1327:
998:
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151:
1229:
993:{\displaystyle \left\lbrace {\begin{array}{cl}U(r)&=a+\displaystyle {{\frac {1}{1+4t/Re}}e^{-r^{2}/(1+4t/Re)}},\\V(r)&=0,\\W(r)&=q\displaystyle {\frac {1-e^{-r^{2}/(1+4t/Re)}}{r}},\end{array}}\right.}
583:
44:
approximation. The physical reasoning behind this approximation is the assumption that the axial gradient of the flow field of interest is of much smaller magnitude than the radial gradient.
1287:
416:{\displaystyle {\begin{array}{cl}U(r)&=U_{\infty }+{\frac {W_{0}}{(R/R_{0})^{2}}}e^{-(r/R)^{2}},\\V(r)&=0,\\W(r)&=qW_{0}{\frac {1-e^{-(r/R)^{2}}}{(r/R_{0})}}.\end{array}}}
1049:
142:
721:
454:
1252:
610:
512:
483:
630:
1105:
1121:
1072:
677:
652:
104:
84:
64:
1368:
1310:
1387:
723:. Using the same symbols for the dimensionless variables, the Batchelor vortex can be expressed in terms of the dimensionless variables as
1361:
1402:
519:
37:
1354:
659:
Note that the radial component of the velocity is zero and that the axial and azimuthal components depend only on
1397:
1314:
1270:
Batchelor, G. K. (1964). Axial flow in trailing line vortices. Journal of Fluid
Mechanics, 20(4), 645-658.
1112:
1006:
654:, the swirl strength, given as a ratio between the maximum tangential velocity and the core velocity.
1326:
109:
684:
432:
1280:
28:
in a 1964 article, have been found useful in analyses of airplane vortex wake hazard problems.
1392:
1237:
25:
1224:{\displaystyle W_{\Theta }(r)={\frac {\Gamma }{2\pi r}}\left(1-e^{-r^{2}/R_{c}^{2}}\right)}
588:
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461:
1075:
615:
1084:
1054:
1338:
1334:
662:
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69:
49:
41:
17:
1381:
739:
681:
We now write the system above in dimensionless form by scaling time by a factor
156:
46:
The axial, radial and azimuthal velocity components of the vortex are denoted
1108:
1107:
and considers an infinitely large swirl number then the
Batchelor
106:
respectively and can be represented in cylindrical coordinates
1313:(Authored by Xueri Mao and Spencer Sherwin and published by
987:
1342:
36:
The
Batchelor vortex is an approximate solution to the
485:, the velocity scale (used for nondimensionalization),
1281:"Theoretical and numerical analysis of wake vortices"
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1124:
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1057:
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909:
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618:
591:
585:, a measure of the core size, with initial core size
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493:
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435:
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112:
92:
72:
52:
514:, the length scale (used for nondimensionalization),
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578:{\displaystyle R=R(t)={\sqrt {R_{0}^{2}+4\nu t}}}
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8:
1051:denotes the free stream axial velocity and
1369:
1355:
1311:Continuous spectra of the Batchelor vortex
426:The parameters in the above equations are
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1194:
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1123:
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91:
71:
51:
1263:
7:
1323:
1321:
1044:{\displaystyle a=U_{\infty }/W_{0}}
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456:, the free-stream axial velocity,
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137:{\displaystyle (x,r,\theta )}
1341:. You can help Knowledge by
1115:for the azimuthal velocity:
1388:Equations of fluid dynamics
716:{\displaystyle R_{0}/W_{0}}
449:{\displaystyle U_{\infty }}
1419:
1320:
1315:Imperial College London
1247:{\displaystyle \Gamma }
632:representing viscosity,
38:Navier–Stokes equations
1337:–related article is a
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605:{\displaystyle R_{0}}
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478:{\displaystyle W_{0}}
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24:, first described by
1403:Fluid dynamics stubs
1254:is the circulation.
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625:{\displaystyle \nu }
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1100:{\displaystyle a=0}
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22:Batchelor vortices
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1113:Lamb–Oseen vortex
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672:{\displaystyle r}
647:{\displaystyle q}
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99:{\displaystyle W}
79:{\displaystyle V}
59:{\displaystyle U}
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1343:expanding it
1332:
1291:. Retrieved
1275:
1266:
1233:
1081:If one lets
1080:
1002:
657:
425:
35:
21:
15:
144:as follows:
1382:Categories
1293:2015-07-29
1258:References
1242:Γ
1182:−
1174:−
1157:π
1150:Γ
1131:Θ
1022:∞
925:−
917:−
805:−
620:ν
568:ν
442:∞
349:−
341:−
242:−
182:∞
129:θ
32:The model
1393:Vortices
1074:is the
1234:where
1109:vortex
1003:where
1333:This
1288:ESAIM
1284:(PDF)
1339:stub
612:and
86:and
679:.
16:In
1384::
1286:.
1078:.
20:,
1370:e
1363:t
1356:v
1345:.
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1296:.
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1205:c
1201:R
1196:/
1190:2
1186:r
1178:e
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1167:(
1160:r
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1142:)
1139:r
1136:(
1127:W
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1092:=
1089:a
1062:e
1059:R
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1018:U
1014:=
1011:a
980:,
975:r
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966:e
963:R
959:/
955:t
952:4
949:+
946:1
943:(
939:/
933:2
929:r
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907:q
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896:r
893:(
890:W
883:,
880:0
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872:)
869:r
866:(
863:V
855:,
849:)
846:e
843:R
839:/
835:t
832:4
829:+
826:1
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819:/
813:2
809:r
801:e
794:e
791:R
787:/
783:t
780:4
777:+
774:1
770:1
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749:r
746:(
743:U
736:{
709:0
705:W
700:/
694:0
690:R
667:r
642:q
598:0
594:R
571:t
565:4
562:+
557:2
552:0
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539:)
536:t
533:(
530:R
527:=
524:R
500:0
496:R
471:0
467:W
438:U
407:.
401:)
396:0
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387:/
383:r
380:(
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355:r
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330:0
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319:=
314:)
311:r
308:(
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298:,
295:0
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284:r
281:(
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271:,
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256:R
252:/
248:r
245:(
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126:,
123:r
120:,
117:x
114:(
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74:V
66:,
54:U
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
