0//8 OUTLINE FOR Chapr - AERODYNAMIC W-- Basc Laws Analss of an problm n fld mchancs ncssarl nclds samn of h basc laws gornng h fld moon. Th basc laws, whch applcabl o an fld, ar: Consraon of mass Conn Eqaon Consraon of anglar Momnm - Nwon s scond law of moon. Consraon of Enrg Th frs law of hrmodnamcs
0//8 BAIC LAW FOR A YTEM - Consraon of Momnm Nwon s cond Law ang ha h sm of all rnal lforc acng on h ssm s qal o h m ra of chang of lnar momnm of h ssm. dp F d ssm DP D D D Whr h lnar momnm of h ssm ssm d FOX Ch 4 TRE FIELD Th forcs ac on fld parcls: Bod forcs gra, lcromagnc. Th graaonal bod forc acng on an lmn of olm, d, s gn b whr ρ s h dns, g s local graaonal acclraon. rfac forcs prssr, scos. Th srfac forcs on a fld parcl lad o srsss: Normal o srfac F n >> normal srss σ n Tangn o srfac srfac F >> shar srss τ n n Fn lm A n 0 A n n F lm A n 0 A n FOX Ch 4
0//8 Th rss a a Pon Th sa of srss a a pon can b dscrbd compll b spcfng h srsss acng on an hr mall prpndclar plan hrogh h pon. Th srss a a pon s spcfd b h nn componns. FOX Ch 4 Noaon for rss I Th forc on h lmn δa s rsold no F, F, F X Th srsss on h lmn δa ar dfnd as: F F F lm lm lm A 0 A 0 A 0 A A A whr h frs sbscrp ndcas h plan on whch h srss acs, h scond sbscrp ndcas h drcon n whch h srss acs. FOX Ch 4
0//8 Noaon for rss II Th srsss on h lmn δa ar Th srsss on h lmn δa ar,,,, FOX Ch 4 Noaon for rss Tnsor Th frs nd of spcfs h srfac ornaon, whl h scond nd ndcas h drcon of h srfac forc componn. Two drcons ndcs ndd do spcf, so s a nsor of ran hang 9 componns n oal. Wh ro olm orq, h srss nsor s smmrc, and hnc has onl 6 ndpndn componns... = In gnral, Pn= = P =.g. P = + + B P = n.g. P = n + n + n FOX Ch 4 4
0//8 Wh ro olm orq, Pro h shar srss nsor s smmrc, and hnc has onl 6 ndpndn componns... = Consraon of Momnm I Basc Law for Consraon of Momnm D d Pd fd D whr P s srfac forc.g. prssr, scos srss pr n ara and f s bod forc.g. gra, lcromagnc forc c. pr n mass. Rnolds Transpor Thorm D D d d =ρ D D d P d f d d FOX Ch 4 5
0//8 6 Consraon of Momnm II Consraon of Momnm II d d d d D D f P d nd d P Gass horm d d d f nc h choc of olm b f Conn q s arbrar f ] [ ] [ f ] [ q. Consraon of Momnm III Consraon of Momnm III f ] [ ] [ D D D D snc D D f ] [ F m a Mass Follow a fld parcl and obsr s acclraon pr n olm N srfac forc pr n olm acng on h fld parcl N bod forc pr n olm acng on h fld parcl
0//8 BAIC LAW FOR A YTEM - Consraon of Enrg Th Frs Law of Thrmodnamcs Rqrng ha h nrg of ssm b consan. Q W Q W de de d ssm DE D D D ssm d Whr E s h oal nrg of h ssm and s h oal nrg of h ssm pr n olm s spcfc nrnal nrg, h spd, and g h hgh of a parcl hang mass dm. FOX Ch 4 Consraon of Enrg I D d Q W D Q q nd D D W [ P ds f d ] d Pds fd { [ ] } d Pd fd q nd q nd Pd nd d q nd q n ds q d 7
0//8 8 d q d f d d } ] [ { {} 0 d q q f ] [ ] [ ] [ LH ] [ LH q f q f RH q f f ] [ Consraon of Consraon of f ] [ Momnm Momnm q D D Condcon ha ransfr Conrson of Mchancal nrg no hrmal nrg parall rrsbl, parall rrrsbl Ra of chang of nrnal nrg
0//8 Dscsson of Consraon Eqaons W ha now 5 scalar qaons: Conn, momnm, and nrg qaon. B hr ar 7 nnowns: ρ,,, q and σ 9. Nd Cons rlaons o rla h srss nsor σ 9 o h dformaon nsor loc gradn and o rla h ha fl q o h mprar gradn T. 7 nnowns rdc o 6 whch ar ρ,,, and T. Now w ha 5 scalar qaons for 6 nnowns. W also nd qaon of sa : p =pρ,t, =ρ,t Two mor qaons ; on mor nnown p. Thrfor, w ha 7 qaons for 7 nnown whch ar ρ,,, T and p. MOTION OF A FLUID PARTICLE Fld Translaon: Th parcl mos from on pon o anohr. Fld Roaon: Th parcl roas abo an or all of h,, as. Fld Dformaon: Anglar Dformaon: Th parcl s angls bwn h sds chang. Lnar Dformaon: Th parcl s sds srch or conrac. 9
0//8 Pcoral Rprsnaon Roaon and Dformaon I Ra of shar 0
0//8 Roaon and Dformaon II Roaon and Dformaon II ξxηea Fld Roaon I Fld Roaon I Th parcl roaon s a cor qan and, n gnral. p q, g l l d d lm lm lm 0 0 0 d
0//8 Fld Roaon II Fld Roaon II w w w w Fld Roaon III orc Dfnng orc ζ whch Dfnng orc ζ whch s a masrmn of h s a masrmn of h roaon of a fld lmn roaon of a fld lmn as mos n h flow as mos n h flow fld: fld: w w w crl In clndrcal coordnas ssm: In clndrcal coordnas ssm: r r r r r r r r r
0//8 Ra of Roaon Arag ra of CCW roaon abo h fld lmn s cnrod: w w Prmaon mbol or s r f or,,,,,,, r,s, f ha h sam al whn an ndcs ohrws.., 0 or,,,,,,, r,s, f rs
0//8 Irroaonal and Roaonal flows crl 0 Irroaonal flow crl 0 roaonal flow W 8 ANGULAR ELOCITY AND ORTICITY Dfn anglar loc: Dfn orc 0 roaonal = 0 Irroaonal Roaonal flow Irroaonal flow AERODYNAMIC W--4 4
0//8 EXAMPLE OF ORTICITY AERODYNAMIC W--4. CIRCULATION Dfnon: C Crclaon Lf d From os horm Irroaonal flow Eampl: Crclaon n a nform flow = = 0 Irroaonal flow For arbrar clos cr C rroaonal AERODYNAMIC W--5 5
0//8 TREAMFUNCTION AND ELOCITY POTENTIAL ramfncon : dfnon loc Ponal : dfnon aomacall sasf conn qaon sramfncon proprs: aomacall sasf rroaonal condon loc ponal proprs:.. Rlaonshp bwn sramfncon and loc ponal = 0 AERODYNAMIC W--6 6
0//8 UMMARY. bsanal dra 5. ramfncon : sbsanal dra. ramln local dra conc dra. orc 6. loc Ponal : 0 roaonal = 0 Irroaonal 4. Crclaon Irroaonal flow AERODYNAMIC W--7 REIEW OF GOERNING EQUATION III Momnm Eqaon Phscal prncpl: Forc = m ra of chang of momnm Tm ra of chang of momnm Forc: Bod farcs: gra, lcromagnc forcs, or an ohr forcs whch ac a a dsanc on h fld nsd. rfac forcs: prssr and shar srss acng on h conrol srfac. Th ngral form of momnm qaons Toal scos forcs = AERODYNAMIC W--8 7
0//8 REIEW OF GOERNING EQUATION I For Drgnc horm Elr qaons for nscd flow Th dffrnal form of momnm qaons Nar-os qaons Inscd, Incomprssbl: consan P - For nsad D flow, comprssbl or ncomprssbl, scos or nscd - - P P w w AERODYNAMIC W--9 UMMARY For sad, ncomprssbl, nscd flow Conn qaon:,, w w 0 Momnm qaon: P w P w w w w P w w AERODYNAMIC W--0 8