CHAPER 6 Derie differenil Conini, Momenm nd Energ eions form Inegrl eions for onrol olmes. imlif hese eions for -D sed, isenroi flo ih rible densi CHAPER 8 Wrie he D eions in erms of eloi oenil reding he hree eions of onini, momenm nd energ o one eion ih one deenden rible, he eloi oenil. CHAPER Mehod of Chrerisis e solion o he -D eloi oenil eion.
Gss's heorem - Diergene heorem rnsforms srfe inegrl ino olme inegrl r (Vd ol r ( Vd ol here : r (V is eor (d Grdien of of ol ( d ol ( i eor is slr slr is eor here :( is slr ( j ( k
CONINUIY EQUAION CONERVAIVE INEGRAL FORM V, eloi eor d onrol olme oen hermodnmi ssem region in se V d ol (b onenion mss inflo is d ol ne of ( V d d ol Conini Eion in inegrl (onserie form mss leing he onrol olme. hnge in mss inside he onrol olme ol
CONINUIY EQUAION CONERVAIVE INEGRAL FORM Gss's heorm rnsforms srfe inegrl ino olme inegrl r r Vd Vd ol here, (onrol olme mss ol ol d ol ling Gss's heorm o ( V b onenion mss inflo is. ne mss oflo r d he ne mss oflo erm,
CONINUIY EQUAION CONERVAIVE INEGRAL FORM ol d ol r (V nsed, ol (6.5 3- D, n flid, ( ( ( r (V ( sbsiing, d ol rible densi in, nd
( mv d F d Fores Bod Fore here f is he bod fore onsn Pressre Fore Visos Fore Momenm hnge inside he olme Chnge of MomenmChnge MONENUM EQUAION CONERVAIVE INEGRAL FORM hnge in momenm f Momenm ih ime ( ( V V d V ol ol d d d ol, d ol ( V d V ( V Bod Fore Pressre Fore Visos Fore ol ol f d ol d ol d d onrol olme oen hermodnmi ssem region in se d V, eloi eor
MONENUM EQUAION CONERVAIVE INEGRAL FORM ( ( V V d V sing Gss' s herom (6., Ad o oner he ol ( ( ( V V Vd ol differeniing, ol ol d ol A d ol hree srfe inegrls o olme inegrls. ol ol nd d ol f ol d ol f ( V ( V V f (d d ol d ol ol d ( d ol d ol ol d ol
MOMENUM EQUAION nsed, 3D, n flid, rible densi ( ( f f f f V V V
resriing he momenm eion o Neonin flids for hih he flids sress is of deformion of for D, µ d d he flid - liner fnion of he hnge of he re eloi ih disne. µ µ 3 µ µ 3 µ µ 3 ( V ( V ( V µ µ µ
ENERGY EQUAION CONERVAIVE INEGRAL FORM Firs L Work Fore Veloi W shf Work Work Work He Inernl ressre bod isos ddiion Q E W E W Vol energ, ( ( f ( d V d V Ne Energ ino onrololme Chngein energ inside he onrololme d d olv U shf W isos W ( V d e Vol ressre V e W V bod d ol
( ( ( ( ( ( V V V V V (g V ( V V V (. d olv ( f V d V d d ol V e V e d Q W W W W E E Q W W W W E W E Q Firs L ol ol bod ressre isos shf olme onrol hnge in olme onrol ne in bod ressre isos shf
EQUAION UMMARY - 3D, isos, rible densi ENERGY f f f direions,, MOMENUM CONINUIY
EQUAION UMMARY - 3D, isos, rible densi ENERGY f f f direions,, MOMENUM CONINUIY D sed inomressible, inisid
BOUNDARY LAYER Prndl 94 Diide flo ino o regions ording o he fores h reil BOUNDARY LAYER hin ler ner ll isos fores s imron s ineril fores ignore D lrge, µ rerse inomresible er lrge momenm bondr ler eions, d d d d eions µ d d FREE REAM, µ, Poenil Flo isenroi, friionless irroionl, niform nd rllel
EQUAION UMMARY - 3D, isos, rible densi ENERGY f f f direions,, MOMENUM CONINUIY
-D, sed, inisid (isenroi, rible densi ENERGY direions,, MOMENUM d d d d d d d d CONINUIY ( (
VELOCIY POENIAL rede o one eion C Vdl for irroionl flo isenroi,, µ Vdl is indeenden of h n e differenil, deenden onl on osiion e differenil d( ( ( Vdl Vdl b omrison, ( ( d(vdl d d(vdl d i d j ( ( Vdl Vdl, d d d nd re fnions of he sme slr ni, define s, eloi oenil fnion V dl ( ( CHECK : Greens heorem, C sbsiing, ( ( d, dd d ( (, C
CONINUIY EQUAION - D sed, inisid, rble densi d d d d d d d d onin eion in erms of eloi oenil sbsie : ( d ( ( d ( ( d ( d ( ( d d d d d d,, d d nd ribles, densi ill be elimined b he momenm eions.
MOMENUM EQUAION mli direion b d d sine for irroionl flo, d d d d d d d d d sbsie : d d ( d ( d ( ( ( ( d for he direion eion, d
( (
( ( D (8.7,for onini eion, ino he sbsiing
(.7 d d d d d d d d (.6 d d d d d d d, d nd d e differenils for (.5 (, sbsiing, D (8.7,for
3 simlneos liner eions in ( d d ( d d ( - d d d d d ( d ( d d N D d d on hrerisi of he solion, When boh N nd D re is indeermine, is indeermine. N defines he hrerisi of he solion, C f(,. D defines roeries long he hrersii.
hrerisi longc K (M hrerisi longc K (M - Meer Fnion he Prndl dθ is V dv M d - ( ( d d dd ( dd ( nmeror N, θ υ θ υ θ ± ± ( ( ± ± si hreri si hreri M d d d d ( d d d d ( (d ( dd d ( D,denominor