122:
25:
1275:
554:
1088:
363:
1413:
SST (Menter's shear stress transport) turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in computational fluid dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the
1406:
In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model that is used as a closure for the
Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables,
1392:
K-epsilon (k-ε) turbulence model is the most common model used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two-equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original
1378:
The
Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary
872:
due to turbulence act in the same direction as the shear stresses produced by the averaged flow). It has since been found to be significantly less accurate than most practitioners would assume. Still, turbulence models which employ the
Boussinesq hypothesis have demonstrated significant practical
1446:
Eddy viscosity based closures cannot account for the return to isotropy of turbulence, observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid
Distortion limit, where the turbulent flow essentially behaves like an elastic
1443:
models have significant shortcomings in complex engineering flows. This arises due to the use of the eddy-viscosity hypothesis in their formulation. For instance, in flows with high degrees of anisotropy, significant streamline curvature, flow separation, zones of recirculating flow or flows
1522:
Boussinesq, J. (1903). Thōrie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la thōrie mc̄anique de la lumi_re: Refroidissement et c̄hauffement par rayonnement, conductibilit ̄des tiges, lames et masses cristallines, courants de convection, thōrie mc̄anique de la
673:
1270:{\displaystyle \nu _{t}=\Delta x\Delta y{\sqrt {\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial v}{\partial y}}\right)^{2}+{\frac {1}{2}}\left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)^{2}}}}
898:. For wall-bounded turbulent flows, the eddy viscosity must vary with distance from the wall, hence the addition of the concept of a 'mixing length'. In the simplest wall-bounded flow model, the eddy viscosity is given by the equation:
881:
terms. Beyond this, most eddy viscosity turbulence models contain coefficients which are calibrated against measurements, and thus produce reasonably accurate overall outcomes for flow fields of similar type as used for calibration.
321:. In 1877 Boussinesq proposed relating the turbulence stresses to the mean flow to close the system of equations. Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant
867:
The
Boussinesq hypothesis – although not explicitly stated by Boussinesq at the time – effectively consists of the assumption that the Reynolds stress tensor is aligned with the strain tensor of the mean flow (i.e.: that the
969:
549:{\displaystyle -{\overline {v_{i}^{\prime }v_{j}^{\prime }}}=\nu _{t}\left({\frac {\partial {\overline {v_{i}}}}{\partial x_{j}}}+{\frac {\partial {\overline {v_{j}}}}{\partial x_{i}}}\right)-{\frac {2}{3}}k\delta _{ij}}
810:
233:
1421:
The
Reynolds stress equation model (RSM), also referred to as second moment closure model, is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models like the
1407:
k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).
1393:
impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.
1280:
In the context of Large Eddy
Simulation, turbulence modeling refers to the need to parameterize the subgrid scale stress in terms of features of the filtered velocity field. This field is called
559:
1009:
169:
govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the
173:, which govern the mean flow. However, the nonlinearity of the Navier–Stokes equations means that the velocity fluctuations still appear in the RANS equations, in the nonlinear term
157:
use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.
1951:
Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures".
1856:
Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures".
149:. Turbulent flows are commonplace in most real-life scenarios. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows.
352:
846:
709:
1351:
738:
901:
706:
300:
267:
1038:
1444:
influenced by rotational effects, the performance of such models is unsatisfactory. In such flows, Reynolds stress equation models offer much better accuracy.
1792:
272:
To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the
Reynolds stress term
170:
176:
121:
873:
value. In cases with well-defined shear layers, this is likely due the dominance of streamwise shear components, so that considerable
1410:
89:
745:
2018:
Absi, R. (2021) "Reinvestigating the
Parabolic-Shaped Eddy Viscosity Profile for Free Surface Flows" Hydrology 2021, 8(3), 126.
61:
1504:
1375:
1301:
2012:
Absi, R. (2019) "Eddy
Viscosity and Velocity Profiles in Fully-Developed Turbulent Channel Flows" Fluid Dyn (2019) 54: 137.
68:
42:
977:
1547:
108:
75:
153:
governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows,
2024:
Townsend, A. A. (1980) "The Structure of Turbulent Shear Flow" 2nd Edition (Cambridge Monographs on Mechanics),
1671:
1646:
57:
2049:
2039:
2029:
1564:"About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity"
1417:
860:
viscosity with an eddy viscosity. This can be a simple constant eddy viscosity (which works well for some free
46:
1647:"Smagorinsky, Joseph. "General circulation experiments with the primitive equations: I. The basic experiment"
1379:
layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.
1048:", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small
302:
as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the
1686:
Spalart, Philippe R.; Allmaras, Steven R. (1992). "A one-equation turbulence model for aerodynamic flows".
154:
668:{\displaystyle -{\overline {v_{i}^{\prime }v_{j}^{\prime }}}=2\nu _{t}S_{ij}-{\tfrac {2}{3}}k\delta _{ij}}
1281:
1064:
166:
150:
2069:
1422:
1382:
1305:
1056:
314:
1011:
is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y)
1353:. The S–A model uses only one additional equation to model turbulence viscosity transport, while the
813:
1433:
1396:
1316:
324:
1496:
1369:
The following is a brief overview of commonly employed models in modern engineering applications.
821:
269:. Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.
82:
35:
1329:
716:
1079:
1537:
1488:
681:
275:
242:
1960:
1917:
1865:
1807:
1765:
1722:
1658:
1619:
1016:
2044:
Wilcox, C. D. (1998), "Turbulence Modeling for CFD" 2nd Ed., (DCW Industries, La Cañada),
2034:
Bradshaw, P. (1971) "An introduction to turbulence and its measurement" (Pergamon Press),
8:
2064:
1489:
1326:–omega) models and offers a relatively low cost computation for the turbulence viscosity
1964:
1921:
1869:
1843:
Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure
1811:
1769:
1726:
1662:
1623:
1976:
1933:
1908:
Lumley, John; Newman, Gary (1977). "The return to isotropy of homogeneous turbulence".
1881:
1823:
1738:
1593:
1075:
142:
1082:
models, based on the local derivatives of the velocity field and the local grid size:
358:, has been introduced. Models of this type are known as eddy viscosity models (EVMs).
2045:
2035:
2025:
1980:
1885:
1827:
1742:
1710:
1543:
1500:
1049:
1937:
1597:
318:
1968:
1925:
1873:
1815:
1773:
1730:
1691:
1666:
1627:
1610:
Prandtl, Ludwig (1925). "Bericht uber Untersuchungen zur ausgebildeten Turbulenz".
1583:
1575:
1045:
894:
introduced the additional concept of the mixing length, along with the idea of a
849:
236:
1756:
Wilcox, D. C. (2008). "Formulation of the k-omega Turbulence Model Revisited".
1711:"A Reynolds stress model of turbulence and its application to thin shear flows"
1579:
1060:
964:{\displaystyle \nu _{t}=\left|{\frac {\partial u}{\partial y}}\right|l_{m}^{2}}
895:
891:
134:
1929:
1734:
1588:
856:
In this model, the additional turbulence stresses are given by augmenting the
2058:
1631:
861:
2013:
1793:"Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications"
2019:
1535:
869:
317:
was the first to attack the closure problem, by introducing the concept of
1070:
1972:
1877:
126:
1695:
1563:
146:
1819:
1777:
1414:
boundary layer and switches to the k-epsilon in the free shear flow.
1059:
have evolved over time, with most modern turbulence models given by
24:
1898:
Pope, Stephen. "Turbulent Flows". Cambridge University Press, 2000.
857:
228:{\displaystyle -\rho {\overline {v_{i}^{\prime }v_{j}^{\prime }}}}
1078:
was the first who proposed a formula for the eddy viscosity in
864:
flows such as axisymmetric jets, 2-D jets, and mixing layers).
235:
from the convective acceleration. This term is known as the
1672:
10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
805:{\displaystyle k={\tfrac {1}{2}}{\overline {v_{i}'v_{i}'}}}
877:
errors in flow-normal components are still negligible in
1536:
John J. Bertin; Jacques Periaux; Josef Ballmann (1992),
1287:
1071:
Smagorinsky model for the sub-grid scale eddy viscosity
756:
638:
1332:
1091:
1019:
980:
904:
824:
748:
719:
684:
562:
366:
327:
278:
245:
179:
16:
Use of mathematical models to simulate turbulent flow
1561:
49:. Unsourced material may be challenged and removed.
1539:Advances in Hypersonics: Modeling hypersonic flows
1345:
1269:
1032:
1003:
963:
885:
840:
804:
732:
700:
667:
548:
346:
294:
261:
227:
1688:30th Aerospace Sciences Meeting and Exhibit, AIAA
1495:. Cambridge: Cambridge University Press. p.
2056:
1004:{\displaystyle {\frac {\partial u}{\partial y}}}
171:Reynolds-averaged Navier–Stokes (RANS) equations
1840:
1708:
1685:
1950:
1855:
1300:The Boussinesq hypothesis is employed in the
1993:
1907:
740:is the (kinematic) turbulence eddy viscosity
1841:Hanjalić, Hanjalić; Launder, Brian (2011).
1644:
1519:
1491:Computational fluid dynamics for engineers
2014:https://doi.org/10.1134/S0015462819010014
1670:
1587:
1486:
109:Learn how and when to remove this message
2020:https://doi.org/10.3390/hydrology8030126
1044:This simple model is the basis for the "
120:
1994:Sagaut, Pierre; Cambon, Claude (2008).
1609:
2057:
1790:
1755:
1487:Andersson, Bengt; et al. (2012).
1411:SST (Menter’s Shear Stress Transport)
556:which can be written in shorthand as
356:(kinematic) turbulence eddy viscosity
1471:
47:adding citations to reliable sources
18:
13:
1709:Hanjalic, K.; Launder, B. (1972).
1245:
1237:
1222:
1214:
1173:
1165:
1135:
1127:
1111:
1105:
992:
984:
933:
925:
594:
579:
496:
474:
452:
430:
398:
383:
214:
199:
160:
14:
2081:
309:
141:is the construction and use of a
1364:
23:
1996:Homogeneous Turbulence Dynamics
1987:
1944:
1901:
1892:
1849:
1834:
1784:
1749:
886:Prandtl's mixing-length concept
34:needs additional citations for
1702:
1679:
1638:
1603:
1555:
1529:
1513:
1480:
1465:
1418:Reynolds stress equation model
1:
1453:
347:{\displaystyle \nu _{t}>0}
1645:Smagorinsky, Joseph (1963).
1562:François G. Schmitt (2007),
841:{\displaystyle \delta _{ij}}
797:
600:
489:
445:
404:
220:
7:
1520:Boussinesq, Joseph (1903).
125:A simulation of a physical
10:
2086:
1953:Journal of Fluid Mechanics
1910:Journal of Fluid Mechanics
1858:Journal of Fluid Mechanics
1715:Journal of Fluid Mechanics
1580:10.1016/j.crme.2007.08.004
710:mean rate of strain tensor
315:Joseph Valentin Boussinesq
145:to predict the effects of
1930:10.1017/s0022112077000585
1735:10.1017/S002211207200268X
814:turbulence kinetic energy
2006:
1632:10.1002/zamm.19250050212
1568:Comptes Rendus MĂ©canique
1458:
1346:{\displaystyle \nu _{t}}
733:{\displaystyle \nu _{t}}
1065:Navier–Stokes equations
167:Navier–Stokes equations
1791:Menter, F. R. (1994).
1651:Monthly Weather Review
1472:Pope, Stephen (2000).
1376:Spalart–Allmaras (S–A)
1347:
1282:subgrid-scale modeling
1271:
1034:
1005:
965:
842:
806:
734:
702:
701:{\displaystyle S_{ij}}
669:
550:
348:
296:
295:{\displaystyle R_{ij}}
263:
262:{\displaystyle R_{ij}}
229:
130:
1348:
1272:
1080:Large Eddy Simulation
1040:is the mixing length.
1035:
1033:{\displaystyle l_{m}}
1006:
966:
843:
807:
735:
703:
670:
551:
349:
297:
264:
230:
124:
58:"Turbulence modeling"
1973:10.1017/jfm.2013.343
1878:10.1017/jfm.2013.343
1612:Z. Angew. Math. Mech
1330:
1089:
1017:
978:
902:
822:
746:
717:
682:
560:
364:
325:
276:
243:
177:
43:improve this article
1965:2013JFM...731..639M
1922:1977JFM....82..161L
1870:2013JFM...731..639M
1812:1994AIAAJ..32.1598M
1770:2008AIAAJ..46.2823W
1727:1972JFM....52..609H
1663:1963MWRv...91...99S
1624:1925ZaMM....5..136P
1525:. Gauthier-Villars.
1361:–ω models use two.
960:
795:
782:
598:
583:
402:
387:
218:
203:
139:turbulence modeling
1696:10.2514/6.1992-439
1589:20.500.12210/73178
1343:
1288:Spalart–Allmaras,
1267:
1076:Joseph Smagorinsky
1050:pressure gradients
1030:
1001:
961:
946:
838:
802:
783:
770:
765:
730:
698:
665:
647:
584:
569:
546:
388:
373:
344:
292:
259:
225:
204:
189:
143:mathematical model
131:
2070:Turbulence models
1764:(11): 2823–2838.
1574:(9–10): 617–627,
1506:978-1-107-01895-2
1265:
1252:
1229:
1203:
1180:
1142:
1057:turbulence models
999:
940:
800:
764:
646:
603:
528:
510:
492:
466:
448:
407:
223:
119:
118:
111:
93:
2077:
2000:
1999:
1991:
1985:
1984:
1948:
1942:
1941:
1905:
1899:
1896:
1890:
1889:
1853:
1847:
1846:
1838:
1832:
1831:
1806:(8): 1598–1605.
1797:
1788:
1782:
1781:
1753:
1747:
1746:
1706:
1700:
1699:
1683:
1677:
1676:
1674:
1642:
1636:
1635:
1607:
1601:
1600:
1591:
1559:
1553:
1552:
1533:
1527:
1526:
1517:
1511:
1510:
1494:
1484:
1478:
1477:
1469:
1352:
1350:
1349:
1344:
1342:
1341:
1302:Spalart–Allmaras
1276:
1274:
1273:
1268:
1266:
1264:
1263:
1258:
1254:
1253:
1251:
1243:
1235:
1230:
1228:
1220:
1212:
1204:
1196:
1191:
1190:
1185:
1181:
1179:
1171:
1163:
1153:
1152:
1147:
1143:
1141:
1133:
1125:
1118:
1101:
1100:
1039:
1037:
1036:
1031:
1029:
1028:
1010:
1008:
1007:
1002:
1000:
998:
990:
982:
970:
968:
967:
962:
959:
954:
945:
941:
939:
931:
923:
914:
913:
847:
845:
844:
839:
837:
836:
811:
809:
808:
803:
801:
796:
791:
778:
768:
766:
757:
739:
737:
736:
731:
729:
728:
707:
705:
704:
699:
697:
696:
674:
672:
671:
666:
664:
663:
648:
639:
633:
632:
620:
619:
604:
599:
597:
592:
582:
577:
567:
555:
553:
552:
547:
545:
544:
529:
521:
516:
512:
511:
509:
508:
507:
494:
493:
488:
487:
478:
472:
467:
465:
464:
463:
450:
449:
444:
443:
434:
428:
421:
420:
408:
403:
401:
396:
386:
381:
371:
353:
351:
350:
345:
337:
336:
301:
299:
298:
293:
291:
290:
268:
266:
265:
260:
258:
257:
234:
232:
231:
226:
224:
219:
217:
212:
202:
197:
187:
114:
107:
103:
100:
94:
92:
51:
27:
19:
2085:
2084:
2080:
2079:
2078:
2076:
2075:
2074:
2055:
2054:
2009:
2004:
2003:
1992:
1988:
1949:
1945:
1906:
1902:
1897:
1893:
1854:
1850:
1839:
1835:
1820:10.2514/3.12149
1795:
1789:
1785:
1778:10.2514/1.36541
1754:
1750:
1707:
1703:
1684:
1680:
1643:
1639:
1608:
1604:
1560:
1556:
1550:
1534:
1530:
1518:
1514:
1507:
1485:
1481:
1474:Turbulent Flows
1470:
1466:
1461:
1456:
1451:
1450:
1367:
1337:
1333:
1331:
1328:
1327:
1315:–epsilon), and
1298:
1259:
1244:
1236:
1234:
1221:
1213:
1211:
1210:
1206:
1205:
1195:
1186:
1172:
1164:
1162:
1158:
1157:
1148:
1134:
1126:
1124:
1120:
1119:
1117:
1096:
1092:
1090:
1087:
1086:
1073:
1063:similar to the
1061:field equations
1046:law of the wall
1024:
1020:
1018:
1015:
1014:
991:
983:
981:
979:
976:
975:
955:
950:
932:
924:
922:
918:
909:
905:
903:
900:
899:
888:
850:Kronecker delta
829:
825:
823:
820:
819:
787:
774:
769:
767:
755:
747:
744:
743:
724:
720:
718:
715:
714:
689:
685:
683:
680:
679:
656:
652:
637:
625:
621:
615:
611:
593:
588:
578:
573:
568:
566:
561:
558:
557:
537:
533:
520:
503:
499:
495:
483:
479:
477:
473:
471:
459:
455:
451:
439:
435:
433:
429:
427:
426:
422:
416:
412:
397:
392:
382:
377:
372:
370:
365:
362:
361:
332:
328:
326:
323:
322:
312:
304:closure problem
283:
279:
277:
274:
273:
250:
246:
244:
241:
240:
237:Reynolds stress
213:
208:
198:
193:
188:
186:
178:
175:
174:
163:
161:Closure problem
155:CFD simulations
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2083:
2073:
2072:
2067:
2053:
2052:
2042:
2032:
2022:
2016:
2008:
2005:
2002:
2001:
1986:
1943:
1900:
1891:
1848:
1833:
1783:
1748:
1721:(4): 609–638.
1701:
1678:
1637:
1602:
1554:
1548:
1528:
1512:
1505:
1479:
1463:
1462:
1460:
1457:
1455:
1452:
1449:
1448:
1432:model and the
1415:
1408:
1394:
1380:
1372:
1371:
1366:
1363:
1340:
1336:
1297:
1286:
1278:
1277:
1262:
1257:
1250:
1247:
1242:
1239:
1233:
1227:
1224:
1219:
1216:
1209:
1202:
1199:
1194:
1189:
1184:
1178:
1175:
1170:
1167:
1161:
1156:
1151:
1146:
1140:
1137:
1132:
1129:
1123:
1116:
1113:
1110:
1107:
1104:
1099:
1095:
1072:
1069:
1042:
1041:
1027:
1023:
1012:
997:
994:
989:
986:
958:
953:
949:
944:
938:
935:
930:
927:
921:
917:
912:
908:
896:boundary layer
892:Ludwig Prandtl
887:
884:
870:shear stresses
854:
853:
835:
832:
828:
816:
799:
794:
790:
786:
781:
777:
773:
763:
760:
754:
751:
741:
727:
723:
712:
695:
692:
688:
662:
659:
655:
651:
645:
642:
636:
631:
628:
624:
618:
614:
610:
607:
602:
596:
591:
587:
581:
576:
572:
565:
543:
540:
536:
532:
527:
524:
519:
515:
506:
502:
498:
491:
486:
482:
476:
470:
462:
458:
454:
447:
442:
438:
432:
425:
419:
415:
411:
406:
400:
395:
391:
385:
380:
376:
369:
343:
340:
335:
331:
319:eddy viscosity
311:
310:Eddy viscosity
308:
289:
286:
282:
256:
253:
249:
222:
216:
211:
207:
201:
196:
192:
185:
182:
162:
159:
135:fluid dynamics
129:airplane model
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2082:
2071:
2068:
2066:
2063:
2062:
2060:
2051:
2047:
2043:
2041:
2037:
2033:
2031:
2027:
2023:
2021:
2017:
2015:
2011:
2010:
1997:
1990:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1954:
1947:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1911:
1904:
1895:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1852:
1844:
1837:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1794:
1787:
1779:
1775:
1771:
1767:
1763:
1759:
1752:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1716:
1712:
1705:
1697:
1693:
1689:
1682:
1673:
1668:
1664:
1660:
1657:(3): 99–164.
1656:
1652:
1648:
1641:
1633:
1629:
1625:
1621:
1617:
1613:
1606:
1599:
1595:
1590:
1585:
1581:
1577:
1573:
1569:
1565:
1558:
1551:
1549:9780817636630
1545:
1541:
1540:
1532:
1524:
1516:
1508:
1502:
1498:
1493:
1492:
1483:
1475:
1468:
1464:
1445:
1442:
1440:
1436:
1431:
1429:
1425:
1419:
1416:
1412:
1409:
1405:
1403:
1399:
1395:
1391:
1389:
1385:
1381:
1377:
1374:
1373:
1370:
1365:Common models
1362:
1360:
1356:
1338:
1334:
1325:
1321:
1319:
1314:
1310:
1308:
1303:
1295:
1291:
1285:
1283:
1260:
1255:
1248:
1240:
1231:
1225:
1217:
1207:
1200:
1197:
1192:
1187:
1182:
1176:
1168:
1159:
1154:
1149:
1144:
1138:
1130:
1121:
1114:
1108:
1102:
1097:
1093:
1085:
1084:
1083:
1081:
1077:
1068:
1066:
1062:
1058:
1055:More general
1053:
1051:
1047:
1025:
1021:
1013:
995:
987:
974:
973:
972:
956:
951:
947:
942:
936:
928:
919:
915:
910:
906:
897:
893:
883:
880:
876:
871:
865:
863:
859:
851:
833:
830:
826:
817:
815:
792:
788:
784:
779:
775:
771:
761:
758:
752:
749:
742:
725:
721:
713:
711:
693:
690:
686:
678:
677:
676:
660:
657:
653:
649:
643:
640:
634:
629:
626:
622:
616:
612:
608:
605:
589:
585:
574:
570:
563:
541:
538:
534:
530:
525:
522:
517:
513:
504:
500:
484:
480:
468:
460:
456:
440:
436:
423:
417:
413:
409:
393:
389:
378:
374:
367:
359:
357:
341:
338:
333:
329:
320:
316:
307:
305:
287:
284:
280:
270:
254:
251:
247:
238:
209:
205:
194:
190:
183:
180:
172:
168:
158:
156:
152:
151:The equations
148:
144:
140:
136:
128:
123:
113:
110:
102:
99:November 2016
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1995:
1989:
1956:
1952:
1946:
1913:
1909:
1903:
1894:
1861:
1857:
1851:
1842:
1836:
1803:
1800:AIAA Journal
1799:
1786:
1761:
1758:AIAA Journal
1757:
1751:
1718:
1714:
1704:
1687:
1681:
1654:
1650:
1640:
1615:
1611:
1605:
1571:
1567:
1557:
1542:, Springer,
1538:
1531:
1521:
1515:
1490:
1482:
1473:
1467:
1438:
1434:
1427:
1423:
1420:
1401:
1397:
1387:
1383:
1368:
1358:
1354:
1323:
1317:
1312:
1306:
1299:
1293:
1289:
1279:
1074:
1054:
1043:
889:
878:
874:
866:
855:
360:
355:
313:
303:
271:
164:
138:
132:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1959:: 639–681.
1916:: 161–178.
1864:: 639–681.
127:wind tunnel
2065:Turbulence
2059:Categories
2050:0963605100
2040:0080166210
2030:0521298199
1618:(2): 136.
1454:References
147:turbulence
69:newspapers
1981:122537381
1886:122537381
1828:120712103
1743:122631170
1430:–epsilon)
1390:–epsilon)
1335:ν
1296:–ω models
1246:∂
1238:∂
1223:∂
1215:∂
1174:∂
1166:∂
1136:∂
1128:∂
1112:Δ
1106:Δ
1094:ν
993:∂
985:∂
934:∂
926:∂
907:ν
858:molecular
827:δ
798:¯
722:ν
654:δ
635:−
613:ν
601:¯
595:′
580:′
564:−
535:δ
518:−
497:∂
490:¯
475:∂
453:∂
446:¯
431:∂
414:ν
405:¯
399:′
384:′
368:−
330:ν
221:¯
215:′
200:′
184:ρ
181:−
1938:39228898
1598:32637068
879:absolute
875:relative
793:′
780:′
1961:Bibcode
1918:Bibcode
1866:Bibcode
1808:Bibcode
1766:Bibcode
1723:Bibcode
1659:Bibcode
1620:Bibcode
1523:lumi_re
1447:medium.
1441:–omega)
1404:–omega)
1357:–ε and
1304:(S–A),
1292:–ε and
890:Later,
848:is the
812:is the
708:is the
83:scholar
2048:
2038:
2028:
1979:
1936:
1884:
1826:
1741:
1596:
1546:
1503:
971:where
675:where
354:, the
85:
78:
71:
64:
56:
2007:Other
1977:S2CID
1934:S2CID
1882:S2CID
1824:S2CID
1796:(PDF)
1739:S2CID
1594:S2CID
1459:Notes
862:shear
90:JSTOR
76:books
2046:ISBN
2036:ISBN
2026:ISBN
1544:ISBN
1501:ISBN
1437:–ω (
1426:–ε (
1400:–ω (
1386:–ε (
818:and
339:>
165:The
62:news
1969:doi
1957:731
1926:doi
1874:doi
1862:731
1816:doi
1774:doi
1731:doi
1692:doi
1667:doi
1628:doi
1584:hdl
1576:doi
1572:335
133:In
45:by
2061::
1975:.
1967:.
1955:.
1932:.
1924:.
1914:82
1912:.
1880:.
1872:.
1860:.
1822:.
1814:.
1804:32
1802:.
1798:.
1772:.
1762:46
1760:.
1737:.
1729:.
1719:52
1717:.
1713:.
1690:.
1665:.
1655:91
1653:.
1649:.
1626:.
1614:.
1592:,
1582:,
1570:,
1566:,
1499:.
1497:83
1320:–ω
1309:–ε
1284:.
1067:.
1052:.
306:.
239:,
137:,
1998:.
1983:.
1971::
1963::
1940:.
1928::
1920::
1888:.
1876::
1868::
1845:.
1830:.
1818::
1810::
1780:.
1776::
1768::
1745:.
1733::
1725::
1698:.
1694::
1675:.
1669::
1661::
1634:.
1630::
1622::
1616:5
1586::
1578::
1509:.
1476:.
1439:k
1435:k
1428:k
1424:k
1402:k
1398:k
1388:k
1384:k
1359:k
1355:k
1339:t
1324:k
1322:(
1318:k
1313:k
1311:(
1307:k
1294:k
1290:k
1261:2
1256:)
1249:x
1241:v
1232:+
1226:y
1218:u
1208:(
1201:2
1198:1
1193:+
1188:2
1183:)
1177:y
1169:v
1160:(
1155:+
1150:2
1145:)
1139:x
1131:u
1122:(
1115:y
1109:x
1103:=
1098:t
1026:m
1022:l
996:y
988:u
957:2
952:m
948:l
943:|
937:y
929:u
920:|
916:=
911:t
852:.
834:j
831:i
789:i
785:v
776:i
772:v
762:2
759:1
753:=
750:k
726:t
694:j
691:i
687:S
661:j
658:i
650:k
644:3
641:2
630:j
627:i
623:S
617:t
609:2
606:=
590:j
586:v
575:i
571:v
542:j
539:i
531:k
526:3
523:2
514:)
505:i
501:x
485:j
481:v
469:+
461:j
457:x
441:i
437:v
424:(
418:t
410:=
394:j
390:v
379:i
375:v
342:0
334:t
288:j
285:i
281:R
255:j
252:i
248:R
210:j
206:v
195:i
191:v
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
