Boundary Layer Flow Solution

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1 == Boundary Layer for Flow Past a Wedge ==
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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. The angle <math>\beta \pi</math> represents the wedge angle.
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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.
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5 [[Image(wedge_bl.png)]]
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7 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 \beta:
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9 u_{e}(x)= U_{0} \left( x/L \right) ^{m}
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11 where L is a characteristic length and ''m'' is a dimensionless constant that depends on \beta:
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14 {\beta} = \frac{2m}{m + 1}
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17 The condition ''m = 0'' gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.
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19 We then define a similarity variable \eta that combines the local streamwise and cross-flow coordinates ''x'' and ''y'' (defined relative to the surface of the wedge):
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22 {\eta} = y \sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{2}}
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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:
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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
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31 This non-linear equation is not amenable to an exact solution (even for the Blausius solution \beta=0, 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 \eta.
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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/])]] ===
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36 [[File(NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl.cdf)]]
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38 [[FootNote]]