{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T02:49:48Z","timestamp":1771469388655,"version":"3.50.1"},"reference-count":42,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2013,7,21]],"date-time":"2013-07-21T00:00:00Z","timestamp":1374364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2013,7,21]]},"abstract":"<jats:p>Physics based simulation of the dynamics of water spray - water droplets dispersed in air - is a means to increase the visual plausibility of computer graphics modeled phenomena such as waterfalls, water jets and stormy seas. Spray phenomena are frequently encountered by the visual effects industry and often challenge state of the art methods. Current spray simulation pipelines typically employ a combination of Lagrangian (particle) and Eulerian (volumetric) methods - the Eulerian methods being used for parts of the spray where individual droplets are not apparent. However, existing Eulerian methods in computer graphics are based on gas solvers that will for example exhibit hydrostatic equilibrium in certain scenarios where the air is expected to rise and the water droplets fall. To overcome this problem, we propose to simulate spray in the Eulerian domain as a two-way coupled two-continua of air and water phases co-existing at each point in space. The fundamental equations originate in applied physics and we present a number of contributions that make Eulerian two-continua spray simulation feasible for computer graphics applications. The contributions include a Poisson equation that fits into the operator splitting methodology as well as (semi-)implicit discretizations of droplet diffusion and the drag force with improved stability properties. As shown by several examples, our approach allows us to more faithfully capture the dynamics of spray than previous Eulerian methods.<\/jats:p>","DOI":"10.1145\/2461912.2461918","type":"journal-article","created":{"date-parts":[[2013,7,16]],"date-time":"2013-07-16T18:06:45Z","timestamp":1373998005000},"page":"1-10","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":40,"title":["A two-continua approach to Eulerian simulation of water spray"],"prefix":"10.1145","volume":"32","author":[{"given":"Michael B.","family":"Nielsen","sequence":"first","affiliation":[{"name":"Aarhus University"}]},{"given":"Ole","family":"\u00d8sterby","sequence":"additional","affiliation":[{"name":"Aarhus University"}]}],"member":"320","published-online":{"date-parts":[[2013,7,21]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050401"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/cav.v21:3\/4"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1275808.1276502"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","volume-title":"Fundamentals of Multiphase Flow","author":"Brennen C.","DOI":"10.1017\/CBO9780511807169"},{"key":"e_1_2_2_5_1","unstructured":"Bridson R. 2008. Fluid Simulation for Computer Graphics. AK Peters.   Bridson R. 2008. Fluid Simulation for Computer Graphics . AK Peters."},{"key":"e_1_2_2_6_1","volume-title":"A multigrid tutorial","author":"Briggs W. L.","edition":"2"},{"key":"e_1_2_2_7_1","volume-title":"Proceedings of the 2010 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation, Eurographics Association, Aire-la-Ville, Switzerland, SCA '10, 197--206","author":"Chentanez N."},{"key":"e_1_2_2_8_1","volume-title":"An Introduction to Ordinary Differential Equations","author":"Coddington E. A."},{"key":"e_1_2_2_9_1","doi-asserted-by":"crossref","unstructured":"Dupont T. F. and Liu Y. 2007. Back and forth error compensation and correction methods for semi-lagrangian schemes with application to level set interface computations. Math. Comput. 76 258 647--668.  Dupont T. F. and Liu Y. 2007. Back and forth error compensation and correction methods for semi-lagrangian schemes with application to level set interface computations. Math. Comput. 76 258 647--668.","DOI":"10.1090\/S0025-5718-06-01898-9"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/383259.383260"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/15886.15894"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/1278780.1278862"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/1179849.1179874"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.01.030"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-012-0697-9"},{"key":"e_1_2_2_16_1","unstructured":"Integrator 2013. http:\/\/integrals.wolfram.com.  Integrator 2013. http:\/\/integrals.wolfram.com."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759732455X"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01638.x"},{"key":"e_1_2_2_19_1","volume-title":"Proceedings of the 2006 ACM SIGGRAPH\/Eurographics symposium on Computer animation, Eurographics Association, Aire-la-Ville, Switzerland, SCA '06, 335--344","author":"Kim J."},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.12.036"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.gmod.2008.04.002"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/1179352.1141960"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2008.37"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01362.x"},{"key":"e_1_2_2_25_1","volume-title":"EUROGRAPHICS 2002 Short Presentations, 405--410","author":"Miyazaki R."},{"key":"e_1_2_2_26_1","volume-title":"Proceedings of the 11th Pacific Conference on Computer Graphics and Applications, IEEE Computer Society, Washington, DC, USA, PG '03, 440--444","author":"Mizuno R."},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01640.x"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/1599470.1599499"},{"key":"e_1_2_2_29_1","unstructured":"Nielsen M. B. 2013. https:\/\/code.google.com\/p\/two-continua-eulerian-spray-simulation\/.  Nielsen M. B. 2013. https:\/\/code.google.com\/p\/two-continua-eulerian-spray-simulation\/."},{"key":"e_1_2_2_30_1","volume-title":"Proceedings of Computer Animation, IEEE Computer Society","author":"O'Brien J. F."},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/15922.15893"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1145\/1661412.1618467"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/1186822.1073282"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1137\/070679065"},{"key":"e_1_2_2_35_1","volume-title":"Fluid Dynamics and Transport of Droplets and Sprays","author":"Sirignano W."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/1037957.1037962"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/311535.311548"},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-8659.00686"},{"key":"e_1_2_2_39_1","volume-title":"Proceedings of the 2006 ACM SIGGRAPH\/Eurographics symposium on Computer animation, Eurographics Association, Aire-la-Ville, Switzerland, SCA '06, 157--164","author":"Th\u00fcrey N."},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901396533"},{"key":"e_1_2_2_41_1","doi-asserted-by":"crossref","unstructured":"Villermaux E. and Boss B. 2009. Single-drop fragmentation determines size distribution of raindrops. Nature Physics.  Villermaux E. and Boss B. 2009. Single-drop fragmentation determines size distribution of raindrops. Nature Physics .","DOI":"10.1038\/nphys1340"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2011.0153"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2461912.2461918","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2461912.2461918","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T08:35:48Z","timestamp":1750235748000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2461912.2461918"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,21]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,7,21]]}},"alternative-id":["10.1145\/2461912.2461918"],"URL":"https:\/\/doi.org\/10.1145\/2461912.2461918","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,7,21]]},"assertion":[{"value":"2013-07-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}