WOLFRAM NOTEBOOK

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

​
particle trajectories
boundary layer
β
0.095
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
U
X
+ν
2
∂
V
X
∂
2
Y
∂
V
X
∂X
+
∂
V
Y
∂Y
=0
V
X
(Y=0)=
V
Y
(Y=0)=0
V
X
(Y→∞)=U(X)
Here,
X
and
Y
are the coordinates parallel and perpendicular to the body surface, respectively.
For a semi-infinite wedge with an angle of taper
ω=βπ
, one can prove that far from the wedge the potential flow is given by
U(X)=A
m
X
, where
m=
βπ
2π-βπ
or
β=
2m
m+1
, and
A
is a scale factor.
The above boundary-layer equations admit a self-similar solution such that the velocity profiles at different distances
X
can be made congruent with suitable scale factors for
X
and
Y
. This reduces the boundary-layer equations to one ordinary differential equation.
Let us introduce a function
f(η)
such that:
V
X
=A
m
X
f'(η)
where
η=
(m+1)A
2ν
m-1
2
X
Y
.
Then, we have from continuity equation:
V
Y
=-
1
2
(m+1)Aν
m-1
2
X
f(η)+
m-1
m+1
ηf'(η)
.
The boundary-layer equations can be written as follows:
f'''(η)+f(η)f''(η)+β1-
2
(f'(η))
=0
with
f(η=0)=f'(η=0)=0
and
f'(η→∞)=1
The above equation can be solved for a user-set value of parameter
β
when
A=1
and
ν=1
using the shooting technique. The limiting case
β=0
is flow over a flat plate (Blasius problem). Using the following definitions of
V
X
=
X
t
and
V
Y
=
Y
t
, 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
δ(X)
(red curve) in a separate plot for any value of the wedge angle. The evolution of the
V
X
-velocity component and its congruent properties with the growth of the boundary layer is also shown.

References

[1] L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport, Cambridge: Cambridge University Press, 2007.
[2] A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Boca Raton, FL: CRC Press, 2002.

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