Ordinary Differential Equations
Autor: riteshk2017 • October 13, 2015 • Course Note • 536 Words (3 Pages) • 692 Views
THE LAMINAR BOUNDARY LAYER ALONG A FLAT PLATE
[pic 1]
The boundary layer along a flat plate at zero incidence
- Momentum Transfer (Blasius’ solution)
The boundary layer equations for momentum transfer are as followings:
[pic 2]
[pic 3]
with the boundary conditions: at y = 0, u = v = 0; and at y = ∞, u = u∞
Similarity Transformation: Transforming the independent variables x and y to η, and the dependent variable Ψ (stream function) to f(η) as the following definitions:
[pic 4] and [pic 5]
Then, the boundary layer equations become a single ordinary differential equation as:
[pic 6] (Blasius’s equation)
with the boundary conditions as: atη= 0, f = f’ = 0 and atη= ∞, f’ = 1.
The differential equation is solved numerically and the solution is
[pic 7]
Velocity distribution in the boundary layer along a flat plate, after Blasius [2]
The function [pic 8]for the boundary layer a flat plate at zero incidence, after L. Howarth [16]
[pic 9] | f | f[pic 10] | f” |
0.0 | 0 | 0 | 0.33206 |
0.2 | 0.00664 | 0.6641 | 0.33199 |
0.4 | 0.02656 | 0.13277 | 0.33147 |
0.6 | 0.05974 | 0.19894 | 0.33008 |
0.8 | 0.10611 | 0.26471 | 0.32739 |
1.0 | 0.16557 | 0.32979 | 0.32301 |
1.2 | 0.23795 | 0.39378 | 0.31659 |
1.4 | 0.32298 | 0.45627 | 0.30787 |
1.6 | 0.42032 | 0.51676 | 0.29667 |
1.8 | 0.52952 | 0.57477 | 0.28293 |
2.0 | 0.65003 | 0.62977 | 0.26675 |
2.2 | 0.78120 | 0.67132 | 0.24835 |
2.4 | 0.92230 | 0.72899 | 0.22809 |
2.6 | 1.07252 | 0.77246 | 0.20646 |
2.8 | 1.23099 | 0.81152 | 0.18401 |
3.0 | 1.39682 | 0.84605 | 0.16136 |
3.2 | 1.56911 | 0.87609 | 0.13913 |
3.4 | 1.74696 | 0.90199 | 0.11788 |
3.6 | 1.92954 | 0.92333 | 0.09809 |
3.8 | 2.11605 | 0.94112 | 0.08013 |
4.0 | 2.30576 | 0.95552 | 0.06424 |
4.2 | 2.49806 | 0.96696 | 0.05052 |
4.4 | 2.69238 | 0.97587 | 0.03897 |
4.6 | 2.88826 | 0.98269 | 0.02948 |
4.8 | 3.08534 | 0.98779 | 0.02187 |
5.0 | 3.28329 | 0.99155 | 0.01591 |
5.2 | 3.48189 | 0.99425 | 0.01134 |
5.4 | 3.68094 | 0.99616 | 0.00793 |
5.6 | 3.88031 | 0.99748 | 0.00543 |
5.8 | 4.07990 | 0.99838 | 0.00365 |
6.0 | 4.27964 | 0.99898 | 0.00240 |
6.2 | 4.47948 | 0.99937 | 0.00155 |
6.4 | 4.67938 | 0.99961 | 0.00098 |
6.6 | 4.87931 | 0.99977 | 0.00061 |
6.8 | 5.07928 | 0.99987 | 0.00037 |
7.0 | 5.27926 | 0.99992 | 0.00022 |
7.2 | 5.47925 | 0.99996 | 0.00013 |
7.4 | 5.67924 | 0.99998 | 0.00007 |
7.6 | 5.87924 | 0.99999 | 0.00004 |
7.8 | 6.07923 | 1.00000 | 0.00002 |
8.0 | 6.27923 | 1.0000 | 0.00001 |
8.2 | 6.47923 | 1.00000 | 0.00001 |
8.4 | 6.67923 | 1.00000 | 0.00000 |
8.6 | 6.87923 | 1.00000 | 0.00000 |
8.8 | 7.07923 | 1.00000 | 0.00000 |
Boundary layer thickness δ: The boundary layer thickness is defined as the distance from surface at u/u∞ = 0.99. From the above table, it is atη= 5; thus,
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