ME3560 Flid Mechanics Fall 08 ME 3560 Flid Mechanics Analsis of Flid Flo Analsis of Flid Flo
ME3560 Flid Mechanics Fall 08 6. Flid Elemen Kinemaics In geneal a flid paicle can ndego anslaion, linea defomaion oaion and angla defomaion. Analsis of Flid Flo
ME3560 Flid Mechanics Tanslaion Fall 08 The simples pe of moion ha a flid elemen can ndego is anslaion. In a small ime ineal δ a paicle locaed a poin O ill moe o poin O. If all poins in he elemen hae he same eloci (hich is onl e if hee ae no eloci gadiens), hen he elemen ill simpl anslae fom one posiion o anohe. In geneal, hee ae eloci gadiens hich esl in defomaion and oaion of he elemen as i moes. Analsis of Flid Flo 3
ME3560 Flid Mechanics Linea Defomaion Fall 08 Conside he effec of a single eloci gadien, /, on a small cbe haing sides δ,δ, and δ. Le he componen of eloci of O and B is. Then a neab poins A and C he componen of he eloci can be epessed as +( / )δ. This diffeence in eloci cases a seching of he olme elemen b an amon ( / )(δ)(δ) ding he sho ime ineal δ in hich line OA seches o OA andbc o BC. Analsis of Flid Flo
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 5 V Changein The coesponding change in he oiginal olme, V= is d V d V ) / ( lim ) ( 0 The ae a hich he olme, V, is changing pe ni olme de o he gadien / is d V d V ) (
ME3560 Flid Mechanics Fall 08 Simila analsis fo he and diecions ih eloci gadiens / and /, especiel esls in he folloing epession fo he ae a hich he olme, V, is changing pe ni olme: d( V ) V d V This ae of change of he olme pe ni olme is called he olmeic dilaaion ae. The olme of a flid ma change as he elemen moes fom one locaion o anohe in he flo field. Fo an incompessible flid he olmeic dilaaion ae is eo Analsis of Flid Flo 6
ME3560 Flid Mechanics Fall 08 Vaiaions in he eloci in he diecion of he componen iself, as epesened b he deiaies /, /, and /, simpl case a linea defomaion of he elemen. Coss deiaies, sch as / and /, ill case he elemen o oae and geneall o ndego an angla defomaion, hich changes he shape of he elemen. Analsis of Flid Flo 7
ME3560 Flid Mechanics Angla Moion and Defomaion Conside moion in he plane (esls can be eended o 3 D). The eloci aiaion ha cases oaion and angla defomaion is shon Fige (a). In δ he line line segmens OA and OB ill oae hogh he angles δα and δβ o he ne posiions OA and OB, (Fige (b)). The angla eloci of line OA, OA, is Fo small angles: an Analsis of Flid Flo OA ( lim 0 / ) Fall 08 8
ME3560 Flid Mechanics Theefoe he angla eloci of segmen OA is OA ( / ) lim 0 OA If / is posiie OA is coneclockise. Fall 08 Analsis of Flid Flo 9
ME3560 Flid Mechanics Fo segmen OB he angla eloci is Whee an OB lim 0 ( / ) ( / ) OB lim 0 OB If / is posiie OB is clockise. Fall 08 Analsis of Flid Flo 0
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 The oaion,, of he elemen abo he ais is defined as he aeage of he angla elociies OA and OB of he o mall pependicla lines OA and OB. Ths, if coneclockise oaion is consideed o be posiie: In a simila analsis he angla elociies in he and diecions can be fond o be:
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 k j i V V cl k j i / / / k j i V is kno as he oici: If V=0: he angla eloci and oici ae eo and an ioaional flo is pesen. If = 0 (Ioaional flo) and hen i is a POTENTIAL FLOW hee: V is a scala fncion.
ME3560 Flid Mechanics In addiion o he oaion associaed ih he deiaies / and /, hese deiaies can case he flid elemen o ndego an angla defomaion, hich esls in a change in shape of he elemen. The change in he oiginal igh angle fomed b he lines OA and OB is emed he sheaing sain, δγ. Analsis of Flid Flo Fall 08 δγ is posiie if he oiginal igh angle is deceasing. The ae of change of δγ is called he ae of sheaing sain o he ae of angla defomaion: lim 0 ( / lim 0 ) ( / ) 3
ME3560 Flid Mechanics 6. Conseaion of Mass Mass accmlaed pe ni ime ihin he CV = Mass flo ae eneing he CV Mass accmlaed pe ni ime ihin he CV Mass flo ae leaing he CV Fall 08 m = Ne mass flo ae cossing he CS Analsis of Flid Flo ( ) ( ) ( ) = 4
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 5 Ths, he diffeenial fom of he mass conseaion eqaion is: 0 ) ( ) ( ) ( 0 ) ( V If he flo is incompessible: 0 V 0
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 6 Conseaion of Mass Clindical Coodinaes k j i V Veloci in Caesian coodinaes: e e e V Veloci in Clindical coodinaes: 0 ) ( ) ( ) ( Conseaion of mass in clindical coodinaes 0 ) ( Fo incompessible flo:
ME3560 Flid Mechanics The Seam Fncion 0 Fall 08 Assming sead, incompessible, o-dimensional flo, he conini eqaion becomes: Inodce a coninos scala fncion ψ(, ), seam fncion, sch ha: This definiion of and in ems of saisfies he conini eqaion. d d d If = consan, he eqaion of seam lines is obained: Ths, = consan lines ae STREAMLINES d d d d d Analsis of Flid Flo 7
ME3560 Flid Mechanics Fall 08 The acal nmeical ale associaed ih a paicla seamline is no of paicla significance. Hoee he change in he ale of ψ is elaed o he olme floae. Conside o closel spaced seamlines, he loe seamline is ψ, he ppe one ψ + dψ. Le dq epesen he olme ae of flo (pe ni idh pependicla o he plane) passing beeen he o seamlines. Noe ha flo nee cosses seamlines, since b definiion he eloci is angen o he seamline. Analsis of Flid Flo 8
ME3560 Flid Mechanics Fom conseaion of mass, he inflo, dq, cossing he abia sface AC ms eqal he ne oflo hogh sfaces AB and BC: dq dq d d d d dq d dq d The olmeic flo ae is he elaie diffeence beeen o seam fncion ales. Fall 08 Analsis of Flid Flo 9
ME3560 Flid Mechanics Seam Fncion: Sead, Compessible Flo Fall 08 Seam Fncion Pola Coodinaes =(, ) In clindical coodinaes he conini eqaion fo incompessible, plane, o-dimensional flo edces o: ( ) 0 And is defined as: Analsis of Flid Flo 0
ME3560 Flid Mechanics Fall 08 6.3 Conseaion of Linea Momenm As deemined in he peios chape, he momenm eqaion is: c V dv cs V ( V n) da Ne, his eqaion is applied o an infiniesimal elemen of flid. The igh hand side of he eqaion can be ien as V ( V ) V F Analsis of Flid Flo
ME3560 Flid Mechanics Fall 08 6.3 Conseaion of Linea Momenm Sface foces acing on a diffeenial elemen of flid in he diecion: Analsis of Flid Flo F s
ME3560 Flid Mechanics A simila analsis fo he and diecions ields: F F s s Fall 08 Analsis of Flid Flo 3
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 4 6.3 Conseaion of Linea Momenm The momenm applied o a conol olme saes ha he foces acing on he CV eqal he momenm accmlaed in he CV mins he ne fl of momenm cossing he CS. g g g
ME3560 Flid Mechanics Fall 08 6.4 Iniscid Flo Sheaing sesses deelop in a moing flid becase of he iscosi of he flid. Fo some common flids (ai and ae), he iscosi is small. I is easonable o assme ha nde some cicmsances iscos effecs can be negleced (and hs sheaing sesses). Flo fields in hich he sheaing sesses ae assmed o be negligible ae said o be iniscid, noniscos, o ficionless. Fo flids in hich hee ae no sheaing sesses he nomal sess a a poin is independen of diecion ha is, σ = σ = σ. Analsis of Flid Flo p 5
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 6 Fo an iniscid flo, = 0, and σ = σ = σ = p, he geneal eqaions of moion become Ele s Eqaions of Moion p g p g p g V V V p g ) (
ME3560 Flid Mechanics Poenial Flo Fo an ioaional flo he angla eloci is eo = ½V = 0. Theefoe i is possible o define he eloci in ems of a scala fncion (,,, )as Noice ha fo ioaional flo: V V 0 Theefoe his definiion of he eloci in ems of saisfies he condiion of ioaionalli. is called eloci poenial. Fall 08 Analsis of Flid Flo 7
ME3560 Flid Mechanics Fall 08 The eloci poenial is a conseqence of he ioaionali of he flo field. The seam fncion is a conseqence of conseaion of mass. The eloci poenial can be defined fo a geneal hee-dimensional flo. The seam fncion is esiced o o-dimensional flos. Fo an Incompessible flo: V = 0 0 The peios eqaion is he goening eqaion fo poenial flo. ( )= ( ) is called Laplacian opeao. 0 Analsis of Flid Flo 8
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 9 e e e () () () () In clindical coodinaes he Del opeao is: Ths: e e e e e e V Since I follos Ths, he Laplace eqaion in clindical coodinaes is: 0
ME3560 Flid Mechanics 6.5 Some Basic, Plane Poenial Flos Laplace's eqaion is a linea paial diffeenial eqaion. Fall 08 Since i is linea, aios solions can be added o obain ohe solions ha is, if (,, ) and (,, ) ae o solions o Laplace's eqaion, hen 3 = + is also a solion. Ths, basic solions can be combined o obain moe complicaed and ineesing solions. Fo a D incompessible flo 0 Analsis of Flid Flo 30
ME3560 Flid Mechanics If he flo is ioaional ( = / / = 0) Analsis of Flid Flo 0 0 0 As shon ne (, ) and (, ) ae ohogonal fncions d d d d d d d d d d Fall 08 3
ME3560 Flid Mechanics Fo (, ) = cons and (, ) = cons d d d 0 d d d 0 d d d d Fall 08 Ths, fo = cons and = cons he lines ae pependicla geneaing a Flo Ne Analsis of Flid Flo 3
ME3560 Flid Mechanics Fall 08 Unifom Flo U ( cos sin) U ( cos sin) Analsis of Flid Flo 33
ME3560 Flid Mechanics Fall 08 Soce and Sink (Radial Flo) m 0 m is he soce/sink sengh Is nis ae m /s (olme flo ae pe ni idh 0 0 m ln c m c Analsis of Flid Flo 34
ME3560 Flid Mechanics Ideal Voe 0 k Analsis of Flid Flo k k is he oe sengh 0 k ln Fall 08 The angenial eloci aies inesel ih he disance fom he oigin. A singlai occs a = 0 (hee he eloci becomes infinie). 35
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 The ciclaion, Γ, is a concep commonl associaed ih oe moion Γ is defined as he line inegal of he angenial componen of he eloci aken aond a closed ce in he flo field Fo a fee oe V ds c d 0 V ds d d d d d d c d kd k Then and ae epessed as: ln is ofen sefl hen ealaing he foces deeloped on bodies immesed in moing flids. 36
ME3560 Flid Mechanics Doble Fall 08 A doble is fomed b combining a soce and sink ih eqal sengh. The soce and he sink ae appoached in a a ha mdisance = cons Ths, he disance is deceased hile m inceases. Resling in: k sin k cos k ma Analsis of Flid Flo 37
ME3560 Flid Mechanics Fall 08 Sd Speposiion of Basic Poenial Flos: Soce + Unifom Flo Half bod Soce + Sink + Unifom Flo Rankine Oal Unifom Flo + Doble Flo aond a clinde (no lif). Unifom Flo + Doble + Voe Flo aond a clinde ih lif Analsis of Flid Flo 38
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 39 6.8 Viscos Flo Fo incompessible Neonian flids i is knon ha he sesses ae lineal elaed o he aes of defomaion and can be epessed in Caesian coodinaes as: Nomal sesses: ; p ; p p Shea sesses: ; ; ;
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 40 g g g Sbsiing hese peios epessions ino he momenm eqaion (diffeenial fom) shon belo: The Naie Sokes Eqaions ae deemine as: V g p D V D
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 4 g p g p g p
ME3560 Flid Mechanics 6.9 Some Simple Solions fo Lamina, Viscos, Incompessible Flids Sead, Lamina Flo beeen Fied Paallel Plaes Flo beeen he o hoional, infinie paallel plaes. Flid paicles moe in he diecion paallel o he plaes = = 0. Fom conini eqaion: / = 0. Thee old be no aiaion of in he diecion fo infinie plaes Fo sead flo / = 0 so ha = (). Unde hese condiions he Naie Sokes eqaions edce o Fall 08 Analsis of Flid Flo 4
ME3560 Flid Mechanics Analsis of Flid Flo Fall 08 43 V g p D V D g p g p g p
ME3560 Flid Mechanics Coee Flo Anohe simple paallel-plae flo can be deeloped b fiing one plae and leing he ohe plae moe ih a consan eloci, U. The Naie Sokes eqaions edce o: 0 0 p (0) 0 ( b) U Fall 08 Analsis of Flid Flo 44
ME3560 Flid Mechanics Sead, Lamina Flo in Cicla Tbes Fall 08 Conside a sead, incompessible, lamina flo hogh a saigh cicla be of consan coss secion of adis R (Hagen Poiseille o Poiseille flo). Assme ha he flo is paallel o he alls so ha = 0 and = 0. Fo sead, aismmeic flo, he eloci is onl a fncion of. Unde he condiions descibed aboe, he conini eqaion: ( Analsis of Flid Flo ) ( ) ( ) Becomes: 0 0 45
ME3560 Flid Mechanics Wheeas he momenm eqaion in clindical coodinaes Fall 08 Resls in: : : p 0 g p 0 g : p 0 Analsis of Flid Flo 46