9. Flow of Real Fluids

Transcription

1 9. Flo of Rel Flds Rel Fld Flo Vscost ests. Veloct rdent ests. 流體粒子 Prtcle 之間以及流體粒子與邊界 bondr 之間有 frcton sher force 存在, 造成流體流動之阻力 流體須克服摩擦力而產生流動, 損失之能量變成熱, 因此會有水頭損失 h L 真實流體之流動存在層流 Lmnr flo 與亂流 Trblent flo 兩種不同流況機制 Veloct Dstrbton nd Correcton Fctors Sher stress of rel flds Non-nform eloct dstrbton nform eloct dstrbton of del fld flo Need correcton!! Idel Rel Correcton fctor α Veloct hed correcton For del fld flo: here V K.E. Weht mv m V d Men eloct Bt for rel fld flo: Tre K.E. V Need correcton α Weht here αcorrecton fctor Veloct hed

2 Tre K.E. Weht α V Tre K.E./tme Weht/tme d d V d d d d d V α d d Correcton fctor β Inert force correcton Momentm/tme QV Tre Momentm/tme β QV Tre Momentm/tme β V d d β V V d For nform flo:α β Non-nform flo:α > β > V. Lmnr s. Trblent Flo Edence of estence of to flo remes Renolds eerment 88: d d

3 Lmnr: for Re < Trblent: for Re >> Trnston: < Re < 4 Vd Re ν here V Men eloct n the e Q / d Pe dmeter ν Knemtc scost Lmnr flo ers to moe n rllel ths. d Sher stress:τ d Dstrbnce 擾動 Rdl dmed 消減 b scost! Trblent flo fld rtcles don't remn n lers. 特色 : Rndom moton 隨機運動 b Eddes 旋渦

4 . Edd Vscost for Trblent Flo Orn of Trblence: Frcton effects t sold bondr. Intercton of fld strems mon th dfferent eloctes. Men eloct nd eloct flctton ' ' t Trblent Tme scle Tme here Instntneos eloct 瞬時速度 Tme men eloct 時間平均速度 Veloct flctton 速度跳動 T T dt Renolds Stress τ for T >> t t.. t s Trblent Sher Stress Renolds Stress ' B ' τ trb. ε ε 通常 ε >> here ε Edd scost 旋渦黏滯性 4

5 . Bondr Ler 邊界層 Deeloment of Bondr Ler Crtcl R ν 5 R : R 5 cr hen R > R cr Trnston to Trblent B.L. R < R cr Lmnr B.L. If totll roh bondr, trblent B.L. ll form from the bennn. Bondr Ler Thckness, δ 邊界層厚度 δ hen δ. 99 or.999 Momentm Eqn. for Bondr Ler 5

6 4. Ner-Stokes Eqton DV V V Derton see end Dt 簡化 : For ncomressble fld V DV V Dt For del fld DV Dt For nform flofeld V constnt 5. Ect Soltons of N-S Eqn for Lmnr Flo Lmnr flo beteen rllel ltes rllel ltes ssmtons: sted flo t rllel flo, ncom. fld V 4 nfntel de lte Contnt eqn: Eqn. of Moton: 方向 : t 6

7 7 方向 : t b 方向 : t Eqn : 由 b 知 fct 積分 C C Emle : Coette flo 不受壓 Sol: C C. B.C constnt τ Emle : Poselle flo < 受壓力

8 8 Sol: C C B.C. rbolc 拋物線 8 m d m τ 6. Nmercl Soltons of N-S Eqn Fnte-dfference scheme 有限差分法 Forrd dfference Bckrd dfference Centrl dfference

9 9 Emle : Coette flo th constnt scost d d d d IF const. d d Fnte-dfference eqn. - } { } ]{ [ b Emle : Coette flo th rble scost ] /.9 e[ j d d d d Fnte-dfference eqn. - / ] / e[.9 ] / e[.9 4. j / / 6

10 [ ]{ } { b} Soln the eqton set Mtr nerson [ ]{ } { b} { } [ ] [ ]{ } [ ] If e kno [ ] then e cn obtn {} { b} [ ] Obtned b mtr nerson rorm! e.. IMSL mth Lbrr, Mthemtc, etc. Grss-Sedel terton method 迭代法 Intl Gess : ,.4,.6,.8 Iterton ntl crter stsfed. ε 4 4 5

11 7. N-S Eqn for Trblent Flo,, here,, re tme-ere eloctes;,, re flcttons Contnt eqn. for trblent flo of ncom. fld Eqn,, Eqn Eqn- : Trblence flcttons lso sf V Eqn. of moton for trblent flo of ncom. fld -comonent of N-S eqn : t Tme ere : t 化簡 t ε ε ε Trblence closre!

12 9. Flo of Rel Flds Rel Fld Flo Vscost ests. Veloct rdent ests. 流體粒子 Prtcle 之間以及流體粒子與邊界 bondr 之間有 frcton sher force 存在, 造成流體流動之阻力 流體須克服摩擦力而產生流動, 損失之能量變成熱, 因此會有水頭損 失 h L P V P V h L 真實流體之流動存在層流 Lmnr flo 與亂流 Trblent flo 兩種不同流況 Veloct Dstrbton nd Correcton Fctors Sher stress of rel flds Non-nform eloct dstrbton nform eloct dstrbton of del fld flo Need correcton!! Correcton fctor Veloct hed correcton For del fld flo: K.E. Weht mv m V Veloct hed here V d Men eloct

13 Bt for rel fld flo: here Tre K.E. Weht V Tre K.E. V Need correcton Weht Correcton fctor Tre K.E./tme Weht/tme d d V d d d d d d V d Correcton fctor Inert force correcton Momentm/tme QV Tre Momentm/tme QV Tre Momentm/tme d d V V V d V d d For nform flo: Non-nform flo: 4

14 . Lmnr s. Trblent Flo Edence of estence of to flo remes Renolds eerment 88: Lmnr:for : Re < Trblent:for Re >> Trnston: < Re < 4 Vd Re here V Men eloct n the e Q / d Pe dmeter Knemtc scost 5 6

15 Lmnr flo ers to moe n rllel ths. d Sher stress: d Dstrbnce 擾動 Rdl dmed 消減 b scost! Trblent flo fld rtcles don't remn n lers. 特色 : Rndom moton 隨機運動 b Eddes 旋渦 7 8

16 . Edd Vscost for Trblent Flo Orn of Trblence: Frcton effects t sold bondr. Intercton of fld strems mon th dfferent eloctes. Men eloct nd eloct flctton here Instntneos eloct 瞬時速度 Tme men eloct 時間平均速度 Veloct flctton 速度跳動 T T dt for T t t. t. s 9 Renolds Stress Trblent Sher Stress Renolds Stress trb. 通常 here Edd scost 旋渦黏滯性

17 . Bondr Ler 邊界層 Deeloment of Bondr Ler R Ct Crtcl l 5 R : R 5 hen cr R R cr R R cr Lmnr B.L. Trnston to Trblent B.L. If totll roh bondr, trblent B.L. ll form from the bennn. Bondr Ler Thckness, 邊界層厚度 hen.99 or.999 Momentm Eqn. for Bondr Ler

18 4. Ner-Stokes Eqton DV V V Dt Derton see end 簡化 : For ncomressble fld V DV V Dt For del fld D V Dt For nform flofeld V constnt 5. Ect Soltons of N-S Eqn for Lmnr Flo Lmnr flo beteen rllel ltes ssmtons: sted flo t rllel flo, ncom. fld V 4 nfntel de lte 4

19 Contnt eqn: Eqn. of Moton: t t b 5 t Eqn : b fct C C Emle : Coette flo Sol: C C. B.C 6 constnt

20 Emle : Poselle flo Sol: C C B.C. rbolc 7 8 m d m 6. Nmercl Soltons of N 6. Nmercl Soltons of N-S Eqn S Eqn Fnte-dfference scheme 有限差分法 Forrd dfference Bckrd dfference Centrl dfference 8

21 9

22 7. N 7. N-S Eqn for Trblent Flo S Eqn for Trblent Flo,, here re tme-ere eloctes; re flcttons,,,, Contnt eqn. for trblent flo of ncom. fld Eqn

23 ,, Eqn Eqn- : Trblence flcttons lso sf V Eqn. of moton for trblent flo of ncom. fld -comonent of N-S eqn : t t Tme ere : t 化簡 4 Trblence closre!