{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T05:00:51Z","timestamp":1776142851855,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T00:00:00Z","timestamp":1585008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both these problems are presented by modifying the original lattices without increasing the number of discrete velocities and without altering the numerical algorithm. In order to increase the Mach number, we employ shifted lattices while the magnitude of lattice speeds is increased in order to extend the temperature range. Accuracy and efficiency of the shifted lattices are demonstrated with simulations of the supersonic flow field around a diamond-shaped and NACA0012 airfoil, the subsonic, transonic, and supersonic flow field around the Busemann biplane, and the interaction of vortices with a planar shock wave. For the lattices with extended temperature range, the model is validated with the simulation of the Richtmyer\u2013Meshkov instability. We also discuss some key ideas of how to reduce the number of discrete speeds in three-dimensional simulations by pruning of the higher-order lattices, and introduce a new construction of the corresponding guided equilibrium by entropy minimization.<\/jats:p>","DOI":"10.3390\/e22030370","type":"journal-article","created":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T13:04:04Z","timestamp":1585055044000},"page":"370","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Theory, Analysis, and Applications of the Entropic Lattice Boltzmann Model for Compressible Flows"],"prefix":"10.3390","volume":"22","author":[{"given":"Nicol\u00f2","family":"Frapolli","sequence":"first","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich, CH-8092 Zurich, Switzerland"}]},{"given":"Shyam","family":"Chikatamarla","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich, CH-8092 Zurich, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9996-9679","authenticated-orcid":false,"given":"Ilya","family":"Karlin","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich, CH-8092 Zurich, Switzerland"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.paerosci.2005.03.003","article-title":"High-order accurate, low numerical diffusion methods for aerodynamics","volume":"41","author":"Ekaterinaris","year":"2005","journal-title":"Progr. Aerospace Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1146\/annurev-fluid-122109-160718","article-title":"Numerical methods for high-speed flows","volume":"43","author":"Pirozzoli","year":"2011","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_3","unstructured":"Succi, S. (2001). The Lattice-Boltzmann Equation, Oxford University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1209\/epl\/i1999-00370-1","article-title":"Perfect entropy functions of the lattice Boltzmann method","volume":"47","author":"Karlin","year":"1999","journal-title":"Europhys. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1098\/rspa.2000.0689","article-title":"Entropic lattice Boltzmann methods","volume":"457","author":"Boghosian","year":"2001","journal-title":"Proc. R. Soc. Lond. Ser. A Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1209\/epl\/i2003-00496-6","article-title":"Minimal entropic kinetic models for hydrodynamics","volume":"63","author":"Ansumali","year":"2003","journal-title":"Europhys. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1103\/PhysRevLett.80.65","article-title":"Discretization of the velocity space in the solution of the Boltzmann equation","volume":"80","author":"Shan","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"R2249","DOI":"10.1103\/PhysRevE.47.R2249","article-title":"Lattice Boltzmann thermohydrodynamics","volume":"47","author":"Alexander","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/BF02179986","article-title":"Stabilization of thermal lattice Boltzmann models","volume":"81","author":"McNamara","year":"1995","journal-title":"J. Stat. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1017\/S0022112005008153","article-title":"Kinetic theory representation of hydrodynamics: A way beyond the Navier\u2013Stokes equation","volume":"550","author":"Shan","year":"2006","journal-title":"J. Fluid Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"056702","DOI":"10.1103\/PhysRevE.73.056702","article-title":"From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models","volume":"73","author":"Philippi","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"016303","DOI":"10.1103\/PhysRevE.68.016303","article-title":"Three-dimensional lattice Boltzmann model for compressible flows","volume":"68","author":"Sun","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"190601","DOI":"10.1103\/PhysRevLett.97.190601","article-title":"Entropy and Galilean invariance of lattice Boltzmann theories","volume":"97","author":"Chikatamarla","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"046701","DOI":"10.1103\/PhysRevE.79.046701","article-title":"Lattices for the lattice Boltzmann method","volume":"79","author":"Chikatamarla","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1017\/S0022112010002740","article-title":"Lattice Boltzmann method for direct numerical simulation of turbulent flows","volume":"656","author":"Chikatamarla","year":"2010","journal-title":"J. Fluid Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"043306","DOI":"10.1103\/PhysRevE.90.043306","article-title":"Multispeed entropic lattice Boltzmann model for thermal flows","volume":"90","author":"Frapolli","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"061301","DOI":"10.1103\/PhysRevE.92.061301","article-title":"Entropic lattice Boltzmann model for compressible flows","volume":"92","author":"Frapolli","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"063302","DOI":"10.1103\/PhysRevE.93.063302","article-title":"Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation","volume":"93","author":"Frapolli","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"010604","DOI":"10.1103\/PhysRevLett.117.010604","article-title":"Lattice kinetic theory in a comoving Galilean reference frame","volume":"117","author":"Frapolli","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","unstructured":"Frapolli, N. (2017). Entropic Lattice Boltzmann Models for Thermal and Compressible Flows. [Ph.D. Thesis, ETH Zurich]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"035701","DOI":"10.1103\/PhysRevE.69.035701","article-title":"Lattice Boltzmann model for the compressible Navier\u2013Stokes equations with flexible specific-heat ratio","volume":"69","author":"Kataoka","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"54003","DOI":"10.1209\/0295-5075\/90\/54003","article-title":"Multiple-relaxation-time lattice Boltzmann approach to compressible flows with flexible specific-heat ratio and Prandtl number","volume":"90","author":"Chen","year":"2010","journal-title":"Europhys. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1007\/BF01023275","article-title":"A model kinetic equation for a gas with rotational degrees of freedom","volume":"10","author":"Rykov","year":"1975","journal-title":"Fluid Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"035701","DOI":"10.1103\/PhysRevE.77.035701","article-title":"Thermal lattice Boltzmann model for gases with internal degrees of freedom","volume":"77","author":"Nie","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ruggeri, T., and Sugiyama, M. (2015). Rational Extended Thermodynamics beyond the Monatomic Gas, Springer.","DOI":"10.1007\/978-3-319-13341-6"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1140\/epjb\/e2007-00100-1","article-title":"Quasi-equilibrium lattice Boltzmann method","volume":"56","author":"Ansumali","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01029546","article-title":"Generalization of the Krook kinetic relaxation equation","volume":"3","author":"Shakhov","year":"1968","journal-title":"Fluid Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1016\/j.physa.2009.12.032","article-title":"Factorization symmetry in the lattice Boltzmann method","volume":"389","author":"Karlin","year":"2010","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Liepmann, H.W., and Roshko, A. (1957). Elements of Gasdynamics, Courier Corporation.","DOI":"10.1063\/1.3060140"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.paerosci.2010.09.003","article-title":"Supersonic biplane: A review","volume":"47","author":"Kusunose","year":"2011","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.compfluid.2005.07.002","article-title":"Simulations of viscous transonic flows over lifting airfoils and wings","volume":"36","author":"Hafez","year":"2007","journal-title":"Comput. Fluid."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1017\/S0022112098003565","article-title":"Sound generation by shock\u2013vortex interactions","volume":"380","author":"Inoue","year":"1999","journal-title":"J. Fluid Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1146\/annurev.fluid.34.090101.162238","article-title":"The Richtmyer\u2013Meshkov instability","volume":"34","author":"Brouillette","year":"2002","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1017\/S0022112002008844","article-title":"PLIF flow visualization and measurements of the Richtmyer\u2013Meshkov instability of an air\/SF 6 interface","volume":"464","author":"Collins","year":"2002","journal-title":"J. Fluid Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"024104","DOI":"10.1063\/1.2472508","article-title":"High-resolution simulations and modeling of reshocked single-mode Richtmyer\u2013Meshkov instability: Comparison to experimental data and to amplitude growth model predictions","volume":"19","author":"Latini","year":"2007","journal-title":"Phys. Fluid."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1002\/cpa.3160130207","article-title":"Taylor instability in shock acceleration of compressible fluids","volume":"13","author":"Richtmyer","year":"1960","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1017\/S002211208400224X","article-title":"Computation of flow around wings based on the Euler equations","volume":"148","author":"Rizzi","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_38","first-page":"7029","article-title":"On the lambda-shock formation on ONERA M6 wing","volume":"9","author":"Kuzmin","year":"2014","journal-title":"Int. J. Appl. Eng. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"016702","DOI":"10.1103\/PhysRevE.76.016702","article-title":"Lattice Boltzmann method for thermal flow simulation on standard lattices","volume":"76","author":"Prasianakis","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"R4053","DOI":"10.1103\/PhysRevE.58.R4053","article-title":"Equilibria for discrete kinetic equations","volume":"58","author":"Karlin","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"016704","DOI":"10.1103\/PhysRevE.78.016704","article-title":"Lattice Boltzmann method for simulation of compressible flows on standard lattices","volume":"78","author":"Prasianakis","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"053311","DOI":"10.1103\/PhysRevE.94.053311","article-title":"Grid refinement for entropic lattice Boltzmann models","volume":"94","author":"Dorschner","year":"2016","journal-title":"Phys. Rev. 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