Eur. Phys. J. E (2014) 37: 80 DOI 10.1140/epje/i2014-14080-7
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Numerical modeling of wind-blown sand on Mars HaoJie Huang1 , TianLi Bo1 , and XiaoJing Zheng1,2,a 1
2
Key Laboratory of Mechanics on Environment and Disaster in Western China, Ministry of Education, Lanzhou University, Lanzhou 730000, China School of Electronic and Mechanical Engineering, Xidian University, Xi’an, 710071, China Received 26 September 2013 and Received in final form 2 March 2014 c EDP Sciences / Societ` Published online: 19 September 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. Recent observation results show that sand ripples and dunes are movable like those on Earth under current Martian climate. And the aeolian process on Mars therefore is re-attracting the eyes of scientific researchers in different fields. In this paper, the spatial and temporal evolution of wind-blown sand on Mars is simulated by the large-eddy simulation method. The simulations are conducted under the conditions of both friction wind speed higher and lower than the “fluid threshold”, respectively. The fluid entrainment of the sand particles, the processes among saltation sand particles and sand bed, and the negative feedback of sand movement to flow field are considered. Our results show that the “overshoot” phenomenon also exists in the evolution of wind-blown sand on Mars both temporally and spatially; impact entrainment affects the sand transport rate on Mars when the wind speed is smaller or larger than the fluid threshold; and both the average saltation length and height are one order of magnitudes larger than those on Earth. Eventually, the formulas describing the sand transport rate, average saltation length and height on Mars are given, respectively.
1 Introduction Mars, which is widely covered by sand dunes [1], has become a popular target of deep-space exploration. The latest observations show that the Martian sand ripples and dunes are also movable [2,3], rather than immovable, as inferred in previous observations [4]. The migration speeds of sand ripples and dunes are probably equivalent to those on Earth [5]. That is, wind-blown sand movement is prevalent both on Earth and on Mars [5]. For better comprehending the change of the climate on Mars [6], selecting the mission’s landing sites [5] and survey routines [7] and avoiding the encounters of the Spirit Mission [8], the studies on the characteristics and development process of wind-blown sand movement have become critical. Wind-blown sand movement, in which saltation sand particles account for 75% [9–11], will make the landform and geomorphology change [11–15]. Furthermore, it plays a key role in the release of dust [10], even on Mars where dust storms and dust devils are excessive [12,16]. Therefore, it is necessary to have a comprehensive study on wind-blown sand movement on Mars. However, existing wind tunnel experiments [17–19] and observations of the Martian surface [12,13,20] have not revealed all the characteristics of wind-blown sand on Mars, such as saltation a
e-mail: [email protected]
trajectory, sand speed and sand transport rate. Thus, theoretical simulation becomes a very important tool to study the characteristics of wind-blown sand on Mars. Since White et al. [21] and White [17] simulated the sand saltation trajectories under Martian conditions and derived that the Martian saltation trajectories are larger than those on Earth, many researchers focus on this subject. After that, Almeida et al. [22] carried out a detailed two-dimensional numerical study of Martian saltation motions by using the software Fluent. They found that Martian saltation trajectories were two orders of magnitudes larger than those on Earth. They also gave the expression of the relationship between the sand transport rate and the friction wind speed under wind speed when the latter is higher than the impact threshold of sand saltation. In addition, their simulations showed that the impact threshold of sand saltation was substantially below the fluid threshold and the impact threshold in the simulations is 1.12 m/s on Mars. This result is consistent with the dynamic transport thresholds of Claudin and Andreotti [23] obtained by considering the effect of cohesion and viscosity. However, the splash process between saltation sand particles and the sand bed was ignored in their models [14]. The splash process plays an important role in wind-blown sand evolution [24–26] and thus a complete splash process is very important for the simulation of wind-blown sand.
Page 2 of 8
On Mars, the fluid threshold wind speed is very high [27,28], but only a few times in a decade during gusts of extreme aeolian activity the Martian wind velocities can be higher than it [12,18,29–34]. Since recent observations indicate that the sand ripples and dunes on the Martian surface are movable [2,3], a problem arises: How could the sand grains sustain the saltation motions under the current Martian climate, in which most of the time the Martian wind speeds are below the fluid threshold? Following the idea of Almeida et al. [22] that saltation on Mars, once initiated, can be sustained under much lower wind speeds than previously thought, Kok [14] proposed a new hypothesis by considering the hysteresis phenomenon and parameterized it into his numerical model. The results indicated that the hysteresis has a significant effect on the threshold speed and causes saltation to occur and be sustained at much lower wind speed than previous conclusions. And the impact threshold required for sustaining saltation on Mars was an order of magnitude smaller than the fluid threshold. He pointed out that the saltation could be sustained under the current Martian conditions since the impact threshold is very low. His findings provide a reasonable explanation for the recent observations that the sand ripples and dunes on the Martian surface are movable. But the evolution of wind-blown sand was simplified as a one-dimensional model in his paper. The model ignored the influence of streamwise turbulence in the development of wind-blown sand. And the spatial variation of wind-blown sand cannot be obtained by the model. It is an important issue to understand the spatial variation of wind-blown sand. Especially, the spatial variation of the mass flux is essential to measure the sand transport rate in the wind tunnel experiment, even to predict soil wind erosion under field conditions [35–37]. Moreover, the wind-blown sand directly affects the wavelength and the formation of the dunes [38]. In order to learn more about the wind-blown sand on Mars, we need to carry out a quantitative research on the spatial variation of the mass flux. Also, the saltation characteristics on Mars, such as saltation trajectory, sand speed, and sand transport rate, need to be investigated further. In this paper, two-dimensional numerical modeling of Martian wind-blown sand is carried out and the evolution of wind-blown sand along the wind direction is studied. The large-eddy simulation approach is used to simulate wind field and the negative feedback of sand movement to flow field is considered. The movements of each particle are tracked and the splashing function is used to describe the collision between sand particles and sand bed. Section 2 gives the model and the validation under the terrestrial conditions. In sect. 3, the evolution of wind-blown sand on Mars is shown and the relationship between the sand transport rate and the friction wind speed, as well as the relationship between the average saltation length and height and friction wind speed on Mars are studied. Finally, sect. 4 is a summary of the main conclusions.
Eur. Phys. J. E (2014) 37: 80
2 Numerical method Actually, in the wind-blown sand, the saltation sand grain falls back onto the ground at a speed due to the gravity, impacts the sand bed, and ejects neighboring sand grains. In this paper, we model the evolution of wind-blown sand by three main processes: i) the movement of sand particles; ii) the coupling interaction between the sand particles and the wind field; iii) the collision process, which contains rebound and ejection, between the saltation sand particles and bed surface. 2.1 Basic equations In this paper, the large-eddy simulation (LES) approach is used to simulate the wind field in wind-blown sand. In LES, the speed field is separated into two fields, a resolved (large-scale) and an unresolved (small-scale) field, by a spatial filtering operation. The feedback of unresolved motions on the resolved-scale flow is achieved by the action of a sub-grid-scale viscosity. The LES filtered continuity equation and filtered momentum equation, including the feedback of sand movement are shown as follows [6,39,40]: ∂u ¯i = 0, (1) VP ∂xi ∂u ¯i u ∂u ¯i 1 ∂ p¯ ∂2u ¯j ¯i + VP = VP − +ν ∂t ∂xj ρf ∂xi ∂xj ∂xj ∂τij Fi − − δi2 g + , (2) ∂xj ρf where, i = 1 and 2 corresponds to the flow and wallnormal directions (i.e., x1 = x, x2 = z, u1 = u, u2 = w), respectively; u ¯i and p¯ represent the filtered wind speed and pressure, respectively; ν is the kinematic viscosity; ρf is the air density; g is the gravity; Fi is the feedback force per unit grid of sand acted on the air flow; VP = k=n 1− k=1 (1/6)πD3 is the volume ratio of the sand particles in the air flow, in which n is the sand particle number in the unit volume and D is the sand grain diameter; δij = 1 ¯i u ¯j are the if i = j, otherwise δij = 0; τij = ui uj − u sub-grid-scale (SGS) stresses [41]. In order to close the above equations, the sub-gridscale stresses can be modeled as follows: ∂u ¯i 1 ∂u ¯j , (3) + τij − τkk δij = −νt 3 ∂xj ∂xi ∂u ¯ ∂u ¯ ∂u ¯i ∂u ¯i where νt = (Csgs Δ)2 0.5( ∂x + ∂xji ).( ∂x + ∂xji ) is the j j turbulent viscosity of the sub-grid-scale; Δ is the grid scale; Csgs = 0.19–0.26, which depends on the detailed case [42]. The initial conditions of the above governing equations are u z ∗ ln , (4) u ¯(0, z) = k z0 u ¯(x, z) = w(x, ¯ z) = 0. (5)
Eur. Phys. J. E (2014) 37: 80
Page 3 of 8
Here k=0.41 is the von K´ arm´an constant; z0 = D/30 is the aerodynamic surface roughness [43]; u∗ is the friction speed of inflow. The boundary conditions of the above governing equations are opening conditions at the upper boundary (z = H): ¯ H)/∂z = 0. (6) ∂u ¯(x, H)/∂z = u∗ /kz, ∂ w(x,
1) Calculate the LES governing equations (1) and (2) till convergence at the first time step Δt and output the wind velocities ui,j , wi,j of each node.
Our wind profile has been given based on experiments in a wild natural environment and in wind tunnel. And eq. (6) is derived by the derivation of the logarithmic profile. No-slip conditions at lower boundary (z = 0) are:
(k = 1, 2, . . ., 200, and is the discrete position of the sand bed) which means that they scale with u2∗ − u2∗t and that they are numbered by N0k (0 represents the initial time) [39]. And the scaling with u3∗ − u3∗t can u3 be described as Na,k = 1.74 × 10−3 u2∗,k (1 − u3∗t )D−3 .
u ¯(x, 0) = w(x, ¯ 0) = 0.
(7)
Fully developed conditions at the outlet boundary (x = L) are: ∂u ¯(L, z)/∂x = 0, w ¯ = 0. (8) The saltation trajectory is mainly influenced by gravity and drag force. And we neglected the other smaller terms acting on the sand grains, such as the Magnus force, the Saffman force [44]. It is worth noting that although recent studies show that the collisions of the ejected sand grain in the air need to be considered [45,46], but in our model, we neglected the process like most previous models did [14,22]. Then the motion of each sand grain in wind-blown sand can be described as 2 dx CD πD2 ρf d2 x u− , (9) mp 2 = FDx = dt 8 dt πρp D3 d2 z mp 2 = −Fg + FDz = − dt 6 2 CD πD2 ρf dz + w− , (10) 8 dt where mp and ρp are the mass and density of sand parti2 cles, respectively; CD = (0.63 + 4.8/Re0.5 p ) is the drag coefficient [47]; ρf is the air density; Rep = (Vf ρp D/μ)[(u − dx/dt)2 + (w − dz/dt)2 ]1/2 is the particle Reynolds numk=n ber [26]; Vf = 1 − k=1 VP /ΔV is the volume fraction that is the total sand volumes within one grid to the grid volume; ΔV is the control volume size; μ is the air kinetic viscosity coefficient. 2.2 Calculation procedures In this paper, the calculation of wind field is based on the finite-volume method [42]. We need to mesh the computational domain 22 m × 1.5 m under Earth conditions [48] (see fig 1). In order to guarantee the efficiency and accuracy of calculation, we mesh the computational domain 80 m × 5 m under Mars conditions with 3000 × 300 grids, although it is larger than that on Earth, the sand particles still can be recognized, because the wind speed on Mars is much larger than on Earth. So the grid size we chose cannot influence the simulation results. The simulation parameters on Earth and Mars are taken as in table 1 [9, 49–51]. The detailed steps are shown as follows:
2) According to the friction speed, the number of lifting sand particles in the unit area near the sand bed is u2 calculated by Na,k = 1.74 × 10−3 u∗,k (1 − u2∗t )D−3 ∗,k
∗,k
The probability density distribution of the initial lifting speed follows: p(vej ) = exp(−vej /vej )/vej , where vej is the ejection speed, the overbar represents the mean value and it can be deserved by the momentum and energy conservation [49,52]. And the lifting angle is 90◦ . It is worth noting that when we simulate the Martian wind-blown sand under the condition of the friction speed below the fluid threshold, u2 Na,k = 1.74 × 10−3 u∗,k (1 − u2∗t )D−3 is no longer ap∗,k
propriate, we give some lift-off particles to saltation movement [14]. Then the coordinates and velocities of k k , x˙ k0,n , z˙0,n (n is the numall sand particles xk0,n , z0,n ber of sand particles) can be obtained. The motion of the sand trajectory, eqs. (9) and (10), is calculated in x z , Fi,j a fluid time step Δt and the reaction forces Fi,j of sand particles to eqs. (2) are recorded. Here, u∗,k is the friction wind speed at the k discrete point of the sand bed and u∗,t is the fluid threshold friction wind speed. 3) The new wind speeds ui,j , wi,j of each node are recalx z , Fi,j at the new time culated by the reaction forces Fi,j step. If collision happens, we assume the rebound probability as preb = 0.95(1 − e−λvimp ), where vimp is the impact speed, λ is 2 m/s according to the simulation result [49], the rebound speed is 0.55 times the impact speed, the rebound√angle is 40◦ [53]. The ejection number is N ≈ avimp / gD [25,54], and the ejection angle distributes randomly between 50◦ and 60◦ [55]. The new saltation sand particles are programmed into Ntk and the position of sand particles and their speeds k k , x˙ kt,n , z˙t,n are reset. xkt,n , zt,n 4) Repeat steps 1)-3) until the simulation time is finished.
2.3 Model verification In this section, we compare our simulation results under two different aerodynamic entrainment laws with the wind tunnel experimental results. Figure 2 shows the variation sand transport rate along with the wind direction. After the wind-blown sand developed to stable, we count the grains in the grids of all heights at the different horizontal positions to derive fig. 2. From fig. 2 we can find that there exists the “overshoot” phenomenon. When enough sand
Page 4 of 8
Eur. Phys. J. E (2014) 37: 80 Table 1. Simulation parameters used in the model. Gravity g (m/s2 )
Air density ρa (kg/m2 )
Viscosity ν (m2 /s)
Earth
9.81
1.225
1.45e-5
2650
250
0.26
Mars
3.71
0.02
6.5e-4
3000
500
1.8(a)
(a)
Sand density ρs (kg/m2 )
Diameter d (μm)
Fluid threshold u∗ (m/s)
Iversen and White [55] obtained the Martian fluid threshold friction speed between 1.4 m/s and 2.8 m/s in
the 4 case experiments when the grain diameter is 500 μm. Shao and Li [39] gave the theoretical expression of the fluid threshold friction speed which is 1.96 m/s when using our simulation parameters. Here, it is reasonable to use 1.8 m/s as the fluid threshold friction speed in our model.
Fig. 1. Schematic diagram of wind-blown sand on Earth.
Fig. 2. The sand transport rate varying with the wind direction on Earth, comparison between our different aerodynamic scaling law results and the experiment results of Shao and Raupach [56].
particles lift-off and fly a certain distance, the reaction of the sand particles to the flow field will reduce the wind speed, which reduces the kinetic energy of the saltation layer. Thus, the ejection number decreases, and macroscopically, the sand transport rate reduces, until the liftoff and the deposit reach a balance. This agrees well with the experimental result of Shao and Raupach [56] qualitatively and quantitatively, that is, the sand flux reaches the maximum and saturation at 7 m and 15 m, respectively. It indicates that our modeling can reflect the real development of wind-blown sand on Earth. Besides, we can also
see that the different aerodynamic entrainment laws only influence the distribution law of the sand transport rate along the stream, but has no effect on the relationship between the sand transport rate and the friction wind speed in the stable stage. All these prove that our simulation results do not depend on the selection of aerodynamic entrainment scaling laws and, therefore, our results are reliable. Then we compare the results of the sand transport rate for different friction speeds (d = 250 μm) with both the experiment results [9,57–60] (see fig. 3). From fig. 3 we can see, approximately, that the sand transport rate increases with the friction speed exponentially, which is consistent with the previous experimental results qualitatively and quantitatively. The wind tunnel experiment results of Zhou et al. [60] indicate that the previous experimental results have each effective region. And from fig. 3 we can see that our results coincide with Kawamura [57] at small friction speed, and agree well with Bagnold [9], Lettau and Lettau [58], White and Mounla [59] at large friction speed. Based on all of the above, not only the development process but also the quantitative one, our simulation results are in accordance with the experimental results, thus indicating that our modeling can reflect the actual wind-blown sand process. From fig. 3, we can also see that the differences of sand transport rate on Earth at two different aerodynamic entrainment laws are less obvious.
3 Result and analysis Figure 4 is the variation of wind-blown sand with wind direction and time on Mars. From fig. 4 we can find that the “overshoot” phenomenon exists in both the temporal and spatial dimensions, which is the same with the situation on Earth qualitatively. However, the saturation curve of wind-blown sand on Mars is much steeper than that on Earth, which means the maximum erosion position moves forward. This may be due to the low density of air and low gravity on Mars [17,22]. Under the same lift-off conditions, sand grains on Mars can fly higher than on Earth. Then more sand grains in the air fall back to the bed after a short distance. Moreover, the saturation length of windblown sand on Mars is longer than that on Earth, which is in agreement with previous studies [61–64]. This can
Eur. Phys. J. E (2014) 37: 80
Page 5 of 8
Fig. 3. The sand transport rate varying with friction wind speed on Earth, comparison between our simulated results and the existing experiment.
mainly be explained by the larger ratio between particle and fluid density [65]. Therefore, the grains can fly farther and need a long distance to reach a steady state. In addition, the saturation time of wind-blown sand is much longer than that on Earth. In detail, it takes about 2– 3 seconds to reach saturation on Earth [56,66], while on Mars, our results show that it takes about 7–8 seconds. The sand grains can fly a long time because of the high wind speed and the low gravity on Mars under the same lift-off condition. Figure 5 is the relation between the sand transport rate and friction speed. From fig. 5 we can see that when the friction speed is larger than the fluid threshold (i.e., u∗t = 1.8 m/s), our simulated mass flux increases exponentially, which is the same as the results of Almeida et al. [22] and Kok [14] qualitatively but smaller than their results quantitatively. Also, we find that when the friction speed is smaller than the fluid threshold but larger than the impact threshold, the saltation can still be sustained, which is consistent with the results of Claudin and Andreotti [23], Almeida et al. [22] and Kok [14]. The numerical solutions of Werner [52] give rise to more general arguments in support of the conclusions that the fluid entrainment of grains is unimportant in steady-state saltation on Earth. Impact entrainment is dominant when the friction speed is smaller than fluid threshold on Mars [14, 22,23,67]. However, when the friction speed is larger than fluid threshold, the wind-blown sand may be also maintained by impact entrainment, and the fluid entrainment can be ignored. Almeida et al. [22] considered that not only the wind speed and the gravity, but also atmospheric environment and the properties of sand grains affect the sand transport rate. In their formulations, they introduce the normalized quantity d/lv , where, the length-scale lv = (v 2 /g)1/3 . Based on it, we refer to their results and give our formulations. The sand flux is proportional to (u∗ − u∗im )2 (see fig. 6). Thereby, the variation of sand
Fig. 4. (a) The sand transport rate along wind direction on Mars under different friction wind velocities. (b) The sand transport rate varying with time on Mars under different friction wind velocities at the streamwise distance of 5.8 meters.
Fig. 5. The sand transport rate varying with friction wind speed on Mars, comparison between our simulated results and the existing model simulation results.
Page 6 of 8
Eur. Phys. J. E (2014) 37: 80
Fig. 7. The relation between the average saltation length and height and friction wind speed, comparison between our simulated results and the simulation results of Kok [15]. Fig. 6. The normalized sand transport rate on Mars and Earth. The straight line corresponds to the equation y = (u∗ − u∗im )2 .
flux with the friction wind speed can be described by dρf u∗im (u∗ − u∗im )2 , qs = 26.4 (vg)2/3 u∗ > u∗im ,
r2 = 0.92
(11)
where u∗im is the impact threshold. From fig. 5, we can see that the simulation results can be fitted by eq. (11). In fig. 6, the horizontal axis means u∗ –u∗im and the impact threshold is 0.26 m/s on Earth [28,39,49] and 0.7 m/s on Mars [14]. The vertical axis means the normalized sand transport rate. In fig. 6, the normalized sand transport rates are QMars = qs /[26.4 × u∗im × dρfluid /(vg)2/3 ] and QEarth = qs /[23.0 × u∗im × dρfluid /(vg)2/3 ] on Mars and Earth, respectively. At the same time, the variation of the average saltation height and length with the friction wind speed on Mars were obtained (as shown in fig. 7). From fig. 7 we can find that the average saltation height and length increase with friction wind speed, which is consistent with Almeida et al. [22] and Kok [15]. The magnitudes of the average saltation length and height are 100 m and 10−1 m, which is less than that of Almeida et al. [22] by one order of magnitude. These maybe due to the fact that the model of Almeida et al. [22] disregards the details of the splash in the sandbed collision process. We can also find Lsalt and Hsalt are proportional to the length-scale lv = (v 2 /g)1/3 both on Mars and Earth and the two quantities obey the Exponential relationship with u∗ , see fig. 8. Then the Martian normalized results can be described by 2 1/3 v u∗ , r2 = 0.89, (12) exp Lsalt ≈ 91.8 g 2u∗im Hsalt ≈ 6.1
v2 g
1/3
exp
u∗ 2u∗im
,
r2 = 0.95.
(13)
Fig. 8. The normalized average saltation height and length both on Mars and Earth. The straight line corresponds to the equation y = exp(u∗ /2u∗im ).
In fig. 8, the horizontal axis means u∗ /u∗im , and the impact threshold is 0.26 m/s on Earth and 0.7 m/s on Mars [14,49]. The vertical axis means the normalized average saltation height and length. On Mars, the normalized length SMars = Lsalt /[91.8 × (v 2 /g)1/3 ] and the normalized height SMars = Hsalt /[6.1 × (v 2 /g)1/3 ] as a function of u∗ , and on Earth the normalized length SEarth = Lsalt /[170.1 × (v 2 /g)1/3 ] and the normalized height SEarth = Hsalt /[26.4 × (v 2 /g)1/3 ] also as a function of exp (u∗ ). Generally, the same rule holds for the average saltation height and length on Mars and on Earth. From the results, we can infer the Martian saltation trajectories are larger than those on Earth because of the low gravity and high speed of the Martian wind-blown sand which is in agreement with the conclusions of Parteli and Herrmann [68], Almeida et al. [22] and Kok [14].
Eur. Phys. J. E (2014) 37: 80
4 Conclusion In this paper, numerical simulation of the evolution of wind-blown sand on Mars is achieved and some saltation characteristics are studied, such as saltation trajectory, sand speed and sand transport rate. In the numerical modeling, the large-eddy simulation approach is used to calculate the flow field, and the negative feedback of the sand movement to the flow field is considered. The movements of each particle are tracked and the splashing function is used to describe the collision between sand particles and sand bed. The development of wind-blown sand along height and wind direction on Mars is analyzed when the friction wind speed is higher and lower than the “fluid threshold”. The results display that the “overshoot” phenomenon existed in the time and spatial dimension. Moreover, our results show that the saturation curve of wind-blown sand on Mars is much steeper than that on Earth and the saturation time and length are much bigger and longer than those on Earth. Then the relationship between the sand transport rate and friction wind speed on Mars is given. The average saltation height and length increase with friction wind speed which satisfies a log-linear law, and the relationship between the average saltation length and height and friction wind speed are given. This research was supported by a grant from the National Natural Science Foundation of China (No. 11072097, No. 11232006, No. 11202088, No. 10972164, No. 11121202), National Key Technology R&D Program (2013BAC07B01), the Science Foundation of Ministry of Education of China (No. 308022), Fundamental Research Funds for the Central Universities (lzujbky-2009-k01) and the Project of the Ministry of Science and Technology of China (No. 2009CB421304). The authors express their sincere appreciation to the supports.
References 1. J.R. Zimbelman, Geophys. Res. Lett. 27, 1069 (2000). 2. M.C. Bourke, K.S. Edgett, B.A. Cantor, Geomorphology 94, 247 (2008). 3. C.J. Hansen et al., Science 331, 575 (2011). 4. P.E. Geissler, J. Geophys. Res. 110, E02001 (2005). 5. N.T. Bridges et al., Geology 40, 31 (2012). 6. E. Gardin et al., Planet. Space Sci. 60, 314 (2012). 7. M. Golombek et al., Space Sci. Rev. 170, 641 (2012). 8. R.A. Kerr, Science 324, 998 (2009). 9. R.A. Bagnold, The Physics of Blown Sand and Desert Dunes (Chapman and Hall, 1956). 10. Y.P. Shao, Physics and modelling of wind erosion (Springer, 2008). 11. X.J. Zheng, Mechanics of wind-blown sand movements (Springer, 2009). 12. M. Balme, R. Greeley, Rev. Geophys. 44, RG3003 (2006). 13. R. Sullivan et al., J. Geophys. Res. 113, E06S07 (2008). 14. J.F. Kok, Phys. Rev. Lett. 104, 074502 (2010). 15. J.F. Kok, Geophys. Res. Lett. 37, L12202 (2010). 16. B.A. Cantor, P.B. James, M. Caplinger, M.J. Wolff, J. Geophys. Res. 106, 23653 (2001). 17. B.R. White, J. Geophys. Res. 84, 4643 (1979).
Page 7 of 8 18. R. Greeley, J.D. Iversen, Wind as a geological process on Earth, Mars, Venus and Titan (Cambridge, 1985). 19. R. Greeley, Planet. Space Sci. 50, 151 (2002). 20. R. Greeley et al., J. Geophys. Res. 111, E02S09 (2006). 21. B.R. White, R. Greeley, J.D. Iversen, J.B. Pollack, J. Geophys. Res. 81, 5643 (1976). 22. M.P. Almeida, E.J. Parteli, J.S. Andrade, H.J. Herrmann, P. Natl. Acad. Sci. U.S.A. 105, 6222 (2008). 23. P. Claudin, B. Andreotti, Earth Planet. Sci. Lett. 252, 30 (2006). 24. J.E. Ungar, P.K. Haff, Sedimentology 34, 289 (1987). 25. B. Androtti, J. Fluid. Mech. 510, 47 (2004). 26. J.F. Kok, N.O. Renno, J. Geophys. Res. 114, D17204 (2009). 27. R. Greeley et al., Geophys. Res. Lett. 7, 121 (1980). 28. J.D. Iversen, B.R. White, Sedimentology 29, 111 (1982). 29. J.L. Sutton, C.B. Leovy, J.E. Tillman, J. Atmos. Sci. 35, 2346 (1978). 30. R.E. Arvidson et al., Science 222, 463 (1983). 31. R. Sullivan et al., J. Geophys. Res. 105, 24547 (2000). 32. L.K. Fenton, Geophys. Res. Lett. 33, L20201 (2006). 33. D.J. Jerolmack et al., J. Geophys. Res. 111, E12S02 (2006). 34. V. Schatz et al., J. Geophys. Res. 111, E04006 (2006). 35. W. Chepil, J. Soil Water Conserv. 14, 214 (1959). 36. N. Woodruff, F. Siddoway, Soil Sci. Soc. Am. J. 29, 602 (1965). 37. D.W. Fryrear, A. Saleh, Soil Sci. 161, 398 (1996). 38. B. Andreotti, P. Claudin, O. Pouliquen, Geomorphology 123, 343 (2010). 39. Y.P. Shao, A. Li, Bound-Lay. Meteorol. 91, 199 (1999). 40. T.L. Bo, H. Zhang, W. Zhu, X. Zheng, J. Geophys. Res. 118, 4494 (2013). 41. J. Smagorinsky, Mon. Weather Rev. 91, 99 (1963). 42. H.K. Versteeg, W. Malalasekera, An introduction to computational fluid dynamics: the finite volume method (Prentice Hall, 2007). 43. J.F. Kok, E.J.R. Parteli, T.I. Michaels, D.B. Karam, Rep. Prog. Phys. 75, 106901 (2012). 44. P.J. Murphy, H. Hooshiari, J. Hydraul. Div 108, 1251 (1982). 45. M.V. Carneiro, T. P¨ ahtz, H.J. Herrmann, Phys. Rev. Lett. 107, 098001 (2011). 46. M.V. Carneiro, N.A.M. Araujo, T. Pahtz, H.J. Herrmann, Phys. Rev. Lett. 111, 058001 (2013). 47. N.S. Cheng, J. Hydraul. Eng. 123, 149 (1997). 48. G.S. Ma, X.J. Zheng, Eur. Phys. J. E 34, 1 (2011). 49. R.S. Anderson, M. Sørensen, B.B. Willetts, Acta. Mech. Suppl. 1, 1 (1991). 50. K.S. Edgett, P.R. Christensen, J. Geophys. Res. 96, 22765 (1991). 51. L.K. Fenton, A.D. Toigo, M.I. Richardson, J. Geophys. Res. 110, E06005 (2005). 52. B.T. Werner, J. Geol. 98, 1 (1990). 53. Y.H. Zhou, W.Q. Li, X.J. Zheng, J. Geophys. Res. 111, D15108 (2006). 54. D. Beladjine, M. Ammi, L. Oger, A. Valance, Phys. Rev. E 75, 061305 (2007). 55. M.A. Rice, B.B. Willetts, I.K. McEwan, Sedimentology 42, 695 (1995). 56. Y.P. Shao, M.R. Raupach, J. Geophys. Res. 97, 20559 (1992).
Page 8 of 8 57. R. Kawamura, Tokyo Daigaku Rikogaku Kenkyusho Hokoku 5, 95 (1951). 58. H. Lettau, K. Lettau, Exploring the world’s driest climate (Center for Climatic Research, Institute for Environmental Studies, University of Wisconsin-Madison, 1978). 59. B.R. White, H. Mounla, Acta. Mech. Suppl. 1, 145 (1991). 60. Y.H. Zhou, X. Guo, X.J. Zheng, Phys. Rev. E 66, 021305 (2002). 61. K. Kroy, S. Fischer, B. Obermayer, J. Phys-Condens. Mat. 17, S1229 (2005). 62. E.J.R. Parteli, O. Dur´ an, H.J. Herrmann, Phys. Rev. E 75, 011301 (2007).
Eur. Phys. J. E (2014) 37: 80 63. E.J.R. Parteli, H.J. Herrmann, Phys. Rev. Lett. 98, 198001 (2007). 64. T. P¨ ahtz, J.F. Kok, E.J.R. Parteli, H.J. Herrmann, Phys. Rev. Lett. 111, 218002 (2013). 65. T. P¨ ahtz, E.J.R. Parteli, J.F. Kok, H.J. Herrmann, Phys. Rev. E 89, 052213 (2014). 66. R.S. Anderson, P.K. Haff, Acta. Mech. Suppl. 1, 21 (1991). 67. T. Pahtz, J.F. Kok, H.J. Herrmann, New J. Phys. 14, 043035 (2012). 68. E.J.R. Parteli, H.J. Herrmann, Phys. Rev. E 76, 041307 (2007).
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Numerical modeling of wind-blown sand on Mars HaoJie Huang1 , TianLi Bo1 , and XiaoJing Zheng1,2,a 1
2
Key Laboratory of Mechanics on Environment and Disaster in Western China, Ministry of Education, Lanzhou University, Lanzhou 730000, China School of Electronic and Mechanical Engineering, Xidian University, Xi’an, 710071, China Received 26 September 2013 and Received in final form 2 March 2014 c EDP Sciences / Societ` Published online: 19 September 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. Recent observation results show that sand ripples and dunes are movable like those on Earth under current Martian climate. And the aeolian process on Mars therefore is re-attracting the eyes of scientific researchers in different fields. In this paper, the spatial and temporal evolution of wind-blown sand on Mars is simulated by the large-eddy simulation method. The simulations are conducted under the conditions of both friction wind speed higher and lower than the “fluid threshold”, respectively. The fluid entrainment of the sand particles, the processes among saltation sand particles and sand bed, and the negative feedback of sand movement to flow field are considered. Our results show that the “overshoot” phenomenon also exists in the evolution of wind-blown sand on Mars both temporally and spatially; impact entrainment affects the sand transport rate on Mars when the wind speed is smaller or larger than the fluid threshold; and both the average saltation length and height are one order of magnitudes larger than those on Earth. Eventually, the formulas describing the sand transport rate, average saltation length and height on Mars are given, respectively.
1 Introduction Mars, which is widely covered by sand dunes [1], has become a popular target of deep-space exploration. The latest observations show that the Martian sand ripples and dunes are also movable [2,3], rather than immovable, as inferred in previous observations [4]. The migration speeds of sand ripples and dunes are probably equivalent to those on Earth [5]. That is, wind-blown sand movement is prevalent both on Earth and on Mars [5]. For better comprehending the change of the climate on Mars [6], selecting the mission’s landing sites [5] and survey routines [7] and avoiding the encounters of the Spirit Mission [8], the studies on the characteristics and development process of wind-blown sand movement have become critical. Wind-blown sand movement, in which saltation sand particles account for 75% [9–11], will make the landform and geomorphology change [11–15]. Furthermore, it plays a key role in the release of dust [10], even on Mars where dust storms and dust devils are excessive [12,16]. Therefore, it is necessary to have a comprehensive study on wind-blown sand movement on Mars. However, existing wind tunnel experiments [17–19] and observations of the Martian surface [12,13,20] have not revealed all the characteristics of wind-blown sand on Mars, such as saltation a
e-mail: [email protected]
trajectory, sand speed and sand transport rate. Thus, theoretical simulation becomes a very important tool to study the characteristics of wind-blown sand on Mars. Since White et al. [21] and White [17] simulated the sand saltation trajectories under Martian conditions and derived that the Martian saltation trajectories are larger than those on Earth, many researchers focus on this subject. After that, Almeida et al. [22] carried out a detailed two-dimensional numerical study of Martian saltation motions by using the software Fluent. They found that Martian saltation trajectories were two orders of magnitudes larger than those on Earth. They also gave the expression of the relationship between the sand transport rate and the friction wind speed under wind speed when the latter is higher than the impact threshold of sand saltation. In addition, their simulations showed that the impact threshold of sand saltation was substantially below the fluid threshold and the impact threshold in the simulations is 1.12 m/s on Mars. This result is consistent with the dynamic transport thresholds of Claudin and Andreotti [23] obtained by considering the effect of cohesion and viscosity. However, the splash process between saltation sand particles and the sand bed was ignored in their models [14]. The splash process plays an important role in wind-blown sand evolution [24–26] and thus a complete splash process is very important for the simulation of wind-blown sand.
Page 2 of 8
On Mars, the fluid threshold wind speed is very high [27,28], but only a few times in a decade during gusts of extreme aeolian activity the Martian wind velocities can be higher than it [12,18,29–34]. Since recent observations indicate that the sand ripples and dunes on the Martian surface are movable [2,3], a problem arises: How could the sand grains sustain the saltation motions under the current Martian climate, in which most of the time the Martian wind speeds are below the fluid threshold? Following the idea of Almeida et al. [22] that saltation on Mars, once initiated, can be sustained under much lower wind speeds than previously thought, Kok [14] proposed a new hypothesis by considering the hysteresis phenomenon and parameterized it into his numerical model. The results indicated that the hysteresis has a significant effect on the threshold speed and causes saltation to occur and be sustained at much lower wind speed than previous conclusions. And the impact threshold required for sustaining saltation on Mars was an order of magnitude smaller than the fluid threshold. He pointed out that the saltation could be sustained under the current Martian conditions since the impact threshold is very low. His findings provide a reasonable explanation for the recent observations that the sand ripples and dunes on the Martian surface are movable. But the evolution of wind-blown sand was simplified as a one-dimensional model in his paper. The model ignored the influence of streamwise turbulence in the development of wind-blown sand. And the spatial variation of wind-blown sand cannot be obtained by the model. It is an important issue to understand the spatial variation of wind-blown sand. Especially, the spatial variation of the mass flux is essential to measure the sand transport rate in the wind tunnel experiment, even to predict soil wind erosion under field conditions [35–37]. Moreover, the wind-blown sand directly affects the wavelength and the formation of the dunes [38]. In order to learn more about the wind-blown sand on Mars, we need to carry out a quantitative research on the spatial variation of the mass flux. Also, the saltation characteristics on Mars, such as saltation trajectory, sand speed, and sand transport rate, need to be investigated further. In this paper, two-dimensional numerical modeling of Martian wind-blown sand is carried out and the evolution of wind-blown sand along the wind direction is studied. The large-eddy simulation approach is used to simulate wind field and the negative feedback of sand movement to flow field is considered. The movements of each particle are tracked and the splashing function is used to describe the collision between sand particles and sand bed. Section 2 gives the model and the validation under the terrestrial conditions. In sect. 3, the evolution of wind-blown sand on Mars is shown and the relationship between the sand transport rate and the friction wind speed, as well as the relationship between the average saltation length and height and friction wind speed on Mars are studied. Finally, sect. 4 is a summary of the main conclusions.
Eur. Phys. J. E (2014) 37: 80
2 Numerical method Actually, in the wind-blown sand, the saltation sand grain falls back onto the ground at a speed due to the gravity, impacts the sand bed, and ejects neighboring sand grains. In this paper, we model the evolution of wind-blown sand by three main processes: i) the movement of sand particles; ii) the coupling interaction between the sand particles and the wind field; iii) the collision process, which contains rebound and ejection, between the saltation sand particles and bed surface. 2.1 Basic equations In this paper, the large-eddy simulation (LES) approach is used to simulate the wind field in wind-blown sand. In LES, the speed field is separated into two fields, a resolved (large-scale) and an unresolved (small-scale) field, by a spatial filtering operation. The feedback of unresolved motions on the resolved-scale flow is achieved by the action of a sub-grid-scale viscosity. The LES filtered continuity equation and filtered momentum equation, including the feedback of sand movement are shown as follows [6,39,40]: ∂u ¯i = 0, (1) VP ∂xi ∂u ¯i u ∂u ¯i 1 ∂ p¯ ∂2u ¯j ¯i + VP = VP − +ν ∂t ∂xj ρf ∂xi ∂xj ∂xj ∂τij Fi − − δi2 g + , (2) ∂xj ρf where, i = 1 and 2 corresponds to the flow and wallnormal directions (i.e., x1 = x, x2 = z, u1 = u, u2 = w), respectively; u ¯i and p¯ represent the filtered wind speed and pressure, respectively; ν is the kinematic viscosity; ρf is the air density; g is the gravity; Fi is the feedback force per unit grid of sand acted on the air flow; VP = k=n 1− k=1 (1/6)πD3 is the volume ratio of the sand particles in the air flow, in which n is the sand particle number in the unit volume and D is the sand grain diameter; δij = 1 ¯i u ¯j are the if i = j, otherwise δij = 0; τij = ui uj − u sub-grid-scale (SGS) stresses [41]. In order to close the above equations, the sub-gridscale stresses can be modeled as follows: ∂u ¯i 1 ∂u ¯j , (3) + τij − τkk δij = −νt 3 ∂xj ∂xi ∂u ¯ ∂u ¯ ∂u ¯i ∂u ¯i where νt = (Csgs Δ)2 0.5( ∂x + ∂xji ).( ∂x + ∂xji ) is the j j turbulent viscosity of the sub-grid-scale; Δ is the grid scale; Csgs = 0.19–0.26, which depends on the detailed case [42]. The initial conditions of the above governing equations are u z ∗ ln , (4) u ¯(0, z) = k z0 u ¯(x, z) = w(x, ¯ z) = 0. (5)
Eur. Phys. J. E (2014) 37: 80
Page 3 of 8
Here k=0.41 is the von K´ arm´an constant; z0 = D/30 is the aerodynamic surface roughness [43]; u∗ is the friction speed of inflow. The boundary conditions of the above governing equations are opening conditions at the upper boundary (z = H): ¯ H)/∂z = 0. (6) ∂u ¯(x, H)/∂z = u∗ /kz, ∂ w(x,
1) Calculate the LES governing equations (1) and (2) till convergence at the first time step Δt and output the wind velocities ui,j , wi,j of each node.
Our wind profile has been given based on experiments in a wild natural environment and in wind tunnel. And eq. (6) is derived by the derivation of the logarithmic profile. No-slip conditions at lower boundary (z = 0) are:
(k = 1, 2, . . ., 200, and is the discrete position of the sand bed) which means that they scale with u2∗ − u2∗t and that they are numbered by N0k (0 represents the initial time) [39]. And the scaling with u3∗ − u3∗t can u3 be described as Na,k = 1.74 × 10−3 u2∗,k (1 − u3∗t )D−3 .
u ¯(x, 0) = w(x, ¯ 0) = 0.
(7)
Fully developed conditions at the outlet boundary (x = L) are: ∂u ¯(L, z)/∂x = 0, w ¯ = 0. (8) The saltation trajectory is mainly influenced by gravity and drag force. And we neglected the other smaller terms acting on the sand grains, such as the Magnus force, the Saffman force [44]. It is worth noting that although recent studies show that the collisions of the ejected sand grain in the air need to be considered [45,46], but in our model, we neglected the process like most previous models did [14,22]. Then the motion of each sand grain in wind-blown sand can be described as 2 dx CD πD2 ρf d2 x u− , (9) mp 2 = FDx = dt 8 dt πρp D3 d2 z mp 2 = −Fg + FDz = − dt 6 2 CD πD2 ρf dz + w− , (10) 8 dt where mp and ρp are the mass and density of sand parti2 cles, respectively; CD = (0.63 + 4.8/Re0.5 p ) is the drag coefficient [47]; ρf is the air density; Rep = (Vf ρp D/μ)[(u − dx/dt)2 + (w − dz/dt)2 ]1/2 is the particle Reynolds numk=n ber [26]; Vf = 1 − k=1 VP /ΔV is the volume fraction that is the total sand volumes within one grid to the grid volume; ΔV is the control volume size; μ is the air kinetic viscosity coefficient. 2.2 Calculation procedures In this paper, the calculation of wind field is based on the finite-volume method [42]. We need to mesh the computational domain 22 m × 1.5 m under Earth conditions [48] (see fig 1). In order to guarantee the efficiency and accuracy of calculation, we mesh the computational domain 80 m × 5 m under Mars conditions with 3000 × 300 grids, although it is larger than that on Earth, the sand particles still can be recognized, because the wind speed on Mars is much larger than on Earth. So the grid size we chose cannot influence the simulation results. The simulation parameters on Earth and Mars are taken as in table 1 [9, 49–51]. The detailed steps are shown as follows:
2) According to the friction speed, the number of lifting sand particles in the unit area near the sand bed is u2 calculated by Na,k = 1.74 × 10−3 u∗,k (1 − u2∗t )D−3 ∗,k
∗,k
The probability density distribution of the initial lifting speed follows: p(vej ) = exp(−vej /vej )/vej , where vej is the ejection speed, the overbar represents the mean value and it can be deserved by the momentum and energy conservation [49,52]. And the lifting angle is 90◦ . It is worth noting that when we simulate the Martian wind-blown sand under the condition of the friction speed below the fluid threshold, u2 Na,k = 1.74 × 10−3 u∗,k (1 − u2∗t )D−3 is no longer ap∗,k
propriate, we give some lift-off particles to saltation movement [14]. Then the coordinates and velocities of k k , x˙ k0,n , z˙0,n (n is the numall sand particles xk0,n , z0,n ber of sand particles) can be obtained. The motion of the sand trajectory, eqs. (9) and (10), is calculated in x z , Fi,j a fluid time step Δt and the reaction forces Fi,j of sand particles to eqs. (2) are recorded. Here, u∗,k is the friction wind speed at the k discrete point of the sand bed and u∗,t is the fluid threshold friction wind speed. 3) The new wind speeds ui,j , wi,j of each node are recalx z , Fi,j at the new time culated by the reaction forces Fi,j step. If collision happens, we assume the rebound probability as preb = 0.95(1 − e−λvimp ), where vimp is the impact speed, λ is 2 m/s according to the simulation result [49], the rebound speed is 0.55 times the impact speed, the rebound√angle is 40◦ [53]. The ejection number is N ≈ avimp / gD [25,54], and the ejection angle distributes randomly between 50◦ and 60◦ [55]. The new saltation sand particles are programmed into Ntk and the position of sand particles and their speeds k k , x˙ kt,n , z˙t,n are reset. xkt,n , zt,n 4) Repeat steps 1)-3) until the simulation time is finished.
2.3 Model verification In this section, we compare our simulation results under two different aerodynamic entrainment laws with the wind tunnel experimental results. Figure 2 shows the variation sand transport rate along with the wind direction. After the wind-blown sand developed to stable, we count the grains in the grids of all heights at the different horizontal positions to derive fig. 2. From fig. 2 we can find that there exists the “overshoot” phenomenon. When enough sand
Page 4 of 8
Eur. Phys. J. E (2014) 37: 80 Table 1. Simulation parameters used in the model. Gravity g (m/s2 )
Air density ρa (kg/m2 )
Viscosity ν (m2 /s)
Earth
9.81
1.225
1.45e-5
2650
250
0.26
Mars
3.71
0.02
6.5e-4
3000
500
1.8(a)
(a)
Sand density ρs (kg/m2 )
Diameter d (μm)
Fluid threshold u∗ (m/s)
Iversen and White [55] obtained the Martian fluid threshold friction speed between 1.4 m/s and 2.8 m/s in
the 4 case experiments when the grain diameter is 500 μm. Shao and Li [39] gave the theoretical expression of the fluid threshold friction speed which is 1.96 m/s when using our simulation parameters. Here, it is reasonable to use 1.8 m/s as the fluid threshold friction speed in our model.
Fig. 1. Schematic diagram of wind-blown sand on Earth.
Fig. 2. The sand transport rate varying with the wind direction on Earth, comparison between our different aerodynamic scaling law results and the experiment results of Shao and Raupach [56].
particles lift-off and fly a certain distance, the reaction of the sand particles to the flow field will reduce the wind speed, which reduces the kinetic energy of the saltation layer. Thus, the ejection number decreases, and macroscopically, the sand transport rate reduces, until the liftoff and the deposit reach a balance. This agrees well with the experimental result of Shao and Raupach [56] qualitatively and quantitatively, that is, the sand flux reaches the maximum and saturation at 7 m and 15 m, respectively. It indicates that our modeling can reflect the real development of wind-blown sand on Earth. Besides, we can also
see that the different aerodynamic entrainment laws only influence the distribution law of the sand transport rate along the stream, but has no effect on the relationship between the sand transport rate and the friction wind speed in the stable stage. All these prove that our simulation results do not depend on the selection of aerodynamic entrainment scaling laws and, therefore, our results are reliable. Then we compare the results of the sand transport rate for different friction speeds (d = 250 μm) with both the experiment results [9,57–60] (see fig. 3). From fig. 3 we can see, approximately, that the sand transport rate increases with the friction speed exponentially, which is consistent with the previous experimental results qualitatively and quantitatively. The wind tunnel experiment results of Zhou et al. [60] indicate that the previous experimental results have each effective region. And from fig. 3 we can see that our results coincide with Kawamura [57] at small friction speed, and agree well with Bagnold [9], Lettau and Lettau [58], White and Mounla [59] at large friction speed. Based on all of the above, not only the development process but also the quantitative one, our simulation results are in accordance with the experimental results, thus indicating that our modeling can reflect the actual wind-blown sand process. From fig. 3, we can also see that the differences of sand transport rate on Earth at two different aerodynamic entrainment laws are less obvious.
3 Result and analysis Figure 4 is the variation of wind-blown sand with wind direction and time on Mars. From fig. 4 we can find that the “overshoot” phenomenon exists in both the temporal and spatial dimensions, which is the same with the situation on Earth qualitatively. However, the saturation curve of wind-blown sand on Mars is much steeper than that on Earth, which means the maximum erosion position moves forward. This may be due to the low density of air and low gravity on Mars [17,22]. Under the same lift-off conditions, sand grains on Mars can fly higher than on Earth. Then more sand grains in the air fall back to the bed after a short distance. Moreover, the saturation length of windblown sand on Mars is longer than that on Earth, which is in agreement with previous studies [61–64]. This can
Eur. Phys. J. E (2014) 37: 80
Page 5 of 8
Fig. 3. The sand transport rate varying with friction wind speed on Earth, comparison between our simulated results and the existing experiment.
mainly be explained by the larger ratio between particle and fluid density [65]. Therefore, the grains can fly farther and need a long distance to reach a steady state. In addition, the saturation time of wind-blown sand is much longer than that on Earth. In detail, it takes about 2– 3 seconds to reach saturation on Earth [56,66], while on Mars, our results show that it takes about 7–8 seconds. The sand grains can fly a long time because of the high wind speed and the low gravity on Mars under the same lift-off condition. Figure 5 is the relation between the sand transport rate and friction speed. From fig. 5 we can see that when the friction speed is larger than the fluid threshold (i.e., u∗t = 1.8 m/s), our simulated mass flux increases exponentially, which is the same as the results of Almeida et al. [22] and Kok [14] qualitatively but smaller than their results quantitatively. Also, we find that when the friction speed is smaller than the fluid threshold but larger than the impact threshold, the saltation can still be sustained, which is consistent with the results of Claudin and Andreotti [23], Almeida et al. [22] and Kok [14]. The numerical solutions of Werner [52] give rise to more general arguments in support of the conclusions that the fluid entrainment of grains is unimportant in steady-state saltation on Earth. Impact entrainment is dominant when the friction speed is smaller than fluid threshold on Mars [14, 22,23,67]. However, when the friction speed is larger than fluid threshold, the wind-blown sand may be also maintained by impact entrainment, and the fluid entrainment can be ignored. Almeida et al. [22] considered that not only the wind speed and the gravity, but also atmospheric environment and the properties of sand grains affect the sand transport rate. In their formulations, they introduce the normalized quantity d/lv , where, the length-scale lv = (v 2 /g)1/3 . Based on it, we refer to their results and give our formulations. The sand flux is proportional to (u∗ − u∗im )2 (see fig. 6). Thereby, the variation of sand
Fig. 4. (a) The sand transport rate along wind direction on Mars under different friction wind velocities. (b) The sand transport rate varying with time on Mars under different friction wind velocities at the streamwise distance of 5.8 meters.
Fig. 5. The sand transport rate varying with friction wind speed on Mars, comparison between our simulated results and the existing model simulation results.
Page 6 of 8
Eur. Phys. J. E (2014) 37: 80
Fig. 7. The relation between the average saltation length and height and friction wind speed, comparison between our simulated results and the simulation results of Kok [15]. Fig. 6. The normalized sand transport rate on Mars and Earth. The straight line corresponds to the equation y = (u∗ − u∗im )2 .
flux with the friction wind speed can be described by dρf u∗im (u∗ − u∗im )2 , qs = 26.4 (vg)2/3 u∗ > u∗im ,
r2 = 0.92
(11)
where u∗im is the impact threshold. From fig. 5, we can see that the simulation results can be fitted by eq. (11). In fig. 6, the horizontal axis means u∗ –u∗im and the impact threshold is 0.26 m/s on Earth [28,39,49] and 0.7 m/s on Mars [14]. The vertical axis means the normalized sand transport rate. In fig. 6, the normalized sand transport rates are QMars = qs /[26.4 × u∗im × dρfluid /(vg)2/3 ] and QEarth = qs /[23.0 × u∗im × dρfluid /(vg)2/3 ] on Mars and Earth, respectively. At the same time, the variation of the average saltation height and length with the friction wind speed on Mars were obtained (as shown in fig. 7). From fig. 7 we can find that the average saltation height and length increase with friction wind speed, which is consistent with Almeida et al. [22] and Kok [15]. The magnitudes of the average saltation length and height are 100 m and 10−1 m, which is less than that of Almeida et al. [22] by one order of magnitude. These maybe due to the fact that the model of Almeida et al. [22] disregards the details of the splash in the sandbed collision process. We can also find Lsalt and Hsalt are proportional to the length-scale lv = (v 2 /g)1/3 both on Mars and Earth and the two quantities obey the Exponential relationship with u∗ , see fig. 8. Then the Martian normalized results can be described by 2 1/3 v u∗ , r2 = 0.89, (12) exp Lsalt ≈ 91.8 g 2u∗im Hsalt ≈ 6.1
v2 g
1/3
exp
u∗ 2u∗im
,
r2 = 0.95.
(13)
Fig. 8. The normalized average saltation height and length both on Mars and Earth. The straight line corresponds to the equation y = exp(u∗ /2u∗im ).
In fig. 8, the horizontal axis means u∗ /u∗im , and the impact threshold is 0.26 m/s on Earth and 0.7 m/s on Mars [14,49]. The vertical axis means the normalized average saltation height and length. On Mars, the normalized length SMars = Lsalt /[91.8 × (v 2 /g)1/3 ] and the normalized height SMars = Hsalt /[6.1 × (v 2 /g)1/3 ] as a function of u∗ , and on Earth the normalized length SEarth = Lsalt /[170.1 × (v 2 /g)1/3 ] and the normalized height SEarth = Hsalt /[26.4 × (v 2 /g)1/3 ] also as a function of exp (u∗ ). Generally, the same rule holds for the average saltation height and length on Mars and on Earth. From the results, we can infer the Martian saltation trajectories are larger than those on Earth because of the low gravity and high speed of the Martian wind-blown sand which is in agreement with the conclusions of Parteli and Herrmann [68], Almeida et al. [22] and Kok [14].
Eur. Phys. J. E (2014) 37: 80
4 Conclusion In this paper, numerical simulation of the evolution of wind-blown sand on Mars is achieved and some saltation characteristics are studied, such as saltation trajectory, sand speed and sand transport rate. In the numerical modeling, the large-eddy simulation approach is used to calculate the flow field, and the negative feedback of the sand movement to the flow field is considered. The movements of each particle are tracked and the splashing function is used to describe the collision between sand particles and sand bed. The development of wind-blown sand along height and wind direction on Mars is analyzed when the friction wind speed is higher and lower than the “fluid threshold”. The results display that the “overshoot” phenomenon existed in the time and spatial dimension. Moreover, our results show that the saturation curve of wind-blown sand on Mars is much steeper than that on Earth and the saturation time and length are much bigger and longer than those on Earth. Then the relationship between the sand transport rate and friction wind speed on Mars is given. The average saltation height and length increase with friction wind speed which satisfies a log-linear law, and the relationship between the average saltation length and height and friction wind speed are given. This research was supported by a grant from the National Natural Science Foundation of China (No. 11072097, No. 11232006, No. 11202088, No. 10972164, No. 11121202), National Key Technology R&D Program (2013BAC07B01), the Science Foundation of Ministry of Education of China (No. 308022), Fundamental Research Funds for the Central Universities (lzujbky-2009-k01) and the Project of the Ministry of Science and Technology of China (No. 2009CB421304). The authors express their sincere appreciation to the supports.
References 1. J.R. Zimbelman, Geophys. Res. Lett. 27, 1069 (2000). 2. M.C. Bourke, K.S. Edgett, B.A. Cantor, Geomorphology 94, 247 (2008). 3. C.J. Hansen et al., Science 331, 575 (2011). 4. P.E. Geissler, J. Geophys. Res. 110, E02001 (2005). 5. N.T. Bridges et al., Geology 40, 31 (2012). 6. E. Gardin et al., Planet. Space Sci. 60, 314 (2012). 7. M. Golombek et al., Space Sci. Rev. 170, 641 (2012). 8. R.A. Kerr, Science 324, 998 (2009). 9. R.A. Bagnold, The Physics of Blown Sand and Desert Dunes (Chapman and Hall, 1956). 10. Y.P. Shao, Physics and modelling of wind erosion (Springer, 2008). 11. X.J. Zheng, Mechanics of wind-blown sand movements (Springer, 2009). 12. M. Balme, R. Greeley, Rev. Geophys. 44, RG3003 (2006). 13. R. Sullivan et al., J. Geophys. Res. 113, E06S07 (2008). 14. J.F. Kok, Phys. Rev. Lett. 104, 074502 (2010). 15. J.F. Kok, Geophys. Res. Lett. 37, L12202 (2010). 16. B.A. Cantor, P.B. James, M. Caplinger, M.J. Wolff, J. Geophys. Res. 106, 23653 (2001). 17. B.R. White, J. Geophys. Res. 84, 4643 (1979).
Page 7 of 8 18. R. Greeley, J.D. Iversen, Wind as a geological process on Earth, Mars, Venus and Titan (Cambridge, 1985). 19. R. Greeley, Planet. Space Sci. 50, 151 (2002). 20. R. Greeley et al., J. Geophys. Res. 111, E02S09 (2006). 21. B.R. White, R. Greeley, J.D. Iversen, J.B. Pollack, J. Geophys. Res. 81, 5643 (1976). 22. M.P. Almeida, E.J. Parteli, J.S. Andrade, H.J. Herrmann, P. Natl. Acad. Sci. U.S.A. 105, 6222 (2008). 23. P. Claudin, B. Andreotti, Earth Planet. Sci. Lett. 252, 30 (2006). 24. J.E. Ungar, P.K. Haff, Sedimentology 34, 289 (1987). 25. B. Androtti, J. Fluid. Mech. 510, 47 (2004). 26. J.F. Kok, N.O. Renno, J. Geophys. Res. 114, D17204 (2009). 27. R. Greeley et al., Geophys. Res. Lett. 7, 121 (1980). 28. J.D. Iversen, B.R. White, Sedimentology 29, 111 (1982). 29. J.L. Sutton, C.B. Leovy, J.E. Tillman, J. Atmos. Sci. 35, 2346 (1978). 30. R.E. Arvidson et al., Science 222, 463 (1983). 31. R. Sullivan et al., J. Geophys. Res. 105, 24547 (2000). 32. L.K. Fenton, Geophys. Res. Lett. 33, L20201 (2006). 33. D.J. Jerolmack et al., J. Geophys. Res. 111, E12S02 (2006). 34. V. Schatz et al., J. Geophys. Res. 111, E04006 (2006). 35. W. Chepil, J. Soil Water Conserv. 14, 214 (1959). 36. N. Woodruff, F. Siddoway, Soil Sci. Soc. Am. J. 29, 602 (1965). 37. D.W. Fryrear, A. Saleh, Soil Sci. 161, 398 (1996). 38. B. Andreotti, P. Claudin, O. Pouliquen, Geomorphology 123, 343 (2010). 39. Y.P. Shao, A. Li, Bound-Lay. Meteorol. 91, 199 (1999). 40. T.L. Bo, H. Zhang, W. Zhu, X. Zheng, J. Geophys. Res. 118, 4494 (2013). 41. J. Smagorinsky, Mon. Weather Rev. 91, 99 (1963). 42. H.K. Versteeg, W. Malalasekera, An introduction to computational fluid dynamics: the finite volume method (Prentice Hall, 2007). 43. J.F. Kok, E.J.R. Parteli, T.I. Michaels, D.B. Karam, Rep. Prog. Phys. 75, 106901 (2012). 44. P.J. Murphy, H. Hooshiari, J. Hydraul. Div 108, 1251 (1982). 45. M.V. Carneiro, T. P¨ ahtz, H.J. Herrmann, Phys. Rev. Lett. 107, 098001 (2011). 46. M.V. Carneiro, N.A.M. Araujo, T. Pahtz, H.J. Herrmann, Phys. Rev. Lett. 111, 058001 (2013). 47. N.S. Cheng, J. Hydraul. Eng. 123, 149 (1997). 48. G.S. Ma, X.J. Zheng, Eur. Phys. J. E 34, 1 (2011). 49. R.S. Anderson, M. Sørensen, B.B. Willetts, Acta. Mech. Suppl. 1, 1 (1991). 50. K.S. Edgett, P.R. Christensen, J. Geophys. Res. 96, 22765 (1991). 51. L.K. Fenton, A.D. Toigo, M.I. Richardson, J. Geophys. Res. 110, E06005 (2005). 52. B.T. Werner, J. Geol. 98, 1 (1990). 53. Y.H. Zhou, W.Q. Li, X.J. Zheng, J. Geophys. Res. 111, D15108 (2006). 54. D. Beladjine, M. Ammi, L. Oger, A. Valance, Phys. Rev. E 75, 061305 (2007). 55. M.A. Rice, B.B. Willetts, I.K. McEwan, Sedimentology 42, 695 (1995). 56. Y.P. Shao, M.R. Raupach, J. Geophys. Res. 97, 20559 (1992).
Page 8 of 8 57. R. Kawamura, Tokyo Daigaku Rikogaku Kenkyusho Hokoku 5, 95 (1951). 58. H. Lettau, K. Lettau, Exploring the world’s driest climate (Center for Climatic Research, Institute for Environmental Studies, University of Wisconsin-Madison, 1978). 59. B.R. White, H. Mounla, Acta. Mech. Suppl. 1, 145 (1991). 60. Y.H. Zhou, X. Guo, X.J. Zheng, Phys. Rev. E 66, 021305 (2002). 61. K. Kroy, S. Fischer, B. Obermayer, J. Phys-Condens. Mat. 17, S1229 (2005). 62. E.J.R. Parteli, O. Dur´ an, H.J. Herrmann, Phys. Rev. E 75, 011301 (2007).
Eur. Phys. J. E (2014) 37: 80 63. E.J.R. Parteli, H.J. Herrmann, Phys. Rev. Lett. 98, 198001 (2007). 64. T. P¨ ahtz, J.F. Kok, E.J.R. Parteli, H.J. Herrmann, Phys. Rev. Lett. 111, 218002 (2013). 65. T. P¨ ahtz, E.J.R. Parteli, J.F. Kok, H.J. Herrmann, Phys. Rev. E 89, 052213 (2014). 66. R.S. Anderson, P.K. Haff, Acta. Mech. Suppl. 1, 21 (1991). 67. T. Pahtz, J.F. Kok, H.J. Herrmann, New J. Phys. 14, 043035 (2012). 68. E.J.R. Parteli, H.J. Herrmann, Phys. Rev. E 76, 041307 (2007).
