Lattice-Boltzmann method for complex flows
Abstract
With its roots in kinetic theory and the cellular automaton concept, the lattice-Boltzmann (LB) equation can be used to obtain continuum flow quantities from simple and local update rules based on particle interactions. The simplicity of formulation and its versatility explain the rapid expansion of the LB method to applications in complex and multiscale flows. We review many significant developments over the past decade with specific examples. Some of the most active developments include the entropic LB method and the application of the LB method to turbulent flow, multiphase flow, and deformable particle and fiber suspensions. Hybrid methods based on the combination of the Eulerian lattice with a Lagrangian grid system for the simulation of moving deformable boundaries show promise for more efficient applications to a broader class of problems. We also discuss higher-order boundary conditions and the simulation of microchannel flow with finite Knudsen number. Additionally, the remarkable scalability of the LB method for parallel processing is shown with examples. Teraflop simulations with the LB method are routine, and there is no doubt that this method will be one of the first candidates for petaflop computational fluid dynamics in the near future.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
LBM is well-known for intuitive modeling, intrinsically parallelizable algorithms, the ability to address complex boundary conditions, and multiphysics models.
“LBM is well-known for intuitive modeling, intrinsically parallelizable algorithms, the ability to address complex boundary conditions, and multiphysics models [39,40].”
From Comprehensive Computational Model for Coupled Fluid Flow, Mass Transfer, and Light Supply in Tubular Photobioreactors Equipped with Glass Sponges (2022) · cites this paper as [39]
Cites
Possible generalization of Boltzmann-Gibbs statistics
Tsallis, C. (1988) · Journal of Statistical Physics · doi:10.1007/bf01016429 · Cited by 9,589 · see citing works
The mathematical theory of non-uniform gases
York, N. (1953) · Journal of the Franklin Institute · doi:10.1016/0016-0032(53)90422-0 · Cited by 7,662 · see citing works
LATTICE BOLTZMANN METHOD FOR FLUID FLOWS
Chen, S., & Doolen, G. D. (1998) · Annual Review of Fluid Mechanics · doi:10.1146/annurev.fluid.30.1.329 · Cited by 7,336 · see citing works
Lattice BGK Models for Navier-Stokes Equation
Qian, Y., d’Humières, D., & Lallemand, P. (1992) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/17/6/001 · Cited by 5,326 · see citing works
Peskin, C. S. (2002) · Acta Numerica · doi:10.1017/s0962492902000077 · Cited by 4,603 · see citing works
The Lattice Boltzmann Equation for Fluid Dynamics and Beyond
Succi, S. (2001) · doi:10.1093/oso/9780198503989.001.0001 · Cited by 4,203 · see citing works
Lattice Boltzmann model for simulating flows with multiple phases and components
Shan, X., & Chen, H. (1993) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.47.1815 · Cited by 3,654 · see citing works
The mathematical theory of non-uniform gases
Budden, K. G. (1971) · Journal of Atmospheric and Terrestrial Physics · doi:10.1016/0021-9169(71)90189-9 · Cited by 3,593 · see citing works
A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres
Krieger, I. M., & Dougherty, T. J. (1959) · Transactions of the Society of Rheology · doi:10.1122/1.548848 · Cited by 3,403 · see citing works
The Boltzmann Equation and Its Applications
Cercignani, C. (1988) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-1039-9 · Cited by 3,130 · see citing works
The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond
Yeomans, J. M. (2001) · Cited by 2,692 · see citing works
Ladd, A. J. C. (1994) · Journal of Fluid Mechanics · doi:10.1017/s0022112094001771 · Cited by 2,466 · see citing works
IV. On the dynamical theory of gases
Maxwell, J. C. (1867) · Philosophical Transactions of the Royal Society of London · doi:10.1098/rstl.1867.0004 · Cited by 2,426 · see citing works
Use of the Boltzmann Equation to Simulate Lattice-Gas Automata
McNamara, G. R., & Zanetti, G. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.2332 · Cited by 2,096 · see citing works
The lattice Boltzmann equation: theory and applications
Benzi, R., Succi, S., & Vergassola, M. (1992) · Physics Reports · doi:10.1016/0370-1573(92)90090-m · Cited by 2,089 · see citing works
Multiple–relaxation–time lattice Boltzmann models in three dimensions
d’Humières, D. (2002) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2001.0955 · Cited by 2,051 · see citing works
Investigations on the Theory of the Brownian Movement
Topping, J. (1956) · Physics Bulletin · doi:10.1088/0031-9112/7/10/012 · Cited by 2,050 · see citing works
Investigations on the theory of the Brownian movement
Einstein, A. (1926) · BURJC Digital (King Juan Carlos University) · Cited by 1,859 · see citing works
Lattice Boltzmann model of immiscible fluids
Gunstensen, A. K., Rothman, D. H., Zaleski, S., et al. (1991) · Physical Review A · doi:10.1103/physreva.43.4320 · Cited by 1,631 · see citing works
The stress system in a suspension of force-free particles
Batchelor, G. K. (1970) · Journal of Fluid Mechanics · doi:10.1017/s0022112070000745 · Cited by 1,585 · see citing works
Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
Chen, H., Chen, S., & Matthaeus, W. H. (1992) · Physical Review A · doi:10.1103/physreva.45.r5339 · Cited by 1,562 · see citing works
Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation
Shan, X., & Chen, H. (1994) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.49.2941 · Cited by 1,517 · see citing works
The computer study of transport processes under extreme conditions
Lees, A. W., & Edwards, S. F. (1972) · Journal of Physics C: Solid State Physics · doi:10.1088/0022-3719/5/15/006 · Cited by 1,506 · see citing works
Microflows and Nanoflows: Fundamentals and Simulation
Karniadakis, G. E. (2008) · doi:10.1007/0-387-28676-4 · Cited by 1,481 · see citing works
Lattice Boltzmann simulations of liquid-gas and binary fluid systems
Swift, M. R., Orlandini, E., Osborn, W. R., & Yeomans, J. M. (1996) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.54.5041 · Cited by 1,427 · see citing works
A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit
He, X., Chen, S., & Doolen, G. D. (1998) · Journal of Computational Physics · doi:10.1006/jcph.1998.6057 · Cited by 1,415 · see citing works
Zur Theorie der Brownschen Bewegung
Einstein, A. (1906) · Annalen der Physik · doi:10.1002/andp.19063240208 · Cited by 1,289 · see citing works
Microhydrodynamics: Principles and Selected Applications
Kim, S., & Karrila, S. (1991) · Cited by 1,257 · see citing works
Momentum transfer of a Boltzmann-lattice fluid with boundaries
Bouzidi, M., Firdaouss, M., & Lallemand, P. (2001) · Physics of Fluids · doi:10.1063/1.1399290 · Cited by 1,251 · see citing works
Ladd, A. J. C. (1994) · Journal of Fluid Mechanics · doi:10.1017/s0022112094001783 · Cited by 1,197 · see citing works
Hilbert, D. R. (1902) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1902-00923-3 · Cited by 1,186 · see citing works
Lattice-Boltzmann Simulations of Particle-Fluid Suspensions
Ladd, A. J. C., & Verberg, R. (2001) · Journal of Statistical Physics · doi:10.1023/a:1010414013942 · Cited by 1,183 · see citing works
He, X., Chen, S., & Zhang, R. (1999) · Journal of Computational Physics · doi:10.1006/jcph.1999.6257 · Cited by 1,181 · see citing works
The determination of the bulk stress in a suspension of spherical particles to order c 2
Batchelor, G. K., & Green, J. T. (1972) · Journal of Fluid Mechanics · doi:10.1017/s0022112072002435 · Cited by 1,138 · see citing works
Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation
Shan, X., Yuan, X., & Chen, H. (2006) · Journal of Fluid Mechanics · doi:10.1017/s0022112005008153 · Cited by 1,092 · see citing works
Equations of state in a lattice Boltzmann model
Yuan, P., & Schaefer, L. A. (2006) · Physics of Fluids · doi:10.1063/1.2187070 · Cited by 1,088 · see citing works
The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems
Feng, Z., & Michaelides, E. E. S. (2003) · Journal of Computational Physics · doi:10.1016/j.jcp.2003.10.013 · Cited by 1,053 · see citing works
Microhydrodynamics: principles and selected applications
Ramos, J. I. (1992) · Applied Mathematical Modelling · doi:10.1016/0307-904x(92)90118-m · Cited by 962 · see citing works
FLUID MECHANICS AND RHEOLOGY OF DENSE SUSPENSIONS
Stickel, J. J., & Powell, R. L. (2003) · Annual Review of Fluid Mechanics · doi:10.1146/annurev.fluid.36.050802.122132 · Cited by 910 · see citing works
Lattice gas hydrodynamics in two and three dimensions
Frisch, U., d’Humières, D., Hasslacher, B., et al. (1986) · University of North Texas Digital Library (University of North Texas) · Cited by 903 · see citing works
Theory and Application of the Boltzmann Equation
Cercignani, C., & Aronson, R. (1977) · Physics Today · doi:10.1063/1.3037373 · Cited by 901 · see citing works
Boltzmann Approach to Lattice Gas Simulations
Higuera, F. J., & Jiménez, J. L. (1989) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/9/7/009 · Cited by 881 · see citing works
Principles of the Kinetic Theory of Gases
Grad, H. (1958) · Handbuch der Physik · doi:10.1007/978-3-642-45892-7_3 · Cited by 864 · see citing works
Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction
Wolf-Gladrow, D. A. (2000) · Helmholtz-Zentrum für Polar-und Meeresforschung (Alfred-Wegener-Institut) · doi:10.1007/b72010 · Cited by 862 · see citing works
Nonextensive Entropy: Interdisciplinary Applications
Gell‐Mann, M., & Tsallis, C. (2004) · CaltechAUTHORS (California Institute of Technology) · Cited by 838 · see citing works
An evaluation of lattice Boltzmann schemes for porous medium flow simulation
Pan, C., Luo, L., & Miller, C. T. (2006) · Computers & Fluids · doi:10.1016/j.compfluid.2005.03.008 · Cited by 762 · see citing works
Pressure-driven flow of suspensions: simulation and theory
Nott, P. R., & Brady, J. F. (1994) · Journal of Fluid Mechanics · doi:10.1017/s0022112094002326 · Cited by 755 · see citing works
Extended Boltzmann Kinetic Equation for Turbulent Flows
Chen, H., Kandasamy, S., Orszag, S. A., et al. (2003) · Science · doi:10.1126/science.1085048 · Cited by 730 · see citing works
Lee, T., & Lin, C. (2005) · Journal of Computational Physics · doi:10.1016/j.jcp.2004.12.001 · Cited by 698 · see citing works
A lattice Boltzmann method for incompressible two-phase flows with large density differences
Inamuro, T., Ogata, T., TAJIMA, S., et al. (2004) · Journal of Computational Physics · doi:10.1016/j.jcp.2004.01.019 · Cited by 666 · see citing works
Immiscible cellular-automaton fluids
Rothman, D. H., & Keller, J. M. (1988) · Journal of Statistical Physics · doi:10.1007/bf01019743 · Cited by 665 · see citing works
Theory and Application of the Boltzmann Equation
Cercignani, C., & Berman, A. S. (1976) · Journal of Applied Mechanics · doi:10.1115/1.3423913 · Cited by 661 · see citing works
Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation
Aidun, C. K., Lu, Y., & Ding, E. J. (1998) · Journal of Fluid Mechanics · doi:10.1017/s0022112098002493 · Cited by 626 · see citing works
Self-citationMathematical Methods in Kinetic Theory
Cercignani, C. (1990) · Springer eBooks · doi:10.1007/978-1-4899-7291-0 · Cited by 601 · see citing works
Lattice Boltzmann method for moving boundaries
Lallemand, P., & Luo, L. (2003) · Journal of Computational Physics · doi:10.1016/s0021-9991(02)00022-0 · Cited by 585 · see citing works
Asymptotic Theory of the Boltzmann Equation
Grad, H. (1963) · The Physics of Fluids · doi:10.1063/1.1706716 · Cited by 561 · see citing works
Multireflection boundary conditions for lattice Boltzmann models
Ginzburg, I., & d’Humières, D. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.68.066614 · Cited by 537 · see citing works
Sukop, M. C., & Thorne, D. T. (2006) · Springer eBooks · doi:10.1007/978-3-540-27982-2 · Cited by 515 · see citing works
Curvilinear flows of noncolloidal suspensions: The role of normal stresses
Morris, J. F., & Boulay, F. (1999) · Journal of Rheology · doi:10.1122/1.551021 · Cited by 500 · see citing works
A lattice Boltzmann model for multiphase flows with large density ratio
Zheng, H. W., Shu, C., & Chew, Y. T. (2006) · Journal of Computational Physics · doi:10.1016/j.jcp.2006.02.015 · Cited by 496 · see citing works
(2004) · Oxford University Press eBooks · doi:10.1093/oso/9780195159769.001.0001 · Cited by 496 · see citing works
Generalized lattice Boltzmann method with multirange pseudopotential
Sbragaglia, M., Benzi, R., Biferale, L., et al. (2007) · Physical Review E · doi:10.1103/physreve.75.026702 · Cited by 492 · see citing works
Mathematical Methods in Kinetic Theory
Cercignani, C. (1969) · Springer eBooks · doi:10.1007/978-1-4899-5409-1 · Cited by 486 · see citing works
A Lattice-Boltzmann Method for Partially Saturated Computational Cells
Noble, D. R., & Torczynski, J. R. (1998) · International Journal of Modern Physics C · doi:10.1142/s0129183198001084 · Cited by 485 · see citing works
Lattice Boltzmann method with regularized pre-collision distribution functions
Lätt, J., & Chopard, B. (2006) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2006.05.017 · Cited by 485 · see citing works
Ginzburg, I. (2005) · Advances in Water Resources · doi:10.1016/j.advwatres.2005.03.004 · Cited by 455 · see citing works
Discretization of the Velocity Space in the Solution of the Boltzmann Equation
Shan, X., & He, X. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.65 · Cited by 445 · see citing works
Lubrication corrections for lattice-Boltzmann simulations of particle suspensions
Nguyen, N.-Q., & Ladd, A. J. C. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.66.046708 · Cited by 445 · see citing works
The lattice Boltzmann equation method: theoretical interpretation, numerics and implications
Nourgaliev, R. R., Dinh, N. T., Theofanous, T. G., et al. (2002) · International Journal of Multiphase Flow · doi:10.1016/s0301-9322(02)00108-8 · Cited by 439 · see citing works
Boundary conditions for lattice Boltzmann simulations
Ziegler, D. P. (1993) · Journal of Statistical Physics · doi:10.1007/bf01049965 · Cited by 435 · see citing works
Ahlrichs, P., & Dünweg, B. (1999) · The Journal of Chemical Physics · doi:10.1063/1.480156 · Cited by 423 · see citing works
Force evaluation in the lattice Boltzmann method involving curved geometry
Mei, R., Yu, D., Shyy, W., et al. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.041203 · Cited by 417 · see citing works
Lattice Boltzmann computations for reaction-diffusion equations
Dawson, S. P., Chen, S., & Doolen, G. D. (1993) · The Journal of Chemical Physics · doi:10.1063/1.464316 · Cited by 400 · see citing works
A lattice Boltzmann model for multiphase fluid flows
Grunau, D., Chen, S., & Eggert, K. (1993) · Physics of Fluids A Fluid Dynamics · doi:10.1063/1.858769 · Cited by 397 · see citing works
Boundary flow condition analysis for the three-dimensional lattice Boltzmann model
Ginzbourg, I., & Adler, P. M. (1994) · Journal de Physique II · doi:10.1051/jp2:1994123 · Cited by 397 · see citing works
Perfect entropy functions of the Lattice Boltzmann method
Karlin, I. V., Ferrante, A. S., & Öttinger, H. C. (1999) · Europhysics Letters (EPL) · doi:10.1209/epl/i1999-00370-1 · Cited by 375 · see citing works
Mathematical Methods In Kinetic Theory
Cercignani, C., & Street, R. E. (1970) · Physics Today · doi:10.1063/1.3021788 · Cited by 373 · see citing works
Asymptotic analysis of the lattice Boltzmann equation
Junk, M., Klar, A., & Luo, L. (2005) · Journal of Computational Physics · doi:10.1016/j.jcp.2005.05.003 · Cited by 371 · see citing works
Lattice-Boltzmann Simulations of Fluid Flows in MEMS
Nie, X., Doolen, G. D., & Chen, S. (2002) · Journal of Statistical Physics · doi:10.1023/a:1014523007427 · Cited by 367 · see citing works
Mesoscopic Modeling of Slip Motion at Fluid-Solid Interfaces with Heterogeneous Catalysis
Succi, S. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.064502 · Cited by 364 · see citing works
Kinetic boundary conditions in the lattice Boltzmann method
Ansumali, S., & Karlin, I. V. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.66.026311 · Cited by 362 · see citing works
Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation
Kang, H. S., Chester, S., & Meneveau, C. (2003) · Journal of Fluid Mechanics · doi:10.1017/s0022112002003579 · Cited by 362 · see citing works
A consistent hydrodynamic boundary condition for the lattice Boltzmann method
Noble, D. R., Chen, S., Georgiadis, J. G., et al. (1995) · Physics of Fluids · doi:10.1063/1.868767 · Cited by 361 · see citing works
Multiple-relaxation-time lattice-Boltzmann model for multiphase flow
McCracken, M. E., & Abraham, J. P. (2005) · Physical Review E · doi:10.1103/physreve.71.036701 · Cited by 334 · see citing works
Application of lattice Boltzmann method to simulate microchannel flows
Lim, C. Y., Shu, C., Niu, X., et al. (2002) · Physics of Fluids · doi:10.1063/1.1483841 · Cited by 330 · see citing works
Expanded analogy between Boltzmann kinetic theory of fluids and turbulence
Chen, H., Orszag, S. A., Staroselsky, I., et al. (2004) · Journal of Fluid Mechanics · doi:10.1017/s0022112004001211 · Cited by 327 · see citing works
Diffusion properties of gradient-based lattice Boltzmann models of immiscible fluids
Latva-Kokko, M., & Rothman, D. H. (2005) · Physical Review E · doi:10.1103/physreve.71.056702 · Cited by 321 · see citing works
Schwabl, F. (2002) · Advanced texts in physics · doi:10.1007/978-3-662-04702-6 · Cited by 321 · see citing works
A lattice boltzmann model for multiphase fluid flowsPreprint
Grunau, D., Chen, S., & Eggert, K. (1993) · Cited by 317 · see citing works
Griffith, B. E., & Peskin, C. S. (2005) · Journal of Computational Physics · doi:10.1016/j.jcp.2005.02.011 · Cited by 302 · see citing works
Rarefied Gas Dynamics: Theory and Simulations
(1994) · American Institute of Aeronautics and Astronautics eBooks · doi:10.2514/4.866319 · Cited by 298 · see citing works
A lattice Boltzmann subgrid model for high Reynolds number flows
Hou, S., Sterling, J. D., Chen, S., & Doolen, G. D. (1995) · doi:10.1090/fic/006/12 · Cited by 297 · see citing works
Chapman, S. (1916) · Philosophical Transactions of the Royal Society of London Series A Containing Papers of a Mathematical or Physical Character · doi:10.1098/rsta.1916.0006 · Cited by 290 · see citing works
Minimal entropic kinetic models for hydrodynamics
Ansumali, S., Karlin, I. V., & Öttinger, H. C. (2003) · Europhysics Letters (EPL) · doi:10.1209/epl/i2003-00496-6 · Cited by 274 · see citing works
THOMSON, W. J. (1887) · The London Edinburgh and Dublin Philosophical Magazine and Journal of Science · doi:10.1080/14786448708628078 · Cited by 269 · see citing works
Displacement of a two-dimensional immiscible droplet in a channel
Kang, Q., Zhang, D., & Chen, S. (2002) · Physics of Fluids · doi:10.1063/1.1499125 · Cited by 268 · see citing works
Lattice Boltzmann model for simulating immiscible two-phase flows
Reis, T., & Phillips, T. N. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/14/018 · Cited by 265 · see citing works
Modeling the flow of dense suspensions of deformable particles in three dimensions
Dupin, M. M., Halliday, I., CARE, C. M., Alboul, L., & Munn, L. L. (2007) · Physical Review E · doi:10.1103/physreve.75.066707 · Cited by 263 · see citing works
RECENT ADVANCES IN LATTICE BOLTZMANN COMPUTING
Qian, Y., Succi, S., & Orszag, S. A. (1995) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812830647_0006 · Cited by 255 · see citing works
Lattice Boltzmann Methods for Shallow Water Flows
Zhou, J. G. (2004) · Springer eBooks · doi:10.1007/978-3-662-08276-8 · Cited by 238 · see citing works
Stabilization of thermal lattice Boltzmann models
McNamara, G. R., Garcia, A. L., & Alder, B. J. (1995) · Journal of Statistical Physics · doi:10.1007/bf02179986 · Cited by 234 · see citing works
On the Finite Difference-Based Lattice Boltzmann Method in Curvilinear Coordinates
Mei, R., & Shyy, W. (1998) · Journal of Computational Physics · doi:10.1006/jcph.1998.5984 · Cited by 232 · see citing works
Surface Roughness-Hydrophobicity Coupling in Microchannel and Nanochannel Flows
Sbragaglia, M., Benzi, R., Biferale, L., et al. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.204503 · Cited by 224 · see citing works
Adhikari, R., Stratford, K., Cates, M. E., & Wagner, A. J. (2005) · Europhysics Letters (EPL) · doi:10.1209/epl/i2004-10542-5 · Cited by 221 · see citing works
On the three-dimensional Rayleigh–Taylor instability
He, X., Zhang, R., Chen, S., et al. (1999) · Physics of Fluids · doi:10.1063/1.869984 · Cited by 216 · see citing works
Sedimentation of homogeneous suspensions of non-Brownian spheres
Ladd, A. J. C. (1997) · Physics of Fluids · doi:10.1063/1.869212 · Cited by 213 · see citing works
A Knudsen layer theory for lattice gases
Cornubert, R., d’Humières, D., & Levermore, D. (1991) · Physica D Nonlinear Phenomena · doi:10.1016/0167-2789(91)90295-k · Cited by 210 · see citing works
Particle motions in sheared suspensions
Forgacs, O. L., & Mason, S. G. (1959) · Journal of Colloid Science · doi:10.1016/0095-8522(59)90012-1 · Cited by 201 · see citing works
A Lattice Boltzmann Model of Binary-Fluid Mixtures
Orlandini, E., Swift, M. R., & Yeomans, J. M. (1995) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/32/6/001 · Cited by 201 · see citing works
Lattice Boltzmann simulation of solid particles suspended in fluid
Aidun, C. K., & Lu, Y. (1995) · Journal of Statistical Physics · doi:10.1007/bf02179967 · Cited by 197 · see citing works
Self-citationSimulations of fiber flocculation: Effects of fiber properties and interfiber friction
Schmid, C. F., Switzer, L. H., & Klingenberg, D. J. (2000) · Journal of Rheology · doi:10.1122/1.551116 · Cited by 196 · see citing works
Guo, Z., Zhao, T. S., & Shi, Y. (2006) · Journal of Applied Physics · doi:10.1063/1.2185839 · Cited by 194 · see citing works
Lattice Boltzmann algorithm for surface tension with greatly reduced microcurrents
Lishchuk, S. V., CARE, C. M., & Halliday, I. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.67.036701 · Cited by 193 · see citing works
Dynamics of short polymer chains in solution
Malevanets, A., & Yeomans, J. M. (2000) · Europhysics Letters (EPL) · doi:10.1209/epl/i2000-00428-0 · Cited by 189 · see citing works
Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions
Sbragaglia, M., & Succi, S. (2005) · Physics of Fluids · doi:10.1063/1.2044829 · Cited by 184 · see citing works
Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme
Eggels, J. G. M. (1996) · International Journal of Heat and Fluid Flow · doi:10.1016/0142-727x(96)00044-6 · Cited by 183 · see citing works
Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows
Qi, D. (1999) · Journal of Fluid Mechanics · doi:10.1017/s0022112099004401 · Cited by 181 · see citing works
Entropy and Galilean Invariance of Lattice Boltzmann Theories
Chikatamarla, S. S., & Karlin, I. V. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.190601 · Cited by 176 · see citing works
A lattice spring model of heterogeneous materials with plasticity
Buxton, G. A., CARE, C. M., & Cleaver, D. J. (2001) · Modelling and Simulation in Materials Science and Engineering · doi:10.1088/0965-0393/9/6/302 · Cited by 175 · see citing works
MacMeccan, R., Clausen, J. R., Neitzel, G. P., et al. (2008) · Journal of Fluid Mechanics · doi:10.1017/s0022112008004011 · Cited by 168 · see citing works
Self-citationA lattice Boltzmann model of ternary fluid mixtures
Lamura, A., Gonnella, G., & Yeomans, J. M. (1999) · Europhysics Letters (EPL) · doi:10.1209/epl/i1999-00165-4 · Cited by 167 · see citing works
Statistical mechanics of the fluctuating lattice Boltzmann equation
Dünweg, B., Schiller, U., & Ladd, A. J. C. (2007) · Physical Review E · doi:10.1103/physreve.76.036704 · Cited by 166 · see citing works
Interpolated boundary condition for lattice Boltzmann simulations of flows in narrow gaps
Chun, B., & Ladd, A. J. C. (2007) · Physical Review E · doi:10.1103/physreve.75.066705 · Cited by 163 · see citing works
The dynamics and scaling law for particles suspended in shear flow with inertia
Ding, E. J., & Aidun, C. K. (2000) · Journal of Fluid Mechanics · doi:10.1017/s0022112000001932 · Cited by 160 · see citing works
Self-citationLattice-Boltzmann Simulation of Polymer-Solvent Systems
Ahlrichs, P., & Dünweg, B. (1998) · International Journal of Modern Physics C · doi:10.1142/s0129183198001291 · Cited by 157 · see citing works
Hydrodynamics beyond Navier-Stokes: Exact Solution to the Lattice Boltzmann Hierarchy
Ansumali, S., Karlin, I. V., Arcidiacono, S., et al. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.124502 · Cited by 156 · see citing works
Capturing Knudsen layer phenomena using a lattice Boltzmann model
Zhang, Y., Gu, X.-J., Barber, R. W., et al. (2006) · Physical Review E · doi:10.1103/physreve.74.046704 · Cited by 154 · see citing works
Inamuro, T., Konishi, N., & Ogino, F. (2000) · Computer Physics Communications · doi:10.1016/s0010-4655(00)00090-4 · Cited by 148 · see citing works
Immiscible displacement in a channel: simulations of fingering in two dimensions
Kang, Q., Zhang, D., & Chen, S. (2003) · Advances in Water Resources · doi:10.1016/j.advwatres.2003.10.002 · Cited by 147 · see citing works
Phase separation under shear in two-dimensional binary fluids
Wagner, A. J., & Yeomans, J. M. (1999) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.59.4366 · Cited by 146 · see citing works
Lattice Boltzmann method at finite Knudsen numbers
Toschi, F., & Succi, S. (2005) · Europhysics Letters (EPL) · doi:10.1209/epl/i2004-10393-0 · Cited by 142 · see citing works
Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers
Kim, S., Pitsch, H. G., & Boyd, I. D. (2008) · Journal of Computational Physics · doi:10.1016/j.jcp.2008.06.012 · Cited by 140 · see citing works
Entropic lattice Boltzmann methods
Boghosian, B. M., Yepez, J., Coveney, P. V., et al. (2001) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2000.0689 · Cited by 137 · see citing works
Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics
Martínez, D. O., Matthaeus, W. H., Chen, S., et al. (1994) · Physics of Fluids · doi:10.1063/1.868296 · Cited by 135 · see citing works
Kinetic equations and asymptotic theory
Bouchut, F., Golse, F., & Pulvirenti, M. (2000) · HAL (Le Centre pour la Communication Scientifique Directe) · Cited by 134 · see citing works
Image representation of a spherical particle near a hard wall
Cichocki, B., & Jones, R. B. (1998) · Physica A Statistical Mechanics and its Applications · doi:10.1016/s0378-4371(98)00267-2 · Cited by 129 · see citing works
Extension of the Lattice-Boltzmann Method for Direct Simulation of Suspended Particles Near Contact
Ding, E. J., & Aidun, C. K. (2003) · Journal of Statistical Physics · doi:10.1023/a:1023880126272 · Cited by 128 · see citing works
Self-citationMUPHY: A parallel MUlti PHYsics/scale code for high performance bio-fluidic simulations
Bernaschi, M., Melchionna, S., Succi, S., et al. (2009) · Computer Physics Communications · doi:10.1016/j.cpc.2009.04.001 · Cited by 126 · see citing works
Three‐dimensional immersed boundary conditions for moving solids in the lattice‐Boltzmann method
Strack, O. E., & Cook, B. K. (2007) · International Journal for Numerical Methods in Fluids · doi:10.1002/fld.1437 · Cited by 124 · see citing works
Suspension properties at finite Reynolds number from simulated shear flow
Kulkarni, P. M., & Morris, J. F. (2008) · Physics of Fluids · doi:10.1063/1.2911017 · Cited by 122 · see citing works
On the model coefficients for the standard and the variational multi-scale Smagorinsky model
Meyers, J., & Sagaut, P. (2006) · Journal of Fluid Mechanics · doi:10.1017/s0022112006002850 · Cited by 122 · see citing works
Roughness Induced Boundary Slip in Microchannel Flows
Kunert, C., & Harting, J. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.176001 · Cited by 120 · see citing works
The motion of suspended particles almost in contact
Cox, R. G. (1974) · International Journal of Multiphase Flow · doi:10.1016/0301-9322(74)90019-6 · Cited by 116 · see citing works
Stabilization of the lattice Boltzmann method by theHtheorem: Anumerical test
Ansumali, S., & Karlin, I. V. (2000) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.62.7999 · Cited by 115 · see citing works
An Improved Hydrodynamics Formulation for Multiphase Flow Lattice-Boltzmann Models
Holdych, D. J., Rovas, D., Georgiadis, J. G., et al. (1998) · International Journal of Modern Physics C · doi:10.1142/s0129183198001266 · Cited by 113 · see citing works
Newtonian fluid meets an elastic solid: Coupling lattice Boltzmann and lattice-spring models
Buxton, G. A., Verberg, R., Jasnow, D., & Balazs, A. C. (2005) · Physical Review E · doi:10.1103/physreve.71.056707 · Cited by 112 · see citing works
Wu, J., & Aidun, C. K. (2009) · International Journal for Numerical Methods in Fluids · doi:10.1002/fld.2043 · Cited by 112 · see citing works
Self-citationA GENERAL MULTIPLE-RELAXATION-TIME BOLTZMANN COLLISION MODEL
Shan, X., & Chen, H. (2007) · International Journal of Modern Physics C · doi:10.1142/s0129183107010887 · Cited by 109 · see citing works
Entropic Lattice Boltzmann MethodsPreprint
Boghosian, B. M., Yepez, J., Coveney, P. V., et al. (2001) · Cited by 107 · see citing works
Thirteen-velocity three-dimensional lattice Boltzmann model
d’Humières, D., Bouzidi, M., & Lallemand, P. (2001) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.63.066702 · Cited by 106 · see citing works
Simulation of Combustion Field with Lattice Boltzmann Method
Yamamoto, K., He, X., & Doolen, G. D. (2002) · Journal of Statistical Physics · doi:10.1023/a:1014583226083 · Cited by 103 · see citing works
Lee, T., & Lin, C. (2005) · Physical Review E · doi:10.1103/physreve.71.046706 · Cited by 103 · see citing works
Lattice Boltzmann across scales: from turbulence to DNA translocation
Sauro Succi, A. (2008) · The European Physical Journal B · doi:10.1140/epjb/e2008-00067-3 · Cited by 102 · see citing works
Variational approach to gas flows in microchannels
Cercignani, C., Lampis, M., & Lorenzani, S. (2004) · Physics of Fluids · doi:10.1063/1.1764700 · Cited by 101 · see citing works
Boundary-Fitting and Local Grid Refinement for Lattice-BGK Models
Filippova, O., & Hänel, D. (1998) · International Journal of Modern Physics C · doi:10.1142/s012918319800114x · Cited by 99 · see citing works
Clausen, J. R., Reasor, D. A., & Aidun, C. K. (2010) · Computer Physics Communications · doi:10.1016/j.cpc.2010.02.005 · Cited by 98 · see citing works
Self-citationGalilean-invariant lattice-Boltzmann models withHtheorem
Boghosian, B. M., Love, P. J., Coveney, P. V., Karlin, I. V., Sauro Succi, A., & Yepez, J. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.68.025103 · Cited by 96 · see citing works
Lattice Boltzmann Models with Mid-Range Interactions
Falcucci, G., Bella, G., Shiatti, G., et al. (2007) · Communications in Computational Physics · doi:10.4208/cicp.2007.v2.p1071 · Cited by 95 · see citing works
Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method
Kandhai, D., Koponen, A., Hoekstra, A. G., et al. (1999) · Journal of Computational Physics · doi:10.1006/jcph.1999.6191 · Cited by 95 · see citing works
Ginzburg, I. (2006) · Journal of Statistical Physics · doi:10.1007/s10955-006-9234-4 · Cited by 95 · see citing works
Lees–Edwards Boundary Conditions for Lattice Boltzmann
Wagner, A. J., & Pagonabarraga, I. (2002) · Journal of Statistical Physics · doi:10.1023/a:1014595628808 · Cited by 94 · see citing works
Asymptotic theory of flow of rarefied gas over a smooth boundary. II
Sone, Y. (1974) · Cited by 93 · see citing works
A lattice Boltzmann model for Jeffreys viscoelastic fluid
Giraud, L., d’Humières, D., & Lallemand, P. (1998) · Europhysics Letters (EPL) · doi:10.1209/epl/i1998-00296-0 · Cited by 92 · see citing works
Mesoscopic modelling of heterogeneous boundary conditions for microchannel flows
Benzi, R., Biferale, L., Sbragaglia, M., et al. (2006) · Journal of Fluid Mechanics · doi:10.1017/s0022112005007512 · Cited by 83 · see citing works
Sofonea, V., & Sekerka, R. F. (2005) · Journal of Computational Physics · doi:10.1016/j.jcp.2005.02.003 · Cited by 82 · see citing works
Lallemand, P., Luo, L., & Peng, Y. (2007) · Journal of Computational Physics · doi:10.1016/j.jcp.2007.05.021 · Cited by 81 · see citing works
Entropic lattice Boltzmann method for microflows
Ansumali, S., Karlin, I. V., FROUZAKIS, C. E., & Boulouchos, K. (2005) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2005.04.039 · Cited by 81 · see citing works
Chin, J. (2002) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2001.0953 · Cited by 80 · see citing works
Ginzburg, I. (2002) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2001.0941 · Cited by 77 · see citing works
Surface tension effects on two-dimensional two-phase Kelvin–Helmholtz instabilities
Zhang, R., He, X., Doolen, G. D., et al. (2001) · Advances in Water Resources · doi:10.1016/s0309-1708(00)00067-1 · Cited by 75 · see citing works
Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows
Kim, S., Pitsch, H. G., & Boyd, I. D. (2008) · Physical Review E · doi:10.1103/physreve.77.026704 · Cited by 70 · see citing works
Rigorous Navier–Stokes limit of the lattice Boltzmann equation
Junk, M., & Yong, W. (2003) · Asymptotic Analysis · doi:10.3233/asy-2003-563 · Cited by 69 · see citing works
Long-time tails in angular momentum correlations
Lowe, C. P., Frenkel, D., & Masters, A. J. (1995) · The Journal of Chemical Physics · doi:10.1063/1.469780 · Cited by 69 · see citing works
Lattice Boltzmann models for nonequilibrium gas flows
Tang, G. H., Zhang, Y., & Emerson, D. R. (2008) · Physical Review E · doi:10.1103/physreve.77.046701 · Cited by 69 · see citing works
Boundary forces in lattice Boltzmann: Analysis of Momentum Exchange algorithm
Caiazzo, A., & Junk, M. (2007) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2007.08.004 · Cited by 68 · see citing works
One-point boundary condition for the lattice Boltzmann method
Junk, M., & Yang, Z. (2005) · Physical Review E · doi:10.1103/physreve.72.066701 · Cited by 68 · see citing works
Galilean-invariant lattice-Boltzmann simulation of liquid-vapor interface dynamics
Kalarakis, A. N., Burganos, V. N., & Payatakes, A. C. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.056702 · Cited by 67 · see citing works
Zhang, J., Li, B., & Kwok, D. Y. (2004) · Physical Review E · doi:10.1103/physreve.69.032602 · Cited by 66 · see citing works
Two-dimensional turbulence with the lattice Boltzmann equation
Benzi, R., & Succi, S. (1990) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/23/1/001 · Cited by 66 · see citing works
Lattice Kinetic Theory for Numerical Combustion
Succi, S., Bella, G., & Papetti, F. (1997) · Journal of Scientific Computing · doi:10.1023/a:1025676913034 · Cited by 63 · see citing works
Accuracy and stability of a lattice-Boltzmann model with subgrid scale boundary conditions
Verberg, R., & Ladd, A. J. C. (2001) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.016701 · Cited by 63 · see citing works
Wu, J., & Aidun, C. K. (2009) · International Journal of Multiphase Flow · doi:10.1016/j.ijmultiphaseflow.2009.11.003 · Cited by 61 · see citing works
Self-citationMukherjee, S., & Abraham, J. P. (2007) · Physical Review E · doi:10.1103/physreve.75.026701 · Cited by 61 · see citing works
The viscosity of suspensions of straight, rigid rods
Blakeney, W. R. (1966) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(66)90012-9 · Cited by 61 · see citing works
Dong, Y., & Sagaut, P. (2008) · Physics of Fluids · doi:10.1063/1.2842381 · Cited by 60 · see citing works
Lattice-BGK Model for Low Mach Number Combustion
Filippova, O., & Hänel, D. (1998) · International Journal of Modern Physics C · doi:10.1142/s0129183198001308 · Cited by 55 · see citing works
Ansumali, S., Karlin, I. V., & Succi, S. (2004) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2004.02.013 · Cited by 54 · see citing works
Simulation of semidilute suspensions of non-Brownian fibers in shear flow
Lindström, S. B., & Uesaka, T. (2008) · The Journal of Chemical Physics · doi:10.1063/1.2815766 · Cited by 54 · see citing works
Bonilla, L. L., & Soler, J. (2001) · Mathematical Models and Methods in Applied Sciences · doi:10.1142/s0218202501001410 · Cited by 53 · see citing works
Scaling and hydrodynamic effects in lamellar ordering
Xu, A., Gonnella, G., Lamura, A., et al. (2005) · Europhysics Letters (EPL) · doi:10.1209/epl/i2005-10130-3 · Cited by 52 · see citing works
Lattice Boltzmann models for nonideal fluids with arrested phase-separation
Chibbaro, S., Falcucci, G., Chiatti, G., et al. (2008) · Physical Review E · doi:10.1103/physreve.77.036705 · Cited by 52 · see citing works
A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method
Thömmes, G., Becker, J., Junk, M., et al. (2008) · Journal of Computational Physics · doi:10.1016/j.jcp.2008.10.032 · Cited by 50 · see citing works
Mukherjee, S., & Abraham, J. P. (2007) · Computers & Fluids · doi:10.1016/j.compfluid.2006.12.002 · Cited by 50 · see citing works
Entropic lattice Boltzmann representations required to recover Navier-Stokes flows
Keating, B., Vahala, G., Yepez, J., et al. (2007) · Physical Review E · doi:10.1103/physreve.75.036712 · Cited by 49 · see citing works
Simulating Solid Colloidal Particles Using the Lattice-Boltzmann Method
Heemels, M. W., Hagen, M. H. J., & Lowe, C. (2000) · Journal of Computational Physics · doi:10.1006/jcph.2000.6564 · Cited by 48 · see citing works
Weighted $\mathbb{L}^2$-Stability of the Lattice Boltzmann Method
Junk, M., & Yong, W. (2009) · SIAM Journal on Numerical Analysis · doi:10.1137/060675216 · Cited by 48 · see citing works
A simple lattice Boltzmann scheme for combustion simulation
Chen, S., Liu, Z., Tian, Z., et al. (2007) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2007.08.020 · Cited by 47 · see citing works
Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains
Junk, M., & Yang, Z. (2008) · Numerische Mathematik · doi:10.1007/s00211-008-0196-0 · Cited by 46 · see citing works
Comment on “Discrete Boltzmann Equation for Microfluidics”
Luo, L. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.139401 · Cited by 46 · see citing works
Viscosity of curved fibers in suspension
Joung, C. G., Phan-Thien, N., & Fan, X. J. (2002) · Journal of Non-Newtonian Fluid Mechanics · doi:10.1016/s0377-0257(01)00163-x · Cited by 44 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Computer Simulation of Liquids
Allen, M. P., & Tildesley, D. J. (2017) · doi:10.1093/oso/9780198803195.001.0001 · Cited by 20,945 · see citing works
Feng, Y., Li, H., Li, L., Bu, L., & Wang, T. (2015) · International Journal of Heat and Mass Transfer · doi:10.1016/j.ijheatmasstransfer.2014.10.048 · Cited by 148 · see citing works
A quasi-mechanistic mathematical representation for blood viscosity.
Hund, S., Kameneva, M., & Antaki, J. (2017) · Fluids · doi:10.3390/fluids2010010 · Cited by 65 · see citing works
A 3D Lattice Boltzmann method for light simulation in participating media
Mink, A., Thäter, G., Nirschl, H., & Krause, M. J. (2016) · Journal of Computational Science · doi:10.1016/j.jocs.2016.03.014 · Cited by 39 · see citing works
Inertial dilute particulate fluid flow simulations with an Euler–Euler lattice Boltzmann method
Trunk, R., Henn, T., Dörfler, W., Nirschl, H., & Krause, M. J. (2016) · Journal of Computational Science · doi:10.1016/j.jocs.2016.03.013 · Cited by 31 · see citing works
Mink, A., Schediwy, K., Posten, C., Nirschl, H., Simonis, S., & Krause, M. J. (2022) · Energies · doi:10.3390/en15207671 · Cited by 30 · see citing works
McHardy, C., Horneber, T., & Rauh, C. (2016) · Optics Express · doi:10.1364/oe.24.016999 · Cited by 24 · see citing works
Mink, A., McHardy, C., Bressel, L., Rauh, C., & Krause, M. J. (2020) · Journal of Quantitative Spectroscopy and Radiative Transfer · doi:10.1016/j.jqsrt.2019.106810 · Cited by 19 · see citing works
Gaedtke, M., Hoffmann, T., Reinhardt, V., Thäter, G., Nirschl, H., & Krause, M. J. (2019) · International Journal of Modern Physics C · doi:10.1142/s012918311950013x · Cited by 12 · see citing works
McHardy, C., Horneber, T., & Rauh, C. (2018) · Applied Mathematics and Computation · doi:10.1016/j.amc.2017.01.045 · Cited by 12 · see citing works
- Authors
- Aidun, C. K., & Clausen, J. R.
- Affiliations
- Georgia Institute of Technology (US)
- Venue
- Annual Review of Fluid Mechanics
- Publisher
- Annual Reviews
- Published
- 2010-01-01
- Volume
- 42
- Issue
- 1
- Pages
- 439-472
- ISSN
- 0066-4189 (ISSN-L)
- 1545-4479
- OpenAlex
- W2120900954
- Cited by
- 1,991
- References
- 223
- Citations by year
- 2026: 110 · 2025: 104 · 2024: 103 · 2023: 97 · 2022: 109 · 2021: 119 · 2020: 145 · 2019: 135 · 2018: 169 · 2017: 181 · 2016: 161 · 2015: 147 · 2014: 136 · 2013: 111 · 2012: 77
- FWCI
- 5.40
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Annual Review of Fluid Mechanics (publishedVersion)
- Topics
- Physical Sciences › Engineering › Computational Mechanics › Lattice Boltzmann Simulation Studies
- Physical Sciences › Engineering › Computational Mechanics › Computational Fluid Dynamics and Aerodynamics
- Physical Sciences › Engineering › Computational Mechanics › Fluid Dynamics and Turbulent Flows
- Keywords
- lattice Boltzmann method, entropic lattice Boltzmann method, turbulent flow simulation, multiphase flow, fiber suspensions, hybrid Eulerian-Lagrangian method, higher-order boundary conditions, microchannel flow, parallel scalability, multiscale flow, kinetic theory
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
