
Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
This is the first course of a two-semester fluid mechanics sequence for graduate students in the thermal sciences. This course deals with solutions of these equations, both exact and approximate. Key points of this lecture are: Blasius Laminar, Derivative Vector, Runge-Kutta Solution, Bottom Portion, Root Function
Typology: Study notes
1 / 1
This page cannot be seen from the preview
Don't miss anything!
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.
Now find the correct boundary condition, using the root function, and then re-set Y1guess to this best value:
Middle portion of Zfinal (to find δ): Bottom portion of Zfinal (to check boundary conditions): η Y 1 =f '' Y 2 =f ' Y 3 =f η Y 1 =f '' Y 2 =f ' Y 3 =f
Zfinal
1 2 3 4 1999 2000 2001
9.99 99526·10 -9^1 8. 9.995 19427·10 -9^1 8. 10 42908·10 -9^1 8.
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
6
Blasius BL Similarity Solution
f'', f', and f
eta
Zfinaln 1,
Zfinaln 1,
Zfinaln 1,
Zfinaln 2, , Zfinaln 3,,Zfinaln 4,
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, Y 1 = f'', Y 2 = f', and Y 3 = f. Vector YBC is Y at η = 0. Note that YBC 1 at η = 0 must be guessed in order to satisfy all boundary conditions in the problem.
Y1guess := 1 YBC2 := 0 YBC3 := 0
YBC Y1guess( )
Y1guess YBC YBC
:= Verify the vector:^ YBC Y1guess( )
− c ⋅ Y 3 ⋅Y 1
Y 1
Y 2
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 Y 2 , and the fourth column is Y 3.
Zfinal
1 2 3 4 982 983 984
4.905 0.01855 0.989908 3. 4.91 0.018403 0.99 3. 4.915 0.018256 0.990092 3.