dx dt V x V t V y a Dt Acceleration field z dz dt V dt V v y V u t a Dt

Transcription

1 FUNDAMENTALS OF Chate 6 Flo Analsis FLUID MECHANICS Using Diffeential Methods

2 MAIN TOPICS I. Flid Element Motion II. Conseation of Mass (Continit eqation) and Conseation of Linea Momentm (Naie-Stokes Eqation) III. Iniscid Flo (Benolli eqation) and Potential Flo (Steam fnction) Incomessible Comessible Iniscid iscos Stead Unstead Rotational - Iotational Mathematical eqation?

3 Motion of a Flid Element. Flid Tanslation: : The element moes fom one oint to anothe. 3. Flid Rotation: : The element otates abot an o all of the,, aes. Flid Defomation: 4. Angla Defomation:The element s angles beteen the sides change.. Linea Defomation:The element s sides stetch o contact. 3

4 . Flid Tanslation elocit and acceleation The elocit of a flid aticle can be eessed (,,, t) i j k elocit field The total acceleation of the flid aticle is gien b a D Dt d d, dt dt D a Dt a D Dt t d dt d, dt t d dt d dt Acceleation field is called the mateial, o sbstantial deiatie. d dt 4

5 Phsical Significance a D Dt t Total Acceleation of a aticle a D Dt ( Conectie Acceleation -> Sace ) t Local Acceleation -> Time 5

6 Scala Comonent Scala Comonent ( 외울필요외울필요없음없음 ) Scala Comonent Scala Comonent ( 외울필요외울필요없음없음 ) a j a t a i a Rectangla Rectangla a k a t a j a coodinates sstem coodinates sstem t t a C li d i l C li d i l t a Clindical Clindical coodinates sstem coodinates sstem 6 t a

7 Tanslation All oints in the element hae the same elocit, then the element ill siml tanslate fom one osition to anothe. 7

8 . Linea Defomation / The shae of the flid element, descibed b the angles at its etices, emains nchanged, since all ight angles contine to be ight angles. A A change in the dimension eqies a noneo ale of / A A / A A / 8

9 Linea Defomation / The change in length of the sides ma odce change in olme of the element. The change in ( )( t) ( )( t) The ate at hich the is changing e nit olme de to gadient t / / d If / and / d dt / ae inoled as in -D o 3D cases, ( )( t) ( )( t) ( )( t) ( )( t) olmetic dilatation ate d dt Diegence of Fo an incomessible flid (constant densit), the olmetic dilatation ate is eo. 9

10 3. Angla Rotation /4 The angla elocit ( 각속도 ) of line OA Fo small angles OA OB CCW ( 시계반대방향 ) CW tan t ω OA t lim0 δt δα δt OB 0 CCW Positie diection 0

11 Angla Rotation /4 The otation of the element abot the -ais is defined as the aeage of the angla elocities of the to mtall eendicla lines OA and OB abot the -ais ais. lim t0 t In ecto fom i j k + CCW ( 시계반대방향 )

12 Angla Rotation Angla Rotation 3/4 Angla Rotation 3/4 Angla Rotation 3/4 3/4 k j i k j i cl Defining Defining oticit oticit > Angla Angla otation otation Defining Defining oticit oticit Defining iotation Defining iotation 0 -> Angla Angla otation otation g 0

13 Angla Rotation Angla Rotation 4/4 Angla Rotation 4/4 Angla Rotation 4/4 4/4 k j i cl k j i 3

14 oticit oticit oticit oticit Defining Defining oticit oticit ζ hich ζ hich is a measement of the otation of a measement of the otation of a flid element flid element as it moes in the flo field: as it moes in the flo field: cl k j i In clindical coodinates sstem In clindical coodinates sstem: e e e e e e 4

15 4. Angla Defomation / Angla defomation of a aticle is gien b the sm of the to angla defomation t t t t t t / / ξ(xi)η(eta) Rate of shea stain o the ate of angla defomation lim t t lim t 0 t0 t t... 5

16 Angla Defomation Angla Defomation / Angla Defomation / Angla Defomation / / The ate of angla defomation in lane The ate of angla defomation in lane The ate of angla defomation in lane 6

17 Eamle 6. oticit Fo a cetain to-dimensional i flo field theelocit is gien b 4 i ( ) j Is this flo iotational? 7

18 Eamle 6 Eamle 6 Soltion Eamle 6. Soltion Eamle 6. Soltion Soltion This flo is iotational This flo is iotational 0 8

19 Conseation Eqations Continit eqation Conseation of Mass Momentm eqation (Naie-Stokes Eq.) Conseation of Linea Momentm Angla momentm eqation Conseation of Angla Momentm Eneg eqation Conseation of Eneg Reesentation Integal (contol olme) eesentation Diffeential eesentation 9

20 Conseation of Mass /5 To deie the diffeential eqation fo conseation of mass in ectangla and in clindical coodinate sstem. The deiation is caied ot b aling conseation of mass to a diffeential contol olme. With the contol olme eesentation of the conseation of mass t C d CS nda 0 The diffeential i fom of continit i eqation??? 0

21 Conseation of Mass /5 d t C CS The C chosen is an infinitesimal cbe ith sides of length,, and. nda 0 diffeential contol olme t C d t n n d n i i on the ight sface i i on the left sface Talo seies

22 Conseation of Mass 3/5 Net ate of mass Otflo in -diection Net ate of mass Otflo in -diection Net ate of mass Otflo in -diection

23 Conseation of Mass Conseation of Mass 4/5 Conseation of Mass 4/5 Conseation of Mass 4/5 4/5 Net ate of mass Net ate of mass Otflo The diffeential eqation fo Continit eqation The diffeential eqation fo Continit eqation 0 t t 0 da n d t d t C CS t C nda CS 3

24 Conseation of Mass Conseation of Mass 5/5 Conseation of Mass 5/5 Conseation of Mass 5/5 5/5 Incomessible flid Incomessible flid (densit is constant and nifom) densit is constant and nifom) 0 Stead flo Stead flo 0 ) ( ) ( ) ( 0 ) ( t 4

25 Eamle 6. Continit Eqation The elocit comonents fo a cetain incomessible, ibl stead flo field ae? Detemine the fom of the comonent,, eqied to satisf the continit eqation. 5

26 Eamle 6. Soltion The continit eqation 0 ( 3 ) 3 f (, ) 6

27 Conseation of Linea Momentm Aling Neton s second la to contol olme DP F Psstem dm d Dt M(sstem) (sstem) SYS D m F m Dt t D m ma Neton s nd la Dt Fo a infinitesimal sstem of mass dm,, hat s the the diffeential fom of linea momentm eqation? 7

28 Foces Acting on Element / The foces acting on a flid element ma be classified as bod foces and sface foces; sface foces inclde nomal foces and tangential (shea) foces. F FS FB Fs i F F i F b s b j F j F s b k k Sface foces acting on a flid element can be descibed in tems of nomal and shea stesses. n F lim n t A 0 F lim t0 A F lim t0 A 8

29 Foces Acting on Element / F F F F F F s s s b b b g g g Eqation of Motion 9

30 Doble Sbscit Notation fo Stesses The diection of the stess The diection of the nomal to the lane on hich the stess acts 30

31 Eqation of Motion F ma F ma Geneal eqation of motion g g g g F ma t t t These ae the diffeential eqations of motion fo an flid. Ho to sole,,? -> These can t be soled becase of moe aiables than eqations, hich eqies moe eqations called constittie eqations to sole the eqations in the case of Netonian flids 3

32 Stess Stess-Defomation Relationshi Defomation Relationshi: constittie eqations / / The stesses mst be eessed The stesses mst be eessed in tems of the elocit and in tems of the elocit and esse field esse field 3 esse field esse field. 3 3 Catesian coodinates in Netonian and C ibl fl id Comessible flids 3 3

33 Stess-Defomation Relationshi: constittie eqations / The stesses mst be eessed in tems of the elocit and esse field. Catesian coodinates in Netonian and Incomessible flids 3 33

34 The Naie The Naie Stokes Eqations Stokes Eqations / The Naie The Naie-Stokes Eqations Stokes Eqations / These obtained eqations of motion ae called the Naie These obtained eqations of motion ae called the Naieq Stokes Eqations. Stokes Eqations. Unde Unde incomessible incomessible Netonian Netonian fl flids ids the Naie the Naie- Unde Unde incomessible incomessible Netonian Netonian fl flids ids, the Naie the Naie Stokes eqations ae edced to: Stokes eqations ae edced to: g t g t g t 34

35 The Naie-Stokes Eqations / The Naie-Stokes eqations al to both lamina and tblent flo, bt fo tblent flo each elocit comonent flctates andoml ith esect to time and this added comlication makes an analtical soltion intactable. The eact soltions efeed to ae fo lamina flos in hich the elocit is eithe indeendent of time (stead flo) o deendent on time (nstead flo) in a ell- defined manne. 35

36 Lamina o Tblent Flo / The flo of a flid in a ie ma be Lamina? O Tblent? Osbone Renolds,, a Bitish scientist and mathematician, as the fist to distingish the diffeence beteen these classification of flo b sing a simle aaats as shon. 36

37 Lamina o Tblent Flo / Fo small enogh floate the de steak ill emain as a Fo the de steak ill emain as a ell-defined line as it flos along, ith onl slight bling de to molecla l diffsion i of the de into the sonding ate. Fo a somehat lage intemediate floate the de Fo a somehat lage the de flctates in time and sace, and intemittent i bsts of iegla behaio aea along the steak. Fo lage enogh hfl floate t the de steak almost Fo the de steak almost immediatel become bled and seads acoss the entie ie in a andom fashion. 37

38 Time Deendence of Flid elocit at a Point 38

39 Indication of Lamina o Tblent Flo The tem floate shold be elaced db b Renolds nmbe, R e L /,hee is the aeage elocit in the ie, and L is the chaacteistic dimension of a flo. L is sall D (diamete) in a ie flo. -> a mease of inetial foce to the iscos foce. It is not onl the flid elocit that detemines the chaacte of the flo its densit, iscosit, and the ie sie ae of eqal imotance. Fo geneal engineeing ose, the flo in a ond ie Lamina R e 00 Tansitional Tblent R e >

40 Some Simle Soltions fo iscos, Incomessible Flids A A incial difficlt in soling the Naie-Stokes eqations is becase of thei nonlineait aising fom the conectie acceleation tems. Thee ae no geneal analtical schemes fo soling nonlinea atial diffeential eqations. Thee ae a fe secial cases fo hich the conectie acceleation anishes. In these cases eact soltion ae often ossible. 40

41 Stead, Lamina Flo beteen Fied Stead, Lamina Flo beteen Fied Paallel Plates Paallel Plates / /4.. Schematic: Schematic:.. Assmtions: Incomessible, Netonian, Stead, One dimensional flo Assmtions: Incomessible, Netonian, Stead, One dimensional flo,,,,,, Continit eqation Continit eqation The Naie The Naie-Stokes eqations Stokes eqations g t g 0 0 g t g t 4 t

42 Stead, Lamina Flo beteen Fied Paallel Plates /4 Bonda conditions (B.C.) =0 at =-h =0 at =h (no-sli bonda condition) 5. Bonda conditions (B.C.) 6. Sole the eqations ith B.C. 0 g 0 0 Integating g f Integating c c c? c? c 0, c h h 4

43 Stead, Lamina Flo beteen Fied Stead, Lamina Flo beteen Fied Paallel Plates Paallel Plates 3/4 Shea stess distibtion Shea stess distibtion Shea stess distibtion Shea stess distibtion olme flo ate olme flo ate e nit deth ( diection) e nit deth ( diection) h d h d q h h h h 3 ) ( 3 0 constant esse otlet the is and esse inlet the is hee, 3 h h h q 43 3 esistance Flo 3 3 h R i h q h q

44 Stead, Lamina Flo beteen Fied Paallel Plates 4/4 Aeage elocit e nit deth aeage q h h 3 Point of maimm elocit d at =0 0 d at 0 ma U h 3 aeage 44

45 Coette Flo /3 (HW) Since onl the bonda conditions hae changed, thee is no need to eeat the entie analsis of the both lates stationa case. 45

46 Coette Flo Coette Flo /3 Coette Flo /3 Coette Flo /3 /3 The bonda conditions fo the moing late case ae The bonda conditions fo the moing late case ae =0 at =0 =0 at =0?? c c c c =U at =b =U at =b U?? c c c c 0 c b b U c elocit distibtion elocit distibtion b b U b b U b b U U b P 46 U

47 Coette Flo 3/3 Simlest te of Coette flo 0 U This flo can be aoimated b the flo beteen closel saced concentic clinde is fied and the othe clinde otates ith a constant angla elocit. b U b i o i i /( o i ) Flo in the nao ga of a jonal beaing. 47

48 Stead, Lamina Flo (Hagen-Poiseille Flo) in Cicla Tbes /5. Schematic:. Assmtions: Incomessible, Netonian, Stead, Lamina, One dimensional flo 0, 0, 0 3. Continit eqation 0 4. The Naie-Stokes eqations 5. Bonda Conditions: At =0, the elocit is finite. At =R, the elocit is eo. 6. Sole the eqation ith B.C. 48

49 Fom the Naie-Stokes Eqations in Clindical coodinates Geneal motion of an incomessible Netonian flid is goened b the continit eqation and the momentm eqation Mass conseation Naie-Stokes Eqation in a clindical coodinate Acceleation 49

50 Stead, Lamina Flo in Cicla Tbes /5 Naie Stokes eqation edced to 0 g sin Integating 0 g cos g sin f, 0 Integating 4 c ln c c? g f c? 50

51 Stead, Lamina Flo in Cicla Tbes 3/5 At =0, the elocit is finite. At =R, the elocit is eo. c 0, c R 4 elocit distibtion 4 R 5

52 Stead, Lamina Flo in Cicla Tbes 4/5 The shea stess distibtion d d olme flo ate 4 R R Q d constant / R R D Q

53 Stead, Lamina Flo in Cicla Tbes 5/5 Aeage elocit aeage Q A Q R R 8 Point of maimm elocit d 0 d at =0 R ma aeage 4 R ma 53

54 Stead, Aial, Lamina Flo in an Annls / (HW) Fo stead, lamina flo in annla tbes Bonda conditions = 0, at = o =0, at= i 54

55 Stead, Aial, Lamina Flo in an Annls / The elocit distibtion ib i 4 o i o ln( / ) o i ln o The olme ate of flo o Q ( ) d o i i 8 The maimm elocit occs at = m ( o ) ln( / ) 4 4 i o i 0 m ln( o i o / i ) / 55

56 Iniscid Flo Shea stesses deelo in a moing flid becase of the iscosit of the flid. Fo some common flid, sch as ai, the iscosit it is small, and theefoe it seems easonable to assme that nde some cicmstances e ma be able to siml neglect the effect of iscosit. Flo fields in hich the shea stesses ae assmed to be negligible g ae said to be iniscid, o fictionless. Dfi Define the esse,, as the negatie of fth the nomal stess 56

57 Ele s Eqation of Motion Ele s Eqation of Motion Ele s Eqation of Motion Ele s Eqation of Motion Unde Unde iniscid flos: fictionless condition iniscid flos: fictionless condition, the the eqations of motion ae edced to eqations of motion ae edced to Ele s Eqation Ele s Eqation: g t g t Ele s Eqation Ele s Eqation g t t g D 57 g Dt

58 Benolli Eqation /3 Ele s eqation fo stead flo along a steamline is g ( ) Selecting the coodinate sstem ith the -ais etical so that the acceleation of gait ecto can be eessed as g g g ( ) ecto identit. 58

59 Benolli Eqation Benolli Eqation /3 Benolli Eqation /3 Benolli Eqation /3 /3 eendicla to eendicla to g s d s d ds g ds ds ds With With k d dj di ds tangential ecto on a steamline tangential ecto on a steamline d d d d k d j d i d k j i d d d d d dk dj di k j i ds gd d d d g dk dj di k j i g ds g d d d d dk dj di k j i ds 59 g g j j g g

60 Benolli Eqation 3/3 ds d d ds g ds 0 gd 0 Integating d g constant Fo stead, iniscid, incomessible flid (commonl called ideal flids) along a steamline Benolli eqation is gien b g constant Benolli eqation 60

61 Iotational Flo Iotational Flo / Iotational Flo / Iotational Flo / / Iotation? The iotational condition is 0 In ectangla coodinates sstem 0 In ectangla coodinates sstem 0 In clindical coodinates sstem 0 0 6

62 Iotational Flo / A A geneal flo field old not be iotational flo. A A secial nifom flo field is an eamle of an iotationalal flo 6

63 Benolli Eqation fo Iotational Flo /3 The Benolli eqation fo stead, incomessible, and iniscid flo is g constant The eqation can be alied beteen an to oints on the same steamline. In geneal, the ale of the constant ill a fom steamline to steamline. Unde additional iotational condition, the Benolli eqation? Ele s Stating ith eqation in ecto fom ( ) gk g ( ) t ZERO Regadless of the diection of ds 63

64 Benolli Eqation fo Iotational Flo /3 With iotaional condition 0 ( ) gk gk d d di dj dk d d gk d Not a steamline d d d gd d gd 0 64

65 Benolli Eqation fo Iotational Flo 3/3 Integating fo incomessible flo d g contant g constant This eqation is alid beteen an to oints in a stead, incomessible, iniscid, and iotational flo iesectie of steamlines. g g 65

66 Steam Fnction /6 Steamlines: Lines tangent to the instantaneos elocit ectos at ee oint. Steam fnction Ψ(,) ) [Psi]? Used to eesent the elocit comonent (,,t) and (,,t) of a to-dimensional incomessible flo. Define a fnction Ψ(,), called the steam fnction, hich elates the elocities shon b the fige in the magin as 66

67 Steam Fnction /6 The steam fnction Ψ(,) satisfies the to-dimensional fom of the incomessible continit eqation 0 0 Ψ(,) is still nknon fo a aticla oblem, bt at least e hae simlif the analsis b haing to detemine onl one nknon, Ψ(,), athe than the to nknon fnction (,) and (,). 67

68 Steam Fnction 3/6 Anothe adantage of sing steam fnction is elated to the fact that line along hich Ψ(,) =constant ae steamlines. Ho to oe? Fom the definition iti of the steamline that t the sloe at an oint along a steamline is gien b d d steamline elocit and elocit comonent along a steamline e 68

69 Steam Fnction 4/6 The change of Ψ(,) as e moe fom one oint (,) to a neal oint (+d,+d) is gien b d d d 0 Along a line of constant Ψ d d steamline d d d d d This is the definition fo a steamline. Ths, if e kno the steam fnction Ψ(,) e can lot lines of constant Ψto oide the famil of steamlines that ae helfl in isaliing the atten of flo. Thee ae an infinite nmbe of steamlines that make a aticla flo field, since fo each constant ale assigned to Ψa steamline can be dan. 0 69

70 Steam Fnction 5/6 The actal nmeical ale associated ith a aticla steamline is not of aticla significance, bt the change in the ale of Ψ is elated to the olme ate of flo. dq : olme ate of flo assing beteen the to steamlines. Flo nee cosses steamlines b definition. dq d d d q If q If q d 0, the flo 0, the flo d d is fom left to ight. is fom ight to left. 70

71 Steam Fnction 6/6 Ths the olme flo ate beteen an to steamlines can be itten as the diffeence beteen the constant ales of Ψ defining to steamlines. The elocit ill be elatiel high heee the steamlines ae close togethe, and elatiel lo heee the steamlines ae fa aat. 7

72 Eamle 6.3 Steam Fnction The elocit comonent in a stead, incomessible, ibl to dimensional flo field ae 4 Detemine the coesonding steam fnction and sho on a sketch seeal steamlines. Indicate the diection of glo along the steamlines. 7

73 Eamle 6.3 Soltion Fom the definition of the steam fnction 4 Ψ=0 f () () f () C Fo simlicit, e set C=0 / Ψ 0 73

74 elocit Potential Φ( /3 Φ(,,,t) t) / The steam fnction fo to-dimensional incomessible flo is Ψ(,) Fo an iotational flo, the elocit comonents can be eessed in tems of a scala fnction Φ(,,,t) as hee Φ(,,,t) is called the elocit otential. 0 74

75 elocit Potential Φ( /3 Φ(,,,t) t) / In ecto fom Fo an incomessible flo 0 Also called a otential flo Fo incomessible, iotational flo 0 Lalace s eqation Lalacian oeato 75

76 elocit Potential Φ( 3/3 Φ(,,,t) t) 3/ Iniscid, incomessible, iotational fields ae goened b Lalace s eqation. This te flo is commonl called a otential flo. To comlete the mathematical fomlation of a gien oblem, bonda conditions hae to be secified. These ae sall elocities secified on the bondaies of the flo field of inteest. 76