Chapter 6 Differential Analysis of Fluid Flow

Transcription

1 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall Chaper 6 Differenial Analysis of Flid Flow Flid Elemen Kinemaics Flid elemen moion consiss of ranslaion, linear deformaion, roaion, and anglar deformaion. Types of moion and deformaion for a flid elemen. Linear Moion and Deformaion: Translaion of a flid elemen Linear deformaion of a flid elemen 1

2 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall 006 Change inδ : δ δ x ( δ y δ z ) δ x he rae a which he volme δ is changing per ni volme de o he gradien /x is 1 d( δ ) ( x) δ lim δ d δ 0 δ x If velociy gradiens v/y and w/z are also presen, hen sing a similar analysis i follows ha, in he general case, 1 d ( δ ) v w V δ d x y z This rae of change of he volme per ni volme is called he volmeric dilaaion rae. Anglar Moion and Deformaion For simpliciy we will consider moion in he x y plane, b he resls can be readily exended o he more general case. Anglar moion and deformaion of a flid elemen The anglar velociy of line OA, ω OA, is

3 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall ω OA δα lim δ 0 δ For small angles ( v x) δxδ v anδα δα δ δ x x so ha ( v x) δ v ωoa lim δ 0 δ x Noe ha if v/x is posiive, ω OA will be conerclockwise. Similarly, he anglar velociy of he line OB is δβ ωob lim δ 0 δ y In his insance if /y is posiive, ω OB will be clockwise. The roaion, ω z, of he elemen abo he z axis is defined as he average of he anglar velociies ω OA and ω OB of he wo mally perpendiclar lines OA and OB. Ths, if conerclockwise roaion is considered o be posiive, i follows ha 1 v ω z x y Roaion of he field elemen abo he oher wo coordinae axes can be obained in a similar manner: 1 w v ω x y z 3

4 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall w ω y z x The hree componens, ω x,ω y, and ω z can be combined o give he roaion vecor, ω, in he form: 1 1 ω ωxi ωyj ωzk crlv V since i j k 1 1 V x y z v w 1 w v 1 w 1 v i j k y z z x x y The voriciy, ζ, is defined as a vecor ha is wice he roaion vecor; ha is, ς ω V The se of he voriciy o describe he roaional characerisics of he flid simply eliminaes he (1/) facor associaed wih he roaion vecor. If V 0, he flow is called irroaional. In addiion o he roaion associaed wih he derivaives /y and v/x, hese derivaives can case he flid elemen o ndergo an anglar deformaion, which resls in a change in shape of he elemen. The change in he original righ angle formed by he lines OA and OB is ermed he shearing srain, δγ, 4

5 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall δγ δα δβ The rae of change of δγ is called he rae of shearing srain or he rae of anglar deformaion: δγ ( v x) δ ( y) δ v γ lim lim δ 0δ δ 0 δ x y The rae of anglar deformaion is relaed o a corresponding shearing sress which cases he flid elemen o change in shape. The Coniniy Eqaion in Differenial Form The governing eqaions can be expressed in boh inegral and differenial form. Inegral form is sefl for large-scale conrol volme analysis, whereas he differenial form is sefl for relaively small-scale poin analysis. Applicaion of RTT o a fixed elemenal conrol volme yields he differenial form of he governing eqaions. For example for conservaion of mass ρv A CS CV ρ dv ne oflow of mass rae of decrease across CS of mass wihin CV 5

6 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall Consider a cbical elemen oriened so ha is sides are o he (x,y,z) axes ( ρ) dydz ρ x ole mass flx inle mass flx ρdydz Taylor series expansion reaining only firs order erm We assme ha he elemen is infiniesimally small sch ha we can assme ha he flow is approximaely one dimensional hrogh each face. The mass flx erms occr on all six faces, hree inles, and hree oles. Consider he mass flx on he x faces x ρ ( ρ) dydz x ρdydz ( ) dydz x ρ V flx oflx inflx Similarly for he y and z faces yflx ( ρv)dydz y z flx ( ρw)dydz z 6

7 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall The oal ne mass oflx ms balance he rae of decrease of mass wihin he CV which is ρ dydz Combining he above expressions yields he desired resl ρ ( ρ) ( ρv) ( ρw) dydz 0 x y z dv ρ ( ρ) x ( ρv) y ( ρw) z 0 per ni V differenial form of coniniy eqaions ρ ( ρv) 0 ρ V V ρ Dρ ρ V D 0 D D V Nonlinear 1 s order PDE; ( nless ρ consan, hen linear) Relaes V o saisfy kinemaic condiion of mass conservaion Simplificaions: 1. Seady flow: ( ρv) 0. ρ consan: V 0 7

8 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall v w i.e., 0 x y z 3D x v y 0 D The coniniy eqaion in Cylindrical Polar Coordinaes The velociy a some arbirary poin P can be expressed as V v e v e v e r r θ θ z z The coniniy eqaion: ρ 1 ( rρvr) 1 ( ρvθ ) ( ρv z) 0 r r r θ z For seady, compressible flow 1 ( rρvr) 1 ( ρvθ ) ( ρvz) 0 r r r θ z For incompressible flids (for seady or nseady flow) 1 ( rvr ) 1 vθ vz 0 r r r θ z 8

9 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall The Sream Fncion Seady, incompressible, plane, wo-dimensional flow represens one of he simples ypes of flow of pracical imporance. By plane, wo-dimensional flow we mean ha here are only wo velociy componens, sch as and v, when he flow is considered o be in he x y plane. For his flow he coniniy eqaion redces o v 0 x y We sill have wo variables, and v, o deal wih, b hey ms be relaed in a special way as indicaed. This eqaion sggess ha if we define a fncion ψ(x, y), called he sream fncion, which relaes he velociies as ψ ψ, v y x hen he coniniy eqaion is idenically saisfied: ψ ψ ψ ψ 0 x y y x xy xy Velociy and velociy componens along a sreamline 9

10 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall Anoher pariclar advanage of sing he sream fncion is relaed o he fac ha lines along which ψ is consan are sreamlines.the change in he vale of ψ as we move from one poin (x, y) o a nearby poin (x, y dy) along a line of consan ψ is given by he relaionship: ψ ψ dψ dy v dy 0 x y and, herefore, along a line of consan ψ dy v The flow beween wo sreamlines The acal nmerical vale associaed wih a pariclar sreamline is no of pariclar significance, b he change in he vale of ψ is relaed o he volme rae of flow. Le dq represen he volme rae of flow (per ni widh perpendiclar o he x y plane) passing beween he wo sreamlines. ψ ψ dq dy v dy dψ x y Ths, he volme rae of flow, q, beween wo sreamlines sch as ψ1 and ψ, can be deermined by inegraing o yield: 10

11 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall q ψ d ψ1 ψ ψ ψ 1 In cylindrical coordinaes he coniniy eqaion for incompressible, plane, wo-dimensional flow redces o 1 ( rvr ) 1 vθ 0 r r r θ and he velociy componens, v r and v θ, can be relaed o he sream fncion, ψ(r, θ), hrogh he eqaions 1 ψ ψ vr, vθ r θ r Navier-Sokes Eqaions Differenial form of momenm eqaion can be derived by applying conrol volme form o elemenal conrol volme The differenial eqaion of linear momenm: elemenal flid volme approach 11

12 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall d F ρvd VρV da d 1-D flow approximaion CV CS d ( ρ V)dydz d ( mv i i) ( mv i i) where o m ρav ρdydz x-face mass flx in ( ρv) ( ρvv) ( ρwv) dydz x y z x-face y-face z-face combining and making se of he coniniy eqaion yields DV DV V F ρ dydz V V D D where F F body Fsrface Body forces are de o exernal fields sch as graviy or magneics. Here we only consider a graviaional field; ha is, F and dfgrav ρgdydz g gkˆ for g z body 1

13 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall i.e., f body ρgkˆ Srface forces are de o he sresses ha ac on he sides of he conrol srfaces symmeric (σ ij σ ji ) σ ij - pδ ij τ ij nd order ensor δ ij 1 i j normal pressre viscos sress δ ij 0 i j -pτ xx τ xy τ xz τ yx -pτ yy τ yz τ zx τ zy -pτ zz As shown before for p alone i is no he sresses hemselves ha case a ne force b heir gradiens. x y z df x,srf ( ) ( σ ) ( ) dydz σ xx p ( τxx ) ( τxy ) ( τxz ) dydz x x y z This can be p in a more compac form by defining τ x τxxî τxy ĵ τxzkˆ vecor sress on x-face and noing ha p df x,srf τx dydz x xy σ xz 13

14 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall f x,srf p τx per ni volme x similarly for y and z p f y,srf τy τ y τyxî τyy ĵ τyzkˆ y f z,srf p τz τ z τzxî τzy ĵ τzzkˆ z finally if we define τ τ î τ ĵ τ kˆ hen ij x y z f srf p τij σij σ ij pδij τij Ping ogeher he above resls DV f f body f srf ρ D f body ρgkˆ f srface p τij DV V a V V D ρa ρgkˆ p τij ineria body force force srface srface force de o force de de o viscos graviy o p shear and normal sresses 14

15 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall For Newonian flid he shear sress is proporional o he rae of srain, which for incompressible flow can be wrien τ ij µε ij µ coefficien of viscosiy ε ij rae of srain ensor x v y x w z x v x y v y v w z y w x z w v y z w z d τ µ 1-D flow dy rae of srain ρ a ρgkˆ p µε ( ) ij µ x i ( ε ) µ V ij ρ a ρgkˆ p µ V 15

16 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall ( ) V z p a µ γ ρ Navier-Sokes Eqaion V 0 Coniniy Eqaion For eqaions in for nknowns: V and p Difficl o solve since nd order nonlinear PDE µ ρ z y x x p z w y v x µ ρ z v y v x v y p z v w y v v x v v µ ρ z w y w x w z p z w w y w v x w w 0 z w y v x Navier-Sokes eqaions can also be wrien in oher coordinae sysems sch as cylindrical, spherical, ec. There are abo 80 exac solions for simple geomeries. For pracical geomeries, he eqaions are redced o algebraic form sing finie differences and solved sing compers. Exac solion for laminar flow in a pipe (neglec g for now) x: y: z:

17 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall se cylindrical coordinaes: v x v r v (r) only v θ w 0 r Coniniy: ( rv) 0 rv consan c v c/r v(r 0) 0 c 0 i.e., v 0 Momenm: D p 1 µ D x x r r ρ 1 θ r r w p 1 ρ v µ z r r θ x r r r 1 r r r r 1 p µ x λ r r λ r A λ 4 () r r Aln r B 17

18 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall (r 0) A 0 λ (r r o ) 0 () r ( r r ) 1 p i.e. () r ( r r ) 4 o o parabolic velociy profile 4µ x Differenial Analysis of Flid Flow We now discss a cople of exac solions o he Navier- Sokes eqaions. Alhogh all known exac solions (abo 80) are for highly simplified geomeries and flow condiions, hey are very valable as an aid o or ndersanding of he characer of he NS eqaions and heir solions. Acally he examples o be discssed are for inernal flow (Chaper 8) and open channel flow (Chaper 10), b hey serve o nderscore and display viscos flow. Finally, he derivaions o follow ilize differenial analysis. See he ex for derivaions sing CV analysis. Coee Flow bondary condiions 18

19 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall Firs, consider flow de o he relaive moion of wo parallel plaes Coniniy 0 (y) x v o p p d 0 Momenm 0 µ x y dy or by CV coniniy and momenm eqaions: ρ 1 y ρ y 1 ( ) Fx ρv da ρq 1 0 dp dτ p y p x y τ x τ dy x 0 dy d τ 0 dy d i.e. d µ 0 dy dy d µ 0 dy from momenm eqaion d µ C dy 19

20 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall C y D µ (0) 0 D 0 () U C U y d τ µ U µ µ U consan dy Generalizaion for inclined flow wih a consan pressre gradien Coniniy 0 x x Momenm 0 ( p γz) µ d dy (y) v o p 0 y i.e., d dh µ γ h p/γ z consan dy 0

21 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall dz plaes horizonal 0 dz plaes verical -1 which can be inegraed wice o yield µ d dy γ dh y A dh y µ γ Ay B now apply bondary condiions o deermine A and B (y 0) 0 B 0 (y ) U µ U γ dh A A µ U γ dh γ dh y (y) µ γ dh µ 1 µ U γ µ U y y dh ( ) y This eqaion can be p in non-dimensional form: γ dh y y y 1 U µ U define: P non-dimensional pressre gradien 1

22 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall 006 γ dh p h z µ U γ Y y/ P Y(1 Y) Y U parabolic velociy profile γ 1 dp µ U γ z dz

23 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall U Py Py y q dy 0 q U 0 [ ] dy U 0 P P y y P P 3 y dy U P 6 1 γ 1µ dh U For laminar flow < 1000 ν Re cri 1000 The maximm velociy occrs a he vale of y for which: d d P P y dy dy U 3

24 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall y ( P max P P for U 0, y / ( y ) UP 4 U max max U 4P noe: if U 0: max P 6 P 4 3 The shape of he velociy profile (y) depends on P: dh 1. If P > 0, i.e., < 0 he pressre decreases in he direcion of flow (favorable pressre gradien) and he velociy is posiive over he enire widh γ dh γ d p γ z dp γsin θ dp a) < 0 dp b) < γsin θ dh 1. If P < 0, i.e., > 0 he pressre increases in he direcion of flow (adverse pressre gradien) and he velociy over a porion of he widh can become negaive (backflow) near he saionary wall. In his 4

25 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall case he dragging acion of he faser layers exered on he flid paricles near he saionary wall is insfficien o over come he inflence of he adverse pressre gradien dp dp γsin θ > 0 > γsin θ or γ sin θ < dp dh. If P 0, i.e., 0 he velociy profile is linear U y dp a) 0 and θ 0 dp b) γsin θ is no appropriae U For U 0 he form PY( 1 Y) Y UPY(1-Y)UY γ dh µ γ µ Y( 1 Y) UY dh Now le U 0: Y( 1 Y) Noe: we derived his special case 5

26 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall Shear sress disribion Non-dimensional velociy disribion * ( 1 ) where U γ dh P µ U y Y * P Y Y Y U is he non-dimensional velociy, is he non-dimensional pressre gradien is he non-dimensional coordinae. Shear sress d τ µ dy In order o see he effec of pressre gradien on shear sress sing he non-dimensional velociy disribion, we define he non-dimensional shear sress: Then where * τ τ 1 ρ U * τ µ 1 ( ) ( ) 1 Ud U µ d ρu d y ρu dy µ ( PY P 1) ρu µ ( PY P 1) ρu A PY P 1 ( ) µ A > 0 is a posiive consan. ρu So he shear sress always varies linearly wih Y across any secion. * 6

27 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall A he lower wall ( Y 0) : A he pper wall ( 1) ( 1 ) * τ lw A P * τ ( 1 ) Y : w A P For favorable pressre gradien, he lower wall shear sress is always posiive: 1. For small favorable pressre gradien ( 0< P < 1) : * * τ > 0 and τ > 0 lw. For large favorable pressre gradien ( 1) * lw w τ > 0 and τ < 0 * w P > : τ τ ( 0< P < 1) ( P > 1) For adverse pressre gradien, he pper wall shear sress is always posiive: 1. For small adverse pressre gradien ( 1< P < 0) : * * τ > 0 and τ > 0 lw. For large adverse pressre gradien ( 1) * lw w * τ < 0 and τ > 0 w P < : 7

28 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall τ τ ( 1< P < 0) ( P < 1) For U 0, i.e., channel flow, he above non-dimensional form of velociy profile is no appropriae. Le s se dimensional form: γ dh γ Y( 1 Y) dh y( y) µ µ Ths he flid always flows in he direcion of decreasing piezomeric pressre or piezomeric head becase γ 0, y 0 µ > > and y > 0. So if dh is negaive, is posiive; if dh is posiive, is negaive. Shear sress: d γ dh 1 τ µ y dy Since 1 y > 0, he sign of shear sress τ is always opposie o he sign of piezomeric pressre gradien dh, and he magnide of τ is always maximm a boh walls and zero a cenerline of he channel. 8

29 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall dh For favorable pressre gradien, 0 <, τ > 0 dh For adverse pressre gradien, 0 >, τ < 0 τ τ Flow down an inclined plane dh 0 < dh 0 > niform flow velociy and deph do no change in x-direcion d Coniniy 0 9

30 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall x y-momenm ( p γz) y x-momenm 0 ( p γz) µ d dy 0 hydrosaic pressre variaion d µ γsin θ dy d dy γ µ sin θy c dp 0 γ y sin θ µ Cy D d dy y d 0 γ µ sin θd c c γ sin θd µ (0) 0 D 0 γ y sin θ µ γ sin θdy µ γ µ sin θ y( d y) 30

31 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall gsin θ ν (y) y( d y) q d 3 γ y dy sin θ dy 0 µ 3 d 0 discharge per ni widh 1 γ d 3 sin θ 3 µ V avg q 1 γ gd d sin θ sin θ d 3 µ 3ν in erms of he slope S o an θ sin θ V gd S 3 ν o Exp. show Re cri 500, i.e., for Re > 500 he flow will become rblen p y γ cosθ Re Vd cri 500 ν p γ cosθ y C ( d) p γ cosθd C p o 31

32 57:00 Mechanics of Flids and Transpor Processes Chaper 6 Professor Fred Sern Fall i.e., p γ cosθ( d y) po * p(d) > p o * if θ 0 p γ(d y) p o enire weigh of flid imposed if θ π/ p p o no pressre change hrogh he flid 3