The vertical variation of particle speed and flux density in aeolian saltation: measurement and modeling

Keld R. Rasmussen1 and Michael Sørensen2 1 Department of Earth Sciences, University of Aarhus, Block 520, Ny Munkegade, DK-8000 Aarhus C, Denmark.

2 Department of Applied Mathematics and Statistics, Universitetsparken 5, DK-2100 Copenhagen, Denmark.

ABSTRACT

Particle dynamics in aeolian saltation has been studied in a boundary layer wind tunnel above beds composed of quartz grains having diameters of either 254 µm or 320 µm. The cross section of the tunnel is 600 mm × 900 mm and its thick boundary layer allows precise estimation of the fluid friction speed. Saltation is modelled using a numerical saltation model and predicted grain speeds agree fairly well with experimental results obtained from laser-Doppler anemometry. At 80 mm height the ratio between air speed and grain speed is about 1.1 and from there it increases towards the bed so that at 5 mm it is about 2.0. All grain speed profiles converge towards a common value of about 1 m/s at 2-3 mm height. Flux density profiles, measured with a laser-Doppler appear to be similar to most other density profiles measured with vertical array compartment traps, i.e. two exponential segments will fit data between heights from a few mm to 100-200 mm. The experimental flux density profiles are found to agree well with model predictions. Generally validation rates are low from 30-50 % except at the highest level of 80 mm where they approach 80 %. When flux density profiles based on the validated data are used to estimate the total mass transport rate results are in fair agreement with measured transport rates except for conditions near threshold where as much as 50 % difference is observed.

Keywords: Aeolian dynamics, saltation model, grain speed, flux density, mass transport, wind tunnel.

1 INTRODUCTION

When the wind blowing over an immobile bed of cohesionless grains becomes sufficiently strong, grains are set in motion. Grains in the size range of 100-600 µm are hopping or dancing over the surface in saltation, which is the primary mode of movement (Bagnold 1941). Saltation is an important link by which momentum is transmitted from the air to the bed through grain impact.

Several approaches have been taken to study the saltation process. In a large number of studies grain fluxes and their horizontal variation and wind profiles within the saltation layer have been measured in wind tunnels or in the field (e.g. Horikawa & Shen 1960; Williams 1964; Jensen et al 1984; Rasmussen & Mikkelsen 1991; White and Mounla 1991; Rasmussen et al. 1996; McKenna-Neuman & Maljaars 1997; Iversen & Rasmussen 1999; Namikas 2003; Liu & Dong 2004). Several mathematical models have been proposed to explain such empirical

findings, either numerical models (e.g. Sørensen 1985, Anderson & Hallet 1986, Jensen & Sørensen 1986, Anderson and Haff 1988, 1991, Werner 1990, McEwan & Willetts 1991, 1993, Shao & Li 1999, Spies & McEwan 2000, Spies el al. 2000) or analytical models that result in explicit formulae (e.g. Owen 1964, 1980; Sørensen 1991, 2004; Sauermann et al. 2001).

To verify the assumptions made in such mathematical models, direct observations are needed of fluid shear stress and particle behavior in the saltation layer, including the splash process when a grain impacts on the bed, rebounds and ejects other grains. However, the momentum transfer and the splash process take place in a very shallow layer at the air-bed interface with large grain concentration and velocity gradients where it is very difficult to make reliable measurements. Therefore existing experimental data are not made close to the surface or are made under somewhat artificial conditions. The splash process has been studied by computer simulations (e.g. Werner & Haff 1988; Anderson & Haff 1991) and experimentally in wind tunnel experiments with a controlled very small transport rate (e.g. Willetts & Rice, 1985, 1989; Rice et al. 1996) or by propelling single metal particles (Mitha et al. 1986) or sand grains (Werner, 1990) into a static bed of similar particles. Particle trajectories have been studied using high speed photography (e.g. White and Schultz 1977; Nalpanis 1985) or stroboscope (Mitha et al. 1986) that enable data on the variation of grain speed with height, but unfortunately only for small particle concentrations. In recent years, laser based methods (e.g. Phase Doppler Analyzers, PDA and laser Doppler) have become available for measuring directly particle speed in the saltation layer (Dong et al. 2002; Rasmussen & Sørensen 2005). Measurement of particle speed using optic Doppler sensors can potentially be used at higher particle concentrations nearer to the bed, and despite the fact that misinterpretation of such data is possible (Rasmussen & Sørensen, 2005), they may potentially give precise values for the variation of grain speed with height (Rasmussen 2002; Dong et al. 2002).

The objective of the present study is to gain further insight in the saltation process by combining an experimental study and theoretical modeling. Data on air and particle speed and flux density are obtained at a number of heights above the bed using laser-Doppler technology, and the numerical model of Jensen & Sørensen (1986) further developed and shown to predict the observations well. The model results are also compared to results in Rasmussen and Sørensen (2005) obtained using an approximate analytical model. .

Firstly a description of the experimental setup is given followed by a brief explanation of the mathematical model. Then data on the vertical variation of horizontal grain speed and grain flux density above beds of quarts grains having either diameter Dp=254 µm or Dp=320 µm are presented. For a range of friction speeds observed profiles of wind velocity, grain velocity, and grain flux density are compared to each other, to findings by other researchers, and to predictions made using the mathematical model.

2 INSTRUMENTS AND METHODS

The general set-up of the wind tunnel at Aarhus University is depicted in Figure 1. The working section of the tunnel is 15 m long and has a rectangular cross section of width W=0.60 m and height H=0.90 m. The tunnel is constructed from 22 mm plywood, with large glass panes inserted in the front and at three positions in the rear side of the working section. All front glass panes can

be opened a thus provide easy access to the tunnel interior. A small bell mouth followed by turbulence spires and a 3 m long replaceable array of roughness blocks provides a turbulent boundary layer in fair equilibrium with the boundary layer induced by ongoing saltation in the main part of the working section. Thus it is possible to avoid the overshoot of saltation otherwise typical for wind tunnel studies (Shao, 1993). Except for the larger cross section, the tunnel is almost similar to that described by Rasmussen & Iversen (1993). Grains are being fed into the tunnel 1 m before the end of the roughness array and caught in a 4 m wide expansion (sand collector) before the axial fan at the end of the tunnel. The speed of the air-flow can be varied continuously between zero and approximately 20 m/s. In the experiment presented here, the bed was covered by a 25 mm thick layer of uniform sand grains. During the first part of the study grains having diameter Dp = 320 µm were used while profiles of air and grain speeds together with grain flux densities were measured at three friction speeds. Then the bed was replaced by one composed of 242 µm grains, and a new set of speeds and flux densities were obtained for four friction speeds. While recording a set of speeds and grain fluxes the average mass transport was determined as the total mass of grains collected in the sand collector divided by the duration of the experiment and the effective tunnel width (550 mm). Thus the procedure is similar to the one used in earlier experiment on effective mass transport (Rasmussen and Iversen, 1999).

Air speed above the 320 µm bed was measured at 10, 20, 40, and 80 mm with a pitot-static tube connected to an electronic micro-manometer (Barocell with a 1.5 mm Hg high-resolution differential pressure cell). Above the 242 µm bed air speeds were recorded at 5, 10, 15, 20, 30, 40, 50, 60, 70, and 80 mm. However, in both cases measurements were made only after an equilibrium bed texture, i.e. a bed with a uniform pattern of wind ripples, had developed. The velocity profile was determined as the average of one set of data recorded before and one set recorded after a set of grain speeds and flux densities were measured. From data on the vertical profile of the horizontal wind speed (u) the bed shear stress can be calculated using the logarithmic wind law

0

* lny

y

k

uu = (1)

where u* is friction speed, y0 is aerodynamic roughness height, and k is von Karman’s constant. Grain speed was measured with a 1-D integrated 632.8 nm laser-optics system (LDA)

consisting of a Dantec Flowlite instrument with a Flow Velocity Analyzer (FVA) signal processor. The LDA-probe is configured with a beam separation of 38 mm and focal length of 400 mm which, according to the technical manual, results in a measuring window having height 0.2 mm and width 4.22 mm. Validation of the signal was chosen at -2 dB in order to have a reasonable signal to noise ratio. The grain velocity component in the mean flow direction, i.e. parallel to the axis of the wind tunnel was recorded with the laser head placed outside the glass panes in the tunnel front wall, and at about 3 m upwind of the sand collector. During the first part of the study, where the bed was covered by 320 µm grains, the duration of a run typically lasted from 15-60 seconds. Runs were short for measurements made close to the bed. Here the grain flux is high so that many measurements can be obtained quickly. Moreover, the local height of the sensor relative to the bed may vary when ripples move, which necessitates short runs. Longer runs were made farther away from the bed where grain concentrations are small and the influence

of ripples is negligible. Mostly 2000 to 3000 grain speed values were measured below 20 mm height; several hundreds were recorded at 20 mm and 40 mm, while only 20-100 grain speeds were recorded at higher elevations. Because analysis of the 320 µm data revealed rather large scatter, in particular in the recorded grain rates, multiple runs at the same friction speed were made during the second part of the experiment where the bed was covered by 242 µm grains. Thus more than 500 samples were mostly recorded at the uppermost level of 80 mm while from 500 to more than 10,000 measurements were recorded at lower levels. The processed data contains information about particle arrival and transit time, but only for those data where the particle speed was validated by the Dantec software.

3 MODELING GRAIN SPEED AND FLUX IN AEOLIAN SALTATION

The model of the trajectories of saltating grains used in this paper is a standard model. It is a slight modification of the model used in Jensen & Sørensen (1986). As in most other recent models, lift forces on the grains and effects of turbulent fluctuations are ignored when the grain has left the bed. The magnitude of the influence of lift forces is relatively small, see McEwan & Willetts (1993), and turbulent eddies are mainly of relevance for grains smaller than 100 µm. All sand grains are assumed to be identical. In particular, they have the same shape, so that the drag on a grain is a function only of the relative speed between the grain and the air. The drag on a grain at the relative speed w is denoted by )(wD . The grain mass is denoted by m. Thus the equations of motion are

0)(

))()((

=++−=

ywHgy

xyUwHx

ɺɺɺ

ɺɺɺ

where x and y denote the horizontal and the vertical position of the grain, respectively, D(w)/(mw)H(w) = , and U(y) is the mean wind speed at height y. The reciprocal of the quantity

H(w) has the dimension of time and can be interpreted as the response time of a grain to changes in the wind speed. The wind profile is taken to be logarithmic with friction speed and roughness height as measured during the experiments. Following Bagnold (1936) and many others we calculate the drag D(w) as the drag on an aerodynamically equivalent sphere, i.e. a sphere with the same terminal velocity of fall in air as the grains. Bagnold found the diameter of this sphere to be 0.75 times the diameter of the sand grains. We use the Schiller & Nauman (1933) formula

( )6870150124 ..)R(c RR

+=

for the drag coefficient of a sphere, where νwd.750 =R is the Reynolds number with d denoting the grain diameter and ν the kinematic viscosity of air. This formula has excellent accuracy for Reynolds numbers up to about 700, which is sufficient for aeolian saltation. It follows that

+=687.0

2123.01

5.13)(

νσµ dw

dwH

where σ is the density of the grains, and µ denotes the viscosity of air. The equations of motion were solved numerically by keeping the wind speed and H constant in small height intervals of 1 mm. This corresponds to using the explicit trajectory model in Sørensen (1991) in each of these small height intervals and is a very quick way of solving the equations.

It is assumed that (x(0),y(0))= (0,0) and that the launch velocity ( ))(y),(x 00 ɺɺ is random. The following rather simple assumptions are made about the probability distribution of the launch velocity vector. The vertical component )(y 0ɺ is taken to be gamma distributed. This distribution has the probability density function

0,)(

)(1

>Γ

=−−

wew

wpw

α

βα

βα

and its mean value is αβ . The parameter α is called the shape parameter because it determines the general shape of p(v) . The probability distribution of the vertical lift-off speed was also modeled by a gamma distribution by Anderson & Hallet (19896) and Namikas (2003). In Rasmussen & Sørensen (2005) )(y 0ɺ was taken to be exponential distributed, which is a particular case of the gamma distribution with 1=α . The generalization to the gamma distribution was made because it gives a much better fit to the observed vertical variation of the grain flux. The launch angle (relative to a vector in the flow direction) is assumed to be normal distributed and independent of )(y 0ɺ . The angle is however restricted to be positive and smaller than 150°.

For a given choice of the gamma distribution and the normal distribution that model the launch velocity of the grains, we can at any given height calculate the distribution of the horizontal component of the grain velocity and the grain flux. For a given launch velocity, we can obviously calculate the horizontal component of the grain velocity at a given height. By making a fine discretization of the launch velocity distribution and calculating a grain trajectory for each velocity vector in the discretization, we obtain a very accurate approximation to the probability distribution of the horizontal component of the grain velocity at the given height. A grain with a particular launch velocity makes two contributions to this distribution: one on the way up and one on the way down, but only if the top of the grain trajectory is above the given height. The mass of grains that per time unit move through a plane surface with unit width, perpendicular to the mean wind direction and going from height 1y to height 2y is

)y,y( 21∆φ

where φ is the mass flux of grains from the sand bed into the air, and )y,y( 21∆ is the horizontal displacement of a grain while its altitude is between the heights 1y and 2y on its way up and down. This quantity can easily be calculated using the numerical model. The bar indicates mean value with respect to the distribution of the launch velocity. In particular, the total transport rate is given by ℓφ , where ℓ is the mean jump length of a grain. For details about these calculations see Jensen & Sørensen (1982) and Sørensen (1985).

4 RESULTS AND DISCUSSION

Wind speed

Before any grain data were collected the boundary layer in the tunnel was investigated by recording profiles of wind speed between 10 mm and 200 mm height. The profiles revealed a “slight wake” region around 140 mm, while between 20 mm and 80 mm they strictly followed the “law of the wall” (White, 1991). Although not always visible, below 20 mm wind profiles are expected to be influenced by shear stress partitioning between grains and fluid so that during moderate or high mass transport, low level air speeds deviate from the law of the wall (Owen 1964, Sørensen 1985, 2004; Anderson & Haff, 1991; McEvan & Willetts 1993). Therefore the wind profiles that were collected during the first part of the investigation were only sampled between 20 mm and 80 mm above the bed. However, during the second part of the investigation data were sampled both from lower levels and with smaller vertical increment. This not only increased precision of the estimated friction speeds, but also allowed a more detailed comparison of the measured/predicted air and grain speeds.

Air speed profiles obtained for the four friction speeds at which grain dynamics were studied above the 242 µm grain bed are presented in Figure 2. Firstly, it is noticed that between 20 mm and 80 mm the data follow a log-linear profile with little scatter. Secondly, the aerodynamic roughness length (y0) increases steadily from about 10-4 m at u*=0.27 m/s (i.e. just above the saltation threshold) to about 10-3 m at u*=0.69 m/s. The observed systematic increase in y0 with increasing friction speed corresponds well to predictions by Owen (1964) as well as observations by e.g. Raupach (1991), Rasmussen et al. (1996), and McKenna Neuman and Maljaars (1997). Thirdly, it is noticed that for the three higher friction speeds, the observed air speeds in the region 5-10 mm above the bed are systematically higher than values extrapolated to the same heights from the log-linear wind profile. This probably indicates shear stress partitioning.

Grain speed

Laser-Doppler measurements In most industrial and environmental applications the laser-Doppler instrument is used to measure the velocity of particles which are considerably smaller than saltating quarts grains. Caution must therefore be exerted in the interpretation of the recorded data. Thus early in the experiment it was observed that the horizontal grain velocities recorded at any level contained a small fraction of negative velocities - even when the velocity of the airflow was large. At 40 mm height, for instance, several grains had velocities smaller than -6 m/s when the average air speed was just over 6 m/s. Although negative speeds may occur occasionally (Bagnold, 1941) it seems unphysical that such negative values will be found at relatively large distance above a soft sand bed. However, for the large sand particles it is likely that a grain will only partially penetrate the control volume of the laser beams or that particle spin or facets on grain surfaces may produce artefacts in the recorded signals. Therefore the recorded data are likely to contain some unrealistic large positive or negative speeds. This interpretation of the negative velocities as mainly erroneous is further supported by the fact that negative values are only recorded for those grains that spend the shortest time in the control volume, i.e. particles which have the lowest

transit times (Rasmussen and Sørensen, 2005). Obviously such artefacts will influence the tails of the distribution of grain speed, but since they are not extremely small, they are unlikely to bias the calculated average grain speeds much.

Another cause for concern is the existence of temporal variation of transport characteristics caused by turbulent fluctuation of the wind speed, in particular influences induced by secondary flow because of the rectangular cross section of the tunnel. However, the dimension of the tunnel limit the creation of very large eddies, and for the speeds used in this study an air parcel will typically travel through the tunnel within 3-5 s. The influence from temporal variation at 40 mm height above the 242 µm bed is illustrated by plotting (Fig. 3) the average grain velocity and flux density as a function of the number of validated grains found in each run (as a measure of the duration of the run). The duration of the runs varies from 30 s to 400 s. All runs were made under identical conditions except that a small change in the set-up caused an increase in the average speed between run 1-8 and run 9-30. The scatter of the average grain speed is about 0.5 m/s for runs with less than 1000 grains whereas it is only about half of that value when the number of grains is somewhat larger than 1000 grains. The scatter in grain flux density seems to be around 2-4×106 grains s-1 m-2 for the short runs with few grains, but it also decreases to about half or less that value when the number of grains considerably exceeds 1000 grains. Thus averages tend to stabilize as the sampling time increases, indicating temporal stability of the wind tunnel.

Variation of grain speed in the saltation layer

Grain speed is rarely measured in neither field nor laboratory experiments. However, in the present investigation we have measured profiles of the average horizontal grain speed (Vg) for grains above a 242 µm for four friction speeds (Fig. 4a), and for a 320 µm bed for 3 friction speeds (Fig. 4b). The data represent conditions in different parts of the saltation layer. Thus data recorded in the interval 5-10 mm above the bed represents the intense part of the saltation layer; those in the interval 15-20 mm are a bit above its most vigorous part, while those above 40 mm are in the upper part of the saltation layer where the grain concentration is low. The data for the lowest friction speed u*=0.272 m/s represents conditions not far above the saltation threshold, while the data for the higher values represent conditions at vigorous (u*=0.394 and 0.558 m/s) and intense transport (u*=0.685 m/s). It is interesting to see that all profiles converge towards a particle speed of about 1.3 m/s at 2-3 mm height above the bed. Like wind profiles (Bagnold 1941), grain speed profiles seem to have a focus point. The profiles increase smoothly with height except for the 0.556 m/s profile which is based on fewer measurements and therefore also expected to show more scatter. The ratio between measured air and grain speed is about 1.2 for all friction speeds at the highest elevation of 80 mm (Fig. 5). Below 80 mm the ratio gradually increases to about 2 or slightly more at 5 mm height. Close to the bed there is considerable scatter. This is to be expected because both wind speed and grain speeds are less well determined here. Moreover, because of the passing ripples, the height is also less well determined here so that the wind and grain speed measurements may have been made at slightly different heights. For the two particle sizes used here, there is no apparent influence from grain size in the ratio between air and grain speed.

The probability distribution of the grain speed was calculated for each of the heights where measurements were made and for the different friction speeds and grain diameters using the saltation model. To do so the parameters, i.e. phi (the mass flux of grains from the bed into the

air) and the parameters of the gamma distribution and the normal distribution that model the probability distribution of the launch velocity, were chosen so that the best possible fit was obtained to the observed grain fluxes and to the observed probability distributions of the horizontal velocity component of the grains. Specifically, the distance between measured and theoretical values of the mean and standard deviation of the horizontal grain velocity distribution and the distance between observed and model grain fluxes were minimized. It turned out that the best fit was in almost all cases obtained for a value of the shape parameter of the gamma distribution close to three. Therefore the shape parameter was in all cases taken to equal three. This differs from the results of Anderson & Hallet (1986) and Namikas (2003) who, using only flux data, obtained the best fit with the shape parameter equal to one (the exponential distribution). With the shape parameter fixed, the general picture was that the magnitude of the mean value of the vertical component of the launch velocity was mainly determined by the vertical variation of the grain flux, the mean launch angle was then determined essentially by the mean values of the horizontal grain speeds, and the variance of the launch angle by the variances of the horizontal grain speeds. The observed variances of the horizontal grain speeds appear to have a considerable random variation, and it is not unlikely that the actual variances vary much less with height. It was therefore not attempted to fit in detail the vertical variation of the observed variances. The quantity phi was chosen so that the sum of the observed fluxes was equal to the sum of the model fluxes. The parameters which produced the best fit to the observed grain speeds and grain flux densities are given in Table 1.

Dp=242 µm Friction speed (m/s) Number

Of grains Mean vertical

launch speed (m/s) Mean vertical launch angle

Variance on launch angle

0.274 16632 0.36 25 89 0.394 46222 0.36 20 53 0.558 17265 0.38 19 35 0.685 31978 0.38 16 30

Dp=320 µm

0.27 1708 0.40 19 25 0.47 12202 0.37 20 50 0.74 20826 0.38 20 65

Table 1. Model parameters

The estimated values of the mean vertical launch velocity are consistently close to 0.38 m/s in good accordance with the values found in Rasmussen & Sørensen (2005) and with those found by White & Schulz (1977) and Nalpanis (1985) using high speed film. Sørensen (1985) and Namikas (2003) found mean vertical lift-off speeds that were somewhat larger. The mean launch angles are smaller than those found in other studies. White & Schulz (1977) found a mean angle of 50 degrees, Nalpanis (1985) reported mean angles in the range 34 – 41 degrees, and Sørensen (1985) in the range 43 – 46 degrees. Willetts & Rice (1985) found the range 21 – 33 degrees for ricochet angles and 52 – 54 for ejected grains. Our smaller values might be due to the fact that we

have observations very close to the bed (5 mm above the bed). It should be noted that the estimated mean values of the vertical component of the launch velocity and of the launch angle do not vary much with the friction speed. This is in accordance with findings in Namikas (2003) and gives support to the arguments in Sørensen (2004) that the empirical parameters in the transport rate formula in that paper do not depend on the friction speed.

Particle diameter Dp=242 µm

Particle diameter Dp=320 µm

u* u*

Height (m)

0.274 (m/s)

0.394 (m/s)

0.558 (m/s)

0.685 (m/s)

0.27 (m/s)

0.47 (m/s)

0.74 (m/s)

0.005 1.63 1.079

- 2.14 1.562

2.41 1.607

1.93 1.532

1.89 1.312

2.07 1.418

0.01 2.02 1.005

2.32 1.079

2.61 1.176

2.85 1.192

2.23 0.978

2.26 1.209

2.49 1.127

0.02 2.55 0.973

2.96 0.997

3.39 1.053

3.70 1.003

2.82 1.037

2.88 1.059

3.18 0.994

0.03 - 3.42 1.009

3.88 0.997

4.22 0.929

- - -

0.04 3.19 0.967

3.81 1.000

4.39 0.883

4.84 0.931

3.55 0.997

3.76 0.957

4.20 0.871

0.08 4.04 0.988

4.77 0.977

5.70 0.973

6.17 0.994

4.52 0.954

4.75 0.921

5.50 0.841

Table 2. Model predictions of the mean horizontal grain speed (Vgp) in m/s and ratios between predicted and measured mean particle speeds (Vgp/Vg).

With the parameters given above the calculated mean horizontal grain speed for the two particle sizes are given in Table 2, together with ratios between predicted and measured speeds. Overall there is reasonable agreement between the values. However, below 10 mm height the model systematically predicts higher grain speed than the measured speeds, while at the highest levels the model predictions are systematically lower than measured values. Possibly this is due to deficiencies in the model, but it may also result from measurements error in the grain laden layer close to the bed.

Initially, grain speeds were also predicted assuming that the vertical lift-off speed of the grains is exponentially distributed (gamma distribution with shape parameter one). For this distribution values the tendency for overestimation at low levels and overestimation at high levels is the same, but the numerical differences between measured and estimated values were found to be larger, i.e. about 20-70 % underestimation at the lowest levels and 10 % overestimation at the highest levels.

Particle diameter - Dp=242 µm Dp=320 µm Height (mm)

u* u*

0.274 (m/s)

0.394 (m/s)

0.558 (m/s)

0.685 (m/s)

0.27 (m/s)

0.47 (m/s)

0.74 (m/s)

σ2

σp2/σ2

σ2 σp

2/σ2 σ2

σp2/σ2

σ2 σp

2/σ2 σ2

σp2/σ2

σ2 σp

2/σ2 σ2

σp2/σ2

0.5 1.11 1.018

- 1.17 0.966

1.17 1.128

0.93 0.968

1.15 1.009

1.27 1.126

1 1.22 0.943

1.18 0.983

1.29 0.891

1.42 0.937

1.15 0.809

1.28 0.914

1.49 0.960

2 1.25 0.912

1.25 0.968

1.39 0.835

1.59 0.836

1.18 0.839

1.41 0.879

1.65 0.915

3 1.33 0.932

1.41 0.851

1.65 0.83

4 1.15 0.957

1.19 1.008

1.65 0.733

1.69 0.781

1.00 1.050

1.42 0.887

1.77 0.842

8 0.94 1.213

1.08 1.056

1.27 0.884

1.61 0.714

0.59 1.763

1.47 0.844

1.50 0.933

Table 3. Observed standard errors (σ) and the ratio between predicted and observed standard errors (σp/σ) for different friction speeds and two particle sizes.

Values of the observed standard deviations (σ) of the horizontal grain speed distribution and the ratio between predicted and observed standard deviations (σp/σ).are given in Table 3 for the two grains sizes. There is much scatter in the table, which is to be expected since standard deviations are less well determined than means and are much more sensitive to erroneous extreme measurements (outliers) that we know are present. There is a clear tendency for the standard deviations to increase with the friction speed. Perhaps there is a tendency that the standard deviations are largest at the middle heights (around 4 cm). The observed variation of the standard deviation with height is not well predicted by the model because the model values vary in a much more smooth way. However, it should be noted that for the majority of the observations the discrepancy between the predicted and measured variance is less than 15 %.

The observed probability distribution of the horizontal grain speed is presented for two heights and two friction speeds for the 242 µm grains in Figure 6 together with model calculations of the same distribution. The examples plotted in Figure 6 were chosen to illustrate what happens at a high and a low wind speed and in a height with intense saltation and a height with relatively low grain concentration. Generally there is fair agreement between the observed and predicted distributions. For the low friction speed, the majority of grains have speeds between 1 and 3 m/s. While grains having speeds between -1.0 and 1.0 m/s are also quite common at the low height, these grains are rarely found at 20 mm height where the second most common group are grains with speeds between 3 and 5 m/s. Grains in the range 1-3 m/s are also the most common group at 5 mm height for the high friction speed, but at 40 mm height these grains are rare and the velocity spectrum is dominated by grains having speeds in the wider range from 3 and 7 m/s. Qualitatively the observed variation agrees with the perception that during the

splash only a limited number of grains will receive a large vertical momentum through the conversion of forward momentum of the impinging grain, while several grains will receive a relatively small momentum and make low, short jumps. The few grains that jump as high as 40 or 80 mm are all accelerated during ascent by the increasing wind speed further away from the bed. As the friction speed increases, saltating grains will receive an increasing amount of forward momentum which will increase not only the chance that more grains will be set in motion by the collision of an impinging grain, but also the chance that the ejected grains will jump higher. The result is that the distribution near the bed is not changed much while the distribution far from the bed changes much more.

We believe that observation quite close to the bed poses a series of technical difficulties, so in the present experiment data were generally not recorded below 5 mm height. However, while the majority of grains have positive speeds at both 5 mm and higher a small proportion of the measured grain speeds are negative (i.e grains having ejection angle > 90°). Even though the model allows grains to start at an angle of 150° (i.e. 60° against the wind direction), it predicts that negative grain speeds are extremely unlikely even at 5 mm above the bed. As discussed above we therefore think that most of the measured negative grain speeds are erroneous and due to grain spin, grain facets and effects of the relatively large particle diameter. The issue has been further investigated by measuring the grain velocity distribution at 3 mm height above the bed composed of 242 µm grains. For a friction speed of 0.27 m/s only 2% of the grains were found to have a negative speed so that it is probably not a serious problem.

Previous experimental studies as well as numerical modelling indicate that immediately above the bed the speed of an impinging grain is typically in the range of 2-8 m/s (e.g. Willetts and McEwan, 1991; Anderson and Haff, 1991). Comparison of the data in Fig. 4 shows that the speeds we have measured at 5 mm height are within the range of expected speeds for an impinging grain. Our data includes ejected grains too, so it is not surprising that our mean speeds are in the lower end of the interval. Experimental data on the variation of grain speed close to the bed are few, but in a recent study Dong et al. (2002) presents data on the distribution of impact speed for different grain sizes and free stream velocities. For the same particle sizes their study generally records impact speeds which are much lower than the bulk speeds found here - typically an order of magnitude. Apart from the serious problems connected with measurements very close to the bed, the difference might partly be due to the fact that close to the bed the proportion of (slow) ejected grains is much larger than 5 mm from the bed. We take the consistent behaviour of our measurements and model predictions combined with the fact that our experimental data are in fair agreement with observed particle trajectories (White & Schultz, 1976; Nalpanis, 1985) and numerical predictions (Anderson & Haff ,1991) as an indication that the grain speed data recorded with the laser-Doppler instrument are not severely influenced by particle spin, influence from large particle diameter, grain facets and other measurement errors which may significant influence results if not handled properly.

Grain flux density

Vertical profiles of mass transport are commonly recorded in aeolian research for estimation of the total sand transport by wind although this is difficult given the non-linearity of the profile and the vigorous nature of transport close to the bed (Butterfield, 1999; Butterfield, 1991; Rice et al. 1995, 1996). Vertical profiles are also commonly used for calibration of numerical saltation

models. Profiles of the raw rate of grains for which the velocity estimation was validated by the LDA- software are presented in Fig. 7a for the 242 µm grains and in Fig. 7b for the 320 µm grains. For transport rate purposes the grain flux density is found as the rate divided with the nominal area of the measuring window, and those values are typically in the range from about 1-5×106 grains s-1 m-2 at 80 mm height to 108 grains s-1 m-2 or more at 5 mm height. Also model predictions of the grain flux are plotted in Fig. 7 and are in good agreement with the measurements.

In the LDA results there are large discrepancies between the number of grains apparently encountered by the laser (attempted samples) and the number of grains for which the estimation of particle speed was successful (validated samples). The ratio between validated and attempted samples typically change from about 35% at 5 mm and about 50% at 20 mm height to about 85% at 80 mm height. To investigate to what extent this implies that we underestimate grain fluxes, the LDA results were compared to the directly observed total transport rate. To do so, we have fitted two exponential segments to the recorded flux density profiles and using this curve calculated the total mass transport for the different friction speeds and plotted the data versus measured mass transport rate (Fig. 8). Generally measured and profile based mass transport results for the 242 µm particles agree well at all heights, although the flux profile slightly underestimates the total transport rate at the high friction speed. For the 320 µm particles, the two transport rate determinations are reasonably close at the low levels where many grains have been recorded, but the values differ considerably at the higher levels where much fewer observations have been recorded. Nevertheless there is no general tendency that the profile based transport rates underestimate the directly measured transport rates, and there is no doubt that the validation level chosen in the present experiment is reasonable and the number of validated samples reflects fairly well the actual flux whereas the number of attempted samples overestimates the flux.

Generally flux density profiles sampled with vertically segmented sand traps up to 100-200 mm above the bed, show profiles with two regions where the flux density decays exponentially (Butterfield, 1999, Rasmussen and Mikkelsen, 1998). In the profiles sampled in the present investigation a single exponential curve (or power function) cannot be fitted well through the data. Two segments fit the data well, but the array of measurement points is too sparse to give information about the detailed shape of the profiles near the bed.

Extrapolation based on flux density values measured above 30 mm shows that for the 320 µm grains the horizontal flux above 250 mm height is very small, while for the 242 µm grains the flux most likely is insignificant above 150-200 mm height. Vertical profiles of flux density have been observed in several experiments, primarily in order to assess total transport rate, and the profiles recorded in the present investigation are comparable to profiles recorded in both field and laboratory investigations. Thus for 250 µm saltating grains on a beach, Namikas (2003) recorded about 95 % of the total flux below 200 mm height while (Rasmussen et al. 1985) for another beach recorded the same fraction below 150 mm. Similarly in wind tunnel studies about 95 % of the flux was measured below respectively 100 mm (McKenna Neuman and Nickling, 1994), and 150 mm (Horikawa and Shen, 1960). Contrary to this, in a wind tunnel study with particles of a diameter similar to those used here, Dong et al. (2002) found than many grains jumped considerably higher that mentioned above. Depending on free stream velocity they found that sampling must be made as high as 250-500 mm in order to record about 95 % of the total flux.

However, a fully developed saltation layer may not have formed in their experiment since they used no sand feed and saltation was measured downwind of a sand bed which was only 4 m long.

5 CONCLUSIONS

The dynamics of saltating particles has been investigated by measurements in a wind tunnel and by simulations using a numerical saltation model. The height of the boundary layer is more than 150 mm and wind profiles follow the log-linear law up to at least 80 mm. Turbulence fluctuations impede the recording of precise vertical profiles of grain speed and grain flux density and requires that about 2000 samples must be taken before a reasonably small uncertainty is achieved. The recorded profiles of average horizontal grain speed are almost log linear, but not quite. Thus the ratio between wind and grain speed is of the order of 2 at some millimetres above the bed, but decreases to approximately 1.1-1.2 above 40 mm height. For the two bed particle diameters Dp=242 µm and 320 µm investigated in the present experiment there is no particular difference in these ratios. Grain speed profiles recorded at different friction speeds seem to converge towards a focal point at about 2-4 mm above the bed somewhat similar to the convergence of wind profiles in a saltation cloud (Bagnold, 1941). Horizontal grain speeds were predicted quite well using the saltation model. Thus measured and calculated mean values differ by less than 10 % except at 5 mm height where in some cases as much as about 50 % deviation is observed. However, near the bed both our experimental data as wellas our numerical predictions differ significantly from data published recently by Dong et al (2002) who find much lower values for the horizontal grain speed. Overall measured and modelled probability distributions of grain speed are fairly similar, except for a tendency of the model to predict more grains at the dominant grain speed and fewer at the higher speeds than actually observed. At low friction speeds the majority of grains have horizontal speeds in the interval 1-3 m/s in most of the saltation cloud. However, at moderate to high friction speeds the upper part of the cloud is characterized by faster grains in the wider range 3-7 m/s. Grain rates increases with increasing friction speed and decrease with height. They are well predicted by the numerical model. The observed decrease of flux density with height can be well approximated by two regions where the flux density decays exponentially. When flux density profiles based on the validated data are used to estimate the total mass transport rate, results are in fair agreement with measured transport rates and certainly do not underestimate these.

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Figure Captions.

Figure 1 The horizontal wind tunnel with main sections indicated: 1 – entry with screen (S) and bell mouth (B); 2 – boundary layer modification with turbulence spires (T), roughness array (R) and sand feed (F); 3 – working section with laser Doppler (L) and Pitot-static tube (P); 4 – expansion box (sand collector) with screens (S); 5 –

Figure 2 Profiles of air speed recorded at four different friction speeds above a bed composed

of 242 µm quarts grains. Figure 3 Grain speed and flux density at 40 mm above a bed composed of 242 µm quarts

grains as function of number of validated samples. A change in set-up between run 1-8 and run 9-30 decreased the average grain velocity.

Figure 4 Mean Horizontal grain speed profiles above a bed of a) 242 µm quarts grains for 4

different friction speeds, and above a bed of b) 320 µm quarts grains for 3 different friction speeds.

Figure 5 Ratio between air speed and mean grain speed as a function of height for two

different grain sizes and different friction speeds. Figure 6 Predicted (grey) and measured (black) distributions of horizontal grain speed of 242

µm grains for two friction speeds and different heights. Figure 7 Profiles of horizontal grain rates measured and modelled above beds of: a) 242 µm

and b) 320 µm quarts grains. Individual symbols refer to different friction speeds. Figure 8 Mass transport calculated from the flux density profile and plotted versus measured

mass transport for the 242 µm bed.

Figur 1 - modificeres

Figur 2

Figure 3.

Figure 4

Figur 5

Figur 6

Figure 7

Figure 8 Mass transport calculated from the flux density profile and plotted versus measured mass

transport for the 242 µm bed.