Here the function Rkadapt is used, which is similar to rkfixed except it internally uses adaptable spacing
instead of fixed spacing (more accuracy where needed). It reports at fixed spacing however.
ηstart 0:= ηend 10:= num_steps 2000:= Z Y1guess( ) Rkadapt YBC Y1guess()ηstart,ηend,num_steps,D,
()
:=
Now find the correct boundary condition, using the root function, and then re-set Y1guess to this best value:
best root Z Y1guess()
num_steps 1+3,1−Y1guess,
()
:= best 0.332057=
Zfinal Rkadapt YBC best()ηstart,ηend,num_steps,D,
()
:=
Middle portion of Zfinal (to find δ): Bottom portion of Zfinal (to check boundary conditions):
η Y1=f '' Y2=f ' Y3=f η Y1=f '' Y2=f ' Y3=f
Zfinal
1234
1999
2000
2001
9.99 99526·10
-9 1 8.269213
9.995 19427·10
-9 1 8.274213
10 42908·10
-9 1 8.279213
=
Now generate a plot of the similarity variables: n 1 num_steps..:=
0 0.5 1 1.5 2
0
1
2
3
4
5
6Blasius BL Similarity Solution
f'', f', and f
eta
Zfinaln1,
Zfinaln1,
Zfinaln1,
Zfinaln2,Zfinaln3,
,Zfinaln4,
,
Blasius flat plate boundary layer similarity solution by the Runge-Kutta method
The equation to solve is f''' + cff'' = 0, where prime denotes d/dη. Here, let c = 1/2. c 0.5:= ORIGIN 1:=
We will define a vector Y which contains three unknowns, Y1 = f'', Y2 = f', and Y3 = f. Vector YBC is Y at η = 0.
Note that YBC1 at η = 0 mus t be guessed in order to satisfy all boundary condit ions in t he problem.
Y1guess 1:= YBC2 0:= YBC3 0:=
YBC Y1guess()
Y1guess
YBC2
YBC3
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
:= Verify the vector: YBC Y1guess()
1
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
DηY,
()
c−Y3
⋅Y1
⋅
Y1
Y2
⎛
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
:=
Define the derivative vector D which contains the first derivative with respect to η of each
variable in the Y vector. This derivative vector D is needed for the Runge-Kutta solution.
Now calculate the solution as η marches from ηstart to ηend, using Runge-Kutta. Here Z is the solution matrix,
where the first column is η, the second column is Y 1, the third column is Y2, and the fourth column is Y3.
Zfinal
1234
982
983
984
4.905 0.01855 0.989908 3.189153
4.91 0.018403 0.99 3.194103
4.915 0.018256 0.990092 3.199053
=
1
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