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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
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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,
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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
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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
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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
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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
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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
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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
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1281:{\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}}}}
909:. 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:
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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.
332:. 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
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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
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560:{\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}}
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244:
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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
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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).
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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.
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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
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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
184:, 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
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use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.
1962:
Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures".
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Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures".
160:. 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.
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influenced by rotational effects, the performance of such models is unsatisfactory. In such flows, Reynolds stress equation models offer much better accuracy.
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To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the
Reynolds stress term
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value. In cases with well-defined shear layers, this is likely due the dominance of streamwise shear components, so that considerable
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Absi, R. (2021) "Reinvestigating the
Parabolic-Shaped Eddy Viscosity Profile for Free Surface Flows" Hydrology 2021, 8(3), 126.
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Absi, R. (2019) "Eddy
Viscosity and Velocity Profiles in Fully-Developed Turbulent Channel Flows" Fluid Dyn (2019) 54: 137.
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governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows,
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325:
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Townsend, A. A. (1980) "The Structure of Turbulent Shear Flow" 2nd Edition (Cambridge Monographs on Mechanics),
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1575:"About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity"
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viscosity with an eddy viscosity. This can be a simple constant eddy viscosity (which works well for some free
1658:"Smagorinsky, Joseph. "General circulation experiments with the primitive equations: I. The basic experiment"
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layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.
1059:", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small
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as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the
1697:
Spalart, Philippe R.; Allmaras, Steven R. (1992). "A one-equation turbulence model for aerodynamic flows".
679:{\displaystyle -{\overline {v_{i}^{\prime }v_{j}^{\prime }}}=2\nu _{t}S_{ij}-{\tfrac {2}{3}}k\delta _{ij}}
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is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y)
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1364:. The S–A model uses only one additional equation to model turbulence viscosity transport, while the
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The following is a brief overview of commonly employed models in modern engineering applications.
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280:. Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.
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Wilcox, C. D. (1998), "Turbulence Modeling for CFD" 2nd Ed., (DCW Industries, La Cañada),
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Bradshaw, P. (1971) "An introduction to turbulence and its measurement" (Pergamon Press),
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1337:–omega) models and offers a relatively low cost computation for the turbulence viscosity
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Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure
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Lumley, John; Newman, Gary (1977). "The return to isotropy of homogeneous turbulence".
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models, based on the local derivatives of the velocity field and the local grid size:
369:, has been introduced. Models of this type are known as eddy viscosity models (EVMs).
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1991:
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872:
1948:
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Prandtl, Ludwig (1925). "Bericht uber Untersuchungen zur ausgebildeten Turbulenz".
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introduced the additional concept of the mixing length, along with the idea of a
137:
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17:
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Wilcox, D. C. (2008). "Formulation of the k-omega Turbulence Model Revisited".
1722:"A Reynolds stress model of turbulence and its application to thin shear flows"
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975:{\displaystyle \nu _{t}=\left|{\frac {\partial u}{\partial y}}\right|l_{m}^{2}}
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In this model, the additional turbulence stresses are given by augmenting the
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1804:"Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications"
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was the first to attack the closure problem, by introducing the concept of
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boundary layer and switches to the k-epsilon in the free shear flow.
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have evolved over time, with most modern turbulence models given by
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Pope, Stephen. "Turbulent Flows". Cambridge University Press, 2000.
239:{\displaystyle -\rho {\overline {v_{i}^{\prime }v_{j}^{\prime }}}}
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was the first who proposed a formula for the eddy viscosity in
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flows such as axisymmetric jets, 2-D jets, and mixing layers).
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from the convective acceleration. This term is known as the
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10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
816:{\displaystyle k={\tfrac {1}{2}}{\overline {v_{i}'v_{i}'}}}
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errors in flow-normal components are still negligible in
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John J. Bertin; Jacques Periaux; Josef Ballmann (1992),
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Smagorinsky model for the sub-grid scale eddy viscosity
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Use of mathematical models to simulate turbulent flow
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1506:. Cambridge: Cambridge University Press. p.
1015:{\displaystyle {\frac {\partial u}{\partial y}}}
182:Reynolds-averaged Navier–Stokes (RANS) equations
1311:The Boussinesq hypothesis is employed in the
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751:is the (kinematic) turbulence eddy viscosity
1852:Hanjalić, Hanjalić; Launder, Brian (2011).
1502:Computational fluid dynamics for engineers
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1055:This simple model is the basis for the "
2005:Sagaut, Pierre; Cambon, Claude (2008).
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1498:Andersson, Bengt; et al. (2012).
1422:SST (Menter’s Shear Stress Transport)
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567:which can be written in shorthand as
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58:adding citations to reliable sources
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358:{\displaystyle \nu _{t}>0}
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1531:Boussinesq, Joseph (1903).
136:A simulation of a physical
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1964:Journal of Fluid Mechanics
1921:Journal of Fluid Mechanics
1869:Journal of Fluid Mechanics
1726:Journal of Fluid Mechanics
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156:to predict the effects of
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1579:Comptes Rendus MĂ©canique
1357:{\displaystyle \nu _{t}}
744:{\displaystyle \nu _{t}}
1076:Navier–Stokes equations
178:Navier–Stokes equations
1802:Menter, F. R. (1994).
1662:Monthly Weather Review
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1976:2013JFM...731..639M
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1823:1994AIAAJ..32.1598M
1781:2008AIAAJ..46.2823W
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1536:. Gauthier-Villars.
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941:
938:
932:
928:
923:
919:
907:boundary layer
903:Ludwig Prandtl
898:
895:
881:shear stresses
865:
864:
846:
843:
839:
827:
810:
805:
801:
797:
792:
788:
784:
774:
771:
765:
762:
752:
738:
734:
723:
706:
703:
699:
673:
670:
666:
662:
656:
653:
647:
642:
639:
635:
629:
625:
621:
618:
613:
607:
602:
598:
592:
587:
583:
576:
554:
551:
547:
543:
538:
535:
530:
526:
517:
513:
509:
502:
497:
493:
487:
481:
473:
469:
465:
458:
453:
449:
443:
436:
430:
426:
422:
417:
411:
406:
402:
396:
391:
387:
380:
354:
351:
346:
342:
330:eddy viscosity
322:
321:Eddy viscosity
319:
300:
297:
293:
267:
264:
260:
233:
227:
222:
218:
212:
207:
203:
196:
193:
173:
170:
146:fluid dynamics
140:airplane model
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2093:
2082:
2079:
2077:
2074:
2073:
2071:
2062:
2058:
2054:
2052:
2048:
2044:
2042:
2038:
2034:
2032:
2028:
2026:
2022:
2021:
2017:
2008:
2001:
1998:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1958:
1955:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1922:
1915:
1912:
1906:
1903:
1898:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1863:
1860:
1855:
1848:
1845:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1805:
1798:
1795:
1790:
1786:
1782:
1778:
1774:
1770:
1763:
1760:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1716:
1713:
1708:
1704:
1700:
1693:
1690:
1684:
1679:
1675:
1671:
1668:(3): 99–164.
1667:
1663:
1659:
1652:
1649:
1644:
1640:
1636:
1632:
1628:
1624:
1617:
1614:
1610:
1606:
1601:
1596:
1592:
1588:
1584:
1580:
1576:
1569:
1566:
1562:
1560:9780817636630
1556:
1552:
1551:
1543:
1540:
1535:
1527:
1524:
1519:
1513:
1509:
1504:
1503:
1494:
1491:
1486:
1479:
1476:
1469:
1464:
1456:
1453:
1451:
1447:
1442:
1440:
1436:
1430:
1427:
1423:
1420:
1416:
1414:
1410:
1406:
1402:
1400:
1396:
1392:
1388:
1385:
1384:
1381:
1376:Common models
1375:
1373:
1371:
1367:
1349:
1345:
1336:
1332:
1330:
1325:
1321:
1319:
1314:
1306:
1302:
1298:
1296:
1294:
1271:
1266:
1259:
1251:
1242:
1236:
1228:
1218:
1211:
1208:
1203:
1198:
1193:
1187:
1179:
1170:
1165:
1160:
1155:
1149:
1141:
1132:
1125:
1119:
1113:
1108:
1104:
1096:
1095:
1094:
1092:
1088:
1081:
1079:
1077:
1073:
1069:
1066:More general
1064:
1062:
1058:
1036:
1032:
1024:
1006:
998:
985:
984:
983:
967:
962:
958:
953:
947:
939:
930:
926:
921:
917:
908:
904:
894:
891:
887:
882:
876:
874:
870:
862:
844:
841:
837:
828:
826:
803:
799:
795:
790:
786:
782:
772:
769:
763:
760:
753:
736:
732:
724:
722:
704:
701:
697:
689:
688:
687:
671:
668:
664:
660:
654:
651:
645:
640:
637:
633:
627:
623:
619:
616:
600:
596:
585:
581:
574:
552:
549:
545:
541:
536:
533:
528:
524:
515:
511:
495:
491:
479:
471:
467:
451:
447:
434:
428:
424:
420:
404:
400:
389:
385:
378:
370:
368:
352:
349:
344:
340:
331:
327:
320:
318:
316:
298:
295:
291:
281:
265:
262:
258:
249:
220:
216:
205:
201:
194:
191:
183:
179:
171:
169:
167:
163:
162:The equations
159:
155:
151:
147:
139:
134:
124:
121:
113:
110:November 2016
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2006:
2000:
1967:
1963:
1957:
1924:
1920:
1914:
1905:
1872:
1868:
1862:
1853:
1847:
1814:
1811:AIAA Journal
1810:
1797:
1772:
1769:AIAA Journal
1768:
1762:
1729:
1725:
1715:
1698:
1692:
1665:
1661:
1651:
1626:
1622:
1616:
1582:
1578:
1568:
1553:, Springer,
1549:
1542:
1532:
1526:
1501:
1493:
1484:
1478:
1449:
1445:
1438:
1434:
1431:
1412:
1408:
1398:
1394:
1379:
1369:
1365:
1334:
1328:
1323:
1317:
1310:
1304:
1300:
1290:
1085:
1065:
1054:
900:
889:
885:
877:
866:
371:
366:
324:
314:
282:
175:
149:
143:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
1970:: 639–681.
1927:: 161–178.
1875:: 639–681.
138:wind tunnel
2076:Turbulence
2070:Categories
2061:0963605100
2051:0080166210
2041:0521298199
1629:(2): 136.
1465:References
158:turbulence
80:newspapers
1992:122537381
1897:122537381
1839:120712103
1754:122631170
1441:–epsilon)
1401:–epsilon)
1346:ν
1307:–ω models
1257:∂
1249:∂
1234:∂
1226:∂
1185:∂
1177:∂
1147:∂
1139:∂
1123:Δ
1117:Δ
1105:ν
1004:∂
996:∂
945:∂
937:∂
918:ν
869:molecular
838:δ
809:¯
733:ν
665:δ
646:−
624:ν
612:¯
606:′
591:′
575:−
546:δ
529:−
508:∂
501:¯
486:∂
464:∂
457:¯
442:∂
425:ν
416:¯
410:′
395:′
379:−
341:ν
232:¯
226:′
211:′
195:ρ
192:−
1949:39228898
1609:32637068
890:absolute
886:relative
804:′
791:′
1972:Bibcode
1929:Bibcode
1877:Bibcode
1819:Bibcode
1777:Bibcode
1734:Bibcode
1670:Bibcode
1631:Bibcode
1534:lumi_re
1458:medium.
1452:–omega)
1415:–omega)
1368:–ε and
1315:(S–A),
1303:–ε and
901:Later,
859:is the
823:is the
719:is the
94:scholar
2059:
2049:
2039:
1990:
1947:
1895:
1837:
1752:
1607:
1557:
1514:
982:where
686:where
365:, the
96:
89:
82:
75:
67:
2018:Other
1988:S2CID
1945:S2CID
1893:S2CID
1835:S2CID
1807:(PDF)
1750:S2CID
1605:S2CID
1470:Notes
873:shear
101:JSTOR
87:books
2057:ISBN
2047:ISBN
2037:ISBN
1555:ISBN
1512:ISBN
1448:–ω (
1437:–ε (
1411:–ω (
1397:–ε (
829:and
350:>
176:The
73:news
1980:doi
1968:731
1937:doi
1885:doi
1873:731
1827:doi
1785:doi
1742:doi
1703:doi
1678:doi
1639:doi
1595:hdl
1587:doi
1583:335
144:In
56:by
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1986:.
1978:.
1966:.
1943:.
1935:.
1925:82
1923:.
1891:.
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1825:.
1815:32
1813:.
1809:.
1783:.
1773:46
1771:.
1748:.
1740:.
1730:52
1728:.
1724:.
1701:.
1676:.
1666:91
1664:.
1660:.
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1625:.
1603:,
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1510:.
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1331:–ω
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250:,
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2009:.
1994:.
1982::
1974::
1951:.
1939::
1931::
1899:.
1887::
1879::
1856:.
1841:.
1829::
1821::
1791:.
1787::
1779::
1756:.
1744::
1736::
1709:.
1705::
1686:.
1680::
1672::
1645:.
1641::
1633::
1627:5
1597::
1589::
1520:.
1487:.
1450:k
1446:k
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1370:k
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1333:(
1329:k
1324:k
1322:(
1318:k
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1301:k
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1171:(
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927:=
922:t
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845:j
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