zbMATH — the first resource for mathematics

A modal discontinuous Galerkin method for simulating dusty and granular gas flows in thermal non-equilibrium in the Eulerian framework. (English) Zbl 1436.76023
Summary: A modal discontinuous Galerkin method was developed for computing compressible rarefied gaseous flows interacting with rigid particles and granular medium. In contrast to previous particle-based models that were developed to handle rarefied flows or solid phase particles, the present computational method employs full continuum-based models. This work is one of the first attempts to apply the modal discontinuous Galerkin method to a two-fluid model framework, which covers a wide range of gas and solid phase regimes, from a continuum to non-equilibrium gas, and from dusty to collisional regimes. The rarefaction effects were taken into account by applying the second-order Boltzmann-Curtiss-based constitutive relationship in a two-fluid system of equations. For the dust phase, computational models were developed based on the kinetic theory of the granular flows. Due to the orthogonal property of the basis functions in the method, no specific treatment of the source terms, commonly necessary in the conventional finite volume method, was required. Moreover, a high-fidelity approach was selected to treat the non-strictly hyperbolic equations of a dusty gas. This allows the same inviscid numerical flux functions to be applied to both the gaseous Euler and solid pressureless-Euler system of equations. Further, we observed that, for the discretization of the viscous fluxes in multiphase cases, the local discontinuous Galerkin is superior to the first method by Bassi and Rebay. After a verification and validation study, the new computational model was used to simulate the impingement of an underexpanded jet on a dusty surface in a rarefied condition. A surface erosion model based on viscous erosion associated with aerodynamic entrainment was implemented at a solid surface. Simulation cases in the near-field of the nozzle flow were tested to evaluate the capabilities of the present computational model in handling the challenging problems of multi-scale multiphase flows.
76M10 Finite element methods applied to problems in fluid mechanics
76T25 Granular flows
76T15 Dusty-gas two-phase flows
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
Full Text: DOI
[1] Miura, H.; Glass, I. I., On a dusty-gas shock tube, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 373-388 (1982)
[2] Igra, O.; Ben-Dor, G.; Rakib, Z., The effect of dust and water droplets on the relaxation zone developed behind strong normal shock waves, Int. J. Multiph. Flow, 11, 121-132 (1985)
[3] Toumi, I.; Kumbaro, A., An approximate linearized Riemann solver for a two-fluid model, J. Comput. Phys., 124, 286-300 (1996) · Zbl 0847.76056
[4] Tiselj, I.; Petelin, S., Modelling of two-phase flow with second-order accurate scheme, J. Comput. Phys., 136, 503-521 (1997) · Zbl 0918.76050
[5] Sainsaulieu, L., Finite volume approximation of two phase-fluid flows based on an approximate Roe-type Riemann solver, J. Comput. Phys., 121, 1-28 (1995) · Zbl 0834.76070
[6] Igra, O.; Elperin, I.; Ben-Dor, G., Dusty gas flow in a converging-diverging nozzle, J. Fluids Eng., 121, 908-913 (1999)
[7] Saito, T., Numerical analysis of dusty-gas flows, J. Comput. Phys., 176, 129-144 (2002) · Zbl 1120.76350
[8] Saito, T.; Marumoto, M.; Takayama, K., Numerical investigations of shock waves in gas-particle mixtures, Shock Waves, 13, 299-322 (2003) · Zbl 1063.76063
[9] Pelanti, M.; LeVeque, R. J., High-resolution finite volume methods for dusty gas jets and plumes, SIAM J. Sci. Comput., 28, 1335-1360 (2006) · Zbl 1123.76042
[10] Vié, A.; Pouransari, H.; Zamansky, R.; Mani, A., Particle-laden flows forced by the disperse phase: comparison between Lagrangian and Eulerian simulations, Int. J. Multiph. Flow, 79, 144-158 (2016)
[11] van der Hoef, M. A.; Ye, M.; van Sint Annaland, M.; Andrews, A. T.; Sundaresan, S.; Kuipers, J. A.M., Multiscale modeling of gas-fluidized beds, Adv. Chem. Eng., 31, 65-149 (2006)
[12] Anderson, T. B.; Jackson, R., Fluid mechanical description of fluidized beds. Equations of motion, Ind. Eng. Chem. Fundam., 6, 527-539 (1967)
[13] Tsuo, Y. P.; Gidaspow, D., Computation of flow patterns in circulating fluidized beds, AIChE J., 36, 885-896 (1990)
[14] Kuipers, J.; Van Duin, K.; Van Beckum, F.; Van Swaaij, W. P.M., A numerical model of gas-fluidized beds, Chem. Eng. Sci., 47, 1913-1924 (1992)
[15] Lun, C.; Savage, S. B.; Jeffrey, D.; Chepurniy, N., Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield, J. Fluid Mech., 140, 223-256 (1984) · Zbl 0553.73098
[16] Reeks, M., On the constitutive relations for dispersed particles in nonuniform flows. I: Dispersion in a simple shear flow, Phys. Fluids A, Fluid Dyn., 5, 750-761 (1993) · Zbl 0803.76008
[17] Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions (1994), Academic Press · Zbl 0789.76001
[18] Gantt, J. A.; Gatzke, E. P., Kinetic theory of granular flow limitations for modeling high-shear mixing, Ind. Eng. Chem. Res., 45, 6721-6727 (2006)
[19] Schneiderbauer, S.; Aigner, A.; Pirker, S., A comprehensive frictional-kinetic model for gas-particle flows: analysis of fluidized and moving bed regimes, Chem. Eng. Sci., 80, 279-292 (2012)
[20] Houim, R. W.; Oran, E. S., A multiphase model for compressible granular-gaseous flows: formulation and initial tests, J. Fluid Mech., 789, 166-220 (2016) · Zbl 1374.76234
[21] Busch, A.; Johansen, S. T., On the validity of the two-fluid-KTGF approach for dense gravity-driven granular flows, Powder Technol. (2020)
[22] Desjardins, O.; Fox, R. O.; Villedieu, P., A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys., 227, 2514-2539 (2008) · Zbl 1261.76027
[23] Fox, R. O., Higher-order quadrature-based moment methods for kinetic equations, J. Comput. Phys., 228, 7771-7791 (2009) · Zbl 1391.82049
[24] Sabat, M.; Larat, A.; Vié, A.; Massot, M., On the development of high order realizable schemes for the Eulerian simulation of disperse phase flows on unstructured grids: a convex-state preserving discontinuous Galerkin method, J. Comput. Multiph. Flows, 6, 3, 247-270 (2014)
[25] Liu, C.; Wang, Z.; Xu, K., A unified gas-kinetic scheme for continuum and rarefied flows VI: dilute disperse gas-particle multiphase system, J. Comput. Phys., 386, 264-295 (2019)
[26] Metzger, P. T.; Smith, J.; Lane, J. E., Phenomenology of soil erosion due to rocket exhaust on the Moon and the Mauna Kea lunar test site, J. Geophys. Res., Planets, 116 (2011)
[27] Gorman, A., Dr Space Junk vs The Universe: Archaeology and the Future (2019), The MIT Press
[28] Abouali, O.; Saadabadi, S.; Emdad, H., Numerical investigation of the flow field and cut-off characteristics of supersonic/hypersonic impactors, J. Aerosol Sci., 42, 65-77 (2011)
[29] Shu, C. W., Discontinuous Galerkin methods for time-dependent convection dominated problems: basics, recent developments and comparison with other methods, (Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (2016), Springer), 371-399 · Zbl 1357.65179
[30] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279 (1997) · Zbl 0871.76040
[31] Cockburn, B.; Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224 (1998) · Zbl 0920.65059
[32] Kontzialis, K.; Ekaterinaris, J. A., High order discontinuous Galerkin discretizations with a new limiting approach and positivity preservation for strong moving shocks, Comput. Fluids, 71, 98-112 (2013) · Zbl 1365.76120
[33] You, H.; Kim, C., Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes I. Volume integration, J. Comput. Phys., 395, 223-246 (2019)
[34] Giraldo, F. X.; Warburton, T., A high-order triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Methods Fluids, 56, 899-925 (2008) · Zbl 1290.86002
[35] Iannelli, J., An implicit Galerkin finite element Runge-Kutta algorithm for shock-structure investigations, J. Comput. Phys., 230, 260-286 (2011) · Zbl 1427.74077
[36] Li, L.; Zhang, Q., A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes, Comput. Fluids, 159, 316-326 (2017) · Zbl 1390.76331
[37] Boscheri, W.; Balsara, D. S., High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes with WENO Adaptive-Order reconstruction on unstructured meshes, J. Comput. Phys., 398, Article 108899 pp. (2019)
[38] Cockburn, B.; Karniadakis, G. E.; Shu, C. W., The development of discontinuous Galerkin methods, (Discontinuous Galerkin Methods (2000), Springer), 3-50 · Zbl 0989.76045
[39] Rivière, B.; Wheeler, M. F.; Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Comput. Geosci., 3, 337-360 (1999) · Zbl 0951.65108
[40] Sun, S.; Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Appl. Numer. Math., 52, 273-298 (2005) · Zbl 1079.76584
[41] Klieber, W.; Rivière, B., Adaptive simulations of two-phase flow by discontinuous Galerkin methods, Comput. Methods Appl. Mech. Eng., 196, 404-419 (2006) · Zbl 1120.76327
[42] Owkes, M.; Desjardins, O., A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows, J. Comput. Phys., 249, 275-302 (2013) · Zbl 1427.76218
[43] Diehl, D.; Kremser, J.; Kröner, D.; Rohde, C., Numerical solution of Navier-Stokes-Korteweg systems by local discontinuous Galerkin methods in multiple space dimensions, Appl. Math. Comput., 272, 309-335 (2016) · Zbl 1410.76166
[44] Dumbser, M.; Loubère, R., A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes, J. Comput. Phys., 319, 163-199 (2016) · Zbl 1349.65447
[45] Moortgat, J.; Firoozabadi, A., Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids, J. Comput. Phys., 315, 476-500 (2016) · Zbl 1349.76241
[46] Le, N. T.P.; Xiao, H.; Myong, R. S., A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases, J. Comput. Phys., 273, 160-184 (2014) · Zbl 1351.76065
[47] Xiao, H.; Myong, R. S., Computational simulations of microscale shock-vortex interaction using a mixed discontinuous Galerkin method, Comput. Fluids, 105, 179-193 (2014) · Zbl 1391.76135
[48] Raj, L. P.; Singh, S.; Karchani, A.; Myong, R. S., A super-parallel mixed explicit discontinuous Galerkin method for the second-order Boltzmann-based constitutive models of rarefied and microscale gases, Comput. Fluids, 157, 146-163 (2017) · Zbl 1390.76350
[49] Singh, S.; Karchani, A.; Myong, R. S., Non-equilibrium effects of diatomic and polyatomic gases on the shock-vortex interaction based on the second-order constitutive model of the Boltzmann-Curtiss equation, Phys. Fluids, 30, Article 016109 pp. (2018)
[50] Ejtehadi, O.; Rahimi, A.; Karchani, A.; Myong, R. S., Complex wave patterns in dilute gas-particle flows based on a novel discontinuous Galerkin scheme, Int. J. Multiph. Flow, 104, 125-151 (2018)
[51] Ejtehadi, O.; Rahimi, A.; Myong, R. S., Numerical investigation of the counter-intuitive behavior of Mach disk movement in underexpanded gas-particle jets, J. Comput. Fluid Engrg., 24, 19-28 (2019)
[52] Myong, R. S., A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gasdynamics, J. Comput. Phys., 168, 47-72 (2001) · Zbl 1074.76573
[53] Myong, R. S., A generalized hydrodynamic computational model for rarefied and microscale diatomic gas flows, J. Comput. Phys., 195, 655-676 (2004) · Zbl 1115.76405
[54] Myong, R. S., Numerical simulation of hypersonic rarefied flows using the second-order constitutive model of the Boltzmann equation, (Advances in Some Hypersonic Vehicles Technologies (2018), InTech)
[55] Rana, A.; Ravichandran, R.; Park, J.; Myong, R. S., Microscopic molecular dynamics characterization of the second-order non-Navier-Fourier constitutive laws in the Poiseuille gas flow, Phys. Fluids, 28, Article 082003 pp. (2016)
[56] Stubbs, T. J.; Vondrak, R. R.; Farrell, W. M., Impact of dust on lunar exploration (2007), 4075
[57] He, X.; He, B.; Cai, G., Simulation of rocket plume and lunar dust using DSMC method, Acta Astronaut., 70, 100-111 (2012)
[58] Morris, A.; Goldstein, D.; Varghese, P.; Trafton, L., Modeling the interaction between a rocket plume, scoured regolith, and a plume deflection fence, (Earth and Space 2012: Engineering, Science, Construction, and Operations in Challenging Environments (2012)), 189-198
[59] Morris, A. B.; Goldstein, D. B.; Varghese, P. L.; Trafton, L. M., Approach for modeling rocket plume impingement and dust dispersal on the moon, J. Spacecr. Rockets, 52, 362-374 (2015)
[60] Burt, J.; Boyd, I., Development of a two-way coupled model for two phase rarefied flows, (42nd AIAA Aerospace Sciences Meeting and Exhibit (2004)), 1351
[61] Gallis, M.; Torczynski, J.; Rader, D., An approach for simulating the transport of spherical particles in a rarefied gas flow via the direct simulation Monte Carlo method, Phys. Fluids, 13, 3482-3492 (2001) · Zbl 1184.76171
[62] Gimelshein, S.; Alexeenko, A.; Wadsworth, D.; Gimelshein, N., The influence of particulates on thruster plume/shock layer interaction at high altitudes, (43rd AIAA Aerospace Sciences Meeting and Exhibit (2004)), 766
[63] Liu, D.; Yang, S.; Wang, Z.; Liu, H.; Cai, C.; Wu, D., On rocket plume, lunar crater and lunar dust interactions, (48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2010)), 1161
[64] Morris, A.; Goldstein, D.; Varghese, P.; Trafton, L., Plume impingement on a dusty lunar surface, (AIP Conference Proceedings (2011), AIP), 1187-1192
[65] Wright, M. J.; Candler, G. V.; Bose, D., Data-parallel line relaxation method for the Navier-Stokes equations, AIAA J., 36, 1603-1609 (1998)
[66] A. Rahimi, O. Ejtehadi, K.H. Lee, R.S. Myong, Near-field plume-surface interaction and regolith erosion and dispersal during the lunar landing, in review.
[67] Ishii, R.; Umeda, Y.; Yuhi, M., Numerical analysis of gas-particle two-phase flows, J. Fluid Mech., 203, 475-515 (1989)
[68] Kim, S. W.; Chang, K. S., Reflection of shock wave from a compression corner in a particle-laden gas region, Shock Waves, 1, 65-73 (1991) · Zbl 0806.76035
[69] Igra, O.; Hu, G.; Falcovitz, J.; Wang, B., Shock wave reflection from a wedge in a dusty gas, Int. J. Multiph. Flow, 30, 1139-1169 (2004) · Zbl 1136.76535
[70] Curtiss, C., The classical Boltzmann equation of a gas of diatomic molecules, J. Chem. Phys., 75, 376-378 (1981)
[71] Myong, R. S., Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects, Contin. Mech. Thermodyn., 21, 389-399 (2009) · Zbl 1234.76048
[72] Myong, R. S., On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules, Phys. Fluids, 26, Article 056102 pp. (2014)
[73] Eu, B. C., Kinetic Theory and Irreversible Thermodynamics (1992), John Wiley and Sons, Inc.
[74] Eu, B. C.; Ohr, Y. G., Generalized hydrodynamics, bulk viscosity, and sound wave absorption and dispersion in dilute rigid molecular gases, Phys. Fluids, 13, 744-753 (2001) · Zbl 1184.76149
[75] Onsager, L., Reciprocal relations in irreversible processes. I, Phys. Rev., 37, 405 (1931) · JFM 57.1168.10
[76] Chapman, S.; Cowling, T. G., The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (1970), Cambridge University Press · JFM 65.1541.01
[77] Hansen, J. P.; McDonald, I. R., Theory of Simple Liquids (1990), Elsevier · Zbl 0756.00004
[78] Carnahan, N. F.; Starling, K. E., Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51, 635 (1969)
[79] Ma, D.; Ahmadi, G., An equation of state for dense rigid sphere gases, J. Chem. Phys., 84, 3449-3450 (1986)
[80] Song, Y.; Mason, E.; Stratt, R. M., Why does the Carnahan-Starling equation work so well?, J. Phys. Chem., 93, 6916-6919 (1989)
[81] Alder, B.; Wainwright, T., Phase transition in elastic disks, Phys. Rev., 127, 359 (1962)
[82] Woodcock, L.; Woodcock, L. V., Ann. N.Y. Acad. Sci.. Ann. N.Y. Acad. Sci., Ann. N.Y. Acad. Sci., 371, 274 (1981)
[83] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation, (Technical Report LA-UR-73-479 (1973), Los Alamos Scientific Lab: Los Alamos Scientific Lab N. Mex. (USA))
[84] Cockburn, B.; Shu, C. W., The Runge-Kutta local projection P^1-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO. Modél. Math. Anal. Numér., 25, 337-361 (1988) · Zbl 0732.65094
[85] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comput., 52, 411-435 (1989) · Zbl 0662.65083
[86] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390 (1991) · Zbl 0742.76059
[87] Rusanov, V. V.e., Calculation of Interaction of Non-steady Shock Waves with Obstacles (1962), NRC, Division of Mechanical Engineering
[88] Nishikawa, H.; Kitamura, K., Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers, J. Comput. Phys., 227, 2560-2581 (2008) · Zbl 1388.76185
[89] Liou, M. S.; Jr, C. J. Steffen, A new flux splitting scheme, J. Comput. Phys., 107, 23-39 (1993) · Zbl 0779.76056
[90] Liou, M. S., A sequel to AUSM, Part II: AUSM+-up for all speeds, J. Comput. Phys., 214, 137-170 (2006) · Zbl 1137.76344
[91] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, D., Discontinuous Galerkin methods for elliptic problems, (Discontinuous Galerkin Methods (2000), Springer), 89-101 · Zbl 0948.65127
[92] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080
[93] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118
[94] Toro, E., Riemann-problem-based techniques for computing reactive two-phased flows, (Numerical Combustion (1989), Springer), 472-481
[95] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150, 425-467 (1999) · Zbl 0937.76053
[96] Powell, K.; Roe, P.; Myong, R. S.; Gombosi, T., An upwind scheme for magnetohydrodynamics, (12th Computational Fluid Dynamics Conference (1995)), 1704 · Zbl 0900.76344
[97] Powell, K. G.; Roe, P. L.; Linde, T. J.; Gombosi, T. I.; De Zeeuw, D. L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154, 284-309 (1999) · Zbl 0952.76045
[98] Janhunen, P., A positive conservative method for magnetohydrodynamics based on HLL and Roe methods, J. Comput. Phys., 160, 649-661 (2000) · Zbl 0967.76061
[99] Jung, S.; Myong, R. S., A second-order positivity-preserving finite volume upwind scheme for air-mixed droplet flow in atmospheric icing, Comput. Fluids, 86, 459-469 (2013) · Zbl 1290.76157
[100] Zhang, X.; Shu, C. W., Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, J. Comput. Phys., 230, 1238-1248 (2011) · Zbl 1391.76375
[101] Zhang, X.; Shu, C. W., On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 3091-3120 (2010) · Zbl 1187.65096
[102] Barth, T. J.; Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, (27th Aerospace Sciences Meeting (1989), American Institute of Aeronautics and Astronautics)
[103] Fedorov, A.; Kharlamova, Y. V., Reflection of a shock wave in a dusty cloud, Combust. Explos. Shock Waves, 43, 104-113 (2007)
[104] Lewis, C.; Carlson, D., Normal shock location in underexpanded gas and gas-particle jets, AIAA J., 2, 776-777 (1964)
[105] Crist, S.; Glass, D.; Sherman, P., Study of the highly underexpanded sonic jet, AIAA J., 4, 68-71 (1966)
[106] Sommerfeld, M., The structure of particle-laden, underexpanded free jets, Shock Waves, 3, 299-311 (1994)
[107] Avduevskii, V. S.; Ivanov, A.; Karpman, I. M.; Traskovskii, V. D.; Yudelovich, M. Y., Flow in supersonic viscous under expanded jet, Fluid Dyn., 5, 409-414 (1970)
[108] Franquet, E.; Perrier, V.; Gibout, S.; Bruel, P., Free underexpanded jets in a quiescent medium: a review, Prog. Aerosp. Sci., 77, 25-53 (2015)
[109] Land, N. S.; Clark, L. V., Experimental Investigation of Jet Impingement on Surfaces of Fine Particles in a Vacuum Environment (1965), National Aeronautics and Space Administration
[110] Morris, A. B., Simulation of rocket plume impingement and dust dispersal on the lunar surface (2012), The University of Texas at Austin
[111] Mitchell, J.; Houston, W.; Scott, R.; Costes, N.; Carrier, W.; Bromwell, L., Mechanical properties of lunar soil: density, porosity, cohesion and angle of internal friction, (Lunar and Planetary Science Conference Proceedings (1972)), 3235
[112] Colwell, J.; Batiste, S.; Horányi, M.; Robertson, S.; Sture, S., Lunar surface: dust dynamics and regolith mechanics, Rev. Geophys., 45 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.