The Partial Differential Equation for the Blasius Equation

8/18/2019 The Partial Differential Equation for the Blasius Equation

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Advanced Fluid MechanicsCoding Assignment - 1

1. Problem Defnition To solve the partial dierential euation !or the "lasius #uation

f ' ' ' (η )+

1

2f

'' (η ) f ( η )=0

This is the governing euation !or a laminar $o% past a semi&infnite $at

plate %hich is derived !rom the continuit' euation and mass momentum

euation b' introducing single composite dimensionless variable (. )n this

assignment* the approach is to solve "lasius euation numericall' using

+unge-,utta Method and e%ton +aphson Method.

. /overning #uation"lasius !ound a classical approach to fnd the sel!&similar solution o!

Prandtl0s problem arising !rom laminar $o% past a semi&infnite $at plate

leads to a one&parameter !amil' o! problems involving a third order

nonlinear ordinar' dierential euation on the semi-infnite domain .

Fig.1 Boundary layer formation on flat plate

For laminar $o% past the $at plate* the boundar' la'er euations given

belo% can be solved eactl' !or u and v being velocities in and ' aisdirection* assuming that the !ree-stream velocit' 2 is constant 3d24d 567.


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g(η)=f ' (η)

h (η )=g ' (η )= f ' ' (η )

so*h

' ( η )=−12

× f (η)× h (η )

a. Runge-Kutta Method

The +unge&,utta method is an important !amil' o! implicit and eplicit

iterative methods* %hich are used in temporal discreti. )nitial Conditions This is a third order partial dierential euation %ith t%o initial conditions

and one boundar' conditions. The boundar' and initial conditions are


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At y=0, f (0 )=f , (0 )=0

At y=ϖ , f , (ϖ , )=1.0

?. Mathematical Formulation"lasius #uation !or a $o% past a $at plate is given b'

f ' ' ' (η )+

1

2f

'' (η ) f ( η )=0

;et us assume* g(η)=f ' (η)

h (η )=g'

(η )= f ' '

(η )

=ence* the above third order euation can be converted into the !ollo%ing

three linear ordinar' single order dierential euations.

f ' (η )=g (η)

g' (η )=h (η )

h'

(η )=−12 × f (η)× h (η )

The above three linear ordinar' dierential euation can be solved b' +,?

method b' having the initial conditions as

f (0 )=0

f ' (0 )=g (0)=0

f , (ϖ )=g (ϖ)=1.0

As there is no initial valve !or third euation* the initial value !or h is

evaluated using e%ton +aphson method !or the root o! the euation

g (ϖ )−1.0=0

The epression !or e%ton +aphson is given b'


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10

¿10

¿¿10

¿g ¿¿

g(¿¿ i−1)×(h (10 )i−h (10)i−1)

¿h(10)i+1=h(10)i−¿

h(0)=h(10)i+1

. Flo% Chart


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B. +esults The code !or solving the "lasius #uation is %ritten in MAT;A" verison

61b. The code is eecuted and the result is given belo%. The value !or

h367 is !ound to be 6.>>6 using e%ton +aphson Method.


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'*'!!!!! +*/+31 !*/,5.,1 !*!3./,3

'*5!!!!! +*,/!13' !*/,/51' !*!33/.1

'*!!!!! +*...+'. !*/.+.3 !*!+/'.'

'*,!!!!! +*/.5, !*/.5'+, !*!+5'5,

'*.!!!!! 3*!.53+1 !*/.,,./ !*!+1.,1

'*/!!!!! 3*1.'+!' !*/./.15 !*!1./. 5*!!!!!! 3*+.3+,' !*//15'+ !*!15/!,

5*1!!!!! 3*3.+5!3 !*//3!!. !*!13'5

5*+!!!!! 3*'.1.. !*//'+'5 !*!113'+

5*3!!!!! 3*5.13' !*//5+. !*!!/5!

5*'!!!!! 3*.!/1/ !*//155 !*!!,/+.

5*5!!!!! 3*,.!5,+ !*//.,/ !*!!5,/

5*!!!!! 3*..!+/1 !*//,',. !*!!5'3+

5*,!!!!! 3*/.!!' !*//,/,1 !*!!''3

5*.!!!!! '*!,/..+ !*//.3,5 !*!!3'.

5*/!!!!! '*1,/,3, !*//.,!5 !*!!+/.

*!!!!!! '*+,/+1 !*//./,3 !*!!+'!+ *1!!!!! '*3,/53! !*///1./ !*!!1/35

*+!!!!! '*',/'5, !*///33 !*!!155!

*3!!!!! '*5,/'!1 !*///5!1 !*!!1+3

*'!!!!! '*,/35, !*///1+ !*!!!/.1

*5!!!!! '*,,/3++ !*///// !*!!!,,'

*!!!!! '*.,/+/ !*///,. !*!!!!.

*,!!!!! '*/,/+, !*///.++ !*!!!',5

*.!!!!! 5*!,/+! !*///.' !*!!!3,!

*/!!!!! 5*1,/+'. !*///./ !*!!!+.

,*!!!!!! 5*+,/+3/ !*////++ !*!!!++!

,*1!!!!! 5*3,/+3+ !*////'1 !*!!!1/ ,*+!!!!! 5*',/++, !*////5 !*!!!1+/

,*3!!!!! 5*5,/++3 !*////, !*!!!!/.

,*'!!!!! 5*,/++! !*////,5 !*!!!!,'

,*5!!!!! 5*,,/+1. !*////.+ !*!!!!55

,*!!!!! 5*.,/+1, !*////., !*!!!!'1

,*,!!!!! 5*/,/+15 !*/////! !*!!!!31

,*.!!!!! *!,/+15 !*/////3 !*!!!!+3

,*/!!!!! *1,/+1' !*/////5 !*!!!!1,

.*!!!!!! *+,/+1' !*///// !*!!!!1+

.*1!!!!! *3,/+13 !*/////, !*!!!!!/

.*+!!!!! *',/+13 !*/////. !*!!!!! .*3!!!!! *5,/+13 !*////// !*!!!!!5

.*'!!!!! *,/+13 !*////// !*!!!!!3

.*5!!!!! *,,/+13 !*////// !*!!!!!+

.*!!!!! *.,/+13 1*!!!!!! !*!!!!!+

.*,!!!!! */,/+13 1*!!!!!! !*!!!!!1

.*.!!!!! ,*!,/+13 1*!!!!!! !*!!!!!1

.*/!!!!! ,*1,/+13 1*!!!!!! !*!!!!!1

/*!!!!!! ,*+,/+13 1*!!!!!! !*!!!!!!

/*1!!!!! ,*3,/+1+ 1*!!!!!! !*!!!!!!

/*+!!!!! ,*',/+1+ 1*!!!!!! !*!!!!!!

/*3!!!!! ,*5,/+1+ 1*!!!!!! !*!!!!!! /*'!!!!! ,*,/+1+ 1*!!!!!! !*!!!!!!


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cic)0

c)%'&h)4!(0

end

dis&srint)&2 eta ) g h2((0

dis&srint)&291!) 91!) 91!) 91!)24!4!4!4h)((0

disla&%'&h)41((0

ubroutine !or +ung-,utta Method3+,?7 is given belo% and has to savedin the name +,?.m in the same !older o! the main program.

)unction C %'&;4"(

an&1(!0

a)&1(!0

ag&1(!0

ah&1(;0

d)g0

dgh0dh-!*57)7h0

i10

hh!*10

)or co!:hh:1!

k)1ag&i(0

kg1ah&i(0

kh1-!*57a)&i(7ah&i(0

k)+&ag&i(

kg+&ah&i(

kh+-!*57&a)&i(k)3&ag&i(

kg3&ah&i(

kh3-!*57&a)&i( k)'&ag&i(


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lot&an4a)422(0

hold on

lot&an4ag42r2(0

lot&an4ah42g2(0

title&2=olution )or #lasius >$uation )or a Flat "late2(

?lael&2eta2( lael&2)2(

end

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The Partial Differential Equation for the Blasius Equation