×

Numerical methods for dispersed multiphase flows. (English) Zbl 1391.76174

Bodnár, Tomáš (ed.) et al., Particles in flows. Based on the summer course and workshop, Prague, Czech Republic, August 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-60281-3/hbk; 978-3-319-60282-0/ebook). Advances in Mathematical Fluid Mechanics, 327-396 (2017).
Summary: This article gives an overview of numerical methods for the calculation of dispersed multi-phase flows. At the beginning, a brief introduction is given on the different flow regimes observed for multi-phase flows in general. Then a characterisation and classification of dispersed multi-phase flows is introduced based on inter-particle spacing and volume fraction. As an introduction to the subject, the numerical methods used for single-phase flows are briefly described based on the turbulent scales being resolved by the numerical grid. Since even dispersed multi-phase flows are extremely complex, the hierarchy of the different numerical methods is highlighted ranging from macro-scale numerical simulations for an entire industrial process down to micro-scale simulations required for analysing particle scale phenomena. Due to constraints in computational power and storage availability, macro-scale simulations can only be done with a limited grid resolution and the assumption of particles being treated as point-masses. Consequently, all transport phenomena occurring on scales smaller than the grid cell and on the scale of the particle have to be considered through additional closures and models. Therefore, essential elements in this multi-scale problem are direct numerical simulations that fully resolve the particles and the flow around them. The different methods for such resolved simulations are briefly described. The major part of this article is focused on the modelling of dispersed multi-phase flows relying on the point-particle assumption. The multi-fluid method or Euler/Euler model is briefly described in order to demark its applicability and limitations. The hybrid Euler/Lagrange approach based on tracking a large number of point particles and its different variants are introduced in more detail, emphasising the two-way coupling approaches for unsteady flows. The importance of accurately modelling particle-scale phenomena is highlighted and an estimate for the significance of particle-wall and inter-particle collisions is given. Finally, three application examples are introduced, emphasising the potential of Euler/Lagrange simulations. For a particle-laden swirling flow the semi-unsteady approach is used for analysing unsteady particle roping phenomena. The simulations of particle suspension in a stirred vessel highlight the importance of inter-particle collisions even at relatively low volume fractions up to 5%. Finally, it is demonstrated that the Euler/Lagrange approach may also be used to study an industrial filtration process where it allows the prediction of particle deposits and filter cake formation. In this respect extensions are possible which provide more information on the internal filter cake structure.
For the entire collection see [Zbl 1381.35003].

MSC:

76F25 Turbulent transport, mixing
76F65 Direct numerical and large eddy simulation of turbulence
76T10 Liquid-gas two-phase flows, bubbly flows
76T15 Dusty-gas two-phase flows
76T20 Suspensions
82C22 Interacting particle systems in time-dependent statistical mechanics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Abrahamson, Collision rates of small particles in a vigorously turbulent fluid. Chem. Eng. Sci. 30, 1371-1379 (1975) · doi:10.1016/0009-2509(75)85067-6
[2] B. Abramzon, W. Sirignano, Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transf. 32, 1605-1618 (1989) · doi:10.1016/0017-9310(89)90043-4
[3] A. Bakker, Applied computational fluid dynamics: Lecture 10: Turbulence models (2002). http://www.bakker.org
[4] G. Balzer, O. Simonin, Extension of Eulerian gas-solid flow modelling to dense fluidised bed prediction. in Proceedings of the 5th International Symposium on Refined Flow Modelling and Turbulence Measurements (1993)
[5] G. Balzer, A. Boelle, O. Simonin, Eulerian gas-solid flow modelling of dense fluidized bed. Fluidization 8, 1125-1134 (1995)
[6] A. Berlemont, P. Desjonqueres, G. Gouesbet, Particle Lagrangian simulation in turbulent flows. Int. J. Multiphase Flow 16, 19-34 (1990) · Zbl 1134.76493 · doi:10.1016/0301-9322(90)90034-G
[7] C.-U. Böttner, M. Sommerfeld, Numerical calculation of electrostatic powder painting using the Euler/Lagrange approach. Powder Technol. 125, 206-216 (2002) · doi:10.1016/S0032-5910(01)00508-3
[8] E. Bourloutski, M. Sommerfeld, Euler/Lagrange calculations of gas-liquid-solid-flows in bubble columns with phase interaction, in Bubbly Flows: Analysis, Modelling and Calculation, ed. by M. Sommerfeld (Springer, Berlin, 2004), pp. 243-259 · doi:10.1007/978-3-642-18540-3_19
[9] T.M. Burton, J.K. Eaton, Fully resolved simulations of particle-turbulence interaction. J. Fluid Mech. 545, 67-111 (2005) · Zbl 1085.76565 · doi:10.1017/S0022112005006889
[10] P. Chen, M.P. Duduković, J. Sanyal, Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. AIChE J. 51, 696-712 (2005) · doi:10.1002/aic.10381
[11] X.Y. Chen, Ch. Focke, H. Marschall, D. Bothe, Investigation of elementary processes of non-newtonian droplets inside spray processes by means of direct numerical simulation, in Process-Spray (Springer, Cham, 2016)
[12] E. Climent, M.R. Maxey, The force coupling method: a flexible approach for the simulation of particulate flows. Theoretical Methods for Micro Scale Viscous Flows (Transworld Research Network, Trivandrum, 2009), pp. 173-193
[13] J. Cousin, A. Berlemont, T. Ménard, S. Grout, Primary breakup simulation of a liquid jet discharged by a low-pressure compound nozzle. Comput. Fluids 63, 165-173 (2012) · Zbl 1365.76080 · doi:10.1016/j.compfluid.2012.04.013
[14] C.T. Crowe, On the relative importance of particle-particle collisions in gas-particle flows, in Proceedings of the Conference on Gas Borne Particles, Paper 78/81 (1981), pp. 135-137
[15] C.T. Crowe (ed.), Multiphase Flow Handbook (CRC Press/Taylor & Francis Group, Boca Raton, 2006) · Zbl 1098.76001
[16] C.T. Crowe, M.P. Sharma, D.E. Stock, The particle-source-in-cell (psi-cell) model for gas-droplet flows. J. Fluids Eng. 99, 325-332 (1977) · doi:10.1115/1.3448756
[17] C.T. Crowe, J.D. Schwarzkopf, M. Sommerfeld, Y. Tsuji, Multiphase Flows with Droplets and Particles, 2nd edn. (CRC Press/Taylor & Francis Group, Boca Raton, 2012)
[18] G.T. Csanady, Turbulent diffusion of heavy particles in the atmosphere. Atmos. Sci. 20, 201-208 (1963) · doi:10.1175/1520-0469(1963)020<0201:TDOHPI>2.0.CO;2
[19] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. Geotechnique 29, 47-65 (1979) · doi:10.1680/geot.1979.29.1.47
[20] S. Decker, Zur Berechnung von gerührten Suspensionen mit dem Euler-Lagrange-Verfahren. Dissertation Martin-Luther-Universität Halle-Wittenberg, Fachbereich Ingenieurwissen-schaften (2005)
[21] N.G. Deen, M. Van Sint Annaland, M.A. Van der Hoef, J.A.M. Kuipers, Review of discrete particle modeling of fluidized beds. Chem. Eng. Sci. 62, 28-44 (2007) · Zbl 1388.76241 · doi:10.1016/j.ces.2006.08.014
[22] J.J. Derksen, Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence. AIChE J. 58, 2589-2600 (2012) · doi:10.1002/aic.12761
[23] O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization. J. Comput. Phys. 227, 8395-8416 (2008) · Zbl 1256.76051 · doi:10.1016/j.jcp.2008.05.027
[24] M.T. Dhotre, N.G. Deen, B. Niceno, Z. Khan, J.B. Joshi, Large eddy simulation for dispersed bubbly flows: a review. Int. J. Chem. Eng. 2013, Article ID 343276 (2013)
[25] M. Dietzel, M. Sommerfeld, Numerical calculation of flow resistance for agglomerates with different morphology by the lattice-Boltzmann method. Powder Technol. 250, 122-137 (2013) · doi:10.1016/j.powtec.2013.09.023
[26] M. Dietzel, M. Ernst, M. Sommerfeld, Application of the lattice-Boltzmann method for particle-laden flows: point-particles and fully resolved particles. Flow Turbul. Combust. 97, 539-570 (2016) · doi:10.1007/s10494-015-9698-x
[27] J. Ding, D. Gidaspow, A bubbling fluidization model using kinetic theory of granular flow. AIChE J. 36, 523-538 (1990) · doi:10.1002/aic.690360404
[28] S. Elghobashi, On predicting particle-laden turbulent flows. Appl. Sci. Res. 52, 309-329 (1994) · doi:10.1007/BF00936835
[29] H. Enwald, E. Peirano, A.-E. Almstedt, Eulerian two-phase flow theory applied to fluidisation. Int. J. Multiphase Flow, Suppl. 22, 21-66 (1996) · Zbl 1135.76409
[30] M. Ernst, Analyse des Clustering-, Kollisions- und Agglomerationsverhalten von Partikeln in laminaren und turbulenten Strömungen, Dissertation Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg (2016)
[31] M. Ernst, M. Sommerfeld, On the volume fraction effects on inertial colliding particles in homogeneous isotropic turbulence. J. Fluids Eng. Trans. ASME 134, 031302 (2012) · doi:10.1115/1.4005681
[32] M. Ernst, M. Sommerfeld, Resolved numerical simulation of particle agglomeration, in Colloid Process Engineering, Proceedings of Topical problems of Fluid Mechanics 2014 (Springer, Cham, 2015), pp. 45-71
[33] M. Ernst, M. Dietzel, M. Sommerfeld, A lattice Boltzmann method for simulating transport and agglomeration of resolved particles. Acta Mech. 224, 2425-2449 (2013) · Zbl 1398.76182 · doi:10.1007/s00707-013-0923-1
[34] P. Fede, O. Simonin, A. Ingram, 3D numerical simulation of a lab-scale pressurized dense fluidized bed focussing on the effect of the particle-particle restitution coefficient and particle-wall boundary conditions. Chem. Eng. Sci. 142, 215-235 (2016) · doi:10.1016/j.ces.2015.11.016
[35] J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, 3rd edn. (Springer, Berlin, 2002) · Zbl 0998.76001 · doi:10.1007/978-3-642-56026-2
[36] O. Filippova, D. Hänel, Grid refinement for lattice-BGK models. J. Comput. Phys. 147, 219-228 (1998) · Zbl 0917.76061 · doi:10.1006/jcph.1998.6089
[37] J. Fröhlich, Large Eddy Simulation turbulenter Strömungen (Teubner Verlag, Wiesbaden, 2006)
[38] H. Gao, H. Li, L.-P. Wang, Lattice Boltzmann simulation of turbulent flow laden with finite-size particles. Comput. Math. Appl. 65, 194-210 (2013) · Zbl 1268.76045 · doi:10.1016/j.camwa.2011.06.028
[39] D. Gidaspow, Multiphase Flow and Fluidization-Continuum and Kinetic Theory Descriptions (Academic, Boston, 1994) · Zbl 0789.76001
[40] R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169, 363-426 (2001) · Zbl 1047.76097 · doi:10.1006/jcph.2000.6542
[41] A.D. Gosman, I.E. Ioannides, Aspects of computer simulation of liquid-fueled combustors. Aerospace Science Meeting, Paper No. AIAA-81-0323 (1981)
[42] G. Gouesbet, A. Berlemont, Eulerian and Lagrangian approaches for predicting the behaviour of discrete particles in turbulent flows. Prog. Energy Combust. Sci. 25, 133-159 (1999) · doi:10.1016/S0360-1285(98)00018-5
[43] Z. Guo, C. Zheng, B. Shi, An extrapolation method for boundary conditions in lattice Boltzmann method. Phys. Fluids 14, 2007-2010 (2002) · Zbl 1185.76156 · doi:10.1063/1.1471914
[44] T.J. Hanratty, Physics of Gas-Liquid Flows (Cambridge University Press, Cambridge, 2013) · doi:10.1017/CBO9781139649421
[45] C.A. Ho, M. Sommerfeld, Modelling of micro-particle agglomeration in turbulent flow. Chem. Eng. Sci. 57, 3073-3084 (2002) · doi:10.1016/S0009-2509(02)00172-0
[46] A. Hölzer, M. Sommerfeld, Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles. Comput. Fluids 38, 572-589 (2009) · doi:10.1016/j.compfluid.2008.06.001
[47] S. Horender, Y. Hardalupas, Turbulent particle mass flux in a two-phase shear flow. Powder Technol. 192, 203-216 (2009) · Zbl 1136.76520 · doi:10.1016/j.powtec.2008.12.013
[48] H.H. Hu, N.A. Patankar, M.Y. Zhu, Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique. J. Comput. Phys. 169, 427-462 (2001) · Zbl 1047.76571 · doi:10.1006/jcph.2000.6592
[49] Y. Igci, A.T. Andrews IV, S. Sundaresan, S. Pannala, T. O‘Brian, Filtered two-fluid models for fluidized gas-particle suspensions. AIChE J. 54, 1431-1448 (2008)
[50] R. Issa, Simulation of intermittent flow in multiphase oil and gas pipelines, in Seventh International Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia, 9-11 December 2009 (2013)
[51] T.K. Kjeldby, R. Henkes, O.J. Nydal, Slug tracking simulation of severe slugging experiments. Int. J. Mech. Aerospace Ind. Mechatron. Manuf. Eng. 5, 1156-1161 (2011)
[52] G. Kohnen, M. Rüger, M. Sommerfeld, Convergence behaviour for numerical calculations by the Euler/Lagrange method for strongly coupled phases, in Numerical Methods in Multiphase Flows 1994, FED-vol. 185, ed. by C.T. Crowe et al. (ASME, New York, 1994)
[53] E. Krepper, Th. Frank, D. Lucas, H.M. Prasser, Ph.J. Zwart, Inhomogeneous MUSIG model-a population balance approach for polydispersed bubbly flows, in Proceedings of The 12th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12), 30 September - 4 October 2007, Pittsburgh, Pennsylvania, Log No. 60 (2007)
[54] S. Kriebitzsch, R. Rzehak, Baseline model for bubbly flows: simulation of monodisperse flow in pipes of different diameters. Comput. Fluid Dyn. 1, 1-28 (2016)
[55] J.G.M. Kuerten, Point-particle DNS and LES of particle-laden turbulent flow - a state-of-the-art review. Flow Turbul. Combust. 97, 689-713 (2016) · doi:10.1007/s10494-016-9765-y
[56] S. Lain, M. Sommerfeld, Euler/Lagrange computations of pneumatic conveying in a horizontal channel with different wall roughness. Powder Technol. 184, 76-88 (2008) · doi:10.1016/j.powtec.2007.08.013
[57] S. Lain, M. Sommerfeld, Numerical calculation of pneumatic conveying in horizontal channels and pipes: Detailed analysis of conveying behaviour. Int. J. Multiphase Flow 39, 105-120 (2012) · doi:10.1016/j.ijmultiphaseflow.2011.09.006
[58] S. Lain, M. Sommerfeld, Characterisation of pneumatic conveying systems using the Euler/Lagrange approach. Powder Technol. 235, 764-782 (2013) · doi:10.1016/j.powtec.2012.11.029
[59] S. Lain, D. Bröder, M. Sommerfeld, M.F. Göz, Modelling hydrodynamics and turbulence in a bubble column using the Euler-Lagrange procedure. Int. J. Multiphase Flow 28, 1381-1407 (2002) · Zbl 1137.76648 · doi:10.1016/S0301-9322(02)00028-9
[60] G.L. Lane, M.P. Schwarz, G.M. Evans, Predicting gas-liquid flow in a mechanically stirred tank. Appl. Math. Model. 26, 223-235 (2002) · Zbl 1205.76168 · doi:10.1016/S0307-904X(01)00057-9
[61] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3, 269-289 (1974) · Zbl 0277.76049 · doi:10.1016/0045-7825(74)90029-2
[62] M. Lesieur, O. Metais, P. Comte, Large Eddy Simulations of Turbulence (Cambridge University Press, Cambridge, 2005) · Zbl 1101.76002 · doi:10.1017/CBO9780511755507
[63] J. Li, J.A.M. Kuipers, Gas-particle interactions in dense gas-fluidized beds. Chem. Eng. Sci. 58, 711-718 (2003) · doi:10.1016/S0009-2509(02)00599-7
[64] J. Li, J.A.M. Kuipers, Effect of competition between particle-particle and gas-particle interactions on flow patterns in dense gas-fluidized bed. Chem. Eng. Sci. 62, 3429-3442 (2007) · doi:10.1016/j.ces.2007.01.086
[65] Y. Liao, R. Rzehak, D. Lucas, E. Krepper, Baseline closure model for dispersed bubbly flow: bubble coalescence and breakup. Chem. Eng. Sci. 122, 336-349 (2015) · doi:10.1016/j.ces.2014.09.042
[66] J. Lipowsky, Zur instationären Euler/Lagrange-Simulation partikelbeladener Drallströmungen, Dissertation, Zentrum Für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg (2013)
[67] J. Lipowsky, M. Sommerfeld, Time dependent simulation of a swirling two-phase flow using an anisotropic turbulent dispersion model, in Proceedings of the ASME Fluids Engineering Summer Conference, Houston, Texas (2005)
[68] J. Lipowsky, M. Sommerfeld, LES-simulation of the formation of particle strands in swirling flows using an unsteady Euler-Lagrange approach, in Proceedings of the 6th International Conference on Multiphase Flow, ICMF2007 (2007)
[69] C. Loha, H. Chattopadhyay, P.K. Chatterjee, Assessment of drag models in simulating bubbling fluidized bed hydrodynamics. Chem. Eng. Sci. 75, 400-407 (2012) · doi:10.1016/j.ces.2012.03.044
[70] S. Lomholt, M.R. Maxey, Force-coupling method for particulate two-phase flow: stokes flow. J. Comput. Phys. 184, 381-405 (2003) · Zbl 1047.76100 · doi:10.1016/S0021-9991(02)00021-9
[71] E. Loth, Numerical approaches for motion of dispersed particles, droplets and bubbles. Prog. Energy Combust. Sci. 26, 161-223 (2000) · doi:10.1016/S0360-1285(99)00013-1
[72] K. Luo, J. Tan, Z. Wang, J. Fan, Particle-resolved direct numerical simulation of gas-solid dynamics in experimental fluidized beds. AIChE J. 62, 1917-1932 (2016) · doi:10.1002/aic.15186
[73] M.V. Lurie, E. Sinaiski, Modeling of Oil Product and Gas Pipeline Transportation (Wiley-VCH, Weinheim, 2008) · doi:10.1002/9783527626199
[74] J.M. MacInnes, F.V. Braco, Stochastic particle dispersion and the tracer particle limit. Phys. Fluids A 4, 2809-2824 (1992) · doi:10.1063/1.858337
[75] T. Ménard, S. Tanguy, A. Berlemont, Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet. Int. J. Multiphase Flow 33, 510-524 (2007) · doi:10.1016/j.ijmultiphaseflow.2006.11.001
[76] R. Mittal, G. Iaccarino, Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239-261 (2005) · Zbl 1117.76049 · doi:10.1146/annurev.fluid.37.061903.175743
[77] Chr. Mundo, M. Sommerfeld, C. Tropea, Droplet-wall collisions: experimental studies of the deformation and breakup process. Int. J. Multiphase Flow 21, 151-173 (1995) · Zbl 1134.76617
[78] Chr. Mundo, M. Sommerfeld, C. Tropea, On the modelling of liquid sprays impinging on surfaces. Atomization Sprays 8, 625-652 (1998)
[79] A.F. Nassar, G. Zivkovic, B. Genenger, F. Durst, PDA measurements and numerical simulation of turbulent two-phase flow in stirred vessels, in Bubbly Flows: Analysis, Modelling and Calculation, ed. by M. Sommerfeld (Springer, Berlin, 2004), pp. 337-352 · doi:10.1007/978-3-642-18540-3_26
[80] D. Oechsle, W. Baur, Praxiserfahrungen mit einem neuen Horizontalfilter. Brauwelt, Jahrg. 129, 2176-2180 (1989)
[81] B. Oesterlé, A. Petitjean, Simulation of particle-to-particle interactions in gas-solid flows. Int. J. Multiphase Flow 19, 199-211 (1993) · Zbl 1144.76434 · doi:10.1016/0301-9322(93)90033-Q
[82] A. Ozel, P. Fede, O. Simonin, Development of filtered Euler-Euler two-phase model for circulating fluidised bed: high resolution simulation, formulation and a priori analyses. Int. J. Multiphase Flow 55, 43-63 (2013) · doi:10.1016/j.ijmultiphaseflow.2013.04.002
[83] J.-F. Parmentier, O. Simonin, A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. AIChE J. 58, 1084-1098 (2012) · doi:10.1002/aic.12647
[84] C.S. Peskin, The fluid dynamics of heart valves: experimental, theoretical, and computational methods. Annu. Rev. Fluid Mech. 14, 235-259 (1982) · Zbl 0488.76129 · doi:10.1146/annurev.fl.14.010182.001315
[85] A. Prosperetti, G. Tryggvason, Computational Methods for Multiphase Flow (Cambridge University Press, Cambridge, 2009) · Zbl 1166.76004
[86] M.A. Rizk, S.E. Elghobashi, A two-equation turbulence model for dispersed dilute confined two-phase flow. Int. J. Multiphase Flow 15, 119-133 (1989) · doi:10.1016/0301-9322(89)90089-X
[87] M. Rudman, Volume-tracking methods for interfacial flow calculations. Int. J. Numer. Methods Fluids 24, 671-691 (1997) · Zbl 0889.76069 · doi:10.1002/(SICI)1097-0363(19970415)24:7<671::AID-FLD508>3.0.CO;2-9
[88] M. Rüger, S. Hohmann, M. Sommerfeld, G. Kohnen, Euler/Lagrange calculations of turbulent sprays: the effect of droplet collisions and coalescence. Atomization Sprays 10, 47-81 (2000) · doi:10.1615/AtomizSpr.v10.i1.30
[89] S. Sazhin, Modelling of heating, evaporation and ignition of fuel droplets: combined analytical, asymptotic and numerical analysis. J. Phys. Conf. Ser. 22, 174-193 (2005) · doi:10.1088/1742-6596/22/1/012
[90] L. Schiller, A. Naumann, Über die grundlegende Berechnung bei der Schwerkraftauf-bereitung. Ver. Dtsch. Ing. 44, 318-320 (1933)
[91] S. Schneiderbauer, S. Pirker, A coarse-grained two-fluid model for gas-solid fluidized beds. J. Comput. Multiphase Flows 6, 29-47 (2014) · doi:10.1260/1757-482X.6.1.29
[92] O. Simonin, Prediction of the dispersed phase turbulence in particle laden jet, in Gas-Solid Flows, ed. by D.E. Stock et al. ASME-JSME Fluids Engineering Conference, FED-vol. 121 (ASME, New York, 1991), pp. 197-206
[93] O. Simonin, Statistical and continuum modelling of turbulent reactive particulate flows. Part I: Theoretical derivation of dispersed phase Eulerian modelling from probability density function kinetic equation. Theoretical and Experimental Modelling of Particulate Flow. VKI Lecture Series 2000-06 (von Karman Institute for Fluid Dynamics, Brussels, 2000)
[94] M. Sommerfeld, Particle dispersion in turbulent flow: the effect of particle size distribution. Part. Part. Syst. Charact. 7, 209-220 (1990) · doi:10.1002/ppsc.19900070135
[95] M. Sommerfeld, Modelling of particle/wall collisions in confined gas-particle flows. Int. J. Multiphase Flow 18, 905-926 (1992) · Zbl 1144.76457 · doi:10.1016/0301-9322(92)90067-Q
[96] M. Sommerfeld, Modellierung und numerische Berechnung von partikelbeladenen turbulenten Strömungen mit Hilfe des Euler/Lagrange-Verfahrens. Habilitationsschrift, Universität Erlangen-Nürnberg (Shaker Verlag, Aachen, 1996)
[97] M. Sommerfeld, Modelling and numerical calculation of turbulent gas-solid flows with the Euler/Lagrange approach. KONA (Powder and Particle), No. 16 (1998), pp. 194-206
[98] Sommerfeld, M.: Analysis of isothermal and evaporating sprays using phase-Doppler anemometry and numerical calculations. Int. J. Heat Fluid Flow 19, 173-186 (1998) · doi:10.1016/S0142-727X(97)10022-4
[99] M. Sommerfeld, Validation of a stochastic Lagrangian modelling approach for inter-particle collisions in homogeneous isotropic turbulence. Int. J. Multiphase Flows 27, 1828-1858 (2001) · Zbl 1137.76744 · doi:10.1016/S0301-9322(01)00035-0
[100] M. Sommerfeld, Bewegung fester Partikel in Gasen und Flüssigkeiten, in VDI-Wärmeatlas, 9. Auflage, Kapitel Lca 1-9 (Springer, Berlin, 2002)
[101] M. Sommerfeld (ed.), in Proceedings of the 11th Workshop on Two-Phase Flow Predictions (CD-Rom), Merseburg, April 2005, Fachbereich Ingenieurwissenschaften, University of Halle (2005)
[102] M. Sommerfeld, Particle motion in fluids, in VDI-Buch: VDI Heat Atlas, Part 11 (Springer, Berlin, 2010), pp. 1181-1196
[103] M. Sommerfeld (ed.), in Proceedings of the 12th Workshop on Two-Phase Flow Predictions (CD-Rom), Merseburg, März 2010, Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg (2010)
[104] M. Sommerfeld, Report on the 13th Workshop on Two-Phase Flow Predictions. ERCOFTAC Bulletin, No. 95 (2013)
[105] M. Sommerfeld, S. Decker, State of the art and future trends in CFD simulation of stirred vessel hydrodynamics. Chem. Eng. Technol. 27, 215-224 (2004) · doi:10.1002/ceat.200402007
[106] M. Sommerfeld, N. Huber, Experimental analysis and modelling of particle-wall collisions. Int. J. Multiphase Flow 25, 1457-1489 (1999) · Zbl 1137.76743 · doi:10.1016/S0301-9322(99)00047-6
[107] M. Sommerfeld, M. Kuschel, Modelling droplet collision outcomes for different substances and viscosities. Exp. Fluids 57, 187 (2016) · doi:10.1007/s00348-016-2249-y
[108] M. Sommerfeld, S. Lain, From elementary processes to the numerical prediction of industrial particle-laden flows. Multiph. Sci. Technol. 21, 123-140 (2009) · doi:10.1615/MultScienTechn.v21.i1-2.100
[109] M. Sommerfeld, S. Lain, Parameters influencing dilute-phase pneumatic conveying through pipe systems: a computational study by the Euler/Lagrange approach. Can. J. Chem. Eng. 93, 1-17 (2015) · doi:10.1002/cjce.22105
[110] M. Sommerfeld, S. Lain, Euler/Lagrange methods. in Multiphase Flow Handbook, 2nd edn. (CRC Press, Boca Raton, 2017)
[111] M. Sommerfeld, H.-H. Qiu, Characterization of particle laden, confined swirling flows by phase-doppler anemometry and numerical calculation. Int. J. Multiphase Flow 9, 1093-1127 (1993) · Zbl 1144.76458 · doi:10.1016/0301-9322(93)90080-E
[112] M. Sommerfeld, S. Schmalfuß, Numerical analysis of carrier particle motion in dry powder inhaler. ASME J. Fluid Eng. 138, 041308-1 to 041308-12 (2016)
[113] M. Sommerfeld, S. Stübing, A novel Lagrangian agglomerate structure model. Powder Technol. 319, 34-52 (2017) · doi:10.1016/j.powtec.2017.06.016
[114] M. Sommerfeld, G. Zivkovic, Recent advances in the numerical simulation of pneumatic conveying through pipe systems. in Computational Methods in Applied Science (Elsevier, Amsterdam, 1992)
[115] M. Sommerfeld, G. Kohnen, H.-H. Qiu, Spray evaporation in turbulent flow: numerical calculations and detailed experiments by phase-doppler anemometry. Rev. Inst. Fr. Pétrol. 48, 677-695 (1993) · doi:10.2516/ogst:1993038
[116] M. Sommerfeld, G. Kohnen, M. Rüger, Some open questions and inconsistencies of Lagrangian particle dispersion models, in Ninth Symposium on Turbulent Shear Flows, Kyoto Japan (1993)
[117] M. Sommerfeld, S. Decker, G. Kohnen, Time-dependent calculation of bubble columns based on the time-averaged Navier-Stokes equations with turbulence model, in Proceedings of the Japanese-German Symposium on Multi-Phase Flow, Tokyo, Japan (1997), pp. 323-334
[118] M. Sommerfeld, E. Bourloutski, D. Bröder, Euler/Lagrange calculations of bubbly flows with consideration of bubble coalescence. Can. J. Chem. Eng. 81, 508-518 (2003) · doi:10.1002/cjce.5450810324
[119] M. Sommerfeld, B. van Wachem, R. Oliemans, Best practice guidelines for computational fluid dynamics of dispersed multiphase flows, in ERCOFTAC: European Research Community on Flow, Turbulence and Combustion, Brussels (2008)
[120] K.D. Squires, J.K. Eaton, Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169-1178 (1991) · doi:10.1063/1.858045
[121] K.D. Squires, J.K. Eaton, Effect of selective modification of turbulence on two-equation models for particle-laden turbulent flows. Trans. ASME, J. Fluids Eng. 116, 778-784 (1994)
[122] S. Sundaresan, Modeling the hydrodynamics of multiphase flow reactors: Current status and challenges. AIChE J. 46, 1102-1105 (2000) · doi:10.1002/aic.690460602
[123] T. Tanaka, Y. Tsuji, Numerical simulation of gas-solid two-phase flow in a vertical pipe: on the effect of inter-particle collision. In Gas-Solid Flows (ASME, New York, 1991)
[124] L. Tang, F. Wen, Y. Yang C.T. Crowe, J.N. Chung, T.R. Troutt, Self-organizing particle dispersion mechanism in free shear flows. Phys. Fluids A4, 2244-2251 (1992) · doi:10.1063/1.858465
[125] S. Tenneti, S. Subramaniam, Particle-resolved direct numerical simulation for gas-solid flow model development. Annu. Rev. Fluid Mech. 46, 199-230 (2014) · Zbl 1297.76179 · doi:10.1146/annurev-fluid-010313-141344
[126] Y. Tsuji, T. Kawaguchi, T. Tanaka, Discrete particle simulation of two-dimensional fluidized bed. Powder Technol. 77, 79-87 (1993) · doi:10.1016/0032-5910(93)85010-7
[127] M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows. J. Comput. Phys. 209, 448-476 (2005) · Zbl 1138.76398 · doi:10.1016/j.jcp.2005.03.017
[128] S. Unverdi, G. Tryggvason, Front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 25-37 (1992) · Zbl 0758.76047 · doi:10.1016/0021-9991(92)90307-K
[129] B.G.M. Van Wachem, J.C. Schouten, C.M. Van den Bleek, R. Krishna, J.L. Sinclair, Comparative analysis of CFD models of dense gas-solid systems. AIChE J. 47, 1035-1051 (2001) · doi:10.1002/aic.690470510
[130] A.W. Vreman, Particle-resolved direct numerical simulation of homogeneous isotropic turbulence modified by small fixed spheres. J. Fluid Mech. 796, 40-85 (2016) · Zbl 1462.76081 · doi:10.1017/jfm.2016.228
[131] G.B. Wallis, One-Dimensional Two-Phase Flow, 2nd edn. (McGraw Hill, New York, 1979)
[132] Q.G. Wang, W. Yao, Computation and validation of the interphase force models for bubbly flow. Int. J. Heat Mass Transf. 98, 799-813 (2016) · doi:10.1016/j.ijheatmasstransfer.2016.03.064
[133] D.C. Wilcox, Turbulence Modelling for CFD, 3rd edn. (DCW Industries, La Cañada Flintridge, 2006)
[134] G.H. Yeoh, J.Y. Tu, Numerical modelling of bubbly flows with and without heat and mass transfer. Appl. Math. Modell. 30, 1067-1095 (2005) · Zbl 1163.76425 · doi:10.1016/j.apm.2005.06.012
[135] G.H. Yeoh, J.Y. Tu, Computational Techniques for Multi-Phase Flows (Elsevier, Amsterdam, 2010)
[136] M. Zastawny, G. Mallouppas, F. Zhao, B. van Wachem, Derivation of drag and lift force and torque coefficients for non-spherical particles in flows. Int. J. Multiphase Flow 39, 227-239 (2012) · doi:10.1016/j.ijmultiphaseflow.2011.09.004
[137] D.H. Zhang, N.G. Deen, J.A.M. Kuipers, Euler-Euler modelling of flow, mass transfer, and chemical reaction in a bubble column. Ind. Eng. Chem. Res. 48, 47-57 (2009) · doi:10.1021/ie800233y
[138] W. Zhong, Y.Q. Xiong, Z.L. Yuan, M.Y. Zhang, Dem simulation of gas-solid flow behaviours in spout-fluid bed. Chem. Eng. Sci. 61, 1571-1584 (2006) · doi:10.1016/j.ces.2005.09.015
[139] Q. Zhou, M.A. Leschziner, A time-correlated stochastic model for particle dispersion in anisotropic turbulence, in 8th Symposium on Turbulent Shear Flows, TU Munich, vol. 1 (1991), p. 1031
[140] N. Zuber, J. Findlay, Average volumetric concentration in two-phase systems. Trans. ASME J. Heat Transf. 87, 453-468 (1965) · doi:10.1115/1.3689137
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.