Department of Mechanical Engineering

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1 Department o Mehanial Engineering AMEE41 / ATO4 Aerodynamis Instrutor: Marios M. Fyrillas eng.m@it.a.y Homework Assignment #4 QESTION 1 Consider the boundary layer low on a lat plate o width b (shown on the Figure below). [ ] a. sing mass onservation show that h u[, y] d y. Page 554 in tetbook b. sing the momentum equation show that the Drag ore on the plate is given by [ ] D b u[, y] ( u[, y]) d y. Page 554 in tetbook. Employing the shear stress on the wall show that the Drag ore an be also u obtained through D b w[ ] d, where w[ ] (, y ). y Page 553 in tetbook (Eq. 9.19)

2 d. Assuming that the veloity proile is approimated as u y/ [ ] or y [ ] and u or y [ ] as shown in the Figure below, determine the shear stress. From (b) we obtain that y y 1 [ ] [ ] d D [ ] 1 d [ ] b d d From () we obtain that dd u b w[ ] b (, y ) b d y [ ] [ ] D b d y b [ Equating above two epressions we get 1 d [ ] b b d [ ] Solving this dierential equation we obtain 1 [ ] The loal skin rition oeiient is deined as w[ ] 1 1 [ ] The rition drag oeiient or a lat plate o length and width b is b [ ] d w D CD d d A b Re l ]

3 QESTION I the boundary layer on the hood o your ar behaves as one on a lat plate, estimate how ar rom the ront edge o the hood the boundary beomes 5 turbulent i the transition point ours at Re 51. How thik is the boundary layer at this loation i the boundary layer thikness is given by =5? I the.4 momentum thikness is given by =, use the momentum integral Re d equation or low over a lat plate ( = ) to obtain the drag ore per unit width. w d The Reynolds number or a lat plate is deined as Re /. The transition ours at Re 5. 5 Re Hene 1.5 The boundary layer thikness or the Blasius's ase is given by =5 =

4 QESTION 3 To obtain an epression o the boundary layer thikness o a turbulent boundary layer it is suggested to use the Blasius s ormula or the rition ator o a smooth-pipe: where the rition ator is deined as 1/4.79 Re, 1 u, and the Reynolds number Re is deined with respet to the radius o the pipe. I the radius o the pipe is assumed equivalent to the boundary layer thikness on a lat plate, obtain an epression or. Obtain an epression or the total drag ore F D on one side o a plate o length and unit width. For a pipe o radius R.79 onst u ma ma 1 w u u R u R sine or a partiular veloity proile u is proportional to u. ma I R is now assumed equivalent to the boundary layer thikness on the lat plate,, we have w onst um where =. um Beore we employ the integral momentum equation ( ) 1 u u u( ) u( ) onsider 1 d y ( ) 1 d m onst1 um w d d y where the dimensionless oordinate. Hene onst 1. Combining the equations in the boes we obtain: onst u d d 1/4 d whih simpliies to onst. Integrating with respet to yields d um 4 5 5/4 onst um 1/4 4/5 onst. um C. I we neglet the onstant C, we obtain

5 QESTION 4 Consider an airoil with hord length o 1.5 m. For the ase where the Re 3.1 1, estimate: (a) the laminar boundary layer thikness at the trailing edge and (b) the net laminar skin-rition drag oeiient or the airoil. The boundary-layer thikness or inompressible laminar low over a lat plate at zero angle o attak is 5. Re Applying above equation at the trailing edge, where, we hav m Re 3.11 Notie, how thin the boundary layer is; at the traling edge, where its thikness is the largest, the boundary layer is only 4.3 mm..4 The loal skin rition oeiient is. Re Letting C denote the skin rition drag oeiient, we obtain C d.4 d V V C 1.38 Re Hene, the skin rition drag oeiient is C This is or a single surae. Taking both suraes into aount C.15

6 Do you think it was reasonable to assume a laminar boundary layer? Estimate the boundary layer thikness and the skin-rition drag or the ase o a turbulent boundary layer. For the relatively high Reynolds number o 3.11, the boundary layer over the airoil will be turbulent, not laminar. So we epet that the eperimental skin-rition drag oeiient to be higher. We replae the airoil with a lat plate at zero angle o attak. The boundary-layer thikness at the trailing edge, where and Re = Re is =.79 m Re /5 1/5 The turbulent boundary layer is.79 m at the trailing edge, and it is muh thiker than the laminar boundary layer thikness. The skin-rition oeiient is given by C.37 Re /5 1/5 This is or a single surae. Taking both suraes into aount C.744 This result is a ator o ive larger than or the laminar boundary layer. It demonstrates the onsiderable inrease in skin rition aused by the a turbulent boundary layer in omparison to that aused by a laminar boundary layer. What is the eet o the turbulent boundary layer on the drag ore? For streamlined bodies, the drag oeiient inreases when the boundary layer beomes turbulent beause most o the drag is due to the shear ore, whih is greater or turbulent low than or laminar low. On the other hand, the drag oeiient or a relatively blunt objet, suh as a ylinder or sphere, atually dereases when the boundary layer beomes turbulent. A turbulent boundary layer an travel urther along the surae into the adverse pressure gradient on the rear portion o the ylinder beore separation ours. This is beause the loation o separation, the width o the wake region behind the objet, and the pressure distribution on the surae depend on the nature o the boundary layer low. Compared with a laminar boundary layer, a turbulent boundary layer low has more kineti energy and momentum assoiated with it beause: (1) the veloity proile is uller, more nearly like the ideal uniorm proile, and () there an be onsiderable energy assoiated with the swirling, random omponents o the veloity. The result is a thinner wake and smaller pressure drag or turbulent boundary layer low.

Chapter 3 Lecture 7. Drag polar 2. Topics. Chapter-3

Chapter 3 Lecture 7. Drag polar 2. Topics. Chapter-3 hapter 3 eture 7 Drag polar Topis 3..3 Summary of lift oeffiient, drag oeffiient, pithing moment oeffiient, entre of pressure and aerodynami entre of an airfoil 3..4 Examples of pressure oeffiient distributions