Corresponding author: N. Huang, Key Laboratory of Mechanics on Disaster and Environment in Western China, Lanzhou University, Ministry of Education, Lanzhou 730000, China. (email@example.com)
 In this work, we numerically simulated the saltating particles in a turbulent boundary layer over flat bed and sand ripples. By using natural sand grains in a wind tunnel, we obtained the initial conditions for the simulation and also verified the correctness of the numerical model. We carefully analyzed the numerically simulated saltating particle movement over the two sand beds, and we found the following. (1) The aeolian sand transport is a dynamic equilibrium process on both sand beds, and it took longer to reach equilibration on the sand ripples than on the flat bed. (2) According to the mass flux profile at the trough of the sand ripples, there is a maximum mass flux at about 4 cm height in the leeward section. The mass flux increases with height below 4 cm and decreases with height above 4 cm. (3) The wind profile near the surface is modified by saltating particles on the two different sand beds, and the flow field characteristics of the sand ripples are more complex than that of the flat bed.
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 Sand transport by wind is a type of movement of many sand grains that can cause environmental problems such as soil erosion, sand storms and desertification as well as invoke health hazard for human being. The sand particle movements can be divided into three modes, i.e., creep, saltation and suspension, among which saltation is predominant in the near surface sand particle transportation [Bagnold, 1941].
 Many experimental studies have been developed to understand the saltation process [Rasmussen and Mikkelsen, 1991, 1998; McKenna Neuman and Maljaars, 1997; Iversen and Rasmussen, 1999; Zou et al., 2001; Namikas, 2003; Zheng et al., 2003]. Meanwhile, some mathematical and theoretical models have also been established to study the motion of saltating grains [Ungar and Haff, 1987; Anderson and Haff, 1988, 1991; Werner, 1990; Shao and Li, 1999; Vinkovic et al., 2006; Kang and Guo, 2006; Huang et al., 2006]. Although various parameters concerning saltating grain movement have been calculated from these models, the physical process of saltation is inadequately recognized and some assumptions and simplifications have to be made to enable calculation. The accuracy of the theoretical calculation depends on the available initial conditions, the validity of assumptions, and the simplifications of the models. The current theoretic and numerical models of wind-blown sand mostly consider ideal circumstances such as steady wind velocity, flat sand surface, etc. However, field measurements of the airflows over aeolian bed forms [White, 1996; Wiggs, 1993; Wiggs et al., 1996] and saltating particles over the surfaces of complex microtopology [Leenders et al., 2005; Bowker et al., 2006, 2007] showed that the wind-blown sand movement occurs in much more complicated environment. For example, the morphology of sand dunes and sand ripples, both of which are basic forms of desert landscape, has great influence on the initiation and transportation of sand particles. Therefore, current theoretical models and empirical formulas based on experimental results are still far from being able to reliably and quantitatively predict sand transport. It is necessary to further study about the wind fields and sand movement over the surfaces with complex micro-topography.
 We aim to elucidate the sophisticated dynamic processes of aeolian sand transport on different bed morphology, and in this paper we propose a theoretical model of sand saltation process considering the coupling between wind field and sand grains on two bed forms, i.e., the flat bed and the sand ripples. The flow field was solved by the large eddy simulation (LES) precisely to obtain the wind velocity, and the discrete element method (DEM) was used to obtain the velocity and position of moving particles. To obtain the initial condition of the model and verify our simulation results, we carried out some experimental investigations about the saltation particles in a wind tunnel. The inlet wind speed was recorded by hot-wire anemometer and the sand flux was recorded using the sand trap. We also used the particle tracking velocimetry (PTV) system to measure the velocities of the saltating particles. We then analyzed the particle velocity profiles in detail.
2. Experimental Methods and Results
 The experiment was performed in a blow-type non-circulating wind tunnel. The working section of the wind tunnel was 20 m long with a cross-section which was 1.3 m wide and 1.45 m high. The roughness elements and wedge were placed in front of the working section in order to produce a thick boundary layer. The wind boundary layer thickness at the measurement position reaches about 0.4 m. The free-stream wind velocity in the wind tunnel can be changed continuously to 40 m/s.
 The wind velocity profiles were measured by an I-type hot-wire probe (DANTEC 55P11), which was connected to a constant temperature hot-wire anemometer. The velocity of sand particles was obtained by the PTV (Particle Tracking Velocimetry) method [Zhang et al., 2007, 2008; Dong et al., 2010], which is a technique to analyze the velocity of particles from the spatial correlation performed by individual particle tracking. Particle images were acquired at a frame rate of 32 frames per second using a CCD (Charge Coupled Device) camera (Power View Plus 2MP) of 1,600 × 1,200 pixels. A schematic drawing on the experimental apparatus is shown in Figure 1. The sand bed is 20 cm high and 12 m long so that a fully developed saltation flow can be developed. The transition section surface between smooth part and sand bed was 0.5 m long, fully glued by sand particles. The hot-wire probe was put at X = 5.5 m from the entrance of the test section, which is long enough to have a steady state flow. The position of CCD camera and sand trap are set at the 7 m and 8.5 m separately on sand bed downstream.
Figure 2shows the mean streamwise velocity profiles of the oncoming flow in the wind tunnel test section (X = 5.5 m). Usually the free-stream wind velocity can be translated to friction velocity [Zhou et al., 2002], the friction velocities corresponding to the four velocities (8, 10, 12 and 14 m/s) in our experiment are 0.340, 0.415, 0.488, 0.553 m/s, respectively. The sand sample was collected from the western edge of Tengger Desert. Sand particles' average diameter is 0.295 mm and its distribution is shown in Figure 3. The sand size-distribution was measured by particle size analyzer (Microtrac 3500).Figure 4 shows the variation of the particle's horizontal velocity with height at different wind speeds. The vertical origin of the PTV measurement is 1 cm from the surface of sand bed for the concentration of the sand particles nearer ground is very high and it is difficult to analyze the experimental data there. It can be seen from Figure 4 that the particle velocity obviously increases with the free stream velocity of wind because the airflow provides the driving force for particle motion. Figure 5 displays the sand flux at different heights as recorded by the sand trap, which plotted by dots and the lines are the fitting results. Figure 6 shows the comparison of the mass flux (Q) with the prediction of Bagnold's results. It can be seen from Figures 5 and 6 that our experimental results are basically identical with that of Bagnold's  and the exponential function is better to fit the experimental results. The experimental results of Figures 4 and 5 will be used to compare with simulation results in section 4.1.
where ui is the fluid velocity, p is the total pressure, ν is the molecular kinematic viscosity, ρ is the density, fiis the volumetric fluid-particle interaction force,Vf is the volume fraction of the fluid and it is a dimensionless parameter, and Sij is the strain tensor. Sij, Vf and fi are defined as:
where ΔV is the volume of a computational cell, n is the total number of particles in the cell. FDkis the interaction force between fluid and the i-th sand particle. The interaction force can be calculated byequation (17). VP is the volume of particles within this cell defined as:
 Because this simulation model is for 2-D flows, we set ΔV = Δx · Δy · dp, where Δx and Δy are the lengths of the computational cell in X and Y directions, and dp is the diameter of sand particles.
 Following the work of Germano et al. , we use the large eddy simulation (LES) turbulent model to solve the fluid turbulence. We denote the grid filtering operation with filter G:
 Now, application of the filter to the Navier–Stokes equations gives:
 In the determination of the SGS stress, τij is parameterized using an eddy viscosity hypothesis:
 In this paper, the wavelength and height of sand ripple are 15 cm and 2 cm, respectively. The windward slope and downstream slope are 11° and 21.8°. This kind shape of sand ripples is very common in desert areas [Werner et al., 1986]. A simple immersed boundary method [Patankar et al., 1978] which can deal with the complex boundary problem was used to generate the ripple terrain.
3.2. The Motion of Solid Particles
 The motion processes of all sand particles are simulated by the DEM (discrete element method) [Gethin et al., 2006] approach. The behavior of the sand particles in a turbulent flow can be characterized by the Stokes number:
Where τp and τl are the particles' relaxation time and Kolmogorov time scale as follows:
where U is the wind velocity, δ is the boundary layer thickness and ν is the kinematic viscosity of the air [Tanière et al., 1997]. When the stokes number of the sand particles is largely greater than 1, the motion of particles is mainly governed by gravity and the drag force. So a simplified equation of particle motion including the drag and gravity forces which was well used in the simulation of sand drift can be considered as [Ungar and Haff, 1987; Anderson and Haff, 1991]:
here, mp and Up are the mass and velocity of sand particles, separately. The fluid drag force FD can be described as:
where CD and Vr are the fluid drag coefficient [Ungar and Haff, 1987] and the relative velocity, respectively, which are defined as follows:
where up and wp are horizontal velocity and vertical velocity of sand particles.
 The particle Reynolds number can be expressed as:
 The equation for the drag coefficient (equation 18) is valid for:
 The initial velocity and angle of the new particles ejected into the flow from the sand bed depend on the impacting sand grain under the relationship [Vinkovic et al., 2006]:
 The number of new ejected sand particles (n) is determined according to the analytic model of Werner :
here, αim and Vim are the impact angle and impact velocity of the sand particle, separately.
3.4. The Simulation Condition and the Computational Scheme
 The computational domain in the horizontal and vertical directions are Lx = 1.5 m and Ly = 1 m for both flat and sand ripple beds. The region is meshed by 300 × 100 grids, which in the horizontal direction are uniform with the scale 0.005 m of one grid. A logarithmically decreasing law was used in the vertical direction to ensure sufficient resolution near the ground. For a typical windblown sand flow, the time to reach saturations is about 2 s and the saltation time of sand particles is about 0.05 s [Bagnold, 1941], so we take the time step 1 × 10−4 s which is less than τp (particles' relaxation time) for moving particles and 0.5 × 10−3 s for wind flow. There are 5 integration steps for moving sand particle in every time step for fluid motion. The finite volume method [Versteeg and Malalasekera, 1995] is used to solve equations of the gas phase based on the non-staggered rectangular grid. The particle size was selected from experimental result (Figure 3), it worth to say that we only use the mean value (295 μm) of the particles' diameter in our simulation work, and the density of the sand particles is 2650 kg/m3. The aerodynamic roughness length in our simulation was chose as Dp/30 [Bagnold, 1941; Anderson and Haff, 1988, 1991; Zheng, 2009].
 The boundary conditions are given as follows. For the fluid phase:
 The periodic boundary condition is used for the sand particles. When one particle exits from the computational domain, it will enter from the other side with its original velocity and height, even though this kind of condition may exclude the fetch effect, it can effectively save the computing resources.
 The initial conditions are as follows.
 1. The initial inlet wind velocity is obtained from the experimentally determined velocity profile in Figure 2. In the numerical simulation, all the particles have the same diameter 0.295 mm. That is, the average diameter of sand particles in Figure 3.
 2. The time step for calculating the fluid phase is 0.5 × 10−3 s and the time step for particle motion is 1 × 10−4 s. The whole time for the computation is taken as 6 s, which is longer than the evolution time of sand flux [Anderson and Haff, 1988, 1991].
 3. For particles, there are some sand grains called “induced-particle” at the sand bed with random position, which are used to initiate the saltation flow. The vertical velocities of induced particles are randomly chosen from 0.5 m/s to 1 m/s [Zhou et al., 2006].
 In our model, we solve the flow field by LES and obtain the particle motion through DEM. The whole computation scheme can be described as follows.
 1. Set the experimentally measured wind velocity as the initial condition for all gas variables and assign the initial position and velocity of induction-sand particles.
 2. During the first time step, the equations of gas phase are calculated to convergence.
 3. Use the wind velocity from step (2) to calculate the particle motion (equation (16)) and the drag force (equation (17)) between the particles and the gas.
 4. Identify the particles that impact the sand bed and apply the splash function (equations (22a)–(24)).
 5. Go to step 2 and calculate the equations of gas phase with fluid-particle interaction force for the next time step.
 6. Terminate calculation when stop time is reached.
4. Simulation Results
 The simulations in this paper have been realized by the code developed by our research team in Key Laboratory of Mechanics on Disaster and Environment in Western China. The code had been used to simulate the sand flux and parts of the results have been detailed in Huang et al.  and Ma and Zheng .
4.1. Comparison of Simulated and Experimentally Measured Sand Flux and Particle Velocity on the Flat Bed
 As we are concerned with the correctness of our simulation, we contrast the results from the simulation and the experiment. We first focus on the variation of sand flux with height at different oncoming flow. Figure 5 shows the experimentally measured sand flux recorded by the sand trap, which measures the mean values of sand flux from the height of 3 cm to 34 cm.
Figure 7 compares the simulated and measured variation of sand flux with height at two different inlet wind velocities. It can be seen that the simulated sand flux is basically identical with the experimental results at Ue = 8 m/s and Ue = 10 m/s.
 Due to the shooting region of the CCD camera in the PIV system, the profile of the particle's mean horizontal velocity is under 10 cm height. The solid and hollow points in Figure 8 represent the measured and simulated particle velocity respectively. It can be seen that the simulated and measured data are very close and both exhibit the same trend.
 From Figures 7 and 8, we conclude that our model can effectively simulate the saltating sand particles in the boundary layer.
4.2. Particle Spatial Distribution on the Flat Bed
 When sand particles are entrained into air, they will be accelerated by the wind flow and splash more particles from the sand bed as they impact the bed. As the number of sand particles in the air increases, they will generate a retardatory effect to the wind. The increasing retardatory force of sand particles on the wind will then decrease wind speed and the number of sand particles in the air will decrease correspondingly. The wind-sand interaction will reach a dynamic equilibrium state when the number of moving sand particles fluctuate around a certain value for a given wind velocity. Then we call that the sand flux attain to steady state.
 The particle spatial distribution in steady state at different inlet wind speeds is shown in Figure 9. It can be seen that the particle concentration increases with wind velocity. Besides, as the inlet wind velocity increases, the height that the particles can reach also increases.
4.3. The Kinetic Variation of Sand Flux on the Flat Bed
 In Figures 10 and 11, sand flux is the integral over the y level vertical variable. Figure 10 shows how the sand flux varies with time at different inlet wind speeds at the position X = 0.95 m. The mass flux increases rapidly as time elapses at first, which indicates that there are more and more particles jumping into the airflow. Then there is a small reduction of mass flux due to the interactions between the sand particles and the gas phase. At last, the mass flux will fluctuate around a certain value for a given wind velocity and reach a dynamic equilibrium after a very short period. This process was called “Overshoot” phenomenon [Anderson and Haff, 1991], which has already proved by the experiments [Butterfield, 1998, 1999].
 Analysis of the simulated mass flux (Figure 11) shows that the mass flux fluctuates with time and changes rapidly at all simulated wind speeds. While the sand particles are accelerated by the wind flow, they will generate some retardatory effect on the wind. The coupling interaction between the wind flow and the sand particles in saltation layer leads to a dynamic stabilization. In all cases, the maximum, mean and minimum mass fluxes all increase with rising inlet wind speed. Note that the spread between the maximum and minimum mass flux also grows with rising wind velocity.
4.4. The Variation of Wind Velocity Profile on the Flat Bed
Figure 12shows the comparison between the free wind profile and the wind profile of saturated wind-blown sand particles at steady state, in which the solid and hollow points represent the free wind profile and that with sand particles, respectively. It can be seen that the wind profile near the surface is modified by saltating particles. This kind of coupling interaction between wind flow and sand particles leads to a “self-regulating feedback mechanism” [Owen, 1964]. The effective height of influence increases with rising inlet wind speed because more energy is transferred to sand grains from the air and the grains can jump higher. The numerical results are similar to the observation by Bagnold , who extensively measured the wind profile in the presence of saltating sand in wind tunnel. It has been previously noted from experiments that the wind velocity profile does not perfectly follow the logarithmic law in the saltation layer [Spies et al., 1995; Ni et al., 2003], which was also observed in some numerical models [Anderson and Haff, 1991; McEwan and Willetts, 1991; Shao and Li, 1999]. In this paper, this characteristic of wind velocity profile in the layer with saltating sand particles was also reproduced.
4.5. Particle Spatial Distribution on the Sand Ripples
Figure 13shows the spatial distribution of the wind-blown sand at steady state (the calculation time T = 4.5 s) over the sand ripples at Ue = 8 m/s and 10 m/s. It can be concluded that the particle concentration increases with rising wind velocity, and the height that the particles can reach also stretches. Compared with the flat bed scenarios at the same Ue (Figure 9), it can be clearly seen that the number of moving particles in the air increased, because the sand ripples not only changed the particle saltating height but also enhanced the flow field velocity, thus giving more energy to the sand particles. This is why the concentration of wind-blown sand is larger on sand ripples than on the flat bed.
4.6. Variation of Sand Flux on Sand Ripples
Figure 14compares how sand flux varies with time on the sand ripples (The sand flux on the crest of the sand ripple, X = 0.95 m) and the flat bed at the same Ue. It can be seen that the equilibration time (from beginning to the stable state) of sand ripples increased due to its higher bed form complexity. The mass flux of the stable wind-blown sand on the sand ripples is about 3–4 times more than that of the flat bed, indicating the substantial effect of the micro-topography on the wind-blown sand motion.
Figure 15 shows the variation of mass flux with height at different wind speeds on different bed forms. The mass flux pattern at the crest (X = 0.95 m) of the sand ripples (blue line) resembles that of the saltating particles on the flat bed (black line), in which the mass flux decreased exponentially with rising height. The mass flux at the trough (X = 1.0 m) of the sand ripples indicates the presence of a maximum mass flux at about 4 cm height in the leeward section. That is, the mass flux increases with rising height below this region and decreases with rising height at above this region.
4.7. Variation of Wind Velocity Profile on Sand Ripples
Figure 16shows the variation of wind velocity profile at different Ue when the wind-blown sand reaches steady state. It can be seen that the inlet wind velocity, free wind velocity and sand-coupled wind velocity have very similar patterns at the same position. Because of the sand ripples bed form, the wind speed is enhanced on the crest and a recirculation region exists in the leeward area. After the sand transport starts, the wind velocity decreases due to the drag force exerted by the particles, and the wind velocity in the saltation layer does not perfectly follow the logarithmic law. In fact, the wind velocity profile is affected by the number of saltating sand grains that are present. Larger inlet wind velocity will bring many more sand particles into the air, thus resulting in a progressively self-regulating wind profile.
 Aeolian sand transport is a complex gas–solid two-phase flow process. Our current model did not incorporate the complex momentum exchanges among the particles, the bed form and the wind, and we aim to improve our model in such direction in future work.
 In this paper, the motion of saltating wind-blown sand particles over two different bed forms is simulated and analyzed in detail. We conclude the following.
 1. The numerical simulation was validated by comparing with the experimentally measured sand flux and particle velocity. The concentration of saltating particles increases with wind velocity on both the flat bed and the sand ripples. As the inlet wind velocity increases, the particle saltation height also increases because the particles receive more energy from the air.
 2. The aeolian sand transport is a dynamic equilibrium process on both the flat bed and the sand ripples. On the flat bed, as the free wind velocity increases, the equilibration time of the aeolian sand transport process also increases. The equilibration time is affected by the bed form, and the sand ripples have a longer equilibration time than the flat bed.
 3. The mass flux of the stable wind-blown sand on the sand ripples is about 3–4 times of that on the flat bed. The mass flux at the trough of the sand ripples has a maximum value at about 4 cm height in the leeward section. Therefore sand ripples have obvious effects on the transportation of sand flux. The effects are related with the oncoming flow condition and the shape of the ripples. Meanwhile, the mass flux increases with height below this region and decreases above this region.
 4. The wind profile near the surface is modified by saltating particles and the effective height of influence increases with rising inlet wind speed. The wind velocity profile in the saltation layer does not perfectly follow the logarithmic law. The sand ripples have more complex flow field characteristics than the flat bed.
 These conclusions indicate that our model can provide the theoretical prediction of the saltation particles over both flat bed and some complex micro-topography, like the sand ripples. In order to tackle the realistic problems of geological and environmental interest, much work still remains. For example, numerical simulations of three dimensional sand saltation over dune-like forms embedded in the planetary boundary layer will no doubt take great leaps forward in the future works.
 This work was supported by National Key Basic Research Program (grant 2009CB421304), the National Creative Research Groups Science Foundation of China (11121202), and National Natural Science Foundation of China (grants 10811130470 and 40971009).