## 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:

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

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):

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:

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]}

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