Boundary Layer Flow Solution

Search

Search pages

Wiki

Tools

Version 3
by (unknown)
Version 22
by (unknown)

Deletions or items before changed

Additions or items after changed

1 == Boundary Layer for Flow Past a Wedge ==
2
3 -
The Blausius boundary layer velocity solution is a special case of a larger class of problems for flow over a wedge, as shown in the following figure. The angle <math>\beta \pi</math> represents the wedge angle.
+
The Blausius boundary layer velocity solution is a special case of a larger class of problems for[http://en.wikipedia.org/wiki/Blasius_boundary_layer flow over a wedge], as shown in the following figure in which <math>\beta \pi</math> represents the wedge angle.
4
5 [[Image(wedge_bl.png)]]
6
7 -
The general solution is called the Faulkner-Skan boundary layer solution, which gives the velocity as a function of local coordinates ''x'' and ''y'' defined parallel and perpendicular to the wedge surface, respectively:
 +
The general solution is called the Falkner-Skan boundary layer solution, which starts with a recognition that the free-stream velocity will accelerate for non-zero values of <math>\beta</math>:
8 +
9 +
<math>u_{e}(x)= U_{0} \left( x/L \right) ^{m}</math>
10 +
11 +
where <math> L </math> is a characteristic length and ''m'' is a dimensionless constant that depends on <math>\beta</math>:
12 +
13 +
<math>
14 +
{\beta} = \frac{2m}{m + 1}
15 +
</math>
16 +
17 +
The condition ''m = 0'' gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.
18 +
19 +
We then define a similarity variable <math>\eta</math> that combines the local streamwise and cross-flow coordinates ''x'' and ''y'' (defined relative to the surface of the wedge):
20 +
21 +
<math>
22 +
{\eta} = y \sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{2}}
23 +
</math>
24 +
25 +
Then, defining a function ''f'' that relates to the streamwise and cross-flow velocities, a single ordinary differential equation ensues from boundary layer momentum and mass conservation:
26 +
27 +
<math>
28 +
\frac{\mathrm{d}^3 f}{\mathrm{d} \eta ^3}+f\frac{\mathrm{d}^2 f}{\mathrm{d} \eta^2}+ \beta \left[1-\left(\frac{\mathrm{d}f}{\mathrm{d}\eta}\right)^2 \right]=0
29 +
</math>
30 +
31 +
This non-linear equation is not amenable to an exact solution (even for the Blausius solution <math>\beta=0</math>, which eliminates the last term on the right side). The [http://demonstrations.wolfram.com/NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl/ following Mathematic CDF file] solves the equation numerically and provides the streamwise velocity normalized by the local freestream velocity as a function of <math>\eta</math>.
32 +
33 +
34 +
=== CDF Tool [[FootNote(Numerical Solution of the Falkner-Skan Equation for Various Wedge Angles, from the Wolfram Demonstrations Project [ http://demonstrations.wolfram.com/NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl/])]] ===
35 +
36 +
[[File(NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl.cdf)]]
37 +
38 +
[[FootNote]]