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Home TOPICS CDF Tools for Heat Transfer Boundary Layer Flow Solution

Boundary Layer Flow Solution

Boundary Layer for Flow Past a Wedge

The Blausius boundary layer velocity solution is a special case of a larger class of problems forflow over a wedge, as shown in the following figure in which βπ represents the wedge angle.


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 β:

u_{e}(x)= U_{0} \left( x/L \right) ^{m}

where L is a characteristic length and m is a dimensionless constant that depends on β:

{\beta} = \frac{2m}{m + 1}

The condition m = 0 gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.

We then define a similarity variable η that combines the local streamwise and cross-flow coordinates x and y (defined relative to the surface of the wedge):

{\eta} = y \sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{2}}

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:

\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

This non-linear equation is not amenable to an exact solution (even for the Blausius solution β = 0, which eliminates the last term on the right side). The following Mathematic CDF file solves the equation numerically and provides the streamwise velocity normalized by the local freestream velocity as a function of η.

CDF Tool [1]

  1. ^ Numerical Solution of the Falkner-Skan Equation for Various Wedge Angles, from the Wolfram Demonstrations Project //

Created on , Last modified on, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.