151:(1911—2011) showed that thermal diffusion stabilized the curved flame front and proposed a relation between the critical wavelength for stability of the flame front, called the Markstein length, and the thermal thickness of the flame. Phenomenological Markstein numbers with respect to the combustion products are obtained by means of the comparison between the measurements of the flame radii as a function of time and the results of the analytical integration of the linear relation between the flame speed and either flame stretch rate or flame curvature. The burning velocity is obtained at zero stretch, and the effect of the flame stretch acting upon it is expressed by a Markstein length. Because both flame curvature and aerodynamic strain contribute to the flame stretch rate, there is a Markstein number associated with each of these components.
470:
1040:
314:
303:
1575:
Clavin, Paul, and F. A. Williams. "Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity." Journal of fluid mechanics 116 (1982):
855:
1164:
1344:
91:
668:
465:{\displaystyle {\mathcal {J}}=\int _{1}^{r}{\frac {\lambda (\theta )}{\theta }}d\theta ,\quad {\mathcal {I}}=\int _{1}^{r}{\frac {\lambda (\theta )}{\theta }}\ln {\frac {r-1}{\theta -1}}\,d\theta .}
181:
1463:
Chrystie R.S.M.; Burns I.S.; Hult J.; Kaminski C.F. (2008). "On the improvement of two-dimensional curvature computation and its application to turbulent premixed flame correlations".
821:
753:
582:
1273:
1242:
1199:
788:
1360:
1594:
Clavin, P., & Garcia, P. (1983). The influence of the temperature dependence of diffusivities on the dynamics. Journal de Mécanique Théorique et
Appliquée, 2(2), 245-263.
719:
118:
1606:
Clavin, Paul, and Geoff Searby. Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and
Explosion of Stars. Cambridge University Press, 2016.
145:
1644:
1069:
534:
504:
847:
1035:{\displaystyle {\mathcal {J}}={\frac {1}{n}}(r^{n}-1),\quad {\mathcal {I}}={\frac {1}{n}}(r^{n}-1)\ln(r-1)-\int _{1}^{r}\theta ^{n-1}\ln(\theta -1)d\theta .}
1659:
1506:
Chakraborty N, Cant RS (2005). "Influence of Lewis number on curvature effects in turbulent premixed flame propagation in the thin reaction zones regime".
1585:
Clavin, Paul. "Dynamic behavior of premixed flame fronts in laminar and turbulent flows." Progress in Energy and
Combustion Science 11.1 (1985): 1–59
1383:
Markstein, G. H. (1988). Experimental and theoretical studies of flame-front stability. In
Dynamics of curved fronts (pp. 413-423). Academic Press.
147:
is the characteristic laminar flame thickness. The larger the
Markstein length, the greater the effect of curvature on localised burning velocity.
1077:
1428:
Karpov V. P.; Lipanikov A. N.; Wolanski P. (1997). "Finding the markstein number using the measurements of expanding spherical laminar flames".
1281:
1624:
Clavin, P., & Graña-Otero, J. C. (2011). Curved and stretched flames: the two
Markstein numbers. Journal of fluid mechanics, 686, 187-217.
1615:
Bechtold, J. K., & Matalon, M. (2001). The dependence of the
Markstein length on stoichiometry. Combustion and flame, 127(1-2), 1906-1913.
50:
1541:
Haq MZ, Sheppard CG, Woolley R, Greenhalgh DA, Lockett RD (2002). "Wrinkling and curvature of laminar and turbulent premixed flames".
597:
1654:
298:{\displaystyle {\mathcal {M}}={\frac {r}{r-1}}{\mathcal {J}}+{\frac {\beta (Le_{\mathrm {eff} }-1)}{2(r-1)}}{\mathcal {I}},}
168:
1365:
1649:
793:
509:
726:
549:
1247:
1216:
1172:
764:
32:
who first proposed the notion in 1951) characterizes the effect of local heat release of a propagating
1550:
1515:
1472:
1437:
1402:
683:
673:
539:
99:
1639:
148:
29:
1488:
164:
123:
1048:
171:. The formula was extended to include temperature dependences on the thermal conductivities by
1558:
1523:
1480:
1445:
1410:
519:
483:
826:
1484:
1554:
1519:
1476:
1441:
1406:
1414:
1562:
1449:
1633:
41:
1492:
1275:
are not same in real flames. In that case, one defines a second
Markstein number as
587:
1202:
172:
160:
37:
36:
on variations in the surface topology along the flame and the associated local
1355:
590:
of the deficient reactant (either fuel or oxidizer or a combination of both);
175:
and Pedro Luis Garcia Ybarra in 1983. The Clavin–Williams formula is given by
17:
1159:{\displaystyle {\mathcal {J}}=\ln r,\quad {\mathcal {I}}=-\mathrm {Li_{2}} }
159:
The
Markstein number with respect to the unburnt gas mixture was derived by
21:
1339:{\displaystyle {\mathcal {M}}_{2}={\mathcal {M}}_{c}-{\mathcal {M}}_{s}.}
1393:
Oran E. S. (2015). "A tribute to Dr. George H. Markstein (1911–2011)".
1527:
723:
is the ratio of temperature to its unburnt value, defined such that
86:{\displaystyle {\mathcal {M}}={\frac {\mathcal {L}}{\delta _{L}}}}
33:
663:{\displaystyle \lambda (\theta )=\rho D_{T}/\rho _{u}D_{T,u}}
1322:
1305:
1288:
1254:
1223:
1106:
1083:
907:
861:
376:
320:
287:
215:
187:
105:
67:
56:
1213:
In general, Markstein number for the curvature effects
1284:
1250:
1219:
1175:
1080:
1051:
858:
829:
796:
767:
729:
686:
600:
552:
522:
486:
317:
184:
126:
102:
53:
1338:
1267:
1236:
1193:
1158:
1063:
1045:In the constant transport coefficient assumption,
1034:
841:
815:
782:
747:
713:
662:
576:
528:
498:
464:
297:
139:
112:
85:
8:
1602:
1600:
1327:
1321:
1320:
1310:
1304:
1303:
1293:
1287:
1286:
1283:
1259:
1253:
1252:
1249:
1228:
1222:
1221:
1218:
1184:
1176:
1174:
1125:
1117:
1105:
1104:
1082:
1081:
1079:
1050:
990:
980:
975:
932:
915:
906:
905:
886:
869:
860:
859:
857:
828:
807:
795:
766:
728:
705:
696:
685:
676:product to its value in the unburnt gas;
648:
638:
629:
623:
599:
561:
560:
551:
521:
485:
452:
426:
399:
393:
388:
375:
374:
343:
337:
332:
319:
318:
316:
286:
285:
240:
239:
223:
214:
213:
195:
186:
185:
183:
131:
125:
104:
103:
101:
75:
66:
64:
55:
54:
52:
1645:Dimensionless numbers of fluid mechanics
1376:
1361:Matalon–Matkowsky–Clavin–Joulin theory
7:
816:{\displaystyle \lambda =\theta ^{n}}
790:, in most cases, is simply given by
748:{\displaystyle 1\leq \theta \leq r}
577:{\displaystyle Le_{\mathrm {eff} }}
1660:Dimensionless numbers of chemistry
1465:Measurement Science and Technology
1415:10.1016/j.combustflame.2014.07.005
1268:{\displaystyle {\mathcal {M}}_{s}}
1237:{\displaystyle {\mathcal {M}}_{c}}
1181:
1177:
1122:
1118:
568:
565:
562:
247:
244:
241:
14:
1194:{\displaystyle \mathrm {Li_{2}} }
783:{\displaystyle \lambda (\theta )}
477:
44:Markstein number is defined as:
1103:
904:
714:{\displaystyle \theta =T/T_{u}}
373:
1485:10.1088/0957-0233/19/12/125503
1153:
1150:
1138:
1132:
1020:
1008:
965:
953:
944:
925:
898:
879:
777:
771:
610:
604:
411:
405:
355:
349:
279:
267:
259:
229:
113:{\displaystyle {\mathcal {L}}}
1:
1563:10.1016/S0010-2180(02)00383-8
1450:10.1016/S0010-2180(96)00166-6
169:activation energy asymptotics
120:is the Markstein length, and
512:defined with density ratio;
140:{\displaystyle \delta _{L}}
1676:
1064:{\displaystyle \lambda =1}
1071:, in which case, we have
849:, in which case, we have
1514:(10): 105105–105105–20.
672:is the ratio of density-
1209:Second Markstein number
155:Clavin–Williams formula
1366:Clavin–Garcia equation
1340:
1269:
1238:
1195:
1160:
1065:
1036:
843:
817:
784:
749:
715:
664:
578:
530:
529:{\displaystyle \beta }
500:
499:{\displaystyle r>1}
466:
299:
141:
114:
87:
1655:Dimensionless numbers
1341:
1270:
1239:
1196:
1161:
1066:
1037:
844:
842:{\displaystyle n=0.7}
818:
785:
750:
716:
665:
579:
531:
501:
467:
300:
142:
115:
88:
1543:Combustion and Flame
1430:Combustion and Flame
1395:Combustion and Flame
1282:
1248:
1217:
1203:dilogarithm function
1173:
1078:
1049:
856:
827:
794:
765:
727:
684:
674:thermal conductivity
598:
550:
520:
484:
315:
182:
124:
100:
51:
1555:2002CoFl..131....1H
1520:2005PhFl...17j5105C
1477:2008MeScT..19l5503C
1442:1997CoFl..109..436K
1407:2015CoFl..162....1O
1244:and strain effects
985:
510:gas expansion ratio
398:
342:
149:George H. Markstein
30:George H. Markstein
1336:
1265:
1234:
1191:
1156:
1061:
1032:
971:
839:
813:
780:
745:
711:
660:
574:
526:
496:
462:
384:
328:
295:
165:Forman A. Williams
137:
110:
83:
1528:10.1063/1.2084231
1508:Physics of Fluids
923:
877:
759:
758:
586:is the effective
540:Zel'dovich number
450:
418:
362:
283:
211:
81:
1667:
1625:
1622:
1616:
1613:
1607:
1604:
1595:
1592:
1586:
1583:
1577:
1573:
1567:
1566:
1538:
1532:
1531:
1503:
1497:
1496:
1460:
1454:
1453:
1425:
1419:
1418:
1390:
1384:
1381:
1345:
1343:
1342:
1337:
1332:
1331:
1326:
1325:
1315:
1314:
1309:
1308:
1298:
1297:
1292:
1291:
1274:
1272:
1271:
1266:
1264:
1263:
1258:
1257:
1243:
1241:
1240:
1235:
1233:
1232:
1227:
1226:
1200:
1198:
1197:
1192:
1190:
1189:
1188:
1165:
1163:
1162:
1157:
1131:
1130:
1129:
1110:
1109:
1087:
1086:
1070:
1068:
1067:
1062:
1041:
1039:
1038:
1033:
1001:
1000:
984:
979:
937:
936:
924:
916:
911:
910:
891:
890:
878:
870:
865:
864:
848:
846:
845:
840:
822:
820:
819:
814:
812:
811:
789:
787:
786:
781:
754:
752:
751:
746:
720:
718:
717:
712:
710:
709:
700:
669:
667:
666:
661:
659:
658:
643:
642:
633:
628:
627:
583:
581:
580:
575:
573:
572:
571:
535:
533:
532:
527:
505:
503:
502:
497:
478:
471:
469:
468:
463:
451:
449:
438:
427:
419:
414:
400:
397:
392:
380:
379:
363:
358:
344:
341:
336:
324:
323:
304:
302:
301:
296:
291:
290:
284:
282:
262:
252:
251:
250:
224:
219:
218:
212:
210:
196:
191:
190:
146:
144:
143:
138:
136:
135:
119:
117:
116:
111:
109:
108:
92:
90:
89:
84:
82:
80:
79:
70:
65:
60:
59:
26:Markstein number
20:engineering and
1675:
1674:
1670:
1669:
1668:
1666:
1665:
1664:
1630:
1629:
1628:
1623:
1619:
1614:
1610:
1605:
1598:
1593:
1589:
1584:
1580:
1574:
1570:
1540:
1539:
1535:
1505:
1504:
1500:
1462:
1461:
1457:
1427:
1426:
1422:
1392:
1391:
1387:
1382:
1378:
1374:
1352:
1319:
1302:
1285:
1280:
1279:
1251:
1246:
1245:
1220:
1215:
1214:
1211:
1180:
1171:
1170:
1121:
1076:
1075:
1047:
1046:
986:
928:
882:
854:
853:
825:
824:
803:
792:
791:
763:
762:
725:
724:
701:
682:
681:
644:
634:
619:
596:
595:
556:
548:
547:
518:
517:
482:
481:
439:
428:
401:
345:
313:
312:
263:
235:
225:
200:
180:
179:
167:in 1982, using
157:
127:
122:
121:
98:
97:
71:
49:
48:
40:curvature. The
12:
11:
5:
1673:
1671:
1663:
1662:
1657:
1652:
1650:Fluid dynamics
1647:
1642:
1632:
1631:
1627:
1626:
1617:
1608:
1596:
1587:
1578:
1568:
1533:
1498:
1471:(12): 125503.
1455:
1420:
1385:
1375:
1373:
1370:
1369:
1368:
1363:
1358:
1351:
1348:
1347:
1346:
1335:
1330:
1324:
1318:
1313:
1307:
1301:
1296:
1290:
1262:
1256:
1231:
1225:
1210:
1207:
1187:
1183:
1179:
1167:
1166:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1128:
1124:
1120:
1116:
1113:
1108:
1102:
1099:
1096:
1093:
1090:
1085:
1060:
1057:
1054:
1043:
1042:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
999:
996:
993:
989:
983:
978:
974:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
935:
931:
927:
922:
919:
914:
909:
903:
900:
897:
894:
889:
885:
881:
876:
873:
868:
863:
838:
835:
832:
810:
806:
802:
799:
779:
776:
773:
770:
757:
756:
744:
741:
738:
735:
732:
721:
708:
704:
699:
695:
692:
689:
678:
677:
670:
657:
654:
651:
647:
641:
637:
632:
626:
622:
618:
615:
612:
609:
606:
603:
592:
591:
584:
570:
567:
564:
559:
555:
544:
543:
536:
525:
514:
513:
506:
495:
492:
489:
473:
472:
461:
458:
455:
448:
445:
442:
437:
434:
431:
425:
422:
417:
413:
410:
407:
404:
396:
391:
387:
383:
378:
372:
369:
366:
361:
357:
354:
351:
348:
340:
335:
331:
327:
322:
306:
305:
294:
289:
281:
278:
275:
272:
269:
266:
261:
258:
255:
249:
246:
243:
238:
234:
231:
228:
222:
217:
209:
206:
203:
199:
194:
189:
156:
153:
134:
130:
107:
94:
93:
78:
74:
69:
63:
58:
13:
10:
9:
6:
4:
3:
2:
1672:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1637:
1635:
1621:
1618:
1612:
1609:
1603:
1601:
1597:
1591:
1588:
1582:
1579:
1572:
1569:
1564:
1560:
1556:
1552:
1549:(1–2): 1–15.
1548:
1544:
1537:
1534:
1529:
1525:
1521:
1517:
1513:
1509:
1502:
1499:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1459:
1456:
1451:
1447:
1443:
1439:
1435:
1431:
1424:
1421:
1416:
1412:
1408:
1404:
1400:
1396:
1389:
1386:
1380:
1377:
1371:
1367:
1364:
1362:
1359:
1357:
1354:
1353:
1349:
1333:
1328:
1316:
1311:
1299:
1294:
1278:
1277:
1276:
1260:
1229:
1208:
1206:
1204:
1185:
1147:
1144:
1141:
1135:
1126:
1114:
1111:
1100:
1097:
1094:
1091:
1088:
1074:
1073:
1072:
1058:
1055:
1052:
1029:
1026:
1023:
1017:
1014:
1011:
1005:
1002:
997:
994:
991:
987:
981:
976:
972:
968:
962:
959:
956:
950:
947:
941:
938:
933:
929:
920:
917:
912:
901:
895:
892:
887:
883:
874:
871:
866:
852:
851:
850:
836:
833:
830:
808:
804:
800:
797:
774:
768:
761:The function
742:
739:
736:
733:
730:
722:
706:
702:
697:
693:
690:
687:
680:
679:
675:
671:
655:
652:
649:
645:
639:
635:
630:
624:
620:
616:
613:
607:
601:
594:
593:
589:
585:
557:
553:
546:
545:
541:
537:
523:
516:
515:
511:
507:
493:
490:
487:
480:
479:
476:
459:
456:
453:
446:
443:
440:
435:
432:
429:
423:
420:
415:
408:
402:
394:
389:
385:
381:
370:
367:
364:
359:
352:
346:
338:
333:
329:
325:
311:
310:
309:
292:
276:
273:
270:
264:
256:
253:
236:
232:
226:
220:
207:
204:
201:
197:
192:
178:
177:
176:
174:
170:
166:
162:
154:
152:
150:
132:
128:
76:
72:
61:
47:
46:
45:
43:
42:dimensionless
39:
35:
31:
28:(named after
27:
24:studies, the
23:
19:
1620:
1611:
1590:
1581:
1571:
1546:
1542:
1536:
1511:
1507:
1501:
1468:
1464:
1458:
1433:
1429:
1423:
1398:
1394:
1388:
1379:
1212:
1168:
1044:
760:
588:Lewis number
474:
307:
158:
95:
25:
15:
173:Paul Clavin
161:Paul Clavin
38:flame front
1640:Combustion
1634:Categories
1436:(3): 436.
1401:(1): 1–2.
1372:References
1356:G equation
18:combustion
1317:−
1145:−
1136:−
1115:−
1095:
1053:λ
1027:θ
1015:−
1012:θ
1006:
995:−
988:θ
973:∫
969:−
960:−
951:
939:−
893:−
805:θ
798:λ
775:θ
769:λ
740:≤
737:θ
734:≤
688:θ
636:ρ
617:ρ
608:θ
602:λ
524:β
457:θ
444:−
441:θ
433:−
424:
416:θ
409:θ
403:λ
386:∫
368:θ
360:θ
353:θ
347:λ
330:∫
274:−
254:−
227:β
205:−
129:δ
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