ME 304 FLUID MECHANICS II

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1 ME 304 LUID MECHNICS II Pof. D. Haşme Tükoğlu Çankaya Uniesiy aculy of Engineeing Mechanical Engineeing Depamen Sping, 07

2 y du dy y n du k dy y du k dy n du du dy dy ME304

3 The undamenal Laws Epeience hae shown ha all fluid moion analysis mus be consisen wih he following fundamenal laws of naue. The law of conseaion of mass: Mass can be neihe ceaed no desoyed. I can only be anspoed o soed. Newon s hee laws of moion: - mass emains in a sae of equilibium, ha is, a es o moing a consan elociy, unless aced on by an unbalanced foce. - The ae of change of linea momenum of mass is equal o he ne foce acing on he mass. - ny foce acion has a foce eacion equal in magniude and opposie in diecion. The fis law of hemodynamics (law of conseaion of enegy) Enegy, like mass, can be neihe ceaed no desoyed. Enegy can be anspoed, changed in fom, o soed. The second law of hemodynamics: The enopy of he uniese mus incease o, in he ideal case, emain consan in all naual pocesses. The sae of posulae (law of popey elaions): The aious popeies of a fluid ae elaed. If a ceain minimum numbe (usually wo) of fluid s popeies ae specified, he emainde of he popeies can be deemined. Diffeenial esus Inegal omulaion We mus now conside he leel of deail of he esuling flow analysis. We mus choose beween a deailed poin by poin descipion and a global o lumped descipion. When a poin by poin (local) descipion is desied, fundamenal laws ae applied o an infiniesimal conol olume. The esul will be a se of diffeenial equaions wih he fluid elociy and pessue as dependen aiables and he locaion (, y, ) and ime as independen aiables. Soluion of hese diffeenial equaions, ogehe wih bounday condiions, will be wo funcions (, y,, ), and P(, y,, ) ha can ell us he elociy and pessue a eey poin. When global infomaion such as flow ae, foce and empeaue change beween inle and oule is desied, he fundamenal laws ae applied o a finie conol olume. The esul will be a se of inegal equaions. ME304 3

4 LUID STTICS Basic Equaion of fluid saics gad P g 0 dp d g p p 0 gh HYDOSTTIC OCE CTING ON PLNE SUBMEGED SUCE Inegal Mehod pd y ypd y pd pd Diecion of ypd pd Magniude of is nomal and owad he suface lgebaic mehod gh c ' I y yc c c y' y I y c c y c Pessue Pism Mehod k P d P P k and y Pd P ypd P ghd P yghd P P d X P yd Y G G ME304 4

5 ME304 5 BSIC EQUTIONS O INITE CONTOL OLUME (Equaions In Inegal om) 0 C CS d d Equaion of Conseaion of Mass (Coninuiy Equaion): C CS B S d d Linea Momenum Equaion: C g p s Benoulli Equaion: CS C shaf C s d d T d g Momen of Momenum Equaion: m T shaf Eule Tubine Equaion: Benoulli equaion subjec o esicions:. Seady flow. No ficion 3. Incompessible flow 4. low along a seamline C CS ohe shea s d p e d e W W W Q Enegy Equaion: h f g p g g p Eended Benouilli Equaion: eynolds Tanspo Equaion: C CS Sysem d d N

6 HED LOSS (PESSUE DOP) The head loss (pessue loss) in closed conduis can be died ino wo pa: ) Majo (ficion) loss: Losses due o iscous effecs on he duc wall. ) Mino (local) losses): Losses due o he flow hough ales, ees, elbows and ohe non-consan coss-secional aea poions of he sysem. Majo Head Loss (Pessue Dop) P f L d o lamina flow, ficion faco, L h f f d g f 64 e o ubulen flow, ficion faco, f f e, s / d h f P g icion facos fo ubulen flows ae gien in chas (Mood Diagam) o as coelaions. Mino (Local) Head Loss The mino head losses in aiable aea pas ae popoional o he elociy head of he fluid, i.e. h f k g Mino head losses fo ales, fiings and bends can be calculaed using he equialen lengh echnique, which may be gien by he following equaion: h f f L d e g Toal loss = (Majo loss) + (Mino loss) ME304 6

7 INTODUCTION TO DIEENTIL NLYSIS O LUID MOTION (Chape 5) In couse luid Mechanics I, we deeloped he basic equaions in inegal fom fo a finie conol olume. The inegal equaions ae paiculaly useful when we ae ineesed in he goss behaio of a flow and is effec on aious deices. Howee, he inegal appoach does no enable us o obain deailed poin by poin daa of he flow field. To obain his deailed knowledge, we mus apply he equaions of fluid moion in diffeenial fom. In his chape, we will deie fundamenal equaions in diffeenial fom and apply his equaions o simple flow poblems. EQUTION O CONSETION O MSS (CONTINUITY EQUTION) The applicaion of he pinciple of conseaion of mass o fluid flow yields an equaion which is efeed as he coninuiy equaion. We shall deie he diffeenial equaion fo conseaion of mass in ecangula and in cylindical coodinaes. ME304 7

8 ecangula Coodinae Sysem The diffeenial fom of he coninuiy equaion may be obained by applying he pinciple of conseaion of mass o an infiniesimal conol olume. The sies of he conol olume ae d, dy, and d. We conside ha, a he cene, O, of he conol olume, he densiy is and he elociy is uı j wk o he conol oule, equaion of conseaion of mass in inegal fom is d 0 d CS C To ealuae he fis em d in his equaion, we mus ealuae he mass CS flow ae oe each face of he conol olume. To be compleed in class The alues of he mass flues a each of si faces of he conol olume may be obained by using a Taylo seies epansion of he densiy and elociy componens abou poin O. o eample, a he igh face, Neglecing highe ode ems, we can wie and similaly, d d! d u d u d u d d ME304 8

9 Coesponding ems a he lef face ae To be compleed in class Table. Mass flu hough he conol suface of a ecangula diffeenial conol olume 9 d d d d u u d u u u d ME304

10 The ne ae of mass flu ou hough conol suface is CS u w d ddyd y The ae of change of mass inside he conol olume is gien by d C ddyd Theefoe, he coninuiy equaion in ecangula coodinae is u w 0 y Since he eco opeao,, in ecangula coodinaes, is gien by ı j k y 0 The coninuiy equaion may be simplified fo wo special cases.. o an incompessible flow, he densiy is consan, he coninuiy equaion becomes, 0. o a seady flow, he paial deiaies wih espec o ime ae eo, ha is. Then,. ME304 0

11 Eample: o a -D flow in he y plane, he elociy componen in he y diecion is gien by y y a) Deemine a possible elociy componen in he diecion fo seady flow of an incompessible fluid. How many possible componens ae hee? b) Is he deemined elociy componen in he -diecion also alid fo unseady flow of an incompessible fluid? To be compleed in class ME304

12 Eample: compessible flow field is descibed by k ai byj e Deemine he ae of change of he densiy a poin =3 m, y= m and = m fo =0. ME304

13 Deiaion of Coninuiy Equaion Cylindical Coodinae Sysem In cylindical coodinaes, a suiable diffeenial conol olume is shown in he figue. The densiy a cene, O, is and he elociy hee is e e e igue. Diffeenial conol olume in cylindical coodinaes. To ealuae CS d, we mus conside he mass flu hough each of he si faces of he conol suface. The popeies a each of he si faces of he conol suface ae obained fom Taylo seies epansion abou poin O. ME304 3

14 Table. Mass flu hough he conol suface of a cylindical diffeenial conol olume The ne ae of mass flu ou hough he conol suface is gien by The ae of change of mass inside he conol olume is gien by In cylindical coodinaes he coninuiy equaion becomes d dd d CS d dd d C 0 4 ME304

15 Diiding by gies o In cylindical coodinaes he eco opeao is gien by Then he coninuiy equaion can be wien in eco noaion as The coninuiy equaion may be simplified fo wo special cases:. o an incompessible flow, he densiy is consan, i.e.,. o a seady flow, 0 0 ) ( ) ( ) ( e e e 0 e e e e and Noe: 5 ME304

16 Deiaion of Coninuiy Equaion in Cylindical Coodinae syem Using eco om of he Equaion To be compleed in class ME304 6

17 Eample: Conside one-dimensional adial flow in he plane, chaaceied by = f() and = 0. Deemine he condiions on f() equied fo incompessible flow. ME304 7

18 STEM UNCTION O TWO-DIMENSIONL INCOMPESSIBLE LOW o a wo-dimensional flow in he y plane of he Caesian coodinae sysems, he coninuiy equaion fo an incompessible fluid educes o u 0 y If a coninuous funcion (, y, ), called seam funcion, is defined such ha u and y Then coninuiy equaion is saisfied eacly, since u 0 y y y Seamlines ae angen o he diecion of flow a eey poin in he flow field. Thus, if d is an elemen of lengh along a seamline, he equaion of seamline is gien by d 0 ( uı j) ( dı dyj) ( udy d) k Subsiuing fo he elociy componens of u and, in ems of he seam funcion udy d 0 a ceain insan of ime, 0, he seam funcion may be epessed as, y, ). his insan, he seamfuncion ( 0 d dy 0 y Compaing equaions () and (B), we see ha along insananeous seamline = consan. In he flow field, -, depends only on he end poins of inegaion, since he diffeenial equaion of is eac. d d d 0 dy y () (B) ME304 8

19 Now, conside he wo-dimensional flow of an incompessible fluid beween wo insananeous seamlines, as shown in he igue. The olumeic flow ae acoss aeas B, BC, DE, and D mus be equal, since hee can be no flow acoss a seamline. o a uni deph, he flow ae acoss B is Q y y udy y y dy y long B, = consan and Q y y dy y d dy y o a uni deph, he flow ae acoss BC is d. Theefoe, Q d d long BC, y = consan and Q d d. Theefoe, d d Thus, he olumeic flow ae pe uni deph beween any wo seamlines, can be epessed as he diffeence beween consan alues of defining he wo seamlines. ME304 9

20 In plane of he cylindical coodinae sysem, he incompessible coninuiy equaion educes o 0 The seamfuncion (,,) hen is defined such ha Eample: Conside he seam funcion gien by = y. ind he coesponding elociy componens and show ha hey saisfy he diffeenial coninuiy equaion. Then skech a few seamlines and sugges any pacical applicaions of he esuling flow field. ME304 0

Control Volume Derivation

Control Volume Derivation School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass