Visualization and LASER measurements on flow field and sand movement on sand dune

Visualization and LASER measurements on flow field and sand movement on sand dune Yusuke Sakamoto¹, Daisuke Aoshima¹, Itsuki Nakamura¹, Takahiro Tsukahara¹, Makoto Yamamoto¹, Yasuo Kawaguchi¹ Department of Mechanical Engineering, Tokyo University of Science, Chiba, Japan, yasuo@rs.noda.tus.ac.jp Abstract The installation of obstacles around sand dune is one of the promising methods to suppress sand movement. We focus on the effect of a small fence installed on a sand dune to clarify the mechanism of the sand movement. A model dune was installed in a horizontal wind tunnel. The flow field around the dune and the process of sand movement were measured using some laser systems, such as LDV, PIV and laser-sheet visualization. In our previous study, the effects of fence height and position on a dune erosion were investigated. A non-porous fence suppressed sand movements upstream of fence, but enhanced if downstream of the fence. The intensive erosion downstream of the fence was caused by separated shere flow from the edge of the fence. In the present study, the influence on flow and erosion by fence porosity was discussed. Four types of porosity were tested:, 1, 3, 5% open area. The shapes of eroded sand dunes were founded to depend on the porosity rate. The relationship between the sand dune erosion and the flow field around the dune was described with schematic diagrams. The flow around a fence on a flat plane was measured by PIV. It can be seen the scale and number of vertices from the edge of the fence were significantly different depending on the fence porosity. The critical friction velocity, which was estimated from the equation proposed by Shimazu et al. (28) and the velocity along the plane surface were compared. According to the present results, the porous fence is effective to suppress the sand movement downstream of the fence. It is found that the effective fence porosity for dune erosion depends on the fence condition. 1. Introduction The dynamics of sand particle in atmosphere have been investigated for many years with the aim of combating the desertification. The desertification is one of serious world-wide environmental problems. One of the main causes of the desertification is wind-blown sand movements in desert. This phenomenon occurs as a result of complicated combinations of various conditions such as sand-dune surface, ambient air flow, sand particles, and so on. This phenomenon is supposed to be the most significant source which gives rise to sand storm and yellow dust, and it finally induces more desertification, health damage and aerial pollution. Hence it is important to clarify the mechanism of the wind-blown sand movement. Many researchers have studied on the dynamics of sand dunes. For example, Bagnold (1956) classified types of sand movement in atmosphere into three: saltation, suspension, and surface creep. In their study, a critical friction velocity was introduced as threshold of an air-flow velocity about sand movement. When an air-flow velocity becomes higher than the critical friction velocity, sand particles start to move. Walker & Nickling (23) measured flow field behind a dune with numerical calculations. Andreotti et al. (22) investigated surface waves on the barchans, which were generated by dune collisions and changes in wind direction. In desert, the installation of obstacles around sand dunes is a promising method to suppress sand movements. The sand movement depends on the flow conditions of sand surface, the air-flow velocity and its fluctuation near the sand bed. Therefore, an obstacle installed on the sand bed can be expected to play an important role in the control of the flow field near the sand-dune surface. For example, sandbags, paving, planting, and fences have been proposed as the obstacles. Sakamoto et al. (25) researched the effect of a wind-break fence, which was framed with log on the top of an artificial embankment. Li et al. (22) used the method of straw checkerboards and planting xerophytic shrubs to fix sand dunes. There also exist some researches on effects of obstacles, such - 1 -

as a fence, on sand movements in desert. The validity of them has been confirmed, but a mechanism of this phenomenon has not been completely analyzed. Kim & Patel (2) and Lee and co-workers (1998, 21, 22) investigated the effect of a porous wind fence, the height of which was the same level as that of a relevant dune, on the wind erosion of the dune. However it is practically difficult to apply such a fence to real scale, since an actual sand-dune height is often over 3 m. Hence, we have studied the erosion of a sand dune, on which a small fence is installed. In our previous study, we investigated the effect of a small fence on wind-blown sand movements on a sand dune. The process of erosion was found to depend on the position and height of the fence. The dune erosion was suppressed in the upstream of the fence, but enhanced in the downstream of the fence. When a non-porous fence was installed at the top of the dune, sand movement was hardly occurred. On the other hand, the sand dune with a fence installed at the toe of the dune was collapsed drastically. It can be believed that the complicated flow behind the fence is accompanied by strong turbulent motions with a separated shear layer as well as a turbulent boundary layer and that the turbulent flow field should have close relation with the sand erosion. Thus it is important to suppress the separated flow, which induces high turbulent intensity. If the sand movement downstream of a fence is suppressed, the fence installed at the toe of the dune becomes more effective to avoid the strong erosion of the dune. In this study, we employed a porous fence to suppress the separated flow from the fence edge. The effect of porosity rate on the flow field around the fence and sand movement was also studied. To analyze the relationship between flow field and sand movement, a process of sand-dune erosion and a flow field around the fence were investigated with laser systems: i.e. laser-sheet visualization, laser-doppler velocimetry (LDV), and particle image velocimetry (PIV). 2. Experimental set up The experimental set-up is shown in Fig. 1. The blower and rectification part was 2.91 m long, and its internal dimensions at the outlet were 25 25 mm 2. The maximum air velocity was 2 m/s, and its minimum relative turbulence intensity was.5%. The developing section was 1. m long, and the test section was 3. m long. The spanwise width of each section was 25 mm, and the upper surface of both sections was movable. The developing section in the upstream of a test section provides variety of turbulent intensity and scale by setting several roughness blocks on the bottom plate. The test section was equipped with the measuring windows for illuminating laser beams and taking pictures. A model dune was installed at 5 mm downstream from the entrance of the test section. For the observation of sand deposition and erosion, the model dune itself was made of sand. The nominal mean sand-particle diameter was 115 µm, which was categorized into the saltation type of sand movement under the present range of the wind velocity. The initial shape of model dune was triangular prism as shown in Fig. 1. This shape was chosen as a typical shape of an actual dune in desert. Owing to the temporal change in the dune shape, it was essential that another non-erosive model dune should be applied to investigate the flow field around the dune. The other model dune was build with metallic boards for measuring mean velocities and turbulent intensities around the model dune. The coordinate parallel to the upstream-side dune surface was named x l. The fence height was 2mm, which corresponded to 1/4 of the height of the dune, and was installed at only x l = 1 mm. We tested four types of porous fences, which had different porosities: ε = % (nonporous fence), 1%, 3%, and 5%. These fences had a geometric porosity, and the porosity rate was defined based on the area ratio of the pore of the fence. The porosity rates and the pore diameters of the fences we used are listed in Table 1. The approaching mean air velocity was fixed at 7.1 m/s. - 2 -

y Fence 8 mm x l x Model dune (Sand or metal) 15 3 424 mm y x Blower and rectification part Developing section 1 mm Test section 3 mm Fig. 1 Experimental set up. Fig. 2 Measurement points for LDV. Table 1: Specification of porous fence Porosity ε Pore diameter d Pitch p 1% 1.8 mm 3% 3.1 mm 5% 4. mm 5. mm The air-flow velocity around the dune surface was measured by laser-doppler velocimetry (LDV) system. The two-color four-beam LDV system (Dantec dynamics, Co.) was employed in the back-scatter mode with a 3 mm focal-length lens and equipped with a 1. W argon-ion laser (Coherent, Co.: INNOVA 38C). Table 2 shows the specification of the LDV system in detail. The oil mist with a mean diameter of 1~3 µm was employed as tracer particles. The LDV probe was fixed on a two-dimensional traverse system. Note that LDV measurement was made only for the metallic prism (without erosion), which simulates the initial stage of the model sand dune. Figure 2 shows the measurement points around the model dune. Two kinds of arrangements of points were applied: one of them was in a wide area above the model dune, and the other a narrow area but with a fine resolution only behind a fence. It is known that a reattachment point in a separate flow behind a fence oscillates periodically. The present data were averaged for 1 seconds to obtain a temporalaveraged velocity. The velocity and turbulent intensity were calculated by a burst spectrum analyzer (Dantec dynamics, Co.: 57N21 BSA enhanced master). The shape of the dune surface was obtained by laser-sheet visualization technique. In this method, a laser sheet irradiates the sand-dune surface so that the measurement line can be detected, as illustrated in Fig. 3. The laser-sheet thickness and its spread angle are 5 mm and 4, respectively. Table 3 shows the specification of the visualization system in detail. Figure 4 displays a typical snapshot of the irradiated sand dune at the beginning of flow, in which the blue line was measured. The initial shape of the sand dune is shown by a dotted line. Such pictures were taken during experiments with blowing wind, and local heights of dune surface were drawn from the measurement lines. These pictures were taken by the single-lens reflex camera (Nikon, Co.: D6). - 3 -

y x Laser sheet Camera z Dune Fig. 3 Schematic illustration of the experimental set-up of dune visualization. Fig. 4 Sand dune irradiated by laser sheet. Table 2 Specification of LDV system. Model INNOVA 38 C Output (maximum) 5.23 W Output (as use) 1. W Beam diameter 1.8 mm Wave length 488 nm, 514.5 nm Focal length 31 mm Beam intersection angle 6.99 Measurement volume diameter 91 µm Measurement volume length 1.49 mm Table 3 S system. Inpu Spre Thickn The effective pixels of this camera was 1.2 million. The time from the beginning of each test is denoted as t (t =.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 min.). Before the blowing wind with an aimed velocity (t < ), a wind speed was well below a threshold value, at which the wind first starts to move the sand grains. At low wind speeds, no sand movement was observed. In order to obtain the detailed flow field influenced by a fence, a two-dimensional measurement of flow field around the fence on flat plane was conducted with a two-frame PIV system, as shown in Fig. 5. Table 4 shows the specification of the PIV system in detail. The oil mist was employed as tracer particles. The PIV system consisted of a double-pulse laser, laser-sheet optics arrangement, CCD camera, synchronizer, and computer for image-sampling and processing. The double-pulse laser (Litron Lasers, Co.: Nano S 65-15) was a combination of a pair of Nd:YAG lasers, each having an output of 4 mj/pulse and the wavelength of 532 nm. The pulse interval was set at 1 µs. The system was able to obtain 5 sets of images per second. The laser sheet thickness can be modified by adjusting the combination of cylindrical lenses. In this experiment, the laser-sheet thickness and the spread angle were set to be mm and 2, respectively. The CCD camera had a resolution of 248 248 pixels and its pixel pitch was of 7.4 7.4 µm. The camera lens was a 5 mm focal length and an aperture of 2.8. The picture frame size is 18 18 mm 2. The sets of two pictures were analyzed with software (Dantec dynamics, Co.: Dynamic Studio). Incorrect vectors were reject and replaced through the following process. First, vectors, which were outside a certain range, were rejected. This range was determined from two components velocities and length of vectors in the flow condition. Next, the blank points were replaced by vector estimated from surrounding vectors. In this experiment, a total of 6 instantaneous velocity vector fields were acquired at each condition. The mean velocity fields were obtained by ensemble averaging the 6-4 -

Table 4 Specification of PIV system. Model Nano S 65-15 Laser Nd: YAG double pulse Output (maximum) 4 mj / pulse Flow Laser sheet Wave length Pulse interval 532 nm 1 µm Measurement area Pulse duration 5 ns Spread angle 2 y Fence Camera resolution Focal length 248 248 pixels 5 mm x Bottom plane of wind tunnel Fig. 5 Measurement area for PIV. Fig. 6 Flow field around dune: (a), (b), distribution of mean streamwise velocity; (c), (d), distribution of streamwise turbulent intencity. (a), (c) and (b), (d) non-porous fence and porous fence (e = 3%), respectively. instantaneous velocity vector fields on the grid points (63 63 grids). 3. Result and discussion Flow Field around Dune First, we present the measured air-flow field around the model dune with a fence and discuss the difference between a non-porous fence and a porous fence (ε = 3%). Figure 6 shows the distributions of the two quantities for each condition in the fence porosity: the streamwise mean velocity (U) in the left column of the figure; the streamwise root-mean-square (RMS) values (Urms) in the right column. The fence was installed at 1 mm from the toe of the dune (x l = 1 mm). When the non-porous fence was installed, there existed a reverse flow behind the fence, see Fig. 6(a). In such a flow, sand which were carried from downstream side accumulated near the fence of - 5 -

y/h 1.4.2 1 2 3 4 x (mm) y/h 1 min 5 min.4.5 min 1 min 6 min 7 min.2 2 min 3 min 8 min 9 min 1 4 min 1 min 2 x (mm) 3 4 Fig. 7 Temporal evolution of dune surface with non-porous fence. Fig. 8 Temporal evolution of dune surface with porous fence (ε = 3%). y/h 1.4.2 Initial shape ε = % ε = 1% ε = 3% ε = 5% 1 2 3 4 x (mm) Fig. 9 Shapes of dune surface with fence at 1min. after from the beginning of air flow. the reverse-flow area. The downstream of the reverse-flow area, the fast favorable flow came over the fence existed. The flow velocity was enough to move sand away. Along border between the reverse flow and the favorable flow, there was a high turbulent intensity region by separated shear flow from edge of the fence. The region spread downstream along the dune surface. On the other hand, bottom counters show the flow field around dune installed a porous fence (ε = 3%). In this condition, the value and area of turbulent intensity of the shear separated flow was small. Especially the value along the dune surface was remarkably small compared with the case of nonporous fence. There wasn t a reverse flow and a calm slow flow existed downstream of the fence. It should be noted that there existed a high turbulent intensity area just behind the fence. According to these features, it seems that the process of dune erosion change with fence porosity. Dune-Erosion Process We measured the processes of erosion, i.e., temporal deformation of the dune shape, with emphasis on the influence of the installed fence. We employed four types of porous fences, which had different porosities: from % (normal plate without permeability) to 5%. At first, we will discuss the influence of the fence porosity rate ε on the process of the sand-dune erosion. Figures 7and 8 show typical evolutions of a sand dune in two cases, where a fence of a normal plate (ε = %) or a porosity fence (ε = 3%) was located at x l = 1 mm, until 1 minutes - 6 -

High turbulent intensity region Main flow Initial dune surface Fence Area3 (a) Area1 Area2 Eroded surface Flow through fence Main flow Fence Area4 (b) Area2 Calm region Area3 Fig. 1 Schematic diagram of the flow field in the downstream of a fence: (a) non-porous fence, (b) 3% porosity fence..42 Fig. 11 Mean stream lines and distribution of U rms around fence on flat plane: (a) non-porous fence, (b) ε = 1%, (c) ε = 3%, (d) ε = 5%. after from the beginning of the test. When the non-porous fence was applied, the top of the dune was subjected to intensive erosion, as given in Fig. 7. In the area of downstream of the fence, the sand dune was eroded constantly in the first few minutes. However, when the top of the sand dune became low as the height of the fence, the erosion had calmed down. At 1 min from the beginning of the test, the dune took the form of a trapezoid. On the other hand, when the 3% porosity fence was installed, the erosion around the top of dune was mild, and thereby the height of the sand dune at 1 min. is higher than that for non-porous one. There was almost no change for more than 5 min. after the beginning of the test. The sand movements are extremely intensive just behind the fence as well as the top of the dune. As previously described, the process of sand-dune erosion depends on the conditions of the installed fence. In Figure 9, the shapes of the eroded surface with different porosity fences are compared. Although a sedimentation area was found close behind the non-porous fence, the sand bed around - 7 -

Fig. 12 Stream lines of instantaneous flow field around a fence: (a) non-porous fence, (b) ε = 1%, (c) ε = 3%, (d) ε = 5%. the porous fence was found to be more eroded as the porosity rate was increased. However, the height of the remaining sand dune at 1 min. was high as the porosity rate was increased. It can be supposed that these characteristic shapes of dune surface were influenced by the air flow around the dune. These shapes of dune surface were different each other. These different features depended on whether the fence had porosity or not. According to the distribution of U and U rms, these features are schematically depicted in Fig.1, which presents a diagram of the deformed dune surface. In this figure, the black solid and dot lines show the initial dune surface and the eroded one, respectively. It seems that the flow condition along the dune surface should affect the sand movements. A significant erosion can be observed at the surface far downstream from the fence, where a highspeed air flow induces sand movements because its velocity is higher than a critical friction velocity. Moreover, another strongly eroded surface can be found below the region (shaded area in the figure), where a high turbulent intensity makes the critical friction velocity small. In such a case, sand particles moved away even in the low-velocity air flow. For instance, the area just behind a porous fence was eroded remarkably under the low magnitude of U. As can be seen in Fig. 1, the dune erosion is apparently influenced by whether or not a fence has the porosity. The flow field around the fence and its dependency on the porosity rate will be discussed in the following section. Flow Field around Fence on Flat Plane T he flow field around a fence on a flat plane was measured using PIV. Here, four porosity rates (ε = %, 1%, 3%, 5%) for the fence were tested. Figure 11 shows the time-averaged flow fields. The horizontal axis (x) is the streamwise distance in the measurement area, and the vertical axis (y) is the wall-normal distance normalized by the height of the fence. The contour shows the distribution of the streamwise turbulent intensity U rms. The solid lines with vectors indicate the streamlines. In the case of the non-porous fence, the separated flow from the fence edge has large values of U rms and it continually grows out of the measurement area. The separated flow became weak and small as the porosity rate was increased. In contrast, the value of U rms just behind fence is large, especially - 8 -

velocity.4.2 U * U *ct /(d p g) 1/2 velocity.2.4 1 x/h f.4.2 U * U *ct /(d p g) 1/2.2.4 1 x/h f.4.2.2.4 1 x/h f (a) (b) (c) (d) velocity U * U *ct /(d p g) 1/2 velocity.4.2 U * U *ct /(d p g) 1/2.2.4 1 x/h f Fig. 13 Relationship between critical friction velocity and streamwise velocity along bottom surface: (a) non-porous fence, (b) ε = 1%, (c) ε = 3%, (d) ε = 5%. for ε = 3% and 5%. This is attributed to a grid turbulence, i.e., separated flows from apertures of the porous fence. Although the turbulent intensity became large with increasing the porosity rate, an area downstream of the high turbulent intensity region was calm comparatively. In Fig. 11(a), a reverse flow existed just behind the fence, which was induced by the separated flow. Okamoto et al. (26) performed a LDV measurement on the flow field around a fence and their obtained distribution of streamlines is consistent with our result. A secondary vortex just behind the foot of the fence is also found in Fig. 11(a). This small vortex is induced by the reverse flow. For the case of the porous fence, the wall-normal velocity around the fence edge became small as the porosity rate was increased. This tendency represents the fluid upstream of the fence through the porous fence. The reverse flow behind the non-porous fence contracted with increasing the porosity, and it finally disappeared. Figure 12 shows the instantaneous velocity field in the same region with that of Fig. 11. The number of streamlines from the left side of the measurement area is same in each case. For the case of the non-porous fence, the density of stream lines was high above the fence, implying the velocity was high there. As the porosity was increased, the velocity gradient became small. Hence, some large-scale eddies were found to occur in the downstream of the porous fence. When the 1% or 3% porous fence was installed, the vortices just behind the fence were few, and the reverse flow behind the fence became small. In the case of the 5% porosity fence, there was no reverse-flow region. Therefore, the flow behind the fence seemed to be rectified by the porous fence. The difference between the flow with a non-porous fence and that with a porous one is obviously significant. It seems that the shape of sand dune surface was influenced by these flow fields around the fence. Critical Friction Velocity Behind Fence In this section, we discuss a relation between the sand movement and the velocity along the dune surface. As mentioned in the introduction, a threshold of the air-flow velocity about sand movements was firstly introduced by Bagnold (1956), and it is called a critical friction velocity. His definition does not consider the influence of the velocity fluctuation on the critical friction velocity. Shimazu et al. (28) had carried out experiments, in which the diameter of test sand grains, the moisture contained in the sand bed, and the turbulence statistics of air flows were measured, and had clarified their effects on the critical friction velocity. In general, the sand particle starts to move under the flow condition of the higher velocity than the critical friction velocity. They clarified that, when the magnitude of the wall-normal velocity fluctuation increased, the sand particles were moved by lower air-flow velocity because of the decreased critical friction velocity. According to their work, the critical friction velocity U *ct is determined as following for particles - 9 -

larger than 1 µm, 1.1 1 d * ' ct p u v' = 2 p g H d p g d p 4. Conclusion 1.2.4 U Fc β. (1) d g τ w The value of β is 1.5 1-1. The capillary force Fc is assumed to be constant in this study. In Fig. 13, the critical friction velocity, which is obtained from Eq. (1), and the measured velocity at 1.6 mm height above are compared the flat plane. The horizontal axis is the distance from the fence along the flat plane. In an area where U > U *ct, sand particles move away. For the cases of the non-porous and the 1%-porosity fence, U *ct is larger than U throughout this measurement area. Hence the sand movement hardly occurred. When the 3%-porosity fence was installed, there exists an area where sand movement occurs because of U > U *ct. In the case of the 5%-porosity fence, a difference of these two velocities is much larger than that for the 3% case, so that the mass flux of sand movement is large. These results are in consistent with the shape of dune surface just behind the fence (see, for instance, Fig. 9). The dune surface just behind the non-porous fence gave rise to a sedimentation area. In the case of the 1%-porosity fence, the dune surface behind the fence was eroded, but there remained the tendency of the sedimentation. For 3% and 5% porosity, the intensive erosion area just behind the fence corresponds to the area where U > U *ct. According to these results, the relation between the flow field and the shape of dune surface around the fence can be explained with the Eq. (1) qualitatively. These results conform to the tendency of the eroded sand dune surface. The erosion of dune surface just behind fence became intensive as the fence porosity became large. We investigated the effect of a porous fence, which was installed on an upstream surface of a model dune, on the dune erosion and the relationship between the erosion and flow field around the dune. The process of erosion was measured by the visualization using a laser sheet. The flow field around the dune and the fence on flat plane were measured by LDV and PIV systems, respectively. It is found that the porous fence was effective to depress the separated flow from the edge of the fence. However, the decelerated air flow through the porous fence had high turbulent intensity just behind fence. Hence, there was intensive erosion area just behind fence, while at the top of the dune the erosion was avoided. This tendency is more remarkable as the porosity is increased. The flow field around a fence was different in each porosity rate. We confirmed that the empirical correlation of the critical friction velocity, proposed by Shimazu et al. (28), can be applied to sand movements influenced by a fence. 5. References B. Andreotti, P. Claudin, and S. Douady. Selection of dune shapes and velocities Part 1: Dynamics of sand, wind and barchans. European Physical Journal B, 28: 321-339, 22a. B. Andreotti, P. Claudin, and S. Douady. Selection of dune shapes and velocities Part 2: A twodimensional modeling. European Physical Journal B, 28: 341-352, 22b. R. A. Bagnold. The Physics of blown sand and desert dunes. Dover publications, Inc., 1956. H. G. Kim and V. C. Patel, Test of turbulence models for wind flow over terrain with separation and recirculation. Boundary-Layer Meteorology, 94: 5-21, 2. - 1 -

S. J. Lee and H. B. Kim. Velocity field measurements of flow around a triangular prism behind a porous fence. Journal of Wind Engineering and Industrial Aerodynamics, 77-78: 521-53, 1998. S. J. Lee and H. C. Lim. A numerical study on flow around a triangular prism located behind a porous fence. Fluid Dynamics Research, 28: 29-221, 21. S. J. Lee, K. C. Park, and C. W. Park, Wind tunnel observations about the shelter effect of porous fences on the sand particle movements. Atmospheric Environment, 36: 1453-1463, 22. X. R. Li, X. P. Wang, T. Li, J. G. Zhang, Microbiotic soil crust and its effect on vegetation and habitat on artificially stabilized desert dunes in Tengger Desert, North China. Biol Fertil Soils, 35: 147-154, 22 S. Okamoto, A. Shimane, N. Kubota, Flow around Two Perforated Plates Placed Perpendicularly on a Ground Plane, Transactions of the Japan Society of Mechanical Engineers. B 72:717:1197-125,26. T. Sakamoto, Y. Ishida, and H. Hagino, Wind tunnel study on the effect of a wind break fence constructed on an embankment. Journal of the Japan Society of Erosion Control Engineering, 58: 49-53, 25. Y. Sakamoto, S. Shimazu, T. Tsukahara, M. Yamamoto and Y. Kawaguchi. Control of sand movement on model dune by fence installation experimental study using LDV. Turbulence, Heat and Mass transfer, 6 (eds. K. Hanjalić et al.), Begell House Inc., 91-94, 29. S. Shimazu, M. Sakai, T, Kimura, M. Yamamoto and Y. Kawaguchi. Experimental investigation of the critical friction velocity for sand movement from the fixed source in the air flow: effect of the critical friction velocity on a turbulence phenomenon. In Proc. of 7th JSME- KSME Thermal and Fluid Engineering Conference, Oct. 13-16, Sapporo, Japan, K142, (CD-ROM) 4 pp., 28 I. J. Walker, W. G. Nickling, Simulation and measurement of surface shear stress over isolated and closely spaced transverse dunes in a wind tunnel. Earth Surf. Process. Landforms 28: 1111-1124, 23-11 -