Aeolian Research 3 (2011) 303–314

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Aeolian Research

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Wind tunnel and computational study of the stoss slope effect on the aeolianerosion of transverse sand dunes

Raquel Faria a, Almerindo D. Ferreira b,⇑, João L. Sismeiro b, João C.F. Mendes a, Antonio C.M. Sousa c

a Department of Mechanical Engineering, ISEC – Polytechnic Institute of Coimbra, Rua Pedro Nunes – Quinta da Nora, 3030-199 Coimbra, Portugalb Department of Mechanical Engineering, FCTUC – University of Coimbra, Polo II, 3030-788 Coimbra, Portugalc Department of Mechanical Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3

a r t i c l e i n f o

Article history:Received 17 January 2011Revised 20 July 2011Accepted 20 July 2011

Keywords:Windward slopeTransverse duneComputational modelingWind tunnelFriction velocityVertical sand flux

1875-9637/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.aeolia.2011.07.004

⇑ Corresponding author. Tel.: +351 239 790 732; faE-mail address: almerindo.ferreira@dem.uc.pt (A.D

a b s t r a c t

To understand aeolian particle entrainment, it is important to take into account the surface slope, as mostnatural sand surfaces are not horizontal. The influence of slope angle on local friction velocity is the sub-ject of this research. Three transverse triangular piles, with stoss slopes of 10�, 20�, and 32�, were testedexperimentally, and modeled computationally. The wind tunnel experiments include two sets of tests:the first one consists of friction velocity measurements across the windward slope and the second setcomprises the measurement of the sand dune longitudinal profile over time. The experimental tests wereconducted at four undisturbed wind speeds ranging from 8.3 to 10.7 m/s.

The computational modeling was performed using a commercial CFD code, and it aimed to replicate theexperimental conditions with the objective of evaluating its ability to predict the friction velocity acrossthe windward slope. The numerical predictions of the friction velocity, for the initial longitudinal profile,show good agreement when compared to the experimental values. The region where the predicted fric-tion velocity exceeds the threshold coincides quite well with the eroded area. The correlation betweenthe vertical sand flux, across the stoss slope, calculated using the first eroded longitudinal profile, againstthe predicted friction velocity showed a cubic relation. The computational model, in view of the predictedresults, seems to be a reliable tool for the estimation of the friction velocity for situations similar to thosestudied in this work.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Aeolian transport of particles may have significant negativeenvironmental impacts affecting, among others, agricultural land,pollination, air quality and built environments. Past studies onthe dispersion of pollutants demonstrated the importance of dustrelease either from transportation or storage yards on air quality(e.g., Ferreira and Oliveira, 2009; Ferreira and Vaz, 2004). More-over, in many parts of the world, aeolian processes have playedan important role in landscape changes through accumulation ormigration of sand yielding the formation of dunes, sand sheetsand tongues (Pye and Tsoar, 2009; Huang et al., 2008).

The key variable in the understanding of aeolian processes andsoil erosion, according several authors, such as Iversen and Ras-mussen (1994), is the threshold friction velocity (u⁄t), which is de-fined as the minimum shear velocity required for the aerodynamicforces to overcome the opposing ones. Considerable research hasbeen conducted on this parameter through theoretical analyses,wind-tunnel experiments and field investigations as reviewed by

ll rights reserved.

x: +351 239 790 771.. Ferreira).

Huang et al. (2008). Threshold friction velocity is affected by anumber of factors which have been studied by several authors.For example, surface moisture is an extremely important variablecontrolling the entrainment process of sands by wind becausethe tensile force between the water molecules and sand grains pro-duces cohesion (Dong et al., 2002). Those capillary forces betweengrains are the main factor responsible for the increase of the winderosion threshold observed when the soil moisture increases(Fécan et al., 1999). The vulnerability of the soil to wind erosion de-pends also on the biological soil crusts (Belnap et al., 2007), crusttype (Williams et al., 1995), and vegetation characteristics, suchas amount and distribution, since verdure is known to affectstrongly the erosion of soil by the wind (Okin, 2008). Among oth-ers, Marticorena and Bergametti (1995) and Marticorena et al.(1997) studied the influence of the surface roughness and the aero-dynamic roughness height, which is the most important parameterthat controls u⁄t, for loose or distributed soils. Also, the presence ofnon-erodible roughness elements on the surface intensely attenu-ates the erosion of soil by wind (Raupach et al., 1993). Additionalparameters, such as soil texture, distribution of grain size, or soilsalt content, should be taken into account when studying theentrainment of particles by the wind in specific cases.

304 R. Faria et al. / Aeolian Research 3 (2011) 303–314

Great advances have been made in understanding the physics ofsand transport since the pioneering work of Bagnold (1941), whoderived the following equation for a flat (subscript ‘‘0’’) bed:

u�t0 ¼ A�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs � q

qgd

rð1Þ

In Eq. (1) u⁄t0 (m/s) is the threshold friction velocity on a levelsurface, qs and q (kg/m3) are the densities of the granular materialand the air, respectively, d (m) is the solid grain diameter, g (m/s2)is the gravitational acceleration and A is a general coefficient,which is nearly constant, with an assigned value of 0.1, if the fluidis air (Bagnold, 1941). When resting particles are exposed to an air-stream they stay subjected to several forces, such as the aerody-namic drag, the aerodynamic lift, gravity force, and interparticlecohesive force. Eq. (1) was derived solely from the balance betweendrag and gravity forces, and thus describes only the behavior of u⁄tfor particles with a diameter greater than 100 lm. Greeley andIversen (1985) also considered the lift and cohesive forces and ob-tained a relation applicable to any range of diameters; however,this relation is rather involved mathematically. The study carriedout by Shao and Lu (2000) derived a simpler expression for u⁄tthrough an explicit treatment of the cohesive force. Recently Bar-chyn and Hugenholtz (2011), instead of using analytical models(e.g., Eq. (1)), compared four methods to calculate the velocitythreshold using field data.

By definition, the friction velocity is related to the wall shearstress (sw), as follows:

u� ¼ffiffiffiffiffiffisw

q

rð2Þ

For non-horizontal situations, the study of particle entrainmentprocesses must consider the surface slope, which may turn to bethe deciding factor; it should be noted that in nature, usually sandsurfaces are not horizontal. The implications of the slope on theshear stress, which is required to initiate the movement of a partic-ular sand grain, are discussed by several researchers (e.g., Tsoaret al., 1996; White and Tsoar, 1998; Iversen and Rasmussen,1999); for a grain laying on an upslope surface, this shear stressis larger than that for a horizontal or down sloping surface, becausegravity acts contrary to the movement of the particle.

A theoretical analysis of the slope effect on threshold frictionvelocity was conducted by Howard (1977); based on this analysis,which was verified by Iversen and Rasmussen (1994) and Huanget al. (2008), the threshold friction velocity for the initiation ofthe movement of a particle resting on a tilted surface, with a slopeangle h, when parameters of no primordial importance, such as theinterparticle cohesive force and the Reynolds number variations,are neglected, is given by:

u�th ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2�t0 cos hþ sin h

tan a

� �sð3Þ

In this equation the angle h is the slope, while a is the static fric-tion angle (or angle of repose), for which a value of 32� is com-monly employed (Iversen and Rasmussen, 1994; Ferreira andLambert, 2011).

As an alternative, instead of using the friction velocity concept,the threshold condition can be expressed through the undisturbedthreshold wind velocity (U0t), which is the minimum wind speedoutside of the boundary layer necessary to drag a particle. Asshown by Bagnold (1941), U0t, for a horizontal bed, can be calcu-lated using the following relation:

U0t ¼ 5:75� u�t0 � lnzk

� �ð4Þ

where u⁄t0 (m/s), is the threshold friction velocity, given by Eq. (1), z(m) is the height above the surface (in fact z = d, as beyond it U0t isassumed constant), and k (m) is the surface roughness parameter.Similarly to the case of a slope surface, the threshold wind velocityis computed as:

U0th ¼ 5:75� u�th � lnzk

� �ð5Þ

The main objective of this work is to evaluate the influence ofthe slope parameter on the friction velocity value, for which bothexperimental and numerical tests were conducted. Three modelsare studied, and the basic geometry consists of a two-dimensionaltransverse dune, with the windward slope equal to 10�, 20� and32�, respectively. The crest height (H) is equal to 75 mm for allthe models, and the leeward slope is equal to the angle of repose,which is taken as 32� in the present work.

Two sets of wind tunnel experiments are reported. The first onecomprises the measurement of the friction velocity distributionacross the stoss slope of rigid triangular models, for several undis-turbed wind speeds. In the second set of experiments, the longitu-dinal profile of a sand dune, initially with the above mentionedtriangular shape, is measured at several erosion time intervals.

The three triangular dune shapes are simulated and for theircomputational modeling is used the commercial CFX code (Ansys,2009). The predicted friction velocity and erosion contours arecompared against the experimental values.

2. Experimental setup

In this section of the paper it is presented the instrumentationused, the geometry of the piles studied, the sand granulometryand the procedure for the various experiments performed.

2.1. Wind tunnel

The wind tunnel used for this study is installed at the IndustrialAerodynamics Laboratory (LAI) of ADAI (Association for the Devel-opment of the Industrial Aerodynamics-University of Coimbra,Portugal). The tunnel has a 2 m � 2 m cross-section nozzle fol-lowed by an open working chamber, which is 5 m long. Such shorttest section precludes the use of mixing devices or roughness ele-ments to thicken or modify the boundary layer profile. The modelsof the piles studied were placed on the floor of the test section, andequidistant to its sidewalls. The crest of all models was positioned2.5 m downstream of the nozzle exit, and perpendicular to themain flow direction (Fig. 1).

The profile of the mean streamwise velocity (u), measured inthe empty working chamber, at a distance of 2.4 m from the exitof the wind tunnel nozzle, at half-width of the wind tunnel work-ing chamber, can be described by the following power law relation:

uU0

zd

� �að6Þ

where u (m/s) is the longitudinal velocity component, U0 (m/s) isthe undisturbed wind speed, and z (m) the vertical distance abovethe ground. Under the described conditions, the boundary layerthickness is d = 0.1 m and the profile is characterized by a = 0.11.The turbulence intensity of the longitudinal velocity component re-mains nearly unchanged with the height, and a reasonable approx-imation is to take its value equal to 10%.

Different undisturbed velocities (U0) were used for the tests,namely 8.3, 9.1, 9.9 and 10.7 m/s, which yielded four differentvelocity profiles as shown in Fig. 2.

When performing experiments for model-scale conditions, dueattention is required to the similarity criteria, namely the equalityof various dimensionless parameters, which normally is not en-

Fig. 1. Schematic view of the wind tunnel test section and placement of the pile model (dimensions in [mm]).

Fig. 2. Incident velocity profile for the various undisturbed wind speeds tested.

R. Faria et al. / Aeolian Research 3 (2011) 303–314 305

tirely satisfied. For the particular situation under analysis, the Rey-nolds number (Re) range needs to be examined. According to somereferences (White, 1996; Stunder and Arya, 1988), the requiredequality between model and full-scale can be relaxed if Re exceedsa minimum critical value of approximately 104. The lowest undis-turbed wind speed used is 8.3 m/s, which yields a Reynolds num-ber of 4.1 � 104; this value is large enough to guarantee theoccurrence of the Reynolds-number independence regime. To calcu-late Re, the initial height of the model (H = 0.075 m) is taken as thecharacteristic length. A detailed discussion of similarity criteria isgiven in White (1996).

The ratio between the pile height and the thickness of the windtunnel boundary layer is an important issue in wind tunnel testsand it would be desirable to keep the tunnel model in the lower20% of the boundary layer (White, 1996). For the present case,the height of the crest represents 75% of the wind tunnel boundarylayer, which does not verify the requirement specified by White(1996). This means that the present results cannot be directlyextrapolated to full-scale conditions; however, the present experi-mental tests can be regarded as data base for the benchmark ofcomputational models. Still, important information can be ex-tracted from the present experimental data, as it is useful to vali-date the computational predictions and to understand thedynamics of the aeolian erosion of a sand dune, and in particularthe slope effect and the modification of the friction velocity distri-bution due to the slope of a tilted surface.

2.2. Geometries studied and pile setup

In this study, three triangular pile geometries (Fig. 3) weretested, with different windward (stoss) slopes. The slope of the up-stream face is equal to 10�, 20�, and 32�, and those models will be,hereafter, named S10, S20, and S32, respectively. The height of allmodels is equal to 75 mm; Table 1 summarizes the characteristicsof the piles. These models were built either with sand or wood, asdescribed later on. For the sand used in the experiments, the angleof repose is approximately equal to 32� (Ferreira and Lambert,

2011), which is the limit slope angle. As a consequence, the slopeof the leeward face is the same for the three models, and equalto the angle of repose. All the models were extruded normal toits plane with a width of 1 m (Fig. 4).

The sand piles were formed by using two wooden guides withthe dimensions given in Table 1. They were placed one meter apart,and then the space between them was filled with dried and sievedsand; afterwards, a ruler supported on the guides swept the sandto obtain the final shape of the dune’s model. Due to the angle ofrepose, the lee side of the dune was naturally obtained. After thesand pile is built, two Perspex plaques were placed on the innersurface of the guides to reduce the edge effects and ensure atwo-dimensional flow (Fig. 4).

2.3. Sand characterization

All the sand piles considered in this study were made out ofsieved sand with a prevailing grain diameter of d = 0.5 mm,according to the granulometry tests conducted by Ferreira et al.(2010). The bulk sand density for this particular granulometry isapproximately equal to 1500 kg/m3. Even so, the valueqs = 2650 kg/m3 is assumed for the sand grain density, as the sandused consists mainly of silica grains, usually in the form of quartz(Bagnold, 1941).

2.4. Experimental procedure

2.4.1. Erosion testsSeveral erosion tests were conducted, in order to analyze the

pile contour evolution as the wind erosion progresses. All the pileswere built using sand of prevailing granulometry d = 0.5 mm. Afterthe pile setup, the pile contour prior to erosion was scannedemploying a distance laser sensor Dimetix-model DLS-B15(Dimetix, 2010), which was placed 2.77 m above the floor of thewind tunnel test section. The distance, from the sensor to the pilesurface, was measured with a typical measuring accuracy of±1.5 mm at a statistical confidence level of 95.4%. That distancesensor was mounted on a traversing system working on a planeparallel to the base of the model. The traversing system carriesthe laser in two perpendicular directions, and is equipped withstepper motors to control the motion and location with high preci-sion. The entire process is controlled from a computer-based plat-form, which was developed by Gonçalves (2008).

The measurement of each profile is processed over the sectionlocated half-way the width of the pile, i.e., along the centerline.The increment in the streamwise direction for each successive datapoint is 10 mm, with the first point coincident with the leadingedge of the model and the last one located 140 mm after the trail-ing edge of the pile prior to erosion. As previously stated, the widthof each pile is 1 m, ensuring that end effects are negligible along itshalf-width, which was confirmed during the experiments.

Fig. 3. View of the three triangular sand piles before erosion (models S32, S20, and S10, from left to right).

Table 1Geometric characteristics of three triangular models studied.

Model Crest height H (mm) Stoss Lee

Slope angle (h) (�) Base length LS (mm) Slope angle c (�) Base length (mm)

S10 75 10 425.4 32 120S20 75 20 206.1 32 120S32 75 32 120.0 32 120

Fig. 4. Sand pile (S10) placed in the wind tunnel test section.

Fig. 5. Close view of the Irwin probes surface-mounted on one of the piles.

306 R. Faria et al. / Aeolian Research 3 (2011) 303–314

The measurement process was executed sequentially, and ateach point the distance registered is taken as the average of fourlaser readings. In addition, the time for each measured point wasrecorded in a plain text format file.

After the initial pile scan, the sand dune was exposed to the de-sired wind speed during a certain time interval; the pile was shel-tered by a large wood board, placed upstream of the sand model,during the wind-tunnel speedup phase. The wind tunnel wasstopped at cumulative times t = 1, 2, 3, 5, 7, 10, 15 and 20 min,although not all the erosion tests have taken that long due to theintense erosion observed in some cases. At each one of these times,the wind tunnel was stopped, and the laser sensor was used forregistering the profile of the pile; such data was then used to deter-

mine the deformation rate of the pile, as it will be described in thefollowing sections.

2.4.2. Friction velocity testsIrwin-type pressure probes (Irwin, 1981) were used to perform

measurements of the shear stress along the stoss slope of the var-ious triangular-shape piles. The rigid models were made out ofnearly smooth wood, and their profile data are reported on Table1. An Irwin probe consists essentially of two concentric pressuretaps, one of them flush with the surface and the other protruding2 mm above the surface (Fig. 5). The pressure difference (DP) be-tween these two taps is used to obtain the wall shear stress, andthen the friction velocity, as desired in this work. In wind tunneltesting, the Irwin sensors are frequently used to estimate the windspeed close to the ground-level (Ferreira et al., 1991; Monteiro andViegas, 1996).

The Irwin sensors are simple in design, do not require alignmentas they are omnidirectional, and allow the measurement at numer-ous closely spaced locations. Based on the calibration of the probes(Ferreira et al., 1991), the wall shear stress, sw [Pa], is related to thepressure difference, DP [Pa], by the following power function:

sw ¼ 0:0373� DP0:768 ð7Þ

The wall shear stress for each of the three triangular modelswas measured at four free stream wind speeds, 8.3, 9.1, 9.9 and10.7 m/s, respectively. For the pile S10, fifteen probes were distrib-uted along the stoss surface of the model, starting at 120 mm fromthe leading edge, and then uniformly distributed up to the crest(Fig. 6). Over the pile S20, only seven probes were mounted, thefirst one placed 80 mm from the leading edge. On the steeper pile(S32) only five probes could be mounted, the first one placed40 mm from the leading edge. After the first probe, and for all mod-els, the distance between two consecutive probes is equal to20 mm, as shown in Fig. 6. In order to reduce the interference,the probes were mounted alternately along three parallel lines,one coincident with the half-width, and the other two placed sym-metrically at a crosswise distance of 20 mm (Fig. 6). The first sen-sors could not be mounted closer to the leading edge due to theprobe size and low height of the surface near the windward footof the pile.

The pressure taps of the Irwin-probes, two per sensor, wereconnected to a 48 channel Scanivalve, and all the measuring pro-cess was controlled by a personal computer-based platform. Pres-sure was measured using a Multur pressure transducer with an

Fig. 6. Distribution of the Irwin probes on the stoss slope of models S10, S20, and S32, respectively, (dimensions along the slope surface, in [mm]).

R. Faria et al. / Aeolian Research 3 (2011) 303–314 307

accuracy of 0.5 Pa; the recorded pressure value is the average of750 samples acquired during 15 s at a frequency of 50 Hz.

3. Numerical setup

3.1. Software employed and assumptions

For the numerical modeling, the computational fluid dynamics(CFD) commercial code CFX (Ansys, 2009) was employed. ThisCFD code solves numerically the 3D Reynolds-averaged Navier–Stokes equations by using a finite volume discretization methodand a segregated procedure; the pressure–velocity coupling is per-formed with the Simplec algorithm (Patankar, 1980). The discreti-zation of the advective terms is accomplished using a second-orderadvection scheme to reduce the occurrence of artificial viscosity.Steady state conditions were assumed as one of the objectives ofthis work is to compare the friction velocity predicted computa-tionally against the experimental data obtained from the measure-ments on wooden triangular piles. Due to the low wind speed, theflow is assumed to be incompressible. For the turbulence modelingthe standard formulation of the k–e model was employed, andnumerical convergence was assumed to be satisfied when all thenormalized residuals were smaller than 10�5.

The geometries of the cases considered in the numerical simu-lations are given on Table 1, and were modeled using the softwareSolidWorks (SolidWorks, 2009), and they were imported by theAnsys Workbench (Ansys, 2009). This software was also used toimprove the organization of each simulation, due to the high num-ber of steps needed to be performed; all applications, such as CFX-Mesh, CFX-Setup and CFX-Solver, were launched from it. The size of

Fig. 7. Size of the domain used in the com

the two-dimensional computational domain for each of the situa-tions modeled is depicted in Fig. 7.

3.2. Mesh generation

As mentioned before, the mesh was generated via the softwareCFX-Mesh. The parameters used in the mesh were the same for allgeometries, and were defined after a few independence/conver-gence grid tests indicate that further refining of the mesh leadsto no appreciable change in the results. The main meshing featuresconsidered were the spacing and inflation. In the first feature, themaximum spacing was set equal to 0.4 mm. The inflation parame-ter is very important in cases like the present one, since in thenear-wall region the boundary layer effects give rise to large veloc-ity gradients in the direction normal to the solid surfaces. Thenumber of inflated layers was set equal to 20 and the expansionfactor equal to 1.2. To define the total height of the inflated layer,the first layer thickness was made equal to 1.5 mm correspondingto 2% of the crest height, which guarantees a distance large enoughto verify the needed conditions for the use of wall functions. Theentire bottom part of the computational domain (base and pile)was considered as an inflated boundary.

3.3. Boundary conditions

As mentioned before, the computational domain was taken astwo-dimensional (2D), assuming symmetry conditions relativelyto the central longitudinal plane. The flow in the working chamberof the wind tunnel is turbulent, and a turbulence intensity value of10% was set at the inlet and top regions (Ferreira and Lambert,

putational simulations (H in [mm]).

Table 3Repeatability level of the friction velocity measurements using the Irwin probes.

Angle (h) (�) Deviation (%)

Maximum Average

10 2.26 0.8920 4.35 1.8232 2.29 0.77

308 R. Faria et al. / Aeolian Research 3 (2011) 303–314

2011). The inlet profile of the streamwise velocity component is gi-ven by Eq. (6), while the vertical component is assumed to be zero.Depending on the simulation, four different undisturbed velocities(U0) were considered: 8.3, 9.1, 9.9 and 10.7 m/s, respectively.

For the outflow, fully developed conditions, i.e., zero-gradientalong the streamwise direction, were assumed for all the variablesand the relative static pressure was set equal to zero.

The solid surfaces (i.e., base and pile) were treated as roughwalls, where a no-slip condition was imposed. The sand grainroughness parameter k appearing in Eqs. (4) and (5) was takenequal to d/30 (Bagnold, 1941), d being the mean sand grain diam-eter (d = 0.5 mm, Section 2.3). Various numerical tests were per-formed considering several values for k, and the best agreementwith the experimental results was achieved with k = d/30, as it willbe discussed in Section 4.2.1.

4. Results and discussion

This section reports on the experimental and computational re-sults. The experimental data include the friction velocity distribu-tion obtained from the measurements performed with the Irwinprobes, and the pile erosion profiles; these experimental resultsare presented and discussed in Section 4.1. In Section 4.2, thenumerical modeling predictions are analyzed and comparedagainst the experimental data.

Henceforth, all the diagrams presented have their axes normal-ized, namely: the local height of the pile is normalized by the initialcrest height (h/H); the local longitudinal coordinate, measuredfrom the leading edge of each pile (xh, as in Fig. 1), is normalizedby the stoss longitudinal length, i.e., the base length in Table 1,(xh/LS), where xh/LS = 1 corresponds to the crest; the local frictionvelocity is normalized by the threshold friction velocity for a givenslope (u⁄/u⁄th).

4.1. Experimental results

4.1.1. Undisturbed threshold wind velocity for each stoss slope angleAs stated before, the main objective of the present work is to

study the influence of the slope on the friction velocity, and thenon the erosion process of a transversal triangular dune model. Asgiven by Eq. (5), erosion on a slope surface occurs only if the undis-turbed wind speed exceeds a specific threshold value, which is afunction of the stoss slope angle. Based on Eqs. (1), (3), and (5),Table 2 presents an estimate for the undisturbed threshold windvelocity (U0th), outside of the boundary layer (i.e., z > 0.1 m), neces-sary to initiate erosion, for the different dunes tested. By examin-ing Table 2 one can conclude that, as expected, the thresholdfriction velocity increases with the slope; this observation is sup-ported by both the experimental and computational results.

4.1.2. Selection of the undisturbed wind speed for the erosion testsAs mentioned in Section 1, each slope is characterized by a spe-

cific threshold fluid velocity. To validate the values obtained fromEq. (5), for the 10� slope, three erosion tests were performed usingthe following undisturbed velocities: 8.3; 9.1 and 9.9 m/s. The low-

Table 2Threshold values of friction velocity, wall shear stress and undisturbed velocity(computed using Eqs. (1), (3), and (5)).

Slope (h) (�) Friction velocity(u⁄th) (m/s)

Wall shear stress(swth) (N/m2)

Undisturbed windspeed (U0th) (m/s)

0 0.326 0.130 7.110 0.367 0.164 8.020 0.398 0.193 8.632 0.425 0.220 9.2

est wind speed, according to Table 2, should had been enough toinitiate the entrainment of particles, since it is larger than therespective threshold value (U0th = 8.0 m/s). In fact, by observingthe experiments it can be noted that erosion of the S10 sand pilewas negligible for U0 = 8.3 m/s, what might be justified by themodification of the wind profile induced by the pile itself. The nextwind speed used for testing the S10 pile was 9.1 m/s. The same pro-cedure was carried out for the other two piles (S20 and S32), andthe undisturbed velocity for the erosion tests of these two slopeswas chosen to be equal to 9.9 m/s.

4.1.3. Friction velocity resultsThis section of the paper presents the results derived from the

measurements performed with the Irwin-type probes; these re-sults can be regarded as wall shear stresses or friction velocities,as both are related through Eq. (2).

First, several wind tunnel tests were conducted to evaluate therepeatability level of the wall shear stress measurements. Afterperforming all the measurements, the average value was computedfor each one of the probes installed, and the deviation relative tothe mean was determined. Table 3 summarizes the results, and,under close scrutiny, it can be noted that their maximum deviationis lower than 5%, and their mean deviation is less than 2%; there-fore, it can be concluded that tests using the Irwin probes are char-acterized by an excellent level of repeatability.

Fig. 8 presents the friction velocity distribution, along the stossslope of the three piles, for four undisturbed flow velocities. Gener-ally speaking, it can be said that the dispersion of points in allgraphs is similar. As expected, the friction velocity increases pro-portionally to the undisturbed wind speed, which indicates thatthe entrainment of particles is enhanced at higher wind speeds.For example, an increase of 10% of the wind speed (from 8.3 to9.1 m/s) implies an average increase of approximately 10% of theu⁄ distribution on S10 pile, 8% on S20, and also 10% on the S32 pile.Similar behavior was observed for the other U0 wind speeds.

Further examination of Fig. 8 indicates the friction velocity in-creases along the stoss surface, a fact more evident on the twosteeper slopes, S20 and S32, respectively, due to the gradual com-pression of the streamlines along the windward slope. These re-sults show a trend similar to those of Walker and Nickling(2003). Such behavior is in agreement with the observations ofHuang et al. (2008), and also Sauermann et al. (2003), who notedconsiderable wind speed changes along a rising slope, and withthe measurements of Qian et al. (2009). Therefore, it can be con-cluded that, depending on the surface slope and on the distanceto the pile’s leading edge, different undisturbed wind velocitiesare required to initiate the local erosion of the particles.

When comparing the values obtained for the three differentslopes, it can be noted that up to approximately xh/LS = 0.6, thethree sets are rather different; however, after this location, theexperimental results nearly overlap. Up to xh/LS = 0.6, pile S10 pre-sents the highest ratio (u⁄/u⁄th), which means that, for the same rel-ative position on the windward surface, increasing slope yieldsincreased resistance to the movement of particles. Moreover, closeto the crest the influence of slope on the initiation of the erosion isless pronounced. In agreement with the observations of Huang

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

u*/u*tθ S10 S20 S32

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

u*/u*tθ S10 S20 S32

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

u*/u*tθ S10 S20 S32

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

u*/u*tθ

xθ /Lsxθ /Ls

xθ /Ls xθ /Ls

S10 S20 S32

Fig. 8. Experimental distribution of the friction velocity along the three stoss slopes, for different undisturbed wind speeds.

R. Faria et al. / Aeolian Research 3 (2011) 303–314 309

et al. (2008), in the present study it was also found that in thevicinity of the dune’s toe the wind speed is weak; however, it in-creases along the windward slope, reaching the maximum valueclose to the crest.

4.1.4. Pile erosion contours and correlation with the friction velocityresults

Several erosion experiments were performed using the proce-dure explained in Section 2.4.1. Succinctly, the pile contour oftransversal sand dune models S10, S20, or S32 (Fig. 3 and Table1), respectively, when exposed to the wind flow, was measuredat different times; the initial profiles of the models are defined inTable 1. The measured profiles, from the leading edge to the crest,i.e., 0 <= xh/LS <= 1, are depicted in Fig. 9. In this figure are also plot-ted the corresponding friction velocity results discussed in the pre-vious section. In addition, it can be noted the erosion occurs mainlyin the region where u⁄/u⁄th > 1. However, it is important to high-light that the friction values were measured over a rigid model,with a fixed triangular shape, while the contour of the sand pileis modified gradually as erosion evolves.

In the first graph shown in Fig. 9 (model S10), most of the valuesof the friction velocity ratio u⁄/u⁄th are higher than one, which sug-gests that the entire windward surface is susceptible to be eroded.In reality, the downstream part of the stoss slope was graduallyeroded, but the pile remained unchanged in the zone xh/LS < 0.4.This observation can be justified based on the modification of thedune’s profile. Due to the dune erosion, the wind pattern overthe initial region of the pile is certainly affected, and consequentlythe local friction velocity is also affected. In case the erosion testhad lasted more than 20 min, the eroded region, most likely, wouldhave extended to the leading region of the stoss surface.

For pile S20, the value of u⁄/u⁄th is larger than one in the instru-mented section – the downwind side of the stoss surface, and thedata correlate well with the eroded profiles. Moreover, as it can benoted in Fig. 9, after 5 min of erosion, the pile is already erodedfrom the location of the first probe onwards. After this time, asthe profile of the pile has already been considerably modified, itis not longer reasonable to correlate the two sets of results.

For the steeper pile (S32) two sets of results are shown in Fig. 9for wind speeds U0 = 9.9 and 10.7 m/s, respectively. It can be ob-served that for the lower velocity (U0 = 9.9 m/s) pile S32 is erodedmore significantly in the region xh/LS > 0.6, at least during the initialerosion process. The pile erosion is in good agreement with thefriction velocity measurements; in the zone xh/LS < 0.6, the frictionvelocity is lower than the threshold limit.

When the undisturbed flow velocity is increased to U0 = 10.7 m/s, the non-dimensional friction velocity is larger than one down-stream of xh/LS � 0.4. Beyond this location, the pile’s profilechanges rapidly due to a high erosion rate. After a significant ero-sion of the dune, the u⁄ values are no longer applicable, and theyare not used to explain the subsequent erosion contours.

4.2. Numerical results

In this section the predictions from the computational modelingare presented and discussed. As mentioned in Section 3.1, onlypiles S10, S20, and S32 were numerically simulated; the computa-tional predictions are compared against the experimental frictionvelocity results, already presented in Section 4.1.3, and are relatedwith the first eroded contour.

Similarly to the experiments, four undisturbed velocities (U0),8.3; 9.1; 9.9 and 10.7 m/s, respectively, were simulated for eachone of the three windward slope angles. In the computational runs,the domain is assumed to be two-dimensional, and the computa-tional domain has the dimensions indicated in Fig. 7, as alreadydiscussed in Section 3.1.

4.2.1. Influence of sand grain roughnessAs stated in Section 3.3, some numerical tests were performed

to evaluate the influence of the sand grain roughness parameter(k) on the predicted friction velocity. Two values of k were tested,respectively d/30 (value recommended by Bagnold, 1941; Ferreiraand Lambert, 2011, among others) and d, where d = 0.5 mm isthe mean sand grain diameter.

The benchmark tests were performed only for the model S32,and the results are presented in Fig. 10, which shows the predicted

Pile S10 ; U0 = 9.1 m/s Pile S20 ; U0 = 9.9 m/s

Pile S32 ; U0 = 9.9 m/s Pile S32 ; U0 = 10.7 m/s

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u*/u*tθh/H

Theoretical Initial Profile 1 min2 min 3 min 5 min7 min 10 min 15 min

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u*/u*tθh/H

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xθ/Lsxθ/Ls

Theoretical Initial Profile 1 min2 min 3 min 5 min

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Theoretical Initial Profile 1 min2 min 3 min u*/u*t θ

Fig. 9. Pile contours (h/H) at several erosion times, and experimental measurements of the friction velocity across the stoss slope (‘‘Theoretical’’ contour corresponds to theideal initial stoss slope).

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u*/ u*tθ

xθ/LS xθ/LS

Experimental Computational

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u*/ u*tθ Experimental Computational

Fig. 10. Influence of the surface roughness parameter (k) on the friction velocity distribution (U0 = 8.3 m/s; model S32).

310 R. Faria et al. / Aeolian Research 3 (2011) 303–314

values and the wind tunnel results. As it can be seen, the smoothercase (k = d/30) gives the best agreement between the predicted andexperimental values; where the maximum deviation between thetwo sets, is 9.9%. For the high roughness case (k = d) the maximumdeviation is 25.7%. The average deviation between the predictionsand the experiments, for the smooth and rough situations, is 5.5%and 16.5%, respectively. Therefore, it can be noted that the sandgrain roughness has some influence on the numerical predictions,especially in the region xh/Ls > 0.5; in the first half of the stoss slopethe influence of the roughness parameter appears to be less pro-

nounced. The value (k = d/30) recommended in previous studieswill be taken for the numerical simulations of the present work;however, it should be noted that several authors have suggestedother relations, as indicated, e.g., in Li et al. (2008).

4.2.2. Numerical prediction of the friction velocityFor the validation of the simulations, the computational predic-

tions for the friction velocity are compared against the experimen-tal data. The maximum and average values of the deviation,between those two sets of results, are presented in Table 4. From

Table 4Maximum and average deviation between numerical and experimental values of the friction velocity.

Model Deviation (%)

U0 (m/s) S10 S20 S32

Maximum Average Maximum Average Maximum Average

8.3 19.7 9.2 13.5 6.5 9.9 5.59.1 20.6 9.2 13.6 5.6 11.0 6.69.9 20.5 9.2 13.8 6.2 11.7 6.2

10.7 19.2 9.0 12.7 5.3 12.3 5.7

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u*/u*tθ

xθ/LS

EXP_8.3m/s COMP_8.3m/s EXP_9.1m/s COMP_9.1m/sEXP_9.9m/s COMP_9.9m/s EXP_10.7m/s COMP_10.7m/s

Fig. 11. Comparison between computational results and experimental measurements of the friction velocity for the various undisturbed wind speeds tested (model S20).

R. Faria et al. / Aeolian Research 3 (2011) 303–314 311

this table, and as an example, it can be noted that for model S20 themaximum absolute deviation ranges between 12.7% and 13.8%,while the average varies between 5.3% and 6.5% for the variouswind speeds tested. The predictions for the different models yieldsimilar results; the highest deviation was observed for the modelS10, and it can be seen that the deviation decreases gradually forsteeper windward surfaces.

Fig. 11 presents the comparison between the numerical andexperimental results of the friction velocity distribution along thewindward slope of pile S20 for different testing velocities. A similarpattern was obtained for the other two triangular piles, S10 andS32. From this figure, it can be observed the good agreement be-tween the computational predictions and the experimental results;both sets indicate that u⁄ increases along the surface, as noted be-fore, due to the flow speedup, and also verified experimentally byQian et al. (2009). However, near the pile crest, it can be seen thatthe computational values are clearly larger than the experimentalones.

From Fig. 11 and Table 4, it can be then concluded that the com-putational modeling accurately predicts the friction velocity distri-bution. Based on such good agreement, the computational valuesof u⁄ will be used, in the future work, to predict numerically thetime evolution of the contour of the sand pile, as its erosionproceeds.

4.2.3. Numerical results and pile erosion contoursThe several graphs of Fig. 12 show the shape of the three piles

eroded, at different times, and also the computational prediction

of the friction velocity for the triangular wooden models. As al-ready mentioned, and by definition, erosion is expected to occurin the regions where u⁄/u⁄th > 1. Looking at the different graphs ofFig. 12, it can be seen that there is a good correlation betweenthe computational results and the erosion experiments. In fact,the expected eroded region is quite well predicted by the compu-tational model.

Independently of the windward slope angle, the lower is thewind speed the better is the correlation between numerical andexperimental results, since the erosion is less intense, which meansa lower change rate of the windward surface slope.

For a given wind speed, an increase of the windward slope leadsto a smaller area eroded, i.e., the region susceptible of erosionmoves closer to the crest. As an example, the two graphs shownin Fig. 12 for U0 = 9.9 m/s, indicate that S20 is susceptible to beeroded mainly from xh/LS = 0.4 onwards, while for S32 erosionstarts at xh/LS = 0.6. On the other hand, as observed in the experi-ments (Fig. 12 for model S32), and as reported by Huang et al.(2008), for the same slope, an increase of the U0 wind speed leadsto a larger eroded region.

Selected experimental friction velocity values are plotted inFig. 12, for U0 = 9.9 m/s and model S32. As observed before, thecomputational results are in good agreement with the experimen-tal measurements.

The numerical simulations reveal a large recirculation regionoccurring in the pile leeward for all the models, as depicted inFig. 13. Inside of the recirculation bubble the shear velocity is quitelow, which corroborates the findings of Walker and Nickling (2002)

Pile S10 ; U0 = 9.1 m/s Pile S20 ; U0 = 9.9 m/s

Pile S32 ; U0 = 9.9 m/s Pile S32 ; U0 = 10.7 m/s

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xθ/LsTheoretical Initial Profile 1 min 2 min3 min 5 min 7 min 10 min15 min 20 min Computational

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Theoretical Initial Profile 1 min

2 min 3 min Computational

Fig. 12. Pile contours (h/H) at several erosion times and computational prediction of the friction velocity on the stoss slope of triangular dune models. (‘‘Theoretical’’ contourcorresponds to the ideal initial stoss slope).

Fig. 13. Computational prediction of the flow topology and recirculation zone inthe leeward region of the three transverse models (U0 = 8.3 m/s).

312 R. Faria et al. / Aeolian Research 3 (2011) 303–314

and, consequently, the local aeolian erosion is almost negligible.The flow separation, and consequent recirculation, leads to the for-mation of a slip face where avalanche occurs when the leewardslope is close to the angle of repose.

The present study confirms the experimental results of Qianet al. (2009), where the length of the recirculation zone increaseswith the windward slope. For all the piles the flow separation oc-curs at the pile crest; the streamwise length of the recirculationzone, measured from the crest and normalized by the crest height(H), is 7.33, 10.27, and 11.47, for piles S10, S20, and S32, respec-tively. The predicted size of the reverse vortex is in good agree-ment with the experimental measurements of Qian et al. (2009),whose experiments indicate a length of approximately 7.5 � Hand 10.5 � H for windward slopes of 10� and 20�, respectively.The maximum slope studied by those authors was 25�. However,the present simulations suggest shorter lengths for the recircula-tion zone than the results reported by Parsons et al. (2004), wherethe separation length is 9.3 � H, 10.2 � H, and 12 � H, for stossslopes of 8.3�, 10.8� and 16�, respectively. Still, and despite of thisdiscrepancy, the work of those authors indicate also a tendency forthe enlargement of the separation bubble for steeper windwardsurfaces, as seen in the present simulations. Walker and Nickling(2003), for a stoss slope of 8.1�, suggest an average recirculationzone length of 7.3 � H.

An interesting finding concerns the location of the reattachmentpoint, which appears to be insensitive to the undisturbed windspeed for all the piles studied. In fact this location practically didnot change with the undisturbed velocity. This observation just

corroborates the wind speeds adopted for the numerical simula-tions fall in the Reynolds independence regime. Under these cir-cumstances, only the results for U0 = 8.3 m/s are presented inFig. 13.

4.2.4. Relation between vertical sand flux and friction velocityFig. 14 shows the time average vertical sand flux, / [kg/(m2 s)],

between instants t = 0 and 1 min, and the relation between thesand flux and predicted friction velocity, for various slopes and dif-

Fig. 14. (a) Vertical sand flux (/ [kg m�2 s�1]) across the stoss slope, between t = 0 and 1 min. (b–d) Relation between measured vertical sand flux and predicted frictionvelocity. (r – regression coefficient).

R. Faria et al. / Aeolian Research 3 (2011) 303–314 313

ferent wind flow conditions. The vertical sand flux was calculatedusing the measured profile contours at those two instants. Fig. 14ashows the variation of / along the stoss slope of models S20 andS32, where it can be seen that the vertical flux increases consider-ably with the wind speed (model S32: U0 = 9.9 and 10.7 m/s), butdecreases with slope (cases S20 and S32, for U0 = 9.9 m/s).

Most of the flux models available in the literature indicate a cu-bic relation between the shear velocity and the horizontal flux inthe saltation layer (e.g., Bagnold, 1941; Greeley and Iversen,1985; Iversen and Rasmussen, 1999; Dong et al., 2003). As / repre-sents the entrained flux (per unit of time and area) released fromthe surface and entering the saltation layer, it is expected a similarrelation between the vertical sand flux and the friction velocity.Fig. 14b to d show the relation, for piles S20 and S32, betweenthe sand flux experimental data and the friction velocity excess(u⁄–u⁄th) predicted numerically. In the present analysis it was con-sidered that the mass flux should be zero in the location where u⁄is lower than the specific threshold u⁄th (indicated in Table 2). Foreach one of the three cases analyzed, the best fitting cubic relationis shown with a dashed line, and it can be seen that those curvesagree reasonably well with the experimental data. Such fact alsoreveals that there is a good relation between the computationalpredictions and the eroded fluxes.

5. Conclusion

This paper aims to contribute to the study of the slope influenceon the entrainment of particles, and to evaluate the capability of a

reasonably simple numerical model to predict the friction velocityon triangular transverse dune models, for which experiments werealso conducted. The geometry of the models studied consists of atwo-dimensional transverse dune, and three different windwardslopes were considered, 10�, 20� and 32�, respectively, all of themwith the same crest height.

All the experiments were performed in a wind tunnel, and fourdifferent undisturbed wind speeds were considered, with the Rey-nolds number ranging between 4.1 � 104 and 5.3 � 104. The firstset of tests comprises the measurement of the friction velocityacross the stoss slope of a triangular wooden dune model. In thesecond experimental set, the longitudinal profile of a sand dunemodel, eroded by the wind flow, was measured at different times.

From the wind tunnel tests, it was concluded that the frictionvelocity increases with the slope, and the preferred area of erosionis located near the crest, extending gradually in the upwind direc-tion as the flow velocity is increased. One further observation wasthat the extension of the region with erosion decreases withincreasing slope.

For the computational modeling only the initial pile shape, i.e.,the triangular wooden profile, was considered. The friction velocitydistribution, predicted computationally, was compared against theexperimental values, and they are in good agreement presenting amaximum deviation of approximately 6%. A good correlation wasalso seen by comparing the shear velocity distribution againstthe erosion contours. The vertical sand flux, computed using theinitial and first eroded contours, increases gradually along the stossslope, and shows a cubic relation with the predicted friction veloc-ity. Therefore, within the experimental conditions and constraints

314 R. Faria et al. / Aeolian Research 3 (2011) 303–314

of the present study, it can be stated the computational model is areliable tool for the estimation of the shear velocity distributionacross the transverse dunes tested.

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable suggestions. This work was supported by the re-search project PTDC/EME-MFE/67631/2006, entitled ‘‘Numericaland experimental modeling of solid particles motion driven bywind’’, which was financed by FCT with funds from ProgramFEDER.

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