Boundary-Layer Flow over a Flat Plate Approximate Method

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1 Bounar-aer lo oer a lat Plate Approimate Metho Transition Turbulent aminar The momentum balance on a control olume o the bounar laer leas to the olloing equation: + () The approimate metho o bounar laer analsis can be summarize as ollos: (a) Assume the elocit proile,, as a unction o an. (b) Ealuate the integrals: an. (c) Ealuate the all-shear stress:. () Substitute in the aboe momentum balance an get an as unctions o the istance,, the elocit ar rom the plate, an the lui properties (, μ). The question no, ho to assume? can be assume as ollos: CHE, Prepare b: r. Usamah Al-Mubaieh Page

2 ( ζ ) here ζ ( ) The unction ( ζ ) must satis the olloing properties:. () ). () ) 3. ζ ζ ;.,. ζ ζ ζ ζ aminar Bounar aer No let us ollo the summar outline preiousl or the approimate metho o bounar laer analsis: (a) The olloing elocit proile ( ζ ) liste beore, hence it can be assume. Sin ζ satisies the properties o the unction (b) Ealuate: sin cos Π Cos Also, Sin A) Cos (using integration ormula in appeni [ ] + Sin (c) Ealuate: CHE, Prepare b: r. Usamah Al-Mubaieh Page

3 the eriatie: Sin Cos an thereore: Cos () Substitute in momentum balance, equation (): Integrate: 3 (. ) (.3 ) ( ) (.3 ).3 eine nols number or bounar laer lo:.3.9 The aboe equation proies the thickness o the bounar laer as a unction o an. No e ealuate the shear stress at the lat plate: call, CHE, Prepare b: r. Usamah Al-Mubaieh Page 3

4 ( )(.9) call the einition o rag Coeicient: C C.38 3 ( )(.38) C.66 Turbulent Bounar aer The preceing analsis as or laminar B.. Hoeer, it is oun eperimentall that hen > 3., the bounar laer becomes turbulent oer a lat plate: < 3. aminar bounar laer > 3. Turbulent bounar laer What is the alue o at hich the bounar laer unergoes transition to turbulent lo? The elocit proile or turbulent bounar laer is base on eperience rom eperimental ata: rom eperience: or turbulent B.. 8 CHE, Prepare b: r. Usamah Al-Mubaieh Page

5 CHE, Prepare b: r. Usamah Al-Mubaieh Page 9! 6 Thereore, e hae a problem that the turbulent elocit proile that is base on eperience, preicts ininite shear stress. Hoeer, rom eperiment:. C. call, the momentum balance on a control olume o the bounar laer: Integrating an simpliing: ( ).36 an ( ).6. C

6 Summar aminar B ( ζ ), ζ C ( ) ( ζ ) Eact Solution Numerical See ig 8. sin ( ζ ).9 ζ o it oursel! ζ ζ o it oursel! ζ o it oursel! 3 3 ζ ζ + ζ 3 ζ o it oursel! C Turbulent B.. >. C.36 ( ) / C.6 ( ) / equation (8.6 o tetbook) equation (8.6 o tetbook) CHE, Prepare b: r. Usamah Al-Mubaieh Page 6

7 Calculation o rag orce Oer a lat Plate rag orce: A Hoeer, A () an () A; A ( Area per unit ith) ( rag orce per unit ith). 3 ( ) ( ) + aminar Turbulent ( ) + ( ) ( C ) + ( C ) aminar Turbulent aminar Turbulent ( ).6 / ( ).6 + /.66.66( ) +.6 ( ) ( ) /.6 + / Note: the aboe analsis or calculation o the rag orce per unit ith is one or the case hen > ( > ). Hoeer, or the case hen < ( < ) the lo oer the entire length o the plate is laminar an the rag orce per unit ith:.66 ( ) ( C ).66( ) aminar aminar CHE, Prepare b: r. Usamah Al-Mubaieh Page

8 Eample Air ( air. lb m /t 3, mair.3 - lb m /t s) los oer a lat plat ith a elocit t/s. Calculate the rug orce oer the lat plate i: (a) The plate is t long along the irection o lo. (b) The plate is t long along the irection o lo. Solution (a) (.)( )( ).8 < 3..3 In this case, the lo oer the entire plate is laminar:.66 ( )..66( ).8 lbm t.3 s t. 3 lb t (b) (.)( )( ) (.) > 3. In this case, the lo unergoes transition rom laminar to turbulent. The transition takes place at: t < t ( ) ( ) ( ) /. (.8.).66( ) 3. lbm t 6 lb.3.86 s t t +.6. ( ) ( ) 6.39 / 3. CHE, Prepare b: r. Usamah Al-Mubaieh Page 8