VI. Computational Fluid Dynamics 1. Examples of numerical simulation

Transcription

1 VI. Comaonal Fld Dnamcs 1. Eamles of nmercal smlaon Eermenal Fas Breeder Reacor, JOYO, wh rmar of coolan sodm. Uer nner srcre Uer lenm Flow aern and emerare feld n reacor essel n flow coas down Core Hh ressre lenm Lower lenm Veloc ecors emerare conors

2 Vore Horse-shoe ore Londnal ore Flow Vorc n bondar laer on a fla lae Horse-shoe ore Flow Vore lnes and edd-scos conors

3 . Conseraon eqaons of mass, momenm, and ener.1 Eqaon of conn rae of ncrease of mass rae of mass n rae of mass o [ ] [ ]

4 or ne rae of mass addon er n olme b conecon rae of ncrease of mass er n olme D Larane derae or D D D D Larane derae Incomressble fld consan dens Sff: reea

5 . Eqaon of moon rae of ncrease of momenm rae of momenm n rae of momenm o eernal force on he fld rae of momenm n - rae of momenm o

6 eernal force on he fld h shear sress ressre and bod force rae of ncrease of momenm

7 hree dmensonal

8 Vecor eresson rae of ncrease of momenm er n olme ne rae of momenm addon er n olme b conecon ne rae of momenm addon er n olme b moleclar ransor eernal force on he fld er n olme D D or D D [ ]

9 3 Sress ensor 3 3

10 D D 3 D 3 D D D 3 D D 3

11 Naer-Soes eqaon D D Eler eqaon when D D

12 D D β β β Dmensonless arables,,,, 1 l l Non-dmensonal eresson 3 1 / / / 1 l l l D D β Re Gr Re 1 D D scos force nera force / / Re l l l V Renolds nmber 3 1 Gr β L Grashof nmber nera force force boanc / Re Gr 1 1 β β l

13 .3 Eqaon of ener f b eernal wor done on ssem rae of h b moleclar wor done on ssem rae of b moleclar hea addon ne rae of ener addon nernal nec and ne rae of l and nec of ncrease rae of ra b forces b sresses mechansms condcon ransor ransor b conece ener addon ener nernal U 1 [ ] [ ] [ ] e e e e e e e e e e e e

14 e e 1 e e U 1 1 U U q [ ] [ ] rae of ncrease of ener er n olme U rae of ncrease of nernal ener er n olme rae of ener addon er n olme b conecon rae of ener addon er n olme b hea condcon rae of wor done on fld er n olme b ressre force [ U] q : ne rae of nernal ener addon er n olme b conecon rae of reersble nernal rae of ener nernal addon ener er n ncrease er olme b n olme hea b condcon comresson rae of wor done on fld er n olme b scos force rreersble rae of nernal ener ncrease er n olme b scos dssaon rae of wor done on fld er n olme b eernal force

15 D c λ λ D D D D c λ D Fld wh consan dens D D c λ c λ Q

16 .4 Dmensonless ros

17 Phscal nerreaon of dmensonless ros

18 3. Calclaon of rblen flow Wae of crclar clnder n waer flow Moon s sreled wh almnm owder.. Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F. 1.7 Insananeos eloc rofles n a rblen bondar laer on a fla lae Re1 5 5f from leadn ede, Measremen wh hdroen bbble mehod.. Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F. 1.3

19 Aal eloc flcaons measred wh a ho-wre Locaon: 56 n. from leadn ede,. n. off srface. Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F. 1.3 Insananeos eloc s searaed o meaerae one and flcaon comonen,, me

20 Relae rblence nenses alon a fla lae. Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F. 1.1

21 Eqaons of conn momenm and ener are me aeraed Eqaons of conn, momenm and ener are me-aeraed Sbsn In case of ncomressble fld Conn Renolds eqaon me-aeraed momenm eqaons D D 1 Renolds sress addonal erms

22 3.1 Prandl mn lenh model Edd dffs of momenm M M lenh eloc l M l m l Mn lenh κ l m on Karman s consan κ.41

23 on Karman Karmans consan κ.41 an Dress hohess near a wall 1/ 1/ 1 e w lm κ, A A 6. B.E. Lander, D. B. Saldn, Lecres n Maehemacal Models of rblence, Academc Press, 197. Nladses formla for e flow l m R R 4.6 1, R Nladse. Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F

24 . Cebec, A. M. O. Smh, "Analss of rblen Bondar Laers," Aled Mahemacs and Mechancs 15, Academc Press F Edd dffs M n e flow Problems of Prandl s mn lenh model - I s dffcl o redc recrclan flow wh ero eloc raden where sress and eloc raden are comlcaed. - I aes no accon of rocesses of conecon or dffson of rblence 4

25 . Renolds aerae rblence model.1 One eqaon model of rblence Renolds sresses R rblence ener 1/ 1 Momenm eqaons for,, are meaeraed and smmed. l R D D

26

27 D D

28 / Aromaon l C σ 1/ Local soro, 3/ l C D C D 3/ l C D D σ

29 . wo-eqaon model of rblence rblence ener rblence model low Renolds nmber model Jones-Lander s model 197 rblence ener 1/ M M M σ Dssaon rae 1 1. f C f C M M M σ Edd dffs f M C Hh Renolds nmber model: las erms nder bar are omed.

30 f 1 1. f f 1..3e e R [.5 1 R 5 ] Renolds nmber of rblence R C C1 C σ Jones-Lander Bondar condons rblence model low Renolds nmber model σ on sold wall rblence model hh Renolds nmber model 3 1 C 1 C on sold wall C κ Veloc rofle near sold wall: Lo law

31 rblence model rblence ener rblence model be cee e M D D Dssaon rae σ D σ f C C D D 1 Renolds sress σ D Edd dffs δ 3 f M C

32 Naano-aawa model where Abe e al. model where f 1 e / 4 1 e f Renolds nmber of rblence R f 6 e 14 R, R 1.3e 6.5 Frcon eloc 5 R 1 e, 3 / 4 R 1 R f 1 e 1.3e 3.1 1/ 4 w 6.5 R

33 Model consans C 1 σ σ C C Naano-aawa Abe e al

34 Problem 5 Dere ma r 1 R 1/ 7 sn he follown eqaons: w , for 3 1 < Re < 1 1/ 4 Re d where Re n,,, w

35 Problem 6 Dere.5ln C sn he follown eqaons: l m l m κκ κ. 4,, w