9. Flow of Real Fluids

Transcription

1 9. Flo of Rel Flds Rel Fld Flo Vscost ests. Veloct grdent 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 Unform eloct dstrbton of del fld flo Need correcton!! Idel Rel Correcton fctor α Veloct hed correcton For del fld flo: here V K.E. Weght mv mg V g d Men eloct Bt for rel fld flo: Tre K.E. V Need correcton α Weght g here αcorrecton fctor Veloct hed

2 Tre K.E. Weght α V Tre K.E./tme Weght/tme d d V g g d d d d d V α g d g 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 regmes 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 Orgn of Trblence: Frcton effects t sold bondr. Intercton of fld strems mong 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 U R ν 5 R : R 5 cr hen R > R cr Trnston to Trblent B.L. R < R cr Lmnr B.L. If totll rogh bondr, trblent B.L. ll form from the begnnng. Bondr Ler Thckness, δ 邊界層厚度 U δ hen δ. 99 U or.999 Momentm Eqn. for Bondr Ler 5

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

7 7 方向 : t g g b 方向 : t g Eqn : 由 b 知 fct 積分 C C Emle : Coette flo 不受壓 Sol: C C U. B.C constnt U U 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. - U U U } { } ]{ [ b Emle : Coette flo th rble scost ] /.9 e[ j d d d d Fnte-dfference eqn. - / ] / e[.9 ] / e[.9 4. U j / / 6 U U U

10 [ ]{ } { b} Solng the eqton set Mtr nerson [ ]{ } { b} { } [ ] [ ]{ } [ ] If e kno [ ] then e cn obtn {} { b} [ ] Obtned b mtr nerson rogrm! e.g. 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-erge 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 : g t Tme erge : t g 化簡 t g g ε ε ε Trblence closre!

12 Rel Fld Flo 9. Flo of Rel Flds Vscost ests. Sher stress ests hen there s eloct grdent. 流體粒子 Prtcle 之間以及流體粒子與邊界 bondr 之間有 scos sher frcton 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, 損失之能量變成熱能, 因此會有水頭損失 h L P V P V h L g g 真實流體之流動存在層流 Lmnr flo 與紊流 Trblent flo 兩種不同流況 Veloct Dstrbton nd Correcton Fctors Vscost of rel flds Non-nform eloct dstrbton Unform eloct dstrbton of del fld flo Need correcton!! Correcton fctor Veloct hed correcton For del fld flo: K.E. Weght mv mg V Veloct hed g here V d Men eloct

13 Bt for rel fld flo: here Tre K.E. Weght V Tre K.E. V Need correcton Weght g Correcton fctor Tre K.E./tme Weght/tme d d V g d g d d d d g d V d g 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 regmes Renolds eerment 88: Lmnr:for Re < Renolds Trblent:for Re > 4 Trnston: < Re < 4 Q Vd Re here V Men eloct n the e Q / d Pe dmeter Knemtc scost 5 6

15 Re < Lmnr Instblt 不穩定 < Re < 4 Edd 渦漩 Re > 4 Fll trblent 7 Lmnr flo Lo Re ers to moe n rllel ths. d Sher stress: d Dstrbnce 擾動 Rdl dmed 消減 b scost! Trblent flo Hgh Re fld rtcles don't remn n lers. 特色 : Rndom moton 隨機跳動 b Eddes 渦漩 8 4

16 τ τ 9. Edd Vscost for Trblent Flo Orgn of Trblence: Frcton effects t sold bondr. Intercton of fld strems mong 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 5

17 6 Renolds Stress Renolds Stress trb. here Edd scost 渦漩黏滯性 通常

18 . Bondr Ler 邊界層 Deeloment of Bondr Ler Prndtl U 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 rogh bondr, trblent B.L. ll form from the begnnng. Bondr Ler Thckness, 邊界層厚度 hen U Momentm Eqn. for Bondr Ler 4 7

19 4. Ner-Stokes Eqton DV g V V Dt Derton see end 簡化 : For ncomressble fld V DV g V Dt For del fld D V g Dt For nform flofeld V constnt g Ner Stokes 5 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 6 8

20 9 Contnt eqn: Eqn. of Moton: 方向 : t g 方向 : t g g b 方向 : g 7 t g Eqn : 由 b 知 fct 積分 C C Emle : Coette flo 不受壓 Sol: C C U. B.C 8 constnt U U

21 Emle : Poselle flo 受壓力 Sol: C C B.C. rbolc 拋物線 9 8 m d m 6. N 6. N-S Eqn Eqn for Trblent Flo for Trblent Flo Renolds erged N Renolds erged N-S Eqn Eqn, RNS, RNS,, here re tme-erge eloctes; re flcttons,,,,, ' Contnt eqn. for trblent flo of ncom. fld Eqn

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

23 6//9 Reltons beteen Stress 應力 nd Rte of Strn 應變率 結合物理學 牛頓流體力學 材料力學 Stress tensor 應力張量 Norml stress,, Sher stress

24 6//9. Norml stress nd lner deformton Norml stress nd lner deformton Hooke s l: E, E, E here,, lner deformtons;,, norml stresses; E elstc modls. Lterl contrcton 側向收縮 : n n / E, n n / E here n Posson s rto.

25 6//9 Totl lner deformton n drecton: [ ]/ n E t t dr dr dr here rte of lner deformton. Smlrl, the totl lner deformtons n nd drecton: [ n ] [ n t t, b E E [ n ] c t t E. Sher stress nd nglr deformton Hooke s l: r G G t r here G r sher modls; For -D flo: Gt scost; r rte of nglr deformton. 4 d d

26 6//9 lne. Mechncs of Mterl 材料力學 Knon relton beteen E, G, nd n: E G n 5 Sbstttng 5 nto : [to elmnte E ] [ n ]/ G n 6 [ n ]/ G n 6b 6c [ n ]/ G n Smmton of 6, 6b, nd 6c: [ n / G n] [ n / G n] P [Defne : / P] 7 4

27 6//9 Sbstttng 7 nd nto 6: [to elmnte n ] V P G PGt P For ncomressble fld: Eqton 8 s redced to: P P V Smlrl, the norml stresses n nd dr: 8 9 P P, P P 9b 9c P P P stress tensor ressre Ner Stokes Eqtons Force n drecton: F m scos stresses D d d d d d d d d d Dt m d d d d d d g 5

28 6//9 D g Dt P g P g P g ssmng constnt D P g Dt P g Smlrl, n nd drecton: D P g Dt D P g Dt DV P g V Dt 6