A CFD study of wind patterns over a desert dune and the effect on seed dispersion

Environ Fluid Mech (2012) 12:23–44DOI 10.1007/s10652-011-9230-3

ORIGINAL ARTICLE

A CFD study of wind patterns over a desert duneand the effect on seed dispersion

Eugéne C. Joubert · Thomas M. Harms ·Annethea Muller · Martin Hipondoka ·Joh R. Henschel

Received: 6 December 2010 / Accepted: 4 November 2011 / Published online: 20 November 2011© Springer Science+Business Media B.V. 2011

Abstract In the Namib Desert seed distribution is greatly influenced by wind patterns.Existing literature regarding wind patterns over dunes focuses on two-dimensional simu-lations of flow over simplified dune structures. The three-dimensional geometries of thesand dunes suggests far more complex flow features exist, which are not captured by two-dimensional simulations. Computational fluid dynamics (CFD) was used to reproduce thethree-dimensional near surface wind patterns around a dune with the aim to learn more aboutseed distribution. Field work included terrain mapping, wind speed, direction and temper-ature metering. The CFD results show the expected two-dimensional flow features of highpressure at the dune toe, low pressure at the crest and flow acceleration up windward slope.Also observed are some three-dimensional flow features such as a spiral vortex near the crestand transverse flow due to crest-line curvature of the dune. It was also observed how thewall shear stress differs due to the three-dimensional shape of the dune. The wall shear stresssuggests that seed accumulation is more likely to occur behind trailing (down-wind) crestedges. Particle tracking showed how seeds tend to move over the dune crest and recirculatetowards the crest on the lee-side. The study showed that adding the third dimension makesthe simulations more complex, adds to computational requirements and increases simulationtime but also provides vital flow information which is not possible with two-dimensionalsimulations.

Keywords Computational fluid dynamics · Atmospheric flow · Dune · Particle tracking

E. C. Joubert (B) · T. M. HarmsDepartment of Mechanical and Mechatronic Engineering, Stellenbosch University,C/o Banhoek and Joubert Street, Stellenbosch 7600, South Africae-mail: [email protected]

A. Muller · M. HipondokaEnvironmental and Geographic Department, University of Namibia, Windhoek, Namibiae-mail: [email protected]

J. R. HenschelGobabeb Training and Research Centre, Desert Research Foundation of Namibia, Windhoek, Namibiae-mail: [email protected]

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24 Environ Fluid Mech (2012) 12:23–44

1 Introduction

Seeds play an important role in the ecosystems of deserts since they influence plant distribu-tion and provide food for insects and therefore the animals that feed on the plants and insects.The seed morphology (Fig. 1) suggests that wind serves as the primary source of transport,however, exactly how the wind allows the seeds to collect in the specific locations was up tonow poorly understood.

There have been a number of studies on the near surface wind patterns over dunes [1–5]but only a few [1,2] included the use of computational fluid dynamics and these were limitedto simple dune geometries, steady state simulations and two-dimensional flow. Some studieshave been done on more complex dunes [4,5] and provided valuable information on the dunedynamics due to sand movement but the study of the near surface wind patterns were limitedto on-site measurements with smoke candles and other measurement techniques. In thesestudies size, complexity and limited computational resources at the time prevented detailedthree-dimensional numerical analysis of the near surface wind patterns.

With the development of high performance computing systems it has become more viableto do large scale simulations over complex geometries. For this reason the current study imple-mented computational fluid dynamics to analyse the three-dimensional flow field around aexisting dune geometry. The study differs from previous studies in that it looks at seed par-ticles rather than sand particles which have completely different aerodynamic properties.It also progresses from previous studies that were limited to simple, two-dimensional dunegeometries by using a complex three-dimensional model and contributes to previous studiesregarding linear dunes [4,5] by using CFD to analyse the flow.

For the CFD simulations a geometry and boundary conditions are required. Since no exist-ing model or inlet conditions were available for the specific dune type (linear dune) in mind,it was necessary to map the dune as well as measure the velocity profiles in the dune fieldover a course of time [6]. The study aimed to investigate how the wind patterns over the dunemay influence where seeds are deposited by introducing seed particles into the flow field andthen track the route and deposition locations. The paper focuses principally on the educateduse of CFD software to describe the wind patterns and particle movement that result.

Fig. 1 Vegitation pattern on linear dune as well as a Stipagrostis sabulicola seed

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Environ Fluid Mech (2012) 12:23–44 25

2 Simulation domain

For this project the finite volume (FV) method was used throughout and therefore a meshwas needed to discretize the flow domain for calculation purposes. At the time of the studythere were no existing dune models available. Therefore, it was decided to create a surfacemesh by mapping a portion of a linear dune. The dune was mapped by walking transverseand contour sections of the dune and recording readings with the differential GPS [6] everyfew steps. After an iterative process of inspection, modification and additional mapping thepoint cloud was improved to a point where the surface generation was possible. The pointcloud along with the initial surface is presented in Fig. 2.

From the point cloud a rough surface (Fig. 2) was generated and then processed to forma workable model. The modelled surface was compared to a typical linear dune in the field.This comparison can be seen in Fig. 3. The comparison shows the modelled surface relateswell to the actual surface as well as some in the vicinity. The general features are visible inboth the model and the actual dune. The regions where the modelled dune surface differsfrom the actual dune were judged acceptable for the scope of this project. This comparisonsupports the choice of the dune site as well as mapping and modelling techniques used.

The next step was to generate the FV mesh. To generate the mesh a built-in automaticmeshing utility called snappyHexMesh in OpenFOAM was used. SnappyHexMesh was usedto generate unstructured meshes in OpenFOAM for complex geometries. Although unstruc-tured meshes are more costly than structured meshes on a cell-by-cell bases they allow forlocal mesh refinement as well as the use of arbitrarily shaped cells. Unstructured meshes facil-itate the transition between regions of varying mesh densities and therefore have the potentialto improve computational accuracy without the penalty of global refinement [7] and so lim-iting increases in computational resources required. Furthermore, arbitrarily shaped meshelements also allow for a geometry fitted mesh which improves the comparability of thenumerical results to the actual occurrence in the field. The finished mesh consisted primarilyof hexahedral cells with some polyhedral and boundary layer cells (prism layers) along thedune surface. The mesh is shown in Fig. 4.

Hexahedral cells are preferred when compared to tetrahedral cells since less are requiredto achieve a numerically converged solution. Furthermore, hexahedral cells have less com-putational faces than that of some polyhedral cells and therefore require less computational

Fig. 2 Point cloud of mapped dune geometry

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Fig. 3 Dune comparison: a modelled dune and b actual dune

Fig. 4 Three-dimensional actual dune mesh

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resources. Also, since a hexahedral mesh can be reasonably aligned with the flow stream-lines, which would generally be impossible with tetrahedral cells, they are less prone to falsediffusion and therefore considered more accurate [8]. Hexahedral cells, however, limit thegeometry fitting capability but this is overcome by combining the use of hexahedral for thebulk mesh with polyhedral along the surface.

The unstructured mesh configuration facilitated refinement of the mesh near the groundsurface and recirculation zone where viscous effects were considered important. Mesh refine-ment also served to limit false diffusion in the wake area, where vortex structures producesangled flows with relation to the mesh configuration. Furthermore, the refined mesh alsoprovided more resolution and so allowed smaller flow structures to be resolved in the refinedregions. Prism layers were used on the dune surface to help improve boundary flow devel-opment and reduce false diffusion.

The final mesh, shown in Fig. 4, was used for all the three-dimensional simulations. Asufficiently long inlet region was included to allow the flow to settle. The inlet section wasset to be 15.5 dune heights from the inlet boundary to the start of the windward slope. Theoutlet was positioned 25.5 dune heights downwind of the lee-side face to allow enough spacefor large flow features created by the 97.3 m high dune geometry to fall within the simulationdomain. The top boundary was set 20.5 dune heights above the ground surface to limit effectof the upper boundary on the flow. Lastly, the domain was set to be 2 dune heights wide.

3 Flow measurements

In order to obtain the velocity profiles at various positions of the dune a 10 m high mast withfour wind anemometers were used. The anemometers were positioned at heights of 2.5 m,5 m, 7.5 m and 10 m above ground level. The data was extrapolation to obtain the velocityprofile for higher flow regions. The wind mast is shown in Fig. 5.

The measurements were conducted at three different locations. The three locations wherethe mast was set up for wind profile measurements are also indicated in Fig. 5. A data-loggerwas used to store 10 min averages of the measured data. The mast was first located at point1 on the interdune, approximately halfway between the two neighbouring dunes, in order tomeasure the inlet velocity profile conditions. Then, the mast was repositioned at location 2and 3 in order to see how the flow changed as it was influenced by the upward slope of thedune.

4 Seed model

The seed selected for modelling and simulation was of the plant species namedStipagrostis sabulicola. This species was selected in collaboration with Muller [9] forits abundance in the Namib Desert, which made it suitable for statistical analysis, as well asfor its morphological properties. A typical seed is shown in Fig. 1.

Muller [9] examined a number of seeds and provided statistical data of the morphologicalfeatures of these seeds. In addition to the measuring of the morphological attributes of theseeds, drop tests were also conducted to determine the fall rate of the seeds in air. The droptests were conducted in still air in a laboratory at room temperature. The drop tests consistedof dropping a number of seed samples over a distance and measuring the time of the fall.Due to the small mass and large flight appendages the seeds reached terminal velocity veryclose to the start of the drop. Therefore, the fall rate calculated over the 2 m high drop was

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Fig. 5 The wind mast and measurement locations

considered to be an accurate enough estimation of the terminal velocity. The terminal velocitywas determined to be 0.57 ± 0.09 m/s [9].

For simulation purposes the seeds were modelled as spherical particles with equivalentmass, drag and effective diameter. The particle drag was determined in a similar manner tothat used by Combrinck [10]. Since the drag coefficient and the particle radius were unknown,an iterative process was required where an initial condition for the drag was first chosen andthen recalculated. The effective radius of the particle was then calculated by rearranging thedrag equation. The drag is given by

CD = D/(ρU 2 A/2) (1)

where CD is the drag coefficient, D is the drag force, ρ is the air density, U is the air flowvelocity and A is the projected area. When the air velocity is taken as the terminal velocitythen the drag force equals the gravitational force on the particle. Therefore the drag forcecan be substituted with

D = F = mg (2)

where m is the particle mass and g is the gravitational constant. Furthermore, the area can bewritten as the cross sectional area of a sphere as

A = πr2 (3)

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where r is the radius of the sphere. By substituting Eqs. 2 and 3 into 1 and rearranging to getthe radius gives

r =√

2mg/(ρU 2πCD) (4)

The first step was to calculate the particle radius with Eq. 4 by choosing the initial valueof the drag coefficient. The next step was to calculate the Reynolds number (Re) with

Re = ρU (2r)/μ (5)

where μ is the dynamic viscosity of air. It was possible to approximate the drag on the par-ticle by using Eqs. 6 and 7 proposed by Flemmer and Banks [11] which are based on theReynolds number. This correlation was chosen since it is reported to be superior to othersimilar approximations [11]. This finding is also supported by the calculations of Combrinck[10]. The Flemmer and Banks [11] approximation is given by:

CD = (24/Re)10E , (6)

E = 0.261Re0.369 − 0.105Re0.431 − 0.124/(1 + (log10 Re)2) (7)

In order to adjust the initial value of the drag coefficient an error value was calculatedwith

E = | CiD − Ci+1

D | (8)

From the calculations the final seed was modelled as a spherical particle with a dragcoefficient of 0.64, an effective diameter of 7.5 mm and a mass of 0.562 mg.

5 Numerical model

Similar to the discretization of the simulation domain into control volumes, the governingequations in CFD are discretized and solved for the values at the centres of the control vol-umes. In order to calculate the flux between control volumes in a FV mesh some values areneeded at cell faces. To calculate these values different differencing schemes can be used.Two of the most common differencing methods in use are upwind and central differencing.Upwind differencing has been shown to be only first order accurate on the basis of the Taylorseries truncation error and therefore suffers significantly from false diffusion [12]. One wayto counter false diffusion is by grid refinement, but the extent to which grid refinement isnecessary usually implies a dramatically increase in the computational resources required.Still, upwind differencing has the advantage that it is inherently stable and simple to extendto multi-dimensional problems.

Central differencing on the other hand has been shown to be second order accurate andthus suffers significantly less from false diffusion compared with upwind differencing [12].The problem with the central differencing scheme is that it is less stable than the upwinddifferencing scheme. While this is countered by the deferred correction and flux blendingimplementations, central differencing has also been shown to result in unrealistic flow fea-tures when the convection term strongly dominates the rest of the equation system [7].

For the steady state simulations in this project an upwind interpolation scheme was usedto initialise the flow field due to the inherent stability. For the final results, however, higherorder differencing schemes were employed. To ensure the stable convergence of the solutionwhile maintaining a higher accuracy, blended schemes were used. The GAMMA switch-ing/blending scheme, first introduced by Jasak [13], uses central differencing in the bulk

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of the solution but switches to a blend of central and upwind differencing to stabilize theequations when the interpolation tends to unboundedness [7]. The GAMMA scheme wastherefore selected for its accuracy while maintaining reasonable stability.

Turbulence models are used in CFD to compensate for the lack of sufficient mesh refine-ment that is needed to resolve small scale eddies in the flow field. Probably the most widelyknown turbulence model is the original k-ε model presented by Launder and Spalding [14].This turbulence model does, however, have a number of drawbacks due to the isotropiceddy viscosity hypothesis that results in reduced accuracy in flows associated with stronganisotropy, stagnation and separation [15]. Since the emergence of the original k-ε modelmany new models were developed to address the inherent shortcomings and a number ofstudies review these different turbulence models [15–18]. A number of turbulence modelswere investigated in this study [6] but the realizable model was chosen due to its simplicity,robustness and reasonable accuracy as was reported by Kim and Boysan [15].

Due to the mesh refinement levels required by low Reynolds number approaches thatincrease the computational resources required it was desired to implement a high Reynoldsnumber approach. High Reynolds number turbulence models use wall functions to approxi-mate the near wall flow by analytically calculating the boundary layer in the near wall regionfrom equations fitted to extensive experimental data which is also known as the law-of-the-wall approach. The wall functions were, however, not modified to incorporate the effect ofsurface roughness since the majority of the dune surface was regarded as smooth.

With the standard release of OpenFOAM-1.6 a Lagrangian particle solver was providedwhich made it possible to simulate and track particles in the flow using this program. Thesolver did, however, need to be converted to incompressible flow first. The way the solverworks is to track Lagrangian particles as they are transported in a flow field. The flow fieldin this case is a single time step or time-averaged results from a previously calculated flowfield. The flow therefore stays constant while the particles are tracked. This method has itsdrawbacks since it does not fully represent the flow dynamics such as turbulence fluctuationsinherent to the time dependent flow. Turbulence fluctuations influence the particles ability tostay suspended as well as to promote erosion. The ideal would have been to implement theparticles in a transient LES solver but this was not possible due to the limited computationalresources and software capabilities available at the time. Since an incompressible, transientflow and Lagrangian solver was not available with the software and the development of sucha solver required deeper understanding of the C++ programming and OpenFOAM libraries,it was decided that, despite the drawbacks of using a single time step to track the particles, theresults would still provide valuable information towards the transportation and distributionof particles in the flow field.

The particle tracking solver requires a dictionary that specifies the particle properties aswell as the injection properties. The particle properties required are the particle mass, effec-tive density and drag model. In this project a sphere drag model was chosen to calculate thedrag on the particles. In the particle properties dictionary the gravitational constant was alsodefined. Furthermore, the wall interaction method was set so that the particles would bouncewhen coming in contact with the wall instead of attaching to it. This wall interaction model,therefore, resembled particle characteristics observed in the field. Particles were injectedinto the flow at heights of 0.25, 0.5, 1 and 1.5 dune heights. At 0.25 and 0.5 dune heights 5particles were injected with 0.5 dune height horizontal spacing. At 1 and 1.5 dune heights15 particles were injected with 0.1 dune height horizontal spacing.

Since the fluid under consideration was air at 20◦C, the kinematic viscosity (ν) was set to1.5 × 10−5 m2/s. The flow model was further defined as Newtonian, turbulent and incom-pressible. Under-relaxation factors in the range of 0.001–0.01 were needed for steady state

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Table 1 Simulation boundaryconditions

Boundary Velocity Pressure

Inlet u = U(x, y, z) ∂p/∂nb = 0

Outlet ∂u/∂nb = (0, 0, 0) p = pout

Wall u = (0, 0, 0) ∂p/∂nb = 0

Symmetry un = (0, 0, 0), ∂u/∂nb = (0, 0, 0) ∂p/∂nb = 0

simulations to improve convergence and stability of the simulation. The steady state solverused in the simulations is based on the semi-implicit method for pressure linked equations(SIMPLE) algorithm which was first introduced by Patankar and Spalding [19].

The types of boundary conditions used at the inlet, outlet, ground and top boundariesincluded a mixture of the Dirichlet and Neumann boundary conditions. The boundary con-ditions are summarised in Table 1.

In order to simulate the effect of flow upstream of the inlet it was decided to implementan equilibrium atmospheric boundary layer (ABL) velocity profile at the inlet that correlatedwith measurements in the field. The implementation of the velocity profile improved the sta-bility of the flow in the region where the inlet and ground boundaries meet. This is becausethe velocity profile starts from a zero at the ground where the alternative block inlet profilestarts from a high velocity.

Richards [20,21] proposed ABL profile equations for k-ε based turbulence models. Theequations include the mean wind speed U, turbulence kinetic energy k and the turbulencedissipation rate ε. The profiles can be expressed by the following equations:

U (y) = (u∗ABL/κ)ln((y + y0)/y0), (9)

k(y) = (u∗ABL)2/

√Cμ, (10)

ε(y) = (u∗ABL)3/κ(y + y0), (11)

κ2 = (Cε2 − Cε1)σε

√Cμ, (12)

where u∗ABL is the ABL friction velocity, κ is the Von Karman constant and y is the vertical

distance from the ground [22]. Also, Cε2, Cε1, σε and Cμ are constants of the original k-εmodel.

For the simulations inlet conditions it was decided to use the west wind data obtained frommeasurements. West wind conditions were common during on-site measurement. Since thedunes are aligned north to south (Fig. 2) the west wind profile could also serve to simulate aneast wind condition. The west wind choice was also made to simplify the flow features and tofacilitate comparison to existing and future data in literature. Therefore, the data was filtereduntil the west wind data was isolated. The data was then averaged to get a representation ofwhat the typical west wind profile looks like.

The inlet conditions were developed by fitting the measured data to a velocity profile withshape of Eq. 13 using the least squares method. Once the values of b and m were known, thefriction velocity u∗

ABL and surface roughness height y0 were calculated with the help of Eqs.13–15. The surface roughness height and the friction velocity were used in the simulationsfor the inlet boundary condition by implementing them in Eqs. 9–12.

y = bmu, (13)

y0 = b, (14)

u∗ABL = κ/ ln(m), (15)

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Fig. 6 Wind profiles fordifferent measurement positionson the dune

Comparing measured west wind profiles with similar free stream velocities in Fig. 6reveals how the profiles change as the flow moves up the slope. The west wind profile on theinterdune is fairly well developed. The profile reveals near surface acceleration as it movesup the windward slope. This is indicated by the increasing near surface velocities of thedifferent profiles.

6 Validation

The numerical model used for the dune simulations was validated using a number of differentcases, each case presenting the opportunity to test different flow characteristics that occur in atypical dune environment. Thus, the validation cases include flow over the three-dimensionaldune, flow around a hydrofoil and flow over a backward-facing step. In all cases the Reynoldsnumbers were set to be in the fully turbulent regime to ensure that the test cases were suitablefor validating the numerical model.

The first validation case involves comparing velocity profiles of the three-dimensionaldune simulation results against the data obtained from field measurements using a 10 m highwind mast. This comparison is shown in Fig. 7. The comparison shows the velocity profilessimulated compare well with measured data which supports the numerical model choice. Thedata shows that the typical distortion of the velocity profile as the air “ramps” up the dune iscaptured correctly. The Reynolds number based on the dune height (h) of 97.3 m and air speedat that height (Uh) of 4.09 m/s was 2.65 × 107. The data is presented non-dimensionalizedrelative to the dune height and the air speed at that height. It should be noted while it wasnot possible to measure the velocity profile at the inlet and a number other areas of intereston the dune, such as in the recirculation zone, at the same time, the typical time-averagedmeasured profiles compare well with the numerically obtained coupled velocity field.

The second validation case is that of flow around a hydrofoil (racing yacht keel). Thetwo-dimensional simulation predictions are compared to wind tunnel particle image veloc-imetry (PIV) results in Fig. 8. The PIV experiment was conducted as part of a project thatlooked at racing yacht keel designs in a wind tunnel [23]. The angle of attack in these cases

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Fig. 7 Comparison of CFD velocity profile results against wind mast measurements [6]

was set to zero and the air flow speed so that a Reynolds number of 8 × 105, based on thefree-stream velocity (U∞) and the chord length (l), was achieved. The hydrofoil presentedan opportunity to test the numerical model for flow characteristics such as stagnation andstreamline curvature without flow separation.

The results compare very well for both the tangential (x) and normal (y) components ofthe flow with the exception of the nose region. The discrepancy near the nose region for thenormal flow component, although small (5% of free-stream velocity), could be due to thelack of eddy-viscosity based model accurately reproduce the flow in the stagnation region[15]. More likely, the discrepancy is a result of reduced accuracy in the PIV measurementsin the nose region. The limited particle seeding equipment had the effect of reducing the sizeof the region that could be accurately measured. Since the goal of the measurement [23] wasto use data from along the hydrofoil and towards the wake region to validate CFD results, thefocus was shifted accordingly, which in turn reduced the accuracy of the measurement of theoncoming flow and nose region. Even so, the data of the nose region is included here since itshows very good comparison with the tangential velocity component. The results from thiscomparison, therefore, supports the choice of numerical model since an error of 5% of thefree-stream value is considered acceptable.

The third comparison involves fully developed channel flow over a backward-facing step.The model was set up to achieve an expansion ratio of 1.125 and a Reynolds number of3.65×104 based on the average inlet velocity (Ure f ) and the step height (hs) as in the exper-iment by Driver and Seegmiller [24]. In this case the flow is relatively steady and includesflow separation, reattachment and recirculation, thus flow features expected in a typical duneenvironment. The comparison of the velocity profiles can be seen in Fig. 9 and the results arein good agreement with the measured data with an error of less than 7% of the inlet velocity.A reattachment length of 6.15 times the step height was measured and compares well withthe 6.13 value reported by Driver and Seegmiller [24] with only a 0.3% error. This validation,

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Fig. 8 Comparison of CFD simulation results against wind tunnel PIV measurements of flow around ahydrofoil profile (Re = 8 × 105) [23]

therefore, shows that the numerical model is capable of reproducing the separating flows thatare present in the dune simulation.

The wall pressure coefficients are given by

Cp = (P − Pref )

/1

2ρU 2

re f (16)

The results for Cp for the backward-facing step are shown in Fig. 10. The pressure dropin the wake region is a result of flow entrainment due to the high velocity gradient at stepheight and the effective viscosity that in effect drags flow away from the step. The results

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Fig. 9 Comparison of two-dimensional CFD velocity field results against experimental results by Driver andSeegmiller [24] of flow over a backward-facing step

Fig. 10 Comparison of two-dimensional CFD wall pressure results against experimental results by Driverand Seegmiller [24] of flow over a backward-facing step

compare well with the measured data of Driver and Seegmiller [24] with slight deviation inthe near wake and far downstream regions with a maximum deviation of 2% of the inletvelocity stagnation pressure, 1

2ρU 2re f . The slight deviation in the wake may be attributed to

a overproduction of turbulent kinetic energy by the eddy-viscosity turbulence model in thehigh shear region similar to what was reported by Kim and Boysan [15]. The effect of theoverproduction is increased eddy-viscosity in the near wake region that causes a higher shearand, therefore, a slightly larger flow entrainment. This slight deviation is, however, judgedacceptable and, therefore, the numerical model pressure prediction is considered validated.

7 Results and discussion

Residual monitoring and grid independence tests ensured that converged numerical solutionshad indeed been achieved. The grid independence study involved three different meshes, onecoarse and two refined meshes. All three meshes included local refinement in the areas of

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high velocity gradients and turbulence activity such as the near surface and in recirculationzone. For the grid independence study the mesh on the lee-side of the dune was refinedrepeatedly. This was done to limit the computational requirements but also since this regionwas expected to be the area where refinement would have the greatest effect. The refinementinvolved consecutive division of a single cells in the x, y and z directions resulting in eightnew hexahedral cells. The grid independence study showed a small change in the velocityprofile for the first refinement but insignificant change for the second refinement [6]. Sincethe first refinement showed grid independence it was selected as the primary grid for thesimulations. The final mesh consisted of 7272639 cells of which the greater majority washexahedral cells. The full mesh statistics are reported in the original study [6].

Since wall functions were used it is important to also achieve the necessary range of y+values for which the wall functions are valid. This required the first cell centre adjacent tothe wall to be placed within the logarithmic part of the boundary layer which relates to ay+ value within the range of 30–500. The simulation y+ values did fall within the requiredrange with limited exceptions near the crest region.

The pressure field is shown in Fig. 11 and show a high pressure at the dune toe on thewindward side, a low pressure at the crest extending to the lee-side and a higher pressure at thelee-side of the dune. The high pressure zone at the dune toe results from the flow stagnationin the region due to the dune geometry in the flow path. The low pressure region near the crestcorresponds to the separation in that area as well as the rapid expansion in local flow area.Flow reattachment on the down-wind side produces a small but important higher pressure inthe region where flow recovery occurs. Between the high and low pressure regions pressuregradients exist. The pressure gradient on the wind-ward slope has the effect that it causesnear surface flow acceleration up the slope of the dune. This effect is also reflected in the flowmeasurements showed in Fig. 6. On the down-wind side of the dune the pressure gradientis much lower but still has an effect by driving the recirculation by accelerating flow nearthe surface towards the dune. It is important to keep in mind that the pressures presented in

Fig. 11 Relative pressure field over a tree-dimensional actual dune geometry

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this report are represented in the kinematic form (P/ρ), as it was used in OpenFOAM by theincompressible solver. The pressure is presented relative to a zero pressure outlet boundary.

The pressure increase at the dune toe region was also investigated in more detail by look-ing at pressure coefficient and total pressure plots. The pressure coefficients showed that thestatic pressure increase was more or less 15 % of what would have occurred if the free-streamstagnated at the dune toe. The total pressures showed that the static pressure increase andvelocity decrease at the dune toe cancelled one another out. Both the pressure coefficientsand total pressures supported the flow fields predicted. These are not included here due tospace limitations.

The velocity field is shown in Fig. 12 and shows that there exists a low velocity zone atthe toe on the windward side, a high velocity at the crest and a low velocity recirculationzone on the lee-side. The velocity field corresponds to the pressure field in such a way thatair approaching the dune from the windward side will decelerate due to the high pressurezone at the toe. From the dune toe the wind accelerates up the dune slope due to the pressuregradient between the high pressure at the toe and the low pressure at the crest. From the crestthe flow decelerates towards the lee-side of the dune.

From Fig. 12 the approach flow appears stable and fully developed. This confirms theuse of the equilibrium ABL inlet conditions as well as the placement of the inlet boundary.Furthermore, the free-stream velocity can be seen to stay the same all along the top boundarywhich supports the placement of the top boundary. Similarly, the effect of the outletboundary was inspected and it was found to be placed sufficiently far from the region ofinterest to have an effect on the flow.

A significant flow characteristic, associated with the curvature of the dune, can be noticed.Flow approaching the dune at a right angle to the general dune direction will cross the dunecrest first at the leading crest edge. Once the flow crosses the dune separation occurs and arecirculation zone is formed. Due to the three-dimensional shape of the dune the recirculation

Fig. 12 Velocity magnitude over a tree-dimensional actual dune geometry

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Fig. 13 Effect of crest line curvature on resurculating flow

zones behind the leading and trailing crest edges would be offset and therefore transverseflow will occur.

Another way to explain this transverse flow is to consider the different pressure zonescaused by the recirculation zones. If the flow features at different cross sections of the duneare compared (Fig. 13), similar flow features are recognised in different sections. Both crosssections in Fig. 13 show separation, recirculation and reattachment but the size of the recir-culation zones differ as well as the offset from the mean crest line. This has the effect thatthe local high and low pressure zones will be slightly offset due to the curvature of thedune. Because of this offset, transverse flow will occur as a result of pressure difference. Thetransverse flow disturbs the recirculation vortex structure and reattachment flow and formsspiral shaped mixing of the flow moving away from the dune.

Therefore, the flow indicated in Fig. 14 has a three-dimensional shape as a result from thecrest-line curvature of the dune. The effect of the crest line curvature on the recirculating flowshows the importance of three-dimensional simulation. This three-dimensional recirculationzone effect is impossible to capture while looking only at the two-dimensional simulations.

A closer inspection of the recirculation region is shown with the help of streamlines inFig. 14. In Fig. 14 two large recirculation zones can be seen behind the trailing crest edges ofthe dune. On the other hand the flow crossing the dune at the leading crest edge flows straightbetween the two vortex structures on either side. Further investigation of the flow behind theleading crest edge shows a smaller recirculation zone in that area. It can also be seen that therecirculation region looks almost continuous but is of different shape and size depending onwhere on the dune it occurs.

Streamlines give a good indication of the path that very light particles might take. Thestreamlines are especially representative of the particle tracks since the particles in this study

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Fig. 14 Streamline presentation of flow over the dune and recirculation zone

are very light weight and have large flight appendages that increase the drag. The streamlinesshow that almost weightless particles will have a greater chance of ending up in recirculationzones when crossing the dune at the trailing crest edge region than for the particles crossingat the leading crest edge. This is expected since the recirculation zone is larger down-wind ofthe trailing crest edge and therefore there is a greater chance of particles losing momentumin these regions. Particles also have a greater chance of being recirculated up the slope dueto the larger size of the recirculation zone.

A second noticeable flow feature observed in post-processing is a helical vortex that formswhen the approaching flow is at an oblique angle to the crest line. The approaching flow canbe separated into two components just before the point of separation. The first is the flowcomponent perpendicular to the crest line that is responsible for flow separation. The secondflow component is parallel to the crest line. This secondary component introduces a sidewaysmomentum to the recirculation zone, causing it to deform into a helical vortex just beyondthe crest.

A similar helix shaped flow feature was observed by Tsoar [4] using smoke flares. Evenwith a perfectly perpendicular approach flow relative to the mean crest line, there are manyregions where the local crest line would be at an oblique angle to the approach flow due tothe curvature of the dune. Therefore, for the flow conditions presented here a helix vortexwould occur in the regions between the leading and trailing crest edges.

Considering the wall shear stress magnitude shown in Fig. 15, valuable informationcan be gained. The first observation is that the wall shear stress is directly related tothe local velocity. High wall shear stress occurs in regions of high near surface veloc-ity and low wall shear stress occurs in regions of low near surface velocity. Keepingthis in mind it can be noticed that there is a significant difference between the wallshear stress on the windward and lee-sides. The windward side appears predictable andorderly where on the other hand the lee-side appears very unstructured. It also is pos-sible to identify the flow acceleration up the windward slope, high velocity region atthe crest and low velocity on the lee-side of the dune. On the windward side, depres-sions on the dune near the crest region, causes low wall shear stress magnitude due tothe local flow expansion. Furthermore, it is possible to identify different flow accelerationrates up different regions of the dune due to the three-dimensional shape of the windwardslope.

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Fig. 15 Wall shear stress of flow over a three-dimensional actual dune geometry

Knowing that separation and reattachment occur where the wall shear stress, and thereforevelocity, is zero, makes it possible to identify where on the dune these flow features are pres-ent. By closer inspection of the crest region, a low wall shear stress can be identified followingthe crest line all along the dune, suggesting separation in this area. Another observation is theregions of low wall shear stress that are present behind the trailing crest edges, suggestingregions of low velocity as well as flow separation and reattachment. High wall shear stress,however, is visible behind the leading crest edge further motivating the observations madepreviously with regards to the effect of the crest line on the recirculation zone.

Wall shear stress has a significant effect on the transportation of particles. Since wall shearstress is directly related to velocity, it is possible to get a feel for the rate with which particleswould be entrained in different regions along the surface. More importantly high wall shearstress regions indicate regions of possible erosion while low wall shear indicate regions ofpossible particle deposition. This would mean that the windward slope would experienceincreasing erosion as one move up the slope towards the crest. The depression regions nearthe crest experience low wall shear stress and would therefore promote deposition of parti-cles. On the lee-side more particles will deposit in the area behind the trailing crest edge thanbehind the leading crest edge due to the low wall shear stress in the region.

In the work by Wiggs et al. [3] it was suggested that the surface shear stress and theresulting particle erosion and deposition is not only influenced by flow acceleration but alsostreamline curvature. Due to the velocity profile and the turbulence structures in the flow,concave streamlines have a destabilizing effect while on the other hand convex streamlinecurvature has a stabilizing effect. The result is that concave streamline curvature along withflow acceleration suggests particle erosion while convex curvature along with flow decel-eration results in particle deposition. This suggest that particle erosion will occur along thegreater part of the windward slope due to the concave streamlines and flow acceleration upthe slope. Near the crest, where the local depressions are found, convex streamline curva-ture would promote particle deposition. Furthermore, particle deposition is promoted in themajority of the recirculation region due to flow deceleration and convex streamline curvaturein the region.

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Now that the flow features have been investigated it is desired to look at how particlesare distributed in the flow field. As mentioned previously, the Lagrangian solver uses thetime-averaged flow field to track particle movement. Due to the small size, mass and num-ber of particles considered it possible to assume one way interaction from the flow to theparticles. This means that the flow field stays constant while the particles move through theflow. Although the ideal would have been to have a transient flow solver with an incorporatedtransient Lagrangian solver, such a solver that suited the flow conditions was not availablein OpenFOAM at the time. The alternative was to use the steady state flow results to drivethe transient Lagrangian particle solver. Although the solver could not fully resolve particlemovement due to transient turbulence related flow structures, it provided acceptable repre-sentation of the time-averaged particle movement and deposition which was the main focusof the study. The transient particle motion over the dune can be seen in Fig. 16.

One observation that can be made is particles that fall to the surface before the dune slopetend to stay at the toe region on the windward slope. On the other hand, particles that fallto the surface almost anywhere on the slope tend to bounce and roll up the slope, cross thecrest and stay in the recirculation region near the crest on the lee-side. These particles thatbounce and roll up the windward slope have different velocities depending on its position onthe windward slope. Furthermore, particles accelerate up the slope due to the acceleratingwind conditions.

It is also observed that particles starting from the ground to as high as 1 dune heightabove ground, if unobstructed, will all cross the dune close to the crest resulting in a highparticle crossing density in this region. These same particles then tend to lose momen-tum and fall to the surface on the lee-side once they enter the low velocity recirculationregion. Most particles will recirculate up towards the crest. Some particles in recirculationzone seem to get caught up in vortex structures and recirculate more than others beforethey drop to the surface. Furthermore, particles in the recirculation zone move in differentdirections and at different velocities showing the three-dimensional flow features present inthe region. Of the particles in the recirculation zone few move down the slope towards theinterdune.

On the other hand particles starting from high in the atmosphere tend to skip the recir-culation region and only drop to the surface on the opposite interdune. Particles high in theatmosphere move faster than the ones lower down due to the ABL present. Once they fall tothe lee-side interdune the particles tend to bounce and roll in the flow away from the dune.In some cases particles on the interdune are observed to move transversely due to the spiralshapes vortex structures behind the trailing crest edges.

From the results it is clear that a great number of particles would come in contact withthe windward surface due to the size of the dune and bounce and roll along the surface.Also, particles ending up in the recirculation zone fall to the surface and also move along thesurface towards the crest due to the presence of recirculating flow. Therefore, the simulationsuggests that a large amount of seeds will move along the surface. This was also observed inthe field and further confirmed by Muller [9]. Furthermore, it is observed that most particlesstill airborne will cross the dune crest near the ground due to the streamline convergence inthis region. This is also supported by the measurements by Muller [9].

For this study it was important to examine where the seed particles would come torest. It was observed that for the given flow field most particles that moved along thewindward slope, if unobstructed, will at some point come to rest in either the smalldepressions near the crest or more likely the recirculation zone. The particles that fallto the ground in the recirculation zone would experience a low velocity towards thecrest line. As the particles move up the windward and lee-side surfaces towards the

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Fig. 16 Transient particle movement through the time-averaged flow field

crest they would experience an increasing slope. Since the gravitational force increaseswith increasing slope a point would be reached where the gravitational and drag forcesare be in equilibrium. For large number of particles in the flow field driven by steadywind conditions would result in particles coming to rest near this equilibrium zone.Because the dune is not straight and wind conditions are not constant the equilibriumzone may not be straight but would more likely reflect the three-dimensional shape of thedune.

Since the majority of particles bounce and roll along the surface makes them vulnerableto obstructions like existing plants or possible sand ridges on the dune. These obstructions

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would prevent particles to move further up the slope. This means that plants would mostlikely occur near existing plants or in lines due to ridges that prevented particles to movefurther up the dune (Fig. 1).

Lastly, in the recirculation zone particles seem to group in areas behind the trailing crestedge. This supports the observations from the surface shear investigation in Fig. 15. It alsoexplains some plant growth observations in the Namib Desert where the seeds accumulatedbehind a large trailing crest edge region of the dune (Fig. 1).

8 Conclusions

Mapping the dune with a differential GPS made it possible to get the necessary accuracywhile still being time and cost effective. The final dune surface compared well with the dunemapped as well as with neighbouring dunes. This supported the choice of the site as well asmapping scheme.

Some typical flow features associated with sand dunes were identified. Flow featuresinclude the high pressure at the dune toe and low pressure at the crest. The high pressureat the toe causes streamwise deceleration of the flow approaching the toe that result in alow velocity in the region. The difference between the high and low pressure zones forma pressure gradient on the windward face resulting in flow acceleration up this slope to amaximum velocity at the crest. Recirculation is also visible on the lee-side of the dune.

It was also clear that the three-dimensional geometry introduces a number of flow featuresthat would be impossible to identify with only two-dimensional simulations. Some featuresinclude cross flow and different recirculation zone sizes due to the crest line curvature. Bylooking at the surface shear plots it was possible identify regions of low shear stress mag-nitude behind trailing crest edges which suggests lower velocities, greater separation andtherefore more seed accumulation in this region of the lee-side face of the dune.

By injecting particles into the steady state RANS turbulence results it was possible toobserve how particles would move in a time-averaged flow field. The results showed thatmost particles, if unobstructed, would eventually end up in the recirculation zone. Since mostparticles fall to the dune surface at some point they tend to roll and bounce and get transportedtowards the recirculation zone. In this rolling motion they can be obstructed by existing plantsor sand ridges. This might explain some of the line groupings of plants as shown in Fig. 1.Furthermore, most particles seem to cross the dune in the first few meters above the groundat the crest. This observation is supported by the work of Muller [9].

Acknowledgements The authors gratefully acknowledge the generous funding received from the SouthAfrican National Research Foundation through its South African Regional Cooperation Fund.

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A CFD study of wind patterns over a desert dune and the effect on seed dispersion

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