Instantaneous velocity field of a round jet

Transcription

1 Fee shea flows

2

3 Instantaneos velocty feld of a ond et 3

4 Aveage velocty feld of a ond et 4

5 Vtal ogn nozzle coe Developng egon elf smla egon 5

6 elf smlaty caled vaables: ~ Q ξ ( ξ, ) y δ ( ) Q Q (, y) ( ) ~ If Q( ξ, ) Qˆ ( ξ ) then Q s selfmla 6

7 Jet cente lne velocty U ( ) ( ),, U / 7 Jet half wdth ( ) (,,)

8 (, ) U ( f ( ξ ) ξ / ( ) ) 8

9 U ( ) Cente lne velocty: elf smlaty of a ond et U B ( ) d J / d ( ) peadng ate: / d const. Jet Reynolds nmbe Re ( ) ( ) U / ν nce U and / Re const. 9

10 Aveage cente lne aal velocty and ms of aal velocty flctatons caled wth the cente lne velocty sqaed the flctatons wll go towads a constant vale of.5

11 Aveage cente lne aal velocty and ms of aal velocty flctatons caled wth the cente lne velocty sqaed the flctatons wll go towads a constant vale of.5

12 Velocty flctatons

13 Velocty flctatons Tblent netc enegy 3

14 Velocty flctatons Components of the Reynolds stess tenso vvv ww vv vv vv www ww vv ww ww vvv www vvv 4

15 The Lmley tangle b δ 3 Thee pncpal nvaants: I b II b III b b [ ( ) ] b b ( b ) b b b 6η 6ξ 3 II 3III b b b b 3 Otsde the tangle: non-ealzable tblent stesses 5

16 Velocty flctatons The Lmley tangle 6

17 Momentm eqatons fo a et y v y v y p y v υ υ ρ y v v v y v v y p y v v v υ υ ρ 7

18 Lateal momentm eqaton p ρ y v v y p v v C ρ Bonday condton: lm p(, y) p ( ) y lm v v y p p v v ρ ρ p ρ dp ρ d v v 8

19 ( ) y v v v y d dp y v υ ρ Aal momentm eqaton y v y y v υ Eqatons fo a ond et: v y v v υ 9

20 Flow ates of mass momentm and netc enegy of a ond et Mass flow ate Momentm flow ate Knetc enegy flow ate m M E ( ) π ρd ( ) π ρ d ( ) π 3 ρ d Intodce self smlaty (, ) U ( f ( ξ ) ξ / ( ) )

21 Flow ates of mass momentm and netc enegy of a ond et Momentm flow ate M ( ) π ρ d πρ( ) / U ξf ( ξ ) dξ Mass flow ate m ( ) π ρd πρ ( ) / / U ξf ( ξ ) dξ Knetc enegy flow ate E 3 πρ 3 dξ ( ) 3 π ρ d ( ) / U ξf ( ξ ) /

22 Flow ates of mass momentm and netc enegy of a ond et Momentm flow ate M ( ) ( U ) M ( ) const. / m Mass flow ate ( ) ( ) ( ) U / / m Knetc enegy 3 / / flow ate E ( ) ( U ) E ( )

23 Knetc enegy of a ond et, t Knetc enegy: ( ) E Decomposton of the mean ~ E, t netc enegy ( ) E Knetc eney of the mean flow (, t ) Tblent netc enegy (, t ) 3

24 Knetc enegy of a ond et Eqaton fo netc enegy p E t E υ υ ρ Rate of stan Aveagng ths eqaton one gets: 4

25 Knetc enegy of a ond et Aveagng ths eqaton one gets: ε ε υ ρ p E E t E ~ ~ ε υ ε υs s Mean ate of stan s Flctatng ate of stan Dsspaton by mean flow Dsspaton by flctatons 5

26 Knetc enegy of a ond et Mean flow netc enegy eqaton ε υ ρ P p E t E P Tblent netc enegy eqaton ε υ ρ P s p t podcton dsspaton Tanspot Convecton by mean flow 6

27 Knetc enegy of a ond et Psedo-dsspaton and the tblent netc enegy eqaton. ε ~ υ ε υs s dsspaton Psedo-dsspaton ~ ν ε ε Tblent netc enegy eqaton sng psedo dsspaton ε υ ρ ~ P p t podcton dsspaton Vscos dffson Tblent tanspot Usally small 7

28 Knetc enegy of a ond et Podcton of tblent netc enegy: Ω otaton ate of stan ate of P P Reynolds stess tenso s symmetc: Ths leads to 8

29 Knetc enegy of a ond et De to the smmaton le dmmy ndces may be enamed: P Hence: 9

30 Knetc enegy of a ond et Now splt the Reynolds stess tenso nto the ansotopc and sotopc pats: P δ 3 a ansotopy tenso δ 3 Consde the sotopc pat: Use that a contacton can be wtten as: δ 3 δ Recognsng the ncompessble mass consevaton and sng the defnton of tblent netc enegy: 3 l l δ 3 l l 3 l l Podcton of tblent netc enegy depends only on the ansotopy and the ate of stan P a

31 Knetc enegy of a ond et Modellng the podcton of The eddy vscosty hypothess s cental fo elatng the Reynolds stess tenso to nown qanttes n most RAN models: δ υ T 3 a P a P 3 a δ ν T a Modelled podcton tem P mod ν ν ( T T ) 3

32 Knetc enegy of a ond et Modelled podcton tem 3 P a Modellng the podcton of ) ( mod y w z v w z v y z w y v P T ν T ν Real podcton tem y w z v a w z a v y a z w a y v a a P T ν What wll ths nd of modellng lead to?

33 Modellng a tblent ond et Fo a tblent ond et the Reynolds stesses can be appomated by: ρ v ρυ T Eddy vscosty υt l l ρ v ρl ( ) υ T η (, ) ( ) U ( ) υ ( η) Obseve that ˆ υ T η and l / / ae faly constant ove most of the et coss secton / ˆT Pandtl mng length model Note that ths type of modellng pedcts zeo Reynolds stesses on the cente lne! 33

34 The modelled eqatons fo a ond et: v v T υ Also note the tblent Reynolds nmbe: ( ) ( ) 35 ˆ Re / T T T U ν ν 34

35 Nomalsed eddy vscosty Mng length 35

36 36

37 Othe self-smla flows Rond et Instantaneos Aveage Mass, momentm, enegy elf-smlaty Plane et Wae Unfom shea

38 The ond et agan U U Re elf smlaty of a ond et: ( ) B ( ) d J / d d U ( ) / ( ) ( ) ν / const. const Flow ate of mass, momentm, enegy: m U m M E ( ) ( ) ( ) / / ( ) ( U ) M ( ) const. / / U 3 E / ( ) ( ) ( ) Y X

39 Plane et o slot et Why not ond? Applcatons W >> H Jet centelne velocty ( ) ( ) U,, H W U U / Jet half-wdth peadng ate U ( ) (,,) y / dy d ( ) /. nozzle Vtal ogn coe Jet Re Re ( ) y ( ) U / ν Y / Y Developng egon elf smla egon X

40 Mass, momentm, enegy flow ate Mass: m Momentm: ( ) dy ρu y f ( ξ ) dξ m ( ) ρ / Enegy: ( ) ρ dy ρu y f ( ξ ) dξ M ( ) const. M / E 3 / 3 3 ( ) ρ dy ρu y f ( ξ ) dξ E ( )

41 Rond vs. Plane et Rond Plane Centelne velocty Jet half wdth peadng ate Jet Re Tblent Re Mass flow ate Momentm flow ate Enegy flow ate / U ( ) ( ) ( / ) U ( ).94. Re ( ) const y / Re ( ) ReT ( ) 35 ReT ( ) 3 m ( ) / m ( ) M ( ) const. E ( ) / M ( ) const. E ( ) / lowe decease of centelne velocty fo plane et!

42 Plane mng laye (these mng layes ae not plane)

43 Plane mng laye U H Hgh- (U H ) and Low-speed (U L ) steams U H U L U C δ Y.9 Y. E.g. edge of a et, atmosphee, etc. velocty scales Chaactestc convecton velocty, U C Chaactestc velocty dffeence, U Wdth of the mng laye δ ( ) y.9( ) y. ( ) U L U

44 Plane mng laye U H U L U H U C Y.9 It s self-smla (see. Fg. 5.) Usally Not symmetc abot y peadng ate constant, bt epement specfc U C dδ.6... Y.5 U d U const Y. δ U δ Re ( ) ν U L U E ( ) Compae to the ond/plane et! Podcton > dsspaton

45 Waes

46 Plane wae U C velocty scales Chaactestc convectve velocty, U C.5U Chaactestc velocty dffeence, U U /U C const U U Y / Half-wdth, y / Not eactly self-smla, asymptotcally selfsmla fo U /U C (fa wae) elf-smla velocty defect: f ( ξ ) ( U U ( )) / U ( ) C

47 Plane wae U C Momentm defct flow-ate U.5U U Y / ρu C U M ( ) ( ) y / ( ) M ρ ( U C U U ) dy C ( ) const. f ( ξ ) f ( ξ ) dξ U U dy/ const. d C U y ( ) / / /

48 Plane vs. asymmetc wae elf.sm. Plane Only fo U /U C Asymm. Only fo U /U C Const. Const. U s /, y / Re / / 3 U ( ) U / /3 y/ / /3 Re ( ) Const. ( ) Faste ecovey fo asymmetc Relamnazaton peadng paamete and tblence level depend on geomety

49 p P t ν ρ ε const. y Homogeneos tblence: tatstcs of velocty and pesse flctatons ae ndependent ove a shft n space The tblent netc enegy eqaton wll then be edced to: ε P t Homogeneos shea flow

50 Homogeneos shea flow Tblent tme scale: P ε t τ ε P. ε τ t Obsevatons show: 7 P ε τ const. ε Then: t P τ ε ( t) () e Knetc eneg gows eponentally n tme.

51 Gd tblence (sotopc tblence) P Why? Isotopc tblence: tatstcs of velocty flctatons ae ndependent nde a coodnate system otaton Dffeent gd desgns?

52 Gd tblence (sotopc tblence) The tblent netc enegy eqaton ε t n t t t ) ( ( ) n t t t n t n t t t ) ε ( ε n M A U

53 Gd tblence (sotopc tblence)