Cartesian tensors. Order (rank) Scalar. Vector. 3x3 matrix

Transcription

1 Caresan ensors Order (rank) a b c d k Scalar ecor 33 mar

2 Caresan ensors Kronecker dela δ = 1 f = 0 f Le- Ca epslon ε k = 1 f,, k are cclc 1 f,, k are ancclc 0 oherse Smmaon conenon (o eqal ncces mples smmaon) a b = d = 3 =3 3 =1 a b = c d = c

3 Caresan ensors Tensor operaons: Addon: s = a + b s = a + b s 3 = a 3 + b 3 s 1 = a 1 + b 1 Tensor prodcs: s = a b s 11 s 1 s 13 s 1 s s 3 s 31 s 3 s 33 = a 1 b 1 a 1 b a 1 b 3 a b 1 a b a b 3 a 3 b 1 a 3 b a 3 b 3 Inner prodcs: =c d lm = b c lm Conracon: a δ = a (race)

4 Caresan ensors Tensor operaons: Graden: p p = p, =, Dergence: Roaon: ε k k = ω Laplacan:

5 Goernng eqaons Compressble: Conseraon of: mass momenm Energ In addon: Sae eqaons Unknons: eloc, pressre, dens, energ (enhalp, emperare) Incompressble: Conseraon of: mass momenm nknons: eloc, pressre

6 Mass conseraon Use Renolds ranspor heorem assmng 1D-flo hrogh a nfnesmal conrol olme dd ddd C Anag d C m d m n 0 d d d d ddd d

7 Mass conseraon: 0 dd dd dd dd d dd d dd d ddd Sm oer all conrol srfaces: Dde b olme: 0 0 0

8 Mass conseraon: 0 0 Saonar (sead) flo: Inkompressbel srömnng: konsan

9 When s ncompressble flo a reasonable assmpon? ds. Dfferenae: d d dp a d dp d Speed of sond dp dp 1 Ma 1 a a Mach nmber Usal lm: Ma 0. 3

10 Momenm conseraon: Use Renolds ranspor heorem assmng 1D-flo hrogh a nfnesmal conrol olme dd d d dd C Anag d C m m n d ddd F d d

11 Momenm conseraon: Use same sraeg as for mass conseraon: F ddd Spl he deraes: F ddd Mass conseraon Maeral derae F ddd D D

12 Frames of reference Lagrangan: Eleran: a,,, a,,, Maeral derae: D D In he Eleran frame acceleraon s descrbed b: D D conece acceleraon Local acceleraon D D

13 Forces: Gra (olme force) df g gddd g g, g, g Srface forces: The sress ensor: p p p

14 Forces: Agan se nfnesmal conrol olme d ddd dd d ddd df s, d ddd dd d d

15 Forces: ddd df s, ddd df s, ddd df s, : : :

16 Forces: df s, d df s, d df s, d d ddd p p p Dde b he olme and se p Kronecker s dela 1om 0 oherse

17 forces p d F d s p g D D p g p g p g g d F d g conece acceleraon Local acceleraon gra Pressre force scos force p g

18 Deformaon of a fld elemen Translaon: Roaon: Shear: olme dlaaon:

19 Kapel 4 Deformaon a e fldelemen d d d d d d d d d d

20 Deformaon of a fld elemen Deformaon rae: d d d d 1 Assme small angles: d d d d d 1 d d d d d 1 d d d d d d d d d d

21 Deformaon of a fld elemen Lå d d d d d 0 For a Neonan fld he sresses a lnearl dependen on he rae of deformaon ble f 0 3ncompress Dnamc scos d d d d d d d d d d Rae of sran: S 1 S

22 Momenm conseraon: p g D D p g p g p g Can for ncompressble flo of a Neonan fld be ren as: p g D D The Naer-Sokes eqaons p g p g

23 Energ conseraon: As for mass conseraon: e Q W s Noe ha 0 W s W ed e nda ss d d for nfnesmal C C CS ddd Q p W p e

24 ddd p e e W Q Mass conseraon Afer some manplaon: ddd p p D De W Q Energ conseraon:

25 Energ conseraon: Hea fl Neglec radaon and assme onl condce ransfer hrogh he CS Forer s la: q kt q k d T d q q d Q q q q ddd qddd Use Forer s la: Q kt ddd

26 scos ork d d d ddd ddd W Energ conseraon:

27 T k p p D De Rere he scos erm: T scos dsspaon, alas pos T For ncompressbel and Neonan: Energ conseraon:

28 Mass Momenm D D 0 g p Energ De p p D kt

29 Mass Momenm Energ 0 p g T k p p e e

30 For ncompressble flo, Neonan fld h consan dens, scos and condc: T k D DT c T p 0 Mass Momenm Energ p g D D Noe ha energ eqaon s no decopled from eqaons for mass and momenm 0 p g p T k T T c

31 orc Defnon: ro,, For D-flo:,,0 0,0,

32 orc orc d d d d d d d d d d

33 Roaon of a fld elemen Anglar eloc: d d d d 1 Assme small angles: d d dd d 1 d d dd d 1 d d d d d d d d d d

34 Le d d d d d 0 d d d d d d d d d d 1 Anglar eloc 1 1 Noe ha for D-flo: 0 ro 1 1 orc: Irroaonal f 0 Roaon of a fld elemen Rae of roaon: 1 k k

35 orc eqaon Take he crl of he Naer-Sokes eqaons

36 Sascal descrpon of rblen flo

37 Trblen e flo

38 0 p g Deermnsc Random

39 Defnon of randomness Eample: A fld mechancs epermen h ell defned bondar conons. An een A: A C 0.6 Three possbles: A s ceran A s mpossble A s random f neher mpossble nor ceran, hen C s a random arable Resl

40 Loren eqaons d d d d d d To cases: Inal ales: (0) (0) (0)

41 Loren eqaons To cases: 3 8

42 Loren eqaons ( 0) ( 0) Dfference

43 Loren eqaons ( 0) ( 0) Dfference

44 Loren eqaons Obseraons: 3 8 Wha can e learn from hs eercse regardng flos?

45 There are alas perrbaons orgnang from bondar condons, nal condons ec. presen n a flo. Trblen flos are acel sense o perrbaons Trblence s onl meanngfl o descrbe n a sascal sense

46 Random arables Tools for characersng random arables Cmlae Dsrbon Fncon (cdf) Probabl ha U b U s a random arable, s he sample space P F P U b b F b F a for b a F 0 F 1 mpossble ceran

47 Random arables Tools for characersng random arables Probabl Dens Fncon (pdf) f Properes: df d f 0 f 1 Probabl ha F a U F f d b a b a b

48 Random arables Momens: Tools for characersng random arables 1s momen, mean flcaon nd momen, arance 3rd momen, skeness 4h momen, flaness (cross) U f U ar U d Mean of U U U f d U f d 4 U f d Sandard deaon

49 Random processes Le he random arable, U be a fncon of me, s he called a random process. F N A each me nsan f ; F, Conans no nformaon abo he coplng n me. Hence, seeral dfferen emporal behaors can hae he same one-me pdf A on N-me cdf s needed ;, ;...;, PU, U,..., U 1, 1 N N 1 1 Normall, mpossble N N

50 Trblen flo Trblen flo Gassan process

51 Sascall saonar: All sascs are naran nder a shf n me Ml-me sascs: Aocoarance Rs s Aocorrelaon fncon s 0 1 s 1 Ths sas somehng abo he se of he mporan me scales,.e. afer ho long me eens a a ceran nsan sll has an effec. s Inegral me scale 0 sds

52 Ml-me sascs: The freqenc conen can be fond b akng he Forer ransform of he aocoarance. Freqenc specrm Aocoarance E s Rse ds R 1 e s s E d 1

53 Aocorrelaon fncon

54 Freqenc specra

55 Random felds U s no a me dependen random ecor feld One-pon, one-ome on cdf 3 1 3,,, ; F f Conans no nformaon abo he coplng n me or space. A on N-pon, N-me pdf s needed Normall, mpossble ) ( ) ( ) ( () () () (1) (1) (1),,,...,,,,,, N N N f

56 Sascall saonar: All sascs are naran nder a shf n me Sascall homogeneos: All sascs are ndependen of poson Homogeneos rblence: The sascs of he eloc flcaons are ndependen of poson Isoropc rblence: The sascs of he eloc flcaons are ndependen of coordnae ssem roaons and reflecons,.e. drecon ndependen

57 To-pon correlaon R r,,, r, Ths sas somehng abo he lengh scales of he flo,.e. ho large are he eddes or ho far aa from an een s he effec fel. Inegral lengh scale L 1 11, R11 e1r, ds R 0,,, 11 0

58 Forer ransform of he correlaon ensor ges nformaon abo he se dsrbon n he flo n erms of ae nmber κr eloc specrm ensor κ, e R r To-pon correlaon Wae nmber ecor: κ Wae lengh: κ, R κ r r, e κ, Energ specrm fncon Eκ, κ, κ κ 0 E 1 1 κ, dκ R 0, dκ dr dκ Trblen knec energ

59 Conracng he eloc specrm ensor one ges he rblen knec energ dsrbon oer he ae nmbers. Energ specrm fncon Eκ, κ, κ κ dκ Inegrang he energ specrm fncon ll resl n he rblen knec energ. 0 E 1 κ, dκ R 0, k 1 Trblen knec energ

60 Aeragng Mean: U f ; d Tme aerage: d 1 T T Ensemble aerage: 1 N N n1 ( n) Qesons: Wha s he dfference beeen mean and aerage? Wh old e need dfferen pes of aerages?

61 1 Aeragng arance: d Sandard deaon or roo-mean-sqare (rms): T 1 T N N ( n) n1 rms

62 Renolds eqaons Noaon ha ll be sed from no on n he lecre noes: Insananeos: Aerage: Flcaon: ' Renolds decomposon ' '