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The characteristic of streamwise mass flux of windblown sand movement
Geomorphology 139–140 (2012) 188–194
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The characteristic of streamwise mass flux of windblown sand movement Jian-Jun Wu, Sheng-Hu Luo ⁎, Li-Hong He Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education of China; School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China
a r t i c l e
i n f o
Article history: Received 12 April 2011 Received in revised form 10 October 2011 Accepted 11 October 2011 Available online 20 October 2011 Keywords: Saltation Streamwise mass flux Stratification pattern
a b s t r a c t Two patterns have been reported in previous literature about the streamwise mass flux of windblown sand movement in steady state; namely, the exponential pattern and the stratification pattern. Although the exponential pattern has been verified by some experimental measurements and numerical simulations, an agreed conclusion is not reached yet. Re-evaluating the previous model, it can be found that the pattern of the PDF (probability density function) distribution of the initial liftoff velocity is the key factor that determines whether the stratification happens or not. If the number of liftoff grains decreases with the increase in liftoff velocity, such as in the case of the exponential distribution, the stratification pattern will not appear. If the number of liftoff grains increases first and decreases later with the increase in liftoff velocity, such as in the case of the log-normal distribution, Gamma distribution and Gaussian distribution, the stratification pattern will come into existence. In addition, the hop height for the initial liftoff velocity, corresponding to the peak value of its PDF distribution, equals the height of the maximum streamwise mass flux. On the basis of some deterministic laws from experimental measurements and numerical simulations of grain/bed impact, a qualitative understanding about the PDF distribution of the liftoff velocity is obtained and the Rayleigh distribution is used for reptating grains. Also included in this paper is a model in which grain/bed impact is discussed and the interaction between the grains and the wind is considered. It is shown that the stratification pattern, composed of a linear increment layer, a saturation layer, and a monotonic decrement layer, indeed exists in the streamwise mass flux of windblown sand movement. Further, as with the average velocity and PDF distribution of the liftoff velocity of the reptating grains, which is determined by the characteristics of grains, the height of the peak value of the streamwise mass flux is also determined by the property of the grains and will be independent of the wind intensity. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Windblown sand movement is responsible for aeolian landforms, deflation and soil erosion. The dominating transport mechanism in windblown sand movement is saltation (e.g., Bagnold, 1941; Anderson et al., 1991; Almeida et al., 2006). The saltating grains can travel a long distance by successive jumps and have sufficient energy to splash other grains when they impact the bed. These splashed, reptating grains, contribute to the augmentation of the sand flux, and some of them can become saltating grains through successive rebounding (e.g., Andreotti, 2004). However, quantitatively, this process is far from being understood (Anderson et al., 1991; Almeida et al., 2006). For more than half a century, many researchers paid their attention to windblown sand movement to understand and predict the process of saltation. For instance, Anderson and Haff (1988, 1991), McEwan and Willetts (1991, 1993), Zheng et al. (2006) and Ren and Huang (2010) have conducted numerical studies on the evolution of the windblown sand movement. Other macroscopic quantities, such as the streamwise sand transport and wind velocity profile, have also ⁎ Corresponding author. Tel.: + 86 931 8914240; fax: + 86 931 8912229. E-mail address: [email protected] (S.-H. Luo). 0169-555X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2011.10.017
been studied in respect of experiment, simulation and theory to explain the empirical findings (e.g. Owen, 1964; Ungar and Haff, 1987; Anderson and Haff, 1988, 1991; Werner, 1990; White and Mounla, 1991; Iversen and Rasmussen, 1999; McEwan et al., 1999; Namikas, 2003; Andreotti, 2004; Dong et al., 2004; Sørensen, 2004; Almeida et al., 2006; Duran and Herrmann, 2006; Huang et al., 2006; Kok and Renno, 2008, 2009; Rasmussen and Sørensen, 2008). Meanwhile, the effect of fluctuations, mid-air collisions, and electrification on windblown sand movement have also been considered (e.g., Anderson, 1987; Sørensen and McEwan, 1996; Zheng et al., 2003; Dong et al., 2005; Shao, 2005; Huang et al., 2007; Kok and Renno, 2008; Kok and Lacks, 2009). The PDF distribution of the liftoff velocity is a bridge connecting studies of the microcosm and macrocosm. Although some attempts have been made to develop physical models of the grain/bed collision (e.g., Willetts and Rice, 1986; Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000) and obtain the PDF distributions of possible liftoff angles and speeds, such as Gamma distribution, exponential distribution, log-normal distribution and Weibull distribution, it is generally acknowledged that the splash function remains ‘poorly defined’ (McEwan and Willetts, 1991), and that the specification of grain liftoff velocity is a key area of uncertainty. Nevertheless, it has been established from previous experiments (e.g.,
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Mitha et al., 1986; Nalpanis et al., 1993; Rioual et al., 2000; Beladjine et al., 2007) and numerical simulations (e.g., Anderson and Haff, 1988, 1991; Werner, and Haff, 1988; Andreotti, 2004; Kok and Renno, 2009) that the average rebound velocity is a fraction of the impact velocity, and the number of reptating grains rather than the reptating velocity increase as the impact speed increases. Generally speaking, the intensity of the windblown sand movement is commonly quantified using the streamwise mass transport, which is the vertical integration of the streamwise mass flux. Chepil (1945) first suggested that the streamwise mass flux in the steady state decays exponentially with height above the surface, and this opinion has been verified by some experimental measurements and numerical simulations (e.g., Anderson and Hallet, 1986; McEwan and Willetts, 1991; Kok and Lacks, 2009). However, another pattern, the stratification pattern, has emerged from experimental measurements (Greeley et al., 1982; Yin, 1989; Dong et al., 2004) and numerical simulations (Anderson and Hallet, 1986; Zheng et al., 2004; Shao, 2005). The stratification pattern was discussed in detail by Zheng et al. (2004), who suggested that the streamwise mass flux is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer, and the height of the peak value of the streamwise mass flux increases with the wind speed. The difference between the stratification pattern and exponential pattern lies in that the region where the linear increment layer appears is very close to the sand surface, and thus it is difficult to measure the streamwise mass flux precisely in experiments. Whether the stratification pattern appears is determined by the PDF distribution pattern of the liftoff velocity, according to the model of Shao (2005), in which the stratification pattern exists in the case of Gaussian distribution rather than exponential distribution. But in the model of Zheng et al. (2004) the stratification appears in both Gamma and exponential distributions. In this paper the model given by Zheng et al. (2004) is employed with the PDF distribution of the liftoff velocity given by Anderson and Hallet (1986) to examine the stratification pattern. Then, based on some deterministic laws of the grain/bed collision and the Constitution Theory, a PDF distribution of the liftoff velocity is obtained and a model is established to show whether the stratification pattern exists or not, and the characteristics of the streamwise mass flux are also expounded. 2. Revaluating the previous model There is a difference between Shao (2005), Anderson and Hallet (1986) and Zheng et al. (2004), that is the stratification in the steady state appears with the exponential distribution of liftoff velocity in Zheng et al. (2004) but not in Shao (2005) and Anderson and Hallet (1986). Hence, we use the model advised by Zheng et al. (2004) to re-examine this problem under the condition of the Gamma
Fig. 1. Probability distribution of liftoff velocity. Friction velocity is 0.5 m s− 1.
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Fig. 2. Gamma distribution at the different friction velocities.
distribution and exponential distribution given by Anderson and Hallet (1986). The distribution functions are: :
: y0 1 exp − 0:63u 0:63u
ð1Þ
:
: 3 : y0 y0 27 1 exp −3 2 0:96u 0:96u 0:96u
ð2Þ
f ðy0 Þ ¼
f ðy0 Þ ¼
It can be found that there is considerable difference in the upper two probability distributions of liftoff velocity (Fig. 1), and the liftoff velocity of grains varies with the friction velocity (Fig. 2) in Gamma distribution. The calculation model and the parameters can be found in Zheng et al. (2004), and the minimal liftoff velocity of the grains selected in the model can hop a grain diameter due to gravity. It is apparent in Fig. 3 that the stratification model exists with a Gamma distribution but not with an exponential distribution. The PDF distribution of liftoff velocity plays an important role in the streamwise mass flux. This finding is consistent with that of Anderson and Hallet (1986) and Shao (2005), but not that of Zheng et al. (2004), who suggested that the stratification happens with both Gamma and exponential distributions. Zheng et al. (2004) also suggested that the height of the peak value of the streamwise mass flux increases with wind speed, and the same result can be found (Fig. 4). It can be found that the velocity has a Gamma distribution (Fig. 2), which corresponds with the peak value of the PDF distribution of liftoff velocity of 0.48 m s − 1 as u* = 0.5 m s − 1 and 0.77 m s − 1 as u* = 0.8 m s − 1, and the hop height is, respectively, 0.00582 m and 0.01716 m in the steady state. It
Fig. 3. The streamwise mass flux with the Gamma and exponential distribution. The friction velocity is 0.5 m s− 1 and the diameter is 0.25 mm.
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important role since most grains lift off at a velocity corresponding with the peak of the PDF distribution of liftoff velocity in a Gamma distribution. The hop height for the grains' liftoff velocity at that velocity then equals the height of the peak value of the streamwise mass flux. 3. Fundamental model
Fig. 4. The streamwise mass flux at different friction velocities. The Gamma distribution is chosen in the model and the diameter is 0.25 mm.
means that the number of grains at these two liftoff velocities are the greatest amongst others. As shown in the Fig. 4, the height corresponding with the peak value of the streamwise mass flux is 0.0058 m and 0.0172 m at the above two friction velocities. It is obvious that these are the same as the hop heights. Just as said by Anderson and Hallet (1986), the grains spend such a proportion of time near the top of their trajectories that the single-trajectory streamwise mass flux peak is sharply bent at the top of the trajectory. Meanwhile, for a grain diameter of 0.00025 m and density of 2650 kg m − 3, the grain mass of 2.175 × 10− 8 kg is very small compared with the grain velocity. Therefore, the reason why the streamwise mass flux varies with height may be that the grain mass flux is at its maximum above the surface, but the velocity of the grains is very small. With increasing height, the grain mass flux decreases rapidly although the grain velocity increases, so that the streamwise mass flux continuously declines. Since the number of liftoff grains decreases with the liftoff velocity in the exponential distribution, there should be no stratification pattern, and this is verified in Fig. 3. However, Zheng et al. (2004) suggested that pffiffiffiffiffiffistratification also exists in the exponential distribution. Here 4 gD m s − 1 is introduced as the minimum liftoff velocity, which is larger than the one used in the above calculation. As shown in Fig. 5, the stratification pattern does exist in the exponential distribution and the height of the peak value of the streamwise mass flux equals the hop height of grains with the minimum liftoff velocity at different friction velocities. This may explain the reason why Zheng et al. (2004) observed the stratification pattern. On the basis of above analyses, it is clear that the stratification pattern is determined by the pattern of the PDF distribution of liftoff velocity; the PDF distribution function of liftoff velocity plays an
Fig. 5. The streamwise mass flux at different friction velocities with exponential distribution chosen. The diameter is 0.25 mm.
Up to now, the classical problem of windblown sand physics, i.e., whether the stratification exists in the streamwise mass flux, is not yet answered. The only fact we are sure of is that the PDF distribution of liftoff velocity plays an important determinant role. Here, on the basis of the grain/bed result, this problem will be further investigated. As in the previous studies (e.g., Bagnold, 1941; Anderson and Haff, 1988, 1991; Werner, 1990; McEwan and Willetts, 1991; Kok and Lacks, 2009), the four distinct sub-processes – aerodynamic entrainment, grain trajectory, wind modification and grain/bed collision – are included in the model. 3.1. Grain trajectory For the sake of simplification, the grains are assumed to be spherical with identical diameter D and uniform density ρg, and the motion of grains is determined mainly by gravity, aerodynamic drag and aerodynamic lift. On the basis of Newton's equation, the fundamental dynamical equation for a single grain can be described as: ::
x¼
3ρ y ¼ −g− a 4 ρg ::
:
3 ρ V ðu− xÞ C a R 4 D ρg D
ð3Þ :
u2top −u2bot V y C D R −C l D D
! ð4Þ
where, x and y are, respectively, the horizontal and vertical coordinates of grains; single and double dots above the variable x and y denote grain speed and acceleration; ρa is air density; u is wind speed; g is the acceleration resulting from gravity; VR is grain velocity relative to wind; CD is the drag coefficient taken as an empirical formula (White, 1974); Cl is the lift coefficient, which is taken to be 0.85 CD (Anderson and Hallet, 1986); and utop and ubot are the wind speed at heights corresponding to the top and bottom of grain. 3.2. Grain/bed collision The grain/bed collision is a key physical process in windblown sand movement (Anderson and Haff, 1991). When the grains impact the bed, some of them can rebound, keeping a part of their energy, while some will be trapped by the bed and some grains will be splashed. A series of PDF distributions of liftoff velocities have been obtained from the experiments and numerical simulations, such as the exponential distribution (e.g., Anderson and Hallet, 1986; Huang et al., 2006), Gamma distribution (e.g., Anderson and Haff, 1988), log-normal distribution (e.g., Nalpanis et al., 1993; Beladjine et al., 2007), PearsonVII distribution (e.g., Zou et al., 2001), Rayleigh distribution (e.g., Werner and Haff, 1988; Werner, 1990; Rioual et al., 2000, 2003) and Weibull distribution (e.g., Dong et al., 2002). Because of the lack of a theoretically based model of collision process and the complexity of the natural conditions, the PDF distribution of liftoff velocities is difficult to obtain, and remains ‘poorly defined’ (McEwan and Willetts, 1991; Namikas, 2003). Although under natural conditions the grain/bed collision is an extremely complex process, some valuable information is obtained in spite of these difficulties. Here we use some deterministic laws which correspond with the average case. From experiments (e.g., Beladjine et al., 2007; Crassous et al., 2007) and numerical
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simulations (e.g., Deboeuf et al., 2009), it can be found that the total kinetic energy of the reptating grains, Eej , is proportional to the impact energy Eimp , and can be expressed as: Eej ¼ kEimp
ð5Þ
C ¼ −∫Ng ðe0 Þ lng ðe0 Þde0
ð6Þ
the requirement of which is that the total kinetic energy of ejected grains should be equal to the sum of the kinetic energy of ejected grains, that is: Eej ¼ ∫Ng ðe0 Þe0 de0
ð7Þ
The integral of PDF distribution of the kinetic energy should equal 1, that is: ∫gðe0 Þde0 ¼ 1
ð8Þ
With the aid of a Lagrangian method to find the extreme value of function (6), the PDF distribution of the kinetic energy can be obtained, that is: g ðe0 Þ ¼ expð−1 þ Ae0 þ BÞ
ð9Þ
1 e exp − 0 he0 i he0 i
! ð11Þ
where, bv02> = 2be0>/m. Up to now, we have obtained the PDF distribution of the liftoff velocity, which is in the form of a Rayleigh distribution. It can be found from the above derivation that only the relationship between the total kinetic energy of the ejected grains and the energy of the incident grain is used, which is a deterministic experiment law. That greatly reduces the impact of other factors on the accuracy of the PDF distribution of the liftoff velocity, and a qualitative understanding about the PDF distribution of the liftoff velocity is thus obtained. When the grains impact the bed, a constant fraction of impact momentum is transferred to the reptating grains (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000; Andreotti, 2004; Beladjine et al., 2007), and the number of reptating grains increases linearly with the impact velocity: 0:02 Neje ¼ pffiffiffiffiffiffi vimp gD
and the rebound angle is almost independent of the impact velocity. We thus take the kinetic energy of rebounding grains to be 45% of the impacting kinetic energy, and the rebound angle to be 45° from the horizontal. Meanwhile, some grains will be trapped when the grains impact with the bed, which is a stochastic process. Following the numerical simulations by Werner (1990) and Anderson and Haff (1988, 1991), the probability of rebound can be expressed as: preb
!! vimp ¼ 0:95 1− exp − pffiffiffiffiffiffiffiffi 10 g D
ð12Þ
From experiments (e.g., Mitha et al., 1986; McEwan et al., 1992; Nalpanis et al., 1993; Rice et al., 1995, 1996; Rioual et al., 2000) and
ð14Þ
3.3. Wind modification According to previous investigations (e.g., Anderson and Haff, 1988, 1991; Almeida et al., 2006; Kok and Lacks, 2009), the sand bed is assumed to be flat. From Newton's third law, the wind is modified by the transfer of momentum between the wind flow and the grains in the air, thus the turbulent Navier–Stokes equation of wind flow in the xdirection can be expressed as (McEwan et al., 1999): ρa
∂u ∂τ ¼ þ F x ðt; yÞ ∂t ∂y
ð15Þ
where τ is the turbulent shear stress, and Fx is the force per unit volume on the wind. They can be calculated solely by:
ð10Þ
where be0> = Eej/N is average kinetic energy of ejected grains. According to the relationship between kinetic energy and velocity, we can obtain the PDF distribution of the liftoff velocity, that is: 2v v 2 f ðv0 Þ ¼ 20 exp − 0 2 v0 v0
ð13Þ
2 2 ∂u ∂u τðt; yÞ ¼ ρa κ y ∂y ∂y
Under the constraints (7) and (8), we obtain: g ðe0 Þ ¼
numerical simulations (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; Andreotti, 2004; Kok and Renno, 2009), it can be found that the rebound velocity is, on average, a fraction of the impact velocity: vreb ¼ γvimp
where, k is a coefficient which is determined by the characteristics of the incident grains and the bed. A stochastic event tends to reach its maximum within certain limitations. The liftoff velocity of each grain in a grain/bed collision process is different from each other, which constitutes its complexity. Here, we assume that the number of ejected grains in a grain/bed collision process is N, and the complexity of the PDF distribution of kinetic energy of the ejected grains, according to Zhang (2003), is C:
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:
Fx ¼
yX max :
y min
ð16Þ
::
::
mx↑ mx↓ Nðt; yÞ : − : y↑ y↓
! ð17Þ
where κ is the von Karman's constant, m is the grain's mass, N is the number of grains at the liftoff velocity v0 and time t, and the arrows depict the upward and downward direction of the trajectory. Substituting Eqs. (16) and (17) into (15), the Navier–Stokes equation of wind flow in the xdirection reduces to: :: :: ! X ∞ mx↑ mx↓ ∂u ∂ 2 2 ∂u ∂u ρa N ðt; v0 Þ : − : ¼ ρ κ y þ y↑ y↓ ∂t ∂y a ∂y ∂y ′
ð18Þ
v0
where v0′ denotes the minimum liftoff velocity of grains that can reach height y. On the basis of the above fundamental equations, the streamwise mass flux is given by: Q ðyÞ ¼
∞ X v0 ′
:
Nðt; v0 Þ
:
mx↑ mx↓ − : : y↑ y↓
! ð19Þ
4. Numerical simulation In the numerical simulation, the aerodynamic roughness is taken as D/30. As shown in previous studies (e.g., Werner, 1990; Anderson and Haff, 1991; McEwan and Willetts, 1991; Kok and Lacks, 2009), some grains are entrained first, then more and more grains are splashed with grain/bed collision. When the variation of the wind
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Fig. 8. The wind profiles in the steady state.
Fig. 6. Comparison between the results in this paper and the empirical formulas given by Bagnold (1941), Kawamura (1951) and Zhou et al. (2002) and experimental result given by Iversen and Rasmussen (1991) in the steady state.
speed and the streamwise mass transport are smaller than a specified value in successive iterations, the steady state is achieved. Here, the numerical simulation is performed to confirm the validity of the model and the program proposed in this paper. On the basis of the model suggested here, through integrating Eq. (19), the streamwise mass transport is given by: Q 0 ¼ ∫Q ðyÞdy
ð20Þ
The well-known empirical formulas of the streamwise mass transport given by Bagnold (1941) and Kawamura (1951) are recognized to be efficient and the most widely quoted (e.g., Sarre, 1987; Shao, 2000; Zheng et al., 2004). When the friction velocity is less than 0.4 m s − 1, the empirical formula given by Kawamura (1951) is close to the experimental results, while when the friction velocity is greater than 0.4 m s− 1 and less than 0.7 m s− 1 the Bagnold's formula is more efficient, but the result using these empirical formulae is smaller than the experimental result with the increase of the friction velocity. Based on experimental results, an effective empirical formula is also given by Zhou et al. (2002). Here, we make a comparison of the streamwise mass transport between our result and those empirical formulas and experiment results given by Iversen and Rasmussen (1999). As shown in Fig. 6, when the friction velocity is less than 0.3 m s− 1, the numerical results are less than the result given by Iversen and Rasmussen (1999) and greater than the result given by Bagnold's empirical formulae. When the friction velocity is greater than 0.4 m s− 1, the numerical
Fig. 7. Evolution of the streamwise mass transport. The diameter is 0.25 mm.
results are greater than the result given by Bagnold (1941), Kawamura (1951) and Iversen and Rasmussen (1999) and very close to the value given by Zhou et al. (2002). This numerical example demonstrates that the model and the algorithm proposed in this paper are feasible. 5. Numerical results and discussions Fig. 7 shows the evolution of the streamwise mass transport at the friction velocity of 0.5 m s − 1. As shown, the time needed for the streamwise mass transport to reach the steady state is 2.6 s, which lies between the corresponding results drawn by Anderson and Haff (1991) and Zheng et al. (2006) (respectively 3 s and 2.5 s). Moreover, the streamwise mass transport in the steady state is much bigger than that given by Anderson and Haff (1991) or Zheng et al. (2006) and closer to experiment data given by Shao and Raupach (1992). This further verifies that the model and the algorithm are feasible. As shown in Fig. 8, the wind in the steady state is modified by the sand grains in the air, and the effective roughness increases with the increase of friction velocity, which can be calculated from projecting an intercept of the gradient of the wind profile down to the height where the wind speed would be zero, which is in accord with numerous measured wind speed profiles during saltation experiments, as summarized by Owen (1964). Obviously, the wind speed in the reptating layer is strongly reduced and varies little at the different friction velocities; and as the friction velocity increases, the wind speed in the reptating layer decreases appreciably, which is consistent with the findings of the previous studies (e.g., Andreotti, 2004). Figs. 9 and 10 show the trajectories of grains at the initial wind pffiffiffiffiffiffi speed and in the steady state at the liftoff velocity of gD m s − 1. It is obvious that the trajectory in the steady state is smaller than that at the initial wind speed; and, contrary to the trajectory in the initial wind speed which increases with the friction velocity, the trajectory in the steady state has an opposite tendency and varies slightly at different friction velocities. The reason for this can be found in Fig. 8.
Fig. 9. Trajectory of grain at the initial wind speed.
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Meanwhile, the hop height for the initial liftoff velocity, corresponding with the peak value of its PDF distribution, equals the height of the maximum streamwise mass flux. Therefore, like the liftoff velocity of the reptating grains and the PDF distribution of liftoff velocity, which are decided by the characteristics of grains, the height of the peak value of the streamwise mass flux also scales according to the property of the grains and will be independent of the wind intensity. 6. Conclusions
Fig. 10. Trajectory of grain in the steady state.
Because the grains with the smaller liftoff velocities travel in the reptating layer; the wind speed in the reptating layer varies little at the different friction velocities in the steady state; and as the friction velocity increases, the wind speed in the reptating layer decreases appreciably. Fig. 11 shows the streamwise mass flux at the different friction velocities in the steady state. It is shown that the streamwise mass flux of windblown sand movement displays the stratification pattern which is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer. This is consistent with Zheng et al. (2004). It also can be found that the peak value of the streamwise mass flux, which is about 4.5 mm below the focus point, is almost invariant at the different friction velocities in the steady state. Most of the grains in the air are the reptating grains, and some of them can be promoted to become the saltating grains (Andreotti, 2004). The grains in air have a exponential probability of surviving (Anderson and Haff, 1988, 1991; Werner, 1990) when they impact the bed, and the velocity of reptating grains is, on average, a constant independent of changes in the impact velocity (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000; Andreotti, 2004). Meanwhile, as shown in Fig. 8, the wind speeds in the reptating layer in the steady state are very close to each other at different friction velocities. Therefore, the motions of the reptating grains at different friction velocities are similar, as shown in Fig. 10, which was also suggested by Andreotti (2004). If the PDF distribution of liftoff velocity of the reptating grains follows the rule that the number of liftoff grains increases with the increase of the liftoff velocity first and decreases later, the PDF distribution of liftoff velocity in the steady state should be in the same form, which also corresponds with the previous studies (e.g., Anderson and Hallet, 1986; Mitha et al., 1986; Werner and Haff, 1988; Rioual et al., 2000; Beladjine et al., 2007; Crassous et al., 2007; Zhang et al., 2007).
Fig. 11. The streamwise mass flux in the steady state.
The numerical simulations of windblown sand movement were conducted to re-evaluate the classical problem of windblown sand physics, i.e., the probability distribution of the streamwise mass flux. With re-evaluing previous models, we established that whether stratification exists or not is determined by the pattern of the PDF distribution of liftoff velocity, and the hop height for the initial liftoff velocity, corresponding with the peak value of its PDF distribution, equalling the height of the maximum streamwise mass flux. The slower the liftoff velocity is, the more grains lift off in the steady state. However, in the case of the liftoff velocity of grains in the exponential distribution, the stratification pattern will not appear. Based on some deterministic laws of the grain/bed collision and the Constitution Theory, a qualitative understanding about the PDF distribution of the liftoff velocity is obtained, and a Rayleigh distribution for reptating grains is employed in the model. It is found that the streamwise mass flux of windblown sand movement displays the stratification pattern in the steady state, which is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer. Also, because the liftoff velocity of the reptating grains is decided by the characteristics of grains, the wind speeds in the reptating layer in the steady state are very close to each other at different friction velocities. Similarly, the height of the peak value of the streamwise mass flux, which does not change with the wind speed, is decided by the characteristics of the grains. Acknowledgments The authors would like to express their sincere appreciation to the National Basic Research Program of China (No. 2009CB421304) and the funding from the projects of the National Natural Science Foundation of China (No. 10772075, No. 10772074). References Almeida, M.P., Andrade Jr., J.S., Herrmann, H.J., 2006. Aeolian transport layer. Physical Review Letters 96, 018001. Anderson, R.S., 1987. Eolian sediment transport as a stochastic process: the effects of a fluctuating wind on particle trajectories. Journal of Geology 95, 497–512. Anderson, R.S., Haff, P.K., 1988. Simulation of eolian saltation. Science 241, 820–823. Anderson, R.S., Haff, P.K., 1991. Wind modification and bed response during saltation of sand in air. Acta Mechanica 1, 21–51 (Suppl.). Anderson, R.S., Hallet, B., 1986. Sediment transport by wind: toward a general model. Geological Society of America Bulletin 97, 523–535. Anderson, R.S., Sørensen, M., Willetts, B.B., 1991. A review of recent progress in our understanding of Aeolian sediment transport. Acta Mechanica 1, 1–19 (Suppl.). Andreotti, B., 2004. A two-species model of aeolian sand transport. Journal of Fluid Mechanics 510, 47–70. Bagnold, R.A., 1941. The Physics of Blown Sand and Desert Dunes. Methuen, New York. Beladjine, D., Madani, A., Luc, O., Alexandre, V., 2007. Collision process between an incident bead and a three-dimensional granular packing. Physical Review E 75 (6), 061305. Chepil, W.S., 1945. Dynamics of wind erosion: I. Nature of movement of soil by wind. Soil Science 60, 305–320. Crassous, J., Beladjine, D., Valance, A., 2007. Impact of projectile on a granular medium described by a collision model. Physical Review Letters 99, 248001. Deboeuf, S., Gondret, P., Rabaud, M., 2009. Dynamics of grain ejection by sphere impact on a granular bed. Physical Review E 79 (4), 041306. Dong, Z.B., Liu, X.P., Li, F., Wang, H.T., Zhao, A.G., 2002. Impact-entrainment relationship in a saltating cloud. Earth Surface Proceedings Land 27, 641–658. Dong, Z., Wang, H., Li, X., Wang, X., 2004. A wind tunnel investigation of the influences of fetch length on the flux profile of a sand cloud blowing over a gravel surface. Earth Surface Proceedings Land 29, 1613–1623.
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Rice, M.A., Willetts, B.B., McEwan, I.K., 1996. Observations of collisions of saltating grains with a granular bed from high-speed cine-film. Sedimentology 43, 21–31. Rioual, F., Valance, A., Bideau, D., 2000. Experimental study of the collision process of a grain on a two-dimensional granular bed. Physical Review E 62, 2450–2459. Rioual, F., Valance, A., Bideau, D., 2003. Collision process of a bead on a twodimensional bead packing: Importance of the inter-granular contacts. Europhysics Letters 61, 194. Sarre, P.D., 1987. Aeolian sand transport. Progress in Physical Geography 11, 157–182. Shao, Y.P., 2000. Physics and Modeling of Wind Erosion. Kluwer Acad, Norwell, Mass. Shao, Y.P., 2005. A similarity theory for saltation and application to aeolian mass flux. Boundary-Layer Meteorology 115, 319–338. Shao, Y., Raupach, M.R., 1992. The overshoot and equilibration of saltation. Journal of Geophysical Research 97, 20559–20564. Sørensen, M., 2004. On the rate of aeolian sand transport. Geomorphology 59, 53–62. Sørensen, M., McEwan, I., 1996. On the effect of mid-air collisions on aeolian saltation. Sedimentology 43, 65–76. Ungar, J., Haff, P.K., 1987. Steady state saltation in air. Sedimentology 34, 289–299. Werner, B.T., 1990. A steady-state model of wind-blown sand transport. Journal of Geology 98, 1–17. Werner, B.T., Haff, P.K., 1988. The impact process in eolian saltation: two dimensional simulations. Sedimentology 35, 189–196. White, F., 1974. Viscous Fluid Flow. McGraw-Hill, New York. White, B.R., Mounla, H., 1991. An experimental study of froude number effect on windtunnel saltation. Acta Mechanica Supplement 1, 145–157. Willetts, B.B., Rice, M.A., 1986. Collisions in aeolian saltation. Acta Mechanica 63, 255–265. Yin, Y.S., 1989. Study on sand drift in strong wind region in gravel desert. Journal of Desert Research 9, 27–36 (in Chinese). Zhang, X.W., 2003. The Constitution Theory. University of Science and Technology of China Press, He Fei. Zhang, W., Kang, J.H., Lee, S.J., 2007. Tracking of saltating sand trajectories over a flat surface embedded in an atmospheric boundary layer. Geomorphology 86, 320–331. Zheng, X.J., Huang, N., Zhou, Y.H., 2003. Laboratory measurement of electrification of windblown sands and simulation of its effect on sand saltation movement. Journal of Geophysical Research 108, 4322. Zheng, X.J., He, H.L., Wu, J.J., 2004. Vertical profile of mass flux for windblown sand movement at steady state. Journal of Geophysical Research 109, 01106. Zheng, X.J., Huang, N., Zhou, Y., 2006. The effect of electrostatic force on the evolution of sand saltation cloud. The European Physical Journal E 19, 129–138. Zhou, Y.H., Xiang, G., Zheng, X.J., 2002. Experimental measurement of wind-sand flux and sand transport for naturally mixed sands. Physical Review E 66, 021305. Zou, X.Y., Wang, Z.L., Hao, Q.Z., Zhang, C.L., Liu, Y.Z., Dong, G.R., 2001. The distribution of velocity and energy of saltating sand grains in a wind tunnel. Geomorphology 36, 155–165.
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The characteristic of streamwise mass flux of windblown sand movement Jian-Jun Wu, Sheng-Hu Luo ⁎, Li-Hong He Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education of China; School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China
a r t i c l e
i n f o
Article history: Received 12 April 2011 Received in revised form 10 October 2011 Accepted 11 October 2011 Available online 20 October 2011 Keywords: Saltation Streamwise mass flux Stratification pattern
a b s t r a c t Two patterns have been reported in previous literature about the streamwise mass flux of windblown sand movement in steady state; namely, the exponential pattern and the stratification pattern. Although the exponential pattern has been verified by some experimental measurements and numerical simulations, an agreed conclusion is not reached yet. Re-evaluating the previous model, it can be found that the pattern of the PDF (probability density function) distribution of the initial liftoff velocity is the key factor that determines whether the stratification happens or not. If the number of liftoff grains decreases with the increase in liftoff velocity, such as in the case of the exponential distribution, the stratification pattern will not appear. If the number of liftoff grains increases first and decreases later with the increase in liftoff velocity, such as in the case of the log-normal distribution, Gamma distribution and Gaussian distribution, the stratification pattern will come into existence. In addition, the hop height for the initial liftoff velocity, corresponding to the peak value of its PDF distribution, equals the height of the maximum streamwise mass flux. On the basis of some deterministic laws from experimental measurements and numerical simulations of grain/bed impact, a qualitative understanding about the PDF distribution of the liftoff velocity is obtained and the Rayleigh distribution is used for reptating grains. Also included in this paper is a model in which grain/bed impact is discussed and the interaction between the grains and the wind is considered. It is shown that the stratification pattern, composed of a linear increment layer, a saturation layer, and a monotonic decrement layer, indeed exists in the streamwise mass flux of windblown sand movement. Further, as with the average velocity and PDF distribution of the liftoff velocity of the reptating grains, which is determined by the characteristics of grains, the height of the peak value of the streamwise mass flux is also determined by the property of the grains and will be independent of the wind intensity. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Windblown sand movement is responsible for aeolian landforms, deflation and soil erosion. The dominating transport mechanism in windblown sand movement is saltation (e.g., Bagnold, 1941; Anderson et al., 1991; Almeida et al., 2006). The saltating grains can travel a long distance by successive jumps and have sufficient energy to splash other grains when they impact the bed. These splashed, reptating grains, contribute to the augmentation of the sand flux, and some of them can become saltating grains through successive rebounding (e.g., Andreotti, 2004). However, quantitatively, this process is far from being understood (Anderson et al., 1991; Almeida et al., 2006). For more than half a century, many researchers paid their attention to windblown sand movement to understand and predict the process of saltation. For instance, Anderson and Haff (1988, 1991), McEwan and Willetts (1991, 1993), Zheng et al. (2006) and Ren and Huang (2010) have conducted numerical studies on the evolution of the windblown sand movement. Other macroscopic quantities, such as the streamwise sand transport and wind velocity profile, have also ⁎ Corresponding author. Tel.: + 86 931 8914240; fax: + 86 931 8912229. E-mail address: [email protected] (S.-H. Luo). 0169-555X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2011.10.017
been studied in respect of experiment, simulation and theory to explain the empirical findings (e.g. Owen, 1964; Ungar and Haff, 1987; Anderson and Haff, 1988, 1991; Werner, 1990; White and Mounla, 1991; Iversen and Rasmussen, 1999; McEwan et al., 1999; Namikas, 2003; Andreotti, 2004; Dong et al., 2004; Sørensen, 2004; Almeida et al., 2006; Duran and Herrmann, 2006; Huang et al., 2006; Kok and Renno, 2008, 2009; Rasmussen and Sørensen, 2008). Meanwhile, the effect of fluctuations, mid-air collisions, and electrification on windblown sand movement have also been considered (e.g., Anderson, 1987; Sørensen and McEwan, 1996; Zheng et al., 2003; Dong et al., 2005; Shao, 2005; Huang et al., 2007; Kok and Renno, 2008; Kok and Lacks, 2009). The PDF distribution of the liftoff velocity is a bridge connecting studies of the microcosm and macrocosm. Although some attempts have been made to develop physical models of the grain/bed collision (e.g., Willetts and Rice, 1986; Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000) and obtain the PDF distributions of possible liftoff angles and speeds, such as Gamma distribution, exponential distribution, log-normal distribution and Weibull distribution, it is generally acknowledged that the splash function remains ‘poorly defined’ (McEwan and Willetts, 1991), and that the specification of grain liftoff velocity is a key area of uncertainty. Nevertheless, it has been established from previous experiments (e.g.,
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Mitha et al., 1986; Nalpanis et al., 1993; Rioual et al., 2000; Beladjine et al., 2007) and numerical simulations (e.g., Anderson and Haff, 1988, 1991; Werner, and Haff, 1988; Andreotti, 2004; Kok and Renno, 2009) that the average rebound velocity is a fraction of the impact velocity, and the number of reptating grains rather than the reptating velocity increase as the impact speed increases. Generally speaking, the intensity of the windblown sand movement is commonly quantified using the streamwise mass transport, which is the vertical integration of the streamwise mass flux. Chepil (1945) first suggested that the streamwise mass flux in the steady state decays exponentially with height above the surface, and this opinion has been verified by some experimental measurements and numerical simulations (e.g., Anderson and Hallet, 1986; McEwan and Willetts, 1991; Kok and Lacks, 2009). However, another pattern, the stratification pattern, has emerged from experimental measurements (Greeley et al., 1982; Yin, 1989; Dong et al., 2004) and numerical simulations (Anderson and Hallet, 1986; Zheng et al., 2004; Shao, 2005). The stratification pattern was discussed in detail by Zheng et al. (2004), who suggested that the streamwise mass flux is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer, and the height of the peak value of the streamwise mass flux increases with the wind speed. The difference between the stratification pattern and exponential pattern lies in that the region where the linear increment layer appears is very close to the sand surface, and thus it is difficult to measure the streamwise mass flux precisely in experiments. Whether the stratification pattern appears is determined by the PDF distribution pattern of the liftoff velocity, according to the model of Shao (2005), in which the stratification pattern exists in the case of Gaussian distribution rather than exponential distribution. But in the model of Zheng et al. (2004) the stratification appears in both Gamma and exponential distributions. In this paper the model given by Zheng et al. (2004) is employed with the PDF distribution of the liftoff velocity given by Anderson and Hallet (1986) to examine the stratification pattern. Then, based on some deterministic laws of the grain/bed collision and the Constitution Theory, a PDF distribution of the liftoff velocity is obtained and a model is established to show whether the stratification pattern exists or not, and the characteristics of the streamwise mass flux are also expounded. 2. Revaluating the previous model There is a difference between Shao (2005), Anderson and Hallet (1986) and Zheng et al. (2004), that is the stratification in the steady state appears with the exponential distribution of liftoff velocity in Zheng et al. (2004) but not in Shao (2005) and Anderson and Hallet (1986). Hence, we use the model advised by Zheng et al. (2004) to re-examine this problem under the condition of the Gamma
Fig. 1. Probability distribution of liftoff velocity. Friction velocity is 0.5 m s− 1.
189
Fig. 2. Gamma distribution at the different friction velocities.
distribution and exponential distribution given by Anderson and Hallet (1986). The distribution functions are: :
: y0 1 exp − 0:63u 0:63u
ð1Þ
:
: 3 : y0 y0 27 1 exp −3 2 0:96u 0:96u 0:96u
ð2Þ
f ðy0 Þ ¼
f ðy0 Þ ¼
It can be found that there is considerable difference in the upper two probability distributions of liftoff velocity (Fig. 1), and the liftoff velocity of grains varies with the friction velocity (Fig. 2) in Gamma distribution. The calculation model and the parameters can be found in Zheng et al. (2004), and the minimal liftoff velocity of the grains selected in the model can hop a grain diameter due to gravity. It is apparent in Fig. 3 that the stratification model exists with a Gamma distribution but not with an exponential distribution. The PDF distribution of liftoff velocity plays an important role in the streamwise mass flux. This finding is consistent with that of Anderson and Hallet (1986) and Shao (2005), but not that of Zheng et al. (2004), who suggested that the stratification happens with both Gamma and exponential distributions. Zheng et al. (2004) also suggested that the height of the peak value of the streamwise mass flux increases with wind speed, and the same result can be found (Fig. 4). It can be found that the velocity has a Gamma distribution (Fig. 2), which corresponds with the peak value of the PDF distribution of liftoff velocity of 0.48 m s − 1 as u* = 0.5 m s − 1 and 0.77 m s − 1 as u* = 0.8 m s − 1, and the hop height is, respectively, 0.00582 m and 0.01716 m in the steady state. It
Fig. 3. The streamwise mass flux with the Gamma and exponential distribution. The friction velocity is 0.5 m s− 1 and the diameter is 0.25 mm.
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important role since most grains lift off at a velocity corresponding with the peak of the PDF distribution of liftoff velocity in a Gamma distribution. The hop height for the grains' liftoff velocity at that velocity then equals the height of the peak value of the streamwise mass flux. 3. Fundamental model
Fig. 4. The streamwise mass flux at different friction velocities. The Gamma distribution is chosen in the model and the diameter is 0.25 mm.
means that the number of grains at these two liftoff velocities are the greatest amongst others. As shown in the Fig. 4, the height corresponding with the peak value of the streamwise mass flux is 0.0058 m and 0.0172 m at the above two friction velocities. It is obvious that these are the same as the hop heights. Just as said by Anderson and Hallet (1986), the grains spend such a proportion of time near the top of their trajectories that the single-trajectory streamwise mass flux peak is sharply bent at the top of the trajectory. Meanwhile, for a grain diameter of 0.00025 m and density of 2650 kg m − 3, the grain mass of 2.175 × 10− 8 kg is very small compared with the grain velocity. Therefore, the reason why the streamwise mass flux varies with height may be that the grain mass flux is at its maximum above the surface, but the velocity of the grains is very small. With increasing height, the grain mass flux decreases rapidly although the grain velocity increases, so that the streamwise mass flux continuously declines. Since the number of liftoff grains decreases with the liftoff velocity in the exponential distribution, there should be no stratification pattern, and this is verified in Fig. 3. However, Zheng et al. (2004) suggested that pffiffiffiffiffiffistratification also exists in the exponential distribution. Here 4 gD m s − 1 is introduced as the minimum liftoff velocity, which is larger than the one used in the above calculation. As shown in Fig. 5, the stratification pattern does exist in the exponential distribution and the height of the peak value of the streamwise mass flux equals the hop height of grains with the minimum liftoff velocity at different friction velocities. This may explain the reason why Zheng et al. (2004) observed the stratification pattern. On the basis of above analyses, it is clear that the stratification pattern is determined by the pattern of the PDF distribution of liftoff velocity; the PDF distribution function of liftoff velocity plays an
Fig. 5. The streamwise mass flux at different friction velocities with exponential distribution chosen. The diameter is 0.25 mm.
Up to now, the classical problem of windblown sand physics, i.e., whether the stratification exists in the streamwise mass flux, is not yet answered. The only fact we are sure of is that the PDF distribution of liftoff velocity plays an important determinant role. Here, on the basis of the grain/bed result, this problem will be further investigated. As in the previous studies (e.g., Bagnold, 1941; Anderson and Haff, 1988, 1991; Werner, 1990; McEwan and Willetts, 1991; Kok and Lacks, 2009), the four distinct sub-processes – aerodynamic entrainment, grain trajectory, wind modification and grain/bed collision – are included in the model. 3.1. Grain trajectory For the sake of simplification, the grains are assumed to be spherical with identical diameter D and uniform density ρg, and the motion of grains is determined mainly by gravity, aerodynamic drag and aerodynamic lift. On the basis of Newton's equation, the fundamental dynamical equation for a single grain can be described as: ::
x¼
3ρ y ¼ −g− a 4 ρg ::
:
3 ρ V ðu− xÞ C a R 4 D ρg D
ð3Þ :
u2top −u2bot V y C D R −C l D D
! ð4Þ
where, x and y are, respectively, the horizontal and vertical coordinates of grains; single and double dots above the variable x and y denote grain speed and acceleration; ρa is air density; u is wind speed; g is the acceleration resulting from gravity; VR is grain velocity relative to wind; CD is the drag coefficient taken as an empirical formula (White, 1974); Cl is the lift coefficient, which is taken to be 0.85 CD (Anderson and Hallet, 1986); and utop and ubot are the wind speed at heights corresponding to the top and bottom of grain. 3.2. Grain/bed collision The grain/bed collision is a key physical process in windblown sand movement (Anderson and Haff, 1991). When the grains impact the bed, some of them can rebound, keeping a part of their energy, while some will be trapped by the bed and some grains will be splashed. A series of PDF distributions of liftoff velocities have been obtained from the experiments and numerical simulations, such as the exponential distribution (e.g., Anderson and Hallet, 1986; Huang et al., 2006), Gamma distribution (e.g., Anderson and Haff, 1988), log-normal distribution (e.g., Nalpanis et al., 1993; Beladjine et al., 2007), PearsonVII distribution (e.g., Zou et al., 2001), Rayleigh distribution (e.g., Werner and Haff, 1988; Werner, 1990; Rioual et al., 2000, 2003) and Weibull distribution (e.g., Dong et al., 2002). Because of the lack of a theoretically based model of collision process and the complexity of the natural conditions, the PDF distribution of liftoff velocities is difficult to obtain, and remains ‘poorly defined’ (McEwan and Willetts, 1991; Namikas, 2003). Although under natural conditions the grain/bed collision is an extremely complex process, some valuable information is obtained in spite of these difficulties. Here we use some deterministic laws which correspond with the average case. From experiments (e.g., Beladjine et al., 2007; Crassous et al., 2007) and numerical
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simulations (e.g., Deboeuf et al., 2009), it can be found that the total kinetic energy of the reptating grains, Eej , is proportional to the impact energy Eimp , and can be expressed as: Eej ¼ kEimp
ð5Þ
C ¼ −∫Ng ðe0 Þ lng ðe0 Þde0
ð6Þ
the requirement of which is that the total kinetic energy of ejected grains should be equal to the sum of the kinetic energy of ejected grains, that is: Eej ¼ ∫Ng ðe0 Þe0 de0
ð7Þ
The integral of PDF distribution of the kinetic energy should equal 1, that is: ∫gðe0 Þde0 ¼ 1
ð8Þ
With the aid of a Lagrangian method to find the extreme value of function (6), the PDF distribution of the kinetic energy can be obtained, that is: g ðe0 Þ ¼ expð−1 þ Ae0 þ BÞ
ð9Þ
1 e exp − 0 he0 i he0 i
! ð11Þ
where, bv02> = 2be0>/m. Up to now, we have obtained the PDF distribution of the liftoff velocity, which is in the form of a Rayleigh distribution. It can be found from the above derivation that only the relationship between the total kinetic energy of the ejected grains and the energy of the incident grain is used, which is a deterministic experiment law. That greatly reduces the impact of other factors on the accuracy of the PDF distribution of the liftoff velocity, and a qualitative understanding about the PDF distribution of the liftoff velocity is thus obtained. When the grains impact the bed, a constant fraction of impact momentum is transferred to the reptating grains (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000; Andreotti, 2004; Beladjine et al., 2007), and the number of reptating grains increases linearly with the impact velocity: 0:02 Neje ¼ pffiffiffiffiffiffi vimp gD
and the rebound angle is almost independent of the impact velocity. We thus take the kinetic energy of rebounding grains to be 45% of the impacting kinetic energy, and the rebound angle to be 45° from the horizontal. Meanwhile, some grains will be trapped when the grains impact with the bed, which is a stochastic process. Following the numerical simulations by Werner (1990) and Anderson and Haff (1988, 1991), the probability of rebound can be expressed as: preb
!! vimp ¼ 0:95 1− exp − pffiffiffiffiffiffiffiffi 10 g D
ð12Þ
From experiments (e.g., Mitha et al., 1986; McEwan et al., 1992; Nalpanis et al., 1993; Rice et al., 1995, 1996; Rioual et al., 2000) and
ð14Þ
3.3. Wind modification According to previous investigations (e.g., Anderson and Haff, 1988, 1991; Almeida et al., 2006; Kok and Lacks, 2009), the sand bed is assumed to be flat. From Newton's third law, the wind is modified by the transfer of momentum between the wind flow and the grains in the air, thus the turbulent Navier–Stokes equation of wind flow in the xdirection can be expressed as (McEwan et al., 1999): ρa
∂u ∂τ ¼ þ F x ðt; yÞ ∂t ∂y
ð15Þ
where τ is the turbulent shear stress, and Fx is the force per unit volume on the wind. They can be calculated solely by:
ð10Þ
where be0> = Eej/N is average kinetic energy of ejected grains. According to the relationship between kinetic energy and velocity, we can obtain the PDF distribution of the liftoff velocity, that is: 2v v 2 f ðv0 Þ ¼ 20 exp − 0 2 v0 v0
ð13Þ
2 2 ∂u ∂u τðt; yÞ ¼ ρa κ y ∂y ∂y
Under the constraints (7) and (8), we obtain: g ðe0 Þ ¼
numerical simulations (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; Andreotti, 2004; Kok and Renno, 2009), it can be found that the rebound velocity is, on average, a fraction of the impact velocity: vreb ¼ γvimp
where, k is a coefficient which is determined by the characteristics of the incident grains and the bed. A stochastic event tends to reach its maximum within certain limitations. The liftoff velocity of each grain in a grain/bed collision process is different from each other, which constitutes its complexity. Here, we assume that the number of ejected grains in a grain/bed collision process is N, and the complexity of the PDF distribution of kinetic energy of the ejected grains, according to Zhang (2003), is C:
191
:
Fx ¼
yX max :
y min
ð16Þ
::
::
mx↑ mx↓ Nðt; yÞ : − : y↑ y↓
! ð17Þ
where κ is the von Karman's constant, m is the grain's mass, N is the number of grains at the liftoff velocity v0 and time t, and the arrows depict the upward and downward direction of the trajectory. Substituting Eqs. (16) and (17) into (15), the Navier–Stokes equation of wind flow in the xdirection reduces to: :: :: ! X ∞ mx↑ mx↓ ∂u ∂ 2 2 ∂u ∂u ρa N ðt; v0 Þ : − : ¼ ρ κ y þ y↑ y↓ ∂t ∂y a ∂y ∂y ′
ð18Þ
v0
where v0′ denotes the minimum liftoff velocity of grains that can reach height y. On the basis of the above fundamental equations, the streamwise mass flux is given by: Q ðyÞ ¼
∞ X v0 ′
:
Nðt; v0 Þ
:
mx↑ mx↓ − : : y↑ y↓
! ð19Þ
4. Numerical simulation In the numerical simulation, the aerodynamic roughness is taken as D/30. As shown in previous studies (e.g., Werner, 1990; Anderson and Haff, 1991; McEwan and Willetts, 1991; Kok and Lacks, 2009), some grains are entrained first, then more and more grains are splashed with grain/bed collision. When the variation of the wind
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Fig. 8. The wind profiles in the steady state.
Fig. 6. Comparison between the results in this paper and the empirical formulas given by Bagnold (1941), Kawamura (1951) and Zhou et al. (2002) and experimental result given by Iversen and Rasmussen (1991) in the steady state.
speed and the streamwise mass transport are smaller than a specified value in successive iterations, the steady state is achieved. Here, the numerical simulation is performed to confirm the validity of the model and the program proposed in this paper. On the basis of the model suggested here, through integrating Eq. (19), the streamwise mass transport is given by: Q 0 ¼ ∫Q ðyÞdy
ð20Þ
The well-known empirical formulas of the streamwise mass transport given by Bagnold (1941) and Kawamura (1951) are recognized to be efficient and the most widely quoted (e.g., Sarre, 1987; Shao, 2000; Zheng et al., 2004). When the friction velocity is less than 0.4 m s − 1, the empirical formula given by Kawamura (1951) is close to the experimental results, while when the friction velocity is greater than 0.4 m s− 1 and less than 0.7 m s− 1 the Bagnold's formula is more efficient, but the result using these empirical formulae is smaller than the experimental result with the increase of the friction velocity. Based on experimental results, an effective empirical formula is also given by Zhou et al. (2002). Here, we make a comparison of the streamwise mass transport between our result and those empirical formulas and experiment results given by Iversen and Rasmussen (1999). As shown in Fig. 6, when the friction velocity is less than 0.3 m s− 1, the numerical results are less than the result given by Iversen and Rasmussen (1999) and greater than the result given by Bagnold's empirical formulae. When the friction velocity is greater than 0.4 m s− 1, the numerical
Fig. 7. Evolution of the streamwise mass transport. The diameter is 0.25 mm.
results are greater than the result given by Bagnold (1941), Kawamura (1951) and Iversen and Rasmussen (1999) and very close to the value given by Zhou et al. (2002). This numerical example demonstrates that the model and the algorithm proposed in this paper are feasible. 5. Numerical results and discussions Fig. 7 shows the evolution of the streamwise mass transport at the friction velocity of 0.5 m s − 1. As shown, the time needed for the streamwise mass transport to reach the steady state is 2.6 s, which lies between the corresponding results drawn by Anderson and Haff (1991) and Zheng et al. (2006) (respectively 3 s and 2.5 s). Moreover, the streamwise mass transport in the steady state is much bigger than that given by Anderson and Haff (1991) or Zheng et al. (2006) and closer to experiment data given by Shao and Raupach (1992). This further verifies that the model and the algorithm are feasible. As shown in Fig. 8, the wind in the steady state is modified by the sand grains in the air, and the effective roughness increases with the increase of friction velocity, which can be calculated from projecting an intercept of the gradient of the wind profile down to the height where the wind speed would be zero, which is in accord with numerous measured wind speed profiles during saltation experiments, as summarized by Owen (1964). Obviously, the wind speed in the reptating layer is strongly reduced and varies little at the different friction velocities; and as the friction velocity increases, the wind speed in the reptating layer decreases appreciably, which is consistent with the findings of the previous studies (e.g., Andreotti, 2004). Figs. 9 and 10 show the trajectories of grains at the initial wind pffiffiffiffiffiffi speed and in the steady state at the liftoff velocity of gD m s − 1. It is obvious that the trajectory in the steady state is smaller than that at the initial wind speed; and, contrary to the trajectory in the initial wind speed which increases with the friction velocity, the trajectory in the steady state has an opposite tendency and varies slightly at different friction velocities. The reason for this can be found in Fig. 8.
Fig. 9. Trajectory of grain at the initial wind speed.
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Meanwhile, the hop height for the initial liftoff velocity, corresponding with the peak value of its PDF distribution, equals the height of the maximum streamwise mass flux. Therefore, like the liftoff velocity of the reptating grains and the PDF distribution of liftoff velocity, which are decided by the characteristics of grains, the height of the peak value of the streamwise mass flux also scales according to the property of the grains and will be independent of the wind intensity. 6. Conclusions
Fig. 10. Trajectory of grain in the steady state.
Because the grains with the smaller liftoff velocities travel in the reptating layer; the wind speed in the reptating layer varies little at the different friction velocities in the steady state; and as the friction velocity increases, the wind speed in the reptating layer decreases appreciably. Fig. 11 shows the streamwise mass flux at the different friction velocities in the steady state. It is shown that the streamwise mass flux of windblown sand movement displays the stratification pattern which is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer. This is consistent with Zheng et al. (2004). It also can be found that the peak value of the streamwise mass flux, which is about 4.5 mm below the focus point, is almost invariant at the different friction velocities in the steady state. Most of the grains in the air are the reptating grains, and some of them can be promoted to become the saltating grains (Andreotti, 2004). The grains in air have a exponential probability of surviving (Anderson and Haff, 1988, 1991; Werner, 1990) when they impact the bed, and the velocity of reptating grains is, on average, a constant independent of changes in the impact velocity (e.g., Anderson and Haff, 1988, 1991; Werner and Haff, 1988; McEwan et al., 1992; Rioual et al., 2000; Andreotti, 2004). Meanwhile, as shown in Fig. 8, the wind speeds in the reptating layer in the steady state are very close to each other at different friction velocities. Therefore, the motions of the reptating grains at different friction velocities are similar, as shown in Fig. 10, which was also suggested by Andreotti (2004). If the PDF distribution of liftoff velocity of the reptating grains follows the rule that the number of liftoff grains increases with the increase of the liftoff velocity first and decreases later, the PDF distribution of liftoff velocity in the steady state should be in the same form, which also corresponds with the previous studies (e.g., Anderson and Hallet, 1986; Mitha et al., 1986; Werner and Haff, 1988; Rioual et al., 2000; Beladjine et al., 2007; Crassous et al., 2007; Zhang et al., 2007).
Fig. 11. The streamwise mass flux in the steady state.
The numerical simulations of windblown sand movement were conducted to re-evaluate the classical problem of windblown sand physics, i.e., the probability distribution of the streamwise mass flux. With re-evaluing previous models, we established that whether stratification exists or not is determined by the pattern of the PDF distribution of liftoff velocity, and the hop height for the initial liftoff velocity, corresponding with the peak value of its PDF distribution, equalling the height of the maximum streamwise mass flux. The slower the liftoff velocity is, the more grains lift off in the steady state. However, in the case of the liftoff velocity of grains in the exponential distribution, the stratification pattern will not appear. Based on some deterministic laws of the grain/bed collision and the Constitution Theory, a qualitative understanding about the PDF distribution of the liftoff velocity is obtained, and a Rayleigh distribution for reptating grains is employed in the model. It is found that the streamwise mass flux of windblown sand movement displays the stratification pattern in the steady state, which is composed of a linear increment layer, a saturation layer, and a monotonic decrement layer. Also, because the liftoff velocity of the reptating grains is decided by the characteristics of grains, the wind speeds in the reptating layer in the steady state are very close to each other at different friction velocities. Similarly, the height of the peak value of the streamwise mass flux, which does not change with the wind speed, is decided by the characteristics of the grains. Acknowledgments The authors would like to express their sincere appreciation to the National Basic Research Program of China (No. 2009CB421304) and the funding from the projects of the National Natural Science Foundation of China (No. 10772075, No. 10772074). References Almeida, M.P., Andrade Jr., J.S., Herrmann, H.J., 2006. Aeolian transport layer. Physical Review Letters 96, 018001. Anderson, R.S., 1987. Eolian sediment transport as a stochastic process: the effects of a fluctuating wind on particle trajectories. Journal of Geology 95, 497–512. Anderson, R.S., Haff, P.K., 1988. Simulation of eolian saltation. Science 241, 820–823. Anderson, R.S., Haff, P.K., 1991. Wind modification and bed response during saltation of sand in air. Acta Mechanica 1, 21–51 (Suppl.). Anderson, R.S., Hallet, B., 1986. Sediment transport by wind: toward a general model. Geological Society of America Bulletin 97, 523–535. Anderson, R.S., Sørensen, M., Willetts, B.B., 1991. A review of recent progress in our understanding of Aeolian sediment transport. Acta Mechanica 1, 1–19 (Suppl.). Andreotti, B., 2004. A two-species model of aeolian sand transport. Journal of Fluid Mechanics 510, 47–70. Bagnold, R.A., 1941. The Physics of Blown Sand and Desert Dunes. Methuen, New York. Beladjine, D., Madani, A., Luc, O., Alexandre, V., 2007. Collision process between an incident bead and a three-dimensional granular packing. Physical Review E 75 (6), 061305. Chepil, W.S., 1945. Dynamics of wind erosion: I. Nature of movement of soil by wind. Soil Science 60, 305–320. Crassous, J., Beladjine, D., Valance, A., 2007. Impact of projectile on a granular medium described by a collision model. Physical Review Letters 99, 248001. Deboeuf, S., Gondret, P., Rabaud, M., 2009. Dynamics of grain ejection by sphere impact on a granular bed. Physical Review E 79 (4), 041306. Dong, Z.B., Liu, X.P., Li, F., Wang, H.T., Zhao, A.G., 2002. Impact-entrainment relationship in a saltating cloud. Earth Surface Proceedings Land 27, 641–658. Dong, Z., Wang, H., Li, X., Wang, X., 2004. A wind tunnel investigation of the influences of fetch length on the flux profile of a sand cloud blowing over a gravel surface. Earth Surface Proceedings Land 29, 1613–1623.
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