A PIV study of flow over interacting Barchan dunes

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1 A PIV study of flow over interacting Barchan dunes Jessica A. Palmer 1, Ricardo Mejia-Alvarez 2, Eric M. Rivera 3, James L. Best 4 and Kenneth T. Christensen 5 1: Department of Geology, University of Illinois, Urbana, USA, jessica.ann.palmer@gmail.com 2: Department of Mechanical Science and Engineering, University of Illinois, Urbana, USA, rmejiaa2@illinois.edu 3: Department of Mechanical Science and Engineering, University of Illinois, Urbana, USA, erivera3@illinois.edu 4: Departments of Geology, Geography and Ven Te Chow Hydrosystems Laboratory, University of Illinois, Urbana, USA, jimbest@illinois.edu 5: Departments of Mechanical Science and Engineering, Aerospace Engineering and Geology, University of Illinois, Urbana, USA, ktc@illinois.edu Abstract Barchan dunes are crescentic planform-shaped dunes that are present in many natural environments, and may occur either in isolation or in groups. This study uses high-resolution particle-image velocimetry (PIV) experiments using fixed-bed models to examine the effects of barchan dune interaction upon the flow field structure. The barchan dune models were based upon an idealized contour map, the shape and dimensions of which were based upon previous empirical studies of dune morphology. The experimental setup comprised two, co-axially aligned barchan dune models that were spaced at different distances apart. In this paper, two volumetric ratios (Vr, upstream barchan dune: downstream barchan dune) of 1.0 and were examined. Models were placed in an Eiffel-type, open-circuit wind tunnel and flow quantification was achieved using PIV at 0.5Hz. PIV measurements of the mean and turbulent flow field were made in the streamwise wallnormal plane, along the centerline of the barchans(s), at an average Reynolds number of 59,000. The presence of an upstream barchan dune of equal volume to the downstream barchan dune (Vr = 1) induces a sheltering effect on the flow, manifested by a significantly shorter separation bubble and both reduced streamwise velocity and turbulence intensity in the downstream barchan dune leeside, as compared to an isolated barchan. The volumetric ratio Vr = shows enhanced turbulence production over the downstream barchan dune leeside, that is proposed to be caused by interacting shear layers from the up- and down- stream dunes. The influence of the upstream dune is greater for a larger volumetric ratio due to the sheltering effect of the upstream bedform. Proper orthogonal decomposition (POD) analysis shows that the distribution of turbulent kinetic energy is shifted to higher modes (i.e. smaller spatial scales) over interacting barchan dunes, which also reflects the role of the leeside free shear layer in dominating the flow field by generation, or redistribution, of TKE to higher modes. 1. Introduction Barchan dunes are common features on the Earth s surface, are characterized by their crescentic planform shape and are formed under the presence of a unidirectional flow and a starved sediment supply (Bagnold, 1941). In deserts, barchan dunes often occur within wind regimes that have a directional variability of less than 15. When a barchan dune is subjected to oblique flows, its horns become asymmetrical, and with sufficient variability in flow direction more complex barchan dune morphologies may emerge. Barchan dunes are found in many aeolian environments on Earth but have also been documented on the continental shelves, as well as in the wind-blown environments of Mars (Figure 1). The simplest type of barchan, the isolated barchan, appears to form under the presence of a restricted sediment supply, and often forms fields of dunes whose individual members may interact with each other. Although the flow field over isolated flow-transverse dunes is fairly well known (Best, 2005), there has been much less attention devoted to the interactions between barchan dunes, despite the fact that most barchans occur in dune fields (Figure 1) where such interactions are the norm

2 Figure 1: Interacting barchan dunes in the Proctor Crater, Mars, illustrating barchans migrating towards the lower left corner and various types of barchan dune interaction (Credits: Malin Space Science Systems, MGS, JPL, NASA). Recent work has begun to document the importance of dune interactions, with consensus in the literature (Endo et al., 2004; Kocurek and Ewing, 2005; Durán et al., 2005) that the relative volumetric ratio, Vr, of the initial size between the approaching upstream barchan dune, Vup, to the downstream barchan dune, Vdown, is key in determining the kinematic behavior and final morphology of a barchan dune. Additionally, this work has generated recent debate on the behavior of interacting dunes (Andreotti et al., 2002; Hersen et al., 2004), and whether such interactions can be viewed as solitary waves (Duran et al., 2005). In a notable continuation, Endo et al. (2004) documented three types of behavior for interacting barchan dunes: (i) absorption; (ii) ejection; and (iii) splitting. The relative volumetric ratio (Vr) increased between behaviors (i) (ii) if the dune maintained a coaxial dune alignment. The absorption behavior was produced by the smallest volumetric ratio (Vr = 0.025), and refers to the absorption of the smaller approaching barchan dune by the larger downstream barchan dune. Ejection behavior was produced by a higher volumetric ratio (Vr = 0.055), and referred to the process by which a barchan dune (smaller than the initial upstream barchan dune) exits the leeside of a larger barchan dune (initially the downstream barchan dune). Splitting behavior was produced by the largest volumetric ratio (Vr = 0.17), and referred to the erosion of the downstream barchan dune by the approaching smaller upstream barchan dune prior to contact, resulting in the generation of two barchan dunes. Although such behaviors undoubtedly control the kinematics of dune movement, little work has been conducted on the fluid flow interactions between barchans. This served as the starting point to the present study that sought to examine the characteristics of the mean and turbulent flow field between two barchan dunes of differing size at differing spacings. These results provide the first step to a more complete understanding of the interactions between these common dune features on the Earth s surface. 2. Methods Experiments were conducted in the Laboratory for Turbulence and Complex Flow at the University of Illinois, Urbana-Champaign. The measurements were collected along the streamwise wallnormal plane in an Eiffel-type, open circuit, boundary-layer wind tunnel with a documented turbulence intensity of 0.16% in the free stream (Wu and Christensen, 2007). The working testsection of the tunnel is 6090 mm long by 914 mm wide by 457 mm high, and planar two

3 Figure 2: Aspect ratio of field measurements of numerous barchan dunes, relative to the aspect ratio of the barchan dune unit model employed for this study. Inset sketch shows definition of W, L and H. dimensional PIV was used to investigate the flow fields around the interacting dunes. The flow was seeded with 1µm olive-oil droplets generated by a Laskin nozzle. The barchan dune unit model used herein was generated from a contour map of an idealized barchan dune presented by Hersen et al. (2004). The aspect ratio of the barchan dune model falls within the range of reported measurements found in nature and the laboratory (Figure 2), thus fulfilling the geometric similarity criteria and justifying the use of the fixed barchan dune model. Physical replicas of the dune unit model were fabricated with a 3D powder deposition printer. In this study, interactions between four different barchan dune sizes were examined (A-D; Table 1), that overlapped with the study of Endo et al. (2004), with the addition of experiments examining interactions with identical size dunes (AA). In the present paper, only interactions between size AA and AB dunes will be detailed. The experiments replicate reported behaviors dependent on the volumetric ratio between coaxially-aligned barchan dunes in tandem (Table 1). For the experiments conducted in this study, two coaxially aligned barchan dune models in tandem were placed in the working section of the wind tunnel, with the exception of the isolated barchan dune experiment, which investigated the flow over one barchan dune unit model (Model A). The upstream barchan dune model was moved progressively upstream (Model A and B), while the downstream barchan dune model was fixed. In every experiment the downstream (and isolated) barchan dune was the largest model, Model A (in AA the dunes were of identical size). For the set of experiments detailed herein, 2,500 paired-images yielding two-dimensional velocity fields (U,V) along the streamwise-wall-normal plane (x,y) were collected for each experimental run using a 4k 2.7k-pixel 12-bit frame-straddle CCD camera (Figures 3 and 4). The velocity fields shown thus comprise the streamwise velocity component, U, and the wall-normal velocity component, V. Time delays used were: for the stoss-side data t=75 µs ( t=100 µs for AB-0) and the leeside data t=140 µs. The leeside required a longer t than the stoss-side in order to resolve the slower velocity in the flow separation region. A 500 µm-thick laser light sheet was produced Table 1: Summary of the barchan dunes studied; only results from barchan dunes A and B are detailed herein

4 Figure 3: Schematic of the experimental set-up. by two Nd:YAG lasers (200 mj/pulse, 5 ns pulse duration), conditioned through an arrangement of spherical and cylindrical lenses, and projected to the working section by a high-energy mirror. In order to minimize reflections, the dune models were painted matte black and then painted with multiple applications of rhodamine 6G dye, and rhodamine B dye for some experiments, dissolved in ethanol and added to transparent acrylic paint. The experiments conducted on the stoss-side of the downstream barchan dune model at the following spacings (AA-0, AB-0, AAtouching, AB-touching, AC-touching, ADtouching) were painted with multiple applications of rhodamine 6G and then with rhodamine B. A band-pass filter (532 ± 2 nm) was mounted in front of the camera lens, allowing only scattered light from the tracer particles to penetrate through the filter, thus protecting the CCD from any fixed reflections produced from the barchan surface. Moreover, the reduction in reflections produced by the laser light sheet drastically improved the quality of the PIV data (Figure 4). Figure 4: (a) PIV image of the barchan leeside without Rhodamine-6G-Acrylic-paint-mixture; (b) PIV image of the barchan leeside with the Rhodamine-6G-Acrylic-Paint-Mixture. (c) A PIV image showing the validated vector field of image (b)

5 3. Results Isolated Barchan: mean flow field Flow over the isolated barchan dune shows the features common to flow over transverse bedforms (see review in Best, 2005). Flow induces flow separation at the crest (Figure 5a-c), generating a region of high vorticity where Kelvin Helmholtz instabilities are shed (the results presented herein are not time resolved, and thus the shedding frequency is unknown). The freestream flow and recirculation region interact, forming a turbulent shear layer, which is the site of fluid and momentum exchange. The flow over an isolated barchan dune is topographically forced on the stoss side (Figure 5b). Within the dune leeside, the flow separates and reattaches at 4.6H (H is the dune crest height; Figure 5). The separation bubble is characterized by upwelling flow directly in the dune leeside (Figure 5b), whereas downstream of the separation bubble, the flow is characterized by downwelling flow, referred to as the expansion zone, as a new internal boundary layer is created in the dune wake. Interacting Barchan Dunes: mean flow field In the case of interacting barchan dunes, two cases are illustrated herein. In both cases, the dunes are spaced as closely together as possible (see inset diagrams on Figure 5d-i), but in one experiment (Figure 5d,e,f) two A dunes are investigated whilst in Figure 5g-i results are shown where the smaller dune B is positioned on the upstream side of dune A (Vr = 0.175). It can clearly be seen that flow over the downstream barchan dune (DBD) is influenced by the flow over and in the leeside of the upstream barchan dune (UBD), manifested by longer reattachment lengths (~5.2H) in the case of Vr = (configuration AB; Figure 5d). The UBD flow regions are visible on the DBD stoss side, such as the UBD wake and separation bubble (Figure 5g,d) and the UBD expansion zone (Figure 5e,h). The DBD stoss side is thus sheltered by the UBD. The region of high Reynolds stresses is focused around the free shear layer in the dune leeside in both upstream and downstream dunes. The presence of the UBD not only introduces strong Reynolds stresses on the stoss side of the DBD, but also amplifies the Reynolds stress within the DBD leeside (Figure 6f). Most noticeably, at the volumetric ratio, the wake of the upstream barchan dune appears to interact with the shear layer of the downstream barchan dune, resulting in increased turbulence intensity and a longer reattachment length in the DBD leeside. Instantaneous flow fields Maps of representative instantaneous fluctuating velocity fields on the stoss side of the DBD for each case are shown in Figure 6 (note that the streamwise origin x = 0 is fixed at the peak height of the DBD). The flow upstream of the isolated dune consists of spanwise vortices adjacent to strong low-momentum regions that are likely associated with the incoming smooth-wall turbulent boundary layer (Adrian et al., 2000). In contrast, the AA-0 (Vr = 1.0) and AB-0 (Vr = 0.175) fields reveal that coherent turbulent structures are periodically shed as a cluster from the UBD, advecting downstream as a group along the free shear layer in the upstream dune leeside. This effect is strongest in the AA-0 case as the UBD is larger than that of the AB-0 case. In particular, the flow on the stoss side (upstream) of the DBD for AA-0 indicates the presence of alternating large-scale low- and high-momentum regions as revealed by the background contours of instantaneous streamwise velocity fluctuation (u ). While these intense instantaneous patterns occur in many of the instantaneous velocity fields for the AA-0 case, they occur much less often in the AB-0 case. These observations indicate that the structures shed by the UBD in the AB-0 case are likely smaller, weaker and less coherent than those shed by the larger UBD in the AA-0 case. Such differences highlight the dependence of the stoss-side flow physics on the volumetric ratio of the dune arrangement

6 Figure 5: Contour maps of the flow over an isolated barchan dune (a c), and interacting barchans dunes at zero spacing but with dunes AA (d f) and AB (g-i): (a,d,g) the mean streamwise velocity component, U/U ; (b,e,h) the mean wall normal velocity component, V/U ; (c,f,g) the dimensionless Reynolds stress, τd (normalized by U, the freestream velocity). The flow features referred to in the text are identified with labels: (U/U plots) the gray and black arrows indicate the maximum upstream velocity in the separation bubble and the flow reattachment point, respectively. The maximum U/U velocity region is labeled a. The 0.6 contour line termination point is labeled Tp. (V/U plots) b stoss side contour inflection; c maximum V/U velocity zone; d crestal jet; e upwelling zone; f leeside contour inflection; g expansion zone; h UBD expansion zone; i UBD upwelling zone. (τd/u 2 plots) The areas of high τd are labeled bb and s; while the area of low τd is labeled aa

7 Figure 6: Instantaneous fluctuating velocity fields on the stoss side of the DBD for the (a) isolated, (b) AA-0 and (c) AB-0 cases. Flow is left to right, the DBD position is shown in black and contours of instantaneous fluctuating streamwise velocity (u ) are shown in the background

8 Figure 7: (a) Fraction of turbulent kinetic energy and (b) cumulative turbulent kinetic energy content versus mode number for flow on the stoss side of the DBD for the isolated, AA-0 and AB-0 cases. POD Analysis To explore the TKE associated with flow on the stoss side of the DBD for the various configurations under consideration, proper orthogonal decomposition (POD) is employed to evaluate TKE content as a function of spatial scale. POD is a technique particularly well-suited for systems that are statistically inhomogeneous in one or more spatial directions, wherein complex infinite-dimensional processes are represented using lower-dimensional approximate descriptions. Analogous to spectral analysis of turbulent flows in statistically homogeneous directions, POD provides a means of quantitatively evaluating the energy content of the flow as a function of length scale. In particular, POD generates a basis for the modal decomposition of ensembles of the instantaneous fluctuating velocity fields and provides the most efficient way of identifying the motions which, on average, contain a majority of the TKE in the flow. The basis functions are optimal in the sense that partial sums of the energy contained in a given number of modes capture more of the TKE of the flow than any other sets of basis functions. In this regard, the modes are arranged in decreasing order of the fractional TKE embodied in the modes as represented by the accompanying eigenvalues garnered from the POD analysis. Further, the lowest-order modes embody larger spatial scales of the flow while higher-order modes represent increasingly smaller spatial scales. Snapshot POD is utilized in the present application and the reader is directed to Sirovich (1987) for a detailed discussion of this methodology. Figure 7(a) presents the fractional TKE content on the stoss side of the DBD as a function of mode number for the three cases (isolated, AA-0 and AB-0). Distinct differences are notable for low mode numbers. For example, the fractional TKE contained in mode 1 is roughly 20% for the isolated case but less than 15% for the AA-0 and AB-0 cases. Thus, the UBD present in these latter two cases enhances the TKE at higher mode numbers via generation and/or redistribution from other modes. In the AA-0 case, additional energy is noted in the next highest mode numbers (particularly modes 2-10) while for the AB-0 case enhanced TKE content is noted at much smaller spatial scales (higher mode numbers). These observations are consistent with the patterns noted in the instantaneous velocity fields where large-scale, alternating low- and high-momentum regions were evident in the AA-0 while smaller-scale patterns dominated the instantaneous fields for the AB-0 case. This enhancement of TKE content in the higher-order modes is also evident in Figure 7(b) that presents the cumulative fractional TKE contained through a given mode number

9 Figure 8. First four POD modes for flow on the stoss side of the DBD for the isolated case. Flow is left to right, the DBD position is shown in black and modal TKE content is included above each plot. Figure 9: First four POD modes for flow on the stoss side of the DBD for the AA-0 case. Flow is left to right, the DBD position is shown in black and modal TKE content is included above each plot. Figures 8 10 present vector plots of the first four POD modes for the isolated, AA-0 and AB-0 cases, respectively, for flow on the stoss side of the DBD. Contours of the fluctuating streamwise velocity component are included in the background to highlight regions of low (blue) and high (red) momentum. While individual POD modes are not representative of actual coherent structures in the flow, they do provide a qualitative glimpse of the dominant flow patterns associated with each mode and their spatial variability. Mode 1, the most energetic mode, is similar in all three cases as it is marked by an elongated, large-scale streamwise velocity event. This pattern is consistent with the individual large-scale regions of low and high momentum evident in the instantaneous velocity fields in Figure 6. In contrast, the next most energetic mode, mode 2, differs considerably between the three cases. While mode 2 for the isolated and AB-0 cases embodies an inclined shear-layer pattern separating a low-momentum region below from a high-momentum region above, mode 2 for the AA-0 case is marked by a spanwise vortex separating a low-momentum region downstream from a high-momentum region upstream. This latter pattern is qualitatively consistent with the alternating large-scale low- and high-momentum regions evident in the AA-0 instantaneous velocity field of Figure 6(b). A similar pattern is also evident in mode 4 of the AA- 0 case, while modes 3 and 4 for the isolated and AB-0 cases are still dominated by inclined shear layers. Thus, while the flow on the stoss side of the DBD for the AA-0 case appears driven by the interaction of the flow with the UBD, the flow on the stoss side of the DBD for the AB-0 case, for which the volumetric ratio is smaller (Vr = 0.175), appears similar to the case for which no - 9 -

10 Figure 10: First four POD modes for flow on the stoss side of the DBD for the AB-0 case. Flow is left to right, the DBD position is shown in black and modal TKE content is included above each plot. Figure 11: First four POD modes for flow on the leeside of the DBD for the isolated case. Flow is left to right, the DBD position is shown in black and modal TKE content is included above each plot. upstream dune is present (isolated). Assuming that the flow on the stoss side of the DBD for the AA-0 case is indeed dominated by interactions between the oncoming flow and the UBD, then the POD modes for the stoss-side AA-0 case should be similar in character to those on the leeside of an isolated dune. To explore this possibility, snapshot POD is applied to the velocity fields acquired on the leeside of the isolated dune (recall that it is the same size as the UBD in the AA-0 case). Figure 11 presents vector plots of the first four POD modes of the leeside flow for an isolated dune accompanied by contours of the fluctuating streamwise velocity. Indeed, these first four modes are strikingly similar to those of Figure 9 for flow on the stoss side of the DBD for the AA-0 case. In particular, the alternating low- and high-momentum regions that mark modes 2 and 4 for the stoss-side AA-0 case are evident in modes 2 and 4 for the leeside isolated barchan case. This consistency lends further support to the idea that the flow on the stoss side of the DBD for case AA-0 is indeed dominated by the interactions between the oncoming flow and the UBD. Thus, the volumetric ratio appears critical

11 for determining whether such interactions will dominate the stoss-side flow of the DBD. 4. Conclusions Interacting barchan dunes are common features of the natural landscape and this experimental study has demonstrated how the volumetric ratio between such adjacent dunes is critical in determining their flow field. Based on past studies that have demonstrated the important kinematic interactions between dunes, we have investigated a simple experimental configuration comparing the flow field of two interacting barchans to the case of a solitary, isolated, barchan dune. The results clearly show the leeside flow of the upstream barchan dune influences the downstream flow field, with this influence being greater for larger upstream dunes that exert a more significant sheltering effect on the downstream dune. Flow on the stoss side of the downstream barchan dune is controlled by the developing internal boundary layer from the upstream dune, which in the case of two identical size dunes is more pronounced. POD analysis shows that the distribution of turbulent kinetic energy is shifted to higher modes (i.e. smaller spatial scales) over interacting barchan dunes, which also reflects the role of the leeside free shear layer in dominating the flow field by generation or redistribution of TKE to higher modes. Ongoing work is investigating the redistribution of TKE in the downstream barchan leeside and the interaction between the two shear layers. 5. References ADRIAN, R. J., MEINHART, C. D. and TOMKINS, C. D Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, ANDREOTTI, B., CLAUDIN, P. and DOUADY, S., 2002a. Selection of dune shapes and velocities part 1: Dynamics of sand, wind and barchans. European Physical Journal B, 28(3), BAGNOLD, R.A., The physics of blown sand and desert dunes. London: Methuen, pp BEST, J., The fluid dynamics of river dunes: A review and some future research directions. Journal of Geophysical Research F: Earth Surface, 110(4), F04S02. DURÁN, O., SCHWÄMMLE, V. and HERRMANN, H., Breeding and solitary wave behavior of dunes. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 72(2), 1-5. ENDO, N., TANIGUCHI, K. and KATSUKI, A., Observation of the whole process of interaction between barchans by flume experiments. Geophysical Research Letters, 31(12), L ELBELRHITI, H., CLAUDIN, P. and ANDREOTTI, B., Field evidence for surface-waveinduced instability of sand dunes. Nature, 437(7059), HERSEN, P., ANDERSEN, K.H., ELBELRHITI, H., ANDREOTTI, B., CLAUDIN, P. and DOUADY, S., Corridors of barchan dunes: Stability and size selection. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 69(11), KOCUREK, G. and EWING, R.C., Aeolian dune field self-organization - Implications for the formation of simple versus complex dune-field patterns. Geomorphology, 72(1-4), SIROVICH, L., Turbulence and the dynamics of coherent structures. Part 1. Coherent structures. Q. Appl. Math. 45, 561. WU, Y. and CHRISTENSEN, K.T., Outer-layer similarity in the presence of highly-irregular surface roughness. Phys. Fluids 19 (8), ,