A wind tunnel experiment of sand transport and its comparison with the Werner model

Abstract

[1] We carried out a wind tunnel experiment on aeolian transport of sand. Fluorescence‐dyed grains of sand were embedded in the sand bed in a wind tunnel, and their dispersion was recorded by a video camera. Dispersion of colored sand both downwind and in the crosswind direction are examined. The concentration of colored sand decreases as an exponential function of the downwind distance y, Ae^−κy, whereas the cross‐sectional distribution is approximated by a Gaussian distribution with the standard deviation proportional to . The characteristic length scale is 1000 times greater in the downwind than in the crosswind direction. A simple stochastic model is proposed to explain these findings. We also found that our experimental results conform to the assumptions of the Werner model, which reproduces aeolian dune dynamics well. We derived conditions on the parameters of the Werner model from the experimental results and suggest the applicability of the Werner model to real problems such as sandy coasts, deserts, and regions whose surface is contaminated.

1. Introduction

[2] Aeolian transport of sand has long been a practical problem in fields such as civil engineering and agricultural meteorology. It is also an intriguing example of physics of granular materials. In recent years, estimating annual transport of sand is one of the important issues from the viewpoints of desertification of arid regions, erosion of coastal dunes, expansion of contaminated areas (such as nuclear test sites), and the decrease of agricultural soil.

[3] In recent years, with the progress in the physics of granular material, computer simulation models have been employed to reproduce behavior of aeolian sand. Two models have been proposed for that purpose, one by B. T. Werner [Landry and Werner, 1994; Werner, 1995; Werner and Kocurek, 1997] and the other by H. Nishimori [Nishimori and Ouchi, 1993; Nishimori et al., 1998; Ouchi and Nishimori, 1995]. These models successfully reproduce the formation and dynamics of dunes and wind ripples.

[4] On the other hand, experiments complementary to such theoretical studies have been rare. In the present study we carry out a wind tunnel experiment and obtain quantitative data of sand transport, from which we derive the parameters in the Werner model.

[5] We first briefly review the basic processes of sand transport. There are four mechanisms in aeolian transport of sand: creeping, saltation, reptation, and avalanching [Bagnold, 1941; Livingstone and Warren, 1996]. Creeping is rolling of sand grains on surfaces. Saltation is the “hop” of a grain along the surface, propelled by wind on a long, low‐angle trajectory. Reptation is the splash of surface grains when a saltating particle hits the ground. The distance of the grain movement due to reptation is generally less than that due to saltation. Larger grains are transported through reptation. Smaller grains are transported through both saltation and reptation. An avalanche occurs when the surface slope exceeds a critical angle, about 31°.

[6] The plan of the paper is as follows. In section 2 we describe our wind tunnel experiment. In section 3 we analyze the results and fit the data to certain functions. In section 4 we relate the experimental results to the parameters of the Werner model. We summarize the present study in section 5.

2. Experimental Setup

[7] We embedded a patch of sand grains dyed fluorescent yellow in the sand bed in the wind tunnel (Figure 1). We observed the movement of the dyed sand by video cameras. The wind tunnel used is at the Institute of Meteorological Research, Tsukuba, Japan. The test section has a cross section of 0.8 × 0.8 m and is 3 m long. The wind is generated by a motor with a rotation variance of ±0.1%. The wind velocity on the axis of the tunnel, v_axis, can be adjusted from 0.1 m s⁻¹ to 12 m s⁻¹ and is measured by the rotation rate of the fan. We calibrated the rotation rate with a coupled‐thermometer anemometer. For example, 794 rpm corresponds to v_axis = 5.00 m s⁻¹. The air in the test section exits through a duct of 0.8 × 0.8 m.

Figure 1

A side view of the wind tunnel. Black curtains cover the test section during the experiments in order to cause the colored sand to glow. The size of the patch of the colored sand is 50 mm × 50 mm and 8 mm in depth. For later analyses the origin is set at the midpoint of the leeward edge of the colored sand.

[8] The sand bed on the floor of the test section is 8 mm thick and about 1.5 m long. Two digital video cameras set on the ceiling recorded the plan view image of the bed. The length of the bed is long compared with the recorded area, which was 0.83 m in the wind direction and 0.57 m in the direction transverse to the wind. We also recorded the temperature and the humidity in the room, which did not change significantly during the experiments.

[9] The sand was coarse fluvial sediment, one of the standard sands; the mean diameter is 0.2 mm, with about 0.07 mm variance. We deposited the sand on the floor of the test section and leveled it manually. We applied uniform pressure to smooth the layer so as not to compact the particles. At the entrance of the test section we tapered the bed thickness. This makes the angle to the wind decrease so that we reduce the effects of the edge of the sand bed and limit unwanted vorticity.

[10] We set a colored‐sand source (50 × 50 mm and 8 mm thick) centered at 0.6 m downwind from the entrance of the test section (Figure 1). We first removed the sand over the patch with a vacuum cleaner and then filled the vacancy with colored sand. The sand was colored by soaking the grains in a solution of yellow fluorescent paint and then drying it at room temperature for weeks.

[11] We adjusted the wind speed as follows. We increased the wind speed quickly until it reached 4.00 m s⁻¹, just below the threshold of sand movement. Then the wind speed was gradually increased at a constant rate up to a goal value. After reaching the goal value the wind speed is maintained for 30 s, and then the fan is abruptly stopped. For the analysis we used images captured 20 s after reaching the goal value.

[12] The sand transport from the colored source was then visualized as the dispersal of the yellow sand. In order to obtain a clear image, we covered the test section with black curtains. Four 40 W black lights (Toshiba FL40S BLB‐A) on the ceiling emitted near‐ultraviolet light, making the colored sand glow, which was captured by the video cameras.

[13] We processed the recorded images as follows. We merged two video images (recorded by the two cameras) with image‐processing software and measured the concentration of the colored sand. The images were recorded in red, green, and blue (RGB) brightnesses at each pixel. The RGB data are integers between 0 and 255. The brightness of the colored sand was about (R, G, B) = (100, 100, 0), whereas that of the uncolored sand was around (R, G, B) = (70, 70, 20). We found that the green brightness was best for representing the concentration of the colored sand. With these data we obtained the concentration of the colored sand on the surface of the sand bed. One pixel corresponds to 0.087 mm in the longitudinal direction and 0.080 mm in the transverse direction.

3. Analyses of Experimental Results

3.1. Concentration Distributions

[14] We now analyze the concentration profiles of the colored sand in the transverse (crosswind) direction and in the longitudinal (downwind) direction. The overall tendency of the profiles is shown in Figure 2. The longitudinal concentration decreases as an exponential function, while the transverse concentration is approximated by a Gaussian distribution.

Figure 2

A schematic distribution of the colored sand in our experiments. The longitudinal concentration is close to an exponential decay, while the transverse concentration is fitted by a Gaussian distribution.

[15] We first analyze the longitudinal distribution. Figure 3 shows the concentration distribution along the tunnel, C(y), where y is the distance from the source. We fit the data with an exponential function

where A and κ are fitting variables. Accordingly, we fit a straight line to the semilogarithmic plots and estimated A and κ by the least squares method (Figure 4). The prefactor A is determined by the intercept of the lines and the index κ by the slope. We found that κ ≃ 1.0 m⁻¹ for all experiments (0.97 m⁻¹ on average), almost independent of the wind velocity. The agreement of the estimates among the four cases is remarkably good. On the other hand, the prefactor A, which is proportional to the overall concentration, decreases rapidly as the wind velocity increases. The estimates of A and κ are discussed in detail in section 3.2.

Figure 3

Semilogarithmic plot of the longitudinal concentration distribution of the colored sand. We show the cases where the axis velocities are 10.0, 9.0, 7.0, and 6.0 m s⁻¹. The slope was determined by the least squares method.

Figure 4

Two fitting parameters defined in equation (1): (a) κ and (b) A. The parameter κ is almost constant for these axis velocities.

[16] The cross‐sectional distribution of the concentration of the colored sand, C(x), is exemplified in Figure 5. The profile becomes closer to a Gaussian distribution as the distance from the source increases. We hence fit the profiles to Gaussians. The mean, μ, and the standard deviation, σ, are plotted for each wind speed in Figure 6. Figure 7 summarizes the dependence of σ on the longitudinal distance y. The standard deviation gradually increases with distance from the source, but they are almost independent of the wind speed. The curves in Figure 7 are well fitted by a function .

Figure 5

Average concentration of the cross‐sectional distribution of the colored sand 20 s after the wind velocity reached the goal value v_axis = 10 m s⁻¹. The concentration is plotted at every quarter of the view of the video cameras.

Figure 6

Fitting the cross‐sectional distribution to the Gaussian distribution. (a) v_axis = 6 m s⁻¹, (b) v_axis = 7 m s⁻¹, (c)v_axis = 9 m s⁻¹, and (d) v_axis = 10 m s⁻¹. The horizontal axis is the distance from the leeward edge of the colored sand, and the vertical axis is the transversal location. Bold lines indicate the fitted mean, μ, of the Gaussian distribution, showing the center of cross‐sectional transport. The thin lines with bars indicate the range μ ± σ, the range of dispersion. In Figures 6c and 6d we estimated μ and σ only close to the source. This is because the overall concentration was small compared to noises in these cases, which makes fitting difficult in the leeward area.

Figure 7

Standard deviation σ of the Gaussian fitting. All four cases show similar increases in σ and are fitted by (see text).

[17] Now we compare the dispersion in both directions. The dispersion in the crosswind direction is much smaller than that in the longitudinal direction. This is shown by comparing the characteristic lengths in both directions. The characteristic length in the cross‐sectional direction is σ, while that in the longitudinal dispersion is 1/κ. The ratio (1/κ)/σ indicates how large the longitudinal dispersion is compared with the transverse dispersion. In our experiments we find (1/κ)/σ ≅ 1000 (σ ≪ 1/κ), which means that longitudinal dispersion dominates. The grains are transported in the direction of wind with very little dispersion in the crosswind direction.

[18] We stress here that the behavior is quantitatively independent of the wind velocity. The dependence on wind speed is seen only in the decrease of the prefactor A.

3.2. Explanations of Experimental Values

[19] The fitting function (1) can be related to the following transport mechanism. We assume the following: a colored particle that is jumping is deposited on the surface with a probability p on average, while the particle goes onto another jump with the probability 1 − p. The jump distance is l, on average. The probability that a particle is deposited after n jumps is then given by

Using y = nl, the concentration at y is written in the form

where

We can confirm the identity in equation (3) by taking the logarithm of the both sides (i.e., a^x ≡ e^{x ln a}).

[20] Setting the proportional constant to A, we have our fitting function (1). Thus the simple exponential decay suggests the above transport mechanism. Our experimental results indicate the relation

[21] As for the estimates of A in the longitudinal profiles, they decrease with the wind speed. We attribute this to the quantity of blown‐out sand. The meaning of A is the total amount of the colored sand captured by the cameras because

When the wind speed is high, more sand is carried away out of the wind tunnel. Naturally, the total amount of the colored sand remaining in the images would decrease. This may be the reason for the decrease in A. However, the estimates of A are not necessary in our subsequent analyses. We use only the parameter κ for the discussions of downwind transport.

[22] We next discuss the transverse profiles. In section 3.1 we showed that the profiles are fit well by Gaussian distributions, with the deviation σ increasing as . This can be explained as follows. We assume the mechanism for dispersion shown in Figure 8. The particles of colored sand jump with angles. Owing to the shear stress between an immobile particle and the jumping particle, the ejection angle has a certain range. The movement, when projected on the transverse axis, is a random walk (with, perhaps, hopping distances fluctuating somewhat). In usual cases the distributions resulting from a random walk are Gaussian distributions, with the deviation given by

where n is the number of jumps [van Kampen, 2001]. If we again assume that the jump distance in the downwind direction is l, on average, we have y = nl, and hence

The best fit in Figure 7 is given by

The added constant 1 may be due to the source length in the y direction. The fit seems to be good.

Figure 8

Stochastic model of the transverse profiles. The inset shows an enlarged view of a moving grain colliding with an immobile one. The transverse dispersion occurs due to the angle that the moving grain takes at the impact.

[23] In summary, the longitudinal concentration basically decreases as Ae^−κy, while the transverse concentration is approximated by a Gaussian distribution with the standard deviation σ proportional to .

3.3. Rouse Number

[24] As for assessing the suspended portion of the total sand, we evaluate the Rouse number. The Rouse number, Z, is a dimensionless number that indicates the ease of suspension. It is defined as

where w_s is the velocity of a particle, k is von Karman's constant (0.40), and u_* is the friction velocity. When Z > 1, suspension scarcely takes place. In our experiment, Z = 2.3 ∼ 3.9, which means that suspension is very unlikely to occur.

3.4. Sand Flux

[25] One of the purposes of this study is to calibrate the Werner model for actual problems. We use the sand flux for comparison between our experiment and the Werner model. In this section we estimate the sand flux from the experimental results. It will be compared with the sand flux obtained from the Werner model in section 4.

[26] The sand flux depends on the friction velocity. We calculate both the friction velocity and the critical friction velocity. The friction velocity, v_*, is defined as

where v(z) is the wind velocity at the height z, k is, again, von Karman's constant, and z₀ is the aerodynamic roughness length.

[27] We estimated the friction velocity v_* from the profiles v(z), measured with a hot‐wire anemometer suspended from the ceiling of the test section at 0.27 m downwind from the entrance. We glued the sand grains all over the flooring of the test section in order to remove the effects of the wind being disturbed by jumping grains and, at the same time, in order to protect the anemometer. The vertical distribution v(z) for several values of the axis velocity is shown in Figure 9. The estimates of v_* and z₀, which were obtained by plotting equation (11) in Figure 9, are given in Table 1. Incidentally, the aerodynamic roughness length z₀ varies among these cases but is consistent with previous knowledge [Blumberg and Greeley, 1993].

Figure 9

Estimation of the friction velocity in the wind tunnel for various wind speeds. The friction velocity, v_*, is estimated from the slope of the straight lines of the plot. The aerodynamic roughness length, z₀, is estimated from the intercept at v_axis = 0.

Table 1. Experimental Quantities Measured and Estimated

Fan Rotation, rpm	vaxis, m s^−1	v*, m s^−1	z0, m	κ, m^−1	A	q, kg m^−1 s^−1
1000	6.00	0.29	8.2 × 10^−5	0.97	144	0.0068
1117	7.00	0.34		0.89	143	0.013
1367	9.00	0.63		1.1	113	0.10
1500	10.00	0.72	4.9 × 10^−5	0.94	83.4	0.15

[28] The critical friction velocity, v_*c, is the friction velocity at which the sand particles start to move. It is estimated by Bagnold [1941] to be

where A′ is an empirical constant (0.1 for dry sand), ρ_p is the particle density (2.7 × 10³ kg m⁻³ for our sand), ρ_a is the air density (1.2 kg m⁻³), D is the mean grain diameter of the surface (0.2 × 10⁻³ m for our sand), and g is the acceleration due to gravity. For our sand, this yields v_*c ≅ 0.21 m s⁻¹. Experimentally, the sand particles started to move for v_axis ≅ 5 m s⁻¹, or v_* ≅ 0.25 m s⁻¹, which is consistent with this estimate.

[29] The specific sand flux, q, the mass of sand transported per unit width per unit time, has been employed by many scientists [e.g., Bagnold, 1941; Lancaster, 1981, 1985; Hsu, 1971]. We choose R. Kawamura's formula because empirical parameters may be dependent on the location and/or sand material; his experiments were all done in Japan [Kawamura, 1951]. In this formula the sand flux q is given by

where K is a dimensionless empirical constant 2.78. The calculated q is shown in Table 1. In section 4 we compare this q with that derived from the Werner model.

4. Calibration of the Werner Model

[30] In this section we show that the results of our experiment conform to the Werner model. We then discuss how the parameters in the Werner model are constrained by the experimental results. We use the sand flux, q, in order to find the parameters of the Werner model. We equate the estimate of the sand flux obtained in section 3.4 to the sand flux calculated directly from the Werner model.

4.1. Werner Model

[31] Now we review the two‐dimensional sand transport model proposed by B. T. Werner [Werner, 1995]. The model reproduces specific dynamics of sand such as wind ripples and dunes [Landry and Werner, 1994; Hatano and Hatano, 1999, 2001]. In this model, dunes are built of sand slabs of size L stacked on a two‐dimensional lattice. The height of the dune is given by the number of sand slabs at the site.

[32] Figure 10 shows the operations in a single time step δt. At the beginning of a time step we choose a slab randomly from all the slabs on the surface. The slab is transported by “wind” in a specific direction over a specific distance. It is called a “hop.” The hopping distance is given by a randomly chosen two‐dimensional vector (m_x, m_y). Here we suppose that the wind blows in the y direction, on average. Therefore the average of m_x is put to zero, while the average of m_y has a finite value m. After one hop the slab is deposited with the deposition probability p; otherwise, with probability 1 − p, the slab hops again.

Figure 10

Procedures of the Werner model. (a) A top view. The field is divided by the grid of size L. The shaded grid indicates a specific slab of sand. The wind, whose direction is (m_x, m_y), transports the slab to the lightly shaded location. In there, the slab is determined by a specific probability, whether it is deposited there or transported farther. (b) A side view of the same process. (c) Definition of the shadow zone. Slabs inside the angle of repose are called the shadow zone. Wind does not pick up the slabs in this zone. (d) An avalanche. When the slabs exceed the critical value ΔH_c in their difference of height, an avalanche occurs. The value of ΔH_c is close to (but not larger than) the angle of repose.

[33] The deposition probability, p, depends on the materials underneath (Figure 10b); if there is sand on the site, the moving slab is more likely to be deposited with the probability p = p_sand. On the other hand, if the site is a bare pavement surface, the slab is more likely to hop again (probability p = p_soil < p_sand).

[34] A region called the “shadow zone” is defined as in Figure 10c. A slab in a shadow zone is not picked up by wind. Once a moving slab is dropped in a shadow zone, it is always deposited at that location.

[35] Avalanching is also taken into account (Figure 10d). If, after deposition of a slab, the height at a site becomes higher by ΔH_c than any of the four neighboring sites, the slab at the top is rolled down to the lowest neighboring site. The hopping and avalanching operations constitute a time step δt.

[36] The Werner model reproduces various sand dunes, such as barchans and linear dunes, as well as their evolution to another form. It is also consistent with observations, such as that barchans are formed in sand‐limited areas. If we wish to put the Werner model to practical use, such as for coastal management, we need to know how the model parameters are related to constants in actual systems. We use our experimental results to estimate these parameters for applications to real coasts or deserts.

4.2. Parameter Estimation

[37] We wish to estimate the following parameters in the Werner model: deposition probability on the sand surface, p_sand; the slab size, L; a single hopping distance, m, in the direction of the wind (in units of L); the time step, δt; and the slab height, h. In order to estimate these parameters, we equate the semiempirical estimates of sand fluxes (obtained in section 3.4) and the Werner formula for sand flux.

[38] In our experiments we fix u = 0 and disregard p_soil. In other words, we assume steady wind direction and p = p_sand everywhere, thereby ignoring the case of bare (nonsandy) substrates.

[39] First, we use the fact that the behavior equation (1) of sand distribution conforms to the Werner model. According to the model, a sand slab is deposited on the ground after n hops (over the distance y = nmL; m is the distance of a single hop, and L is the size of a slab) with probability (1 − p_sand)ⁿ⁻¹p_sand. This mechanism is similar to the one discussed in section 3.2. Thus the concentration of the colored sand should decrease as

Using the estimate κ ≅ 1.0, we have an equation for the quantity mL:

[40] Second, we calculate the theoretical sand flux of the Werner model, q_Werner:

where ρ_bulk = 1.4 × 10³ kg m⁻³ is the bulk density of the sand bed. The mean number of hops is given by

Here, we used

Summing both sides over n, we have

To obtain equation (20), we used the following expansion of a series a_n = a₀b^{n − 1}:

Equation (21) follows from equation (20) after straightforward differentiation. Therefore we have

Equating the experimental estimate of q (Table 1) with q_Werner, we have another equation for mL:

[41] In order to estimate δt/h in equation (23), we carried out the following additional experiment. We prepared the same sand bed, except that the thickness of the colored sand is now 4 mm. Then we started the fan of the wind tunnel and measured the time necessary to remove all the colored sand. We regard δt as the period until the colored sand disappeared from the initial location and h as the height of the colored sand. We tried two axis velocities, 8 m s⁻¹ and 10 m s⁻¹. The estimates δt/h are shown in Table 2. Using the estimates in the case of the axis velocity 10 m s⁻¹, we have

Thus we have two equations between mL and p_sand which may be solved simultaneously.

Table 2. Estimates of the Values of δt/h

vaxis, m s^−1	δt/h, s m^−1
8.00	2.75 × 10^5
10.00	3.03 × 10^5

[42] Figure 11 shows the plots of equations (15) and (24). The crossing point of the two curves gives

In this way we map our experimental result to the Werner model. We found that a sand slab is highly probable to hop (high value of p_sand), and a single travel distance before deposition is estimated to be 3.1 m. (This is the case under the assumption that the sand transport can be modeled by the moves of such slabs. It means that the experimental results correspond to the Werner model when p_sand = 0.96 and mL = 3.1, not meaning that these slabs would jump for such a distance in the real world.) These estimates are for the case v_axis = 10 m s⁻¹, but the procedure is the same in the other cases. The difference is that we use the values of q_exp for the relevant wind speeds.

Figure 11

Estimation of p_sand and mL for v_axis = 10 m s⁻¹. Equations (15) and (24), both described by mL and p_sand, are plotted in the same graph. The crossing point is the solution for mL and p_sand.

[43] Finally, we perform Werner's model simulation for demonstration, using p_sand = 0.96 and mL = 3.1 m. The deposition probability p_soil is set to be the same as p_sand. The values of m and L are arbitrary under the condition ml = 3.1, and we here assume m = 2 and L = 1.55 m. Figure 12 shows a snapshot of the simulation. Two barchan dunes are formed. The length and the width of the barchan are approximately 100 L = 155 m. The size of barchans can be changed by varying the values of m and L. Using the procedure presented in this study, we can connect the parameters of the Werner model to the dunes of the real world.

Figure 12

Simulation result of the Werner model using p_sand = 0.96 and mL = 3.1.

5. Conclusions

[44] From the wind tunnel experiment we found that the sand grains transported downwind decrease in concentration as an exponential function of downwind distance, with a little dispersion in the crosswind direction that is approximated by a Gaussian distribution. We gave a simple explanation for such distributions.

[45] We attempted to map our experimental results to Werner's model, in which a slab of sand jumps with a specific probability. We hope such computer models of sand transport will be more commonly applied to more practical problems, such as simulating the dunes in deserts [Hatano and Hatano, 2001, 1999] or on ocean coasts [Udo, 2003].

Acknowledgments

[46] Y. H. is grateful to the Institute of Meteorological Research for discussions, for its hospitality, and for permission to use the wind tunnel. We also thank the editor and the reviewers for their critical reading of the manuscript and for their useful comments.