# Boundary-Layer Flow Past a Semi-Infinite Wedge: The Falkner-Skan Problem

Boundary-Layer Flow Past a Semi-Infinite Wedge: The Falkner-Skan Problem

In 1904 Prandtl showed that the effects of viscosity at a high Reynolds number could be represented by approximating the Navier–Stokes equations with the celebrated boundary-layer equations, which for two-dimensional steady flow reduce to:

V

X

∂

V

X

∂X

V

Y

∂

V

X

∂Y

U

X

2

∂

V

X

∂

2

Y

∂

V

X

∂X

∂

V

Y

∂Y

V

X

V

Y

V

X

Here, and are the coordinates parallel and perpendicular to the body surface, respectively.

X

Y

For a semi-infinite wedge with an angle of taper , one can prove that far from the wedge the potential flow is given by

ω=βπ

U(X)=A

m

X

m=

βπ

2π-βπ

β=

2m

m+1

A

The above boundary-layer equations admit a self-similar solution such that the velocity profiles at different distances can be made congruent with suitable scale factors for and . This reduces the boundary-layer equations to one ordinary differential equation.

X

X

Y

Let us introduce a function such that: =Af'(η) where .

f(η)

V

X

m

X

η=Y

(m+1)A

2ν

m-1

2

X

Then, we have from continuity equation: =-(m+1)Aνf(η)+ηf'(η).

V

Y

1

2

m-1

2

X

m-1

m+1

The boundary-layer equations can be written as follows:

f'''(η)+f(η)f''(η)+β1-=0

2

(f'(η))

with and

f(η=0)=f'(η=0)=0

f'(η→∞)=1

The above equation can be solved for a user-set value of parameter when and using the shooting technique. The limiting case is flow over a flat plate (Blasius problem). Using the following definitions of = and =, one can obtain the particle trajectories (streamlines) for different starting points. This Demonstration plots such trajectories and also shows the growth of the boundary-layer (red curve) in a separate plot for any value of the wedge angle. The evolution of the -velocity component and its congruent properties with the growth of the boundary layer is also shown.

β

A=1

ν=1

β=0

V

X

X

t

V

Y

Y

t

δ(X)

V

X