1865:
smoothing or regularization parameter in a real fluid. There have been extensive studies on a vortex sheet, most of them by discrete or point vortex approximation, with or without desingularization. Using a point vortex approximation and delta-regularization Krasny (1986) obtained a smooth roll-up of a vortex sheet into a double branched spiral. Since point vortices are inherently chaotic, a
Fourier filter is necessary to control the growth of round-off errors. Continuous approximation of a vortex sheet by vortex panels with arc wise diffusion of circulation density also shows that the sheet rolls-up into a double branched spiral.
1584:
394:
611:
885:
1985:
1579:
1864:
The vortex sheet solution as given by the
Birkoff-Rott equation cannot go beyond the critical time. The spontaneous loss of analyticity in a vortex sheet is a consequence of mathematical modeling since a real fluid with viscosity, however small, will never develop singularity. Viscosity acts a
1017:
1754:. That is, higher the wavenumber of a Fourier mode, the faster it grows. However, a linear theory cannot be extended much beyond the initial state. If nonlinear interactions are taken into account, asymptotic analysis suggests that for large
889:
This nonlinear integro-differential equation is called the
Birkoff-Rott equation. It describes the evolution of the vortex sheet given initial conditions. Greater details on vortex sheets can be found in the textbook by Saffman (1977).
228:
469:
1874:
744:
2076:
1705:
1412:
A flat vortex sheet with periodic boundaries in the streamwise direction can be used to model a temporal free shear layer at high
Reynolds number. Let us assume that the interval between the periodic boundaries is of length
1401:
1438:
1266:
1868:
In many engineering and physical applications the growth of a temporal free shear layer is of interest. The thickness of a free shear layer is usually measured by momentum thickness, which is defined as
657:
2140:
615:
As a consequence of Kelvin's circulation theorem, in the absence of external forces on the sheet, the circulation between any two material points in the sheet remains conserved, so
1630:
221:
930:
1805:
737:
677:
416:
389:{\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2\pi }}\int \limits _{-\infty }^{\infty }{\frac {\gamma (s',t)\mathrm {d} s'}{z(s,t)-z(s',t)}}}
126:
91:
1859:
1832:
1732:
1289:
1046:
922:
606:{\displaystyle \Gamma (s,t)=\int \limits _{0}^{s}\gamma (s',t)\mathrm {d} s'\qquad \mathrm {and} \qquad {\frac {\mathrm {d} \Gamma }{\mathrm {d} s}}=\gamma (s,t)}
462:
2096:
1980:{\displaystyle \theta =\int \limits _{y=-\infty }^{\infty }\left({\frac {1}{4}}-\left({\frac {\left\langle u\right\rangle }{2U}}\right)^{2}\right)\mathrm {d} y}
1772:
1752:
1431:
1313:
717:
697:
436:
186:
166:
146:
880:{\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2\pi }}\int \limits _{-\infty }^{\infty }{\frac {d\Gamma '}{z(\Gamma ,t)-z(\Gamma ',t)}}}
1861:
decays exponentially. The vortex sheet solution is expected to lose analyticity at the critical time. See Moore (1979), and Meiron, Baker and Orszag (1983).
1591:
Note that the integral in the above equation is a Cauchy principal value integral. The initial condition for a flat vortex sheet with constant strength is
1992:
1635:
2187:
Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge
University Press.
1321:
1574:{\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2}}\int \limits _{0}^{1}\cot \pi (z(\Gamma ,t)-z(\Gamma ',t))\;d\Gamma '}
2208:
223:
denote the strength of the sheet, that is, the jump in the tangential discontinuity. Then the velocity field induced by the sheet is
1054:
52:
2098:
is the freestream velocity. Momentum thickness has the dimension of length and the non-dimensional momentum thickness is given by
1632:. The flat vortex sheet is an equilibrium solution. However, it is unstable to infinitesimal periodic disturbances of the form
1587:
Continuous vortex sheet approximation by panel method. Roll-up of a vortex sheet due to an initial sinusoidal perturbation.
618:
40:
component of the flow velocity is continuous. The discontinuity in the tangential velocity means the flow has infinite
2203:
2101:
1594:
898:
Once a vortex sheet, it will diffuse due to viscous action. Consider a planar unidirectional flow at
1012:{\displaystyle u={\begin{cases}+U,&{\text{for }}y>0\\-U,&{\text{for }}y<0\end{cases}}}
945:
1292:
25:
191:
58:
The formulation of the vortex sheet equation of motion is given in terms of a complex coordinate
1777:
37:
722:
662:
401:
96:
61:
48:
1837:
1810:
1710:
2175:
1274:
21:
1025:
901:
441:
2156:
2081:
1757:
1737:
1416:
1298:
702:
682:
421:
398:
The integral in the above equation is a Cauchy principal value integral. We now define
171:
151:
131:
2197:
29:
2071:{\displaystyle \left\langle u\right\rangle ={\frac {1}{L}}\int _{0}^{L}\ u(x,y,t)dx}
1700:{\displaystyle \sum _{k=-\infty }^{\infty }A_{k}\mathrm {e} ^{\imath 2\pi k\Gamma }}
418:
as the integrated sheet strength or circulation between a point with arc length
2151:
36:
components of the flow velocity are discontinuous across the vortex sheet, the
2142:. Momentum thickness can be used to measure the thickness of a vortex layer.
1396:{\displaystyle \omega _{z}=-{\frac {U}{\sqrt {\pi \nu t}}}e^{-y^{2}/4\nu t}}
41:
1583:
33:
51:, vortex sheets tend to be unstable. In particular, they may exhibit
32:, such as in slippage of one layer of fluid over another. While the
659:. The equation of motion of the sheet can be rewritten in terms of
1582:
1261:{\displaystyle u(y,t)={\frac {U}{2{\sqrt {\pi \nu t}}}}\left.}
1433:. Then the equation of motion of the vortex sheet reduces to
1005:
2176:
McGraw-Hill
Dictionary of Scientific and Technical Terms
1048:. The velocity discontinuity smooths out according to
2104:
2084:
1995:
1877:
1840:
1813:
1780:
1760:
1740:
1713:
1638:
1597:
1441:
1419:
1324:
1301:
1277:
1057:
1028:
933:
904:
747:
725:
705:
685:
665:
621:
472:
444:
424:
404:
231:
194:
174:
154:
134:
99:
64:
1707:. Linear theory shows that the Fourier coefficient
652:{\displaystyle \mathrm {d} \Gamma /\mathrm {d} t=0}
2134:
2090:
2070:
1979:
1853:
1826:
1799:
1766:
1746:
1726:
1699:
1624:
1573:
1425:
1395:
1307:
1295:. The only non-zero vorticity component is in the
1283:
1260:
1040:
1011:
916:
879:
731:
711:
691:
671:
651:
605:
456:
430:
410:
388:
215:
180:
160:
140:
120:
85:
1734:grows exponentially at a rate proportional to
1834:is a critical value, the Fourier coefficient
8:
93:. The sheet is described parametrically by
1559:
1022:impling the presence of a vortex sheet at
2124:
2109:
2103:
2083:
2029:
2024:
2010:
1994:
1969:
1958:
1931:
1913:
1902:
1888:
1876:
1845:
1839:
1818:
1812:
1791:
1779:
1759:
1739:
1718:
1712:
1679:
1674:
1667:
1657:
1643:
1637:
1596:
1494:
1489:
1475:
1452:
1442:
1440:
1418:
1377:
1371:
1363:
1341:
1329:
1323:
1300:
1276:
1242:
1226:
1220:
1200:
1190:
1185:
1170:
1154:
1148:
1128:
1118:
1113:
1088:
1079:
1056:
1027:
988:
959:
940:
932:
903:
814:
808:
800:
781:
758:
748:
746:
724:
704:
684:
664:
635:
630:
622:
620:
571:
561:
558:
546:
532:
503:
498:
471:
443:
423:
403:
324:
298:
292:
284:
265:
242:
232:
230:
193:
173:
153:
133:
98:
63:
699:by a change of variable. The parameter
2168:
2135:{\displaystyle \theta _{ND}=\theta /L}
24:for a surface across which there is a
1408:Vortex sheet with periodic boundaries
148:is the arclength between coordinate
7:
1625:{\displaystyle z(\Gamma ,0)=\Gamma }
438:and the reference material point
1970:
1903:
1898:
1692:
1675:
1658:
1653:
1619:
1604:
1564:
1540:
1518:
1460:
1445:
1243:
1191:
1171:
1119:
858:
836:
821:
809:
804:
766:
751:
726:
666:
636:
627:
623:
572:
566:
562:
553:
550:
547:
533:
473:
405:
325:
293:
288:
250:
235:
14:
557:
545:
2059:
2041:
1613:
1601:
1556:
1553:
1536:
1527:
1515:
1509:
1217:
1204:
1145:
1132:
1073:
1061:
871:
854:
845:
833:
600:
588:
529:
512:
488:
476:
380:
363:
354:
342:
321:
304:
210:
198:
115:
103:
1:
216:{\displaystyle \gamma (s,t)}
168:and a reference point, and
53:Kelvin–Helmholtz instability
2209:Fluid dynamic instabilities
894:Diffusion of a vortex sheet
2225:
1800:{\displaystyle t<t_{c}}
732:{\displaystyle \Gamma }
672:{\displaystyle \Gamma }
411:{\displaystyle \Gamma }
2136:
2092:
2072:
1981:
1907:
1855:
1828:
1801:
1768:
1748:
1728:
1701:
1662:
1626:
1588:
1575:
1499:
1427:
1397:
1309:
1285:
1262:
1042:
1013:
918:
881:
813:
733:
713:
693:
673:
653:
607:
508:
458:
432:
412:
390:
297:
217:
182:
162:
142:
122:
121:{\displaystyle z(s,t)}
87:
86:{\displaystyle z=x+iy}
2137:
2093:
2073:
1982:
1884:
1856:
1854:{\displaystyle A_{k}}
1829:
1827:{\displaystyle t_{c}}
1802:
1769:
1749:
1729:
1727:{\displaystyle A_{k}}
1702:
1639:
1627:
1586:
1576:
1485:
1428:
1398:
1310:
1286:
1263:
1043:
1014:
919:
882:
796:
734:
714:
694:
674:
654:
608:
494:
459:
433:
413:
391:
280:
218:
183:
163:
143:
123:
88:
2157:Burgers vortex sheet
2102:
2082:
1993:
1875:
1838:
1811:
1778:
1758:
1738:
1711:
1636:
1595:
1439:
1417:
1322:
1315:direction, given by
1299:
1284:{\displaystyle \nu }
1275:
1055:
1026:
931:
902:
745:
723:
703:
683:
663:
619:
470:
442:
422:
402:
229:
192:
172:
152:
132:
97:
62:
2178:Retrieved July 2012
2034:
1293:kinematic viscosity
1195:
1123:
1041:{\displaystyle y=0}
917:{\displaystyle t=0}
457:{\displaystyle s=0}
44:on a vortex sheet.
2132:
2088:
2068:
2020:
1977:
1851:
1824:
1797:
1764:
1744:
1724:
1697:
1622:
1589:
1571:
1423:
1393:
1305:
1281:
1258:
1181:
1109:
1038:
1009:
1004:
914:
877:
729:
709:
689:
669:
649:
603:
454:
428:
408:
386:
213:
178:
158:
138:
118:
83:
20:is a term used in
2091:{\displaystyle U}
2037:
2018:
1952:
1921:
1767:{\displaystyle k}
1747:{\displaystyle k}
1483:
1467:
1426:{\displaystyle 1}
1357:
1356:
1308:{\displaystyle z}
1102:
1099:
991:
962:
875:
794:
773:
712:{\displaystyle s}
692:{\displaystyle t}
580:
431:{\displaystyle s}
384:
278:
257:
181:{\displaystyle t}
161:{\displaystyle z}
141:{\displaystyle s}
2216:
2188:
2185:
2179:
2173:
2141:
2139:
2138:
2133:
2128:
2117:
2116:
2097:
2095:
2094:
2089:
2077:
2075:
2074:
2069:
2035:
2033:
2028:
2019:
2011:
2006:
1986:
1984:
1983:
1978:
1973:
1968:
1964:
1963:
1962:
1957:
1953:
1951:
1943:
1932:
1922:
1914:
1906:
1901:
1860:
1858:
1857:
1852:
1850:
1849:
1833:
1831:
1830:
1825:
1823:
1822:
1806:
1804:
1803:
1798:
1796:
1795:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1733:
1731:
1730:
1725:
1723:
1722:
1706:
1704:
1703:
1698:
1696:
1695:
1678:
1672:
1671:
1661:
1656:
1631:
1629:
1628:
1623:
1580:
1578:
1577:
1572:
1570:
1546:
1498:
1493:
1484:
1476:
1468:
1466:
1458:
1457:
1456:
1443:
1432:
1430:
1429:
1424:
1402:
1400:
1399:
1394:
1392:
1391:
1381:
1376:
1375:
1358:
1346:
1342:
1334:
1333:
1314:
1312:
1311:
1306:
1290:
1288:
1287:
1282:
1267:
1265:
1264:
1259:
1254:
1250:
1246:
1241:
1240:
1230:
1225:
1224:
1194:
1189:
1174:
1169:
1168:
1158:
1153:
1152:
1122:
1117:
1103:
1101:
1100:
1089:
1080:
1047:
1045:
1044:
1039:
1018:
1016:
1015:
1010:
1008:
1007:
992:
989:
963:
960:
923:
921:
920:
915:
886:
884:
883:
878:
876:
874:
864:
828:
827:
815:
812:
807:
795:
793:
782:
774:
772:
764:
763:
762:
749:
738:
736:
735:
730:
718:
716:
715:
710:
698:
696:
695:
690:
678:
676:
675:
670:
658:
656:
655:
650:
639:
634:
626:
612:
610:
609:
604:
581:
579:
575:
569:
565:
559:
556:
544:
536:
522:
507:
502:
463:
461:
460:
455:
437:
435:
434:
429:
417:
415:
414:
409:
395:
393:
392:
387:
385:
383:
373:
337:
336:
328:
314:
299:
296:
291:
279:
277:
266:
258:
256:
248:
247:
246:
233:
222:
220:
219:
214:
187:
185:
184:
179:
167:
165:
164:
159:
147:
145:
144:
139:
127:
125:
124:
119:
92:
90:
89:
84:
49:Reynolds numbers
2224:
2223:
2219:
2218:
2217:
2215:
2214:
2213:
2194:
2193:
2192:
2191:
2186:
2182:
2174:
2170:
2165:
2148:
2105:
2100:
2099:
2080:
2079:
1996:
1991:
1990:
1944:
1933:
1927:
1926:
1912:
1908:
1873:
1872:
1841:
1836:
1835:
1814:
1809:
1808:
1787:
1776:
1775:
1756:
1755:
1736:
1735:
1714:
1709:
1708:
1673:
1663:
1634:
1633:
1593:
1592:
1563:
1539:
1459:
1448:
1444:
1437:
1436:
1415:
1414:
1410:
1367:
1359:
1325:
1320:
1319:
1297:
1296:
1273:
1272:
1216:
1196:
1144:
1124:
1108:
1104:
1084:
1053:
1052:
1024:
1023:
1003:
1002:
986:
974:
973:
957:
941:
929:
928:
900:
899:
896:
857:
829:
820:
816:
786:
765:
754:
750:
743:
742:
721:
720:
719:is replaced by
701:
700:
681:
680:
661:
660:
617:
616:
570:
560:
537:
515:
468:
467:
440:
439:
420:
419:
400:
399:
366:
338:
329:
307:
300:
270:
249:
238:
234:
227:
226:
190:
189:
170:
169:
150:
149:
130:
129:
95:
94:
60:
59:
22:fluid mechanics
12:
11:
5:
2222:
2220:
2212:
2211:
2206:
2204:Fluid dynamics
2196:
2195:
2190:
2189:
2180:
2167:
2166:
2164:
2161:
2160:
2159:
2154:
2147:
2144:
2131:
2127:
2123:
2120:
2115:
2112:
2108:
2087:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2032:
2027:
2023:
2017:
2014:
2009:
2005:
2002:
1999:
1976:
1972:
1967:
1961:
1956:
1950:
1947:
1942:
1939:
1936:
1930:
1925:
1920:
1917:
1911:
1905:
1900:
1897:
1894:
1891:
1887:
1883:
1880:
1848:
1844:
1821:
1817:
1794:
1790:
1786:
1783:
1763:
1743:
1721:
1717:
1694:
1691:
1688:
1685:
1682:
1677:
1670:
1666:
1660:
1655:
1652:
1649:
1646:
1642:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1569:
1566:
1562:
1558:
1555:
1552:
1549:
1545:
1542:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1497:
1492:
1488:
1482:
1479:
1474:
1471:
1465:
1462:
1455:
1451:
1447:
1422:
1409:
1406:
1405:
1404:
1390:
1387:
1384:
1380:
1374:
1370:
1366:
1362:
1355:
1352:
1349:
1345:
1340:
1337:
1332:
1328:
1304:
1280:
1269:
1268:
1257:
1253:
1249:
1245:
1239:
1236:
1233:
1229:
1223:
1219:
1215:
1212:
1209:
1206:
1203:
1199:
1193:
1188:
1184:
1180:
1177:
1173:
1167:
1164:
1161:
1157:
1151:
1147:
1143:
1140:
1137:
1134:
1131:
1127:
1121:
1116:
1112:
1107:
1098:
1095:
1092:
1087:
1083:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1037:
1034:
1031:
1020:
1019:
1006:
1001:
998:
995:
987:
985:
982:
979:
976:
975:
972:
969:
966:
958:
956:
953:
950:
947:
946:
944:
939:
936:
913:
910:
907:
895:
892:
873:
870:
867:
863:
860:
856:
853:
850:
847:
844:
841:
838:
835:
832:
826:
823:
819:
811:
806:
803:
799:
792:
789:
785:
780:
777:
771:
768:
761:
757:
753:
728:
708:
688:
668:
648:
645:
642:
638:
633:
629:
625:
602:
599:
596:
593:
590:
587:
584:
578:
574:
568:
564:
555:
552:
549:
543:
540:
535:
531:
528:
525:
521:
518:
514:
511:
506:
501:
497:
493:
490:
487:
484:
481:
478:
475:
464:in the sheet.
453:
450:
447:
427:
407:
382:
379:
376:
372:
369:
365:
362:
359:
356:
353:
350:
347:
344:
341:
335:
332:
327:
323:
320:
317:
313:
310:
306:
303:
295:
290:
287:
283:
276:
273:
269:
264:
261:
255:
252:
245:
241:
237:
212:
209:
206:
203:
200:
197:
177:
157:
137:
117:
114:
111:
108:
105:
102:
82:
79:
76:
73:
70:
67:
30:fluid velocity
13:
10:
9:
6:
4:
3:
2:
2221:
2210:
2207:
2205:
2202:
2201:
2199:
2184:
2181:
2177:
2172:
2169:
2162:
2158:
2155:
2153:
2150:
2149:
2145:
2143:
2129:
2125:
2121:
2118:
2113:
2110:
2106:
2085:
2065:
2062:
2056:
2053:
2050:
2047:
2044:
2038:
2030:
2025:
2021:
2015:
2012:
2007:
2003:
2000:
1997:
1987:
1974:
1965:
1959:
1954:
1948:
1945:
1940:
1937:
1934:
1928:
1923:
1918:
1915:
1909:
1895:
1892:
1889:
1885:
1881:
1878:
1870:
1866:
1862:
1846:
1842:
1819:
1815:
1792:
1788:
1784:
1781:
1761:
1741:
1719:
1715:
1689:
1686:
1683:
1680:
1668:
1664:
1650:
1647:
1644:
1640:
1616:
1610:
1607:
1598:
1585:
1581:
1567:
1560:
1550:
1547:
1543:
1533:
1530:
1524:
1521:
1512:
1506:
1503:
1500:
1495:
1490:
1486:
1480:
1477:
1472:
1469:
1463:
1453:
1449:
1434:
1420:
1407:
1388:
1385:
1382:
1378:
1372:
1368:
1364:
1360:
1353:
1350:
1347:
1343:
1338:
1335:
1330:
1326:
1318:
1317:
1316:
1302:
1294:
1278:
1255:
1251:
1247:
1237:
1234:
1231:
1227:
1221:
1213:
1210:
1207:
1201:
1197:
1186:
1182:
1178:
1175:
1165:
1162:
1159:
1155:
1149:
1141:
1138:
1135:
1129:
1125:
1114:
1110:
1105:
1096:
1093:
1090:
1085:
1081:
1076:
1070:
1067:
1064:
1058:
1051:
1050:
1049:
1035:
1032:
1029:
999:
996:
993:
983:
980:
977:
970:
967:
964:
954:
951:
948:
942:
937:
934:
927:
926:
925:
911:
908:
905:
893:
891:
887:
868:
865:
861:
851:
848:
842:
839:
830:
824:
817:
801:
797:
790:
787:
783:
778:
775:
769:
759:
755:
740:
706:
686:
646:
643:
640:
631:
613:
597:
594:
591:
585:
582:
576:
541:
538:
526:
523:
519:
516:
509:
504:
499:
495:
491:
485:
482:
479:
465:
451:
448:
445:
425:
396:
377:
374:
370:
367:
360:
357:
351:
348:
345:
339:
333:
330:
318:
315:
311:
308:
301:
285:
281:
274:
271:
267:
262:
259:
253:
243:
239:
224:
207:
204:
201:
195:
188:is time. Let
175:
155:
135:
112:
109:
106:
100:
80:
77:
74:
71:
68:
65:
56:
54:
50:
45:
43:
39:
35:
31:
27:
26:discontinuity
23:
19:
2183:
2171:
1988:
1871:
1867:
1863:
1774:and finite
1590:
1435:
1411:
1270:
1021:
897:
888:
741:
614:
466:
397:
225:
57:
46:
18:vortex sheet
17:
15:
2152:Vortex ring
739:. That is,
2198:Categories
2163:References
34:tangential
2122:θ
2107:θ
2022:∫
1924:−
1904:∞
1899:∞
1896:−
1886:∫
1879:θ
1693:Γ
1687:π
1681:ı
1659:∞
1654:∞
1651:−
1641:∑
1620:Γ
1605:Γ
1565:Γ
1541:Γ
1531:−
1519:Γ
1507:π
1504:
1487:∫
1478:ı
1473:−
1461:∂
1454:∗
1446:∂
1386:ν
1365:−
1351:ν
1348:π
1339:−
1327:ω
1279:ν
1235:ν
1202:−
1192:∞
1183:∫
1179:−
1163:ν
1139:−
1130:−
1120:∞
1111:∫
1094:ν
1091:π
990:for
978:−
961:for
859:Γ
849:−
837:Γ
822:Γ
810:∞
805:∞
802:−
798:∫
791:π
784:ı
779:−
767:∂
760:∗
752:∂
727:Γ
667:Γ
628:Γ
586:γ
567:Γ
510:γ
496:∫
474:Γ
406:Γ
358:−
302:γ
294:∞
289:∞
286:−
282:∫
275:π
268:ı
263:−
251:∂
244:∗
236:∂
196:γ
42:vorticity
2146:See also
2004:⟩
1998:⟨
1941:⟩
1935:⟨
1807:, where
1568:′
1544:′
862:′
825:′
542:′
520:′
371:′
334:′
312:′
47:At high
1291:is the
128:where
2036:
1989:where
1271:where
38:normal
2078:and
1785:<
997:<
968:>
679:and
1501:cot
28:in
2200::
924:,
55:.
16:A
2130:L
2126:/
2119:=
2114:D
2111:N
2086:U
2066:x
2063:d
2060:)
2057:t
2054:,
2051:y
2048:,
2045:x
2042:(
2039:u
2031:L
2026:0
2016:L
2013:1
2008:=
2001:u
1975:y
1971:d
1966:)
1960:2
1955:)
1949:U
1946:2
1938:u
1929:(
1919:4
1916:1
1910:(
1893:=
1890:y
1882:=
1847:k
1843:A
1820:c
1816:t
1793:c
1789:t
1782:t
1762:k
1742:k
1720:k
1716:A
1690:k
1684:2
1676:e
1669:k
1665:A
1648:=
1645:k
1617:=
1614:)
1611:0
1608:,
1602:(
1599:z
1561:d
1557:)
1554:)
1551:t
1548:,
1537:(
1534:z
1528:)
1525:t
1522:,
1516:(
1513:z
1510:(
1496:1
1491:0
1481:2
1470:=
1464:t
1450:z
1421:1
1403:.
1389:t
1383:4
1379:/
1373:2
1369:y
1361:e
1354:t
1344:U
1336:=
1331:z
1303:z
1256:.
1252:]
1248:s
1244:d
1238:t
1232:4
1228:/
1222:2
1218:)
1214:y
1211:+
1208:s
1205:(
1198:e
1187:0
1176:s
1172:d
1166:t
1160:4
1156:/
1150:2
1146:)
1142:y
1136:s
1133:(
1126:e
1115:0
1106:[
1097:t
1086:2
1082:U
1077:=
1074:)
1071:t
1068:,
1065:y
1062:(
1059:u
1036:0
1033:=
1030:y
1000:0
994:y
984:,
981:U
971:0
965:y
955:,
952:U
949:+
943:{
938:=
935:u
912:0
909:=
906:t
872:)
869:t
866:,
855:(
852:z
846:)
843:t
840:,
834:(
831:z
818:d
788:2
776:=
770:t
756:z
707:s
687:t
647:0
644:=
641:t
637:d
632:/
624:d
601:)
598:t
595:,
592:s
589:(
583:=
577:s
573:d
563:d
554:d
551:n
548:a
539:s
534:d
530:)
527:t
524:,
517:s
513:(
505:s
500:0
492:=
489:)
486:t
483:,
480:s
477:(
452:0
449:=
446:s
426:s
381:)
378:t
375:,
368:s
364:(
361:z
355:)
352:t
349:,
346:s
343:(
340:z
331:s
326:d
322:)
319:t
316:,
309:s
305:(
272:2
260:=
254:t
240:z
211:)
208:t
205:,
202:s
199:(
176:t
156:z
136:s
116:)
113:t
110:,
107:s
104:(
101:z
81:y
78:i
75:+
72:x
69:=
66:z
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