ME 425: Aerodynamics

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1 ME 45: Aerodnamics Dr. A.B.M. Toiqe Hasan Proessor Deparmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET, Dhaka Lecre-7 Fndamenals so Aerodnamics oiqehasan.be.ac.bd oiqehasan@me.be.ac.bd Fndamenal principles in Aerodnamics Flid Mechanics:. Conseraion o mass. Conseraion o momenm 3. Conseraion o energ i compressible lows

2 Mass can neiher be creaed nor desroed. Consider a small olme o space conrol olme hrogh which a lid is lowing. For simplici, a D low is considered and he conrol olme is bonded b he sraces and asshowninigre. Accordingohelaw,he ne olow o mass hrogh he sraces srronding he olme ms be eqal o he decrease o mass wihin he olme. The mass low rae is eqal o he prodc o densi eloci componen normal o srace and he area o ha Conseraion o Mass The mass low rae is eqal o he prodc o densi, eloci componen normal o srace and he area o ha srace. In ecor orm; ρ, ρ da m s nˆ V A irs-order Talor series is sed o ealae he low properies a he aces o he elemen, since he properies are a ncion o posiion coninm approach. The ne olow o mass per ni o ime per ni deph is olow +e olow +e area olow +e olow +e inlow e Dr. A.B.M. Toiqe Hasan BUET 3 L-4 T-, Dep. o ME ME 45: Aerodnamics Jl 8 inlow e inlow e area inlow e ! h h h Talor series Conseraion o Mass which ms be eqal he rae a which he mass conained wihin he elemen decreases mass in de o decrease e ; mass in de o decrease e ; mass= densi olme Eqaing he aboe wo epressions and diiding b - I -dimension is considered, he dierenial orm o he aboe epression comes as w Dr. A.B.M. Toiqe Hasan BUET 4 L-4 T-, Dep. o ME ME 45: Aerodnamics Jl 8 w which is known as dierenial conini eqaion in ecor orm. w V V,, operaor, del and,, where ;

3 Conseraion o Mass In case o sead lows, he conini eqaion becomes as- w V di V w V div Compressible lows Incompressible lows 5 Conseraion o Momenm Linear Momenm Eqaion The ne orce acing on a lid paricle is eqal o he ime rae o change o he linear momenm o he lid paricle. As lid elemen moes in space, is eloci, densi, shape and olme ma change, b is mass is consered. Conseraion o momenm can be wrien as- F m DV D D direcion : F m D D direcion : F m D Dw direcion i : F m D ; V,, w and F F, F, F The eloci o a lid paricle is, in general, an eplici ncion o ime as well as o is posiion,,. Frhermore, he posiion coordinaes,, o he lid paricle are hemseles a ncion o ime,. The deriaie in he aboe epression is reqenl ermed as paricle, oal or sbsanial deriaie D/D o eloci. 6 3

4 Conseraion o Momenm Since,,,,,, w w,,, D D D D oal local w ; conecie,, w Similarl D D Dw D w w w w w w A > A Area=A A < A 3 Sead low Veloci increases o Veloci decreases o 3 a Conecie acceleraion 7 Conseraion o Momenm The principal orces wih which we are concerned are hose which ac direcl on he mass o he lid elemen, he bod orce, and hose which ac on is srace, hepressre orces and shear orces. The sress ssem acing on an elemen o he srace is shown in igre: There is a oal o 6 shear sresses and 3 normal sresses acing on a lid elemen. The properies o mos lids hae no preerred direcion in space; ha is, lids are isoropic. Asa resl ace shear shear ace ace normal 8 4

Conseraion o Momenm In general, he arios sresses change rom poin o poin coninm approach. Ths, he prodce ne orces on he lid paricle, which case i o accelerae.

5 Conseraion o Momenm In general, he arios sresses change rom poin o poin coninm approach. Ths, he prodce ne orces on he lid paricle, which case i o accelerae. To simpli he illsraion o he orce balance on he lid paricle, consider a D low, as indicaed in igre. The reslan orce in -direcion or a ni deph in he -direcion is where is he bod orce per ni mass in - direcion. Incldinglowinhe-direcion, he reslan orce in he -direcion- F 9 Conseraion o Momenm Use his epression in eqn. or -direcion: F D D D D Similarl, or - and -direcions D d Dw d These are he basic orm o Naier-Sokes eqaions NS eqaions. NS eqaions are he mos amos eqaions or adanced analsis in lid dnamics. 5

Conseraion o Momenm Now, we need o relae he sresses o he moion o lid. The normal sress is in he orm o a pressre hdrosaic. For ideal lid low iniscid, he shear sress anishes and can be se o ero.

Sreamline o a low A sreamline is an imaginar cre whose angen a an poin is in he direcion o he eloci ecor a ha poin. Across he sreamline, here is no low.

6 Conseraion o Momenm Now, we need o relae he sresses o he moion o lid. The normal sress is in he orm o a pressre hdrosaic. For ideal lid low iniscid, he shear sress anishes and can be se o ero. Ths p normal sress Shear sress D D D D Dw D Ths he eqaions o moion or iniscid, incompressible lid low comes as- p p p DV p D Vecor orm These eqaions are known as Eler Eqaions. Sreamline o a low A sreamline is an imaginar cre whose angen a an poin is in he direcion o he eloci ecor a ha poin. Across he sreamline, here is no low. Le ds be a direced elemen o he sreamline, sch as shown a poin in Fig..9. The eloci a poin isv, and b deiniion o a sreamline, V is parallel o ds. From he ecor deiniion- ii ds V iˆ d ˆj d kˆ d w iˆ wd d ˆ j d wd kˆ d d Ths wd d d wd d d d d w d w d d d Dierenial eqaion or a sreamline 6

7 Bernolli s Eqaion Bernolli s eqaion is probabl he mos amos and sel eqaion in lid dnamics. I relaed he pressre and eloci in an iniscid, incompressible low. Consider -componen o he momenm eqaion or iniscid low wih no bod orces- D p D p w p w p d d w d d ; mlipling b d Consider he low along a sreamline, he dierenial eqaions are d wd d d hen se hese aboe epressions ino eqaion a a p d d d d b 3 Bernolli s Eqaion Since or sead low,, d d d d,, oal deriaie w w,, This is eacl he erm in parenheses in he las epression o -momenm eqaion b, hen p d d d p d In a similar ashion, he - and -momenm eqaions will ake he ollowing orms d d w p d p d

8 Bernolli s Eqaion Adding he eqaions, and 3 p p p d w d d d dp p p p d V ; since dp d d d dp VdV ; V,, w V w oal deriaie I applies o an iniscid low wih no bod orce, and i relaes he change in eloci along a sreamline, dv o he change in pressre, dp along he same sreamline. On inegraing he aboe Eler s Eqaion along a sreamline p V dp V dv ; or incompressible low ρ consan p V p V p This is he amos Bernolli s Eqaion. V 5 Bernolli s Eqaion p V Consan Ths i can be wrien asalong a sreamline p V Consan In deriing he aboe Bernolli s eqaion, no siplaion has been made as o wheher he low is roaional or irroaional circlaion=. For a general, roaional low, he ale o consan in he aboe eqaion will change rom one sreamline o he ne. Howeer, i he low is irroaional circlaion=, hen Bernolli s eqaion holds beween an wo poins in he low, no necessaril js on he same sreamline. Ths or irroaional low- hrogh o he low The phsical signiicance o Bernolli s eqaion is: when he eloci increases, he pressre decreases and when he eloci decreases, he pressre increases along a sreamline in a lid low. 6 8

9 Bernolli s Eqaion The sraeg or soling mos problems in sead, iniscid, incompressible low is as ollows:. Obain he eloci ield rom he goerning eqaions.. Once he eloci ield is known, obain he corresponding pressre ield rom Bernolli s eqaion. 7 9

ME 425: Aerodynamics

ME 425: Aerodynamics 3/4/18 ME 45: Aerodnamics Dr. A.B.M. Toiqe Hasan Proessor Deparmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET Dhaka Lecre-6 3/4/18 Fndamenals so Aerodnamics eacher.be.ac.bd/oiqehasan/