Fig. 1.
Fig. 3. Variations of the average saltation length and height of total sand particles with the wind friction velocity based on the estimated mean launch velocity in Fig. 2. Table 1 Comparison of the vertical slopes between the simulated and the experimental (Ni et al. [31]) sand mass flux distribution in Fig. 1. u∗ (m s−1 )
Vertical slopes
0.47 0.77 1.11 1.53 2.31
Experiment
Simulation
Difference (%)
−0.113 −0.101 −0.097 −0.090 −0.081
−0.108 −0.110 −0.107 −0.102 −0.091
−4.8 8.8 10.5 13.2 12.6
launch velocity obtained from the simulation as a function of the wind friction velocity is shown in Fig. 2. After estimating the mean launch velocity, we calculated the average saltation length (Ls0 ) and height (Hs0 ) by the following formulas: ∞
Z
p(v0i )
Ls0 = 0
Z Hs0 = 0
π
Z
p(α0i )Lsi dv0 dα0
(11)
p(α0i )Hsi dv0 dα0
(12)
0
∞
p(v0i )
π
Z 0
where Lsi and Hsi are the maximum jump length and height of the i-th particle with a launch speed of v0i and a launch angle of α0i , respectively. Fig. 3 illustrates the variations of average saltation length and height with wind friction velocity. Fig. 1 showed that reasonable agreement between the simulated and experimental results has been achieved. The vertical slopes of the best-fit exponential function for the experimental and simulated mass flux above the saltation layer (z > 4 cm) were obtained following Namikas [28], and the comparison results are shown in Table 1. From Table 1, the difference
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between the simulated and the experimental slopes increased with increasing wind velocity, which implies that the errors of simulation would increase as the wind velocity becomes greater. Fig. 2 shows a mild increase of the mean launch velocity (v¯ 0 ) from 1.2 m s−1 to 1.5 m s−1 as the wind friction velocity increases from 0.47 m s−1 to 2.31 m s−1 , while an apparent decrease of the relative mean launch velocity (v¯ 0 /u∗) occurs correspondingly from 2.7 to 0.5. Note that the mean launch velocities of 1.22 m s−1 and 1.23 m s−1 corresponded to the two runs with the lowest wind velocities u∗ = 0.47 m s−1 and u∗ = 0.77 m s−1 (see Fig. 2), respectively. Thus, the mean launch velocity remained approximately constant under the two lowest u∗ conditions (within experimental error). This finding agreed well with the results reported by Namikas [28] that v¯ 0 was largely independent of u∗. In Namikas’ [28] data set, u∗ ranged from about 0.27 m s−1 to 0.63 m s−1 , roughly corresponding to the two lowest values mentioned above. The simulated mean launch velocities reported for the three highest u∗ experiments were different from those for the two smallest u∗ experiments. This may be explained as follows: sand would not saltate under these conditions, and it would fly into suspension. Thus, modelling saltation under such conditions seems inappropriate. While the wind speeds of these conditions were relatively rare and represented unusual extremes rather than typical conditions, they certainly occurred in the natural environment, and especially in wind tunnel experiments with a saltation cloud. The results in Fig. 2, under the two lowest u∗ conditions, disagreed with those of Bagnold [2], who suggested that the mean launch velocity scaled with the wind friction velocity. The mean launch velocities have previously been found to be 1.8u∗, 3.3u∗, 5.8u∗, and 3.5(4.4)u∗ in Refs. [14–16,18], respectively. Anderson and Hallet [3] proposed a mean launch velocity of 0.63u∗. McDonald and Anderson [9] obtained good results with a value of 1.0u∗. Nishimura and Hunt [21] found that the relative mean launch velocity of ice particles decreased from 5.4 to 2.5 with increasing wind friction velocity (0.35–0.65 m s−1 ). However, the three largest u∗ experiments showed a clear trend of an increase in v¯ 0 with u∗. As the wind friction velocity increases, saltating sand particles obtain more momentum from the wind flow, and impact the sand surface with a larger velocity. Both the rebound velocity of impact particles and the ejected velocity of surface particles increase accordingly. This physical explanation may account for this study’s results, that the mean launch velocity was positively proportional to the wind friction velocity. By considering the negative dependence of the relative mean launch velocity on the wind friction velocity, we may infer that more factors in addition to wind friction velocity are involved in influencing the mean launch velocity. Fig. 3 showed that the average saltation length (Ls0 ) increased from about 0.25 m to 0.65 m with increasing wind friction velocity. Many experiments have been carried out to measure the saltation trajectory characteristics of sand particles [14– 18,26]. Willetts and Rice [16] reported that Ls0 was about 0.64–0.8 m under the wind friction velocity of 0.39 m s−1 . Nalpanis et al. [18] found that Ls0 varied between 0.064 m and 0.08 m for the given u∗ values ranging from 0.18 m s−1 to 0.2 m s−1 . Zhang et al. [26] obtained Ls0 values of 0.036–0.06 m under a u∗ of 0.21–0.39 m s−1 . Sørensen’s [17] results revealed that there was a clear increase of Ls0 with increasing u∗, which was similar to the present results. We can explain this by the fact that the sand saltation length was determined by the flight time and the particle horizontal velocity. Saltating particles in air are dragged by the wind to the downstream position. They will absorb more momentum if the wind flow velocity increases; the particle horizontal velocity increases accordingly, and this leads to an increase in the saltation length. Fig. 3 also shows that the average saltation height (Hs0 ) was almost constant (about 0.026 m) and independent of wind friction velocity, with the exception of high wind speed conditions of a larger value (about 0.031 m). The values of Hs0 have been experimentally reported to be about 0.048–0.056 m, 0.005–0.006 m, 0.002–0.004 m in Refs. [16,18,26], respectively. In contrast to the present study, some investigators have proposed that the saltation height is positively proportional to the wind friction velocity (e.g., Refs. [2,29]). The disagreement may be attributed to the fact that the saltation height increases with an increase in the launch velocity of sand particles. In general, the vertical wind velocity is irrelevant to steady aeolian sand transport, and the vertical motion of sand particles is mainly determined by the launch velocity. Fig. 2 reveals that the difference in the magnitude of the mean launch velocity was small under different wind velocities, except for the largest one (u∗ = 2.31 m s−1 ). Thus, the variation of the estimated mean launch velocities resulted in the effects on the average saltation height. There were some uncertainties in the present simulation study. Although it is generally accepted that the mean grain size of the surface sand should be used as the input parameter to simulate the vertical profile of the mass flux of total sand, and that this treatment gives the overall characteristics of total sand transport, this simplification does not represent the actual conditions, and a more appropriate grain size for the simulation needs to be better specified. The mean launch velocity in this section represented the average launch velocity of total sand particles. 4. Launch velocity characteristics of different-sized sand particles Li et al. [32] investigated the vertical profiles of the sand mass flux of different size gradings by re-analyzing the wind tunnel data of Ni et al. [31], and suggested that the vertical profile of the sand mass flux depended on the size grading. The data for the four size gradings of Li et al. [32] were compared with the simulated results of the present model: 0.18–0.25 mm, 0.25–0.355 mm, 0.355–0.5 mm, and 0.5–0.71 mm. The input parameters are as follows. Eqs. (6) and (9) were used in the size gradings of 0.18–0.25 mm and 0.25–0.355 mm. Eqs. (6) and (7) were used in the size gradings of 0.355–0.5 mm and 0.5–0.71 mm. The sand diameter used the mean grain diameter of each size grading, and its density was 2650 kg m−3 . The mean launch angle was 35◦ , and its standard deviation was 0.5. The mean launch velocity was also used as a primary variable
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Table 2 Comparison of the vertical slopes between the simulated and the experimental (Li et al. [32]) sand mass flux distributions in Fig. 4. Grain size grading (mm)
Vertical slopes u∗ = 1.11 (m s−1 )
0.18–0.25 0.25–0.355 0.355–0.5 0.5–0.71
u∗ = 1.53 (m s−1 )
u∗ = 2.31 (m s−1 )
Exp.
Sim.
Dif. (%)
Exp.
Sim.
Dif. (%)
Exp.
Sim.
−0.025 −0.075 −0.166 −0.353
−0.030 −0.068 −0.172 −0.377
16.6 −8.4 3.7 6.9
−0.033 −0.061 −0.136 −0.198
−0.030 −0.060 −0.146 −0.228
−8.9 −0.8
−0.016 −0.048 −0.105 −0.169
−0.017 −0.045 −0.113 −0.170
7.9 14.9
Dif. (%) 3.8
−5.1 7.6 0.9
Fig. 4. Comparison of the experimental data of Li et al. [32] and the simulated results using Eqs. (6) and (7) with a mean launch angle of 35◦ for the grain size grading of 0.355–0.5 mm and 0.5–0.71 mm, and using Eqs. (6) and (9) with a mean launch angle of 35◦ and its standard deviation of 0.5 for the grain size grading of 0.18–0.25 mm and 0.25–0.355 mm.
in the simulation. After estimating the mean launch velocity, the average saltation length and height were calculated as a function of wind friction velocity and grain size. A comparison of the simulated sand flux with the experimental data of Li et al. [32] for different size gradings is shown in Fig. 4 (also following Namikas [28]). The discrepancy of the vertical slopes of the sand mass flux between the experimental data of Li et al. [32] and the simulated results for the outer saltation layer are shown in Table 2. The comparison results showed that the agreement between the simulated results and the experimental data was good. Fig. 4 shows that both the simulated and experimental sand mass fluxes of the grain size grading ≤0.355 mm decayed exponentially with height whereas those of the grain size grading >0.355 mm represented a remarkably positive deviation in the near-bed region. This result may support the idea that the creep of coarse sand particles causes the deviation of the sand mass flux from the exponential distribution near the sand surface. The dependence of the mean launch velocity on both the wind friction velocity and the sand grain size is shown in Fig. 5. The variation of mean launch velocity with the wind friction velocity in Fig. 5 was similar to that of Fig. 2. This may indicate that the wind friction velocity exerts a similar effect on the launch velocity of different-sized sand particles. Fig. 5 also shows that the (relative) mean launch velocity decreased with an increase in grain size. Rice et al. [19] used highspeed photographic techniques to observe the launch velocity of different-sized particles on a sand surface of multiple grain sizes that was divided into 0.15–0.25 mm, 0.3–0.35 mm, and 0.425–0.6 mm gradings. Rice et al. [19] found that the mean rebound velocities of coarse sand particles were larger than those of fine particles, and the mean ejected velocities of all particles were almost the same. Therefore, the experimental data of Rice et al. were similar to the present simulated results: the mean launch velocities of coarse sand particles were larger than those of fine grains. In addition, the present results agreed with the results presented by Namikas [37], that the launch velocity was limited by the inertia of bed grains. This may be explained by the fact that the number of sand particles ejected from the bed after impact with coarse particles is larger than that caused by fine particles [8]. So coarse particles lose more momentum in the grain–bed collision process and then leave the bed at lower launch velocities than fine grains, but with approximately the same kinetic energy level (i.e., the launch velocity is inversely proportional to the grain mass). If the impact energy exceeds the inertia of the bed grains, the bed grain is set into motion and the impactor slams into more grains and transfers more energy, until it finally hits a grain with sufficient inertia to allow rebound [37]. However, the grain–bed collision process is too complex to be understood only by using such theoretical models. A great deal of effort has been made to investigate the launch velocity of total sand particles on a surface containing multiple grain sizes, but few research studies have been done on the launch velocities of different-sized sand particles, which is the particular focus of this section.
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Fig. 5. Variations of the mean launch velocity of the different-sized sand particles as a function of the wind friction velocity and the sand grain size obtained from the simulated results in Fig. 4.
Fig. 6. Variations of the average saltation length and height of the different-sized sand particles with the wind friction velocity and the sand grain size based on the estimated mean launch velocity in Fig. 5.
Fig. 6 illustrates the variations of average saltation length and height with the wind friction velocity and sand grain size. It shows that the average saltation length increased with increasing wind friction velocity, whereas the average saltation heights are almost the same value for u∗ = 1.11 m s−1 and 1.53 m s−1 , which are smaller than that for u∗ = 2.31 m s−1 . This result was similar to that of Fig. 3. Fig. 6 also shows that all of the saltation lengths and heights decreased with increases in grain size. We may infer that the motion of coarse sand particles is more easily influenced by other particles than that of fine grains. Nalpanis et al. [18] obtained similar results in wind tunnel experiments containing different-sized sand beds. Usually, the height representing 50% of the cumulative percentage of the total sand transport rate is defined as the average saltation height [1]. Using this definition, Dong et al. [38] analyzed the experimental data of mass flux and found that the average saltation height increased with increasing grain size, which was different from the present results. The discrepancy between the present results and those of Dong et al.’s may be explained by two factors. First, the definition of average saltation height for the present study was different from that in Dong et al. [38], although the two different definitions may reflect similar characteristics of saltation trajectory. Second, Dong et al.’s conclusion merely reflected the saltation height characteristics obtained assuming uniform bed sand when the particle size varied in a series of separate tests. Whether the characteristics remained similar for non-uniform bed sand is another matter. Thus, it can be seen that the behavior of average saltation length and height shown in Fig. 6 reflected the results shown in Fig. 5. The simulated results further revealed that the launch velocity distribution was determined by the sand particle size, which supports the assumption of Shao and Mikami [30] that the profiles of saltation fluxes of particles of different sizes differed, but these profiles can be universal if the initial velocity probability density function is given. However, the present results only reflected the launch velocity characteristics obtained using the experimental data from high wind velocity (u∗ ≥ 1.11 m s−1 ). Whether the characteristics would remain similar for low wind velocity was another matter. In addition, we assumed that the motion in air of different-sized sand particles was independent, and midair inter-particle collisions were ignored. Actually, there should be interaction among saltating sand particles [39]. Thus, future studies need to be conducted on the characteristics of non-uniform sand in aeolian saltation.
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5. Conclusions Based on the two typical probability distribution functions of the launch velocity of saltating sand particles, a numerical model was established to simulate the vertical profile of the sand mass flux. The mean launch velocity (v¯ 0 ) was used as an important parameter in the simulation, which was also of significance in the calculation of the average saltation length and height. The simulated sand mass fluxes agreed well with both the experimental results on total sand particles by Ni et al. [31] and those on different-sized sand particles by Li et al. [32]. The mean launch velocity and average saltation height of total sand particles was essentially constant and independent of the shear velocity under lower wind velocity conditions, although they increased with increasing wind friction velocity (u∗) under higher wind velocity conditions. The relative mean launch velocity (v¯ 0 /u∗) decreased with increasing wind friction velocity but the average saltation length increased. The simulated results on launch velocity depending on particle size agreed with Namikas’ claim in [37] on the launch velocity being limited by the inertia of bed grains. Both the absolute and the relative mean launch velocities of different-sized sand particles varied with the wind friction velocity; so did the average saltation length and height. On the other hand, the above four parameters decreased with increasing particle size. Acknowledgements The project was supported by the National Natural Science Foundation of China under Grant No. 49625101, 40371011. The reviewers’ valuable comments for improving the manuscript are especially acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
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