Force distribution within a barchan dune
aa r X i v : . [ phy s i c s . g e o - ph ] J a n Force distribution within a barchan dune
Force distribution within a barchan duneThis article may be downloaded for personal use only. Any other use requires priorpermission of the author and AIP Publishing. This article appeared in Phys. Fluids33, 013313 (2021) and may be found at https://doi.org/10.1063/5.0033964
Carlos A. Alvarez and Erick M. Franklin a) School of Mechanical Engineering, UNICAMP - University of Campinas,Rua Mendeleyev, 200, Campinas, SP, Brazil (Dated: 26 January 2021)
Barchan dunes, or simply barchans, are crescent-shaped dunes found in diverse environ-ments such as the bottom of rivers, Earth’s deserts and the surface of Mars. In a recentpaper [Phys. Rev. E 101, 012905 (2020)], we investigated the evolution of subaqueousbarchans by using computational fluid dynamics - discrete element method (CFD-DEM),and our simulations captured well the evolution of an initial pile toward a barchan dune inboth the bedform and grain scales. The numerical method having shown to be adequate,we obtain now the forces acting on each grain, isolate the contact interactions, and in-vestigate how forces are distributed and transmitted in a barchan dune. We present forcemaps and probability density functions (PDFs) for values in the streamwise and spanwisedirections, and show that stronger forces are experienced by grains at neither the crest norleading edge of the barchan, but in positions just upstream the dune centroid on the periph-ery of the dune. We show also that a great part of grains undergo longitudinal forces ofthe order of 10 − N, with negative values around the crest, resulting in decelerations andgrain deposition in that region. These data show that the force distribution tends to route agreat part of grains toward the crest and horns of subaqueous barchans, being fundamentalto comprehend their morphodynamics. However, to the best of the authors’ knowledge,they are not accessible from current experiments, making of our results an important steptoward understanding the behavior of barchan dunes. a) Electronic mail: [email protected]; Corresponding author
I. INTRODUCTION
Barchan dunes, or simply barchans, are crescent-shaped dunes resulting from the transport ofgrains, usually sand, by a one-directional fluid flow in a mode of transport called bedload, inwhich grains roll, slide or effectuate small jumps maintaining contact with a fixed part of thebed . For an initial granular pile, grains are eroded on regions upstream the pile crest and settleat the crest. The fluid flow then separates at the crest region and a recirculation bubble that formsjust downstream the crest shapes a lee face with a curved base and horns pointing downstream .Under one-directional flow and bedload conditions, barchan dunes are robust bedforms that canbe found in different environments and scales . For example, they are found in oil and waterpipelines, rivers, Earth’s deserts, and other planetary environments, their scales varying from thedecimeter and minute under water to the kilometer and millennium on Mars , passing bythe hundred of meters and year on Earth’s deserts .The large time scales of aeolian and martian barchans compared to the aquatic case make ofsubaqueous barchans the ideal object of study. Therefore, experiments have been conducted inwater tanks and channels, where measurements were made at both the dune and grainscales , and from which length and time scales of barchans and typical trajectories and ve-locities of moving grains were obtained. In particular, Wenzel and Franklin and Alvarez andFranklin are the only experimental studies showing the distribution of grain velocities over thebarchan surface and trajectories of grains migrating to different parts of the barchan dune. For theforces on individual grains within a barchan dune, there are currently no experimental results, theacquisition of these data being highly difficult.In addition to experiments in water, numerical simulations have been employed for the studyof dunes. Also for numerical investigations, in particular those at the grain scale, subaqueousbarchans are interesting since the number of involved grains is much smaller than in other en-vironments. The first numerical investigations were on aeolian dunes and made use, initially, ofcontinuum models for the grains , and more recently of simplified discrete models such asthe cellular automaton . The most recent investigations were focused on subaqueous bedloadand bedforms and used Euler-Lagrange methods, such as computational fluid dynamics - discreteelement method (CFD-DEM) . In particular, Kidanemariam and Uhlmann used directnumerical simulations (DNS) for the fluid and DEM for the grains, both coupled by immersedboundary (IB), which is, currently, the most accurate technique, fully solving the flow around each2orce distribution within a barchan dunegrain. However, while it captures all turbulence scales down to Kolmogorov scale, the computa-tional cost is exceedingly high and the time required for obtaining developed barchans is seldomreached .In a recent paper , we presented numerical computations of the growth and evolution of sub-aqueous barchans carried out at the grain scale, where we coupled DEM with large eddy simulation(LES). LES, although less accurate than DNS and needing turbulence models, is able to computethe flow around dunes at a much lower computational cost. The simulations captured well theevolution of an initial pile toward a barchan dune in both the bedform and grain scales, with thesame characteristic time and lengths observed in previous experiments . In addition to repro-ducing accurately previous experimental data, the numerical results revealed in detail quantitiesnot accessible from experiments, such as the resultant force acting on each grain. However, al-though such quantities are important to understand the behavior of barchans, an analysis of theforces experienced by each grain within a barchan dune is still missing.In this paper, we investigate how forces on grains are transmitted within a subaqueous barchan.Based on the CFD-DEM computations of Alvarez and Franklin , we plot maps showing the dis-tribution of grain forces in an isolated barchan and compute probability density functions (PDFs)for values in the streamwise and spanwise directions. We show that force distributions tend toroute a great part of grains toward the crest and horns of subaqueous barchans, with the strongerforces experienced by grains occurring at neither the crest nor leading edge of the barchan, butin positions just upstream the dune centroid on the periphery of the dune. We show also that agreat part of grains undergo forces in the streamwise direction of the order of 10 − N, and thaton the crest they have negative values, resulting in decelerations and grain deposition in the crestregion. Finally, we found that around 9% of grains migrate to each horn and 13% to the dunecrest. Our results show, for the first time, the values of the resultant force acting on each particlefor all grains within a barchan dune. To the authors’ knowledge, these data are not accessible fromcurrent experiments and represent a new step for understanding the motion of grains over the dunesurface as well as the load experienced by grains below the bedload layer. The present data arethus fundamental to comprehend the morphodynamics of barchans, being linked to the distributionof grains within a barchan dune. 3orce distribution within a barchan dune
II. METHODS
Our simulations coupled DEM with LES to compute numerically the growth and evolutionof single barchans from a conical pile. For that, we used the open-source code CFDEM . We describe briefly in the following the used model and numericalsetup, a detailed description being found in Alvarez and Franklin .The DEM part computes the dynamics of solid particles in a Lagrangian framework by usingthe linear and angular momentum equations, Eqs. 1 and 2, respectively, m p d ~ u p dt = ~ F p , (1) I p d ~ ω p dt = ~ T c , (2)where, for each grain, m p is the mass, ~ u p is the velocity, I p is the moment of inertia, ~ ω p is theangular velocity, ~ T c is the resultant of contact torques between solids, and ~ F p is the resultant force, ~ F p = ~ F f p + ~ F c + m p ~ g , (3)where ~ F f p is the resultant of fluid forces acting on a grain, ~ F c is the resultant of contact forcesbetween solids, and ~ g is the acceleration of gravity. In our simulations, we consider that ~ F f p isthe sum of components given by the fluid drag, fluid stresses and added mass, and we neglectthe Basset, Saffman and Magnus forces once they are usually considered of lesser importance inCFD-DEM simulations . For the angular momentum, Eq. 2, we neglect torques caused by thedirect action of the fluid since the term due to contacts is usually much higher .The CFD part computes the dynamics of the fluid phase in an Eulerian framework, by solvingthe incompressible mass and momentum equations, Eqs. 4 and 5, respectively, ∇ · ~ u f = , (4) ∂ρ f ~ u f ∂ t + ∇ · ( ρ f ~ u f ~ u f ) = − ∇ P + ∇ · ~~ τ + ρ f ~ g − ~ f f p , (5)where ~ u f is the fluid velocity, ρ f is the fluid density, and ~ f f p is the resultant of fluid forces actingon each grain, ~ F f p , by unit of fluid volume. 4orce distribution within a barchan duneFor the DEM, we considered a Hertzian model for which we used the parameters listed in Tab.I. The boundary conditions for the grains were solid walls at the top and bottom walls, no massentering at the inlet, and free exit at the outlet. As initial condition, grains were poured from above,falling freely in still water until they settled completely. TABLE I. Parameters used in the DEM computationsInitial number of particles 4 × Particle diameter d (mm) 0.5Particle density ρ p (kg/m ) 2500Restitution coefficient e µ f r E (MPa) 5Poisson ratio σ × − For the CFD, we used LES with the wall-adapting local eddy-viscosity (WALE) model , andthe domain was set to 0.3 × × x , wall-normal, y , and spanwise, z ,directions, respectively. The x and z directions were divided into 150 and 160 segments that wereuniform in size, whereas the y direction was divided into 150 unevenly spaced segments. Oursimulations were performed for two different flow conditions, corresponding to channel Reynoldsnumbers based on the cross-sectional mean velocity U and channel height 2 δ , Re = U δν − , of1.47 × and 1.82 × , and to Reynolds numbers based on the shear velocity u ∗ , Re ∗ = u ∗ δν − ,of 420 and 506, where ν is the kinematic viscosity of the fluid. The resulting Shields numbers, θ = u ∗ / (cid:16)(cid:16) ρ p ρ − f − (cid:17) gd (cid:17) , were of 0.04 and 0.06, where ρ p and d are the density and diameterof solid particles. For the two flow conditions, the grid spacings in the streamwise and spanwisedirections scaled in inner wall units ( ν u − ∗ ) were of 33.6 and 40.4, and 16.8 and 20.2, respectively,and the normalized grid spacings in the wall-normal direction at the first point were of 0.91 and1.10. The boundary conditions for the fluid were impermeability and no-slip conditions at the topand bottom walls, and periodic conditions in the longitudinal and transverse directions. The initialcondition was based on single phase flows computed prior to simulations with grains, from whichthe final realization was used as the initial condition.From the simulations, we obtained the forces on each grain, and we investigate now how forces5orce distribution within a barchan dune -2024681012 10 -7 (a) -2-10123 10 -7 (b)FIG. 1. Maps of the resultant force on grains composing an evolving barchan dune ( t < . t c ) in the (a)streamwise and (b) spanwise directions, F px and F pz , respectively. This figure corresponds to Re = 1.47 × , and the solid circle in the maps indicates the crest position. Values in the colorbar are in N. are distributed within the barchan. A layout of the numerical setup is available in the supple-mentary material and numerical data (in terms of forces) from our simulations are available inMendeley Data . III. RESULTS
We present next values of forces acting on each grain within a barchan dune. Because data forforces experienced by individual grains within a barchan dune are still missing, we cannot directlycompare our results with previous works in terms of forces. Instead, after presenting the valuesof forces, we analyze our results in terms of trajectories, obtained experimentally by Alvarez andFranklin .Figures 1 and 2 show maps of the resultant force on grains composing a barchan dune, ~ F p ,in both the streamwise and spanwise directions for growing ( t < . t c ) and developed ( t > . t c )barchans, respectively, where t c is a characteristic time for the displacement of barchans computedas the length of the bedform divided by its celerity . Figures 1(a) and 2(a) correspond to theresultant force in the streamwise direction, F px , and Figs. 1(b) and 2(b) in the spanwise direction, F pz . The maps of Figs. 1 and 2 are in polar coordinates with origin at the dune centroid, the waterflow direction is 270 ◦ , and R is the radius of the initial conical pile . The mean forces wereaveraged over all grains, including those inside the barchan, and from the t = -2-10123456 10 -7 (a) -1.5-1-0.500.511.5 10 -7 (b)FIG. 2. Maps of the resultant force on grains composing a developed barchan dune ( t > . t c ) in the (a)streamwise and (b) spanwise directions, F px and F pz , respectively. This figure corresponds to Re = 1.47 × , and the solid circle in the maps indicates the crest position. Values in the colorbar are in N. until t = . t c for Fig. 1 and from t = . t c to t = . t c for Fig. 2. For F px , positive valuespoint downstream and negative upstream, while for F pz positive values point to the left (180 ◦ ) andnegative to the right (0 ◦ ), and they were computed for Re = 1.47 × .From Figs. 1(a) and 2(a), we observe that forces on grains point upstream at the crest, indicatingthat grains eroded upstream and migrating toward the crest decelerate and settle there. In theregion downstream the crest, in between evolving or developed horns (215 ◦ < ∼ α < ∼ ◦ , where α is the angle formed with 0 ◦ ), forces on grains point upstream, showing that grains arriving inthat region are subjected to the recirculation bubble and stay trapped close to the lee face (formingor already formed). Figures 1(a) and 2(a) show also that forces in the downstream direction arestronger neither at the leading edge of the bedform nor along the symmetry line in the streamwisedirection, but toward the lateral flanks of the dune, in the regions within 120 ◦ < ∼ α < ∼ ◦ and 60 ◦ < ∼ α < ∼ ◦ . Together, the maps for both F px and F pz of evolving and developed barchans showthat, for distances farther than 0.6 R from the centroid, grains within 10 ◦ < ∼ α < ∼ ◦ and 100 ◦ < ∼ α < ∼ ◦ are forced outwards (with respect to the streamwise centerline) and downstream, whilethose within 0 ◦ < ∼ α < ∼ ◦ and 210 ◦ < ∼ α < ◦ are forced inwards and downstream. Thiscorroborates our previous experimental findings that in the subaqueous case barchan horns areformed and sustained with grains coming from upstream regions on the periphery of the bedform.Although previous works on morphodynamics considered the fluid flow and inertial effects toexplain the growth of bedforms based on the flux of grains , the resultant forces on individual7orce distribution within a barchan dune (a) (b)FIG. 3. PDFs of the resultant forces in the (a) streamwise and (b) spanwise directions. Re = 1.47 × . grains, responsible for their trajectories, had never been shown before.We observe forces that tend to entrain grains to the crest and horns of both evolving anddeveloped barchans, which corroborates the observations made in experiments on subaqueousbarchans (see the supplementary material for a movie from our numerical simulations showingthe instantaneous values of the resultant force on each grain). We estimated the granular flux bycomputing the number of grains crossing a barchan cross section ( x plane) at x = t c . We then counted the number of grains going to the crest andeach of the horns, and computed the respective percentages. We found that around 9% of grainsmigrate to each horn (so that 18% of grains migrate to horns) and 13% to the crest, the remaindergrains migrating to either the regions between the crest and one of the horns, where they settlebefore falling by avalanches, or going around the dune until reaching the horn tips, from wherethey are entrained downstream.We computed, for all grains, the probability density functions (PDFs) of the resultant forcefrom t =0 to t =5.0 t c , presented in Figs. 3(a) and 3(b) for values in the streamwise and spanwisedirections, respectively. We observe that the PDF of F pz is symmetric, peaked at zero and witha width approximately equal to that of F px , indicating that in average grains have considerablespanwise forces, but with a zero mean (and also most probable value). This corroborates the largespanwise displacements of grains observed in subaqueous barchans , following circular trajec-tories and migrating toward the horns (symmetrically with respect to the streamwise centerline).For F px , the PDF is not symmetric and is peaked at a value of the order of 10 − N, which is thesame order of magnitude of the mean value. In addition, Fig. 3(a) shows a probability of 86.3%8orce distribution within a barchan dune (a) (b)FIG. 4. PDFs of contact forces in the (a) streamwise and (b) spanwise directions. Re = 1.47 × . of finding F px within -0.5 × − and 0.5 × − N, and that 5.5% of grains undergo values of F px higher than their relative weight (of approximately 1 × − N), corresponding to highly ac-celerated and decelerated grains. In order to investigate the transmission of forces by contacts, weseparated the contact from the other forces (Eq. 3) for all grains within the bedform and computedthe PDFs for values in the streamwise ( F cx ) and spanwise ( F cz ) directions, presented in Figs. 4(a)and 4(b), respectively. Both PDFs are peaked at zero, as expected since the action of one grainis the reaction on others, with most of values ranging from -0.5 × − to 0.5 × − N, suchextremes being of the order of the grain’s weight. In addition, we observe that the PDF in thespanwise direction has a width comparable to that in the streamwise direction. Because very fewgrains experience resultant forces in the streamwise direction stronger than their weight, whereasall of them undergo gravity, most of values transmitted by contacts and presented in Figs. 4(a) and4(b) are due to gravity. However, while F cx presents a small asymmetry, which could be inferredto contact forces with the wall being neglected in the PDF, F cz has a relatively larger asymmetry.We believe that these asymmetries are caused by irregularities in the contact network and the factthat Figs. 4(a) and 4(b) are based upon one particular simulation. Networks of contact forcescorroborate that supposition, and are available in the supplementary material.Finally, Fig. 5 shows examples of Lagrangian trackings of the resultant force acting on in-dividual grains migrating to the crest, to one of the horns, and falling by avalanches. Figure 5presents one example for each trajectory, but they represent well other grains following similartrajectories. For the grain migrating to the horn, it took 1.8 t c to reach the horn, and was trackedfor additional 0.7 t c (2.5 t c being the characteristic time for growth of subaqueous barchans ), while9orce distribution within a barchan dune -6 FIG. 5. Resultant force on some individual grains migrating to the crest, to one of the horns, and falling byavalanches, as functions of the normalized time. Re = 1.47 × . the grain migrating to the crest spent approximately 1.6 t c to arrive there. The grain tracked duringan avalanche started falling at 2.5 t c and took 1.1 t c to arrive at the dune base, after which it wasburied. In particular, we note large variations in F px for grains migrating to horns and crest, withpositive average values. These variations express the intermittent motion of grains in subaque-ous bedload . For grains involved in avalanches, fluctuations are smaller and due mainly tocontacts during their fall.Our results consist in new information, not accessible from previous experiments or simula-tions, on the resultant force acting on each grain, and, therefore, on how forces are distributedwithin a subaqueous barchan dune. However, differences with respect to aeolian and martiandunes are expected since grains within a bedload layer move differently depending on the fluidstate. When the fluid is a liquid, grains move by rolling and sliding and follow closely the fluidflow, while for gases grains move by saltation and reptation, those in saltation following ballisticflights in the main flow direction. In spite of these differences, our results represent an importantstep toward understanding the morphodyamics of barchans.10orce distribution within a barchan dune IV. CONCLUSIONS
Based on CFD-DEM computations, we measured the resultant force acting on each grain forall grains composing a barchan dune. We obtained maps of force distributions within evolvingand developed barchans, PDFs of the magnitude of the resultant and contact forces, and the timeevolution of forces on tracked grains. We confirmed that the resultant force on grains acts in theirentrainment toward the crest and horns of subaqueous barchans. In particular, we showed thatstronger forces on grains occur at neither the crest nor leading edge of the barchan, but in positionsjust upstream the dune centroid on the periphery of the dune, which corroborates the trajectories ofgrains migrating to horns reported in the literature . We showed also that a great part of grainsundergo longitudinal forces of the order of 10 − N (around 86% of grains experience resultantforces within -0.5 × − and 0.5 × − N), with negative values around the crest, resultingin decelerations and grain deposition in that region. Finally, we found that around 18% of grainsmigrate to horns and 13% to the crest, the remainder grains migrating to either the regions betweenthe crest and one of the horns (where they settle before falling by avalanches) or going around thedune until reaching the horn tips (from where they are entrained downstream). The present resultsprovide new insights into barchan morphology and how grains are distributed within the dune.
SUPPLEMENTARY MATERIAL
See the supplementary material for a layout of the numerical setup, PDFs of the fluid forces ongrains, networks of contact forces within the bedform, and a movie from our numerical simulationsshowing the instantaneous values of the resultant force on each grain.
DATA AVAILABILITY
The data that support the findings of this study are openly available in Mendeley Data athttp://dx.doi.org/10.17632/g666hxgrty.1.
ACKNOWLEDGMENTS
REFERENCES R. A. Bagnold,
The Physics of Blown Sand and Desert Dunes (Chapman and Hall, London,1941). B. Andreotti, P. Claudin, and S. Douady, “Selection of dune shapes and velocities. part 1: Dy-namics of sand, wind and barchans,” Eur. Phys. J. B , 321–329 (2002). F. Charru, B. Andreotti, and P. Claudin, “Sand ripples and dunes,” Ann. Rev. Fluid Mech. ,469–493 (2013). S. Courrech du Pont, “Dune morphodynamics,” C. R. Phys. , 118 – 138 (2015). H. J. Herrmann and G. Sauermann, “The shape of dunes,” Physica A (Amsterdam) , 24–30(2000). P. Hersen, “On the crescentic shape of barchan dunes,” Eur. Phys. J. B , 507–514 (2004). C. A. Alvarez and E. M. Franklin, “Birth of a subaqueous barchan dune,”Phys. Rev. E , 062906 (2017). C. A. Alvarez and E. M. Franklin, “Role of transverse displacements in the formation of sub-aqueous barchan dunes,” Phys. Rev. Lett. , 164503 (2018). P. Hersen, S. Douady, and B. Andreotti, “Relevant length scale of barchan dunes,”Phys. Rev. Lett. , 264301 (2002). P. Claudin and B. Andreotti, “A scaling law for aeolian dunes on Mars, Venus, Earth, and forsubaqueous ripples,” Earth Plan. Sci. Lett. , 20–44 (2006). E. M. Franklin and F. Charru, “Morphology and displacement of dunes in a closed-conduit flow,”Powder Technology , 247–251 (2009). E. M. Franklin and F. Charru, “Subaqueous barchan dunes in turbulent shear flow. Part 1. Dunemotion,” J. Fluid Mech. , 199–222 (2011). E. J. R. Parteli and H. J. Herrmann, “Dune formation on the present mars,”Phys. Rev. E , 041307 (2007). 12orce distribution within a barchan dune N. Endo, H. Kubo, and T. Sunamura, “Barchan-shaped ripple marks in a wave flume,” EarthSurf. Process. Landforms , 31–42 (2004). N. Hori, A. Yamada, Y. Oshiro, and O. Sano, “Formation of barchans and ripples due to steadyviscous flow in an annular channel,” J. Phys. Soc. Jpn. , 024401 (2007). E. Reffet, S. Courrech du Pont, P. Hersen, and S. Douady, “Formation and stability of transverseand longitudinal sand dunes,” Geology , 491–494 (2010). J. L. Wenzel and E. M. Franklin, “Velocity fields and particle trajectories for bed load oversubaqueous barchan dunes,” Granular Matter , 321–334 (2019). C. A. Alvarez and E. M. Franklin, “Horns of subaqueous barchan dunes: A study at the grainscale,” Phys. Rev. E , 042904 (2019). G. Sauermann, K. Kroy, and H. J. Herrmann, “Continuum saltation model for sand dunes,”Phys. Rev. E , 031305 (2001). K. Kroy, G. Sauermann, and H. J. Herrmann, “Minimal model for aeolian sand dunes,”Phys. Rev. E , 031302 (2002). K. Kroy, G. Sauermann, and H. J. Herrmann, “Minimal model for sand dunes,”Phys. Rev. Lett. , 054301 (2002). K. Kroy, S. Fischer, and B. Obermayer, “The shape of barchan dunes,” J. Phys. Condens. Matter , S1229–0S1235 (2005). V. Schwämmle and H. J. Herrmann, “A model of barchan dunes including lateral shear stress,”Eur. Phys. J. E , 57–65 (2005). E. J. R. Parteli, O. Durán, M. C. Bourke, H. Tsoar, T. Pöschel, and H. Herrmann, “Origins ofbarchan dune asymmetry: Insights from numerical simulations,” Aeol. Res. , 121–133 (2014). C. Narteau, D. Zhang, O. Rozier, and P. Claudin, “Setting the length and time scales of a cellularautomaton dune model from the analysis of superimposed bed forms,” J. Geophys. Res.: EarthSurf. (2009). D. Zhang, X. Yang, O. Rozier, and C. Narteau, “Mean sediment residence time in barchandunes,” J. Geophys. Res.: Earth Surf. , 451–463 (2014). M. W. Schmeeckle, “Numerical simulation of turbulence and sediment transport of mediumsand,” J. Geophys. Res. Earth Surf. , 1240–1262 (2014). A. G. Kidanemariam and M. Uhlmann, “Direct numerical simulation of pattern formation insubaqueous sediment,” J. Fluid Mech. , R2 (2014).13orce distribution within a barchan dune A. G. Kidanemariam and M. Uhlmann, “Interface-resolved direct numerical simulation of theerosion of a sediment bed sheared by laminar channel flow,” Int. J. Multiphase Flow , 174–188(2014). A. G. Kidanemariam and M. Uhlmann, “Formation of sediment patterns in channel flow: mini-mal unstable systems and their temporal evolution,” J. Fluid Mech. , 716–743 (2017). D. Liu, X. Liu, X. Fu, and W. G., “Quantification of the bed load effects on turbulent open-channel flows,” J. Geophys. Res. Earth Surf. , 767–789 (2016). R. Sun and H. Xiao, “SediFoam: A general-purpose, open-source CFD-–DEM solver forparticle-laden flow with emphasis on sediment transport,” Comput. Geosci. , 207–219 (2016). T. Pähtz and O. Durán, “Fluid forces or impacts: What governs the entrainment of soil particlesin sediment transport mediated by a newtonian fluid?” Phys. Rev. Fluids , 074303 (2017). T. Pähtz and O. Durán, “Unification of aeolian and fluvial sediment transport rate from granularphysics,” Phys. Rev. Lett. , 168001 (2020). M. Colombini, “A decade’s investigation of the stability of erodible stream beds,” J. Fluid Mech. , 1–4 (2014). C. A. Alvarez and E. M. Franklin, “Shape evolution of numerically obtained subaqueous barchandunes,” Phys. Rev. E , 012905 (2020). C. Goniva, C. Kloss, N. G. Deen, J. A. M. Kuipers, and S. Pirker, “Influence of rolling frictionon single spout fluidized bed simulation,” Particuology , 582–591 (2012). C. Kloss and C. Goniva, “LIGGGHTS: a new open source discrete element simulation software,”in
Proc. 5th Int. Conf. on Discrete Element Methods (London, UK, 2010). R. Berger, C. Kloss, A. Kohlmeyer, and S. Pirker, “Hybrid parallelization of the LIGGGHTSopen-source DEM code,” Powder Technology , 234–247 (2015). Z. Y. Zhou, S. B. Kuang, K. W. Chu, and A. B. Yu, “Discrete particle simulation of particle–fluidflow: model formulations and their applicability,” J. Fluid Mech. , 482–510 (2010). Y. Tsuji, T. Tanaka, and T. Ishida, “Lagrangian numerical simulation of plug flow of cohesionlessparticles in a horizontal pipe,” Powder Technology , 239–250 (1992). Y. Tsuji, T. Kawaguchi, and T. Tanaka, “Discrete particle simulation of two-dimensional flu-idized bed,” Powder Technology , 79–87 (1993). F. Nicoud and F. Ducros, “Subgrid-scale stress modelling based on the square of the velocitygradient tensors,” Flow Turbul Combust , 183–200 (1999).14orce distribution within a barchan dune C. A. Alvarez and E. M. Franklin, “Numerical data for ’Force distribution within a barchandune’,” Mendeley Data (2020), http://dx.doi.org/10.17632/g666hxgrty.1. F. Engelund and J. Fredsoe, “Sediment ripples and dunes,” Ann. Rev. Fluid Mech. , 13–37(1982). E. Lajeunesse, L. Malverti, and F. Charru, “Bed load transport in turbulent flow at the grainscale: Experiments and modeling,” J. Geophys. Res. , F04001 (2010). M. R. M. Penteado and E. M. Franklin, “Velocity fields of a bed-load layer under a turbulentliquid flow,” Exp. Therm. Fluid Sci.78