Numerical modelling of flow structures over idealized transverse aeolian dunes of varying geometry

Geomorphology 59 (2004) 149 – 164 www.elsevier.com/locate/geomorph

Numerical modelling of flow structures over idealized transverse aeolian dunes of varying geometry Daniel R. Parsons a,*, Ian J. Walker b, Giles F.S. Wiggs c b

a School of Earth Sciences, University of Leeds, Woodhouse Lane, Leeds LS2 9JJT, UK Department of Geography, University of Victoria, P.O. Box 3050, Station CSC, Victoria, British Columbia, Canada V8W3P5 c Department of Geography, University of Sheffield, Western Bank, Sheffield S10 2TN, UK

Accepted 16 July 2003

Abstract A Computational Fluid Dynamics (CFD) model (PHOENICSk 3.5) previously validated for wind tunnel measurements is used to simulate the streamwise and vertical velocity flow fields over idealized transverse dunes of varying height (h) and stoss slope basal length (L). The model accurately reproduced patterns of: flow deceleration at the dune toe; stoss flow acceleration; vertical lift in the crest region; lee-side flow separation, re-attachment and reversal; and flow recovery distance. Results indicate that the flow field over transverse dunes is particularly sensitive to changes in dune height, with an increase in height resulting in flow deceleration at the toe, streamwise acceleration and vertical lift at the crest, and an increase in the extent of, and strength of reversed flows within, the lee-side separation cell. In general, the length of the separation zone varied from 3 to 15 h from the crest and increased over taller, steeper dunes. Similarly, the flow recovery distance ranged from 45 to >75 h and was more sensitive to changes in dune height. For the range of dune shapes investigated in this study, the differing effects of height and stoss slope length raise questions regarding the applicability of dune aspect ratio as a parameter for explaining airflow over transverse dunes. Evidence is also provided to support existing research on: streamline curvature and the maintenance of sand transport in the toe region; vertical lift in the crest region and its effect on grainfall delivery; relations between the turbulent shear layer and downward forcing of flow re-attachment; and extended flow recovery distances beyond the separation cell. Field validation is required to test these findings in natural settings. Future applications of the model will characterize turbulence and shear stress fields, examine the effects of more complex isolated dune forms and investigate flow over multiple dunes. D 2003 Elsevier B.V. All rights reserved. Keywords: Aeolian; Dunes; Computational Fluid Dynamics (CFD); Flow acceleration; Flow separation; Flow reversal; Flow recovery; Aspect ratio

1. Introduction The role of secondary flow structures in the morphology, dynamics and spacing of desert sand dunes * Corresponding author. Tel.: +44-113-343-6624; fax: +44-113343-5259. E-mail address: [email protected] (D.R. Parsons). 0169-555X/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2003.09.012

has been the focus of much recent research (McKenna Neuman et al., 1997, 2000; Wiggs, 2001; Walker and Nickling, 2002) and has been complimentary to similar investigations of bedforms in fluvial environments (e.g., Nelson et al., 1993; Bennett and Best, 1995). Building upon earlier work over low-angled hills (e.g., Jackson and Hunt, 1975; Bowen and Lindley, 1977; Bradley, 1980; Britter et al., 1981; Zeman and

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Jensen, 1987; Raithby et al., 1987), this recent research has greatly improved our understanding of dune form – flow interactions (stoss flow acceleration, crestal separation, lee re-circulation, re-attachment, etc.). This progress has been achieved with field studies of windward flow dynamics (Lancaster et al., 1996; Frank and Kocurek, 1996a; Wiggs et al., 1996; McKenna Neuman et al., 2000) and lee-side flow separation and recovery (Frank and Kocurek, 1996b; Walker and Nickling, 2002, in press (a,b)). However, while field and laboratory studies have succeeded in providing some imprecise relationships between dune aspect ratio and flow acceleration (e.g., Lancaster, 1994) and have provided detailed empirical relationships characterizing the flow field over transverse dunes and related these to dune height and lee-side flow re-attachment and recovery (e.g., Frank and Kocurek, 1996b; Walker and Nickling, in press (a)), questions remain as to the presence and the sensitivity of these secondary flow structures to changes in dune geometry. Progress in this regard is hampered by paucity of additional field and laboratory studies to validate such relationships and by the relatively small number of dune geometries investigated. In particular, the complex turbulent structure in the lee side of dunes has generally precluded the measurement of flow structure in this region, largely because of limitations in instrumentation (Nickling and McKenna Neuman, 1999; McKenna Neuman, 2002). Mathematical modelling of airflow over dunes has provided additional data for the investigation of secondary flow regimes, but studies to date have also suffered from an inability to simulate the highly turbulent flow in the lee of dunes (e.g., Walmsley et al., 1982; Raithby et al., 1987; Stam, 1997). For example, Stam (1997) applied an analytical flow model based on Jackson and Hunt’s (1975) boundary layer model, which is unable to solve the reverse flow lee-side eddy. This limits the calculation of flow structures to low angle dunes where lee-side eddies are not present. Stam (1997) notes that numerical techniques are required successfully to simulate flows over a greater range of dune forms. Numerical flow models have been widely applied in engineering disciplines for many years. In the last few years, there has been a proliferation of the use of Computational Fluid Dynamics (CFD) in the fields of

geomorphology and hydrology (see Bates and Lane, 1998). These models provide spatially rich data on flow field properties that facilitate considerable insight and understanding of the distribution of complex flow processes. Indeed, these models can provide details of the flow field that are often difficult to measure and offer controlled conditions in which certain aspects of the experimental set up can be varied rapidly. This paper applies a CFD model to flow over idealized transverse aeolian dunes and describes the sensitivity of different elements of the flow field to variations in geomorphic parameters. The model used is capable of simulating the highly turbulent reverse flow vortex in the lee of the dune and so is able to provide an acceptable solution of the downwind distance to flow re-attachment given variations in dune height, windward slope length and, thus, aspect ratio.

2. Methods 2.1. Numerical model This paper employs a numerical model based upon the PHOENICSk 3.5 code, which is one of several commercially available CFD programs. The model solves the elliptic form of the Reynolds-averaged Navier – Stokes equations in two dimensions with a finite volume method: a cuboidal grid in a Cartesian frame. The form of the dune was represented within the model using a relatively new ‘cut-cell’ porosity treatment, where the intersections of the inserted geometry with the grid lines are determined and the areas and volumes of partially blocked cells are calculated to a high degree of accuracy (Spalding and Zhang, 1996; Yang et al., 1997a,b). The equation formulation is modified to account for the local nonorthogonal intersection of the dune with the grid cells, resulting in significantly enhanced predictions of nearsurface flow dynamics. The hybrid-upwind interpolation scheme (Peclet number = 2) applied in the model is only first order accurate and can suffer from numerical diffusion when flow is highly skewed relative to the grid. Nevertheless, it is more stable than higher-order schemes, and investigations analogous to this present one have indicated that errors due to the interpolation

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scheme are not likely to be significant (e.g., Waterson, 1994). The pressure and momentum equations were coupled through the SIMPLEST algorithm (a variation of SIMPLE; Pantaankar and Spalding, 1972) where pressure and velocity fields were iteratively calculated until continuity errors in mass and momentum were adequately small (residuals were < 0.01% of inlet flux). Turbulence closure was achieved through application of a two-equation k – e model, modified by renormalisation group theory (Yakhot et al., 1992). This turbulence model is recommended for simulating flows with significant mean strain and shear. For example, it has been shown to perform better in the prediction of sheared and re-circulating flows over backward facing steps (e.g., Bradbrook et al., 1998). 2.2. Model application and assessment Fig. 1. Simulated incoming plane bed velocity profile.

The model was initially applied to the experimental set-up of Walker and Nickling (in press (a)) (see Parsons et al., in press, for full details). Mass flux values were specified for each grid cell in the upstream inflow, providing an incoming velocity profile for the model. In order to simulate upwind effects of the dune (e.g., pressure stagnation and flow deceleration), this profile had to be specified far enough upstream of the dune. A modelled inflow profile was specified that implicitly produced the measured plane bed boundary layer (with a free stream velocity of 13 m s 1) at the point of dune intersection (see Walker and Nickling, in press (a)). This inflow profile was used in all the experiments in this paper (Fig. 1). At the outlet profile, a zero pressure boundary condition was applied, and, thus, calculated pressure values for all cells in the domain were defined relative to this. The length of the simulation domain was 960 cm and flow depth was 76 cm, with the dune toe positioned at 500 cm into the domain, matching the conditions and dimensions of the wind tunnel simulation of Walker and Nickling (in press (a)). In the cell at the fluid– solid interface, it was necessary to prescribe conditions for the velocity and turbulence parameters. For this purpose, the universal ‘Law of the Wall’ was applied in the interface cells. In this experiment, smooth wall conditions were applied, matching the roughness experimental set-up of Walker and Nickling (in press (a)).

Full verification and validation of the numerical model to these experimental conditions and the flow measurements obtained was demonstrated and discussed by Parsons et al. (in press). Validation was based on 415 predicted points within the model domain, which coincided with the locations of measurements taken in the wind tunnel experiment. Although excellent agreement between the measured and predicted velocities was established, with correlation coefficients for streamwise and vertical velocity components of 0.97 and 0.83, respectively (Table 1; Fig. 2), there are significant zones of disagreement, particularly for the lower velocities (Fig. 2). Parsons et al. (in press) identified that the majority of these points are from the lee separation zone where, due to design limitations, the measuring probe (a TSIR IFA 300 constant temperature hot-film anemometer) applied in the wind tunnel experiment (Walker and Nickling, in press (a)) was unable to resolve the highly turbulent and negative velocities within this region. Thus, although the validation process identified some notable differences between the measured and modelled results, they are primarily due to the limitations of the measuring instrument rather than that of the numerical model. In regions where the instrument is known to perform well, the match is very good. Indeed, the removal of validation points in the lee re-circulation zone improves the relationships

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Table 1 Linear regression results between predicted and measured variables for all validation points (n = 415) and for all points excluding those within the dune lee separation zone (212) Variable

b Coefficient

Correlation coefficient

Streamwise velocity (all points) Vertical velocity (all points) Streamwise velocity (excluding separation zone) Vertical velocity (excluding separation zone)

1.41

0.98

1.03

0.83

1.29

0.95

1.03

0.88

(Table 1), with a considerable increase in the vertical velocity correlation, and although there is slight decline in the streamwise velocity correlation, there is a significant movement of the regression line towards that of equality. Furthermore, qualitative assessment of the predicted flow patterns and the indications given by flow streamers (see Walker and Nickling, in press (a)) confirm the presence and the extent of the separation zone, which is successfully simulated by the numerical model. The model is able to simulate areas of flow stagnation at the toe, acceleration up the stoss slope

Table 2 Geometric properties of Experiments 1 – 9 Experiment number

Dune height (h)

Stoss base length (L)

Stoss angle

Aspect ratio (h/L)

Lee base length

Lee slope angle

1 2 3 4 5 6 7 8 9

8.00 8.00 8.00 8.00 8.00 4.00 16.00 6.00 12.00

56.00 112.00 28.00 84.00 42.00 56.00 56.00 56.00 56.00

8.13 4.09 15.95 5.44 10.78 4.09 15.95 6.12 12.10

0.143 0.071 0.286 0.095 0.190 0.071 0.286 0.107 0.214

12.80 12.80 12.80 12.80 12.80 6.40 25.61 9.60 19.20

32.0 32.0 32.0 32.0 32.0 32.0 32.0 32.0 32.0

and flow reversal in the lee, which closely match measurements obtained in the wind tunnel. The model is therefore deemed able to provide a realistic and complete 2D picture of the flow structure over idealized dune forms, providing prediction fields that are spatially much richer than results produced by current wind tunnel experiments and field studies. The use of numerical modelling allows rapid alteration the geometry of the dune under controlled conditions, permitting analysis of the interactive effect of dune form on the flow field.

Fig. 2. Comparison of modelled to measured velocities (a) streamwise (U) and (b) vertical (V) for the wind tunnel data of Walker and Nickling (2002) (Experiment 1 in Table 2).

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2.3. Experiments



Based on the successful verification and validation of the model (Parsons et al., in press), it was deemed appropriate to use the model to test the effect of simple dune geometry variations on streamwise and vertical velocity flow fields. In particular, certain elements of the flow field were investigated including: 

velocity profiles at the dune toe, dune crest and at three dune heights downstream of the crest;  streamwise and vertical velocity profiles at 1 cm above the dune toe, crest and three dune heights downstream of the crest;

Fig. 3. Isovel contour plots of streamwise velocity (U, m s in Table 2).

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lee-side separation zone length; and lee-side distance to flow recovery.

Simulated velocity profiles and the streamwise velocities near the bed are of interest for predicting the effects of dune aspect ratio on stoss flow acceleration and the strength of flow within the separation cell, particularly for examining the implications for sediment transport. Vertical velocities provide insight on the presence of streamwise curvature effects (i.e., flow stabilization) as well as the occurrence and magnitude of vertical lift or downdrafts over different dune forms. The last two parameters were of particular interest due to the ability of the model to predict

) calculated for five different dune geometry scenarios (Experiments 1, 2, 5, 6 and 7

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separated and reversed flow in the highly turbulent lee-side eddy. The lee-side separation zone length was determined from the location where the simulated near-surface streamwise velocity at 1 cm changed from negative (upstream) to positive (downstream) in orientation. Distance to flow recovery in the lee was determined as the point at which the simulated near-surface streamwise velocity at 1 cm above the surface was within 99% of its unperturbed upwind value. This point was not necessarily the point where the full boundary layer had recovered; nevertheless, it does provide an indication of flow recovery. A fixed 1-cm distance from the surface was applied in each experiment as the uniform grid resolution used in the modelling precluded the variable setting this nearsurface distance. Details of the differing dune geometries that were used in the model runs are shown in Table 2. All units are in centimetres and degrees, allowing testing and comparisons with simulated experiment of the wind tunnel data of Walker and Nickling (in press (a)).

3. Results The model output for streamwise velocity (U, m s 1) is shown as isovel contour plots in Fig. 3. The results shown here correspond to Experiments 1, 2, 5, 6 and 7 in Table 2 and cover a range of dune height, stoss length and aspect ratios simulated in this investigation. Flow separation length and distance to flow recovery results are summarized in Table 3.

Table 3 Distances from dune crest to flow re-attachment and flow recovery over each of the experimental dune geometries Experiment number

Length to flow re-attachment, cm (x/h) from crest

Length to flow recovery, cm (x/h) from crest

1 2 3 4 5 6 7 8 9

73 59 96 65 82 13 234 34 148

558 526 594 544 568 306 724 424 624

(9.13) (7.34) (12.00) (8.13) (10.25) (3.25) (14.63) (5.67) (12.33)

(69.75) (65.75) (74.25) (68.00) (71.00) (76.50) (45.25) (70.67) (52.00)

In each experiment, it is clear that the intrusion of the dune into the simulated boundary layer has a large effect on flow structure. In each case, the model predicts flow deceleration immediately upwind of the dune followed by windward slope acceleration to a maximum velocity at the crest. These results correspond to findings in previous investigations (e.g., Jackson and Hunt, 1975; Bowen and Lindley, 1977; Lancaster et al., 1996; McKenna Neuman et al., 1997; Walker and Nickling, 2002; Wiggs, 1993; Wiggs et al., 1996). In the lee of the dunes, the simulations shown in Fig. 3 show large disturbances in streamwise velocity with flow re-attachment occurring within approximately 3 –15 dune heights and flow recovery occurring several tens of dune heights downwind (Table 3). The dune in Experiment 7 was steep-sided, and, here, near-bed flow recovery occurs just within the boundaries of the simulation. Another interesting observation is the convergence in the upper (faster) isovels of the flow field, which corresponds to a zone or ‘jet’ of accelerated, overshot flow extending from the crest above the flow separation cell (Walker and Nickling, 2002). This effect is observable for Experiments 1, 2 and 5 where dune height (h) has been maintained and is less apparent for Experiments 6 and 7. Fig. 4 shows the vertical velocity field (V, m s 1) for the same group of experiments. For each experiment, a zone of positive V exists on the upper stoss increasing toward a maximum at the crest. This relates to the topographic (upward) forcing of the dune on near-surface streamlines. Small pockets of positive V are also evident on the mid-lee slope indicating vertical lift in this region. A zone of strong downward flow delineated by the 0.8 m s 1 isovel exists in the lee extending above the flow separation region from the base of the lee slope to beyond the flow re-attachment point. This zone of downward flow can be seen to shift further downstream with increases in dune height (Fig. 4) and appears to vary with the size of the lee-side separation cell. Indeed, this zone of downdraft aligns closely above the point of flow re-attachment as was measured in Walker and Nickling’s (in press (a)) study. Interestingly, the height of this zone extends to approximately 5 h for all runs (with the exception of the steep dune in Experiment 7). These observations confirm that dune height (and not necessarily aspect ratio) plays an important role in perturbing the pres-

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Fig. 4. Isovel contour plots of vertical velocity (V, m s Table 2).

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) calculated for five different dune geometry scenarios (Experiments 1, 2, 5, 6 and 7 in

sure filed over the dune and influencing the vertical velocity distributions and flow re-attachment (Walker and Nickling, in press (a)). The sensitivity of flow patterns to changing dune geometry are highlighted in more detail in Figs. 5 and 6 and in Figs. 7– 12. Full velocity profiles at the dune toe, crest and lee with changing dune stoss slope length and changing dune height are provided in Figs. 5 and 6, respectively. Figs. 7– 10 identify changes in the near-bed velocity components with changes in the dune stoss length and the dune height. The variability of separation re-attachment length and distance to flow recovery with changing dune geometry are detailed in Figs. 11 and 12.

Figs. 6 and 7 indicate that both deceleration at the toe and acceleration at the crest are sensitive to changes in dune height while maintaining dune stoss slope length (i.e., as dune stoss angle increases). Both figures indicate that increasing dune height appears to have a greater impact on acceleration at the crest than on deceleration at the toe. Furthermore, near-bed flow velocities at the toe decelerate almost linearly with increasing dune height as near-bed acceleration at the crest follows a power function (Fig. 7). Streamwise velocity in the lee of the dune decreases rapidly with increasing dune height, before becoming negative as the separated lee-side eddy reverses flow at the foot of the lee slope (Figs. 6 and 7). The magnitude of the

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Fig. 5. Velocity profiles at the dune toe, crest and lee for changing dune stoss slope lengths (Experiments 1, 6, 7, 8 and 9 in Table 2).

lee-side sheltering effect near the bed increases dramatically with dune height for the range of dune sizes simulated here (Fig. 7). When the flow is reversed, increasing dune height has the effect of increasing the velocity of near-surface reversed flow at 3 h downstream of the crest in the separation cell, although this increase appears rather minor. The effect of changing stoss slope basal length (L) while maintaining dune height is shown in Figs. 5 and 8. Increasing stoss slope length appears to have a negligible impact upon streamwise velocities at the crest, although minor effects are clear for velocities at the toe and in the lee (i.e., near-bed lee-side velocities become less negative). Such results are expected given that stoss slope angle is less sensitive to a change in stoss slope length than a change in dune. A steepening of this windward angle leads to both an increase in flow acceleration at the dune crest and flow deceleration in the upwind toe region due to increased

streamline compression and flow stagnation effects respectively (Wiggs et al., 1996). Figs. 9 and 10 are similar to Figs. 7 and 8, except that they focus on vertical velocity at the crest, toe and lee side of each of the experimental dune geometries. Dune height is shown to have a significant impact on vertical velocity in the crestal regions of the dunes due to topographic forcing and a small, but significant, effect on vertical velocities at the toe (Fig. 9). This increase in V in the toe region with increasing dune height provides some support for the streamline curvature model of Wiggs et al. (1996). Sediment transport is maintained through this toe region, despite a reduction in time-averaged streamwise flow velocity, as a consequence of concave streamline curvature resulting in increased turbulence intensity and Reynolds stresses (Wiggs et al., 1996). The small increase in vertical velocity with increasing dune height shown in these experiments (Fig. 9) corresponds to an

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Fig. 6. Velocity profiles at the dune toe, crest and lee for changing dune heights (Experiments 1, 6, 7, 8 and 9 in Table 2).

Fig. 7. Streamwise velocity as a function of dune height.

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Fig. 8. Streamwise velocity as a function of dune stoss slope length.

increase in streamline angle from 2.5j to 8.0j in the toe region. As expected, dune height appears to have only a small effect on vertical velocities close to the surface in the lee (at 3 h downstream of the crest) (Fig. 9). At low dune heights (4 and 6 cm), the negative vertical velocities indicate that downward flow occurs closer to the form as flow reattaches (Fig. 4). At dune heights above 8 cm, there is no evidence of a vertical velocity component in the flow structure, with the zone of

negative vertical velocity shifting downstream. This is attributed to re-attachment lengths being much greater at higher dune heights (Fig. 11) than the measurement point 3 h downstream of the crest, and, hence, nearsurface flow at 3 h is dominated by the reversed streamwise component in larger dunes. Fig. 10 shows that vertical velocities decline rapidly with increasing stoss length. It appears that changes in stoss slope length have a greater impact on vertical velocities at the crest than on streamwise

Fig. 9. Vertical velocity as a function of dune height.

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Fig. 10. Vertical velocity as a function of dune stoss slope length.

velocities (Fig. 8). This is expected given that changes in stoss slope length have an immediate impact on streamline angles at the dune crest. In addition, shorter stoss slope lengths result in steeper windward slopes, and, hence, higher vertical velocities at the crest (Fig. 10). This appears to have a lesser effect at the dune toe where V decreases only slightly with increasing stoss slope length. Similar to Fig. 9, the results in Fig. 10 suggest that vertical velocity at the bed in the separation cell is independent of dune geometry.

The length of the flow separation zone with changing dune aspect ratio is shown in Fig. 11. In general, flow re-attaches within 3 – 15 dune heights downwind (Table 3), which fits within previously documented estimates of 4 –10 h (Frank and Kocurek, 1996b; Walker and Nickling, 2002). These data confirm Walker and Nickling’s (2002) suggestion that an increase in dune height (i.e., an increase in aspect ratio) causes a corresponding increase in the extent of the lee-side separation cell

Fig. 11. Separation zone length (cm) as a function of dune aspect ratio.

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(i.e., flow re-attachment occurs further downwind from the dune crest). The wide range in values from 3 < x/h < 15 (i.e., 3 –15 dune heights downwind from the crest) indicates the sensitivity of the structure of the lee-side eddy to this geometrical parameter. The data also show that a similar, though less steep, relation occurs if dune height is maintained at a constant but stoss slope length is decreased (i.e., also an increase in aspect ratio). The different gradients of the relations evident in Fig. 11 for changing dune height and changing stoss length indicate that the length of the separation zone is more sensitive to the former. Similarly, Fig. 12 shows the effect of dune aspect ratio on the downwind distance to flow recovery in the lee side of the dune. In general, streamwise velocities at the surface do not recover to 99% of upwind values until approximately 52 – 77 dune heights downwind. However, even at these distances, only the near-surface velocity values have recovered, with the full boundary layer profile often still recovering. Although actual recovery distances increase with aspect ratio, interestingly, the shortest height normalised recovery distances occur over the tallest dunes with the steepest stoss slope angles (Experiments 7 and 9, Table 3). These flow recovery lengths exceed Lancaster’s (1988) estimate of 10 –15 h and Walker and Nickling’s (in press (b)) estimate of 25 – 30 h, and they are closer to distances of 30 –50 h for

flow over a backward-facing step and sub-aqueous dunes (Bradshaw and Wong, 1972; McLean and Smith, 1986, respectively). The relationship shown (Fig. 12) for increasing dune height in the aspect ratio demonstrates a power function increase in recovery distance. Similarly, a decrease in stoss slope length (resulting in an increase in the aspect ratio) increases recovery distance in a linear relation that is less steep than that for changing dune height (Fig. 12). The results in Figs. 11 and 12 are interesting in that while variations in dune height and stoss slope length both influence the aspect ratio of the dune, they have differing impacts on the downwind distance to flow recovery and the separation zone length. In both cases, the airflow structure is more sensitive to changes in dune height than changes in stoss slope length.

4. Discussion To date, logistical and instrumentation limitations have prevented effective characterization of flow field response over transverse dunes of varying geometry. Only recently have CFD models become available and validated for use in simulating the flow field over isolated transverse dunes (see Parsons et al., in press). It is clear from this simulation and other previous wind tunnel and field studies that variations in stream-

Fig. 12. Distance to flow recovery (cm) as a function of dune aspect ratio.

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wise and vertical velocities over transverse dunes are the outcome of the intrusion of the dune into the atmospheric boundary layer. The result is a perturbation in the near-surface pressure field resulting in fluid momentum changes that, in turn, cause secondary flow effects such as: flow stagnation and deceleration at the upwind toe; streamline compression, flow acceleration and vertical lift up the windward slope; and, in the lee, flow separation, streamline expansion, flow re-attachment and reversal, a zone of downward vertical flow and a lengthy flow recovery distance. These secondary flow effects are shown in this study to vary significantly with dune geometry—namely dune height, stoss basal length and, thus, aspect ratio. 4.1. Streamwise velocity distribution The CFD model used in this study reliably predicts streamwise flow deceleration immediately upwind of the dune followed by windward slope acceleration to a maximum velocity at the crest. At the dune toe, velocities increase only slightly with stoss slope length (i.e., for less steep dunes) and decrease only slightly with increases in dune height for the range of forms investigated. This is likely the result of a reduced stagnation effect imposed on the flow by dune with lesser aspect ratios (i.e., less steep dunes) (Wiggs et al., 1996). Toward the crest, flow acceleration is more sensitive to increases in dune height than to stoss slope length. This is primarily due to stoss slope angle being more sensitive to changes in dune height than basal length and thus dune height exerting a greater perturbation on the windward flow field, which results in enhanced streamline convergence and, hence, flow acceleration over taller dunes. The model also characterizes lee-side streamwise velocity variations very effectively (Parsons et al., in press). Reversed near-surface velocities inside the separation zone increase slightly (i.e., become less negative) with increasing stoss slope length (Figs. 5 and 8) but are highly sensitive to changes in dune height (Figs. 6 and 7). This difference is highlighted through comparison of the lee profiles in Figs. 5 and 6. Taller dunes having a larger lee-side separation cell and stronger reversed flow near the surface can be explained for two main reasons. First, stoss flow is accelerated more toward, and overshot faster from,

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the crest of taller, steeper dunes. Second, the lee-side velocity gradient in the separation zone is steeper, and as a result, momentum exchange and resultant recycling of fluid mass back toward the dune is greater (Walker and Nickling, in press (a)). Therefore, for the range of dune forms investigated in this study, taller dunes with greater aspect ratios (i.e., h/L>0.14) have larger lee-side flow separation regions (Fig. 11) and stronger near-surface reversed flows (Figs. 5 and 6). Dune height also has a greater effect than stoss slope length on streamwise flow velocity recovery distance (Fig. 12). Interestingly, the shortest normalised recovery distances occur over the tallest dunes with the steepest stoss slope angles (Experiments 7 and 9, Table 3) while longer distances (up to 76.5 h) are required for shorter dunes. This is mainly due to the effects normalising for height (Table 3) and Fig. 12 demonstrates the effect of h and L and, thus, aspect ratio on the actual recovery distances. The power law function with increases in dune height shows that recovery distance increases begin to diminish with larger heights (Experiments 1 and 9). This may relate to enhanced turbulent momentum exchange and dissipation in the larger flow separation region in the lee of steeper obstacles, thereby reducing the distance for boundary layer recovery. It may also be due to increases in negative vertical velocity magnitudes in the lee forcing higher streamwise velocity towards the surface and producing recovery sooner. This will be explored further in a related paper. 4.2. Vertical velocity distribution This study shows that vertical velocity variations on the windward face are most affected by dune height and stoss slope length (i.e., increasing stoss slope angle), especially at the crest. This confirms the widely held view that dunes with steeper windward slopes experience greater topographic forcing, streamline convergence and flow acceleration toward the crest (Lancaster, 1985; Tsoar, 1985; Wiggs, 1993; Lancaster et al., 1996). It is shown here that as dune height increases, flow is overshot with greater vertical velocity (upward lift) from the crest (Figs. 3 and 9). In terms of sand transport at the dune toe, the decrease in streamwise velocity (and hence sand

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transport) with increasing dune height (Fig. 7) may be offset by an increase in streamline angle (controlled by stoss slope length) and a rise in vertical velocity (Figs. 9 and 10). This, in turn, may result in higher turbulence intensities, Reynolds stresses and vertical lift in the toe region that might be sufficient to maintain sand transport in this region, which supports recent wind tunnel and field studies by McKenna Neuman et al. (2000) and Walker and Nickling (in press (a,b)) and the streamline curvature model of Wiggs et al. (1996). In the lee, near-surface vertical velocities are only slightly sensitive to changes in dune height for the range of forms simulated. The effect is greatest for shorter dunes (i.e., h < 8 cm) where the 1-cm measurement height 3 h downstream of the crest at in the dune lee shows greater downward velocity as it is closer to the upper boundary of the smaller separation cell (Figs. 4 and 10). This may also reflect a small pocket of vertical lift (i.e., positive V) evident over the mid lee slope for most runs (Fig. 4). This occurs due to a steep favourable (negative) pressure gradient that causes slower lee flow to rise toward the faster flow in the shear layer bounding the separation cell (Walker and Nickling, in press (a)). This phenomenon is found in flow over roofs (Ginger and Letchford, 1993) and over low-angle fluvial dunes (Best and Kostaschuk, 2002). Walker and Nickling (in press (a)) suggest that this effect (combined with slope effects and impact from fallout grains, though slight) is able to reduce transport thresholds in the lee slope region. They also indicated that vertical updrafts in the upper lee enhance modified suspension of grains into the lee to distances well beyond typical saltation trajectories (cf. Nickling et al., 2002). This reinforces the thought that saltation is not likely the dominant mechanism for sediment delivery into the lee and that secondary lee-side airflow patterns have a significant effect on dune sedimentary dynamics (Walker and Nickling, in press (a)). The flow field simulation (Fig. 4) shows that beyond the lift region immediately leeward of the crest, flow becomes downward (i.e., vertical velocities become negative) beyond the lee slope. The progressive shift from stoss-upward to lee-downward motion reflects a wave-like, dune-generated perturbation in the flow field and confirms measured patterns over sub-aqueous dunes (Best and Kostaschuk, 2002) and

over an idealized aeolian form (Walker and Nickling, in press (a)). This wave-like influence of dune form on vertical velocity extends to a height of approximately 3 –5 h and the shift to downward flow translates further downwind over the lee-slope as dune height increases (Fig. 4). In general, the extent of the downward flow zone appears to be relatively independent of stoss slope length and, hence, aspect ratio. It extends from the crest (for shorter dunes) to tens of dune heights downwind (e.g., >14 h for the taller dune in Experiment 7) and the flow re-attachment point is found near the downwind edge of the faster core of is zone. Though this paper does not characterize turbulence, this finding corroborates Walker and Nickling’s (in press (a)) claim that downward flow from a turbulent shear zone (their zone ‘G’) overlying the separation region drives flow re-attachment at the surface. This study also shows that, like the separation cell, the extent of this zone is more dependent on dune height and not necessarily explained the aspect ratio alone. Therefore, the influence of dune form on vertical velocity plays an important role in sediment delivery into the lee via grainfall in separated airflows (Nickling et al., 2002) as well as in determining the point of flow re-attachment, boundary layer recovery and subsequent saltation development distance (Walker and Nickling, in press (a,b)).

5. Conclusion Analysis of CFD-derived flow structures over idealized transverse dunes has shown the potential to quickly and reliably test relations between dune geometry and wind flow structure. Data confirm that the flow field over transverse dunes is particularly sensitive to changes in dune height, with an increase in height resulting in flow deceleration at the toe, acceleration at the crest and an increase in the size of the lee-side separation zone. Evidence is provided to support the streamline curvature model of Wiggs et al. (1996) that explains the maintenance of sand transport in the toe region of the dune despite declining streamwise velocities. This study also confirms patterns of vertical lift in the crest region and downward flow beyond the lee slope documented by Walker and Nickling (in press (a)), which respectively have influence on lee-side sediment delivery via grainfall (Nick-

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ling et al., 2002) and flow re-attachment, boundary layer recovery and the re-development of saltation at distances tens of dune heights downwind of the dune (Walker and Nickling, in press (b)). The structure of lee-side airflow has been shown to be very sensitive to changes in dune height (h) and stoss slope basal length (L). The length of the lee-side separation zone varied from approximately 3 –15 h downwind, increasing with height or shorter stoss slope lengths. Similarly, the downwind distance to flow recovery ranged from 45 to >75 dune heights downwind with changes in dune height. The differing effects of height and stoss slope length in these experiments raises questions regarding the applicability of dune aspect ratio as a parameter for explaining airflow over transverse dunes. Though changes in either L or h can produce the same aspect ratio (h/L), this study shows that dune height has a greater effect on streamwise flow perturbations (e.g., flow deceleration at the dune toe, flow acceleration at the dune crest, reversed lee-side surface flow) and on lee-side flow structure (e.g., separation cell length, flow recovery distance). In general, steeper (and not longer) dunes of the same aspect ratio have a greater effect on the flow field, particularly in the lee, for the range of dune shapes investigated in this study. This is because the important factor is the size of the dune in relation to the boundary layer rather than the actual shape of the dune, which is described by the aspect ratio. Further experimentation using CFD for calculating airflow over transverse dunes is planned and extended analyses will include shear stress development and turbulent momentum exchange on simple triangular dune profiles before extending experimentation to more complex and realistic dune geometries consisting of concave– convex-shaped windward slopes and multi-dune geometries.

Acknowledgements This work was undertaken while Daniel Parsons was in receipt of a NERC studentship GR16/99/FS/2 with additional financial support from the British Geomorphological Research Group to attend the International Conference on Aeolian Research 5 at Lubbock, TX.

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