CONCEPT OF DRAG AND LIFTMain TopicsThe Boundary-Layer ConceptBoundary-Layer ThicnessesLa!inar F"at-P"ate Boundary Layer# E$act %o"utionMo!entu! Inte&ra" E'uation(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure GradientPressure Gradients in Boundary-Layer F"o*Dra&Li)tThe Boundary-Layer ConceptThe Boundary-Layer ConceptBoundary Layer ThicnessesBoundary Layer ThicnessesDistur,ance Thicness- Displacement Thickness, *Momentum Thickness, La!inar F"at-P"ateBoundary Layer# E$act %o"utionGo.ernin& E'uationsLa!inar F"at-P"ateBoundary Layer# E$act %o"utionBoundary ConditionsLa!inar F"at-P"ateBoundary Layer# E$act %o"utionE'uations are Coup"ed- Non"inear- Partia" Di/erentia" E'uationsB"asius %o"ution#Trans)or! to sin&"e- hi&her-order- non"inear- ordinary di/erentia" e'uationLa!inar F"at-P"ateBoundary Layer# E$act %o"utionResu"ts o) Nu!erica" Ana"ysisMo!entu! Inte&ra" E'uationPro.ides Appro$i!ate A"ternati.e to E$act 0B"asius1 %o"utionMo!entu! Inte&ra" E'uationE'uation is used to esti!ate the ,oundary-"ayer thicness as a )unction o) x#O,tain a 2rst appro$i!ation to the )reestrea! .e"ocity distri,ution- U0x13The pressure in the ,oundary "ayer is re"ated to the )reestrea! .e"ocity- U0x1- usin& the Bernou""i e'uationAssu!e a reasona,"e .e"ocity-pro2"e shape inside the ,oundary "ayer Deri.e an e$pression )or * usin& the resu"ts o,tained )ro! ite! 4(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure Gradient%i!p"i)y Mo!entu! Inte&ra" E'uation0Ite! 51The Momentum Integral Equation becomes(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure GradientLa!inar F"o*E$a!p"e# Assu!e a Po"yno!ia" 6e"ocity Pro2"e 0Ite! 41 The wall shear stress w is then (Item 3)(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure GradientLa!inar F"o* Resu"ts0Po"yno!ia" 6e"ocity Pro2"e1Compare to Eact (!lasius) results"(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure GradientTur,u"ent F"o* E$a!p"e# 578-Po*er La* Pro2"e 0Ite! 41(se o) the Mo!entu! E'uation )or F"o* *ith +ero Pressure GradientTur,u"ent F"o* Resu"ts0578-Po*er La* Pro2"e1Pressure Gradients in Boundary-Layer F"o*Dra&Dra& Coe9cientwithorDra&Pure Friction Dra F"at P"ate Para""e" to the F"o*Pure Pressure Dra F"at P"ate Perpendicu"ar to the F"o*Friction and Pressure Dra F"o* o.er a %phere and Cy"inder%trea!"inin&Dra&F"o* o.er a F"at P"ate Para""e" to the F"o*# Friction Dra&!oun#ar$ %a$er can be &''( laminar, partl$ laminar an# partl$ turbulent, or essentiall$ &''( turbulent) hence se*eral #i++erent #rag coe++icients are a*ailableDra&F"o* o.er a F"at P"ate Para""e" to the F"o*# Friction Dra& 0Continued1%aminar !%,Turbulent !%,- plus others +or transitional +lowDra&F"o* o.er a F"at P"ate Perpendicu"ar to the F"o*# Pressure Dra&Drag coe++icients are usuall$ obtaine# empiricall$Dra&F"o* o.er a F"at P"ate Perpendicu"ar to the F"o*# Pressure Dra& 0Continued1Dra&F"o* o.er a %phere and Cy"inder# Friction and Pressure Dra&Dra&F"o* o.er a %phere and Cy"inder# Friction and Pressure Dra& 0Continued1%trea!"inin&(sed to Reduce :ae and hence Pressure Dra&Li)tMost"y app"ies to Air)oi"s.ote, !ase# on plan+orm area ApLi)t E$a!p"es# NACA 4; NACA ??4-45=Li)tInduced Dra&Li)tInduced Dra& 0Continued1/e#uction in E++ecti*e 0ngle o+ 0ttack,1inite 2ing Drag Coe++icient,Li)tInduced Dra& 0Continued1