Lecture 13 Introduction to Boundary Layer...
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1 10.539 Lecture 13 Introduction to Boundary-Layer Theory Prof. Dean Wang 1. Boundary-Layer Theory Boundary-layer theory is a collection of perturbative methods for obtaining an asymptotic approximation to the solution of a differential equation whose highest derivative is multiplied by a small parameter . Solutions to such equations usually develop regions of rapid variations as → 0. If the thickness of these regions approaches 0 as → 0, they are called boundary layers, and boundary-layer theory may be used to approximate . Example 1. Exactly soluble boundary-layer problem. Consider the differential equation !! + 1 + ! + = 0 0 = 0 1 = 1 (1) Solution: The exact solution of this equation is = ! !! !! !!/! ! !! !! !!/! (2) In the limit → 0 +, this solution becomes discontinuous at = 0, as shown in the following figure. There are two standard approximations that one makes in boundary-layer theory. In the outer region (away from a boundary layer) is slowly varying, so it is valid to neglect any derivative of which are multiplied by . Inside a boundary layer the derivative of are large, but the boundary layer is so narrow that we may approximate the coefficient functions of the differential equation by constants. Thus, we can replace a single differential equation by a sequence of much simpler approximate equations in the inner and outer regions. x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y 0 0.5 1 1.5 2 2.5 3 0 = 0.1 0 = 0.0025 0 = 0.0005 x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 y 0 0.5 1 1.5 2 2.5 3 0 = 0.1 0 = 0.0025 0 = 0.0005
Transcript of Lecture 13 Introduction to Boundary Layer...
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10.539Lecture13IntroductiontoBoundary-LayerTheory
Prof.DeanWang
1. Boundary-LayerTheoryBoundary-layertheoryisacollectionofperturbativemethodsforobtaininganasymptoticapproximationtothesolution𝑦 𝑥 ofadifferentialequationwhosehighestderivativeismultipliedbyasmallparameter𝜀.Solutionstosuchequationsusuallydevelopregionsofrapidvariationsas𝜀 → 0.Ifthethicknessoftheseregionsapproaches0as𝜀 → 0,theyarecalledboundarylayers,andboundary-layertheorymaybeusedtoapproximate𝑦 𝑥 .Example1.Exactlysolubleboundary-layerproblem.Considerthedifferentialequation
𝜀𝑦!! + 1+ 𝜀 𝑦! + 𝑦 = 0𝑦 0 = 0 𝑦 1 = 1
(1)
Solution:Theexactsolutionofthisequationis 𝑦 𝑥 = !!!!!!!/!
!!!!!!!/! (2)
Inthelimit𝜀 → 0+,thissolutionbecomesdiscontinuousat𝑥 = 0,asshowninthefollowingfigure.
Therearetwostandardapproximationsthatonemakesinboundary-layertheory.Intheouterregion(awayfromaboundarylayer)isslowlyvarying,soitisvalidtoneglectanyderivativeof𝑦 𝑥 whicharemultipliedby𝜀.Insideaboundarylayerthederivativeof𝑦 𝑥 arelarge,buttheboundarylayerissonarrowthatwemayapproximatethecoefficientfunctionsofthedifferentialequationbyconstants.Thus,wecanreplaceasingledifferentialequationbyasequenceofmuchsimplerapproximateequationsintheinnerandouterregions.
x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y
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30 = 0.10 = 0.00250 = 0.0005
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y
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K!
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! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! GMH!
L$%<!4567!6+646*32;*3/)!9$%:3)<D!?)!?675!4%!0)4)$<6+)!*!3)*06+-2%$0)$!9)$4/$:*46;)!*99$%=6<*46%+!4%!! ! !*7!! ! !!'!N345%/-5!4567!67!%+31!*!&6$742%$0)$!06&&)$)+46*3!)>/*46%+D!64!67!+%+36+)*$!*+0!67!</85!4%%!06&&68/34!4%!7%3;)!6+!83%7)0!&%$<'!!@%3/46%+I!@4)9!"I!O%?);)$D!6+!$)-6%+7!?5)$)!!!*+0!!!!*$)!+%4!3*$-)!G7/85!$)-6%+7!*$)!8*33)0!%/4)$!$)-6%+7H'!A4!67!;*360!4%!+)-3)84!!""!!8%<9*$)0!?645!!!! '!J5/7D!6+!%/4)$!$)-6%+7!?)!*99$%=6<*4)!45)!7%3/46%+!4%!GMH!:1!45)!7%3/46%+!4%!45)!%/4)$!)>/*46%+!
!!!"#! ! !!!"# ! !!!!! GPH!@4)9!KI!GPH!67!)*71!4%!7%3;)!:)8*/7)!64!67!36+)*$'!J5)!7%3/46%+!?5685!7*467&6)7!!!!" ! !!!!!67!
!!"# ! ! !! !"! !!!! ! GQH!@4)9!M'!A+!45)!:%/+0*$1!3*1)$!G45)!6++)$!$)-6%+HD!!!67!7<*33!7%!64!67!;*360!4%!*99$%=6<*4)!!!!!:1!"'!L/$45)$<%$)D!76+8)!!!;*$6)7!$*96031!6+!45)!+*$$%?!:%/+0*$1!3*1)$D!?)!<*1!+)-3)84!!"!8%<9*$)0!?645!!"!'!O)+8)D!6+!45)!6++)$!$)-6%+!?)!*99$%=6<*4)!45)!7%3/46%+!4%!GMH!:1!45)!7%3/46%+!4%!45)!6++)$!$)-6%+
! ! !!!" !!"! ! !! GRH!!@4)9!P'!GRH!67!*!36+)*$!)>/*46%+!6&!?)!$)-*$0!!!*7!45)!0)9)+0)+4!;*$6*:3)'!A47!7%3/46%+!67!
! ! ! !!" ! ! ! !!!!" !! GSH!@4)9!Q'!F!67!0)4)$<6+)0!:1!*71<94%468*331!<*4856+-!45)!%/4)$!*+0!6++)$!7%3/46%+7!GQH!*+0!GSH'!
!!"# ! ! !! !"! !!!! !! !!"# ! !!! !"!! &%$!!!67!7<*33!!GTH!! ! ! !!" ! ! ! !!!!" ! !! !!!!!!" !!! &%$!! ! !!! GUH!
@4)9!R'!GTH!*+0!GUH!-6;)7!! ! !
!! G"CH!!
!
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Example3.Second-orderlinearboundary-valueproblem.Letusfindanapproximatesolutiontotheboundary-valueproblem
𝜀𝑦!! 𝑥 + 𝑎 𝑥 𝑦! 𝑥 + 𝑏 𝑥 𝑦 𝑥 = 0 0 ≤ 𝑥 ≤ 1𝑦 0 = 𝐴 𝑦 1 = 𝐵
(11)
Solution:Step1:Intheouterregionagoodapproximationto(11)isthefirst-orderlinearequation 𝑎 𝑥 𝑦!"#! 𝑥 + 𝑏 𝑥 𝑦!"# 𝑥 = 0 (12)Step2:Thesolutionto(12)is
𝑦!"# 𝑥 = 𝑐𝑒! !! ! !"!! , (13)
where,𝑐isanintegrationconstant.Step3:Requiringthat𝑦!"# 1 = 𝐵gives𝑐 = 𝐵,sowehave
𝑦!"# 𝑥 = 𝐵𝑒! !! ! !"!! , (14)
Step4:Todeterminethebehaviorof𝑦 𝑥 when𝑥 → 0+,wemayapproximatethefunctionsof𝑎 𝑥 and𝑏 𝑥 by
𝑎 0 = 𝛼 ≠ 0𝑏 0 = 𝛽 (15)
Step5:Intheboundarylayer,𝑦ismuchsmallerthan𝑦′because𝑦israpidlyvarying.Sowemayneglect𝑦comparedwith𝑦′.Sotheinnerapproximationto(11)istheconstantcoefficientdifferentialequation 𝜀𝑦!"!! + 𝛼𝑦!"! = 0 (16)Step6:thegeneralsolutionto(16)is 𝑦!" 𝑥 = 𝐶! + 𝐶!𝑒!!"/! (17)Step7:Applying𝑦!" 0 = 𝑦 0 = 𝐴gives 𝑦!" 𝑥 = 𝐴 + 𝐶! 𝑒!!"/! − 1 (18)Step8:Since𝑦!" 𝑥 in(18)variesrapidlywhen𝑥 = 𝑂 𝜀 ,weconcludethattheboundary-layerthickness𝛿isoforder𝜀.Theasymptoticmatchoftheinnerandoutersolutionstakesplacebetweentherightmostedgeoftheinnerregionandtheleftmostedgeoftheouterregion,sayforvaluesof𝑥 = 𝑂 𝜀!/! .Forsuchvaluesof𝑥,
𝑦!" 𝑥 = 𝐴 − 𝐶!
𝑦!"# 𝑥 ~𝑦!"# 0 = 𝐵𝑒! !! ! !"!!
𝜀 → 0+ (19)
Step9:Byequatingthetwoequationsin(19),wehave
𝐶! = 𝐴 − 𝐵𝑒! !! ! !"!! (20)
Step10:Tosummarize,theboundary-layerapproximationis
𝑦!" 𝑥 ~𝐴𝑒!! ! !/! + 𝐵 1− 𝑒!! ! !/! 𝑒
! !! ! !"!! , 𝑥 = 𝑂 𝜀 , 𝜀 → 0+
𝑦!"# 𝑥 ~𝐵𝑒! !! ! !"!! , 0 < 𝑥 ≤ 1, 𝜀 → 0+
(21)
Step11:Wemayproceedfurtherbycombiningtheabovethetwoexpressionsintoasingle,uniformapproximation𝑦!"#$ ,validforall0 ≤ 𝑥 ≤ 1.Asuitableexpressionis 𝑦!"#$ 𝑥 = 𝑦!" 𝑥 + 𝑦!"# 𝑥 − 𝑦!"#$! 𝑥 , (22)
where,𝑦!"#$! 𝑥 = 𝐴 − 𝐶! = 𝐵𝑒! !! ! !"!!
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Step12:Finally,wehave𝑦!"#$ 𝑥 as
𝑦!"#$ 𝑥 = 𝐵𝑒! !! ! !"!! + 𝐴 − 𝐵𝑒
! !! ! !"!! 𝑒!! ! !/! (24)
Exercise1.Wewishtoobtainanapproximatesolutiontotheboundary-valueproblem
𝜀𝑦!! 𝑥 + 1+ 𝑥 𝑦! 𝑥 + 𝑦 𝑥 = 0 0 ≤ 𝑥 ≤ 1𝑦 0 = 1 𝑦 1 = 1
(25)
Solution:𝑎 𝑥 = 1+ 𝑥and𝑏 𝑥 = 1.Thesolutionto(25)is 𝑦!"#$ 𝑥 = !
!!!− 𝑒!!/! (26)
Thefollowingfiguresshowcomparisonswiththeexactsolutionsfordifferentvaluesof𝜀.Itcanbeseenthattheagreementbecomesbetteras𝜀becomessmaller.
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1.80 = 0.05
exactyunif
x0 0.2 0.4 0.6 0.8 1
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exactyunif
x0 0.2 0.4 0.6 0.8 1
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exactyunif
